A distributionally robust optimization for blood supply network considering disasters

Transportation Research Part E 134 (2020) 101840

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A distributionally robust optimization for blood supply network considering disasters

T

Changjun Wanga, , Shutong Chena,b ⁎

a b

Glorious Sun School of Business and Management, Donghua University, Shanghai 200051, China Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, VA 24061, United States

ARTICLE INFO

ABSTRACT

Keywords: Blood supply network Disaster relief Stochastic distributionally robust optimization Transshipment Semidefinite programming

We study blood supply network optimization considering disasters where only a small number of historical observations exist. A two-stage distributionally robust optimization (DRO) model is proposed, in which uncertain distributions of blood demand are described by a moment-based ambiguous set, to optimize blood inventory prepositioning and relief activities together. To solve this intractable DRO with integer recourse, an approximate way is developed to transform it into a semidefinite program. A case study, based on the Longmenshan Fault in China, validates that our approach outperforms typical benchmarks, including deterministic, stochastic and robust programming. Sensitivity analysis provides helpful managerial insights.

1. Introduction The occurrence of natural disasters has become more frequent and caused a great loss of assets and numerous victims in recent years. To mitigate the consequences of disasters, a challenging issue is to develop an efficient supply network to provide the required relief goods promptly (Fahimnia et al., 2017; Fazli-Khalaf et al., 2017). Among these relief goods, blood is significantly different from other ordinary commodities. Specifically, over-supply of human blood would result in huge waste due to its preciousness, while shortage would lead to high costs for society (Beliën and Forcé, 2012). Moreover, it is not easy to efficiently match blood supply and demand in the context of disasters. On the supply side, human blood, as a scarce and perishable asset, cannot be rapidly produced and heavily stored. On the other hand, the blood demand in a suddenly-occurring disaster is uncertain, and we often lack adequate historical information to describe such uncertainty exactly. All these bring extra difficulties to the design of a BSN. Hence, the BSN optimization considering disasters has received considerable attention in recent years (Fahimnia et al., 2017; Jabbarzadeh et al. 2014; Ramezanian and Behboodi, 2017; Salehi et al., 2017; etc.). Demand uncertainty is a major concern in the BSN-related studies. To address this uncertainty, previous works based on stochastic programming (SP) assume that a precise distribution can be obtained from historical data (Birge and Louveaux, 2011; King and Wallace, 2012). However, a specific region is not frequently subject to critical disasters such as earthquakes. Then, firstly, the number of historical observations of emergency blood demand is usually limited. Second, each disaster is unique. Thus, the assumption of known distribution may be too rigid (Balcik and Yanıkoglu, 2019; Dalal and Üster, 2018; Samani et al., 2018; Shi et al., 2019; Snyder, 2006). Motivated by this observation, many works consider the robust optimization (RO) approach. Some robust models still require the scenario set of uncertain parameters (Mulvey et al., 1995; Aghezzaf et al., 2010), which has the same difficulties as SP. Besides, some robust models only require the worst bound for uncertain parameters and optimize the so-called worst-case objective over this

⁎

Corresponding author. E-mail address: [email protected] (C. Wang).

https://doi.org/10.1016/j.tre.2020.101840 Received 10 April 2019; Received in revised form 4 January 2020; Accepted 4 January 2020 1366-5545/ © 2020 Elsevier Ltd. All rights reserved.

Transportation Research Part E 134 (2020) 101840

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bound (Ben-Tal et al., 2009). However, they neglect potential information contained in the limited historical observations, and sometimes, lead to unnecessarily conservative solutions (Agrawal et al., 2012; Goh and Sim, 2010; Liu et al., 2019). Moreover, to handle the imprecise data, triangular or trapezoidal fuzzy numbers are used in the fuzzy-robust approach to study the BSN problems (Samani and Hosseini-Motlagh, 2018; Zahiri et al. 2015). But, obtaining the appropriate fuzzy numbers is still a problematic and cumbersome task. Thus, when only a small number of historical disaster observations are available, we concern how to use the limited data to describe future uncertain demand and implement optimization for the BSN problems considering disasters. The above analysis reveals the opportunities that this paper targets and the aforementioned issues can be rectified by adopting a distributionally robust optimization (DRO) approach. In DRO, uncertain parameters can follow one of a collection of possible probability distributions within an ambiguity set. Considering such distributional uncertainty, DRO optimizes the worst expected cost over these distributions, instead of a specific distribution, the worst bound or a fuzzy number. One of the benefits of DRO is that, the ambiguity sets, such as moment-based, can be easily constructed by limited historical observations. Thus, DRO is data-driven and suitable for the scarce data environment (Delage and Ye, 2010; Nakao et al., 2017). Then, we consider it as an alternative to address the optimization of our BSN in the context of disasters. To deal with the uncertain blood demand in an unexpected disaster, we consider a two-stage BSN model. Before a disaster strikes, i.e., the preparedness stage, we focus on the daily storage and distribution of human blood in a disaster-prone region, which should not only satisfy the known day-to-day demand but also prepare for a possible future disaster. When the disaster occurs, called the response stage, the prepositioning inventory maybe not enough for an emergency realization. Thus, a recourse policy, i.e., the decisions on the location of temporary donation facilities and blood transshipment, should be determined. Such issue from preparedness to response leads naturally to min-max-min formulations, which correspond nicely to two-stage optimization (Matthews et al., 2019). Thus, our job is to develop a novel two-stage DRO-based BSN model to address the above decisions. The goal of the model is to minimize the sum of the cost in the first stage and the worst expected cost in the second stage. The ambiguity set of the stochastic blood demand is characterized by the first and second moments that can be derived from the historical data directly. Before we apply our model in practice, it is worth noting that the binary decision variables (e.g., the temporary facility location decisions) exist in the response stage (Acar and Kaya, 2019; Fazli-Khalaf et al., 2017; Habibi-Kouchaksaraei et al., 2018; Liu et al., 2019; Ramezanian and Behboodi, 2017). It means that the model we present is a two-stage DRO with integer recourse. However, most DRO-related studies focus on single-stage settings. Only a few works consider two-stage DRO but take linear recourse into account (e.g., Liu et al., 2019; Nakao et al., 2017; Zhang et al., 2016). Two-stage DRO with integer recourse is intractable and rarely explored. To fill this gap, we design an approximate approach. Specifically, we dualize the DRO model to get a bilevel semidefinite programming (SDP) model with a binary lower-level program at first. Next, we construct a convex hull (see Sherali and Adams, 1994) to linearize the lower-level integer variables, and then, use Karush-Kuhn-Tucker (KKT) conditions to obtain a tractable single-level SDP model. Our method can be applied to solve two-stage DRO, regardless of linear or integer recourse, in other settings. The rest of this paper proceeds as follows. Section 2 presents the literature review. In Section 3, we formulate a two-stage mixedinteger BSN model by moment-based DRO. Section 4 shows a computationally tractable way to transform our DRO with integer recourse into an approximate mixed-integer SDP. In Section 5, we implement a case study based on earthquakes in the Longmenshan earthquake fault in China. Computational results of the proposed DRO model and several typical benchmarks are compared to validate the applicability of our model and approach. Finally, we conclude the paper and point out the future research in Section 6. 2. Literature review The literature review will be provided in three parts: (i) papers concerning the BSN problems under uncertainty; (ii) SP and RO in the BSN problems; and (iii) DRO and moment-based DRO. 2.1. The BSN problems under uncertainty In recent years, lots of studies focus on designing and operating a BSN. Some works do not specify that they focus on disasters, but take uncertain demand into account, to study BSN-related strategic and operational decisions, such as blood facility location; blood production; inventory; routing; distribution and transshipment (see Duan and Liao, 2014; Ensafian and Yaghoubi, 2017; Gunpinar and Centeno, 2015; Jafarkhan and Yaghoubi, 2018; Samani et al., 2019). In these studies, random demand is assumed to follow a known distribution (e.g., Duan and Liao, 2014), be bounded by the worst scenario (e.g., Samani et al., 2019), or be described by a fuzzy number (e.g., Zahiri et al., 2015). Moreover, typical characteristics of blood, such as perishability; multi-products, are also taken into account (e.g., Duan and Liao, 2014; Ensafian and Yaghoubi, 2017; Gunpinar and Centeno, 2015; Samani et al., 2019). However, all these papers do not focus on disasters. With the consideration of disasters, the BSN-related decisions can be implemented before and after disasters respectively, i.e., the preparedness stage and the response stage (Altay and Iii, 2006; Galindo and Batta, 2013; Ye et al., 2019). The coordination of these two stages is important to supply-demand match in disasters, because enhancing preparedness can, to some extent, facilitate disaster relief, while relief activities can remedy the pre-disaster plan (Pérez-Rodríguez and Holguín-Veras, 2016; Vanajakumari et al., 2016). Thus, a two-stage model is required. However, in the BSN-related optimization research, relatively few studies focus on two-stage modeling in the setting of disasters. For example, Jabbarzadeh et al. (2014) consider the permanent facility location before disasters and the temporary facility location, donor assignment, delivery decisions, etc., when disasters occur. Fahimnia et al. (2017) also present a two-stage model to determine the number of blood facilities in the preparedness stage, and the donor assignment, delivery as well as inventory in the crisis response stage. Generally, the purpose of these studies is to satisfy blood demand in disasters by two2

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stage activities involved in the collection, production, stock and distribution echelons of a BSN, while at the same time reducing wastage and minimizing costs (Beliën and Forcé, 2012). 2.2. SP and RO in the BSN studies considering disasters In the formulation of the two-stage models, the consideration of disaster parameters, especially uncertain demand, is a crucial issue. Two mainstream stochastic mathematical programming methods, SP and RO, address this challenge in different ways. In the SP-based models, the uncertain parameters, which are realized after disasters, are assumed to follow a known set of discrete scenarios. Thus, the goal is to optimize the performance sum of the first stage and the expected value in the second one if a two-stage model is adopted (Fahimnia et al., 2017). In the previous BSN research, not limited to the disaster setting, the discrete scenarios are given directly (Fahimnia et al., 2017; Hamdan and Diabat, 2019) or assumed to follow a known distribution, such as Poisson (Gunpinar and Centeno, 2015) or gamma (Gunpinar and Centeno, 2015). Different from SP, RO addresses stochastic problems by taking the worst-case into account (Ben-Tal et al., 2009). According to the classification way in Govindan et al. (2017), the BSN works based on RO considering disasters can be divided into three subgroups accordingly. The first one is still based on discrete scenarios. In this setting, many works adopt the models developed by Mulvey et al. (1995) or Aghezzaf et al. (2010), which consider the so-called solution robustness and model robustness (Habibi-Kouchaksaraei et al., 2018; Mulvey et al., 1995). For example, based on Mulvey et al. (1995), Jabbarzadeh et al. (2014) aim to find a solution with the smallest expected performance deviation. Khalilpourazari and Khamseh (2017) further extend this work to a multi-objective setting. Salehi et al. (2017) follow the way of Aghezzaf et al. (2010) and formulate a two-stage multi-period RO model for a three-layer BSN with the consideration of a possible earthquake. It optimizes the expected cost under all the scenarios and the highest variability. Similar works can refer to Habibi-Kouchaksaraei et al. (2018). The second subgroup considers the uncertain parameters continuously vary within a pre-defined interval, i.e., internal-uncertainty. An often-used approach is developed by Bertsimas and Sim (2004) which introduces a budget parameter to limit the uncertainty within the interval, and thus, controls the conservatism level of robust solutions (Zhang and Tang, 2018). For instance, Rahmani (2018) displays a two-stage RO model with the budget constraints to address dynamic BSN design problems. This approach is also adopted by Ramezanian and Behboodi (2017). Third, some BSN studies recently employ fuzzy-robust mathematical programming where the imprecise or ambiguous data is modeled through the possibility distributions in the form of triangular or trapezoidal fuzzy numbers (Zahiri et al., 2015). For example, Samani and Hosseini-Motlagh (2018) consider the epistemic uncertainty by using a triangular fuzzy number and develop a robust model based on the fuzzy measure called Me. Similar works include Fazli-Khalaf et al. (2017). Moreover, the aforementioned BSN works show that the binary decisions, such as the location of temporary blood collection facilities, donor assignment, should be taken into consideration after disasters. Such integer recourse, in terms of solving, usually increases the calculational burden. However, the SP and most RO models with integer recourse have the same tractability of their deterministic equivalents. Specifically, for the SP models with integer resource, if their deterministic equivalents are mixed-integer linear programming (MILP), these SP models are still MILP although the number of integer variables would increase in proportion to the size of scenarios set. Then, such models are amenable to standard solvers (Jabbarzadeh et al., 2014) or algorithms such as Lagrangian relaxation (Fahimnia et al., 2017). Regarding the robust models, taking the RO with budget parameters as the example, the models with integer recourse have the similar computational complexity with their deterministic equivalents because only the worst bound, rather than all scenarios, is considered. Their counterparts also could be calculated by off-the-shelf solvers or existing algorithms (Rahmani, 2018). 2.3. DRO and moment-based DRO In reality, compared with the scenarios or the enumeration set of random parameters, moment information or uncertainty about the distribution itself is relatively easy to know (Govindan et al., 2017). Hence, based on distributional uncertainty within an ambiguity set, DRO is proposed to optimize the expected objective functions under the worst probability distribution (Bertsimas et al., 2010; Delage and Ye, 2010). Its advantages lie in two aspects. Firstly, DRO allows uncertain variables to follow an arbitrary distribution defined in the ambiguity set, instead of a given distribution, interval or fuzzy number. Thus, such a way reduces the modeling difficulty for uncertain parameters. Second, most basic DRO variants, especially the single-stage, are tractable, if their deterministic equivalents can be solved optimally. Therefore, it provides an alternative for addressing stochastic problems. There are two prevailing ways to construct the ambiguity sets in DRO: statistical-distance-based and moment-based. The former defines the ambiguity sets by choosing a Wasserstein distance (Esfahani and Kuhn, 2018) or ϕ-divergence (Ben-Tal et al., 2013) to describe the deviation from a pre-defined distribution. The latter considers the uncertain distributions which follow the moments (such as mean and covariance) given by historical observations (Bertsimas et al., 2010; Delage and Ye, 2010; Popescu, 2007). Here, we focus on the latter because it is easier to obtain the moment information from the limited disaster data than a nominal distribution. Bertsimas et al. (2010) and Delage and Ye (2010) are two of the pioneering papers concerning moment-based DRO. In their papers, a solving way is developed by equivalently transforming a moment-based DRO into a tractable semi-definite program via dualization. Moment-based DRO has been successfully applied to many areas, including investment portfolio (Delage and Ye, 2010); newsvendor problems (Fu et al., 2018); shortest path problems (Zhang et al., 2017) and surgery block allocation (Wang et al., 2019). 3

