logistic

OR Spectrum (2013) 35:905–933DOI 10.1007/s00291-011-0268-x

REGULAR ARTICLE

A multi-objective robust stochastic programming modelfor disaster relief logistics under uncertainty

Ali Bozorgi-Amiri · M. S. Jabalameli ·S. M. J. Mirzapour Al-e-Hashem

Published online: 13 August 2011© Springer-Verlag 2011

Abstract Humanitarian relief logistics is one of the most important elements of arelief operation in disaster management. The present work develops a multi-objectiverobust stochastic programming approach for disaster relief logistics under uncertainty.In our approach, not only demands but also supplies and the cost of procurement andtransportation are considered as the uncertain parameters. Furthermore, the model con-siders uncertainty for the locations where those demands might arise and the possibilitythat some of the pre-positioned supplies in the relief distribution center or suppliermight be partially destroyed by the disaster. Our multi-objective model attempts tominimize the sum of the expected value and the variance of the total cost of the reliefchain while penalizing the solution’s infeasibility due to parameter uncertainty; at thesame time the model aims to maximize the affected areas’ satisfaction levels throughminimizing the sum of the maximum shortages in the affected areas. Considering theglobal evaluation of two objectives, a compromise programming model is formulatedand solved to obtain a non-dominating compromise solution. We present a case studyof our robust stochastic optimization approach for disaster planning for earthquakescenarios in a region of Iran. Our findings show that the proposed model can help inmaking decisions on both facility location and resource allocation in cases of disasterrelief efforts.

Keywords Humanitarian relief logistics · Preparedness and response phase ·Pre-positioning · Robust stochastic programming · Multi-objective optimization

A. Bozorgi-Amiri (B) · M. S. Jabalameli · S. M. J. Mirzapour Al-e-HashemDepartment of Industrial Engineering, Iran University of Science and Technology,Narmak, 1684613114 Tehran, Irane-mail: [email protected]

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906 A. Bozorgi-Amiri et al.

1 Introduction

Every year, natural disasters, such as earthquake, flood, drought, hurricane, landslide,volcanic eruption, fire, tsunami, avalanche, extreme cold, heat wave, and cyclone killthousands of people and destroy millions-of-dollars’ worth of habitats and assets (VanWassenhove 2006). For example, a major tsunami affected 12 countries in 2004; mas-sive earthquakes struck Bam, Iran in 2003, Pakistan in 2005, China in 2008 and Haitiin 2010; and an extensive flood devastated Pakistan in 2010. The rapid growth in worldpopulation and increased human concentrations in dangerous environment have led torises in both the frequency and severity of natural disasters; consequently, the numberof people affected by natural disasters continues to rise. For example, between 2000and 2007, the number of reported natural disasters was approximately 460 disastersper year, indicating a dramatic increase, and also the number of victims is generallybetween 100 million and 400 million per year around the world (Haghani and Afshar2009).

The unpredictable nature of such events, in addition to the large number of casual-ties at stake, makes the field of humanitarian logistics a critical aspect of any disasterrelief operation, and it represents one of the main levers to achieve improvements interms of cost, time, and quality (Blecken et al. 2009). The timely and effective mobi-lization of resources is essential in aiding people who are made vulnerable by naturaldisasters. The supply shortage may render emergency response ineffective and resultin increased suffering (Knott 1987, 1988). Hence, it is important to develop strategiesto accelerate supply response or when dealing with the unpredictability of demand.The pre-positioning of commodities at or near the location of consumption is one ofthe most important logistical strategies to reduce delivery time and operating costs(Yi and Kumar 2007; Balcik and Beamon 2008). Pre-positioning allows not only afaster response but also better procurement planning and improved distribution costs;however, it requires an additional investment before the event occurs.

The complex nature and dynamics of the relationships among the different actorsin a relief chain imply an important degree of uncertainty in relief chain planningdecisions. In such environments, there are two main characteristics of the disasterplanning problems that a decision maker will be faced with: (1) conflicting objectivesthat may arise from the nature of operations (e.g., to minimize costs and at the sametime to increase affected areas’ satisfaction) and the structure of the relief chain whereit is often difficult to align the goals of the different parties within the relief chain;(2) lack of knowledge of data (e.g., demand, supply or cost). In real humanitarianoperations, it is often seen that demand, supply, and cost are uncertain during the firststage of disaster response (Mert and Adivar 2010). Uncertainty in supply is causedby the variability brought about by how the supplier operates because of the faults ordelays in the supplier’s deliveries. It is often unknown which resources are available,and even the involvement and contribution of suppliers is unpredictable (Tomasini andVan Wassenhove 2004). On the other hand, per-positioned assets can be destroyed bya disaster. Uncertainty in the cost of the operations generally happens because of theuncertainty associated with routes, suppliers, etc. Finally, demand uncertainty, accord-ing to (Davis 1993), is the most important of the three and is presented as demandvolatility or inaccurate assessments. For example, in the Haiti earthquake, during the

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A multi-objective robust stochastic programming model 907

first 2–3 days estimates of the numbers of victims ranged from 30,000 to 100,000 (VanWassenhove et al. 2010). Thus, it is important that models addressing problems in thisarea should be designed to handle the foregoing two complexities, that is, “conflictingobjectives” and “data uncertainty”.

The purpose of this paper was to model disaster planning and response capturing theinherent uncertainty in demand, supply, and cost resulting from a disaster. To cope withthe challenging problem resulting from the analysis of three sources of uncertainty, anovel robust optimization model is therefore proposed. The model tackles the disas-ter relief logistics problem as a multi-objective, stochastic, mixed-integer, nonlinearprogramming model. The problem is then solved using the Lp-metrics technique.

The main contributions of this paper can be summarized as follows:

• Achieving a model which contemplates the different sources of uncertainty affect-ing relief chains in an integrated fashion. This is important because the interactionbetween these sources of uncertainty creates decision making challenges.

• Develop a novel robust optimization model to tackle the disaster relief logisticsproblem as a multi-objective, stochastic, mixed-integer, nonlinear programmingproblem. This formulation not only takes into account the expected total cost ofrelief logistics but also considers the risk reflected by the variability of the totalcost and the satisfaction of affected areas (AAs).

• Applying the model to a real-world disaster relief chain dedicated to the supplyof relief commodities to the AAs.

We also assess the potential improvement of using robust stochastic optimizationwith all sources of uncertainty (demand/supply/cost) in the relief chain planning pro-cess compared with the models with lower degrees of uncertainty and report the costsaving quantities.

The rest of this paper is organized as follows: In Sect. 2, the relevant literature isreviewed. In Sect. 3, the concept of the robust optimization is described. The generalproblem description statement is given in Sect. 4. A robust optimization model isdeveloped to formulate the relief logistics network in Sect. 5. In Sect. 6, a solutionmethod is presented. Then, the robustness and effectiveness of the proposed model aredemonstrated by a case analysis and experimental result in Sect. 7. Our conclusionsand future research plans are presented in the final section.

2 Literature review

Although a broad literature based on humanitarian aid and disaster relief exists, thereare only limited research results on the specific area of humanitarian logistics (VanWassenhove 2006; Kovacs and Spens 2007). There are a number of articles provid-ing overviews on humanitarian logistics (Altay and Green 2006; Kovacs and Spens2007). In this section, we review the literature on disaster relief logistics problems.This review is divided into two contrasting categories: literatures on disaster relieflogistics and those on the management of uncertainties in disaster relief logistics. Wediscuss some of the key papers in each category.

