# Vehicle Routing Problem in Relief Supply under a Crisis Condition considering Blood Types

Authors: Mahsa Rezaei Kallaj; Milad Abolghasemian; Samaneh Moradi Pirbalouti; Majid Sabk Ara; Adel Pourghader Chobar
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Despite the advances achieved in Medical Sciences, no substitute has been found for blood as a vital factor. Therefore, preparing sufficient and healthy blood in crisis conditions is a challenge that health systems encounter. Along with examining the conducted investigations in this field, the main contribution of current research is to develop a biobjective Mixed-Integer Linear Programming (MILP) model for relief supply under crisis condition. For this purpose, this paper proposes a model for routing of bus blood receiver under crisis conditions considering different blood groups. Besides, hours of unnecessary travel by bloodmobiles (buses) between each blood station (BS) and the crisis-stricken city for dispatching the collected blood is prevented thanks to considering a helicopter. The mentioned model has two objectives: maximizing the amount of blood collected by bloodmobiles and minimizing the arrival time of the blood receiver buses and a helicopter to a crisis-stricken city after the collected blood is used up. The model is coded by CPLEX software, and the results obtained from solving the model indicate that, without considering a helicopter, the demand is not supplied within the critical period after crisis. Given that blood cannot be artificially produced, its primary resource is blood donors. Concerning the importance of this issue under crisis conditions, this research investigates the relief vehicles’ routing problem, including bus and helicopter, in a crisis considering supply and transfer of different blood groups to a crisis-stricken city for maximum relief supply and blood transfer within the shortest period.


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## Body

## 1. Introduction

Today, supply chain network design has been a demanding question and attracted great interest in a wide range of fields, such as medical industry. The important aspect of the entire discipline of the medical industry is the concept of blood management which recently brought striking attention among managers and decision-makers [1]. Blood transfusion is one of the most vital medical actions and interventions. However, blood is a highly scarce resource. Since blood cannot artificially be produced, the only blood resource is blood donors. It is reported that the demand for blood is 92 million units per year worldwide, while regular blood donors merely account for 5% of the world population [2]. Generally, the blood network consists of main centers, temporary facilities, bloodmobiles, and laboratory centers. The schematic of the blood supply chain is indicated in Figure 1. The main centers with preservation capacity and equipped with high technologies are responsible for collecting blood. Creating these centers requires very high investment. However, temporary facilities with a limited level of facilities for supplying demand and increasing the number of donors are highly flexible. Effective use of the bloodmobiles may be helpful and increase the number of donors. The bloodmobiles are vehicles equipped with essential equipment for processing the received transfusion blood from donors. Compared to fixed blood donation centers, these vehicles can attract more donors by staying in crowded regions. Besides, employing this system prevents blood scarcity in blood transfusion centers and hospitals in a crisis and reduces blood wastes and system costs [3]. Blood donation starts when an individual visits a blood transfusion center or a service-provider bus. Bloodmobiles start their route from a particular location. Afterward, they follow their route to more centers in each city or even neighboring cities to collect more blood. It may take them several days to continue their routes. In order to prevent the blood collected by bloodmobiles from spoilage during their routes or deliver the collected blood to the patients on time, a helicopter visits bloodmobiles at the end of a day and dispatches the blood collected by bloodmobiles to the crisis-stricken city so that the bloodmobiles do not have to return to the crisis-stricken city at the end of the day. In the blood collecting centers, the blood units are processed and its components are separated. Ultimately, it is transferred to a hospital or a clinic according to the demand [4]. Even though blood units as a product may not be highly subject to spoilage, their separated components can be completely spoiled and destructed within a particular time [4]. Blood units are broken down into blood products, including white blood cells, platelets, and plasma. Red blood cells and platelets can be stored for 42 and 5 days, respectively. Each parameter can be divided into four groups of O, B, A, and AB, which can also be classified into other groups with positive or negative RH depending on the existence of special antigens [4]. Thereby, eight blood groups must be controlled simultaneously. Also, there are recipients (as chain customers) who are in dire need of various blood units and will suffer serious injury and death if they do not receive blood in the time of need. Therefore, not only are the different levels in such networks of a greater importance than those in other supply networks, but also the availability of network flow can determine the recipients’ life at lower levels of the chain [1]. Consequently, for making better use of available resources, advanced approaches for decision-making are needed. Also, better planning for receiving and collecting leads to better handling in times of scarcity such as crisis, i.e., COVID-19 pandemic, earthquake, etc. [5]. Therefore, one of important decisions in blood transfusion is the substitution of blood groups. Ideally, the transfused blood group must be identical to the patient’s blood type, while this is not always possible. When a blood group is not available at the time of the request, a compatible blood group with the desired blood group must be provided [6]. According to Table 1, the donated blood types are categorized based on the blood receiver and donor [3].Figure 1
Routing-inventory of the bloodmobiles and the helicopter in blood supply chain.Table 1
Blood groups that are compatible with each other.
Receptor blood typeDonor blood type+AB−AB+B−B+A−A+O−O−O∗+O∗∗−A∗∗+A∗∗∗∗−B∗∗+B∗∗∗∗−AB∗∗∗∗+AB∗∗∗∗∗∗∗∗In general, the term “crisis” refers to any situation that temporarily limits the capability of services to preserve, store, and supply blood more than usual or creates a situation that increases the sudden demand for blood more than usual [6]. Disasters often lead to many problems, such as damage, financial losses, fatalities, and transportation problems in the damaged regions. Eighty-three thousand disasters have occurred from 2000 to 2012 worldwide [2]; for instance, demand is usually more than supply during the first 24 hours after the earthquake. Another concern regarding the design of blood supply chain networks is the limitation in the storage and transportation of blood products [7]. Compared to normal conditions, there is more demand for water, food, shelter, medical equipment, and other vital requirements, such as blood as a vital product after a disaster [2]. In order to tackle the challenges mentioned above, a novel and effective collecting system is proposed in this research for the vehicle routing problem, such as bloodmobiles and relief supplier helicopters after a crisis employing the concept of routing and using mathematical modeling. Besides, it is assumed that crises happened in a city, and the blood receiving bloodmobiles collect blood from certain cities. Accordingly, in the mathematical model, a bloodmobile is allowed to stop at each visited city in order for the collected blood to be maximized. In addition, given that a crisis has taken place, the arrival time of the bloodmobiles and helicopters to the crisis-stricken city is minimized. Besides, the relief helicopters collect and dispatch the blood left in the bloodmobiles in the cities to the crisis-stricken city. Indeed, this investigation carries out the routing of blood receiving bloodmobiles and relief helicopters in the crisis-stricken location such that the maximum amount of blood is collected within the shortest period. Since the number of injured individuals suddenly increases in a crisis, patients wait to receive blood when the blood is being collected. Thus, the earlier each donation vehicle arrives at the blood bank, the more patients receive blood and are healed. Hence, this research accords a high priority to the minimum arrival time of the blood receiving bloodmobile and helicopter to the crisis-stricken city. The following suppositions are the main objectives of the paper:(i)
Developing a blood network to consider both amount of collected blood and arrival time to service(ii)
Maximizing the amount of collected blood and minimizing the arrival time of service provider into the crisis-stricken city(iii)
Utilizing deterministic programming to evaluate the model behavior in the real-world case studyThe following of this article is arranged as follows: In Section 2, a historical review of the issue is presented. In Section 3, along with introducing the methodology, the problem assumptions and mathematical model are presented. The problem-solving results are indicated in Section 4. Finally, in Section 5, a conclusion and some suggestion for future researches are presented.