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Table 1 The literature on two-stage BSN optimization in the setting of disasters. Methods

References

Features ①

Decisions The first stage

② L

SP

RO

Fahimnia et al. (2017) Samani et al., 2018 Scenario -based Jabbarzadeh et al. (2014) Khalilpourazari and Khamseh (2017) Salehi et al. (2017) Habibi-Kouchaksaraei et al. (2018) Internal-uncertainty Ramezanian and Behboodi (2017) Rahmani (2018) Fuzzy-robust Zahiri et al. (2015) Fazli-Khalaf et al. (2017) Samani and Hosseini-Motlagh (2018) This paper

√ √ √

√

√ √ √ √ √ √ √ √

√ √ √

√ √ √ √ √ √

I

The second stage

D/R

√

√ √ √

√ √ √ √

L

√ √ √

√

√

D/R √ √ √ √ √ √ √ √ √ √ √ √

S

W

A

√ √ √ √

√

√ √ √

√ √

√

T

√ √

√ √ √ √ √ √

√

①Perishability; ②Uncertain demand. L: Location; I: Inventory prepositioning; D/R: Distribution flow/Routing; S: Shortage; W: Wastage; A: Assignment; T: Transshipment.

But DRO has not been used in addressing the BSN problems and most works merely consider single-stage settings. Some studies explore the applications of moment-based DRO with two-stage structure. For example, Nakao et al. (2017) discuss transportation network optimization to design arc capacities in the first stage and distribution flow in the second stage. Liu et al. (2019) present a two-stage model to address emergency medical service station location and sizing problems in the first stage and demand allocation decisions in the second one. Zhang et al. (2016) study two-stage lot-sizing problems. Both of two papers use the first and second moments to define the ambiguity sets of uncertain demand which is known in the second stage. Notice the two-stage DRO models are min-max-min problems which are hard to solve, obtaining tractable reformulations becomes the main tasks. Hence, Nakao et al. (2017) present an approach to approximately transform the proposed model into an intermediate mixed-integer model which can be solved by the cutting-plane algorithm. Liu et al. (2019) develop a way to obtain an approximate second-order cone program. Zhang et al. (2016) develop a shortest path-based reformulation. However, only linear recourse is considered in their works. Bansal et al. (2018) and Xie and Ahmed (2018) are the only two considering DRO with integer recourse. But the limitation of the former lies in that, to facilitate the utilization of a decomposition approach based on the Benders’ algorithm, the stochastic scenarios defined by the ambiguity set must be finite. The latter considers the simple integer resource of only one decision variable. To the best of our knowledge, except these two papers, there is no further research on this topic. 2.4. Summary of literature review According to the literature review in Section 2.1, an efficient BSN can rapidly respond to disasters with well-designed preparations and relief activities. One crucial issue is how to deal with uncertain demand. Thus, the stochastic optimization methodologies with the two-stage structure are used in the aforementioned studies (see Section 2.2). To facilitate our analysis of the reviewed works, we present the previous two-stage BSN works considering disasters in Table 1. There are several drawbacks to their proposed models and approaches, which are summarized as follows. At first, inventory prepositioning is a widely applied disaster preparedness strategy (Balcik et al., 2019; Sabbaghtorkan et al., 2019; Ye et al., 2019). But, as shown in Table 1, it is considered in only a few BSN studies. Besides, blood transshipment is also rarely explored in the BSN optimization considering disasters (Jafarkhan and Yaghoubi, 2018). It should be noted that these relief activities, as well as blood’s perishability and preciousness, would interact with each other and complicate the problems. For example, due to the perishability, too much prepositioning blood inventory may result in wastage. The transshipment may help to alleviate wastage and shortage, meanwhile, reduce systemwide prepositioning inventory due to risk pooling. Hence, it leaves us a research opportunity to consider these elements together. Second, in terms of uncertainty modeling, although the SP models show better performance than the deterministic models (Alem et al., 2016; Fahimnia et al., 2017), the assumption that exact distributions are available is usually unrealistic, due to the uniqueness of each disaster and the scarcity of the historical observations. Scenario-based RO, which depends on a set of discrete scenarios, has the similar problem. The fuzzy-robust approach can address the so-called epistemic uncertainty. But estimations for the uncertain parameters depend on the field experts’ professional experience, which may be subjective. In RO with budget constraints, only the worst bound is considered, while the possible statistical inferences of uncertain parameters hidden in the limited historical data are discarded. Furthermore, in terms of the quality of solutions, most robust models aim at searching a solution that performs well in all possible realizations, which sometimes results in over-conservative results (Agrawal et al., 2012; Goh and Sim, 2010; Liu et al., 2019). At last, as previously mentioned, two-stage moment-based DRO is a possible alternative for scarce data environment but has not been used for the BSN problems. Besides, the binary recourse problems should be taken into account. However, only a few papers focus on two-stage DRO, in which most of them merely consider linear recourse (Liu et al., 2019; Nakao et al., 2017; Zhang et al., 4

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2016). The integer recourse problems have rarely been studied (Bansal et al., 2018; Xie and Ahmed, 2018), and a more general solving approach is required. As a result, the first two points inspire us to develop an appropriate two-stage DRO model to address BSN problems considering disasters. The last one leads to a motivation to present a computationally tractable approach. 2.5. Contributions of our paper Thus, the main contributions of our paper are as follows: (1) A novel two-stage DRO model with mixed-integer recourse is developed for the BSN optimization considering disasters. The uncertain demand is described by a data-driven ambiguity set based on the first and second moments. Several distinctive features of blood and disaster relief, such as perishability, multi-products, wastage and shortage, prepositioning and transshipment are taken into account. To the best of our knowledge, it is the first time to explore the application of DRO in the BSN problems. (2) Through duality theory, we reformulate the original model into an equivalent semi-infinite programming model with mixedinteger programs in constraints. For tractability, we use Schur’s complementary, convex hull construction technique and KKT conditions in turn, to derive a semidefinite program reformulation as an approximation. Our method relaxes the limitations on the number of scenarios or decision variables in Bansal et al. (2018) and Xie and Ahmed (2018), and then, can be applied to solve general two-stage moment-based DRO with linear or integer recourse in other settings. (3) To demonstrate the applicability, the proposed model and approach are implemented in a real case, i.e., the Longmenshan Fault in China, where only seven but serious earthquakes have occurred since 1933. Sensitivity analysis based on critical parameters, such as unit shortage cost and replenishment cycle, is studied. 3. Problem definition and model formulation In a disaster-prone region, we consider a two-echelon BSN, which is composed of collection facilities and blood transfusion centers. The BSN supplies human blood to meet the demand in both the preparedness and disaster response stage, also named the first and the second stage respectively. The collection facilities, including permanent and temporary ones, are in the first layer in such BSN which collects, tests and distributes blood products. The permanent collection facilities (PCFs) are those already existing in the BSN, while the temporary collection facilities (TCFs) would be determined from candidates in the second stage. The transfusion centers (TRs) compose the second layer in the BSN where the daily and emergency demands occur. To mitigate the blood shortage when a disaster occurs, a certain amount of blood products would be prepositioned in TRs. However, the blood wastage also should be avoided due to its perishability. Based on the above background, we illustrate our problem by taking the case in Fig. 1 as an example. The decisions are made in two stages. The first stage focuses on optimizing the prepositioning inventory in the TRs and the daily distribution flow from the PCFs to the TRs in repetitive cycles (shown as the full line in Fig. 1(1)). When a disaster occurs, the injured would get first-time treatment in the TRs. Here, we only consider the emergency demand caused by disasters. In other words, daily medical treatments like surgeries could be rescheduled after the disaster-aid stage. With the revelation of disaster information, the decision-maker (DM) would decide whether one or more TCFs are required (for example, TCF candidate 2 and 4 are activated in Fig. 1(2)), and then, assign the selected TCF to a specific TR (bold lines). Besides, the distribution flow (light line) from the PCFs to the TRs and the transshipment flow (dashed line) between TRs would be decided to improve the blood supply-demand match in the whole BSN. The objective is to minimize the total cost, including the preparedness stage and the response stage of the BSN within a given planning horizon. TCF candidate 3

TR 1 TCF candidate 2 PCF 2

PCF 1

TCF 2

PCF 1

TR 2 PCF 3

TCF candidate 3

TR 1

PCF 2

TR 2 PCF 3

TR 3

TR 3

TCF candidate 4

TCF candidate 1

TCF 4

(1). The first stage

(2). The second stage

Fig. 1. The illustration of the structure of a two-echelon blood supply network. 5

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3.1. Notation The sets, the parameters, and the variables used in the models are given as below. Sets and indices I H C i h c

Set of PCFs Set of TRs Set of TCF candidates Indices to PCFs, i ∈ I Indices to TRs, h ∈ H Indices to TCF candidates, c∈C

Parameters T uc hc rh fcc opc

caph1

capc2

eih ech ehh1 l si UBc dh ξh

Time horizon in the first stage Unit shortage cost (RMB/U) Unit surplus cost (RMB/U) Unit prepositioning cost per cycle of TR h (RMB /(U cycle), U is the abbreviation of blood product units. One unit represents the blood product extracted from 200 ml whole blood)) Setup cost of activating TCF c (RMB) Unit operational cost of collecting blood in TCF c (RMB/U) The capacity of blood product in TR h (U) The capacity of blood product in TCF c (U) Unit transportation fee from PCF i to TR h (RMB/U) Unit transportation fee from TCF c to TR h (RMB/U) Unit transshipment fee from TR h to TR h1(RMB/U) Lifespan of blood product (cycle) Supply of blood product in PCF i (U/cycle) Maximum supply of blood product in TCF c (U) Daily demand for blood product per hour in TR h in the first stage (U/cycle) Emergency demand for blood product in TR h in the second stage (U)

Decision variables The first stage ssh

The inventory level of blood product in TR h (U). Distribution flow of blood product from PCF i to TR h before disaster (U/cycle).

wihp

The second stage yc tgch qc wihr zch v hh1 osh ovh spi

1 if the TCF c is activated to collect blood when disaster occurs; 0 otherwise. 1 if TCF c supply to transfusion center h; 0 otherwise. Collection amount of blood product in TCF c (U). Distribution flow of blood product from PCF i to TR h after disaster (U). Distribution flow of blood product from TCF c to TR h after disaster (U). Transshipment flow of blood product from TR h to TR h1 after disaster (U). Shortage amount of blood product in TR h after disaster (U). Surplus amount of blood product in TR h after disaster (U). Surplus amount of blood product in PCF i after disaster (U).

3.2. Formulation of the BSN problems under blood demand uncertainty In the first stage, the objective of the BSN optimization is to minimize the total cost within T daily cycles, including (1) the inventory cost and (2) the transportation cost from the PCFs to the TRs, which is shown as formula (1-1):

minp T ss, w

eih wihp

rh ssh + h H

.

