A Two-Objective Model for the Multilevel Supply Chain of Blood Products with the Approach of Reducing the Rate of Contagion under the (COVID-19) Epidemic Outbreak Conditions

Te conditions of the coronavirus epidemic have put much pressure on the healthcare system. Tis disease has hurt the blood supply through the reduction of blood donation and the reduction of access to suitable collection facilities due to dysfunction. Considering the importance of the subject, the purpose of this paper is to design a two-level supply chain network for blood products with the approach of reducing costs and the rate of contagion under the conditions of epidemic outbreaks (COVID-19). After examining the solution methods for multilevel supply chain networks of blood products under the conditions of the spread of the COVID-19 virus, three exact solution methods, including LP-metric, an improved version of the augmented ε -constraint (AUGMECON2), and an improved weighted Chebyshev, are proposed. Tey are used to solve the model in small dimensions. In order to compare the methods in the obtained solutions, several numerical examples of diferent sizes are generated and solved. Ten, using the statistical assumption test, the obtained results are compared in all numerical examples by Tukey’s technique. Also, the TOPSIS is applied to select the best method. Finally, in order to investigate the reaction of the objectives to the changes in the contagion probability parameter, a sensitivity analysis has been performed. Te results emphasize that improving the performance of the blood supply chain (BSC) can lead to a reduction in BSC costs and improved service to patients. Also, the adaptation of diferent components of the BSC and regular coordination between them play an efcient role in controlling and improving this disease and reducing the costs of the BSC. Also, receiving the plasma product of recovered people from type (II) donors can play a vital role in reducing the percentage of disease transmission.


Introduction
Te occurrence of unpredictable events such as natural disasters with a short term but high demand or widespread diseases such as the COVID-19 virus with a long-term and high contagion rate will cause many disruptions all over the world, and, as a result, the supply chain.It creates various problems, including the BSC [1].
Te widespread coronavirus 2019 (COVID- 19) is one of the recent crises around the world at the end of 2019.Te epidemic of COVID-19 virus caused a decrease in the supply of blood and its products.As a result, it had destructive efects on the activities of organizations and blood facilities in many countries.Also, the epidemic led to the lack of proper response to the demand of other patients and blood applicants.Due to the government's initial intervention, such as general quarantine and strategies to prevent the gathering of people during the outbreak of COVID-19, many blood centers and facilities were suspended, the most signifcant efect of which is a considerable reduction in the number of blood donors [2].
Wang et al. [3] from the First Hospital Afliated with the University School of Medicine in China announced that the primary concern of blood donors was the fear of contracting the COVID-19 virus when donating blood.On the other hand, people who have recovered from COVID-19 develop natural defence systems against this disease in their blood.Terefore, the blood plasma of these people contains COVID-19 antibodies, which can be used to prepare the plasma product of recovered people for the treatment of other COVID-19 patients [4].Nevertheless, the statistics indicated that only 1.88% of those who recovered donated their blood plasma.
According to a report from the World Health Organization, there is no evidence anywhere in the world based on the transmission of the coronavirus or other respiratory viruses through blood transfusion [5].However, there are concerns and misleading rumors about this disease in the minds of donors.Accordingly, adopting educational policies to inform people correctly will have a signifcant efect on improving the current conditions and increasing the donation rate again.Meanwhile, the role of blood facilities and related managers in coordinating with new changes and providing a safe environment for donors is undeniable [6].
While the application of some policies is helpful in order to grow the donation of all blood products, it should be kept in mind that blood is not a commonplace commodity, and the unique characteristics of blood and its supply chain (including the fact that the supply of donated blood is relatively irregular, the demand for blood products is uncertain, it is not easy to match supply and demand efciently, and blood products are perishable) cause more complexity of the problem and ultimately leading to blood shortage as a high cost to the society through increasing the mortality rate.On the other hand, the wastage and expiration of blood are often not accepted, because blood donors are infrequent resources of society.For each donor, a specifc period must be considered between two donations, depending on the type of donation.Even after applying for volunteers, only a tiny percentage (5%) of people is eligible to donate.All these conditions emphasize the need to particularly study the issues related to the BSC afected by the crisis of the spread of the coronavirus.
Te main goal of the current study is to design a twolevel supply chain network for blood products, focusing on the conditions of the spread of the coronavirus.In such a way that taking into account the critical conditions, i.e., the outbreak of the widespread disease of COVID-19, not only the amount of supply of blood products in two dimensions related to regular patients and patients with corona will increase, and as a result, the rate of contagion of this disease will decrease, but also, by managing the entire supply chain network, cost wastage and, more importantly, wastage or lack of blood products can be prevented.Some of the special features and innovations of the proposed formulation are as follows: (a) comprehensive minimization of all types of costs of the entire network is considered as the frst objective.(b) For the frst time, the objective of minimizing the spread of the coronavirus has been developed.Tis goal is pursued by increasing the supply of plasma from recovered people to reduce the shortage of this product at various demand points.Also, the minimization of the spread of the disease through the maximization of the satisfaction of the applicants of this type of plasma is also followed in the second objective function.

