Research ArticleStochastic versus Deterministic Approach to CoordinatedSupply Chain Scheduling

Tadeusz Sawik

Department of Operations Research AGH University of Science and Technology Al Mickiewicza 30 30-059 Krakow Poland

Correspondence should be addressed to Tadeusz Sawik ghsawikcyf-kredupl

Received 9 January 2017 Accepted 11 May 2017 Published 19 June 2017

Academic Editor Anna Pandolfi

Copyright copy 2017 Tadeusz SawikThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The purpose of this paper is to consider coordinated selection of supply portfolio and scheduling of production and distributionin supply chains under regional and local disruption risks Unlike many papers that assume the all-or-nothing supply disruptionpattern in this paper only the regional disruptions belong to the all-or-nothing disruption category while for the local disruptionsall disruption levels can be considered Two biobjective decision-making models stochastic based on the wait-and-see approachand deterministic based on the expected value approach are proposed and compared to optimize the trade-off between expectedcost and expected service The main findings indicate that the stochastic programming wait-and-see approach with its ability tohandle uncertainty by probabilistic scenarios of disruption events and the much simpler expected value problem in which therandom parameters are replaced by their expected values lead to similar expected performance of a supply chain under multileveldisruptions However the stochastic approach which accounts for all potential disruption scenarios leads to a more diversifiedsupply portfolio that will hedge against a variety of scenarios

1 Introduction

Unexpected disruptions of material flows have become amajor source of concerns in global supply chains over therecent years and coordinated decision-making of suppliesproduction and distribution operations under disruptedflows appears to be a crucial issue (eg Blackhurst et al[1] and Hoffmann et al [2]) While the probability of flowdisruptions is very low their business impact can be hugeFor example flow disruptions in the electronics supply chainsdue to the great East Japan earthquake of 11 March 2011 andthen the catastrophic October flooding in Thailand wheremany component manufacturers were concentrated resultedin huge losses of major electronics producers (eg Parket al [3] and Haraguchi and Lall [4]) Similar losses wereexperienced by the automotive industry (eg Fujimoto andPark [5] Matsuo [6] and Marszewska [7]) For example twomonths after the earthquake Toyota North America whichreceived up to 15 of its parts from Japan experienced ashortage of 150 critical parts and was forced to operate at only30 of its capacity

The business practices ofmany companies (eg Zeng andXia [8]) provide a typical decision-making environment ofsupply chains under disruptions Suppliers are often locatedin different geographic regions and differ in wholesale pricesdelivery lead time and reliability while their disruption pro-files contain parameters such as disruption probability andfulfillment rate or the percentage of an order that is actuallydelivered A popular approach to decision-making in supplychains with disrupted flows is stochastic programmingwhich is capable of incorporating probabilistic disruptionscenarios and finding supply chain coordinated schedulesfor all potential scenarios with respect to various conflictingobjective functions In this paper we present an applicationof stochastic mixed integer programming (stochasticMIP) tocoordinated selection of supply portfolio and scheduling ofproduction and distribution in supply chains with partially orfully disrupted supplies The suppliers are located in differentgeographic regions and the supplies are subject to partial(multilevel) local disruptions of each supplier individuallyand to all-or-nothing (two-level) regional disruptions of allsuppliers in the same region Two equally important and

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 3460721 15 pageshttpsdoiorg10115520173460721

2 Mathematical Problems in Engineering

conflicting objectives are simultaneously optimized expectedcost and expected service level In this paper the stochas-tic MIP approach is compared with a deterministic MIPapproach in which all potential disruption scenarios arereplaced by a single scenario which is obtained by replacingthe stochastic parameters by their expected values

The paper is organized as follows The review of relevantliterature is presented in Section 2 The problem of coordi-nated decision-making in a supply chain subject to partiallocal disruptions and all-or-nothing regional disruptions isdescribed in Section 3 The stochastic mixed integer pro-gram with the objective of minimizing the weighted sumof expected cost and expected service level is developed inSection 4 and the corresponding deterministic mixed integerprogram is proposed in Section 5 Numerical examples com-putational results and some comparison of the two solutionapproaches are provided in Section 6 Finally conclusionsand managerial implications as well as directions for furtherresearch are presented in Section 7

2 Literature Review

The literature on coordinated decision-making in produc-tion-distribution planning and scheduling is mostly limitedto deterministic models with the supply operation consid-ered separately (eg Erenguc et al [9]) For example anintegrated production inventory and distribution routingproblem and a MIP approach combined with a heuristicrouting algorithm to coordinate the production inventoryand transportation operations was considered by Lei et al[10] Kaur et al [11] proposed a graph theoretic approach forsupply chain coordination to model various mechanisms ofcoordination and their interdependencies A digraph repre-senting the supply chain coordination is converted to itsadjacency matrix whose permanent function gives a com-posite index of coordination In Choi et al [12] a single- ormultisupplier and single-manufacturer supply chain schedul-ing and coordination problem was formulated as a two-machine common-due-window flow shop scheduling prob-lem The authors developed scheduling algorithms for boththe decentralized (with the manufacturer as a decision-maker) and centralized supply chains Liu and Papageorgiou[13] developed a multiobjective MIP approach to addressproduction distribution and capacity planning of globalsupply chains considering cost responsiveness and servicelevel simultaneously Chen [14] presented a review of existingmodels that integrate production and outbound distributionoperations at the detailed scheduling levelThemodels aim atoptimizing detailed order-by-order production and deliveryscheduling jointly by taking into account relevant revenuescosts and service levels at the individual order level Sawik[15] proposed a MIP approach for integrated schedulingof material manufacturing material supply and productassembly in a customer driven supply chain A monolithicapproach where the manufacturing supply and assemblyschedules are determined simultaneously was comparedwitha hierarchical approach Numerical examples modeled afterreal-world scheduling in the electronics supply chain werereported

Mitigation and contingencyrecovery actions were stud-ied by Tomlin [16] in a dual sourcing setting one unreliablesupplier and another reliable and more expensive one Abuyer that suffers a supply shortage can buy from a moreexpensive alternate supplier or produce less and its decisiondepends on its inventory If the reliable supplier has volumeflexibility contingent rerouting by temporarily increasing itsproduction may prove to be an effective way to speed uprecovery processThe author established that alongwith costpercentage of supplier uptime disruption length capacityand flexibility play an important role in determining a buyerrsquosdisruption management strategy The recent developmentsin the field of supply chain disruption management from amultidisciplinary perspective were summarized by Ivanov etal [17ndash19]who studied theRipple effect in supply chainsTheyemphasized that the Ripple effect can consolidate research insupply chain disruptionmanagement similar to the bullwhipeffect regarding demand and lead time fluctuations

In the literature on supply uncertainty the supply is sub-ject to either complete disruptions or yield uncertainty Yielduncertainty occurs when the quantity of supply deliveredis a random variable modeled as either a random additiveor multiplicative quantity whereas disruptions occur whensupply is subject to partial or complete failure Typicallydisruptions are modeled as events which occur randomlyand may have a random length Schmitt and Snyder [20]considered inventory systems subject to both supply disrup-tions and yield uncertainty They compared single-periodversus multiperiod models and showed that the former canlead to selecting the wrong strategy for mitigating supplyrisk Schmitt et al [21] investigated optimal system designin a multilocation system under supply disruptions Theyexamined the expected costs and cost variances of the systemin both centralized and decentralized inventory systemsThey showed that when demand is deterministic and supplyis disrupted a decentralized inventory reduces cost variancethrough the risk diversification effect and that a decentralizedinventory may also be selected when supply is disruptedand demand is stochastic A recent literature review onORMS models for supply chain disruptions was presentedby Snyder et al [22] They discussed 180 scholarly workson the topic organized into six categories evaluating supplydisruptions strategic decisions sourcing decisions contractsand incentives inventory and facility location

Sawik [23 24] proposed a new stochastic MIP approachto integrated selection of supply portfolio and scheduling ofcustomer orders in a supply chain under all-or-nothing dis-ruption risks The stochastic MIP formulations were furtherenhanced by Sawik [25] to jointly optimize supply portfolioand production and distribution of finished products Forthe distribution of products three shipping methods wereconsidered and compared

This paper differs from the previous research in thefollowing two aspects First unlike many articles that assumethe all-or-nothing supply disruption pattern in this paperonly the regional disruptions belong to the all-or-nothingdisruption category For the local disruptions however alldisruption levels can be considered within three categoriesminor disruption major disruption and complete shutdown

Mathematical Problems in Engineering 3

(eg [8 26]) Disruption profiles contain parameters suchas probability of disruption at all levels and fulfillment rateor the percentage of an order that is actually deliveredSecond in this paper a stochastic programming wait-and-see approach with its ability to handle uncertainty by prob-abilistic scenarios of disruption events is compared with adeterministic programming approach in which the randomparameters are replaced by their corresponding expectedvalues to achieve the so-called expected value problem (egKall and Mayer [27]) The expected value problem is a MIPand is often used in practice as the related stochastic mixedinteger program is in general much harder to solve since itconsiders multiple scenarios (eg Durbach and Stewart [28]and Maggioni and Wallace [29]) The objective of both thewait-and-see approach and the expected value approach is tooptimize expected performance of a supply chain under thetwo types of disruptionswith respect to two conflicting objec-tive functions expected cost and expected service Whilethe stochastic approach aims at optimizing the expectedperformance of a supply chain over all possible disruptionscenarios the deterministic approach accounts only for a sin-gle scenario representing the expected disruption conditionsThe stochastic programming approach determines a subset ofnondominated solutions for all disruption scenarios whereasthe deterministic approach produces a nondominated solu-tion for a single scenario only The solution of the expectedvalue problem does not take into account any distributioninformation and remains the same as long as the expectationsdo not change Unlike the expected value problem stochasticprogramming provides a recommendation for selection ofsupply portfolio that will hedge against a variety of disruptionscenarios The two approaches and the corresponding solu-tions are compared and some managerial insights derived

3 Problem Description

Consider a three-echelon supply chain (see Figure 1) inwhich a single producer of one product type assembles anddelivers products to multiple distribution centers to meetcustomer demand using a critical part type that can bemanufactured and provided by many suppliers

Let 119868 = 1 119868 be the set of 119868 suppliers let 119869 = 1 119869be the set of 119869 customers let 119870 = 1 119870 be the set of 119870distribution centers and let 119879 = 1 119879 be the set of 119879planning periods (for notations see Notations)

The orders for parts are assumed to be placed at the begin-ning of the planning horizon and the parts ordered fromsupplier 119894 are delivered in period120590119894 Each customer is suppliedwith the ordered products via exactly one distribution centerThe products for each customer 119895 isin 119869119896 are delivered tothe distribution center 119896 in a single delivery which cannotbe scheduled before all customer orders 119895 isin 119869119896 have beencompleted The products shipped in period 119905 to distributioncenter 119896 are delivered in period 119905 + 120591119896 minus 1

The suppliers of parts are located in119877 geographic regionsThe supplies are subject to random local disruptions of differ-ent levels 119897 isin 119871 = 0 119871 where the disruption level refersto the fraction of an order that can be delivered (fulfillment

rate) Level 119897 = 0 represents complete shutdown of a supplierthat is no order delivery while level 119897 = 119871 represents normalconditions with no disruption that is full order deliveryThe intermediate disruption levels 119897 = 1 119871 minus 1 representdifferent fractions of an order that can be delivered Thesmaller 119897 isin 119871 the smaller portion of an order that can bedelivered due to the smaller fraction of the supplier capacityavailable The fraction of an order that can be deliveredby supplier 119894 under disruption level 119897 is described by theassociated fulfillment rate 120574119894119897

120574119894119897 =

0 if 119897 = 0isin (0 1) if 119897 = 1 119871 minus 11 if 119897 = 119871

(1)

Denote by 119901119894119897 the probability of disruption level 119897 isin 119871for supplier 119894 that is the parts ordered from supplier 119894 aredelivered fully with probability119901119894119871 partially at different levelsof supplier output 120574119894119897 with probability 119901119894119897 119897 = 1 119871 minus 1 ornot at all with probability 1199011198940

In addition to independent local disruptions of eachsupplier there are potential regional disasters that may resultin complete shutdown of all suppliers in the same regionsimultaneously For example regional disaster events mayinclude an earthquake and flooding Let 119901119903 be the probabilityof regional disruptions of all suppliers 119894 isin 119868119903 in region 119903 isin 119877

Denote by 119878 = 1 119878 the index set of all disruptionscenarios where each scenario 119904 isin 119878 can be represented byan integer-valued vector 120582119904 = (1205821119904 120582119868119904) where 120582119894119904 isin 119871 isthe disruption level of an order delivery from supplier 119894 isin 119868under scenario 119904 isin 119878 All potential disruption scenarios willbe considered that is 119878 = (119871 + 1)119868

The probability 119875119904 for disruption scenario 119904 isin 119878 with thesubset 119868119904 of nonshutdown suppliers (that can deliver partsunder scenario 119904) is [26]

119875119904 = prod119903isin119877

119875119903119904 (2)

where 119875119903119904 is the probability of realizing of disruption scenario119904 for suppliers in 119868119903119875119903119904 =

(1 minus 119901119903)prod119894isin119868119903

(119901119894120582119894119904) if 119868119903 cap 119868119904 = 0119901119903 + (1 minus 119901119903)prod

119894isin1198681199031199011198940 if 119868119903 cap 119868119904 = 0 (3)

and 119901119894120582119894119904 is the probability of occurrence of the disruptionat level 119897 = 120582119894119904 of an order delivery from supplier 119894 underscenario 119904

The objective of the coordinated decision-making in asupply chain under multilevel disruptions is to allocate thetotal demand for parts among a subset of selected suppliers(ie to determine the supply portfolio) and to schedule foreach disruption scenario the customer orders for productsand the delivery of products to distribution centers tooptimize the trade-off between expected cost and expectedservice level

4 Mathematical Problems in Engineering

Supplier

Supplier

Producer

ProductsPartsDC

DC

Customer

Customer

Customer

Customer

Figure 1 A three-echelon supply chain

4 Problem Formulation Stochastic Approach

In this section a stochastic MIP model WCS is presentedfor the coordinated decision-making in the presence ofsupply chain under multilevel disruption risksThe followingdecisions are jointly made using the proposedmodel [25 26]

(i) Supply portfolio selection 119906119894 = 1 if supplier 119894 isselected otherwise 119906119894 = 0 and V119894 isin [0 1] the fractionof total demand for parts ordered from supplier 119894

(ii) Production scheduling 119908119904119895119905 = 1 if under disruptionscenario 119904 customer order 119895 is scheduled for period 119905otherwise 119908119904119895119905 = 0

(iii) Distribution scheduling 119909119904119896119905 = 1 if under disruptionscenario 119904 a shipment of products to distributioncenter 119896 is scheduled for period 119905 otherwise 119909119904119896119905 = 0

(iv) Customer order nondelayed delivery 119910119904119895 = 1 if underdisruption scenario 119904 customer order 119895 is delivered byits due date otherwise 119910119904119895 = 0

Thedemand allocation vector (V1 V119868) wheresum119894isin119868 V119894 =1 and 0 le V119894 le 1 119894 isin 119868 defines the supply portfoliointroduced by Sawik [30]

Let 1198641 be the minimized expected cost per product andlet 1198642 be the maximized expected service level

1198641 = sum119894isin119868 119890119894119906119894 + sum119904isin119878 119875119904 (sum119894isin119868119904 119861119900119894120574119894120582119894119904V119894 + sum119895isin119869 119892119895119887119895 (sum119905isin119879119908119904119895119905 minus 119910119904119895) + sum119895isin119869 ℎ119895119887119895 (1 minus sum119905isin119879119908119904119895119905))119861 (4)

1198642 = sum119895isin119869sum119904isin119878 119875119904119887119895119910119904119895119861 (5)

where 120582119894119904 is disruption level of supplier 119894 under scenario 119904 and120574119894120582119894119904 is the corresponding fulfillment rate that is the fractionof an order delivered by supplier 119894 under disruption scenario119904

The expected cost 1198641 (see (4)) constitutes fixed order-ing cost sum119894isin119868 119890119894119906119894 expected purchasing cost for deliveredparts sum119904isin119878 119875119904sum119894isin119868119904 119861119900119894120574119894120582119894119904V119894 expected penalty for delayedcustomer demand sum119904isin119878 119875119904sum119895isin119869 119892119895119887119895(sum119905isin119879119908119904119895119905 minus 119910119904119895) andexpected penalty for unsatisfied (rejected) customer demandsum119904isin119878 119875119904sum119895isin119869 ℎ119895119887119895(1 minus sum119905isin119879119908119904119895119905)

Denote by

1198911 = 1198641 minus 11986411198641 minus 1198641 (6)

the normalized (scaled into the interval [0 1]) expected costper product (1198641 1198641 are the minimum and the maximumvalues of 1198641 resp) and by

1198912 = 1198642 minus 11986421198642 minus 1198642 (7)

the normalized expected service level (1198642 1198642 are the mini-mum and the maximum values of 1198642 resp)Model WCS It consists in supplier selection customer orderand distribution scheduling to minimize Weighted sum ofnormalized expected Cost and expected Service level

Minimize

1205721198911 + (1 minus 120572) 1198912 (8)

where 0 le 120572 le 1 subject to (4)ndash(7)Supply Portfolio Selection Constraints

(i) The total demand for parts must be fully allocatedamong the selected suppliers

(ii) Demand for parts cannot be assigned to nonselectedsuppliers

sum119894isin119868

V119894 = 1 V119894 le 119906119894 119894 isin 119868 (9)

Mathematical Problems in Engineering 5

Customer Order Scheduling Constraints

(i) For each disruption scenario 119904 each customer order119895 is either scheduled during the planning hori-zon (sum119905isin119879119908119904119895119905 = 1) or unscheduled and rejected(sum119905isin119879119908119904119895119905 = 0)

(ii) For any period 119905 and each disruption scenario 119904the total demand on capacity of all customer ordersscheduled in period 119905 must not exceed the producercapacity

sum119905isin119879

119908119904119895119905 le 1 119895 isin 119869 119904 isin 119878sum119895isin119869

119887119895119908119904119895119905 le 119862 119905 isin 119879 119904 isin 119878 (10)

Supply-Production-Distribution Coordinating Constraints

(i) For each disruption scenario 119904 and each planningperiod 119905 the cumulative demand for parts of all cus-tomer orders scheduled in period 1 through 119905 cannotexceed the cumulative deliveries of parts in period 1through 119905 minus 1 from the nonshutdown suppliers 119894 isin 119868119904

(ii) For each disruption scenario shipment to distribu-tion center 119896 can be scheduled only after the latestcompletion period of scheduled customer orders 119895 isin119869119896

sum119895isin119869

sum1199051015840isin1198791199051015840le119905

1198871198951199081199041198951199051015840 le 119861 sum119894isin119868119904 120590119894le119905minus1

120574119894120582119894119904V119894 119905 isin 119879 119904 isin 119878 (11)

sum119905isin119879119870

119905119909119904119896119905 ge sum119905isin119879

(119905 + 1)119908119904119895119905119896 isin 119870 119895 isin 119869119896 119904 isin 119878

(12)

where 119879119870 = min119894isin119868120590119894 + 2 119879 + 1 is the set ofshipping periods

Shipping Constraints

(i) For each disruption scenario at most one shipmentcan be scheduled to each distribution center

sum119905isin119879119870

119909119904119896119905 le 1 119896 isin 119870 119904 isin 119878 (13)

Customer Due Date Meeting Constraints

(i) For each disruption scenario 119904 isin 119878 customer order119895 isin 119869119896 can be delivered without delay (ie 119910119904119895 =1) if it is scheduled not later than 119889119895 minus 120591119896 andshipped to distribution center 119896 not later than 119889119895 minus120591119896 + 1 otherwise the customer order is delayed orunscheduled (ie 119910119904119895 = 0)

119910119904119895 le sum119905isin119879119905le119889119895minus120591119896

119908119904119895119905119896 isin 119870 119895 isin 119869119896 119904 isin 119878

119910119904119895 le sum119905isin119879119870119905le119889119895minus120591119896+1

119909119904119896119905119896 isin 119870 119895 isin 119869119896 119904 isin 119878

sum119905isin119879119905le119889119895minus120591119896

119908119904119895119905 + sum119905isin119879119870119905le119889119895minus120591119896+1

119909119904119896119905 minus 1 le 119910119904119895119896 isin 119870 119895 isin 119869119896 119904 isin 119878

(14)

Nonnegativity and Integrality Conditions

119906119894 isin 0 1 119894 isin 119868 (15)

V119894 isin [0 1] 119894 isin 119868 (16)

119908119904119895119905 isin 0 1 119895 isin 119869 119905 isin 119879 119904 isin 119878 (17)

119909119904119896119905 isin 0 1 119896 isin 119870 119905 isin 119879119870 119904 isin 119878 (18)

119910119904119895 ge 0 119895 isin 119869 119904 isin 119878 (19)

Note that119910119904119895 does not need to be restricted to being binarysince for any feasible solution satisfying constraints (14) 119910119904119895is always binary

Model WCS is a deterministic equivalent mixed integerprogram of stochastic mixed integer program When thisproblem is solved a recommendation is obtained for selec-tion of supply portfolio (119906119894 V119894) thatwill hedge against a varietyof disruption scenarios in which fulfillment rates of certainsuppliers are not sufficient to satisfy demand for parts Thescheduling variables 119908119904119895119905 119909119904119896119905 (and 119910119904119895) are decisions that willbe implemented in the future when scenario 119904 isin 119878 is finallyrealized

Model WCS illustrates the wait-and-see approach (eg[27]) Basically this approach is based on perfect informationabout the future Model WCS can be decomposed into atwo-stage stochastic mixed integer program with recourseThe supply portfolio selection variables (119906119894 V119894) are referredto as first-stage decisions and the scheduling variables(119908119904119895119905 119909119904119896119905 119910119904119895) are referred to as recourse or second-stagedecisions Unlike the first-stage decisions the latter variablesare dependent on the scenario 119904 isin 119878

Stochastic mixed integer programs are usually hard tosolve because they are large-scale optimization problemswhen applied to real-world problems A common approach(eg [27]) is to consider a simpler deterministic programknown as expected value problem in which the randomparameters are replaced by their expected values or toconsider several deterministic programs each of which cor-responds to one particular scenario and then to combine theobtained solutions into a single heuristic solution

5 Problem FormulationDeterministic Approach

In this section the expected value problem EWCS is pre-sented for the coordinated supply chain scheduling under

6 Mathematical Problems in Engineering

expected supply conditions In model WCS where the ran-domness is characterized by a set of disruption scenariosthe only random parameters are suppliers fulfillment rates120574119894120582119894119904 which appear both in the objective function (4) and inconstraints (11)

In model EWCS suppliers probabilistic fulfillment ratesdefined for each disruption scenario 120574119894120582119894119904 119894 isin 119868 119904 isin 119878 orequivalently for each disruption level 120574119894119897 119894 isin 119868 119897 isin 119871 (1)have been replaced by the expected fulfillment rates of eachsupplier

Γ119894 = sum119904isin119878

119875119904120574119894120582119894119904 119894 isin 119868 (20)

or equivalently

Γ119894 = (1 minus 119901119903)sum119897isin119871

119901119894119897120574119894119897 119894 isin 119868119903 119903 isin 119877 (21)

Accordingly stochastic binary decision variables 119908119904119895119905 119909119904119896119905 119910119904119895(17)ndash(19) defined for each disruption scenario 119904 isin 119878 havebeen replaced by their deterministic equivalents 119882119895119905 119883119896119905119884119895

Now the expected cost per product 1198641 (see (22)) andthe expected service level 1198642(see (23)) are defined as fol-lows

1198641 = sum119894isin119868 119890119894119906119894 + sum119894isin119868 119861119900119894Γ119894V119894 + sum119895isin119869 119892119895119887119895 (sum119905isin119879119882119895119905 minus 119884119895) + sum119895isin119869 ℎ119895119887119895 (1 minus sum119905isin119879119882119895119905)119861 (22)

1198642 = sum119895isin119869 119887119895119884119895119861 (23)

Model EWCS is presented below

Model EWCS

Minimize (8)subject to (6) (7) (9) (22) (23)

sum119905isin119879

119882119895119905 le 1 119895 isin 119869sum119895isin119869

119887119895119882119895119905 le 119862 119905 isin 119879sum119895isin119869

sum1199051015840isin1198791199051015840le119905

1198871198951198821198951199051015840 le 119861 sum119894isin119868120590119894le119905minus1

Γ119894V119894 119905 isin 119879sum119905isin119879119870

119905119883119896119905 ge sum119905isin119879

(119905 + 1)119882119895119905119896 isin 119870 119895 isin 119869119896

sum119905isin119879119870

119883119896119905 le 1 119896 isin 119870119884119895 le sum119905isin119879119905le119889119895minus120591119896

119882119895119905 119896 isin 119870 119895 isin 119869119896119884119895 le sum119905isin119879119870119905le119889119895minus120591119896+1

119883119896119905 119896 isin 119870 119895 isin 119869119896sum

119905isin119879119905le119889119895minus120591119896

119882119895119905 + sum119905isin119879119870119905le119889119895minus120591119896+1

119883119896119905 minus 1le 119884119895 119896 isin 119870 119895 isin 119869119896119906119894 isin 0 1 119894 isin 119868V119894 isin [0 1] 119894 isin 119868

119882119895119905 isin 0 1 119895 isin 119869 119905 isin 119879119883119896119905 isin 0 1 119896 isin 119870 119905 isin 119879119870119884119895 ge 0 119895 isin 119869

(24)

Notice that unlike the stochastic programming modelWCS which is formulated to determine optimal schedulesfor all potential disruption scenarios model EWCS accountsfor a single scenario only representing the expected suppliesExcept for the expected values of the random parametersthis model does not take into account any distributioninformation and the solution remains the same as long as theexpectations do not change In contrast tomodelWCS wherethe selection of supply portfolio (V1 V119868) is combinedwith supply chain scheduling for all disruption scenariosconsidered now the portfolio is determined along with asingle schedule

If randomparameters appear only in the constraints then[27]

EV le WS (25)

where EV is the optimal solution value of the expected valueproblem EWCS and WS is the optimal solution value ofthe wait-and-see problem WCS On the other hand whenuncertainty is limited to the objective function of the prob-lem the solution obtained by simply replacing the randomparameters with their expected values provides already arobust alternative (Delage et al [31])

6 Computational Examples

In this section some computational examples are presentedto illustrate possible applications of the proposedMIPmodelsand to compare the wait-and-see and the expected value

Mathematical Problems in Engineering 7

approaches The examples are modeled in part after a real-world electronics supply chain (eg Sawik [32]) The fol-lowing parameters have been selected for the computationalexamples119868 = 4 suppliers 119869 = 20 customer orders 119870 = 2 dis-tribution centers 119871 = 3 partial disruption levels 119877 = 2geographic regions and 119879 = 7 planning periods1198681 = 1 2 1198682 = 3 41198691 = 1 10 1198692 = 11 20

Shipping times from suppliers 120590 = (1 1 3 3)Shipping times to distribution centers 120591 = (1 2)Customer demand 119887119895 isin 2000 3000 10000 for all119895 isin 119869 and total demand 119861 = 100000Due dates 119889119895 isin 2 + min119894isin119868(120590119894) + min119896isin119870(120591119896) 119879 +

max119896isin119870(120591119896) for all 119895 isin 119869Fixed ordering costs for suppliers 119890 = (8000 6000 1200013000)Unit purchasing prices from suppliers 119900 = (14 12 8 9)Unit penalties for delayed unfulfilled customer orders119892119895 = lceilmax119894isin119868119900119894350rceil = 1 ℎ119895 = 2max119894isin119868119900119894 = 28 respectively

for all 119895 isin 119869Producer capacity 119862 = 45000Local disruption levels and the associated fulfillment rates

(the percentage of an order that can be delivered) are shownbelow119871 = 0 1 2 3 where 119897 = 0 complete shutdown1205741198940 = 0 forall119894 isin 119868 that is 0 of an order delivered 119897 = 1major disruption 1205741198941 isin [001 050] forall119894 isin 1198681 and 1205741198941 isin[001 030] forall119894 isin 1198682 that is 1 to 50 and 1 to 30 ofan order delivered respectively 119897 = 2 minor disruption

1205741198942 isin [0 51 099] forall119894 isin 1198681 and 1205741198942 isin [0 31 099] forall119894 isin 1198682that is 51 to 99 and 31 to 99 of an order deliveredrespectively 119897 = 119871 = 3 no disruption 1205741198943 = 1 forall119894 isin 119868 thatis 100 of an order delivered

The total number of all potential scenarios is 119878 = (119871 +1)119868 = 44 = 256 scenarios where each scenario 119904 isin 119878 is repre-sented by vector 120582119904 = (1205821119904 1205824119904) where 120582119894119904 isin 119871 119894 isin 119868 (seeTable 1)

The local nondisruption probability (level 119897 = 3) 1199011198943was uniformly distributed over [089 099] and [079 089]respectively for suppliers 119894 isin 1198681 and 119894 isin 1198682 that is theprobabilities were drawn independently from U[089 099]and U[079 089] respectively

Given local nondisruption probabilities 1199011198943 119894 isin 119868 theprobabilities for the remaining local disruption levels 119897 =0 1 2 were calculated as follows

Probability of complete shutdown (level 119897 = 0) 1199011198940 =01(1 minus 1199011198943)Probability of major disruption (level 119897 = 1) 1199011198941 =03(1 minus 1199011198943)Probability of minor disruption (level 119897 = 2) 1199011198942 =06(1 minus 1199011198943) for all suppliers 119894 isin 119868

Thus 1199011198940 le 1199011198941 le 1199011198942 le 1199011198943 which reflects a real-world relation among probabilities of disruption occurrenceat different levels (eg [8])

Regional disruption probabilities are 1199011 = 0001 and1199012 =001The probability of realizing of disruption scenario 119904 for

suppliers in region 119903 = 1 2 is calculated as follows

119875119903119904

=(1 minus 119901119903)( prod

119894isin119868119903 120582119894119904=0

01 (1 minus 1199011198943))( prod119894isin119868119903 120582119894119904=1

03 (1 minus 1199011198943))( prod119894isin119868119903120582119894119904=2

06 (1 minus 1199011198943))( prod119894isin119868119903 120582119894119904=3

1199011198943) if sum119894isin119868119903120582119894119904 gt 0

119901119903 + (1 minus 119901119903)prod119894isin119868119903

01 (1 minus 1199011198943) if sum119894isin119868119903120582119894119904 = 0

(26)

and the probability for disruption scenario 119904 isin 119878 is given by119875119904 = 1198751119904 1198752119904 Figure 2 presents basic characteristics of all suppliers

probability of complete shutdown119901119903+(1minus119901119903)1199011198940 119894 isin 119868119903 119903 isin119877 expected fulfillment rate Γ119894 = sum119897=123(1 minus 119901119903)119901119894119897120574119894119897 119894 isin119868119903 119903 isin 119877 and purchasing price 119900119894 119894 isin 119868Table 2 presents a subset of nondominated solutions for

the wait-and-see problem WCS for a subset of trade-offparameter120572 isin 0 01 02 03 04 05 06 07 08 09 1Theresults indicate that most of nondominated supply portfoliosconsist of the two suppliers only the second most expensiveand most reliable supplier 119894 = 2 from region 119903 = 1 andthe cheapest and most unreliable supplier 119894 = 3 from region119903 = 2 The most expensive and most reliable supplier 119894 = 1in region 119903 = 1 and the second cheapest and most unreliablesupplier 119894 = 4 in region 119903 = 2 are rarely selected For 120572 = 1

(minimization of cost) the cheapest supplier 119894 = 3 is selectedonly For 120572 = 0 (maximization of service level) the totaldemand for parts is allocated among the two most reliableand most expensive suppliers 119894 = 1 2 The above solutionshows that the service-oriented supply portfolio (120572 close to 0)is more diversified than the cost-oriented portfolio (120572 closeto 1) Table 2 also shows the associated expected fractionof fulfilled demand 1198643 = sum119904isin119878sum119895isin119869sum119905isin119879 119875119904119887119895119908119904119895119905119861 that isdemand fulfilled on time or delayed The solution resultsdemonstrate that a large expected service level is sometimesassociated with a small expected fraction of fulfilled demandThus the maximization of expected service level leads torejection of customer orders that cannot be fulfilled bycustomer requested due dates

Figure 3 shows the expected supply production andshipping schedules respectively for 120572 = 0 (ie for the

8 Mathematical Problems in Engineering

Shutdown probability

Supplier

Expected yield rate

Purchasing price

4321

Supplier4321

Supplier4321

6

8

10

12

14

16

09

092

094

096

098

0

0005

001

0015

002

0025

Figure 2 Suppliers

1 2 3 4 5 6 7Period

Expected supply schedules

Alpha = 0Alpha = 05

Alpha = 1

1 2 3 4 5 6 7Period

Expected production schedules

1 2 3 4 5 6 7Period

Expected shipping schedules

0

20000

40000

60000

80000

100000

Part

s

0

10000

20000

30000

40000

50000

Prod

ucts

Alpha = 0Alpha = 05

Alpha = 1

0

10000

20000

30000

40000

50000

Prod

ucts

Alpha = 0Alpha = 05

Alpha = 1

Figure 3 Expected schedules for modelWCS

Mathematical Problems in Engineering 9

Table 1 Disruption scenarios

119904 119894 = 1 2 3 41 0 0 0 02 0 0 0 13 0 0 0 24 0 0 0 35 0 0 1 06 0 0 1 17 0 0 1 28 0 0 1 39 0 0 2 010 0 0 2 111 0 0 2 212 0 0 2 313 0 0 3 014 0 0 3 115 0 0 3 216 0 0 3 317 0 1 0 018 0 1 0 119 0 1 0 220 0 1 0 321 0 1 1 022 0 1 1 123 0 1 1 224 0 1 1 325 0 1 2 026 0 1 2 127 0 1 2 228 0 1 2 329 0 1 3 030 0 1 3 131 0 1 3 232 0 1 3 333 0 2 0 034 0 2 0 135 0 2 0 236 0 2 0 337 0 2 1 038 0 2 1 139 0 2 1 240 0 2 1 341 0 2 2 042 0 2 2 143 0 2 2 244 0 2 2 345 0 2 3 046 0 2 3 147 0 2 3 248 0 2 3 349 0 3 0 0

Table 1 Continued

119904 119894 = 1 2 3 450 0 3 0 151 0 3 0 252 0 3 0 353 0 3 1 054 0 3 1 155 0 3 1 256 0 3 1 357 0 3 2 058 0 3 2 159 0 3 2 260 0 3 2 361 0 3 3 062 0 3 3 163 0 3 3 264 0 3 3 365 1 0 0 066 1 0 0 167 1 0 0 268 1 0 0 369 1 0 1 070 1 0 1 171 1 0 1 272 1 0 1 373 1 0 2 074 1 0 2 175 1 0 2 276 1 0 2 377 1 0 3 078 1 0 3 179 1 0 3 280 1 0 3 381 1 1 0 082 1 1 0 183 1 1 0 284 1 1 0 385 1 1 1 086 1 1 1 187 1 1 1 288 1 1 1 389 1 1 2 090 1 1 2 191 1 1 2 292 1 1 2 393 1 1 3 094 1 1 3 195 1 1 3 296 1 1 3 397 1 2 0 098 1 2 0 1

10 Mathematical Problems in Engineering

Table 1 Continued

119904 119894 = 1 2 3 499 1 2 0 2100 1 2 0 3101 1 2 1 0102 1 2 1 1103 1 2 1 2104 1 2 1 3105 1 2 2 0106 1 2 2 1107 1 2 2 2108 1 2 2 3109 1 2 3 0110 1 2 3 1111 1 2 3 2112 1 2 3 3113 1 3 0 0114 1 3 0 1115 1 3 0 2116 1 3 0 3117 1 3 1 0118 1 3 1 1119 1 3 1 2120 1 3 1 3121 1 3 2 0122 1 3 2 1123 1 3 2 2124 1 3 2 3125 1 3 3 0126 1 3 3 1127 1 3 3 2128 1 3 3 3129 2 0 0 0130 2 0 0 1131 2 0 0 2132 2 0 0 3133 2 0 1 0134 2 0 1 1135 2 0 1 2136 2 0 1 3137 2 0 2 0138 2 0 2 1139 2 0 2 2140 2 0 2 3141 2 0 3 0142 2 0 3 1143 2 0 3 2144 2 0 3 3145 2 1 0 0146 2 1 0 1147 2 1 0 2148 2 1 0 3149 2 1 1 0150 2 1 1 1

Table 1 Continued

119904 119894 = 1 2 3 4151 2 1 1 2152 2 1 1 3153 2 1 2 0154 2 1 2 1155 2 1 2 2156 2 1 2 3157 2 1 3 0158 2 1 3 1159 2 1 3 2160 2 1 3 3161 2 2 0 0162 2 2 0 1163 2 2 0 2164 2 2 0 3165 2 2 1 0166 2 2 1 1167 2 2 1 2168 2 2 1 3169 2 2 2 0170 2 2 2 1171 2 2 2 2172 2 2 2 3173 2 2 3 0174 2 2 3 1175 2 2 3 2176 2 2 3 3177 2 3 0 0178 2 3 0 1179 2 3 0 2180 2 3 0 3181 2 3 1 0182 2 3 1 1183 2 3 1 2184 2 3 1 3185 2 3 2 0186 2 3 2 1187 2 3 2 2188 2 3 2 3189 2 3 3 0190 2 3 3 1191 2 3 3 2192 2 3 3 3193 3 0 0 0194 3 0 0 1195 3 0 0 2196 3 0 0 3197 3 0 1 0198 3 0 1 1199 3 0 1 2200 3 0 1 3201 3 0 2 0202 3 0 2 1

Mathematical Problems in Engineering 11

Table 1 Continued

119904 119894 = 1 2 3 4203 3 0 2 2204 3 0 2 3205 3 0 3 0206 3 0 3 1207 3 0 3 2208 3 0 3 3209 3 1 0 0210 3 1 0 1211 3 1 0 2212 3 1 0 3213 3 1 1 0214 3 1 1 1215 3 1 1 2216 3 1 1 3217 3 1 2 0218 3 1 2 1219 3 1 2 2220 3 1 2 3221 3 1 3 0222 3 1 3 1223 3 1 3 2224 3 1 3 3225 3 2 0 0226 3 2 0 1227 3 2 0 2228 3 2 0 3229 3 2 1 0230 3 2 1 1231 3 2 1 2232 3 2 1 3233 3 2 2 0234 3 2 2 1235 3 2 2 2236 3 2 2 3237 3 2 3 0238 3 2 3 1239 3 2 3 2240 3 2 3 3241 3 3 0 0242 3 3 0 1243 3 3 0 2244 3 3 0 3245 3 3 1 0246 3 3 1 1247 3 3 1 2248 3 3 1 3249 3 3 2 0250 3 3 2 1251 3 3 2 2252 3 3 2 3

Table 1 Continued

119904 119894 = 1 2 3 4253 3 3 3 0254 3 3 3 1255 3 3 3 2256 3 3 3 3

maximum expected service level) 120572 = 05 and 120572 = 1 (ie forthe minimum expected cost) The expected schedules werecomputed using the formulae presented below

(i) Expected schedule of supplies of parts to the pro-ducer

sum119904isin119878

sum119894isin119868119904120590119894=119905

119875119904119861120574119894120582119894119904V119894 119905 isin 119879 (27)

(ii) Expected production schedule

sum119904isin119878

sum119895isin119869

119875119904119887119895119908119904119895119905 119905 isin 119879 (28)

(iii) Expected schedule of shipping of products from theproducer to the distribution centers

sum119904isin119878

sum119896isin119870

sum119895isin119869119896

119875119904119887119895( sum1199051015840isin1198791199051015840lt119905

1199081199041198951199051015840)119909119904119896119905 119905 isin 119879 (29)

As 120572 increases that is the decision-makerrsquos priority shiftsfrom the maximum service level to minimum cost and moreparts are ordered from less reliable and lower cost suppliersthe expected supply schedules and the corresponding pro-duction schedules are more delayed as well as the delivery ofproducts to the customers Note that despite constraint (13)that ensures a feasible schedule with at most one shipmentof products to each distribution center for each disruptionscenario the expected shipping schedule (29) may be splitinto more smaller size shipments (cf Figure 3 where twomajor and one small-size shipments are indicated for eachconfidence level)

For comparison Table 3 presents a subset of non-dominated solutions obtained for the expected value problemEWCS and Figure 4 shows supply production and shippingschedules Unlike the stochastic programming approachwhich accounts for all potential disruption scenarios tooptimize an expected performance of the supply chain thesolution obtained using the deterministic approach is basedon aggregate information on suppliers expected fulfillmentIn general the results are similar for both models and thecorresponding optimal solution values are close to each otherwhich indicates that the expected value problem can be usedin practice when stochastic mixed integer programs are hardto solve The optimal solution values for the expected valueproblem frequently outperform the corresponding solutionvalues of the wait-and-see problem which is in line withthe proposition EV le WS in [27] compare Section 5for example the minimum cost 1198641 for EWCS (see Table 3)

12 Mathematical Problems in Engineering

Table 2 Nondominated solutions for modelWCS

120572 0 01 02 03 04 05 and 06 07 08 and 09 1Var = 38077 Bin = 33076 Cons = 28769 Nonz = 348325(a)

Exp cost 1731 (1198641) 1620 1547 1206 1200 1176 1070 1065 1051 (1198641)Exp service level(b) 8552 (1198642) 8487 8452 7719 7676 7595 5863 5871 3822 (1198642)Exp fulfilled demand(c) 8552 8530 8452 9370 9361 9353 9134 9120 9068

Suppliers selected( of total demand)

1 (76) 1 (35)2 (24) 2 (55) 2 (87) 2 (55) 2 (55) 2 (55) 2 (12) 2 (10)3 (10) 3 (13) 3 (22) 3 (32) 3 (45) 3 (88) 3 (90) 3 (100)4 (23) 4 (13)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119904isin119878 sum119895isin119869 119875119904119887119895119910

119904119895119861 times

100 (c)sum119904isin119878 sum119895isin119869 sum119905isin119879 119875119904119887119895119908119904119895119905119861 times 100

Table 3 Nondominated solutions for model EWCS

120572 0 01 02 and 03 04 and 05 06 and 07 08 09 and 1Var = 162 Bin = 136 Cons = 125 Nonz = 1430(a)

Exp cost 1691 (1198641) 1427 1214 1167 1045 (1198641)Exp service level(b) 87 (1198642) 86 81 79 62 (1198642)Exp fulfilled demand(c) 87 90 91 93 91

Suppliers selected( of total demand)

1 (89) 2 (94) 2 (49) 2 (50) 2 (5)3 (6) 3 (51) 3 (50) 3 (95)4 (11)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119895isin119869 119887119895119884119895119861 times 100(c)sum119895isin119869 sum119905isin119879 119887119895119882119895119905119861 times 100

1 2 3 4 5 6 7Period

Supply schedules

1 2 3 4 5 6 7Period

Production schedules

1 2 3 4 5 6 7Period

Shipping schedules

0

10000

20000

30000

40000

50000

Prod

ucts

0

10000

20000

30000

40000

50000

Prod

ucts

0

20000

40000

60000

80000

100000

Part

s

Alpha = 0Alpha = 05

Alpha = 1 Alpha = 0Alpha = 05

Alpha = 1

Alpha = 0Alpha = 05

Alpha = 1

Figure 4 Schedules for model EWCS

Mathematical Problems in Engineering 13

1045 le 1051 forWCS (see Table 2) or themaximum servicelevel1198642 for EWCS (see Table 3) 8700 ge 8552 forWCS (seeTable 2)

The stochastic wait-and-see approach however leads toa more diversified supply portfolio For the deterministicapproach most of nondominated supply portfolios consistof two suppliers only 119894 = 2 3 while suppliers 119894 = 1 4 areselected only for the maximum service level objective (thatis for 120572 = 0) Comparison of Figures 3 and 4 indicatesthat the expected schedules for model WCS computed asexpectations over all schedules for all potential disruptionscenarios (27)ndash(29) are similar to the corresponding singleschedules determined by model EWCS The main differencesobserved aremore delayed expected production and shippingschedules for model WCS when minimization of cost is con-sidered (that is for120572 = 1) Finally it is interesting to note thatin the multiple sourcing environment considered both thewait-and-see approach and in particular the expected valueapproach frequently select a dual sourcing supply portfoliowith one main supplier and one supporting supplier

The computational experiments were performed usingthe AMPL programming language and the CPLEX 1262solver on a MacBook Pro laptop with Intel Core i7 processorrunning at 28 GHz and with 16GB RAM The solver wascapable of finding proven optimal solution for all exampleswith CPU time ranging from several minutes to several hoursfor the stochastic model WCS and fraction of a second forthe deterministic model EWCS Since the stochastic pro-gramming model WCS needs to determine nondominatedschedules for all potential disruption scenarios while modelEWCS deals with a single scenario only the difference in thecomputational effort required is obvious

7 Conclusions

In this paper two biobjective MIP formulations stochasticand deterministic have been proposed and compared for thecoordinated decision-making in supply chains under partiallocal disruptions and all-or-nothing regional disruptionsTheproblem objective has been to jointly schedule supplies pro-duction and distribution to optimize the trade-off betweenexpected cost and expected service levelWhile the stochasticprogramming approach aims at optimizing the expectedperformance of a supply chain over all possible disruptionscenarios the deterministic approach accounts only for asingle scenario representing average disruption conditionsAs a result the stochastic programming approach determinesnondominated solutions for all disruption scenarios whereasthe deterministic approach produces a single solution onlyIn particular in the stochastic programming approach theselection of supply portfolio is combined with supply chainscheduling for all disruption scenarios considered In con-trast the deterministic expected value approach provides theportfolio along with a single expected schedule of productionand distribution

The expected schedules obtained for the stochastic pro-gramming model WCS as expectations over all schedulesfor all disruption scenarios have been compared with thecorresponding schedules determined by the deterministic

model EWCS based on an expected disruption scenarioThecomparison has demonstrated that the two approaches leadto similar expected solutions

The main findings are in line with other research and arelisted as follows

(i) The two decision-making approaches stochastic anddeterministic lead to similar expected performanceof a supply chain under multilevel disruptions

(ii) The optimal solution values for the expected valueproblem frequently outperform the correspondingsolution values of the wait-and-see problem

(iii) Despite the multiple sourcing environment consid-ered both the wait-and-see approach and in partic-ular the expected value approach frequently selecta dual sourcing supply portfolio with one mainsupplier and one supporting supplier

(iv) The stochastic approach which accounts for allpotential disruption scenarios may lead to a morediversified supply portfolio that will hedge against avariety of scenarios