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2.1 Disaster relief logistics

The related academic literature in this context falls into three streams: facility location,inventory management, and network flow problems. Research on facility locationsemphasizes the pre-positioning of emergency stock in the pre-disaster phase. One ofthe earliest studies directed at the locations of emergency facilities is by Toregas et al.(1971), who modeled the problem as a set-covering problem and used linear program-ming as the method of solution. Parentela and Nambisan (2000) developed emergencyresponse plans that combined all the information in connection with the location andcapacities of resource suppliers, the spatial distribution of the victims, the environmentand the economy. Brotcorne et al. (2003) classified the location and relocation modelsof ambulances and other emergency vehicles into three categories: deterministic mod-els, probabilistic queuing models, and dynamic models. Akkihal (2006) consideredoptimal locations for warehousing non-consumable inventories required for the initialaid deployment. Tzeng et al. (2007) proposed a multi-objective deterministic modelto distribute commodities to demand points, considering the cost, response time, anddemand satisfaction, and solved it using a fuzzy multi-objective programming method.Balcik and Beamon (2008) presented a maximally covered location model of pre-posi-tioning relief commodities in order to determine the number, locations, and capacitiesof the relief distribution centers (RDCs) as well as when the demand for relief com-modities can be met by suppliers and warehouses. The model considered pre- andpost-disaster budget constraints but did not consider the possibility of either inventorybeing destroyed or the shortage costs. Ukkusuri and Yushimito (2008) developed amodel for selecting the optimal locations for the pre-positioning of supplies in sucha way to maximize the probability that demand points can be reached from a singlesupply facility in the presence of transportation network disruptions.

Research on inventory management focuses on determining the item quantitiesrequired at various RDCs along the relief chain, procurement quantities, and order fre-quency; it also identifies the appropriate amount of safety stock to maintain. Whybark(2007) argued that disaster planning is centered on disaster inventories and, there-fore, acquisition, storage, and the distribution of products are significant. However,little research is known on the inventory in disaster relief logistics. Ozbay and Ozguven(2007) developed a time-dependent inventory control model for safety stock levels thatcould be used for the development of efficient pre- and post-disaster plans. Beamonand Kotleba (2006) formulated a stochastic inventory control model that determinedoptimal order quantities and reorder points for a pre-positioned warehouse during thecourse of a long-term emergency relief response.

Given the decisions regarding the location and replenishment, the next step is net-work flows, classified in two categories: victim evacuation and relief distribution.Some research work on victim evacuation can be found in Bakuli and Smith (1996),Barbarosoglu et al. (2002), Han et al. (2006), Yi and Ozdamar (2007), Yi and Kumar(2007), Regnier (2008), Stepanov and Smith (2009) and Saadatseresht et al. (2009).Models for relief distribution have been proposed by Haghani and Oh (1996), Fiedrichet al. (2000), Viswanath and Peeta (2003), Sakakibara et al. (2004), Ozdamar et al.(2004), Barbarosoglu and Arda (2004), Amiri (2006), Jia et al. (2007a), Lodree andTaskin (2008), Campbell et al. (2008), Sheu (2007, 2010). Each of these models

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chooses an optimization goal such as minimum response time, maximum populationcoverage, or overall cost. Relief distribution models plan the allocation of resources tostochastic AAs using available transportation networks that are subject to predefinedimpact scenarios.

2.2 Stochastic optimization approach for disaster relief logistics

The significance of uncertainty has motivated a number of researchers to address sto-chastic optimization in disaster relief planning involving the distribution of emergencycommodities and necessity items by probabilistic scenarios representing disasters andtheir outcomes (see, e.g., Cormican et al. 1998; Barbarosoglu and Arda 2004; Beraldiet al. 2004; Chang et al. 2007; Beraldi and Bruni 2009; Mete and Zabinsky 2010; Rawlsand Turnsquist 2010). Research addressing the design of disaster planning is limitedto those that modeled the stochastic situation under demand uncertainty and thosethat modeled it under demand and supply uncertainty (or demand/cost uncertainty).Jia et al. (2007a) presented a single-objective, uncapacitated facility location modelto locate emergency service facilities such as fire stations or resources such as ambu-lances in the event of a large-scale emergency. As in the problem we address here, theyconsidered that the capability of a facility of providing Emergency Medical Service(EMS) may be disrupted during a large-scale emergency due to road damage and/or thedestruction of the facility. They did not consider the inventory decision and capacityconstraint of facility. Jia et al. (2007b) also proposed heuristic methods and addressedthe demand uncertainty and shortage of supplies by requiring multiple supply points.

There are a few studies that address a comprehensive (strategic and tactical issuessimultaneously) disaster design (emergency) relief logistics using two-stage stochasticprogramming. Chang et al. (2007) modeled locating and distributing rescue resourcesin a flood emergency under possible flood scenarios using two-stage stochastic pro-gramming; the model could serve as a decision-making tool for the government agen-cies in the planning of flood emergency logistics under demand uncertainty. Salmeronand Apte (2010) developed a two-stage stochastic optimization model for planning theallocation of budget for acquiring and positioning relief assets; in this model, the first-stage decisions represented the “aid pre-positioning” by the expansion of resourcessuch as warehouses, medical facilities, ramp spaces, and shelters, whereas the secondstage concerned the logistics of the problem under demand/cost uncertainty; how-ever, they did not consider the relationship between relief locations (backup coverage)and the possibility of inventory being destroyed. Mete and Zabinsky (2010) intro-duced a stochastic optimization model for disaster preparedness and response underdemand/cost uncertainty, in order to assist in deciding on the location and alloca-tion of medical supplies to be used during emergencies in the Seattle area, whichis known to be vulnerable to earthquakes. They also offered an MIP transportationmodel that is potentially useful in routing decisions during the response phase. Rawlsand Turnsquist (2010) developed a two-stage, stochastic, mixed-integer program thatdetermined the locations and quantities of various types of emergency commodities;their model also considered the transportation network availability following a disasterunder demand/cost uncertainty.

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Although stochastic programming has been applied in Chang et al. (2007);Salmeron and Apte (2010); Mete and Zabinsky (2010); Rawls and Turnsquist (2010),conventional stochastic programming models are severely limited due to the inabilityto handle risk aversion or decision-makers’ preferences in a direct manner with a con-sequence of excluding many important domains of application (Azaron et al. 2008).Mulvey et al. (1995) presented an improved stochastic programming, called robust pro-gramming, that is capable of tackling the preferred risk aversion of the decision-mak-ers. In this method, the variability term was simply added to the main objective functionwith an associated weighting parameter representing the risk tolerance of the modeler.

To overcome the disadvantages of traditional, stochastic relief-chain designapproaches, we developed a robust stochastic programming approach for the designof disaster relief logistics under uncertainty. In our approach, not only demands butalso supplies and the cost of procuring and transportation are all considered as theuncertain parameters. To the best of knowledge, it is the first time these three sourcesof uncertainty are considered simultaneously in the disaster planning problem. Wecoped with uncertainty by using a scenario-based approach that attempts to captureuncertainty by representing it in terms of a number of discrete realizations of sto-chastic quantities that constitute distinct scenarios. We further assumed that a disasterspecifically disrupts the capability of the facility (suppliers and RDCs) by damagingthe roads and/or destroying the facility, which is a realistic scenario in some areasfaced with frequent, sudden-onset disasters, such as Iran.

The goal is to select optimum numbers, locations, and capacity levels of the RDCfor delivering the commodities to the AAs with minimum costs while satisfying thedesired service level to the beneficiaries (or AAs). Our multi-objective model includestwo objectives. The first objective function is the minimization of the sum of the costsfor the first-stage setup, inventory procurement, and transportation and the costs for thesecond-stage procurement, transportation, holding, and shortage. To develop a robustmodel, two additional terms are added to the first objective: cost variability and penaltyof infeasibility. The variability of the total cost should be considered in the model,because when we focus merely on the expected total cost, the obtained solution maybe sub-optimal if the total cost substantially varies because of randomness. The secondobjective function is the maximization of the AAs’ satisfaction by minimizing the sumof the maximum shortage in AAs. In other words, we conceptualized the satisfactionof fairness in formulating the multi-objective functions to avoid the possibility of aseverely unfair relief distribution to certain AAs in the relief distribution process.