Literature Review
In this context, the review of the research background shows that, for example, Hsieh [7] has presented the BSC in two levels of collection and distribution with the goals of minimizing costs and maximizing the satisfaction of applicants.Te results indicated an improvement in the cost and quality indicators of health services.In another study, Habibi et al. [8] solved the problem of the location of facilities to optimize costs and blood shortage by presenting the BSC network in three levels: collection, processing, and distribution in crisis conditions.
Zahiri et al. [9] also investigated the reliability, efectiveness, and efciency in a three-level supply chain of blood products to optimize the total cost and also the useful life of the products.Ramazanian and Behbodhi [10] have designed a dynamic location problem in a three-level blood supply network including donors, blood donation bases (fxed and mobile), and blood centers, considering only one objective of usefulness from the donors' point of view.Te results of solving the mixed integer programming model indicated that in order to reduce costs, bases close to blood centers must be selected for construction.
Heydari Fathian and Pasandideh [11] also proposed a sustainable BSC network for three products: red cells, platelets, and plasma, at all levels and with the goals of minimizing costs and minimizing the environmental efects caused by the activities of the blood chain network.Diabat et al. [12] designed a blood product supply chain network including red blood cells, platelets, and plasma in four donation levels to reduce the time and cost of transporting the product by taking into account the disruption in the facilities and the route between them.Tey have designed the BSC in four levels: donation, collection, product production, and distribution.Kamran et al. [13] formulated a new stochastic multiobjective, multiperiod, and multicommodity simulation-optimization model for the COVID-19 vaccine's production, distribution, location, allocation, and inventory control decisions.Teir supply chain network includes four echelons of manufacturers, hospitals, vaccination centers, and volunteer vaccine students.Ghasemi et al. [14] developed a novel multiobjective mathematical model for a plasma supply chain network during the COVID-19 outbreak conditions to maximize the coverage of blood donors during periods and minimize the blood transportation costs between diferent nodes, relocation cost of temporary mobile facilities, inventory holding cost of the blood, and the costs of newly established blood centers.
Numerous factors can afect the process of blood transfusion, blood donation, and rate of contagion under the conditions of epidemic outbreaks .In this regard, it is necessary to plan and create strategies to improve the efciency of BSC and reduce the rate of contagion under the conditions of epidemic outbreaks (COVID-19).Accordingly, an efcient approach is designed in the present 2 Journal of Mathematics study to optimize the overall cost and the contagion rate under the conditions of epidemic outbreaks (COVID-19).Indeed, an attractive environment can help to remove barriers to blood donation and improve blood health.

Mathematical Modeling
After examining the solution methods for multilevel supply chain networks of blood products under the conditions of the spread of the coronavirus, three exact solution methods, including LP-metric, augmented improved ε-constrained, and improved weighted Chebyshev, are presented.Tey are applied to solve the proposed mathematical formulation in small dimensions.In order to compare the methods in production responses, 20 numerical test examples are presented.Ten, due to the statistical hypothesis test, the results of the methods are compared with each other in all numerical test examples using Tukey's technique.Also, TOPSIS is applied to select the best method.Finally, in order to investigate the reaction of the objectives to the changes in the contagion probability parameter, a sensitivity analysis is performed.
Table 1 shows the proposed mathematical model in an overview, where the objective functions, features, and solution method can also be seen.
It should be noted that as the second objective, the percentage of the spread of the COVID-19 virus is minimized along with the costs of the entire network.Since this epidemic has afected many sections and subsystems of the blood supply network after the outbreak, in the proposed model, the rate of the spread of this disease has been taken into consideration and added as a separate objective function to the proposed mathematical formulation.Te purpose of this objective is to minimize the spread of corona disease in the presence of blood donors in mobile, regional, and local blood centers in the BSC.