(v) The expected schedules are more delayed for thestochastic approach

(vi) The service-oriented supply portfolio is more diversi-fied andmay combine both high-cost reliable suppli-ers and low-cost unreliable suppliers while the cost-oriented portfolio depends mainly on low-cost andless reliable suppliers

Overall the results of computational experiments indi-cate that the proposed approach and developed MIP modelsare flexible and efficient tools for coordinated supply chainscheduling The portfolio approach leads to MIP formula-tions with strong LP relaxations and has been proven to becomputationally very efficient CPU time required to findproven optimal solutions for realistic size examples usingcommercially available software for MIP is acceptable for areal-world supply chain disruption management (see Sawik[33])

Since the probability distribution of supply disruptionsfrom each supplier is usually unknown (eg [8]) local multi-level disruptions in the stochastic programming modelWCShave a multinomial discrete distribution while the two-levelregional disruptions have a binomial discrete distributionand all disruption events are independent As part of futureresearch we propose to enhance the stochastic programmingmodel formore general scenarios with finitelymany elementssingled out and all the probability concentrated in them Forexample rather than complete shutdown suppliers within aregion may have correlated disruptions Another importantstream of future research is the study of robustness andsensitivity in relation to input data changes in supply chainscheduling under disruption risks (eg [33]) Future researchcan also focus on the following improvement of the proposedmodels to consider the minimization of time and cost ofrecovery (eg Whitney et al [34])

14 Mathematical Problems in Engineering

Notations

Indices

119894 Supplier 119894 isin 119868119895 Customer 119895 isin 119869119896 Distribution center 119896 isin 119870119897 Disruption level 119897 isin 119871119903 Region 119903 isin 119877119904 Disruption scenario 119904 isin 119878119905 Planning period 119905 isin 119879Input Parameters

119887119895 Size (number of products) of customer order 119895119861 Total demand for partsproducts 119861 = sum119895isin119869 119887119895119862 Capacity of producer119889119895 Due date for customer order 119895119890119894 Fixed cost of ordering parts from supplier 119894119892119895 Per unit penalty cost of delayed customer order 119895ℎ119895 Per unit penalty cost of unfulfilled customer order 119895119868119903 Subset of suppliers in region 119903119869119896 Subset of customers serviced by distribution center 119896119900119894 Per unit price of parts purchased from supplier 119894119901119894119897 Probability of disruption level 119897 for supplier 119894119901119903 Regional disruption probability for region 119903120574119894119897 Fraction of an order delivered by supplier 119894 underdisruption level 119897 (fulfillment rate)Γ119894 Expected fraction of an order delivered by supplier 119894(expected fulfillment rate)120590119894 Delivery lead time from supplier 119894120591119896 Transportation time to distribution center 119896

Conflicts of Interest

The author declares that there are no conflicts of interest

Acknowledgments

This work has been supported by NCN research grant (noDEC-201311BST804458) and by AGH (no 1111200324)

References

[1] J V Blackhurst K P Scheibe and D J Johnson ldquoSupplierrisk assessment and monitoring for the automotive industryrdquoInternational Journal of Physical Distribution amp Logistics Man-agement vol 38 no 2 pp 143ndash165 2008

[2] P Hoffmann H Schiele and K Krabbendam ldquoUncertaintysupply risk management and their impact on performancerdquoJournal of Purchasing amp Supply Management vol 19 no 3 pp199ndash211 2013

[3] Y Park P Hong and J J Roh ldquoSupply chain lessons from thecatastrophic natural disaster in Japanrdquo Business Horizons vol56 no 1 pp 75ndash85 2013

[4] M Haraguchi and U Lall ldquoFlood risks and impacts A casestudy of Thailandrsquos floods in 2011 and research questions forsupply chain decisionmakingrdquo International Journal of DisasterRisk Reduction vol 14 pp 256ndash272 2015

[5] T Fujimoto and Y W Park ldquoBalancing supply chain com-petitiveness and robustness through ldquovirtual dual sourcingrdquoLessons from the Great East Japan Earthquakerdquo InternationalJournal of Production Economics vol 147 pp 429ndash436 2014

[6] HMatsuo ldquoImplications of the Tohoku earthquake for Toyotarsquoscoordination mechanism supply chain disruption of auto-motive semiconductorsrdquo International Journal of ProductionEconomics vol 161 pp 217ndash227 2015

[7] J R Marszewska ldquoImplications of seismic hazard in Japan ontoyota supply chain disruption risksrdquo in Proceedings of the 13thInternational Conference on Industrial Logistics (ICIL rsquo16) pp178ndash185 Zakopane Poland October 2016

[8] A Z Zeng and Y Xia ldquoBuilding a mutually beneficial partner-ship to ensure backup supplyrdquo Omega vol 52 pp 77ndash91 2015

[9] S S Erenguc N C Simpson and A J Vakharia ldquoIntegratedproductiondistribution planning in supply chains an invitedreviewrdquo European Journal of Operational Research vol 115 no2 pp 219ndash236 1999

[10] L Lei S Liu A Ruszczynski and S Park ldquoOn the integratedproduction inventory and distribution routing problemrdquo IIETransactions vol 38 no 11 pp 955ndash970 2006

[11] A Kaur A Kanda and S G Deshmukh ldquoA graph theoreticapproach for supply chain coordinationrdquo International Journalof Logistics Systems and Management vol 2 no 4 pp 321ndash3412006

[12] T ChoiW Yeung andT C E Cheng ldquoScheduling and co-ordi-nation of multi-suppliers single-warehouse-operator single-manufacturer supply chains with variable production rates andstorage costsrdquo International Journal of Production Research vol51 no 9 pp 2593ndash2601 2013

[13] S Liu and L G Papageorgiou ldquoMultiobjective optimisation ofproduction distribution and capacity planning of global supplychains in the process industryrdquo Omega vol 41 no 2 pp 369ndash382 2013

[14] Z L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[15] T Sawik ldquoCoordinated supply chain schedulingrdquo InternationalJournal of Production Economics vol 120 no 2 pp 437ndash4512009

[16] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[17] D Ivanov B Sokolov and A Dolgui ldquoThe Ripple effect in sup-ply chains Trade-off rsquoefficiency-flexibility- resiliencersquo in disrup-tionmanagementrdquo International Journal of ProductionResearchvol 52 no 7 pp 2154ndash2172 2014

[18] D Ivanov A Pavlov and B Sokolov ldquoOptimal distribution(re)planning in a centralized multi-stage supply network underconditions of the ripple effect and structure dynamicsrdquo Euro-pean Journal of Operational Research vol 237 no 2 pp 758ndash770 2014

[19] D Ivanov A Pavlov A Dolgui D Pavlov and B SokolovldquoDisruption-driven supply chain (re)-planning and perfor-mance impact assessment with consideration of pro-active andrecovery policiesrdquo Transportation Research Part E Logistics andTransportation Review vol 90 pp 7ndash24 2016

[20] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

Mathematical Problems in Engineering 15

[21] A J Schmitt S A Sun L V Snyder and Z-J M Shen ldquoCen-tralization versus decentralization Risk pooling risk diversifi-cation and supply chain disruptionsrdquo Omega vol 52 pp 201ndash212 2015

[22] L V Snyder Z Atan P Peng Y Rong A J Schmitt andB Sinsoysal ldquoORMS models for supply chain disruptions areviewrdquo IIE Transactions vol 48 no 2 pp 89ndash109 2016

[23] T Sawik ldquoIntegrated selection of suppliers and scheduling ofcustomer orders in the presence of supply chain disruptionrisksrdquo International Journal of Production Research vol 51 no23-24 pp 7006ndash7022 2013

[24] T Sawik ldquoOptimization of cost and service level in the presenceof supply chain disruption risks Single vs multiple sourcingrdquoComputers amp Operations Research vol 51 pp 11ndash20 2014

[25] T Sawik ldquoIntegrated supply production and distributionscheduling under disruption risksrdquoOmega vol 62 pp 131ndash1442016

[26] T Sawik ldquoIntegrated supply chain scheduling under multi-leveldisruptionsrdquo IFAC-PapersOnLine vol 28 no 3 pp 1515ndash15202015

[27] P Kall and J Mayer Stochastic Linear Programming ModelsTheory and Computation vol 156 of International Series inOperations Research amp Management Science Springer NewYork 2011

[28] I N Durbach and T J Stewart ldquoUsing expected values to sim-plify decision making under uncertaintyrdquo Omega vol 37 no 2pp 312ndash330 2009

[29] F Maggioni and S W Wallace ldquoAnalyzing the quality of theexpected value solution in stochastic programmingrdquo Annals ofOperations Research vol 200 pp 37ndash54 2012

[30] T Sawik ldquoSelection of supply portfolio under disruption risksrdquoOmega vol 39 no 2 pp 194ndash208 2011

[31] E Delage S Arroyo and Y Ye ldquoThe value of stochastic mod-eling in two-stage stochastic programs with cost uncertaintyrdquoOperations Research vol 62 no 6 pp 1377ndash1393 2014

[32] T Sawik Scheduling in Supply Chains Using Mixed Integer Pro-gramming John Wiley amp Sons Inc Hoboken NJ 2011

[33] T Sawik Supply Chain DisruptionManagement Using StochasticMixed Integer Programming vol 256 of International Seriesin Operations Research amp Management Science Springer NewYork 2017 httpwwwspringercomgpbook9783319588223

[34] D E Whitney J Luo and D A Heller ldquoThe benefits and con-straints of temporary sourcing diversification in supply chaindisruption and recoveryrdquo Journal of Purchasing and SupplyManagement vol 20 no 4 pp 238ndash250 2014

Mathematical Problems in Engineering 3

(eg [8 26]) Disruption profiles contain parameters suchas probability of disruption at all levels and fulfillment rateor the percentage of an order that is actually deliveredSecond in this paper a stochastic programming wait-and-see approach with its ability to handle uncertainty by prob-abilistic scenarios of disruption events is compared with adeterministic programming approach in which the randomparameters are replaced by their corresponding expectedvalues to achieve the so-called expected value problem (egKall and Mayer [27]) The expected value problem is a MIPand is often used in practice as the related stochastic mixedinteger program is in general much harder to solve since itconsiders multiple scenarios (eg Durbach and Stewart [28]and Maggioni and Wallace [29]) The objective of both thewait-and-see approach and the expected value approach is tooptimize expected performance of a supply chain under thetwo types of disruptionswith respect to two conflicting objec-tive functions expected cost and expected service Whilethe stochastic approach aims at optimizing the expectedperformance of a supply chain over all possible disruptionscenarios the deterministic approach accounts only for a sin-gle scenario representing the expected disruption conditionsThe stochastic programming approach determines a subset ofnondominated solutions for all disruption scenarios whereasthe deterministic approach produces a nondominated solu-tion for a single scenario only The solution of the expectedvalue problem does not take into account any distributioninformation and remains the same as long as the expectationsdo not change Unlike the expected value problem stochasticprogramming provides a recommendation for selection ofsupply portfolio that will hedge against a variety of disruptionscenarios The two approaches and the corresponding solu-tions are compared and some managerial insights derived

3 Problem Description

Consider a three-echelon supply chain (see Figure 1) inwhich a single producer of one product type assembles anddelivers products to multiple distribution centers to meetcustomer demand using a critical part type that can bemanufactured and provided by many suppliers

Let 119868 = 1 119868 be the set of 119868 suppliers let 119869 = 1 119869be the set of 119869 customers let 119870 = 1 119870 be the set of 119870distribution centers and let 119879 = 1 119879 be the set of 119879planning periods (for notations see Notations)

The orders for parts are assumed to be placed at the begin-ning of the planning horizon and the parts ordered fromsupplier 119894 are delivered in period120590119894 Each customer is suppliedwith the ordered products via exactly one distribution centerThe products for each customer 119895 isin 119869119896 are delivered tothe distribution center 119896 in a single delivery which cannotbe scheduled before all customer orders 119895 isin 119869119896 have beencompleted The products shipped in period 119905 to distributioncenter 119896 are delivered in period 119905 + 120591119896 minus 1

The suppliers of parts are located in119877 geographic regionsThe supplies are subject to random local disruptions of differ-ent levels 119897 isin 119871 = 0 119871 where the disruption level refersto the fraction of an order that can be delivered (fulfillment

rate) Level 119897 = 0 represents complete shutdown of a supplierthat is no order delivery while level 119897 = 119871 represents normalconditions with no disruption that is full order deliveryThe intermediate disruption levels 119897 = 1 119871 minus 1 representdifferent fractions of an order that can be delivered Thesmaller 119897 isin 119871 the smaller portion of an order that can bedelivered due to the smaller fraction of the supplier capacityavailable The fraction of an order that can be deliveredby supplier 119894 under disruption level 119897 is described by theassociated fulfillment rate 120574119894119897

120574119894119897 =

0 if 119897 = 0isin (0 1) if 119897 = 1 119871 minus 11 if 119897 = 119871

(1)

Denote by 119901119894119897 the probability of disruption level 119897 isin 119871for supplier 119894 that is the parts ordered from supplier 119894 aredelivered fully with probability119901119894119871 partially at different levelsof supplier output 120574119894119897 with probability 119901119894119897 119897 = 1 119871 minus 1 ornot at all with probability 1199011198940

In addition to independent local disruptions of eachsupplier there are potential regional disasters that may resultin complete shutdown of all suppliers in the same regionsimultaneously For example regional disaster events mayinclude an earthquake and flooding Let 119901119903 be the probabilityof regional disruptions of all suppliers 119894 isin 119868119903 in region 119903 isin 119877

Denote by 119878 = 1 119878 the index set of all disruptionscenarios where each scenario 119904 isin 119878 can be represented byan integer-valued vector 120582119904 = (1205821119904 120582119868119904) where 120582119894119904 isin 119871 isthe disruption level of an order delivery from supplier 119894 isin 119868under scenario 119904 isin 119878 All potential disruption scenarios willbe considered that is 119878 = (119871 + 1)119868

The probability 119875119904 for disruption scenario 119904 isin 119878 with thesubset 119868119904 of nonshutdown suppliers (that can deliver partsunder scenario 119904) is [26]

119875119904 = prod119903isin119877

119875119903119904 (2)

where 119875119903119904 is the probability of realizing of disruption scenario119904 for suppliers in 119868119903119875119903119904 =

(1 minus 119901119903)prod119894isin119868119903

(119901119894120582119894119904) if 119868119903 cap 119868119904 = 0119901119903 + (1 minus 119901119903)prod

119894isin1198681199031199011198940 if 119868119903 cap 119868119904 = 0 (3)

and 119901119894120582119894119904 is the probability of occurrence of the disruptionat level 119897 = 120582119894119904 of an order delivery from supplier 119894 underscenario 119904

The objective of the coordinated decision-making in asupply chain under multilevel disruptions is to allocate thetotal demand for parts among a subset of selected suppliers(ie to determine the supply portfolio) and to schedule foreach disruption scenario the customer orders for productsand the delivery of products to distribution centers tooptimize the trade-off between expected cost and expectedservice level

4 Mathematical Problems in Engineering

Supplier

Supplier

Producer

ProductsPartsDC

DC

Customer

Customer

Customer

Customer

Figure 1 A three-echelon supply chain

4 Problem Formulation Stochastic Approach

In this section a stochastic MIP model WCS is presentedfor the coordinated decision-making in the presence ofsupply chain under multilevel disruption risksThe followingdecisions are jointly made using the proposedmodel [25 26]

(i) Supply portfolio selection 119906119894 = 1 if supplier 119894 isselected otherwise 119906119894 = 0 and V119894 isin [0 1] the fractionof total demand for parts ordered from supplier 119894

(ii) Production scheduling 119908119904119895119905 = 1 if under disruptionscenario 119904 customer order 119895 is scheduled for period 119905otherwise 119908119904119895119905 = 0

(iii) Distribution scheduling 119909119904119896119905 = 1 if under disruptionscenario 119904 a shipment of products to distributioncenter 119896 is scheduled for period 119905 otherwise 119909119904119896119905 = 0

(iv) Customer order nondelayed delivery 119910119904119895 = 1 if underdisruption scenario 119904 customer order 119895 is delivered byits due date otherwise 119910119904119895 = 0

Thedemand allocation vector (V1 V119868) wheresum119894isin119868 V119894 =1 and 0 le V119894 le 1 119894 isin 119868 defines the supply portfoliointroduced by Sawik [30]

Let 1198641 be the minimized expected cost per product andlet 1198642 be the maximized expected service level

1198641 = sum119894isin119868 119890119894119906119894 + sum119904isin119878 119875119904 (sum119894isin119868119904 119861119900119894120574119894120582119894119904V119894 + sum119895isin119869 119892119895119887119895 (sum119905isin119879119908119904119895119905 minus 119910119904119895) + sum119895isin119869 ℎ119895119887119895 (1 minus sum119905isin119879119908119904119895119905))119861 (4)

1198642 = sum119895isin119869sum119904isin119878 119875119904119887119895119910119904119895119861 (5)

where 120582119894119904 is disruption level of supplier 119894 under scenario 119904 and120574119894120582119894119904 is the corresponding fulfillment rate that is the fractionof an order delivered by supplier 119894 under disruption scenario119904

The expected cost 1198641 (see (4)) constitutes fixed order-ing cost sum119894isin119868 119890119894119906119894 expected purchasing cost for deliveredparts sum119904isin119878 119875119904sum119894isin119868119904 119861119900119894120574119894120582119894119904V119894 expected penalty for delayedcustomer demand sum119904isin119878 119875119904sum119895isin119869 119892119895119887119895(sum119905isin119879119908119904119895119905 minus 119910119904119895) andexpected penalty for unsatisfied (rejected) customer demandsum119904isin119878 119875119904sum119895isin119869 ℎ119895119887119895(1 minus sum119905isin119879119908119904119895119905)

Denote by

1198911 = 1198641 minus 11986411198641 minus 1198641 (6)

the normalized (scaled into the interval [0 1]) expected costper product (1198641 1198641 are the minimum and the maximumvalues of 1198641 resp) and by

1198912 = 1198642 minus 11986421198642 minus 1198642 (7)

the normalized expected service level (1198642 1198642 are the mini-mum and the maximum values of 1198642 resp)Model WCS It consists in supplier selection customer orderand distribution scheduling to minimize Weighted sum ofnormalized expected Cost and expected Service level

Minimize

1205721198911 + (1 minus 120572) 1198912 (8)

where 0 le 120572 le 1 subject to (4)ndash(7)Supply Portfolio Selection Constraints

(i) The total demand for parts must be fully allocatedamong the selected suppliers

(ii) Demand for parts cannot be assigned to nonselectedsuppliers

sum119894isin119868

V119894 = 1 V119894 le 119906119894 119894 isin 119868 (9)

Mathematical Problems in Engineering 5

Customer Order Scheduling Constraints

(i) For each disruption scenario 119904 each customer order119895 is either scheduled during the planning hori-zon (sum119905isin119879119908119904119895119905 = 1) or unscheduled and rejected(sum119905isin119879119908119904119895119905 = 0)

(ii) For any period 119905 and each disruption scenario 119904the total demand on capacity of all customer ordersscheduled in period 119905 must not exceed the producercapacity

sum119905isin119879

119908119904119895119905 le 1 119895 isin 119869 119904 isin 119878sum119895isin119869

119887119895119908119904119895119905 le 119862 119905 isin 119879 119904 isin 119878 (10)

Supply-Production-Distribution Coordinating Constraints

(i) For each disruption scenario 119904 and each planningperiod 119905 the cumulative demand for parts of all cus-tomer orders scheduled in period 1 through 119905 cannotexceed the cumulative deliveries of parts in period 1through 119905 minus 1 from the nonshutdown suppliers 119894 isin 119868119904

(ii) For each disruption scenario shipment to distribu-tion center 119896 can be scheduled only after the latestcompletion period of scheduled customer orders 119895 isin119869119896

sum119895isin119869

sum1199051015840isin1198791199051015840le119905

1198871198951199081199041198951199051015840 le 119861 sum119894isin119868119904 120590119894le119905minus1

120574119894120582119894119904V119894 119905 isin 119879 119904 isin 119878 (11)

sum119905isin119879119870

119905119909119904119896119905 ge sum119905isin119879

(119905 + 1)119908119904119895119905119896 isin 119870 119895 isin 119869119896 119904 isin 119878

(12)

where 119879119870 = min119894isin119868120590119894 + 2 119879 + 1 is the set ofshipping periods

Shipping Constraints

(i) For each disruption scenario at most one shipmentcan be scheduled to each distribution center

sum119905isin119879119870

119909119904119896119905 le 1 119896 isin 119870 119904 isin 119878 (13)

Customer Due Date Meeting Constraints

(i) For each disruption scenario 119904 isin 119878 customer order119895 isin 119869119896 can be delivered without delay (ie 119910119904119895 =1) if it is scheduled not later than 119889119895 minus 120591119896 andshipped to distribution center 119896 not later than 119889119895 minus120591119896 + 1 otherwise the customer order is delayed orunscheduled (ie 119910119904119895 = 0)

119910119904119895 le sum119905isin119879119905le119889119895minus120591119896

119908119904119895119905119896 isin 119870 119895 isin 119869119896 119904 isin 119878

119910119904119895 le sum119905isin119879119870119905le119889119895minus120591119896+1

119909119904119896119905119896 isin 119870 119895 isin 119869119896 119904 isin 119878

sum119905isin119879119905le119889119895minus120591119896

119908119904119895119905 + sum119905isin119879119870119905le119889119895minus120591119896+1

119909119904119896119905 minus 1 le 119910119904119895119896 isin 119870 119895 isin 119869119896 119904 isin 119878

(14)

Nonnegativity and Integrality Conditions

119906119894 isin 0 1 119894 isin 119868 (15)

V119894 isin [0 1] 119894 isin 119868 (16)

119908119904119895119905 isin 0 1 119895 isin 119869 119905 isin 119879 119904 isin 119878 (17)

119909119904119896119905 isin 0 1 119896 isin 119870 119905 isin 119879119870 119904 isin 119878 (18)

119910119904119895 ge 0 119895 isin 119869 119904 isin 119878 (19)

Note that119910119904119895 does not need to be restricted to being binarysince for any feasible solution satisfying constraints (14) 119910119904119895is always binary

Model WCS is a deterministic equivalent mixed integerprogram of stochastic mixed integer program When thisproblem is solved a recommendation is obtained for selec-tion of supply portfolio (119906119894 V119894) thatwill hedge against a varietyof disruption scenarios in which fulfillment rates of certainsuppliers are not sufficient to satisfy demand for parts Thescheduling variables 119908119904119895119905 119909119904119896119905 (and 119910119904119895) are decisions that willbe implemented in the future when scenario 119904 isin 119878 is finallyrealized

Model WCS illustrates the wait-and-see approach (eg[27]) Basically this approach is based on perfect informationabout the future Model WCS can be decomposed into atwo-stage stochastic mixed integer program with recourseThe supply portfolio selection variables (119906119894 V119894) are referredto as first-stage decisions and the scheduling variables(119908119904119895119905 119909119904119896119905 119910119904119895) are referred to as recourse or second-stagedecisions Unlike the first-stage decisions the latter variablesare dependent on the scenario 119904 isin 119878

Stochastic mixed integer programs are usually hard tosolve because they are large-scale optimization problemswhen applied to real-world problems A common approach(eg [27]) is to consider a simpler deterministic programknown as expected value problem in which the randomparameters are replaced by their expected values or toconsider several deterministic programs each of which cor-responds to one particular scenario and then to combine theobtained solutions into a single heuristic solution

5 Problem FormulationDeterministic Approach

In this section the expected value problem EWCS is pre-sented for the coordinated supply chain scheduling under

6 Mathematical Problems in Engineering

expected supply conditions In model WCS where the ran-domness is characterized by a set of disruption scenariosthe only random parameters are suppliers fulfillment rates120574119894120582119894119904 which appear both in the objective function (4) and inconstraints (11)

In model EWCS suppliers probabilistic fulfillment ratesdefined for each disruption scenario 120574119894120582119894119904 119894 isin 119868 119904 isin 119878 orequivalently for each disruption level 120574119894119897 119894 isin 119868 119897 isin 119871 (1)have been replaced by the expected fulfillment rates of eachsupplier

Γ119894 = sum119904isin119878

119875119904120574119894120582119894119904 119894 isin 119868 (20)

or equivalently

Γ119894 = (1 minus 119901119903)sum119897isin119871

119901119894119897120574119894119897 119894 isin 119868119903 119903 isin 119877 (21)

Accordingly stochastic binary decision variables 119908119904119895119905 119909119904119896119905 119910119904119895(17)ndash(19) defined for each disruption scenario 119904 isin 119878 havebeen replaced by their deterministic equivalents 119882119895119905 119883119896119905119884119895

Now the expected cost per product 1198641 (see (22)) andthe expected service level 1198642(see (23)) are defined as fol-lows

1198641 = sum119894isin119868 119890119894119906119894 + sum119894isin119868 119861119900119894Γ119894V119894 + sum119895isin119869 119892119895119887119895 (sum119905isin119879119882119895119905 minus 119884119895) + sum119895isin119869 ℎ119895119887119895 (1 minus sum119905isin119879119882119895119905)119861 (22)

1198642 = sum119895isin119869 119887119895119884119895119861 (23)

Model EWCS is presented below

Model EWCS

Minimize (8)subject to (6) (7) (9) (22) (23)

sum119905isin119879

119882119895119905 le 1 119895 isin 119869sum119895isin119869

119887119895119882119895119905 le 119862 119905 isin 119879sum119895isin119869

sum1199051015840isin1198791199051015840le119905

1198871198951198821198951199051015840 le 119861 sum119894isin119868120590119894le119905minus1

Γ119894V119894 119905 isin 119879sum119905isin119879119870

119905119883119896119905 ge sum119905isin119879

(119905 + 1)119882119895119905119896 isin 119870 119895 isin 119869119896

sum119905isin119879119870

119883119896119905 le 1 119896 isin 119870119884119895 le sum119905isin119879119905le119889119895minus120591119896

119882119895119905 119896 isin 119870 119895 isin 119869119896119884119895 le sum119905isin119879119870119905le119889119895minus120591119896+1

119883119896119905 119896 isin 119870 119895 isin 119869119896sum

119905isin119879119905le119889119895minus120591119896

119882119895119905 + sum119905isin119879119870119905le119889119895minus120591119896+1

119883119896119905 minus 1le 119884119895 119896 isin 119870 119895 isin 119869119896119906119894 isin 0 1 119894 isin 119868V119894 isin [0 1] 119894 isin 119868

119882119895119905 isin 0 1 119895 isin 119869 119905 isin 119879119883119896119905 isin 0 1 119896 isin 119870 119905 isin 119879119870119884119895 ge 0 119895 isin 119869

(24)

Notice that unlike the stochastic programming modelWCS which is formulated to determine optimal schedulesfor all potential disruption scenarios model EWCS accountsfor a single scenario only representing the expected suppliesExcept for the expected values of the random parametersthis model does not take into account any distributioninformation and the solution remains the same as long as theexpectations do not change In contrast tomodelWCS wherethe selection of supply portfolio (V1 V119868) is combinedwith supply chain scheduling for all disruption scenariosconsidered now the portfolio is determined along with asingle schedule

If randomparameters appear only in the constraints then[27]

EV le WS (25)

where EV is the optimal solution value of the expected valueproblem EWCS and WS is the optimal solution value ofthe wait-and-see problem WCS On the other hand whenuncertainty is limited to the objective function of the prob-lem the solution obtained by simply replacing the randomparameters with their expected values provides already arobust alternative (Delage et al [31])

6 Computational Examples

In this section some computational examples are presentedto illustrate possible applications of the proposedMIPmodelsand to compare the wait-and-see and the expected value

Mathematical Problems in Engineering 7

approaches The examples are modeled in part after a real-world electronics supply chain (eg Sawik [32]) The fol-lowing parameters have been selected for the computationalexamples119868 = 4 suppliers 119869 = 20 customer orders 119870 = 2 dis-tribution centers 119871 = 3 partial disruption levels 119877 = 2geographic regions and 119879 = 7 planning periods1198681 = 1 2 1198682 = 3 41198691 = 1 10 1198692 = 11 20

Shipping times from suppliers 120590 = (1 1 3 3)Shipping times to distribution centers 120591 = (1 2)Customer demand 119887119895 isin 2000 3000 10000 for all119895 isin 119869 and total demand 119861 = 100000Due dates 119889119895 isin 2 + min119894isin119868(120590119894) + min119896isin119870(120591119896) 119879 +

max119896isin119870(120591119896) for all 119895 isin 119869Fixed ordering costs for suppliers 119890 = (8000 6000 1200013000)Unit purchasing prices from suppliers 119900 = (14 12 8 9)Unit penalties for delayed unfulfilled customer orders119892119895 = lceilmax119894isin119868119900119894350rceil = 1 ℎ119895 = 2max119894isin119868119900119894 = 28 respectively

for all 119895 isin 119869Producer capacity 119862 = 45000Local disruption levels and the associated fulfillment rates

(the percentage of an order that can be delivered) are shownbelow119871 = 0 1 2 3 where 119897 = 0 complete shutdown1205741198940 = 0 forall119894 isin 119868 that is 0 of an order delivered 119897 = 1major disruption 1205741198941 isin [001 050] forall119894 isin 1198681 and 1205741198941 isin[001 030] forall119894 isin 1198682 that is 1 to 50 and 1 to 30 ofan order delivered respectively 119897 = 2 minor disruption

1205741198942 isin [0 51 099] forall119894 isin 1198681 and 1205741198942 isin [0 31 099] forall119894 isin 1198682that is 51 to 99 and 31 to 99 of an order deliveredrespectively 119897 = 119871 = 3 no disruption 1205741198943 = 1 forall119894 isin 119868 thatis 100 of an order delivered

The total number of all potential scenarios is 119878 = (119871 +1)119868 = 44 = 256 scenarios where each scenario 119904 isin 119878 is repre-sented by vector 120582119904 = (1205821119904 1205824119904) where 120582119894119904 isin 119871 119894 isin 119868 (seeTable 1)

The local nondisruption probability (level 119897 = 3) 1199011198943was uniformly distributed over [089 099] and [079 089]respectively for suppliers 119894 isin 1198681 and 119894 isin 1198682 that is theprobabilities were drawn independently from U[089 099]and U[079 089] respectively

Given local nondisruption probabilities 1199011198943 119894 isin 119868 theprobabilities for the remaining local disruption levels 119897 =0 1 2 were calculated as follows

Probability of complete shutdown (level 119897 = 0) 1199011198940 =01(1 minus 1199011198943)Probability of major disruption (level 119897 = 1) 1199011198941 =03(1 minus 1199011198943)Probability of minor disruption (level 119897 = 2) 1199011198942 =06(1 minus 1199011198943) for all suppliers 119894 isin 119868

Thus 1199011198940 le 1199011198941 le 1199011198942 le 1199011198943 which reflects a real-world relation among probabilities of disruption occurrenceat different levels (eg [8])

Regional disruption probabilities are 1199011 = 0001 and1199012 =001The probability of realizing of disruption scenario 119904 for

suppliers in region 119903 = 1 2 is calculated as follows

119875119903119904

=(1 minus 119901119903)( prod

119894isin119868119903 120582119894119904=0

01 (1 minus 1199011198943))( prod119894isin119868119903 120582119894119904=1

03 (1 minus 1199011198943))( prod119894isin119868119903120582119894119904=2

06 (1 minus 1199011198943))( prod119894isin119868119903 120582119894119904=3

1199011198943) if sum119894isin119868119903120582119894119904 gt 0

119901119903 + (1 minus 119901119903)prod119894isin119868119903

01 (1 minus 1199011198943) if sum119894isin119868119903120582119894119904 = 0

(26)

and the probability for disruption scenario 119904 isin 119878 is given by119875119904 = 1198751119904 1198752119904 Figure 2 presents basic characteristics of all suppliers

probability of complete shutdown119901119903+(1minus119901119903)1199011198940 119894 isin 119868119903 119903 isin119877 expected fulfillment rate Γ119894 = sum119897=123(1 minus 119901119903)119901119894119897120574119894119897 119894 isin119868119903 119903 isin 119877 and purchasing price 119900119894 119894 isin 119868Table 2 presents a subset of nondominated solutions for

the wait-and-see problem WCS for a subset of trade-offparameter120572 isin 0 01 02 03 04 05 06 07 08 09 1Theresults indicate that most of nondominated supply portfoliosconsist of the two suppliers only the second most expensiveand most reliable supplier 119894 = 2 from region 119903 = 1 andthe cheapest and most unreliable supplier 119894 = 3 from region119903 = 2 The most expensive and most reliable supplier 119894 = 1in region 119903 = 1 and the second cheapest and most unreliablesupplier 119894 = 4 in region 119903 = 2 are rarely selected For 120572 = 1

(minimization of cost) the cheapest supplier 119894 = 3 is selectedonly For 120572 = 0 (maximization of service level) the totaldemand for parts is allocated among the two most reliableand most expensive suppliers 119894 = 1 2 The above solutionshows that the service-oriented supply portfolio (120572 close to 0)is more diversified than the cost-oriented portfolio (120572 closeto 1) Table 2 also shows the associated expected fractionof fulfilled demand 1198643 = sum119904isin119878sum119895isin119869sum119905isin119879 119875119904119887119895119908119904119895119905119861 that isdemand fulfilled on time or delayed The solution resultsdemonstrate that a large expected service level is sometimesassociated with a small expected fraction of fulfilled demandThus the maximization of expected service level leads torejection of customer orders that cannot be fulfilled bycustomer requested due dates

Figure 3 shows the expected supply production andshipping schedules respectively for 120572 = 0 (ie for the

8 Mathematical Problems in Engineering

Shutdown probability

Supplier

Expected yield rate

Purchasing price

4321

Supplier4321

Supplier4321

6

8

10

12

14

16

09

092

094

096

098

0

0005

001

0015

002

0025

Figure 2 Suppliers

1 2 3 4 5 6 7Period

Expected supply schedules

Alpha = 0Alpha = 05

Alpha = 1

1 2 3 4 5 6 7Period

Expected production schedules

1 2 3 4 5 6 7Period

Expected shipping schedules

0

20000

40000

60000

80000

100000

Part

s

0

10000

20000

30000

40000

50000

Prod

ucts

Alpha = 0Alpha = 05

Alpha = 1

0

10000

20000

30000

40000

50000

Prod

ucts

Alpha = 0Alpha = 05

Alpha = 1

Figure 3 Expected schedules for modelWCS

Mathematical Problems in Engineering 9

Table 1 Disruption scenarios

119904 119894 = 1 2 3 41 0 0 0 02 0 0 0 13 0 0 0 24 0 0 0 35 0 0 1 06 0 0 1 17 0 0 1 28 0 0 1 39 0 0 2 010 0 0 2 111 0 0 2 212 0 0 2 313 0 0 3 014 0 0 3 115 0 0 3 216 0 0 3 317 0 1 0 018 0 1 0 119 0 1 0 220 0 1 0 321 0 1 1 022 0 1 1 123 0 1 1 224 0 1 1 325 0 1 2 026 0 1 2 127 0 1 2 228 0 1 2 329 0 1 3 030 0 1 3 131 0 1 3 232 0 1 3 333 0 2 0 034 0 2 0 135 0 2 0 236 0 2 0 337 0 2 1 038 0 2 1 139 0 2 1 240 0 2 1 341 0 2 2 042 0 2 2 143 0 2 2 244 0 2 2 345 0 2 3 046 0 2 3 147 0 2 3 248 0 2 3 349 0 3 0 0

Table 1 Continued

119904 119894 = 1 2 3 450 0 3 0 151 0 3 0 252 0 3 0 353 0 3 1 054 0 3 1 155 0 3 1 256 0 3 1 357 0 3 2 058 0 3 2 159 0 3 2 260 0 3 2 361 0 3 3 062 0 3 3 163 0 3 3 264 0 3 3 365 1 0 0 066 1 0 0 167 1 0 0 268 1 0 0 369 1 0 1 070 1 0 1 171 1 0 1 272 1 0 1 373 1 0 2 074 1 0 2 175 1 0 2 276 1 0 2 377 1 0 3 078 1 0 3 179 1 0 3 280 1 0 3 381 1 1 0 082 1 1 0 183 1 1 0 284 1 1 0 385 1 1 1 086 1 1 1 187 1 1 1 288 1 1 1 389 1 1 2 090 1 1 2 191 1 1 2 292 1 1 2 393 1 1 3 094 1 1 3 195 1 1 3 296 1 1 3 397 1 2 0 098 1 2 0 1

10 Mathematical Problems in Engineering

Table 1 Continued

119904 119894 = 1 2 3 499 1 2 0 2100 1 2 0 3101 1 2 1 0102 1 2 1 1103 1 2 1 2104 1 2 1 3105 1 2 2 0106 1 2 2 1107 1 2 2 2108 1 2 2 3109 1 2 3 0110 1 2 3 1111 1 2 3 2112 1 2 3 3113 1 3 0 0114 1 3 0 1115 1 3 0 2116 1 3 0 3117 1 3 1 0118 1 3 1 1119 1 3 1 2120 1 3 1 3121 1 3 2 0122 1 3 2 1123 1 3 2 2124 1 3 2 3125 1 3 3 0126 1 3 3 1127 1 3 3 2128 1 3 3 3129 2 0 0 0130 2 0 0 1131 2 0 0 2132 2 0 0 3133 2 0 1 0134 2 0 1 1135 2 0 1 2136 2 0 1 3137 2 0 2 0138 2 0 2 1139 2 0 2 2140 2 0 2 3141 2 0 3 0142 2 0 3 1143 2 0 3 2144 2 0 3 3145 2 1 0 0146 2 1 0 1147 2 1 0 2148 2 1 0 3149 2 1 1 0150 2 1 1 1

Table 1 Continued

119904 119894 = 1 2 3 4151 2 1 1 2152 2 1 1 3153 2 1 2 0154 2 1 2 1155 2 1 2 2156 2 1 2 3157 2 1 3 0158 2 1 3 1159 2 1 3 2160 2 1 3 3161 2 2 0 0162 2 2 0 1163 2 2 0 2164 2 2 0 3165 2 2 1 0166 2 2 1 1167 2 2 1 2168 2 2 1 3169 2 2 2 0170 2 2 2 1171 2 2 2 2172 2 2 2 3173 2 2 3 0174 2 2 3 1175 2 2 3 2176 2 2 3 3177 2 3 0 0178 2 3 0 1179 2 3 0 2180 2 3 0 3181 2 3 1 0182 2 3 1 1183 2 3 1 2184 2 3 1 3185 2 3 2 0186 2 3 2 1187 2 3 2 2188 2 3 2 3189 2 3 3 0190 2 3 3 1191 2 3 3 2192 2 3 3 3193 3 0 0 0194 3 0 0 1195 3 0 0 2196 3 0 0 3197 3 0 1 0198 3 0 1 1199 3 0 1 2200 3 0 1 3201 3 0 2 0202 3 0 2 1

Mathematical Problems in Engineering 11

Table 1 Continued

119904 119894 = 1 2 3 4203 3 0 2 2204 3 0 2 3205 3 0 3 0206 3 0 3 1207 3 0 3 2208 3 0 3 3209 3 1 0 0210 3 1 0 1211 3 1 0 2212 3 1 0 3213 3 1 1 0214 3 1 1 1215 3 1 1 2216 3 1 1 3217 3 1 2 0218 3 1 2 1219 3 1 2 2220 3 1 2 3221 3 1 3 0222 3 1 3 1223 3 1 3 2224 3 1 3 3225 3 2 0 0226 3 2 0 1227 3 2 0 2228 3 2 0 3229 3 2 1 0230 3 2 1 1231 3 2 1 2232 3 2 1 3233 3 2 2 0234 3 2 2 1235 3 2 2 2236 3 2 2 3237 3 2 3 0238 3 2 3 1239 3 2 3 2240 3 2 3 3241 3 3 0 0242 3 3 0 1243 3 3 0 2244 3 3 0 3245 3 3 1 0246 3 3 1 1247 3 3 1 2248 3 3 1 3249 3 3 2 0250 3 3 2 1251 3 3 2 2252 3 3 2 3

Table 1 Continued

119904 119894 = 1 2 3 4253 3 3 3 0254 3 3 3 1255 3 3 3 2256 3 3 3 3

maximum expected service level) 120572 = 05 and 120572 = 1 (ie forthe minimum expected cost) The expected schedules werecomputed using the formulae presented below

(i) Expected schedule of supplies of parts to the pro-ducer

sum119904isin119878

sum119894isin119868119904120590119894=119905

119875119904119861120574119894120582119894119904V119894 119905 isin 119879 (27)

(ii) Expected production schedule

sum119904isin119878

sum119895isin119869

119875119904119887119895119908119904119895119905 119905 isin 119879 (28)

(iii) Expected schedule of shipping of products from theproducer to the distribution centers

sum119904isin119878

sum119896isin119870

sum119895isin119869119896

119875119904119887119895( sum1199051015840isin1198791199051015840lt119905

1199081199041198951199051015840)119909119904119896119905 119905 isin 119879 (29)

As 120572 increases that is the decision-makerrsquos priority shiftsfrom the maximum service level to minimum cost and moreparts are ordered from less reliable and lower cost suppliersthe expected supply schedules and the corresponding pro-duction schedules are more delayed as well as the delivery ofproducts to the customers Note that despite constraint (13)that ensures a feasible schedule with at most one shipmentof products to each distribution center for each disruptionscenario the expected shipping schedule (29) may be splitinto more smaller size shipments (cf Figure 3 where twomajor and one small-size shipments are indicated for eachconfidence level)

For comparison Table 3 presents a subset of non-dominated solutions obtained for the expected value problemEWCS and Figure 4 shows supply production and shippingschedules Unlike the stochastic programming approachwhich accounts for all potential disruption scenarios tooptimize an expected performance of the supply chain thesolution obtained using the deterministic approach is basedon aggregate information on suppliers expected fulfillmentIn general the results are similar for both models and thecorresponding optimal solution values are close to each otherwhich indicates that the expected value problem can be usedin practice when stochastic mixed integer programs are hardto solve The optimal solution values for the expected valueproblem frequently outperform the corresponding solutionvalues of the wait-and-see problem which is in line withthe proposition EV le WS in [27] compare Section 5for example the minimum cost 1198641 for EWCS (see Table 3)

12 Mathematical Problems in Engineering

Table 2 Nondominated solutions for modelWCS

120572 0 01 02 03 04 05 and 06 07 08 and 09 1Var = 38077 Bin = 33076 Cons = 28769 Nonz = 348325(a)

Exp cost 1731 (1198641) 1620 1547 1206 1200 1176 1070 1065 1051 (1198641)Exp service level(b) 8552 (1198642) 8487 8452 7719 7676 7595 5863 5871 3822 (1198642)Exp fulfilled demand(c) 8552 8530 8452 9370 9361 9353 9134 9120 9068

Suppliers selected( of total demand)

1 (76) 1 (35)2 (24) 2 (55) 2 (87) 2 (55) 2 (55) 2 (55) 2 (12) 2 (10)3 (10) 3 (13) 3 (22) 3 (32) 3 (45) 3 (88) 3 (90) 3 (100)4 (23) 4 (13)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119904isin119878 sum119895isin119869 119875119904119887119895119910

119904119895119861 times

100 (c)sum119904isin119878 sum119895isin119869 sum119905isin119879 119875119904119887119895119908119904119895119905119861 times 100

Table 3 Nondominated solutions for model EWCS

120572 0 01 02 and 03 04 and 05 06 and 07 08 09 and 1Var = 162 Bin = 136 Cons = 125 Nonz = 1430(a)

Exp cost 1691 (1198641) 1427 1214 1167 1045 (1198641)Exp service level(b) 87 (1198642) 86 81 79 62 (1198642)Exp fulfilled demand(c) 87 90 91 93 91

Suppliers selected( of total demand)

1 (89) 2 (94) 2 (49) 2 (50) 2 (5)3 (6) 3 (51) 3 (50) 3 (95)4 (11)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119895isin119869 119887119895119884119895119861 times 100(c)sum119895isin119869 sum119905isin119879 119887119895119882119895119905119861 times 100

1 2 3 4 5 6 7Period

Supply schedules

1 2 3 4 5 6 7Period

Production schedules

1 2 3 4 5 6 7Period

Shipping schedules

0

10000

20000

30000

40000

50000

Prod

ucts

0

10000

20000

30000

40000

50000

Prod

ucts

0

20000

40000

60000

80000

100000

Part

s

Alpha = 0Alpha = 05

Alpha = 1 Alpha = 0Alpha = 05

Alpha = 1

Alpha = 0Alpha = 05

Alpha = 1

Figure 4 Schedules for model EWCS

Mathematical Problems in Engineering 13

1045 le 1051 forWCS (see Table 2) or themaximum servicelevel1198642 for EWCS (see Table 3) 8700 ge 8552 forWCS (seeTable 2)

The stochastic wait-and-see approach however leads toa more diversified supply portfolio For the deterministicapproach most of nondominated supply portfolios consistof two suppliers only 119894 = 2 3 while suppliers 119894 = 1 4 areselected only for the maximum service level objective (thatis for 120572 = 0) Comparison of Figures 3 and 4 indicatesthat the expected schedules for model WCS computed asexpectations over all schedules for all potential disruptionscenarios (27)ndash(29) are similar to the corresponding singleschedules determined by model EWCS The main differencesobserved aremore delayed expected production and shippingschedules for model WCS when minimization of cost is con-sidered (that is for120572 = 1) Finally it is interesting to note thatin the multiple sourcing environment considered both thewait-and-see approach and in particular the expected valueapproach frequently select a dual sourcing supply portfoliowith one main supplier and one supporting supplier

The computational experiments were performed usingthe AMPL programming language and the CPLEX 1262solver on a MacBook Pro laptop with Intel Core i7 processorrunning at 28 GHz and with 16GB RAM The solver wascapable of finding proven optimal solution for all exampleswith CPU time ranging from several minutes to several hoursfor the stochastic model WCS and fraction of a second forthe deterministic model EWCS Since the stochastic pro-gramming model WCS needs to determine nondominatedschedules for all potential disruption scenarios while modelEWCS deals with a single scenario only the difference in thecomputational effort required is obvious

7 Conclusions

In this paper two biobjective MIP formulations stochasticand deterministic have been proposed and compared for thecoordinated decision-making in supply chains under partiallocal disruptions and all-or-nothing regional disruptionsTheproblem objective has been to jointly schedule supplies pro-duction and distribution to optimize the trade-off betweenexpected cost and expected service levelWhile the stochasticprogramming approach aims at optimizing the expectedperformance of a supply chain over all possible disruptionscenarios the deterministic approach accounts only for asingle scenario representing average disruption conditionsAs a result the stochastic programming approach determinesnondominated solutions for all disruption scenarios whereasthe deterministic approach produces a single solution onlyIn particular in the stochastic programming approach theselection of supply portfolio is combined with supply chainscheduling for all disruption scenarios considered In con-trast the deterministic expected value approach provides theportfolio along with a single expected schedule of productionand distribution