(1-1)

i I h H

The constraints in the first stage are represented as follows:

wihp

si , i

I,

(1-2)

h H

6

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Inventory(U)

dh

ssh

t=n

t=n+2

t=n+1

t=n+3

Period

Fig. 2. The updated process of the prepositioning inventory in the first stage – an example.

wihp

dh , h

H,

(1-3)

i I

min{caph1

ssh

wihp,

ssh

0, i

d h , dh ( l

I; h

1)}, h

(1-4)

H,

(1-5)

H,

Constraints (1-2) ensure that the distribution flow from each PCF does not exceed the total supply. Constraints (1-3) require that the daily demand should be satisfied by the PCFs. Constraints (1-4) mean that the prepositioning stock should be limited by both the capacity of the corresponding TR and its lifespan. Taking blood product with a lifespan of four cycles as an example, Fig. 2 shows the daily inventory change in a TR, in which, the inventory would be consumed by the rule of First-In-First-Out. Thus, to avoid the wastage, the amount of the prepositioning inventory should be no more than (l–1) times of the daily demand (dh). Finally, constraints (1-5) specify the domains of the decision variables. Given a set of the first-stage decisions ss, the second stage aims to minimize the total cost of (1) the activating cost of selected TCFs, (2) the transportation cost from the PCFs and the TCFs to the TRs, (3) the transshipment cost between the TRs, (4) the blood collection cost, (5) the shortage cost and (6) the surplus cost. Hence, the resource problem, defined as 1 (ss, ) , can be formulated as: 1 (ss ,

=

): min

y, tg , q, w r , z, os, sp, ov, v

i I

spi +

eih wihr + i I h H

ech z ch + c C h H

h H h1 H & h1 h

ehh1 vhh1 +

ovh , ssh,

, ssH

yc = 0, c

]T ,

qc

qc

= [ 1,

, h,

,

H

C,

0, c

UBc , c

MM · tgch qc

(2-3)

C,

(2-4)

0, c

z ch = 0, c

C; h

(2-5)

H,

C,

(2-6)

h H

ssh +

wihr + osh

z ch + c C

ovh

i I

wihr + spi = si , i

vhh1 + h1 H & h1 h

v h1 h = h1 H & h1 h

I,

{0, 1}, c

C; h

qc , wihr, z ch , spi , osh, ovh, vhh1

h,

h

H,

(2-7) (2-8)

h H

yc , tgch

osh + hc h H

(2-2)

C, z ch

opc qc + uc

]T .

h H

yc capc2

c C

(2-1)

h H

in which, ss = [ss1,

tgch

fcc yc +

c C

H, 0, i

I; c

C; h, h1

H &h1

h.

(2-9)

Formula (2-1) is the objective of the second stage under some demand scenario . Constraints (2-2) ensure that each TCF is assigned to one TR. Constraints (2-3) and (2-4) display that blood collection should not exceed the capacity of the corresponding TCF 7

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and the available maximum supply amount, respectively. Constraints (2-5)–(2-6) guarantee that the distribution flow from a TCF to the TRs should be less than the collection amount in this TCF. Constraints (2-7) specify the relationship between the supply, shortage, surplus amount, transshipment flow and the demand realization. Constraints (2-8) calculate the surplus amount in each PCF, and the domains of the decision variables are specified in formula (2-9). Denote fξ as the distribution of the demand vector ξ. Then, a two-stage SP formulation with integer recourse is given as:

minp T

eih wihp + E f

rh ssh +

ss, w

h H

1 (ss ,

)

,

(3)

i I h H

s. t. Constraints (1

2)

(1

5).

where E f is the expected value under the distribution function fξ. As mentioned before, it is necessary to consider the uncertainty of fξ in the setting of disasters. Hence, we next employ DRO to address the BSN optimization with the scarce data. 3.3. Formulation of the DRO-BSN based on exact moments From the historical data, the marginal mean vector μ and covariance matrix Ʃ can be obtained as formula (4) and (5):

µ = [µ1,

=

1 M

, µh ,

1 M

, µH ]T =

m1

µ1

mh

µh

mH

µH

M

[

m1,

,

mh,

,

T mH ] ,

(4)

m=1

M

[

µ1 ,

m1

,

µh ,

mh

,

µH ],

mH

m=1

(5)

where M, mh and µh are the number of the historical data samples, the emergency blood demand in TR h in the mth historical sample, and the mean of emergency blood demand in TR h obtained from all historical samples, respectively. The marginal moment-based ambiguity set of the uncertain fξ for DRO-BSN is formulated as:

df ( ) = 1 (a) df ( ) = µ (b)

D ( , µ, ) = f : (

µ )(

µ )Tdf

.

( )=

(c )

(6)

Specifically, constraints (6a) ensure that D only contains valid distributions over the support set Ω of ξ, while constraints (6b) and (6c) restrict these distributions to follow the moment information given by the historical demands. Hence, the moment-based DRO-BSN variant of formula (3) is represented as (7), in which the objective is to minimize the sum of the cost of the first stage and the expected cost of the second stage under the worst distribution fξ within D. It is a DRO with integer recourse.

minp T ss, w

eih wihp + sup E f

rh ssh + h H

f

i I h H

s. t. Constraints (1

2)

(1

D

5), (6a)

1 (ss ,

)

,

(7)

(6c)

4. Reformulation of the DRO-BSN To convert the two-stage DRO-BSN model (7) to a tractable one, we first dualize the inner maximization problem (Section 4.1), and then, approximately transform the dualized model into a deterministic bilevel SDP problem, in which the lower-level problem is a MILP (Section 4.2). Finally, in Section 4.3, we equivalently linearize the lower-level binary variables of the bilevel SDP problem by reformulating its convex hull form, and then, utilize the KKT conditions to turn it into a tractable single-level mixed-integer SDP problem. 4.1. Dualizing the second-stage problem The inner maximization problem of model (7) can be formulated as:

max f

D

1 (ss ,

) df ( ),

(8) 8

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s. t. Constraints (6

1)

(6

3).

By implementing the standard Lagrange dual for the moment constraints (6–1)–(6–3), we can obtain its equivalent model (9) (refer to Appendix A for more details): Tµ

min + , ,Q

T

+

TQ

+

µTQµ ,

+ Q·

(9-1)

2 TQµ

1 (ss ,

),

(9-2)

,

H H×H where ‘·’ refers to the Frobenius inner products between matrices, θ ,ρ ,Q are the dual multipliers of constraints (61)–(6-3), respectively. Replacing (8) by (9), the original DRO-BSN model (7) can be equivalently reformulated as:

min

ss, w p, , , Q

T

eih wihp +

rh ssh + h H

+

Tµ

µT Qµ ,

+ Q·

(10)

i I h H

s. t. Constraints (1

2)

(1

5), (9

2).

Such a model remains intractable due to constraints (9-2). Its difficulties lie in two points. First, constraints (9-2) are semi-infinite due to the random demand . Second, there is a MILP problem 1 (ss, ) included in (9-2). We address these two issues in the next two sections respectively. 4.2. Formulation of the bilevel SDP In this section, we focus on eliminating in constraints (9-2) to obtain the corresponding deterministic model. First, for 1 (ss, ) , note that exists in its equality constraints (2-7). Thus, we move from the constraints to the objective and reformulate 1 (ss, ) as (11-1): 2 (ss ,

): =min

c C

fcc yc +

i I

+

h H

c C

eih wihr +

opc qc + hc (

c C i I

ech z ch +

h H

spi +

h H

h H

ovh) + uc

e v h1 H & h1 h hh1 hh1 + h H h

(11-1)

with + + h

=

ssh

h

wihr + ovh +

z ch c C

i I

vhh1 h1 H & h1 h

+

v h1 h

,h

H.

(11-2)

h1 H & h1 h

r

z ch , spi, ovh, v hh1) (c ∈ C, i ∈ I, Here, (a) = max{0, a}. For simplicity, Δ=(y, tg, q, w , z, sp, ov,v) is defined to replace h,h1 ∈ H & h1 ≠ h) and its feasible region Λ in (11-1) includes constraints (2-2)–(2-6), (2-8) and (2-9). Next, to simplify our reformulation, we ignore the transshipment cost in the second stage. The assumption is reasonable for two reasons. First, the unit shortage cost usually is far larger than the transshipment cost due to the importance of human lives. Thus, when the shortage incurs in several TRs while other TRs have surplus blood, the transshipment would be certainly implemented to reduce the shortage. Hence, this assumption is feasible in practice. Furthermore, compared with the long-term prepositioning cost in the preparedness stage, the impact caused by the transshipment cost on the here-and-now decisions, i.e., the prepositioning inventory, could be ignored. Therefore, the mild assumption does not sacrifice much modeling power. Thus, we can further reformulate (11-1) depending on whether the shortage occurs in the whole BSN or not, respectively: (i) When the blood shortage occurs in the BSN, 2 (ss, ) is equivalent to formula (12): uc

h

+

(yc, tgc, qc, wihr ,

2,1 (ss, 1),

(12)

h H

in which, 2,1 (ss, 1):

= min

c C

fcc yc + + hc

1

i I

i I

h H

eih wihr +

spi + uc [

h H

c C

(ssh +

h H c C

ech z ch +

z ch +

i I

c C

wihr)]

opc qc

,

(13)

where the shortage amount is the difference between the total demand and the supply in the BSN. (ii) When the total supply is sufficient to cover the emergency demand, the shortage amount is zero and the wastage may occur. Thus, 2 (ss, ) is equivalent to 2,2 (ss, 2) : 2,2 (ss, 2):

= min 2

c C

fcc yc + +

c C

i I

h H

eih wihr +

opc qc + hc (

h H

c C

ovh +

h H i I

spi )

ech z ch

.

(14)

Δ1 and Δ2 in formulas (13) and (14) are introduced to specify the decision sets of the optimization problems in the cases (i) and (ii) respectively. Therefore, constraints (9-2) hold when (15-1) and (15-2) hold simultaneously: 9

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+

T

+

TQ

2 TQµ

+

T

+

TQ

2 TQµ

uc· IHT

2,1 (ss ,

2,2 (ss ,

2),

1),

ifIHT

(IHT ss + ECH · z + EIH · w r )

0

(15-1) (15-2)

otherwise,

H C×H I×H in which, IH , ECH and EIH are the matrix of all ones. Notice that (15-1) and (15-2) are conditional semi-infinite constraints which hold under the specified range of the random parameter . Generally, S-Lemma (Delage and Ye, 2010; Yakubovich, 1977) can be used. However, for our case, it would result in a set of nonlinear constraints. For tractability, we relax constraints (15) by allowing take arbitrary value in the whole support set as below:

+

T

+

TQ

2 TQµ

+

T

+

TQ

2 TQµ

uc· IHT

2,1 (ss ,

2,2 (ss ,

(16-1)

1),

(16-2)

2),

Obviously, constraints (16), which must be satisfied over the whole , are much stricter than (15). Thus, the corresponding solution would be relatively conservative. Then, for standard semi-infinite constraints (16), Schur’s complement (Boyd and Vandenberghe, 2004; Delage and Ye, 2010) can equivalently transform them into semidefinite matrix constraints (17-1) and (17-2) respectively: 1 ( 2

Q 1 ( 2

2Qµ

2Qµ )T

uc· I )

2,1 (ss ,

1 ( 2

Q 1 ( 2

2Qµ

uc·I )T

2Qµ ) 2,2 (ss ,

0,

1)

(17-1)

0.

2)

(17-2)

Thus, the problem (10) is turned into an approximate but deterministic SDP model as problem (18):

min

ss, w p, , , Q

T

eih wihp +

rh ssh + h H

Tµ

+ Q·

1), (17

2).