## 2. Literature Review

The investigations into the management of spoilable products’ supply chain and blood products specifically started in 1960 and culminated in the early 1980s [8]. During these years, various studies have been conducted on the blood supply chain, addressing different problems, such as the location of regional centers, blood banks, inventory, and routing problems of vehicles for blood collection and distribution. In recent years, some investigations into blood supply chain management have been conducted, which are introduced in the following. For instance, considering the location and allocation of them to the closest blood bank, Prastacos [9] proposed a mathematical model. This model aimed to minimize the total costs of the supply chain. In 2017, Haghjoo et al. [10] proposed a random bistage mathematical model for the design of a blood supply chain network and inventory management. Sha and Huang [11] proposed a multiperiod location-allocation mathematical model for the design of a blood supply chain network under emergency circumstances. They employed the Lagrange Relaxation method to solve the proposed mathematical model. Jabbarzadeh et al. [7] proposed a robust optimization mathematical model for the design of the blood supply chain network in an earthquake. They aimed to design a blood supply chain network with costs of three levels, including blood donors, blood centers, and blood collecting centers. The blood centers aimed to minimize the total costs of the supply chain. Zahiri and Pishvaee [12] proposed a biobjective mathematical model aiming to minimize the total costs of the supply chain and the unsupplied demands. This article designs a supply chain network by considering the compatibility of blood groups. For this purpose, a biobjective mathematical model is developed that minimizes the total costs and addresses the maximum undesired demand. Gunpinar and Centeno [13] modeled a vehicle routing problem employing an integer programming approach to simultaneously identify the number of blood transfusion buses and minimize the traveled distance. Besides, this model is developed to consider the uncertainty in the blood donation potential and the variable length in the visitors of the bloodmobiles. Rodríguez-Espíndola et al. [14] proposed a collaborative humanitarian approach for disaster preparedness for logistic resource management when a natural disaster happens. In this model, the multiobjective optimization model is employed for making emergency logistic management decisions. Paul and Wang [15] proposed a robust network of allocation of relief supply pieces of equipment for earthquake preparedness. This robust network can optimize the number of equipment, capacities, and distribution centers and reduce social costs. Parameters with uncertainty in this model are listed as follows: Damaged facilities, casualties because of the intensity of an incidence, and the travel time of the relief supplier vehicles. In a recent study, Adarang et al. [16] proposed a temporary robust biobjective location-routing model for providing emergency medical services. This model aimed to minimize the relief supplying time and total cost. Total costs are the sum of spatial costs and the cost of covering the route by vehicle (ambulance and helicopter). Arani et al. [17] presented a novel mixed-integer programming model for designing a sustainable lateral resupply blood supply chain network. In this paper, we addressed a blood inventory-routing problem with supply and demand uncertainties. Also, we developed a scenario-based stochastic optimization model and revised multichoice goal programming approach to solve. Goodarzian et al. [18] considered an economic green medicine supply chain network under uncertainty. In this study, we developed multiperiod, three-echelon, multiproduct, and multimodal transportation based fuzzy biobjective Mixed-Integer Linear Programming. Davoodi and Goli [19] presented an integrated model for location, allocation, and routing in disaster relief response. In this paper, minimizing last visit time is used as a suitable objective function in rural areas disasters. For faster convergence Benders decomposition was developed using metaheuristics. Finally, we analyzed a real case to demonstrate the applicability of the research methodology. Samani and Hosseini-Motlagh [20] presented a novel capacity sharing mechanism to collaborative activities in the blood collection process during the COVID-19 outbreak. According to this research, a two-stage optimization tool for coordinating activities to mitigate the shortage in this urgent situation is considered. In the first stage, a blood collection plan considering disruption risk in supply to minimize the unmet demand will be solved. Then, 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. Tirkolaee et al. [21] presented a robust biobjective mathematical model for disaster rescue unit’s allocation and scheduling with learning effect. For this purpose, the effect of learning in the disaster management problem is considered. Then, a biobjective robust MILP model for rescue units allocation and scheduling is designed. Finally, a multichoice goal programming (MCGP) with utility functions to cope with the biobjectiveness of the model is applied. Wang et al. [22] presented a framework for optimization of warehouse location and resources distribution for emergency rescue under uncertainty. For this purpose in this paper, a mixed-integer programming (MIP) model based on time cost under uncertainty is proposed, which helps solve the emergency warehouse location and distribution problem. Mamashli et al. [23] presented a heuristic-based multichoice goal programming for the stochastic sustainable-resilient routing-allocation problem in relief logistics. The proposed model aims at minimizing total traveling time, total environmental impacts, and total demand loss. The fuzzy robust stochastic optimization approach is utilized to cope with uncertain data arising in disaster conditions. Then, due to the complexity of the research problem, a hybrid approach based on the multichoice goal programming method and a heuristic algorithm is developed to solve the problem in a reasonable time. Mousavi et al. [1] evaluated an integrated sustainable medical supply chain network during the COVID-19 pandemic. Three hybrid metaheuristic algorithms, namely, ant colony optimization, fish swarm algorithm, and firefly algorithm, are suggested, hybridized with variable neighborhood search used to solve the sustainable medical supply chain network model.Although several studies on vehicle routing under crisis conditions have been introduced in the literature review, a study that addresses the defined condition mentioned in this study has not been found despite the carried-out examinations. Therefore, in this research, the routing of the blood receiver bus and relief supplier helicopter in the crisis-stricken area is carried out such that the maximum amount of blood is collected within the shortest possible period. In this case, the arrival time of the blood receiver bus and the relief supplier helicopter from the defined cities to the crisis-stricken city is minimized.


