logistical support scheduling under stochastic travel times given an emergency repair work schedule

Computers & Industrial Engineering 67 (2014) 20–35

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Logistical support scheduling under stochastic travel times given anemergency repair work schedule q

0360-8352/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.cie.2013.10.007

q This manuscript was processed by Area Editor Joseph Geunes.⇑ Corresponding author. Tel.: +886 3 422 7151x34141; fax: +886 3 425 2960.

E-mail address: [email protected] (S. Yan).

Shangyao Yan a,⇑, Chih-Kang Lin b, Sheng-Yu Chen a

a Department of Civil Engineering, National Central University, Chungli 32001, Taiwanb Department of Transportation Technology and Logistics Management, Chung Hua University, Hsinchu 30012, Taiwan

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 September 2012Received in revised form 18 June 2013Accepted 14 October 2013Available online 26 October 2013

Keywords:Logistical support schedulingStochastic travel timeEmergency repair work scheduleTime–space networkHeuristic

Stochastic factors during the operational stage could have a significant influence on the planning resultsof logistical support scheduling for emergency roadway repair work. An optimal plan might thereforelose its optimality when applied in real world operations where stochastic disturbances occur. In thisstudy we employ network flow techniques to construct a logistical support scheduling model understochastic travel times. The concept of time inconsistency is also proposed for precisely estimating theimpact of stochastic disturbances arising from variations in vehicle trip travel times during the planningstage. The objective of the model is to minimize the total operating cost with an unanticipated penaltycost for logistical support under stochastic traveling times in short term operations, based on anemergency repair work schedule, subject to related operating constraints. This model is formulated asa mixed-integer multiple-commodity network flow problem and is characterized as NP-hard. To solvethe problem efficiently, a heuristic algorithm, based on problem decomposition and variable fixingtechniques, is proposed. A simulation-based evaluation method is also presented to evaluate theschedules obtained using the manual method, the deterministic model and the stochastic model in theoperation stage. Computational tests are performed using data from Taiwan’s 1999 Chi-Chi earthquake.The preliminary test results demonstrate the potential usefulness of the proposed stochastic model andsolution algorithm in actual practice.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Earthquakes are one of the most destructive types of naturaldisasters because they happen so quickly, affect such a wide area,and are impossible to predict accurately. Systematic monitoringhas shown that the majority of earthquakes have been distributedalong a belt-shaped area, the Circum-Pacific seismic zone, in whichprobably 80% of earthquakes occur. Taiwan is located in this zone,so is one of the areas in the world with the most earthquakes withover 50 destructive earthquakes being recorded before 1900 and asmany as 83 after 1900. The island’s characteristic mountainoustopography accounts for more than 30% of the total area. Whenstrong earthquakes take place, the resulting damage can disrupttraffic and lifeline systems in mountain areas, obstructing theoperation of rescue machines, rescue vehicles, ambulances and re-lief workers. The roadway system in remote mountain areas is thekey channel for transportation. The most important task after anatural disaster is to repair the damaged roadways in the leastamount of time.

The successful accomplishment of all roadway repair tasks thatneed to be performed by the repair work teams requires the in-time supply of various materials, such as fuel oil, needed machineparts, food and water, for rescue workers. Up to now, logistical sup-port planning for emergency repair work in Taiwan has been donemanually based on the decision-makers’ experience, a methodwhich is neither effective nor efficient. This could have a negativeimpact on emergency rescue effectiveness and damage reduction.Thus improvements to the manual method of scheduling the effi-cient delivery of materials and sustenance to work teams is animportant issue.

Some methods have been proposed in previous studies, butthese deal only with emergency repair and/or relief distributionscheduling. Studies on emergency repair include Kemball-Cookand Stephenson (1984), Knott (1988), Brown and Vassiliou(1993), Tamura, Sugimoto, and Kamimae (1994), Sato and Ichii(1996), Arimura, Tamura, and Saito (1999), Chen and Tzeng(1999), Fiedrich, Gehbauer, and Rickers (2000), Feng and Wang(2003) and Feng and Wang (2005). Relief distribution has also beenextensively researched, for example, by Ardekani and Hobeika(1988), Hobeika, Ardekani, and Han (1988), Ardekani (1992), Ohand Haghani (1996), Haghani and Oh (1996) and Barbarosoglu,}Ozdamar, and Çevik (2002). Yan and Shih (2009) proposed an

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S. Yan et al. / Computers & Industrial Engineering 67 (2014) 20–35 21

integrated model which incorporates emergency repair and reliefdistribution within the same framework. Studies concerning thescheduling of logistical support for emergency repair work after anatural disaster are sparse. Yan, Lin, and Chen (2012) recently pro-posed a logistical support scheduling model for a given emergencyrepair work schedule. The objective is to minimize the total short-term operating costs subject to time constraints and related operat-ing constraints. In this model an average travel time is used for eachtrip in actual operations, meaning that stochastic disturbances aris-ing from variations in vehicle travel times have been neglected.However, stochastic factors during the operational stage could havea significant influence on the planning results, especially for emer-gency repair work. An optimal plan might therefore lose its optimal-ity when applied in real world operations where stochasticdisturbances occur. When real stochastic travel times are not con-sidered in traditional deterministic models (such as Yan et al.(2012)’s model) based on average travel times, resources can beused too tightly, resulting in an overly optimistic ‘‘optimal’’ logisti-cal support schedule. In particular, the extra cost incurred foradjusting a planned logistical support schedule in real operationsis not typically incorporated into the traditional deterministic mod-els in advance. Thus, although the schedule appears good in theplanning stage, its performance may deteriorate during operationsdue to disturbances that arise from stochastic travel times, whichoften occur in real operations. In particular, an optimisticallyplanned schedule often needs to be repeatedly adjusted during ac-tual operation, incurring an additional cost (called the unantici-pated penalty cost). As a result, the deterministic model proposedby Yan et al. (2012) may not be effective in real operations.

Stochastic programming problems have been studied by many.In particular, the expected optimization concepts have recentlybeen employed to deal with planning problems subject to stochas-tic issues. For examples, see Mulvey and Ruszczynski (1995),Kenyon and Morton (2003), Yan, Chi, and Tang (2006) and Li, Tian,and Leung (2010). The objective functions of the expected optimi-zation models are generally designed to optimize the expected va-lue of all scenarios in order to reduce stochastic delays that occurin actual operations. These works, however, do not deal with thesame problem as faced in this study. Although they cannot be di-rectly applied to our problem, their concepts are useful referencesfor this current work. In this study, we employ network flow tech-niques to construct an expected optimization logistical supportscheduling model under stochastic travel times. The objective ofthe model is to minimize the expected total operating cost withan unanticipated penalty cost for logistical support given stochas-tic travel times in short-term operations, based on an emergencyrepair work schedule, subject to related operating constraints. Inparticular, the probability of a stochastic travel time for each tripis considered when developing a stochastic model, an extensionof Yan et al.’s (2012) model. The concept of time inconsistency be-tween the scheduled travel time and the stochastic model is alsoproposed for precisely estimating the impact of stochastic distur-bances arising from variations in vehicle trip travel times duringthe planning stage. An unanticipated penalty cost for schedulingeach vehicle trip is proposed to reflect the cost incurred in realoperations for having to adjust for possible early or late arrivalsat times other than those in the planned schedule. The problemis formulated as an integer multiple-commodity network flowproblem, and is characterized as NP-hard (Garey & Johnson,1979), making it difficult to optimally solve for realistic large-scaleproblems. To facilitate solution, we propose a heuristic algorithmbased on problem decomposition and variable fixing techniques.The method should help the authorities plan more effectivelogistical support schedules. Since it is not fair to compare theschedules obtained from stochastic and deterministic models in

the planning stage, because of their different arc designs and objec-tive functions, a simulated-based evaluation method is alsopresented to evaluate the schedules obtained from the manualmethod, the deterministic model and stochastic model in theoperation stage. It is expected that the simulation-based evalua-tion method will be a useful planning tool for logistical supporttransportation planners and operators to evaluate their plannedschedules and to further improve their operations.

The rest of the paper is organized as follows: First, the problem isdescribed and assumptions made. The stochastic and deterministicmodels and the solution algorithm are then proposed. A simulated-based evaluation method is also developed. Thereafter, a case studyregarding the Chi-Chi earthquake using data from real practices,Yan and Shih (2009) and Yan et al. (2012), is performed to evaluatethe proposed stochastic and deterministic models and the solutionalgorithm. Finally, conclusions and suggestions are given.

One example is the Chi-Chi earthquake that occurred in NantouCounty, Taiwan on September 21, 1999. It had a magnitude of 7.3,and took more than two thousand lives overnight.