+

µT Qµ ,

i I h H

s. t. Constraints (1

2)

(1

5), (17

(18)

Note 2,1 (ss, 1) and 2,2 (ss, 2) in constraints (17) are MILP. Thus, this intermediate model has a bilevel structure consisting of a continuous upper-level problem and a mixed integer lower-level problem that is controlled by the upper-level variable ss. 4.3. Transforming the bilevel SDP into the single-level SDP The combinatorial nature of bilevel programming makes it more difficult to solve compared with single-level programming, especially when integer variables exist in the lower-level of the model (Talbi, 2013). To address this difficulty, in this paper, by utilizing the convex hull construction method (Sherali and Adams, 1994), we first transform the lower-level MILP problems 2,1 (ss , 1) and 2,2 (ss , 2) into equivalent linear programming. Then, the two-level model can be easily simplified to a single-level SDP model by employing the KKT conditions. 4.3.1. Linearization of discrete variables For an integer programming, if the feasible region (Ʊ) contains all the vertices of its corresponding vertex polyhedral convex hull (conv(Ʊ)), then the optimal integer solution is equivalent to its linear programming relaxation (Wolsey, 1998). Inspired by this idea, we could linearize our discrete lower-level problem by utilizing the reformulation/linearization method proposed by Sherali and Adams (1994). Thus, a polyhedral convex hull conv(Ʊ) is constructed to equivalently replace the original feasible region. We focus on linearizing the binary constraints (2-2), (2-3) and (2-5). Taking (2-2) as an example, the complexity of the polyhedral convex envelope largely depends on the column of constraints. With the increase of |H|, the column of constraints (2-2) would become larger which would make the linearization more difficult. Thus, to reduce the column of constraints (2-2), we provide their equivalent formulas (19-1) and (19-2):

yc

tgch

0, c

C; h

H

1

tgch1

tgch2

0, c

C ; h1, h2

(19-1)

H &h1

h2

(19-2)

in which, constraints (19-1) indicate that if TCF c is assigned to provide blood for TR h (tgch = 1), then the TCF c must be activated (yc = 1). Besides, for a candidate TCF c*, when it is not assigned for any TR h (tgc*h = 0, h H ), yc* must be zero because its setup cost can be saved. Thus, we have the statement:

if

tgc *h = 0, h

H , then y c * = 0, c * C

holds, then its contrapositive statement: 10

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C. Wang and S. Chen

if y c * = 1, then

tgc *h = 1, h

H , c*

C

also holds. It can be represented as constraints (19-2) which restrict that one TCF c (if yc = 1) can only be assigned to one TR. Thus, we can use two 2-column constraints (19-1) and (19-2) to equivalently replace constraints (2-2). Furthermore, to formulate the convex hull representation of constraints (19), constraints (20) are first introduced to ensure the variables yc and tgch in (19) are 0–1 integers.

yc2 = yc , c

(20-1)

C

2 tgch = tgch, c

C; h

(20-2)

H.

Then, we introduce the corresponding polynomial factors: yctgch, yc(1–tgch), tgch(1–yc) and (1–yc)(1–tgch). Multiplying constraints (19-1) with these polynomial factors and further linearizing the squared terms by equations (20), we have

yc tgch yc

yc tgch

yc tgch

yc tgch yc

0, c

C; h

yc tgch + yc tgch

yc tgch

(21-1)

H 0, c

C; h

tgch + yc tgch

0, c

C; h

tgch + tgch

yc tgch

0, c

yc + yc tgch

(21-2)

H

(21-3)

H C; h

(21-4)

H

Note that constraints (21-1) and (21-4) always hold. Hence, only constraints (21-2) and (21-3) remain which can be further simplified as:

yc

yc tgch

0, c

tgch + yc tgch

C; h 0, c

(22-1)

H

C; h

(22-2)

H

Introducing an auxiliary variable αch to replace the bilinear term yctgch, we have

yc

0, c

ch

tgch +

C; h 0, c

ch

(23-1)

H

C; h

(23-2)

H.

Besides, the following additional constraints should be satisfied: ch

0, c

tgch

ch

yc 1

yc

H

0, c

C; h

0, c

ch

(24-1)

C; h

tgch +

C; h

(24-3)

H

0, c

ch

(24-2)

H

C; h

(24-4)

H

Note that constraints (23-2) and (24-2) restrict that αch = tgch. We finally obtain the convex hull (25), which is a linear equivalent of the original 0–1 integer constraints (19-1).

yc

tgch

0, c

C; h

H

(25-1)

0

tgch

1, c

C; h

H

(25-2)

0

yc

1, c

(25-3)

C

The same procedure can be applied to constraints (19-2), in which we introduce the second degree polynomial factors: tgch1 tgch2 , tgch1 (1-tgch2 ), tgch2 (1-tgch1) , (1-tgch1 )(1-tgch2 ) . Finally, the equivalent reformulation of constraints (19-2) can be obtained as below:

1

tgch1

tgch2

0

tgch

1, c

0, c

C ; h1, h2

C; h

H &h1

(26-1)

h2

(26-2)

H.

Similarly, constraints (2-3) and (2-5) can be transformed into their corresponding continuous convex hulls, shown as constraint sets (27) and (28), respectively (see Appendix B for transformation procedure).

yc capc2

qc

0, c

C

(27-1)

yc UBc

qc

0, c

C

(27-2)

UBc

qc

yc UBc

0, c

MM · tgch

0, c

(27-3)

C

(27-4)

C z ch

0, c

C; h

(28-1)

H 11

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0

z ch

tgch

0, c

1, c

C; h

C; h

(28-2)

H

(28-3)

H

Therefore, based on the above procedure, we can equivalently transform MILP problems responding linear programs with constraints (25)–(28).

2,1 (ss, 1)

and

2,2 (ss, 2)

into the cor-

4.3.2. Reformulating the lower-level problem with the KKT conditions After the above reformulation, the lower-level models are now linear w.r.t. their variables, and thus can be equivalently replaced by their KKT conditions. Specifically, see Appendix C, constraints (41-1)–(41-22), (41-24), (42-1)–(42-2) can be used to replace 2,1 (ss, 1) and constraints (43-1)–(43-27) can be used to replace 2,2 (ss, 2) respectively. Combining these constraints with the upperlevel problem, i.e., the problem (18), we have the single-level mixed-integer SDP, which is amenable to the off-the-shelf solvers, such as MOSEK and SEDUMI, based on the interior point algorithm. 5. Computational study The computational study is based on the Longmenshan Fault in China where has suffered from seven serious earthquakes since 1933, resulting in the huge loss of property and human lives. The historical data on hand is very limited. To confirm the applicability of the proposed DRO model and approach, the benchmarks are introduced in Section 5.1, including (i) the deterministic model (denoted as det), (ii) the two-stage stochastic programming-BSN model (denoted as SP-BSN) and (iii) the two-stage robust optimization-BSN model (denoted as RO-BSN). Section 5.2 implements the case study by using the real earthquake data from the Longmenshan Fault in China. Based on the mean and covariance information obtained from the historical data, we implement Monte Carlo simulation to randomly generate the out-of-sample datasets to test the performance of these models. In Section 5.3, sensitivity analysis is conducted and finally the corresponding managerial insights are revealed in Section 5.4. 5.1. Benchmark models First, the deterministic model can be simply obtained by replacing the random demand with the empirical value. The following two subsections focus on SP-BSN and RO-BSN respectively. 5.1.1. Two-stage SP-BSN model We consider the typical prior distributions to sample the emergency demand realizations based on the mean and variance obtained from the historical data. The sample average approximation (SAA) method (Kleywegt et al., 2002) is employed to formulate the SP-BSN model. In SAA, Monte Carlo simulation is implemented to randomly sample demand realization hs following the predefined distributions. Here, three kinds of distributions: the uniform, normal, and gamma distribution, are considered by using the parameters listed in Table 2. And then, three corresponding SP models, denoted as sp(uni), sp(norm) and sp(gam), can be obtained respectively. In each SP model, let Ξ be the set of demand scenarios, and each scenario is indexed by s with the probability of 1/|Ξ|. The detailed formulation of the two-stage SP-BSN model is left to Appendix D.1. 5.1.2. Two-stage RO-BSN model We consider two kinds of RO models based on min-max regret (denoted as ro(s)) and budget uncertainty (denoted as ro(box)). (i) RO-BSN based on min-max regret

¯ s, w ¯ s, p) and the obBased on the optimization objective, each scenario s(s ∈ Φ) has a corresponding optimal decision Xs* = (ss jective value Os* . Thus, we have |Φ| kinds of optimal decisions. If the decision Xs* is applied to another scenario s1(∈Φ&s1 ≠ s), there exists a corresponding objective value Os,s1. Thus, Os,s1 Os* represents the regret value of applying decision Xs* to scenario s1. The DMs should compare the “maximum regret value” of each decision and choose the one that has the minimum “maximum regret value”, Table 2 The input parameters for demand sample realization in each distribution. Type of distribution

Input parameter

Uniform distribution

lower bound max {0, µh 3 mean µh shape

Normal distribution Gamma distribution

µh2

* µh and

h

2 h

h}

upper bound µh + 3 h standard deviation h

scale 2 h µh

are mean demand and standard deviation of TR h obtained from historical data. 12

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which can be evaluated by formula (29):

minmax(Os, s1

Os1 ).

Xs s1

(29)

Refer to Appendix D.2 for the detailed formulation. (ii) RO-BSN based on budget uncertainty The second robust benchmark, which is based on box uncertainty set, aims at minimizing the total cost of the two-stage under the worst-case blood demand. The set of the emergency demand in TR h can be represented as h [µ^h uh ^h, µ^h + uh ^h], in which µh , h , and uh denote the nominal demand, the maximum deviation from the nominal demand and the degree of deviation in TR h, respectively. Besides, the budget parameter Г proposed by Bertsimas and Sim (2004) is introduced here to adjust the level of protection that takes the value in the interval [0, |H|]. That is, Г = 0 represents that the TRs have the nominal demands, while Г = |H| denotes the worst-case. When Г ∈ (0, |H|), the DMs can make a trade-off between robustness and optimization. Thus, we define the box uncertainty set Ω1 = { h H uh , 0 uh 1}. The two-stage RO-BSN based on the budget of demand uncertainty can be represented as:

min T

ss, w p

eih wihp + max 1 (ss , )

rh ssh + h H

i I h H

s. t. Constraints (1

2)

ssh +

(1

wihr + osh

z ch + c C

(30-1)

1

5), (2

1)

ovh

i I

(2

6), (2

8)

vhh1 + h1 H & h1 h

(2

9)

v h1 h = µh + uh h, h

H.

(30-2)

h1 H & h1 h

Note that the transshipment cost is ignored in the DRO model. Hence, we also exclude this cost item in these benchmarks to make the comparison sensible. 5.2. Case study 5.2.1. Data The experiments in this study are based on the earthquakes in the Longmenshan Fault in China. Before calculation, we give the parameters in the preparedness stage and the response stage at first. (i) The parameters in the preparedness stage In the Longmenshan Fault, the TRs in Beichuan, Pingwu, Wenchuan, and Lushan are considered as four demand nodes. The population densities of the four districts covered by the TRs are (77, 30, 27, 88)/km2 (see http://data.stats.gov.cn). Considering the population and distances to the TRs, four PCFs are correspondingly chosen as the daily supply nodes in the BSN, which are in Mianyang, Jiangyou, Dujiangyan and Ya’an. Based on the distances obtained from Google Maps, the unit transportation fee (eih) between each PCF and TR is displayed in Table 3. To provide management insights for blood perishability, we consider two kinds of blood products with different shelf lives, i.e., frozen plasma and red blood cells, which can be calculated by our way respectively. From “The Ministry of Health of the People’s Republic of China (2012)” information, the lifespans of frozen plasma and red blood cells are 365 and 21 days respectively. In the first stage, the replenishment cycle is 1 week, and the planning horizon of decisions is 10 years (i.e., T ≈ 520 cycles). In accordance with the donation rate in China (Liang et al., 2016), the population in the PCFs and the blood collection data (Deng et al., 2011), the blood supply of each PCF per cycle is shown in Table 4. It also displays the daily demand of two blood products in each TR per cycle (Wang et al., 2016). Considering the potential earthquakes in this fault, the inventory is prepositioned in each TR. Referring to Jabbarzadeh et al. (2014), the unit prepositioning cost (rh) in TRs is set to be 0.26 RMB/(U cycle). The capacities of frozen plasma and red blood cells in each TR are 1600 U and 2500 U respectively. Besides, during the preparedness stage, five TCF candidates are considered in the Longmenshan Fault. It should be noted that the transportation time between each TR and each blood collection node, including PCF Table 3 The transportation fee (eih) between the PCFs and the TRs (RMB/U). TR h PCF i

Beichuan

Pingwu

Wenchuan

Lushan

Mianyang Jiangyou Dujiangyan Ya’an

6.23 5.74 20.36 30.83

16.02 11.99 28.44 41.22

22.48 26.87 2.44 17.82

29.17 33.54 18.99 3.54

13

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Table 4 The blood supply of PCF i (si) and the blood demand of TR h (dh) (U/cycle). PCF i