Although the ideas of variability, infeasibility penalty function, and fair distributionhave been considered in other areas, to the best of our knowledge, this is the first timethat they are combined in a multi-objective scheme for designing the robust disasterrelief logistics under three sources of uncertainty (demand/supply/cost). Furthermore,our model incorporates the idea of backup coverage.

3 Robust optimization

Robust optimization, one of the most popular topics in the field of optimization andcontrol science the late 1990s, deals with an optimization problem involving uncer-tain parameters. Mulvey et al. (1995) introduce a model for robust optimization that

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involves two types of robustness: “solution robustness” (the solution is nearly optimalin all scenarios) and “model robustness” (the solution is nearly feasible in all sce-narios). The definition of “nearly” is left up to the modeler; their objective functionhas general penalty functions for both model and solution robustness, weighted by aparameter intended to capture the modeler’s preference between the two. The robustoptimization method developed by Mulvey et al. (1995), in fact, extends stochasticprogramming by replacing traditional expected cost minimization objective by onethat explicitly addresses cost variability. In the following, the framework of robustoptimization is briefly described (Mirzapour Al-e-hashem et al. 2011). Consider thefollowing linear programming model that includes random parameters:

Min f (x, y) = cT x + dT y (1)

s.t. Ax = b (2)

Bx + Cy = e (3)

x, y ≥ 0 (4)

where x is a vector of the design variables that should be determined under the uncer-tainty of model parameters and y is a vector of the control variables. A, B, C areparameter matrices, while b, e are parameter vectors. A, b are known deterministi-cally, while B, C, e are uncertain. A specific realization of an uncertain parameteris called a scenario. Assume a finite set of scenarios � = {1, 2, . . . , s} to model theuncertain parameters; with each scenario s ε � we associate the subset {ds, Bs, Cs, es}and the probability of the scenario ps (

∑s ps = 1).

Also, the control variable y, which is subject to adjustment when one scenario isrealized, can be denoted as ys for scenario s. Because of parameter uncertainty, themodel may be infeasible for some scenarios. Therefore, error vector δs presents theinfeasibility of the model under scenario s. If the model is feasible δs will be equal to0; otherwise, δs will be assigned a positive value according to Eq. (7). A robust opti-mization model based on the mathematical problem (1)–(4) is formulated as follows:

Min σ(x, y1, y2, . . . , ys) + γρ(δ1, δ2, . . . , δs) (5)

s.t. Ax = b (6)

Bs x + Cs ys + δs = es ∀s ∈ � (7)

x, ys ≥ 0 ∀s ∈ � (8)

There are two terms in the above objective function: The first term represents thesolution robustness that captures the firm’s desire for lower costs and its degree of riskaversion, while the second term represents the model robustness, penalizing solutionsthat fail to meet demand in a scenario or violate other physical constraints such ascapacity. Using the weight γ , the trade-offs between the solution robustness measuredfrom the first term σ(◦) and the model robustness measured from the penalty termρ(◦) can be modeled under the multi-criteria decision-making process.

We use ξ to represent f (x, y), which is a cost or benefit function, ξs = f (x, ys),for scenario s. A high variance for ξs = f (x, ys) means that the solution is a high-riskdecision. Mulvey et al. (1995) applied a quadratic form of variance to capture the

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concept of risk and represent the solution robustness. To cope with the computationalcomplexity due to its nonlinearity, however, Yu and Li (2000) proposed an absolutedeviation in place of the quadratic term, which is presented as follows:

σ(◦) =∑

s∈�

psξs + λ∑

s∈�

ps

∣∣∣∣∣ξs −

∑

s′∈�

ps′ξs′

∣∣∣∣∣

(9)

where λ indicates the weight placed on the solution variance in which the solution isless sensitive to change in data under all scenarios as λ increases. In order to minimizeobjective (9) efficiently, they proposed an efficient method. The framework of Yu andLi’s model is designed to minimize the objective function in (10):

Min∑

s∈�

psξs + λ∑

s∈�

ps

[(

ξs −∑

s′∈�

ss′ξs′

)

+ 2θs

]

(10)

s.t. ξs −∑

s∈�

psξs + θs ≥ 0 s ∈ � (11)

θs ≥ 0 s ∈ � (12)

It can be interpreted as the amount by which ξs is greater than∑

s∈� psξs ,then θs = 0, while as the amount by which

∑s∈� psξs is greater than ξs , then

θs = ∑s∈� psξs − ξs .

Based on this discussion, the objective function can be formulated as follows:

Min∑

s∈�

psξs + λ∑

s∈�

ps

[(

ξs −∑

s′∈�

ps′ξs′

)

+ 2θs

]

+ γ∑

s∈�

psδs (13)

With this background in mind, we will present the detailed problem formulation inthe following sections.

4 Problem description

In this paper, we assume that disaster relief logistics network consists of three stagesand two echelons (see, Fig. 1). The first stage is the set of Suppliers, the second stagecontains RDCs, and the last stage consists of areas affected by disaster. Suppliers (e.g.,aid agencies, governments, private sector, etc.) play the critical role in our relief chainand provide the required commodities to people in devastated areas; those peopleplay the role of the customer in our physical distribution. Concerning the selectionof the warehouse locations from a set of candidate RDCs, certain issues have to beaddressed, namely (i) the storage capacity of the warehouses and (ii) the distance tothe affected people that keeps the transportation costs at minimum. In other words,the RDCs in our network are positioned close to both the AAs and the suppliers inorder to distribute the goods to affected people efficiently. In our paper, a third issuealso is of considerable practical relevance, namely (iii) the safety of warehouses withrespect to the risk of destruction or stealing (Doerner and Hartl 2008).

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Fig. 1 General schema of relief distribution chain

Before the formulation is considered, we make the following assumptions on theproblem:

(1) The capability of suppliers and candidate RDCs may be partially disrupted by adisaster through damage to the roads and/or destruction to the facility.

(2) Both the level of demand for the AA and the cost parameters are uncertain anddepend on various factors including the disaster scenario and the impact of thedisaster. To represent uncertain parameters, we make use of discrete scenariosfrom a set S of possible disaster situations. We assume that the probability distri-bution of scenarios can be devised by subject matter experts or disaster planners.

(3) More than one kind of relief commodity must be delivered, and each commodityis associated with a different volume and a different cost of procurement, storage,and transportation.

(4) All nodes are candidates for the pre-positioning of RDCs.(5) An RDC can be opened in only one of three possible configurations with distinct

storage capacity (small, medium, or large), subject to the associated setup cost.(6) Each AA may be served by multiple RDCs, provided no RDC is being opened

in the area.(7) Each RDC may be supplied by either suppliers (with limited capacity) or other

RDCs (backup coverage).(8) Inventory may be stored at RDCs, but such inventory is penalized.

With the above assumptions in place, we consider two objectives for our model:total cost and customer satisfaction. Customer satisfaction and its prerequisite, fair dis-tribution, play a significant role in disaster relief distribution systems because the mainpurpose of the system is to satisfy the demand of the victims as much as possible; forthis reason, this objective is modeled here as a mini-max rather than a mini-sum prob-lem. The purpose of the design is to resolve for each of candidate locations as the RDC,so that we can identify and investigate the efficiency of the optimal distribution systems.First-stage variables and their related parameters such as RDC setup cost are assumedto be certain. On the other hand, second-stage variables and related parameters, such

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as demand from AAs, are considered to be uncertain parameters. The need to considerthe different sources of uncertainty (demand, supply, and cost) in the relief chain arisesfrom the fact that the pre-positioning of the RDCs, the procurement of relief commod-ities, and all of the decisions to be made for disaster relief are affected by uncertainty.On the other hand, modeling complexity, variability, and uncertainty in model parame-ters will also make the design and management of efficient relief logistics systems verychallenging and, therefore, deserve particular attention by the system planners. Theuncertainty is delineated from a finite sample of scenarios designed by experts basedon historical data and geographical faults. Each scenario is associated with a proba-bility level representing the experts’ expectations of the occurrence of the particularscenario. Next, we present the mathematical model formulation of this problem.