## 3. Methodology

### 3.1. Problem Statement

This study aims to determine which of the equipped buses visit which stations. Also, it determines in what order and how much time the stations should be visited. Besides, it specifies in what order the helicopter visits the stations to take delivery of the collected blood. With such a case, the collected blood by the bloodmobiles (buses and helicopters) is maximized until the end of the day. Since the condition is critical, the lower the blood collection time is, the more the individuals are saved. Accordingly, this is a problem with two objective functions. The first objective is to maximize the collected blood by considering the blood type. The second objective is to minimize the longest time for arrival of the equipped buses and the helicopter to the crisis-stricken city. Since the amount of collected blood is inversely proportional to the blood collection time, our objective is to determine the variables such that a balance is struck between the blood collection time and collected blood amount.


### 3.2. Model Structure

Generally, proposed blood supply chain consists of main centers, mobile centers, and laboratory centers. Mobile centers are a few numbers of equipped bloodmobiles (bus). Blood buses are equipped with several beds, the pieces of equipment required for receiving blood from a donor, and a lab for monitoring the blood accuracy, a few nurses, and one physician. Shuttle, which is indeed a helicopter, prevents unnecessary hourly bus travels for receiving the collected blood between each BS and the crisis-stricken city. In this case, the buses are not required to transfer the collected blood to the crisis area, and a helicopter will take this responsibility. Thus, we assume that a crisis has happened in a city. This problem consists of a blood bank located in the crisis-stricken city and several nodes that are the cities and the defined villages of the city. The blood bank is equipped with a specific number of bloodmobiles (a number of buses) and a relief helicopter. The equipped buses (bloodmobiles) start moving from the blood bank and visit the blood stations (BS) for blood collection; they should return to the blood bank until the end of the day. Concerning the proposed model, the buses decide for how much time (hours) they make a stop in each BS to maximize the collected blood. Besides, an excessive stop in a city confines the opportunity of visiting other cities. Therefore, making more stops in a city does not necessarily mean more blood collection. Since a shortage would be at the cost of more human lives, it is not allowed in the problem, and all of the demands should be satisfied. The buses can provide the collected blood from each BS to that station by considering the blood preservation condition until the helicopter reaches that point. The main objective of the helicopter is to visit the buses leaving or that have left the stations each hour, taking delivery of their collected blood. In this case, they can deliver the collected blood to the crisis-stricken city. This approach enables the buses to continue the predefined routes and make complete use of the other blood stations without having to return to each station of the crisis-stricken city at the end of being in service time. However, at the end of each route, the buses deliver the remaining collected blood to the blood bank. Therefore, the helicopter does not visit every blood station (BS).


