A COMPUTATIONAL METHOD FOR MEASURING TRANSPORT RELATED CARBON EMISSIONS IN A HEALTHCARE SUPPLY NETWORK UNDER MIXED UNCERTAINTY : AN EMPIRICAL STUDY

Measuring carbon emissions is an essential step in taking required action to fight global warming. This research presents a computational method for measuring transport related carbon emissions in a healthcare supply network. The network configuration significantly impacts carbon emissions. First, a multi-objective mathematical programing model is developed for designing a healthcare supply network in the form of a two-graph location routing problem under demand and fuel consumption uncertainty. Objective functions are minimizing total cost and minimizing total fuel consumption. In the presented model, the demand of each customer must be completely satisfied in each time period, and backlog is not permitted. The number and capacity of vehicles are determined, and vehicles are heterogeneous. Furthermore, fuel consumption depends on traveling distance, vehicle and road conditions, and the load of a vehicle. The centroid method is applied to face demand uncertainty. Next, a multi-objective non-dominated ranked genetic algorithm (M-NRGA) is proposed to solve the model. Then, a Monte Carlo based approach is presented for measuring transport-related carbon emissions based on fuel consumption in supply network. Finally, the proposed approach is applied to the case of a healthcare supply network in the Fars province in Iran. The obtained results illustrate that the proposed approach is a practical tool in designing healthcare supply networks and measuring transport-related carbon emissions in the network.


INTRODUCTION
Environmental changes and global warming are among the most important challenges humans have faced in the last hundred years [1,2].Earth system simulation shows that the current trend of greenhouse gas emissions may affect most ecosystems and the lives of over 3.5 billion people worldwide as early as 2050 [3].Carbon dioxide (CO 2 ) and carbon monoxide (CO) are the primary greenhouse gases emitted through human activities.As the most important cause of long-term climate change, carbon emissions provide a good baseline to assess progress and evaluate the consequences.The current emission trends continue to follow scenarios that lead to the highest global temperature increases [4].Carbon emissions produced by human activities usually come from burning fossil fuels, e.g.oil, natural gas, coal, and wood [5].
To overcome this challenge, governments and industry sectors are working to understand their own carbon footprint and explore the required actions.Reducing the global temperature and associated climate change impacts can be achieved by decreasing the emissions of greenhouse gases as soon as possible [6].The experiences of the largest global corporations as well as those of start-up companies indicate how these companies can profitably reduce greenhouse gas emissions in their supply chains [7].Approximately 45% of greenhouse gas emissions is caused by production and transportation of goods [8].Hence, it is important to restrict such emissions in a supply network.
In the past decades, many researchers have focused on modeling the uncertainty in designing supply networks.For this aim, different stochastic modeling approaches have been successfully applied to supply chain production planning problems considering the randomness concept.However, in some applications, the probability distributions are not available or reliable [9].On the other hand, in some real applications, the nature of retailer's demand is imprecise.In this case, fuzzy set theory (FST) provides a proper decision-making with regard to procurement, production, and inventory management using relatively simple and widely used models.
The transportation of goods accounts for a large amount of carbon releasing all over the world.TRCE cannot be measured easily because of the large amount of data required and the use of different standards for carbon footprint calculation [14].Several researchers proposed various methods for computing carbon emissions in a supply chain [15][16][17][18][19][20].For instance, they calculated the carbon footprint of different feedstock for dairy cattle by applying life cycle assessment (LCA) [18].Others proposed a fast and vigorous technique applying a novel approach based upon multi-gene genetic programming (MGGP) to determine carbon dioxide minimum miscibility pressure (CO 2 MMP) for carbon dioxide injection processes [19].In [20], a "corrected average emission model" was provided, i.e., an improved average speed model that accurately computed CO 2 emissions on the road.
Transportation consumes energy severely worldwide and is overwhelmingly oil-oriented.For this reason, TRCE is one of the main sources of global carbon emissions and also contributes to air quality concerns, particularly in and around major population centers [21].In this regard, [22] showed that most of the imported oil in the USA is consumed only in the transportation sector.Additionally, one third of total greenhouse gas (GHG) emissions is produced by this sector.For reducing GHG emissions and oil consumption, different policy scenarios must be considered in the US transportation sector.Therefore, proper green transportation network design is a suitable solution for decreasing carbon emission [23], [24].
Recently, in the period of global competition, the network design has been considered as one of the strategic decision problems.A network design problem focuses on the number and locations of raw material suppliers, manufacturing plants, and inventory warehouses.In an extended time horizon, the decisions in this regard include selecting the distribution channel from suppliers to customers as well as determining the transportation volume among distributed facilities [25].The supply network design problem is well documented in the literature, and the readers are referred to the review the paper [26].
The vehicle routing problem (VRP) is the problem of designing routes for delivery vehicles with known capacities operating from a single depot to supply a number of customers with known locations and known demands for a certain commodity.Vehicle routes are designed to minimize several objectives, like the total distance traveled [27].For instance, the authors in [28] presented a new mathematical model in order to measure and evaluate the efficiency of the periodic vehicle routing problem (PVRP) in a competitive environment.A general model for dynamic vehicle routing framework for handling uncertainty or vagueness in a dataset.In decision sciences, fuzzy set theory has a great impact on preference modeling and multi-criteria assessment by taking into account the user needs in optimization techniques [10].Hence, in the proposed method, trapezoidal fuzzy numbers are used to characterize the fuzzy demand for products, vehicle performance, and road conditions.Note that in most real applications of a supply chain problem there is not enough historical data for the previous demand values.In these cases, using fuzzy numbers can be used to describe the decision models for the external demand of a network.Although there is no conclusive methodology for measuring carbon emissions at present [11,12], this research provides a computational method for measuring transport related carbon emissions (TRCE) in a supply network.First, a mathematical model to design proper supply network is developed, and then a computational method is proposed for measuring TRCE in this network, based on the Monte Carlo approach.In Section 2, a literature overview of supply network design and measuring carbon emissions is given.The mathematical model to design the supply network is presented in Section 3, followed by the solution approach and TRCE measuring approach.The case study, i.e., Fars healthcare supply network, is presented in Section 5.At the end, the conclusion and future study considerations finalize the paper.

