key: cord-0844275-is94f9ri
authors: Samani, Mohammad Reza Ghatreh; Hosseini-Motlagh, Seyyed-Mahdi
title: A novel capacity sharing mechanism to collaborative activities in the blood collection process during the COVID-19 outbreak
date: 2021-08-13
journal: Appl Soft Comput
DOI: 10.1016/j.asoc.2021.107821
sha: e4a6f7f7b4ef8cf3cf822be35c9c928c43e96b1f
doc_id: 844275
cord_uid: is94f9ri

Because of government intervention, such as quarantine and cancellation of public events at the peak of the COVID-19 outbreak and donors’ health scare of exposure to the virus in medical centers, the number of blood donors has considerably decreased. In some countries, the rate of blood donation has reached lower than 30%. Accordingly, in this study, to fill the lack of blood product during COVID-19, especially at the outbreak’s peak, we propose a novel mechanism by providing a two-stage optimization tool for coordinating activities to mitigate the shortage in this urgent situation. In the first stage, a blood collection plan considering disruption risk in supply to minimize the unmet demand will be solved. Afterward, in the second stage, the collected units will be shared between regions by applying the capacity sharing concept to avoid the blood shortage in health centers. Moreover, to tackle the uncertainty and disruption risk, a novel stochastic model combining the mixed uncertainty approach is tailored. A rolling horizon planning method is implemented under an iterative procedure to provide and share the limited blood resources to solve the proposed model A real-world case study of Iran is investigated to examine the applicability and performance of the proposed model; it should be noted that the designed mechanism is not confined just to this case. Obtained computational results indicate the applicability of the model, the superior performance of the capacity sharing concept, and the effectiveness of the designed mechanism for mitigating the shortage and wastage during the COVID-19 outbreak.

capacity in blood supply in different regions. In this regard, the capacity sharing concept and transshipment strategy between regions is a practical approach that can enhance the BSN's flexibility.

In light of the above discussion, the main goal of this study is to develop an analytical approach by presenting a well-adapted mathematical formulation to the planning and managing of BSN during the outbreak of COVID-19, taking into account the particular characteristics of blood such as perishability, demand uncertainty, various collection methods, and disruption risk in blood supply. The proposed approach in this study has excellent potential for making the appropriate decision on the blood collection process in different regions with different statuses of the COVID-19 outbreak. Also, the capacity-sharing strategy between regions at the peak of the outbreak can mitigate the blood shortage and wastage in this urgent situation. Accordingly, this study aims to provide a collaborative mechanism in BSN to coordinate activities between regions to mitigate the shortage in this network. Motivated by a real case study in Iran, this study tries to recommend the appropriate answer to the following main research questions:

• Which approach should be adopted to mitigate the blood shortage, especially at the peak of the outbreak?

• What policy should be considered in regions for blood collection procedures at the peak and non-peak times of the outbreak?

• What strategy should be applied for blood sharing between regions when the actual demand is realized in each period?

• Which policy should be used to deal with disruption and operational risks in blood supply and demand during the outbreak?

In this paper, to answer the first question, we utilize mathematical optimization programming to model BSN activities to coordinate the blood banks for collecting blood units from donors to mitigate shortages. To answer the second question, we develop a novel mechanism based on a two-stage optimization tool. In the first stage, blood collection centers try to collect the blood from donors as much as possible, and in the second stage, blood centers are coordinated to share the blood units between regions to respond to the demand of health centers. To answer the third question and dynamic nature of this network, by applying a capacity sharing concept, the sharing process between regions will be done. For avoiding the level of wastage and shortage, the proposed model will be solved by a rollinghorizon strategy. Finally, to answer the last question, we propose a novel mixed possibilistic-flexible programming method to cope with uncertainty (operational risk), and a new strategy is also devised to deal with disruption risk in this network. To the best of our knowledge, this study is the first research in BSN literature that proposes a novel real case-based collaborative mechanism to coordinate blood collection activities between regions in the outbreak of COVID-19. The main contributions of this study that differentiate it from previous studies are provided as follows:

• Developing a medium-term plan to manage the blood collection process during the outbreak of COVID-19 by applying transshipment between regions to mitigate the fluctuation of demand and supply, especially at the peak of the outbreak;

• Proposing a novel two-stage optimization tool combining mixed uncertainty approach and rolling-horizon strategy to tackle the uncertain and dynamic essence of BSN;

• Applying a reactive model to update the collection planning for fulfilling the blood shortage in regions in the peak time of the outbreak;

• Selecting a real-world case study of Iran to evaluate the practicality of the proposed model in collaboration with IBTO;

• Some useful managerial insights are concluded by implementing our collaborative mechanism for a real-life case study.

This study is organized as follows. Section 2 provides a systematic background on BSN. Section 3 extends this study using mathematical formulations and methodology for managing the concerned BSN is proposed in Section 4. The practicality of the proposed methodology by a real case study is dedicated to Section 5. The computational results and sensitivity analysis of this study are provided in Section 6. Section 7 provides research implications and future directions, and finally, Section 8 presents the final remarks of this study.

disaster. Therefore, the related literature can be divided into two main categories: BSN in disaster conditions and blood collection management. Consequently, we address these two categories separately in the following related literature. First, in Section 2.1, a targeted review on BSN in the disaster condition is provided. Then, in Section 2.2, a review on blood collection management in BSN is presented.

Also, a comprehensive review on BSN in disaster and normal situations in recent years is provided and presented in Table 1 in Appendix A.

Several studies in the literature investigated BSN management under disaster conditions. Sha and Huang [61] have developed a multiperiod location-allocation mathematical formulation for emergency blood supply to minimize the total transportation cost, reposition cost of temporary facilities, and punishment cost (the penalty cost for unfulfilled demand). They rendered a Lagrangian relaxation-based algorithm to solve the proposed mixed-integer programming model. The applicability of their proposed model was illustrated by implementing a real case in Beijing. In another effort, Fahimnia et al. [35] developed a mathematical model for blood collection in disaster using a bi-objective two-stage stochastic mathematical formulation. Total network cost and the average delivery time are minimized over multiple planning horizons, and the -constraint and Lagrangian relaxation methods are applied to solve the proposed model. Habibi et al. [28] addressed a bi-objective robust model for the multi-echelon BSN in disaster relief cases, aiming to reduce the total cost and shortage.

The authors handled the plurality of objective functions by employing a goal programming method. Also, they tested the performance of the proposed model through a real-world case study. In another study, Samani et al. [41] presented a mixed-integer multi-objective model to design and plan for BSN in a disaster condition. They did a tradeoff between total network cost, perishability, and reliability of blood facilities. A mixed two-stage stochastic-possibilistic approach is tailored to handle the uncertainty of the parameter. A real data study from

Mashhad city in the northwest of Iran is applied to explore the model applicability.