Mianyang

Jiangyou

Dujiangyan

Ya’an

Frozen plasma Red blood cells

1266.11 1202.84

1721.95 1635.89

3021.77 2870.75

535.78 509

TR h

Beichuan

Pingwu

Wenchuan

Lushan

Frozen plasma Red blood cells

614.4 778.3

242.45 307.1

218.5 276.8

699.15 885.6

and TCF, in our case is less than 6 h which meets the requirements of frozen fresh plasma production. Besides, the capacity of each TCF is set to be 1500 U. Table 5 shows the unit transportation fee from the TCFs to the TRs. (ii) The parameters in the response stage For each selected TCF, we consider two types of cost: the TCFs’ setup cost and the unit operational cost. The former is 2000 (RMB) for each TCF, mainly including the labor cost, depreciation expenses, etc. The latter is 198.57 (RMB/U), containing the costs of collecting materials, blood tests, packages, etc.. Moreover, the upper limit of blood donation in each TCF is displayed in Table 6, which can be estimated as below. First, each TCF is assumed to be responsible for the blood collection service within an approximate 2-kilometer vicinity. Taking the districts’ population density into account, the number of people covered by each TCF is shown in Table 6. In addition, according to Yan et al. (2008), the post-disaster donation rate is on average 2.96 times that of the daily donation rate (0.94% mentioned in Liang et al., 2016), and we assume that each donor donates 1 U of blood. Moreover, an average deferral rate of 13% is considered, indicating the proportion of donors who are rejected to give blood due to various medical reasons such as high/low blood pressure, colds and anemia. Furthermore, we list the historical data, including the injured number, of the seven earthquakes in Table 7 (Chen, 2015; Sun, 2010). The emergency demand in each TR can be calculated by the following three steps. First, we consider the blood transfusion rate (BTR), which estimates the people who need to be transfused among all the injured. The RTR is set to be 0.12085 here (Lin et al., 2010). Thus, we can obtain the numbers of transfused people (denoted as No.Tran) as shown in Table 7. Second, we sort the historical transfusion data and No.Tran in the recent earthquakes that occurred in the Longmenshan Fault (Lin et al., 2010) to estimate the average blood demand per person. With the No.Tran and the average demand per person, the total emergency blood demand can be obtained. Finally, due to the lack of detailed information on the emergency blood demand in each TR, we divide the total demand to each TR in accordance with the population density of the district each TR lies in. Thus, we have Table 8, which shows the demand for two blood products in each TR in the seven historical earthquakes. Based on formulas (4) and (5), the empirical mean (μ) and covariance (Σ) of two blood products’ emergency demand in each TR can be given in Tables 9 and 10. The unit shortage cost (uc) and the unit surplus cost (hc) are set to be 657 RMB/U and 361.35 RMB/ U, respectively. Finally, we summarize the data used by the benchmark models and the datasets for performance testing. Table 11 displays the demand-related parameters of DRO-BSN (denoted as dro later) and the benchmarks. Specifically, µh and Σ for dro are shown in Tables 9 and 10 respectively, and the standard deviation ( h ) used in SP-BSN is shown in Table 12. For each probability distribution used in the SP-BSN model, we generate 50 demand scenarios, and each scenario has a probability ^ of 1/50. For ro(s), mh is displayed in Table 8. For ro(box), the mean µh is shown in Table 9, and the deviation h (= hmax µ^h ) is the difference between the worst historical demand and the mean. hmax for frozen plasma and red blood cells are given in Table 8. Similar to Rahmani (2018) and Ramezanian and Behboodi (2017), we give the degree of deviation uh in ro(box) directly. Specifically, we set uh = 0.5 for all h H . Thus, the budget parameter Г is 2. For the deterministic model, we only consider the nominal demand µh . Other parameters are the same in all the models. Furthermore, we construct the so-called out-of-sample datasets to test the performance of the decisions generated from all models. Thus, the random demand datasets, which follow the moments given in Tables 9, 10 and 12 in the TRs, are generated by Monte Carlo Table 5 The transportation fee from the TCFs to the TRs (RMB/U). TR h TCF c

1

2

3

4

1 2 3 4 5

20.83 14.29 24.68 7.27 16.38

31.22 24.68 35.32 18.97 11.43

7.84 7.42 11.67 12.57 38.17

14.29 19.17 9.91 24.7 49.92

14

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Table 6 Population and upper limit of blood donation in TCFs. TCF

No. People covered

Donation(U)

1 2 3 4 5

4807 5821 4736 3651 1539

116.359 140.898 114.643 88.389 37.246

Table 7 Earthquakes in the Longmenshan Fault from 1933 to the present. Time

Magnitude

No. injured

1933.08.25 1976.08.16 2001.02.23 2008.05.12 2008.08.01 2013.04.20 2014.11.22

7.5 7.2 6.0 8.0 6.1 7.0 6.3

20,000 756 154 375,783 347 13,019 78

Table 8 ^ The blood demand (U) of each TR ( mh ) in the historical earthquakes in the Longmenshan Fault. Historical earthquake

1933.08.25 1976.08.16 2001.02.23 2008.05.12 2008.08.01 2013.04.20 2014.11.22

Frozen plasma

Red blood cells

TR 1

TR 2

TR 3

TR 4

TR 1

TR 2

TR 3

TR 4

1170.77 44.26 9.01 21997.85 20.31 762.12 4.57

462.00 17.46 3.56 8680.52 8.02 300.74 1.80

416.38 15.74 3.21 7823.43 7.22 271.04 1.62

1332.24 50.36 10.26 25031.57 23.11 867.22 5.20

1571.76 59.41 12.10 29532.00 27.27 1023.14 6.13

620.23 23.44 4.78 11653.56 10.76 403.74 2.42

558.99 21.13 4.30 10502.91 9.70 363.87 2.18

1788.52 67.61 13.77 33604.76 31.03 1164.24 6.98

Table 9 The mean emergency blood demand in each TR (µh ) (U). TR

Frozen plasma

Red blood cells

1 2 3 4

3429.84 1353.44 1219.81 3902.85

4604.54 1816.99 1637.58 5239.56

Table 10 The covariance (Σ) of emergency blood demand in four TRs. Frozen plasma

6.73 2.65 2.39 7.65

2.65 2.39 7.65 1.05 0.94 3.02 0.94 0.85 2.72 3.02 2.72 8.71

Red blood cells

1.21 0.48 0.43 1.38

× 107

0.48 0.43 1.38 0.19 0.17 0.54 0.17 0.15 0.49 0.54 0.49 1.57

× 108

Table 11 The demand-related parameters of all models. Models

dro

SP-BSN

Demand-relatedparameters

µh , Σ

µh ,

h

15

ro(s) mh

ro(box)

det

µh ,

µh

hmax

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Table 12 The standard deviation ( h ) (U) of two blood products’ emergency demand in TRs. Blood product

TR 1

TR 2

TR 3

TR 4

Frozen plasma Red blood cells

8200.738 11009.45

3236.076 4344.416

2916.552 3915.457

9331.701 12527.76

simulation using MATLAB. During the generation, three distributions, i.e., the normal, uniform, and gamma distribution, are considered, denoted as Set 1, Set 2 and Set 3, respectively. Each dataset contains 5000 demand samples, totaling 15,000 random samples, to represent all the demand realizations in disasters. If any sampled demand value is less than 0, we resample it. 5.2.2. Computational results We calculate all the models in MATLAB on a PC with a CPU at 2.5 GHz with 8 GB memory, using the toolbox YALMIP with solver MOSEK 8.1 (the SDP module for DRO and the LP/QP module for the benchmark models) to obtain the optimal solutions with the default gap tolerance (1 0 −4). For each model, the total cost (in_Total), the prepositioning cost (in_SC), the second-stage cost (in_2nd) ^ of two blood products are given in Table 13. as well as the prepositioning decision ss The results show that the two RO models present relatively conservative prepositioning decisions, resulting in the high total costs. The decisions and the costs of the three SP models are significantly distinguished from each other. It means that the performance of the SP model heavily depends on the distributions, which is, however, not easy to identify from the scarce data. det has the lowest total cost since it assumes the disaster occurs as expected, which means a slight shortage (in_2nd). dro is positioned in between all models in terms of both the prepositioning inventory level and the cost. However, since the calculations above are based on the in-sample data, it is important to evaluate how these decisions, especially the prepositioning inventory in the first stage, perform under the disasters described by the out-of-sample datasets. Hence, we ^ of dro and the benchmarks under each of the 15,000 optimize the relief activities in the second stage with the prepositioning level ss out-of-sample data. In each dataset, the calculation results of 5000 out-of-sample data are first averaged to obtain the expected out-of-sample costs (i.e., the total cost (out_2nd), the shortage cost (out_uc), the operational cost (out_op), the setup cost (out_st), the transportation cost (out_trans) and the surplus cost (out_hc)). Then, in order to confirm the models’ ability to estimate the loss caused by the disasters, refer to Nakao et al. (2017), the estimation gap between the in_2nd and the out_2nd is calculated by [(out_2nd) –(in_2nd)]/(in_2nd) (denoted as (O–I)/I). For frozen plasma and red blood cells, the expected costs and the estimation gap of each model among each dataset are shown in Tables 14 and 15, respectively. Furthermore, notice that the three datasets constitute the whole out-of-sample set, and the comprehensive estimation performance of each model is displayed in Table 16, in which the absolute value of (O–I)/I among the three distribution sets are averaged. For example, regarding frozen plasma, the (O–I)/I of dro in the three distribution sets is –29.04%, –17.54% and –41.89%. Thus, the expected estimation gap of dro is

1 ( 3

29.04% +

17.54% +

41.89% ) = 29.49%

It is obvious that the smaller the gap is, the better the estimation performance is. Thus, in accordance with Tables 14–16, dro dominates other models (with the smallest gap of 29.49% of frozen plasma and 25.42% of red blood cells). Specifically, det is too optimistic and causes the largest average deviation, although it always results in the lowest total cost. Besides, the deviations of two RO models are negative because of their higher in-sample total costs compared with other models. Furthermore, the estimation gaps of Table 13 The computational results of dro and the benchmarks. Blood products

Frozen plasma

Red blood cells

Model

dro sp(uni) sp(norm) sp(gam) det ro(s) ro(box) dro sp(uni) sp(norm) sp(gam) det ro(s) ro(box)

in_Total (million)

18.62 12.61 20.50 12.54 6.70 41.06 20.24 26.34 19.29 19.87 16.53 11.44 58.75 30.79

in_SC (× 104)

in_2nd (million)

50.8 62.57 62.57 0 45.45 62.57 62.57 57.23 58.69 58.69 0 58.69 58.69 58.69

11.89 5.76 13.65 6.32 0.027 34.22 13.39 16.53 9.46 10.04 7.29 1.61 48.91 20.96

16

ss (U) TR 1

TR 2

TR 3

TR 4

760.16 985.6 985.6 0 985.6 985.6 985.6 1526.4 1556.6 1556.6 0 1556.6 1556.6 1556.6

1136.48 1357.55 1357.55 0 92.38 1357.55 1357.55 589.7 614.2 614.2 0 614.2 614.2 614.2

1159.94 1381.5 1381.5 0 1381.5 1381.5 1381.5 529.8 553.6 553.6 0 553.6 553.6 553.6

699.22 900.85 900.85 0 900.85 900.85 900.85 1585.12 1614.4 1614.4 0 1614.4 1614.4 1614.4

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Table 14 The comparison of the second-stage’s costs and the estimation gap of each model in out-of-samples (frozen plasma). Set

Set 1

Set 2

Set 3

Model

dro sp(uni) sp(norm) sp(gam) det ro(s) ro(box) dro sp(uni) sp(norm) sp(gam) det ro(s) ro(box) dro sp(uni) sp(norm) sp(gam) det ro(s) ro(box)

in_2nd (million)

out_2nd (million)

out_uc (million)

out_op (× 104)

out_st

11.89 5.76 13.65 6.32 0.027 34.22 13.39 11.89 5.76 13.65 6.32 0.027 34.22 13.39 11.89 5.76 13.65 6.32 0.027 34.22 13.39

8.44 8.13 8.13 10.03 8.59 8.13 8.13 9.81 9.44 9.44 11.58 9.98 9.44 9.44 6.91 6.98 6.98 6.75 6.88 6.98 6.98

7.82 7.42 7.42 9.71 8.01 7.42 7.42 9.28 8.85 8.85 11.28 9.48 8.85 8.85 4.74 4.59 4.59 5. 5 4.81 4.59 4.59

7.08 6.84 6.84 8.09 7.19 6.84 6.84 7.08 7.46 7.46 8.45 7.74 7.46 7.46 2.75 2.63 2.63 3.46 2.83 2.63 2.63

7169.2 6923.6 6923.6 8190.79 7281.2 6923.6 6923.6 7762 7548.39 7548.39 8556 7838 7548.39 7548.39 2786 2660.4 2660.4 3498.4 2862 2660.4 2660.4

out_trans (× 104)

out_hc (× 104)

os(U)

(O–I)/I

1

2

3

4

2.56 2.48 2.48 2.88 2.59 2.48 2.48 2.69 2.64 2.64 2.96 2.73 2.64 2.64 1.06 1.01 1.01 1.43 1.09 1.01 1.01

51.27 60.34 60.34 20.94 47.41 60.34 60.34 41.20 48.38 48.38 16.92 38.12 48.38 48.38 212.67 235.42 235.42 119.76 202.48 235.42 235.42

