A New Multiechelon Mathematical Modeling for Pre-and Postdisaster Blood Supply Chain: Robust Optimization Approach

Disaster management is one of the most important actions to protect the property and lives of the victims. Failure to pay attention to logistical decisions of disaster can have irreversible consequences. Therefore, a multiechelon mathematical model for blood supply chain management in disaster situations is proposed in this research. The proposed supply chain includes supplier, central warehouse, reliable distributor, unreliable distributor, distributor


Introduction
One unit of blood can be used once for a patient or several times.In addition to urgent cases, blood therapeutically supports (not for treatment) diseases such as hemoglobin, coagulation, and bone marrow disorders.Having a stable blood supply system requires good services at the level of the blood transfusion organization, a suitable platform such as volunteer donors (donating blood without receiving money and altruistically), rigorous testing systems, a desirable quality system, and a proper monitoring process.Having such a system requires the commitment and support of national health o cials and the use of appropriate human, nancial, and technological resources [1,2].e most important challenge after a major event such as an earthquake is locating relief distribution centers and sending essential goods to the a ected areas in order to meet the basic needs of the a ected [3].Disaster relief logistics is one of the main activities in disaster management.What is important in relief logistics is the value of time and delivering the necessary goods and services to the a ected in the shortest possible time [4].One of the important logistic strategies to improve performance and reduce latency is to locate and establish relief distribution centers near these vulnerable areas [5][6][7].us, the existence of distribution centers in suitable locations of the network, which can properly cover the demand created in these conditions, is of great importance in the successful implementation of relief and rescue operations, and in all cases, the weakness in choosing the right location will increase the likelihood of wasting capital and will ultimately lead to many casualties [8,9].e goals pursued in the issue of locating relief distribution centers are to get the location of relief distribution centers closer to the affected areas and reduce the distance (objective minimization function) to deliver services as fast as possible, increase the level of services, and attempt to balance the distance between distribution centers and demand areas (affected people).
e golden time to rescue the injured in an earthquake is only 72 hours [10].As it is clear, not paying attention to logistical decisions can have harmful financial and human consequences.us, minimizing relief time is one of the most important goals of logistics measures.Also, uncertainties in disaster situations make the issue of relief more complicated.
erefore, in this research, a multiechelon mathematical model for blood supply chain management in disaster situations is proposed.e proposed supply chain includes supplier, central warehouse, reliable distributor, unreliable distributor, distributor, and affected areas.How the proposed model performs is explained as follows: blood is sent from the supplier to warehouses and distribution centers.Also, the capacity of suppliers is limited.e main objective of the mathematical model is to minimize supply chain costs while maximizing the level of satisfaction in order to meet the demand of the affected area.Hence, this research seeks to decide whether or not to establish a reliable distributor, unreliable distributor, and central warehouse.
e amount of blood sent to the centers will also be calculated.e amount of blood sent to the centers will also be calculated.
e main contributions of the study are as follows: (i) Considering the pre-and postdisaster modes simultaneously.(ii) Locating and investigating the flow between centers.(iii) Solving the proposed model using robust optimization approach.