2. The model

2.1. Notations

For the ease of description, the notations used in the modeldevelopment are listed as follows:

Decision variables
xn
ij
the arc (i, j) flow in the nth vehicle-flow network(units: logistical support vehicle)
yn;dij

the arc (i, j) flow in the (n,d)th logistics-flow network(associated with the nth vehicle and the dth kind ofmaterials); note that the food-flow time–spacenetworks can be numbered in ascending orderfollowing all other material-flow time–spacenetworks
Parameters
CRnij
the arc (i, j) cost in the nth vehicle-flow network
CMn;dij

the arc (i, j) cost in the (n,d)th logistics-flow network

C
an external vehicle use cost URn
ij
the arc (i, j) flow’s upper bounds in the nth vehicle-flow network
UMn;dij

the arc (i, j) flow’s upper bounds in the (n,d)thlogistics-flow network

cn;d;sij;s

an index indicating the corresponding relationshipbetween the stochastic travel arc (i, j) and thestochastic scenario s for vehicle trip s in the (n,d)th

logistics-flow network; if cn;d;sij;s equals 1, which means

that the stochastic travel arc (i, j) corresponds to the

stochastic scenario s for the trip s; otherwise (cn;d;sij;s

equals 0), the arc (i, j) does not correspond to thestochastic scenario s for the trip s

ani
the ith node supply in the nth vehicle-flow network
qdw

the total supply of the dth kind of materials for thewth work station

bdj

the dth material demand for the jth demand timepoint

Ed
the number of standard material equivalents per unitfor the dth kind of materials (unit: standard materialequivalent)
En
the carrying capacity of the nth vehicle (unit:standard material equivalent)
(continued on next page)

22 S. Yan et al. / Computers & Industrial Engineering 67 (2014) 20–35

N
the set of all logistical support vehicles NE the set of all external vehicles NP the set of all kinds of support materials W the set of all work stations L the set of all vehicle trips NRn the set of all nodes in the nth vehicle-flow network NMn,d the set of all nodes associated with work stations and
intersections in the (n,d)th logistics-flow network
DSd the set of all supply nodes for the dth kind of
materials
NDn,d the set of all nodes associated with demand time
points in the (n,d)th logistics-flow network

SSn;dw

the set of all first nodes associated with the wth workstation in the (n,d)th logistics-flow network

VAn
the set of all arcs in the nth vehicle-flow network VRn the set of all stochastic travel arcs in the nth vehicle-
flow network
SAn,d the set of all arcs in the (n,d)th logistics-flow network SRn,d the set of all stochastic delivery arcs in the (n,d)th
logistics-flow network
COn,d the set of all collection arcs in the (n,d)th logistics-
flow network

DPn;dj

the set of all arcs pointing to the jth demand timepoint from the last delivery time point (or the startingtime point if there is no previous delivery time point)in the (n,d)th logistics-flow network

DNn
the set of dispatch arcs pointing to the first nodesassociated with all work stations in the nth vehicle-flow network
TPn;d;ss

the unanticipated penalty cost for the stochasticscenario s in the (n,d)th logistics-flow network for avehicle trip s

pws
the probability of early or late arrival scenario w for a
vehicle trip s
tw;ss the inconsistent time (in minutes) between the early
or late arrival scenario w and the scenario s whosetravel time is scheduled for a vehicle trip s, w – s

ld;w;ss
the unanticipated penalty cost of the dth kind of
materials per minute between the early or late arrivalscenario w and the scenario s whose travel time isscheduled for a vehicle trip s

Ss
the set of all stochastic travel times (or scenarios) fora vehicle trip s
DVRn
the set of all travel arcs in the nth vehicle-flownetwork
DSRn,d
the set of all delivery arcs arc in the (n,d)th logistics-flow network
2.2. Problem description and model assumptions

This study focuses on the development of a logistical supportscheduling model in Taiwan. For ease of modeling, we first definecommon modeling terms similar to those used in Yan and Shih(2009). Roadways refer to the main arteries in rural areas inTaiwan. An intersection is a point where two or more main arteriescross. A segment denotes a piece of roadway that links twointersections. A repair point located in a segment represents thelocation of a damaged area that needs repair by a work team. Morethan one kind of material may be needed for repairs at each repairpoint. A work team schedule covers a sequence of contiguousintersections and segments in which repair points may beincluded. In actuality, the goal of repair work after a devastatingearthquake is to reconnect the roadway-network in remotemountain areas as soon as possible. The emergency roadway repair

work typically involves making a deck or excavating a path thatwill allow the most urgently needed work/rescue teams to passthrough. This means that it is not necessary to include materialssuch as sandstone, cement, asphalt or concrete for logistical sup-port planning of emergency roadway repair work. Because bothstochastic model proposed in this study and Yan et al.’s (2012)model deal with similar logistical support scheduling problemsafter natural disasters, the related assumptions considered in theirmodel are also considered in the stochastic model. The main infor-mation and conditions of the disaster area based on real practicesin Taiwan, are given as follows:

(1) Given the mountainous topography and road width con-straints in the area, at most two work teams can access aroadway segment from either end to repair a segment.

(2) Two types of support materials are needed, those requiredfor machine maintenance (such as fuel oil and machineparts) and those for the workers (such as food and water).Note that these support materials, coupled with severaltypes of vehicles (such as trucks and jeeps), are allocatedto work stations for the convenience of logical support anddistribution. For simplicity, all types of logistical supportmaterials are transformed into equivalent units.

(3) For simplicity, it is assumed that all vehicles will be suppliedby the relevant governmental engineering department. Iftheir own vehicles are insufficient, authorities can assign ahigher cost for logistical support vehicles dispatched fromother government units. Note that there could be severalvehicle types (such as trucks and jeeps) with differentcarrying capacities and operating costs in use.

(4) Transshipment of logistical support materials betweendifferent vehicles is avoided.

(5) In order to reduce the length of time needed for the convey-ance of support materials, support vehicles do not need toreturn to the same work stations from where they depart.

(6) The total demand for the logistical support plan is given,because, based on the emergency repair work schedule,the quantity of support materials demanded for each repairpoint can be estimated in advance. It is also assumed thatevery work station can adequately supply each kind of sup-port material.

(7) The travel time distribution for each vehicle trip follows aspecific probability distribution based on real operations,which is adjustable for other applications. Note that road-way segments can become damaged and cluttered after anearthquake, and conditions on every segment will be differ-ent, making the probability distribution of travel times asso-ciated with these road segments more or less independent ofeach other. Hence, for simplicity, travel time probability dis-tributions are assumed to be independent.

(8) To apply the stochastic travel times to the time–space net-work, some of the possible travel times can be substitutedby a stochastic travel arc for each vehicle trip, within theacceptable error. The probability of each stochastic travelarc is simultaneously determined by looking at the numberof possible travel times for this stochastic travel arc out oftotal possible travel times. Note that the scheduled travelarc for each vehicle trip can be decided on from all possibletravel arcs by the planner.

(9) The purpose of logistical support scheduling is to transportthe support materials to each repair point before the timespecified when a work team should start work at a repairpoint. Only the following are considered in the modeling:total short-term operating cost, including vehicle deadhead-ing cost (which includes the costs of fuel cost consumption,vehicle maintenance, vehicle insurance, etc.); the cost of


transporting the material (mainly the incremental cost offuel consumption); the cost of using external vehicles (thatare directly related with this short-term operation); andpenalty costs for delays/early arrivals resulting from sto-chastic travel times that will be discussed later. Other fixedcosts, such as salary for drivers who are governmentemployees, vehicle ownership costs, and so on, that are inde-pendent of this short-term operation, are excluded from themodeling.

To summarize, the general assumptions based on the logisticalsupport planning for emergency repair work are considered inpoints (2), (6)–(8); the specific assumptions based on real practicesin Taiwan are considered in points (1), (3)–(5). In addition, points(7) and (8) are specific to the stochastic model.

2.3. Modeling approach

A time–space network technique is utilized in this study toformulate the vehicle/support materials scheduling and routingdue to its flexibility and effectiveness reported in literature. Forexamples, see Haghani and Oh (1996), Oh and Haghani (1996),Yan and Tseng (2002), Yan and Chen (2002), Chardaire, Mckeown,Verity-Harrison, and Richardson (2005), Yan and Shih (2009) andYan et al. (2012). The major elements in the modeling, which in-clude the vehicle-flow time–space networks, the logistics-flowtime–space networks, the constraints, and the mathematical for-mulation, are described below.

2.3.1. The vehicle-flow time–space networksIn practice, each work station may hold various types of

vehicles that can be used to deliver support material. Sometimesit is necessary to dispatch extra vehicles from other governmentunits (external vehicles), when the authority’s own vehicles (inter-nal vehicles) are insufficient. Thus, the vehicle-flow time–spacenetworks comprise both internal and external vehicle-flow time–space networks, which are slightly different in design. The twokinds of vehicle-flow time–space networks are described below.