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

11907.4 11300.8 11300.8 14773.0 12190.3 11300.8 11300.8 14,130 13469.4 13469.4 17174.2 14436.6 13469.4 13469.4 7217.8 6984 6984 8367.2 7327.6 6984 6984

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

–29.04% 41.14% –40.47% 58.85% 31659.42% –76.25% –39.29% –17.54% 63.94% –30.85% 83.26% 36800.84% –72.41% –29.49% –41.89% 21.23% –48.87% 6.82% 25345.91% –79.60% –47.86%

Table 15 The comparison of the second-stage’s costs and the estimation gap of each model in out-of-samples (red blood cells). Set

Set 1

Set 2

Set 3

Model

dro sp(uni) sp(norm) sp(gam) det ro(s) ro(box) dro sp(uni) sp(norm) sp(gam) det ro(s) ro(box) dro sp(uni) sp(norm) sp(gam) det ro(s) ro(box)

in_2nd (million)

out_2nd (million)

out_uc (million)

out_op (× 104)

out_st

16.53 9.46 10.04 7.29 1.61 48.91 20.96 16.53 9.46 10.05 7.3 1.64 48.91 20.96 16.53 9.46 10.05 7.3 1.64 48.91 20.96

13.05 13.01 13.01 15.11 13.01 13.01 13.01 15.05 14.99 14.99 17.27 14.99 14.99 14.99 8.87 8.88 8.88 8.83 8.88 8.88 8.88

12.55 12.49 12.49 14.85 12.50 12.50 12.50 14.64 14.58 14.58 17.04 14.58 14.58 14.58 6.76 6.74 6.74 7.70 6.74 6.74 6.74

7.82 7.80 7.80 8.63 7.80 7.80 7.80 8.25 8.23 8.23 8.91 8.23 8.23 8.23 3.03 3.01 3.01 3.79 3.01 3.01 3.01

7916 7895.6 7895.6 8736.4 7895.6 7895.6 7895.6 8348.4 8331.6 8331.6 9020.4 8331.6 8331.6 8331.6 3065.2 3046.4 3046.4 3838.4 3046.4 3046.4 3046.4

out_trans (× 104)

out_hc (× 104)

os(U) 1

2

3

4

2.16 2.16 2.16 2.38 2.16 2.16 2.16 2.26 2.25 2.25 2.43 2.25 2.25 2.25 0.91 0.91 0.91 1.25 0.91 0.91 0.91

38.52 39.31 39.31 13.45 39.31 39.31 39.31 29.75 30.38 30.38 10.35 30.38 30.38 30.38 206.86 209.55 209.55 106.99 209.55 209.55 209.55

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

19107.8 19023.1 19023.1 22604.4 19023.1 19023.1 19023.1 22277.3 22187.8 22187.8 25937.9 22187.8 22187.8 22187.8 10290.4 10257.9 10257.9 11719.5 10257.9 10257.9 10257.9

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

Table 16 The expected estimation performance of each model. Model

Frozen plasma

Red blood cells

dro sp(uni) sp(norm) sp(gam) det ro(s) ro(box)

29.49% 42.11% 40.07% 49.64% 31268.72% 76.08% 38.88%

25.42% 34.10% 30.14% 88.49% 662.23% 74.87% 41.33%

17

(O–I)/I

–21.03% 37.55% 29.48% 107.33% 706.23% –73.42% –37.91% –8.92% 58.66% 49.35% 137.01% 829.97% –69.34% –28.39% –46.31% –6.08% –11.59% 21.11% 450.50% –81.85% –57.61%

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Table 17a The expected estimation gap of each model under varying rh (frozen plasma) (%). Model

The change ratio of the unit inventory cost rh 0.2

0.4

0.6

0.8

1

1.2

1.4

dro sp(uni) sp(norm) sp(gam) det ro(s) ro(box)

30.58 42.11 40.07 48.69 31268.72 76.08 38.88

30.58 42.11 40.07 49.64 31268.72 76.08 38.88

30.58 42.11 40.07 49.64 31268.72 76.08 38.88

30.58 42.11 40.07 49.64 31268.72 76.08 38.88

29.49 42.11 40.07 49.64 31268.72 76.08 38.88

27.87 42.11 40.07 49.64 31268.72 300.41 38.88

27.01 42.11 40.07 49.64 31268.72 300.41 38.88

dro sp(uni) sp(norm) sp(gam) det ro(s) ro(box)

1.6 24.84 42.11 40.07 49.64 31268.72 300.41 38.88

1.8 24.75 42.11 40.07 49.64 6506.77 300.41 38.88

2 24.76 42.11 40.07 49.64 6050.24 300.41 38.88

2.2 24.76 42.11 40.07 49.64 6050.24 300.41 38.88

2.4 24.76 42.11 40.07 49.64 6050.24 300.41 38.88

2.6 24.76 42.76 40.07 49.64 6050.24 300.41 38.88

three SP models are not stable due to different distribution functions, and all are worse than that of dro. 5.3. Results of sensitivity analysis We conduct a sensitivity analysis by varying the unit inventory prepositioning cost, the unit shortage cost, and the replenishment cycle. 5.3.1. Sensitivity to unit prepositioning cost (rh) We set rh as 0.2, 0.4, …, 2.6 times the default cost for two blood products. First, we calculate the prepositioning results of each model under different rh and then obtain the estimation gap (see Table 17a for frozen plasma and Table 17b for red blood cells). As seen, dro always shows the best estimation gap among all the models under any rh. Furthermore, it is found that the estimation gap of dro decreases as rh increases. This is because the growth of rh would result in a smaller prepositioning inventory. Thus, the negative deviation between the out_2nd and the in_2nd decreases, which causes a declining estimation gap. Moreover, det yields the worst results since it only considers the mean emergency demand, and thus leads to the small in_2nd and the large (O-I)/I. Next, we further observe the prepositioning decisions of dro for frozen plasma (see Fig. 3a) and red blood cells (see Fig. 3b). It is obvious that with the increase of rh, the prepositioning inventory in each TR shows a downward trend. Taking frozen plasma as an example, when rh varies from 0.2 to 0.8 times, no significant change could be observed. However, when rh exceeds 0.8, the inventory becomes extremely sensitive to rh and decreases significantly to avoid the overly high inventory cost. Finally, when rh is large enough, the inventory is close to zero. Table 17b The expected estimation gap of each model under varying rh (red blood cells) (%). Model

The change ratio of the unit inventory cost rh 0.2

0.4

0.6

0.8

1

1.2

1.4

dro sp(uni) sp(norm) sp(gam) det ro(s) ro(box)

25.71 34.10 30.14 86.39 662.23 74.87 41.33

25.68 34.10 30.14 88.49 662.23 74.87 41.33

25.23 34.10 30.14 88.49 662.23 74.87 41.33

25.77 34.10 30.14 88.49 662.23 74.87 41.33

25.42 34.10 30.14 88.49 662.23 74.87 41.33

25.65 34.10 30.14 88.49 662.23 513.01 41.33

25.70 34.10 30.14 88.49 662.23 513.01 41.33

dro sp(uni) sp(norm) sp(gam) det ro(s) ro(box)

1.6 23.85 34.10 30.14 88.49 662.23 513.01 41.33

1.8 22.42 34.10 30.14 88.49 662.23 513.01 41.33

2 21.62 34.10 30.14 88.49 662.23 513.01 41.33

2.2 21.57 34.10 30.14 88.49 662.23 513.01 41.33

2.4 22.03 34.10 30.14 88.49 662.23 513.01 41.33

2.6 21.60 34.10 30.14 88.49 662.23 71.90 41.33

18

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a. The prepositioning inventory of frozen plasma

b. The prepositioning inventory of red blood cells

Fig. 3. The prepositioning inventory of dro under varying rh.

Moreover, we observe the prepositioning cost (in_SC), the second-stage cost (in_2nd) as well as the total cost of dro for two blood products as shown in Fig. 4. When rh is relatively low, since the prepositioning inventory remains stable (see Fig. 3), the in_2nd remains unchanged while the in_SC linearly increases with rh. When rh increases continuously, the prepositioning inventory is reduced to alleviate the higher inventory cost, which results in the smaller in_SC but the larger in_2nd. Finally, when rh is large enough, the in_SC tends to be zero and the in_2nd remains unchanged. Accordingly, with the increase of rh, the total cost first increases slowly and then approaches a fixed value. Obviously, the decisions and the costs generated from dro are significantly sensitive to the unit inventory cost. The overestimation of rh would lead to insufficient blood storage to future emergency demand while the underestimation may lead to the unnecessary prepositioning inventory cost. 5.3.2. Sensitivity to unit shortage cost (uc) It is difficult to determine the appropriate penalty of unmet blood needs in disasters. Here, we vary uc as 0.2, 0.4, …, 1.8 times of the default shortage cost. Other parameters are set to be the default values. First, the estimation gap results of all the models for two blood products are displayed in Table 18. We find that, only when uc is very low, dro has a larger estimation gap compared with the SP and ro(box) models. However, dro shows its advantage when uc is over 0.8 times (frozen plasma) and 1 time (red blood cells). Notice that uc here represents the loss of relief failure, which would not be very small. Hence, dro is an ideal model under different uc. Besides, det always incurs the largest deviation among all the models. Next, Fig. 5 displays the prepositioning levels of frozen plasma and red blood cells given by dro. When uc is relatively small, i.e., 0.2–0.6 times, the prepositioning cost dominates the decisions, so the inventory level remains zero. With the increase of uc, the shortage cost increases and begins to take effect, which makes the prepositioning inventory level bigger accordingly, finally reaching up to the corresponding inventory capacity. The related costs of dro under different uc are further shown in Fig. 6. When uc is relatively low, the in_SC remains unchanged due to the constant prepositioning level (see Fig. 5), and the in_2nd increases as uc increases. With the growth of uc from 0.6 times, the prepositioning inventory levels of two products are significantly raised to control the deteriorating shortage penalty. Accordingly, it would lead to the synchronous growth of the in_SC. Moreover, with the rapid growth of uc, although the shortage amount is reduced due to the increasing inventory, the in_2nd still increases. Finally, because of the limit of the inventory capacity and the lifespan, the prepositioning level (cost) of frozen plasma and red blood

a. The cost results of frozen plasma

b. The cost results of red blood cells

Fig. 4. The in_SC/in_2nd/total costs of dro under varying rh. 19

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Table 18 The expected estimation gap of each model under varying uc (%). Blood product

Model

The change ratio of unit shortage cost uc 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Frozen plasma

dro sp(uni) sp(norm) sp(gam) det ro(s) ro(box)

57.33 33.67 29.02 8.42 402.72 68.65 29.68

43.26 28.41 29.30 28.93 14769.61 76.00 26.28

34.54 44.05 35.18 39.18 20269.4 150.80 33.23

29.26 42.87 38.21 45.43 25769.05 225.61 36.75

29.49 42.11 40.07 49.64 31268.78 76.08 38.88

27.59 41.57 41.31 52.67 36768.46 76.67 40.31

26.17 41.17 42.21 54.95 42268.14 77.08 41.33

24.20 40.86 42.89 54.24 47767.82 77.39 42.10

22.47 40.62 43.42 53.31 53267.50 77.64 42.69

Red blood cells

dro sp(uni) sp(norm) sp(gam) det ro(s) ro(box)

56.60 29.04 19.62 32.18 228.73 69.83 34.86

40.74 31.55 29.81 61.06 778.10 160.15 24.46

31.66 32.22 25.86 75.87 787.60 280.04 28.09

28.73 32.09 28.51 84.71 792.87 399.94 29.92

25.42 34.10 30.14 90.59 796.22 70.45 31.02

22.54 35.46 31.25 94.77 798.53 70.78 31.76

20.14 36.46 32.04 94.84 800.22 71.02 32.29

19.47 37.21 32.65 91.36 801.51 71.20 32.68

19.47 37.80 33.12 91.04 802.53 71.34 32.99

b. The prepositioning inventory of red blood cells

a. The prepositioning inventory of frozen plasma

Fig. 5. The prepositioning inventory of dro under varying uc.

a. The cost results of frozen plasma

b. The cost results of red blood cells

Fig. 6. The in_SC/in_2nd/total costs of dro under varying uc.

cells remains unchanged, while the in_2nd increases continuously. 5.3.3. Sensitivity to the replenishment cycle Different from the two exogenous variables above, the replenishment cycle is adjustable and its change would influence the daily demand per cycle as well as the turnover ratio of blood inventory. Specifically, if the replenishment cycle changes to x times its default value, the daily demand per cycle will be x· dh (U) in TR h, h H , and the blood lifespan would change from l to l·(1 x ) (cycles) since the duration of each cycle changes. Hence, the inventory limit on the right-hand side of formula (1-4) should be updated as: 20

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Table 19 The expected estimation gap of each model under varying replenishment cycles (%). Blood product