Fazli-Khalaf et al. [29] proposed a mixed possibilistic-flexible robust model for designing an emergency BSN. Their model aims to minimize the total cost of the network and the total time transportation between blood facilities and maximize the collected blood reliability. In another study in the context of disaster, Fereiduni and Shahanaghi [62] proposed a robust optimization model for controlling BSN after a striking natural disaster. Their model aims to minimize the total network cost and find the allocation pattern, location, and blood distribution decisions for a multi-period planning horizon. In another effort, Ma et al. [63] proposed a blood supply chain to find an optimal blood allocation strategy after striking disasters assuming blood group compatibilities. The computational results revealed that the concept of blood compatibility singularly enhances network efficiency. also, they used a greedy heuristic algorithm for solving the proposed mathematical model in large-size instances. Khalilpourazari and Khamesh [33] applied a bi-objective optimization model to select the optimal allocation pattern, location, inventory, and transportation decisions in the context of an earthquake event. The authors tailored multiple transportation modes, and the objectives function aim to minimize the total blood network expenses along with the delivery time of collected blood to blood centers. Sharma et al. [30] modeled a Min-Max problem to optimize the location-allocation pattern for mobile blood facilities after a disaster condition. They applied a mixed approach in which the Tabu search algorithm fixes the location of mobile blood facilities. The priority of locations for mobile blood facilities is determined by using the Bayesian belief network.

Salehi et al. [34] developed a robust-stochastic programming model to manage a BSN in pre and post-disaster phases by considering compatibility between blood groups and safety stock in blood inventories. The first stage of the model decides on the location decisions and the safety stock level, and then blood allocation decisions are made in the second stage. In another research, Cheraghi and Hosseini-Motlagh [64] proposed a relief-based BSN, which considered the priority level of patients, equity in blood distribution, and disruption and operational risks in a disaster situation. In addition, they applied a multi-objective model to minimize the total cost and the shortage level between demand zones. There are several studies recently published that addressed the application of analytical techniques during the 

This subsection reviews the related literature on published papers on BSN, focusing on the blood collection process. Blood collection management is considered by Zahiri et al. [20] through a mixed-integer linear model in both strategic and tactical levels. The location of temporary blood facilities besides permanent blood centers as strategic decisions, and the allocation of donors to blood facilities and blood volume, distributed from blood facilities to demand zones in each period as tactical decisions, are determined via the proposed model.

Also, data uncertainty was tackled by applying a robust fuzzy approach, and model application is evaluated by applying a real case of Babol city in Iran. Şahinyazan et al. [21] developed a bi-objective model for blood systems in the context of the selective vehicle routing problem, including bloodmobiles, shuttles, and a specific depot in which the collected blood is transferred by the shuttle at the end of each period.

The model calculates both bloodmobile tours and shuttles, reduces transportation costs, and optimizes the number of collected blood. They employed a two-stage heuristic method based on integer programming for solving the proposed model in the large-size instance. The model and its solution technique are examined using real data of the Turkish Red Crescent.

In another study, Alfonso et al. [25] proposed two mathematical models, i.e., annually and weekly, for a bloodmobile collection system.

The annual blood collection planning aims to ensure each region's self-sufficiency for red blood cell supply and minimize the total red blood cell supplied by other regions. The weekly blood collection planning aims to minimize total working time, including setup, collection, and transportation time. Blood donation was also estimated through a proposed model for forecasting donation based on donor availability and population. To evaluate the performance of the proposed model, they used data from the French Blood Service. Chen et al. [68] developed a dynamic programming model for a joint decision-making problem of blood collection and platelet inventory control based on different demand priorities and freshness requirements. The model's practicality is validated by the real data obtained from most countries, especially the developing ones.

Various collection approaches are evaluated by a simulation model proposed by Lowalekar and Ravichandran [69] . Both fixed and variable quantities collected by the collection policies and the donation times are assumed in the proposed study. The obtained results revealed that blood collecting as much as possible is not necessarily beneficial. Considering social aspects, Ramezanian and Behboodi [40] designed a BSN in the collection phase. The distance between donors and blood facilities, donors' experience, and advertisement in blood facilities, are assumed to formulate the donors' utility function. For modeling the stochastic nature of the parameters, a robust optimization approach is utilized. In continue, Hosseini-Motlagh et al. [54] proposed advertisement, education, and medical credits to increase the donors' utility and control the supply amount more efficiently. They applied an augmented version of the data envelopment analysis model to find the optimal location of blood collection facilities. A mixed stochastic-possibilistic robust approach is developed to cope with operational and disruption risks simultaneously. A real case study in Mashhad city in the northwest of Iran is used to validate the applicability of the proposed model.

Samani et al. [48] outlined a multi-attribute group decision-making technique to evaluate the best-fit candidate location to establish blood collection facilities based on qualitative criteria. This study presented a multi-objective model for designing an integrated BSN based on quantitative factors. Then, to deal with the uncertainty of input data, they used a robust formulation technique. Larimi and Yaghoubi [70] investigated the impact of different types of donors, the number of non-scheduled and scheduled donors, social impacts, and different production technologies on platelet donation. They developed a bi-objective stochastic-robust model to deal with the uncertainty of critical parameters. The first objective aims to minimize the total cost of the network, and the second one aims to maximize the number of delivered platelets. In another study, Ensafian et al. [32] rendered a discrete Markov chain process to evaluate the number of donors for the platelet supply network. To mitigate the shortage in this network, the possibility of ABO-RH group compatibility is considered in the model. For handling the data uncertainty, a two-stage stochastic programming model is applied. Reviewing the related literature on blood collection management in BSN reveals that no research focused on the impact of disruption risk in blood supply and using the capacity sharing concept to mitigate the shortage in disaster situations like the COVID-19 outbreak. Also, the presented study considers the fluctuations of blood demand in this situation and applying a rolling horizon planning mechanism to fulfill the demand in an efficient way.