LITERATURE REVIEW
Generally, the causes of carbon emissions can be divided into natural and human sources.The main carbon emissions due to human activities includes industries like cement and electricity production, commercial and residential causes, deforestation, agriculture, and transportation.Also, the major carbon footprint applications in the UK can be categorized by seven areas, including national emissions inventories and trade, emission drivers, economic sectors, supply chains, organizations, household consumption and lifestyles, as well as sub-national emission inventories.Researchers in [8,13] state that to restrict the emissions of greenhouse gases, companies should take into account their business practices and operational policies along with focusing on their physical processes (inefficient equipment and facilities, redesigning products and packaging, finding less polluting sources of energy, or instituting energy saving programs).They attempted to explore how operational decisions across the supply chain affect the carbon footprint of these supply chains and the extent to which concerns about carbon emissions are covered by adjusting operational decisions and improving collaboration among supply chain partners.They also showed how carbon emission concerns can be integrated into operational prove their idea, they created a probabilistic technique for measuring and evaluating that impact by using the Markov theorem.
In this paper, we simultaneously take into account the environmental concerns and uncertainty conditions in a healthcare supply network.A multi-objective non-dominated ranked genetic algorithm (M-NRGA) is proposed to solve the developed multi-objective mathematical model.A Monte Carlo based approach is presented to measure transport-related carbon emissions based on fuel consumption in a supply network.We also represent the application of the proposed approach to the case of the healthcare supply network in the Fars province in Iran.

