A novel option contract integrated with supplier selection and inventory prepositioning for humanitarian relief supply chains

Socio-Economic Planning Sciences xxx (xxxx) xxx

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A novel option contract integrated with supplier selection and inventory prepositioning for humanitarian relief supply chains Mojtaba Aghajani , S. Ali Torabi *, Jafar Heydari School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran

A R T I C L E I N F O

A B S T R A C T

Keywords: Humanitarian logistics Relief supply management Option contract Supplier selection Two-stage possibilistic-stochastic programming

This paper proposes a novel two-period option contract integrated with supplier selection and inventory prep­ ositioning. A two-stage scenario-based mixed possibilistic-stochastic programming model is developed to cope with various uncertainties. The first stage’s decisions include supplier selection and capacity reservation level at each supplier/period and the level of inventory prepositioning. Furthermore, decisions regarding the time and exercised amount are made in the second stage. Applicability of the model is validated through a real case study. Finally, several sensitivity analyses are conducted to examine the effect of important parameters on the solutions to gain useful managerial insights.

1. Introduction In recent years, several natural disasters such as earthquakes, hur­ ricanes, and floods have occurred frequently around the world which have caused severe fatalities, social costs and economic loss. According to the Emergency Event Database (EM-DAT, www.emdat.be), the number of natural disasters globally between 1980 and 2015 was 11,495, resulting in over 2.5 million deaths, affecting 6 billion people and led to 2.71 trillion dollars global economic loss. The increasing number of casualties caused by these disastrous events creates a growing need for providing relief supplies in an effective while efficient manner for saving the lives of affected people. After a disaster strikes, relief supplies such as water, shelter and health products should be available at the right time and in the right amount. One of the main reasons for ineffective emergency response is due to the supply shortage and delay [1]. Therefore, it is essential for relief organizations (e.g. Red Crescent Societies) to develop appropriate relief supply decision models (which are capable of generating reasonable supply plans) for reducing the impact of disasters and maintaining social stability. Humanitarian organizations (HOs) may suffer from having too much or too little inventory in their relief supplies management, and hence are exposed to inventory shortage/surplus risks. Governmental (i.e. large) HOs typically procure relief supplies with low unit prices and preposition them at pre-disaster to decrease the response time at postdisaster [2]. However, pre-positioning of relief items could be so costly as this relief strategy is exposed to inventory shortage/surplus

risks due to a high level of demand uncertainty [3]. When potential disasters are not struck or if the demand is low, most of the pre­ positioned relief items are not used within their durability/expiry period, which leads to redundancy and waste and causes financial loss. Alternatively, pre-positioning of relief supplies in large amounts can be financially prohibitive for many relief organizations due to budgetary and capacity limitations. Hence, it can be very ineffective if the demand surge is high, which may lead to stock-out risk. Therefore, in order to reduce the inventory-related risks, and increasing the flexibility and effectiveness in responding to the uncertain demand raised by disasters, using a mixture of relief pre-positioning and suitable supply contracts (between relief organizations and suppliers) at the pre-disaster phase would be more beneficial than using just inventory pre-positioning. Noteworthy, procurement of relief supplies in many relief organiza­ tions such as the International Federation of Red Cross (IFRC) and Red Crescent Societies are done through pre-established supply contracts at pre-disaster, which can increase the response capacity and timely response and concurrently decrease the procurement cost [4]. Among various supply contracts, it has been proved and demon­ strated that option contract improves the relief chain performance and is efficient in managing risk, including demand uncertainty, supply unre­ liability, and price volatility as well as reducing supply costs [3,5,6], which are the main challenges of disastrous events [7]. Usually, an option contract includes two important parameters, namely the option price o and the exercise price e. In the relief setting, the option price is an allowance paid by the HO to the supplier for reserving one unit of the

* Corresponding author. E-mail addresses: [email protected] (M. Aghajani), [email protected] (S.A. Torabi), [email protected] (J. Heydari). https://doi.org/10.1016/j.seps.2019.100780 Received 10 February 2019; Received in revised form 21 December 2019; Accepted 27 December 2019 Available online 3 January 2020 0038-0121/© 2020 Elsevier Ltd. All rights reserved.

Please cite this article as: Mojtaba Aghajani, Socio-Economic Planning Sciences, https://doi.org/10.1016/j.seps.2019.100780

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geographical coverage, commitment requirements and pricing sched­ ules, which directly affect disaster response performance. The multiple sourcing strategy (i.e. selecting two or more suppliers) can help to reduce and mitigate the impact of supply disruptions in the humani­ tarian supply chains [11]. In this regard, HO would face the supplier selection and order allocation (SS&OA) problem by which the most suitable suppliers and their optimal order quantities are simultaneously determined [12]. Despite its importance, there is no study in the relief literature investigating this problem integrated with an option contract. In this regard, we develop a mathematical programming model to address this issue. In the supplier selection sub-problem, in addition to general criteria such as quality and cost, we also consider some other criteria such as the coverage degree of each supplier in regards to each potential demand point, supplier’s capacity in the fast and normal de­ livery modes and the option and exercise prices offered by the supplier. The number and geographical locations of selected suppliers highly impact the coverage degree of affected areas, which in turn affect the responsiveness, equity and fairness measures (which are the most sig­ nificant criteria in relief logistics). Notably, although there is a rich literature that addresses covering problems, this topic has not received much attention in the domain of relief chains, especially in the supplier selection phase. In this way, we aim to establish a procedure to prevent imbalance coverage of potential demand points. Uncertainty in the required data is one of the main challenges of logistical planning in relief chains [13]. Galindo and Batta [14] stated that using two-stage stochastic scenario-based models would help to deal with the random uncertainty, but due to specific characteristics of disasters, where in most cases, historical data is not available or enough, accounting for uncertainty of scenario-based data is required within each scenario. Since in the relief logistics, fuzziness (especially in scenario-dependent parameters) and randomness are two sources of uncertainty that simultaneously appear, combining the traditional two-stage stochastic programming framework with fuzzy numbers would be helpful to jointly consider these different uncertainties involved in the problem [15]. Therefore, we develop a two-stage sce­ nario-based mixed possibilistic-stochastic programming model to take into account both random and possibilistic (imprecise) data in the developed modeling framework.

supplier capacity before a disaster strikes. When a disaster strikes, the HO has the right (not the obligation) to exercise the option contract after the actual demand is realized and purchase the required relief supplies with unit price e. Therefore, relief organizations can use option contracts to reduce the risk of demand uncertainty and ensure relief supplies availability at reasonable prices. Also, an option contract provides flexibility in ordering relief supplies because it allows HO to determine the exact purchase amounts of needed relief supplies after further de­ mand information becomes available at post-disaster. It can also attract supplier; since she/he can earn some money (through the option price) paid by the HO at the beginning of an option contract before sending any supplies [8,9]. On the other hand, the supplier can enjoy early commitment from the HO so that it can make better capacity and ma­ terial planning. Consequently, an option contract generates mutual benefits for both the HO and the supplier to interact. In this paper, according to the opinion of procurement managers of IFRC, and motivated by the significant benefits described above, we focus on the option contract as the most suitable format which can fit the specific requirements of relief chains [6]. Although there is a rich literature on using option contract in the commercial setting, surprisingly, there is very limited literature in the area of relief logistics. To fill this gap, this paper aims to develop a novel option contract model while considering the priority of relief items to improve the process of relief supply in the context of humanitarian operations. In the relief logistics, there would be considerable demand for several relief items with different response times while it is not possible to send all of them to affected areas simultaneously. For example, after the Tsunami that occurred in several Asian regions in December 2004, the oversupply of different relief items exceeded the capacity of aid agencies in the field to sort, store, and quickly deliver, which caused several logistical problems [10]. Also, concerning response times, relief supplies have different priorities. That is, some items (with higher priority) must be distributed within the early hours/days at post-disaster (such as shelters and water) while others (with lower priority) can be distributed in later hours/days at post-disaster (such as cleaning supplies). Therefore, to have an appro­ priate supply policy, we divide the post-disaster horizon into the critical (i.e. the first 48 h after the disaster occurs) and non-critical (i.e. later hours or days at post-disaster) periods according to the response time of each classified item. In light of the above discussion, this paper aims to develop two different types of two-period option contract models each of which has its specific capacity reservation strategy based on a two-step delivery policy with delivery time-dependent option and exercise prices. The proposed option contract models can provide mutual benefits to both relief organizations and suppliers. A most obvious benefit to the relief organizations is that they can reserve supplier capacity to guar­ antee the availability of relief supplies in sufficient quantities when they are needed. Indeed, the proposed two-step delivery policy allows relief organizations to receive the requirements of each period in the same period which results in the timely use of relief items. With the increased frequency of delivery and reduced order sizes, it also enables relief or­ ganizations to mitigate the problem caused by limited logistical capac­ ities (for sorting, storing and delivery activities). It is one of the most significant benefits of using this approach because resource capacities are the main constraints in relief logistics. With considering the possi­ bility of splitting the order into two parts of smaller sizes, in addition to reducing supplier’s inventory holding cost, even small suppliers who cannot preposition large quantities of relief items, can participate in the procurement process. This allows relief organizations to assess a broader range of candidate suppliers. Suppliers play an essential role in the successful and efficient disaster response. Evaluating candidate suppliers is a challenging issue for relief organizations due to the high uncertainty in disaster location and de­ mand size. For instance, the availability and ability of suppliers for quick delivery of relief supplies depend on the supplier’s distance from the affected area. Also, candidate suppliers may have different capacities,

2. Literature review In this section, we review the literature in two separate but relevant research streams including procurement in humanitarian logistics and covering problems in relief logistics. 2.1. Procurement in humanitarian logistics Sourcing and procurement planning are among the significant issues in relief operations which can affect the optimal decisions of humani­ tarian logistics such as transportation, storage, and distribution of relief supplies. Besides, Falasca and Zobel [16] state that most expenditures in disaster relief logistics (about 65%) are related to the procurement processes. However, despite the importance of disaster relief procure­ ment, there is limited research in this area. Trestrail et al. [17] developed a two-stage mixed-integer program­ ming model from the perspective of bidders to improve the bid pricing strategies of the US Department of Agriculture (USDA). Shokr and Torabi [18] proposed an enhanced reverse auction framework and presented two novel possibilistic programming models under the epistemic uncertainty of input data which could be used in both pre- and post-disaster to support the bidders and the auctioneer. Also, some other studies have suggested auction-based frameworks for post-disaster relief procurement, which typically consist of announcement construction, bid construction and bid evaluation phases [19–21]. Falasca and Zobel [16] proposed a two-stage stochastic decision model with recourse by which procurement decisions such as order lot-sizes are made in the first stage 2

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and after observing the uncertain demand, recourse action is performed as the second-stage decisions to compensate the decisions made in the first stage. Hu et al. [23] proposed a scenario-based two-stage stochastic programming model for integrating pre-positioning and post-disaster procurement with supplier selection in humanitarian relief. Hu and Dong [24] addressed the joint decision-making of the relief supplies pre-positioning and supplier selection under disruption risks using a two-stage stochastic programming model. Supply contracts have received more attention in recent years to improve the efficiency of procurement activities in supply chains. Cachon [25] provides a comprehensive review of supply chain coordi­ nation contracts. Consequently, several types of coordination contracts have been investigated intensively, such as wholesale price contracts, buyback contracts, revenue sharing contracts and quantity-flexibility contracts [26–28]. Among other varieties of contracts, option contract has attracted a significant amount of research [25]. In recent years, designing the procurement contracts in the context of relief logistics is increasingly capturing the attention of researchers. Among them, option contracts [3,6,29] and quantity flexibility contracts [4,11,30] are the two most applied methods. Liang et al. [6] presented an option contract mechanism to cope with the issues of inaccurate demand forecasting and overstock in relief material management. They proposed an alternative binominal option pricing model to determine a feasible range of prices in which both the buyer and supplier will prefer option contracts to wholesale contracts. Wang et al. [3] developed an option contract for pre-purchasing of relief items and demonstrated that it dominates both the pre-purchasing with buyback contract and instant purchasing with the return policy in a relief supply chain. Torabi et al. [11] developed a novel integrated pre-positioning and post-disaster procurement model based on a quantity flexibility contract using two-stage scenario-based mixed fuzzy-stochastic programming. Balcik and Ak [4] proposed a quantity flexibility contract in the context of humanitarian relief chains and developed a scenario-based stochastic programming model to select the most appropriate suppliers in the first stage and placing the required orders in the second stage. Nikkhoo et al. [30] presented a quantity flexibility contract framework to coordinate purchasing and ordering activities in a three-echelon relief chain that consists of a relief organi­ zation, supplier and affected areas.