In this section, a real problem that the IBTO in the COVID-19 outbreak is considered a case study of this paper. A two-stage optimization model with a multi-period planning horizon using a rolling planning horizon approach is developed to this aim. Figure 1 illustrates the schematic view of the proposed two-stage optimization model. As shown from this figure, there are two stages in each iteration. In the first stage, a blood collection plan in each region considering disruption risk in blood supply in the event of peak outbreak to minimize the total unfulfilled demand will be solved. In this stage, the collection procedure is tailored by the whole-blood collection mechanism. Therefore, by solving the first-stage model, the quantity of collected blood in each region in each period is determined. It should be noted that the collection procedure is planned based on the total nominal demand of regions in this stage. Then in the second stage of the model, based on sharing strategy concept, the quantity of collected blood from regions in blood collection centers is shared between central blood banks in different regions to fulfill the demand. It is worth mentioning that, due to the maximum allowable distance between blood collection centers and central blood banks, the assignment procedure is done based on the maximum coverage radius between each pair of nodes. In the second stage model, the quantity of blood units which is delivered to each central blood bank is determined based on solving the first- List of sequences

The sequence of events in each period are listed below:

1. In each period, the quantity of blood collection in each region by considering disruption risk is determined;

2. Blood collection centers transport the collected units to central blood banks;

3. Central blood banks receive the collected units and kept as inventory of blood banks after the testing process;

4. The inventory level in each period uses to fulfill the demand of regions; otherwise, leftover units is transferred to the inventory of the next period;

5. If the actual demand is greater than the inventory level in each period, the demand is fulfilled by the apheresis method in central blood banks;

6. When each period ends, the inventory of each central blood bank is updated after realizing the actual demand;

7. Based on updated inventory at the end of each period in central blood banks, the collection plan is updated from next periods;

The main assumptions of the first and second stage models made in this research are as follows:

• The location of blood collection centers and central blood banks are predetermined;

• The capacity of blood collection centers and central blood banks in each period is considered to be limited;

The considered model is a multi-period problem in which each period equals one week;

• Donation of blood from each region is considered to be the supply amount of each region as it seems impracticable to plan for blood collection separately for each donor;

• As blood is a critical product for the lifesaving of patients and shortage can result in death, the quantity of shortage at the end of each period is fulfilled by the apheresis collection method. In other words, the shortage is not allowed in regions in each period;

• Both disruption and operational risk are counted in the model. Disruption risk is considered in blood supply, and the demand parameter is assumed to be uncertain. These parameters are handled according to Section 4.1 and Section 4.2, respectively. 

The population of region ∈ ℐ;

The minimum blood donation rate in region ∈ ℐ in period ∈ ;

The maximum blood donation rate in region ∈ ℐ in period ∈ ;

The maximum capacity of blood collection center ∈ ;

ℯ The maximum capacity of central blood bank ∈ ;

The distance between blood collection centers ∈ and region ∈ ℐ; ′ The distance between blood collection centers ∈ and central blood bank ∈ ;

The distance between central blood bank ∈ and region ∈ ℐ;

The maximum coverage radius between blood collection centers and regions; ′ The maximum coverage radius between blood collection centers and central blood banks;

The maximum coverage radius between central blood banks and regions;

The available (non-disrupted) fraction of blood donation in region ∈ ℐ with prevalence level of ∈ ℛ in period ∈ in scenario ∈ ;

The travel time between blood collection centers ∈ and central blood bank ∈ ; ′ The travel time between central blood bank ∈ and region ∈ ℐ;

The historical discard rate per unit in testing and production process in central blood bank;

The capacity of central blood bank ∈ ;

The demand in hospitals of region ∈ ℐ in period ∈ (uncertain parameter);

Transportation cost per unit from collection center ∈ ℐ to central blood bank ∈ ;

J o u r n a l P r e -p r o o f

Transportation cost per unit from central blood bank ∈ to region ∈ ℐ;

Expiration cost per unit in central blood bank;

ℎ Inventory holding cost per unit in central blood bank;

Collection cost of blood by apheresis method in central blood bank;

ℊ Collection cost of blood by whole blood method;

The probability of scenario ∈ ;

ℳ A very large number.

It is equal to 1; if donors' region ∈ ℐ is assigned to blood collection center ∈ ; and 0 otherwise; ′ It is equal to 1; if blood collection center ∈ is assigned to central blood bank ∈ ; and 0 otherwise; ′′ It is equal to 1; if region ∈ ℐ is assigned to central blood bank ∈ ; and 0 otherwise;

Quantity of blood donated by donors' region ∈ ℐ to blood collection center ∈ in period ∈ in scenario ∈ ;

Quantity of collected blood in collection center ∈ transferred to central blood bank ∈ in period ∈ in scenario ∈ ; ′ Inventory level with shelf-life of ∈ ℒ in central blood bank ∈ at the beginning of period ∈ in scenario ∈ ;

Inventory level with shelf-life of ∈ ℒ in central blood bank ∈ at the end of period ∈ in scenario ∈ ;

Total inventory level in central blood bank ∈ at the end of period ∈ in scenario ∈ ;

Unfulfilled demand in region ∈ ℐ in period ∈ in scenario ∈ ;

Expired units in central blood bank ∈ at the end of period ∈ in scenario ∈ ;

Quantity of units in central blood bank ∈ used to fulfill the demand of region ∈ ℐ with shelf-life of ∈ ℒ in period ∈ in scenario ∈ ;

Quantity of unfulfilled demand which is covered by apheresis method in region ∈ ℐ in period ∈ in scenario ∈ .

In this section, the concerned problem is modeled by applying a two-stage optimization tool in Section 3.2 and Section 3.3.

In this subsection, the mathematical model is formulated to present the first stage decisions discussed earlier in this paper.

subject to

.

.

,

The aim of the first stage model is to minimize the total unfulfilled demand in regions in each period. This objective is shown in Equation (1). Constraint (2) shows that each region can be assigned to a blood collection center only if the distance between the region and the facility is not more than the maximum standard distance between them. Constraint (3) allows blood collection only if donors'

regions are assigned to a blood collection facility in each period. In this constraint, if equals to zero, the blood collection process in region is not allowed. Each region can donate blood no more than its potential capacity. The potential capacity of each region is calculated by multiplying the population of regions ( ) and percentage blood donation rate ( ). This subject is bounded by Constraint (4). In this constraint ( ) shows the non-disrupted fraction of blood donation in each region. Constraint (5) limits the capacity of each blood collection center in each period. Equation (6) guarantees the quantity of blood collected units for demand satisfaction in each period. In this equation ( * , −1 ) determines the updated inventory level in the previous period by applying the rolling horizon mechanism. The domain of decision variables is specified by Constraints (7) and Constraint (8).

In this subsection, the mathematical model is formulated to present the second stage decisions, which were previously discussed in this paper. In this stage, the quantity of � is considered as input according to solving the first-stage optimization model.

min � (� .

, , + � .