PROBLEM DEFINITION 3.1 Healthcare supply network
In the presented healthcare supply network, a number of distribution centers should be located among candidate sites.Furthermore, delivery routes between the central depot and these distribution centers as well as delivery routes for a set of customers must be established in such a way that the total system cost is minimized.Therefore, the network configuration includes a two-graph location routing problem.In one graph, a fleet of vehicles with known capacities is operating from a central depot to supply a number of distribution centers which should be located among candidate sites.All vehicle routes start and end at the central depot.The demand of each distribution center must be completely satisfied, and backlog is not permitted.All distribution centers are visited exactly once by exactly one vehicle in each time period, so split delivery is not permitted.The sum of distribution centers' demand for any vehicle route may not exceed vehicle capacity.In the other graph, the network configuration is a multi-depot vehicle routing problem in which a fleet of vehicles with known capacities operate from a central depot to supply a number of customers with known locations.Each vehicle route starts and ends at the same distribution center.The demand of each customer must be completely satisfied, and backlog is not permitted.All customers are visited exactly once by exactly one vehicle in each time period, so split delivery is not permitted; the sum of customers' demand for any vehicle route may not exceed the vehicle capacity.A direct delivery from the central depot to the customers is not permitted, as well as a delivery from one distribution center to the other.In summary, we face a two-graph location routing problem, i.e., to simultaneously determine the number and locations of distribution centers, assignment of customers to distribution centers, and vehicle routes.Figure 1 illustrates the healthcare supply network.In the proposed model, there are multiple time periods in the planning problem with time windows (DVRPTW) considering the minimization of the total distance as the main objective was presented in [29].A review paper highlighted major exact algorithms in the VRP literature [30].They focused on mathematical formulations, relaxations, and recent exact methods for two main VRP variants, including the capacitated VRP (CVRP) and the VRP with time windows (VRPTW).For detailed information about VRP, please refer to the review paper [30].
As one type of the network design problem, the location routing problem (LRP) deals with the combination of both the facility location problem (FLP) and the vehicle routing problem (VRP).Since these two categories of problems belong to the class of NP-hard problems, the LRP is also an NP-hard problem.In the LRP, the entire consumer demand should be satisfied in such a way that a facility's operating and fixed costs are minimized while vehicle capacities must be considered and routing costs must be reduced simultaneously [31].Several researchers have developed heuristic and meta-heuristic algorithms for LRP [32][33][34].
A new heuristic algorithm for the capacitated location routing problem (CLRP) was presented in [35], called granular variable tabu neighborhood search (GVTNS).In order to solve the periodic location routing problem (PLRP), a large neighborhood search (LNS) algorithm was presented in [36].
Recently, researchers and practitioners have taken into account the environmental concerns in designing supply networks.In this respect, decisions on supply chain design have to be integrated with those related to environmental concerns.In other words, a green supply chain network design problem contains an initial investment into environmental protection equipment or techniques to ensure its long-term benefit to environmental indicators [37].Fortunately, due to the importance of the green supply chain concept, a lot of research has been focused on this area.Please see the comprehensive survey provided by [38] for more detailed information.Most research on designing a green supply network has considered a deterministic behavior for the supply network.However, in most practical situations, we may face numerous sources of technical and/or commercial uncertainty in the design phase.
The concept of uncertainty in LRP problems is considered in several studies [33], [39][40][41][42][43].The authors in [44] described the impact of network type on uncertainty in demand estimation.They suggested that the configuration of the network can affect the final accumulation of uncertainty in the supply chain.The authors in [45] identified the impact of the accumulation of individual delivery time uncertainties on overall delivery time uncertainty.Also, they stated that the type of network and their structures have a significant influence on delivery time uncertainty.In order to Assumptions -The selection of the central depot and customers' location is beyond the scope of this paper, but the locations of distribution centers are determined by the presented model -The vehicles are heterogeneous -Split delivery is not permitted -The capacities of distribution centers and vehicles are definite -The routing process is performed between customer -distribution center echelons and central depot -distribution center echelons.Hence, the VRP is a multi-graph problem.-It is not possible to stock pharmaceutical substances for future periods in distribution centers -Demand is considered to be uncertain -Fuel consumption per distance unit is considered to be uncertain -Fuel consumption per load unit is considered to be uncertain -Time windows for distribution centers and customers are not considered.