5 Model formulation

The following notation is used to formulate the facility location and supply pre-posi-tioning model:

5.1 Explanation of parameters and variables

Sets/ indices

I Set of suppliers indexed by i ∈ IJ Set of candidate RDCs indexed by j ∈ JK Set of AAs by disaster indexed by k ∈ KL Set of size of RDCs indexed by l ∈ LS Set of possible scenarios indexed by s ∈ SC Set of commodities indexed by c ∈ C

Parameters

ps Occurrence probability of scenario sFjl Fixed cost for opening a RDC of size l at location jϕic Procuring cost of a unit commodity c from supplier iϕics Procuring cost of a unit commodity c from supplier i in scenario sCi jc Transportation cost for a unit commodity c from supplier i to RDC jCi jcs Transportation cost of a unit commodity c from supplier i to RDC j in

scenario sC jkcs Transportation cost of a unit commodity c from RDC j to AA k in

scenario shkc Inventory holding cost for a unit commodity c at AA kπc Inventory shortage cost for a unit commodity cvc Required unit space for commodity cD jcs Amount of demand for commodity c at AA j in scenario sSic Amount of commodity c that could be supplied from supplier iCapl Type of capacity of RDC that has been openedρ jcs Fraction of stocked material of commodity c at RDC j

that remains usable in scenario s(0 ≤ ρ jcs ≤ 1)

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ρics Fraction of stocked material of commodity cat supplier i that remains usable in scenario s(0 ≤ ρics ≤ 1)

λ1, λ2 Weight assigned to cost variabilityγ (Gamma) Weight assigned to model infeasibility penaltyM A very large number

Continuous and Binary variables

Qi jc Amount of commodity c procured from supplier i and stored at theRDC j

Xi jcs Amount of commodity c transferred from supplier i to RDC j inscenario s

Y jkcs Amount of commodity c transferred from RDC j to AA k inscenario s

Ikcs Amount of inventory held at AA k in scenario sbkcs Amount of shortage at AA k in scenario sZ jl 1 if RDC with capacity category l is located at candidate RDC j ;

0 otherwise

5.2 Mathematical formulation

Now, we present a novel multi-objective, robust, stochastic optimization approachbased on Mulvey’s model in which uncertainty is represented by a set of finite discretescenarios (s). Equations (14)–(21) are defined for the convenience of formulation.

∑

j∈J

∑

l∈L

Fjl · Z jl SC (Setup costs) (14)

∑

i∈I

∑

j∈J,

∑

c∈C

ϕic · Qi jc PC (Procuring costs) in pre-disaster (15)

∑

i∈I

∑

j∈J

∑

c∈C

ci jc · Qi jc TC (Transportation costs from suppliers to RDCs)

in pre-disaster (16)∑

i∈I

∑

j∈J

∑

c∈C

ϕics · Xi jcs PCs (Procuring costs) in post-disaster (17)

∑

i∈I

∑

j∈J

∑

c∈C

ci jcs · Xi jcs TCSs (Transportation costs from suppliers to RDCs)

in post-disaster (18)∑

j∈J

∑

k∈K

∑

c∈C

c jkcs · Y jkcs TCRCs (Transportation costs from RDCs to AAs)

in post-disaster (19)∑

k∈K

∑

c∈C

hkc · Ikcs ICs (Inventory holding costs in AAs) (20)

∑

k∈K

∑

c∈C

πc · bkcs SCs (Shortage costs in AAs) (21)

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We recast the above discussion in the following multi-objective, disaster reliefplanning formulation:

Min Obj1

= SC + PC + TC +∑

s∈S

ps (PCs + TCSs + TCRCs + ICs + SCs)

+λ1 ·∑

s∈S

ps

[((PCs + TCSs + TCRCs + ICs + SCs)

−∑

s′∈S

ps′ (PCs′ + TCSs′ + TCRCs′ + ICs′ + SCs′))

+ 2θ1s

]

+γ∑

j∈J

∑

c∈C

∑

s∈S

psδ jcs (22)

Min Obj2 =∑

s∈S

ps ·(

∑

c∈C

maxk∈K

{bkcs})

+λ2 ·∑

s∈S

ps ·[(

∑

c∈C

maxk∈K

{bkcs} −∑

s′∈S

ps′ ·∑

c∈C

maxk∈K

{bkcs′ })

+2θ2s

]

(23)

s.t :∑

i∈I

Xi jcs+ρ jcs ·∑

i∈I

Qi jc +⎛

⎝∑

k �= j∈J

Yk jcs

⎞

⎠

(∑

l∈L

Z jl

)

−∑

k �= j∈J

Y jkcs = δ jcs

∀ j ∈ J, c ∈ C, s ∈ S (24)

Ykkcs +⎛

⎝∑

j �=k∈J

Y jkcs

⎞

⎠

(

1 −∑

l∈L

Zkl

)

− Dkcs = Ikcs − bkcs ∀k ∈ K , c∈C, s ∈ S

(25)

Y jkcs ≤ M

(∑

l∈L

Zkl + Dkcs

)

∀ j ∈ J, k ∈ K , c ∈ C, s ∈ S (26)

Y j jcs ≤ M.D jcs ·∑

j∈L

Z jl ∀ j ∈ J, c ∈ C, s ∈ S (27)

∑

i∈I

Xi jcs ≤ M ·∑

l∈L

Z jl ∀i ∈ I, j ∈ J, c ∈ C, s ∈ S (28)

∑

k∈K

Y jkcs ≤ M ·∑

l∈L

Z jl ∀ j ∈ J, k ∈ K , c ∈ C, s ∈ S (29)

∑

i∈I

∑

c∈C

vc · Qi jc ≤∑

l∈L

Capl · Z jl ∀ j ∈ J (30)

∑

c∈C

vc · I jcs ≤∑

l∈L

Capl · Z jl ∀ j ∈ J, s ∈ S (31)

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∑

j∈J

Qi jc ≤ Sic ∀i ∈ I, c ∈ C (32)

∑

j∈J

Xi jcs ≤ ρics · Sic ∀i ∈ I, c ∈ C, s ∈ S (33)

∑

l∈L

Z jl ≤ 1 ∀ j ∈ J (34)

(PCs + TCSs + TCRCs + ICs)

−∑

s′∈S

ps′ (PCs′ + TCSs′ + TCRCs′ + ICs′ + SCs′) + θ1s ≥ 0 ∀s ∈ S (35)

∑

c∈C

maxk∈K

{bkcs} −∑

s′∈S

ps′ ·(

∑

c∈C

maxk∈K

{bkcs′ })

+ θ2s ≥ 0 ∀s ∈ S (36)

Z jl ∈ {0, 1} Qi jc, Xi jcs, Y jkcs, I jcs, bkcs, δ jcs, θ1s, θ2s ≥ 0

∀i ∈ I, j ∈ J, k ∈ K , l ∈ L , c ∈ C, s ∈ S (37)

The two objectives in our model are indicated by Eqs. (22) and (23) and describedas follows:

Objective # 1: The first objective function (Eq. 22) minimizes the three terms. Thefirst term is related to the expected value of total cost or the sum of the first stage andthe expected value of second-stage costs. The first-stage costs include the preparednessphase costs (associated with setup, procurement and transportation from suppliers toRDCs). The second-stage costs include the response phase costs (associated with pro-curement, transportation from suppliers to RDCs, transportation from RDCs to AAs,inventory holding and shortage).