### 3.3. Model Assumption and Notation

In this section, all elements of the assumption are presented in Table2.Table 2
Detail of the proposed model.
Assumption(i) The start time of the blood donation operations is 8:00, and the end time is at most 20:00.(ii) The stop time has a constraint with a minimum and maximum limit.(iii) The collected blood in each BS depends on the city population and the stop time in that station.(iv) There is a specific demand for blood that should be satisfied; the shortage is not allowed. Therefore, it is a vital constraint.(v) Each equipped bus and the helicopter start their operations from the crisis-stricken city; after visiting various areas, it returns to the crisis-stricken city. The number of equipped buses is limited and specified.ParametersNotationDescriptionNMaximum number of points chosen as the stationKThe number of bloodmobiles (buses)DThe donors’ populationαThe blood type percentage in that population (the normality of universal blood type)RbThe demand of blood typestijThe interval (time) that a bus travels from cityi to city jttijThe interval (time) a helicopter travels from cityi to city jBetaThe population of donors frequently donating bloodBinary variableXijk=1,busktravels from cityitoj,0,otherwise.Zij=1,the helicopter travels from cityitoj,0,otherwise.Yik=1,buskvisits cityi,0,otherwise.YDik=1,after receiving the collected blood, if buskleaves them at cityi,0,otherwise.wj=1,the helicopter visits cityj,0,otherwise.Nonnegative variableNotationDescriptionTTiThe required time for arriving cityiSikThe stop time (hours) of bus at citykSSiThe stop time (hours) of the helicopter at cityQ1kbThe amount of collected blood of blood types by bloodmobiles (buses)Q2bThe amount of collected blood of blood types by the helicopterQbThe total amount of collected blood by the bloodmobiles (buses) and the helicopter per blood typeFbSik=Sik.Di.α.betaFunction of the amount of collected blood typeb by bus k at city i, when it stopped Sik hours


### 3.4. Model Notation and Formulation

In current section, objectives and constraints of the mentioned problem are modeled as follows. The objective function includes two sections:(1)MaxZ1=∑b=14Qb.Equation (1) formulates the first objective. This objective firstly maximizes the amount of collected blood.(2)MinZ2=maxTB,TM,(3)TB=maxTn+1k∗Yn+1k,∀kϵ1,…,K,(4)TS=maxTTn+1∗Wn+1.Equation (2) indicates second objective, that is, minimizing the arrival time of the service-provider bus and helicopter to a crisis-stricken city that separate under assumptions intended for TB and TS in equations (3) and (4), respectively. Thus, the objective of the model is to minimize the arrival time of the service-provider bus and the helicopter to the crisis-stricken city if the collected blood is maximized by considering the blood type.(5)Xiik=0,∀i∈1,…,n+1,k∈K.Constraint (5) indicates that bloodmobile k (buses) cannot travel from city i to city i (loop is not allowed).(6)X1n+1k=0,∀k∈K.Constraint (6) indicates that bloodmobiles k (buses) cannot travel directly from the starting city to the crisis-stricken city.(7)∑i=0nXijk=∑i=1n+1Xjik=Yjk,∀j∈1,…,n,K∈1,…,k.Constraint (7) is the flow balance; based on this constraint, if bloodmobile k (buses) enters the BS of city j from each point, it should drive to other stations merely from the BS of city j only if the mentioned bus (bus k) has visited city i. In order words, for each input, one output should exist.(8)Yik≤∑j=2nX1jk,∀iϵ1,…,n+1,kϵ1,…,k.Constraint (8): bloodmobile k (buses) should start visiting from starting city.(9)Yik≤∑j=2nXjn+1k,∀iϵ1,…,n+1,kϵ1,…,k.Constraint (9) confines the bloodmobile k (buses) at the end of their route to return to the crisis-stricken city.(10)Xijk≤Yik,∀i,jϵ1,…,n+1,kϵ1,…,k.(11)Xijk≤Yjk,∀i,jϵ1,…,n+1,kϵ1,…,k.Constraints (10) and (11) guarantee that when bloodmobile k (buses) travels from city i to city j, it should visit city i at first and then visit city j.(12)∑k=1KYik≤1,∀iϵ2,…,n.Constraint (12) guarantees that the maximum number of buses visiting a city is 1.(13)∑i=2n+1X1ik≤1,∀kϵ1,…,k.Constraint (13) indicates that, at the time of operation start, each bloodmobile k (buses) can either be used or not. It is obvious that each blood mobile k (bus) travels to merely one city at first.(14)YDik≤Yik,∀i∈2,…,n,kϵ1,…,k.Constraint (14) indicates that if bloodmobile k (buses) visits city i, bus k can either transport the collected blood from city i or leave it there to be transported by the helicopter to the same point.(15)TiK+SiK+tij≤Tjk+1−XijkM,∀i∈1,…,n,j∈2,…,n+1,K∈1,…,k.Constraint (15) is related to the schedule of each bloodmobile (buses). Indeed, it strikes a balance between the entrance and exit time of bloodmobile (buses) from cities.(16)Zii=0,∀i∈1,…,n+1,kϵ1,…,k.Similar to constraint (5), constraint (16) guarantees that the helicopter cannot travel from city i to i (loop is not allowed).(17)Z1n+m=0,∀kϵ1,…,k.Similar to constraint (6), constraint (17) indicates that the helicopter cannot travel directly from starting city to the crisis-stricken city.(18)∑i=0nZij=∑i=1n+1Zji=Wj,∀j∈1,…,n.Constraint (18) is the flow balance constraint of the helicopter, such that if the helicopter enters from each city to city j, it should travel to other cities only from city j (only if it has visited city j).(19)Wi≤∑j=2nZ1j,∀i∈1,…,n+1.Constraint (19) guarantees that the helicopter starts routing from the starting city.(20)Wi≤∑j=2nZjn+1,∀i∈1,…,n+1.Constraint (20) guarantees that the helicopter visits the crisis-stricken city at the end of the route.(21)Zij≤Wi,∀i,j∈1,…,n+1,(22)Zij≤Wj,∀i,j∈1,…,n+1.Constraints (21) and (22) guarantee that when the helicopter is going to travel from city i to city j, it should visit city j; afterward, it visits city j.(23)∑j=2n=1Z1j≤1.Constraint (23) guarantees that when the operation starts, the helicopter can either be used or not. It is clear that the helicopter can travel to only one city at first.(24)Wi=∑kYDik,∀iϵ1,…,n+1.Constraint (24) indicates that if the bloodmobiles (buses) left the collected blood in a city, the helicopter would visit that city.(25)TTi+SSi+ttij≤TTj+1−ZijM,∀i∈1,…,n,j∈2,…,n+1,K∈1,…,k.Constraint (25) is related to the helicopter’s schedule. Indeed, it strikes a balance between the entrance and exit of the helicopter from cities.(26)Sik+Tik.YDik≤TTi.Wi,kϵ1,…,K,iϵ1,…,N.Constraint (26) is about the helicopter’s schedule for collecting blood, such that the helicopter should arrive at that location after delivering the collected blood to city i by bus if city i is not the final station of the route.(27)Q1kb=∑i=1nFbSik.1−YDik,∀bϵ1,…,4,kϵ1,…,K.Constraint (27) is the total blood type b delivered to the crisis-stricken city by bloodmobile k (buses).(28)Q2b=∑k=1K∑i=0nFbSik.YDik.Wi,∀bϵ1,…,4.Constraint (28) is the total blood type b delivered to the crisis-stricken city by the helicopter.(29)Qb=∑k=1KQ1kb+Q2b,∀bϵ1,…,4.Constraint (29) is the buses’ total collected blood and the helicopter per blood type.(30)Q1≥R1,(31)Q1+Q2≥R2,(32)Q1+Q3≥R3,(33)Q1+Q3+Q2+Q4≥R4.Constraints (30) to (33): the collected blood by the blood mobiles (buses) and the helicopter should be equal or greater than the demand of each group. In fact, constraints emphasize that shortage is not allowed.(34)Xijk,Yik,YDik,Wi,Zij∈0,1,∀i,jϵ1,…,n,∀Kϵ1,…,k.In the end, constraint (34) specifies the limits of variables.