Mathematical model for healthcare supply network design
The proposed mathematical programing formulation is presented below:  [46].
In Equation 1, if d 2 =d 3 , then t u is a triangular fuzzy number.Also, if d 1 =d 2 =t 3 =t 4 , then t u is a crisp number.Figure 2 illustrates the membership function of the trapezoidal fuzzy number.
The number and capacity of vehicles and candidate locations for distribution centers are determined.Vehicles are heterogeneous and can take only one tour in each time period.The objectives are minimizing the total cost of the system and minimizing total fuel consumption.Fuel consumption depends on traveling distance, vehicle and road conditions, and the load of a vehicle.All of these quotients are trapezoidal fuzzy numbers.
Objective function 2 shows the sum of fixed costs of opening distribution centers in candidate locations, the sum of fixed costs for using vehicles at the central depot, the sum of fixed costs for using vehicles at distribution centers, and transportation costs.Objective function 3 represents fuel consumption considering traveling distance, vehicle and road conditions, and the load of a vehicle.
Constraint 4 assures customers are allocated to distribution centers within their capacity.Constraint 5 states that the demand of distribution center j is equal to sum of the demands of customers which are allocated to distribution center j.Constraint

SOLUTION APPROACH
The solution approach is presented in 3 steps.First, we present an approach for facing uncertainty; then we describe a method for solving the proposed optimization model for the network design problem; and, finally, we propose a Monte Carlo approach based approach for measuring TRCE.
and then the solution is selected with the probability of P r from the selected front, as presented in Equation 25.
where NF is the number of fronts, Rank F is the rank of front F, NSF F is the number of solutions in front F, and Rank sF is the rank of solution s in front F. 6) If the end condition is satisfied, the solution approach is over, otherwise go back to step 3.
In the second phase, the best solution is identified by PROMETHEE-II from the best Pareto front obtained in the previous phase.The PROMETHEE was first developed by Brans and Vincke in 1985 [50].PROMETH-EE-I can provide a partial ranking, while PROMETHEE-II can drive total ranking of the solutions.Therefore, in this paper, we use PROMETHEE-II.The method we applied is exactly as it described in [51].

Measuring TRCE
Measuring TRCE based on fuel consumption is not a straightforward processs.The amount of carbon emission depends on many factors such as the quality of fuel, vehicle conditions, weather conditions, the load of vehicle, etc.Therefore, in the mathematical formulation based on fuel consumption for TRCE, the carbon emission quotient is a probabilistic parameter.Due to Equation 3, TRCE can be measured by Equation 26.
where } t is a probabilistic parameter probability density function f(y) for the carbon emission quotient.The cumulative distribution function F Y is defined as Equation 27.
To calculate TRCE, we applied a Monte Carlo based approach.The suggested algorithm is as follows: In this paper, the centroid method is applied for the defuzzification of the trapezoidal fuzzy number.The centroid of the trapezoidal fuzzy number t u is shown in Equation 23 [47].