, , ,

subject to

,

The first aim of the second stage model in Equation (9) minimizes the total delivery time of whole blood units from blood collection centers to central blood banks (∑ .

, ,

) and blood units from central blood banks to demand zones (∑ ′ .

, , ,

). The second aim of the model in Equation (10), including six terms, is to minimize the total cost of the network. In the first term (∑ . (21) and Constraint (22) . Equation (23) calculates the number of blood units used to fulfill the demand of each region. Equation (24) represents that the quantity of unfulfilled demand in each region is fulfilled by the apheresis method. Finally, Constraint (25) and Constraint (26) display the type of decision variables 4. Solution methodology

Supply chain components are surrounded by various risks, and accordingly, the supply chains are planned in a stochastic environment and disruption risk. The BSN is no exception, and they encounter different kinds of risks such as uncertainty in demand and disruption in supply. In order to prevent supply chain malfunctions, appropriate strategies should be taken by policy-makers, and in this study, a hybrid stochastic possibilistic-flexible programming approach is presented to handle risks in the proposed BSN.

Experience of COVID-19 outbreak implied that there would be a meaningful impact on blood supply due to the reduction in blood donors. The COVID-19 epidemic can decrease the blood supply and its by-products and negatively affect blood network activities. When a region faces a peak of the COVID-19 outbreak, one of the major impacts of this situation on blood transfusion services is that it leads to remarkable shortages in blood donation due to donors' fear of exposure to the virus in a medical center. Generally, in this situation, blood banks should deal with a disruption in blood supply. The solution to cope with supply disruption can include advertisement, encouraging healthy donors to visit blood centers and other motivational initiatives. It should be noted that in this situation, several time periods are required for disappearing the effect of disruption and blood donation returns to the normal status.

This study is proposed during the multiple periods (several weeks) that are long enough to make disruptions independent from each other in a temporary state. That is, the striking disruption in one period has no effect on the occurrence probability of disruption risk in the next periods. In such a problem, disruption in blood supply parameters has a Bernoulli distribution. Furthermore, disruption risk is a particular kind of yield uncertainty in which the yield is considered a Bernoulli random variable [71] . It is presumed that the disruption probability for blood supply in each region within a period at the peak of the outbreak follows a Bernoulli distribution with parameter 0 < < 1, which addresses disruption probability in region ∈ . In this regard, if a disruption in region with a prevalence level of ∈ ℛ under scenario within period happens, parameter is equal to 1, and 0 otherwise [72, 73] .

According to a normal distribution ( , ) that ∈ , it is assumed that a percentage of maximum blood supply will be interrupted if a disruption happens. Indeed, if a disruption occurs at the peak of the outbreak during the period under scenario , the percentage of available blood supply in region with a prevalence level of ∈ ℛ will be gone, which is presented by . When a disruption occurs in blood supply for a region at the outbreak's peak, initiative strategies should be executed to return its original supply. By considering a linear recovery function, if the recovery time equals to , the percentage of the recovered supply for a region in a period equals to 1 × 100%. Figure 2 shows the linear recovery of blood supply over a planning horizon at the outbreak's peak. In the following equation, parameter represents the available blood supply in region with a prevalence level of during period under scenario . As can be seen, min{(1 − ), ( 1 )} is the recovered blood supply at the beginning of period . ( 

) denotes the fraction of the available supply when a disruption occurs in the period . It is worth mentioning that parameter is always a positive value between zero and one. 

To deal with uncertainty, there are many different methods in the literature. On the other hand, fuzzy programming is considered as a frequently used approach to tackle epistemic uncertainties [74] . When access to sufficient information is not available, or the available information is ill-known, this approach is adopted. The fuzzy programming approach can be divided into two groups: flexible programming used to handle flexible constraints and objective functions and possibilistic programming used to handle data uncertainty [75, 76] . Herein, in order to cope with different sources of uncertainty, fuzzy and stochastic approaches have been utilized simultaneously in the form of proposed hybrid stochastic possibilistic flexible programming, which are provided as follows.

Step 1: The compact form of the first and second stage models proposed in Section 3.2 and Section 3.3 are as follows. , , , ′ , , ′ and are matrices of the models, which consist of deterministic parameters and ℳ denotes a very large positive number. and ′ are binary variables associated with the first and second stages, which are independent scenario variables, respectively. Also, and are continuous variables in the first-stage model and � , , and show the continuous scenario-based variables in the second stage model.

Additionally, denotes disruption scenarios, and its occurrence probability is (∑ = 1). In these models, the last constraint of the first stage model is a flexible constraint. Violation is allowed in this constraint, controlled according to the decision-makers' opinions by assigning a penalty.

Step 2: The amount of violation in flexible constraint is shown by ̃ which is triangular fuzzy numbers. Besides, for flexible constraint, a parameter is assigned that indicates the minimum satisfaction level ( in this model). The above model changes to the following form [77, 78] : Step 3. ̃ is shown in the triangular fuzzy forms of ̃= ( 1 , 2 , 3 ), which can be defuzzified as follows [79, 80] :

which , ′ are calculated as follows:

Step 4: According to the above-mentioned descriptions, the crisp equivalent of Model (29) is as follows:

is the violation of the flexible constraint.

Step 5: In Model (32), the value of is determined based on decision-makers' opinions.

Step 6: According to the compact models of first and second stage models, the basic hybrid stochastic possibilistic programming model is as follows:

In this model, the parameter is considered as triangular fuzzy numbers in which the actual value of this parameter is realized in the second stage model at the end of each period. addresses the confidence level of fuzzy constraint so that 0.5< ≤1.

Step 7: To eradicate the mentioned shortcomings, the hybrid stochastic possibilistic-flexible programming model is presented as follows: To cope with disruption and operational risks

Step 1: Determine the disruption scenario, flexible constraint, and uncertain parameters in the mathematical formulation

Step 2: Determine the amount of violation in flexible constraint by triangular fuzzy numbers

Step 3: Defuzzify the triangular fuzzy form

Step 4: Convert the flexible model to the crisp equivalent model

Step 5: Determine the allowable violation of the flexible constraint by decision-maker

Step 6: Consider the uncertain parameters as triangular fuzzy numbers and defuzzify by possibilistic programming

Step 7: Solve the hybrid stochastic possibilistic-flexible programming model

An appropriate plan must be flexible enough to solve uncertainties in a system at multiple levels. It is identified as a necessary requirement for responding to input data about the past (i.e., random parameter realization) and the future (i.e., frequently updated constraints and forecast) in the planning process of a dynamic environment. The original plan is always open to revision while rolling forward the planning horizon in a standard manner. In an uncertain environment, it needs continuing planning since the database constantly updates. The rolling horizon approach is an integrated procedure to deal with the planning and scheduling optimization problem iteratively [81] . In each iteration, the detailed model is only made for a part of the planning horizon, and the remaining portion is illustrated from a general aspect. The rolling horizon has a coherent solution framework within which each scheduling sub-horizon is successively solved, and its leftover demands are carried over the next sub-horizon. Consequently, practical solutions to planning and scheduling are achieved with a marked decrease in computational requirements [82, 83] .