2.3.1.1. The internal vehicle-flow time–space networks. Each net-work, as shown in Fig. 1, is used to formulate the movement ofone internal logistical support vehicle within a specific time periodand among some certain locations in the disaster area. The hori-zontal axis represents the work stations, the intersections and re-pair points in order; whereas the vertical axis stands for the timeduration. A node, with the exception of a collection node, standsfor a work station, an intersection, or a repair point at a specifictime. In the network, the supply node is a specific node associatedwith a work station at the starting time (with a supply of 1) for theassociated vehicle (or associated with a repair point at a middletime for an already-dispatched vehicle). For ease of modeling,any arc in the network linking two nodes (such as a work station,an intersection, a repair point, a collection node, or a supply node)is called a vehicle trip. The collection node is a demand node (witha demand of 1) and is used to ensure flow conservation. Two spe-cific types of time points associated with each repair point in thegiven work team schedule are defined as follows: the ‘‘demandtime point’’ denotes the last time point when the material requiredfor the repair work has to be delivered; and the ‘‘repaired timepoint’’ indicates the time point when vehicles can begin to passthrough the repair point, meaning that the repair work has beencompleted. Note that repair points that do not require any supportmaterials do not have a demand time point (or their material de-mands are zero). In addition, every repair point may have zero ormore than one demand time point for each associated type of sup-port material, but exactly one repaired time point. An arc denotes

the activity of the associated vehicle. The arc flows express thevehicle flows in the network.

The arc flow’s lower and upper bounds are defined as the min-imum and maximum number of allowable flow units in the arc.The three types of arcs are defined below.

1. Stochastic travel arc

A stochastic travel arc, as shown in Fig. 1(a)-(d), represents themovement of a vehicle with a stochastic travel time: (a) from awork station to an intersection or vice versa; (b) from an intersec-tion to a repair point or vice versa; (c) from an intersection to an-other intersection; or (d) from a repair point to another repairpoint. All possible stochastic travel arcs, each with a possible traveltime for a vehicle trip, are formed in the network. The time blockfor a stochastic travel arc denotes the time window between thetrip starting time and the trip ending time which is equal to thetrip starting time plus the possible travel time. To take into consid-eration stochastic travel times for a vehicle trip, we set several pos-sible travel arcs for a trip, each associated with a travel time and aprobability. The total probability for all stochastic travel arcs, asso-ciated with a vehicle trip, equals one. These arcs are designed todetermine a suitable travel time for each vehicle trip in theplanned schedule. The concept of time inconsistency between thescheduled travel time and the stochastic one is discussed below.If a stochastic travel time (occurring in real operations) is differentthan the scheduled travel time, then the vehicle trip is consideredinconsistent with the scheduled travel time. For example, in Fig. 2,suppose that there are only four possible travel times (30, 45, 60and 75 min, respectively), each with a probability, for a vehicle trip.There are thus four travel arcs, x0, x1, x2 and x3, set for this vehicletrip. Also assume that x2 is the planned travel arc for this vehicletrip (a travel time of 60 min is given to this vehicle trip in theplanned schedule). A comparison of all four possible travel arcsshows that the times for x0, x1 and x3 are inconsistent with thatfor x2, differing by �30, �15 and 15 min, respectively (in minimi-zation problems a positive value denotes delay, and vice versa).The penalty costs associated with early or late transportation ofthe required materials as a consequence of variations in traveltimes will be discussed in Section 2.3.2.1.

Note that all stochastic travel arcs for the vehicle trip connectedto a repair point after the last of its demand time points can beremoved, because after this time no material/food needs to betransported by the associated vehicle. Removing these stochastictravel arcs will cause the vehicle trip to be scheduled earlier andthus producing a more conservative schedule so as to prevent de-lays for repairing the repair point. The arc cost is the short-termoperating cost of transporting with the vehicle associated with thismovement (and the associated travel time). The arc flow’s upperbound is one (meaning that at most one vehicle moves withinthe associated travel time). The arc flow’s lower bound is zero.

2. Holding arc

A holding arc, shown as (e)-(g) in Fig. 1, indicates the holding of avehicle: (e) at a work station; (f) at an intersection; or (g) at a repairpoint within a specific time window. The arc cost represents the costof holding a vehicle with its engine idling in this time window. Thiscost is generally much smaller than the traveling arc cost. The arcflow, which is a binary variable, denotes the number of vehicles (0or 1) held at a work station, an intersection, or a repair point withina time window. The holding arc flow’s upper bound is one, meaningthat at most one vehicle is held at the station, intersection, or repairpoint during the specific time window. The arc flow’s lower bound iszero, indicating that no vehicle is held at the station, intersection, orrepair point in this time window.

Fig. 1. Internal vehicle-flow time–space networks.


3. Collection arc

A collection arc, shown as (h) in Fig. 1, connects the last nodeassociated with a work station and the node associated with a re-pair point in every time point to the collection node. All collectionarcs are used to ensure flow conservation at the collection nodeand other related nodes. The arc cost is zero if it is connected toa work station, and is the cost of deadheading a vehicle back tothe nearest work station if it connects to a repair point. The arcflow’s upper bound is one and the arc flow’s lower bound is zero,meaning that the arc flow is a binary variable.

To summarize, the stochastic travel arc and the holding arc areconsidered for real activities in actual operations; the collection arcis considered for an auxiliary activity used to ensure flow conserva-tion in network design.

2.3.1.2. The external vehicle-flow time–space network. An externalvehicle-flow time–space network is used to formulate the dispatch

of an extra vehicle from other government units in cases where theauthority’s own vehicles are insufficient. All external vehicles canbe numbered in ascending order following internal vehicles. Thisnetwork is similar to the internal vehicle-flow time–space networkexcept for the new dispatch node (with supply of 1) and new dis-patch/retrieval arcs. There are two additional types of arcs definedbelow.

1. Dispatch arc

A dispatch arc connects the dispatch node to the first node,associated with a work station at the starting time, and is usedto decide whether to dispatch an external vehicle to a suitablework station or not. The arc cost represents the cost of using anexternal vehicle, which is normally much higher than the cost ofusing an internal vehicle. The arc flow’s upper bound is one andthe arc flow’s lower bound is zero, meaning that the arc flow is abinary variable.

Intersection/ Repair point Repair point

9:30

8:00

09:45

10:00

10:15

10:30

10:45

P1=0.25

P3 =0.15P

4 =0.15

x1 (30 minutes)

x0 (45 minutes)

x2 (60 minutes)

x3 (75 minutes)

Stochastic travel arcs: x0, x1, x2, x3

P2=0.4

Fig. 2. Stochastic travel arcs in the vehicle-flow time–space network.


2. Retrieval arc

A retrieval arc connects the dispatch node to the collectionnode, and is used to decide when the associated external vehicleshould be used or not. The arc cost is zero, because no cost is in-curred if the associated vehicle is not used (when the arc flow isone). The arc flow’s upper bound is one and the arc flow’s lowerbound is zero, meaning that the arc flow is a binary variable.

Note that external networks are designed to avoid yieldinginfeasible solutions for the logistical support schedule. Some exter-nal vehicle-flow time–space networks can be added to the modelbefore solving this problem.

Similar to the internal vehicle-flow time–space networks, thestochastic travel arc, the holding arc and the dispatch arc in theexternal vehicle-flow time–space network are considered for realactivities in actual operations; the collection arc and the retrievalarc are considered for auxiliary activities used to ensure flow con-servation in network design.

2.3.2. The logistics-flow time–space networksThe time–space network technique is also applied to formulate

the movement of materials and food in the logistical support vehi-cles. For ease of modeling, the networks are designed to corre-spond to the vehicle-flow time–space networks. The definition ofa vehicle trip is the same as those in the vehicle-flow time–spacenetworks. There are two types of support materials that need tobe considered in the model: materials for machine maintenancesand materials for human sustenance. The support materials formachine maintenances mainly include fuel oil, machine parts,and the like. That for human sustenance includes lunch and dinnerboxes and drinking water. Specific support materials/food is trans-ported through the time–space networks without transshipment.

In total, two types of logistics-flow time–space networks, material-and food-flow time–space networks are designed.

2.3.2.1. The material-flow time–space network. Each material-flowtime–space network, as shown in Fig. 3, corresponds to one specifickind of material on one vehicle within a certain time period be-tween specified locations. A group of material-flow time–spacenetworks is designed for one kind of material, each associated witha vehicle. Suppose that there are four kinds of material that need tobe transported and three vehicles are arranged for conveyingmaterials. There will then be 12 (=4 � 3) material-flow time–spacenetworks. In these networks, the horizontal and vertical axes arethe same as those in the vehicle-flow time–space networks. Herea node also represents a work station, an intersection, or a repairpoint. There are also a number of supply nodes and a collectionnode associated with each group of material-flow networks,depending on the kind of material. These are used to ensure flowconservation for the associated kind of material. It is assumed thatevery supply node can provide adequate amounts of material.Thus, it must be decided at each supply node how much and whichvehicle the material will be transported by. Material remainingafter delivery is finished is collected at the collection node. Twospecific types of time points for each repair point, the ‘‘demandtime point’’ and the ‘‘repaired time point’’, are designed. They arethe same as in the vehicle-flow time–space networks. The arc flowsexpress the flow of materials in the network. Altogether, there arefour types of arcs.

