key: cord-0842380-v1ixe8u1
authors: Mosallanezhad, Behzad; Chouhan, Vivek Kumar; Paydar, Mohammad Mahdi; Hajiaghaei-Keshteli, Mostafa
title: Disaster relief supply chain design for personal protection equipment during the COVID-19 pandemic
date: 2021-08-18
journal: Appl Soft Comput
DOI: 10.1016/j.asoc.2021.107809
sha: cf7951b8752597d58463950bc9a6c3cb4e1a1af9
doc_id: 842380
cord_uid: v1ixe8u1

The global epidemic caused by novel coronavirus continues to be a crisis in the world and a matter of concern. The way the epidemic has wreaked havoc on the international level has become difficult for the healthcare systems to supply adequately personal protection equipment for medical personnel all over the globe. In this paper, considering the COVID-19 outbreak, a multi-objective, multi-product, and multi-period model for the personal protection equipment demands satisfaction aiming to optimize total cost and shortage, simultaneously, is developed. The model is embedded with instances and validated by both modern and classic multi-objective metaheuristic algorithms. Moreover, the Taguchi method is exploited to set the metaheuristic into their best performances by finding their parameters’ optimum level. Furthermore, fifteen test examples are designed to prove the established PPE supply chain model and tuned algorithms’ applicability. Among the test examples, one is related to a real case study in Iran. Finally, metaheuristics are evaluated by a series of related metrics through different statistical analyses. It can be concluded from the obtained results that solution methods are practical and valuable to achieve the efficient shortage level and cost.

One of the leading environmental measures is frequently cleaning and decontamination of surfaces with disinfectants like alcohol. Measuring body temperature, remote working, preventing conferences and meetings, etc., are categorized as organizational measures (Cirrincione et al., 2020) . Moreover, using Personal Protection Equipment (PPE) like face masks, social distancing, washing hands carefully and regularly, avoiding travel to busy high-risk areas, reduction of gathering like parties, and keeping away from hugging and kissing are some of the measures at individual levels (Sohrabi et al., 2020; Hosseini et al., 2021) .

Among mentioned measures, PPE meets the standard requirements to reduce the infection inhibition and losses caused by contagious viral diseases such as the COVID-19 and helps healthcare systems control the growth of the COVID-19 among their personnel. A group of PPEs is made to control personnel healthcare. The PPEs are gloves, gowns, respiratory protection, eye protection, and testing kits (Cao et al., 2020) .

On the other side, the current stock of PPE might be quickly in shortage for many items like a medical mask or respiratory protection. This crisis is begun by the growing number of COVID-19 cases globally, alongside bulk buying of PPE due to misinformation and fear (WHO, 2020) . Moreover, PPE production has encountered diminished capacity, and increasing daily demand in the hospital must be met to care for contaminated patients (Livingston et al., 2020) . Although it could be recommended numerous avenues to handle PPE demand befittingly, this study aims to propose a mathematical supply chain model in which optimal PPE flow in the network during the COVID-19 pandemic is critical.

Generally, emergency management has been defined in many ways. However, it can be summarized in three main aspects: being prepared before a disaster happens, immediate response to disasters, and reestablishing detrimental damages caused by natural or humanorigin disasters (Ghaffari et al., 2020) . Improper and meager care to injured people by disaster could bring about longstanding health impacts and fatality growth, specifically COVID-19. During a health disaster, adequate and timely supply of necessary commodities such as food, medicine, medical equipment has a tremendous impact on the hectic situation of the public health emergency (Jensen & Hertz, 2016) .

In supply chain (SC) theories, the situation mentioned above is called supply chain disruption. Supply chain disruption is an unexpected occasion that disables one or more components in the supply chain and disrupts common goods and commodities distribution within the network (Haghjoo et al., 2020; Garvey and Carnovale, 2020) . Epidemic outbreaks are named as one of the particular cases of supply chain disruption that is consist of three features as follows: permanent presence and vague disruption level; concurrent disruption significantly spreads in supply chain and occurs over a broad population; it begins on a small scale and immediately dominates numerous geographic areas (Ivanov, and Dolgui, 2020) . Emergency management practices' performance is intensively associated with relief supply chain design (Hong et al., 2015) . producers, warehouses, and stores to produce and dispense the proper goods. At the same time, it optimizes costs and makes a competitive advantage available for all cooperated partners over the supply chain (Fattahi et al., 2018 , Abdi et al., 2021 . So, designing a relief SC is the center of focus, especially during a natural disaster or any uncertain medical emergency like COVID-19. The right step not taken on time becomes the reason for a large number of losses of lives.