As a decision-making process, the rolling horizon procedure is based on forecasting the future and adjusting to disruptive influences.

It is regarded as a flexible instrument for making adaptations to a planning horizon with uncertain data containing unpredictable parameters to plan a schedule at the lowest error rate. The rolling horizon decision-making process proves to be a flexible planning tool in environments with low data availability and planning horizons, including parameters with a varying level of performance. It would be better to say; the rolling horizon sets a connecting link between past incidents and their updated status. It enables to update input data for a positive reaction to any change and investigates all the possible scenarios. As a result, it offers decision-makers the possibility for using the data acquired over time. Rolling horizon simulation aims to analyze the results achieved by adopting optimal solutions over a planning horizon. It is worth mentioning that the optimal policies for period are the given optimal decisions that must be made by solving the stochastic programming problem with |T |− t +1 periods before uncertainty realization at period t. The schematic of the rolling horizon approach is depicted in Figure 3 .

In this study, due to the uncertainty in demand and disruption risk in supply, a rolling horizon approach is adopted to perform continuous runs for models to implement the decisions with the capability to reconsidering them. To preserve the dynamic essence of the decision-making process, the presented uncertainty approach is integrated into a rolling horizon approach, that is, while the decisions are made over a planning horizon. The process is continually repeated in every period of the planning horizon. Preliminary data per period such as supply and demand, leftover demand of each region are generated by decisions made in its previous period. In Algorithm 2, the steps of the rolling horizon planning approach to solve the mathematical formulation are presented. Step 1: Calculate the disruption scenario for the next periods

Step 2: Estimate the nominal blood demand and formulate the uncertainty approach

Step 3: Run the first stage model to determine the collection plan based on initial inventory at the current period

Step 4: Implement the sharing strategy of collected blood units between regions

Step 5: Obtain the actual blood demand in each region at the end of period

Step 6: Update the inventory level of blood banks according to sharing strategy and actual demand

Step 7: Use the apheresis mechanism to cover the unfulfilled demand at the end of period 

Various methods for handling multiple objectives have been applied in the literature. However, the techniques of the compromise programming approach, goal programming approach, and epsilon constraint approach are three of the most well-known methodologies in the literature [85] . In this study, the compromise programming approach is applied to handle the multiple objectives of the second stage model. We consider that two objective functions of the second stage model are called 1 and 2 . According to the compromise programming approach, the model should be optimized for each objective function separately. We consider that the optimal values for two objective function are 1 * and 2 * , the converted single objective model can now be formulated as follows:

where 0 ≤ ≤ 1 is the weight of each objective function that is given by the decision-maker. The procedure of the compromise programming approach to solve the bi-objective model is described in Algorithm 3. According to the above-mentioned strategies in this section, the flowchart of the proposed solution methodology of this study is depicted in Figure 4 .

The procedure of the compromise programming approach for solving the bi-objective model

Step 1: Determine the ideal solution of the first objective function called 1 *

Step 2: Determine the ideal solution of the second objective function called 2 *

Step 3: Covert the bi-objective model to a single objective function by min � 1 . 1 − 1 * 

Managing blood supply is a vital activity; hence blood transfusions are considered as a lifesaving process in many situations. For example, the SARS-COV outbreak in 2003 hurt blood supply. In an outbreak situation like COVID-19, blood banks confront challenges in preparing an adequate and safe blood supply due to decreasing blood donors. WHO predicted that the COVID-19 outbreak caused about 20% to 30% reduction in countries, and it was noted that the donation rate has dropped by approximately 10-30% in the US and by about 30% at Canadian Blood centers [7] . The blood centers in many countries reported their lowest blood supply levels since the beginning of the COVID-19 outbreak. Blood donation proceeded to be canceled since many organizations, schools, and businesses remained closed due to quarantine roles. In some other countries, such as Saudi Arabia, were reported the blood supply and donation at blood collection centers showed a fall of 39.5% [8] . In Malaysia, the supply of blood at blood banks throughout the country had decreased by 40% compared with the same years before the COVID-19 outbreak. Also, in Iran, the blood donation rate and RBC inventory decreased about 29.5% during this outbreak [10] .

As a result, blood bank managers should take precautionary measures to reduce any changes in blood stocks as much as possible.

Blood and its products are continuously required during the COVID-19 outbreak for patients suffering from cancers, trauma, blood diseases, and also emergency surgeries. It is clear that not adopting an appropriate management approach of blood demand and supply in BSN, hospitals, and clinics will face a shortage to service the patients who need blood, and consequently, many patients may suffer or even die unnecessarily. In this section, about an actual problem that the IBTO in the COVID-19 outbreak is deal with, the proposed model and approach for BSN is applied for the whole of Iran through a collaboration with the IBTO's specialists, and experts will be then provided with the outcomes of it.

Iran is the seventeenth most populous country globally, with an area of over 1,648,000 km 2 . Iran, with a population of approximately 82 million, comprises 31 provinces and 430 counties [86, 87] . The geographical depiction of counties in Iran is depicted in Figure 5 .