1. Stochastic delivery arc

A stochastic delivery arc, as shown in Fig. 3(a)-(d), representsthe transportation of the dth material by the specified vehicle witha stochastic travel time: (a) from a work station to an intersection;(b) from an intersection to a repair point and vice versa; (c) from anintersection to another intersection; or (d) from a repair point toanother repair point. The material-flow time–space networks aredesigned to correspond to the vehicle-flow time–space networksand all support materials are transported by vehicles. Therefore,the stochastic delivery arcs in each material-flow time–space net-work correspond to the stochastic travel arcs in the vehicle-flowtime–space network associated with the same vehicle. The timeblock required for a trip is calculated as from the time when trans-portation of the material begins to the time when the material ar-rives and is unloaded. Note that there is no arc connecting anintersection to a work station in the network, because it is assumedthat all materials will be unloaded from the vehicle at the repairpoints or collected at the collection node. Also, any stochasticdelivery arcs connected to a repair point after the last time associ-ated with its demand time points can be removed, because afterthis time no material is needed.

The arc cost is the cost of transporting the material associatedwith the specific vehicle, plus an unanticipated penalty cost thatreflects the cost incurred in real operations for adjusting possibleearly or late arrivals other than the planned schedule for the vehi-cle trip. Early delivery of materials required for repair work can sig-nificantly reduce the time and cost for preparation tasks. Moreover,if the demand of the repair work team is not supplied at the righttime, the work could be delayed, which would not only affect therescue efficiency but might also increase human injuries. In prac-tice, work teams will be stuck at the repair point where the deliv-ery of the support materials is late. To reflect stochasticdisturbances arising due to variations in travel times, we designan unanticipated penalty cost for late or early arrival of the re-quired materials. In particular, a positive inconsistent time meansthat the delivery of support materials is late, and the cost forspeeding up the work will be increased. On the other hand, if the

Fig. 3. Material-flow time–space networks for the dth material.


inconsistent time for a scenario is negative, it means that the deliv-ery of support materials is early, and the cost for preparation tasksbefore repair work beginning can be decreased. Note that if thematerial delivery time is later than the scheduled time, then thework team will be stuck at the repair point due to the lack of re-quired materials. To suitably avoid the late delivery of the requiredmaterials, a large crash cost can be set to reflect the cost incurredfor temporarily ceasing the work team’s work at the repair pointaccording to real practices. If, in other applications, there are sparematerials for a repair point, then the crash cost can be adjusted tobe a smaller penalty to reflect the delay of materials delivered.Consequently, the unanticipated penalty cost will only happenfor a travel arc from an intersection to a repair point or from a re-pair point to another repair point, so as to reflect the cost incurredin real operations for adjusting for possible early or late delivery ofmaterials other than the planned schedule for the vehicle trip. Thatis, the unanticipated penalty cost will not be added to a stochasticdelivery arc from a work station to an intersection and from anintersection to another intersection. Note that for the purpose ofthe model, a scenario denotes a set of stochastic travel times asso-ciated with all vehicle trips that form all vehicle routes, which arerandomly generated from realizations of all stochastic events. Inparticular, for a vehicle trip, which is a part of a vehicle route, a sce-

nario corresponds to a stochastic travel time. The unanticipatedpenalty cost is calculated as follows:

TPn;d;ss ¼

Xw2Ss ;w–s

pws tw;s

s ld;w;ss ð1Þ

Eq. (1) denotes the calculation of the unanticipated penalty cost fora vehicle trip. For example, as shown in Fig. 4, to calculate the TPn;d;s

s ,we assume that the probability of each possible delivery arc (i.e.,each scenario) for this trip is 0.25, 0.4, 0.15 and 0.2 in increasingorder of travel time. A comparison of all possible delivery arcsshows that, with respect to x2, travel times are inconsistent fordelivery arcs x0, x1, x2 and x3 by �30, �15, 0 and 15 min, respec-tively. For simplicity, let l+/l� be the unanticipated penalty costper minute for late/early arrival. The unanticipated penaltycost for x2 ðTPn;d;2

s Þ can thus be calculated by 0.25 � (�30) � l�

+ 0.4 � (�15) � l� + 0.15 � (15) � l+. Besides, the stochasticdelivery arc flow’s upper bound is the associated vehicle capacity(in equivalent units). The arc flow’s lower bound is zero.

2. Holding arc

A holding arc, shown as (e)-(g) in Fig. 4, indicates the holding ofmaterials: (e) at a work station; (f) at an intersection; or (g) at a


repair point in a specific time window. The holding arc cost is zero(or very small) because the cost incurred for holding materials atone place in short-term operations is very small. The arc flow’supper bound is infinity. The arc flow’s lower bound is set to be zero,denoting that no material is held during the time window.

3. Collection arc

A collection arc, shown as (h) in Fig. 4, connects the last nodeassociated with a work station for each network associated withthe dth material to the collection node. It is used to ensure flowconservation of all collection arc flows of the dth material in thesenetworks. The arc cost is zero. The arc flow’s upper bound is the to-tal amount of the dth material supplied from all work stations. Thearc flow’s lower bound is zero, meaning that no material remainsat the associated work station.

4. Supply arc

A supply arc, shown as (k) in Fig. 4, connects a supply node tothe first node associated with a work station for each networkassociated with the dth material. It is used to decide how muchof the dth material to supply. The arc cost is zero. The arc flow’supper bound is the total amount of the dth material supplied toall work stations. The arc flow’s lower bound is zero, meaning thatno material is supplied to the associated work station.

To summarize, the stochastic delivery arc, the holding arc andthe supply arc are considered for real activities in actual opera-tions; the collection arc is considered for an auxiliary activity usedto ensure flow conservation in network design.

2.3.2.2. The food-flow time–space network. Each food-flow time–space network corresponds to the food transported on one vehiclewithin a certain time period between specified locations. The net-work is similar to the material-flow time–space network, except

9:30

8:00

09:45

10:00

10:15

10:30

10:45

Fig. 4. Example of setting delivery penalty cost for a stochastic delivery arc.

that food, specifically lunch/dinner boxes and water, must be deliv-ered to each repair point before a specific time. Realistically, thelunch/dinner boxes can be kept for 2 h. There are thus two specifictime windows set for food delivery to the repair point (10:00–12:00, i.e., the 17th time to the 25th time for lunch and 16:00–18:00, i.e., the 41th time to the 49th time for dinner). Note that atime unit is 15 min long in this study. In other words, there areat most two demand points for food at each repair point in oneday. Finally, all arcs in these two time windows which are the sameas those in the material-flow time–space networks are generated.The arcs outside these two time windows are removed.

Similar to the material-flow time–space network, the stochasticdelivery arc, the holding arc and the supply arc are considered forreal activities in actual operations; the collection arc be consideredfor an auxiliary activity used to ensure flow conservation in net-work design.

2.4. Model formulation

Based on the vehicle- and logistics-flow time–space networksintroduced above, as well as the operating constraints, the modelis formulated as an integer multiple-commodity network flowproblem. The model is based on the expected optimization con-cept. According the design of both penalty costs, the model is for-mulated as follows:

MinimizeE½zs� ¼Xn2N

Xij2VAn

ðCRnij � xn

ijÞ þ C �Xn2NE

Xij2DNn

xnij

þXn2N

Xd2NP

Xij2SAn;d

CMn;dij � yn;d

ij

� �

þXn2N

Xd2DP

Xs2L

Xij2SR

Xs2Ss

Xw2Ss ;w–s

pws � tw;s

s � ld;w;ss

!