Obviously, designing such relief SC networks under particular conditions like pandemic is inevitable to better response time and reliability. Safaei et al. (2018) developed an efficient and cost-effective optimization framework to distribute relief items considering dynamic conditions adequately. Reportedly, the Chinese government built new medical facilities less than two weeks after understanding the COVID-19 outbreak in this country (Ran et al., 2020) . Similarly, the same approach has been implemented in several countries, like Iran, using churches, mosques, stadiums, etc., to be proactive to reach the following advantages:

• Supplying and delivering materials and drugs in the right quantity.

• Providing and planning the right number of nurses and doctors.

• Optimal transportation planning.

• Timely and well-organized responding to the patients.

• Proactive to possible future cases.

• Minimizing the total costs while servicing and covering all potential cases.

• Centralized decision-making.

Hence, this paper develops a multi-objective model including medical supply manufacturers, distributors, medical centers, and world donators to minimize both total costs and unsatisfied demand to provide better distribution of PPEs.

A detailed review of previous related works is conducted and reported in Section 2. The proposed model is formulated in Section 3. Section 4 provides the proposed solution approaches and encoding scheme. We develop applied examples and computational results and also investigate a real case in Section 5. In Section 6, computational outputs are elaborated. Lastly, Section 7 embodies the conclusion is presented, and future directions are proposed.

Recently, COVID-19 caused millions of death, and also this crisis has influenced global activities. Besides, economic troubles and malfunctions appeared in many sectors, including production, supply chain, logistics (Barman et al., 2021) . Since the universal supply chain has been dependent on China over the past decades, many large supply chain sectors, including those related to electronics and automotive industries, pharmaceutical manufacturers, airlines, transportation, etc., have been incredibly damaged by the COVID-19 outbreak (Belhadi et al., 2021) .

As an authentic experience, before WHO's declaration about COVID-19 widespread presence, a global shortage in supplying PPE was abundantly noticeable. For instance, the storage of N95 masks in the United States reported by Health and Human Services (HHS) was only 12 million, considerably lower than what is expected to be the US needs. Moreover, the US government could not properly use domestic manufacturers to handle the PPE shortage of N95 masks. On the other hand, countries like South Korea made their domestic manufacturers allied to satisfy healthcare PPE demands (Dai et al., 2021) .

Regarding the pandemic disruptions in the SC sector and mitigating severe and harmful impacts on different supply chains, researchers and scholars sought the presence of supply chains that can cope with the unpleasant troubles and problems in this field. As we elaborately discuss recent studies in the supply chain in following, they encompass a variety of subjects such as supply chain design (Motevalli-Taher, and Paydar, 2021; Zahedi et al., 2021) , forecasting and planning (Nikolopoulos et al., 2021) , supply management (Baveja et al., 2020; Iyengar et al., 2020) , transportation and logistics management (Chiaramonti and Maniatis, 2020; Kumar et al., 2020) . Nagurney (2021) developed a supply chain network using mathematical modeling concentrated on labor availability during the COVID-19 pandemic for industries and businesses. In other researches considering COVID-19 circumstances, she and her collaborators also carried out studies on a variety of subjects, including presenting a model for demand competition on medical supplies and designing supply chain networks for perishable food (Nagurney, 2021a; Nagurney, 2021b; Nagurney, 2020; Nagurney et al., 2020) . Nandi et al. (2021) suggested a resilient SC for COVID-19, which is subjected to transparency and sustainability. In this study, localization, agility, and digitization features have been considered in terms of SC. Ivanov (2020) utilized simulation research toolkits to study the impacts of the virus on the chain worldwide. This research is divided into two parts. First, they clearly expressed why the epidemic led to a supply chain disruption then provided evidence supporting the practicability of simulation-based studies for predicting epidemic effects. Goodarzian et al. (2021) considered a sustainable chain related to COVID-19 conditions. They suggested a model facilitating medicine distribution to patients while finding the best way to produce and store medicine under perishability constraints. Since the proposed model is NP-hardness for large-size models, the authors decided to solve with hybrid meta-heuristics. In one of the devoting studies for the COVID-19, Abbasi et al. (2020) designed an allocation model for a vaccination with both operational constraints (capacity of hospitals, transportation, and storage) and non-operational constraints (exposure risks).

In this work, we formulate an optimization problem for personal protection equipment during the pandemic. Hence, a brief survey on the relief supply chain design is presented, and the more relevant literature on relief supply chain design is accessible in Table 1. . Habibi-Kouchaksaraei et al. (2018) implemented a mathematical model for blood donation during disasters like earthquakes, floods. They utilized the goal programming technique to address their model. In some relief supply chain networks, finding the best location for distributing relief goods and telecommunication facilities is indispensable. Mohamadi et al. (2019) figured this problem out as an optimization model based on multiple cases in a fuzzy environment. Their proposed model identifies the relevant places for telecommunication towers and shelters and assists decision-makers in amplifying the emergency services.