According to the investigations of IBTO's experts, the BSN in Iran in the COVID-19 outbreak deserves special consideration to reschedule the blood collection plan because the number of donors dramatically decreased, especially in the cities that are at the peak of the outbreak J o u r n a l P r e -p r o o f (http://fna.ir/f0r603; https://tn.ai/2407890). Based on the professional experts' knowledge in IBTO, this condition arises from the fact that donors in cities at the peak of COVID-19 with quarantine situations are less inclined to donate their blood than the people in cities with normal situations [88] . Consequently, blood inventories have decreased almost in the counties with the outbreak's peak, and shortages occurred in some hospitals (www.irna.ir/news/84423945/). It should be mentioned that currently, the system for announcing the status of blood inventory in Iran has been set up (https://www.isna.ir/news/1400011103961/), and IBTO is upgrading its system to share the blood units between the regions of the country to mitigate the shortage and wastage in this outbreak situation Regarding this challenge during the COVID-19 outbreak, IBTO has decided to prepare a plan for BSN to coordinate the blood collection processes among the counties of the country. Therefore, blood collection centers in the counties with a normal situation in terms of COVID-19 outbreak collect blood from donors as much as possible and, after fulfilling the needs of their city, share their leftover collected blood to the counties that are at the peak of the outbreak. It should be noted that the counties at the peak of the outbreak mostly face shortages to fulfill the demand of their hospitals. The characteristics of blood collection centers and central blood banks in Iran are provided in Table   1 and Table 2 , respectively. Also, the geographical schematic of central blood banks in the provinces of Iran is presented in Figure 6 . According to IBTO protocols, people in the range of 18-60 can donate blood, which approximately 60% of the population of Iran are in this age range. Another hand, regarding the documented donation in IBTO, the rate of donation is between 10-15%. Consequently, it is considered that between 6%-9% of the people can donate their blood. The population of each county in Iran is collected from the Statistical Center of Iran. Accordingly, the minimum and maximum supply of blood in each county are estimated by multiplying the blood donation rate and population of each county. In this research, the maximum shelf life of blood is considered three weeks. The demand for blood in each county is gathered based on documentation of hospitals in each county and assumed as fuzzy numbers based on experts' opinions.

The geographical coordinates of each county, blood collection centers, central blood banks, and the transportation time between these J o u r n a l P r e -p r o o f nodes are calculated based on Google Map. As reported by IBTO, the wastage rate of blood over the production process is assumed 7%.

The reference of parameters in this research is reported in Table 3 . On the other hand, the minimum affected rate in blood donation happens at a low rate of COVID-19 prevalence.

Applying the formulation for disruption risk in blood supply which is proposed in Section 4.1, the mathematical model was solved, and then, results were reported. It is worth mentioning that due to the large-size number of outputs for all counties, for simplicity and better-displaying results, the output of the counties of each province are aggregated and presented in the form of a result obtained for a province. The number of collected units in each province is computed based on the first-stage optimization model. Notably, due to the limited capacity of blood supply, it is possible that all of the demand may not be fulfilled completely. Therefore, as shown in this table, the provinces with more population have more collected units based on the available capacity for blood donation rate. This table shows that Tehran and Ilam have the highest and lowest portion for blood collection among provinces, respectively. In Table 4 , columns 3-8 demonstrate the number of collected units in each period in each province, column 9 shows the average collected units per period in each province, and finally, column 10 reflects the participation ratio of each province for blood collection in comparison to the total collected units in the country. J o u r n a l P r e -p r o o f The allocation pattern of transferring for collected units between different provinces is shown in Table 5 . Due to the limited transfer time of collected units (Maximum 8 hours), the possibility of transferring is determined based on the maximum coverage radius in the model. It can be realized from this pattern that the sharing of collected units is usually done between neighboring provinces to prevent the shortage. As can be seen from Table 5 , the provinces located in central Iran, due to their geographical location, have more ability to share the blood units than provinces located near border areas. Table 6 represents the number of transferred units from central blood banks to each province in each period. It is determined according to the second stage model. Results show that Tehran and Ilam receive the largest and lowest portion of blood units transferred from central blood banks. In Table 6 , columns 3-8 demonstrate the number of transferred units to each province in each period, column 9 shows the average of arrived units per period in each province, and finally, column 10

shows the ratio of received units in each province to the total units transferred from all central blood banks in the country. The average collection and consumption rates in each province during the planning horizon are depicted in Figure 8 . Figure 8 . The average collection and consumption rates in each province during the planning horizon

In the continuation, Table 7 demonstrates the total number of unfulfilled units in each province in each period. The results show that by applying the capacity sharing mechanism, the demand is completely fulfilled at most periods and provinces. In 27 provinces, no unfulfilled demand has occurred. From this table, it can be seen that the shortage has occurred mostly in the border provinces (except Kerman) of Iran. Due to the geographical location, the far distance of Hormozgan province from other neighboring provinces, the severity of the COVID-19 outbreak, and more disruption in blood supply, the largest portion of shortage have occurred in this province. It should be noted that the number of unfulfilled demand in each province should be covered from the apheresis collection method in the same province in each period. J o u r n a l P r e -p r o o f

In this section, the sensitivity analysis on critical aspects and important parameters of BSN is presented. In continue, the analyses on solution methodology, uncertainty approach, coverage radius, cost parameters, and multi-objective programming are investigated.

In Table 8 , the results of the adopted approach in this study are compared to the non-capacity sharing mechanism and non-rolling horizon approach, which are common approaches in BSN literature. As shown in the table, the total network cost of the BSN by the adopted method in comparison to the others decrease significantly, in which the total cost decreases by about 61.03% in comparison to the noncapacity sharing status, and about 9.11% reduces in comparison to the non-rolling horizon approach status. Furthermore, the unmet demand that occurs in our approach is significantly lower than the unmet demand of other statuses. Therefore, by applying the capacity sharing and rolling horizon mechanisms in the BSN, the total network cost and unfulfillment rate decrease remarkably. Finally, between two other statuses were used for comparison in this study, the capacity sharing strategy with the non-rolling horizon strategy outperforms another status in the context of cost and unfulfilled demand measures. In status #2, because the sharing of blood units is impossible, the inventory range compared to other statuses decreases. On the other hand, in status #1, in comparison to status #3, by applying the rolling horizon approach and the possibility of updating the inventory level at the end of each period, the inventory rate in central blood banks decreases by about 5.63%. In the last column of Table 8 , the running time of each status is presented. The computational times of all statuses are almost equal and slightly different together, which is negligible. Noteworthy, all computational times have been carried out with reasonable processing time, and therefore, we did not develop heuristic or metaheuristic algorithms to solve the model. of Table 9 shows that the HSPFP model with parameter setting of , = 0.9 has minimum total network cost, and by increasing (or decreasing) in the value of confidence level parameters, the total cost increases (decreases). Therefore, it is clear that if the stakeholders of BSN focus on minimizing the total network cost, the parameter setting of ( , = 0.9) can achieve a minimum cost in an uncertain environment of COVID-19 outbreak. On the other hand, because in deterministic conditions, a lower level of blood units collected, the deterministic model has a better performance in total delivery time. In Figure 9 , the comparison between the performance of various models in terms of total network cost and total delivery time is depicted.