� cn;d;sij;s � yn;d

ij ð2Þsubject toX

j2NRn

xnij �

Xk2NRn

xnki ¼ an

i ; 8i 2 NRn; n 2 N ð3Þ

Xj2NMn;d

yn;dij �

Xk2NMn;d

yn;dki ¼ 0; 8i 2 NMn;d; n 2 N ; d 2 NP

ð4ÞXn2N

Xj2SSn;d

w

yn;dij ¼ qd

w; 8i 2 DSd; d 2 NP ;w 2W ð5Þ

�Xn2N

Xij2DPn;d

j

yn;dij ¼ �bd

j ; 8j 2[

n

NDn;d; d 2 NP ð6Þ

�Xn2N

Xij2COn;d

yn;dij ¼ �

Xw2W

qdw þ

Xj 2 [

nNDn;d

bdj ; 8d 2 NP ð7Þ

Xd2NP

ðEd � yn;dij Þ 6 ðEn � xn

ijÞ; 8ij 2 VRn; n 2 N ð8Þ

xnij ¼ 0 or 1; 8ij 2 VAn

; n 2 N ð9Þ

0 6 yn;dij 6 UMn;d

ij ; 8ij 2 SAn;d; n 2 N; d 2 NP ð10Þ

yn;dij 2 INT; 8ij 2 SAn;d

; n 2 N; d 2 NP: ð11Þ

The objective function (Eq. (2)), used to optimize the expectedshort-term operating cost with a unanticipated penalty cost forlogistical support for all possible travel times, includes a summa-tion of vehicle operating cost, external vehicle use cost, materialtransportation cost and penalty cost. Eqs. (3) and (4) ensure flowconservation at every node in each vehicle- and logistics-flow net-work. Eq. (5) denotes that the total supply of all materials is ade-quate for every work station. Eq. (6) denotes that total demand


of every kind of materials for each demand time point is satisfied.Eq. (7) ensures flow conservation at the collection node associatedwith the logistics-flow networks for each kind of material. Eq. (8)ensures that the sum of all materials transported in each vehicledoes not exceed that vehicle’s capacity for every trip betweentwo locations. Eq. (9) indicates that all arc flows in the vehicle-flownetworks are either zero or one. Eq. (10) holds all the arc flows inthe logistics-flow networks within their upper and lower bounds.Eq. (11) ensures the integrality of the arc flows in the logistics-flownetworks.

2.5. Model discussions

In this sub-section, the deterministic model which was simpli-fied from the proposed stochastic model is first discussed. Then,two techniques for setting the probability functions associatedwith a vehicle route recently used for the development of stochas-tic models with the expected optimization concepts are discussed.Finally, the incorporation of trip buffer times into the deterministicmodel to resolve stochastic trip travel times is discussed.

The deterministic logistical support scheduling model (DM),which uses the average travel times, is very similar to the abovestochastic model (called SM), except for the objective function,the stochastic travel arcs and the stochastic delivery arcs. TheDM is a simplification of the SM. All stochastic travel arcs in thevehicle-flow networks and all stochastic delivery arcs in the mate-rial-flow networks can be replaced by a travel arc and a deliveryarc, which are both based on the average travel times. The objec-tive function is replaced by a deterministic objective function.The unanticipated penalty cost terms in the DM are removed. Tosave space, we only list those modifications from the DM, whichare different from the SM, as follows.

(1) The objective function, Eq. (2) is replaced by Eq. (12) in theDM. The objective value of this equation contains only thefirst three terms in the SM.

Minimize z ¼Xn2N

Xij2VAn

CRnij � xn

ij

� �þ C �

Xn2NE

Xij2DNn

xnij

þXn2N

Xd2NP

Xij2SAn;d

CMn;dij � yn;d

ij

� �ð12Þ

(2) Two notations/symbols are modified for use in the DM. VRn,used in the vehicle-flow networks is replaced by DVRn; andSRn,d, used in the material-flow networks is replaced byDSRn,d.

(3) The other constraints in the DM are the same as those in theSM.

Note that the DM, a simplification of the SM, is exactly equal toYan et al.’s (2012) model, meaning that their model is a special caseof the SM. The major differences between Yan et al.’s (2012) model(i.e., the DM) and the SM include the designs for the trip traveltime, the time inconsistency for scheduling each trip, and theunanticipated penalty cost for scheduling each trip. In particular,in the DM an average travel time is used for formulating each trip,while several stochastic travel times, each with a probability, isused for formulating each trip. That is, a travel arc is extended toseveral stochastic travel arcs associated with a vehicle trip in theinternal and external vehicle-flow time–space networks, and adelivery arc is extended to several stochastic delivery arcs associ-ated with a material/food delivery trip in the material-flow andfood-flow time–space networks. In addition, the concept of timeinconsistency between the scheduled travel time and the stochas-tic travel time for a vehicle trip or a material/food delivery trip isalso proposed for precisely estimating the impact of stochastic dis-

turbances in vehicle and material/food delivery trip travel timesduring the planning stage. Moreover, an unanticipated penalty costfor scheduling each vehicle or material/food delivery trip, as shownin Eq. (1), is proposed to reflect the cost incurred in real operationsfor adjusting to possible early or late arrivals other than theplanned schedule for each vehicle trip. Finally, the objective func-tion of the DM, the short-term operating cost, is extended to theexpected short-term operating cost with an unanticipated penaltycost. It should be mentioned that a simulation-based evaluationmethod is developed to evaluate the real performance of the pro-posed schedules obtained from the DM and the SM in the operationstage (to be addressed in Section 3.2). Numerical tests will also beperformed later as discussed in Section 4.

Note that stochastic programming problems with the expectedoptimization concept usually assume that model parameters fol-low probability distributions and find a solution that is feasiblein all the scenarios. Concerning the probability function associatedwith a vehicle route, in the past the ‘‘route-based’’ (also calledpath-based) method is used to set a probability function. For exam-ple, Li et al. (2010) recently proposed a stochastic programmingmodel for a stochastic vehicle routing problem. A probability func-tion associated with a vehicle route is set which is indeed a vari-able related to the route decision. With the probability function,the penalty coupled with the decision on the vehicle route canbe accordingly measured. However, the probability function, whichshould be a joint probability function of the vehicle’s trips along aroute, could be complicated to estimate in practice. Additionally, alarge number of complicated probability functions for all vehicleroutes should be accurately estimated in advance and incorporatedinto the problem. Both requirements are difficult to do in practice,especially for our problems. Moreover, with these variable proba-bility functions, the problem becomes a nonlinear integer programwhich is more difficult to solve than traditional vehicle routingproblems (formulated as integer linear programs) which are char-acterized as NP-hard (Garey & Johnson, 1979). As a result, the mod-el is difficult to formulate and to efficiently solve especially forrealistically large scales. To cope with this issue, instead of estimat-ing complicated conditional probability distributions for traveltimes along a vehicle route, we adopt a ‘‘trip-based’’ (also calledarc-based) method to set the probability for a vehicle trip with atravel time in order to construct a stochastic model. Several sto-chastic travel arcs associated with a vehicle trip are formed in thismodel. Note that the trip-based design of probability functions isvalid only if the trip travel times along a route are independentof each other, which is a suitable assumption for our model. Withthis trip-based design, the probability function associated with anarc can be set as a constant and our model becomes an integer lin-ear program. As a result, not only is the model easy to formulatewith an easy collection of roadway data that fits in with logisticalsupport practices, but it is also more tractable in terms ofoptimization.

Additionally, in the past, in deterministic models a suitable buf-fer time has been added to each trip in order to absorb stochasticdisturbances arising from variations in vehicle travel times duringreal-time operations; for example, see Yan and Chen (2002). How-ever, determining the most suitable buffer time length for each tripso as to resolve the stochastic travel times is a complicated issue,correlated with the trip characteristics, e.g., trip distance, trip den-sity, and others (e.g., see Yan, Tang, & Shieh, 2005). In theory, evenfor the same trip with different departure times and travel times,the effective buffer time could be different, due to the complicatedconnections of trips with all possible departure times and traveltimes. That is, finding the most effective buffer time for each triprequires a very complicated analysis of system optimization. Inparticular, if a buffer time is formulated as a variable and addedto a trip in the DM, then the extended model becomes a nonlinear


integer program, which is a very complicated problem to solvethan the DM (and even the SM). This also means that, without acomplete analysis, which could involve an infinite number of buf-fer time combinations, the true effectiveness cannot be ensured byusing a limited set of buffer times coupled with the DM to resolvestochastic disturbances arising from variations in vehicle traveltimes. Nevertheless, how to incorporate the variable trip buffertimes into the DM, coupled with the tests in comparison with theSM is suggested as a future research topic.

3. Solution algorithm and evaluation method

In this section, the solution algorithm proposed to efficientlysolve the SM/DM is discussed. Since the DM is a simplified problemof the SM, the solution algorithm developed based on the SM canalso be used to solve the DM. Since it is not fair to compare thesolutions obtained from the SM and the DM in the planning stagedue to different designs of the arcs and objective functions, an eval-uation method is also developed to evaluate the performance ofthe SM/DM in the simulated operation stage.

3.1. Solution algorithm

The SM and the DM can both be regarded as integer multiple-commodity network flow problems characterized as NP-hard (Garey& Johnson, 1979). It could be difficult to optimally solve realisticallylarge problems, making the model not applicable in emergent condi-tions. Referring to Yan et al. (2012), a heuristic algorithm based on aproblem decomposition technique is proposed, which, coupled withthe mathematical problem solver, CPLEX, can be used to efficientlysolve realistically large-scale problems within a reasonable solutiontime. Note that according to the characteristics of the logistical sup-port scheduling problem (i.e., point (3) in the modeling assump-tions), the SM/DM is mathematically feasible.