The second term of first objective function (Eq. 22) is related to the total cost var-iability. The last term in (22) measures the model’s robustness with respect to theinfeasibility associated with control constraints (24) under scenario s.

Objective # 2: The second objective maximizes the AAs’ satisfaction by minimizingthe sum of maximum shortage at devastated demand points. The idea of the mini-maxmethod is employed to sum up the least satisfactory value of each relief item so as toreduce the number of objective equations.

The first and second terms in Eq. (23) are the expected value and variability of theobjective function, respectively. This objective is explicitly nonlinear, and the linearequivalent equations could be rewritten with the help of an auxiliary variable Wcs ≥ 0as follows:

Min Obj2 =∑

s∈S

ps ·(

∑

c∈C

Wcs

)

+λ2 ·∑

s∈S

ps ·[

∑

c∈C

Wcs −∑

s′∈S

ps′ ·(

∑

c∈C

Wcs′

)

+ 2θ2s

]

(38)

s.t. Wcs ≥ bkcs ∀k ∈ K , c ∈ C, s ∈ S (39)

Wcs ≥ 0 ∀c ∈ C, s ∈ S (40)

123


Our model attempts to determine the best nodes for opening the RDC among thepotential sites while attempting to minimize total cost and maximize demand coverage.We will now explain the constraints:

Constraint (24) is a control balance equation for each RDC that is used to determinethe amount of commodity supplied to a specific RDC from suppliers in preparednessphase, a similar quantity from suppliers and other RDCs (backup plan) in responsephase, and the amount of commodity transferred to AAs from the RDCs. Note that,if the amount of commodity supplied in the preparedness phase plus that supplied inthe response phase is greater than the amount of commodity transferred to AAs, thedeviation (δ jcs) is indicating an increased commodity inventory penalized by the lastterm of the first objective function (γ

∑j∈J

∑c∈C

∑s∈S psδ jcs). We designate this

penalty function as expected overloading. This constraint has an explicit non-linearterm that could be rewritten as a linear term with the help of the following constraints:

∑

i∈I

Xi jcs+ρ jcs.

∑

i∈I

Qi jc + G1 jcs −∑

k �= j∈J

Y jkcs = δ jcs ∀ j ∈ J, c ∈ C, s ∈ S

(41)

G1 jcs ≤∑

k �= j∈J

Yk jcs ∀ j ∈ J, c ∈ C, s ∈ S (42)

G1 jcs ≤ M ·∑

l∈L

Z jl ∀ j ∈ J, c ∈ C, s ∈ S (43)

G1 jcs ≥∑

k �= j∈J

Yk jcs − M ·(

1 −∑

l∈L

Z jl

)

∀ j ∈ J, c ∈ C, s ∈ S (44)

Constraint (25) is an inventory balance equation for AAs. This constraint has anexplicit non-linear term that could be transformed to a linear term with the help of aninteger variable as follows:

Ykkcs + G2kcs − Dkcs = Ikcs − bkcs ∀k ∈ K , c ∈ C, s ∈ S (45)

G2kcs ≤∑

j �=k∈J

Y jkcs ∀k ∈ K , c ∈ C, s ∈ S (46)

G2kcs ≤ M ·(

1 −∑

l∈L

Zkl

)

∀k ∈ K , c ∈ C, s ∈ S (47)

G2kcs ≥∑

j �=k∈J

Y jkcs − M ·∑

l∈L

Zkl ∀k ∈ K , c ∈ C, s ∈ S (48)

Constraint (26) guarantees that a RDC could transfer commodity to other nodesonly if there exist either another RDC or an AA on that node. Constraint (27) indi-cates that an RDC could transfer commodity to its own area only if its demand pointsare affected. Constraints (28) and (29) prevent suppliers and RDCs from transferringcommodity to demand points where no RDC has been opened. The capacity limitsof RDCs are represented by (30) and (31). Constraint (32) ensures that the amount

123


of commodity c procured from supplier i cannot exceed the supplier’s capacity. Con-straint (33) ensures that the dispatched commodity from no supplier can exceed theamount of commodity c at supplier i and remain usable in scenario s. Note that theparameter ρics with a value between zero and one is to represent the degree of accessi-bility of suppliers. Constraint (34) prevents more than one RDC from being placed atany nodes. Constraints (35) and (36) are auxiliary constraints for linearization definedin (11). Finally, Constraint (37) determines the type of the variables.

6 Solution procedure

The methods of multi-objective optimization have been successfully applied in the lit-erature. The techniques of Goal Programming (GP), Compromise Programming (CP),and the Reference Point Method (RPM) are three of the most well-known multi-objec-tive decision-making methodologies (Romero et al. 1998). Among them, CompromiseProgramming is still frequently employed in multi-objective optimization problems.In this paper, the Compromise Programming is used to solve our multi-objective opti-mization problem.

In this method, a multi-objective problem is solved by regarding each objectivefunction separately and then reformulating a single objective that aims to minimizethe sum over the normalized difference between each objective and correspondingoptimum value. For our proposed model, we assume that two objective functions arecalled Obj1 and later Obj2. Based on the Lp-metrics method, our model should besolved for each of the two objectives separately. Assuming that the optimum valuesfor these two problems are Obj∗1 and Obj∗2, the Lp-metrics objective function can nowbe formulated as follows:

Min Obj3 =[

w · Obj1 − Obj∗1Obj∗1

+ (1 − w) · Obj2 − Obj∗2Obj∗2

]

(49)

where 0 ≤ w ≤ 1 is the relative weight of the components of the objective functiongiven by the decision maker, namely, Lp-metrics coefficient. Using this Lp-metricsobjective function and considering our model constraints, we arrive at a single-objec-tive, mixed-integer programming model, which can be solved efficiently using a linearprogramming solver.

7 Case study

Iran is a country prone to many types of natural disasters, such as earthquakes. Therehave been a number of earthquakes with magnitude 2 or above on the Richter scalethat caused heavy causalities as well as widespread economic loses. We present acase study in a region of Iran to demonstrate the effectiveness of our proposed model.It is based on discussions with subject matter experts of Tehran Municipality UrbanPlanning and Research Center.

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Fig. 2 Seismicity of Tehran and surrounding areas; 1909–2008 (Nowroozi 2010). (Faults are from Geo-logical map of Iran, National Iranian Oil Co. 1978; and Nazari et al. 2007)

7.1 Case description

The main study area lies in a region of Iran (Fig. 2). This region is located near thefoothills of the southern Central Alborz, which is surrounded by several active faultswith a population about 27 million people and with the size of about 1,300 km indiameter. Many historical and recent earthquakes have affected this region, mostly onthe Mosha, North-Tehran, Garmsar, Eshtehard, Kahrizak, Parchin, Ipak, North-Ray,and South-Ray faults, producing earthquakes greater than Ms = 6.5 (Berberian andYeats 2001; Jafari 2010).

Studies of recent destructive earthquakes in Iran show that social and economiclosses have been mainly due to the failure of buildings. For example, Nateghi (2001)demonstrated that for a 0.35 g scenario in Tehran about 640,000 residences out of1,100,000 would collapse or suffer serious damage, 1,450,000 people would be killed,and about 4,330,000 would suffer injuries.