The obtained results in Table 9 are calculated based on an average of 10 random realizations. The performance of HSPFP models and deterministic model are compared and evaluated by uniformly generating 10 random realizations from the respective uncertainty intervals of uncertain demand. The performance of both deterministic and HSPFP models in each realization are reported in Table 10 . As can be seen, HSPFP ( , = 0.8) outweighed other models in terms of the average cost of the network. Also, the superiority of the HSPFP model with the confidence level of , = 1.0 in terms of standard deviation can be seen in the last row of Table 10 . The trend of average cost and standard deviation in different models is shown in Figure 10 . Figure 11 . In Figure 11 -a, the effect of increasing coverage radius on total unfulfilled demand is depicted. Figure 11 -b and Figure 11 -c demonstrate the effect of this variation on total delivery time and total network cost. Figure 11 -a shows that blood units can be transshipped between provinces with farther distance by increasing the maximum coverage radius. By increasing the coverage radius, more blood units are capable of being shared between provinces. Increasing shared blood units can lead to a lower number of unfulfilled demand in provinces, and for this reason, the total network cost of the BSN decreases because the cost of unfulfilled demand is more than transportation cost in this network. This fact is shown in Figure 11 -c. It can be realized from Figure 11 -b that further increase in the coverage radius imposes increases in the total delivery time of blood units. Because in this situation, more blood units are delivered to provinces, and consequently, the total delivery time increases. Figure 11 . The impact of coverage radius on performance measures

In this subsection, the role of considering different cost parameters is analyzed. Table 11 analyzes the impact of variation in cost parameters on the total unfulfilled demand, total network cost, and total delivery time. As can be seen in this table, an increase in , decreases the total unfulfilled demand of blood units in the BSN. This model prefers fewer blood collections by the apheresis method to meet the demand, and the whole blood collection mechanism covers more percentage of the demand. On the other hand, by increasing blood collection by the whole blood mechanism, total delivery time increases because more blood units are collected and need more time to be delivered to the provinces. Since the collection cost by the apheresis method dominates the transportation cost of the network, the total network cost increases in this situation. In the second row of Table 11 , the opposite trend is visible for decreasing in parameter . The changes of parameters , and ℎ have not significant impact on the quantity of unfulfilled demand and total delivery time because varying this parameter does not affect the quantity of blood collection. Thus, unfulfilled demand and delivery time remain constant. Increasing (decreasing) in parameters , and ℎ results in more (less) cost in the expired units and inventory units, and consequently, the total network cost increase (decreases). When the parameters of , and increase, the rate of collection unit by whole blood mechanism decreases, and for this reason, the number of unfulfilled demand increases. Consequently, provinces receive fewer blood units by sharing mechanism, and the total delivery time decreases. Also, the total network cost leads to an increase in this status. Another general observation can be seen in decreasing in parameters , and in Table 11 . The obtained results reveal that the capacity sharing concept can enhance the service level of the BSN and lower the amount of shortage during the COVID-19 outbreak. The proposed two-stage optimization tool is resilient to cope with fluctuation in blood donation in this situation. Applying a rolling horizon mechanism due to the dynamic nature of BSN and both disruption risk in blood supply and uncertainty in blood demand during the outbreak can prevent low blood service levels in regions of the country. The following key insights for managers of BSN concluded by applying the capacity sharing concept and results of this research:

• Stakeholders of BSNs can benefit from capacity sharing concept due to the reduction in blood donors during the COVID-19

outbreak; as it results in a more service level in hospitals, resilient network, and thus, quicker response level to the blood demand;

• Applying an HSPFP approach to tackle disruption risk in blood supply and uncertain demand in BSN, especially in outbreak peak, can result in better performance and more service level than the deterministic condition and considering just one aspect of risk measures. • Employing motivational aspects for blood donation during the COVID-19 outbreak and planning for blood collection via an apheresis method from booked donors can be beneficial to cope with disruption in blood supply. On the other side, the possibility of blood sharing between blood banks by maximum possible distance can mitigate the shortage and result in less wastage and more service quality for patients.

• Although the proposed model and methodology in this research were applied for real data of IBTO in Iran, our designed mechanism can be beneficial for other BSN in countries that face the challenge of reducing blood donation during the COVID-19

outbreak.

Several directions for future study can be suggested. Due to the limited blood supply and donors in disaster situations like the COVID-19 outbreak, motivational initiatives can be considered to encourage blood donations in blood collection centers. Furthermore, different exact or heuristics solution approaches can be developed for this problem, resulting in optimal solutions quickly for large-sized problems.

The proposed model could be applied to another future research avenue by taking other uncertainty approaches such as data-driven or distributionally robust optimization approaches. Another domain for future research could also be extended this model by considering different blood groups and compatibility rules to satisfy the patient's demands and reduce the shortage level during the epidemic outbreak.

Moreover, when a disaster strikes, preparing a scheduling plan for blood collection via an apheresis mechanism from booked donors can help this network to cover the unfulfilled demand in hospitals. Finally, as further extensions for this study, transshipment of blood units between hospitals and collaboration to sharing the leftover blood units in blood banks to cover the total unfulfilled demand can significantly mitigate the shortage in this urgent situation.

A considerable reduction in blood donations was observed in epidemic outbreaks like the SARS epidemic in 2003; Similarly, during the Influenza pandemic in 2009 and in the recent COVID-19 outbreak. Monitoring the supply and demand for blood and its by-products should be escalated during and after the epidemic outbreaks like COVID-19 due to their effects on patients so that adequate blood is maintained to support ongoing critical needs. Some reasons such as fear of infection, inconvenient location, weakened immune system, and avoidance of public places has been lead to a reduction in blood donation in regions at the peak of the outbreak. While in this situation, regions with a low prevalence rate of the epidemic frequently have no challenges in preparing the required blood units. A proper strategy in blood collection management should be implemented to reduce the shortage of blood and its by-products in regions of the country. In this research, by conducting a capacity sharing concept, a two-stage optimization tool to coordinate the blood collection activities in BSN was proposed to lower the shortage and wastage during the COVID-19 outbreak. To prevent BSN malfunctions in this urgent situation, a novel hybrid stochastic possibilistic-flexible robust programming approach was developed. Due to the dynamic nature of the COVID-19 outbreak, a rolling horizon mechanism was adopted to implement the decisions by the capability to reconsideration in blood collection activities. In collaboration with IBTO, the applicability and performance of the proposed model and methodology were implemented on real data in Iran country. The computational results and sensitivity analysis revealed that the capacity sharing concept has a good potential to mitigate the shortage level of blood units in regions, especially at the peak of the COVID-19 outbreak.