The expected schedules obtained for the stochastic pro-gramming model WCS as expectations over all schedulesfor all disruption scenarios have been compared with thecorresponding schedules determined by the deterministic

model EWCS based on an expected disruption scenarioThecomparison has demonstrated that the two approaches leadto similar expected solutions

The main findings are in line with other research and arelisted as follows

(i) The two decision-making approaches stochastic anddeterministic lead to similar expected performanceof a supply chain under multilevel disruptions

(ii) The optimal solution values for the expected valueproblem frequently outperform the correspondingsolution values of the wait-and-see problem

(iii) Despite the multiple sourcing environment consid-ered both the wait-and-see approach and in partic-ular the expected value approach frequently selecta dual sourcing supply portfolio with one mainsupplier and one supporting supplier

(iv) The stochastic approach which accounts for allpotential disruption scenarios may lead to a morediversified supply portfolio that will hedge against avariety of scenarios

(v) The expected schedules are more delayed for thestochastic approach

(vi) The service-oriented supply portfolio is more diversi-fied andmay combine both high-cost reliable suppli-ers and low-cost unreliable suppliers while the cost-oriented portfolio depends mainly on low-cost andless reliable suppliers

Overall the results of computational experiments indi-cate that the proposed approach and developed MIP modelsare flexible and efficient tools for coordinated supply chainscheduling The portfolio approach leads to MIP formula-tions with strong LP relaxations and has been proven to becomputationally very efficient CPU time required to findproven optimal solutions for realistic size examples usingcommercially available software for MIP is acceptable for areal-world supply chain disruption management (see Sawik[33])

Since the probability distribution of supply disruptionsfrom each supplier is usually unknown (eg [8]) local multi-level disruptions in the stochastic programming modelWCShave a multinomial discrete distribution while the two-levelregional disruptions have a binomial discrete distributionand all disruption events are independent As part of futureresearch we propose to enhance the stochastic programmingmodel formore general scenarios with finitelymany elementssingled out and all the probability concentrated in them Forexample rather than complete shutdown suppliers within aregion may have correlated disruptions Another importantstream of future research is the study of robustness andsensitivity in relation to input data changes in supply chainscheduling under disruption risks (eg [33]) Future researchcan also focus on the following improvement of the proposedmodels to consider the minimization of time and cost ofrecovery (eg Whitney et al [34])

14 Mathematical Problems in Engineering

Notations

Indices

119894 Supplier 119894 isin 119868119895 Customer 119895 isin 119869119896 Distribution center 119896 isin 119870119897 Disruption level 119897 isin 119871119903 Region 119903 isin 119877119904 Disruption scenario 119904 isin 119878119905 Planning period 119905 isin 119879Input Parameters

119887119895 Size (number of products) of customer order 119895119861 Total demand for partsproducts 119861 = sum119895isin119869 119887119895119862 Capacity of producer119889119895 Due date for customer order 119895119890119894 Fixed cost of ordering parts from supplier 119894119892119895 Per unit penalty cost of delayed customer order 119895ℎ119895 Per unit penalty cost of unfulfilled customer order 119895119868119903 Subset of suppliers in region 119903119869119896 Subset of customers serviced by distribution center 119896119900119894 Per unit price of parts purchased from supplier 119894119901119894119897 Probability of disruption level 119897 for supplier 119894119901119903 Regional disruption probability for region 119903120574119894119897 Fraction of an order delivered by supplier 119894 underdisruption level 119897 (fulfillment rate)Γ119894 Expected fraction of an order delivered by supplier 119894(expected fulfillment rate)120590119894 Delivery lead time from supplier 119894120591119896 Transportation time to distribution center 119896

Conflicts of Interest

The author declares that there are no conflicts of interest

Acknowledgments

This work has been supported by NCN research grant (noDEC-201311BST804458) and by AGH (no 1111200324)

References

[1] J V Blackhurst K P Scheibe and D J Johnson ldquoSupplierrisk assessment and monitoring for the automotive industryrdquoInternational Journal of Physical Distribution amp Logistics Man-agement vol 38 no 2 pp 143ndash165 2008

[2] P Hoffmann H Schiele and K Krabbendam ldquoUncertaintysupply risk management and their impact on performancerdquoJournal of Purchasing amp Supply Management vol 19 no 3 pp199ndash211 2013

[3] Y Park P Hong and J J Roh ldquoSupply chain lessons from thecatastrophic natural disaster in Japanrdquo Business Horizons vol56 no 1 pp 75ndash85 2013

[4] M Haraguchi and U Lall ldquoFlood risks and impacts A casestudy of Thailandrsquos floods in 2011 and research questions forsupply chain decisionmakingrdquo International Journal of DisasterRisk Reduction vol 14 pp 256ndash272 2015

[5] T Fujimoto and Y W Park ldquoBalancing supply chain com-petitiveness and robustness through ldquovirtual dual sourcingrdquoLessons from the Great East Japan Earthquakerdquo InternationalJournal of Production Economics vol 147 pp 429ndash436 2014

[6] HMatsuo ldquoImplications of the Tohoku earthquake for Toyotarsquoscoordination mechanism supply chain disruption of auto-motive semiconductorsrdquo International Journal of ProductionEconomics vol 161 pp 217ndash227 2015

[7] J R Marszewska ldquoImplications of seismic hazard in Japan ontoyota supply chain disruption risksrdquo in Proceedings of the 13thInternational Conference on Industrial Logistics (ICIL rsquo16) pp178ndash185 Zakopane Poland October 2016

[8] A Z Zeng and Y Xia ldquoBuilding a mutually beneficial partner-ship to ensure backup supplyrdquo Omega vol 52 pp 77ndash91 2015

[9] S S Erenguc N C Simpson and A J Vakharia ldquoIntegratedproductiondistribution planning in supply chains an invitedreviewrdquo European Journal of Operational Research vol 115 no2 pp 219ndash236 1999

[10] L Lei S Liu A Ruszczynski and S Park ldquoOn the integratedproduction inventory and distribution routing problemrdquo IIETransactions vol 38 no 11 pp 955ndash970 2006

[11] A Kaur A Kanda and S G Deshmukh ldquoA graph theoreticapproach for supply chain coordinationrdquo International Journalof Logistics Systems and Management vol 2 no 4 pp 321ndash3412006

[12] T ChoiW Yeung andT C E Cheng ldquoScheduling and co-ordi-nation of multi-suppliers single-warehouse-operator single-manufacturer supply chains with variable production rates andstorage costsrdquo International Journal of Production Research vol51 no 9 pp 2593ndash2601 2013

[13] S Liu and L G Papageorgiou ldquoMultiobjective optimisation ofproduction distribution and capacity planning of global supplychains in the process industryrdquo Omega vol 41 no 2 pp 369ndash382 2013

[14] Z L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[15] T Sawik ldquoCoordinated supply chain schedulingrdquo InternationalJournal of Production Economics vol 120 no 2 pp 437ndash4512009

[16] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[17] D Ivanov B Sokolov and A Dolgui ldquoThe Ripple effect in sup-ply chains Trade-off rsquoefficiency-flexibility- resiliencersquo in disrup-tionmanagementrdquo International Journal of ProductionResearchvol 52 no 7 pp 2154ndash2172 2014

[18] D Ivanov A Pavlov and B Sokolov ldquoOptimal distribution(re)planning in a centralized multi-stage supply network underconditions of the ripple effect and structure dynamicsrdquo Euro-pean Journal of Operational Research vol 237 no 2 pp 758ndash770 2014

[19] D Ivanov A Pavlov A Dolgui D Pavlov and B SokolovldquoDisruption-driven supply chain (re)-planning and perfor-mance impact assessment with consideration of pro-active andrecovery policiesrdquo Transportation Research Part E Logistics andTransportation Review vol 90 pp 7ndash24 2016

[20] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

Mathematical Problems in Engineering 15

[21] A J Schmitt S A Sun L V Snyder and Z-J M Shen ldquoCen-tralization versus decentralization Risk pooling risk diversifi-cation and supply chain disruptionsrdquo Omega vol 52 pp 201ndash212 2015

[22] L V Snyder Z Atan P Peng Y Rong A J Schmitt andB Sinsoysal ldquoORMS models for supply chain disruptions areviewrdquo IIE Transactions vol 48 no 2 pp 89ndash109 2016

[23] T Sawik ldquoIntegrated selection of suppliers and scheduling ofcustomer orders in the presence of supply chain disruptionrisksrdquo International Journal of Production Research vol 51 no23-24 pp 7006ndash7022 2013

[24] T Sawik ldquoOptimization of cost and service level in the presenceof supply chain disruption risks Single vs multiple sourcingrdquoComputers amp Operations Research vol 51 pp 11ndash20 2014

[25] T Sawik ldquoIntegrated supply production and distributionscheduling under disruption risksrdquoOmega vol 62 pp 131ndash1442016

[26] T Sawik ldquoIntegrated supply chain scheduling under multi-leveldisruptionsrdquo IFAC-PapersOnLine vol 28 no 3 pp 1515ndash15202015

[27] P Kall and J Mayer Stochastic Linear Programming ModelsTheory and Computation vol 156 of International Series inOperations Research amp Management Science Springer NewYork 2011

[28] I N Durbach and T J Stewart ldquoUsing expected values to sim-plify decision making under uncertaintyrdquo Omega vol 37 no 2pp 312ndash330 2009

[29] F Maggioni and S W Wallace ldquoAnalyzing the quality of theexpected value solution in stochastic programmingrdquo Annals ofOperations Research vol 200 pp 37ndash54 2012

[30] T Sawik ldquoSelection of supply portfolio under disruption risksrdquoOmega vol 39 no 2 pp 194ndash208 2011

[31] E Delage S Arroyo and Y Ye ldquoThe value of stochastic mod-eling in two-stage stochastic programs with cost uncertaintyrdquoOperations Research vol 62 no 6 pp 1377ndash1393 2014

[32] T Sawik Scheduling in Supply Chains Using Mixed Integer Pro-gramming John Wiley amp Sons Inc Hoboken NJ 2011

[33] T Sawik Supply Chain DisruptionManagement Using StochasticMixed Integer Programming vol 256 of International Seriesin Operations Research amp Management Science Springer NewYork 2017 httpwwwspringercomgpbook9783319588223

[34] D E Whitney J Luo and D A Heller ldquoThe benefits and con-straints of temporary sourcing diversification in supply chaindisruption and recoveryrdquo Journal of Purchasing and SupplyManagement vol 20 no 4 pp 238ndash250 2014

4 Mathematical Problems in Engineering

Supplier

Supplier

Producer

ProductsPartsDC

DC

Customer

Customer

Customer

Customer

Figure 1 A three-echelon supply chain

4 Problem Formulation Stochastic Approach

In this section a stochastic MIP model WCS is presentedfor the coordinated decision-making in the presence ofsupply chain under multilevel disruption risksThe followingdecisions are jointly made using the proposedmodel [25 26]

(i) Supply portfolio selection 119906119894 = 1 if supplier 119894 isselected otherwise 119906119894 = 0 and V119894 isin [0 1] the fractionof total demand for parts ordered from supplier 119894

(ii) Production scheduling 119908119904119895119905 = 1 if under disruptionscenario 119904 customer order 119895 is scheduled for period 119905otherwise 119908119904119895119905 = 0

(iii) Distribution scheduling 119909119904119896119905 = 1 if under disruptionscenario 119904 a shipment of products to distributioncenter 119896 is scheduled for period 119905 otherwise 119909119904119896119905 = 0

(iv) Customer order nondelayed delivery 119910119904119895 = 1 if underdisruption scenario 119904 customer order 119895 is delivered byits due date otherwise 119910119904119895 = 0

Thedemand allocation vector (V1 V119868) wheresum119894isin119868 V119894 =1 and 0 le V119894 le 1 119894 isin 119868 defines the supply portfoliointroduced by Sawik [30]

Let 1198641 be the minimized expected cost per product andlet 1198642 be the maximized expected service level

1198641 = sum119894isin119868 119890119894119906119894 + sum119904isin119878 119875119904 (sum119894isin119868119904 119861119900119894120574119894120582119894119904V119894 + sum119895isin119869 119892119895119887119895 (sum119905isin119879119908119904119895119905 minus 119910119904119895) + sum119895isin119869 ℎ119895119887119895 (1 minus sum119905isin119879119908119904119895119905))119861 (4)

1198642 = sum119895isin119869sum119904isin119878 119875119904119887119895119910119904119895119861 (5)

where 120582119894119904 is disruption level of supplier 119894 under scenario 119904 and120574119894120582119894119904 is the corresponding fulfillment rate that is the fractionof an order delivered by supplier 119894 under disruption scenario119904

The expected cost 1198641 (see (4)) constitutes fixed order-ing cost sum119894isin119868 119890119894119906119894 expected purchasing cost for deliveredparts sum119904isin119878 119875119904sum119894isin119868119904 119861119900119894120574119894120582119894119904V119894 expected penalty for delayedcustomer demand sum119904isin119878 119875119904sum119895isin119869 119892119895119887119895(sum119905isin119879119908119904119895119905 minus 119910119904119895) andexpected penalty for unsatisfied (rejected) customer demandsum119904isin119878 119875119904sum119895isin119869 ℎ119895119887119895(1 minus sum119905isin119879119908119904119895119905)

Denote by

1198911 = 1198641 minus 11986411198641 minus 1198641 (6)

the normalized (scaled into the interval [0 1]) expected costper product (1198641 1198641 are the minimum and the maximumvalues of 1198641 resp) and by

1198912 = 1198642 minus 11986421198642 minus 1198642 (7)

the normalized expected service level (1198642 1198642 are the mini-mum and the maximum values of 1198642 resp)Model WCS It consists in supplier selection customer orderand distribution scheduling to minimize Weighted sum ofnormalized expected Cost and expected Service level

Minimize

1205721198911 + (1 minus 120572) 1198912 (8)

where 0 le 120572 le 1 subject to (4)ndash(7)Supply Portfolio Selection Constraints

(i) The total demand for parts must be fully allocatedamong the selected suppliers

(ii) Demand for parts cannot be assigned to nonselectedsuppliers

sum119894isin119868

V119894 = 1 V119894 le 119906119894 119894 isin 119868 (9)

Mathematical Problems in Engineering 5

Customer Order Scheduling Constraints

(i) For each disruption scenario 119904 each customer order119895 is either scheduled during the planning hori-zon (sum119905isin119879119908119904119895119905 = 1) or unscheduled and rejected(sum119905isin119879119908119904119895119905 = 0)

(ii) For any period 119905 and each disruption scenario 119904the total demand on capacity of all customer ordersscheduled in period 119905 must not exceed the producercapacity

sum119905isin119879

119908119904119895119905 le 1 119895 isin 119869 119904 isin 119878sum119895isin119869

119887119895119908119904119895119905 le 119862 119905 isin 119879 119904 isin 119878 (10)

Supply-Production-Distribution Coordinating Constraints

(i) For each disruption scenario 119904 and each planningperiod 119905 the cumulative demand for parts of all cus-tomer orders scheduled in period 1 through 119905 cannotexceed the cumulative deliveries of parts in period 1through 119905 minus 1 from the nonshutdown suppliers 119894 isin 119868119904

(ii) For each disruption scenario shipment to distribu-tion center 119896 can be scheduled only after the latestcompletion period of scheduled customer orders 119895 isin119869119896

sum119895isin119869

sum1199051015840isin1198791199051015840le119905

1198871198951199081199041198951199051015840 le 119861 sum119894isin119868119904 120590119894le119905minus1

120574119894120582119894119904V119894 119905 isin 119879 119904 isin 119878 (11)

sum119905isin119879119870

119905119909119904119896119905 ge sum119905isin119879

(119905 + 1)119908119904119895119905119896 isin 119870 119895 isin 119869119896 119904 isin 119878

(12)

where 119879119870 = min119894isin119868120590119894 + 2 119879 + 1 is the set ofshipping periods

Shipping Constraints

(i) For each disruption scenario at most one shipmentcan be scheduled to each distribution center

sum119905isin119879119870

119909119904119896119905 le 1 119896 isin 119870 119904 isin 119878 (13)

Customer Due Date Meeting Constraints

(i) For each disruption scenario 119904 isin 119878 customer order119895 isin 119869119896 can be delivered without delay (ie 119910119904119895 =1) if it is scheduled not later than 119889119895 minus 120591119896 andshipped to distribution center 119896 not later than 119889119895 minus120591119896 + 1 otherwise the customer order is delayed orunscheduled (ie 119910119904119895 = 0)

119910119904119895 le sum119905isin119879119905le119889119895minus120591119896

119908119904119895119905119896 isin 119870 119895 isin 119869119896 119904 isin 119878

119910119904119895 le sum119905isin119879119870119905le119889119895minus120591119896+1

119909119904119896119905119896 isin 119870 119895 isin 119869119896 119904 isin 119878

sum119905isin119879119905le119889119895minus120591119896

119908119904119895119905 + sum119905isin119879119870119905le119889119895minus120591119896+1

119909119904119896119905 minus 1 le 119910119904119895119896 isin 119870 119895 isin 119869119896 119904 isin 119878

(14)

Nonnegativity and Integrality Conditions

119906119894 isin 0 1 119894 isin 119868 (15)

V119894 isin [0 1] 119894 isin 119868 (16)

119908119904119895119905 isin 0 1 119895 isin 119869 119905 isin 119879 119904 isin 119878 (17)

119909119904119896119905 isin 0 1 119896 isin 119870 119905 isin 119879119870 119904 isin 119878 (18)

119910119904119895 ge 0 119895 isin 119869 119904 isin 119878 (19)

Note that119910119904119895 does not need to be restricted to being binarysince for any feasible solution satisfying constraints (14) 119910119904119895is always binary

Model WCS is a deterministic equivalent mixed integerprogram of stochastic mixed integer program When thisproblem is solved a recommendation is obtained for selec-tion of supply portfolio (119906119894 V119894) thatwill hedge against a varietyof disruption scenarios in which fulfillment rates of certainsuppliers are not sufficient to satisfy demand for parts Thescheduling variables 119908119904119895119905 119909119904119896119905 (and 119910119904119895) are decisions that willbe implemented in the future when scenario 119904 isin 119878 is finallyrealized

Model WCS illustrates the wait-and-see approach (eg[27]) Basically this approach is based on perfect informationabout the future Model WCS can be decomposed into atwo-stage stochastic mixed integer program with recourseThe supply portfolio selection variables (119906119894 V119894) are referredto as first-stage decisions and the scheduling variables(119908119904119895119905 119909119904119896119905 119910119904119895) are referred to as recourse or second-stagedecisions Unlike the first-stage decisions the latter variablesare dependent on the scenario 119904 isin 119878

Stochastic mixed integer programs are usually hard tosolve because they are large-scale optimization problemswhen applied to real-world problems A common approach(eg [27]) is to consider a simpler deterministic programknown as expected value problem in which the randomparameters are replaced by their expected values or toconsider several deterministic programs each of which cor-responds to one particular scenario and then to combine theobtained solutions into a single heuristic solution

5 Problem FormulationDeterministic Approach

In this section the expected value problem EWCS is pre-sented for the coordinated supply chain scheduling under

6 Mathematical Problems in Engineering

expected supply conditions In model WCS where the ran-domness is characterized by a set of disruption scenariosthe only random parameters are suppliers fulfillment rates120574119894120582119894119904 which appear both in the objective function (4) and inconstraints (11)

In model EWCS suppliers probabilistic fulfillment ratesdefined for each disruption scenario 120574119894120582119894119904 119894 isin 119868 119904 isin 119878 orequivalently for each disruption level 120574119894119897 119894 isin 119868 119897 isin 119871 (1)have been replaced by the expected fulfillment rates of eachsupplier

Γ119894 = sum119904isin119878

119875119904120574119894120582119894119904 119894 isin 119868 (20)

or equivalently

Γ119894 = (1 minus 119901119903)sum119897isin119871

119901119894119897120574119894119897 119894 isin 119868119903 119903 isin 119877 (21)

Accordingly stochastic binary decision variables 119908119904119895119905 119909119904119896119905 119910119904119895(17)ndash(19) defined for each disruption scenario 119904 isin 119878 havebeen replaced by their deterministic equivalents 119882119895119905 119883119896119905119884119895

Now the expected cost per product 1198641 (see (22)) andthe expected service level 1198642(see (23)) are defined as fol-lows

1198641 = sum119894isin119868 119890119894119906119894 + sum119894isin119868 119861119900119894Γ119894V119894 + sum119895isin119869 119892119895119887119895 (sum119905isin119879119882119895119905 minus 119884119895) + sum119895isin119869 ℎ119895119887119895 (1 minus sum119905isin119879119882119895119905)119861 (22)

1198642 = sum119895isin119869 119887119895119884119895119861 (23)

Model EWCS is presented below

Model EWCS

Minimize (8)subject to (6) (7) (9) (22) (23)

sum119905isin119879

119882119895119905 le 1 119895 isin 119869sum119895isin119869

119887119895119882119895119905 le 119862 119905 isin 119879sum119895isin119869

sum1199051015840isin1198791199051015840le119905

1198871198951198821198951199051015840 le 119861 sum119894isin119868120590119894le119905minus1

Γ119894V119894 119905 isin 119879sum119905isin119879119870

119905119883119896119905 ge sum119905isin119879

(119905 + 1)119882119895119905119896 isin 119870 119895 isin 119869119896

sum119905isin119879119870

119883119896119905 le 1 119896 isin 119870119884119895 le sum119905isin119879119905le119889119895minus120591119896

119882119895119905 119896 isin 119870 119895 isin 119869119896119884119895 le sum119905isin119879119870119905le119889119895minus120591119896+1

119883119896119905 119896 isin 119870 119895 isin 119869119896sum

119905isin119879119905le119889119895minus120591119896

119882119895119905 + sum119905isin119879119870119905le119889119895minus120591119896+1

119883119896119905 minus 1le 119884119895 119896 isin 119870 119895 isin 119869119896119906119894 isin 0 1 119894 isin 119868V119894 isin [0 1] 119894 isin 119868

119882119895119905 isin 0 1 119895 isin 119869 119905 isin 119879119883119896119905 isin 0 1 119896 isin 119870 119905 isin 119879119870119884119895 ge 0 119895 isin 119869

(24)

Notice that unlike the stochastic programming modelWCS which is formulated to determine optimal schedulesfor all potential disruption scenarios model EWCS accountsfor a single scenario only representing the expected suppliesExcept for the expected values of the random parametersthis model does not take into account any distributioninformation and the solution remains the same as long as theexpectations do not change In contrast tomodelWCS wherethe selection of supply portfolio (V1 V119868) is combinedwith supply chain scheduling for all disruption scenariosconsidered now the portfolio is determined along with asingle schedule

If randomparameters appear only in the constraints then[27]

EV le WS (25)

where EV is the optimal solution value of the expected valueproblem EWCS and WS is the optimal solution value ofthe wait-and-see problem WCS On the other hand whenuncertainty is limited to the objective function of the prob-lem the solution obtained by simply replacing the randomparameters with their expected values provides already arobust alternative (Delage et al [31])

6 Computational Examples

In this section some computational examples are presentedto illustrate possible applications of the proposedMIPmodelsand to compare the wait-and-see and the expected value

Mathematical Problems in Engineering 7

approaches The examples are modeled in part after a real-world electronics supply chain (eg Sawik [32]) The fol-lowing parameters have been selected for the computationalexamples119868 = 4 suppliers 119869 = 20 customer orders 119870 = 2 dis-tribution centers 119871 = 3 partial disruption levels 119877 = 2geographic regions and 119879 = 7 planning periods1198681 = 1 2 1198682 = 3 41198691 = 1 10 1198692 = 11 20

Shipping times from suppliers 120590 = (1 1 3 3)Shipping times to distribution centers 120591 = (1 2)Customer demand 119887119895 isin 2000 3000 10000 for all119895 isin 119869 and total demand 119861 = 100000Due dates 119889119895 isin 2 + min119894isin119868(120590119894) + min119896isin119870(120591119896) 119879 +

max119896isin119870(120591119896) for all 119895 isin 119869Fixed ordering costs for suppliers 119890 = (8000 6000 1200013000)Unit purchasing prices from suppliers 119900 = (14 12 8 9)Unit penalties for delayed unfulfilled customer orders119892119895 = lceilmax119894isin119868119900119894350rceil = 1 ℎ119895 = 2max119894isin119868119900119894 = 28 respectively

for all 119895 isin 119869Producer capacity 119862 = 45000Local disruption levels and the associated fulfillment rates

(the percentage of an order that can be delivered) are shownbelow119871 = 0 1 2 3 where 119897 = 0 complete shutdown1205741198940 = 0 forall119894 isin 119868 that is 0 of an order delivered 119897 = 1major disruption 1205741198941 isin [001 050] forall119894 isin 1198681 and 1205741198941 isin[001 030] forall119894 isin 1198682 that is 1 to 50 and 1 to 30 ofan order delivered respectively 119897 = 2 minor disruption

1205741198942 isin [0 51 099] forall119894 isin 1198681 and 1205741198942 isin [0 31 099] forall119894 isin 1198682that is 51 to 99 and 31 to 99 of an order deliveredrespectively 119897 = 119871 = 3 no disruption 1205741198943 = 1 forall119894 isin 119868 thatis 100 of an order delivered

The total number of all potential scenarios is 119878 = (119871 +1)119868 = 44 = 256 scenarios where each scenario 119904 isin 119878 is repre-sented by vector 120582119904 = (1205821119904 1205824119904) where 120582119894119904 isin 119871 119894 isin 119868 (seeTable 1)

The local nondisruption probability (level 119897 = 3) 1199011198943was uniformly distributed over [089 099] and [079 089]respectively for suppliers 119894 isin 1198681 and 119894 isin 1198682 that is theprobabilities were drawn independently from U[089 099]and U[079 089] respectively

Given local nondisruption probabilities 1199011198943 119894 isin 119868 theprobabilities for the remaining local disruption levels 119897 =0 1 2 were calculated as follows

Probability of complete shutdown (level 119897 = 0) 1199011198940 =01(1 minus 1199011198943)Probability of major disruption (level 119897 = 1) 1199011198941 =03(1 minus 1199011198943)Probability of minor disruption (level 119897 = 2) 1199011198942 =06(1 minus 1199011198943) for all suppliers 119894 isin 119868

Thus 1199011198940 le 1199011198941 le 1199011198942 le 1199011198943 which reflects a real-world relation among probabilities of disruption occurrenceat different levels (eg [8])

Regional disruption probabilities are 1199011 = 0001 and1199012 =001The probability of realizing of disruption scenario 119904 for

suppliers in region 119903 = 1 2 is calculated as follows

119875119903119904

=(1 minus 119901119903)( prod

119894isin119868119903 120582119894119904=0

01 (1 minus 1199011198943))( prod119894isin119868119903 120582119894119904=1

03 (1 minus 1199011198943))( prod119894isin119868119903120582119894119904=2

06 (1 minus 1199011198943))( prod119894isin119868119903 120582119894119904=3

1199011198943) if sum119894isin119868119903120582119894119904 gt 0

119901119903 + (1 minus 119901119903)prod119894isin119868119903

01 (1 minus 1199011198943) if sum119894isin119868119903120582119894119904 = 0

(26)

and the probability for disruption scenario 119904 isin 119878 is given by119875119904 = 1198751119904 1198752119904 Figure 2 presents basic characteristics of all suppliers

probability of complete shutdown119901119903+(1minus119901119903)1199011198940 119894 isin 119868119903 119903 isin119877 expected fulfillment rate Γ119894 = sum119897=123(1 minus 119901119903)119901119894119897120574119894119897 119894 isin119868119903 119903 isin 119877 and purchasing price 119900119894 119894 isin 119868Table 2 presents a subset of nondominated solutions for

the wait-and-see problem WCS for a subset of trade-offparameter120572 isin 0 01 02 03 04 05 06 07 08 09 1Theresults indicate that most of nondominated supply portfoliosconsist of the two suppliers only the second most expensiveand most reliable supplier 119894 = 2 from region 119903 = 1 andthe cheapest and most unreliable supplier 119894 = 3 from region119903 = 2 The most expensive and most reliable supplier 119894 = 1in region 119903 = 1 and the second cheapest and most unreliablesupplier 119894 = 4 in region 119903 = 2 are rarely selected For 120572 = 1

(minimization of cost) the cheapest supplier 119894 = 3 is selectedonly For 120572 = 0 (maximization of service level) the totaldemand for parts is allocated among the two most reliableand most expensive suppliers 119894 = 1 2 The above solutionshows that the service-oriented supply portfolio (120572 close to 0)is more diversified than the cost-oriented portfolio (120572 closeto 1) Table 2 also shows the associated expected fractionof fulfilled demand 1198643 = sum119904isin119878sum119895isin119869sum119905isin119879 119875119904119887119895119908119904119895119905119861 that isdemand fulfilled on time or delayed The solution resultsdemonstrate that a large expected service level is sometimesassociated with a small expected fraction of fulfilled demandThus the maximization of expected service level leads torejection of customer orders that cannot be fulfilled bycustomer requested due dates

Figure 3 shows the expected supply production andshipping schedules respectively for 120572 = 0 (ie for the

8 Mathematical Problems in Engineering

Shutdown probability

Supplier

Expected yield rate

Purchasing price

4321

Supplier4321

Supplier4321

6

8

10

12

14

16

09

092

094

096

098

0

0005

001

0015

002

0025

Figure 2 Suppliers

1 2 3 4 5 6 7Period

Expected supply schedules

Alpha = 0Alpha = 05

Alpha = 1

1 2 3 4 5 6 7Period

Expected production schedules

1 2 3 4 5 6 7Period

Expected shipping schedules

0

20000

40000

60000

80000

100000

Part

s

0

10000

20000

30000

40000

50000

Prod

ucts

Alpha = 0Alpha = 05

Alpha = 1

0

10000

20000

30000

40000

50000

Prod

ucts

Alpha = 0Alpha = 05

Alpha = 1

Figure 3 Expected schedules for modelWCS

Mathematical Problems in Engineering 9

Table 1 Disruption scenarios

119904 119894 = 1 2 3 41 0 0 0 02 0 0 0 13 0 0 0 24 0 0 0 35 0 0 1 06 0 0 1 17 0 0 1 28 0 0 1 39 0 0 2 010 0 0 2 111 0 0 2 212 0 0 2 313 0 0 3 014 0 0 3 115 0 0 3 216 0 0 3 317 0 1 0 018 0 1 0 119 0 1 0 220 0 1 0 321 0 1 1 022 0 1 1 123 0 1 1 224 0 1 1 325 0 1 2 026 0 1 2 127 0 1 2 228 0 1 2 329 0 1 3 030 0 1 3 131 0 1 3 232 0 1 3 333 0 2 0 034 0 2 0 135 0 2 0 236 0 2 0 337 0 2 1 038 0 2 1 139 0 2 1 240 0 2 1 341 0 2 2 042 0 2 2 143 0 2 2 244 0 2 2 345 0 2 3 046 0 2 3 147 0 2 3 248 0 2 3 349 0 3 0 0

Table 1 Continued

119904 119894 = 1 2 3 450 0 3 0 151 0 3 0 252 0 3 0 353 0 3 1 054 0 3 1 155 0 3 1 256 0 3 1 357 0 3 2 058 0 3 2 159 0 3 2 260 0 3 2 361 0 3 3 062 0 3 3 163 0 3 3 264 0 3 3 365 1 0 0 066 1 0 0 167 1 0 0 268 1 0 0 369 1 0 1 070 1 0 1 171 1 0 1 272 1 0 1 373 1 0 2 074 1 0 2 175 1 0 2 276 1 0 2 377 1 0 3 078 1 0 3 179 1 0 3 280 1 0 3 381 1 1 0 082 1 1 0 183 1 1 0 284 1 1 0 385 1 1 1 086 1 1 1 187 1 1 1 288 1 1 1 389 1 1 2 090 1 1 2 191 1 1 2 292 1 1 2 393 1 1 3 094 1 1 3 195 1 1 3 296 1 1 3 397 1 2 0 098 1 2 0 1

10 Mathematical Problems in Engineering

Table 1 Continued

119904 119894 = 1 2 3 499 1 2 0 2100 1 2 0 3101 1 2 1 0102 1 2 1 1103 1 2 1 2104 1 2 1 3105 1 2 2 0106 1 2 2 1107 1 2 2 2108 1 2 2 3109 1 2 3 0110 1 2 3 1111 1 2 3 2112 1 2 3 3113 1 3 0 0114 1 3 0 1115 1 3 0 2116 1 3 0 3117 1 3 1 0118 1 3 1 1119 1 3 1 2120 1 3 1 3121 1 3 2 0122 1 3 2 1123 1 3 2 2124 1 3 2 3125 1 3 3 0126 1 3 3 1127 1 3 3 2128 1 3 3 3129 2 0 0 0130 2 0 0 1131 2 0 0 2132 2 0 0 3133 2 0 1 0134 2 0 1 1135 2 0 1 2136 2 0 1 3137 2 0 2 0138 2 0 2 1139 2 0 2 2140 2 0 2 3141 2 0 3 0142 2 0 3 1143 2 0 3 2144 2 0 3 3145 2 1 0 0146 2 1 0 1147 2 1 0 2148 2 1 0 3149 2 1 1 0150 2 1 1 1

Table 1 Continued

119904 119894 = 1 2 3 4151 2 1 1 2152 2 1 1 3153 2 1 2 0154 2 1 2 1155 2 1 2 2156 2 1 2 3157 2 1 3 0158 2 1 3 1159 2 1 3 2160 2 1 3 3161 2 2 0 0162 2 2 0 1163 2 2 0 2164 2 2 0 3165 2 2 1 0166 2 2 1 1167 2 2 1 2168 2 2 1 3169 2 2 2 0170 2 2 2 1171 2 2 2 2172 2 2 2 3173 2 2 3 0174 2 2 3 1175 2 2 3 2176 2 2 3 3177 2 3 0 0178 2 3 0 1179 2 3 0 2180 2 3 0 3181 2 3 1 0182 2 3 1 1183 2 3 1 2184 2 3 1 3185 2 3 2 0186 2 3 2 1187 2 3 2 2188 2 3 2 3189 2 3 3 0190 2 3 3 1191 2 3 3 2192 2 3 3 3193 3 0 0 0194 3 0 0 1195 3 0 0 2196 3 0 0 3197 3 0 1 0198 3 0 1 1199 3 0 1 2200 3 0 1 3201 3 0 2 0202 3 0 2 1

Mathematical Problems in Engineering 11

Table 1 Continued

119904 119894 = 1 2 3 4203 3 0 2 2204 3 0 2 3205 3 0 3 0206 3 0 3 1207 3 0 3 2208 3 0 3 3209 3 1 0 0210 3 1 0 1211 3 1 0 2212 3 1 0 3213 3 1 1 0214 3 1 1 1215 3 1 1 2216 3 1 1 3217 3 1 2 0218 3 1 2 1219 3 1 2 2220 3 1 2 3221 3 1 3 0222 3 1 3 1223 3 1 3 2224 3 1 3 3225 3 2 0 0226 3 2 0 1227 3 2 0 2228 3 2 0 3229 3 2 1 0230 3 2 1 1231 3 2 1 2232 3 2 1 3233 3 2 2 0234 3 2 2 1235 3 2 2 2236 3 2 2 3237 3 2 3 0238 3 2 3 1239 3 2 3 2240 3 2 3 3241 3 3 0 0242 3 3 0 1243 3 3 0 2244 3 3 0 3245 3 3 1 0246 3 3 1 1247 3 3 1 2248 3 3 1 3249 3 3 2 0250 3 3 2 1251 3 3 2 2252 3 3 2 3

Table 1 Continued

119904 119894 = 1 2 3 4253 3 3 3 0254 3 3 3 1255 3 3 3 2256 3 3 3 3

maximum expected service level) 120572 = 05 and 120572 = 1 (ie forthe minimum expected cost) The expected schedules werecomputed using the formulae presented below

(i) Expected schedule of supplies of parts to the pro-ducer

sum119904isin119878

sum119894isin119868119904120590119894=119905

119875119904119861120574119894120582119894119904V119894 119905 isin 119879 (27)

(ii) Expected production schedule

sum119904isin119878

sum119895isin119869

119875119904119887119895119908119904119895119905 119905 isin 119879 (28)

(iii) Expected schedule of shipping of products from theproducer to the distribution centers

sum119904isin119878

sum119896isin119870

sum119895isin119869119896

119875119904119887119895( sum1199051015840isin1198791199051015840lt119905

1199081199041198951199051015840)119909119904119896119905 119905 isin 119879 (29)

As 120572 increases that is the decision-makerrsquos priority shiftsfrom the maximum service level to minimum cost and moreparts are ordered from less reliable and lower cost suppliersthe expected supply schedules and the corresponding pro-duction schedules are more delayed as well as the delivery ofproducts to the customers Note that despite constraint (13)that ensures a feasible schedule with at most one shipmentof products to each distribution center for each disruptionscenario the expected shipping schedule (29) may be splitinto more smaller size shipments (cf Figure 3 where twomajor and one small-size shipments are indicated for eachconfidence level)

For comparison Table 3 presents a subset of non-dominated solutions obtained for the expected value problemEWCS and Figure 4 shows supply production and shippingschedules Unlike the stochastic programming approachwhich accounts for all potential disruption scenarios tooptimize an expected performance of the supply chain thesolution obtained using the deterministic approach is basedon aggregate information on suppliers expected fulfillmentIn general the results are similar for both models and thecorresponding optimal solution values are close to each otherwhich indicates that the expected value problem can be usedin practice when stochastic mixed integer programs are hardto solve The optimal solution values for the expected valueproblem frequently outperform the corresponding solutionvalues of the wait-and-see problem which is in line withthe proposition EV le WS in [27] compare Section 5for example the minimum cost 1198641 for EWCS (see Table 3)

12 Mathematical Problems in Engineering

Table 2 Nondominated solutions for modelWCS

120572 0 01 02 03 04 05 and 06 07 08 and 09 1Var = 38077 Bin = 33076 Cons = 28769 Nonz = 348325(a)

Exp cost 1731 (1198641) 1620 1547 1206 1200 1176 1070 1065 1051 (1198641)Exp service level(b) 8552 (1198642) 8487 8452 7719 7676 7595 5863 5871 3822 (1198642)Exp fulfilled demand(c) 8552 8530 8452 9370 9361 9353 9134 9120 9068

Suppliers selected( of total demand)

1 (76) 1 (35)2 (24) 2 (55) 2 (87) 2 (55) 2 (55) 2 (55) 2 (12) 2 (10)3 (10) 3 (13) 3 (22) 3 (32) 3 (45) 3 (88) 3 (90) 3 (100)4 (23) 4 (13)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119904isin119878 sum119895isin119869 119875119904119887119895119910

119904119895119861 times

100 (c)sum119904isin119878 sum119895isin119869 sum119905isin119879 119875119904119887119895119908119904119895119905119861 times 100

Table 3 Nondominated solutions for model EWCS

120572 0 01 02 and 03 04 and 05 06 and 07 08 09 and 1Var = 162 Bin = 136 Cons = 125 Nonz = 1430(a)

Exp cost 1691 (1198641) 1427 1214 1167 1045 (1198641)Exp service level(b) 87 (1198642) 86 81 79 62 (1198642)Exp fulfilled demand(c) 87 90 91 93 91

Suppliers selected( of total demand)

1 (89) 2 (94) 2 (49) 2 (50) 2 (5)3 (6) 3 (51) 3 (50) 3 (95)4 (11)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119895isin119869 119887119895119884119895119861 times 100(c)sum119895isin119869 sum119905isin119879 119887119895119882119895119905119861 times 100

1 2 3 4 5 6 7Period

Supply schedules

1 2 3 4 5 6 7Period

Production schedules

1 2 3 4 5 6 7Period

Shipping schedules

0

10000

20000

30000

40000

50000

Prod

ucts

0

10000

20000

30000

40000

50000

Prod

ucts

0

20000

40000

60000

80000

100000

Part

s

Alpha = 0Alpha = 05

Alpha = 1 Alpha = 0Alpha = 05

Alpha = 1

Alpha = 0Alpha = 05

Alpha = 1

Figure 4 Schedules for model EWCS

Mathematical Problems in Engineering 13

1045 le 1051 forWCS (see Table 2) or themaximum servicelevel1198642 for EWCS (see Table 3) 8700 ge 8552 forWCS (seeTable 2)

The stochastic wait-and-see approach however leads toa more diversified supply portfolio For the deterministicapproach most of nondominated supply portfolios consistof two suppliers only 119894 = 2 3 while suppliers 119894 = 1 4 areselected only for the maximum service level objective (thatis for 120572 = 0) Comparison of Figures 3 and 4 indicatesthat the expected schedules for model WCS computed asexpectations over all schedules for all potential disruptionscenarios (27)ndash(29) are similar to the corresponding singleschedules determined by model EWCS The main differencesobserved aremore delayed expected production and shippingschedules for model WCS when minimization of cost is con-sidered (that is for120572 = 1) Finally it is interesting to note thatin the multiple sourcing environment considered both thewait-and-see approach and in particular the expected valueapproach frequently select a dual sourcing supply portfoliowith one main supplier and one supporting supplier

The computational experiments were performed usingthe AMPL programming language and the CPLEX 1262solver on a MacBook Pro laptop with Intel Core i7 processorrunning at 28 GHz and with 16GB RAM The solver wascapable of finding proven optimal solution for all exampleswith CPU time ranging from several minutes to several hoursfor the stochastic model WCS and fraction of a second forthe deterministic model EWCS Since the stochastic pro-gramming model WCS needs to determine nondominatedschedules for all potential disruption scenarios while modelEWCS deals with a single scenario only the difference in thecomputational effort required is obvious

7 Conclusions

In this paper two biobjective MIP formulations stochasticand deterministic have been proposed and compared for thecoordinated decision-making in supply chains under partiallocal disruptions and all-or-nothing regional disruptionsTheproblem objective has been to jointly schedule supplies pro-duction and distribution to optimize the trade-off betweenexpected cost and expected service levelWhile the stochasticprogramming approach aims at optimizing the expectedperformance of a supply chain over all possible disruptionscenarios the deterministic approach accounts only for asingle scenario representing average disruption conditionsAs a result the stochastic programming approach determinesnondominated solutions for all disruption scenarios whereasthe deterministic approach produces a single solution onlyIn particular in the stochastic programming approach theselection of supply portfolio is combined with supply chainscheduling for all disruption scenarios considered In con-trast the deterministic expected value approach provides theportfolio along with a single expected schedule of productionand distribution

The expected schedules obtained for the stochastic pro-gramming model WCS as expectations over all schedulesfor all disruption scenarios have been compared with thecorresponding schedules determined by the deterministic

model EWCS based on an expected disruption scenarioThecomparison has demonstrated that the two approaches leadto similar expected solutions

The main findings are in line with other research and arelisted as follows

(i) The two decision-making approaches stochastic anddeterministic lead to similar expected performanceof a supply chain under multilevel disruptions

(ii) The optimal solution values for the expected valueproblem frequently outperform the correspondingsolution values of the wait-and-see problem

(iii) Despite the multiple sourcing environment consid-ered both the wait-and-see approach and in partic-ular the expected value approach frequently selecta dual sourcing supply portfolio with one mainsupplier and one supporting supplier

(iv) The stochastic approach which accounts for allpotential disruption scenarios may lead to a morediversified supply portfolio that will hedge against avariety of scenarios

(v) The expected schedules are more delayed for thestochastic approach

(vi) The service-oriented supply portfolio is more diversi-fied andmay combine both high-cost reliable suppli-ers and low-cost unreliable suppliers while the cost-oriented portfolio depends mainly on low-cost andless reliable suppliers

Overall the results of computational experiments indi-cate that the proposed approach and developed MIP modelsare flexible and efficient tools for coordinated supply chainscheduling The portfolio approach leads to MIP formula-tions with strong LP relaxations and has been proven to becomputationally very efficient CPU time required to findproven optimal solutions for realistic size examples usingcommercially available software for MIP is acceptable for areal-world supply chain disruption management (see Sawik[33])

Since the probability distribution of supply disruptionsfrom each supplier is usually unknown (eg [8]) local multi-level disruptions in the stochastic programming modelWCShave a multinomial discrete distribution while the two-levelregional disruptions have a binomial discrete distributionand all disruption events are independent As part of futureresearch we propose to enhance the stochastic programmingmodel formore general scenarios with finitelymany elementssingled out and all the probability concentrated in them Forexample rather than complete shutdown suppliers within aregion may have correlated disruptions Another importantstream of future research is the study of robustness andsensitivity in relation to input data changes in supply chainscheduling under disruption risks (eg [33]) Future researchcan also focus on the following improvement of the proposedmodels to consider the minimization of time and cost ofrecovery (eg Whitney et al [34])

14 Mathematical Problems in Engineering

Notations

Indices

119894 Supplier 119894 isin 119868119895 Customer 119895 isin 119869119896 Distribution center 119896 isin 119870119897 Disruption level 119897 isin 119871119903 Region 119903 isin 119877119904 Disruption scenario 119904 isin 119878119905 Planning period 119905 isin 119879Input Parameters

119887119895 Size (number of products) of customer order 119895119861 Total demand for partsproducts 119861 = sum119895isin119869 119887119895119862 Capacity of producer119889119895 Due date for customer order 119895119890119894 Fixed cost of ordering parts from supplier 119894119892119895 Per unit penalty cost of delayed customer order 119895ℎ119895 Per unit penalty cost of unfulfilled customer order 119895119868119903 Subset of suppliers in region 119903119869119896 Subset of customers serviced by distribution center 119896119900119894 Per unit price of parts purchased from supplier 119894119901119894119897 Probability of disruption level 119897 for supplier 119894119901119903 Regional disruption probability for region 119903120574119894119897 Fraction of an order delivered by supplier 119894 underdisruption level 119897 (fulfillment rate)Γ119894 Expected fraction of an order delivered by supplier 119894(expected fulfillment rate)120590119894 Delivery lead time from supplier 119894120591119896 Transportation time to distribution center 119896

Conflicts of Interest

The author declares that there are no conflicts of interest

Acknowledgments

This work has been supported by NCN research grant (noDEC-201311BST804458) and by AGH (no 1111200324)

References

[1] J V Blackhurst K P Scheibe and D J Johnson ldquoSupplierrisk assessment and monitoring for the automotive industryrdquoInternational Journal of Physical Distribution amp Logistics Man-agement vol 38 no 2 pp 143ndash165 2008