cooperative covering model with budget considerations to maximize the benefits to the affected population under demand uncertainty. 2.3. Gap analysis Making long-term agreements with suppliers in the pre-disaster phase can guarantee the availability and cost-effective procurement of relief supplies and at the same time, would be helpful to decrease the holding cost and increase the reliability of on-time delivery at postdisaster. As the literature indicates, there are few studies developing an integrated supplier selection and procurement planning model based on a supply contract in the relief logistics setting. In this regard, we can refer to the research works carried out by Balcik and Ak [4] and Torabi et al. [11], which are somewhat similar to our work. However, our research differs from these works in different aspects. First, to make the model closer to reality, we account for some specific criteria (e.g. time-dependent prices, supplier’s capacity for the fast and normal de­ livery modes and supplier coverage levels) in the supplier selection part, which were not considered in these studies. Second, we consider the disruption risk of disasters and assume that both warehouses and sup­ pliers might be destroyed in a disastrous event, while those models do not account for disruption risks. Third, we consider an option contract between the relief organization and suppliers, while they have used the quantity flexibility contract in their studies. Although there is a rich literature on option contract design in the commercial context, few studies addressed this topic in the context of relief logistics. Different from the existing option contract models for the relief supply chains proposed by Liang et al. [6] and Wang et al. [3], which are basically single-period models and try to find a feasible range and acceptable prices for both suppliers and HOs, this paper aims to develop a novel two-period option contract model integrated with the supplier selection and inventory prepositioning to handle the dynamic nature of disastrous events. Also, despite the importance and applica­ bility of the demand coverage requirement in the supplier selection phase (i.e. considering responsiveness, equity and fairness), surpris­ ingly, this issue has not received any attention in the relief logistics literature. To fill these gaps, we develop a two-stage scenario-based mixed possibilistic-stochastic programming model to cope with various uncertainties at pre- and post-disaster phases, including imprecise/­ possibilistic parameters and random scenarios’ probabilities. The developed model determines the amount of capacity reservation of each supplier in each period based on their locations (i.e. coverage level), available capacity and delivery time-dependent option and exercise prices. Also, in contrast to the previous research in the context of relief chains, which consider the maximal covering approach in the facility location part, in this study, we address this issue in the supplier selection process. It should be noted that although considering mixed uncertainty including random and possibilistic data in the context of relief logistics is not new (see Ref. [15]), however, their tailoring, combination and application under an option contract mechanism are unique here. In summary, the main contributions of this study are briefly as fol­ lows: (I) proposing a novel two-period option contract model for relief procurement with delivery-time dependent option and exercise prices; (II) developing a mixed procurement policy comprising of a novel option contract and inventory pre-positioning while addressing some real concerns (e.g. considering priority of relief supplies, useable ratio of prepositioned inventories and potential disruption of suppliers) (III) incorporating a supplier selection and order allocation problem into an option contract model for relief procurement planning; (IV) considering a new approach based on the maximal covering model in the supplier selection process; (V) developing a tailored two-stage scenario-based possibilistic-stochastic programming model to incorporate various un­ certainties at pre- and post-disaster phases.

2.2. Covering problems in relief logistics Covering models are generally used in the facility location problems, mainly when the response time is a critical performance measure. Ac­ cording to objective type, covering location problems can be divided into two categories: (1) set covering and (2) maximal covering location models. The set covering models try to minimize the number of required facilities to cover all demand points. However, in many real-world problems, allocated resources (e.g. budget) are not sufficient to cover all demand points with acceptable coverage distance. Therefore, Church and ReVelle [44] introduced the maximal covering location model to maximize the number of covered demands given a fixed number of fa­ cilities or budget limitations. Although the literature on covering location problems is extensive (see relevant studies in Refs. [45–48]), few studies can be found in the context of relief logistics. Mohammadi et al. [49] presented a multi-objective stochastic programming model for considering an earthquake response plan, which integrates pre-and post-disaster de­ cisions. They constructed a three-objective model to maximize the total expected demand coverage, minimize the total expected cost, and minimize the difference in the satisfaction rates of demand points to ensure equity. Balcik and Beamon [50] developed a scenario-based mathematical model that integrated facility location and inventory de­ cisions in the domain of relief logistics with considering multiple-item, budget constraints and capacity restrictions. They formulated a maximal covering location model to determine the number, locations and capacities of distribution centers. Li et al. [51] proposed a maximal 3

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3. Problem description

non-critical period. Hence, the HO can meet the demand of non-critical period from suppliers with the coverage levels 1 and 2. Moreover, due to some constraints such as budget limitation, it is not possible to apply a set covering model for fully satisfying all the needed relief items. Therefore, the HO should apply a maximal covering-type model for supplier selection of relief items to maximize the number of covered affected people subject to resource limitations. Consequently, we develop a mathematical model in the form of a maximal covering model for making the supplier selection and capacity reservation related de­ cisions through a novel option contract model as well as determining the amounts of inventory prepositioning at pre-disaster and order execution decisions at the post-disaster horizon. In this way, our model integrates an option contract with the supplier selection and inventory preposi­ tioning decisions while considering multiple types of relief items with different priorities and budgetary and capacity constraints.

We consider a relief supply chain including a HO (who is responsible for relief procurement) and multiple suppliers. In the pre-disaster phase (i.e. the first stage), the HO makes a two-period option contract with each selected supplier to supply multiple relief items at post-disaster (i.e. the second stage). The option contract allows the HO to reserve each supplier’s capacity through paying an option price (o), and, in return, the supplier promises to deliver supplies according to the pre-specified agreement terms. In the general form of an option contract in the re­ lief setting, decision making is done in two stages (in accordance with pre- and post-disaster phases). In the first stage, the HO reserves q units at unit price o from the selected supplier. Then, when the actual demand is realized at post-disaster, a recourse decision is taken to determine the purchasing amount (no more than q units) at unit exercise price e. Within the contract lifetime, if no disaster happens, the option contract will not be exercised. Now, different from the general form of an option contract, we develop two different types of two-period option contract models which are elaborated in sub-section 3.1. Notably, as mentioned before, we divide the post-disaster horizon into the critical (i.e. the first 48 h after the disaster occurs) and non-critical (i.e. later hours or days at post-disaster) periods in accordance to the different response times needed for relief items. Furthermore, each contract model applies a specific capacity reservation strategy based on a two-step delivery policy with delivery time-dependent option and exercise prices. To start with details of the problem under investigation, suppose that there are I candidate suppliers denoted by i, with different capacities, geographical coverage and pricing (i.e. different option and exercise prices). With considering three coverage levels for each supplier denoted by c, c ¼ {1,2,3}, we distinguish the suppliers by dividing them into separate sets and define SUc as the set of suppliers that provide coverage level c to a disaster region. A demand point is covered by the coverage level c if it is located within a specified response distance/time from a supplier. For instance, in Fig. 1, region 1 (denoted by R1) is covered by the coverage level 1 of supplier 1 and the coverage level 3 of supplier 3. Also, region 2 (denoted by R2) is located within the coverage level 1 of supplier 2 and the coverage level 2 of supplier 3. Notably, the idea of using multiple coverage levels is adopted from Berman and Krass [46]. In relief logistics, the occurrence of any shortage and delay in providing critical supplies could result in more fatalities and causes huge social cost. When the selected suppliers are located far from the disaster area, the golden time (typically the first 72 h) during which the lives could be saved, is lost which in turn could result in a considerable in­ crease in the fatality rate. So, to have an appropriate response in the early post-disaster, the HO must have enough pre-contracted suppliers or prepositioned items within the first coverage level of each region so that all the demands of the critical period (i.e. the so-called golden time) could be satisfied. Nevertheless, there would be enough time to satisfy the demand of

3.1. The proposed option contract models In this section, we develop two types of option contracts based on two different capacity reservation strategies. In both models, the postdisaster phase is divided into two periods, and the HO makes de­ cisions at two stages (pre- and post-disaster phases). 3.1.1. The first option contract model The first model is characterized by seven parameters ðo1 ;e1 ;o2 ;e2 ;e3 ; cap1 ; cap2 Þ. Two option prices are denoted by o1 and o2 and three ex­ ercise prices are denoted by e1 , e2 , and e3 , where subscripts 1 and 2 correspond to the first and second periods, respectively. Also, e3 is related to the exercise price of additional purchasing (when the realized demand in the first period is more than the reserved capacity). Each supplier will charge a higher/lower price for the fast/slow delivery modes. To ensure that the model works reasonably, we require that o1 > o2 ; e3 > e1 > e2 . The capacity of each supplier in two periods is limited and denoted by cap1 and cap2 . At the first stage (i.e. pre-disaster), to meet the forecasted demand of the first and second periods of the postdisaster phase, HO reserves q1 units (for the first period) at option price o1 and q2 units (for the second period) at option price o2 from each selected supplier, where q1 þ q2 ¼ Q. Therefore, the reservation cost in this model is equal to q1 :o1 þ q2 :o2 . This reservation gives the right (not the obligation) to the HO to order up to q1 units in the first period and q2 units in the second period with exercise prices e1 and e2 , respectively. In the second stage, according to the updated demand information of the first and second periods ðd1 and d2 Þ, the HO determines the exercise order quantities of the first and second periods (g1 and g2 while g1 þ g2 � Q). In this way, our proposed option contract provides flexibility for the HO to purchase additional items in each period more than the reserved capacity of that period (up to Q units) when higher demand than the forecasted size is realized. Although reservation quantity for meeting the first-period demand is q1 units, the HO has the chance of placing an additional order at a higher price if demand realization of the first period is more than q1 . For any order quantity in the intervals ½0; q1 � and ðq1 ; Q q1 �, the unit exercise price e1 and e3 is charged by the supplier, respectively. Indeed, e3 is an incentive to motivate the supplier to prepare additional items for delivering in the first period. Therefore, for example, if demand for the first period is higher than the total reserved quantity ðd1 > QÞ, the exercise cost of the first period is expressed as: q1 :e1 þ ðQ

q1 Þe3

(1)

Also, in the developed model, it is assumed that the unused reserved capacity of the first period can be transferred and exercised in the second period. Concerning the required times and demand natures (i.e. recurring and non-recurring demands), we consider four types of relief supplies. The first type is related to those critical items needed in the first period, such as medical kits and shelters ðd2 ¼ 0; 0 < q1 � cap1 ; q2 ¼ 0Þ. The

Fig. 1. Coverage example. 4

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second type is needed in the second period, such as hot meals and cleaning supplies ðd1 ¼ 0; 0 < q2 � cap2 ; q1 ¼ 0Þ. The third type is related to those items that their required distribution time is unknown at pre-disaster as it depends on the disaster location and time ðd1 ¼ 0; 0 < q2 � cap2 ; q1 ¼ 0 or d2 ¼ 0; 0 < q1 � cap1 ; q2 ¼ 0Þðd1 ¼ 0; 0 < q2 � cap2 ; q1 ¼ 0 or d2 ¼ 0; 0 < q1 � cap1 ; q2 ¼ 0Þ. For example, blanket and warm clothes are needed in the first period in a cold region or at a cold season ðd2 ¼ 0; 0 < q1 � cap1 ; q2 ¼ 0Þ while they are needed in the second period for the non-cold regions or at non-cold seasons ðd1 ¼ 0; 0 < q2 � cap2 ;q1 ¼ 0Þ. The last type of relief items is related to those with recurring demands over the planning horizon (such as water) which are needed in both periods ð0 < q1 � cap1 ; 0 < q2 � cap2 Þ. In summary, we name the relief items needed in the first and second pe­ riods as the critical and non-critical items, respectively. Our proposed models can apply to all types of relief items mentioned above. In the following, by considering different values of d1 and d2 , we show how this model works in some cases:

demand for the first period is higher than the total reserved quantity ðd1 > QÞ, the exercise cost of the first period can be expressed as Q:e1 . In contrary to the previous model, in this model, the maximum amount ðQÞ that can be exercised (order quantity) during the entire planning horizon (i.e. the first and second periods) is determined without separating the first and second periods’ reservation quantities. A more detailed com­ parison between these models is described in the next sub-section. Here, we investigate three cases of this model by considering two different values of demand. Case 1. d1 þ d2 > Q;d1 < Q: In this case, the reserved quantity cannot meet the total demand and shortage will occur in the second period. We will have v1 ¼ d1 , v2 ¼ Q d1 and the exercise cost can be written as: v1 :e1 þ ðQ

Case 2. d1 þ d2 > Q;d1 > Q: In this case, the reserved quantity cannot meet the total demand and the shortage will occur in both periods. Because of the importance of the first period, the whole of the reserved quantity is exercised in this period. We will have v1 ¼ Q; v2 ¼ 0and the exercise cost can be written as: Q:e1 .