J o u r n a l P r e -p r o o f Journal Pre-proof Shavarani (2019) addressed one of the critical issues in disaster settings. When disaster happens, relief centers and gas stations are highly required not only for emergency services by organizations but also for the daily activities of individuals. He suggested mathematical modeling for a multi-level network. Akbarpour et al. (2020) considered the same chain for pharmaceutical commodities. The robust optimization model has two objectives: one minimizes the cost of the relief network, and the other maximizes the demand coverage. The formulation also has considered the uncertainty of demand, perishability of commodities, and mobility of facilities.

In one of the recent works of relief SC networks, Li et al. (2020) tackled mathematical modeling to distribute injured parties and develop response networks. Ghasemi and Khalili-Damghani (2021) presented a hybrid technique including simulation and mathematical modeling in uncertainty conditions for preparedness goals in the relief supply chain. This model cop with decisions regarding the setting up and allocation of distribution centers.

Global COVID-19 outbreaks proved that a well-founded network must overcome unforeseen incidents and handle consequent events associated with the epidemic. One of the most reliable methods and the perfect way by which we can achieve the closest outputs to reality is mathematical modeling. In addition to its superiority in results, it can display a better tool for future prediction.

Literature reviews on COVID-19 supply chain studies and relief SC are presented in the previous section and Table 1 to ensure the developed model is unique in terms of novelty and application. Most of the previous works in the COVID-19 supply chain are focus on subjects comprising labor availability of supply chain, medical supply chain, and allocation model for vaccination. On the other hand, designed relief supply chain literature has paid attention to problems such as blood donation, emergency services, and pharmaceutical distributions.

The proposed bi-objective, multi-period, and multi-product deals with the total cost and unsatisfied demands minimization. No study reported the fair distribution of PPEs among medical centers of a region, considering medical supply manufacturer, distributor, medical center, and world donator. Therefore, there is no similar model for PPE distribution in this context; this model can help prominent associations and organizations to control the COVID-19 pandemic situation. 

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Generally, there are three primary sources, medical supply manufacturers, humanitarian world aids, and local volunteer aids, to supply and satisfy the PPE demand of the healthcare system, as shown in Fig. . These purchased commodities from medical supply manufacturers (MSMs) or globally dispatched medical aids are directly moved to the distributors. Generally, the distributors are authorized institutes or organizations by governments or the ministry of health and medical education. However, local aids are straightly donated to outpatient clinics or hospitals. They are responsible for allocating needed PPE to be occupied and selected hospitals for COVID-19 patients. One of the pillars of any SC network, especially a relief SC network, is controlling costs or making more profit. According to the literature review, cost minimization is one of the organizations' essential goals, so most studies in this context have employed a cost minimization objective (Molladavoodi et al., 2020 , Mousavi et al., 2021 ). Moreover, a relief chain may confront unsatisfied demands, known as a shortage of demands. With this in mind, to model the relief supply chain existence of fairness for demand shortage is essential (Hajiaghaei-Keshteli et al., 2011 , Cheraghalipour et al., 2017 . Consequently, for the second objective function of this study, we aim to minimize unsatisfied demands.

In this section, the proposed network is formulated by a mathematical modeling approach. 

Number of PPE c shipped from MSM i to distributor j in period t Number of PPE c shipped from world donator w to distributor j in period t ℎ Number of PPE c shipped from distributor j to medical center h in period t

The number of PPE c in distributor j in period t ℎ Unsatisfied demand for PPE c in period t. Assignment Decision Variables ℎ Is 1 if Distributor j is assigned to medical center h otherwise 0.

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

Equation (1), as the first objective function, is to minimize the cost of the PPE supply. It includes four parts: the first to third part concentrate on transportation costs between different network components, while the fourth calculates holding cost in distribution spots. The second objective function (Equation 2) makes the model distribute the PPE in the network equitably to optimize the maximum amount of unsatisfied demands. Interested readers might refer to the Appendix for linearizing the second objective function (Equation A.1 and A.2).