[2] P Hoffmann H Schiele and K Krabbendam ldquoUncertaintysupply risk management and their impact on performancerdquoJournal of Purchasing amp Supply Management vol 19 no 3 pp199ndash211 2013

[3] Y Park P Hong and J J Roh ldquoSupply chain lessons from thecatastrophic natural disaster in Japanrdquo Business Horizons vol56 no 1 pp 75ndash85 2013

[4] M Haraguchi and U Lall ldquoFlood risks and impacts A casestudy of Thailandrsquos floods in 2011 and research questions forsupply chain decisionmakingrdquo International Journal of DisasterRisk Reduction vol 14 pp 256ndash272 2015

[5] T Fujimoto and Y W Park ldquoBalancing supply chain com-petitiveness and robustness through ldquovirtual dual sourcingrdquoLessons from the Great East Japan Earthquakerdquo InternationalJournal of Production Economics vol 147 pp 429ndash436 2014

[6] HMatsuo ldquoImplications of the Tohoku earthquake for Toyotarsquoscoordination mechanism supply chain disruption of auto-motive semiconductorsrdquo International Journal of ProductionEconomics vol 161 pp 217ndash227 2015

[7] J R Marszewska ldquoImplications of seismic hazard in Japan ontoyota supply chain disruption risksrdquo in Proceedings of the 13thInternational Conference on Industrial Logistics (ICIL rsquo16) pp178ndash185 Zakopane Poland October 2016

[8] A Z Zeng and Y Xia ldquoBuilding a mutually beneficial partner-ship to ensure backup supplyrdquo Omega vol 52 pp 77ndash91 2015

[9] S S Erenguc N C Simpson and A J Vakharia ldquoIntegratedproductiondistribution planning in supply chains an invitedreviewrdquo European Journal of Operational Research vol 115 no2 pp 219ndash236 1999

[10] L Lei S Liu A Ruszczynski and S Park ldquoOn the integratedproduction inventory and distribution routing problemrdquo IIETransactions vol 38 no 11 pp 955ndash970 2006

[11] A Kaur A Kanda and S G Deshmukh ldquoA graph theoreticapproach for supply chain coordinationrdquo International Journalof Logistics Systems and Management vol 2 no 4 pp 321ndash3412006

[12] T ChoiW Yeung andT C E Cheng ldquoScheduling and co-ordi-nation of multi-suppliers single-warehouse-operator single-manufacturer supply chains with variable production rates andstorage costsrdquo International Journal of Production Research vol51 no 9 pp 2593ndash2601 2013

[13] S Liu and L G Papageorgiou ldquoMultiobjective optimisation ofproduction distribution and capacity planning of global supplychains in the process industryrdquo Omega vol 41 no 2 pp 369ndash382 2013

[14] Z L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[15] T Sawik ldquoCoordinated supply chain schedulingrdquo InternationalJournal of Production Economics vol 120 no 2 pp 437ndash4512009

[16] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[17] D Ivanov B Sokolov and A Dolgui ldquoThe Ripple effect in sup-ply chains Trade-off rsquoefficiency-flexibility- resiliencersquo in disrup-tionmanagementrdquo International Journal of ProductionResearchvol 52 no 7 pp 2154ndash2172 2014

[18] D Ivanov A Pavlov and B Sokolov ldquoOptimal distribution(re)planning in a centralized multi-stage supply network underconditions of the ripple effect and structure dynamicsrdquo Euro-pean Journal of Operational Research vol 237 no 2 pp 758ndash770 2014

[19] D Ivanov A Pavlov A Dolgui D Pavlov and B SokolovldquoDisruption-driven supply chain (re)-planning and perfor-mance impact assessment with consideration of pro-active andrecovery policiesrdquo Transportation Research Part E Logistics andTransportation Review vol 90 pp 7ndash24 2016

[20] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

Mathematical Problems in Engineering 15

[21] A J Schmitt S A Sun L V Snyder and Z-J M Shen ldquoCen-tralization versus decentralization Risk pooling risk diversifi-cation and supply chain disruptionsrdquo Omega vol 52 pp 201ndash212 2015

[22] L V Snyder Z Atan P Peng Y Rong A J Schmitt andB Sinsoysal ldquoORMS models for supply chain disruptions areviewrdquo IIE Transactions vol 48 no 2 pp 89ndash109 2016

[23] T Sawik ldquoIntegrated selection of suppliers and scheduling ofcustomer orders in the presence of supply chain disruptionrisksrdquo International Journal of Production Research vol 51 no23-24 pp 7006ndash7022 2013

[24] T Sawik ldquoOptimization of cost and service level in the presenceof supply chain disruption risks Single vs multiple sourcingrdquoComputers amp Operations Research vol 51 pp 11ndash20 2014

[25] T Sawik ldquoIntegrated supply production and distributionscheduling under disruption risksrdquoOmega vol 62 pp 131ndash1442016

[26] T Sawik ldquoIntegrated supply chain scheduling under multi-leveldisruptionsrdquo IFAC-PapersOnLine vol 28 no 3 pp 1515ndash15202015

[27] P Kall and J Mayer Stochastic Linear Programming ModelsTheory and Computation vol 156 of International Series inOperations Research amp Management Science Springer NewYork 2011

[28] I N Durbach and T J Stewart ldquoUsing expected values to sim-plify decision making under uncertaintyrdquo Omega vol 37 no 2pp 312ndash330 2009

[29] F Maggioni and S W Wallace ldquoAnalyzing the quality of theexpected value solution in stochastic programmingrdquo Annals ofOperations Research vol 200 pp 37ndash54 2012

[30] T Sawik ldquoSelection of supply portfolio under disruption risksrdquoOmega vol 39 no 2 pp 194ndash208 2011

[31] E Delage S Arroyo and Y Ye ldquoThe value of stochastic mod-eling in two-stage stochastic programs with cost uncertaintyrdquoOperations Research vol 62 no 6 pp 1377ndash1393 2014

[32] T Sawik Scheduling in Supply Chains Using Mixed Integer Pro-gramming John Wiley amp Sons Inc Hoboken NJ 2011

[33] T Sawik Supply Chain DisruptionManagement Using StochasticMixed Integer Programming vol 256 of International Seriesin Operations Research amp Management Science Springer NewYork 2017 httpwwwspringercomgpbook9783319588223

[34] D E Whitney J Luo and D A Heller ldquoThe benefits and con-straints of temporary sourcing diversification in supply chaindisruption and recoveryrdquo Journal of Purchasing and SupplyManagement vol 20 no 4 pp 238ndash250 2014

Mathematical Problems in Engineering 5

Customer Order Scheduling Constraints

(i) For each disruption scenario 119904 each customer order119895 is either scheduled during the planning hori-zon (sum119905isin119879119908119904119895119905 = 1) or unscheduled and rejected(sum119905isin119879119908119904119895119905 = 0)

(ii) For any period 119905 and each disruption scenario 119904the total demand on capacity of all customer ordersscheduled in period 119905 must not exceed the producercapacity

sum119905isin119879

119908119904119895119905 le 1 119895 isin 119869 119904 isin 119878sum119895isin119869

119887119895119908119904119895119905 le 119862 119905 isin 119879 119904 isin 119878 (10)

Supply-Production-Distribution Coordinating Constraints

(i) For each disruption scenario 119904 and each planningperiod 119905 the cumulative demand for parts of all cus-tomer orders scheduled in period 1 through 119905 cannotexceed the cumulative deliveries of parts in period 1through 119905 minus 1 from the nonshutdown suppliers 119894 isin 119868119904

(ii) For each disruption scenario shipment to distribu-tion center 119896 can be scheduled only after the latestcompletion period of scheduled customer orders 119895 isin119869119896

sum119895isin119869

sum1199051015840isin1198791199051015840le119905

1198871198951199081199041198951199051015840 le 119861 sum119894isin119868119904 120590119894le119905minus1

120574119894120582119894119904V119894 119905 isin 119879 119904 isin 119878 (11)

sum119905isin119879119870

119905119909119904119896119905 ge sum119905isin119879

(119905 + 1)119908119904119895119905119896 isin 119870 119895 isin 119869119896 119904 isin 119878

(12)

where 119879119870 = min119894isin119868120590119894 + 2 119879 + 1 is the set ofshipping periods

Shipping Constraints

(i) For each disruption scenario at most one shipmentcan be scheduled to each distribution center

sum119905isin119879119870

119909119904119896119905 le 1 119896 isin 119870 119904 isin 119878 (13)

Customer Due Date Meeting Constraints

(i) For each disruption scenario 119904 isin 119878 customer order119895 isin 119869119896 can be delivered without delay (ie 119910119904119895 =1) if it is scheduled not later than 119889119895 minus 120591119896 andshipped to distribution center 119896 not later than 119889119895 minus120591119896 + 1 otherwise the customer order is delayed orunscheduled (ie 119910119904119895 = 0)

119910119904119895 le sum119905isin119879119905le119889119895minus120591119896

119908119904119895119905119896 isin 119870 119895 isin 119869119896 119904 isin 119878

119910119904119895 le sum119905isin119879119870119905le119889119895minus120591119896+1

119909119904119896119905119896 isin 119870 119895 isin 119869119896 119904 isin 119878

sum119905isin119879119905le119889119895minus120591119896

119908119904119895119905 + sum119905isin119879119870119905le119889119895minus120591119896+1

119909119904119896119905 minus 1 le 119910119904119895119896 isin 119870 119895 isin 119869119896 119904 isin 119878

(14)

Nonnegativity and Integrality Conditions

119906119894 isin 0 1 119894 isin 119868 (15)

V119894 isin [0 1] 119894 isin 119868 (16)

119908119904119895119905 isin 0 1 119895 isin 119869 119905 isin 119879 119904 isin 119878 (17)

119909119904119896119905 isin 0 1 119896 isin 119870 119905 isin 119879119870 119904 isin 119878 (18)

119910119904119895 ge 0 119895 isin 119869 119904 isin 119878 (19)

Note that119910119904119895 does not need to be restricted to being binarysince for any feasible solution satisfying constraints (14) 119910119904119895is always binary

Model WCS is a deterministic equivalent mixed integerprogram of stochastic mixed integer program When thisproblem is solved a recommendation is obtained for selec-tion of supply portfolio (119906119894 V119894) thatwill hedge against a varietyof disruption scenarios in which fulfillment rates of certainsuppliers are not sufficient to satisfy demand for parts Thescheduling variables 119908119904119895119905 119909119904119896119905 (and 119910119904119895) are decisions that willbe implemented in the future when scenario 119904 isin 119878 is finallyrealized

Model WCS illustrates the wait-and-see approach (eg[27]) Basically this approach is based on perfect informationabout the future Model WCS can be decomposed into atwo-stage stochastic mixed integer program with recourseThe supply portfolio selection variables (119906119894 V119894) are referredto as first-stage decisions and the scheduling variables(119908119904119895119905 119909119904119896119905 119910119904119895) are referred to as recourse or second-stagedecisions Unlike the first-stage decisions the latter variablesare dependent on the scenario 119904 isin 119878

Stochastic mixed integer programs are usually hard tosolve because they are large-scale optimization problemswhen applied to real-world problems A common approach(eg [27]) is to consider a simpler deterministic programknown as expected value problem in which the randomparameters are replaced by their expected values or toconsider several deterministic programs each of which cor-responds to one particular scenario and then to combine theobtained solutions into a single heuristic solution

5 Problem FormulationDeterministic Approach

In this section the expected value problem EWCS is pre-sented for the coordinated supply chain scheduling under

6 Mathematical Problems in Engineering

expected supply conditions In model WCS where the ran-domness is characterized by a set of disruption scenariosthe only random parameters are suppliers fulfillment rates120574119894120582119894119904 which appear both in the objective function (4) and inconstraints (11)

In model EWCS suppliers probabilistic fulfillment ratesdefined for each disruption scenario 120574119894120582119894119904 119894 isin 119868 119904 isin 119878 orequivalently for each disruption level 120574119894119897 119894 isin 119868 119897 isin 119871 (1)have been replaced by the expected fulfillment rates of eachsupplier

Γ119894 = sum119904isin119878

119875119904120574119894120582119894119904 119894 isin 119868 (20)

or equivalently

Γ119894 = (1 minus 119901119903)sum119897isin119871

119901119894119897120574119894119897 119894 isin 119868119903 119903 isin 119877 (21)

Accordingly stochastic binary decision variables 119908119904119895119905 119909119904119896119905 119910119904119895(17)ndash(19) defined for each disruption scenario 119904 isin 119878 havebeen replaced by their deterministic equivalents 119882119895119905 119883119896119905119884119895

Now the expected cost per product 1198641 (see (22)) andthe expected service level 1198642(see (23)) are defined as fol-lows

1198641 = sum119894isin119868 119890119894119906119894 + sum119894isin119868 119861119900119894Γ119894V119894 + sum119895isin119869 119892119895119887119895 (sum119905isin119879119882119895119905 minus 119884119895) + sum119895isin119869 ℎ119895119887119895 (1 minus sum119905isin119879119882119895119905)119861 (22)

1198642 = sum119895isin119869 119887119895119884119895119861 (23)

Model EWCS is presented below

Model EWCS

Minimize (8)subject to (6) (7) (9) (22) (23)

sum119905isin119879

119882119895119905 le 1 119895 isin 119869sum119895isin119869

119887119895119882119895119905 le 119862 119905 isin 119879sum119895isin119869

sum1199051015840isin1198791199051015840le119905

1198871198951198821198951199051015840 le 119861 sum119894isin119868120590119894le119905minus1

Γ119894V119894 119905 isin 119879sum119905isin119879119870

119905119883119896119905 ge sum119905isin119879

(119905 + 1)119882119895119905119896 isin 119870 119895 isin 119869119896

sum119905isin119879119870

119883119896119905 le 1 119896 isin 119870119884119895 le sum119905isin119879119905le119889119895minus120591119896

119882119895119905 119896 isin 119870 119895 isin 119869119896119884119895 le sum119905isin119879119870119905le119889119895minus120591119896+1

119883119896119905 119896 isin 119870 119895 isin 119869119896sum

119905isin119879119905le119889119895minus120591119896

119882119895119905 + sum119905isin119879119870119905le119889119895minus120591119896+1

119883119896119905 minus 1le 119884119895 119896 isin 119870 119895 isin 119869119896119906119894 isin 0 1 119894 isin 119868V119894 isin [0 1] 119894 isin 119868

119882119895119905 isin 0 1 119895 isin 119869 119905 isin 119879119883119896119905 isin 0 1 119896 isin 119870 119905 isin 119879119870119884119895 ge 0 119895 isin 119869

(24)

Notice that unlike the stochastic programming modelWCS which is formulated to determine optimal schedulesfor all potential disruption scenarios model EWCS accountsfor a single scenario only representing the expected suppliesExcept for the expected values of the random parametersthis model does not take into account any distributioninformation and the solution remains the same as long as theexpectations do not change In contrast tomodelWCS wherethe selection of supply portfolio (V1 V119868) is combinedwith supply chain scheduling for all disruption scenariosconsidered now the portfolio is determined along with asingle schedule

If randomparameters appear only in the constraints then[27]

EV le WS (25)

where EV is the optimal solution value of the expected valueproblem EWCS and WS is the optimal solution value ofthe wait-and-see problem WCS On the other hand whenuncertainty is limited to the objective function of the prob-lem the solution obtained by simply replacing the randomparameters with their expected values provides already arobust alternative (Delage et al [31])

6 Computational Examples

In this section some computational examples are presentedto illustrate possible applications of the proposedMIPmodelsand to compare the wait-and-see and the expected value

Mathematical Problems in Engineering 7

approaches The examples are modeled in part after a real-world electronics supply chain (eg Sawik [32]) The fol-lowing parameters have been selected for the computationalexamples119868 = 4 suppliers 119869 = 20 customer orders 119870 = 2 dis-tribution centers 119871 = 3 partial disruption levels 119877 = 2geographic regions and 119879 = 7 planning periods1198681 = 1 2 1198682 = 3 41198691 = 1 10 1198692 = 11 20

Shipping times from suppliers 120590 = (1 1 3 3)Shipping times to distribution centers 120591 = (1 2)Customer demand 119887119895 isin 2000 3000 10000 for all119895 isin 119869 and total demand 119861 = 100000Due dates 119889119895 isin 2 + min119894isin119868(120590119894) + min119896isin119870(120591119896) 119879 +

max119896isin119870(120591119896) for all 119895 isin 119869Fixed ordering costs for suppliers 119890 = (8000 6000 1200013000)Unit purchasing prices from suppliers 119900 = (14 12 8 9)Unit penalties for delayed unfulfilled customer orders119892119895 = lceilmax119894isin119868119900119894350rceil = 1 ℎ119895 = 2max119894isin119868119900119894 = 28 respectively

for all 119895 isin 119869Producer capacity 119862 = 45000Local disruption levels and the associated fulfillment rates

(the percentage of an order that can be delivered) are shownbelow119871 = 0 1 2 3 where 119897 = 0 complete shutdown1205741198940 = 0 forall119894 isin 119868 that is 0 of an order delivered 119897 = 1major disruption 1205741198941 isin [001 050] forall119894 isin 1198681 and 1205741198941 isin[001 030] forall119894 isin 1198682 that is 1 to 50 and 1 to 30 ofan order delivered respectively 119897 = 2 minor disruption

1205741198942 isin [0 51 099] forall119894 isin 1198681 and 1205741198942 isin [0 31 099] forall119894 isin 1198682that is 51 to 99 and 31 to 99 of an order deliveredrespectively 119897 = 119871 = 3 no disruption 1205741198943 = 1 forall119894 isin 119868 thatis 100 of an order delivered

The total number of all potential scenarios is 119878 = (119871 +1)119868 = 44 = 256 scenarios where each scenario 119904 isin 119878 is repre-sented by vector 120582119904 = (1205821119904 1205824119904) where 120582119894119904 isin 119871 119894 isin 119868 (seeTable 1)

The local nondisruption probability (level 119897 = 3) 1199011198943was uniformly distributed over [089 099] and [079 089]respectively for suppliers 119894 isin 1198681 and 119894 isin 1198682 that is theprobabilities were drawn independently from U[089 099]and U[079 089] respectively

Given local nondisruption probabilities 1199011198943 119894 isin 119868 theprobabilities for the remaining local disruption levels 119897 =0 1 2 were calculated as follows

Probability of complete shutdown (level 119897 = 0) 1199011198940 =01(1 minus 1199011198943)Probability of major disruption (level 119897 = 1) 1199011198941 =03(1 minus 1199011198943)Probability of minor disruption (level 119897 = 2) 1199011198942 =06(1 minus 1199011198943) for all suppliers 119894 isin 119868

Thus 1199011198940 le 1199011198941 le 1199011198942 le 1199011198943 which reflects a real-world relation among probabilities of disruption occurrenceat different levels (eg [8])

Regional disruption probabilities are 1199011 = 0001 and1199012 =001The probability of realizing of disruption scenario 119904 for

suppliers in region 119903 = 1 2 is calculated as follows

119875119903119904

=(1 minus 119901119903)( prod

119894isin119868119903 120582119894119904=0

01 (1 minus 1199011198943))( prod119894isin119868119903 120582119894119904=1

03 (1 minus 1199011198943))( prod119894isin119868119903120582119894119904=2

06 (1 minus 1199011198943))( prod119894isin119868119903 120582119894119904=3

1199011198943) if sum119894isin119868119903120582119894119904 gt 0

119901119903 + (1 minus 119901119903)prod119894isin119868119903

01 (1 minus 1199011198943) if sum119894isin119868119903120582119894119904 = 0

(26)

and the probability for disruption scenario 119904 isin 119878 is given by119875119904 = 1198751119904 1198752119904 Figure 2 presents basic characteristics of all suppliers

probability of complete shutdown119901119903+(1minus119901119903)1199011198940 119894 isin 119868119903 119903 isin119877 expected fulfillment rate Γ119894 = sum119897=123(1 minus 119901119903)119901119894119897120574119894119897 119894 isin119868119903 119903 isin 119877 and purchasing price 119900119894 119894 isin 119868Table 2 presents a subset of nondominated solutions for

the wait-and-see problem WCS for a subset of trade-offparameter120572 isin 0 01 02 03 04 05 06 07 08 09 1Theresults indicate that most of nondominated supply portfoliosconsist of the two suppliers only the second most expensiveand most reliable supplier 119894 = 2 from region 119903 = 1 andthe cheapest and most unreliable supplier 119894 = 3 from region119903 = 2 The most expensive and most reliable supplier 119894 = 1in region 119903 = 1 and the second cheapest and most unreliablesupplier 119894 = 4 in region 119903 = 2 are rarely selected For 120572 = 1

(minimization of cost) the cheapest supplier 119894 = 3 is selectedonly For 120572 = 0 (maximization of service level) the totaldemand for parts is allocated among the two most reliableand most expensive suppliers 119894 = 1 2 The above solutionshows that the service-oriented supply portfolio (120572 close to 0)is more diversified than the cost-oriented portfolio (120572 closeto 1) Table 2 also shows the associated expected fractionof fulfilled demand 1198643 = sum119904isin119878sum119895isin119869sum119905isin119879 119875119904119887119895119908119904119895119905119861 that isdemand fulfilled on time or delayed The solution resultsdemonstrate that a large expected service level is sometimesassociated with a small expected fraction of fulfilled demandThus the maximization of expected service level leads torejection of customer orders that cannot be fulfilled bycustomer requested due dates

Figure 3 shows the expected supply production andshipping schedules respectively for 120572 = 0 (ie for the

8 Mathematical Problems in Engineering

Shutdown probability

Supplier

Expected yield rate

Purchasing price

4321

Supplier4321

Supplier4321

6

8

10

12

14

16

09

092

094

096

098

0

0005

001

0015

002

0025

Figure 2 Suppliers

1 2 3 4 5 6 7Period

Expected supply schedules

Alpha = 0Alpha = 05

Alpha = 1

1 2 3 4 5 6 7Period

Expected production schedules

1 2 3 4 5 6 7Period

Expected shipping schedules

0

20000

40000

60000

80000

100000

Part

s

0

10000

20000

30000

40000

50000

Prod

ucts

Alpha = 0Alpha = 05

Alpha = 1

0

10000

20000

30000

40000

50000

Prod

ucts

Alpha = 0Alpha = 05

Alpha = 1

Figure 3 Expected schedules for modelWCS

Mathematical Problems in Engineering 9

Table 1 Disruption scenarios

119904 119894 = 1 2 3 41 0 0 0 02 0 0 0 13 0 0 0 24 0 0 0 35 0 0 1 06 0 0 1 17 0 0 1 28 0 0 1 39 0 0 2 010 0 0 2 111 0 0 2 212 0 0 2 313 0 0 3 014 0 0 3 115 0 0 3 216 0 0 3 317 0 1 0 018 0 1 0 119 0 1 0 220 0 1 0 321 0 1 1 022 0 1 1 123 0 1 1 224 0 1 1 325 0 1 2 026 0 1 2 127 0 1 2 228 0 1 2 329 0 1 3 030 0 1 3 131 0 1 3 232 0 1 3 333 0 2 0 034 0 2 0 135 0 2 0 236 0 2 0 337 0 2 1 038 0 2 1 139 0 2 1 240 0 2 1 341 0 2 2 042 0 2 2 143 0 2 2 244 0 2 2 345 0 2 3 046 0 2 3 147 0 2 3 248 0 2 3 349 0 3 0 0

Table 1 Continued

119904 119894 = 1 2 3 450 0 3 0 151 0 3 0 252 0 3 0 353 0 3 1 054 0 3 1 155 0 3 1 256 0 3 1 357 0 3 2 058 0 3 2 159 0 3 2 260 0 3 2 361 0 3 3 062 0 3 3 163 0 3 3 264 0 3 3 365 1 0 0 066 1 0 0 167 1 0 0 268 1 0 0 369 1 0 1 070 1 0 1 171 1 0 1 272 1 0 1 373 1 0 2 074 1 0 2 175 1 0 2 276 1 0 2 377 1 0 3 078 1 0 3 179 1 0 3 280 1 0 3 381 1 1 0 082 1 1 0 183 1 1 0 284 1 1 0 385 1 1 1 086 1 1 1 187 1 1 1 288 1 1 1 389 1 1 2 090 1 1 2 191 1 1 2 292 1 1 2 393 1 1 3 094 1 1 3 195 1 1 3 296 1 1 3 397 1 2 0 098 1 2 0 1

10 Mathematical Problems in Engineering

Table 1 Continued

119904 119894 = 1 2 3 499 1 2 0 2100 1 2 0 3101 1 2 1 0102 1 2 1 1103 1 2 1 2104 1 2 1 3105 1 2 2 0106 1 2 2 1107 1 2 2 2108 1 2 2 3109 1 2 3 0110 1 2 3 1111 1 2 3 2112 1 2 3 3113 1 3 0 0114 1 3 0 1115 1 3 0 2116 1 3 0 3117 1 3 1 0118 1 3 1 1119 1 3 1 2120 1 3 1 3121 1 3 2 0122 1 3 2 1123 1 3 2 2124 1 3 2 3125 1 3 3 0126 1 3 3 1127 1 3 3 2128 1 3 3 3129 2 0 0 0130 2 0 0 1131 2 0 0 2132 2 0 0 3133 2 0 1 0134 2 0 1 1135 2 0 1 2136 2 0 1 3137 2 0 2 0138 2 0 2 1139 2 0 2 2140 2 0 2 3141 2 0 3 0142 2 0 3 1143 2 0 3 2144 2 0 3 3145 2 1 0 0146 2 1 0 1147 2 1 0 2148 2 1 0 3149 2 1 1 0150 2 1 1 1

Table 1 Continued

119904 119894 = 1 2 3 4151 2 1 1 2152 2 1 1 3153 2 1 2 0154 2 1 2 1155 2 1 2 2156 2 1 2 3157 2 1 3 0158 2 1 3 1159 2 1 3 2160 2 1 3 3161 2 2 0 0162 2 2 0 1163 2 2 0 2164 2 2 0 3165 2 2 1 0166 2 2 1 1167 2 2 1 2168 2 2 1 3169 2 2 2 0170 2 2 2 1171 2 2 2 2172 2 2 2 3173 2 2 3 0174 2 2 3 1175 2 2 3 2176 2 2 3 3177 2 3 0 0178 2 3 0 1179 2 3 0 2180 2 3 0 3181 2 3 1 0182 2 3 1 1183 2 3 1 2184 2 3 1 3185 2 3 2 0186 2 3 2 1187 2 3 2 2188 2 3 2 3189 2 3 3 0190 2 3 3 1191 2 3 3 2192 2 3 3 3193 3 0 0 0194 3 0 0 1195 3 0 0 2196 3 0 0 3197 3 0 1 0198 3 0 1 1199 3 0 1 2200 3 0 1 3201 3 0 2 0202 3 0 2 1

Mathematical Problems in Engineering 11

Table 1 Continued

119904 119894 = 1 2 3 4203 3 0 2 2204 3 0 2 3205 3 0 3 0206 3 0 3 1207 3 0 3 2208 3 0 3 3209 3 1 0 0210 3 1 0 1211 3 1 0 2212 3 1 0 3213 3 1 1 0214 3 1 1 1215 3 1 1 2216 3 1 1 3217 3 1 2 0218 3 1 2 1219 3 1 2 2220 3 1 2 3221 3 1 3 0222 3 1 3 1223 3 1 3 2224 3 1 3 3225 3 2 0 0226 3 2 0 1227 3 2 0 2228 3 2 0 3229 3 2 1 0230 3 2 1 1231 3 2 1 2232 3 2 1 3233 3 2 2 0234 3 2 2 1235 3 2 2 2236 3 2 2 3237 3 2 3 0238 3 2 3 1239 3 2 3 2240 3 2 3 3241 3 3 0 0242 3 3 0 1243 3 3 0 2244 3 3 0 3245 3 3 1 0246 3 3 1 1247 3 3 1 2248 3 3 1 3249 3 3 2 0250 3 3 2 1251 3 3 2 2252 3 3 2 3

Table 1 Continued

119904 119894 = 1 2 3 4253 3 3 3 0254 3 3 3 1255 3 3 3 2256 3 3 3 3

maximum expected service level) 120572 = 05 and 120572 = 1 (ie forthe minimum expected cost) The expected schedules werecomputed using the formulae presented below

(i) Expected schedule of supplies of parts to the pro-ducer

sum119904isin119878

sum119894isin119868119904120590119894=119905

119875119904119861120574119894120582119894119904V119894 119905 isin 119879 (27)

(ii) Expected production schedule

sum119904isin119878

sum119895isin119869

119875119904119887119895119908119904119895119905 119905 isin 119879 (28)

(iii) Expected schedule of shipping of products from theproducer to the distribution centers

sum119904isin119878

sum119896isin119870

sum119895isin119869119896

119875119904119887119895( sum1199051015840isin1198791199051015840lt119905

1199081199041198951199051015840)119909119904119896119905 119905 isin 119879 (29)

As 120572 increases that is the decision-makerrsquos priority shiftsfrom the maximum service level to minimum cost and moreparts are ordered from less reliable and lower cost suppliersthe expected supply schedules and the corresponding pro-duction schedules are more delayed as well as the delivery ofproducts to the customers Note that despite constraint (13)that ensures a feasible schedule with at most one shipmentof products to each distribution center for each disruptionscenario the expected shipping schedule (29) may be splitinto more smaller size shipments (cf Figure 3 where twomajor and one small-size shipments are indicated for eachconfidence level)

For comparison Table 3 presents a subset of non-dominated solutions obtained for the expected value problemEWCS and Figure 4 shows supply production and shippingschedules Unlike the stochastic programming approachwhich accounts for all potential disruption scenarios tooptimize an expected performance of the supply chain thesolution obtained using the deterministic approach is basedon aggregate information on suppliers expected fulfillmentIn general the results are similar for both models and thecorresponding optimal solution values are close to each otherwhich indicates that the expected value problem can be usedin practice when stochastic mixed integer programs are hardto solve The optimal solution values for the expected valueproblem frequently outperform the corresponding solutionvalues of the wait-and-see problem which is in line withthe proposition EV le WS in [27] compare Section 5for example the minimum cost 1198641 for EWCS (see Table 3)

12 Mathematical Problems in Engineering

Table 2 Nondominated solutions for modelWCS

120572 0 01 02 03 04 05 and 06 07 08 and 09 1Var = 38077 Bin = 33076 Cons = 28769 Nonz = 348325(a)

Exp cost 1731 (1198641) 1620 1547 1206 1200 1176 1070 1065 1051 (1198641)Exp service level(b) 8552 (1198642) 8487 8452 7719 7676 7595 5863 5871 3822 (1198642)Exp fulfilled demand(c) 8552 8530 8452 9370 9361 9353 9134 9120 9068

Suppliers selected( of total demand)

1 (76) 1 (35)2 (24) 2 (55) 2 (87) 2 (55) 2 (55) 2 (55) 2 (12) 2 (10)3 (10) 3 (13) 3 (22) 3 (32) 3 (45) 3 (88) 3 (90) 3 (100)4 (23) 4 (13)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119904isin119878 sum119895isin119869 119875119904119887119895119910

119904119895119861 times

100 (c)sum119904isin119878 sum119895isin119869 sum119905isin119879 119875119904119887119895119908119904119895119905119861 times 100

Table 3 Nondominated solutions for model EWCS

120572 0 01 02 and 03 04 and 05 06 and 07 08 09 and 1Var = 162 Bin = 136 Cons = 125 Nonz = 1430(a)

Exp cost 1691 (1198641) 1427 1214 1167 1045 (1198641)Exp service level(b) 87 (1198642) 86 81 79 62 (1198642)Exp fulfilled demand(c) 87 90 91 93 91

Suppliers selected( of total demand)

1 (89) 2 (94) 2 (49) 2 (50) 2 (5)3 (6) 3 (51) 3 (50) 3 (95)4 (11)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119895isin119869 119887119895119884119895119861 times 100(c)sum119895isin119869 sum119905isin119879 119887119895119882119895119905119861 times 100

1 2 3 4 5 6 7Period

Supply schedules

1 2 3 4 5 6 7Period

Production schedules

1 2 3 4 5 6 7Period

Shipping schedules

0

10000

20000

30000

40000

50000

Prod

ucts

0

10000

20000

30000

40000

50000

Prod

ucts

0

20000

40000

60000

80000

100000

Part

s

Alpha = 0Alpha = 05

Alpha = 1 Alpha = 0Alpha = 05

Alpha = 1

Alpha = 0Alpha = 05

Alpha = 1

Figure 4 Schedules for model EWCS

Mathematical Problems in Engineering 13

1045 le 1051 forWCS (see Table 2) or themaximum servicelevel1198642 for EWCS (see Table 3) 8700 ge 8552 forWCS (seeTable 2)

The stochastic wait-and-see approach however leads toa more diversified supply portfolio For the deterministicapproach most of nondominated supply portfolios consistof two suppliers only 119894 = 2 3 while suppliers 119894 = 1 4 areselected only for the maximum service level objective (thatis for 120572 = 0) Comparison of Figures 3 and 4 indicatesthat the expected schedules for model WCS computed asexpectations over all schedules for all potential disruptionscenarios (27)ndash(29) are similar to the corresponding singleschedules determined by model EWCS The main differencesobserved aremore delayed expected production and shippingschedules for model WCS when minimization of cost is con-sidered (that is for120572 = 1) Finally it is interesting to note thatin the multiple sourcing environment considered both thewait-and-see approach and in particular the expected valueapproach frequently select a dual sourcing supply portfoliowith one main supplier and one supporting supplier

The computational experiments were performed usingthe AMPL programming language and the CPLEX 1262solver on a MacBook Pro laptop with Intel Core i7 processorrunning at 28 GHz and with 16GB RAM The solver wascapable of finding proven optimal solution for all exampleswith CPU time ranging from several minutes to several hoursfor the stochastic model WCS and fraction of a second forthe deterministic model EWCS Since the stochastic pro-gramming model WCS needs to determine nondominatedschedules for all potential disruption scenarios while modelEWCS deals with a single scenario only the difference in thecomputational effort required is obvious

7 Conclusions

In this paper two biobjective MIP formulations stochasticand deterministic have been proposed and compared for thecoordinated decision-making in supply chains under partiallocal disruptions and all-or-nothing regional disruptionsTheproblem objective has been to jointly schedule supplies pro-duction and distribution to optimize the trade-off betweenexpected cost and expected service levelWhile the stochasticprogramming approach aims at optimizing the expectedperformance of a supply chain over all possible disruptionscenarios the deterministic approach accounts only for asingle scenario representing average disruption conditionsAs a result the stochastic programming approach determinesnondominated solutions for all disruption scenarios whereasthe deterministic approach produces a single solution onlyIn particular in the stochastic programming approach theselection of supply portfolio is combined with supply chainscheduling for all disruption scenarios considered In con-trast the deterministic expected value approach provides theportfolio along with a single expected schedule of productionand distribution

The expected schedules obtained for the stochastic pro-gramming model WCS as expectations over all schedulesfor all disruption scenarios have been compared with thecorresponding schedules determined by the deterministic

model EWCS based on an expected disruption scenarioThecomparison has demonstrated that the two approaches leadto similar expected solutions

The main findings are in line with other research and arelisted as follows

(i) The two decision-making approaches stochastic anddeterministic lead to similar expected performanceof a supply chain under multilevel disruptions

(ii) The optimal solution values for the expected valueproblem frequently outperform the correspondingsolution values of the wait-and-see problem

(iii) Despite the multiple sourcing environment consid-ered both the wait-and-see approach and in partic-ular the expected value approach frequently selecta dual sourcing supply portfolio with one mainsupplier and one supporting supplier

(iv) The stochastic approach which accounts for allpotential disruption scenarios may lead to a morediversified supply portfolio that will hedge against avariety of scenarios

(v) The expected schedules are more delayed for thestochastic approach

(vi) The service-oriented supply portfolio is more diversi-fied andmay combine both high-cost reliable suppli-ers and low-cost unreliable suppliers while the cost-oriented portfolio depends mainly on low-cost andless reliable suppliers

Overall the results of computational experiments indi-cate that the proposed approach and developed MIP modelsare flexible and efficient tools for coordinated supply chainscheduling The portfolio approach leads to MIP formula-tions with strong LP relaxations and has been proven to becomputationally very efficient CPU time required to findproven optimal solutions for realistic size examples usingcommercially available software for MIP is acceptable for areal-world supply chain disruption management (see Sawik[33])

Since the probability distribution of supply disruptionsfrom each supplier is usually unknown (eg [8]) local multi-level disruptions in the stochastic programming modelWCShave a multinomial discrete distribution while the two-levelregional disruptions have a binomial discrete distributionand all disruption events are independent As part of futureresearch we propose to enhance the stochastic programmingmodel formore general scenarios with finitelymany elementssingled out and all the probability concentrated in them Forexample rather than complete shutdown suppliers within aregion may have correlated disruptions Another importantstream of future research is the study of robustness andsensitivity in relation to input data changes in supply chainscheduling under disruption risks (eg [33]) Future researchcan also focus on the following improvement of the proposedmodels to consider the minimization of time and cost ofrecovery (eg Whitney et al [34])

14 Mathematical Problems in Engineering

Notations

Indices

119894 Supplier 119894 isin 119868119895 Customer 119895 isin 119869119896 Distribution center 119896 isin 119870119897 Disruption level 119897 isin 119871119903 Region 119903 isin 119877119904 Disruption scenario 119904 isin 119878119905 Planning period 119905 isin 119879Input Parameters

119887119895 Size (number of products) of customer order 119895119861 Total demand for partsproducts 119861 = sum119895isin119869 119887119895119862 Capacity of producer119889119895 Due date for customer order 119895119890119894 Fixed cost of ordering parts from supplier 119894119892119895 Per unit penalty cost of delayed customer order 119895ℎ119895 Per unit penalty cost of unfulfilled customer order 119895119868119903 Subset of suppliers in region 119903119869119896 Subset of customers serviced by distribution center 119896119900119894 Per unit price of parts purchased from supplier 119894119901119894119897 Probability of disruption level 119897 for supplier 119894119901119903 Regional disruption probability for region 119903120574119894119897 Fraction of an order delivered by supplier 119894 underdisruption level 119897 (fulfillment rate)Γ119894 Expected fraction of an order delivered by supplier 119894(expected fulfillment rate)120590119894 Delivery lead time from supplier 119894120591119896 Transportation time to distribution center 119896

Conflicts of Interest

The author declares that there are no conflicts of interest

Acknowledgments

This work has been supported by NCN research grant (noDEC-201311BST804458) and by AGH (no 1111200324)

References

[1] J V Blackhurst K P Scheibe and D J Johnson ldquoSupplierrisk assessment and monitoring for the automotive industryrdquoInternational Journal of Physical Distribution amp Logistics Man-agement vol 38 no 2 pp 143ndash165 2008

[2] P Hoffmann H Schiele and K Krabbendam ldquoUncertaintysupply risk management and their impact on performancerdquoJournal of Purchasing amp Supply Management vol 19 no 3 pp199ndash211 2013

[3] Y Park P Hong and J J Roh ldquoSupply chain lessons from thecatastrophic natural disaster in Japanrdquo Business Horizons vol56 no 1 pp 75ndash85 2013

[4] M Haraguchi and U Lall ldquoFlood risks and impacts A casestudy of Thailandrsquos floods in 2011 and research questions forsupply chain decisionmakingrdquo International Journal of DisasterRisk Reduction vol 14 pp 256ndash272 2015

[5] T Fujimoto and Y W Park ldquoBalancing supply chain com-petitiveness and robustness through ldquovirtual dual sourcingrdquoLessons from the Great East Japan Earthquakerdquo InternationalJournal of Production Economics vol 147 pp 429ndash436 2014

[6] HMatsuo ldquoImplications of the Tohoku earthquake for Toyotarsquoscoordination mechanism supply chain disruption of auto-motive semiconductorsrdquo International Journal of ProductionEconomics vol 161 pp 217ndash227 2015

[7] J R Marszewska ldquoImplications of seismic hazard in Japan ontoyota supply chain disruption risksrdquo in Proceedings of the 13thInternational Conference on Industrial Logistics (ICIL rsquo16) pp178ndash185 Zakopane Poland October 2016

[8] A Z Zeng and Y Xia ldquoBuilding a mutually beneficial partner-ship to ensure backup supplyrdquo Omega vol 52 pp 77ndash91 2015

[9] S S Erenguc N C Simpson and A J Vakharia ldquoIntegratedproductiondistribution planning in supply chains an invitedreviewrdquo European Journal of Operational Research vol 115 no2 pp 219ndash236 1999

[10] L Lei S Liu A Ruszczynski and S Park ldquoOn the integratedproduction inventory and distribution routing problemrdquo IIETransactions vol 38 no 11 pp 955ndash970 2006

[11] A Kaur A Kanda and S G Deshmukh ldquoA graph theoreticapproach for supply chain coordinationrdquo International Journalof Logistics Systems and Management vol 2 no 4 pp 321ndash3412006

[12] T ChoiW Yeung andT C E Cheng ldquoScheduling and co-ordi-nation of multi-suppliers single-warehouse-operator single-manufacturer supply chains with variable production rates andstorage costsrdquo International Journal of Production Research vol51 no 9 pp 2593ndash2601 2013

[13] S Liu and L G Papageorgiou ldquoMultiobjective optimisation ofproduction distribution and capacity planning of global supplychains in the process industryrdquo Omega vol 41 no 2 pp 369ndash382 2013

[14] Z L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[15] T Sawik ldquoCoordinated supply chain schedulingrdquo InternationalJournal of Production Economics vol 120 no 2 pp 437ndash4512009

[16] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[17] D Ivanov B Sokolov and A Dolgui ldquoThe Ripple effect in sup-ply chains Trade-off rsquoefficiency-flexibility- resiliencersquo in disrup-tionmanagementrdquo International Journal of ProductionResearchvol 52 no 7 pp 2154ndash2172 2014

[18] D Ivanov A Pavlov and B Sokolov ldquoOptimal distribution(re)planning in a centralized multi-stage supply network underconditions of the ripple effect and structure dynamicsrdquo Euro-pean Journal of Operational Research vol 237 no 2 pp 758ndash770 2014

[19] D Ivanov A Pavlov A Dolgui D Pavlov and B SokolovldquoDisruption-driven supply chain (re)-planning and perfor-mance impact assessment with consideration of pro-active andrecovery policiesrdquo Transportation Research Part E Logistics andTransportation Review vol 90 pp 7ndash24 2016

[20] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

Mathematical Problems in Engineering 15

[21] A J Schmitt S A Sun L V Snyder and Z-J M Shen ldquoCen-tralization versus decentralization Risk pooling risk diversifi-cation and supply chain disruptionsrdquo Omega vol 52 pp 201ndash212 2015

[22] L V Snyder Z Atan P Peng Y Rong A J Schmitt andB Sinsoysal ldquoORMS models for supply chain disruptions areviewrdquo IIE Transactions vol 48 no 2 pp 89ndash109 2016

[23] T Sawik ldquoIntegrated selection of suppliers and scheduling ofcustomer orders in the presence of supply chain disruptionrisksrdquo International Journal of Production Research vol 51 no23-24 pp 7006ndash7022 2013

[24] T Sawik ldquoOptimization of cost and service level in the presenceof supply chain disruption risks Single vs multiple sourcingrdquoComputers amp Operations Research vol 51 pp 11ndash20 2014

[25] T Sawik ldquoIntegrated supply production and distributionscheduling under disruption risksrdquoOmega vol 62 pp 131ndash1442016

[26] T Sawik ldquoIntegrated supply chain scheduling under multi-leveldisruptionsrdquo IFAC-PapersOnLine vol 28 no 3 pp 1515ndash15202015

[27] P Kall and J Mayer Stochastic Linear Programming ModelsTheory and Computation vol 156 of International Series inOperations Research amp Management Science Springer NewYork 2011

[28] I N Durbach and T J Stewart ldquoUsing expected values to sim-plify decision making under uncertaintyrdquo Omega vol 37 no 2pp 312ndash330 2009

[29] F Maggioni and S W Wallace ldquoAnalyzing the quality of theexpected value solution in stochastic programmingrdquo Annals ofOperations Research vol 200 pp 37ndash54 2012

[30] T Sawik ldquoSelection of supply portfolio under disruption risksrdquoOmega vol 39 no 2 pp 194ndash208 2011

[31] E Delage S Arroyo and Y Ye ldquoThe value of stochastic mod-eling in two-stage stochastic programs with cost uncertaintyrdquoOperations Research vol 62 no 6 pp 1377ndash1393 2014

[32] T Sawik Scheduling in Supply Chains Using Mixed Integer Pro-gramming John Wiley amp Sons Inc Hoboken NJ 2011

[33] T Sawik Supply Chain DisruptionManagement Using StochasticMixed Integer Programming vol 256 of International Seriesin Operations Research amp Management Science Springer NewYork 2017 httpwwwspringercomgpbook9783319588223

[34] D E Whitney J Luo and D A Heller ldquoThe benefits and con-straints of temporary sourcing diversification in supply chaindisruption and recoveryrdquo Journal of Purchasing and SupplyManagement vol 20 no 4 pp 238ndash250 2014

6 Mathematical Problems in Engineering

expected supply conditions In model WCS where the ran-domness is characterized by a set of disruption scenariosthe only random parameters are suppliers fulfillment rates120574119894120582119894119904 which appear both in the objective function (4) and inconstraints (11)

In model EWCS suppliers probabilistic fulfillment ratesdefined for each disruption scenario 120574119894120582119894119904 119894 isin 119868 119904 isin 119878 orequivalently for each disruption level 120574119894119897 119894 isin 119868 119897 isin 119871 (1)have been replaced by the expected fulfillment rates of eachsupplier

Γ119894 = sum119904isin119878

119875119904120574119894120582119894119904 119894 isin 119868 (20)

or equivalently

Γ119894 = (1 minus 119901119903)sum119897isin119871

119901119894119897120574119894119897 119894 isin 119868119903 119903 isin 119877 (21)

Accordingly stochastic binary decision variables 119908119904119895119905 119909119904119896119905 119910119904119895(17)ndash(19) defined for each disruption scenario 119904 isin 119878 havebeen replaced by their deterministic equivalents 119882119895119905 119883119896119905119884119895

Now the expected cost per product 1198641 (see (22)) andthe expected service level 1198642(see (23)) are defined as fol-lows

1198641 = sum119894isin119868 119890119894119906119894 + sum119894isin119868 119861119900119894Γ119894V119894 + sum119895isin119869 119892119895119887119895 (sum119905isin119879119882119895119905 minus 119884119895) + sum119895isin119869 ℎ119895119887119895 (1 minus sum119905isin119879119882119895119905)119861 (22)