Model

The change ratio of the replenishment cycle 0.1

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

Frozen plasma

dro sp(uni) sp(norm) sp(gam) det ro(s) ro(box)

29.52 50.51 39.42 49.64 31268.72 76.31 36.39

29.41 49.05 39.53 49.64 31268.72 76.28 36.82

29.61 46.70 39.73 49.64 31268.72 76.23 37.57

29.53 44.35 39.91 49.64 31268.72 76.16 38.24

29.49 42.11 40.07 49.64 31268.72 76.08 38.88

29.54 39.98 40.21 49.64 31268.81 75.99 39.45

29.71 37.85 40.32 49.64 31268.73 75.88 39.91

28.92 35.79 40.43 49.64 19862.26 75.78 40.41

28.35 33.89 40.55 49.64 5729.77 75.66 40.83

27.82 32.15 40.67 49.64 1888.11 75.52 41.18

Red blood cells

dro sp(uni) sp(norm) sp(gam) det ro(s) ro(box)

27.54 34.86 30.02 88.49 4035.29 75.34 40.22

27.21 33.95 30.09 88.49 2239.83 75.27 40.44

26.68 33.20 30.17 88.49 1269.32 75.15 40.77

26.18 33.65 30.19 88.49 875.45 75.01 41.06

25.42 34.10 30.14 88.49 662.23 74.87 41.33

25.14 34.55 30.08 88.49 528.73 74.71 41.56

24.57 34.98 30.05 88.49 437.39 74.54 41.71

24.10 35.30 30.03 88.49 371.04 74.38 41.91

23.73 35.50 30.03 88.49 320.71 74.20 42.05

23.09 35.61 30.02 88.49 281.30 74.02 42.15

min{caph1

x· dh , l· dh

x· dh }, h

(31)

H

in which, the two terms are ‘the capacity limit’ and ‘the lifespan limit’. Here, x is set to be 0.1, 0.25, 0.5, …, 2.25. The inventory holding cost per cycle and blood supply per cycle would be proportionally adjusted in accordance with the length of the cycle. Other parameters use the default values as shown in Section 5.2.1. As shown in Table 19, in terms of the estimation gap, dro always performs better than other models, while det is the worst one, under different replenishment cycle values. Next, the decisions of dro on the prepositioning inventory are given in Fig. 7. For frozen plasma (see Fig. 7a), when x is small (less than 1.5 times), the daily demand per cycle ( x· dh ) is also small. Thus, the constraints (1–4) would not take effect, and thus the inventory level can be set up optimally to handle the emergency demand. However, as x increases, the inventory level begins to be restricted by ‘the capacity limit’ (see the dot-dash line), which may lead to more shortages in disasters. Besides, as shown in Fig. 7b, for red blood cells, due to the short shelf life, its inventory is strongly restricted by ‘the lifespan limit’ (see the full line) and insufficient to most of the demand realizations especially when x increases. On the contrary, it also means the smaller cycle can help red blood cells to relax ‘the lifespan limit’ for the prepositioning inventory. Thus, it could bring the smaller out_2nd, and then, generate a better estimation gap as shown in Table 19. Furthermore, we observe the in_SC and the in_2nd of the dro model under different replenishment cycles (see Fig. 8).

a. The prepositioning inventory of frozen plasma

b. The prepositioning inventory of red blood cells Fig. 7. The prepositioning inventory of dro under varying replenishment cycles. 21

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a. The in_SC/in_2nd of frozen plasma

b. The in_SC/in_2nd of red blood cells

Fig. 8. The in_SC/in_2nd of dro under varying replenishment cycles.

The changing trend of the in_SC, i.e., the prepositioning cost, is closely related to the corresponding inventory level. Thus, the in_SC of frozen plasma first remains relatively stable and then decreases when the cycle becomes larger than 1.5 times, whereas the in_SC of red blood cells decreases continuously. Regarding the in_2nd, the in_2nd of red blood cells always increases along with the cycle because more shortages are incurred by the declining prepositioning inventory. However, the in_2nd of frozen plasma first remains relatively stable because its inventory level is not restricted by the limit constraints, and thus, the extent of the supply shortage could be mitigated. Then, as x becomes longer than 1.5 times, its in_SC decreases due to the lower prepositioning inventory. 5.4. Findings and managerial implications The above results generate the following findings with managerial implications. (1) Our paper focus on humanitarian operations in the data-scarce disaster setting. To be specific, the case explored only has seven historical observations. Such limited information brings extra difficulties in scenario modelling of SP and RO methods. However, the moment-based ambiguous set can be easily constructed by the scarce historical observations and the better relief results can be obtained by our DRO model (see Tables 16–19). This finding exhibits the biggest advantage of our work. By utilizing our model, the DMs can implement high-quality humanitarian operations optimization without suffering from the difficulties of historical data processing. (2) The results show the correlation of the preparedness stage and the response stage while a natural disaster may be unavoidable. For example, the increase of the unit prepositioning cost in preparedness would lead to the smaller prepositioning inventory as shown in Fig. 3, and then, result in larger disaster loss as shown in Fig. 4. This finding implies that, to address disaster relief, the DMs should consider the two-stage operations in an integrated way. Such implication coincides with the prior works (e.g., Salehi et al., 2017; Ye et al., 2019) and the practice (e.g., Sichuan in China). But, compared to the traditional stochastic optimization methods, our model can help to get better here-and-now decisions which can effectively mitigate the suddenly-occurring disasters. (3) It is found that the underestimation of the shortage penalty would incur inadequate prepositioning inventory, which would result in huge disaster loss, while overestimation may lead to unnecessary inventory as shown in Fig. 5. The misestimate of the unit prepositioning cost would result in the opposite consequences as shown in Fig. 3. Rahmani (2018) emphasizes the importance of forecasting critical parameters. Our finding further implies the importance of accurate estimation of the unit prepositioning cost and the unit shortage cost, especially when these two exogenous parameters are close to the turning points which can be identified by our work. However, as we know, the accurate estimation of the shortage penalty is a challenging problem to the DMs because the loss of human lives is not only an economic issue, but also social-related. It deserves more efforts in practice. (4) Our results show that the prepositioning inventory of plasma is limited by the inventory capacity especially when the replenishment cycle is relatively big. Moreover, the red blood cells inventory is often restricted by the lifespan, which has a linear relationship with the replenishment cycle, as shown in Fig. 7. Thus, the replenishment cycle can influence the inventory-related decisions, and then, the relief effects as shown in Fig. 8. However, such practical issue is ignored in previous blood-related studies in the context of disaster. It is worth noting that, different from the two exogenous parameters above, the length of the replenishment cycle can be adjusted by the DMs. Thus, the DMs in a disaster-prone region may consider setting different replenishment cycles for the blood products with distinct shelf lives. Specifically, for the blood products with a short lifespan, the DMs should shorten the replenishment cycle to avoid the high loss of a possible disaster. Besides, for the blood products with a long lifespan, enlarging the inventory capacity sometimes is more useful than adjusting the replenishment cycle. Our work (see Fig. 7a) can help the DMs to determine the appropriate capacity.

22

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6. Conclusions 6.1. Summary of this paper In this paper, we focus on the BSN design problem considering random demand incurred by disasters. Due to the scarce data of disasters, we propose a two-stage mixed-integer model, in which random demand can follow uncertain distributions defined by a moment-based ambiguous set, to optimize the inventory prepositioning before disasters, as well as the temporary collection facility location and the transshipment after disasters. To address this DRO model with binary recourse, we develop an approximate approach, in which the Lagrange dual and Schur’s complement are firstly used to obtain a deterministic intermediate model, and then, the convex hull reformulation and the KKT conditions are employed to get a tractable SDP model. Finally, the case study based on the limited historical observations of the Longmenshan Fault in China is implemented and the proposed DRO model is compared with the deterministic equivalent, the SP and RO models under different parameter settings. Several observations and implications are obtained from the results. First, all the calculational results indicate the best estimation performance of our proposed DRO model compared with the benchmarks. The deterministic model generates a poor estimation since it only considers the nominal demand and always leads to the highest relief loss. The RO model based on the box ambiguity set is generally conservative, and the solutions of the min-max regret RO model are also conservative in most of the cases. The SP models heavily depend on the distribution functions followed, which implicates the poor performance when we cannot accurately identify the distribution. Moreover, it is found the decisions and related cost results are very sensitive to the unit inventory cost, the unit shortage cost, and the replenishment cycle. Thus, more efforts should be made to accurately measure the first two exogenous parameters and appropriately determine the replenishment cycles for the blood products with different lifespans. 6.2. Future studies In the future, we can extend our study in the following aspects. First, in this paper, the tractable reformulation of the two-stage DRO model can be divided into two steps. The first step, which transforms the original model to the intermediate two-level SDP model, is approximate. The main reason is that we consider the shortage case and the wastage case under the semi-infinite setting, which makes the equivalent transformation difficult. Such an issue is hard but deserves further effort. Second, besides demands, humanitarian settings involve a variety of uncertainties, such as supplies. Different from demands, uncertain blood supplies during disaster response can be influenced by social announcements (Larimi and Yaghoubi, 2019; Yan and Pedraza-Martinez, 2019). Thus, how to incorporate social media decisions into the DRO model should be considered in future. In this way, the DMs may obtain a better match between the fluctuating blood demand and supply in the setting of disasters. Moreover, the blood supply chain studied in this paper considers the collection, inventory and distribution decisions. To realize orchestrated decisions management of the whole blood system, the test, production echelons and associated practical elements, such as the processing time and blood extraction technology of the blood products, also should be taken in account. At last, novel heuristic or metaheuristic methods need to be developed for solving large-scale problems, which could not be handled using commercial solvers such as MOSEK in a reasonable CPU time. Acknowledgment The authors would like to thank the anonymous referees and the Co-Editor in Chief, Professor Tsan-Ming Choi, for their constructive comments which have improved the presentation of the paper. This research was partly supported by the National Natural Science Foundation of China (Grant no. 71971053, 71832001), the MOE (Ministry of Education in China) Project of Humanities and Social Sciences (Grant no. 18YJA630129), the Shanghai Philosophy and Social Science Program (Grant no. 2019BGL036) and the Fundamental Research Funds for the Central Universities. Declaration of Competing Interest None. Appendix A A. Formulation of the Lagrange dual problem In this section, we focus on formulating the dual problem of Problem (8) and displaying a more detailed transformation process. To take the dual of problem (8) with the primal variable fξ, dual multipliers θ ∈ , ρ ∈ |H|, Q ∈ H × H are introduced to relax constraints (6001)–(6003), respectively. Q is a symmetric matrix. Thus, the optimization problem (8) can be transformed into its Lagrangian function (9):

L (f , , , Q ) =

1 (ss ,

) df ( ) +

1

df ( ) +

T

23

µ

df ( ) + Q·

(

µ )(

µ )T df ( )

(32)

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C. Wang and S. Chen

where “•” represent the Frobenius inner product between matrices. Then, the Lagrange dual function (33) is obtained:

G ( , , Q ) = maxL (f , , , Q) = max f

D

f

D

T

[ 1 (ss, )

TQ

+2 TQµ ] df ( ) +

+

Tµ

+ Q·

µTQµ

(33)

From the duality theorem, the dual problem of the primal problem (8) can be represented as problem (34) according to the Lagrange dual function (33):

minG ( , , Q) = minmaxL (f , , , Q ) , ,Q

, ,Q f

D

= min max , ,Q

f

[ 1 (ss, )

D

T

TQ

+2 TQµ ] df ( ) +

+

Tµ

+ Q·

µTQµ

(34)

We then obtain the dual problem (35) of the primal problem (8): Tµ

min + , ,Q

T

+

µT Qµ

+ Q· TQ

+

(35-1)

TQµ

2

1 (ss ,

(35-2)

),

B. Convex hull construction In this part, we display the convex hull construction of constraints (2-3) and (2-5). (i) Constraints (2-3): yc capc2 qc 0, c C Different from purely integer linear constraints, constraints (2-3) are mixed-binary linear constraints. Similar to the pure zero-one case, the constraints (2-3) are first multiplied by one-degree polynomial factors yc and (1–yc), and the problem variables are linearized via introducing new variables. In addition, the interval [0, UBc] of the linear continuous term qc in constraints (2-3) should also be multiplied with the polynomial factors. Thus, we have

yc (yc capc2

qc )

yc )(yc capc2

(1

yc (UBc (1

qc )

yc )(UBc

yc qc (1

0, c

0, c

C

qc )

0, c

(36-1) (36-2)

C

(36-3)

0, c

C

qc )

0, c

(36-4)

C

(36-5)

C

yc ) qc

0, c

(36-6)

C

Then, equation (2-1) is used to replace the squared term

yc capc2 yc qc

yc qc qc

0, c

0, c

yc2

in constraints (36-1) and (36-2), and we have (37-1)

C

(37-2)

C

Since constraints (37-2) and (36-6) hold simultaneously, we have qc = ycqc and finally obtain the equivalent linear convex hull (38) to replace constraints (2-3).

yc capc2

qc

0, c

C

(38-1)

yc UBc

qc

0, c

C

(38-2)

UBc

qc

yc UBc

0, c

0, c

(38-3)

C

(38-4)

C

(ii) Constraints (2-5): MM · tgch z ch 0, c C ; h ∈ H Based on the same idea, we construct one-degree polynomial factors tgch, (1–tgch), multiplying constraints (2-5) and z ch we obtain the following reformulation constraints:

MM · tgch

tgch z ch

z ch + tgch z ch

tgch z ch z ch

0, c

tgch z ch

0, c

0, c

C; h 0, c

C; h

C; h

(39-1)

H

(39-2)

H

(39-3)

H C; h

0 . Then,

(39-4)

H

In accordance with to constraints (39-2) and (39-4), we have zch = tgchzch, and constraints (2-5) can be replaced as the linear 24

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C. Wang and S. Chen

Table 20 The KKT conditions of

2,1 (ss , 1) .