### 3.5. Solution Method

In this paper, weighted sum method (WSM) is used for solution method to find the optimal solution. WSM is a multiobjective optimization method in which there will be multiobjective and we have to determine the best solution. WSM combines all multiobjective functions into single scalar, composite objective function using weighted sum according to the following equation:(35)Fx=w1f1+⋯+wnfn.An important issue there is in assigning the weighting coefficients vectorw1,…,wn in the WSM because the optimal solution strongly depends on the chosen weighting coefficients. Also, these weights have be positive and should be according to the following equation:(36)∑i=1nwi=1,wi∈0,1.


## 3.1. Problem Statement

This study aims to determine which of the equipped buses visit which stations. Also, it determines in what order and how much time the stations should be visited. Besides, it specifies in what order the helicopter visits the stations to take delivery of the collected blood. With such a case, the collected blood by the bloodmobiles (buses and helicopters) is maximized until the end of the day. Since the condition is critical, the lower the blood collection time is, the more the individuals are saved. Accordingly, this is a problem with two objective functions. The first objective is to maximize the collected blood by considering the blood type. The second objective is to minimize the longest time for arrival of the equipped buses and the helicopter to the crisis-stricken city. Since the amount of collected blood is inversely proportional to the blood collection time, our objective is to determine the variables such that a balance is struck between the blood collection time and collected blood amount.


## 3.2. Model Structure

Generally, proposed blood supply chain consists of main centers, mobile centers, and laboratory centers. Mobile centers are a few numbers of equipped bloodmobiles (bus). Blood buses are equipped with several beds, the pieces of equipment required for receiving blood from a donor, and a lab for monitoring the blood accuracy, a few nurses, and one physician. Shuttle, which is indeed a helicopter, prevents unnecessary hourly bus travels for receiving the collected blood between each BS and the crisis-stricken city. In this case, the buses are not required to transfer the collected blood to the crisis area, and a helicopter will take this responsibility. Thus, we assume that a crisis has happened in a city. This problem consists of a blood bank located in the crisis-stricken city and several nodes that are the cities and the defined villages of the city. The blood bank is equipped with a specific number of bloodmobiles (a number of buses) and a relief helicopter. The equipped buses (bloodmobiles) start moving from the blood bank and visit the blood stations (BS) for blood collection; they should return to the blood bank until the end of the day. Concerning the proposed model, the buses decide for how much time (hours) they make a stop in each BS to maximize the collected blood. Besides, an excessive stop in a city confines the opportunity of visiting other cities. Therefore, making more stops in a city does not necessarily mean more blood collection. Since a shortage would be at the cost of more human lives, it is not allowed in the problem, and all of the demands should be satisfied. The buses can provide the collected blood from each BS to that station by considering the blood preservation condition until the helicopter reaches that point. The main objective of the helicopter is to visit the buses leaving or that have left the stations each hour, taking delivery of their collected blood. In this case, they can deliver the collected blood to the crisis-stricken city. This approach enables the buses to continue the predefined routes and make complete use of the other blood stations without having to return to each station of the crisis-stricken city at the end of being in service time. However, at the end of each route, the buses deliver the remaining collected blood to the blood bank. Therefore, the helicopter does not visit every blood station (BS).