Literature Review
Shaw et al. [11] provide a mathematical model for location and resource allocation in disaster condition.e uncertainty of the model was fuzzy and was solved using chanceconstrained approaches.Minimizing total delivery time and maximizing total affected coverage are the most important goals of their model.Considering loading and unloading times is one of the innovations of this research.e results indicate the proper performance of the proposed model.
Alizadeh et al. [12] provided a multiperiod model for locating relief facilities in natural disaster situations.e main objective of their research is to maximize the coverage of hospitals and distribution centers.e Lagrangian approach is used to solve the proposed model.e results of the case study indicate that an increasing demand reduces the level of coverage of the areas.Alinaghian et al. [13] addressed locating and allocating disaster relief centers in the pre-and postearthquake conditions.Consideration of temporary relief centers to prevent crowding of the injured is one of the innovations of this research.In order to solve the proposed model, the harmonic algorithm and the forbidden search along with the neighborhood search are used.Numerous numerical examples solved with robust optimization indicate the proper performance of the proposed model.Manopiniwes and Irohara [14] proposed a mathematical model for locating temporary relief centers in flood conditions.e main objective of the research is to minimize supply chain costs such as location and transportation.erefore, the proposed model is considered as a multiperiod model, as investigated in different periods of demand.One of the considered innovations is the consideration of several modes of transportation.e results of solving the case study indicate the proper performance of the proposed model.Dunn and Gonzalez [15] proposed the development of an adaptive algorithm to increase resilience and the location and allocation of relief bases in flood conditions.Considering possible uncertainty and reliability for the centers are among the innovations considered.ArcGIS software is used to locate the centers.e main objective of the research is to minimize the costs of the whole system, including location costs.e proposed system is solved for various examples and its performance results are satisfactory.Cavdur et al. [16] proposed new strategies for distributing relief goods and human resources in times of disaster.erefore, a possible two-level model for locating and allocating temporary relief centers is provided.
e first level is the predisaster phase and the second level is the postdisaster phase.Minimizing chain costs is done in the first phase and allocating resources to centers is done in the second phase.e results of the sensitivity analysis of the case study indicate that with increasing demand, allocation costs increase sharply.Zhou et al. [17] proposed a mathematical model for blood supply chain management in conditions of uncertainty.e proposed model is implemented using two scenarios of LIFO and FIFO.Considering fairness and minimizing shortage are among the objectives of this research.One of the contributions of the proposed model is demand estimation using discrete event system simulation.Finally, the proposed model is solved using the estimated withdrawal and aging approach.Arani et al. [18] proposed a mixed-integer programming mathematical model for managing blood databases.e considered innovation includes sustainable lateral resupply.Considering inventory-routing decisions is one of the approaches of this research.e proposed model is solved using the multichoice goal programming approach.Shokouhifar et al. [19] proposed a fuzzy mathematical model for blood inventory management.Considering the short lifetime of blood platelets and the cost of corruption is one of the contributions considered.Minimizing the costs of shortages and wastage costs in the supply chain is one of the objectives of the research.e results show that lateral transshipment reduced costs by 3.4%.Shirazi et al. [2] proposed a mathematical model for blood chain management in the Covid-19 outbreak.ey first estimated the blood demand using discrete simulation and then the value of this parameter was entered into the mathematical model.e proposed model is solved by the approaches of strength pareto evolutionary algorithm 2 (SPEA-II) and multi-objective gray wolf optimizer (MOGWO).e results show that with increasing demand, the number of established relief bases increases sharply.Khalili-Damghani et al. [20] proposed a mathematical model for cascade disaster management under uncertainty conditions.e considered innovation includes location using the GIS approach.e main objective of the research is to maximize the coverage level.e proposed model is solved using the invasive weed optimization algorithm approach.Table 1 lists the summary of literature review.
One of the research gaps is the lack of attention to the pre-and postdisaster situations simultaneously.Also, locating and examining the flow between the centers are among the research gaps considered.Lack of attention to solving the proposed model using a sustainable approach is another research gap in this research.

Problem Statement
is section presents problem assumptions and mathematical modeling as follows: (iii) Doing transactions between LDCs is allowed.Relief items can be sent from reliable LDCs to unreliable items in case of shortage (ancillary transportation).(iv) Distribution of relief items in the response phase is assumed for several periods.(v) Suppliers can procure postdisaster relief items and distribute them in LDC.