,

Constraint (3) checks the demand flow between the distribution centers and medical centers. ℎ is a positive decision variable for the shortage of demand in this flow, provided the transported PPE to the medical center does not match the exact amount of demand. Constraints (4) and (5) control the distribution centers' inventory level and balance the PPE J o u r n a l P r e -p r o o f Journal Pre-proof input and output. Constraints (6)-(8) are capacity constraints. Constraint (6) imposes that the total amount of PPE transported from world donators to distribution cannot be unlimited, and it must be less than or equal to the maximum donation by world donators. Constraint (7) denotes that the total amount of PPE moved between the medical manufacturer and distribution centers are limited to each manufacturer's highest production capacity. Constraint (8) is a space constraint that ensures that distribution centers have sufficient space to conserve PPE packages. Constraint (9) make at least one distribution center to be established. The products might be transported to medical centers from distribution spots only if they are opened, guaranteed by constraint (10). Correspondingly, constraint (11) ensures that there will be PPE transportation flow between distribution spots and medical centers if assigned to each other. Constraint (12) specifies PPE transportation between nodes, inventory variables for distribution centers, and demand shortage in medical centers. Also, constraint (13) ensures the assignment of binary variables.

The following section is divided into two subsections: encoding strategy and multiobjective algorithms. The first subsection deals with the utilized encoding scheme known as the random-key method (Chouhan et al., 2020; Mosallanezhad et al., 2021) . In the second subsection, four metaheuristics are developed, including some famous and recent algorithms in addition to hybrid ones.

The random-key method is a popular technique to generate the chromosome size for any problem. The generated chromosome will be in the form of a matrix that represents the set of candidate solutions. These chromosomes are then sorted, and priorities are assigned. The matrix obtained after that is known as the priority-based matrix. Initially, the set of candidate solutions are generated in the interval (0,1) based on the considered problem size. This proposed chromosome matrix, as shown in Fig. , is generated by assuming the following data: Medical supply manufacturers (i) =2, Humanitarian world aids (w) =3, Distributors (j) =3, Medical centers (h) =2, and, time period=4.

J o u r n a l P r e -p r o o f The chromosome matrix designed has two components according to the designed chromosome structure + + 2 × + ℎ. This chromosome matrix is then converted to the priority-based matrix by assigning the priorities to the random keys, as shown in Figure 3 . Based on these priorities, the flow decisions will be taken. From Figure 3 (a), component one depicts the flow of relief aids from second medical supply manufacturers and humanitarian world aids to the third distributors. This process goes on till one of the capacities is exhausted. When this happens, a second facility is selected, and the procedure is repeated for the other. Similarly, Figure 3 

Generally, these algorithms are to find an optimal solution between two or more conflicting objectives. The weighted sum approach is a famous solution approach in a multi-objective perspective (Nurjanni et al., 2017; Sarvestani et al., 2019; Sarma et al., 2020; Zahedi et al. 2021) . Under this method, initially, all the objectives are converted to single-objective by assigning some weight, making it easier to apply traditional algorithms. The decision-makers choose the objective function and weight assignments according to the preference of the i+w i+w J o u r n a l P r e -p r o o f utility. In contrast, using the Pareto optimal concept achieves a non-dominance solution rather than converting the original problem to a single-objective one Fathollahi-Fard et al. 2020; Goodarzian et al. 2021a; Goodarzian et al. 2021b ). These nondominance solutions can be determined through comparisons of solutions obtained from the Pareto Front.

The Pareto optimal solution shows good trade-offs between objectives, due to which it is seen as a preferred solution. Pareto fronts are particularly non-dominating solutions that are necessary for this trade-off. Like if we consider two solutions, X and Y. Furthermore, the crowding distance calculation is used to select a good solution from the obtained front.

The MOKA is a well-known metaheuristics algorithm. The algorithm was first introduced and developed by Aminnayeri (2013, 2014) . In the proposed algorithm, the merging of the population is done based on the sorting using the crowding distance. The procedure is shown in Figure 4 . In this algorithm, Keshtel is referred to as randomly generated solutions, the food source is the solution, and the lake is the feasible region. More explanations can be found in Cheraghalipour et al. (2018) , and the pseudo-code is presented in Figure 5 . Land N Keshtel and calculate fitness 2

Do non-dominate shorting and calculate crowding distance 3

Again, sort the non-dominated Keshtels based on crowding distance 4

Find lucky Keshtels 5

Find best Lucky Keshtel ( ) 6

For each lucky Keshtels ( 1 ) 6.1 Nearest Keshtel ( ) keep on Swirl around the 6.2 In case better food is obtained, replaces . For new go to step 6.1 6.3 Excess food available attracts do swirling else go to step 8 7 explores the unexplored regions for the existence of food. 8

The Keshtels with the least food are replaced with new Keshtels ( 3 ) 9

Move the remaining Keshtels in the lake between the other Keshtels ( 2 ) 10 Merge the population [ 1 ; 2 ; 3 ] 11 Do non-dominate shorting and crowding distance 12 Again, sort the non-dominated Keshtels based on crowding distance 13 Select ( ) better Keshtels from this merged population for the next generation 14 Do steps 11 and 12 for the new population 15 Stop if stopping criteria meet; otherwise, go to step 5 

The process of this algorithm, which its basic version is firstly proposed by Kirkpatrick et al. (1983) , can be defined as firstly keeping the maximum temperature for the heat bath at which the solid melts. At this temperature, the particles arrange them randomly, and later, the temperature is decreased slowly and carefully. At last, the solid structure is arranged in the optimal lattice structure with minimum energy. Here is the maximum temperature of the heat bath, is the Boltzmann constant, and the accepting rule is known as the Metropolis criteria. The interested readers may refer to Cheraghalipour et al. (2018) . The pseudo-code of the MOSA is illustrated in Figure 6 .