Assumptions.
Te main assumptions of the proposed model are as follows: (1) Mobile facilities are in local blood centers and potential areas while there are hospitals as well (2) Donors are a group of people dispersed in an area, and it is impossible to design a blood collection program for each person (3) Donors are divided into regular and improved groups (4) Donors who have recovered from COVID-19 only go to apheresis centers to donate plasma (5) Mobile facilities receive blood from diferent donor groups only by apheresis (6) Regional blood centers are equipped with both simple and apheresis methods (7) It is possible to store diferent products in hospitals and regional blood centers, while it is not possible in mobile facilities and local centers (8) Te age of blood products will be specifed from the production time in regional centers (9) Blood products with a lifespan of fewer than two days will not be sent from regional centers to the regional points of demand (10) Depending on the type of each center and the data collected from the Blood Transfusion Organization (BTO) of a given country, the model is solved with the capacity parameters of each facility determined in advance (11) Blood expiration has a penalty cost (12) In the face of some uncertain parameters, the stochastic programming approach is used.So, scenarios with a defnite probability are used in a discrete case, and normal log distribution is practiced in a continuous case 3.2.Notations.Te indices, parameters, and decision variables applied in the mathematical formulation are represented as follows.

Sets and Indices
I � I′ ∪ I ″ Set of regular donors and those having recovered from coronary heart disease (i ′ , i ″ ∈ I) M Set of candidate locations for mobile facilities (m ∈ M) L Set of candidate locations for local blood centers (l ∈ L) R Set of candidate locations for regional blood centers (r ∈ R) M ∪ L ∪ R � W Te whole set of centers responsive to donors (w ∈ W) P 1 ∪ P 2 ∪ P 3 ∪ P 4 ∪ P 5 � P Whole blood (P 1 ), red blood cells (P 2 ), platelets (P 3 ), plasma type (I) (belonging to normal people) (P 4 ), and type (II) plasma (belonging to people having recovered from coronary heart disease) (P 5 lifetimes of whole blood products, red blood cells, and platelets

Parameters
Fa l Te cost of setting up local blood center l Fb r Te cost of setting up regional blood center r Fc m′m″t Te cost of moving each mobile device from place m ′ to place m ″ during t

Journal of Mathematics
Te objective ( 1) is to minimize all costs of the entire chain network.Tey include the costs of establishing the local and regional blood centers, transportation of mobile blood collection vehicles to each location, blood collection in each of the regional and local mobile blood collection centers, blood transfusion from one center to another, product production in regional blood centers, storage of blood in regional blood centers and demand areas, and penalty costs for blood expiration in some areas.
Objective (2) calculates the spread of the disease in the presence of blood donors in mobile, regional, and local blood centers.In the second objective, by using parameter α, which represents the percentage of disease transmission from a person with COVID-19 disease to healthy people, the percentage of disease transmission in the presence of blood donors in mobile, regional, and local blood centers is calculated (the determining method of parameter α is explained in Section 4).
Constraints (3) state that donating blood in any mobile blood collection device is possible if the donors are assigned to that device.Constraints (4) emphasize that whole blood donation in local blood centers is possible if regular donors are assigned to this facility.Constraints (5) explain that it is possible to donate whole blood or its products in regional blood centers if the donors are assigned to this facility.Constraints ( 6) and (7) represent that allocating mobile blood collection devices to the regional blood centers is possible if both facilities are established.Constraints ( 8) and ( 9) demonstrate that allocating the local centers to the regional centers is possible if both regional and local blood centers are established.Constraints (10) also indicate that hospitals are allocated only to regional blood centers that have been established.Constraints (11) to (13) indicate that donors are served based on the maximum coverage radius.Constraints ( 14) and ( 15) indicate that only one mobile blood collection device should go to each place, and also, each mobile device can only go to one place.Constraints (16) emphasize that the movement of the mobile device from the frst place to the second is possible if that device was located in the frst place in the previous period.Constraints (17) and (18) state that each donor group can be assigned to only one of the related blood centers.Constraints (19) and ( 20) calculated the production amount of each blood product by the standard method and apheresis method and collected in each regional center.Constraints ( 21) and (22), respectively, determine the capacity of mobile blood collection devices and the capacity of local blood centers to collect each of the blood products in each time period.Constraints (23) specify the capacity of regional blood centers to collect and produce blood products by standard or apheresis method in each period.Constraints (24) and (25) express the capacity of each warehouse in regional centers and demand points to store each blood product separately by production method.Constraints (26) cause balance in the fow of input and output of mobile blood collection devices.Constraint (27) causes balance in the fow of input and output of local blood centers.Constraints (28) and (29) establish the inventory balance for regional blood centers and demand points, respectively.Constraint (30) deals with calculating the amount of shortage of each blood product according to the production method, at the demand points at the end of each time period.Constraints (31) indicate the minimum demand that must be met.Finally, constraints (32) and (33) represent the status of the decision variables.