Problem Assumptions
In this section, we will model the designed network and describe sets, then parameters, and then variables, and the problem indices are as follows: (i) I: Supplier J: Central warehouse K: Reliable distributor L: Unreliable distributor M: Distributor D: Affected areas T: Period c: Product Now, we have described the problem indices, and we will examine the problem parameters as follows: g j : Fixed cost of building a warehouse j fr k : Fixed cost of building a reliable distributor k fu l : Fixed cost of building an unreliable distributor l aq c : Cost of maintenance of relief good c pc ic : Cost of supplying relief product c before the disaster by supplier i pc cit : Cost of supply of relief product c by supplier i after the disaster in period t tc ij : Cost of transportation from supplier i to distributor j tc jm : Cost of transportation from warehouse j to central distributor m tc md : Cost of transferring from the central distributor m to the affected area d tc kl : Cost of transferring the product from a reliable distributor k to an unreliable distributor l π cdt : Cost of fining a unit of relief product c in affected center d in period t cap ic : Supply capacity of relief product c by supplier i w cm : Capacity of distributor warehouse m for relief product c v cj : Central warehouse capacity j for relief goods c v c : Volume of relief product c φ m : Percentage of storage capacity that is likely to deteriorate d cdt : Demand for relief product c by affected area d in period t Now, the problem parameters have been described, and we will examine the research variables as follows: qc cij : Transfer of product from supplier i to warehouse j ql cim : Transfer of product c from supplier i to distributor m before the disaster q cimt : Transfer of product c from supplier i to distributor m in period t after the disaster y cjmt : Transfer of product from warehouse j to distributor m in period t x cmdt : Transfer of product from distributor m to affected area d in period t u cklt : Transfer of product from a reliable center k to unreliable center l in period t it cmt : e amount of distributor m from relief product c in period t b cdt : Deficiency of product c in affected area d in period t zr k : Takes a value of 1 if center k is constructed and otherwise zero.zu l : Takes a value of 1 if center l is constructed.δ j : Takes a value of 1 if center j is constructed.Now, the indices, parameters, and variables of the research have been described, and we will provide the mathematical model of the research as follows: (1) e objective function (1) in the first part will minimize the cost of establishing a local collection center and charity center (consisting fixed cost of building a warehouse, fixed cost of building an unreliable distributor, and fixed cost of building a reliable distributor), and in other parts, it will minimize the cost of sending goods between different levels of the network (consisting supplying relief product c before the disaster by supplier, maintenance of relief good, transportation from supplier to distributor, and cost of fining relief product in affected center): e second objective function of the problem will maximize the level of satisfaction in order to meet the demand of the affected area: Constraints ( 4) and ( 5) will address the issue that no relief goods will be sent until the warehouse and distribution center are activated.
Constraints ( 6) and ( 7) are inventory equilibrium constraints, and constraint (8) states that the amount of goods sent plus the shortage must be equal to the demand of the affected area: Constraints ( 9) and ( 10) will examine the capacity of the supplier and the central warehouse, which ensures that the maximum number of goods sent is equal to their capacity: Constraints ( 11) and ( 12) ensure that it is not possible to send to the distribution center unless reliable and unreliable centers are active: Constraints ( 13) and ( 14) state that the maximum amount sent to the affected area must be equal to the supply capacity: Constraints ( 15) to (17) examine the capacity of the distribution center before and after the disaster: Constraints ( 18) to ( 21) state that no goods must be sent until the distribution center is active.

Solution Method: Scenario-Based Robust Optimization.
Suppose U is a set of all possible scenarios that may occur in the future and S is a specific scenario of this set.ese two concepts are in fact the stability of the model and response.In the response stability, the optimal response of the model for each scenario is close to the desired value while the response of the model for each scenario in the stability of the model is nearly practical: e first part is the mean and the second part is the variance of the objective function.A is the weight of variance depending on the decision-maker and is obviously chosen by the degree of risk aversion in the decision-maker.However, since the above function is nonlinear in nature, according to Yu [21], the following function will be used in the problem: e following constraint will be used to linearize the above equation: erefore, the model under the scenario will be as follows: Discrete Dynamics in Nature and Society (25)

Results
In this section, we will first examine the problem in small dimensions in certain conditions and then we will solve the problem in conditions of uncertainty.For this purpose, we should first describe the dimensions of the problem, then initialize the various parameters of the model, and then examine the research variables and how they perform in the model (Table 2).Now, the dimensions of the problem have been determined, and we will initialize the parameters for this purpose.
e first parameter that will be determined is the amount of product use, which is specified based on Table 3.
As can be seen in the table above, the fixed cost of building a central warehouse is described.After this, we need to describe the fixed cost of building a reliable and unreliable distribution center and the cost of maintaining relief goods, as listed in Table 4.

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Discrete Dynamics in Nature and Society Now, we have to describe the cost of supplying relief goods before the disaster, as listed in Table 5.Now, the cost of supplying relief goods before the disaster has been determined, and we must now address the cost of supplying relief goods after the disaster, as listed in Table 6.Now, after the cost of supplying the relief goods has been determined, we have to address the transfer cost of the relief goods, as listed in Table 7.
Table 8 lists the cost of transferring relief goods from supplier to central distributor.
Table 9 lists the transfer cost of relief goods from reliable distributor to unreliable distributor.Now, we need to determine the cost of transporting relief goods from the distributor to the affected area, as listed in Table 10.

Examination of the Problem in Conditions of Uncertainty.
In order to examine the problem in conditions of uncertainty, it is first necessary to determine values of the parameters in the conditions of uncertainty in the research and then examine the results.For this purpose, it is first necessary to determine scenarios for the occurrence of an accident with certain probabilities, as listed in Table 11.Now, the different scenarios have been identified, and we need to determine the values of the parameters that have uncertainty in the research, as listed in Table 12.Now, in order to understand what effect robust optimization has on the objective function of the problem, we will examine different objective functions under different scenarios, and the results are listed in Table 13.Now, after examining the objective function under different scenarios, the worst value of the objective function must be determined under uncertainty conditions, which is 5398653, that is, the worst answer of the system.