1198642 = sum119895isin119869 119887119895119884119895119861 (23)

Model EWCS is presented below

Model EWCS

Minimize (8)subject to (6) (7) (9) (22) (23)

sum119905isin119879

119882119895119905 le 1 119895 isin 119869sum119895isin119869

119887119895119882119895119905 le 119862 119905 isin 119879sum119895isin119869

sum1199051015840isin1198791199051015840le119905

1198871198951198821198951199051015840 le 119861 sum119894isin119868120590119894le119905minus1

Γ119894V119894 119905 isin 119879sum119905isin119879119870

119905119883119896119905 ge sum119905isin119879

(119905 + 1)119882119895119905119896 isin 119870 119895 isin 119869119896

sum119905isin119879119870

119883119896119905 le 1 119896 isin 119870119884119895 le sum119905isin119879119905le119889119895minus120591119896

119882119895119905 119896 isin 119870 119895 isin 119869119896119884119895 le sum119905isin119879119870119905le119889119895minus120591119896+1

119883119896119905 119896 isin 119870 119895 isin 119869119896sum

119905isin119879119905le119889119895minus120591119896

119882119895119905 + sum119905isin119879119870119905le119889119895minus120591119896+1

119883119896119905 minus 1le 119884119895 119896 isin 119870 119895 isin 119869119896119906119894 isin 0 1 119894 isin 119868V119894 isin [0 1] 119894 isin 119868

119882119895119905 isin 0 1 119895 isin 119869 119905 isin 119879119883119896119905 isin 0 1 119896 isin 119870 119905 isin 119879119870119884119895 ge 0 119895 isin 119869

(24)

Notice that unlike the stochastic programming modelWCS which is formulated to determine optimal schedulesfor all potential disruption scenarios model EWCS accountsfor a single scenario only representing the expected suppliesExcept for the expected values of the random parametersthis model does not take into account any distributioninformation and the solution remains the same as long as theexpectations do not change In contrast tomodelWCS wherethe selection of supply portfolio (V1 V119868) is combinedwith supply chain scheduling for all disruption scenariosconsidered now the portfolio is determined along with asingle schedule

If randomparameters appear only in the constraints then[27]

EV le WS (25)

where EV is the optimal solution value of the expected valueproblem EWCS and WS is the optimal solution value ofthe wait-and-see problem WCS On the other hand whenuncertainty is limited to the objective function of the prob-lem the solution obtained by simply replacing the randomparameters with their expected values provides already arobust alternative (Delage et al [31])

6 Computational Examples

In this section some computational examples are presentedto illustrate possible applications of the proposedMIPmodelsand to compare the wait-and-see and the expected value

Mathematical Problems in Engineering 7

approaches The examples are modeled in part after a real-world electronics supply chain (eg Sawik [32]) The fol-lowing parameters have been selected for the computationalexamples119868 = 4 suppliers 119869 = 20 customer orders 119870 = 2 dis-tribution centers 119871 = 3 partial disruption levels 119877 = 2geographic regions and 119879 = 7 planning periods1198681 = 1 2 1198682 = 3 41198691 = 1 10 1198692 = 11 20

Shipping times from suppliers 120590 = (1 1 3 3)Shipping times to distribution centers 120591 = (1 2)Customer demand 119887119895 isin 2000 3000 10000 for all119895 isin 119869 and total demand 119861 = 100000Due dates 119889119895 isin 2 + min119894isin119868(120590119894) + min119896isin119870(120591119896) 119879 +

max119896isin119870(120591119896) for all 119895 isin 119869Fixed ordering costs for suppliers 119890 = (8000 6000 1200013000)Unit purchasing prices from suppliers 119900 = (14 12 8 9)Unit penalties for delayed unfulfilled customer orders119892119895 = lceilmax119894isin119868119900119894350rceil = 1 ℎ119895 = 2max119894isin119868119900119894 = 28 respectively

for all 119895 isin 119869Producer capacity 119862 = 45000Local disruption levels and the associated fulfillment rates

(the percentage of an order that can be delivered) are shownbelow119871 = 0 1 2 3 where 119897 = 0 complete shutdown1205741198940 = 0 forall119894 isin 119868 that is 0 of an order delivered 119897 = 1major disruption 1205741198941 isin [001 050] forall119894 isin 1198681 and 1205741198941 isin[001 030] forall119894 isin 1198682 that is 1 to 50 and 1 to 30 ofan order delivered respectively 119897 = 2 minor disruption

1205741198942 isin [0 51 099] forall119894 isin 1198681 and 1205741198942 isin [0 31 099] forall119894 isin 1198682that is 51 to 99 and 31 to 99 of an order deliveredrespectively 119897 = 119871 = 3 no disruption 1205741198943 = 1 forall119894 isin 119868 thatis 100 of an order delivered

The total number of all potential scenarios is 119878 = (119871 +1)119868 = 44 = 256 scenarios where each scenario 119904 isin 119878 is repre-sented by vector 120582119904 = (1205821119904 1205824119904) where 120582119894119904 isin 119871 119894 isin 119868 (seeTable 1)

The local nondisruption probability (level 119897 = 3) 1199011198943was uniformly distributed over [089 099] and [079 089]respectively for suppliers 119894 isin 1198681 and 119894 isin 1198682 that is theprobabilities were drawn independently from U[089 099]and U[079 089] respectively

Given local nondisruption probabilities 1199011198943 119894 isin 119868 theprobabilities for the remaining local disruption levels 119897 =0 1 2 were calculated as follows

Probability of complete shutdown (level 119897 = 0) 1199011198940 =01(1 minus 1199011198943)Probability of major disruption (level 119897 = 1) 1199011198941 =03(1 minus 1199011198943)Probability of minor disruption (level 119897 = 2) 1199011198942 =06(1 minus 1199011198943) for all suppliers 119894 isin 119868

Thus 1199011198940 le 1199011198941 le 1199011198942 le 1199011198943 which reflects a real-world relation among probabilities of disruption occurrenceat different levels (eg [8])

Regional disruption probabilities are 1199011 = 0001 and1199012 =001The probability of realizing of disruption scenario 119904 for

suppliers in region 119903 = 1 2 is calculated as follows

119875119903119904

=(1 minus 119901119903)( prod

119894isin119868119903 120582119894119904=0

01 (1 minus 1199011198943))( prod119894isin119868119903 120582119894119904=1

03 (1 minus 1199011198943))( prod119894isin119868119903120582119894119904=2

06 (1 minus 1199011198943))( prod119894isin119868119903 120582119894119904=3

1199011198943) if sum119894isin119868119903120582119894119904 gt 0

119901119903 + (1 minus 119901119903)prod119894isin119868119903

01 (1 minus 1199011198943) if sum119894isin119868119903120582119894119904 = 0

(26)

and the probability for disruption scenario 119904 isin 119878 is given by119875119904 = 1198751119904 1198752119904 Figure 2 presents basic characteristics of all suppliers

probability of complete shutdown119901119903+(1minus119901119903)1199011198940 119894 isin 119868119903 119903 isin119877 expected fulfillment rate Γ119894 = sum119897=123(1 minus 119901119903)119901119894119897120574119894119897 119894 isin119868119903 119903 isin 119877 and purchasing price 119900119894 119894 isin 119868Table 2 presents a subset of nondominated solutions for

the wait-and-see problem WCS for a subset of trade-offparameter120572 isin 0 01 02 03 04 05 06 07 08 09 1Theresults indicate that most of nondominated supply portfoliosconsist of the two suppliers only the second most expensiveand most reliable supplier 119894 = 2 from region 119903 = 1 andthe cheapest and most unreliable supplier 119894 = 3 from region119903 = 2 The most expensive and most reliable supplier 119894 = 1in region 119903 = 1 and the second cheapest and most unreliablesupplier 119894 = 4 in region 119903 = 2 are rarely selected For 120572 = 1

(minimization of cost) the cheapest supplier 119894 = 3 is selectedonly For 120572 = 0 (maximization of service level) the totaldemand for parts is allocated among the two most reliableand most expensive suppliers 119894 = 1 2 The above solutionshows that the service-oriented supply portfolio (120572 close to 0)is more diversified than the cost-oriented portfolio (120572 closeto 1) Table 2 also shows the associated expected fractionof fulfilled demand 1198643 = sum119904isin119878sum119895isin119869sum119905isin119879 119875119904119887119895119908119904119895119905119861 that isdemand fulfilled on time or delayed The solution resultsdemonstrate that a large expected service level is sometimesassociated with a small expected fraction of fulfilled demandThus the maximization of expected service level leads torejection of customer orders that cannot be fulfilled bycustomer requested due dates

Figure 3 shows the expected supply production andshipping schedules respectively for 120572 = 0 (ie for the

8 Mathematical Problems in Engineering

Shutdown probability

Supplier

Expected yield rate

Purchasing price

4321

Supplier4321

Supplier4321

6

8

10

12

14

16

09

092

094

096

098

0

0005

001

0015

002

0025

Figure 2 Suppliers

1 2 3 4 5 6 7Period

Expected supply schedules

Alpha = 0Alpha = 05

Alpha = 1

1 2 3 4 5 6 7Period

Expected production schedules

1 2 3 4 5 6 7Period

Expected shipping schedules

0

20000

40000

60000

80000

100000

Part

s

0

10000

20000

30000

40000

50000

Prod

ucts

Alpha = 0Alpha = 05

Alpha = 1

0

10000

20000

30000

40000

50000

Prod

ucts

Alpha = 0Alpha = 05

Alpha = 1

Figure 3 Expected schedules for modelWCS

Mathematical Problems in Engineering 9

Table 1 Disruption scenarios

119904 119894 = 1 2 3 41 0 0 0 02 0 0 0 13 0 0 0 24 0 0 0 35 0 0 1 06 0 0 1 17 0 0 1 28 0 0 1 39 0 0 2 010 0 0 2 111 0 0 2 212 0 0 2 313 0 0 3 014 0 0 3 115 0 0 3 216 0 0 3 317 0 1 0 018 0 1 0 119 0 1 0 220 0 1 0 321 0 1 1 022 0 1 1 123 0 1 1 224 0 1 1 325 0 1 2 026 0 1 2 127 0 1 2 228 0 1 2 329 0 1 3 030 0 1 3 131 0 1 3 232 0 1 3 333 0 2 0 034 0 2 0 135 0 2 0 236 0 2 0 337 0 2 1 038 0 2 1 139 0 2 1 240 0 2 1 341 0 2 2 042 0 2 2 143 0 2 2 244 0 2 2 345 0 2 3 046 0 2 3 147 0 2 3 248 0 2 3 349 0 3 0 0

Table 1 Continued

119904 119894 = 1 2 3 450 0 3 0 151 0 3 0 252 0 3 0 353 0 3 1 054 0 3 1 155 0 3 1 256 0 3 1 357 0 3 2 058 0 3 2 159 0 3 2 260 0 3 2 361 0 3 3 062 0 3 3 163 0 3 3 264 0 3 3 365 1 0 0 066 1 0 0 167 1 0 0 268 1 0 0 369 1 0 1 070 1 0 1 171 1 0 1 272 1 0 1 373 1 0 2 074 1 0 2 175 1 0 2 276 1 0 2 377 1 0 3 078 1 0 3 179 1 0 3 280 1 0 3 381 1 1 0 082 1 1 0 183 1 1 0 284 1 1 0 385 1 1 1 086 1 1 1 187 1 1 1 288 1 1 1 389 1 1 2 090 1 1 2 191 1 1 2 292 1 1 2 393 1 1 3 094 1 1 3 195 1 1 3 296 1 1 3 397 1 2 0 098 1 2 0 1

10 Mathematical Problems in Engineering

Table 1 Continued

119904 119894 = 1 2 3 499 1 2 0 2100 1 2 0 3101 1 2 1 0102 1 2 1 1103 1 2 1 2104 1 2 1 3105 1 2 2 0106 1 2 2 1107 1 2 2 2108 1 2 2 3109 1 2 3 0110 1 2 3 1111 1 2 3 2112 1 2 3 3113 1 3 0 0114 1 3 0 1115 1 3 0 2116 1 3 0 3117 1 3 1 0118 1 3 1 1119 1 3 1 2120 1 3 1 3121 1 3 2 0122 1 3 2 1123 1 3 2 2124 1 3 2 3125 1 3 3 0126 1 3 3 1127 1 3 3 2128 1 3 3 3129 2 0 0 0130 2 0 0 1131 2 0 0 2132 2 0 0 3133 2 0 1 0134 2 0 1 1135 2 0 1 2136 2 0 1 3137 2 0 2 0138 2 0 2 1139 2 0 2 2140 2 0 2 3141 2 0 3 0142 2 0 3 1143 2 0 3 2144 2 0 3 3145 2 1 0 0146 2 1 0 1147 2 1 0 2148 2 1 0 3149 2 1 1 0150 2 1 1 1

Table 1 Continued

119904 119894 = 1 2 3 4151 2 1 1 2152 2 1 1 3153 2 1 2 0154 2 1 2 1155 2 1 2 2156 2 1 2 3157 2 1 3 0158 2 1 3 1159 2 1 3 2160 2 1 3 3161 2 2 0 0162 2 2 0 1163 2 2 0 2164 2 2 0 3165 2 2 1 0166 2 2 1 1167 2 2 1 2168 2 2 1 3169 2 2 2 0170 2 2 2 1171 2 2 2 2172 2 2 2 3173 2 2 3 0174 2 2 3 1175 2 2 3 2176 2 2 3 3177 2 3 0 0178 2 3 0 1179 2 3 0 2180 2 3 0 3181 2 3 1 0182 2 3 1 1183 2 3 1 2184 2 3 1 3185 2 3 2 0186 2 3 2 1187 2 3 2 2188 2 3 2 3189 2 3 3 0190 2 3 3 1191 2 3 3 2192 2 3 3 3193 3 0 0 0194 3 0 0 1195 3 0 0 2196 3 0 0 3197 3 0 1 0198 3 0 1 1199 3 0 1 2200 3 0 1 3201 3 0 2 0202 3 0 2 1

Mathematical Problems in Engineering 11

Table 1 Continued

119904 119894 = 1 2 3 4203 3 0 2 2204 3 0 2 3205 3 0 3 0206 3 0 3 1207 3 0 3 2208 3 0 3 3209 3 1 0 0210 3 1 0 1211 3 1 0 2212 3 1 0 3213 3 1 1 0214 3 1 1 1215 3 1 1 2216 3 1 1 3217 3 1 2 0218 3 1 2 1219 3 1 2 2220 3 1 2 3221 3 1 3 0222 3 1 3 1223 3 1 3 2224 3 1 3 3225 3 2 0 0226 3 2 0 1227 3 2 0 2228 3 2 0 3229 3 2 1 0230 3 2 1 1231 3 2 1 2232 3 2 1 3233 3 2 2 0234 3 2 2 1235 3 2 2 2236 3 2 2 3237 3 2 3 0238 3 2 3 1239 3 2 3 2240 3 2 3 3241 3 3 0 0242 3 3 0 1243 3 3 0 2244 3 3 0 3245 3 3 1 0246 3 3 1 1247 3 3 1 2248 3 3 1 3249 3 3 2 0250 3 3 2 1251 3 3 2 2252 3 3 2 3

Table 1 Continued

119904 119894 = 1 2 3 4253 3 3 3 0254 3 3 3 1255 3 3 3 2256 3 3 3 3

maximum expected service level) 120572 = 05 and 120572 = 1 (ie forthe minimum expected cost) The expected schedules werecomputed using the formulae presented below

(i) Expected schedule of supplies of parts to the pro-ducer

sum119904isin119878

sum119894isin119868119904120590119894=119905

119875119904119861120574119894120582119894119904V119894 119905 isin 119879 (27)

(ii) Expected production schedule

sum119904isin119878

sum119895isin119869

119875119904119887119895119908119904119895119905 119905 isin 119879 (28)

(iii) Expected schedule of shipping of products from theproducer to the distribution centers

sum119904isin119878

sum119896isin119870

sum119895isin119869119896

119875119904119887119895( sum1199051015840isin1198791199051015840lt119905

1199081199041198951199051015840)119909119904119896119905 119905 isin 119879 (29)

As 120572 increases that is the decision-makerrsquos priority shiftsfrom the maximum service level to minimum cost and moreparts are ordered from less reliable and lower cost suppliersthe expected supply schedules and the corresponding pro-duction schedules are more delayed as well as the delivery ofproducts to the customers Note that despite constraint (13)that ensures a feasible schedule with at most one shipmentof products to each distribution center for each disruptionscenario the expected shipping schedule (29) may be splitinto more smaller size shipments (cf Figure 3 where twomajor and one small-size shipments are indicated for eachconfidence level)

For comparison Table 3 presents a subset of non-dominated solutions obtained for the expected value problemEWCS and Figure 4 shows supply production and shippingschedules Unlike the stochastic programming approachwhich accounts for all potential disruption scenarios tooptimize an expected performance of the supply chain thesolution obtained using the deterministic approach is basedon aggregate information on suppliers expected fulfillmentIn general the results are similar for both models and thecorresponding optimal solution values are close to each otherwhich indicates that the expected value problem can be usedin practice when stochastic mixed integer programs are hardto solve The optimal solution values for the expected valueproblem frequently outperform the corresponding solutionvalues of the wait-and-see problem which is in line withthe proposition EV le WS in [27] compare Section 5for example the minimum cost 1198641 for EWCS (see Table 3)

12 Mathematical Problems in Engineering

Table 2 Nondominated solutions for modelWCS

120572 0 01 02 03 04 05 and 06 07 08 and 09 1Var = 38077 Bin = 33076 Cons = 28769 Nonz = 348325(a)

Exp cost 1731 (1198641) 1620 1547 1206 1200 1176 1070 1065 1051 (1198641)Exp service level(b) 8552 (1198642) 8487 8452 7719 7676 7595 5863 5871 3822 (1198642)Exp fulfilled demand(c) 8552 8530 8452 9370 9361 9353 9134 9120 9068

Suppliers selected( of total demand)

1 (76) 1 (35)2 (24) 2 (55) 2 (87) 2 (55) 2 (55) 2 (55) 2 (12) 2 (10)3 (10) 3 (13) 3 (22) 3 (32) 3 (45) 3 (88) 3 (90) 3 (100)4 (23) 4 (13)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119904isin119878 sum119895isin119869 119875119904119887119895119910

119904119895119861 times

100 (c)sum119904isin119878 sum119895isin119869 sum119905isin119879 119875119904119887119895119908119904119895119905119861 times 100

Table 3 Nondominated solutions for model EWCS

120572 0 01 02 and 03 04 and 05 06 and 07 08 09 and 1Var = 162 Bin = 136 Cons = 125 Nonz = 1430(a)

Exp cost 1691 (1198641) 1427 1214 1167 1045 (1198641)Exp service level(b) 87 (1198642) 86 81 79 62 (1198642)Exp fulfilled demand(c) 87 90 91 93 91

Suppliers selected( of total demand)

1 (89) 2 (94) 2 (49) 2 (50) 2 (5)3 (6) 3 (51) 3 (50) 3 (95)4 (11)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119895isin119869 119887119895119884119895119861 times 100(c)sum119895isin119869 sum119905isin119879 119887119895119882119895119905119861 times 100

1 2 3 4 5 6 7Period

Supply schedules

1 2 3 4 5 6 7Period

Production schedules

1 2 3 4 5 6 7Period

Shipping schedules

0

10000

20000

30000

40000

50000

Prod

ucts

0

10000

20000

30000

40000

50000

Prod

ucts

0

20000

40000

60000

80000

100000

Part

s

Alpha = 0Alpha = 05

Alpha = 1 Alpha = 0Alpha = 05

Alpha = 1

Alpha = 0Alpha = 05

Alpha = 1

Figure 4 Schedules for model EWCS

Mathematical Problems in Engineering 13

1045 le 1051 forWCS (see Table 2) or themaximum servicelevel1198642 for EWCS (see Table 3) 8700 ge 8552 forWCS (seeTable 2)

The stochastic wait-and-see approach however leads toa more diversified supply portfolio For the deterministicapproach most of nondominated supply portfolios consistof two suppliers only 119894 = 2 3 while suppliers 119894 = 1 4 areselected only for the maximum service level objective (thatis for 120572 = 0) Comparison of Figures 3 and 4 indicatesthat the expected schedules for model WCS computed asexpectations over all schedules for all potential disruptionscenarios (27)ndash(29) are similar to the corresponding singleschedules determined by model EWCS The main differencesobserved aremore delayed expected production and shippingschedules for model WCS when minimization of cost is con-sidered (that is for120572 = 1) Finally it is interesting to note thatin the multiple sourcing environment considered both thewait-and-see approach and in particular the expected valueapproach frequently select a dual sourcing supply portfoliowith one main supplier and one supporting supplier

The computational experiments were performed usingthe AMPL programming language and the CPLEX 1262solver on a MacBook Pro laptop with Intel Core i7 processorrunning at 28 GHz and with 16GB RAM The solver wascapable of finding proven optimal solution for all exampleswith CPU time ranging from several minutes to several hoursfor the stochastic model WCS and fraction of a second forthe deterministic model EWCS Since the stochastic pro-gramming model WCS needs to determine nondominatedschedules for all potential disruption scenarios while modelEWCS deals with a single scenario only the difference in thecomputational effort required is obvious

7 Conclusions

In this paper two biobjective MIP formulations stochasticand deterministic have been proposed and compared for thecoordinated decision-making in supply chains under partiallocal disruptions and all-or-nothing regional disruptionsTheproblem objective has been to jointly schedule supplies pro-duction and distribution to optimize the trade-off betweenexpected cost and expected service levelWhile the stochasticprogramming approach aims at optimizing the expectedperformance of a supply chain over all possible disruptionscenarios the deterministic approach accounts only for asingle scenario representing average disruption conditionsAs a result the stochastic programming approach determinesnondominated solutions for all disruption scenarios whereasthe deterministic approach produces a single solution onlyIn particular in the stochastic programming approach theselection of supply portfolio is combined with supply chainscheduling for all disruption scenarios considered In con-trast the deterministic expected value approach provides theportfolio along with a single expected schedule of productionand distribution

The expected schedules obtained for the stochastic pro-gramming model WCS as expectations over all schedulesfor all disruption scenarios have been compared with thecorresponding schedules determined by the deterministic

model EWCS based on an expected disruption scenarioThecomparison has demonstrated that the two approaches leadto similar expected solutions

The main findings are in line with other research and arelisted as follows

(i) The two decision-making approaches stochastic anddeterministic lead to similar expected performanceof a supply chain under multilevel disruptions

(ii) The optimal solution values for the expected valueproblem frequently outperform the correspondingsolution values of the wait-and-see problem

(iii) Despite the multiple sourcing environment consid-ered both the wait-and-see approach and in partic-ular the expected value approach frequently selecta dual sourcing supply portfolio with one mainsupplier and one supporting supplier

(iv) The stochastic approach which accounts for allpotential disruption scenarios may lead to a morediversified supply portfolio that will hedge against avariety of scenarios

(v) The expected schedules are more delayed for thestochastic approach

(vi) The service-oriented supply portfolio is more diversi-fied andmay combine both high-cost reliable suppli-ers and low-cost unreliable suppliers while the cost-oriented portfolio depends mainly on low-cost andless reliable suppliers

Overall the results of computational experiments indi-cate that the proposed approach and developed MIP modelsare flexible and efficient tools for coordinated supply chainscheduling The portfolio approach leads to MIP formula-tions with strong LP relaxations and has been proven to becomputationally very efficient CPU time required to findproven optimal solutions for realistic size examples usingcommercially available software for MIP is acceptable for areal-world supply chain disruption management (see Sawik[33])

Since the probability distribution of supply disruptionsfrom each supplier is usually unknown (eg [8]) local multi-level disruptions in the stochastic programming modelWCShave a multinomial discrete distribution while the two-levelregional disruptions have a binomial discrete distributionand all disruption events are independent As part of futureresearch we propose to enhance the stochastic programmingmodel formore general scenarios with finitelymany elementssingled out and all the probability concentrated in them Forexample rather than complete shutdown suppliers within aregion may have correlated disruptions Another importantstream of future research is the study of robustness andsensitivity in relation to input data changes in supply chainscheduling under disruption risks (eg [33]) Future researchcan also focus on the following improvement of the proposedmodels to consider the minimization of time and cost ofrecovery (eg Whitney et al [34])

14 Mathematical Problems in Engineering

Notations

Indices

119894 Supplier 119894 isin 119868119895 Customer 119895 isin 119869119896 Distribution center 119896 isin 119870119897 Disruption level 119897 isin 119871119903 Region 119903 isin 119877119904 Disruption scenario 119904 isin 119878119905 Planning period 119905 isin 119879Input Parameters

119887119895 Size (number of products) of customer order 119895119861 Total demand for partsproducts 119861 = sum119895isin119869 119887119895119862 Capacity of producer119889119895 Due date for customer order 119895119890119894 Fixed cost of ordering parts from supplier 119894119892119895 Per unit penalty cost of delayed customer order 119895ℎ119895 Per unit penalty cost of unfulfilled customer order 119895119868119903 Subset of suppliers in region 119903119869119896 Subset of customers serviced by distribution center 119896119900119894 Per unit price of parts purchased from supplier 119894119901119894119897 Probability of disruption level 119897 for supplier 119894119901119903 Regional disruption probability for region 119903120574119894119897 Fraction of an order delivered by supplier 119894 underdisruption level 119897 (fulfillment rate)Γ119894 Expected fraction of an order delivered by supplier 119894(expected fulfillment rate)120590119894 Delivery lead time from supplier 119894120591119896 Transportation time to distribution center 119896

Conflicts of Interest

The author declares that there are no conflicts of interest

Acknowledgments

This work has been supported by NCN research grant (noDEC-201311BST804458) and by AGH (no 1111200324)

References

[1] J V Blackhurst K P Scheibe and D J Johnson ldquoSupplierrisk assessment and monitoring for the automotive industryrdquoInternational Journal of Physical Distribution amp Logistics Man-agement vol 38 no 2 pp 143ndash165 2008

[2] P Hoffmann H Schiele and K Krabbendam ldquoUncertaintysupply risk management and their impact on performancerdquoJournal of Purchasing amp Supply Management vol 19 no 3 pp199ndash211 2013

[3] Y Park P Hong and J J Roh ldquoSupply chain lessons from thecatastrophic natural disaster in Japanrdquo Business Horizons vol56 no 1 pp 75ndash85 2013

[4] M Haraguchi and U Lall ldquoFlood risks and impacts A casestudy of Thailandrsquos floods in 2011 and research questions forsupply chain decisionmakingrdquo International Journal of DisasterRisk Reduction vol 14 pp 256ndash272 2015

[5] T Fujimoto and Y W Park ldquoBalancing supply chain com-petitiveness and robustness through ldquovirtual dual sourcingrdquoLessons from the Great East Japan Earthquakerdquo InternationalJournal of Production Economics vol 147 pp 429ndash436 2014

[6] HMatsuo ldquoImplications of the Tohoku earthquake for Toyotarsquoscoordination mechanism supply chain disruption of auto-motive semiconductorsrdquo International Journal of ProductionEconomics vol 161 pp 217ndash227 2015

[7] J R Marszewska ldquoImplications of seismic hazard in Japan ontoyota supply chain disruption risksrdquo in Proceedings of the 13thInternational Conference on Industrial Logistics (ICIL rsquo16) pp178ndash185 Zakopane Poland October 2016

[8] A Z Zeng and Y Xia ldquoBuilding a mutually beneficial partner-ship to ensure backup supplyrdquo Omega vol 52 pp 77ndash91 2015

[9] S S Erenguc N C Simpson and A J Vakharia ldquoIntegratedproductiondistribution planning in supply chains an invitedreviewrdquo European Journal of Operational Research vol 115 no2 pp 219ndash236 1999

[10] L Lei S Liu A Ruszczynski and S Park ldquoOn the integratedproduction inventory and distribution routing problemrdquo IIETransactions vol 38 no 11 pp 955ndash970 2006

[11] A Kaur A Kanda and S G Deshmukh ldquoA graph theoreticapproach for supply chain coordinationrdquo International Journalof Logistics Systems and Management vol 2 no 4 pp 321ndash3412006

[12] T ChoiW Yeung andT C E Cheng ldquoScheduling and co-ordi-nation of multi-suppliers single-warehouse-operator single-manufacturer supply chains with variable production rates andstorage costsrdquo International Journal of Production Research vol51 no 9 pp 2593ndash2601 2013

[13] S Liu and L G Papageorgiou ldquoMultiobjective optimisation ofproduction distribution and capacity planning of global supplychains in the process industryrdquo Omega vol 41 no 2 pp 369ndash382 2013

[14] Z L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[15] T Sawik ldquoCoordinated supply chain schedulingrdquo InternationalJournal of Production Economics vol 120 no 2 pp 437ndash4512009

[16] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[17] D Ivanov B Sokolov and A Dolgui ldquoThe Ripple effect in sup-ply chains Trade-off rsquoefficiency-flexibility- resiliencersquo in disrup-tionmanagementrdquo International Journal of ProductionResearchvol 52 no 7 pp 2154ndash2172 2014

[18] D Ivanov A Pavlov and B Sokolov ldquoOptimal distribution(re)planning in a centralized multi-stage supply network underconditions of the ripple effect and structure dynamicsrdquo Euro-pean Journal of Operational Research vol 237 no 2 pp 758ndash770 2014

[19] D Ivanov A Pavlov A Dolgui D Pavlov and B SokolovldquoDisruption-driven supply chain (re)-planning and perfor-mance impact assessment with consideration of pro-active andrecovery policiesrdquo Transportation Research Part E Logistics andTransportation Review vol 90 pp 7ndash24 2016

[20] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

Mathematical Problems in Engineering 15

[21] A J Schmitt S A Sun L V Snyder and Z-J M Shen ldquoCen-tralization versus decentralization Risk pooling risk diversifi-cation and supply chain disruptionsrdquo Omega vol 52 pp 201ndash212 2015

[22] L V Snyder Z Atan P Peng Y Rong A J Schmitt andB Sinsoysal ldquoORMS models for supply chain disruptions areviewrdquo IIE Transactions vol 48 no 2 pp 89ndash109 2016

[23] T Sawik ldquoIntegrated selection of suppliers and scheduling ofcustomer orders in the presence of supply chain disruptionrisksrdquo International Journal of Production Research vol 51 no23-24 pp 7006ndash7022 2013

[24] T Sawik ldquoOptimization of cost and service level in the presenceof supply chain disruption risks Single vs multiple sourcingrdquoComputers amp Operations Research vol 51 pp 11ndash20 2014

[25] T Sawik ldquoIntegrated supply production and distributionscheduling under disruption risksrdquoOmega vol 62 pp 131ndash1442016

[26] T Sawik ldquoIntegrated supply chain scheduling under multi-leveldisruptionsrdquo IFAC-PapersOnLine vol 28 no 3 pp 1515ndash15202015

[27] P Kall and J Mayer Stochastic Linear Programming ModelsTheory and Computation vol 156 of International Series inOperations Research amp Management Science Springer NewYork 2011

[28] I N Durbach and T J Stewart ldquoUsing expected values to sim-plify decision making under uncertaintyrdquo Omega vol 37 no 2pp 312ndash330 2009

[29] F Maggioni and S W Wallace ldquoAnalyzing the quality of theexpected value solution in stochastic programmingrdquo Annals ofOperations Research vol 200 pp 37ndash54 2012

[30] T Sawik ldquoSelection of supply portfolio under disruption risksrdquoOmega vol 39 no 2 pp 194ndash208 2011

[31] E Delage S Arroyo and Y Ye ldquoThe value of stochastic mod-eling in two-stage stochastic programs with cost uncertaintyrdquoOperations Research vol 62 no 6 pp 1377ndash1393 2014

[32] T Sawik Scheduling in Supply Chains Using Mixed Integer Pro-gramming John Wiley amp Sons Inc Hoboken NJ 2011

[33] T Sawik Supply Chain DisruptionManagement Using StochasticMixed Integer Programming vol 256 of International Seriesin Operations Research amp Management Science Springer NewYork 2017 httpwwwspringercomgpbook9783319588223

[34] D E Whitney J Luo and D A Heller ldquoThe benefits and con-straints of temporary sourcing diversification in supply chaindisruption and recoveryrdquo Journal of Purchasing and SupplyManagement vol 20 no 4 pp 238ndash250 2014

Mathematical Problems in Engineering 7

approaches The examples are modeled in part after a real-world electronics supply chain (eg Sawik [32]) The fol-lowing parameters have been selected for the computationalexamples119868 = 4 suppliers 119869 = 20 customer orders 119870 = 2 dis-tribution centers 119871 = 3 partial disruption levels 119877 = 2geographic regions and 119879 = 7 planning periods1198681 = 1 2 1198682 = 3 41198691 = 1 10 1198692 = 11 20

Shipping times from suppliers 120590 = (1 1 3 3)Shipping times to distribution centers 120591 = (1 2)Customer demand 119887119895 isin 2000 3000 10000 for all119895 isin 119869 and total demand 119861 = 100000Due dates 119889119895 isin 2 + min119894isin119868(120590119894) + min119896isin119870(120591119896) 119879 +

max119896isin119870(120591119896) for all 119895 isin 119869Fixed ordering costs for suppliers 119890 = (8000 6000 1200013000)Unit purchasing prices from suppliers 119900 = (14 12 8 9)Unit penalties for delayed unfulfilled customer orders119892119895 = lceilmax119894isin119868119900119894350rceil = 1 ℎ119895 = 2max119894isin119868119900119894 = 28 respectively

for all 119895 isin 119869Producer capacity 119862 = 45000Local disruption levels and the associated fulfillment rates

(the percentage of an order that can be delivered) are shownbelow119871 = 0 1 2 3 where 119897 = 0 complete shutdown1205741198940 = 0 forall119894 isin 119868 that is 0 of an order delivered 119897 = 1major disruption 1205741198941 isin [001 050] forall119894 isin 1198681 and 1205741198941 isin[001 030] forall119894 isin 1198682 that is 1 to 50 and 1 to 30 ofan order delivered respectively 119897 = 2 minor disruption

1205741198942 isin [0 51 099] forall119894 isin 1198681 and 1205741198942 isin [0 31 099] forall119894 isin 1198682that is 51 to 99 and 31 to 99 of an order deliveredrespectively 119897 = 119871 = 3 no disruption 1205741198943 = 1 forall119894 isin 119868 thatis 100 of an order delivered

The total number of all potential scenarios is 119878 = (119871 +1)119868 = 44 = 256 scenarios where each scenario 119904 isin 119878 is repre-sented by vector 120582119904 = (1205821119904 1205824119904) where 120582119894119904 isin 119871 119894 isin 119868 (seeTable 1)

The local nondisruption probability (level 119897 = 3) 1199011198943was uniformly distributed over [089 099] and [079 089]respectively for suppliers 119894 isin 1198681 and 119894 isin 1198682 that is theprobabilities were drawn independently from U[089 099]and U[079 089] respectively

Given local nondisruption probabilities 1199011198943 119894 isin 119868 theprobabilities for the remaining local disruption levels 119897 =0 1 2 were calculated as follows

Probability of complete shutdown (level 119897 = 0) 1199011198940 =01(1 minus 1199011198943)Probability of major disruption (level 119897 = 1) 1199011198941 =03(1 minus 1199011198943)Probability of minor disruption (level 119897 = 2) 1199011198942 =06(1 minus 1199011198943) for all suppliers 119894 isin 119868

Thus 1199011198940 le 1199011198941 le 1199011198942 le 1199011198943 which reflects a real-world relation among probabilities of disruption occurrenceat different levels (eg [8])

Regional disruption probabilities are 1199011 = 0001 and1199012 =001The probability of realizing of disruption scenario 119904 for

suppliers in region 119903 = 1 2 is calculated as follows

119875119903119904

=(1 minus 119901119903)( prod

119894isin119868119903 120582119894119904=0

01 (1 minus 1199011198943))( prod119894isin119868119903 120582119894119904=1

03 (1 minus 1199011198943))( prod119894isin119868119903120582119894119904=2

06 (1 minus 1199011198943))( prod119894isin119868119903 120582119894119904=3

1199011198943) if sum119894isin119868119903120582119894119904 gt 0

119901119903 + (1 minus 119901119903)prod119894isin119868119903

01 (1 minus 1199011198943) if sum119894isin119868119903120582119894119904 = 0

(26)

and the probability for disruption scenario 119904 isin 119878 is given by119875119904 = 1198751119904 1198752119904 Figure 2 presents basic characteristics of all suppliers

probability of complete shutdown119901119903+(1minus119901119903)1199011198940 119894 isin 119868119903 119903 isin119877 expected fulfillment rate Γ119894 = sum119897=123(1 minus 119901119903)119901119894119897120574119894119897 119894 isin119868119903 119903 isin 119877 and purchasing price 119900119894 119894 isin 119868Table 2 presents a subset of nondominated solutions for

the wait-and-see problem WCS for a subset of trade-offparameter120572 isin 0 01 02 03 04 05 06 07 08 09 1Theresults indicate that most of nondominated supply portfoliosconsist of the two suppliers only the second most expensiveand most reliable supplier 119894 = 2 from region 119903 = 1 andthe cheapest and most unreliable supplier 119894 = 3 from region119903 = 2 The most expensive and most reliable supplier 119894 = 1in region 119903 = 1 and the second cheapest and most unreliablesupplier 119894 = 4 in region 119903 = 2 are rarely selected For 120572 = 1

(minimization of cost) the cheapest supplier 119894 = 3 is selectedonly For 120572 = 0 (maximization of service level) the totaldemand for parts is allocated among the two most reliableand most expensive suppliers 119894 = 1 2 The above solutionshows that the service-oriented supply portfolio (120572 close to 0)is more diversified than the cost-oriented portfolio (120572 closeto 1) Table 2 also shows the associated expected fractionof fulfilled demand 1198643 = sum119904isin119878sum119895isin119869sum119905isin119879 119875119904119887119895119908119904119895119905119861 that isdemand fulfilled on time or delayed The solution resultsdemonstrate that a large expected service level is sometimesassociated with a small expected fraction of fulfilled demandThus the maximization of expected service level leads torejection of customer orders that cannot be fulfilled bycustomer requested due dates

Figure 3 shows the expected supply production andshipping schedules respectively for 120572 = 0 (ie for the

8 Mathematical Problems in Engineering

Shutdown probability

Supplier

Expected yield rate

Purchasing price

4321

Supplier4321

Supplier4321

6

8

10

12

14

16

09

092

094

096

098

0

0005

001

0015

002

0025

Figure 2 Suppliers

1 2 3 4 5 6 7Period

Expected supply schedules

Alpha = 0Alpha = 05

Alpha = 1

1 2 3 4 5 6 7Period

Expected production schedules

1 2 3 4 5 6 7Period

Expected shipping schedules

0

20000

40000

60000

80000

100000

Part

s

0

10000

20000

30000

40000

50000

Prod

ucts

Alpha = 0Alpha = 05

Alpha = 1

0

10000

20000

30000

40000

50000

Prod

ucts

Alpha = 0Alpha = 05

Alpha = 1

Figure 3 Expected schedules for modelWCS

Mathematical Problems in Engineering 9

Table 1 Disruption scenarios

119904 119894 = 1 2 3 41 0 0 0 02 0 0 0 13 0 0 0 24 0 0 0 35 0 0 1 06 0 0 1 17 0 0 1 28 0 0 1 39 0 0 2 010 0 0 2 111 0 0 2 212 0 0 2 313 0 0 3 014 0 0 3 115 0 0 3 216 0 0 3 317 0 1 0 018 0 1 0 119 0 1 0 220 0 1 0 321 0 1 1 022 0 1 1 123 0 1 1 224 0 1 1 325 0 1 2 026 0 1 2 127 0 1 2 228 0 1 2 329 0 1 3 030 0 1 3 131 0 1 3 232 0 1 3 333 0 2 0 034 0 2 0 135 0 2 0 236 0 2 0 337 0 2 1 038 0 2 1 139 0 2 1 240 0 2 1 341 0 2 2 042 0 2 2 143 0 2 2 244 0 2 2 345 0 2 3 046 0 2 3 147 0 2 3 248 0 2 3 349 0 3 0 0

Table 1 Continued

119904 119894 = 1 2 3 450 0 3 0 151 0 3 0 252 0 3 0 353 0 3 1 054 0 3 1 155 0 3 1 256 0 3 1 357 0 3 2 058 0 3 2 159 0 3 2 260 0 3 2 361 0 3 3 062 0 3 3 163 0 3 3 264 0 3 3 365 1 0 0 066 1 0 0 167 1 0 0 268 1 0 0 369 1 0 1 070 1 0 1 171 1 0 1 272 1 0 1 373 1 0 2 074 1 0 2 175 1 0 2 276 1 0 2 377 1 0 3 078 1 0 3 179 1 0 3 280 1 0 3 381 1 1 0 082 1 1 0 183 1 1 0 284 1 1 0 385 1 1 1 086 1 1 1 187 1 1 1 288 1 1 1 389 1 1 2 090 1 1 2 191 1 1 2 292 1 1 2 393 1 1 3 094 1 1 3 195 1 1 3 296 1 1 3 397 1 2 0 098 1 2 0 1

10 Mathematical Problems in Engineering

Table 1 Continued

119904 119894 = 1 2 3 499 1 2 0 2100 1 2 0 3101 1 2 1 0102 1 2 1 1103 1 2 1 2104 1 2 1 3105 1 2 2 0106 1 2 2 1107 1 2 2 2108 1 2 2 3109 1 2 3 0110 1 2 3 1111 1 2 3 2112 1 2 3 3113 1 3 0 0114 1 3 0 1115 1 3 0 2116 1 3 0 3117 1 3 1 0118 1 3 1 1119 1 3 1 2120 1 3 1 3121 1 3 2 0122 1 3 2 1123 1 3 2 2124 1 3 2 3125 1 3 3 0126 1 3 3 1127 1 3 3 2128 1 3 3 3129 2 0 0 0130 2 0 0 1131 2 0 0 2132 2 0 0 3133 2 0 1 0134 2 0 1 1135 2 0 1 2136 2 0 1 3137 2 0 2 0138 2 0 2 1139 2 0 2 2140 2 0 2 3141 2 0 3 0142 2 0 3 1143 2 0 3 2144 2 0 3 3145 2 1 0 0146 2 1 0 1147 2 1 0 2148 2 1 0 3149 2 1 1 0150 2 1 1 1

Table 1 Continued

119904 119894 = 1 2 3 4151 2 1 1 2152 2 1 1 3153 2 1 2 0154 2 1 2 1155 2 1 2 2156 2 1 2 3157 2 1 3 0158 2 1 3 1159 2 1 3 2160 2 1 3 3161 2 2 0 0162 2 2 0 1163 2 2 0 2164 2 2 0 3165 2 2 1 0166 2 2 1 1167 2 2 1 2168 2 2 1 3169 2 2 2 0170 2 2 2 1171 2 2 2 2172 2 2 2 3173 2 2 3 0174 2 2 3 1175 2 2 3 2176 2 2 3 3177 2 3 0 0178 2 3 0 1179 2 3 0 2180 2 3 0 3181 2 3 1 0182 2 3 1 1183 2 3 1 2184 2 3 1 3185 2 3 2 0186 2 3 2 1187 2 3 2 2188 2 3 2 3189 2 3 3 0190 2 3 3 1191 2 3 3 2192 2 3 3 3193 3 0 0 0194 3 0 0 1195 3 0 0 2196 3 0 0 3197 3 0 1 0198 3 0 1 1199 3 0 1 2200 3 0 1 3201 3 0 2 0202 3 0 2 1

Mathematical Problems in Engineering 11

Table 1 Continued

119904 119894 = 1 2 3 4203 3 0 2 2204 3 0 2 3205 3 0 3 0206 3 0 3 1207 3 0 3 2208 3 0 3 3209 3 1 0 0210 3 1 0 1211 3 1 0 2212 3 1 0 3213 3 1 1 0214 3 1 1 1215 3 1 1 2216 3 1 1 3217 3 1 2 0218 3 1 2 1219 3 1 2 2220 3 1 2 3221 3 1 3 0222 3 1 3 1223 3 1 3 2224 3 1 3 3225 3 2 0 0226 3 2 0 1227 3 2 0 2228 3 2 0 3229 3 2 1 0230 3 2 1 1231 3 2 1 2232 3 2 1 3233 3 2 2 0234 3 2 2 1235 3 2 2 2236 3 2 2 3237 3 2 3 0238 3 2 3 1239 3 2 3 2240 3 2 3 3241 3 3 0 0242 3 3 0 1243 3 3 0 2244 3 3 0 3245 3 3 1 0246 3 3 1 1247 3 3 1 2248 3 3 1 3249 3 3 2 0250 3 3 2 1251 3 3 2 2252 3 3 2 3

Table 1 Continued

119904 119894 = 1 2 3 4253 3 3 3 0254 3 3 3 1255 3 3 3 2256 3 3 3 3

maximum expected service level) 120572 = 05 and 120572 = 1 (ie forthe minimum expected cost) The expected schedules werecomputed using the formulae presented below

(i) Expected schedule of supplies of parts to the pro-ducer

sum119904isin119878

sum119894isin119868119904120590119894=119905

119875119904119861120574119894120582119894119904V119894 119905 isin 119879 (27)

(ii) Expected production schedule

sum119904isin119878

sum119895isin119869

119875119904119887119895119908119904119895119905 119905 isin 119879 (28)

(iii) Expected schedule of shipping of products from theproducer to the distribution centers

sum119904isin119878

sum119896isin119870

sum119895isin119869119896

119875119904119887119895( sum1199051015840isin1198791199051015840lt119905

1199081199041198951199051015840)119909119904119896119905 119905 isin 119879 (29)

As 120572 increases that is the decision-makerrsquos priority shiftsfrom the maximum service level to minimum cost and moreparts are ordered from less reliable and lower cost suppliersthe expected supply schedules and the corresponding pro-duction schedules are more delayed as well as the delivery ofproducts to the customers Note that despite constraint (13)that ensures a feasible schedule with at most one shipmentof products to each distribution center for each disruptionscenario the expected shipping schedule (29) may be splitinto more smaller size shipments (cf Figure 3 where twomajor and one small-size shipments are indicated for eachconfidence level)

For comparison Table 3 presents a subset of non-dominated solutions obtained for the expected value problemEWCS and Figure 4 shows supply production and shippingschedules Unlike the stochastic programming approachwhich accounts for all potential disruption scenarios tooptimize an expected performance of the supply chain thesolution obtained using the deterministic approach is basedon aggregate information on suppliers expected fulfillmentIn general the results are similar for both models and thecorresponding optimal solution values are close to each otherwhich indicates that the expected value problem can be usedin practice when stochastic mixed integer programs are hardto solve The optimal solution values for the expected valueproblem frequently outperform the corresponding solutionvalues of the wait-and-see problem which is in line withthe proposition EV le WS in [27] compare Section 5for example the minimum cost 1198641 for EWCS (see Table 3)