MOSA pseudo-code 1

Initialize and evaluate the fitness value ( , ( ))) Where, ∆ = ( ′ ) − ( ), = 1, 2 … …J 2

Best solution = ( , ( ))) 3 For = 1 3.1 Do mutation operator ′ 3.2

Calculate the fitness value and (∆ ) 3.2.1 If ∆ 1 ≤ 0 && ∆ 2 ≥ 0

Update the best solution = ( ′ , ( ′ )) Update the solution = ′ 3.2.2 Else if ∆ 1 ≥ 0 && ∆ 2 ≥ 0 ∥ ∆ 1 ≤ 0 && ∆ 2 ≤ 0

Keep the solution Pareto set 3.3.3 Else ∆ 1 ≤ 0 && ∆ 2 ≤ 0 1 = ( −∆ 1 ) , 2 = ( −∆ 2 ) , ℎ = If ℎ < 1 && ℎ < 2 Update the solution = ′ 4

Update temperature ( = * ) 5

Do non-dominated sorting of the Pareto set 6

Stop if stopping criteria meet otherwise, go to step 3.1 Fig. 6 . The MOSA Pseudo-code.

Genetic Algorithm (GA) is known as one of the most efficient algorithms to solve a wide range of the problem developed by Holland (1975) . The multiple objective genetic algorithms MOGA is a particular case when the problem has various objectives. The diversity in the algorithm is maintained using the crowding distance sorting to the set of made solutions (Golmohamadi et al., 2017) . Finally, it converges and returns the optimal Pareto front. To study more, see Deb et al. (2002) . Figure 7 illustrates the pseudo-code of this algorithm.

NSGA-II pseudo-code 1

Initialize the random population 2

Calculate the fitness value 3

Non-dominance ranking is assigned to the individuals 4

While ≤ 5

Reproduce the population of offspring 6

Do selection, crossover, and mutation 7

Fitness values are calculated 8 populations are merged 9

Non-dominance ranking is assigned to the individuals 10 Non-dominated Pareto fronts sets are created 11

Between the points on each front, crowding distances are calculated 12

Based on the ranks and crowding distance, new parents are selected 13

Pareto optimal solutions are stored 14 = + 1 15 End while 

The proposed hybrid MOKASA is designed to enrich the search phase of the KA by hybridizing with the SA algorithm intelligently. The generated population in KA gets divided into three parts such as 1 , 2 , 3 . The 1 the population is known as the best solution and 3 is known as the worst solution. The algorithm has an intense exploration phase and during the exploitation phase the 3 the population is generated randomly using the SA. The better solution witnessed will be accepted or rejected based on the Metropolis criteria. The interested readers may also refer to Chouhan et al. (2020) . Figure 8 illustrates the pseudocode of this algorithm.

Land N Keshtel and calculate fitness 2 Do non-dominate shorting and calculate crowding distance 3

Again sort the non-dominated Keshtels based on crowding distance 4

Find lucky Keshtels 5

Find best Lucky Keshtel ( ) 6

For each lucky Keshtels ( 1 ) 6.1 Nearest Keshtel ( ) keep on Swirl around the 6.2

In case better food is obtained, replaces . For new go to step 6.1 6.3

Excess food available attracts do swirling else go to step 8 7 explores the unexplored regions for the existence of food. 8

The Keshtels with the least food are replaced with new Keshtels ( 3 ) 9

Accept ( ′ ) as a new solution if its objective is better than ( ) 10 % SA Main loop 10.1 Else if ∆ 1 ≤ 0 && ∆ 2 ≥ 0 10.2

Update the best solution = ( ′ , ( ′ )) 10.3

Update the solution = ′ 10.4

Else if ∆ 1 ≥ 0 && ∆ 2 ≥ 0 ∥ ∆ 1 ≤ 0 && ∆ 2 ≤ 0 10.5

Keep the solution Pareto set 10.6

Else ∆ 1 ≤ 0 && ∆ 2 ≤ 0 10.7 1 = ( −∆ 1 ) , 2 = ( −∆ 2 ) , ℎ = 10.8 If ℎ < 1 && ℎ < 2 10.9