Analysis of Objective Function Relative to Standard Deviation of Response A 2 .
In order to investigate how robust optimization works, we intend to perform a sensitivity analysis on the effect of robust optimization on the objective function.For this purpose, we will analyze the objective function based on Table 14, and since the problem is minimization, it has to have an ascending value with an increasing standard deviation of the objective function.
As can be seen from Figure 1, by increasing the robustness coefficient, the response of the objective function increases, which indicates the proper performance of the model.

Changes of the Objective Function
Relative to the Amount of the Fine.Now, in order to determine how the objective function changes relative to the amount of the fine, the following experiment has been designed, in which the constant coefficients and the degree of violation of the objective function have increased (Table 15 and Figure 2).

Managerial Insight and Practical Implication
e benefits of the proposed model in this study can be useful for the Blood Transfusion Organization, the Red Cross, the Crisis Relief Organization, the municipality, and ultimately all people.e main objective of the mathematical model is to minimize supply chain costs while maximizing the level of satisfaction in order to meet the demand of the affected area.Hence, this research seeks to decide whether or  erefore, this study intends to provide a model to meet the needs of managers in these conditions.Also, considering that in response conditions, the main goal is usually a quick response to minimize costs and maximize the satisfaction of the injured; in the presented model, an attempt has been made to provide a plan taking into account the real logistical conditions.

Conclusion
In this research, a mathematical model to minimize logistics costs in the blood supply chain in disaster situations is proposed.One of the innovations of the proposed model is to consider the pre-and postdisaster modes simultaneously.Locating and examining the flow between centers are also the other contributions of this study.Solving the proposed model using the robust optimization approach is another innovation considered in this research.e benefits of the proposed model in this study can be useful for the Blood Transfusion Organization, the Red Cross, the Crisis Relief Organization, the municipality, and ultimately all people.
In order to examine the problem in conditions of uncertainty, it is first necessary to determine values of the parameters in the conditions of uncertainty in the research and then to examine the results.For this purpose, it is first necessary to determine scenarios for the occurrence of an accident with certain probabilities.Also, we will analyze the objective function, and since the problem is minimization, it has to have an ascending value with an increasing standard deviation of the objective function.Finally, the robust optimization method was used to solve the problem, which examined the problem in conditions of uncertainty.Future suggestions are as follows: It is interesting to extend the proposed model to perishable relief items.Also, the dangers of disruptions in roads and distribution of the last item and the problem of routing for vehicles can be studied in the future.e resilient supply chain may not be the least expensive supply chain, but the resilient supply chain can overcome uncertainties and disruptions in the business environment.e competitive advantage of the supply chain not only depends on low costs, high quality, reduced latency, and high level of service but also on the ability of the chain to avoid disasters and overcome critical situations, and this is the resilience of the supply chain.Resilience is the ability of the supply chain to overcome unpredictable events.e purpose of creating resilience in the supply chain is to prevent the chain from moving toward adverse conditions.Incidents such as the eruption in Iceland made companies aware of how little control they have over the risks they face, but today, there are companies that are able to return to normal or even get better than before after severe disturbances and fluctuations.erefore, it is suggested that we try to increase the resilience of the problem by considering redundancy [2,9]. is research, like other research, is not without limitation and assumptions.erefore, the limitation of the research is expressed as follows: as there was no official database for some parts of cost elements, the driver's estimations and blood transfusion center officers were asked to help.
(i) Costs related to the shortage of relief items in postdisaster are considered.(ii) Transportation costs depend on items and scenario.

Figure 1 :
Figure 1: Changes in the objective function relative to the robustness factor.

Figure 2 :
Figure 2: Changes in the objective function relative to the robustness factor.

Table 2 :
Dimensions of the problem.

Table 3 :
Fixed cost of building central warehouse j.

Table 4 :
Fixed cost of building a reliable and unreliable distribution center and maintenance cost of relief goods.

Table 5 :
Cost of supply of relief goods before the disaster.

Table 6 :
Cost of supply of relief goods after the disaster.

Table 7 :
Cost of transferring relief goods from supplier to central warehouse.

Table 8 :
Cost of transferring relief goods from supplier to central distributor.

Table 9 :
Transfer cost of relief goods from reliable distributor to unreliable distributor.

Table 10 :
Transfer cost of relief goods from distributor to the affected area.

Table 11 :
Total probabilities of possible scenarios.

Table 12 :
Different parameters under possible scenarios.

Table 13 :
Response of the objective function under different scenarios.

Table 14 :
Robustness analysis of the objective function.

Table 15 :
Robustness analysis of the objective function relative to the amount of the fine.