The Calculating Method of Parameter α
Te method of calculating the parameter α t (probability of being infected with the coronavirus by another person in period t) in the second objective (equation (2) of the proposed mathematical formulation (for example), whose value is considered between [0.02, 0.03], is represented in Figure 1 [15].

The Proposed Exact Solution Methods
To exact solve the proposed mathematical model, three solution methods are examined, and the best and most efcient one is selected.Te LP-metric, AUGMECON2, and improved weighted Chebyshev are three famous, most popular, and widely used methods for solving multiobjective problems.Since the three mentioned methods are used to solve diferent problems, these were chosen for comparison to choose the best and most efcient and accurate solution approach for the presented mathematical model.It should be noted that several exact solution approaches have been introduced and presented to solve multiobjective problems.However, the three mentioned methods are the most compatible for solving the proposed multiobjective mathematical model.

8
Journal of Mathematics 5.1.Te Improved Weighted Chebyshev Method.Te improved weighted Chebyshev is categorized among the methods for solving multiobjective problems.It uses a precise approach to fnd Pareto-optimal solutions [16,17].Te main structure of the applied approach is described in equation ( 35).
Here, ω is a parameter that takes small and positive values, and η is a free variable.Also, the preference of the objective r is specifed by using the weighting factor y i , in such a way that  r i�1 y i � 1.

Te Improved Version of Augmented Ɛ-Constraint (AUGMECON2
). Te AUGMECON2 had a signifcant drawback in that it was very time-consuming to solve any problem with more than two objectives.Tis weakness led to the AUGMECON2 [18].Furthermore, the authors of [19] represent this algorithm by introducing a bypass coefcient and also a kind of Lexicographic optimization for all objectives and creating the augmented Ɛ-constraint algorithm.
Using the bypass coefcient, the augmented Ɛ-constraint uses the information provided by the redundant (auxiliary) variables of the objectives, which are in the form of constraints, to avoid unnecessary iterations and speed up the solution.Also, this method can identify the exact Pareto set [20].Accordingly, the AUGMECON2 is considered as an approach that can compete with meta-heuristic multiobjective Pareto-based approaches [19].How the AUG-MECON2 works is as follows: In equation ( 36), Z k (x) represents the objectives to be optimized.In addition, x is the element of the problem space, and e takes a value between 10 − 6 and 10 − 3 .Also, ε k is the parameter on the right side of the equation for the k th objective considering that k ∈ 2, . . ., P { }.In addition, r 2 , r 3 , . . ., r P are the domain parameters for the second, third to P th objective function, respectively.In addition, the surplus variables of the problem are represented by S 2 , S 3 , • • • , S P .It should be noted that in the improved version of the augmented Ɛ-constraint (II), S i /r i should be placed instead of S i .Tis avoids scaling issues.Te general steps of the AUGMECON2 are as follows [21]: Step 1: We establish the payof table by performing lexicographic optimization.
Step 2: We calculate the ranges (r k ) and determine a lower bound (lb k ) for the objective k according to the payof table.
Step 3: We create equal intervals (g k ) by dividing the domain of the k th objective.
Step 4: Using ε k � lb k + i k × step k , we obtain the right side of the constraint of the specifc objective, where i k is the counter of the k th objective and step k is determined using step k � r k /g k .
Step 5: We solve the problem.
Step 6: We check the S k associated with the innermost objective for each iteration by applying the bypass coefcient: b � int(S k /step k ).When S k is greater than step k , the same solution is set for the next iteration, the only diference being the redundant (auxiliary) variable.Tis causes redundant iteration, so it may be bypassed, while no new Pareto-optimal solution is created.
Step 7: We determine the Pareto set due to the bypass coefcient and the number of grid points.

