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Journal of Engineering Science and Technology Special Issue on 4th International Technical Conference 2014, June (2015) 89 - 101 © School of Engineering, Taylor’s University

89

A RECOVERY MODEL FOR A SUPPLY CHAIN SYSTEM WITH MULTIPLE SUPPLIERS SUBJECT TO SUPPLY DISRUPTION

HAWA HISHAMUDDIN1,*, RUHUL SARKER

2, DARYL ESSAM

2

1Department of Mechanical and Materials Engineering, UKM, Malaysia 2School of Engineering and IT, University of New South Wales Canberra, Australia

*Corresponding Author: [email protected]

Abstract

The study of a real-time procurement and production mechanism for a multi-stage supply chain system with multiple suppliers subject to an unexpected disruption is presented in this paper. Specifically, a mathematical model is developed for the problem of optimizing replenishment and production decisions for each node after a supply disruption occurrence. The system considered in this research is a three stage supply chain system that consists of three suppliers, one manufacturer and one retailer. The problem that will be considered is an instance of the class of inventory management problems under disruptions with a finite horizon. The solution approach will utilize a heuristic that we have developed in previous works. In addition, an experiment was conducted to study the effects of disruption on the system using predefined scenarios, where supplier prioritization of disruption mitigation strategies was explored. Various disruption scenarios were predefined by combining different disruption locations, as well as different combinations of suppliers. It can be shown that supply disruption at the suppliers with higher inventory holding costs causes higher recovery costs for the overall system. Therefore, it is important for the manufacturer to focus on the suppliers with higher holding cost when planning appropriate strategies for risk management. This optimization study has enabled an increased understanding of the impacts of random disruption events on the total system behaviour, as well as determining priorities in risk mitigation efforts.

Keywords: Supply chain, Multiple suppliers, Disruption, Recovery.

1. Introduction

For business continuity, it is important for a firm to know and understand the different risk properties of each of their partners in their supply chain (SC). In the literature on supply chain disruption, there are works that explore the task of

optimizing the inventory replenishment decisions in single and two-stage supply

90 H. Hishamuddin et al.

Journal of Engineering Science and Technology Special Issue 6/2015

Nomenclatures AMO Ordering cost for the manufacturer ($/order)

AMS Setup cost for the manufacturer ($/setup) AR Ordering cost for the retailer ($/order) ASi Setup cost for supplier i ($/setup) BqM Back order quantity for the manufacturer BqR Back order quantity for the retailer BSD,BM,BR Unit back order cost per unit time for the disrupted supplier,

manufacturer and retailer respectively ($/unit/time) D Demand rate for the finished product (units/year)

DSi Demand rate for parts from supplier i (units/year) f1 Penalty function for delay in recovering the original schedule in

the first stage f2 Penalty function for delay in recovering the original schedule of

the second stage handled by the first stage f3 Penalty function for delay in recovering the original schedule in

the second stage HM Annual inventory cost for a finished product at the manufacturer

($/unit/year) HMi Annual inventory cost for part i at the manufacturer’s site

($/unit/year) HR Annual inventory cost for a finished product at the retailer

($/unit/year) HSi Annual inventory cost for part i at supplier i ($/unit/year) Ii inventory level at the end of cycle i in the recovery window LqM Lost sales quantity for the manufacturer

LqR Lost sales quantity for the retailer LSD,LM,LR Unit lost sales cost for the disrupted supplier, manufacturer and

retailer respectively ($/unit) PM Production rate of the manufacturer (units/year) PSi Production rate of supplier i (units/year) QMO Ordering lot size for the manufacturer in the original schedule (units) QMP Production lot size for the manufacturer in the original schedule

(units)

QR Ordering lot size for the retailer in the original schedule (units) QSi Production lot size for supplier i in the original schedule (units) Td

M Disruption period for the manufacturer

TdS Disruption period for disrupted supplier

te Start of recovery time window tf End of recovery time window TM Production cycle time of a normal cycle for the manufacturer (Q/D) TMj Production time for cycle j in the recovery window for the manufacturer