Case 1. d1 > q1 ; d2 < q2 ; d1 þ d2 � Q: In this case, the total reserved quantity is higher than the total demand and we will have: g1 ¼ q1 þ ð d1

(2)

q1 Þ; g2 ¼ d2

The exercise cost can also be written as: q1 :e1 þ ðd1

Case 3. d1 þ d2 � Q: In this case, the reserved quantity can meet the total demand and we will have v1 ¼ d1 , v2 ¼ d2 . The exercise cost can be written as: v1 :e1 þ v2 :e2 .

(3)

q1 Þe3 þ g2 :e2

As mentioned before, the HO pays higher exercise price e3 > e1 for purchasing extra items in the first period (more than the reserved quantity in this case, i.e.d1 q1 ).

3.1.3. Comparison of two option contract models In the second model, since the reserved capacity for each period is not separated and the HO reserves Q units with the option price o1 ðo1 � o2 Þ, the reservation cost is higher than that of the first model ðQ:o1 > q1 :o1 þ q2 :o2 Þ. In the first model, the HO pays the unit exercise price e3 ðe3 > e1 > e2 Þ for the order quantity more than q1 units in the first period, but in the second model, the HO can exercise all the Q units with exercise price e1 in the first period, which leads to lower exercise costs. In other words, the second model gives higher flexibility to the HO for order quantity after disaster strikes; since this model reserves all units in the faster delivery mode (delivery in the first period). For relief supplies that are needed only in the first period ðd2 ¼ 0;q2 ¼ 0;q1 ¼ QÞ, both models will have the same behavior. In this situation, the reser­ vation and exercise costs in both models will be equal to Q:o1 and g1 :e1 ¼ v1 :e1 , respectively. For clarifying the difference between the two proposed option con­ tract models further, a simple example is presented here. Assume the HO is willing to reserve Q ¼ 1000 ðq1 ¼ 400 and q2 ¼ 600Þ from a supplier with the contract parameters: o1 ¼ 50; o2 ¼ 30; e1 ¼ 100; e2 ¼ 80; e3 ¼ 120. Then, the reservation cost of the first and second models (Rc1 and Rc2 ) can be stated as:

Case 2. Q > d1 > q1 ; d2 < q2 ; d1 þ d2 > Q or Q > d1 > q1 ; d2 > q2 ; d1 þ d2 > Q: In this case, the total reserved quantity is lower than the total demand, and therefore the shortage is inevitable. Because of the higher importance of the first period’s demand satisfaction, the HO uses the reserved quantity of both periods for satisfying the first period’s demand for minimizing the shortage in this period. Hence, the shortage will occur only in the second period and we will have: g1 ¼ q1 þ ðd1

q1 Þ; g2 ¼ Q

g1 < d2

(4)

The exercise cost can also be written as: q1 :e1 þ ðd1

q1 Þe3 þ ðQ

g1 Þe2

(6)

d1 Þe2

(5)

Case 3. d1 < q1 ; d2 > q2 ; d1 þ d2 � Q: In this case, the total reserved quantity is higher than the total demand and the HO uses the unexer­ cised amount of the first-period reservation quantity in the second period to satisfy this period’s demand and prevent any shortage. In the first period, demand is lower than the reserved quantity and an unex­ ercised amount can be transported and used in the second period without extra price. So, we will have g1 ¼ d1 ; g2 ¼ d2 and the exercise cost is equal to: g1 :e1 þ g2 :e2 .

Rc1 ¼ q1 :o1 þ q2 :o2 ¼ 400*50 þ 600*30 ¼ 38000; Rc2 ¼ Q:o1 ¼ 1000*50 ¼ 50000; Furthermore, if d1 ¼ 700and d2 ¼ 300, the exercise cost (Ec) of these models are calculated as:

Case 4. d1 < q1 ; d2 < q2 : In this case, the reserved quantity can meet the total demand and we will have g1 ¼ d1 ; g2 ¼ d2 . The exercise cost can be written as: g1 :e1 þ g2 :e2 .

g1 ¼ q1 þ ðd1

3.1.2. The second option contract model The second model is characterized by four parameters ðo1 ; e1 ; e2 ; cap1 Þ. In the first stage, HO reserves Q units ðQ � cap1 Þ at option price o1 from each selected supplier. Therefore, the reservation cost in this model is Q:o1 . During the second stage, upon demand realization, the HO de­ termines the order quantity of the first period ðv1 Þ and the second period ðv2 Þ with exercise prices e1 and e2 , respectively. This reservation strategy gives the right (not the obligation) to order up to Q units in each period subject to the reservation quantity constraint v1 þ v2 � Q. Indeed, as needed, the HO can exercise all the reserved capacity in the first period without paying the higher exercise price. Therefore, for example, if

q1 Þ; g2 ¼ d2 ; Ec1 ¼ q1 :e1 þ ðd1 q1 Þe3 þ g2 :e2 ¼ 400 * 100 þ 300 * 120 þ 300 * 80 ¼ 100000;

v1 ¼ d1 ; v2 ¼ d2 ; Ec2 ¼ v1 :e1 þ v2 :e2 ¼ 700*100 þ 300*80 ¼ 94000 As mentioned before and shown in this simple example, the first model has a less reservation cost but a higher exercise cost than the second model. For the first model, when the uncertainty level of demand is low, lower exercise cost is expected due to the reduced need for additional purchasing with exercise price e3. Generally speaking, the performance of the first model is better for those relief items that the level of their demand uncertainty is low, while the second model works better under a high level of demand uncertainty. Fig. 2 shows the 5

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Fig. 2. Sequence of events in the proposed two-period option contract.

sequence of general events corresponding to the proposed two-period option contract models. Also, Fig. 3 depicts the detailed comparison of these models with their related parameters, decisions, and costs.

� Some of the contracted suppliers might be partially disrupted by the disaster. � Vulnerability of storage facilities is taken into account, and therefore some of the prepositioned inventories may be destroyed by the disaster. � Making a contract with each selected supplier incurs a fixed agree­ ment cost to the HO. � Due to incompleteness or unavailability of historical data for most of the input parameters, they are estimated subjectively based on the experts’ imprecise data reflecting their experiences/feelings. There­ fore, we quantify such imprecise/judgmental data by appropriate

(Please insert Figs. 2 and 3 around here) 3.2. Model formulation The following assumptions are made for the problem formulation:

Fig. 3. Comparison of two proposed option contract models. 6

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(continued ) Indices and sets: Importance weight of satisfying demand for the first period in the second objective function (0 � θ1 � 1) θ2 Importance weight of satisfying demand for the second period in the second objective function (0 � θ2 � 1)), θ1 þ θ2 ¼ 1 Unit transportation cost of item k from supplier i to the demand point under teciks scenario s First-stage decision variables: wik 1; if supplier i is selected by the HO for supplying relief item k, 0 otherwise 1; if the first option contract model is selected by the HO for supplying relief Rik 1 item k from supplier i, 0 otherwise 1; if the second option contract model is selected by the HO for supplying Rik 2 relief item k from supplier i, 0 otherwise Reserved quantity of relief item k from supplier i for delivering in the first qik 1 period (in the first option contract model) Reserved quantity of relief item k from supplier i for delivering in the second qik 2 period (in the first option contract model) Qik Reserved quantity of relief item k from supplier i (in the second option contract model) Ijk Prepositioned inventory level of item k in warehouse j Second-stage decision variables: ik giks Quantity of relief item k which can be exercised with ~eik 1 1 from supplier i (� q1 ) under scenario s in the first period (in the first option contract model)

~ ¼ ðal ; am ; au Þ. Fig. 4. Triangular fuzzy number a

possibility distributions in the form of triangular fuzzy numbers (TFNs). TFNs are the most common tool used in the literature and practical situations because of their intuitiveness, and the conceptual and computation simplicity [52]. Interested readers are referred to Ref. [53] to see more details of theoretical justifications of triangular ~ ¼ ðal ;am ; fuzzy numbers. Fig. 4 shows the triangular fuzzy number a au Þ, where al , am and au are the smallest possible value, the most possible value, and the largest possible value of this imprecise parameter, respectively. � At the beginning of the second stage (i.e. post-disaster), demand data is fully known with certainty. � Different disaster scenarios are defined based on the severity, loca­ tion and time of demand. (Please insert Fig. 4 around here)

xiks 1

giks 2 viks 1 viks 2 ziks 1

ziks 2

Following indices, parameters and variables are used to formulate the defined problem mathematically, where the possibilistic/imprecise parameters are associated with the tilde sign (e) and represented by triangular fuzzy numbers.

Quantity of relief item k which can be exercised with ~eik 2 from supplier i under scenario s in the second period (in the second option contract model) Exercise amount (order quantity) in the first period Exercise amount (order quantity) in the second period

i2I k2K

k2K j2J

"

X XX � iks ik e1 þ xiks giks ps eik3 þ 1 þ v1 ~ 1 :~ s2S

i2SU1 k2K

Available capacity of supplier i for supplying relief item k in the second period

~ks d 2 ps

Demand level for the relief item k in the second period under scenario s P Occurrence probability (i.e. likelihood) of scenario s ( ps ¼ 1)

Useable inventory ratio of warehouse j under scenario s

b~ u2s

Available budget at post-disaster under scenario s

~ oik 2

Unit option price of item k supplied by supplier i in the second period

~eik 2

Unit exercise price of item k supplied by supplier i in the second period

Available budget at pre-disaster

~ oik 1

Unit option price of item k supplied by supplier i in the first period

~eik 1

Unit exercise price of item k supplied by supplier i in the first period

~eik 3

Unit exercise prices of additional item k supplied by supplier i in the first period. This cost is related to purchasing those quantities more than reserved capacity in the first period

� ik e2 ziks 2 ~

P

ziks 1

þ

ziks 2

�

teciks (7)

maxz2 ¼

" P X θ1 k2K ps s2S

Unit volume of relief item k Volume capacity of warehouse j Unit procurement cost of item k for prepositioning in the first period Available (non-disrupted) fraction of capacity of supplier i under scenario s

b~ u1

P

i2SU1 [ SU2 k2K

s2S

~γjs

X

i2SU1 [ SU2 k2K

þ

Demand level for the relief item k in the first period under scenario s

Unit inventory holding cost of item k at warehouse j

X #

Available capacity of supplier i for supplying relief item k in the first period

Cik 2

~kj h uk capj pck ~ βis

Quantity of relief item k which can be exercised with ~eik 1 from supplier i under scenario s in the first period (in the second option contract model)

Clearly, to ensure the model is reasonable, we require that pck <

I Set of contracted suppliers, indexed by i J Set of warehouses, indexed by j S Set of disaster scenarios at post-disaster, indexed by s SUc Set of suppliers which provide the coverage level c to a disaster region WAc Set of warehouses which provide the coverage level c to a disaster region c Index of coverage levels, c ¼ {1,2,3} k Index of relief items K Set of relief items/services, indexed by k 2K Parameters: ~f Fixed agreement cost with supplier i for supplying relief item k ik

~ks d 1

Quantity of relief item k which can be exercised with ~eik 2 from supplier i under scenario s in the second period (in the first option contract model)

o~ik eik oik oik eik eik eik 1 þ~ 1; ~ 1 >~ 2; ~ 3 >~ 1 >~ 2 for all i and k. According to the above notations, the problem under consideration is formulated as a biobjective two-stage mixed possibilistic-stochastic programming model as follows: XX XX � ~ ~f ik :wik þ ~ hkj þ pck :Ijk þ minz1 ¼ oik1 :Qik þ ~ oik1 :qik1 þ ~ oik2 :qik2 þ

Indices and sets:

Cik 1

Quantity of relief item k which can be exercised with ~eik 3 from supplier i under scenario s in the first period (in the first option contract model)

þ

θ2

�P

P k2K

�P

iks i2SU1 z1

P

þ

~ks k2K d 1

iks i2SU1 [ SU2 z2

þ

P

γ js j2WA1 Ijk :~

P

�

γ js j2WA1 [ WA2 Ijk :~

�#

(8)

P

~ks k2K d2

Two objective functions are considered in the proposed model. The first objective function (Eq. (7)) contains both pre-disaster and postdisaster cost elements. The pre-disaster cost elements include the cost of making agreements with suppliers; capacity reservation costs for the first and second periods and the cost of prepositioned relief items including the procurement and inventory holding costs. Also, the postdisaster related costs include the exercise cost in the first and second periods. The second objective function (Eq. (8)) maximizes the total expected demand for the critical and non-critical periods covered by suppliers within the acceptable coverage levels:

θ1 (continued on next column)

Rik1 þ Rik2 ¼ wik 7

}i; k

(9)

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Constraint (9) ensures that only one type of option contract can be applied for each selected supplier. 8i 2 SU1 ; 8k; s

(10)

8i 2 SU1 [ SU2 ; 8k; s

(11)

iks iks iks giks 1 þ x1 þ v1 ¼ z1 iks iks giks 2 þ v2 ¼ z2

X wik � 1

Constraint (22) shows using the multiple sourcing policy for pro­ curing the relief supplies at pre-disaster.