According to Fig. 3, we consider five suppliers, named Sup1, . . ., Sup5 (Sari, Qaz-vin, Tehran, Arak, and Isfahan), and 15 demand points spread geographically overthe entire map. In dealing with disasters, three types of emergency supplies, namely,water, food, and shelter are considered in this case study. We consider four scenarios,s1,s2,s3, and s4, with occurrence probabilities of 0.45, 0.3, 0.1, and 0.15, respectively.Note that these scenarios and their associated probabilities are devised by the sub-ject matter experts or disaster planners on the basis of historical records and knowngeological faults. Nodes, potential RDCs, and potential demand points are shown in

123


Node Suppliers

Caspian Sea

Sari

Semnan

GorganRasht

Qazvin

KarajTehran

Gom

Arak

Kashan

Isfahan

Varamin

Islamshahr

Robatkarim

Shahriar

Fig. 3 Map of case study: nodes and suppliers

Table 1 Nodes, potentialRDCs, and potential DPs

Gorgan Karaj Shahriar

Semnan Tehran Gom

Sari Varamin Arak

Rasht Robatkarim Isfahan

Qazvin Islamshahr Kashan

Table 2 Capacity of Suppliersfor each commodity (103 units)

Suppliers (Water, food, shelter)

Sari (450, 450, 150)

Qazvin (450, 450, 150)

Tehran (510, 510, 170)

Arak (450, 450, 150)

Isfahan (450, 450, 150)

Table 1. The capacities of five suppliers for each commodity are shown in Table 2. Theunit cost of opening a new facility (F) that depends on its capacity is listed in Table 3.

Table 4 contains the pre-disaster phase data for the volume (v) occupied by each1,000 units of commodity; the procurement price for the same (ϕ); and the transporta-tion cost per unit distance for the same (transport). In the response phase, the 1000-unitprocurement prices are estimated to be the same as the commodity procurement price

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Table 3 Facility setup costdepending on its storagecapacity

Size F (103 $) Cap (103 m3)

Small 500 10

Medium 800 16

Large 1,200 24

Table 4 Unit procure price,transportation cost, and volumeoccupied by commodity

Commodity ϕ v Transport.(103) (103$/unit) (m3/unit) (103$/unit-km)

Water 0.5 4.5 0.6

Food 2 2 0.15

Shelter 20 120 1.8

Table 5 Demand (103 units)

DP Gorgan Semnan Sari Rasht Qazvin(water, food (water, food, (water, food, (water, food, (water, food,shelter) shelter) shelter) shelter) shelter)

s1 (319, 319, 106) (34, 34, 11) (579, 579, 193) (524, 524, 175) (68, 68, 23)

s2 (159, 159, 53) (69, 69, 23) (521, 521, 174) (429, 429, 143) (169, 169, 56)

s3 (96, 96, 32) (29, 29, 10) (289, 289, 96) (238, 238, 79) (158, 158, 53)

s4 (287, 287, 96) (57, 57, 19) (579, 579, 193) (476, 476, 159) (113, 113, 38)

DP Karaj Tehran Varamin Robatkarim Islamshahr(water, food, (water, food, (water, food, (water, food, (water, food,shelter) shelter) shelter) shelter) shelter)

s1 (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0)

s2 (293, 293, 98) (1418, 1418, 473) (103, 103, 34) (138, 138, 46) (98, 98, 33)

s3 (222, 222, 74) (1654, 1654, 551) (81, 81, 27) (100, 100, 33) (76, 76, 25)

s4 (256, 256, 85) (1339, 1339, 446) (76, 76, 25) (94, 94, 31) (67, 67, 22)

DP Shahriar Gom Arak Isfahan Kashan(water, food, (water, food, (water, food, (water, food, (water, food,shelter) shelter) shelter) shelter) shelter)

s1 (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0)

s2 (188, 188, 63) (228, 228, 76) (22, 22, 7) (225, 225, 75) (30, 30, 10)

s3 (177, 177, 59) (187, 187, 62) (75, 75, 25) (990, 990, 330) (30, 30, 10)

s4 (146, 146, 49) (166, 166, 55) (22, 22, 7) (225, 225, 75) (21, 21, 7)

in the pre-disaster phase, and the unit transportation cost is estimated to be 1.8 timesthat of the pre-disaster phase; these data are assumed to be fixed among scenarios.The cost for unmet demand penalty is estimated to be ten times the procurement priceof the corresponding commodity, and the holding cost is estimated according to thecurrent procurement price of the commodity. The cost of transportation between nodesis estimated on the basis of distance.

Table 5 shows that for each scenario, there is a four-element vector representingdemand at 15 potential demand points in detail. Demand for water and that for food at

123


Table 6 Percentage of commodity that remains usable after disaster (ρ jcs )

Node Gorgan Semnan Sari Rasht Qazvin(water, food, (water, food, (water, food, (water, food, (water, food,shelter) shelter) shelter) shelter) shelter)

s1 (80, 77, 85) (93, 90, 98) (75, 72, 80) (82, 79, 87) (100, 100,100)

s2 (90, 87, 95) (88, 85, 93) (82, 79, 87) (82, 79, 87) (85, 82, 90)

s3 (94, 91, 99) (95, 92, 100) (90, 87, 95) (92, 89, 97) (87, 84, 92)

s4 (82, 79, 87) (90, 87, 95) (80, 77, 85) (81, 78, 86) (90, 87, 95)

Node Karaj Tehran Varamin Robatkarim Islamshahr(water, food, (water, food, (water, food, (water, food, (water, food,shelter) shelter) shelter) shelter) shelter)

s1 (100, 100, 100) (100, 100, 100) (100, 100, 100) (100, 100, 100) (100, 100, 100)

s2 (83, 80, 88) (82, 79, 87) (81, 78, 86) (78, 75, 83) (78, 75, 83)

s3 (87, 84, 92) (79, 76, 84) (85, 82, 90) (84, 81, 89) (83, 80, 89)

s4 (85, 82, 90) (83, 80, 88) (86, 83, 91) (85, 82, 90) (85, 80, 90)

Node Shahriar Gom Arak Isfahan Kashan(water, food, (water, food, (water, food, (water, food, (water, food,shelter) shelter) shelter) shelter) shelter)

s1 (100, 100, 100) (100, 100, 100) (100, 100, 100) (100, 100, 100) (100, 100, 100)

s2 (82, 79, 83) (78, 75, 83) (95, 92, 100) (95, 92, 100) (90, 87, 95)

s3 (83, 80, 87) (82, 79, 87) (83, 80, 88) (78, 75, 83) (90, 87, 95)

s4 (86, 83, 89) (84, 81, 89) (95, 92, 100) (95, 92, 100) (93, 90, 98)

each demand point for a given scenario is estimated on the basis of the population den-sity multiplied by the vulnerability probability of the demand point. This probabilitydepends on the following factors: (1) disaster type, (2) disaster intensity, and (3) urbanfabrics. Because the shelter under consideration has a capacity for accommodatingthree people, the estimated demand for shelter is set to the number of affected peopledivided by three.

The capability of a facility (supplier or RDC) may be disrupted partially by a disas-ter due to damage to the roads and/or the facility itself. The percentage of commodityc at facility j that remains usable under scenario s is shown in Table 6. We use the Lp-metrics formulation (49) with equal relative weights (0.5) to solve this multi-objectivedisaster relief problem.