TR Production cycle time of a normal cycle for the retailer (Q/D) TRk Production time for cycle k in the recovery window for the retailer TSDi Production time for cycle i in the recovery window for the

disrupted supplier

TSi Production cycle time of a normal cycle for supplier i (Q/D) u Production down time for a normal cycle W Warehouse capacity for stage 2 (units)

A Recovery Model for a Supply Chain System with Multiple Suppliers 91


ρ Production up time for a normal cycle (Q/P)

Decision Variables

m Number of suppliers in the supply chain system n1 Number of cycles in the recovery window for the disrupted

supplier and manufacturer n2 Number of cycles in the recovery window for the manufacturer

and retailer SMi Manufacturer’s order lot size of cycle i in the recovery schedule

for the disrupted supplier’s parts (units) SRi Retailer’s order lot size of cycle i in the recovery schedule for

the finished goods (units) XMi Production lot size of cycle i in the recovery schedule for the

manufacturer (units) XMi Production lot size of cycle i in the recovery schedule for the

manufacturer (units) XSDi Production lot size of cycle i in the recovery schedule for the

disrupted supplier (units) XSDi Production lot size of cycle i in the recovery schedule for the

disrupted supplier (units) z1 Number of optimal production lots in the recovery window for

the disrupted supplier z2 Number of optimal production lots in the recovery window for

the manufacturer

chain structures, in the aftermath of a disruption [1-4]. However, the design of today’s supply chains is more complex in nature and consists of multiple stages

with multiple entities in each stage. In addition, several research works have studied models with multiple unreliable suppliers that supply the same parts, where supply uncertainty is considered in their model. Most of the works have the objective of selecting which supplier to order from and in what quantities, in

order to hedge against disruptions, given different supplier assumptions [5-8]. Tehrani et al. [9] studied the impact of correlation between suppliers for two types of systems, a multi-source system and an assembly system. Sawik [10] proposed a supply portfolio approach, where selection and protection of part suppliers was combined with order quantity allocation for a supply chain system with disruptions.

In a typical SC system, different suppliers exist with varying system and operating costs. Moreover, each plant faces plant-specific, stationary demand.

Each supplier in a SC may have unequal optimal production lot sizes (Qi) depending on the differing cost parameters of the plant. Therefore, when a disruption arises, it is obvious that the effects of the disruption on each plant differ depending on these parameters. Despite the importance of understanding

the properties of each node in the SC, there is a lacking of quantitative models for system analysis and decision support. Additionally, it has been emphasized that one of the most important dimensions in discussing risk is the outcome of risk impact [11].

This paper investigates the recovery properties of differing suppliers in a semi-integrated lot sizing problem for a three stage supply chain system with multiple suppliers that is subject to random disruption occurrences. This paper is



an extension to the study of the two stage model in the works of Hishamuddin et al. [12]. The system under study consists of a partially integrated system of three different parts suppliers, a manufacturer and a retailer. In addition to optimizing the replenishment and manufacturing decisions of the intended system, one of the

goals of this research is to obtain insights on priority selection for risk management strategies at the most critical suppliers. As we assume that the suppliers have been pre-selected and work on a pre-determined supply portfolio, our work focuses more on the decision of prioritizing suppliers for protection

against disruption by analysing the impact of different supplier disruptions on the total system recovery costs.

2. Problem Description

Consider a manufacturer that has three suppliers, each supplier supplying different parts or raw material to the manufacturer, depending on the type of production system in hand. The manufacturer could be an assembly manufacturer who assembles the parts into finished goods, or a pure manufacturer that converts the raw material into the end product through its manufacturing processes. To avoid confusion, the term parts will consistently be used throughout this paper.

The finished goods are delivered to the retailer, who only has inventory and no production. Each supplier has a coordinated procurement system with the manufacturer. Under this policy, suppliers form a long–term partnership with the manufacturer and both parties share demand information. Meanwhile, information

may or may not be shared between suppliers, or in other words, there exists a non-coordinated system between suppliers. Table 1 describes the specific disruption types that the system may experience and the designated location for disruption.

Table 1. The different disruption types.

Disruption Type Description

1 Supply Disruption at Supplier 1



4 Supply Disruption at Manufacturer

Following a supply disruption at a supplier, the manufacturer and retailer will experience supply disruption of the same magnitude (Td) as a direct consequence.