Constraints (10) and (11) indicate the total exercise amount in the first and second periods, respectively. X Ijk :uk � capj 8j (12) k2K

Constraint (12) ensures that for each warehouse, the inventory level cannot exceed the respective capacity. Notably, the aim of this paper is designing an option contract model which is also integrated with sup­ plier selection and inventory pre-positioning decisions to have a comprehensive decision framework and to avoid trapping in suboptimality arising from separately treating these interrelated prob­ lems. However, we do not go through the details of the inventory prepositioning network and assume that the location, number, and the size of warehouses are known a priori. ik ~ giks 1 � q1 :βis

wik ; Rik1 ; Rik2 2 f0; 1g:8i; k

(23)

Ijk � 0:8j; k

(24)

qik1 ; qik2 ; Qik � 0 8i; k

(25)

iks iks iks iks giks 1 ; g2 ; x1 ; v1 ; v2 � 0 8i; k; s

(26)

Constraint (23) defines binary variables and constraints (24)–(26) are non-negativity constraints. 4. Solution approach The developed model is a bi-objective scenario-based mixed possibilistic-stochastic model, which includes both random disaster scenarios and possibilistic (imprecise) scenario-dependent and scenarioindependent data. As the proposed model has several complexities due to its several modeling features; therefore, it is necessary to propose a step-by-step solution procedure to solve it efficiently. The proposed solution procedure involves the following five steps:

(13)

8i 2 SU1 ; 8k; s

Constraint (13) represents that the quantity of relief item k which is

exercised with ~eik 1 from supplier i, should be less than the available reserved amount of that supplier for the first period. � ik ik ~ xiks giks 8i 2 SU1 ; 8k; s (14) 1 � q1 þ q2 βis 1

Step 1, Use the weighted augmented ε-constraint method to convert the original bi-objective model of the second stage into a singleobjective problem. Step 2, Apply the credibility-based fuzzy mathematical programming approach to defuzzify the first stage’s possibilistic model as well as the resulting single-objective possibilistic model of the second stage. Step 3, Convert the defuzzified two-stage stochastic programming model into its auxiliary crisp equivalent model. Step 4, Solve the equivalent single-objective model to find an effi­ cient solution of the original bi-objective problem using a simulated annealing (SA) algorithm. Step 5, Repeat Steps 1–4 with a new ε vector to obtain the most preferred efficient solution interactively with the decision maker.

Constraint (14) indicates the maximum quantity of relief item k which is exercised with ~eik 3 in the first period from supplier i. � � iks ik ik ~ iks 8i 2 SU1 [ SU2 ; 8k; s g2 � q1 þ q2 βis g1 þ xiks (15) 1 Constraint (15) indicates the maximum quantity of relief item k

which is exercised with ~eik 2 in the second period from supplier i. qik1 � Cik1 :Rik1

(16)

8i; k

Constraint (16) is related to the first option contract model and states that the reservation quantity of relief item k from supplier i in the first period does not exceed the available capacity of that supplier in the same period. qik2 � Cik2 :Rik1

8i; k

(17)

Qik � Cik1 :Rik2

8i; k

(18)

We now elaborate on the details of steps 1 to 4 through sub-sections 4.1 to 4.4, respectively. 4.1. Converting the bi-objective model

Constraints (17–18) are similar to the constraint (16) but for the second period’s reservation and the second option contract model, respectively. iks ~ viks 1 þ v2 � Qik :βis

The models of stages 1 and 2 can be stated as follow: The model of stage 1: XX XX � ~ ~f ik :wik þ ~ hkj þ pck :Ijk minz1 ¼ oik1 :Qik þ o~ik1 :qik1 þ ~ oik2 :qik2 þ

(19)

8i 2 SU1 [ SU2 ; 8k; s

Constraint (19) represents that the total exercise amount of relief item k from supplier i in the first and second periods should be less than the total available reserved quantity of that supplier. This constraint is related to the second contract model. XX ik XX o~1 :Qik þ ~ oik1 :qik1 þ ~ oik2 :qik2 þ pck :Ijk � b~u1 (20) i2I k2K

XX i2SU1 k2K

i2I k2K

X

X

� iks

z2 :~eik2 � b~ u2s

8s

k2K j2J

þ E½x; g; u; v; s� (27) s.t. (9), (12), (16)–(18), (20), (22)–(25)

k2K j2J

� iks ik e1 þ xiks giks eik3 þ 1 þ v1 ~ 1 :~

(22)

8k

i

where E½x; g; u; v; s� is the expected value of the second stage’s objective function with respect to the possible disaster scenarios. The model of stage 2: XX X X � � ik iks ik e1 þ xiks e2 obj1s ¼ giks ziks eik3 þ 1 þ v1 ~ 1 :~ 2 ~

(21)

i2SU1 [ SU2 k2K

Constraints (20) and (21) are related to budget constraints. Constraint (20) ensures that the total procurement costs do not exceed the available budget in the first stage (i.e. pre-disaster). Constraint (21) assures that the exercise costs do not exceed the available budget in the second stage (i.e. post-disaster).

i2SU1 k2K

X X þ i2SU1 [SU2 k2K

8

� iks e ziks 1 þ z2 tciks

i2SU1 [SU2 k2K

(28)

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M. Aghajani et al.

� �# P P P �P " P �P iks iks θ1 k2K γ js θ2 k2K γjs i2SU1 z1 þ j2WA1 Ijk :~ i2SU1 [SU2 z2 þ j2WA1 [WA2 Ijk :~ obj2s ¼ þ P ~ks P ~ks k2K d 1 k2K d 2

s.t.

rp ¼ f max p

(10), (11), (13)–(15), (19), (21), (26) Since the second stage is a bi-objective model, we apply an improved version of the ε-constraint method named the “weighted augmented ε-constraint method” [42] by which the bi-objective model of the second stage is converted to a single-objective counterpart. It is noteworthy that there are some studies in the relief logistics literature similar to our work, which formulate multi-objective mathematical models. The weighted sum [31–33] and ε-constraint [15,34–41] methods are the commonly used methods in these studies. However, the ε-constraint method has significant advantages over the weighted sum method. Unlike the weighted sum method, which can produce only extreme efficient solutions, the ε-constraint method has the capability of gener­ ating non-extreme efficient solutions as well. The weighted sum method requires the scaling of objective functions, but the ε-constraint does not need scaling. The ε-constraint method can control the number of generated efficient solutions by adjusting the epsilon vector’s elements by selecting specific grid points for each objective function [42,43]. Also, the ε-constraint method works well for non-convex multiple ob­ jectives problems and assures the efficiency of the achieved solutions [38]. Hence, by relying on the literature and the strengths of the ε -constraint method, it has been selected as a proper choice to solve the proposed bi-objective problem. In this method, the most important objective function acts as the objective function of the single-objective counterpart, while the other objective is added to the constraints in a constrained form [54]. The following formulation represents the general form of the weighted augmented ε-constraint method for a classic multi-objective problem (i. e., minimizing P objectives simultaneously subject to a feasible decision space X): � � �� s2 s3 sp min w1 f1 ðxÞ r1 � δ � w2 þ w3 þ … þ wp r2 r3 rp

r2 �l q

l ¼ 0; …; q

1; 8p 6¼ 1

(31)

(32)

s:t: obj2s ðxÞ

s2s ¼ ε2s

x 2 X; s2 2 Rþ The model (32) is solved for each value of ε2 , which is obtained from (31) to generate a distinct Pareto-optimal solution. 4.2. Applying the credibility-based fuzzy mathematical programming model As mentioned before, most of the parameters in our proposed twostage problem are tainted with a high degree of epistemic uncertainty (i.e. lack of knowledge about the precise values of imprecise parame­ ters), which is modeled by appropriate possibility distributions in the form of triangular fuzzy numbers. There are different methods in the literature to cope with epistemic uncertainty and transform a possibil­ istic programming model into an equivalent crisp model. Among them, credibility-based chance-constrained programming is one of the most adopted and effective methods [55]. Literature review shows that the three distinct credibility-based fuzzy mathematical programming models, including the expected value, the chance-constrained pro­ gramming and the dependent chance-constrained programming models are the most appropriate choices to convert a possibilistic model to an equivalent crisp one. However, each method has some weaknesses and strengths. The expected value model needs low computational complexity but without any control on the minimum confidence level of chance constraints’ satisfaction. On the contrary, the chance-constrained programming can control the minimum satisfaction level of chance constraints, but adding a constraint for each objective function leads to higher computational complexity. Also, for deter­ mining the right-hand side of the added constraints, additional infor­ mation about the ideal values of objective functions is needed in this model. Furthermore, the dependent-chance-constrained programming model has a similarity to the chance-constrained programming model, but as it gives more importance to confidence levels, it is more suitable for conservative decision-makers. Nevertheless, in order to form a credibility-based fuzzy mathematical programming framework for the proposed model, we use an efficient hybrid method based on a combi­ nation of the expected value and the chance-constrained programming models introduced by Pishvaee et al. [55], which benefits from the ad­ vantages of both methods. In this method, the expected value and the chance-constrained programming approaches are applied to model the objective functions and chance constraints, respectively. Consider ξ as a fuzzy variable with membership function μ(x) and r be a real number. Based on Liu and Liu [56], the credibility measure (Cr) as an average of the possibility (Pos) and necessity (Nec) measures is defined as follows:

8p ¼ 2; …; P (30)

x 2 X; sp 2 Rþ

max l f min p ; εp ¼ f p

where f min and f max denote the minimum and maximum (i.e. the ideal p p and nadir) values of objective p, respectively and l is the number of grid points. With applying the weighted augmented ε-constraint method, our problem can be formulated as: � � � � �� w2 s2s � min obj1s ðxÞ r1s � δ � w1 r2s

s.t. fp ðxÞ þ sp ¼ εp ;

(29)

where X is the feasible space of the main problem, x is the vector of decision variables and fp ðxÞðp ¼ 1; …; PÞ denotes the p-th objective function. Moreover, δ is a small number usually between 10 6 and 10 3, wp is the importance weight of p-th objective function determined by the P wp ¼ 1), rp is the range of decision-maker’s preferences (where p

objective function p obtained from the payoff table and sp is the slack variable of p-th constrained objective function. Also, different Paretooptimal solutions could be obtained by parametric variation of the right-hand-side (i.e. ε vector) of constrained objective functions. Different values for ε vector should be determined by calculating the ranges of p-1 constrained objective functions. The well-known approach is to obtain these ranges from the payoff table, which is constructed by solving p-1 single-objective problems for constrained objective functions individually. Then, by calculating the range of each constrained objec­ tive and dividing it to q equal intervals, different values for εp can be determined as follows:

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1 Crfξ � rg ¼ ðPosfξ � rg þ Nec fξ � rgÞ 2

(33)