## 3.3. Model Assumption and Notation

In this section, all elements of the assumption are presented in Table2.Table 2
Detail of the proposed model.
Assumption(i) The start time of the blood donation operations is 8:00, and the end time is at most 20:00.(ii) The stop time has a constraint with a minimum and maximum limit.(iii) The collected blood in each BS depends on the city population and the stop time in that station.(iv) There is a specific demand for blood that should be satisfied; the shortage is not allowed. Therefore, it is a vital constraint.(v) Each equipped bus and the helicopter start their operations from the crisis-stricken city; after visiting various areas, it returns to the crisis-stricken city. The number of equipped buses is limited and specified.ParametersNotationDescriptionNMaximum number of points chosen as the stationKThe number of bloodmobiles (buses)DThe donors’ populationαThe blood type percentage in that population (the normality of universal blood type)RbThe demand of blood typestijThe interval (time) that a bus travels from cityi to city jttijThe interval (time) a helicopter travels from cityi to city jBetaThe population of donors frequently donating bloodBinary variableXijk=1,busktravels from cityitoj,0,otherwise.Zij=1,the helicopter travels from cityitoj,0,otherwise.Yik=1,buskvisits cityi,0,otherwise.YDik=1,after receiving the collected blood, if buskleaves them at cityi,0,otherwise.wj=1,the helicopter visits cityj,0,otherwise.Nonnegative variableNotationDescriptionTTiThe required time for arriving cityiSikThe stop time (hours) of bus at citykSSiThe stop time (hours) of the helicopter at cityQ1kbThe amount of collected blood of blood types by bloodmobiles (buses)Q2bThe amount of collected blood of blood types by the helicopterQbThe total amount of collected blood by the bloodmobiles (buses) and the helicopter per blood typeFbSik=Sik.Di.α.betaFunction of the amount of collected blood typeb by bus k at city i, when it stopped Sik hours