LP-Metric Method. Te LP-metric method is a part of the frst category of multiobjective decision-making problems, especially in cases where the decision-maker provides
Total population of a country = 83500000 people Total population of a large city = 13,260,000 people Te total population with critical conditions at the specifed time t = 4129 people Total at-risk personnel inside the hospital in the selected large city at a given time t = 22000 people α t = (4129*(13260000/83500000))/22000=0.0298 Figure 1: Te method of calculating the parameter α t [1].
Journal of Mathematics all the needed information before solving the problem [22].Tis method minimizes the deviations of objectives in a multiobjective formulation compared to their ideal solution.In other words, LP-metric is applied to measure the proximity of an ideal solution, and its relationship is as follows: where w i shows the degree of importance (weight) for the i th objective function.Te 1 ≤ p ≤ ∞ represents the defning parameters of the LP family.Te value of p specifes the degree of emphasis on existing deviations so that the larger p is, the more emphasis will be placed on the largest deviation.
If p � ∞, it will mean that the largest deviation of the existing deviations is investigated for optimization.Here, in the process of single targeting by the LP-metric method, the value of p � 1 has been considered so that, as mentioned, each deviation has its own weight.Also, for each objective, the value of w i is considered equal to 1.

Design of Numerical Instances and Results of Solving Them
In order to evaluate the developed mathematical formulation and solution methods, several numerical examples are examined and analyzed.For each example, the value of the objective functions and the duration of the formulation solution are determined by each method.Te values for each numerical test example are listed in Table 2.At frst, the indicators are defned.Tey include the values of the frst and second objectives and the CPU time of the model by each method.By generating diferent numerical example problems in small dimensions, as represented in Table 2, the proposed methods are compared.Other parameter values applied in numerical test examples are shown in Table 3, all of which are due to the uniform distribution.In addition, the needed weights for the methods of LP-metric and improved (modifed) weighted Chebyshev for the frst and second objectives are considered the same, and their value is 0.5.
To solve the mathematical formulation by proposed solution methods, GAMS software version 24.1.3and CPLEX solver have been used in a system with the specifcations of CPU � Cori7 6700 HQ and RAM � 16 GIG DDR4, and the results are compiled in Table 4.

. Statistical Analysis
In order to analyze the results of three solution methods and compare them with each other, Tukey's method has been applied.Tis method is used when more than two samples are compared with each other, and it shows a suitable performance by comparing each pair of average results with each other [22].Considering the confdence level of 95%, the statistical comparison test of the average results of the three proposed methods is performed for all three defned evaluation indicators.So that in each comparison, the null hypothesis (H 0 ) is equal to the equality of the averages of the results obtained from the three methods and the opposite hypothesis (H 1 ) seeks to reject this hypothesis.Using MINITAB version 21.1.1.0software, the obtained results are shown in Tables 5-7 and Figure 2. Due to observational analysis, some technical remarks can be concluded as follows: (i) As shown in Tables 5 and 6, considering that the P values for the indicators of the frst and second objective function values are higher and lower than the signifcance level, respectively (0.05 < 0.999 for the indicator of the frst objective function and 0.001 < 0.05 for the indicator of the second objective function), so the null hypothesis is accepted for the frst indicator and rejected for the second indicator.Tis means that based on the 95% confdence level, there is no signifcant diference between the answers obtained from the three proposed solution methods regarding the value indicator of the frst objective.However, there is a diference in terms of the value indicator of the second objective function.(ii) Te null hypothesis is accepted concerning the CPU time indicator.According to the results of Table 7, the P value in this indicator is equal to 0.647, which is greater than 0.05, and as a result, it has led to the acceptance of the null hypothesis.In other words, there is no signifcant diference between the CPU times of the three proposed solution methods regarding the solution time indicator.It emphasizes that the computational efciency of the methods is similar.
Te "Tukey simultaneous control limits" is a graph that shows the control limits of the diference of the averages of all the examined samples for all pairs in multiple comparisons.If the diference interval of at least one of the examined pairwise comparisons does not include the zero line, it means a signifcant diference in the average results of the two methods and fnally rejects the null hypothesis.Te graphs displayed in parts (A) and (C) of Figure 3 are related to the indicators of the value of the frst objective and the CPU time of three proposed solution methods, respectively.It is considered that for the CPU time and the frst objective function, all pairwise comparisons include the zero line, and the null assumption is accepted for two indicators.No signifcant diference is observed between the results of the three proposed solution methods in producing the values of the objective functions.Considering the graph in part (B) of Figure 3 and regarding the indicators of the value of the second objective function, there is a signifcant diference between the results of the AUGMECON2 and LP-metric methods, as well as the AUGMECON2 and improved weighted Chebyshev methods.It causes the null hypothesis to be rejected.