Primal feasibility

1r 1 I h H wih + spi = si , i 1 1 qc C h H z ch = 0, c qc1 UBc + c1 = 0, c C 1 2 tgch yc1 + ch = 0, c C; h H 1 1 3 tgch + tg 1 + ch = 0, c ch2 1 1 h2 qc1 yc1 capc2 + c4 = 0, c C qc1 UBc yc1 + c5 = 0, c C 1 1 6 z ch MM ·tgch + ch = 0, c C; h 7 1r wih = 0, i I ; h H ih 8 1 z ch = 0, c C; h H ch 9 1 q C c c = 0, c 10 1 sp = 0, i I i i 11 1 y = 0, c C c c yc1 1 + c12 = 0, c C 13 1 tgch = 0, c C; h H ch 1 14 tgch 1 + ch = 0, c C; h H

Stationarity

fcc eih ech opc

hc 2 ch

(41-1) (41-2) (41-3) (41-4)

C ; h1, h2

(41-5)

h2

(41-6) (41-7) (41-8)

H

2 4 2 5 11 ch c capc c UBc c + 7 + 1 = 0, i I ; h H ih i 2 6 8 uc + ch C; h H ch + c = 0, c 9 2 + c1 + c4 + c5 C c + c = 0, c 10 1 I i + i = 0, i 3 6 13 + h H & h h chh MM · ch ch + 1 1 1 h H

H &h1

(41-9) (41-10) (41-11) (41-12) (41-13) (41-14) (41-15) (41-16) 12 c

= 0, c

(41-17)

C

(41-18)

uc

(41-19) (41-20) (41-21) 14 ch

= 0, c

C; h

H

(41-22)

Complementary slackness

t1 t1

= 0 , t1 = 1, …, 14

(41-23)

Dual feasibility

t1, t 1

0 , t1 = 1, …, 14

(41-24)

convex hull (40).

MM · tgch

z ch

0, c

C; h

0

1, c

C; h

H

tgch

z ch

0, c

C; h

(40-1)

H

(40-2) (40-3)

H

C. The KKT conditions of lower-level problems This section displays the KKT conditions of two lower-level problems. We introduce the KKT multiplier vectors 1, 2 , , 14 = ( c1 , 2 14 14 , ..., ch ), , ch ) and χ1, χ2=( i1, c2 ) of inequality and equality constraints as well as the slack variables 1, 2 ,…, 14 = ( c1, ch i ∈ I, c ∈ C, h ∈ H for inequality constraints in 2,1 (ss, 1) . The equivalent KKT conditions of 2,1 (ss, 1) are displayed in Table 20. The complementarity condition constraints (41-23) involve bilinear terms, which could be further linearized as constraints (42): 2 ch ,…

t1

t1

(42-1)

MM ·Y t1, t1 = 1, ...,14

MM·(1

(42-2)

Y t1), t1 = 1, ...,14

in which are 0–1 auxiliary variables. 2 , …, h15 ), the slack variables 1, 2 ,…, 15 , the 0–1 auxiliary Similarly, we introduce the Lagrange multiplier 1, 2 , …, 15 = ( c1, ch variables X t2 (t2 = 1, …, 15) for the inequality constraints and the multiplier ς1, ς2=( i1, c2 ), (i ∈ I, c ∈ C, h ∈ H) for the equality constraints in the lower-level problem 2,2 (ss , 2). The KKT conditions are then shown in Table 21.

Y t1

D. Formulation of benchmark models D.1 Two-stage SP-BSN model The decision variables in the response stage are displayed in Table 22. Thus, the objective function and constraints are represented as follows:

25

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C. Wang and S. Chen

Table 21 The KKT conditions of

2,2 (ss, 2) .

Primal feasibility

2r 2 I h H wih + spi = si , i 2 2 qc C h H z ch = 0, c qc2 UBc + c1 = 0, c C 2 2 tgch yc2 + ch = 0, c C; h H 2 2 3 tgch + tg 1 + ch = 0, c C; h1, ch2 1 h2 1 qc2 yc2 capc2 + c4 = 0, c C qc2 UBc yc2 + c5 = 0, c C 2 2 6 = 0, c z ch MM ·tgch + ch C; h H 2r 7 wih = 0, i I ; h H ih 2 8 z ch = 0, c C; h H ch 9 2 q C c c = 0, c 10 2 sp = 0, i I i i 11 yc2 = 0, c C c yc2 1 + c12 = 0, c C 2 13 tgch = 0, c C ; h H ch 2 14 tgch 1 + ch = 0, c C; h H 2 15 ovh = 0, h H h

Stationarity

fcc eih

ech + opc + hc 2 ch

+

hc Complementary slackness

(43-2) (43-3) (43-4)

h2

11 c

H &h1

(43-5)

h2

(43-6) (43-7) (43-8) (43-9) (43-10) (43-11) (43-12) (43-13) (43-14) (43-15) (43-16) (43-17)

+

12 c

= 0, c

(43-18)

C

(43-19) (43-20)

H

(43-21)

C

(43-22) 13 ch

+

14 ch

= 0, c

C; h

(43-23)

H

(43-24) (43-25)

MM·X t2 , t2 = 1, …, 15

t2

Dual feasibility

2 4 2 5 c capc c UBc h H ch 7 1 + = 0, i I ; h H ih i 6 8 2 C; h ch ch c = 0, c 1 4 5 9 2 + + + c c c c c = 0, c 10 1 I i + i = 0, i 3 6 MM · ch h1 H & h1 h chh1 15 H h = 0, h

(43-1)

t2

MM·(1

t2 ,

t2

(43-26)

X t2) , t2 = 1, …, 15

(43-27)

0 , t2 = 1, …, 15

Table 22 Decision variables in the response stage.

min T

ss, w p,

Symbol

Description

ycs

1 if TCF c is activated to collect blood in scenario s; 0 otherwise.

qcs

Collection amount of blood product in TCF c in scenario s.

s z ch

Distribution flow of blood product from TCF c to TR h (U) in scenario s.

oshs

Shortage amount of blood product in TR h in scenario s.

ovhs

Amount of surplus blood product in TR h in scenario s (U).

s tgch

1 if TCF c is supplied to transfusion center h in scenario s; 0 otherwise.

wihsr

Distribution flow of blood product from PCF i to TR h (U) in scenario s.

s vhh 1

Transshipment flow from TR h to TR h1 (U) in scenario s.

spis

Amount of surplus blood product in PCF i in scenario s (U).

rh ssh + h H s ech z ch +

+ c C h H

s. t. constraints(1

h H

i I h H

eih wihp +

tgchs

ycs = 0, s

h H h1 H & h1 h

2)

(1

;c

1 s

c C

fcc ycs + uc

s ehh1 vhh 1

oshs + h H

c C

opc qcs + hc

i I

spis +

ovhs + h H

eih wihsr i I h H

(44-1)

5)

C

(44-2) 26

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C. Wang and S. Chen

ycs · capc2 qcs

qcs

0, s

UBc , s

MM · tgchs

;c

s z ch

qcs

;c

s z ch

(44-3)

C

(44-4)

C

0, s

;c

= 0, s

C; h

;c

(44-5)

H

C

(44-6)

h H s z ch +

ssh + c C

wihsr + oshs

ovhs

s vhh + 1

i I

h1 H & h1 h

wihsr + spis = si , s

;i

v hs1 h =

s h,

s

;h

H

(44-7)

h1 H & h1 h

I

(44-8)

h H

ycs ,

tgchs

{0, 1}, s

;c

C; h

s s qcs , wihsr , z ch , oshs, ovhs, spis , vhh 1

H 0, s

;i

I; c

C; h , h1

H &h1

(44-9)

h

Formula (44-1) considers the prepositioning cost, the transportation cost in the first stage and the expected cost of all scenarios in the second-stage, including the setup cost of PCFs, the transportation cost from the TCFs and the PCFs to the TRs, the collection cost in the TCFs, the shortage cost, the surplus cost in both the PCFs and the TRs as well as the transshipment cost between the TRs. Similarly, constraints (1-2)–(1-5) restrict the blood distribution flow and the prepositioning stock in the preparedness stage. Constraints (44-2) ensure that each TCF is assigned to a TR in order to provide a timely supply. Constraints (44-3)–(44-4) display that blood collection should not only be within the capacity of the TCF but also less than the available maximum supply amount in any scenario. Constraints (44-5)–(44-6) guarantee that the distribution flow from a TCF to all the TRs should be less than the collection amount in this TCF. Constraints (44-7) specify the relationship between the supply, shortage, transshipment flow and actual emergency demand. Constraints (44-8) calculate the surplus amount in each PCF, and domains of the decision variables are specified in constraints (44-9). D.2 Two-stage RO-BSN model Here, we provide the formulation of the ro(s) model. Specifically, the decisions variables are shown in Table 21. We consider the prepositioning cost and the transportation cost in the preparedness stage as well as all the costs in the response stage in the objective function (45-1). For scenario s ∈ Φ, we obtain the optimum value Os* by model (45):

min T

ss, w p,

rh ssh + h H

i I h H

eih wihp +

s ech z ch + c C h H

h H h1 H & h1 h

s. t. Constraints(1

h H

tgchs

ycs · capc2 qcs

ycs = 0, c

C

qcs

C

UBc , c

MM · tgchs

0, c

(1

fcc ycs + uc

oshs + h H

c C

opc qcs + hc

i I

spis +

ovhs + h H

s ehh1 vhh 1

5) (45-2) (45-3) (45-4)

0, c

s z ch = 0, c

C; h

(45-5)

H

C

(45-6)

h H s z ch +

ssh + c C

wihsr + oshs

ovhs

i I

wihsr + spis = si , i

s vhh + 1 h1 H & h1 h

v hs1 h =

s h,

h

H

h1 H & h1 h

I

tgchs

qcs ,

wihsr ,

{0, 1}, c s z ch ,

oshs,

ovhs,

C; h spis ,

(45-7) (45-8)

h H

ycs ,

eih wihsr + i I h H

(45-1)

C

s z ch

qcs

2)

c C

H s vhh 1

0, i

I; c

C; h , h1

H &h1

h

The explanations of constraints (45-2)–(45-9) are similar to those of constraints (44-2)–(44-9). 27

(45-9)

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C. Wang and S. Chen

To test the performance of each optimal decision in another scenario s1, we propose the model (46) to obtain Os, s1 , and further obtain the regret value Os, s1 Os .

rh ss ¯ hs +

T h H

c C h H

i I h H s1 ech z ch +

eih w¯ ihps + min

c C

fcc ycs1 + uc

h H

oshs1 +

c C

opc qcs1 + hc

i I

spis1 +

h H

ovhs1 +

s1 ehh1 vhh 1

h H h1 H & h1 h

i I h H

eih wihs1 r +

(46-1)

s.t. Constraints (1-2)–(1-5) h H

tgchs1

ycs1 · capc2

qcs1

ycs1 = 0, c

C

qcs1

C

UBc , c

MM · tgchs1

(46-3) (46-4)

C

s1 z ch s1 z ch

qcs1

0, c

(46-2)

0, c

C; h

= 0, c

(46-5)

H

C

(46-6)

h H

s1 z ch +

ss ¯ hs + c C

wihs1 r + oshs1

ovhs1

i I

h1 H & h1 h

wihs1 r + spis1 = si, i

s1 vhh + 1

h1 H & h1 h

v hs11h =

s1 h ,

h

H

I

(46-8)

h H

ycs1 ,

tgchs1

{0, 1}, c

s1 qcs1, wihs1 r , z ch ,

oshs1,

C; h ovhs1,

(46-7)

H

s1 spis1, vhh 1

0, i

I; c

C ; h, h1

H &h1

h

(46-9)

Appendix B. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.tre.2020.101840.

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