## 3.4. Model Notation and Formulation

In current section, objectives and constraints of the mentioned problem are modeled as follows. The objective function includes two sections:(1)MaxZ1=∑b=14Qb.Equation (1) formulates the first objective. This objective firstly maximizes the amount of collected blood.(2)MinZ2=maxTB,TM,(3)TB=maxTn+1k∗Yn+1k,∀kϵ1,…,K,(4)TS=maxTTn+1∗Wn+1.Equation (2) indicates second objective, that is, minimizing the arrival time of the service-provider bus and helicopter to a crisis-stricken city that separate under assumptions intended for TB and TS in equations (3) and (4), respectively. Thus, the objective of the model is to minimize the arrival time of the service-provider bus and the helicopter to the crisis-stricken city if the collected blood is maximized by considering the blood type.(5)Xiik=0,∀i∈1,…,n+1,k∈K.Constraint (5) indicates that bloodmobile k (buses) cannot travel from city i to city i (loop is not allowed).(6)X1n+1k=0,∀k∈K.Constraint (6) indicates that bloodmobiles k (buses) cannot travel directly from the starting city to the crisis-stricken city.(7)∑i=0nXijk=∑i=1n+1Xjik=Yjk,∀j∈1,…,n,K∈1,…,k.Constraint (7) is the flow balance; based on this constraint, if bloodmobile k (buses) enters the BS of city j from each point, it should drive to other stations merely from the BS of city j only if the mentioned bus (bus k) has visited city i. In order words, for each input, one output should exist.(8)Yik≤∑j=2nX1jk,∀iϵ1,…,n+1,kϵ1,…,k.Constraint (8): bloodmobile k (buses) should start visiting from starting city.(9)Yik≤∑j=2nXjn+1k,∀iϵ1,…,n+1,kϵ1,…,k.Constraint (9) confines the bloodmobile k (buses) at the end of their route to return to the crisis-stricken city.(10)Xijk≤Yik,∀i,jϵ1,…,n+1,kϵ1,…,k.(11)Xijk≤Yjk,∀i,jϵ1,…,n+1,kϵ1,…,k.Constraints (10) and (11) guarantee that when bloodmobile k (buses) travels from city i to city j, it should visit city i at first and then visit city j.(12)∑k=1KYik≤1,∀iϵ2,…,n.Constraint (12) guarantees that the maximum number of buses visiting a city is 1.(13)∑i=2n+1X1ik≤1,∀kϵ1,…,k.Constraint (13) indicates that, at the time of operation start, each bloodmobile k (buses) can either be used or not. It is obvious that each blood mobile k (bus) travels to merely one city at first.(14)YDik≤Yik,∀i∈2,…,n,kϵ1,…,k.Constraint (14) indicates that if bloodmobile k (buses) visits city i, bus k can either transport the collected blood from city i or leave it there to be transported by the helicopter to the same point.(15)TiK+SiK+tij≤Tjk+1−XijkM,∀i∈1,…,n,j∈2,…,n+1,K∈1,…,k.Constraint (15) is related to the schedule of each bloodmobile (buses). Indeed, it strikes a balance between the entrance and exit time of bloodmobile (buses) from cities.(16)Zii=0,∀i∈1,…,n+1,kϵ1,…,k.Similar to constraint (5), constraint (16) guarantees that the helicopter cannot travel from city i to i (loop is not allowed).(17)Z1n+m=0,∀kϵ1,…,k.Similar to constraint (6), constraint (17) indicates that the helicopter cannot travel directly from starting city to the crisis-stricken city.(18)∑i=0nZij=∑i=1n+1Zji=Wj,∀j∈1,…,n.Constraint (18) is the flow balance constraint of the helicopter, such that if the helicopter enters from each city to city j, it should travel to other cities only from city j (only if it has visited city j).(19)Wi≤∑j=2nZ1j,∀i∈1,…,n+1.Constraint (19) guarantees that the helicopter starts routing from the starting city.(20)Wi≤∑j=2nZjn+1,∀i∈1,…,n+1.Constraint (20) guarantees that the helicopter visits the crisis-stricken city at the end of the route.(21)Zij≤Wi,∀i,j∈1,…,n+1,(22)Zij≤Wj,∀i,j∈1,…,n+1.Constraints (21) and (22) guarantee that when the helicopter is going to travel from city i to city j, it should visit city j; afterward, it visits city j.(23)∑j=2n=1Z1j≤1.Constraint (23) guarantees that when the operation starts, the helicopter can either be used or not. It is clear that the helicopter can travel to only one city at first.(24)Wi=∑kYDik,∀iϵ1,…,n+1.Constraint (24) indicates that if the bloodmobiles (buses) left the collected blood in a city, the helicopter would visit that city.(25)TTi+SSi+ttij≤TTj+1−ZijM,∀i∈1,…,n,j∈2,…,n+1,K∈1,…,k.Constraint (25) is related to the helicopter’s schedule. Indeed, it strikes a balance between the entrance and exit of the helicopter from cities.(26)Sik+Tik.YDik≤TTi.Wi,kϵ1,…,K,iϵ1,…,N.Constraint (26) is about the helicopter’s schedule for collecting blood, such that the helicopter should arrive at that location after delivering the collected blood to city i by bus if city i is not the final station of the route.(27)Q1kb=∑i=1nFbSik.1−YDik,∀bϵ1,…,4,kϵ1,…,K.Constraint (27) is the total blood type b delivered to the crisis-stricken city by bloodmobile k (buses).(28)Q2b=∑k=1K∑i=0nFbSik.YDik.Wi,∀bϵ1,…,4.Constraint (28) is the total blood type b delivered to the crisis-stricken city by the helicopter.(29)Qb=∑k=1KQ1kb+Q2b,∀bϵ1,…,4.Constraint (29) is the buses’ total collected blood and the helicopter per blood type.(30)Q1≥R1,(31)Q1+Q2≥R2,(32)Q1+Q3≥R3,(33)Q1+Q3+Q2+Q4≥R4.Constraints (30) to (33): the collected blood by the blood mobiles (buses) and the helicopter should be equal or greater than the demand of each group. In fact, constraints emphasize that shortage is not allowed.(34)Xijk,Yik,YDik,Wi,Zij∈0,1,∀i,jϵ1,…,n,∀Kϵ1,…,k.In the end, constraint (34) specifies the limits of variables.


## 3.5. Solution Method

In this paper, weighted sum method (WSM) is used for solution method to find the optimal solution. WSM is a multiobjective optimization method in which there will be multiobjective and we have to determine the best solution. WSM combines all multiobjective functions into single scalar, composite objective function using weighted sum according to the following equation:(35)Fx=w1f1+⋯+wnfn.An important issue there is in assigning the weighting coefficients vectorw1,…,wn in the WSM because the optimal solution strongly depends on the chosen weighting coefficients. Also, these weights have be positive and should be according to the following equation:(36)∑i=1nwi=1,wi∈0,1.


## 4. Numerical Results

In order to solve the developed mathematical model, it is assumed that there has been an earthquake in Avaj, located in Qazvin Province in Iran. There are several injured individuals in need of receiving blood. Blood collection should be carried out from a number of neighboring cities to save their lives. In order to solve the model, IBM ILOG CPLEX Optimization Studio software is used. Some cities are considered for blood collection, such as Qazvin, Alvand, Takestan, Abyek, Buin Zahra, Danesfahan, Lak, and Avaj. The considered interval (time) for the cities is indicated in Table3, and the arrival time is considered to be 30% of that of buses.Table 3
The distance between cities (hours).
QazvinAlvandTakestanAbyekBuin ZahraDanesfahanLakAvajQazvin00.260.370.480.541.101.411.31Alvand0.2600.470.450.451.081.571.47Takestan0.370.4701.100.580.381.141.05Abyek0.480.451.1001.191.422.132.04Buin Zahra0.540.450.581.1900.292.331.24Danesfahan1.101.080.381.420.2901.141.05Lak1.411.561.142.131.331.1400.54Avaj1.311.471.052.041.241.050.540In order to execute the model, the input parameters of the problem are determined as follows. The number of bloodmobiles (buses) is considered 3, and the helicopter 1. The demand for each blood type is considered [31895, 22865, 25965, and 21156]. After executing the model, almost 102000 blood units were collected during 11.84 hours, and all demands were satisfied. The optimum results of the model are depicted in Table4. Also, in Table 5 routing results are shown. The route of the first bus is demonstrated in Table 5 from Qazvin to Alvand, then to Buin Zahra, and at the end to Avaj. The second bus route is from Qazvin to Lak and Danesfahan and, in the end, to Avaj. Finally, the third bus route is from Qazvin to Takestan and then to Abyek and Avaj. Also, the route of the helicopter has been from Qazvin to Danesfahan, Lak, and Avaj.Table 4
Optimum result.
Blood quantity (blood units)Time collection (hours)Number of vehiclesBusHelicopter10200011.8431Table 5
Routing results.
VehicleQuantityAssigned route to vehicleTime (hours)VehicleRouteBus3Bus 1Qazvin > Alvand > Buin Zahra > Avaj1.95Bus 2Qazvin > Lak > Danesfahan > Avaj3.60Bus 3Qazvin > Takestan > Abyek > Avaj3.51Helicopter1Helicopter 1Qazvin > Danesfahan > Lak > Avaj2.78