Crfξ � rg � α ⇔ r � ð2α

Using Eqs. (39) and (40), the fuzzy chance constraints can be con­ verted into their equivalent crisp ones. According to the above-mentioned descriptions and justifications, the first stage’s possibilistic model, as well as the resulting singleobjective possibilistic model of the second stage, can be defuzzified by considering the expected value of triangular fuzzy numbers and using Eqs. (39) and (40). The defuzzified models of the first and second stages are as follows: � � XX� � ik � � o1ð1Þ þ 2oik1ð2Þ þ oik1ð3Þ fikð1Þ þ 2fikð2Þ þ fikð3Þ minE z1 ¼ wik þ Qik 4 4 i2I k2K � ik � � � o1ð1Þ þ 2oik1ð2Þ þ oik1ð3Þ ik oik2ð1Þ þ 2oik2ð2Þ þ oik2ð3Þ ik þ q1 þ q2 þ 4 4 � � � � XX hjkð1Þ þ 2hjkð2Þ þ hjkð3Þ þ þ pck Ijk þ E x; g; u; v; s 4 k2K j2J

Notably, the expected value of ξ based on the credibility measure can be determined as follows [56]: Z ∞ Z 0 E½ξ� ¼ Cr fξ � rgdr Crfξ � rgdr: (34) 0

∞

Thus, according to Eq. (34), the expected value of a triangular fuzzy number ξ ¼ ðξ1 ; ξ2 ; ξ3 Þ is ðξ1 þ2ξ2 þξ3 Þ=4 and the corresponding possi­ bility and necessity measures are as follows: 8 9 1 if r � ξ2 > > > > > > > ξ ξ 2 3 > > > > : ; 0 if r � ξ3 8 1 > > > > ξ1 > : 0

if r � ξ1 ξ2 ξ2

if ξ1 � r � ξ2 if r � ξ2

9 > > > = > > > ;

(40)

2αÞξ2 :

1Þξ1 þ ð2

(41)

(36) s.t. (9), (12), (16)–(18), (22)–(25) XX� � � ð2 2αÞoik1ð2Þ þ ð2α 1Þoik1ð3Þ Qik þ ð2

Accordingly, credibility measures of ξ � r and ξ � r can be written as follows:

i2I k2K

� þ ð2

2αÞoik2ð2Þ þ ð2α

2αÞoik1ð2Þ þ ð2α

XX � 1Þoik2ð3Þ qik2 þ pck :Ijk � ð2α

� 1Þoik1ð3Þ qik1 1Þbu1ð1Þ

k2K j2J

2αÞbu1ð2Þ

þ ð2

(42)

minE½z2� ¼

� � ik � � X X�eik2ð1Þ þ 2eik2ð2Þ þ eik2ð3Þ � � X X��eik1ð1Þ þ 2eik1ð2Þ þ eik1ð3Þ �� e3ð1Þ þ 2eik3ð2Þ þ eik3ð3Þ iks iks þ giks x1 þ ziks 1 þ v1 2 4 4 4 i2SU1 k2K i2SU1 [SU2 k2K � �� � � � � � �� X X tciksð1Þ þ 2tciksð2Þ þ tciksð3Þ w2 s2s iks þ ziks r1s � δ � � 1 þ z2 4 w r2s 1 i2SU1 [SU2 k2K

8 1 > > > > > 2ξ ξ1 r > 2 > > < 2ðξ ξ1 Þ 2 Cr fξ � rg ¼ > r ξ3 > > > > > > 2ðξ2 ξ3 Þ > : 0

9 > > > > > > if ξ1 � r � ξ2 > > =

8 0 > > > > > r ξ1 > > > < 2ðξ ξ1 Þ 2 Cr fξ � rg ¼ > 2ξ ξ r > 2 3 > > > > 2ðξ2 ξ3 Þ > > : 1

9 > > > > > > if ξ1 � r � ξ2 > > =

s.t.

if r � ξ1

> > > if ξ2 � r � ξ3 > > > > > ; if r � ξ3

(10), (11), (26)

(37)

ik giks 1 � q1 ½ð2α

2αÞξ2 þ ð2α

1Þξ3 :

(44)

2αÞβisð2Þ �; 8i 2 SU1 ; 8k; s

ik xiks 1 � q1 þ q2 ½ð2α

1Þβisð1Þ þ ð2

2αÞβisð2Þ �

� ik ik giks 2 � q1 þ q2 ½ð2α

1Þβisð1Þ þ ð2

2αÞβisð2Þ �

giks 1

8i 2 SU1 ; 8k; s

iks giks 1 þ x1

�

(38)

iks viks 1 þ v2 � Qik ½ð2α

XX

(46) 1Þβisð1Þ þ ð2

�� iks giks ð2 1 þ v1

i2SU1 k2K

(45)

8i

2 SU1 [ SU2 ; 8k; s

According to (37) and (38), it can be shown [57] that if ξ is a triangular fuzzy number, then we would have the following equiva­ lencies for the minimum confidence level α 2 ½0:5; 1�: Crfξ � rg � α ⇔ r � ð2

1Þβisð1Þ þ ð2 � ik

if r � ξ1

> > > if ξ2 � r � ξ3 > > > > > ; if r � ξ3

(43)

2αÞβisð2Þ � 8i 2 SU1 [ SU2 ; 8k; s � � 1Þeik1ð3Þ þ xiks 1 ð2

2αÞeik1ð2Þ þ ð2α X

þ ð2α

� 1Þeik3ð3Þ þ

� ð2α

1Þbu2sð1Þ þ ð2

X

�� ziks ð2 2

(47)

2αÞeik3ð2Þ

2αÞeik2ð2Þ þ ð2α

1Þeik2ð3Þ

�

i2SU1 [ SU2 k2K

2αÞbu2sð2Þ

8s (48)

(39)

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h ks θ1 ð2α 1Þdks 2ð1Þ þ ð2 2αÞd 2ð2Þ

�X� X k2K

X

ziks 1 þ

i2SU1

X

Ijk ð2α 1Þγjsð1Þ þð2 2αÞγjsð2Þ

j2WA1

Ijk ð2α 1Þγjsð1Þ þð2 2αÞγ jsð2Þ

þ

���i

h�X �

j2WA1 [ WA2

h ks θ1 ð2 2αÞdks 2ð2Þ þð2α 1Þd 2ð3Þ

X þ

��

ks þθ2 ð2α 1Þdks 1ð1Þ þð2 2αÞd 1ð2Þ

ks ð2 2αÞdks 1ð2Þ þð2α 1Þd 1ð3Þ

k2K

�X� X k2K

i2SU1

ziks 1 þ

X

Ijk ð2 2αÞγjsð2Þ þð2α 1Þγjsð3Þ

���i

j2WA1 [ WA2

h�X �

X

ziks 2

i2SU1 [ SU2

k2K

ks ð2 2αÞdks 2ð2Þ þð2α 1Þd 2ð3Þ

i �� ðε2s þs2s Þ 8s

49

k2K

Ijk ð2 2αÞγjsð2Þ þð2α 1Þγjsð3Þ

j2WA1

���X

��X�

��

ks þθ2 ð2 2αÞdks 1ð2Þ þð2α 1Þd 1ð3Þ

ks ð2α 1Þdks 1ð1Þ þð2 2αÞd 1ð2Þ

k2K

���X

��X� k2K

X

ziks 2

i2SU1 [ SU2

ks ð2α 1Þdks 2ð1Þ þð2 2αÞd 2ð2Þ

i �� ðε2s þs2s Þ 8s

(50)

k2K

s.t. (9)–(12), (16)–(18), (22)–(26), (42), (44)–(50) 4.3. Treatment of randomness It is worth noting that there are several methods to cope with random data in the two-stage stochastic programming models, such as the Lshaped method [58], Benders decomposition [59] and scenario decomposition [60]. For more details, interested readers can refer to Birge [61]. Since we are dealing with a finite number of scenarios in the concerned problem, the deterministic single-stage method is more popular [15]. In this regard, the equivalent auxiliary crisp model can be written as follows:

4.4. Simulated annealing algorithm The proposed possibilistic scenario-based model is so complicated and could be very large especially for real cases. As a result, using com­ mercial optimization solvers such as GAMS for applying exact optimi­ zation methods would be ineffective because of high computation time. Therefore, due to the high complexity of the proposed model, we use a

� ik � ik � ik � � � � � XX�fikð1Þ þ2fikð2Þ þfikð3Þ � o1ð1Þ þ2oik1ð2Þ þoik1ð3Þ o1ð1Þ þ2oik1ð2Þ þoik1ð3Þ ik o2ð1Þ þ2oik2ð2Þ þoik2ð3Þ ik XX hjkð1Þ þ2hjkð2Þ þhjkð3Þ wik þ Qik þ q1 þ q2 þ þpck Ijk 4 4 4 4 4 i2I k2K k2K j2J � �� � � � � � �� � �� ik ik ik ik ik ik ik ik ik X X e1ð1Þ þ2e1ð2Þ þe1ð3Þ X X e2ð1Þ þ2e2ð2Þ þe2ð3Þ X e3ð1Þ þ2e3ð2Þ þe3ð3Þ iks iks þ ps þ giks x1 þ ziks 1 þv1 2 4 4 4 s2S i2SU1 k2K i2SU1 [ SU2 k2K � � � � � � ��� �� X X tciksð1Þ þ2tciksð2Þ þtciksð3Þ w2 s2s iks þ r1s �δ� � (51) ziks 1 þz2 4 w1 r2s i2SU1 [ SU2 k2K

minZ¼

simulated annealing (SA) algorithm to find near-optimal solutions within a reasonable computational time. SA is a stochastic optimization algo­ rithm which was proposed by Kirkpatrick et al. [62]. SA has been widely used to solve many combinatorial problems due to the ease of use and the ability to find good solutions [63]. SA starts with a set of initial feasible solutions to generate neighboring solutions randomly. It randomizes the local search procedure and to reduce the probability of getting trapped in a local optimum, in addition to accepting better solutions, accepting a worse solution (which is worse than the current solution) with a certain probability in each temperature will be allowed. There will be more probability of accepting worse solutions at higher temperatures while this probability is decreased during the algorithm’s evolution. Initial temperature (T0), cooling rate (α), and the number of iterations before a temperature change (N) are the three main parameters of this algorithm. T0 is a controlled parameter by which the acceptance probability of worse solutions is determined, and it affects the global search performance of the SA algorithm. The higher value for T0 might result in better perfor­ mance of the global search, but it will increase the computation time and vice versa. During the algorithm’s evolution, the temperature decreases by Tkþ1 ¼ αTk . This process is the cooling strategy, which enforces the convergence of the search and affects the overall performance of the al­ gorithm. The number of new neighbor solutions produced in each tem­ perature is determined by N and this procedure is repeated for a certain number of iterations or it is stopped if the algorithm reaches the final temperature. Fig. 5 shows the pseudo-code of the applied SA algorithm considering a minimization problem.

Fig. 5. The pseudo-code of SA algorithm. 11

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Socio-Economic Planning Sciences xxx (xxxx) xxx

Fig. 6. Suppliers locations and seismic map of Iran from 1990 to 2017.

5. Computational experiments

5.1. The case description

In this section, we provide a case study inspired by the real practice of the supply department at the Iranian Red Crescent Society (IRCS) to demonstrate the applicability and validity of the proposed model. In addition, several sensitivity analyses are performed on the main pa­ rameters of the proposed model to provide practical insights that can help the decision-makers of humanitarian logistics.