Although in real life the magnitude may be larger due to the “ripple effect”, we choose not to consider this feature for ease of calculation purposes. The assembly line at the manufacturer will be interrupted due to parts shortages from the disrupted supplier. Since there is no information sharing between suppliers, the

non-disrupted suppliers will still deliver the current lot to the manufacturer despite the fact that one of the suppliers is disrupted. This will cause inventory build-up of parts from the non-disrupted suppliers at the manufacturer site. Thus, there will be additional cost incurred by the manufacturer for holding this extra inventory. It is only when the disruption is resolved that the manufacturer can continue its normal operations. In the case of disruption, the non-disrupted suppliers will be requested to delay the next shipment of parts to the manufacturer until the parts inventory at the manufacturer is zero. This adheres to the zero-

inventory ordering policy that the system adopts.



3. Mathematical Representation

In this subsection, the model’s cost functions will be derived based on the

assumptions stated previously. The mathematical model for the case of supply disruption will be formulated, where we will model the cost functions following the sequence of: disrupted supplier, non-disrupted suppliers,

manufacturer, and retailer.

3.1. Suppliers’ cost function

The decision that has to be made in the event of a disruption at one of the

suppliers is the optimal recovery schedule for the production quantities of the disrupted supplier. The associated costs for recovery include machine setup cost, inventory holding cost, penalty costs for late recovery, and shortage costs due to

stock outs. This can be written as:

TCSD

= TCsetup

+ TCinventory

+ TCpenalty

+ TCbackorders

+ TClost sales (1)

The expression for the total cost, TCSD, can be expressed as:

( )( )

( )( )MSDM

Si

Si

Si

SDid

Si

SD

SD

z

i Si

SDISDiid

Si

SiSi

SDi

SD

LqLBq

D

Q

P

XT

P

quB

Tn

nfnfP

XXIqTq

PHzA

znXTC

+

−++++

+

++

⋅

++++⋅

=

∑=

−

2

)()(2

1

2

1

),,(1

2

12

2

111

1

2

1

11

(2)

The total production cost of the non-disrupted suppliers, Si, may be formulated as:

( )

+⋅=∑ 11

2zQT

HzATC SiSi

Si

Si

m

i

Si where ( )Si

dSD

T

TTnz

−= 1

1 (3)

The total recovery cost for all the suppliers, TCS is the total of Eqs. (2) and (3),

which is

TCS = TC

SD + TC

Si (4)

3.2. Manufacturer’s cost function

When the supplier experiences a supply disruption, the manufacturer also has to decide the optimal ordering quantities of raw material from the disrupted supplier in the new recovery schedule. The costs consist of the ordering, inventory

holding, penalty and shortage costs, as denoted by the following

CM1 = TCordering + TCinventory + TCpenalty + TCbackorders + TClost sales (5)

and can be converted to the following expression:

( )( ) ( )

( )( )MMM

Si

MO

Si

SDd

Si

M

SD

z

iMiMMMO

Si

MiMO

Mi

M

LqLBq

D

Q

P

XT

P

quB

Tn

nfSBqSQD

HzA

znSTC

+

−+++

+

+

+−++−⋅

=∑−

=

2

)(2

1

),,(

1

1

2

13

1

2

22

1

2

1

111 (6)



The manufacturer’s ordering cost from the non-disrupted suppliers, i, is

formulated as

( ) ( )

++⋅= ∑ dSiMiSiSi

MiMOi

m

i

MOi TQHzQTH

zATC 112

where ( )Si

dSD

T

TTnz

−= 1

1 (7)

The inventory buildup cost of parts from a non-disrupted supplier at the manufacturer is given by

= ∑

m

iSidM

inv QTHTC (8)

for i = all suppliers except the disrupted supplier

Note that in the case where the manufacturer experiences a disruption and the supplier is not disrupted (disruption type 7), TC

SD, TC

M1 and TCinv reduces to zero,

as there are no costs associated with recovering the supplier’s production schedule and the manufacturer’s ordering schedule, and there is no inventory build-up at

the manufacturer’s site.