7.2 Results

In this section, we present computational results and analyze the behavior of the pro-posed model. We solve the problem using Lingo 10 running on a PC Pentium IV-3GHz with 4 GB of RAM (DDR 3) under the Windows XP SP3 environment. We ranthe model, and the results are given in Tables 7, 8 and 9. Table 7 shows that nineRDCs are opened and distributed widely across the network. Four of the nine RDCs

123


Table 7 Results of the storage amount of commodities in RDCs

RDC Facility size Water (103units) Food (103units) Shelter (103units)

Semnan Large 2,025 2,310 85

Sari Large – – 200

Qazvin Small – – 83

Karaj Medium – – 133

Varamin Small – – 78

Isfahan Small – – –

Kashan Large 284 – 189

Table 8 Amount of RDCs’ relief commodities procured from suppliers in post-disaster phase (Xi jcs )

Supply RDC Semnan Sari Qazvin Karaj Varamin Isfahanpoint Item (s1, s2, (s1, s2 (s1, s2 (s1, s2, (s1, s2, (s1, s2,

s3, s4) s3, s4) s3, s4) s3, s4) s3, s4) s3, s4)

Sari Water (0, 0, 0, 360) (0, 369, 405, 0) – – – –

Food (0, 0, 0, 346) (0, 0, 391, 0) – – – (0, 355, 0, 0)

Shelter (0, 0, 0, 127) (0, 130, 142, 0) – – – –

Qazvin Water – – (0, 0, 391, 405) – – –

Food – – (0, 369, 378, 155) – – –

Shelter (0, 135, 0, 142) – (0, 0, 138, 0) – – –

Tehran Water – – (0, 0, 0, 423) (0, 418, 402, 0) – –

Food – (0, 402, 0, 0) – (0, 0, 387, 408) – –

Shelter – (0, 147, 0, 149) – (0, 0, 142, 0) – –

Arak Water – – (0, 427, 0, 0) – – (0, 0, 373, 427)

Food – (0, 414, 0, 0) (0, 0, 0, 414) – – (0, 0, 360, 0)

Shelter (0, 0, 0, 150) – – – (0, 0, 25, 0) (0, 0, 106, 0)

Isfahan Water – – – – – (0, 427, 351, 0)

Food – – – – – (0, 414, 337, 220)

Shelter (0, 0, 0, 30) – – – – (0, 150, 124, 414)

are specialized: one facility stores only one of the three commodity types (shelter);another facility (Semnan) stores only two of the commodity types (water and shelter);yet another facility stores all three commodity types; and one facility stores nothingin pre-disaster phase. In total, approximately 2.3 million units of water are pre-posi-tioned, along with 2.3 million units of food and approximately 0.58 million unitsof shelter. The total cost of the pre-disaster phase for this solution that is not sub-jected to uncertainty is approximately 35.1 million dollars. The expected value oftotal cost for the post-disaster phase in this solution is about 55.5 million dollars. Theexpected value of the sum of maximum shortage for all AAs in this solution is 52 1,000units.

Table 8 shows the amount of each commodity procured from the suppliers andtransported to RDCs under various scenarios in the post-disaster phase. Under sce-

123


Tabl

e9

Am

ount

ofre

lief

com

mod

ities

tran

sfer

red

from

RD

Cs

tode

man

dpo

ints

(Yjk

cs)

RD

CIt

emG

orga

nSe

mna

nSa

riR

asht

Qaz

vin

Kar

ajTe

hran

Var

amin

Rob

atE

slam

Shah

Gom

Ara

kIs

faha

nK

asha

n

Sem

nan

Wat

er22

434

1,22

3–

––

653

1,91

5–

––

––

––

Food

232

3282

6–

––

1,31

91,

311

––

––

––

–

Shel

ter

9614

39–

––

311

51–

––

––

––

Sari

Wat

er18

1–

479

––

–28

4–

––

––

––

–

Food

167

–48

052

4–

279

62–

––

––

––

2

Shel

ter

6218

162

151

––

212

––

–63

–7

––

Qaz

vin

Wat

er–

–63

140

411

471

217

181

––

–19

828

––

Food

––

–33

311

6–

847

277

––

––

38–

–

Shel

ter

––

–12

441

1841

––

––

–7

––

Kar

ajW

ater

––

199

––

240

1,12

114

368

6815

316

6–

––

Food

––

––

–24

3–

–84

6614

710

4–

–21

Shel

ter

––

139

––

8492

–29

2554

––

1,31

3–

Var

amin

Wat

er–

––

––

–1,

574

7072

61–

167

19–

–

Food

––

––

68–

1,56

370

26–

174

158

826

455

2

Shel

ter

––

––

114

1,31

915

527

46–

49–

––

–

Isfa

han

Wat

er–

––

–23

2–

––

––

––

–46

342

3

Food

––

––

–16

755

7–

––

––

2246

6

Shel

ter

–38

––

––

––

––

–75

–15

81,

313

Kas

han

Wat

er–

–66

6–

592

–22

5–

––

––

–24

520

Food

––

1,10

3–

––

––

––

––

––

13

Shel

ter

––

––

––

1072

6–

––

5620

104

7

123


(b)(a)

DP

Karaj

TehranKashan

Relief Distribution

DP

Semnan

Varamin Varamin

Qazvin

Kashan

Backup Relief Distribution

Fig. 4 Difference between relief distribution and backup coverage

nario 2 (s2), for example, Semnan RDC provides shelters received in various quantitiesfrom several suppliers (127, 142, 150, and 30 1,000 units from Sari, Qazvin, Arak,and Isfahan, respectively). Note that the supplier cities in which a RDC is located cantake advantage of and provide part of its relief commodities from its own supplier tothe RDC with zero transportation cost. When an RDC has a commodity shortfall toserve the closest demand points, the second closest RDC is assigned to service them.For example, Semnan RDC provides food for AAs Gorgan, Semnan, Sari, Tehran,and Varamin with the average amount of 232, 32, 826, 1319, and 1311 1,000 units,respectively, in all scenarios (see Table 9). It should be noted that whenever an RDCprovides a service to a specific AA with an existing RDC, the service is considered abackup coverage for this existing RDC (see Fig. 4a); otherwise, the service is regardedas a relief distribution for that demand point (see Fig. 4b). In other words, if a certainAA has a designated RDC, the relief distribution is carried out by its own RDC.

To emphasize the importance of simultaneously considering the total cost and thesatisfaction of affected people, as incorporated in our proposed model, the followingthree models are defined for sensitivity analysis:

1. Model1 consists of the total cost of relief supply chain (Obj1) and its relatedconstraints;

2. Model2 consists of the sum of the maximum shortage at all demand points (Obj2)and its related constraints;

3. Lp-metrics model consists of the objective function (Obj3) calculated by Eq. (49)and its related constraints.

A series of multi-objective solutions for the model were obtained by varying thepossible number of RDCs (n). Figures 5 and 6 graphically show the objective functionvalues Obj1 and Obj2 for these solutions, respectively, as a function of the possiblenumber of selected RDCs ranging from 1 to 9.

As can be seen in Fig. 5, the value of Obj1decreases when the possible numberof RDCs increases in all three models. Thus, it can be concluded that the best valuefor the possible number of RDCs is equal to 7. According to Fig. 6, the best value ofObj2 is obtained when n = 6, 7, 8, and 9 in the same way as Obj1. Although the valueof Obj1decreases when the possible number of RDCs increases both in Model2 andLp-metrics model, the value of Obj2 obtained from solving Model1 does not decreasewhen the possible number of RDCs increases regularly. According to Figs. 5 and 6,the possible number of RDCs is assumed to be 7, and the solution reported in theprevious section is based on this assumption. In addition, Figs. 5 and 6 show that the

123


50000

70000

90000

110000

130000

150000

170000

190000

210000

230000

250000

270000

1 2 3 4 5 6 7 8 9

Obj

1(1

03 $)

N=Number of selected RDC

Model1 Model 2 Lp-metric

Fig. 5 Total cost versus the possible number of selected RDCs

40

50

60

70

80

90

100

110

120

130

140

Obj

1(1

03 $)


Model2 Lp-metric

0

200

400

600

800

1000

1200

1400

1600

1800

2000

1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9

Obj

2(1

03U

nits

)


Model1 Model2 Lp-metric

Fig. 6 Sum of maximum shortage in all AAs versus the possible number of selected RDCs

proposed Lp-metrics model behaves in such a way that the values of Obj1 and Obj2follow their best values, Obj∗1 and Obj∗2, as possible.