Iran is an earthquake-prone country, which has experienced a large number of devastating earthquakes over the past decades, causing heavy causalities and widespread economic loses [11]. Therefore, designing and implementing reliable relief supply plans to enhance the perfor­ mance of humanitarian operations is essential for the country. Our purpose is to help the decision-makers in IRCS by providing useful in­ sights and recommendations to enable them to make optimized de­ cisions regarding several aforementioned strategic and operational decisions. Most of the data have been gathered from the Institute of Geophysics, University of Tehran (IGUT) website (http://www.geophys

Table 1 Warehouse capacities and coverage level. Warehouse

Location

Capacity (103 m3)

Regions within the coverage level 1

Regions within the coverage level 2

1 2 3

MZ GL GZ

11.2 8.32 6

(EA,HM) (EA,HM) (NK)

4 5 6 7 8 9 10

HM EA NK SK FA BU HO

5.4 80 12.28 11.48 32.4 3.88 6.6

11

KR

23

12

SB

54.14

(MZ,NK,GL,GZ) (MZ,GL,GZ) (GZ,MZ,HM,GL, EA) (GL, HM,GZ) (EA) (MZ,NK) (SK,KR,SB) (FA,BU,HO) (BU,FA,HO) (HO,FA,BU,KR, SB) (HO,FA,SK,KR, SB) (SB,KR,SK,HO)

(EA,MZ) (GL,MZ,HM,GZ) (GL,SK,GZ) (HO,FA,NK) (SK, SB, KR) (KR, SB) (NK,SK) (BU) (FA,BU)

Fig. 7. Conditional probability of a 6-magnitude and above earthquake. 12

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Table 2 Demand and probability data of earthquake scenarios. Scenario

Disaster region, impact and time

Probability

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

MZ, L, C MZ, L, W MZ, H, C MZ, H, W GL, L, C GL, L, W GL, H, C GL, H, W EA, L, C EA, L, W EA, H, C EA, H, W GZ, L, C GZ, L, W GZ, H, C GZ, H, W HM, L, C HM, L, W HM, H, C HM, L, w BU, L, C BU, L, W BU, H, C BU, H, W FA, L, C FA, L, W FA, H, C FA, H, W HO, L, C HO, L, W HO, H, C HO, H, W KR, L, C KR, L, W KR, H, C KR, H, W SB, L, C SB, L, W SB, H, C SB, H, W SK, L, C SK, L, W SK, H, C SK, H, W NK, L, C NK, L, W NK, H, C NK, H, W

0.017 0.017 0.0035 0.0035 0.0175 0.0175 0.007 0.007 0.0355 0.0355 0.0035 0.0035 0.0175 0.0175 0.0035 0.0035 0.0175 0.0175 0.0035 0.0035 0.0175 0.0175 0.0035 0.0035 0.05 0.05 0.0035 0.0035 0.05 0.05 0.0035 0.0035 0.085 0.085 0.0035 0.0035 0.105 0.105 0.007 0.007 0.0175 0.0175 0.007 0.007 0.0175 0.0175 0.0035 0.0035

Demand in the first period (103 units)

Demand in the second period (103 units)

Water

Food

kit

Shelter

Blanket

Water

Food

kit

Shelter

Blanket

640 640 1280 1280 500 500 1000 1000 720 720 1440 1440 240 240 480 480 360 360 720 720 240 240 480 480 960 960 1920 1920 360 360 720 720 640 640 1280 1280 540 540 1080 1080 152 152 304 304 172 172 344 344

640 640 1280 1280 500 500 1000 1000 720 720 1440 1440 240 240 480 480 360 360 720 720 240 240 480 480 960 960 1920 1920 360 360 720 720 640 640 1280 1280 540 540 1080 1080 152 152 304 304 172 172 344 344

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

40 40 80 80 32 32 64 64 45 45 90 90 15 15 30 30 22 22 44 44 15 15 30 30 60 60 120 120 22 22 44 44 40 40 80 80 34 34 68 68 10 10 20 20 11 11 22 22

320 0 640 0 248 0 512 0 360 0 760 0 120 0 240 0 196 0 392 0 120 0 240 0 480 0 960 0 196 0 392 0 320 0 640 0 272 0 524 0 80 0 160 0 88 0 196 0

960 960 1920 1920 750 750 1500 1500 1080 1080 2160 2160 300 300 600 600 425 425 850 850 300 300 600 600 1200 1200 2400 2400 425 425 850 850 960 960 1920 1920 675 675 1350 1350 190 190 380 380 215 215 430 430

960 960 1920 1920 750 750 1500 1500 1080 1080 2160 2160 300 300 600 600 425 425 850 850 300 300 600 600 1200 1200 2400 2400 425 425 850 850 960 960 1920 1920 675 675 1350 1350 190 190 380 380 215 215 430 430

160 160 320 320 124 124 256 256 180 180 380 380 60 60 120 120 88 88 196 196 60 60 120 120 240 240 480 480 88 88 196 196 160 160 320 320 136 136 262 262 40 40 80 80 44 44 88 88

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 320 0 640 0 248 0 512 0 360 0 760 0 120 0 240 0 196 0 392 0 120 0 240 0 480 0 960 0 196 0 392 0 320 0 640 0 272 0 524 0 80 0 160 0 88 0 196

L: Low impact, H: High impact, C: Cold season, W: Warm season.

ics.ut.ac.ir) and through interviews with experts of IRCS. Notably, some parameters have been generated randomly while considering their practical situations whenever adequate information is lacking. Iran consists of 31 states that most of which are earthquake-prone. The seismic map in Fig. 6 shows the considered geographic regions. We used available data on the earthquakes greater than 6.0 magnitudes that had taken place between 1990 and 2017 to define the disaster scenarios. According to Fig. 6, twelve states experiencing such a situa­ tion are considered as the demand points for procurement planning. These states (i.e. potential demand nodes) include: Mazandaran (MZ), Gilan (GL), East Azarbaijan (EA), Gazvin (GZ), Hamedan (HM), North Khorasan (NK), South Khorasan (SK), Bushehr (BU), Fars (FA), Kerman (KR), Hormozgan (HO) and Sistan va Baluchestan (SB). Each state has a central warehouse for relief prepositioning purposes which is managed by the IRCS. Available capacity and coverage levels of each warehouse are shown in Table 1. Fig. 7 represents the conditional probabilities of earthquakes greater than 6 magnitudes. To simplify the model descrip­ tion, we used four potential earthquake scenarios for each region, con­ cerning the magnitude (low and high) and time of the earthquake (cold

and warm seasons), which leads to 48 disaster scenarios. Low and high impact disasters are used to represent the 6–7 and 7–8 magnitude earthquakes, respectively. Hence, using the data from Fig. 7, we assume that the probability of occurrence of low- and high-impact disasters is 0.89 and 0.11, respectively. Also, we consider that earthquakes can occur in a cold (winter and fall) or warm season (spring and summer), which leads to the different priorities of some relief items (i.e. blanket in our case study). As mentioned before, the blanket is needed in the first and second periods during the cold and warm seasons, respectively. The resulting demands of each scenario, as well as their probabilities, are shown in Table 2. The demand for each region is estimated based on the population size and the predicted damage in each disaster scenario. The probabilities of scenarios are based on historical data (i.e. those earth­ quakes experienced in the past). Five different types of relief items that are commonly distributed among disaster victims (i.e. water, food, shelter, blanket and hygiene kit) are considered in this case study. Water and food are consumable items that their demands occur periodically over the planning horizon (in both periods of the post-disaster phase). Similar to Bozorgi-Amiri and Khorsi [64], we assume that each person 13

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Socio-Economic Planning Sciences xxx (xxxx) xxx

shelter is considered for a family of four people over the planning ho­ rizon. Two blankets (as a non-consumable item) are considered for each person. Also, for a hygiene kit that is needed in the non-critical period, the distribution quantity is one kit per person. Other features of the relief items are summarized in Table 3. The planning horizon includes 5 days (2 and 3 days for the first and second post-disaster periods, respectively) after the disaster. We consider 9, 4, 2 and 2 suppliers for water and food, hygiene kit, blanket and shelter, respectively. As Fig. 6 shows, they are geographically dispersed over the entire geographic regions. The ca­ pacities of available suppliers and their coverage levels for each region are shown in Table 4. For the scenario definition, based on consultation with experts of IRCS, we assume capacities of suppliers and warehouses located in the affected region for low and high impact disasters damaged by 20% and 50%, respectively. For example, in the first and third sce­ narios, in which MZ is the affected region, the third supplier and the first warehouse which have been located in this region, are damaged by 20% and 50% for the low and high impact disasters, respectively (i.e. β~31 ¼ 80%; ~γ 11 ¼ 80%; β~33 ¼ 50%; ~γ13 ¼ 50%). It is noteworthy that sym­ metric TFNs are used to handle all the imprecise parameters in the case ~ ¼ ðal ; am ; au Þ, if the left and right spreads are the study. For the TFN a same (am al ¼ au am ), then the triangular fuzzy number is called a symmetric TFN. For the fuzzy parameters, the provided data indicates the central value of the respective possibility distribution (i.e. the most possible value). Furthermore, for the construction of the related fuzzy numbers, 10% spread on both sides of the central values are considered. According to the experts’ opinions, in our case study, the importance weights of satisfying demand in the first and second periods (i:e: θ1 and θ2 ) in the second objective function are set to (0.65, 0.35). The numerical experiment is done at the confidence level α ¼ 0.8.

Table 3 Characteristics of relief items. Parameter

Relief items Water

Food

Shelter

Hygiene kit

Blanket

pck (103$/ unit)

1.5

4

200

8

40

~kj (103$/ h unit)

3 ~eik 1 (10 $/ unit)

pck�U [0.18, 0.22] pck�U [0.09, 0.11] pck�U [0.06, 0.08] pck�U [1.3, 1.5]

pck�U [0.20, 0.25] pck�U [0.09, 0.11] pck�U [0.06, 0.08] pck�U [1.3, 1.5]

pck�U [0.22, 0.25] pck�U [0.09, 0.11] pck�U [0.06, 0.08] pck�U [1.3, 1.5]

pck�U [0.20, 0.22] pck�U [0.09, 0.11] pck�U [0.06, 0.08] pck�U [1.3, 1.5]

pck�U [0.20, 0.25] pck�U [0.09, 0.11] pck�U [0.06, 0.08] pck�U [1.3, 1.5]

3 ~eik 2 (10 $/ unit)

pck�U [1, 1.2]

pck�U [1, 1.2]

pck�U [1, 1.2]

pck�U [1, 1.2]

pck�U [1, 1.2]

3 ~eik 3 (10 $/ unit)

pck�U [1.6, 1.9]

pck�U [1.6, 1.9]

pck�U [1.6, 1.9]

pck�U [1.6, 1.9]

pck�U [1.6, 1.9]

0.004

0.002

1

0.018

0.05

3 ~ oik 1 (10 $/ unit) 3 ~ oik 2 (10 $/ unit)

uk (m3/ unit)

Where U[a, b] is a random number within the range [a, b] to generate the center of symmetric fuzzy parameters.

needs 3L of drinking water per day. Food is in the form of meals-ready-to-eat (MREs) and its distribution quantity is considered 2 MREs per person per day. The shelter is a non-consumable item whose demand occurs once at the beginning of the planning horizon and one Table 4 Available capacity (103 units) and coverage level of candidate suppliers. Supplier

capacity

Kit (cap2)

Water (cap1,cap2)

Food (cap1,cap2)

Shelter (cap1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

(0,0) (0,0) (2000,2300) (2000,2300) (1100,1300) (2200,2200) (1300,2000) (1000,1500) (1100,1700) (1400,1800) (1100,1700) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0)

(0,0) (0,0) (1000,1200) (1700, 1900) (1500,2000) (2300,2500) (2000,2500) (2200,2300) (2100,2500) (1900,2500) (2100,2500) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0)

(220) (146) (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) (0)

(0) (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) (300) (400) (500) (450) (0) (0)

Blanket (cap1,cap2)

Region in coverage level 1

(250, 300) (180, 220)

(MZ,GL,GZ,HM) (GZ,HM,FA,BU,KR) (MZ,NK,GL,GZ) (MZ,GL,GZ,HM) (GZ,HM,FA,BU,KR) (FA,BU,HO) (HO,FA,BU,KR) (GZ,MZ,HM,GL,EA) (HO,FA,SK,KR,SB) (EA) (GL, HM,GZ) (MZ,GL,GZ,HM) (GZ,HM,FA,BU,KR) (HO,FA,SK,KR,SB) (GZ,HM,FA,BU,KR) (MZ,GL,GZ,HM) (GZ,HM,FA,BU,KR)

Region in coverage level 2

(EA,NK,SK) (GL,MZ,EA,SB,HO) (EA,HM) (EA,NK,SK) (GL,MZ,EA,SB,HO) (SK, SB, KR) (NK, SB,SK) (NK) (BU) (GL,MZ,HM,GZ) (EA,MZ) (EA,SK,NK) (GL,MZ,EA,SB,HO) (BU) (GL,MZ,EA,SB,HO) (EA,NK,SK) (GL,MZ,EA,SB,HO)

Table 5 Effectiveness of SA in small- and medium-sized problems. Size jIj � jKj � jSj

GAMS

SA

Gap1

Gap2

101

1.25

0

157

0.5

0

84%

239

2.67

2.3

20.65

85%

226

1.98

3.5

5040

26.41

81%

271

3.42

4.9

82%

11,970

47.75

80%

365

4.63

53.12

82%

14,580

54.38

82%

393

2.38

68.19

83%

17,930

70.74

80%

345

3.75

3.7

2.57

2.11

Z1

Z2

CPU time (s)