As mentioned in the previous section, a supply disruption at the supplier will have a direct effect to the manufacturer and retailer in the form of supply disruption as well. Thus, the manufacturer and retailer will incur penalty and

stock out costs as a consequence. In order to obtain the manufacturer’s new production quantity in the recovery schedule, the costs of setup, inventory holding, penalty, and shortages are considered:

TCM2 = TCsetup + TCinventory + TCpenalty + TCbackorders + TClost sales (9)

This total cost can be derived as

( )( )

( )( )RMRMP

M

Md

M

M

M

z

i M

MiMiid

M

MMS

Mi

M

LqLBq

D

Q

P

XT

P

quB

Tn

nfnfP

XXIqTq

PHzA

znXTC

+

−++++

+

++

⋅

++++⋅

=

∑=

−

2

)()(2

1

2

1

),,(

1

2

2

22

2

211

12

2

222 (10)

Thus, by combining Eqs. (6), (7), (8) and (10), one can obtain the total

recovery cost for the manufacturer as

TCM

= TCM1

+ TCMOi

+ TCinv

+ TCM2 (11)

3.3. Retailer’s cost function

The new ordering schedule for the retailer is determined by considering the costs

of ordering, inventory holding, penalty, and shortages, as follows:

TCR = TC

ordering + TC

inventory + TC

penalty + TC

backorders + TC

lost sales (12)

This can be expressed as



( )( ) ( )

( )( )RRRR

M

Md

M

R

M

z

iRiRRR

RR

Ri

R

LqLBq

D

Q

P

XT

P

quB

Tn

nfSBqSQD

HzA

znSTC

+

−+++

+

+

+−++−⋅

=∑−

=

2

)(2

1

),,(

1

2

2

23

1

2

22

1

2

3

22 (13)

3.4. Total Cost Function of the entire supply chain system under

supply disruption

The total recovery cost of the system, TC, for the three stage integrated supply

chain system subject to supply disruption may be written as

TC = TCS

+ TCM

+ TCR (14)

Combining all of the above equations will give the total cost function of the

complete system as:

( )( )

( )( ) ( )

( )( ) ( )+

+

+−++−⋅

+

+⋅++

−++++

++

⋅

++++⋅

=

∑

∑

∑

−

=

=−

SD

z

iMiMMMO

Si

MiMO

SiSiSi

Si

m

iMSD

M

Si

Si

Si

SDid

Si

SD

SD

z

i Si

SDISDiid

Si

SiSi

RiMiMiSDi

Tn

nfSBqSQD

HzA

zQTH

zALqLBq

D

Q

P

XT

P

quB

Tn

nfnfP

XXIqTq

PHzA

znSXznSXTC

1

2

13

1

2

22

1

2

1

11

1

2

12

2

111

12

1

2211

)(2

1

22

)()(2

1

2

1

),,,,,,,(

( )( ) ( )

( )( )

( ) ( )

( )( )

( )( )

( )( ) ( )

( )( )RRRR

M

Md

M

R

M

z

iRiRRR

RR

RMRMP

M

Md

M

M

M

z

i M

MiMiid

M

MMS

m

iSidMdSiMiSiSi

MiMOi

m

i

MMM

Si

MO

Si

SDd

Si

M

SD

z

iMiMMMO

Si

MiMO

LqLBq

D

Q

P

XT

P

quB

Tn

nfSBqSQD

HzA

LqLBq

D

Q

P

XT

P

quB

Tn

nfnfP

XXIqTq

PHzA

QTHTQHzQTH

zA

LqLBq

D

Q

P

XT

P

quB

Tn

nfSBqSQD

HzA

+

−+++

+

+

+−++−⋅

+

+

−+++

+

++

⋅

++++⋅

+

+

++⋅

++

−+++

+

+

+−++−⋅

∑

∑

∑∑

∑

−

=

=−

−

=

2

)(2

1

2

)()(2

1

2

1

2

2

)(2

1

1

2

2

23

1

2

22

1

2

3

1

2

2

22

2

211

1

2

2

11

1

1

2

13

1

2

22

1

2

1

(15)



By solving the above model (15) for 2211 ,,,,,,, znSXznSX RiMiMiSDi subject to

the system constraints, one can obtain the optimal recovery plan for the three

stage supply chain system under supply disruption.