12 Mathematical Problems in Engineering

Table 2 Nondominated solutions for modelWCS

120572 0 01 02 03 04 05 and 06 07 08 and 09 1Var = 38077 Bin = 33076 Cons = 28769 Nonz = 348325(a)

Exp cost 1731 (1198641) 1620 1547 1206 1200 1176 1070 1065 1051 (1198641)Exp service level(b) 8552 (1198642) 8487 8452 7719 7676 7595 5863 5871 3822 (1198642)Exp fulfilled demand(c) 8552 8530 8452 9370 9361 9353 9134 9120 9068

Suppliers selected( of total demand)

1 (76) 1 (35)2 (24) 2 (55) 2 (87) 2 (55) 2 (55) 2 (55) 2 (12) 2 (10)3 (10) 3 (13) 3 (22) 3 (32) 3 (45) 3 (88) 3 (90) 3 (100)4 (23) 4 (13)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119904isin119878 sum119895isin119869 119875119904119887119895119910

119904119895119861 times

100 (c)sum119904isin119878 sum119895isin119869 sum119905isin119879 119875119904119887119895119908119904119895119905119861 times 100

Table 3 Nondominated solutions for model EWCS

120572 0 01 02 and 03 04 and 05 06 and 07 08 09 and 1Var = 162 Bin = 136 Cons = 125 Nonz = 1430(a)

Exp cost 1691 (1198641) 1427 1214 1167 1045 (1198641)Exp service level(b) 87 (1198642) 86 81 79 62 (1198642)Exp fulfilled demand(c) 87 90 91 93 91

Suppliers selected( of total demand)

1 (89) 2 (94) 2 (49) 2 (50) 2 (5)3 (6) 3 (51) 3 (50) 3 (95)4 (11)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119895isin119869 119887119895119884119895119861 times 100(c)sum119895isin119869 sum119905isin119879 119887119895119882119895119905119861 times 100

1 2 3 4 5 6 7Period

Supply schedules

1 2 3 4 5 6 7Period

Production schedules

1 2 3 4 5 6 7Period

Shipping schedules

0

10000

20000

30000

40000

50000

Prod

ucts

0

10000

20000

30000

40000

50000

Prod

ucts

0

20000

40000

60000

80000

100000

Part

s

Alpha = 0Alpha = 05

Alpha = 1 Alpha = 0Alpha = 05

Alpha = 1

Alpha = 0Alpha = 05

Alpha = 1

Figure 4 Schedules for model EWCS

Mathematical Problems in Engineering 13

1045 le 1051 forWCS (see Table 2) or themaximum servicelevel1198642 for EWCS (see Table 3) 8700 ge 8552 forWCS (seeTable 2)

The stochastic wait-and-see approach however leads toa more diversified supply portfolio For the deterministicapproach most of nondominated supply portfolios consistof two suppliers only 119894 = 2 3 while suppliers 119894 = 1 4 areselected only for the maximum service level objective (thatis for 120572 = 0) Comparison of Figures 3 and 4 indicatesthat the expected schedules for model WCS computed asexpectations over all schedules for all potential disruptionscenarios (27)ndash(29) are similar to the corresponding singleschedules determined by model EWCS The main differencesobserved aremore delayed expected production and shippingschedules for model WCS when minimization of cost is con-sidered (that is for120572 = 1) Finally it is interesting to note thatin the multiple sourcing environment considered both thewait-and-see approach and in particular the expected valueapproach frequently select a dual sourcing supply portfoliowith one main supplier and one supporting supplier

The computational experiments were performed usingthe AMPL programming language and the CPLEX 1262solver on a MacBook Pro laptop with Intel Core i7 processorrunning at 28 GHz and with 16GB RAM The solver wascapable of finding proven optimal solution for all exampleswith CPU time ranging from several minutes to several hoursfor the stochastic model WCS and fraction of a second forthe deterministic model EWCS Since the stochastic pro-gramming model WCS needs to determine nondominatedschedules for all potential disruption scenarios while modelEWCS deals with a single scenario only the difference in thecomputational effort required is obvious

7 Conclusions

In this paper two biobjective MIP formulations stochasticand deterministic have been proposed and compared for thecoordinated decision-making in supply chains under partiallocal disruptions and all-or-nothing regional disruptionsTheproblem objective has been to jointly schedule supplies pro-duction and distribution to optimize the trade-off betweenexpected cost and expected service levelWhile the stochasticprogramming approach aims at optimizing the expectedperformance of a supply chain over all possible disruptionscenarios the deterministic approach accounts only for asingle scenario representing average disruption conditionsAs a result the stochastic programming approach determinesnondominated solutions for all disruption scenarios whereasthe deterministic approach produces a single solution onlyIn particular in the stochastic programming approach theselection of supply portfolio is combined with supply chainscheduling for all disruption scenarios considered In con-trast the deterministic expected value approach provides theportfolio along with a single expected schedule of productionand distribution

The expected schedules obtained for the stochastic pro-gramming model WCS as expectations over all schedulesfor all disruption scenarios have been compared with thecorresponding schedules determined by the deterministic

model EWCS based on an expected disruption scenarioThecomparison has demonstrated that the two approaches leadto similar expected solutions

The main findings are in line with other research and arelisted as follows

(i) The two decision-making approaches stochastic anddeterministic lead to similar expected performanceof a supply chain under multilevel disruptions

(ii) The optimal solution values for the expected valueproblem frequently outperform the correspondingsolution values of the wait-and-see problem

(iii) Despite the multiple sourcing environment consid-ered both the wait-and-see approach and in partic-ular the expected value approach frequently selecta dual sourcing supply portfolio with one mainsupplier and one supporting supplier

(iv) The stochastic approach which accounts for allpotential disruption scenarios may lead to a morediversified supply portfolio that will hedge against avariety of scenarios

(v) The expected schedules are more delayed for thestochastic approach

(vi) The service-oriented supply portfolio is more diversi-fied andmay combine both high-cost reliable suppli-ers and low-cost unreliable suppliers while the cost-oriented portfolio depends mainly on low-cost andless reliable suppliers

Overall the results of computational experiments indi-cate that the proposed approach and developed MIP modelsare flexible and efficient tools for coordinated supply chainscheduling The portfolio approach leads to MIP formula-tions with strong LP relaxations and has been proven to becomputationally very efficient CPU time required to findproven optimal solutions for realistic size examples usingcommercially available software for MIP is acceptable for areal-world supply chain disruption management (see Sawik[33])

Since the probability distribution of supply disruptionsfrom each supplier is usually unknown (eg [8]) local multi-level disruptions in the stochastic programming modelWCShave a multinomial discrete distribution while the two-levelregional disruptions have a binomial discrete distributionand all disruption events are independent As part of futureresearch we propose to enhance the stochastic programmingmodel formore general scenarios with finitelymany elementssingled out and all the probability concentrated in them Forexample rather than complete shutdown suppliers within aregion may have correlated disruptions Another importantstream of future research is the study of robustness andsensitivity in relation to input data changes in supply chainscheduling under disruption risks (eg [33]) Future researchcan also focus on the following improvement of the proposedmodels to consider the minimization of time and cost ofrecovery (eg Whitney et al [34])

14 Mathematical Problems in Engineering

Notations

Indices

119894 Supplier 119894 isin 119868119895 Customer 119895 isin 119869119896 Distribution center 119896 isin 119870119897 Disruption level 119897 isin 119871119903 Region 119903 isin 119877119904 Disruption scenario 119904 isin 119878119905 Planning period 119905 isin 119879Input Parameters

119887119895 Size (number of products) of customer order 119895119861 Total demand for partsproducts 119861 = sum119895isin119869 119887119895119862 Capacity of producer119889119895 Due date for customer order 119895119890119894 Fixed cost of ordering parts from supplier 119894119892119895 Per unit penalty cost of delayed customer order 119895ℎ119895 Per unit penalty cost of unfulfilled customer order 119895119868119903 Subset of suppliers in region 119903119869119896 Subset of customers serviced by distribution center 119896119900119894 Per unit price of parts purchased from supplier 119894119901119894119897 Probability of disruption level 119897 for supplier 119894119901119903 Regional disruption probability for region 119903120574119894119897 Fraction of an order delivered by supplier 119894 underdisruption level 119897 (fulfillment rate)Γ119894 Expected fraction of an order delivered by supplier 119894(expected fulfillment rate)120590119894 Delivery lead time from supplier 119894120591119896 Transportation time to distribution center 119896

Conflicts of Interest

The author declares that there are no conflicts of interest

Acknowledgments

This work has been supported by NCN research grant (noDEC-201311BST804458) and by AGH (no 1111200324)

References

[1] J V Blackhurst K P Scheibe and D J Johnson ldquoSupplierrisk assessment and monitoring for the automotive industryrdquoInternational Journal of Physical Distribution amp Logistics Man-agement vol 38 no 2 pp 143ndash165 2008

[2] P Hoffmann H Schiele and K Krabbendam ldquoUncertaintysupply risk management and their impact on performancerdquoJournal of Purchasing amp Supply Management vol 19 no 3 pp199ndash211 2013

[3] Y Park P Hong and J J Roh ldquoSupply chain lessons from thecatastrophic natural disaster in Japanrdquo Business Horizons vol56 no 1 pp 75ndash85 2013

[4] M Haraguchi and U Lall ldquoFlood risks and impacts A casestudy of Thailandrsquos floods in 2011 and research questions forsupply chain decisionmakingrdquo International Journal of DisasterRisk Reduction vol 14 pp 256ndash272 2015

[5] T Fujimoto and Y W Park ldquoBalancing supply chain com-petitiveness and robustness through ldquovirtual dual sourcingrdquoLessons from the Great East Japan Earthquakerdquo InternationalJournal of Production Economics vol 147 pp 429ndash436 2014

[6] HMatsuo ldquoImplications of the Tohoku earthquake for Toyotarsquoscoordination mechanism supply chain disruption of auto-motive semiconductorsrdquo International Journal of ProductionEconomics vol 161 pp 217ndash227 2015

[7] J R Marszewska ldquoImplications of seismic hazard in Japan ontoyota supply chain disruption risksrdquo in Proceedings of the 13thInternational Conference on Industrial Logistics (ICIL rsquo16) pp178ndash185 Zakopane Poland October 2016

[8] A Z Zeng and Y Xia ldquoBuilding a mutually beneficial partner-ship to ensure backup supplyrdquo Omega vol 52 pp 77ndash91 2015

[9] S S Erenguc N C Simpson and A J Vakharia ldquoIntegratedproductiondistribution planning in supply chains an invitedreviewrdquo European Journal of Operational Research vol 115 no2 pp 219ndash236 1999

[10] L Lei S Liu A Ruszczynski and S Park ldquoOn the integratedproduction inventory and distribution routing problemrdquo IIETransactions vol 38 no 11 pp 955ndash970 2006

[11] A Kaur A Kanda and S G Deshmukh ldquoA graph theoreticapproach for supply chain coordinationrdquo International Journalof Logistics Systems and Management vol 2 no 4 pp 321ndash3412006

[12] T ChoiW Yeung andT C E Cheng ldquoScheduling and co-ordi-nation of multi-suppliers single-warehouse-operator single-manufacturer supply chains with variable production rates andstorage costsrdquo International Journal of Production Research vol51 no 9 pp 2593ndash2601 2013

[13] S Liu and L G Papageorgiou ldquoMultiobjective optimisation ofproduction distribution and capacity planning of global supplychains in the process industryrdquo Omega vol 41 no 2 pp 369ndash382 2013

[14] Z L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[15] T Sawik ldquoCoordinated supply chain schedulingrdquo InternationalJournal of Production Economics vol 120 no 2 pp 437ndash4512009

[16] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[17] D Ivanov B Sokolov and A Dolgui ldquoThe Ripple effect in sup-ply chains Trade-off rsquoefficiency-flexibility- resiliencersquo in disrup-tionmanagementrdquo International Journal of ProductionResearchvol 52 no 7 pp 2154ndash2172 2014

[18] D Ivanov A Pavlov and B Sokolov ldquoOptimal distribution(re)planning in a centralized multi-stage supply network underconditions of the ripple effect and structure dynamicsrdquo Euro-pean Journal of Operational Research vol 237 no 2 pp 758ndash770 2014

[19] D Ivanov A Pavlov A Dolgui D Pavlov and B SokolovldquoDisruption-driven supply chain (re)-planning and perfor-mance impact assessment with consideration of pro-active andrecovery policiesrdquo Transportation Research Part E Logistics andTransportation Review vol 90 pp 7ndash24 2016

[20] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

Mathematical Problems in Engineering 15

[21] A J Schmitt S A Sun L V Snyder and Z-J M Shen ldquoCen-tralization versus decentralization Risk pooling risk diversifi-cation and supply chain disruptionsrdquo Omega vol 52 pp 201ndash212 2015

[22] L V Snyder Z Atan P Peng Y Rong A J Schmitt andB Sinsoysal ldquoORMS models for supply chain disruptions areviewrdquo IIE Transactions vol 48 no 2 pp 89ndash109 2016

[23] T Sawik ldquoIntegrated selection of suppliers and scheduling ofcustomer orders in the presence of supply chain disruptionrisksrdquo International Journal of Production Research vol 51 no23-24 pp 7006ndash7022 2013

[24] T Sawik ldquoOptimization of cost and service level in the presenceof supply chain disruption risks Single vs multiple sourcingrdquoComputers amp Operations Research vol 51 pp 11ndash20 2014

[25] T Sawik ldquoIntegrated supply production and distributionscheduling under disruption risksrdquoOmega vol 62 pp 131ndash1442016

[26] T Sawik ldquoIntegrated supply chain scheduling under multi-leveldisruptionsrdquo IFAC-PapersOnLine vol 28 no 3 pp 1515ndash15202015

[27] P Kall and J Mayer Stochastic Linear Programming ModelsTheory and Computation vol 156 of International Series inOperations Research amp Management Science Springer NewYork 2011

[28] I N Durbach and T J Stewart ldquoUsing expected values to sim-plify decision making under uncertaintyrdquo Omega vol 37 no 2pp 312ndash330 2009

[29] F Maggioni and S W Wallace ldquoAnalyzing the quality of theexpected value solution in stochastic programmingrdquo Annals ofOperations Research vol 200 pp 37ndash54 2012

[30] T Sawik ldquoSelection of supply portfolio under disruption risksrdquoOmega vol 39 no 2 pp 194ndash208 2011

[31] E Delage S Arroyo and Y Ye ldquoThe value of stochastic mod-eling in two-stage stochastic programs with cost uncertaintyrdquoOperations Research vol 62 no 6 pp 1377ndash1393 2014

[32] T Sawik Scheduling in Supply Chains Using Mixed Integer Pro-gramming John Wiley amp Sons Inc Hoboken NJ 2011

[33] T Sawik Supply Chain DisruptionManagement Using StochasticMixed Integer Programming vol 256 of International Seriesin Operations Research amp Management Science Springer NewYork 2017 httpwwwspringercomgpbook9783319588223

[34] D E Whitney J Luo and D A Heller ldquoThe benefits and con-straints of temporary sourcing diversification in supply chaindisruption and recoveryrdquo Journal of Purchasing and SupplyManagement vol 20 no 4 pp 238ndash250 2014

8 Mathematical Problems in Engineering

Shutdown probability

Supplier

Expected yield rate

Purchasing price

4321

Supplier4321

Supplier4321

6

8

10

12

14

16

09

092

094

096

098

0

0005

001

0015

002

0025

Figure 2 Suppliers

1 2 3 4 5 6 7Period

Expected supply schedules

Alpha = 0Alpha = 05

Alpha = 1

1 2 3 4 5 6 7Period

Expected production schedules

1 2 3 4 5 6 7Period

Expected shipping schedules

0

20000

40000

60000

80000

100000

Part

s

0

10000

20000

30000

40000

50000

Prod

ucts

Alpha = 0Alpha = 05

Alpha = 1

0

10000

20000

30000

40000

50000

Prod

ucts

Alpha = 0Alpha = 05

Alpha = 1

Figure 3 Expected schedules for modelWCS

Mathematical Problems in Engineering 9

Table 1 Disruption scenarios

119904 119894 = 1 2 3 41 0 0 0 02 0 0 0 13 0 0 0 24 0 0 0 35 0 0 1 06 0 0 1 17 0 0 1 28 0 0 1 39 0 0 2 010 0 0 2 111 0 0 2 212 0 0 2 313 0 0 3 014 0 0 3 115 0 0 3 216 0 0 3 317 0 1 0 018 0 1 0 119 0 1 0 220 0 1 0 321 0 1 1 022 0 1 1 123 0 1 1 224 0 1 1 325 0 1 2 026 0 1 2 127 0 1 2 228 0 1 2 329 0 1 3 030 0 1 3 131 0 1 3 232 0 1 3 333 0 2 0 034 0 2 0 135 0 2 0 236 0 2 0 337 0 2 1 038 0 2 1 139 0 2 1 240 0 2 1 341 0 2 2 042 0 2 2 143 0 2 2 244 0 2 2 345 0 2 3 046 0 2 3 147 0 2 3 248 0 2 3 349 0 3 0 0

Table 1 Continued

119904 119894 = 1 2 3 450 0 3 0 151 0 3 0 252 0 3 0 353 0 3 1 054 0 3 1 155 0 3 1 256 0 3 1 357 0 3 2 058 0 3 2 159 0 3 2 260 0 3 2 361 0 3 3 062 0 3 3 163 0 3 3 264 0 3 3 365 1 0 0 066 1 0 0 167 1 0 0 268 1 0 0 369 1 0 1 070 1 0 1 171 1 0 1 272 1 0 1 373 1 0 2 074 1 0 2 175 1 0 2 276 1 0 2 377 1 0 3 078 1 0 3 179 1 0 3 280 1 0 3 381 1 1 0 082 1 1 0 183 1 1 0 284 1 1 0 385 1 1 1 086 1 1 1 187 1 1 1 288 1 1 1 389 1 1 2 090 1 1 2 191 1 1 2 292 1 1 2 393 1 1 3 094 1 1 3 195 1 1 3 296 1 1 3 397 1 2 0 098 1 2 0 1

10 Mathematical Problems in Engineering

Table 1 Continued

119904 119894 = 1 2 3 499 1 2 0 2100 1 2 0 3101 1 2 1 0102 1 2 1 1103 1 2 1 2104 1 2 1 3105 1 2 2 0106 1 2 2 1107 1 2 2 2108 1 2 2 3109 1 2 3 0110 1 2 3 1111 1 2 3 2112 1 2 3 3113 1 3 0 0114 1 3 0 1115 1 3 0 2116 1 3 0 3117 1 3 1 0118 1 3 1 1119 1 3 1 2120 1 3 1 3121 1 3 2 0122 1 3 2 1123 1 3 2 2124 1 3 2 3125 1 3 3 0126 1 3 3 1127 1 3 3 2128 1 3 3 3129 2 0 0 0130 2 0 0 1131 2 0 0 2132 2 0 0 3133 2 0 1 0134 2 0 1 1135 2 0 1 2136 2 0 1 3137 2 0 2 0138 2 0 2 1139 2 0 2 2140 2 0 2 3141 2 0 3 0142 2 0 3 1143 2 0 3 2144 2 0 3 3145 2 1 0 0146 2 1 0 1147 2 1 0 2148 2 1 0 3149 2 1 1 0150 2 1 1 1

Table 1 Continued

119904 119894 = 1 2 3 4151 2 1 1 2152 2 1 1 3153 2 1 2 0154 2 1 2 1155 2 1 2 2156 2 1 2 3157 2 1 3 0158 2 1 3 1159 2 1 3 2160 2 1 3 3161 2 2 0 0162 2 2 0 1163 2 2 0 2164 2 2 0 3165 2 2 1 0166 2 2 1 1167 2 2 1 2168 2 2 1 3169 2 2 2 0170 2 2 2 1171 2 2 2 2172 2 2 2 3173 2 2 3 0174 2 2 3 1175 2 2 3 2176 2 2 3 3177 2 3 0 0178 2 3 0 1179 2 3 0 2180 2 3 0 3181 2 3 1 0182 2 3 1 1183 2 3 1 2184 2 3 1 3185 2 3 2 0186 2 3 2 1187 2 3 2 2188 2 3 2 3189 2 3 3 0190 2 3 3 1191 2 3 3 2192 2 3 3 3193 3 0 0 0194 3 0 0 1195 3 0 0 2196 3 0 0 3197 3 0 1 0198 3 0 1 1199 3 0 1 2200 3 0 1 3201 3 0 2 0202 3 0 2 1

Mathematical Problems in Engineering 11

Table 1 Continued

119904 119894 = 1 2 3 4203 3 0 2 2204 3 0 2 3205 3 0 3 0206 3 0 3 1207 3 0 3 2208 3 0 3 3209 3 1 0 0210 3 1 0 1211 3 1 0 2212 3 1 0 3213 3 1 1 0214 3 1 1 1215 3 1 1 2216 3 1 1 3217 3 1 2 0218 3 1 2 1219 3 1 2 2220 3 1 2 3221 3 1 3 0222 3 1 3 1223 3 1 3 2224 3 1 3 3225 3 2 0 0226 3 2 0 1227 3 2 0 2228 3 2 0 3229 3 2 1 0230 3 2 1 1231 3 2 1 2232 3 2 1 3233 3 2 2 0234 3 2 2 1235 3 2 2 2236 3 2 2 3237 3 2 3 0238 3 2 3 1239 3 2 3 2240 3 2 3 3241 3 3 0 0242 3 3 0 1243 3 3 0 2244 3 3 0 3245 3 3 1 0246 3 3 1 1247 3 3 1 2248 3 3 1 3249 3 3 2 0250 3 3 2 1251 3 3 2 2252 3 3 2 3

Table 1 Continued

119904 119894 = 1 2 3 4253 3 3 3 0254 3 3 3 1255 3 3 3 2256 3 3 3 3

maximum expected service level) 120572 = 05 and 120572 = 1 (ie forthe minimum expected cost) The expected schedules werecomputed using the formulae presented below

(i) Expected schedule of supplies of parts to the pro-ducer

sum119904isin119878

sum119894isin119868119904120590119894=119905

119875119904119861120574119894120582119894119904V119894 119905 isin 119879 (27)

(ii) Expected production schedule

sum119904isin119878

sum119895isin119869

119875119904119887119895119908119904119895119905 119905 isin 119879 (28)

(iii) Expected schedule of shipping of products from theproducer to the distribution centers

sum119904isin119878

sum119896isin119870

sum119895isin119869119896

119875119904119887119895( sum1199051015840isin1198791199051015840lt119905

1199081199041198951199051015840)119909119904119896119905 119905 isin 119879 (29)

As 120572 increases that is the decision-makerrsquos priority shiftsfrom the maximum service level to minimum cost and moreparts are ordered from less reliable and lower cost suppliersthe expected supply schedules and the corresponding pro-duction schedules are more delayed as well as the delivery ofproducts to the customers Note that despite constraint (13)that ensures a feasible schedule with at most one shipmentof products to each distribution center for each disruptionscenario the expected shipping schedule (29) may be splitinto more smaller size shipments (cf Figure 3 where twomajor and one small-size shipments are indicated for eachconfidence level)

For comparison Table 3 presents a subset of non-dominated solutions obtained for the expected value problemEWCS and Figure 4 shows supply production and shippingschedules Unlike the stochastic programming approachwhich accounts for all potential disruption scenarios tooptimize an expected performance of the supply chain thesolution obtained using the deterministic approach is basedon aggregate information on suppliers expected fulfillmentIn general the results are similar for both models and thecorresponding optimal solution values are close to each otherwhich indicates that the expected value problem can be usedin practice when stochastic mixed integer programs are hardto solve The optimal solution values for the expected valueproblem frequently outperform the corresponding solutionvalues of the wait-and-see problem which is in line withthe proposition EV le WS in [27] compare Section 5for example the minimum cost 1198641 for EWCS (see Table 3)

12 Mathematical Problems in Engineering

Table 2 Nondominated solutions for modelWCS

120572 0 01 02 03 04 05 and 06 07 08 and 09 1Var = 38077 Bin = 33076 Cons = 28769 Nonz = 348325(a)

Exp cost 1731 (1198641) 1620 1547 1206 1200 1176 1070 1065 1051 (1198641)Exp service level(b) 8552 (1198642) 8487 8452 7719 7676 7595 5863 5871 3822 (1198642)Exp fulfilled demand(c) 8552 8530 8452 9370 9361 9353 9134 9120 9068

Suppliers selected( of total demand)

1 (76) 1 (35)2 (24) 2 (55) 2 (87) 2 (55) 2 (55) 2 (55) 2 (12) 2 (10)3 (10) 3 (13) 3 (22) 3 (32) 3 (45) 3 (88) 3 (90) 3 (100)4 (23) 4 (13)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119904isin119878 sum119895isin119869 119875119904119887119895119910

119904119895119861 times

100 (c)sum119904isin119878 sum119895isin119869 sum119905isin119879 119875119904119887119895119908119904119895119905119861 times 100

Table 3 Nondominated solutions for model EWCS

120572 0 01 02 and 03 04 and 05 06 and 07 08 09 and 1Var = 162 Bin = 136 Cons = 125 Nonz = 1430(a)

Exp cost 1691 (1198641) 1427 1214 1167 1045 (1198641)Exp service level(b) 87 (1198642) 86 81 79 62 (1198642)Exp fulfilled demand(c) 87 90 91 93 91

Suppliers selected( of total demand)

1 (89) 2 (94) 2 (49) 2 (50) 2 (5)3 (6) 3 (51) 3 (50) 3 (95)4 (11)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119895isin119869 119887119895119884119895119861 times 100(c)sum119895isin119869 sum119905isin119879 119887119895119882119895119905119861 times 100

1 2 3 4 5 6 7Period

Supply schedules

1 2 3 4 5 6 7Period

Production schedules

1 2 3 4 5 6 7Period

Shipping schedules

0

10000

20000

30000

40000

50000

Prod

ucts

0

10000

20000

30000

40000

50000

Prod

ucts

0

20000

40000

60000

80000

100000

Part

s

Alpha = 0Alpha = 05

Alpha = 1 Alpha = 0Alpha = 05

Alpha = 1

Alpha = 0Alpha = 05

Alpha = 1

Figure 4 Schedules for model EWCS

Mathematical Problems in Engineering 13

1045 le 1051 forWCS (see Table 2) or themaximum servicelevel1198642 for EWCS (see Table 3) 8700 ge 8552 forWCS (seeTable 2)

The stochastic wait-and-see approach however leads toa more diversified supply portfolio For the deterministicapproach most of nondominated supply portfolios consistof two suppliers only 119894 = 2 3 while suppliers 119894 = 1 4 areselected only for the maximum service level objective (thatis for 120572 = 0) Comparison of Figures 3 and 4 indicatesthat the expected schedules for model WCS computed asexpectations over all schedules for all potential disruptionscenarios (27)ndash(29) are similar to the corresponding singleschedules determined by model EWCS The main differencesobserved aremore delayed expected production and shippingschedules for model WCS when minimization of cost is con-sidered (that is for120572 = 1) Finally it is interesting to note thatin the multiple sourcing environment considered both thewait-and-see approach and in particular the expected valueapproach frequently select a dual sourcing supply portfoliowith one main supplier and one supporting supplier

The computational experiments were performed usingthe AMPL programming language and the CPLEX 1262solver on a MacBook Pro laptop with Intel Core i7 processorrunning at 28 GHz and with 16GB RAM The solver wascapable of finding proven optimal solution for all exampleswith CPU time ranging from several minutes to several hoursfor the stochastic model WCS and fraction of a second forthe deterministic model EWCS Since the stochastic pro-gramming model WCS needs to determine nondominatedschedules for all potential disruption scenarios while modelEWCS deals with a single scenario only the difference in thecomputational effort required is obvious

7 Conclusions

In this paper two biobjective MIP formulations stochasticand deterministic have been proposed and compared for thecoordinated decision-making in supply chains under partiallocal disruptions and all-or-nothing regional disruptionsTheproblem objective has been to jointly schedule supplies pro-duction and distribution to optimize the trade-off betweenexpected cost and expected service levelWhile the stochasticprogramming approach aims at optimizing the expectedperformance of a supply chain over all possible disruptionscenarios the deterministic approach accounts only for asingle scenario representing average disruption conditionsAs a result the stochastic programming approach determinesnondominated solutions for all disruption scenarios whereasthe deterministic approach produces a single solution onlyIn particular in the stochastic programming approach theselection of supply portfolio is combined with supply chainscheduling for all disruption scenarios considered In con-trast the deterministic expected value approach provides theportfolio along with a single expected schedule of productionand distribution

The expected schedules obtained for the stochastic pro-gramming model WCS as expectations over all schedulesfor all disruption scenarios have been compared with thecorresponding schedules determined by the deterministic

model EWCS based on an expected disruption scenarioThecomparison has demonstrated that the two approaches leadto similar expected solutions

The main findings are in line with other research and arelisted as follows

(i) The two decision-making approaches stochastic anddeterministic lead to similar expected performanceof a supply chain under multilevel disruptions

(ii) The optimal solution values for the expected valueproblem frequently outperform the correspondingsolution values of the wait-and-see problem

(iii) Despite the multiple sourcing environment consid-ered both the wait-and-see approach and in partic-ular the expected value approach frequently selecta dual sourcing supply portfolio with one mainsupplier and one supporting supplier

(iv) The stochastic approach which accounts for allpotential disruption scenarios may lead to a morediversified supply portfolio that will hedge against avariety of scenarios

(v) The expected schedules are more delayed for thestochastic approach

(vi) The service-oriented supply portfolio is more diversi-fied andmay combine both high-cost reliable suppli-ers and low-cost unreliable suppliers while the cost-oriented portfolio depends mainly on low-cost andless reliable suppliers

Overall the results of computational experiments indi-cate that the proposed approach and developed MIP modelsare flexible and efficient tools for coordinated supply chainscheduling The portfolio approach leads to MIP formula-tions with strong LP relaxations and has been proven to becomputationally very efficient CPU time required to findproven optimal solutions for realistic size examples usingcommercially available software for MIP is acceptable for areal-world supply chain disruption management (see Sawik[33])

Since the probability distribution of supply disruptionsfrom each supplier is usually unknown (eg [8]) local multi-level disruptions in the stochastic programming modelWCShave a multinomial discrete distribution while the two-levelregional disruptions have a binomial discrete distributionand all disruption events are independent As part of futureresearch we propose to enhance the stochastic programmingmodel formore general scenarios with finitelymany elementssingled out and all the probability concentrated in them Forexample rather than complete shutdown suppliers within aregion may have correlated disruptions Another importantstream of future research is the study of robustness andsensitivity in relation to input data changes in supply chainscheduling under disruption risks (eg [33]) Future researchcan also focus on the following improvement of the proposedmodels to consider the minimization of time and cost ofrecovery (eg Whitney et al [34])

14 Mathematical Problems in Engineering

Notations

Indices

119894 Supplier 119894 isin 119868119895 Customer 119895 isin 119869119896 Distribution center 119896 isin 119870119897 Disruption level 119897 isin 119871119903 Region 119903 isin 119877119904 Disruption scenario 119904 isin 119878119905 Planning period 119905 isin 119879Input Parameters

119887119895 Size (number of products) of customer order 119895119861 Total demand for partsproducts 119861 = sum119895isin119869 119887119895119862 Capacity of producer119889119895 Due date for customer order 119895119890119894 Fixed cost of ordering parts from supplier 119894119892119895 Per unit penalty cost of delayed customer order 119895ℎ119895 Per unit penalty cost of unfulfilled customer order 119895119868119903 Subset of suppliers in region 119903119869119896 Subset of customers serviced by distribution center 119896119900119894 Per unit price of parts purchased from supplier 119894119901119894119897 Probability of disruption level 119897 for supplier 119894119901119903 Regional disruption probability for region 119903120574119894119897 Fraction of an order delivered by supplier 119894 underdisruption level 119897 (fulfillment rate)Γ119894 Expected fraction of an order delivered by supplier 119894(expected fulfillment rate)120590119894 Delivery lead time from supplier 119894120591119896 Transportation time to distribution center 119896

Conflicts of Interest

The author declares that there are no conflicts of interest

Acknowledgments

This work has been supported by NCN research grant (noDEC-201311BST804458) and by AGH (no 1111200324)

References

[1] J V Blackhurst K P Scheibe and D J Johnson ldquoSupplierrisk assessment and monitoring for the automotive industryrdquoInternational Journal of Physical Distribution amp Logistics Man-agement vol 38 no 2 pp 143ndash165 2008

[2] P Hoffmann H Schiele and K Krabbendam ldquoUncertaintysupply risk management and their impact on performancerdquoJournal of Purchasing amp Supply Management vol 19 no 3 pp199ndash211 2013

[3] Y Park P Hong and J J Roh ldquoSupply chain lessons from thecatastrophic natural disaster in Japanrdquo Business Horizons vol56 no 1 pp 75ndash85 2013

[4] M Haraguchi and U Lall ldquoFlood risks and impacts A casestudy of Thailandrsquos floods in 2011 and research questions forsupply chain decisionmakingrdquo International Journal of DisasterRisk Reduction vol 14 pp 256ndash272 2015

[5] T Fujimoto and Y W Park ldquoBalancing supply chain com-petitiveness and robustness through ldquovirtual dual sourcingrdquoLessons from the Great East Japan Earthquakerdquo InternationalJournal of Production Economics vol 147 pp 429ndash436 2014

[6] HMatsuo ldquoImplications of the Tohoku earthquake for Toyotarsquoscoordination mechanism supply chain disruption of auto-motive semiconductorsrdquo International Journal of ProductionEconomics vol 161 pp 217ndash227 2015

[7] J R Marszewska ldquoImplications of seismic hazard in Japan ontoyota supply chain disruption risksrdquo in Proceedings of the 13thInternational Conference on Industrial Logistics (ICIL rsquo16) pp178ndash185 Zakopane Poland October 2016

[8] A Z Zeng and Y Xia ldquoBuilding a mutually beneficial partner-ship to ensure backup supplyrdquo Omega vol 52 pp 77ndash91 2015

[9] S S Erenguc N C Simpson and A J Vakharia ldquoIntegratedproductiondistribution planning in supply chains an invitedreviewrdquo European Journal of Operational Research vol 115 no2 pp 219ndash236 1999

[10] L Lei S Liu A Ruszczynski and S Park ldquoOn the integratedproduction inventory and distribution routing problemrdquo IIETransactions vol 38 no 11 pp 955ndash970 2006

[11] A Kaur A Kanda and S G Deshmukh ldquoA graph theoreticapproach for supply chain coordinationrdquo International Journalof Logistics Systems and Management vol 2 no 4 pp 321ndash3412006

[12] T ChoiW Yeung andT C E Cheng ldquoScheduling and co-ordi-nation of multi-suppliers single-warehouse-operator single-manufacturer supply chains with variable production rates andstorage costsrdquo International Journal of Production Research vol51 no 9 pp 2593ndash2601 2013

[13] S Liu and L G Papageorgiou ldquoMultiobjective optimisation ofproduction distribution and capacity planning of global supplychains in the process industryrdquo Omega vol 41 no 2 pp 369ndash382 2013

[14] Z L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[15] T Sawik ldquoCoordinated supply chain schedulingrdquo InternationalJournal of Production Economics vol 120 no 2 pp 437ndash4512009

[16] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[17] D Ivanov B Sokolov and A Dolgui ldquoThe Ripple effect in sup-ply chains Trade-off rsquoefficiency-flexibility- resiliencersquo in disrup-tionmanagementrdquo International Journal of ProductionResearchvol 52 no 7 pp 2154ndash2172 2014

[18] D Ivanov A Pavlov and B Sokolov ldquoOptimal distribution(re)planning in a centralized multi-stage supply network underconditions of the ripple effect and structure dynamicsrdquo Euro-pean Journal of Operational Research vol 237 no 2 pp 758ndash770 2014

[19] D Ivanov A Pavlov A Dolgui D Pavlov and B SokolovldquoDisruption-driven supply chain (re)-planning and perfor-mance impact assessment with consideration of pro-active andrecovery policiesrdquo Transportation Research Part E Logistics andTransportation Review vol 90 pp 7ndash24 2016

[20] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

Mathematical Problems in Engineering 15

[21] A J Schmitt S A Sun L V Snyder and Z-J M Shen ldquoCen-tralization versus decentralization Risk pooling risk diversifi-cation and supply chain disruptionsrdquo Omega vol 52 pp 201ndash212 2015

[22] L V Snyder Z Atan P Peng Y Rong A J Schmitt andB Sinsoysal ldquoORMS models for supply chain disruptions areviewrdquo IIE Transactions vol 48 no 2 pp 89ndash109 2016

[23] T Sawik ldquoIntegrated selection of suppliers and scheduling ofcustomer orders in the presence of supply chain disruptionrisksrdquo International Journal of Production Research vol 51 no23-24 pp 7006ndash7022 2013

[24] T Sawik ldquoOptimization of cost and service level in the presenceof supply chain disruption risks Single vs multiple sourcingrdquoComputers amp Operations Research vol 51 pp 11ndash20 2014

[25] T Sawik ldquoIntegrated supply production and distributionscheduling under disruption risksrdquoOmega vol 62 pp 131ndash1442016

[26] T Sawik ldquoIntegrated supply chain scheduling under multi-leveldisruptionsrdquo IFAC-PapersOnLine vol 28 no 3 pp 1515ndash15202015

[27] P Kall and J Mayer Stochastic Linear Programming ModelsTheory and Computation vol 156 of International Series inOperations Research amp Management Science Springer NewYork 2011

[28] I N Durbach and T J Stewart ldquoUsing expected values to sim-plify decision making under uncertaintyrdquo Omega vol 37 no 2pp 312ndash330 2009

[29] F Maggioni and S W Wallace ldquoAnalyzing the quality of theexpected value solution in stochastic programmingrdquo Annals ofOperations Research vol 200 pp 37ndash54 2012

[30] T Sawik ldquoSelection of supply portfolio under disruption risksrdquoOmega vol 39 no 2 pp 194ndash208 2011

[31] E Delage S Arroyo and Y Ye ldquoThe value of stochastic mod-eling in two-stage stochastic programs with cost uncertaintyrdquoOperations Research vol 62 no 6 pp 1377ndash1393 2014

[32] T Sawik Scheduling in Supply Chains Using Mixed Integer Pro-gramming John Wiley amp Sons Inc Hoboken NJ 2011

[33] T Sawik Supply Chain DisruptionManagement Using StochasticMixed Integer Programming vol 256 of International Seriesin Operations Research amp Management Science Springer NewYork 2017 httpwwwspringercomgpbook9783319588223

[34] D E Whitney J Luo and D A Heller ldquoThe benefits and con-straints of temporary sourcing diversification in supply chaindisruption and recoveryrdquo Journal of Purchasing and SupplyManagement vol 20 no 4 pp 238ndash250 2014

Mathematical Problems in Engineering 9

Table 1 Disruption scenarios

119904 119894 = 1 2 3 41 0 0 0 02 0 0 0 13 0 0 0 24 0 0 0 35 0 0 1 06 0 0 1 17 0 0 1 28 0 0 1 39 0 0 2 010 0 0 2 111 0 0 2 212 0 0 2 313 0 0 3 014 0 0 3 115 0 0 3 216 0 0 3 317 0 1 0 018 0 1 0 119 0 1 0 220 0 1 0 321 0 1 1 022 0 1 1 123 0 1 1 224 0 1 1 325 0 1 2 026 0 1 2 127 0 1 2 228 0 1 2 329 0 1 3 030 0 1 3 131 0 1 3 232 0 1 3 333 0 2 0 034 0 2 0 135 0 2 0 236 0 2 0 337 0 2 1 038 0 2 1 139 0 2 1 240 0 2 1 341 0 2 2 042 0 2 2 143 0 2 2 244 0 2 2 345 0 2 3 046 0 2 3 147 0 2 3 248 0 2 3 349 0 3 0 0

Table 1 Continued

119904 119894 = 1 2 3 450 0 3 0 151 0 3 0 252 0 3 0 353 0 3 1 054 0 3 1 155 0 3 1 256 0 3 1 357 0 3 2 058 0 3 2 159 0 3 2 260 0 3 2 361 0 3 3 062 0 3 3 163 0 3 3 264 0 3 3 365 1 0 0 066 1 0 0 167 1 0 0 268 1 0 0 369 1 0 1 070 1 0 1 171 1 0 1 272 1 0 1 373 1 0 2 074 1 0 2 175 1 0 2 276 1 0 2 377 1 0 3 078 1 0 3 179 1 0 3 280 1 0 3 381 1 1 0 082 1 1 0 183 1 1 0 284 1 1 0 385 1 1 1 086 1 1 1 187 1 1 1 288 1 1 1 389 1 1 2 090 1 1 2 191 1 1 2 292 1 1 2 393 1 1 3 094 1 1 3 195 1 1 3 296 1 1 3 397 1 2 0 098 1 2 0 1

10 Mathematical Problems in Engineering

Table 1 Continued

119904 119894 = 1 2 3 499 1 2 0 2100 1 2 0 3101 1 2 1 0102 1 2 1 1103 1 2 1 2104 1 2 1 3105 1 2 2 0106 1 2 2 1107 1 2 2 2108 1 2 2 3109 1 2 3 0110 1 2 3 1111 1 2 3 2112 1 2 3 3113 1 3 0 0114 1 3 0 1115 1 3 0 2116 1 3 0 3117 1 3 1 0118 1 3 1 1119 1 3 1 2120 1 3 1 3121 1 3 2 0122 1 3 2 1123 1 3 2 2124 1 3 2 3125 1 3 3 0126 1 3 3 1127 1 3 3 2128 1 3 3 3129 2 0 0 0130 2 0 0 1131 2 0 0 2132 2 0 0 3133 2 0 1 0134 2 0 1 1135 2 0 1 2136 2 0 1 3137 2 0 2 0138 2 0 2 1139 2 0 2 2140 2 0 2 3141 2 0 3 0142 2 0 3 1143 2 0 3 2144 2 0 3 3145 2 1 0 0146 2 1 0 1147 2 1 0 2148 2 1 0 3149 2 1 1 0150 2 1 1 1

Table 1 Continued

119904 119894 = 1 2 3 4151 2 1 1 2152 2 1 1 3153 2 1 2 0154 2 1 2 1155 2 1 2 2156 2 1 2 3157 2 1 3 0158 2 1 3 1159 2 1 3 2160 2 1 3 3161 2 2 0 0162 2 2 0 1163 2 2 0 2164 2 2 0 3165 2 2 1 0166 2 2 1 1167 2 2 1 2168 2 2 1 3169 2 2 2 0170 2 2 2 1171 2 2 2 2172 2 2 2 3173 2 2 3 0174 2 2 3 1175 2 2 3 2176 2 2 3 3177 2 3 0 0178 2 3 0 1179 2 3 0 2180 2 3 0 3181 2 3 1 0182 2 3 1 1183 2 3 1 2184 2 3 1 3185 2 3 2 0186 2 3 2 1187 2 3 2 2188 2 3 2 3189 2 3 3 0190 2 3 3 1191 2 3 3 2192 2 3 3 3193 3 0 0 0194 3 0 0 1195 3 0 0 2196 3 0 0 3197 3 0 1 0198 3 0 1 1199 3 0 1 2200 3 0 1 3201 3 0 2 0202 3 0 2 1

Mathematical Problems in Engineering 11

Table 1 Continued

119904 119894 = 1 2 3 4203 3 0 2 2204 3 0 2 3205 3 0 3 0206 3 0 3 1207 3 0 3 2208 3 0 3 3209 3 1 0 0210 3 1 0 1211 3 1 0 2212 3 1 0 3213 3 1 1 0214 3 1 1 1215 3 1 1 2216 3 1 1 3217 3 1 2 0218 3 1 2 1219 3 1 2 2220 3 1 2 3221 3 1 3 0222 3 1 3 1223 3 1 3 2224 3 1 3 3225 3 2 0 0226 3 2 0 1227 3 2 0 2228 3 2 0 3229 3 2 1 0230 3 2 1 1231 3 2 1 2232 3 2 1 3233 3 2 2 0234 3 2 2 1235 3 2 2 2236 3 2 2 3237 3 2 3 0238 3 2 3 1239 3 2 3 2240 3 2 3 3241 3 3 0 0242 3 3 0 1243 3 3 0 2244 3 3 0 3245 3 3 1 0246 3 3 1 1247 3 3 1 2248 3 3 1 3249 3 3 2 0250 3 3 2 1251 3 3 2 2252 3 3 2 3

Table 1 Continued

119904 119894 = 1 2 3 4253 3 3 3 0254 3 3 3 1255 3 3 3 2256 3 3 3 3

maximum expected service level) 120572 = 05 and 120572 = 1 (ie forthe minimum expected cost) The expected schedules werecomputed using the formulae presented below

(i) Expected schedule of supplies of parts to the pro-ducer

sum119904isin119878

sum119894isin119868119904120590119894=119905

119875119904119861120574119894120582119894119904V119894 119905 isin 119879 (27)

(ii) Expected production schedule

sum119904isin119878

sum119895isin119869

119875119904119887119895119908119904119895119905 119905 isin 119879 (28)

(iii) Expected schedule of shipping of products from theproducer to the distribution centers

sum119904isin119878

sum119896isin119870

sum119895isin119869119896

119875119904119887119895( sum1199051015840isin1198791199051015840lt119905

1199081199041198951199051015840)119909119904119896119905 119905 isin 119879 (29)

As 120572 increases that is the decision-makerrsquos priority shiftsfrom the maximum service level to minimum cost and moreparts are ordered from less reliable and lower cost suppliersthe expected supply schedules and the corresponding pro-duction schedules are more delayed as well as the delivery ofproducts to the customers Note that despite constraint (13)that ensures a feasible schedule with at most one shipmentof products to each distribution center for each disruptionscenario the expected shipping schedule (29) may be splitinto more smaller size shipments (cf Figure 3 where twomajor and one small-size shipments are indicated for eachconfidence level)

For comparison Table 3 presents a subset of non-dominated solutions obtained for the expected value problemEWCS and Figure 4 shows supply production and shippingschedules Unlike the stochastic programming approachwhich accounts for all potential disruption scenarios tooptimize an expected performance of the supply chain thesolution obtained using the deterministic approach is basedon aggregate information on suppliers expected fulfillmentIn general the results are similar for both models and thecorresponding optimal solution values are close to each otherwhich indicates that the expected value problem can be usedin practice when stochastic mixed integer programs are hardto solve The optimal solution values for the expected valueproblem frequently outperform the corresponding solutionvalues of the wait-and-see problem which is in line withthe proposition EV le WS in [27] compare Section 5for example the minimum cost 1198641 for EWCS (see Table 3)