In order to arrive at an appropriate solution such that the decision maker will beable to make trade-offs of one criterion against another based on the results, the afore-mentioned problem is solved several times while varying the Lp-metrics coefficient(w). By solving this disaster relief logistics problem, it is concluded that the rela-tionship between total cost and satisfaction is clear, and that it is possible to easilydefine a utility function. According to Fig. 7, increasing weight for Obj1 (the totalcost) causes Obj2 (amount of shortage) to increase (less satisfaction). Also, decreas-ing weight for Obj1 causes Obj2 to decrease. Thus, it seems that by raising the goalfor any of the objectives, we create more space for other objectives to be improved. Itis also concluded that there are some positive correlations between the total cost andthe satisfaction.

In Fig. 8, a sensitivity analysis is performed for model robustness and solutionrobustness against the multiplier of model robustness (Gamma) in the Lp-metricsmodel. As Fig. 8a demonstrates, the expected overloading will eventually drop to zero

123


w=0.1

w=0.2

w=0.3w=0.5

w=0.4w=0.6

w=0.7w=0.8

w=0.9

45

50

55

60

65

70

96000 97000 98000 99000 100000 101000 102000 103000 104000 105000 106000

Obj

2(1

03un

its)

Obj1 (103 $)

Obj1 versus Obj2

Fig. 7 Results of sensitivity analysis for Lp-metrics coefficient (w)

50000

55000

60000

65000

70000

75000

80000

85000

90000

Exp

ecte

d co

st(1

03$)

Gamma

Solution Robustness

0

200

400

600

800

1000

1200

1400

1600

1800

0 200 400 600 800 0 200 400 600 800

Exp

ecte

d ov

erlo

adin

g

Gamma

Model Robustness

(a) (b)

Fig. 8 Solution robustness (a) and model robustness (b) versus Gamma (γ )

with an increase in the value of Gamma. On the other hand, expected cost increasesexponentially by increasing the value of Gamma (see Fig. 8b). It means that the DMcould here choose the best value of Gamma based on his/her preferences. A risk-averseDM may tend to select the lower values of Gamma; conversely a risk-seeking DMmay prefer higher values of that.

Managing uncertainty in disaster relief planning is a crucial issue. To highlight therole of uncertainty in disaster relief logistic planning, we compare here, four typicalmodels; each one covers partial degree of uncertainty. This comparison is made tomeasure the effect of not considering all uncertain parameters all together in thosesituations where the environment is actually uncertain. An example of this kind ofmistakes may appear when a deterministic model is used to solve problems which areaffected by uncertain parameters considering just mean values for such parameters.In order to perform the aforementioned comparison, the relief chain designs obtainedby solving the proposed model for the case study are evaluated against the models inwhich some or all sources of uncertainty is neglected. The aforementioned models areas follows:

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Modeli (Deterministic): Assuming: - No demand uncertainty - No supply uncertainty - No cost uncertainty

Modelii: Assuming: - Demand uncertainty - No supply uncertainty - No cost uncertainty

Modeliii: Assuming: - Demand uncertainty - Supply uncertainty - No cost uncertainty

Modeliv: Assuming: - Demand uncertainty - Supply uncertainty - Cost uncertainty

Fig. 9 Typical models with different degrees of uncertainty (Modeli to Modeliv)

86

88

90

92

94

96

98

100

102

Tot

al C

ost

(106 $)

1 2 3 4

DeterministicDemand uncertaintyDemand/supply uncertaintyAll source of uncertainty

Fig. 10 Total cost of Modeli to Modeliv under scenario 1–4

Modeli is not subject to uncertainty (deterministic model). Modelii considersmerely demand uncertainty. Modeliii covers both uncertainty of demand and supply.Finally, Modeliv (the proposed model) considers all sources of uncertainty simulta-neously (see Fig. 9). Note that, in this comparison, we set the second objective functionto a pre-specified level for all models.

To quantify the cost saving by considering the various sources of uncertainty, eachtypical model is solved for the case problem and the results are shown in Fig. 10.

According to Fig. 10, Modeli does not consider uncertainty. Therefore an essentialcost is incurred in disaster relief system and causes total cost to become considerablyunstable dealing with different scenarios. When demand uncertainty is considered inModelii , the complexity of the model increases to some extent, but the capability of themodel to respond to unexpected situations increases as well. Whenever higher degreesof uncertainty are being considered, the modeling complexity (highly constrained andlarge amount of variables) increases accordingly, but total cost will be improved incomparison with existing models which cover lower degrees of uncertainty. As shownin Fig. 10, Modeliv is more reliable than Modeliii and in the same way; Modeliii ismore reliable than Modelii.This fact is conceived by the lower cost of these modelsin comparison with each other under different scenarios. On average, 8.9, 6.8, and

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0

5

10

15

20

25

30

35

40

45

Cos

t (1

06$)

DeterministicDemand/suppply uncertainty

Demand unall sources o

certaintyof uncertainty

Fig. 11 Average component total cost for Modeli to Modeliv

3.8% cost saving is achieved by considering three sources of uncertainty (Modeliv)in comparison with models i, ii and iii, respectively.

Fig. 11 shows the components of the average total cost for different models. As canbe seen, by considering all sources of uncertainty (Modeliv), we have lower cost foreach objective function components in comparison with the other models. It thereforewould seem more reasonable for decision makers to prefer the modeling complexityof considering all sources of uncertainty (the proposed model) to immune themselvesfrom the risk of probable future losses.

8 Conclusions and recommendations

In this paper, we proposed a multi-objective, robust, stochastic programming modelto simultaneously optimize the humanitarian relief operations in both the prepared-ness and response phases. Our model consists of two stages; the first stage determinesthe location of RDCs and the required inventory quantities for each type of reliefitems under storage, and the second stage determines the amount of transportationfrom RDCs to AAs. Our multi-objective model minimizes expected total costs, costvariability, and expected penalty for infeasible solution due to uncertain parameterswhile maximizing customer satisfaction. In our model, the cost parameters for therelief logistics as well as demand are subject to uncertainty. Furthermore, the modelconsiders uncertainty in the locations where those demands might arise as well as thepossibility that some of the pre-positioned supplies at RDC or supplier might be par-tially destroyed by the disaster (supply uncertainty). Finally, we solved our model asa single-objective, linear programming problem applying the Lp-metrics method. Todemonstrate the effectiveness of the proposed model, a case study based on a specificdisaster scenario is presented. Sensitivity analysis is also performed to validate themodel.

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We also assessed the potential improvement of using robust stochastic optimizationwith all sources of uncertainty in the relief chain planning process compared with themodels with lower degrees of uncertainty. The results of our case study show that onaverage, cost savings of at least 3.8% could be achieved by using the proposed model.

Because both the model and the computed results, obtained from a set of real data,have been based on the input and feedback from experts, we have confidence in thestrength and practicability of these conclusions for dealing with uncertain disaster sce-narios. Decision-makers involved in relief planning can gain insights into their ownplanning problem by using our proposed model to analyze their own sets of data.

At the end, we make the following relevant suggestions for further research: (1)Although we have not addressed the vehicle scheduling problem, the constraints ofrescue vehicle types and the limitations of rescue manpower should be consideredin disaster-relief logistics management; (2) for large-scale problems, especially thoseinvolving a concurrent increase in the number of scenarios, the number of commoditiesand the number of potential locations, it would be necessary to employ meta-heuris-tics. Although meta-heuristics have been applied successfully to the deterministicoptimization programming, only a few meta-heuristic solutions have been devisedfor stochastic optimization programming; and (3) it would be interesting to developa new approach based, for example, on fuzzy logic, to determine the probability ofoccurrence behind real scenarios because the model results are highly dependent onthem.

Acknowledgments We are most grateful to The Guest Editor, Professor Van Wassenhove, to ProfessorKarl Doerner, and to the anonymous referees whose detailed reviews and insightful comments led to asignificant improvement in the article.

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