4. Solution Procedure

Due to the complexity of the problem, it is difficult to obtain an optimal solution for Eq. (1) by solving the three stage SC problem in one optimization model. Therefore, our proposed solution approach is partitioned into two phases. The

model will be divided into two sub-models, which will utilize the heuristics we have developed in the work by Hishamuddin et al. [12] as the efficient solution technique. Basically, heuristic P4 is the solution method to solve for the supply disruption case. This heuristic was originally the method to solve the recovery schedule between two nodes, but will be modified and implemented twice to solve for the three stage supply chain system. The steps of the algorithm to solve

the model for disruption types 1-4 is described here.

Phase I: Production-procurement policy between disrupted supplier and

manufacturer

Step 0: Initialize the system parameters

Step 1: If the supplier experiences disruption types 1-3, implement heuristic P4 to

solve for decision variables11,,, znSX MiSDi.

Otherwise for disruption type 4, perform Phase II.

Step 2: Record the solutions from Phase I.

Phase II: Production-procurement policy between manufacturer and retailer

Step 0: Initialize the system parameters

Step 1: For disruption types 1-4, implement heuristic P4 to solve for decision

variables22 ,,, znSX RiMi.

Step 2: Record the solutions from Phase II.

Step 3: Compute the performance measure, the total cost of the system, TC. TC is

computed using Eq. (1).

Illustrative Example

In this section, we present an illustrative example to demonstrate the effectiveness

of the solution procedure proposed in the previous section. Table 2 shows the base data used for the test problem.

Example: Supply disruption at Supplier 1 (Type 1)

Suppose three suppliers, a manufacturer, and a retailer, form a supply chain system with system parameters as in Table 2. Also suppose an unexpected supply disruption occurs at supplier 1 with magnitude 0.003. The optimal

recovery schedule for supplier 1 is determined using heuristic P4. The results are TC1 = 534423.1316, n1 = 3, z1 = 4, BqM = 1200, and LqM = 2653.805. Then,



solving for the manufacturer using heuristic P4 gives the following results: TC2 = 156551.8, n2 = 4, z2 = 5, BqR = 1200, and LqR = 4031.975. The total system cost is computed using Eq. (1). The total recovery duration, total backorders and total lost sales are obtained by adding the individual results from Phase I

and II. The results of the system are TC = 534423.1316, n = 7, Bq = 2400, and Lq = 6685.78.

Table 2. Input parameters for numerical example.

Cost

Unit

Suppliers Manufacturer Retailer

1 2 3

ASi 200 400 200

AMOi 20 50 20 AMS 200 AR 20

HSi 1.2 1.2 4 HM 1.8 HR 2

HMi 1.8 1.8 5

PSi 5000000 5000000 5000000 PM 5000000

DSi 4000000 4000000 4000000 D 4000000

QSi 25252.3 36115.8 14650.4 QMP 22619.19 QR 22619.19

TdS 0.003 0.003 0.003 Td

M 0.003 Td

R 0.003

BSD 1 1 1 BM 1 BR 1

LSD 15 15 15 LM 15 LR 15

5. Computational Studies

A sensitivity study was carried out to explore the effects of supply and

transportation disruption on the solution and total system cost functions. The impact of the supplier’s system parameters, setup and inventory holding costs, in particular (ASi, HSi), were examined by doubling these values. The experiment was designed to have a number of scenarios for the system that were created by setting different combinations of supplier parameters. Different values of the setup cost and inventory holding cost for the suppliers, were combined to obtain varying supplier lot sizes, QSi, which consists of

high, medium and low QSi.

Table 3 depicts the suppliers’ input parameters for the experiment. The other parameters, if not varied, are fixed and equal to that of Table 2 in Section 4. Since the system we consider has three suppliers, we combine different QSi’s for the suppliers and generate 7 scenarios. The design of the

experiment is shown in Table 4. Each scenario was tested for two cases of lost sales cost, namely low and high (LSD,M,R = 2, and LSD,M,R = 15 respectively), which sums to a total of 14 scenarios. For each scenario, we test disruption occurrence at suppliers 1, 2, 3 and the manufacturer, where the corresponding minimum total cost, which is the performance measure of the system, is obtained.