Update the solution = ′ 10.10 Update temperature ( = * ) 11 Merge the population [ 1 ; 2 ; 3 ] 12 Do non-dominate shorting and crowding distance 13 Again, sort the non-dominated Keshtels based on crowding distance 14 Select ( ) better Keshtels from this merged population for the next generation 15 Do steps 12 and 13 for the new population 16 Stop if stopping criteria meet otherwise, go to step 5 

Although the number of cases shockingly increases, according to official Iranian statistics provided by the Ministry of Health and Medical Education 1 , more than 2,006,964 COVID-19 cases are confirmed, in which 63,884 cases are dead, and 1,684,570 cases are recovered until 8 th April 2021. Still, the country has an increasing number of total cases; but the right step was taken at the correct time to ensure the total cases are significantly less as compared to the other countries such as Italy, USA, France, Spain, Germany, UK, Russia, Turkey, etc. 2 Countries such as France and Italy, known for their world-class health care facilities, are also failed to do the needful for the country's people 3 . The instances are probed and investigated to validate the model's pertinence and solution approach in this section. In this regard, we developed fifteen test problems in different sizes ( Table 2) . The third example is evolved from a real case. Khuzestan province in southern Iran is marked as red zones in Iran with 16,632 Positive cases, 613 deaths, and 15,309 recovered cases until 3 rd June 2020. In all examples, we assume that the network is designed to handle PPE demands for four time periods, each of which is seven days.

In the real-case example, two medical supply manufacturers qualified by the ministry of health and medical education are active in Khuzestan, one in Khorramshahr, and one in Ahvaz. In addition, two international channels, the international Ahvaz airport and Bandare-Emam international port exist in Khuzestan to receive international donations. During COVID-19 outbreaks, the government of Iran exclusively appointed medical sciences universities to store and distribute both purchased and global donated PPE. Therefore, in Khuzestan, five medical sciences universities are responsible for supplying necessary PPE for COVID-19 medical centers all over the province. In a similar situation, ten particular medical centers throughout the province are planned to serve the COVID-19 patients. Lastly, the medical centers require a wide range of different PPE; nevertheless, the shortage of apron protection, surgical gloves, face shields, respirator, surgical gowns, and masks are pretty tangible in Khuzestan. Because of the strange situation with which medical centers and related organizations are dealing, the needed data are not provided precisely. As a result, the data are presented in approximate form in Table 3 , retrieved from the Statistical Center of Iran 4 , and the Ministry of Health and Medical Education. Figure 9 is a symbolic scheme for medical supply manufacturers, world donators, distributors, and medical centers in Khuzestan province. 

In the subsequent section, we explore parameter tuning and the behavior of the proposed multi-objective algorithm using the different assessment metrics. To verify the obtained results from the proposed algorithm, statistical techniques have been utilized.

After reaching tuned levels for each algorithm, the proposed algorithms are set with these tuned control parameters to find the optimal results. The performance measures for multi-objective problems are somewhat different from single-objective problems. Multiple studies proposed numerous measures to appraise the multi-objective metaheuristics' performance and help scholars identify intelligent algorithms rather than others. Typically, performance matrices are used to analyze Pareto fronts. Thus, this study utilizes specific metrics to appraise multi-objective metaheuristics added to the next section. The interested reader may also refer to Behnamian and Fatemi Ghomi (2011), Fard et al. (2017) , and Roghanian & Cheraghalipour (2019) .

SNS calculates the spread of ideal and non-dominance solutions by the following formula:

Here, = ‖ ⃗ ⃗ − ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ‖, = , ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ = {min( 1 ) , min( 2 ) … … … min( )}

In order to find the gap between Pareto and the ideal solution, we utilize this index, and obviously, the lesser MID is always preferred to achieve better performance. where represents the entire non-dominated set.

The distance between solutions is calculated by this index concerning the best Pareto front and is given as:

(16)

J o u r n a l P r e -p r o o f represents the total objective considered, , and , represent the maximum and minimum of ℎ function value in respectively, , and , represents the maximum and minimum of ℎ function value in , respectively.

If the points are represented as a solution in the objective space, the solution set's ndimensional space (or volume) is referred to as hypervolume. In other words, hypervolume is a metric that measures the volume covered by Pareto set in the feasible space for the objective function. Hypervolume is calculated according to Equation (17),

where R is the set of Pareto solutions, is volume regarding a reference point T.

An important factor for evaluating an algorithm's performance is the speed of running the algorithm that achieves optimal solutions. The CPU time for any algorithm is the total computational time for that algorithm. According to this metric, always lesser CPU time is preferred.