The Best-Proposed Solution Method
In order to choose the best method among the three proposed solution methods, TOPSIS has been applied.TOPSIS stands for technique for order preference by similarity to the 10 Journal of Mathematics ideal solution.It, as proposed by Huang and Yun [23], is a suitable technique for ranking options.It evaluates m options by identifying negative and positive ideal solutions and using n evaluation criteria.Finally, the option that has the largest distance from the negative ideal solution and the smallest distance from the positive ideal solution is known as the best option.Te main steps of the TOPSIS are detailed in [23].
Due to the results of the numerical test example to implement the TOPSIS, the improved weighted Chebyshev was selected as the most efcient method among the three proposed solution methods.It can be represented in Table 8.

Sensitivity Analysis
In order to evaluate the impact of changes in the main parameters of the mathematical formulation on the results of the objectives, sensitivity analysis has been used.Considering that due to the results of the TOPSIS technique, the improved weighted Chebyshev was chosen as the best method, and the sensitivity analysis has been carried out on the mathematical formulation using the improved weighted Chebyshev method.Te results of the sensitivity analysis are represented in Figure 4.
Here, the sensitivity analysis has been carried out only according to the changes in parameter α.First, only the cost parameter has been added to the mathematical model.Terefore, its efect has been investigated by increasing it from 0.02 to 0.03 (in 11 cases).As can be concluded, with the growth of parameter α, the frst objective has remained constant in all values and has not changed.Tis means that increasing the parameter α does not afect the values of the frst objective.On the other hand, since the frst objective is related to costs, it is logical to remain constant for several values of parameter α.
Nevertheless, the results for the second objective function are not similar.More precisely, the α had a signifcant impact on the second objective function.According to Figure 4, with the increase of parameter α, the values of the second objective have increased.Tis means that the higher   Journal of Mathematics 13 the α, the higher the probability of disease transmission from a corona patient to a healthy person.Terefore, the α has a signifcant efect on the values of the second objective, so a change in this parameter can cause many changes in the second objective.

Conclusions
Te conditions of the coronavirus epidemic have put much pressure on the healthcare system.Tis epidemic has harmed the impact on the blood supply by reducing blood    Journal of Mathematics donation and reducing access to proper collection facilities due to dysfunction.Considering the importance of the subject, in the current study, the network of the multilevel dual-purpose BSC was investigated.
In the present study, for the frst time, the objective of minimizing the spread of the coronavirus through increasing the supply of plasma from recovered people or reducing the shortage of this product at demand points, along with the objective function of minimizing the costs of the entire network, has been studied.In the proposed mathematical formulation, the percentage of the spread of the coronavirus is also minimized along with the costs of the entire network.Te main goal is to minimize the spread of the Coronavirus at the location of blood donors in mobile, regional, and local blood centers in the proposed supply chain.Games software was used to exact solve the mathematical model by the proposed solution methods.In order to analyze the results of the three methods and compare them with each other, Tukey's test was used.Te P value in this indicator is equal to 0.647, which is greater than 0.05, and as a result, it led to the acceptance of the null hypothesis.In other words, a signifcant diference between the results specifed from the AUGMECON2, the LP-metric, and the improved weighted Chebyshev methods means a signifcant superiority of one of the proposed methods.By applying the TOPSIS technique, it is clear that the improved weighted Chebyshev can be chosen as the most efcient method among the three proposed solution methods.
Te analytical results of the present study represent that improving the performance of the BSC can lead to a reduction in BSC costs and improved service to patients.Also, the adaptation of several components of the BSC and regular coordination between them play an infuential role in controlling and improving this disease and reducing the costs of the BSC.Also, receiving the plasma product of recovered people from donors' type (II) can play a critical role in reducing the percentage of disease transmission.

Figure 2 :
Figure 2: Pareto diagram (trade-of between objectives) of the proposed solution methods.

Figure 3 :
Figure 3: Simultaneous control graph of Tukey limits.(a) Te (pairwise comparison) indicator of the frst objective.(b) Te (pairwise comparison) indicator of the second objective.(c) Te (pairwise comparison) indicator of CPU time.

Figure 4 :
Figure 4: Te sensitivity analysis of parameter α on the objective functions.