12 Mathematical Problems in Engineering

Table 2 Nondominated solutions for modelWCS

120572 0 01 02 03 04 05 and 06 07 08 and 09 1Var = 38077 Bin = 33076 Cons = 28769 Nonz = 348325(a)

Exp cost 1731 (1198641) 1620 1547 1206 1200 1176 1070 1065 1051 (1198641)Exp service level(b) 8552 (1198642) 8487 8452 7719 7676 7595 5863 5871 3822 (1198642)Exp fulfilled demand(c) 8552 8530 8452 9370 9361 9353 9134 9120 9068

Suppliers selected( of total demand)

1 (76) 1 (35)2 (24) 2 (55) 2 (87) 2 (55) 2 (55) 2 (55) 2 (12) 2 (10)3 (10) 3 (13) 3 (22) 3 (32) 3 (45) 3 (88) 3 (90) 3 (100)4 (23) 4 (13)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119904isin119878 sum119895isin119869 119875119904119887119895119910

119904119895119861 times

100 (c)sum119904isin119878 sum119895isin119869 sum119905isin119879 119875119904119887119895119908119904119895119905119861 times 100

Table 3 Nondominated solutions for model EWCS

120572 0 01 02 and 03 04 and 05 06 and 07 08 09 and 1Var = 162 Bin = 136 Cons = 125 Nonz = 1430(a)

Exp cost 1691 (1198641) 1427 1214 1167 1045 (1198641)Exp service level(b) 87 (1198642) 86 81 79 62 (1198642)Exp fulfilled demand(c) 87 90 91 93 91

Suppliers selected( of total demand)

1 (89) 2 (94) 2 (49) 2 (50) 2 (5)3 (6) 3 (51) 3 (50) 3 (95)4 (11)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119895isin119869 119887119895119884119895119861 times 100(c)sum119895isin119869 sum119905isin119879 119887119895119882119895119905119861 times 100

1 2 3 4 5 6 7Period

Supply schedules

1 2 3 4 5 6 7Period

Production schedules

1 2 3 4 5 6 7Period

Shipping schedules

0

10000

20000

30000

40000

50000

Prod

ucts

0

10000

20000

30000

40000

50000

Prod

ucts

0

20000

40000

60000

80000

100000

Part

s

Alpha = 0Alpha = 05

Alpha = 1 Alpha = 0Alpha = 05

Alpha = 1

Alpha = 0Alpha = 05

Alpha = 1

Figure 4 Schedules for model EWCS

Mathematical Problems in Engineering 13

1045 le 1051 forWCS (see Table 2) or themaximum servicelevel1198642 for EWCS (see Table 3) 8700 ge 8552 forWCS (seeTable 2)

The stochastic wait-and-see approach however leads toa more diversified supply portfolio For the deterministicapproach most of nondominated supply portfolios consistof two suppliers only 119894 = 2 3 while suppliers 119894 = 1 4 areselected only for the maximum service level objective (thatis for 120572 = 0) Comparison of Figures 3 and 4 indicatesthat the expected schedules for model WCS computed asexpectations over all schedules for all potential disruptionscenarios (27)ndash(29) are similar to the corresponding singleschedules determined by model EWCS The main differencesobserved aremore delayed expected production and shippingschedules for model WCS when minimization of cost is con-sidered (that is for120572 = 1) Finally it is interesting to note thatin the multiple sourcing environment considered both thewait-and-see approach and in particular the expected valueapproach frequently select a dual sourcing supply portfoliowith one main supplier and one supporting supplier

The computational experiments were performed usingthe AMPL programming language and the CPLEX 1262solver on a MacBook Pro laptop with Intel Core i7 processorrunning at 28 GHz and with 16GB RAM The solver wascapable of finding proven optimal solution for all exampleswith CPU time ranging from several minutes to several hoursfor the stochastic model WCS and fraction of a second forthe deterministic model EWCS Since the stochastic pro-gramming model WCS needs to determine nondominatedschedules for all potential disruption scenarios while modelEWCS deals with a single scenario only the difference in thecomputational effort required is obvious

7 Conclusions

In this paper two biobjective MIP formulations stochasticand deterministic have been proposed and compared for thecoordinated decision-making in supply chains under partiallocal disruptions and all-or-nothing regional disruptionsTheproblem objective has been to jointly schedule supplies pro-duction and distribution to optimize the trade-off betweenexpected cost and expected service levelWhile the stochasticprogramming approach aims at optimizing the expectedperformance of a supply chain over all possible disruptionscenarios the deterministic approach accounts only for asingle scenario representing average disruption conditionsAs a result the stochastic programming approach determinesnondominated solutions for all disruption scenarios whereasthe deterministic approach produces a single solution onlyIn particular in the stochastic programming approach theselection of supply portfolio is combined with supply chainscheduling for all disruption scenarios considered In con-trast the deterministic expected value approach provides theportfolio along with a single expected schedule of productionand distribution

The expected schedules obtained for the stochastic pro-gramming model WCS as expectations over all schedulesfor all disruption scenarios have been compared with thecorresponding schedules determined by the deterministic

model EWCS based on an expected disruption scenarioThecomparison has demonstrated that the two approaches leadto similar expected solutions

The main findings are in line with other research and arelisted as follows

(i) The two decision-making approaches stochastic anddeterministic lead to similar expected performanceof a supply chain under multilevel disruptions

(ii) The optimal solution values for the expected valueproblem frequently outperform the correspondingsolution values of the wait-and-see problem

(iii) Despite the multiple sourcing environment consid-ered both the wait-and-see approach and in partic-ular the expected value approach frequently selecta dual sourcing supply portfolio with one mainsupplier and one supporting supplier

(iv) The stochastic approach which accounts for allpotential disruption scenarios may lead to a morediversified supply portfolio that will hedge against avariety of scenarios

(v) The expected schedules are more delayed for thestochastic approach

(vi) The service-oriented supply portfolio is more diversi-fied andmay combine both high-cost reliable suppli-ers and low-cost unreliable suppliers while the cost-oriented portfolio depends mainly on low-cost andless reliable suppliers

Overall the results of computational experiments indi-cate that the proposed approach and developed MIP modelsare flexible and efficient tools for coordinated supply chainscheduling The portfolio approach leads to MIP formula-tions with strong LP relaxations and has been proven to becomputationally very efficient CPU time required to findproven optimal solutions for realistic size examples usingcommercially available software for MIP is acceptable for areal-world supply chain disruption management (see Sawik[33])

Since the probability distribution of supply disruptionsfrom each supplier is usually unknown (eg [8]) local multi-level disruptions in the stochastic programming modelWCShave a multinomial discrete distribution while the two-levelregional disruptions have a binomial discrete distributionand all disruption events are independent As part of futureresearch we propose to enhance the stochastic programmingmodel formore general scenarios with finitelymany elementssingled out and all the probability concentrated in them Forexample rather than complete shutdown suppliers within aregion may have correlated disruptions Another importantstream of future research is the study of robustness andsensitivity in relation to input data changes in supply chainscheduling under disruption risks (eg [33]) Future researchcan also focus on the following improvement of the proposedmodels to consider the minimization of time and cost ofrecovery (eg Whitney et al [34])

14 Mathematical Problems in Engineering

Notations

Indices

119894 Supplier 119894 isin 119868119895 Customer 119895 isin 119869119896 Distribution center 119896 isin 119870119897 Disruption level 119897 isin 119871119903 Region 119903 isin 119877119904 Disruption scenario 119904 isin 119878119905 Planning period 119905 isin 119879Input Parameters

119887119895 Size (number of products) of customer order 119895119861 Total demand for partsproducts 119861 = sum119895isin119869 119887119895119862 Capacity of producer119889119895 Due date for customer order 119895119890119894 Fixed cost of ordering parts from supplier 119894119892119895 Per unit penalty cost of delayed customer order 119895ℎ119895 Per unit penalty cost of unfulfilled customer order 119895119868119903 Subset of suppliers in region 119903119869119896 Subset of customers serviced by distribution center 119896119900119894 Per unit price of parts purchased from supplier 119894119901119894119897 Probability of disruption level 119897 for supplier 119894119901119903 Regional disruption probability for region 119903120574119894119897 Fraction of an order delivered by supplier 119894 underdisruption level 119897 (fulfillment rate)Γ119894 Expected fraction of an order delivered by supplier 119894(expected fulfillment rate)120590119894 Delivery lead time from supplier 119894120591119896 Transportation time to distribution center 119896

Conflicts of Interest

The author declares that there are no conflicts of interest

Acknowledgments

This work has been supported by NCN research grant (noDEC-201311BST804458) and by AGH (no 1111200324)

References

[1] J V Blackhurst K P Scheibe and D J Johnson ldquoSupplierrisk assessment and monitoring for the automotive industryrdquoInternational Journal of Physical Distribution amp Logistics Man-agement vol 38 no 2 pp 143ndash165 2008

[2] P Hoffmann H Schiele and K Krabbendam ldquoUncertaintysupply risk management and their impact on performancerdquoJournal of Purchasing amp Supply Management vol 19 no 3 pp199ndash211 2013

[3] Y Park P Hong and J J Roh ldquoSupply chain lessons from thecatastrophic natural disaster in Japanrdquo Business Horizons vol56 no 1 pp 75ndash85 2013

[4] M Haraguchi and U Lall ldquoFlood risks and impacts A casestudy of Thailandrsquos floods in 2011 and research questions forsupply chain decisionmakingrdquo International Journal of DisasterRisk Reduction vol 14 pp 256ndash272 2015

[5] T Fujimoto and Y W Park ldquoBalancing supply chain com-petitiveness and robustness through ldquovirtual dual sourcingrdquoLessons from the Great East Japan Earthquakerdquo InternationalJournal of Production Economics vol 147 pp 429ndash436 2014

[6] HMatsuo ldquoImplications of the Tohoku earthquake for Toyotarsquoscoordination mechanism supply chain disruption of auto-motive semiconductorsrdquo International Journal of ProductionEconomics vol 161 pp 217ndash227 2015

[7] J R Marszewska ldquoImplications of seismic hazard in Japan ontoyota supply chain disruption risksrdquo in Proceedings of the 13thInternational Conference on Industrial Logistics (ICIL rsquo16) pp178ndash185 Zakopane Poland October 2016

[8] A Z Zeng and Y Xia ldquoBuilding a mutually beneficial partner-ship to ensure backup supplyrdquo Omega vol 52 pp 77ndash91 2015

[9] S S Erenguc N C Simpson and A J Vakharia ldquoIntegratedproductiondistribution planning in supply chains an invitedreviewrdquo European Journal of Operational Research vol 115 no2 pp 219ndash236 1999

[10] L Lei S Liu A Ruszczynski and S Park ldquoOn the integratedproduction inventory and distribution routing problemrdquo IIETransactions vol 38 no 11 pp 955ndash970 2006

[11] A Kaur A Kanda and S G Deshmukh ldquoA graph theoreticapproach for supply chain coordinationrdquo International Journalof Logistics Systems and Management vol 2 no 4 pp 321ndash3412006

[12] T ChoiW Yeung andT C E Cheng ldquoScheduling and co-ordi-nation of multi-suppliers single-warehouse-operator single-manufacturer supply chains with variable production rates andstorage costsrdquo International Journal of Production Research vol51 no 9 pp 2593ndash2601 2013

[13] S Liu and L G Papageorgiou ldquoMultiobjective optimisation ofproduction distribution and capacity planning of global supplychains in the process industryrdquo Omega vol 41 no 2 pp 369ndash382 2013

[14] Z L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[15] T Sawik ldquoCoordinated supply chain schedulingrdquo InternationalJournal of Production Economics vol 120 no 2 pp 437ndash4512009

[16] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[17] D Ivanov B Sokolov and A Dolgui ldquoThe Ripple effect in sup-ply chains Trade-off rsquoefficiency-flexibility- resiliencersquo in disrup-tionmanagementrdquo International Journal of ProductionResearchvol 52 no 7 pp 2154ndash2172 2014

[18] D Ivanov A Pavlov and B Sokolov ldquoOptimal distribution(re)planning in a centralized multi-stage supply network underconditions of the ripple effect and structure dynamicsrdquo Euro-pean Journal of Operational Research vol 237 no 2 pp 758ndash770 2014

[19] D Ivanov A Pavlov A Dolgui D Pavlov and B SokolovldquoDisruption-driven supply chain (re)-planning and perfor-mance impact assessment with consideration of pro-active andrecovery policiesrdquo Transportation Research Part E Logistics andTransportation Review vol 90 pp 7ndash24 2016

[20] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

Mathematical Problems in Engineering 15

[21] A J Schmitt S A Sun L V Snyder and Z-J M Shen ldquoCen-tralization versus decentralization Risk pooling risk diversifi-cation and supply chain disruptionsrdquo Omega vol 52 pp 201ndash212 2015

[22] L V Snyder Z Atan P Peng Y Rong A J Schmitt andB Sinsoysal ldquoORMS models for supply chain disruptions areviewrdquo IIE Transactions vol 48 no 2 pp 89ndash109 2016

[23] T Sawik ldquoIntegrated selection of suppliers and scheduling ofcustomer orders in the presence of supply chain disruptionrisksrdquo International Journal of Production Research vol 51 no23-24 pp 7006ndash7022 2013

[24] T Sawik ldquoOptimization of cost and service level in the presenceof supply chain disruption risks Single vs multiple sourcingrdquoComputers amp Operations Research vol 51 pp 11ndash20 2014

[25] T Sawik ldquoIntegrated supply production and distributionscheduling under disruption risksrdquoOmega vol 62 pp 131ndash1442016

[26] T Sawik ldquoIntegrated supply chain scheduling under multi-leveldisruptionsrdquo IFAC-PapersOnLine vol 28 no 3 pp 1515ndash15202015

[27] P Kall and J Mayer Stochastic Linear Programming ModelsTheory and Computation vol 156 of International Series inOperations Research amp Management Science Springer NewYork 2011

[28] I N Durbach and T J Stewart ldquoUsing expected values to sim-plify decision making under uncertaintyrdquo Omega vol 37 no 2pp 312ndash330 2009

[29] F Maggioni and S W Wallace ldquoAnalyzing the quality of theexpected value solution in stochastic programmingrdquo Annals ofOperations Research vol 200 pp 37ndash54 2012

[30] T Sawik ldquoSelection of supply portfolio under disruption risksrdquoOmega vol 39 no 2 pp 194ndash208 2011

[31] E Delage S Arroyo and Y Ye ldquoThe value of stochastic mod-eling in two-stage stochastic programs with cost uncertaintyrdquoOperations Research vol 62 no 6 pp 1377ndash1393 2014

[32] T Sawik Scheduling in Supply Chains Using Mixed Integer Pro-gramming John Wiley amp Sons Inc Hoboken NJ 2011

[33] T Sawik Supply Chain DisruptionManagement Using StochasticMixed Integer Programming vol 256 of International Seriesin Operations Research amp Management Science Springer NewYork 2017 httpwwwspringercomgpbook9783319588223

[34] D E Whitney J Luo and D A Heller ldquoThe benefits and con-straints of temporary sourcing diversification in supply chaindisruption and recoveryrdquo Journal of Purchasing and SupplyManagement vol 20 no 4 pp 238ndash250 2014

10 Mathematical Problems in Engineering

Table 1 Continued

119904 119894 = 1 2 3 499 1 2 0 2100 1 2 0 3101 1 2 1 0102 1 2 1 1103 1 2 1 2104 1 2 1 3105 1 2 2 0106 1 2 2 1107 1 2 2 2108 1 2 2 3109 1 2 3 0110 1 2 3 1111 1 2 3 2112 1 2 3 3113 1 3 0 0114 1 3 0 1115 1 3 0 2116 1 3 0 3117 1 3 1 0118 1 3 1 1119 1 3 1 2120 1 3 1 3121 1 3 2 0122 1 3 2 1123 1 3 2 2124 1 3 2 3125 1 3 3 0126 1 3 3 1127 1 3 3 2128 1 3 3 3129 2 0 0 0130 2 0 0 1131 2 0 0 2132 2 0 0 3133 2 0 1 0134 2 0 1 1135 2 0 1 2136 2 0 1 3137 2 0 2 0138 2 0 2 1139 2 0 2 2140 2 0 2 3141 2 0 3 0142 2 0 3 1143 2 0 3 2144 2 0 3 3145 2 1 0 0146 2 1 0 1147 2 1 0 2148 2 1 0 3149 2 1 1 0150 2 1 1 1

Table 1 Continued

119904 119894 = 1 2 3 4151 2 1 1 2152 2 1 1 3153 2 1 2 0154 2 1 2 1155 2 1 2 2156 2 1 2 3157 2 1 3 0158 2 1 3 1159 2 1 3 2160 2 1 3 3161 2 2 0 0162 2 2 0 1163 2 2 0 2164 2 2 0 3165 2 2 1 0166 2 2 1 1167 2 2 1 2168 2 2 1 3169 2 2 2 0170 2 2 2 1171 2 2 2 2172 2 2 2 3173 2 2 3 0174 2 2 3 1175 2 2 3 2176 2 2 3 3177 2 3 0 0178 2 3 0 1179 2 3 0 2180 2 3 0 3181 2 3 1 0182 2 3 1 1183 2 3 1 2184 2 3 1 3185 2 3 2 0186 2 3 2 1187 2 3 2 2188 2 3 2 3189 2 3 3 0190 2 3 3 1191 2 3 3 2192 2 3 3 3193 3 0 0 0194 3 0 0 1195 3 0 0 2196 3 0 0 3197 3 0 1 0198 3 0 1 1199 3 0 1 2200 3 0 1 3201 3 0 2 0202 3 0 2 1

Mathematical Problems in Engineering 11

Table 1 Continued

119904 119894 = 1 2 3 4203 3 0 2 2204 3 0 2 3205 3 0 3 0206 3 0 3 1207 3 0 3 2208 3 0 3 3209 3 1 0 0210 3 1 0 1211 3 1 0 2212 3 1 0 3213 3 1 1 0214 3 1 1 1215 3 1 1 2216 3 1 1 3217 3 1 2 0218 3 1 2 1219 3 1 2 2220 3 1 2 3221 3 1 3 0222 3 1 3 1223 3 1 3 2224 3 1 3 3225 3 2 0 0226 3 2 0 1227 3 2 0 2228 3 2 0 3229 3 2 1 0230 3 2 1 1231 3 2 1 2232 3 2 1 3233 3 2 2 0234 3 2 2 1235 3 2 2 2236 3 2 2 3237 3 2 3 0238 3 2 3 1239 3 2 3 2240 3 2 3 3241 3 3 0 0242 3 3 0 1243 3 3 0 2244 3 3 0 3245 3 3 1 0246 3 3 1 1247 3 3 1 2248 3 3 1 3249 3 3 2 0250 3 3 2 1251 3 3 2 2252 3 3 2 3

Table 1 Continued

119904 119894 = 1 2 3 4253 3 3 3 0254 3 3 3 1255 3 3 3 2256 3 3 3 3

maximum expected service level) 120572 = 05 and 120572 = 1 (ie forthe minimum expected cost) The expected schedules werecomputed using the formulae presented below

(i) Expected schedule of supplies of parts to the pro-ducer

sum119904isin119878

sum119894isin119868119904120590119894=119905

119875119904119861120574119894120582119894119904V119894 119905 isin 119879 (27)

(ii) Expected production schedule

sum119904isin119878

sum119895isin119869

119875119904119887119895119908119904119895119905 119905 isin 119879 (28)

(iii) Expected schedule of shipping of products from theproducer to the distribution centers

sum119904isin119878

sum119896isin119870

sum119895isin119869119896

119875119904119887119895( sum1199051015840isin1198791199051015840lt119905

1199081199041198951199051015840)119909119904119896119905 119905 isin 119879 (29)

As 120572 increases that is the decision-makerrsquos priority shiftsfrom the maximum service level to minimum cost and moreparts are ordered from less reliable and lower cost suppliersthe expected supply schedules and the corresponding pro-duction schedules are more delayed as well as the delivery ofproducts to the customers Note that despite constraint (13)that ensures a feasible schedule with at most one shipmentof products to each distribution center for each disruptionscenario the expected shipping schedule (29) may be splitinto more smaller size shipments (cf Figure 3 where twomajor and one small-size shipments are indicated for eachconfidence level)

For comparison Table 3 presents a subset of non-dominated solutions obtained for the expected value problemEWCS and Figure 4 shows supply production and shippingschedules Unlike the stochastic programming approachwhich accounts for all potential disruption scenarios tooptimize an expected performance of the supply chain thesolution obtained using the deterministic approach is basedon aggregate information on suppliers expected fulfillmentIn general the results are similar for both models and thecorresponding optimal solution values are close to each otherwhich indicates that the expected value problem can be usedin practice when stochastic mixed integer programs are hardto solve The optimal solution values for the expected valueproblem frequently outperform the corresponding solutionvalues of the wait-and-see problem which is in line withthe proposition EV le WS in [27] compare Section 5for example the minimum cost 1198641 for EWCS (see Table 3)

12 Mathematical Problems in Engineering

Table 2 Nondominated solutions for modelWCS

120572 0 01 02 03 04 05 and 06 07 08 and 09 1Var = 38077 Bin = 33076 Cons = 28769 Nonz = 348325(a)

Exp cost 1731 (1198641) 1620 1547 1206 1200 1176 1070 1065 1051 (1198641)Exp service level(b) 8552 (1198642) 8487 8452 7719 7676 7595 5863 5871 3822 (1198642)Exp fulfilled demand(c) 8552 8530 8452 9370 9361 9353 9134 9120 9068

Suppliers selected( of total demand)

1 (76) 1 (35)2 (24) 2 (55) 2 (87) 2 (55) 2 (55) 2 (55) 2 (12) 2 (10)3 (10) 3 (13) 3 (22) 3 (32) 3 (45) 3 (88) 3 (90) 3 (100)4 (23) 4 (13)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119904isin119878 sum119895isin119869 119875119904119887119895119910

119904119895119861 times

100 (c)sum119904isin119878 sum119895isin119869 sum119905isin119879 119875119904119887119895119908119904119895119905119861 times 100

Table 3 Nondominated solutions for model EWCS

120572 0 01 02 and 03 04 and 05 06 and 07 08 09 and 1Var = 162 Bin = 136 Cons = 125 Nonz = 1430(a)

Exp cost 1691 (1198641) 1427 1214 1167 1045 (1198641)Exp service level(b) 87 (1198642) 86 81 79 62 (1198642)Exp fulfilled demand(c) 87 90 91 93 91

Suppliers selected( of total demand)

1 (89) 2 (94) 2 (49) 2 (50) 2 (5)3 (6) 3 (51) 3 (50) 3 (95)4 (11)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119895isin119869 119887119895119884119895119861 times 100(c)sum119895isin119869 sum119905isin119879 119887119895119882119895119905119861 times 100

1 2 3 4 5 6 7Period

Supply schedules

1 2 3 4 5 6 7Period

Production schedules

1 2 3 4 5 6 7Period

Shipping schedules

0

10000

20000

30000

40000

50000

Prod

ucts

0

10000

20000

30000

40000

50000

Prod

ucts

0

20000

40000

60000

80000

100000

Part

s

Alpha = 0Alpha = 05

Alpha = 1 Alpha = 0Alpha = 05

Alpha = 1

Alpha = 0Alpha = 05

Alpha = 1

Figure 4 Schedules for model EWCS

Mathematical Problems in Engineering 13

1045 le 1051 forWCS (see Table 2) or themaximum servicelevel1198642 for EWCS (see Table 3) 8700 ge 8552 forWCS (seeTable 2)

The stochastic wait-and-see approach however leads toa more diversified supply portfolio For the deterministicapproach most of nondominated supply portfolios consistof two suppliers only 119894 = 2 3 while suppliers 119894 = 1 4 areselected only for the maximum service level objective (thatis for 120572 = 0) Comparison of Figures 3 and 4 indicatesthat the expected schedules for model WCS computed asexpectations over all schedules for all potential disruptionscenarios (27)ndash(29) are similar to the corresponding singleschedules determined by model EWCS The main differencesobserved aremore delayed expected production and shippingschedules for model WCS when minimization of cost is con-sidered (that is for120572 = 1) Finally it is interesting to note thatin the multiple sourcing environment considered both thewait-and-see approach and in particular the expected valueapproach frequently select a dual sourcing supply portfoliowith one main supplier and one supporting supplier

The computational experiments were performed usingthe AMPL programming language and the CPLEX 1262solver on a MacBook Pro laptop with Intel Core i7 processorrunning at 28 GHz and with 16GB RAM The solver wascapable of finding proven optimal solution for all exampleswith CPU time ranging from several minutes to several hoursfor the stochastic model WCS and fraction of a second forthe deterministic model EWCS Since the stochastic pro-gramming model WCS needs to determine nondominatedschedules for all potential disruption scenarios while modelEWCS deals with a single scenario only the difference in thecomputational effort required is obvious

7 Conclusions

In this paper two biobjective MIP formulations stochasticand deterministic have been proposed and compared for thecoordinated decision-making in supply chains under partiallocal disruptions and all-or-nothing regional disruptionsTheproblem objective has been to jointly schedule supplies pro-duction and distribution to optimize the trade-off betweenexpected cost and expected service levelWhile the stochasticprogramming approach aims at optimizing the expectedperformance of a supply chain over all possible disruptionscenarios the deterministic approach accounts only for asingle scenario representing average disruption conditionsAs a result the stochastic programming approach determinesnondominated solutions for all disruption scenarios whereasthe deterministic approach produces a single solution onlyIn particular in the stochastic programming approach theselection of supply portfolio is combined with supply chainscheduling for all disruption scenarios considered In con-trast the deterministic expected value approach provides theportfolio along with a single expected schedule of productionand distribution

The expected schedules obtained for the stochastic pro-gramming model WCS as expectations over all schedulesfor all disruption scenarios have been compared with thecorresponding schedules determined by the deterministic

model EWCS based on an expected disruption scenarioThecomparison has demonstrated that the two approaches leadto similar expected solutions

The main findings are in line with other research and arelisted as follows

(i) The two decision-making approaches stochastic anddeterministic lead to similar expected performanceof a supply chain under multilevel disruptions

(ii) The optimal solution values for the expected valueproblem frequently outperform the correspondingsolution values of the wait-and-see problem

(iii) Despite the multiple sourcing environment consid-ered both the wait-and-see approach and in partic-ular the expected value approach frequently selecta dual sourcing supply portfolio with one mainsupplier and one supporting supplier

(iv) The stochastic approach which accounts for allpotential disruption scenarios may lead to a morediversified supply portfolio that will hedge against avariety of scenarios

(v) The expected schedules are more delayed for thestochastic approach

(vi) The service-oriented supply portfolio is more diversi-fied andmay combine both high-cost reliable suppli-ers and low-cost unreliable suppliers while the cost-oriented portfolio depends mainly on low-cost andless reliable suppliers

Overall the results of computational experiments indi-cate that the proposed approach and developed MIP modelsare flexible and efficient tools for coordinated supply chainscheduling The portfolio approach leads to MIP formula-tions with strong LP relaxations and has been proven to becomputationally very efficient CPU time required to findproven optimal solutions for realistic size examples usingcommercially available software for MIP is acceptable for areal-world supply chain disruption management (see Sawik[33])

Since the probability distribution of supply disruptionsfrom each supplier is usually unknown (eg [8]) local multi-level disruptions in the stochastic programming modelWCShave a multinomial discrete distribution while the two-levelregional disruptions have a binomial discrete distributionand all disruption events are independent As part of futureresearch we propose to enhance the stochastic programmingmodel formore general scenarios with finitelymany elementssingled out and all the probability concentrated in them Forexample rather than complete shutdown suppliers within aregion may have correlated disruptions Another importantstream of future research is the study of robustness andsensitivity in relation to input data changes in supply chainscheduling under disruption risks (eg [33]) Future researchcan also focus on the following improvement of the proposedmodels to consider the minimization of time and cost ofrecovery (eg Whitney et al [34])

14 Mathematical Problems in Engineering

Notations

Indices

119894 Supplier 119894 isin 119868119895 Customer 119895 isin 119869119896 Distribution center 119896 isin 119870119897 Disruption level 119897 isin 119871119903 Region 119903 isin 119877119904 Disruption scenario 119904 isin 119878119905 Planning period 119905 isin 119879Input Parameters

119887119895 Size (number of products) of customer order 119895119861 Total demand for partsproducts 119861 = sum119895isin119869 119887119895119862 Capacity of producer119889119895 Due date for customer order 119895119890119894 Fixed cost of ordering parts from supplier 119894119892119895 Per unit penalty cost of delayed customer order 119895ℎ119895 Per unit penalty cost of unfulfilled customer order 119895119868119903 Subset of suppliers in region 119903119869119896 Subset of customers serviced by distribution center 119896119900119894 Per unit price of parts purchased from supplier 119894119901119894119897 Probability of disruption level 119897 for supplier 119894119901119903 Regional disruption probability for region 119903120574119894119897 Fraction of an order delivered by supplier 119894 underdisruption level 119897 (fulfillment rate)Γ119894 Expected fraction of an order delivered by supplier 119894(expected fulfillment rate)120590119894 Delivery lead time from supplier 119894120591119896 Transportation time to distribution center 119896

Conflicts of Interest

The author declares that there are no conflicts of interest

Acknowledgments

This work has been supported by NCN research grant (noDEC-201311BST804458) and by AGH (no 1111200324)

References

[1] J V Blackhurst K P Scheibe and D J Johnson ldquoSupplierrisk assessment and monitoring for the automotive industryrdquoInternational Journal of Physical Distribution amp Logistics Man-agement vol 38 no 2 pp 143ndash165 2008

[2] P Hoffmann H Schiele and K Krabbendam ldquoUncertaintysupply risk management and their impact on performancerdquoJournal of Purchasing amp Supply Management vol 19 no 3 pp199ndash211 2013

[3] Y Park P Hong and J J Roh ldquoSupply chain lessons from thecatastrophic natural disaster in Japanrdquo Business Horizons vol56 no 1 pp 75ndash85 2013

[4] M Haraguchi and U Lall ldquoFlood risks and impacts A casestudy of Thailandrsquos floods in 2011 and research questions forsupply chain decisionmakingrdquo International Journal of DisasterRisk Reduction vol 14 pp 256ndash272 2015

[5] T Fujimoto and Y W Park ldquoBalancing supply chain com-petitiveness and robustness through ldquovirtual dual sourcingrdquoLessons from the Great East Japan Earthquakerdquo InternationalJournal of Production Economics vol 147 pp 429ndash436 2014

[6] HMatsuo ldquoImplications of the Tohoku earthquake for Toyotarsquoscoordination mechanism supply chain disruption of auto-motive semiconductorsrdquo International Journal of ProductionEconomics vol 161 pp 217ndash227 2015

[7] J R Marszewska ldquoImplications of seismic hazard in Japan ontoyota supply chain disruption risksrdquo in Proceedings of the 13thInternational Conference on Industrial Logistics (ICIL rsquo16) pp178ndash185 Zakopane Poland October 2016

[8] A Z Zeng and Y Xia ldquoBuilding a mutually beneficial partner-ship to ensure backup supplyrdquo Omega vol 52 pp 77ndash91 2015

[9] S S Erenguc N C Simpson and A J Vakharia ldquoIntegratedproductiondistribution planning in supply chains an invitedreviewrdquo European Journal of Operational Research vol 115 no2 pp 219ndash236 1999

[10] L Lei S Liu A Ruszczynski and S Park ldquoOn the integratedproduction inventory and distribution routing problemrdquo IIETransactions vol 38 no 11 pp 955ndash970 2006

[11] A Kaur A Kanda and S G Deshmukh ldquoA graph theoreticapproach for supply chain coordinationrdquo International Journalof Logistics Systems and Management vol 2 no 4 pp 321ndash3412006

[12] T ChoiW Yeung andT C E Cheng ldquoScheduling and co-ordi-nation of multi-suppliers single-warehouse-operator single-manufacturer supply chains with variable production rates andstorage costsrdquo International Journal of Production Research vol51 no 9 pp 2593ndash2601 2013

[13] S Liu and L G Papageorgiou ldquoMultiobjective optimisation ofproduction distribution and capacity planning of global supplychains in the process industryrdquo Omega vol 41 no 2 pp 369ndash382 2013

[14] Z L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[15] T Sawik ldquoCoordinated supply chain schedulingrdquo InternationalJournal of Production Economics vol 120 no 2 pp 437ndash4512009

[16] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[17] D Ivanov B Sokolov and A Dolgui ldquoThe Ripple effect in sup-ply chains Trade-off rsquoefficiency-flexibility- resiliencersquo in disrup-tionmanagementrdquo International Journal of ProductionResearchvol 52 no 7 pp 2154ndash2172 2014

[18] D Ivanov A Pavlov and B Sokolov ldquoOptimal distribution(re)planning in a centralized multi-stage supply network underconditions of the ripple effect and structure dynamicsrdquo Euro-pean Journal of Operational Research vol 237 no 2 pp 758ndash770 2014

[19] D Ivanov A Pavlov A Dolgui D Pavlov and B SokolovldquoDisruption-driven supply chain (re)-planning and perfor-mance impact assessment with consideration of pro-active andrecovery policiesrdquo Transportation Research Part E Logistics andTransportation Review vol 90 pp 7ndash24 2016

[20] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

Mathematical Problems in Engineering 15

[21] A J Schmitt S A Sun L V Snyder and Z-J M Shen ldquoCen-tralization versus decentralization Risk pooling risk diversifi-cation and supply chain disruptionsrdquo Omega vol 52 pp 201ndash212 2015

[22] L V Snyder Z Atan P Peng Y Rong A J Schmitt andB Sinsoysal ldquoORMS models for supply chain disruptions areviewrdquo IIE Transactions vol 48 no 2 pp 89ndash109 2016

[23] T Sawik ldquoIntegrated selection of suppliers and scheduling ofcustomer orders in the presence of supply chain disruptionrisksrdquo International Journal of Production Research vol 51 no23-24 pp 7006ndash7022 2013

[24] T Sawik ldquoOptimization of cost and service level in the presenceof supply chain disruption risks Single vs multiple sourcingrdquoComputers amp Operations Research vol 51 pp 11ndash20 2014

[25] T Sawik ldquoIntegrated supply production and distributionscheduling under disruption risksrdquoOmega vol 62 pp 131ndash1442016

[26] T Sawik ldquoIntegrated supply chain scheduling under multi-leveldisruptionsrdquo IFAC-PapersOnLine vol 28 no 3 pp 1515ndash15202015

[27] P Kall and J Mayer Stochastic Linear Programming ModelsTheory and Computation vol 156 of International Series inOperations Research amp Management Science Springer NewYork 2011

[28] I N Durbach and T J Stewart ldquoUsing expected values to sim-plify decision making under uncertaintyrdquo Omega vol 37 no 2pp 312ndash330 2009

[29] F Maggioni and S W Wallace ldquoAnalyzing the quality of theexpected value solution in stochastic programmingrdquo Annals ofOperations Research vol 200 pp 37ndash54 2012

[30] T Sawik ldquoSelection of supply portfolio under disruption risksrdquoOmega vol 39 no 2 pp 194ndash208 2011

[31] E Delage S Arroyo and Y Ye ldquoThe value of stochastic mod-eling in two-stage stochastic programs with cost uncertaintyrdquoOperations Research vol 62 no 6 pp 1377ndash1393 2014

[32] T Sawik Scheduling in Supply Chains Using Mixed Integer Pro-gramming John Wiley amp Sons Inc Hoboken NJ 2011

[33] T Sawik Supply Chain DisruptionManagement Using StochasticMixed Integer Programming vol 256 of International Seriesin Operations Research amp Management Science Springer NewYork 2017 httpwwwspringercomgpbook9783319588223

[34] D E Whitney J Luo and D A Heller ldquoThe benefits and con-straints of temporary sourcing diversification in supply chaindisruption and recoveryrdquo Journal of Purchasing and SupplyManagement vol 20 no 4 pp 238ndash250 2014

Mathematical Problems in Engineering 11

Table 1 Continued

119904 119894 = 1 2 3 4203 3 0 2 2204 3 0 2 3205 3 0 3 0206 3 0 3 1207 3 0 3 2208 3 0 3 3209 3 1 0 0210 3 1 0 1211 3 1 0 2212 3 1 0 3213 3 1 1 0214 3 1 1 1215 3 1 1 2216 3 1 1 3217 3 1 2 0218 3 1 2 1219 3 1 2 2220 3 1 2 3221 3 1 3 0222 3 1 3 1223 3 1 3 2224 3 1 3 3225 3 2 0 0226 3 2 0 1227 3 2 0 2228 3 2 0 3229 3 2 1 0230 3 2 1 1231 3 2 1 2232 3 2 1 3233 3 2 2 0234 3 2 2 1235 3 2 2 2236 3 2 2 3237 3 2 3 0238 3 2 3 1239 3 2 3 2240 3 2 3 3241 3 3 0 0242 3 3 0 1243 3 3 0 2244 3 3 0 3245 3 3 1 0246 3 3 1 1247 3 3 1 2248 3 3 1 3249 3 3 2 0250 3 3 2 1251 3 3 2 2252 3 3 2 3

Table 1 Continued

119904 119894 = 1 2 3 4253 3 3 3 0254 3 3 3 1255 3 3 3 2256 3 3 3 3

maximum expected service level) 120572 = 05 and 120572 = 1 (ie forthe minimum expected cost) The expected schedules werecomputed using the formulae presented below

(i) Expected schedule of supplies of parts to the pro-ducer

sum119904isin119878

sum119894isin119868119904120590119894=119905

119875119904119861120574119894120582119894119904V119894 119905 isin 119879 (27)

(ii) Expected production schedule

sum119904isin119878

sum119895isin119869

119875119904119887119895119908119904119895119905 119905 isin 119879 (28)

(iii) Expected schedule of shipping of products from theproducer to the distribution centers

sum119904isin119878

sum119896isin119870

sum119895isin119869119896

119875119904119887119895( sum1199051015840isin1198791199051015840lt119905

1199081199041198951199051015840)119909119904119896119905 119905 isin 119879 (29)

As 120572 increases that is the decision-makerrsquos priority shiftsfrom the maximum service level to minimum cost and moreparts are ordered from less reliable and lower cost suppliersthe expected supply schedules and the corresponding pro-duction schedules are more delayed as well as the delivery ofproducts to the customers Note that despite constraint (13)that ensures a feasible schedule with at most one shipmentof products to each distribution center for each disruptionscenario the expected shipping schedule (29) may be splitinto more smaller size shipments (cf Figure 3 where twomajor and one small-size shipments are indicated for eachconfidence level)

For comparison Table 3 presents a subset of non-dominated solutions obtained for the expected value problemEWCS and Figure 4 shows supply production and shippingschedules Unlike the stochastic programming approachwhich accounts for all potential disruption scenarios tooptimize an expected performance of the supply chain thesolution obtained using the deterministic approach is basedon aggregate information on suppliers expected fulfillmentIn general the results are similar for both models and thecorresponding optimal solution values are close to each otherwhich indicates that the expected value problem can be usedin practice when stochastic mixed integer programs are hardto solve The optimal solution values for the expected valueproblem frequently outperform the corresponding solutionvalues of the wait-and-see problem which is in line withthe proposition EV le WS in [27] compare Section 5for example the minimum cost 1198641 for EWCS (see Table 3)

12 Mathematical Problems in Engineering

Table 2 Nondominated solutions for modelWCS

120572 0 01 02 03 04 05 and 06 07 08 and 09 1Var = 38077 Bin = 33076 Cons = 28769 Nonz = 348325(a)

Exp cost 1731 (1198641) 1620 1547 1206 1200 1176 1070 1065 1051 (1198641)Exp service level(b) 8552 (1198642) 8487 8452 7719 7676 7595 5863 5871 3822 (1198642)Exp fulfilled demand(c) 8552 8530 8452 9370 9361 9353 9134 9120 9068

Suppliers selected( of total demand)

1 (76) 1 (35)2 (24) 2 (55) 2 (87) 2 (55) 2 (55) 2 (55) 2 (12) 2 (10)3 (10) 3 (13) 3 (22) 3 (32) 3 (45) 3 (88) 3 (90) 3 (100)4 (23) 4 (13)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119904isin119878 sum119895isin119869 119875119904119887119895119910

119904119895119861 times

100 (c)sum119904isin119878 sum119895isin119869 sum119905isin119879 119875119904119887119895119908119904119895119905119861 times 100

Table 3 Nondominated solutions for model EWCS

120572 0 01 02 and 03 04 and 05 06 and 07 08 09 and 1Var = 162 Bin = 136 Cons = 125 Nonz = 1430(a)

Exp cost 1691 (1198641) 1427 1214 1167 1045 (1198641)Exp service level(b) 87 (1198642) 86 81 79 62 (1198642)Exp fulfilled demand(c) 87 90 91 93 91

Suppliers selected( of total demand)

1 (89) 2 (94) 2 (49) 2 (50) 2 (5)3 (6) 3 (51) 3 (50) 3 (95)4 (11)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119895isin119869 119887119895119884119895119861 times 100(c)sum119895isin119869 sum119905isin119879 119887119895119882119895119905119861 times 100

1 2 3 4 5 6 7Period

Supply schedules

1 2 3 4 5 6 7Period

Production schedules

1 2 3 4 5 6 7Period

Shipping schedules

0

10000

20000

30000

40000

50000

Prod

ucts

0

10000

20000

30000

40000

50000

Prod

ucts

0

20000

40000

60000

80000

100000

Part

s

Alpha = 0Alpha = 05

Alpha = 1 Alpha = 0Alpha = 05

Alpha = 1

Alpha = 0Alpha = 05

Alpha = 1

Figure 4 Schedules for model EWCS

Mathematical Problems in Engineering 13

1045 le 1051 forWCS (see Table 2) or themaximum servicelevel1198642 for EWCS (see Table 3) 8700 ge 8552 forWCS (seeTable 2)

The stochastic wait-and-see approach however leads toa more diversified supply portfolio For the deterministicapproach most of nondominated supply portfolios consistof two suppliers only 119894 = 2 3 while suppliers 119894 = 1 4 areselected only for the maximum service level objective (thatis for 120572 = 0) Comparison of Figures 3 and 4 indicatesthat the expected schedules for model WCS computed asexpectations over all schedules for all potential disruptionscenarios (27)ndash(29) are similar to the corresponding singleschedules determined by model EWCS The main differencesobserved aremore delayed expected production and shippingschedules for model WCS when minimization of cost is con-sidered (that is for120572 = 1) Finally it is interesting to note thatin the multiple sourcing environment considered both thewait-and-see approach and in particular the expected valueapproach frequently select a dual sourcing supply portfoliowith one main supplier and one supporting supplier

The computational experiments were performed usingthe AMPL programming language and the CPLEX 1262solver on a MacBook Pro laptop with Intel Core i7 processorrunning at 28 GHz and with 16GB RAM The solver wascapable of finding proven optimal solution for all exampleswith CPU time ranging from several minutes to several hoursfor the stochastic model WCS and fraction of a second forthe deterministic model EWCS Since the stochastic pro-gramming model WCS needs to determine nondominatedschedules for all potential disruption scenarios while modelEWCS deals with a single scenario only the difference in thecomputational effort required is obvious

7 Conclusions

In this paper two biobjective MIP formulations stochasticand deterministic have been proposed and compared for thecoordinated decision-making in supply chains under partiallocal disruptions and all-or-nothing regional disruptionsTheproblem objective has been to jointly schedule supplies pro-duction and distribution to optimize the trade-off betweenexpected cost and expected service levelWhile the stochasticprogramming approach aims at optimizing the expectedperformance of a supply chain over all possible disruptionscenarios the deterministic approach accounts only for asingle scenario representing average disruption conditionsAs a result the stochastic programming approach determinesnondominated solutions for all disruption scenarios whereasthe deterministic approach produces a single solution onlyIn particular in the stochastic programming approach theselection of supply portfolio is combined with supply chainscheduling for all disruption scenarios considered In con-trast the deterministic expected value approach provides theportfolio along with a single expected schedule of productionand distribution

The expected schedules obtained for the stochastic pro-gramming model WCS as expectations over all schedulesfor all disruption scenarios have been compared with thecorresponding schedules determined by the deterministic

model EWCS based on an expected disruption scenarioThecomparison has demonstrated that the two approaches leadto similar expected solutions

The main findings are in line with other research and arelisted as follows

(i) The two decision-making approaches stochastic anddeterministic lead to similar expected performanceof a supply chain under multilevel disruptions

(ii) The optimal solution values for the expected valueproblem frequently outperform the correspondingsolution values of the wait-and-see problem

(iii) Despite the multiple sourcing environment consid-ered both the wait-and-see approach and in partic-ular the expected value approach frequently selecta dual sourcing supply portfolio with one mainsupplier and one supporting supplier

(iv) The stochastic approach which accounts for allpotential disruption scenarios may lead to a morediversified supply portfolio that will hedge against avariety of scenarios

(v) The expected schedules are more delayed for thestochastic approach

(vi) The service-oriented supply portfolio is more diversi-fied andmay combine both high-cost reliable suppli-ers and low-cost unreliable suppliers while the cost-oriented portfolio depends mainly on low-cost andless reliable suppliers

Overall the results of computational experiments indi-cate that the proposed approach and developed MIP modelsare flexible and efficient tools for coordinated supply chainscheduling The portfolio approach leads to MIP formula-tions with strong LP relaxations and has been proven to becomputationally very efficient CPU time required to findproven optimal solutions for realistic size examples usingcommercially available software for MIP is acceptable for areal-world supply chain disruption management (see Sawik[33])

Since the probability distribution of supply disruptionsfrom each supplier is usually unknown (eg [8]) local multi-level disruptions in the stochastic programming modelWCShave a multinomial discrete distribution while the two-levelregional disruptions have a binomial discrete distributionand all disruption events are independent As part of futureresearch we propose to enhance the stochastic programmingmodel formore general scenarios with finitelymany elementssingled out and all the probability concentrated in them Forexample rather than complete shutdown suppliers within aregion may have correlated disruptions Another importantstream of future research is the study of robustness andsensitivity in relation to input data changes in supply chainscheduling under disruption risks (eg [33]) Future researchcan also focus on the following improvement of the proposedmodels to consider the minimization of time and cost ofrecovery (eg Whitney et al [34])