Appendix A 

A decision support system for demand management in healthcare supply chains considering the epidemic outbreaks: A case study of coronavirus disease 2019 (COVID-19)

Risk Analysis in Logistics Systems: A Research Agenda During and After the COVID-19 Pandemic

Innovative "bring-service-near-your-home" operations under Corona-virus (COVID-19/SARS-CoV-2) outbreak: Can logistics become the messiah

Viability of intertwined supply networks: extending the supply chain resilience angles towards survivability. A position paper motivated by COVID-19 outbreak

Balancing supply and demand for blood during the COVID-19 pandemic

The blood supply management amid the COVID-19 outbreak

Management of blood supply and demand during the COVID-19 pandemic in King Abdullah Hospital

Blood supply sufficiency and safety management in Iran during the COVID-19 outbreak

Analysis of blood donors' characteristics and deferrals related to COVID-19 in Iran

COVID-19 pandemic and comparative health policy learning in Iran

Coronavirus disease 2019 (COVID-19) and decrease in blood donation: experience of Iranian Blood Transfusion Organization (IBTO)

Supply chain management of blood products: A literature review

A structured review of quantitative models in the blood supply chain: a taxonomic framework for decision-making

Survey on blood supply chain management: Models and methods

Optimization of blood supply chain with shortened shelf lives and ABO compatibility

Dynamic supply chain network design for the supply of blood in disasters: A robust model with real world application

Approximate dynamic programming modeling for a typical blood platelet bank

Stochastic integer programming models for reducing wastages and shortages of blood products at hospitals. Computers & Operations Research

Blood collection management: Methodology and application

Selective vehicle routing for a mobile blood donation system

Mathematical programming models for annual and weekly bloodmobile collection planning

Minimizing operational costs by restructuring the blood sample collection chain. Operations Research for Health Care

The inventory centralization impacts on sustainability of the blood supply chain

Mathematical programming models for annual and weekly bloodmobile collection planning

Threshold-based allocation policies for inventory management of red blood cells

Blood inventory management in hospitals: Considering supply and demand uncertainty and blood transshipment possibility. Operations Research for Health Care

Designing a bi-objective multi-echelon robust blood supply chain in a disaster

Mixed robust possibilistic flexible chance constraint optimization model for emergency blood supply chain network design

Dynamic temporary blood facility location-allocation during and post-disaster periods

Robust platelet logistics planning in disaster relief operations under uncertainty: a coordinated approach

Robust optimization model for integrated procurement, production and distribution in platelet supply chain

Bi-objective emergency blood supply chain network design in earthquake considering earthquake magnitude: a comprehensive study with real world application

Developing a robust stochastic model for designing a blood supply chain network in a crisis: a possible earthquake in Tehran

Supply chain design for efficient and effective blood supply in disasters

Blood supply chain network design considering blood group compatibility under uncertainty

A two-stage stochastic programming model for inventory management in the blood supply chain

Managing blood inventory with multiple independent sources of supply

Raising quality and safety of platelet transfusion services in a patient-based integrated supply chain under uncertainty

Blood supply chain network design under uncertainties in supply and demand considering social aspects

Integrated blood supply chain planning for disaster relief

Simulation-optimization model for production planning in the blood supply chain. Health care management science

Designing the blood supply chain: how much, how and where

Designing and optimizing a sustainable supply chain network for a blood platelet bank under uncertainty

Green-blood supply chain network design: Robust optimization, bounded objective function & Lagrangian relaxation

Designing a robust and dynamic network for the emergency blood supply chain with the risk of disruptions

A two-stage multi-echelon stochastic blood supply chain problem

An enhanced procedure for managing blood supply chain under disruptions and uncertainties

A multilateral perspective towards blood network design in an uncertain environment: Methodology and implementation

Itemized platelet supply chain with lateral transshipment under uncertainty evaluating inappropriate output in laboratories

Robust design of blood supply chains under risk of disruptions using Lagrangian relaxation

A bi-objective integrated model for the uncertain blood network design with raising products quality

Socially optimal contracting between a regional blood bank and hospitals

Robust and stable flexible blood supply chain network design under motivational initiatives

Proactive transshipment in the blood supply chain: A stochastic programming approach

A mixed resilient-efficient approach toward blood supply chain network design

Blood supply chain management: robust optimization, disruption risk, and blood group compatibility (a real-life case)

Toward a coordination of inventory and distribution schedules for blood in disasters

A robust framework for designing blood network in disaster relief: a real-life case

A reactive phase against disruptions for designing a proactive platelet supply network

The multi-period location-allocation problem of engineering emergency blood supply systems

A robust optimization model for distribution and evacuation in the disaster response phase

The multi-period location-allocation problem of engineering emergency blood supply systems

Responsive and reliable injured-oriented blood supply chain for disaster relief: a real case study

Potential challenges faced by blood bank services during COVID-19 pandemic and their mitigative measures: The Indian scenario. Transfusion and Apheresis Science

High throughput and comprehensive approach to develop multiepitope vaccine against minacious COVID-19

The blood supply management amid the COVID-19 outbreak

Joint decisions for blood collection and platelet inventory control

Blood bank inventory management in India

A robust mathematical model for platelet supply chain considering social announcements and blood extraction technologies

Responsive and resilient supply chain network design under operational and disruption risks with delivery lead-time sensitive customers

Risk diversification and risk pooling in supply chain design

OR/MS models for supply chain disruptions: A review

Robust optimization of large-scale systems

Novel robust fuzzy mathematical programming methods

Innovative strategy to design a mixed resilient-sustainable electricity supply chain network under uncertainty

Fuzzy optimization for supply chain planning under supply, demand and process uncertainties. Fuzzy sets and systems

Using fuzzy numbers in linear programming

Ranking fuzzy subsets over the unit interval

Linear programming with fuzzy parameters: an interactive method resolution

An efficient hybrid particle swarm optimization algorithm in a rolling horizon framework for the aircraft landing problem

A two-stage fuzzy optimization model for scarce drugs supply and ration planning under uncertainty: A case study

Aggregate formulations for large-scale process scheduling problems

A rolling horizon optimization framework for the simultaneous energy supply and demand planning in microgrids

Goal programming, compromise programming and reference point method formulations: linkages and utility interpretations

A mixed uncertainty approach to design a bioenergy network considering sustainability and efficiency measures

A bi-level programming approach for improving relief logistics operations: A real case in Kermanshah earthquake

A stochastic bi-objective simulation-optimization model for plasma supply chain in case of COVID-19 outbreak

 Esfahan  409  517  434  318  331  360  395  6.51  5  Alborz  192  246  209  202  181  180  202  3.32  6  Ilam  47  46  45  39  34  38  41  0.68  7  Boushehr  202  203  128  79  65  89  128  2.10  8  Tehran  1012  1025  1172  867  772  877  954 15. 73 9 Chahar Mahal Bakhtiari 62 80