This solution algorithm is primarily solved based on a problemdecomposition method. In particular, the original problem is decom-posed into several smaller sub-problems, each being associated witha partial time window and some repair points. These sub-problemsare solved sequentially, using CPLEX. The first sub-problem is gener-ated by setting the time window from the starting time to a time t1

which is less than the ending time of the planning period, taking intoaccount some repair points. To ensure flow conservation, a numberof temporary arcs need to be added to the time–space network ineach sub-problem, as described in detail below.

(1) A number of temporary arcs, collection arcs, connect the lastnode associated with every work station at time t1 to the col-lection node in all internal/external vehicle-, material- andfood-flow time–space networks;

(2) A number of temporary arcs, collection arcs, connect thenode associated with a repair point in every time pointbefore time t1 to the collection node in all internal/externalvehicle-flow time–space networks.

When solving the first sub-problem by CPLEX, a number of arcvalues in the solution are fixed in the following sub-problems. Therules are described in detail below.

(1) All of the arcs, except the retrieval arcs and the temporarycollection arcs, in all internal/external vehicle-, material-and food-flow time–space networks, need to be fixed if arcvalues are positive.

(2) The retrieval arcs in the external vehicle-flow time–spacenetworks need to be fixed if their values are equal to zero.

(3) All temporary collection arcs added in this sub-problemneed to be removed.

Now one must check to see whether all repair points in the modelhave been satisfied. If the answer is yes, stop the algorithm and thesolution procedure is finished. If the answer is no, the time windowis then extended from another time period which is also less than theending time of the planning period. The second sub-problem is gen-erated for a new time window, in which some variables in the firsttime window are fixed using the aforementioned rules. After addingnew temporary arcs in the second sub-problem, CPLEX can be usedto solve the second sub-problem and fix some variable values basedon the aforementioned rules. The procedure is repeated until the de-mands for all repair points are satisfied. The solution is thus ob-tained. We finally use the flow decomposition method (Yan andYoung, 1996) to decompose the arc flows into arc chains, each repre-senting a vehicle’s route/schedule. It should be mentioned thatalthough the procedure is similar to Yan et al.’s (2012), the sub-prob-lems are different. In particular, the designs of trip travel time, thetime inconsistency for scheduling each trip, the unanticipated pen-alty cost for scheduling each trip, and the objective function in thesub-problems are all different from those in Yan et al.’s (2012)sub-problems, as discussed in Section 2.5.

Additionally, to preliminarily evaluate the performance of theproposed heuristic algorithm, a lower bound solution, called LBS,is developed by relaxing main operational restriction, i.e., notallowing the transshipment of support materials between differentvehicles. First, the original logistics-flow networks representing themovements of each kind of material are compressed into a logis-tics-flow network. Then, the supply nodes in the original logis-tics-flow networks associated with each kind of material can becompressed into a supply node, with arcs pointing to every workstation, as shown in Fig. 5. Based on this design, materials can betransferred between vehicles in the modified logistics-flow net-work associated with each kind of materials, meaning that a trans-shipment strategy of materials is added to modify the problem.Based on the aforementioned relaxations, the problem size wouldbe significantly reduced. In this way the number of logistics-flowtime–space networks can be reduced from |NP| � |N| to |NP|.Although the LBS yielded by this modified model is an infeasiblesolution in practice (because of relaxation of the operationalrestriction), it forms a lower bound solution to the original prob-lem (a minimization problem). Since the problem size for the mod-ified logistics-flow time–space networks has been greatlydecreased, the modified model can be directly solved using CPLEX.

3.2. Evaluation method

We also developed an evaluation method to evaluate the per-formance of the SM, the DM and the current scheduling practicein simulated operations since it is not fair to compare these modelsin the planning stage. First, we set up a number of simulation sce-narios. Then, in each scenario an actual travel time for each vehicletrip is randomly generated according to a given probability distri-bution of travel times. Thereafter, we compare every actual trip ar-rival time for each vehicle route with that in the SM, the DM andthe current scheduling practice. For example, assume that for avehicle route, the first actual trip travel time from point A (anintersection or a repair point) to B (a repair point) is 30 min, whilethe scheduled trip travel time for the DM/SM and the currentscheduling practice is 25, 20 and 35 min, respectively. This meansthe actual arrival time is inconsistent with the scheduled one witha difference of 5, 10, and �5 min. The unanticipated penalty costfor the DM/SM and the current scheduling practice in the first tripis thus 5l+, 10l+ and 5l�, respectively (l+ for late delivery,whereas l� is for early delivery). The vehicle’s actual schedule isalso adjusted according to the actual arrival time. Note that forearly arrivals, the vehicle will not be held. The comparisons of all

Fig. 5. Modified logistics-flow time–space networks.

Fig. 6. Objective values for different numbers of scenarios.


actual arrival times for the later trips with the DM/SM and the cur-rent scheduling practice for this vehicle route are similarly per-formed. When the evaluation of performance for the SM, the DMand the current scheduling practice in this scenario have been fin-ished, the objective value (i.e., the total operating cost plus anunanticipated penalty cost) for the SM, the DM and the currentscheduling practice, can be calculated. Finally, the evaluation pro-

cedure is repeated for all scenarios. The steps are listed in detailbelow:

Step 1: Let the preset number of simulation scenarios = M.Step 2: Set m = 1.Step 3: Randomly generate an actual travel time for each vehicletrip according to a given probability distribution of travel times.Step 4: Compare all actual trip arrival times with those in theDM/SM and the current scheduling practice for all vehicleroutes.Step 5: Calculate the objective value (i.e., the total operatingcost plus an unanticipated penalty cost) of the SM, the DMand the current scheduling practice for the mth scenario.Step 6: If m = M, then go to Step 7; otherwise, m = m + 1, andreturn to Step 3.Step 7: Calculate the statistical results obtained with the SM, theDM, and the current scheduling practice for the M scenarios.Compare the differences in performance between them.

4. Numerical tests

To test how well the proposed stochastic model and theheuristic algorithm perform in the real world, we perform some

Table 1Vans and jeeps at each work station.

Work stations # of vehicles

(No.) Van Jeep

1 3 12 3 03 2 04 3 05 3 26 2 17 2 18 3 09 2 1


numerical tests using data from the 1999 Chi-Chi earthquake inTaiwan. The emergency roadway repair work schedule is providedby Yan and Shih (2009) and the problem configuration data by Yanet al. (2012). The C computer language, coupled with the CPLEX11.1 mathematical programming solver, is used to develop allthe necessary programs for building and solving the models. Thetests are performed on an Intel� Core(TM)2 Duo CPU T72502.00 GHz with 4 GB RAM in the environment of Microsoft Win-dows XP. Finally, we performed some sensitivity/scenario analyseson the models.

4.1. Data analysis

Related data, which comprise roadway-network information,the emergency roadway repair work schedule, material/food sup-plies and demands, support vehicle types with different carryingcapacities and the relevant cost data, are used as input for numer-ical tests. In general, roadway networks can be classified into fourtypes of roadway-network patterns: serial and radial network pat-terns, web network patterns, branching network patterns, and gridnetwork patterns (Cluskey, 1979). The roadway network in NantouCounty, Taiwan can be categorized as a branching network pattern.Therefore, the roadway networks of all tests in this study are de-signed as a branching network pattern to comply with the modelassumptions addressed in Section 2.2. There are 9 work stations,46 intersections and 24 repair points, all located in Nantou County,Taiwan. The roadway-network is located in a mountain area, so thestudy area is large. The emergency roadway repair work scheduleobtained by Yan and Shih (2009), coupled with the repair timefor each repair point, gives repair sequences for 24 repair pointsperformed by 24 work teams from 9 work stations. Altogether,55 time units are designed in the model networks, where each timeunit is 15 min long.

The demand of support materials for each repair point is givenby the responsible engineering department of the Taiwan Govern-ment. Basically, two types of support materials (fuel oil and ma-chine parts) and food need to be transported to each repairpoint. For ease of testing, the standard material equivalent is setto be a volume equivalent to three ten-liter jugs of light distillates.The amount of support materials is calculated in equivalent units.Based on real practices, for each repair point it is assumed that 20units of fuel oil are needed every 40 time units (i.e., 600 min), 20units of machine parts are needed every 20 time units (i.e.,300 min), and 2 units of food for lunch and 2 units of food for din-ner are needed. Altogether, in these tests, the demands are: 60units of fuel oil for three repair points; 160 units of machine partsfor eight repair points; and 16 units of food for eight repair points.It is not necessary to design a demand point for repair points with-out any support material demand in the vehicle- and logistics-flownetworks. Note that each kind of material can be adequately pro-vided by every work station to every repair point.