14 Mathematical Problems in Engineering

Notations

Indices

119894 Supplier 119894 isin 119868119895 Customer 119895 isin 119869119896 Distribution center 119896 isin 119870119897 Disruption level 119897 isin 119871119903 Region 119903 isin 119877119904 Disruption scenario 119904 isin 119878119905 Planning period 119905 isin 119879Input Parameters

119887119895 Size (number of products) of customer order 119895119861 Total demand for partsproducts 119861 = sum119895isin119869 119887119895119862 Capacity of producer119889119895 Due date for customer order 119895119890119894 Fixed cost of ordering parts from supplier 119894119892119895 Per unit penalty cost of delayed customer order 119895ℎ119895 Per unit penalty cost of unfulfilled customer order 119895119868119903 Subset of suppliers in region 119903119869119896 Subset of customers serviced by distribution center 119896119900119894 Per unit price of parts purchased from supplier 119894119901119894119897 Probability of disruption level 119897 for supplier 119894119901119903 Regional disruption probability for region 119903120574119894119897 Fraction of an order delivered by supplier 119894 underdisruption level 119897 (fulfillment rate)Γ119894 Expected fraction of an order delivered by supplier 119894(expected fulfillment rate)120590119894 Delivery lead time from supplier 119894120591119896 Transportation time to distribution center 119896

Conflicts of Interest

The author declares that there are no conflicts of interest

Acknowledgments

This work has been supported by NCN research grant (noDEC-201311BST804458) and by AGH (no 1111200324)

References

[1] J V Blackhurst K P Scheibe and D J Johnson ldquoSupplierrisk assessment and monitoring for the automotive industryrdquoInternational Journal of Physical Distribution amp Logistics Man-agement vol 38 no 2 pp 143ndash165 2008

[2] P Hoffmann H Schiele and K Krabbendam ldquoUncertaintysupply risk management and their impact on performancerdquoJournal of Purchasing amp Supply Management vol 19 no 3 pp199ndash211 2013

[3] Y Park P Hong and J J Roh ldquoSupply chain lessons from thecatastrophic natural disaster in Japanrdquo Business Horizons vol56 no 1 pp 75ndash85 2013

[4] M Haraguchi and U Lall ldquoFlood risks and impacts A casestudy of Thailandrsquos floods in 2011 and research questions forsupply chain decisionmakingrdquo International Journal of DisasterRisk Reduction vol 14 pp 256ndash272 2015

[5] T Fujimoto and Y W Park ldquoBalancing supply chain com-petitiveness and robustness through ldquovirtual dual sourcingrdquoLessons from the Great East Japan Earthquakerdquo InternationalJournal of Production Economics vol 147 pp 429ndash436 2014

[6] HMatsuo ldquoImplications of the Tohoku earthquake for Toyotarsquoscoordination mechanism supply chain disruption of auto-motive semiconductorsrdquo International Journal of ProductionEconomics vol 161 pp 217ndash227 2015

[7] J R Marszewska ldquoImplications of seismic hazard in Japan ontoyota supply chain disruption risksrdquo in Proceedings of the 13thInternational Conference on Industrial Logistics (ICIL rsquo16) pp178ndash185 Zakopane Poland October 2016

[8] A Z Zeng and Y Xia ldquoBuilding a mutually beneficial partner-ship to ensure backup supplyrdquo Omega vol 52 pp 77ndash91 2015

[9] S S Erenguc N C Simpson and A J Vakharia ldquoIntegratedproductiondistribution planning in supply chains an invitedreviewrdquo European Journal of Operational Research vol 115 no2 pp 219ndash236 1999

[10] L Lei S Liu A Ruszczynski and S Park ldquoOn the integratedproduction inventory and distribution routing problemrdquo IIETransactions vol 38 no 11 pp 955ndash970 2006

[11] A Kaur A Kanda and S G Deshmukh ldquoA graph theoreticapproach for supply chain coordinationrdquo International Journalof Logistics Systems and Management vol 2 no 4 pp 321ndash3412006

[12] T ChoiW Yeung andT C E Cheng ldquoScheduling and co-ordi-nation of multi-suppliers single-warehouse-operator single-manufacturer supply chains with variable production rates andstorage costsrdquo International Journal of Production Research vol51 no 9 pp 2593ndash2601 2013

[13] S Liu and L G Papageorgiou ldquoMultiobjective optimisation ofproduction distribution and capacity planning of global supplychains in the process industryrdquo Omega vol 41 no 2 pp 369ndash382 2013

[14] Z L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[15] T Sawik ldquoCoordinated supply chain schedulingrdquo InternationalJournal of Production Economics vol 120 no 2 pp 437ndash4512009

[16] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[17] D Ivanov B Sokolov and A Dolgui ldquoThe Ripple effect in sup-ply chains Trade-off rsquoefficiency-flexibility- resiliencersquo in disrup-tionmanagementrdquo International Journal of ProductionResearchvol 52 no 7 pp 2154ndash2172 2014

[18] D Ivanov A Pavlov and B Sokolov ldquoOptimal distribution(re)planning in a centralized multi-stage supply network underconditions of the ripple effect and structure dynamicsrdquo Euro-pean Journal of Operational Research vol 237 no 2 pp 758ndash770 2014

[19] D Ivanov A Pavlov A Dolgui D Pavlov and B SokolovldquoDisruption-driven supply chain (re)-planning and perfor-mance impact assessment with consideration of pro-active andrecovery policiesrdquo Transportation Research Part E Logistics andTransportation Review vol 90 pp 7ndash24 2016

[20] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

Mathematical Problems in Engineering 15

[21] A J Schmitt S A Sun L V Snyder and Z-J M Shen ldquoCen-tralization versus decentralization Risk pooling risk diversifi-cation and supply chain disruptionsrdquo Omega vol 52 pp 201ndash212 2015

[22] L V Snyder Z Atan P Peng Y Rong A J Schmitt andB Sinsoysal ldquoORMS models for supply chain disruptions areviewrdquo IIE Transactions vol 48 no 2 pp 89ndash109 2016

[23] T Sawik ldquoIntegrated selection of suppliers and scheduling ofcustomer orders in the presence of supply chain disruptionrisksrdquo International Journal of Production Research vol 51 no23-24 pp 7006ndash7022 2013

[24] T Sawik ldquoOptimization of cost and service level in the presenceof supply chain disruption risks Single vs multiple sourcingrdquoComputers amp Operations Research vol 51 pp 11ndash20 2014

[25] T Sawik ldquoIntegrated supply production and distributionscheduling under disruption risksrdquoOmega vol 62 pp 131ndash1442016

[26] T Sawik ldquoIntegrated supply chain scheduling under multi-leveldisruptionsrdquo IFAC-PapersOnLine vol 28 no 3 pp 1515ndash15202015

[27] P Kall and J Mayer Stochastic Linear Programming ModelsTheory and Computation vol 156 of International Series inOperations Research amp Management Science Springer NewYork 2011

[28] I N Durbach and T J Stewart ldquoUsing expected values to sim-plify decision making under uncertaintyrdquo Omega vol 37 no 2pp 312ndash330 2009

[29] F Maggioni and S W Wallace ldquoAnalyzing the quality of theexpected value solution in stochastic programmingrdquo Annals ofOperations Research vol 200 pp 37ndash54 2012

[30] T Sawik ldquoSelection of supply portfolio under disruption risksrdquoOmega vol 39 no 2 pp 194ndash208 2011

[31] E Delage S Arroyo and Y Ye ldquoThe value of stochastic mod-eling in two-stage stochastic programs with cost uncertaintyrdquoOperations Research vol 62 no 6 pp 1377ndash1393 2014

[32] T Sawik Scheduling in Supply Chains Using Mixed Integer Pro-gramming John Wiley amp Sons Inc Hoboken NJ 2011

[33] T Sawik Supply Chain DisruptionManagement Using StochasticMixed Integer Programming vol 256 of International Seriesin Operations Research amp Management Science Springer NewYork 2017 httpwwwspringercomgpbook9783319588223

[34] D E Whitney J Luo and D A Heller ldquoThe benefits and con-straints of temporary sourcing diversification in supply chaindisruption and recoveryrdquo Journal of Purchasing and SupplyManagement vol 20 no 4 pp 238ndash250 2014

12 Mathematical Problems in Engineering

Table 2 Nondominated solutions for modelWCS

120572 0 01 02 03 04 05 and 06 07 08 and 09 1Var = 38077 Bin = 33076 Cons = 28769 Nonz = 348325(a)

Exp cost 1731 (1198641) 1620 1547 1206 1200 1176 1070 1065 1051 (1198641)Exp service level(b) 8552 (1198642) 8487 8452 7719 7676 7595 5863 5871 3822 (1198642)Exp fulfilled demand(c) 8552 8530 8452 9370 9361 9353 9134 9120 9068

Suppliers selected( of total demand)

1 (76) 1 (35)2 (24) 2 (55) 2 (87) 2 (55) 2 (55) 2 (55) 2 (12) 2 (10)3 (10) 3 (13) 3 (22) 3 (32) 3 (45) 3 (88) 3 (90) 3 (100)4 (23) 4 (13)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119904isin119878 sum119895isin119869 119875119904119887119895119910

119904119895119861 times

100 (c)sum119904isin119878 sum119895isin119869 sum119905isin119879 119875119904119887119895119908119904119895119905119861 times 100

Table 3 Nondominated solutions for model EWCS

120572 0 01 02 and 03 04 and 05 06 and 07 08 09 and 1Var = 162 Bin = 136 Cons = 125 Nonz = 1430(a)

Exp cost 1691 (1198641) 1427 1214 1167 1045 (1198641)Exp service level(b) 87 (1198642) 86 81 79 62 (1198642)Exp fulfilled demand(c) 87 90 91 93 91

Suppliers selected( of total demand)

1 (89) 2 (94) 2 (49) 2 (50) 2 (5)3 (6) 3 (51) 3 (50) 3 (95)4 (11)(a)Var number of variables Bin number of binary variables Cons number of constraints Nonz number of nonzero coefficients (b)sum119895isin119869 119887119895119884119895119861 times 100(c)sum119895isin119869 sum119905isin119879 119887119895119882119895119905119861 times 100

1 2 3 4 5 6 7Period

Supply schedules

1 2 3 4 5 6 7Period

Production schedules

1 2 3 4 5 6 7Period

Shipping schedules

0

10000

20000

30000

40000

50000

Prod

ucts

0

10000

20000

30000

40000

50000

Prod

ucts

0

20000

40000

60000

80000

100000

Part

s

Alpha = 0Alpha = 05

Alpha = 1 Alpha = 0Alpha = 05

Alpha = 1

Alpha = 0Alpha = 05

Alpha = 1

Figure 4 Schedules for model EWCS

Mathematical Problems in Engineering 13

1045 le 1051 forWCS (see Table 2) or themaximum servicelevel1198642 for EWCS (see Table 3) 8700 ge 8552 forWCS (seeTable 2)

The stochastic wait-and-see approach however leads toa more diversified supply portfolio For the deterministicapproach most of nondominated supply portfolios consistof two suppliers only 119894 = 2 3 while suppliers 119894 = 1 4 areselected only for the maximum service level objective (thatis for 120572 = 0) Comparison of Figures 3 and 4 indicatesthat the expected schedules for model WCS computed asexpectations over all schedules for all potential disruptionscenarios (27)ndash(29) are similar to the corresponding singleschedules determined by model EWCS The main differencesobserved aremore delayed expected production and shippingschedules for model WCS when minimization of cost is con-sidered (that is for120572 = 1) Finally it is interesting to note thatin the multiple sourcing environment considered both thewait-and-see approach and in particular the expected valueapproach frequently select a dual sourcing supply portfoliowith one main supplier and one supporting supplier

The computational experiments were performed usingthe AMPL programming language and the CPLEX 1262solver on a MacBook Pro laptop with Intel Core i7 processorrunning at 28 GHz and with 16GB RAM The solver wascapable of finding proven optimal solution for all exampleswith CPU time ranging from several minutes to several hoursfor the stochastic model WCS and fraction of a second forthe deterministic model EWCS Since the stochastic pro-gramming model WCS needs to determine nondominatedschedules for all potential disruption scenarios while modelEWCS deals with a single scenario only the difference in thecomputational effort required is obvious

7 Conclusions

In this paper two biobjective MIP formulations stochasticand deterministic have been proposed and compared for thecoordinated decision-making in supply chains under partiallocal disruptions and all-or-nothing regional disruptionsTheproblem objective has been to jointly schedule supplies pro-duction and distribution to optimize the trade-off betweenexpected cost and expected service levelWhile the stochasticprogramming approach aims at optimizing the expectedperformance of a supply chain over all possible disruptionscenarios the deterministic approach accounts only for asingle scenario representing average disruption conditionsAs a result the stochastic programming approach determinesnondominated solutions for all disruption scenarios whereasthe deterministic approach produces a single solution onlyIn particular in the stochastic programming approach theselection of supply portfolio is combined with supply chainscheduling for all disruption scenarios considered In con-trast the deterministic expected value approach provides theportfolio along with a single expected schedule of productionand distribution

The expected schedules obtained for the stochastic pro-gramming model WCS as expectations over all schedulesfor all disruption scenarios have been compared with thecorresponding schedules determined by the deterministic

model EWCS based on an expected disruption scenarioThecomparison has demonstrated that the two approaches leadto similar expected solutions

The main findings are in line with other research and arelisted as follows

(i) The two decision-making approaches stochastic anddeterministic lead to similar expected performanceof a supply chain under multilevel disruptions

(ii) The optimal solution values for the expected valueproblem frequently outperform the correspondingsolution values of the wait-and-see problem

(iii) Despite the multiple sourcing environment consid-ered both the wait-and-see approach and in partic-ular the expected value approach frequently selecta dual sourcing supply portfolio with one mainsupplier and one supporting supplier

(iv) The stochastic approach which accounts for allpotential disruption scenarios may lead to a morediversified supply portfolio that will hedge against avariety of scenarios

(v) The expected schedules are more delayed for thestochastic approach

(vi) The service-oriented supply portfolio is more diversi-fied andmay combine both high-cost reliable suppli-ers and low-cost unreliable suppliers while the cost-oriented portfolio depends mainly on low-cost andless reliable suppliers

Overall the results of computational experiments indi-cate that the proposed approach and developed MIP modelsare flexible and efficient tools for coordinated supply chainscheduling The portfolio approach leads to MIP formula-tions with strong LP relaxations and has been proven to becomputationally very efficient CPU time required to findproven optimal solutions for realistic size examples usingcommercially available software for MIP is acceptable for areal-world supply chain disruption management (see Sawik[33])

Since the probability distribution of supply disruptionsfrom each supplier is usually unknown (eg [8]) local multi-level disruptions in the stochastic programming modelWCShave a multinomial discrete distribution while the two-levelregional disruptions have a binomial discrete distributionand all disruption events are independent As part of futureresearch we propose to enhance the stochastic programmingmodel formore general scenarios with finitelymany elementssingled out and all the probability concentrated in them Forexample rather than complete shutdown suppliers within aregion may have correlated disruptions Another importantstream of future research is the study of robustness andsensitivity in relation to input data changes in supply chainscheduling under disruption risks (eg [33]) Future researchcan also focus on the following improvement of the proposedmodels to consider the minimization of time and cost ofrecovery (eg Whitney et al [34])

14 Mathematical Problems in Engineering

Notations

Indices

119894 Supplier 119894 isin 119868119895 Customer 119895 isin 119869119896 Distribution center 119896 isin 119870119897 Disruption level 119897 isin 119871119903 Region 119903 isin 119877119904 Disruption scenario 119904 isin 119878119905 Planning period 119905 isin 119879Input Parameters

119887119895 Size (number of products) of customer order 119895119861 Total demand for partsproducts 119861 = sum119895isin119869 119887119895119862 Capacity of producer119889119895 Due date for customer order 119895119890119894 Fixed cost of ordering parts from supplier 119894119892119895 Per unit penalty cost of delayed customer order 119895ℎ119895 Per unit penalty cost of unfulfilled customer order 119895119868119903 Subset of suppliers in region 119903119869119896 Subset of customers serviced by distribution center 119896119900119894 Per unit price of parts purchased from supplier 119894119901119894119897 Probability of disruption level 119897 for supplier 119894119901119903 Regional disruption probability for region 119903120574119894119897 Fraction of an order delivered by supplier 119894 underdisruption level 119897 (fulfillment rate)Γ119894 Expected fraction of an order delivered by supplier 119894(expected fulfillment rate)120590119894 Delivery lead time from supplier 119894120591119896 Transportation time to distribution center 119896

Conflicts of Interest

The author declares that there are no conflicts of interest

Acknowledgments

This work has been supported by NCN research grant (noDEC-201311BST804458) and by AGH (no 1111200324)

References

[1] J V Blackhurst K P Scheibe and D J Johnson ldquoSupplierrisk assessment and monitoring for the automotive industryrdquoInternational Journal of Physical Distribution amp Logistics Man-agement vol 38 no 2 pp 143ndash165 2008

[2] P Hoffmann H Schiele and K Krabbendam ldquoUncertaintysupply risk management and their impact on performancerdquoJournal of Purchasing amp Supply Management vol 19 no 3 pp199ndash211 2013

[3] Y Park P Hong and J J Roh ldquoSupply chain lessons from thecatastrophic natural disaster in Japanrdquo Business Horizons vol56 no 1 pp 75ndash85 2013

[4] M Haraguchi and U Lall ldquoFlood risks and impacts A casestudy of Thailandrsquos floods in 2011 and research questions forsupply chain decisionmakingrdquo International Journal of DisasterRisk Reduction vol 14 pp 256ndash272 2015

[5] T Fujimoto and Y W Park ldquoBalancing supply chain com-petitiveness and robustness through ldquovirtual dual sourcingrdquoLessons from the Great East Japan Earthquakerdquo InternationalJournal of Production Economics vol 147 pp 429ndash436 2014

[6] HMatsuo ldquoImplications of the Tohoku earthquake for Toyotarsquoscoordination mechanism supply chain disruption of auto-motive semiconductorsrdquo International Journal of ProductionEconomics vol 161 pp 217ndash227 2015

[7] J R Marszewska ldquoImplications of seismic hazard in Japan ontoyota supply chain disruption risksrdquo in Proceedings of the 13thInternational Conference on Industrial Logistics (ICIL rsquo16) pp178ndash185 Zakopane Poland October 2016

[8] A Z Zeng and Y Xia ldquoBuilding a mutually beneficial partner-ship to ensure backup supplyrdquo Omega vol 52 pp 77ndash91 2015

[9] S S Erenguc N C Simpson and A J Vakharia ldquoIntegratedproductiondistribution planning in supply chains an invitedreviewrdquo European Journal of Operational Research vol 115 no2 pp 219ndash236 1999

[10] L Lei S Liu A Ruszczynski and S Park ldquoOn the integratedproduction inventory and distribution routing problemrdquo IIETransactions vol 38 no 11 pp 955ndash970 2006

[11] A Kaur A Kanda and S G Deshmukh ldquoA graph theoreticapproach for supply chain coordinationrdquo International Journalof Logistics Systems and Management vol 2 no 4 pp 321ndash3412006

[12] T ChoiW Yeung andT C E Cheng ldquoScheduling and co-ordi-nation of multi-suppliers single-warehouse-operator single-manufacturer supply chains with variable production rates andstorage costsrdquo International Journal of Production Research vol51 no 9 pp 2593ndash2601 2013

[13] S Liu and L G Papageorgiou ldquoMultiobjective optimisation ofproduction distribution and capacity planning of global supplychains in the process industryrdquo Omega vol 41 no 2 pp 369ndash382 2013

[14] Z L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[15] T Sawik ldquoCoordinated supply chain schedulingrdquo InternationalJournal of Production Economics vol 120 no 2 pp 437ndash4512009

[16] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[17] D Ivanov B Sokolov and A Dolgui ldquoThe Ripple effect in sup-ply chains Trade-off rsquoefficiency-flexibility- resiliencersquo in disrup-tionmanagementrdquo International Journal of ProductionResearchvol 52 no 7 pp 2154ndash2172 2014

[18] D Ivanov A Pavlov and B Sokolov ldquoOptimal distribution(re)planning in a centralized multi-stage supply network underconditions of the ripple effect and structure dynamicsrdquo Euro-pean Journal of Operational Research vol 237 no 2 pp 758ndash770 2014

[19] D Ivanov A Pavlov A Dolgui D Pavlov and B SokolovldquoDisruption-driven supply chain (re)-planning and perfor-mance impact assessment with consideration of pro-active andrecovery policiesrdquo Transportation Research Part E Logistics andTransportation Review vol 90 pp 7ndash24 2016

[20] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

Mathematical Problems in Engineering 15

[21] A J Schmitt S A Sun L V Snyder and Z-J M Shen ldquoCen-tralization versus decentralization Risk pooling risk diversifi-cation and supply chain disruptionsrdquo Omega vol 52 pp 201ndash212 2015

[22] L V Snyder Z Atan P Peng Y Rong A J Schmitt andB Sinsoysal ldquoORMS models for supply chain disruptions areviewrdquo IIE Transactions vol 48 no 2 pp 89ndash109 2016

[23] T Sawik ldquoIntegrated selection of suppliers and scheduling ofcustomer orders in the presence of supply chain disruptionrisksrdquo International Journal of Production Research vol 51 no23-24 pp 7006ndash7022 2013

[24] T Sawik ldquoOptimization of cost and service level in the presenceof supply chain disruption risks Single vs multiple sourcingrdquoComputers amp Operations Research vol 51 pp 11ndash20 2014

[25] T Sawik ldquoIntegrated supply production and distributionscheduling under disruption risksrdquoOmega vol 62 pp 131ndash1442016

[26] T Sawik ldquoIntegrated supply chain scheduling under multi-leveldisruptionsrdquo IFAC-PapersOnLine vol 28 no 3 pp 1515ndash15202015

[27] P Kall and J Mayer Stochastic Linear Programming ModelsTheory and Computation vol 156 of International Series inOperations Research amp Management Science Springer NewYork 2011

[28] I N Durbach and T J Stewart ldquoUsing expected values to sim-plify decision making under uncertaintyrdquo Omega vol 37 no 2pp 312ndash330 2009

[29] F Maggioni and S W Wallace ldquoAnalyzing the quality of theexpected value solution in stochastic programmingrdquo Annals ofOperations Research vol 200 pp 37ndash54 2012

[30] T Sawik ldquoSelection of supply portfolio under disruption risksrdquoOmega vol 39 no 2 pp 194ndash208 2011

[31] E Delage S Arroyo and Y Ye ldquoThe value of stochastic mod-eling in two-stage stochastic programs with cost uncertaintyrdquoOperations Research vol 62 no 6 pp 1377ndash1393 2014

[32] T Sawik Scheduling in Supply Chains Using Mixed Integer Pro-gramming John Wiley amp Sons Inc Hoboken NJ 2011

[33] T Sawik Supply Chain DisruptionManagement Using StochasticMixed Integer Programming vol 256 of International Seriesin Operations Research amp Management Science Springer NewYork 2017 httpwwwspringercomgpbook9783319588223

[34] D E Whitney J Luo and D A Heller ldquoThe benefits and con-straints of temporary sourcing diversification in supply chaindisruption and recoveryrdquo Journal of Purchasing and SupplyManagement vol 20 no 4 pp 238ndash250 2014

Mathematical Problems in Engineering 13

1045 le 1051 forWCS (see Table 2) or themaximum servicelevel1198642 for EWCS (see Table 3) 8700 ge 8552 forWCS (seeTable 2)

The stochastic wait-and-see approach however leads toa more diversified supply portfolio For the deterministicapproach most of nondominated supply portfolios consistof two suppliers only 119894 = 2 3 while suppliers 119894 = 1 4 areselected only for the maximum service level objective (thatis for 120572 = 0) Comparison of Figures 3 and 4 indicatesthat the expected schedules for model WCS computed asexpectations over all schedules for all potential disruptionscenarios (27)ndash(29) are similar to the corresponding singleschedules determined by model EWCS The main differencesobserved aremore delayed expected production and shippingschedules for model WCS when minimization of cost is con-sidered (that is for120572 = 1) Finally it is interesting to note thatin the multiple sourcing environment considered both thewait-and-see approach and in particular the expected valueapproach frequently select a dual sourcing supply portfoliowith one main supplier and one supporting supplier

The computational experiments were performed usingthe AMPL programming language and the CPLEX 1262solver on a MacBook Pro laptop with Intel Core i7 processorrunning at 28 GHz and with 16GB RAM The solver wascapable of finding proven optimal solution for all exampleswith CPU time ranging from several minutes to several hoursfor the stochastic model WCS and fraction of a second forthe deterministic model EWCS Since the stochastic pro-gramming model WCS needs to determine nondominatedschedules for all potential disruption scenarios while modelEWCS deals with a single scenario only the difference in thecomputational effort required is obvious

7 Conclusions

In this paper two biobjective MIP formulations stochasticand deterministic have been proposed and compared for thecoordinated decision-making in supply chains under partiallocal disruptions and all-or-nothing regional disruptionsTheproblem objective has been to jointly schedule supplies pro-duction and distribution to optimize the trade-off betweenexpected cost and expected service levelWhile the stochasticprogramming approach aims at optimizing the expectedperformance of a supply chain over all possible disruptionscenarios the deterministic approach accounts only for asingle scenario representing average disruption conditionsAs a result the stochastic programming approach determinesnondominated solutions for all disruption scenarios whereasthe deterministic approach produces a single solution onlyIn particular in the stochastic programming approach theselection of supply portfolio is combined with supply chainscheduling for all disruption scenarios considered In con-trast the deterministic expected value approach provides theportfolio along with a single expected schedule of productionand distribution

The expected schedules obtained for the stochastic pro-gramming model WCS as expectations over all schedulesfor all disruption scenarios have been compared with thecorresponding schedules determined by the deterministic

model EWCS based on an expected disruption scenarioThecomparison has demonstrated that the two approaches leadto similar expected solutions

The main findings are in line with other research and arelisted as follows

(i) The two decision-making approaches stochastic anddeterministic lead to similar expected performanceof a supply chain under multilevel disruptions

(ii) The optimal solution values for the expected valueproblem frequently outperform the correspondingsolution values of the wait-and-see problem

(iii) Despite the multiple sourcing environment consid-ered both the wait-and-see approach and in partic-ular the expected value approach frequently selecta dual sourcing supply portfolio with one mainsupplier and one supporting supplier

(iv) The stochastic approach which accounts for allpotential disruption scenarios may lead to a morediversified supply portfolio that will hedge against avariety of scenarios

(v) The expected schedules are more delayed for thestochastic approach

(vi) The service-oriented supply portfolio is more diversi-fied andmay combine both high-cost reliable suppli-ers and low-cost unreliable suppliers while the cost-oriented portfolio depends mainly on low-cost andless reliable suppliers

Overall the results of computational experiments indi-cate that the proposed approach and developed MIP modelsare flexible and efficient tools for coordinated supply chainscheduling The portfolio approach leads to MIP formula-tions with strong LP relaxations and has been proven to becomputationally very efficient CPU time required to findproven optimal solutions for realistic size examples usingcommercially available software for MIP is acceptable for areal-world supply chain disruption management (see Sawik[33])

Since the probability distribution of supply disruptionsfrom each supplier is usually unknown (eg [8]) local multi-level disruptions in the stochastic programming modelWCShave a multinomial discrete distribution while the two-levelregional disruptions have a binomial discrete distributionand all disruption events are independent As part of futureresearch we propose to enhance the stochastic programmingmodel formore general scenarios with finitelymany elementssingled out and all the probability concentrated in them Forexample rather than complete shutdown suppliers within aregion may have correlated disruptions Another importantstream of future research is the study of robustness andsensitivity in relation to input data changes in supply chainscheduling under disruption risks (eg [33]) Future researchcan also focus on the following improvement of the proposedmodels to consider the minimization of time and cost ofrecovery (eg Whitney et al [34])

14 Mathematical Problems in Engineering

Notations

Indices

119894 Supplier 119894 isin 119868119895 Customer 119895 isin 119869119896 Distribution center 119896 isin 119870119897 Disruption level 119897 isin 119871119903 Region 119903 isin 119877119904 Disruption scenario 119904 isin 119878119905 Planning period 119905 isin 119879Input Parameters

119887119895 Size (number of products) of customer order 119895119861 Total demand for partsproducts 119861 = sum119895isin119869 119887119895119862 Capacity of producer119889119895 Due date for customer order 119895119890119894 Fixed cost of ordering parts from supplier 119894119892119895 Per unit penalty cost of delayed customer order 119895ℎ119895 Per unit penalty cost of unfulfilled customer order 119895119868119903 Subset of suppliers in region 119903119869119896 Subset of customers serviced by distribution center 119896119900119894 Per unit price of parts purchased from supplier 119894119901119894119897 Probability of disruption level 119897 for supplier 119894119901119903 Regional disruption probability for region 119903120574119894119897 Fraction of an order delivered by supplier 119894 underdisruption level 119897 (fulfillment rate)Γ119894 Expected fraction of an order delivered by supplier 119894(expected fulfillment rate)120590119894 Delivery lead time from supplier 119894120591119896 Transportation time to distribution center 119896

Conflicts of Interest

The author declares that there are no conflicts of interest

Acknowledgments

This work has been supported by NCN research grant (noDEC-201311BST804458) and by AGH (no 1111200324)

References

[1] J V Blackhurst K P Scheibe and D J Johnson ldquoSupplierrisk assessment and monitoring for the automotive industryrdquoInternational Journal of Physical Distribution amp Logistics Man-agement vol 38 no 2 pp 143ndash165 2008

[2] P Hoffmann H Schiele and K Krabbendam ldquoUncertaintysupply risk management and their impact on performancerdquoJournal of Purchasing amp Supply Management vol 19 no 3 pp199ndash211 2013

[3] Y Park P Hong and J J Roh ldquoSupply chain lessons from thecatastrophic natural disaster in Japanrdquo Business Horizons vol56 no 1 pp 75ndash85 2013

[4] M Haraguchi and U Lall ldquoFlood risks and impacts A casestudy of Thailandrsquos floods in 2011 and research questions forsupply chain decisionmakingrdquo International Journal of DisasterRisk Reduction vol 14 pp 256ndash272 2015

[5] T Fujimoto and Y W Park ldquoBalancing supply chain com-petitiveness and robustness through ldquovirtual dual sourcingrdquoLessons from the Great East Japan Earthquakerdquo InternationalJournal of Production Economics vol 147 pp 429ndash436 2014

[6] HMatsuo ldquoImplications of the Tohoku earthquake for Toyotarsquoscoordination mechanism supply chain disruption of auto-motive semiconductorsrdquo International Journal of ProductionEconomics vol 161 pp 217ndash227 2015

[7] J R Marszewska ldquoImplications of seismic hazard in Japan ontoyota supply chain disruption risksrdquo in Proceedings of the 13thInternational Conference on Industrial Logistics (ICIL rsquo16) pp178ndash185 Zakopane Poland October 2016

[8] A Z Zeng and Y Xia ldquoBuilding a mutually beneficial partner-ship to ensure backup supplyrdquo Omega vol 52 pp 77ndash91 2015

[9] S S Erenguc N C Simpson and A J Vakharia ldquoIntegratedproductiondistribution planning in supply chains an invitedreviewrdquo European Journal of Operational Research vol 115 no2 pp 219ndash236 1999

[10] L Lei S Liu A Ruszczynski and S Park ldquoOn the integratedproduction inventory and distribution routing problemrdquo IIETransactions vol 38 no 11 pp 955ndash970 2006

[11] A Kaur A Kanda and S G Deshmukh ldquoA graph theoreticapproach for supply chain coordinationrdquo International Journalof Logistics Systems and Management vol 2 no 4 pp 321ndash3412006

[12] T ChoiW Yeung andT C E Cheng ldquoScheduling and co-ordi-nation of multi-suppliers single-warehouse-operator single-manufacturer supply chains with variable production rates andstorage costsrdquo International Journal of Production Research vol51 no 9 pp 2593ndash2601 2013

[13] S Liu and L G Papageorgiou ldquoMultiobjective optimisation ofproduction distribution and capacity planning of global supplychains in the process industryrdquo Omega vol 41 no 2 pp 369ndash382 2013

[14] Z L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[15] T Sawik ldquoCoordinated supply chain schedulingrdquo InternationalJournal of Production Economics vol 120 no 2 pp 437ndash4512009

[16] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[17] D Ivanov B Sokolov and A Dolgui ldquoThe Ripple effect in sup-ply chains Trade-off rsquoefficiency-flexibility- resiliencersquo in disrup-tionmanagementrdquo International Journal of ProductionResearchvol 52 no 7 pp 2154ndash2172 2014

[18] D Ivanov A Pavlov and B Sokolov ldquoOptimal distribution(re)planning in a centralized multi-stage supply network underconditions of the ripple effect and structure dynamicsrdquo Euro-pean Journal of Operational Research vol 237 no 2 pp 758ndash770 2014

[19] D Ivanov A Pavlov A Dolgui D Pavlov and B SokolovldquoDisruption-driven supply chain (re)-planning and perfor-mance impact assessment with consideration of pro-active andrecovery policiesrdquo Transportation Research Part E Logistics andTransportation Review vol 90 pp 7ndash24 2016

[20] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

Mathematical Problems in Engineering 15

[21] A J Schmitt S A Sun L V Snyder and Z-J M Shen ldquoCen-tralization versus decentralization Risk pooling risk diversifi-cation and supply chain disruptionsrdquo Omega vol 52 pp 201ndash212 2015

[22] L V Snyder Z Atan P Peng Y Rong A J Schmitt andB Sinsoysal ldquoORMS models for supply chain disruptions areviewrdquo IIE Transactions vol 48 no 2 pp 89ndash109 2016

[23] T Sawik ldquoIntegrated selection of suppliers and scheduling ofcustomer orders in the presence of supply chain disruptionrisksrdquo International Journal of Production Research vol 51 no23-24 pp 7006ndash7022 2013

[24] T Sawik ldquoOptimization of cost and service level in the presenceof supply chain disruption risks Single vs multiple sourcingrdquoComputers amp Operations Research vol 51 pp 11ndash20 2014

[25] T Sawik ldquoIntegrated supply production and distributionscheduling under disruption risksrdquoOmega vol 62 pp 131ndash1442016

[26] T Sawik ldquoIntegrated supply chain scheduling under multi-leveldisruptionsrdquo IFAC-PapersOnLine vol 28 no 3 pp 1515ndash15202015

[27] P Kall and J Mayer Stochastic Linear Programming ModelsTheory and Computation vol 156 of International Series inOperations Research amp Management Science Springer NewYork 2011

[28] I N Durbach and T J Stewart ldquoUsing expected values to sim-plify decision making under uncertaintyrdquo Omega vol 37 no 2pp 312ndash330 2009

[29] F Maggioni and S W Wallace ldquoAnalyzing the quality of theexpected value solution in stochastic programmingrdquo Annals ofOperations Research vol 200 pp 37ndash54 2012

[30] T Sawik ldquoSelection of supply portfolio under disruption risksrdquoOmega vol 39 no 2 pp 194ndash208 2011

[31] E Delage S Arroyo and Y Ye ldquoThe value of stochastic mod-eling in two-stage stochastic programs with cost uncertaintyrdquoOperations Research vol 62 no 6 pp 1377ndash1393 2014

[32] T Sawik Scheduling in Supply Chains Using Mixed Integer Pro-gramming John Wiley amp Sons Inc Hoboken NJ 2011

[33] T Sawik Supply Chain DisruptionManagement Using StochasticMixed Integer Programming vol 256 of International Seriesin Operations Research amp Management Science Springer NewYork 2017 httpwwwspringercomgpbook9783319588223

[34] D E Whitney J Luo and D A Heller ldquoThe benefits and con-straints of temporary sourcing diversification in supply chaindisruption and recoveryrdquo Journal of Purchasing and SupplyManagement vol 20 no 4 pp 238ndash250 2014

14 Mathematical Problems in Engineering

Notations

Indices

119894 Supplier 119894 isin 119868119895 Customer 119895 isin 119869119896 Distribution center 119896 isin 119870119897 Disruption level 119897 isin 119871119903 Region 119903 isin 119877119904 Disruption scenario 119904 isin 119878119905 Planning period 119905 isin 119879Input Parameters

119887119895 Size (number of products) of customer order 119895119861 Total demand for partsproducts 119861 = sum119895isin119869 119887119895119862 Capacity of producer119889119895 Due date for customer order 119895119890119894 Fixed cost of ordering parts from supplier 119894119892119895 Per unit penalty cost of delayed customer order 119895ℎ119895 Per unit penalty cost of unfulfilled customer order 119895119868119903 Subset of suppliers in region 119903119869119896 Subset of customers serviced by distribution center 119896119900119894 Per unit price of parts purchased from supplier 119894119901119894119897 Probability of disruption level 119897 for supplier 119894119901119903 Regional disruption probability for region 119903120574119894119897 Fraction of an order delivered by supplier 119894 underdisruption level 119897 (fulfillment rate)Γ119894 Expected fraction of an order delivered by supplier 119894(expected fulfillment rate)120590119894 Delivery lead time from supplier 119894120591119896 Transportation time to distribution center 119896

Conflicts of Interest

The author declares that there are no conflicts of interest

Acknowledgments

This work has been supported by NCN research grant (noDEC-201311BST804458) and by AGH (no 1111200324)

References

[1] J V Blackhurst K P Scheibe and D J Johnson ldquoSupplierrisk assessment and monitoring for the automotive industryrdquoInternational Journal of Physical Distribution amp Logistics Man-agement vol 38 no 2 pp 143ndash165 2008

[2] P Hoffmann H Schiele and K Krabbendam ldquoUncertaintysupply risk management and their impact on performancerdquoJournal of Purchasing amp Supply Management vol 19 no 3 pp199ndash211 2013

[3] Y Park P Hong and J J Roh ldquoSupply chain lessons from thecatastrophic natural disaster in Japanrdquo Business Horizons vol56 no 1 pp 75ndash85 2013

[4] M Haraguchi and U Lall ldquoFlood risks and impacts A casestudy of Thailandrsquos floods in 2011 and research questions forsupply chain decisionmakingrdquo International Journal of DisasterRisk Reduction vol 14 pp 256ndash272 2015

[5] T Fujimoto and Y W Park ldquoBalancing supply chain com-petitiveness and robustness through ldquovirtual dual sourcingrdquoLessons from the Great East Japan Earthquakerdquo InternationalJournal of Production Economics vol 147 pp 429ndash436 2014

[6] HMatsuo ldquoImplications of the Tohoku earthquake for Toyotarsquoscoordination mechanism supply chain disruption of auto-motive semiconductorsrdquo International Journal of ProductionEconomics vol 161 pp 217ndash227 2015

[7] J R Marszewska ldquoImplications of seismic hazard in Japan ontoyota supply chain disruption risksrdquo in Proceedings of the 13thInternational Conference on Industrial Logistics (ICIL rsquo16) pp178ndash185 Zakopane Poland October 2016

[8] A Z Zeng and Y Xia ldquoBuilding a mutually beneficial partner-ship to ensure backup supplyrdquo Omega vol 52 pp 77ndash91 2015

[9] S S Erenguc N C Simpson and A J Vakharia ldquoIntegratedproductiondistribution planning in supply chains an invitedreviewrdquo European Journal of Operational Research vol 115 no2 pp 219ndash236 1999

[10] L Lei S Liu A Ruszczynski and S Park ldquoOn the integratedproduction inventory and distribution routing problemrdquo IIETransactions vol 38 no 11 pp 955ndash970 2006

[11] A Kaur A Kanda and S G Deshmukh ldquoA graph theoreticapproach for supply chain coordinationrdquo International Journalof Logistics Systems and Management vol 2 no 4 pp 321ndash3412006

[12] T ChoiW Yeung andT C E Cheng ldquoScheduling and co-ordi-nation of multi-suppliers single-warehouse-operator single-manufacturer supply chains with variable production rates andstorage costsrdquo International Journal of Production Research vol51 no 9 pp 2593ndash2601 2013

[13] S Liu and L G Papageorgiou ldquoMultiobjective optimisation ofproduction distribution and capacity planning of global supplychains in the process industryrdquo Omega vol 41 no 2 pp 369ndash382 2013

[14] Z L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[15] T Sawik ldquoCoordinated supply chain schedulingrdquo InternationalJournal of Production Economics vol 120 no 2 pp 437ndash4512009

[16] B Tomlin ldquoOn the value of mitigation and contingency strate-gies for managing supply chain disruption risksrdquo ManagementScience vol 52 no 5 pp 639ndash657 2006

[17] D Ivanov B Sokolov and A Dolgui ldquoThe Ripple effect in sup-ply chains Trade-off rsquoefficiency-flexibility- resiliencersquo in disrup-tionmanagementrdquo International Journal of ProductionResearchvol 52 no 7 pp 2154ndash2172 2014

[18] D Ivanov A Pavlov and B Sokolov ldquoOptimal distribution(re)planning in a centralized multi-stage supply network underconditions of the ripple effect and structure dynamicsrdquo Euro-pean Journal of Operational Research vol 237 no 2 pp 758ndash770 2014

[19] D Ivanov A Pavlov A Dolgui D Pavlov and B SokolovldquoDisruption-driven supply chain (re)-planning and perfor-mance impact assessment with consideration of pro-active andrecovery policiesrdquo Transportation Research Part E Logistics andTransportation Review vol 90 pp 7ndash24 2016

[20] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

Mathematical Problems in Engineering 15

[21] A J Schmitt S A Sun L V Snyder and Z-J M Shen ldquoCen-tralization versus decentralization Risk pooling risk diversifi-cation and supply chain disruptionsrdquo Omega vol 52 pp 201ndash212 2015

[22] L V Snyder Z Atan P Peng Y Rong A J Schmitt andB Sinsoysal ldquoORMS models for supply chain disruptions areviewrdquo IIE Transactions vol 48 no 2 pp 89ndash109 2016

[23] T Sawik ldquoIntegrated selection of suppliers and scheduling ofcustomer orders in the presence of supply chain disruptionrisksrdquo International Journal of Production Research vol 51 no23-24 pp 7006ndash7022 2013

[24] T Sawik ldquoOptimization of cost and service level in the presenceof supply chain disruption risks Single vs multiple sourcingrdquoComputers amp Operations Research vol 51 pp 11ndash20 2014

[25] T Sawik ldquoIntegrated supply production and distributionscheduling under disruption risksrdquoOmega vol 62 pp 131ndash1442016

[26] T Sawik ldquoIntegrated supply chain scheduling under multi-leveldisruptionsrdquo IFAC-PapersOnLine vol 28 no 3 pp 1515ndash15202015

[27] P Kall and J Mayer Stochastic Linear Programming ModelsTheory and Computation vol 156 of International Series inOperations Research amp Management Science Springer NewYork 2011

[28] I N Durbach and T J Stewart ldquoUsing expected values to sim-plify decision making under uncertaintyrdquo Omega vol 37 no 2pp 312ndash330 2009

[29] F Maggioni and S W Wallace ldquoAnalyzing the quality of theexpected value solution in stochastic programmingrdquo Annals ofOperations Research vol 200 pp 37ndash54 2012

[30] T Sawik ldquoSelection of supply portfolio under disruption risksrdquoOmega vol 39 no 2 pp 194ndash208 2011

[31] E Delage S Arroyo and Y Ye ldquoThe value of stochastic mod-eling in two-stage stochastic programs with cost uncertaintyrdquoOperations Research vol 62 no 6 pp 1377ndash1393 2014

[32] T Sawik Scheduling in Supply Chains Using Mixed Integer Pro-gramming John Wiley amp Sons Inc Hoboken NJ 2011

[33] T Sawik Supply Chain DisruptionManagement Using StochasticMixed Integer Programming vol 256 of International Seriesin Operations Research amp Management Science Springer NewYork 2017 httpwwwspringercomgpbook9783319588223

[34] D E Whitney J Luo and D A Heller ldquoThe benefits and con-straints of temporary sourcing diversification in supply chaindisruption and recoveryrdquo Journal of Purchasing and SupplyManagement vol 20 no 4 pp 238ndash250 2014

Mathematical Problems in Engineering 15

[21] A J Schmitt S A Sun L V Snyder and Z-J M Shen ldquoCen-tralization versus decentralization Risk pooling risk diversifi-cation and supply chain disruptionsrdquo Omega vol 52 pp 201ndash212 2015

[22] L V Snyder Z Atan P Peng Y Rong A J Schmitt andB Sinsoysal ldquoORMS models for supply chain disruptions areviewrdquo IIE Transactions vol 48 no 2 pp 89ndash109 2016

[23] T Sawik ldquoIntegrated selection of suppliers and scheduling ofcustomer orders in the presence of supply chain disruptionrisksrdquo International Journal of Production Research vol 51 no23-24 pp 7006ndash7022 2013

[24] T Sawik ldquoOptimization of cost and service level in the presenceof supply chain disruption risks Single vs multiple sourcingrdquoComputers amp Operations Research vol 51 pp 11ndash20 2014

[25] T Sawik ldquoIntegrated supply production and distributionscheduling under disruption risksrdquoOmega vol 62 pp 131ndash1442016

[26] T Sawik ldquoIntegrated supply chain scheduling under multi-leveldisruptionsrdquo IFAC-PapersOnLine vol 28 no 3 pp 1515ndash15202015

[27] P Kall and J Mayer Stochastic Linear Programming ModelsTheory and Computation vol 156 of International Series inOperations Research amp Management Science Springer NewYork 2011

[28] I N Durbach and T J Stewart ldquoUsing expected values to sim-plify decision making under uncertaintyrdquo Omega vol 37 no 2pp 312ndash330 2009

[29] F Maggioni and S W Wallace ldquoAnalyzing the quality of theexpected value solution in stochastic programmingrdquo Annals ofOperations Research vol 200 pp 37ndash54 2012

[30] T Sawik ldquoSelection of supply portfolio under disruption risksrdquoOmega vol 39 no 2 pp 194ndash208 2011

[31] E Delage S Arroyo and Y Ye ldquoThe value of stochastic mod-eling in two-stage stochastic programs with cost uncertaintyrdquoOperations Research vol 62 no 6 pp 1377ndash1393 2014

[32] T Sawik Scheduling in Supply Chains Using Mixed Integer Pro-gramming John Wiley amp Sons Inc Hoboken NJ 2011

[33] T Sawik Supply Chain DisruptionManagement Using StochasticMixed Integer Programming vol 256 of International Seriesin Operations Research amp Management Science Springer NewYork 2017 httpwwwspringercomgpbook9783319588223

[34] D E Whitney J Luo and D A Heller ldquoThe benefits and con-straints of temporary sourcing diversification in supply chaindisruption and recoveryrdquo Journal of Purchasing and SupplyManagement vol 20 no 4 pp 238ndash250 2014

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of