### 4.1. Sensitivity Analysis

In this section, the effect of changing the key parameters including the blood mobiles and helicopter quantity on the decisions of proposed model is examined. In Table6 all considered changing is shown. As depicted in Table 6, changing blood mobiles and helicopter quantity have the remarkable effect on time and collected blood quantity.Table 6
Changing the key parameters.
ScenariosParametersTime (hours)Blood (blood units)Blood mobilesHelicopter12113.658200022014.239200031117.687500041018.206200053012.9598000Figures2 and 3 show a comparison of the optimal situation with each of the scenarios considered for sensitivity analysis. According to Figure 2, it is shown that, based on each of the scenarios, the collection time of blood by the considered vehicles is always longer than the optimal time. Also, Figure 3 shows the amount of blood collected. According to Figure 3, the amount of blood collected based on the considered scenarios is always less than the optimal amount of blood. Therefore, it is impossible to improve the situation if the parameters are manipulated. The favorable condition is the optimal solution.Figure 2
Time comparison between optimal and each scenario.Figure 3
Blood collected comparison between optimal and each scenario.


## 4.1. Sensitivity Analysis

In this section, the effect of changing the key parameters including the blood mobiles and helicopter quantity on the decisions of proposed model is examined. In Table6 all considered changing is shown. As depicted in Table 6, changing blood mobiles and helicopter quantity have the remarkable effect on time and collected blood quantity.Table 6
Changing the key parameters.
ScenariosParametersTime (hours)Blood (blood units)Blood mobilesHelicopter12113.658200022014.239200031117.687500041018.206200053012.9598000Figures2 and 3 show a comparison of the optimal situation with each of the scenarios considered for sensitivity analysis. According to Figure 2, it is shown that, based on each of the scenarios, the collection time of blood by the considered vehicles is always longer than the optimal time. Also, Figure 3 shows the amount of blood collected. According to Figure 3, the amount of blood collected based on the considered scenarios is always less than the optimal amount of blood. Therefore, it is impossible to improve the situation if the parameters are manipulated. The favorable condition is the optimal solution.Figure 2
Time comparison between optimal and each scenario.Figure 3
Blood collected comparison between optimal and each scenario.


## 5. Conclusion

When a crisis strikes, blood transfusion is one of the most vital actions and medical interventions. Despite its importance and vital role, there is a significantly limited resource of blood. Concerning the fact that it cannot be produced artificially, the donors are the only blood source. Concerning the importance of the issue at the time of a crisis, to provide relief operations and transfer maximum blood level during a shorter period to a crisis-stricken city, this study has provided a routing problem for relief vehicles, including a number of buses and a helicopter. Therefore, the main contribution of current research is to develop a biobjective Mixed-Integer Linear Programming (MILP) model for relief supply under crisis condition. In this study, the MILP problem is examined at the time of a crisis strike by considering the supply and transportation of various kinds of blood types to a crisis-stricken city. Given that the routing problem of relief vehicles to provide services under a crisis is an active research area, the contributing factors to it can be investigated from different aspects. Accordingly, concerning the existing research gaps in the former studies, a Mixed-Integer Linear Programming (MILP) model is provided for bloodmobile routing under a crisis and by considering blood types. In the provided model, the stop time duration of each bloodmobile in blood donation stations varies. The amount of collected blood is considered a function of stop time duration. The blood delivery can be carried out by both buses and the helicopter. Also, the blood type compatibility is considered, getting the model closer to the real-life blood supply chain. In the end, the model is solved by using CPLEX software and close data to real life. By using the provided model, the optimal route of each bloodmobile and their stop time in each station are determined such that the total collected blood is maximized. In contrast, the arrival time of the bloodmobiles (the buses) to the crisis-stricken city is minimized. By changing the models’ assumptions, such as lowering the number of bloodmobiles, the blood collection time increases, reaching 44 hours. In addition, we considered a model without any helicopter, observing that the required blood in the crisis-stricken area cannot be transported and supplied on time. The main limitations of this work can be described as follows:(i)
Some limitations to access operational data that must be obtained from indirect sources(ii)
Challenges to determine demand for each blood type



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*Source: 7217182-2021-11-16.xml*