Two types of vehicles, jeeps and vans, are used in the logisticalsupport plan, as shown in Table 1. For each jeep, the maximum car-rying capacity is 2 units of fuel oil, 2 units of machine parts, or 10units of food. For each van, it is 10 units of fuel oil, 10 units of ma-chine parts, or 50 units of food. Note that external vehicles fromother government units (a maximum of 29 external vehicles) canbe added for material delivery if internal vehicles are insufficient.

All cost data are set as reported by the responsible engineeringdepartment of the Taiwan Government. The average vehicle run-ning speed is set to be 30 km/h, which is lower than the normalspeed because of the damage and clutter created by the earth-quake. The cost of deadheading a vehicle is set to be NT$30/kmper van and NT$25/km per jeep. The holding cost for engine idlingis set to be NT$40/h. The cost for use of external vehicles is set to beNT$10,000/per van and NT$8,000/per jeep. The transportation cost

of material (both fuel oil and machine parts) is NT$5.625/per unitper hour for both vans and jeeps, while that for food is NT$4.212/per unit per hour. Based on real practices in Taiwan, we must alsoconsider the crash cost for each repair work which is usually penal-ized per time. For simplicity, in the tests, the late delivery penaltycost for positive inconsistent time for each repair point is set to beNT$10,000/per time (i.e., tw;s

s � ld;w;ss ¼ 10;000 for each vehicle trip

s if tw;ss > 0). On the other hand, the early delivery penalty cost

for a negative inconsistent time per repair point (i.e., the reduciblecost for preparation tasks before beginning repair work) is set to beNT$5.0 per minute. The other costs are set to be zero.

4.2. Test results

Before starting to perform numerical tests, we evaluated a suit-able number of scenarios that can suitably represent the eventpopulation. In each scenario an actual travel time for a supportvehicle trip is randomly generated according to a given probabilitydistribution of travel times. The probability distribution of traveltimes for each support vehicle trip is obtained by referring to theannual report provided by the relevant engineering departmentof the Taiwan Government. A truncated normal distribution ismainly obtained corresponding to the possible travel time for eachvehicle trip. For convenience, the continuous truncated normal dis-tribution is further simplified as a discrete distribution. This isdone by separating the possible travel time range into several timeperiods, each being associated with a discrete travel time and aprobability. The probability of each time period is estimatedaccording to the continuous truncated normal distribution. Be-sides, as suggested by the department staff, each scenario is as-sumed to be independent and the probability for each scenario isthe same. Thus, it is convenient for us to randomly generate eachsupport vehicle trip’s travel times according to its own distribu-tions, for each scenario. We tested 9 situations, from 20 to 100 sce-narios, in 10 scenario increments. As shown in Fig. 6, after 70scenarios, the variation was only about 0.06%, showing that theobjective values varied slightly. The results show that the solutionsdid not change by any significant amount. For ease of testing, thenumber of scenarios are both set to be 70. Note that users can per-form similar tests based on their own operations to determine themost suitable number of scenarios that can suitably represent theevent population.

The size of the test problems for the SM and the DM are verylarge, including 1,399,387 variables and 1,295,603 constraints(437,154 for flow conservation and 858,220 for others) for theSM, and 905,189 variables and 1,008,898 constraints (437,154 forflow conservation and 571,744 for others) for the DM. We triedto use CPLEX to directly solve these problems, but could not findfeasible solutions after one day. We then employed the heuristicalgorithm to solve the problems. The solutions were comparedwith practical arrangements.


Before testing, for clarity, ZP, ZE, the evaluation gap (EG), andimprovement percentage (IP) are defined:ZP, ZE: the objective va-lue for the planning and evaluation stages, respectively;

EG ð%Þ ¼ j ZE� ZPZP

j � 100%; ð13Þ

IP ð%Þ ¼ ZE of DM or the arrangement in practice� ZE of SMZE of SM

� 100%:

ð14Þ

EG is the difference between ZP and ZE. A large EG means thatthere is a larger variation between the planning and the evaluationstages. Based on the measure of EG, the variations in performanceof the planned schedule when applied in real operations can beevaluated. In addition, the IP denotes the difference between thearrangement in practice (AIP) or the DM and the SM for the ZE. Apositive IP denotes that the ZE of the SM is better than that ofthe DM or the current schedule, and vice versa.

Table 2 shows the computational results comprised of the plan-ning and evaluation stages. In the planning stage, the objective va-lue of the DM is less than that of the AIP or the SM. The AIP solutionis only a feasible solution for the DM and the DM did not considerthe stochastic disturbances from vehicle travel times that arise inpractice. Two vans are saved in the DM and the SM, compared withthe AIP. Besides, compared with the lower bound solutions, themaximum error gap for the DM is 1.01% and that for the SM is3.90%, showing that the solutions yielded by the proposed heuristicalgorithm are close to the optimal solutions obtained with the DMand the SM. The computation times for the heuristic algorithm forsolving the DM and the SM are about 375 and 706 s, respectively.These results indicate that the heuristic algorithm can efficientlyand effectively solve these logistical support scheduling models,a very important result in practical application.

To evaluate the performance of the SM, the DM and the AIP inreal operations, numerical tests are performed. In the evaluationstage, as shown in Table 2, the EG for the SM (6.04%) is less thanthat for the DM (43.74%) or the AIP (56.53%), meaning that thereis a smaller variation between the planning and the evaluation

Table 2Test results.

Arrangeme

Planning stageZP 86,221Vehicle operating cost (NT$) 80,573Transportation cost of material (NT$) 5648The unanticipated penalty cost (NT$) –# of total vehicle used 32

# of internal vehicles used 29# of external vehicles used 3(all vans)Computation time (second) –

Gap with the LBSa (%) –

Evaluation stageZE 172,424Vehicle operating cost (NT$) 80,311Transportation cost of material (NT$) 12,113The delivery penalty cost for early arrival –

– the reducible cost for preparation tasks(NT$)The delivery penalty cost for delay arrival 80,000

– the crash cost(NT$)EG (%) 56.53IP (%) 61.75

– Not applicable.a The lower bound solutions of the DM and the SM are NT$51130 and NT$68508, resb A negative value shows that the operating cost is reduced for the same vehicle, com

stages for the obtained SM plan. In addition, a comparison of theIPs for the SM is positive, about 61.75% and 17.35%, respectively.This verifies that the DM plan loses its optimality, becoming infe-rior to the SM, when applied to real operations. Therefore, thereader should not be misled by the ‘‘planning results’’ to assumethat the DM is better than the SM. The actual performance of theplanning models can only be evaluated after being applied in realoperations.

4.3. Sensitivity analyses

Sensitivity analyses offer some guidance as to the effects ofvariations due to the crash cost for repair work, the placing of extrasupport vehicles at each work station and the problem scale. Othersensitivity analyses, which can be similarly preformed in the fu-ture, are not addressed here to save space.

4.3.1. Crash cost for repair workWe evaluate the influence of the crash cost for repair work on

the SM by testing four additional scenarios, 50%, 100%, 150% and200%. For example, 200% indicates that the crash cost is doubled.As shown in Fig. 7, the objective values of the SM in the planningand evaluation stages both increase as the crash cost for repairwork increases. In particular, this crash cost is less sensitive insolution during the planning stage than for the evaluation stage.The results indicate that the simplified crash cost may not have agreat influence on the solution in the planning stage, but could sig-nificantly affect the solution in the evaluation stage. Consequently,this crash cost may be more accurately estimated, particularlyaccording to formula (1), in future to be applied to model real oper-ations. Note that the computation times for all cost scenarios areusually less than 16 min, meaning the proposed algorithm isefficient.

It should be mentioned that according to real practices a largecrash cost is set to reflect the cost incurred for temporarily stop-ping the work team’s work at the repair point to suitably avoid latedelivery of the required materials. Based on the large crash cost, aconservative arrival time may be preferred for each demand pointbefore the demand point time, only if the vehicle and materialrouting/scheduling is feasible, which requires a complicated

nt in practice DM SM

51,647 71,29445,825 62,5305822 9399– �63530 3029 291(van) 1(van)375.06 706.091.01 3.90

91,800 75,87867,100 78,63110,722 9495�6022b �22,248b

20,000 10,000

43.74 6.0417.35 –

pectively.pared with the planning results.

Fig. 7. Sensitivity analysis of the crash cost for each repair work.

Table 3Test results for different problem instances.

DM SM Saving cost (%)

Planning stageCase A

ZP (NT$) 42,828 43,185 –# of variables 1,212,625 1,659,279 –# of constraints 1,326,777 1,374,801 –


analysis by the proposed model. This means that if the vehicle andmaterial routing/scheduling is so constrained that it is infeasible toallow the materials to be delivered to every demand point at a con-servative arrival time, then the most suitable vehicle arrival times(including average travel times) can be obtained by the proposedmodel from a system optimization perspective.