Z1

Z2

CPU time (s)

j2j � j2j � j2j

5.29

90%

193

5.35

90%

j4j � j2j � j3j

8.67

92%

372

8.71

92%

j4j � j3j � j4j

14.86

86%

648

15.25

j6j � j3j � j5j

20.25

88%

2646

j6j � j4j � j6j

25.54

85%

j8j � j4j � j7j

32.64

j10j � j5j � j8j j12j � j5j � j10j Average gap (percent)

14

2.5 0

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Table 6 Effectiveness of SA in large-sized problems. Size jIj � jKj � jSj

GAMS

SA

Gap2

Z1

Z2

CPU time (s)

Z1

Z2

CPU time (s)

j12j � j5j � j12j

74.26

87%

>18,000

73.20

85%

394

1.43

j13j � j5j � j14j

75.76

82%

>18,000

74.04

86%

447

2.27

4.88

j13j � j5j � j16j

71.48

81%

>18,000

71.36

85%

488

1.70

4.94

j14j � j5j � j18j

83.72

87%

>18,000

84.36

86%

616

j14j � j5j � j20j

88.65

84%

>18,000

85.38

88%

666

3.69

j15j � j5j � j22j

84.20

80%

>18,000

80.89

79%

621

3.93

j15j � j5j � j24j

88.36

85%

>18,000

82.84

89%

684

6.25

4.71

j16j � j5j � j26j

86.62

83%

>18,000

84.13

87%

788

2.88

4.82

j16j � j6j � j28j

92.71

89%

>18,000

89.48

86%

1051

3.48

j17j � j6j � j30j

92.53

83%

>18,000

89.77

87%

1102

2.98

4.82

j17j � j6j � j32j

100.76

81%

>18,000

99.83

86%

1601

0.92

6.17

j18j � j6j � j34j

102.23

88%

>18,000

102.58

86%

1193

3.40

j18j � j6j � j36j

105.41

81%

>18,000

106.81

84%

1732

1.33

j19j � j6j � j38j

104.27

80%

>18,000

104.20

81%

1228

0.70

j19j � j6j � j40j

106.85

86%

>18,000

105.36

89%

1439

1.39

3.49

j20j � j6j � j42j

104.49

84%

>18,000

101.21

88%

1855

3.14

4.76

994

1.82

2.14

Average

0.76

Gap2 2.29

1.15 4.76 1.25

3.37

2.27 3.70 2.41

5.2. Implementation

Fig. 8. The Pareto ε-constraint method.

front

found

by

the

weighted

In order to validate the solution procedure, eight test problems are considered in different sizes and solved using GAMS software v.24.1 with CPLEX solver and the proposed SA algorithm (which is coded in the Matlab software version R2011a) for obtaining optimal and nearoptimal solutions, respectively. All the experiments are run on a com­ puter with Intel i7 2.60 GHz CPU and 8 GB RAM. In addition, some parameters of the SA algorithm such as T0, α and N are set as 100, 0.95 and 50, respectively by a trial and error process. A comparative study for the performance of GAMS and SA in terms of their solution quality and computing time is shown in Table 5. It is clear that for the small- and medium-sized instances (those problem instances with a run time less than 18,000 s), the SA algorithm can obtain good results in a reasonable time. Therefore, it can be concluded that the obtained solutions by the SA are reliable and this algorithm can be used for large-sized problems. Although GAMS can reach the optimal solution for the small- and medium-sized problem within a predetermined (18,000 s) run time, the computational time increases when dealing with large-sized problem

augmented

Fig. 9. The pre-positioning quantity of each relief item in the local warehouses. 15

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instances, (i.e. those with a large number of candidate suppliers, relief items and scenarios, which could not be solved within 18,000 s). To evaluate the performance of the proposed SA algorithm in largesized instances, 16 random instances are generated and solved by both SA and GAMS. However, due to computational complexity of these large-sized instances, they cannot be optimally solved by GAMS within a reasonable amount of computational time (5 h in our experiments). Therefore, to save additional computation time, we interrupt the solver within 18,000 s and report the best solution found so far as a basis for comparison with the SA algorithm. Table 6 indicates the comparison between the results obtained from GAMS and the proposed SA algorithm in large-sized problems. As Table 6 shows, the average gap between the best result found by GAMS and SA for the first (minimizes total cost) and second (maximizes covered demand) objective functions are 1.82% and 2.14%, respectively. It is noted that for minimizing (maximizing) objective function, the negative (positive) value indicates the better performance of SA. Thus, the average improvements of SA compared to GAMS is 1.82% and 2.14% for the first and second objective functions, respectively. In other words, these results indicate the better perfor­ mance of SA compared to the best solution found by GAMS within 18,000 s. Also, the average CPU time is 994 s for SA that is a promising result in comparison to the CPU time reported for GAMS. In total, the obtained results about the gaps and CPU times imply that using SA method to solve large-sized problems is preferred to GAMS.

shelter, blanket and hygiene kit suppliers are not significantly changed. This is because of two facts: the limited number of suppliers of shelter and blanket and high (low) importance of quick response of critical (non-critical) supplies. As Fig. 8 demonstrates, the Pareto front is divided into three parts. The first part corresponds to the solutions where increasing the covered demand is not very costly. In this part, when the total cost increases from 1.386Eþ8 to 1.48Eþ8, the percentage of demand coverage drastically changes from 20% to 55%. In contrast, in the second part, increasing the demand coverage is more costly so that for increasing the 20% of de­ mand coverage, the value of the first objective function changes from 1.48Eþ8 to 1.61Eþ8. Also, the last part shows that even small changes in the value of the second objective function incur extra costs to the model. According to Fig. 8, although Pareto solutions between the point numbers 1 and 8 (section 1) have a low cost, they cannot satisfy the coverage requirement of relief operations. This is because, in this sec­ tion, the number of selected suppliers and the amount of inventory prepositioning are low, so that this strategy is not able to provide acceptable demand coverage. Also, in Pareto solutions between the point numbers 13 and 16 (section 3), high demand coverage is met while the relative total cost is high. So, the HO may not afford such an expensive approach due to the limited available budget. Although the decision maker can select the most preferred Pareto solution based on his/her preferences, these analyses show that a Pareto solution between the point numbers 9 and 12 (section 2) could be a good candidate with reasonable satisfac­ tion levels for both objective functions. In the following, we present the details of a Pareto solution in section 2 to provide some managerial insights. Fig. 9 indicates the amounts of inventory pre-positioning of relief items in the local warehouse of each region. As this Figure shows, the pre-positioning levels of relief items that are needed in the first period (i.e. shelter, food and water) are higher than those of relief items that are needed in the second period (i. e. hygiene kit), because they can be supplied from distant suppliers or warehouses. Therefore, in contrast to the results of Torabi et al. [11], a few amounts of non-critical supplies would be prepositioned at pre-disaster. It is because of that HO can use more suppliers to satisfy the demand of non-critical items that leads to a low level of inventory pre-positioning. For example, the HO can supply non-critical items from those suppliers with the coverage levels 1 and 2, while supplying critical supplies from those suppliers with the coverage level 1. Although pro­ curement quantities of critical items at pre-disaster for prepositioning purposes are higher than those of non-critical items, the average in­ ventory level of these items is low in overall. As Fig. 9 shows, the average inventory level of shelter and blanket is higher than those of food and water since the number of shelter suppliers is limited compared to those of food and water. Also, the demand of food and water as consumable items are recurring over the planning horizon so that for the second period’s requirement of these items, the capacity of more suppliers even those with coverage level 2 can be reserved. This issue enables the HO to decrease inventory levels. Furthermore, when there are enough sup­ pliers around a region, the level of inventory prepositioning decreases and vice versa. For example, there are a limited number of shelter and blanket suppliers around states SB and EA while state NK has a limited number of food and water suppliers around. Therefore, by applying an option contract, the HO can decrease the level of prepositioning, espe­ cially in those areas where there are enough suppliers, which leads to

5.3. General results The relationship between the two objective functions is shown in Fig. 8, where the first one aims to minimize the total cost including the pre- and post-disaster costs while the latter seeks to maximize the ex­ pected covered demand. The Pareto-optimal solutions are generated using the weighted augmented ε-constraint method. Fig. 8 indicates that there is an obvious conflict between these objective functions. It is worth noting that the first objective function has a tendency towards central­ ized supplier management and contract handling via selecting the lesser number of suppliers and a low amount of reservation capacities to minimize the total costs (i.e., achieving the cost efficiency). On the other hand, the second objective function has a tendency towards more decentralized supplier management and contract handling via selecting a higher number of suppliers to maximize the total covered demand (i.e., achieving the responsiveness). For example, in the first Pareto-optimal solution (W1 ¼ 1.386Eþ8 and W2 ¼ 20), the minimum demand satis­ faction is achieved by selecting 2, 1, 1 and 1 suppliers for food and water, shelter, blanket and kit, respectively, whereas in the last Paretooptimal solution (W1 ¼ 2.106Eþ8 and W2 ¼ 95), the maximum demand satisfaction is achieved by selecting 7, 2, 2 and 2 suppliers for food and water, shelter, blanket and kit, respectively. As the results show, although the numbers of food and water suppliers are increased by increasing the value of the second objective function, the numbers of

Table 7 Comparison between models. Total cost Inventory cost Pre-disaster procurement cost Post-disaster procurement cost Percentage of demand coverage

Fig. 10. The capacity usage of each supplier at pre-disaster. 16

Model 1

Model 2

Model 3

2.6274Eþ08 7.4023Eþ07 8.2275Eþ07 _ 43%

1.8935Eþ08 2.30788Eþ07 3.6285Eþ07 7.0512Eþ07 80%

1.5733Eþ08 2.24475Eþ07 3.3766Eþ07 6.12901Eþ07 85%

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improved cost-efficiency. Although it might seem that decreasing the inventory level may decrease the responsiveness, nonetheless, this is not true in practice. It is because, under option contract, the HO can reserve suppliers’ capacity to guarantee the availability of relief supplies in sufficient quantities exactly when they needed at post-disaster. There­ fore, an option contract is desirable for the HO because the costs asso­ ciated with acquiring and storing inventory at pre-disaster are mitigated while at the same time responsiveness is increased. Fig. 10 illustrates the capacity usage (the proportion of the reserved quantity to the available capacity) of each supplier. As Fig. 10 shows, two suppliers for shelter (suppliers 1 and 2), six suppliers for water and food (suppliers 3, 4, 5, 7, 8 and 9) and two suppliers for hygiene kit (suppliers 12 and 13) are selected. It is obvious from Fig. 10 that for the critical relief items, more suppliers are selected compared to the noncritical items. In other words, it can be stated that the selection of a large number of suppliers with low reservation capacity is a suitable policy for supplying critical items at the right time and consequently covering more regions. Nevertheless, the supply policy for non-critical items is different, so that the selection of few numbers of suppliers with high capacity reservations is desirable. Because for this type of relief items, the importance of geographic proximity of suppliers to the affected region and necessity of quick response is low. In brief, it can be stated that two types of supplier selection policies namely the central­ ized and decentralized ones can be applied for the non-critical and critical relief items, respectively, in which the former has a lower sup­ plier coordination degree while the latter has a higher responsiveness level. It is because that in supplying the critical and non-critical items, the important issues are quick response and cost-efficiency, respectively. Therefore, by defining a suitable supply policy according to the type of relief items, the HO can reach a balance between the cost-efficiency and responsiveness of the relief chain. Also, as shown in Fig. 10, capacity reservation from suppliers with higher coverage level is more than other suppliers. For example, suppliers 6, 10 and 11 that cover a smaller number of regions rather than other suppliers (as can be seen from Table 4), have not been selected for water and food supply. Thus, through the reservation capacity of suppliers with a high coverage level, a lower number of suppliers can be selected and consequently, coordi­ nation and management costs can be decreased. Although contracting with fewer suppliers would increase efficiency in coordination, it could also lead to low reliability of supply flow due to the probable disruption of suppliers that may cause poor performance of suppliers. Therefore, in addition to selecting suppliers with a wide coverage range, considering their reliability is also important for improving both responsiveness and cost-efficiency measures simultaneously. A comparison between the inventory prepositioning model (Model

Fig. 12. The effect of changing the first stage’s budget on both objectives under the high level of second stage’s budget.