Before further proceeding, to avoid useless and time-consuming runs of metaheuristics, parameters are carefully tuned using a popular design of experiment (DOE) method. Among various DOE techniques, Taguchi experimental design method is selected. This method was developed by Taguchi (1986) . Taguchi method seeks for maximization of controllable factors as well as minimization of noise effect. In various circumstances, this ratio is considered better in one of these three cases: smaller, nominal, and larger. Since our model is a minimization problem, tuning of parameters is attained by "the smaller is better" for each algorithm (see Equation (18)). ⁄ = −10 × log (∑( 2 ) / )

In this study, the response value is computed based on the division of two separated metrics, namely MID and MS (Equation (19)).

This response value considers both the convergence rate of solution (MID) and verity of solution (MS) (Cheraghalipour et al., 2018) . For a start, the level of each factor for different algorithms should be identified. MOSA has three parameters with three levels. NSGA-II and MOKA have four and five parameters with three levels, respectively. Finally, MOKASA contains seven parameters with three levels. Thus, L27 is recommended as a proper array for NSGA-II, MOKA, and MOKASA in addition to L9 for MOSA. For tuning purposes, test number 10 in Table 2 is applied. Then, the orthogonal arrays are implemented in Minitab software. The obtained plot of ⁄ is shown in Figures 10-13 . According to this ⁄ ratio plot, the J o u r n a l P r e -p r o o f level of parameters with the lowest value is preferable. Additionally, tuned values for the tenth test reported in Table 4 are considered for all test problems. 

In this section, a comprehensive comparison is presented to check the quality of performance for each algorithm. To this end, we use five performance metrics. In the beginning, we set the parameters with the tuned values obtained in the previous section, then run each test problem 30 times and reported the results of them as the average of all runs. Figure 14 is related to problems 4, 9, and 14 as representative of different problem sizes. The result of five performance metrics is demonstrated in Table 5 . Firstly, we convert all the metrics value to Relative Deviation Index (RDI) by using Equation (20):

where is the value attained by performance metric, is the best value of performance metric among the applied metaheuristics. The means plot and LSD for each problem size are thoroughly explained in relevant sections.

Besides the abovementioned tool, we use two nonparametric statistical tests. Generally, metaheuristic algorithms have stochastic traits, so nonparametric statistical tests are mandatory for a more precise analysis of archived solutions by algorithm and comparison (Sahebjamnia et al., 2020) . If the nonparametric statistical test only compares two algorithms, they label as pairwise comparisons. One of the appreciated pairwise nonparametric tests is Wilcoxon signed-rank test which we conduct in our study. As a post hoc test, we implement the Holm test. But, if they compare more than two algorithms, they define as multiple comparisons. The Friedman test is of multiple comparison tests, which is applied here (Derrac et al., 2020) . Table 6 encompasses required definitions of terms associated with statistical tests; however, the detailed information can be found in Derrac et al. (2020) .

J o u r n a l P r e -p r o o f Table 6 . Definition of statistical test terms.

The null hypothesis (H0) Declares that two metaheuristics have no difference.

alternative hypothesis (H1) Declares that two metaheuristics have differences.

Statistical significance level (α) The probability of mistakenly rejecting H0. For p-value, less than α H0 is rejected.

The probability of rejecting a correct H0 by at least one comparison test.

The test examines and controls FWER.

According to Figure 15 , MOSA is leading in terms of SNS, CPU, and HV, though MOKA and MOKASA are better, respectively, in terms of MID and MS. To implement Wilcoxon signed ranked test, we use HV values for small-size problems and run the test with a statistical significance level (α) of 0.05 using SPSS software. Table 6 presents the result of the Wilcoxon test. P-values for Wilcoxon tests and Holm tests in each comparison set are less than the significance level (α) of 0.05. Finally, we did multiple statistical comparisons using the Friedman test for the smallsize problem with a statistical significance level (α) of 0.05 using SPSS. This test is performed based on all performance metrics, and Table 7 shows the result of the Friedman test. Results state that MOSA has superior performance in all metrics than other algorithms except in HV J o u r n a l P r e -p r o o f that MOKA precedes other algorithms. It is worthwhile to mention that p-values in all tests are less than the significance level (α) of 0.05. Figure 16 displays the means plot and LSD for medium-size problems. It can be concluded that MS, SNS, and CPU means plot and LSD, MOSA outperformed others. In terms of MID and HV, MOKA and MOKASA had better performance in comparison with other algorithms. The result of the Wilcoxon test is demonstrated in Table 8 . Like small-size problems, Wilcoxon signed a ranked test applied using HV values for medium-size problems with a statistical significance level (α) of 0.05 using SPSS software. P-values for Wilcoxon tests and J o u r n a l P r e -p r o o f Holm tests are less than 0.05, so it determined that there is a difference between the two metaheuristics in each set. Friedman test for medium-size problems with a significance level (α) of 0.05 using SPSS was carried out. All performance metrics have participated in this test for multiple comparisons, and Table 9 shows the achieved results. Excluding SNS that MOKA has been surpassed, results assert that MOSA has been at the top in all performance metrics. 