Table 3. Input parameters for suppliers.

Q Size ASi AMOi HSi HMOi PSi DSi QSi

High Q 400 50 1.2 1.8 500000 400000 36115.8

Medium Q 200 20 1.2 1.8 500000 400000 25252.3

Low Q 200 20 4 5 500000 400000 14650.5



Table 4. Input parameters for suppliers.

Scenario

Case 1

(LSD, M, R=15, BSD, M, R=1)

Supplier 1 Supplier 2 Supplier 3

Scenario 1 (C4S1) High High High

Scenario 2 (C5S1) Low Low Low

Scenario 3 (C6S1) Medium Medium Medium

Scenario 4 (C7S1) Medium High High

Scenario 5 (C8S1) Medium Medium High

Scenario 6 (C9S1) Medium Low Low

Scenario 7 (C10S1) Medium Medium Low

Scenario

Case 2

(LSD, M, R=2, BSD, M, R=1)

Supplier 1 Supplier 2 Supplier 3

Scenario 8 (C4S2) High High High

Scenario 9 (C5S2) Low Low Low

Scenario 10 (C6S2) Medium Medium Medium

Scenario 11 (C7S2) Medium High High

Scenario 12 (C8S2) Medium Medium High

Scenario 13 (C9S2) Medium Low Low

Scenario 14 (C10S2) Medium Medium Low

6. Results and Analysis

Figure 1 shows a plot of the total cost obtained for each scenario and each disruption location for supply disruption, when L is 15. Scenario C6S1 is taken as a benchmark for comparison with the other scenarios. It is easy to show by looking at the histogram for C6S1 and C7S1, that A and H have a

positive relationship with TC. When all other parameters are fixed, it can be shown that as Asi increases from 200 to 400 and AMOi increases from 20 to 50, the total cost, TC, also slightly increases. On the other hand, it can be seen by comparison of C6S1 and C9S1 that TC increases more significantly with the

increase of H. This indicates that H has higher sensitivity to TC as compared to A. However, it can be observed that, in a system that consists of medium Q (low H) and low Q (high H) suppliers, a disruption occurring at the medium Q supplier will not have as much of an impact to the system cost as compared to disruption occurring at the low Q supplier. In other words, the parameters of other suppliers do not have a significant effect on the system’s cost when a

certain supplier is disrupted.



Fig. 1. Total cost of different scenarios of supply disruption for the high L case.

Similar observations on TC patterns were found for the case where L = 2, as depicted in Fig. 2.

Fig. 2. Total cost of different scenarios of supply disruption for the low L case.

From the above observations, we note the following insights: Disruption at the supplier with higher H will cause a bigger impact to the system cost;

hence, the recovery cost will be higher. Therefore, it is important for the manufacturer to focus on the suppliers with higher holding cost when



planning appropriate strategies for risk management. In addition, it can be observed that the degree of sensitivity of L to TC is very low. Figure 3 shows

a plot of comparison between the cases L = 15 and L = 2 for three selected scenarios (6, 7, and 9). The results indicate that as L decreases from 15 to 2, TC slightly decreases for all scenarios.

Fig. 3. Total cost comparison for high and low L case of supply disruption.

In addition, supply disruption at the earlier stages, have more negative impact to the overall supply chain compared to disruption at the later stages.

This finding is in line with the observations of previous supply chain disruption

models in the literature.

7. Conclusion

In this paper, an optimization approach for the rescheduling problem of a three stage supply chain system with multiple suppliers was presented. The effects of disruption at different locations of the supply chain were examined. Furthermore,

suggestions on the prioritization of disruption mitigation strategies for the

suppliers were proposed.

Analysis of the experimental results provided useful managerial insights for a manufacturing firm. The setup cost and inventory holding costs of a supplier play

an important role in determining which supplier requires more emphasis on the implementation of disruption mitigation strategies. The supplier with higher holding costs should have the highest priority, especially in the event of supply disruption. For example, the manager of a manufacturing firm may select an

alternative supplier for the particular parts to avoid shortages during disruption.

Extending the model to a multi-echelon supply chain with multiple retailers is a worthwhile extension and is currently under way.



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