Table 1 :
Characteristics of the proposed mathematical formulation.Te coronavirus epidemic imposes higher investment and operating costs on the health and treatment network, the BSC network, and, subsequently, the society's economy, in diferent ways To solve the proposed mathematical formulation, the exact solution methods, LP-metric, AUGMECON2, and improved weighted Chebyshev, are used.Te model is implemented and compared by them in several sample test problems.Finally, the superior solution method is selected by the TOPSIS technique Minimizing the spread of the coronavirus Te objective is to minimize the spread of the COVID-19 virus in the presence of blood donors in mobile, regional, and local blood centers in the proposed supply chain Ta pt Te cost of transporting each unit of blood product p per kilometer per t Oa ipt Te cost of collecting each unit of blood product p from donor group i during t Pa pt Te cost of production per unit of blood product p in regional blood centers in t Hc pt Te cost of maintaining each unit of blood product p during period t Sa lp 1 t Local blood center capacity l to collect whole blood (p 1 ) during period t Sb pt Te capacity of mobile facilities to collect blood products p during t Sc rpat Te capacity of regional center r to collect and produce blood products p obtained by collection method a during period t Se Dm s hpat Hospital demand h for blood products p obtained by collection method a during t under scenario s τ w Time spent on the blood donation process at the blood center w σ w Advertising budget in the blood center w ϑ w Experience factor in the blood center w Sτ w Donor sensitivity to time τ w Sσ w Donor sensitivity to advertising σ w Sϑ w Donor sensitivity to the experience factor ϑ w Dr s Average donation rate under scenario s Pop w Te population of the area allocated to the blood center w Ub w Te best productivity available in the blood If local blood center l is launched, 1; otherwise, 0 YR r If the center of region r is set up, 1; otherwise, 0 ZA s imt If donor group i is allocated by mobile facility m in period t under scenario s, 1; otherwise, 0 ZB s i′lt If donor group i ′ is allocated to local center l in period t under scenario s, 1; otherwise, 0 ZC s rpat Te capacity of regional center r to store blood products p obtained by collection method a during period t Sk hpat Demand point capacity h to store blood products p obtained by collection method a during period t Da im Te amount of route between donor group i and facility m Db i′l Te amount of route between donor group i ′ and the local center l Dc ir Te amount of route between donor group i and the regional center r Dd mr Te amount of route between facility m and the regional center r De lr Te amount of route between the local center l and the regional center r Dh rh Te amount of route between the regional center r and demand point h Ra m Maximum coverage radius of mobile facility m to serve donors Rb l Maximum coverage radius of the local center l to serve donors Rc r Maximum coverage radius of the regional center r to serve donors Po p Percentage of blood product production p Ea hpat Expiry cost per unit of extra blood product p obtained by collection method a at the demand point h during t irt If donor group i is allocated to regional center r during period t under scenario s, 1; otherwise, 0 MV s m′m″t If the mobile facility passes from place m ′ to m ″ in interval t under scenario s, 1; otherwise, 0 XA s impt Te amount of blood product p donated by donor group i on mobile device m during t under scenario s XB s i′lp 1 t Te amount of whole blood p 1 donated by donor group i ′ at local center l during t under scenario s XC s irpt Te amount of whole blood p donated by donor group i in regional center r during t under scenario s hpat Te amount of extra blood product p obtained from collection method a expired at the point of demand h during period t under scenario s i  m  l  r  p  t  s π s .Oa ipt .XA s impt + XB s i′lp 1 t + XC s irpt   ψ 5 �  m  l  r  h  p  a  k  t  s π s .Ta pt .Dd mr .XD s mrpt + De lr .XE s lrp 1 t + Dh rh .XG s rhpakt   ψ 6 �  r  p≠p 1  t  s π s .Pa pt .PR s rpa′t ψ 7 �  r  h  p  a  k  t  s π s .Hc pt .IR s rpat + IH s hpakt   ψ 8 �  h  p  a  t  s π s .Ea hpat .EX s hpat , (1)

Table 2 :
Numerical example problems generated in small dimensions.

Table 3 :
Values of the parameters applied in the numerical examples.

Table 5 :
Statistical hypothesis test results for the frst objective function.

Table 6 :
Statistical hypothesis test results for the second objective function.

Table 7 :
Statistical hypothesis test results for the CPU time.

Table 8 :
Te fnal results of the TOPSIS technique.