4.3.2. Effect of placing extra support vehicles at each work stationWe examine the effect of placing extra vehicles at each work

station. The number of both types of vehicles is changed at onework station at a time. For example, we add a van and a jeep toeach work station for each scenario. As shown in Fig. 8, the resultsfor the planning and evaluation stages are similar. The improve-ment rates for all work stations, except station No. 5, are less thanzero. The objective values obtained with both the SM and the DMdecrease the most when a van or a jeep is added to station No. 5.This improvement of 13.27% and 18.09% for the SM and the DM,respectively, occurs because station No. 5 is closer to the repairpoints than the other work stations, meaning it is better to placeadditional vehicles there. Note that the computational times forthese scenarios are all less than 12 min. Note that for ease of mod-eling in this study, a holding cost is designed that represents thecost of holding a vehicle with its engine idling (in this caseNT$40/h), which is generally much smaller than the travel arc cost.Although an additional vehicle (with the exception of No. 5) is usu-ally not used at the work station, the obtained objective value withthe additional holding cost is slightly larger than the original onefor all work stations, except No. 5.

4.3.3. Problem scaleIn this section, we evaluate the performance of the stochastic

and deterministic models for different scaled problems with

Fig. 8. Scenario analysis of placing extra support vehicles at each work stationwork.

various numbers of work teams, intersections, demand pointsand time points. Three test cases, obtained from Yan and Shih(2009), are as listed below:

Case A: 7 work stations, 42 intersections, 21 demand points and55 time points.Case B (original case): 9 work stations, 46 intersections, 24demand points and 55 time points.Case C: 12 work stations, 55 intersections, 31 demand pointsand 62 time points.

All the test cases contain 3 kinds of materials/food. The repair/travel time distributions for repair points/roadway segments aswell as other parameters, coupled with the emergency roadway re-pair work schedule, for each case, have also been obtained fromYan and Shih (2009).

Table 3 shows the test results comprised of the planning and eval-uation stages for the three cases. In the planning stage, the ZPs, thenumbers of variables/constraints and the computation times forboth models increase as the problem scale increases. In particular,the ZPs for the three cases are 42,828, 51,647 and 105,421, for theDM, and 43185, 71294 and 129746, for the SM. In addition, the num-bers of variables/constraints for these test cases are 1,212,625/1,326,777, 1,374,809/1,507,644 and 2,310,550/2,640,021, for theDM, and 1,659,279/1,374,801, 1,845,906/1,521,285 and 3,241,793/2,860,339, for the SM. Moreover, the computation times for thesethree test cases are 479, 575, and 1836 s, for the DM and are 551,706, and 2,727 s, for the SM. These results show that the heuristicalgorithm can efficiently solve the DM and SM for large scale prob-lems. In addition, numerical tests are carried out to evaluate the per-formance of the DM and the SM in real operations. In the evaluationstage, the ZEs for the SM are all superior to those for the DM, by11.46%, 17.35% and 11.36%, respectively, for the three cases. Thisshows that the SM is more suitable than the DM for planning thelogistical support schedule for emergency roadway repair work.All in all, these results demonstrate that the proposed stochasticmodel and heuristic algorithm are both efficient and effective forsolving realistic large-scale problems.

Computation time (s) 479.85 551.01 –

Case B (original case)ZP (NT$) 51,647 71,294 –# of variables 1,374,809 1,845,906 –# of constraints 1,507,644 1,521,285 –Computation time (s) 575.06 706.09 –

Case CZP (NT$) 105,421 129,746 –# of variables 2,310,550 3,241,793 –# of constraints 2,640,021 2,860,339 –Computation time (s) 1836.55 2727.66 –

Evaluation stageCase A

ZE (NT$) 66,979 59,301 11.46Case B

ZE (NT$) 91,800 75,873 17.35Case C

ZE (NT$) 181769 161111 11.36

– Not applicable.


5. Conclusions and discussion

In this study we use expected optimization concepts, combinedwith a time–space network technique, to develop a logistical sup-port scheduling model (the SM) which is further simplified to gen-erate the deterministic model (the DM). The simplified DM isexactly equal to Yan et al.’s (2012) model. The major differencesbetween Yan et al.’s (2012) and this paper include the modelingdesign (specifically the designs for the trip travel time, the timeinconsistency for scheduling each trip, the unanticipated penaltycost for scheduling each trip, and the objective function), asimulation-based evaluation method, and numerical tests.Mathematically, the SM/DM is formulated as an integer multiple-commodity network flow problem, which is characterized as NP-hard. It is almost impossible to optimally solve the realisticlarge-scale problems that occur in practice within a limited periodof time. To efficiently solve realistically large-scale problems with-in a reasonable solution time, a heuristic algorithm based on aproblem decomposition technique is proposed, coupled with theuse of the mathematical problem solver, CPLEX.

To test how well the proposed stochastic model and the heuris-tic algorithm may be applied in the real world, numerical tests,based on the 1999 Chi-Chi earthquake data, the emergency road-way repair work schedule from Yan and Shih (2009) and the prob-lem configuration data from Yan et al. (2012), are performed toevaluate the performance of the models. The problem size in theSM/DM can reach the same 714,937 nodes, 1,399,387/905,189 arcs(i.e., variables), and 1,295,603/1,008,898 constraints. The test re-sults show that two vans are saved in the SM/DM, compared withthe arrangement used in practice. The solution algorithm performswell with the worst error gap being 3.90/1.01% between the objec-tive value and the LBS in the SM/DM for the planning stage. Inaddition, in the evaluation stage, the SM is more efficient and effec-tive than the DM and the AIP, with an IP of 61.75% and 17.35%,meaning that the SM is more practical and effective than Yanet al.’s (2012) model and the arrangement used in practice. Threesensitivity analyses are also performed to test the performance ofthe SM in different situations. The overall results show that theproposed SM and heuristic algorithm are both more efficient andeffective than Yan et al.’s (2012) model and the currently usedmanual approach, and could therefore be useful references for bothpractitioners and researchers. Although the preliminary test resultsare good, indicating the potential usefulness of the proposed sto-chastic model and solution algorithm, more tests should be per-formed in future to set rules appropriate for each user’sapplications.

It should be mentioned that the successful application of themodel depends on three tasks in the following order: data collec-tion and configuration, model generation and model solving. Ingeneral, data collection and configuration takes the most timeout of these three tasks. Given the required data sets, the timefor model generation is typically less than half of the model solvingtime, which is less than a couple of minutes. Regarding data collec-tion and configuration, in practice, damages are dynamically re-ported from the scenes to the disaster prevention center in realtime. That is, the disaster prevention center has real-time informa-tion about roadways and repair resources to be used to design a re-pair work schedule (Yan & Shih, 2009), with necessary logisticalsupport resources. Based on this schedule, coupled with the infor-mation on roadways and logistical support resources, the logisticsupport plan is designed and implemented. To effectively use themodel, the method used to collect and configure data should bemore precise and efficient than the current manual method thatis adopted in practice. In particular, all changes in real-time infor-mation for roadways and logistical support resources have to beefficiently updated. For this purpose, a well-designed database

which includes real-time roadway information, up-to-date logisti-cal support resources and related data necessary for generating themodel should be built in the future. Additionally, a user-friendlyinterface should also be constructed as a tool so that the modelcan be efficiently generated with real-time information from thedatabase, and then be solved efficiently using the proposed heuris-tic algorithm. Thus, the process of data collection and configurationwill not be a bottleneck for the application of the model for gener-ating a good logistical support plan. Ultimately, a computerizeddecision support system that integrates a well-designed database,an efficient model generation and optimization procedure, and acomprehensive result demonstration methodology, coupled witha friendly user interface, can be developed to efficiently apply themodel in practice. How to develop such a system can be a directionfor future research.

In future, the proposed stochastic model could be extended toconsider multiple stochastic factors (particularly other regularincidents, such as stochastic variations in the amount of requestedmaterials and so on), to form a more complicated stochastic model.It should also be noted that this study only focuses on the incidentof stochastic travel times that occur in regular operations, ratherthan other types of larger incidents, such as breakdown of road-ways, which can cause larger disturbances. These larger types ofincidents, which do not occur frequently in operations, are typi-cally not considered in the planning stage, but can be handledwhen they occur in real operations (e.g., see Yan & Lin, 1997). How-ever, how to develop suitable models and algorithms for handlingthese larger incidents in real operations could be a direction of fu-ture research.

Acknowledgements

This research was supported by a grant (NSC101-2221-E-008-097-MY3) from the National Science Council of Taiwan. Theauthors would like to thank the area editor and the three anony-mous reviewers for their helpful comments and suggestions onthe presentation of the paper.

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logistical support scheduling under stochastic travel times given an emergency repair work schedule

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