Here, the performance of large-size problems is evaluated through previously mentioned metrics. Firstly, the means plot and LSD are presented in Figure 15 . It reveals that MOSA had overcome other algorithms in terms of MS, SNS, CPU, and HV. But, in terms of MID, MOKASA is better than the others.

Large-size problems are statistically compared using Wilcoxon signed ranked test. In this test, HV values for large-size problems are considered and run the test with a significance level (α) of 0.05. The result illustrated in Table 10 for the Wilcoxon test, and Holm test is less than the significance level (α) of 0.05. As a result, there is a difference between each pair of algorithms. Table 11 shows the results of the Friedman test for the large-size problem with a statistical significance level (α) of 0.05. It can be comprehended that MOSA has superior performance in all metrics. The obtained p-value in all tests is less than 0.05.

In conclusion, the analysis of performance metrics for each problem size affirms that MOSA has a better performance compared to other algorithms. J o u r n a l P r e -p r o o f

Here, we investigate the sensitivity analyses of parameter changes' effect on the proposed model's objective function values. In previous sections, MOSA is found as the most superior algorithm. Hence, MOSA is selected as the algorithm used for sensitivity analysis purposes using medium size problem number 9 in Table 2 . The first experiment evaluates the impact of storage capacity, distributor capacity, and PPE production capacity on the objective functions. In this experiment, the range of storage capacity, distributor capacity, and PPE production capacity change from the lowest value to the highest value in four test problems. Afterward, test problems are solved using the MOSA algorithm, and the results are reported in Table 12 . Figure 18 exhibits that both objective functions Z1 and Z2 increase as the capacity and storage parameters have positive growth. The second experiment monitors how the treatment capacity of the medical center and population of the region affects the behavior of total cost of supply chain and demand shortage. Similar to the previous experiment, this experiment considers the lowest and highest value of both parameters. For the designed test problems, the achieved outputs of MOSA for the second experiment are presented in Table 13 . Furthermore, Figure 18 emphasizes that the total cost of supply chain and demand shortage are increased while the capacity of the medical center and population of the region significantly raised. The third experiment is founded on the behavior of objective functions influenced by variation of recovery and death rate of patients. We consider a wide range for the death rate, from the lowest to the highest, in four experiments, and utilize the MOSA algorithm to solve these test problems. Table 12 and Figure 18 point out that total cost and unsatisfied demand in the supply chain network have considerably positive relations. 

A relief SC network for the pandemic situation such as COVID-19 was designed in this paper. We modeled our proposed SC by using a multi-objective mathematical approach. We tried to investigate and analyze the total cost of the PPE SC and shortage throughout the network. Four multi-objective metaheuristic algorithms were applied, including MOKA, MOSA, NSGA-II, and MOKASA, to investigate and endorse the model's applicability. Afterward, the Taguchi method was used as a tuning assistant in this study to find the best level for the algorithms' parameters results achieving better performance. Among the fifteen J o u r n a l P r e -p r o o f developed test problems, the third test problem is inspired by a real case in Khuzestan province in southern Iran. Then, five evaluation criteria, MID, MS, SNS, HV, and CPU time, were utilized. Considering the evaluation criteria, each algorithm showed different behaviors, but, totally, we can conclude that MOSA showed better performance for the problems we studied in this work.

In this study, we confronted a few limitations which they could be overcome for future researches. Data gathering was the critical limitation in this work. As we earlier alleged, healthcare systems and organizations control the tough situation during the pandemic and try to prevent more casualties; therefore, they could not provide valid data. In future studies, researchers can access more valid data that might bring about closer solutions to reality. Another limitation of this study is that sustainability assumptions should be informed especially environmental and social aspects. In this study, the proposed model has not considered the problem in stochastic and uncertainty settings. Nevertheless, the model might be formulated with uncertainty constraints or parameters.

In general, we suggest the researchers for future studies to design the proposed PPE supply chain model tailored to the specific traits related to other countries' conditions and constraints, improve the proposed model by considering the vital aspect of sustainability, and formulate stochastic models for the PPE supply chain to satisfy the network's demand flow.

In Equation (2), nonlinearity is discernible, so to overcome this issue, we use Equation (A.1) instead of Equation (2) and also add constraint (A.2) to the constraint sets. 

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