1), an integrated pre-positioning and a single-period option contract model (Model 2), and our proposed model (Model 3) is also carried out to show the benefits of the proposed approach. Table 7 shows a summary of the results of these models. It is noteworthy that we consider two parameters (o1, e1) for the single-period option contract model. Pro­ curement from suppliers under the option contract decreases the in­ ventory cost, pre-disaster procurement cost, and the total cost and increases the percentage of demand coverage significantly in Model 2 compared to Model 1. Indeed, in Model 2, by considering contracted suppliers, extra needed relief supplies could be procured at post-disaster which improves the responsiveness and increases the demand coverage. It also decreases the need for pre-positioning a large number of relief supplies in the pre-disaster and consequently reduces the inventory and pre-disaster procurement cost. This result demonstrates that the effec­ tiveness of relief logistics could be improved by selecting suitable sup­ pliers and cooperating with them through proper supply contracts. Moreover, it is evident from the results shown in Table 7 that the Model 3 outperforms Model 2, especially regarding logistics cost. It is because that the two-period option contract model provides more flexibility compared to a single-period one in responding to disasters. In other words, it is efficient for reducing the purchasing cost, because relief organizations do not incur additional payments for receiving the secondperiod requirement in the first period which must be purchased at a higher price. Therefore, it can be concluded that the proposed approach is not only applicable to the real-world disaster relief operations but also can improve the cost-efficiency and responsiveness of relief logistics

Fig. 11. The effect of changing the first stage’s budget on both objectives under the low level of second stage’s budget.

Fig. 13. The effect of changing the ratio of o1/o2 on reservation quantity of relief supplies in the first and second option contract models. 17

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Fig. 16. The effect of changing the ratio of o1þo2/e1þe2 on the first and second stages’ costs.

Fig. 14. The effect of changing the ratio of e3/e1 on reservation quantity of relief supplies in the first and second option contract models.

limitations on contract execution and post-disaster procurement, most of the pre-disaster budget is spent on prepositioning more supplies for delivering more items at post-disaster. Therefore, this leads to an in­ crease in the total cost while the demand coverage is not highly affected, because a wide range of potentially affected areas cannot be covered only by prepositioning inventory. In addition, it is not possible to have enough prepositioned inventory in all potentially affected areas due to the limited budget. However, putting this analysis into perspective, it could be perceived that when the post-disaster budget is low, the HO cannot considerably increase the demand coverage and improve the cost-efficiency by increasing the pre-disaster budget. Fig. 12 represents the changes in the total cost and demand coverage when the pre-disaster budget varies under the high level of post-disaster budget (150% of its base level). The figure shows that when the predisaster budget is in the range of 50–100%, although the demand coverage is increased, it is not considerably improved compared to a similar condition of the low level of post-disaster budget (see Fig. 11). Therefore, increasing the second stage’s budget without increasing that of the first stage is not effective alone. Unlike the previous condition, when the percentage increase in the pre-disaster budget is in the range of 100–150%, the demand coverage significantly increases with a reason­ able increase in the total cost. When both pre- and post-disaster budgets are high, by selecting more suppliers and reserving more capacity from them as well as procuring more relief items in the post-disaster, the HO could reduce the cost of prepositioned inventory and at the same time increase the demand coverage. In summary, in contrast to the situation in which the pre-disaster budget is low, increasing the post-disaster budget when the pre-disaster budget is high; will be beneficial. In addition, it is concluded that it would be more desirable if the HO al­ locates the budget to both pre- and post-disaster in a balanced manner instead of increase one of them. Indeed, opposed to the results obtained by Mohammadi et al. [49], our analysis demonstrated the importance of both pre- and post-disaster budgets in the responsiveness and

simultaneously. 5.4. Sensitivity analyses In this section, we conduct a sensitivity analysis to show the effect of critical parameters on the optimal solution. The pre- and post-disaster budgets are two important parameters that can affect the final solutions. The results of the sensitivity analysis performed on the pre-disaster budget under the low and high level of post-disaster budget are provided in Figs. 11 and 12, respectively. It is noted that the vertical axis shows the expected scaled value of both objectives. We performed our experiments under different pre-disaster budget values ranging from 50% to 150% of the base levels. Fig. 11 shows the changes in the first and second objective functions when the pre-disaster budget varies under a low level of post-disaster budget (50% of its base level). As Fig. 11 indicates both objective functions are increased by growing the value of the pre-disaster budget and vice versa. As it was expected, this observation demonstrates the sensitivity of both objective functions to the available budget. It can be seen that decreasing the pre-disaster budget in the range of 50–100% leads to a significant decrease in demand coverage and the total cost. It means that when the pre-disaster budget is low, low amounts of relief supplies can be prepositioned or reserved from the available suppliers. As Fig. 11 shows, increasing the available budget in the range of 100–150% does not have a significant effect on the demand coverage while it increases the total cost. It means that when the post-disaster budget is low, due to

Table 8 Comparison between the uncertain (possibilistic) and deterministic models.

Fig. 15. The effect of changing the ratio of d1/d2 on reservation quantity of relief supplies in the first and second option contract models. 18

Model

First objective function’s value

Second objective function’s value

Possibilistic model with α ¼ 0.5 Possibilistic model with α ¼ 0.8 Possibilistic model with α ¼1 Deterministic model

1.5254Eþ08

91%

1.5733Eþ08

85%

1.6418Eþ08

82%

2.35018Eþ08

79%

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cost-efficiency of relief logistics. The behavior of the proposed option contract models is different especially for items which have recurring demand (e.g. water and food). Thus, in Fig. 13, we present the sensitivity of solutions related to these items to the variations in the option prices. With increasing the o~1 = ~ o2 ratio, the reservation quantity of relief supplies through the first option contract tends to be increased more than the second model and vice versa. As Fig. 14 shows, this is in contrast to the situation in which the ~e3 =~e1 ratio is increased. Also, changing the ratio of ~e3 = ~e1 has a smaller effect on selecting the best option contract model than o~1 = o~2 . This result o2 or low ratio indicates that those suppliers who have a high ratio of ~ o1 = ~ of ~e3 =~e1 , are more likely to be selected for making the first option con­ tract due to their cost-effectiveness. Also, it would be better if the HO selects the second option contract for those suppliers who have a low ratio of ~ o1 =~ o2 or a high ratio of ~e3 =~e1 . Fig. 15 represents the changes in the reservation quantity of relief supplies through the first and second option contract models under ~1 =d ~2 . As Fig. 15 shows, when d ~1 =d ~2 ratio is decreased, different ratios of d

model makes the model more applicable to real-world disaster situations than a deterministic one since the possible changes of parameters and imperfect information about the future events could be incorporated into the model. The performance of the proposed uncertain model is compared with the equivalent deterministic model in which the expected value of each fuzzy parameter is used to construct the deterministic counterpart (i.e. the crisp mean - value approach). Based on the method used by Liu and Liu [56], the expected value of a triangular fuzzy number ξ ¼ ðξ1 ; ξ2 ; ξ3 Þ can be stated as follows: E½ξ� ¼

ξ1 þ ξ2 þ ξ3 3

52

The results of the comparison between the uncertain and determin­ istic models under different confidence levels ðαÞ are presented in Table 8. The results show that at each level of credibility, the proposed credibility-based possibilistic model outperforms the deterministic one in terms of both objective function values, which represents the use­ fulness of fuzzy mathematical programming for the decision-making process in the concerned problem. Also, Table 8 presents the results of the sensitivity analysis performed on the confidence level. As Table 8 shows, both objective values are improved as the confidence level de­ creases, because decreasing the confidence level leads to increasing the feasible region and therefore creating more chances for yielding better solutions.

the reservation quantity of relief supplies through the first option con­ tract model tends to be increased. It means that when more relief items are needed in the second period, applying the second option contract model results in a higher reservation cost. The reason is that in the second model, reservation of the second period’s requirements is done with price ~ o1 (the whole needed quantity is reserved by price ~o1 ), while in the first option contract model, they are reserved with ~ o2 (i.e. the option prices for the first and second periods’ requirements are different in this model).

6. Conclusions and future research Having an appropriate supply plan at pre-disaster for effective and efficient emergency response at post-disaster is essential for saving lives, lowering the impact of disasters and maintaining social stability. In this study, we propose two varieties of a novel two-period option contract model to improve the level of responsiveness and cost-efficiency, simultaneously. Using the proposed option contract, the HO can decrease the pre-positioning level and reduce the procurement cost without degrading the responsiveness level. Moreover, this study in­ tegrates the option contract with the supplier selection and order allo­ cation related decisions using a maximal covering model while considering the priorities of relief items. A novel mathematical model is developed for the designed option contract, which helps the HO for supplier selection and determining the capacity reservation quantity of each selected supplier at the pre-disaster phase and the exercised amount at the post-disaster phase. A scenario-based mixed possibilisticstochastic programming approach is applied to cope with the inherent fuzziness in input data and randomness of disaster scenarios’ likeli­ hoods. The weighted augmented ε-constraint method is used to find Pareto optimal solutions in the resulting bi-objective model. Also, a case study is provided to show the performance and applicability of the proposed models in practice. In addition, we perform numerical exper­ iments and several sensitivity analyses to understand the effects of agreement terms and some key parameters on the final decisions. A potential direction for future research could involve developing integrated pre-disaster relief pre-positioning and post-disaster procure­ ment by applying other well-known contracts such as buy-back contracts or developing mixed contracts. Also, considering two supply options such as the main and backup suppliers to make the supply flow resilient/ reliable enough and coping with uncertainty and disruption of suppliers is another interesting possible direction for further research. Lastly, it would be interesting to extend the proposed model to a multi-period horizon case in order to consider a longer planning horizon at postdisaster.

(Please insert Fig. 15 around here) In another sensitivity analysis, we show the effect of the ratio of the total option and exercise prices on the first objective function when the second objective function is constant. As Fig. 16 shows, if a supplier decreases the total option price and increases the total exercise price (i.e. when o~1 þ o~2 =~e1 þ ~e2 ratio decreases), it leads to a considerable reduc­ tion in the cost of pre-disaster operations while the cost of post-disaster procurement is increased and vice versa. The reason for this fact is that the HO can reserve more capacities of suppliers and decrease the prepositioning levels, which leads to a considerable reduction in the predisaster cost. Also, the post-disaster procurement cost can be increased by increasing the total exercise price. Therefore, it would be more desirable if the decision makers negotiate with the suppliers for changing their preferences about the option and exercise prices and make decisions concerning the available budget of each stage. 5.5. Comparison of the deterministic and fuzzy models In this section, the performance of the proposed two-stage scenariobased mixed possibilistic-stochastic programming model is compared with the deterministic model to verify the resulting possibilistic model and evaluate the benefits of considering uncertainty. Generally speaking, incorporating inherent uncertainties in a mathematical model provides the number of merits compared to neglecting them in a deterministic model [15,65]. Firstly, using uncertain data improves the flexibility of the model as several solutions could be obtained for the problem under consideration by changing the uncertain parameters’ values within their fluctuating ranges, contrary to the deterministic model which takes into account only one value for each parameter. Also, considering possible changes in parameters during the solution process would lead to the robustness of final solutions. This robustness has particular importance, especially for strategic decisions (e.g. deter­ mining inventory prepositioning levels and relief suppliers at pre-disaster) that must be made for a long-term planning horizon while cannot be changed easily. Secondly, due to intrinsic uncertainties in the relief logistics problems, considering fuzzy parameters in an uncertain

CRediT authorship contribution statement Mojtaba Aghajani: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing 19

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original draft, Writing - review & editing, Visualization. S. Ali Torabi: Conceptualization, Methodology, Validation, Formal analysis, Investi­ gation, Writing - original draft, Writing - review & editing, Visualization. Jafar Heydari: Methodology, Investigation, Writing - review & editing.

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M. Aghajani et al.

Socio-Economic Planning Sciences xxx (xxxx) xxx

Mojtaba Aghajani is a Ph.D. Candidate at the School of Industrial Engineering, University of Tehran. His main research interest is Humanitarian Logistics.

Retail Supply Chain Planning. Professor Torabi has published several papers in peerreviewed international journals such as EJOR, IJPE, TRE, JORS, IJPR, COR, FSS and CIE.

S. Ali Torabi (PhD, MSc, BSc) is a Professor of Operations and Supply Chain Management at the School of Industrial Engineering, University of Tehran. His main research interests include: Supply Chain Resilience, Humanitarian Logistics, Sustainable Operations and

Jafar Heydari is an associate Professor of Operations and Supply Chain Management at the School of Industrial Engineering, University of Tehran. Professor Heydari has pub­ lished several papers in reputable academic journals and conferences.

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