Solution algorithm
To solve the proposed model, we developed a twophase algorithm based on non-dominated ranked genetic algorithm for solving multi-objective optimization problems (NRGA) and preference ranking organization method for enrichment of evaluations (PROMETHEE).In the first phase, the Pareto front is generated by NRGA; in the second phase, the best solution is determined by the PROMETHEE-II method.
NRGA was developed by Al Jadaan et al. [48].In this paper, we develop a modified version of the multi-objective NRGA called M-NRGA.In NRGA, a two-phase selection method is implemented.In the first phase, one of the fronts is selected by a fitness proportionate selection procedure (roulette wheel) based on its rank, and in the next phase one chromosome is selected from this front.Therefore, in NRGA the possibility of selecting a chromosome depends on its front's rank [49].In the proposed M-NRGA, to improve the quality of the solutions in the Pareto front, P e percent of solutions are selected based on elitism from the elitist solutions in the first front.In addition, to increase the diversity of the solutions in the Pareto front, P t percent of solutions are selected based on the tournament method.In the proposed M-NRGA, since P e percent of solutions are selected based on elitism, it is possible to adjust the number of solutions in the Pareto front to be at least equal to a fixed number.The proposed M-NRGA, as described, is as follows: 1) Initialize the first generation of solutions randomly with the population size N. 2) Calculate both objective functions for all solutions.
3) Sort and locate the solution based on the method presented in [48].4) Do the crossover operation for P c and mutation for P m presentation of population.5) Select N solution for the next generation.Three strategies are applied to select the solutions: a) P e percent of solutions are selected based on elitism from the elitist solutions in the first front.b) P t percent of solutions are selected based on the tournament method.Two solutions are selected randomly.The one that belongs to the front with the higher rank wins.If the solutions come from the same front, the one with the higher rank in the front wins.c) P r percent of solutions are selected based on the roulette wheel method.At first, the front is selected with the probability of P F , as shown in Equation 24, Iran with the population of 1.9 million [52].There are 29 customers and 5 candidate locations for distribution centers in Shiraz, Fasa Eqlid, Firuzabad, and Lar.The distances between network nodes are given in Table 1.In this supply network, 2,447 types of products are distributed.The demand for each product is estimated based on recent 36-month demand.The total demand for all products in Fars is 2,405,281,094 annually.We used Equation 23 to calculate the centroid of demands .Cd im t u ^h Some of the results are given in Table 2. Obviously, these 2,447 types of products increase the problem complexity unnecessarily.In order to simplify the problem, we classified the pharmaceutical substances into 47 major groups, as shown in Table 3.
Three different meta-heuristic algorithms, including NSGA-II, MOPSO, and the proposed M-NRGA, are used to solve the proposed optimization model for the network design problem.NSGA-II is one of the most popular multi-objective evolutionary algorithms,

S S S S S S S S S R T S S S S S S S S S S S R T S S S S S S S S S S S R T S S S S S S S S S S S S R T S S S S S S S S S S S R T S S S S S S S S S S S S R T S S S S S S S S S S S V
4) Calculate average and variance of resulted vector.
That would present the average and variance of TRCE.5) Draw the histogram of the transpose of resulted vector.From the sampling frequency of [TRCE 1 ,TRCE 2 ,..., TRCE B ], the probability density function of TRCE can be obtained.

CASE STUDY 5.1 Fars healthcare supply network
In this paper, Fars healthcare supply network is considered as the case study.Fars is one of the thirty-one provinces of Iran and located in the south of the country.Fars has one of the richest cultural heritages in Iran, encompassing many disciplines such as literature, poetry, architecture, and stonemasonry, and it is known as the cultural capital of Iran.In 2015, this province had a population of 4.6 million people [52].Due to the capabilities of Fars healthcare industry, such as a remarkable number of hospitals, reputable physicians, high quality healthcare services, mild weather, tourist attractions, and good hotels, many patients from all over Iran, the Persian Gulf countries, and even Europe travel to Shiraz to receive healthcare services.
We consider five years as the planning horizon for the Fars healthcare supply network divided into 60 one-month periods.In the Fars healthcare supply network, as illustrated in Figure 4, the central depot is located in Shiraz.Shiraz is the capital city of the Fars province and the most populous city in the south of  2) For each solution: a) Select the G-best from the archive; b) Update velocity; c) Update position 3) Update the archive of non-dominated solutions 4) Repeat [55].
In the presented case study, to measure TRCE, a vector with B=1,000,000 array for each probabilistic and fuzzy numbers based on their probability density function or fuzzy membership function is generated by MATLAB 2012a.Then the histogram and Kolmogorov-Smirnov test is calculated to estimate the possibility distribution function of TRCE.

Results
The presented problem is solved by NSGA-II, MOP-SO, and M-NRGA.The input parameters of these algorithms are given in Table 4. Four common measures are used to compare the algorithms.The most important index is the quality of solutions.This index cannot be measured for a Pareto front individually and must be calculated comparing to another Pareto front.Assume P i is the Pareto front resulted by algorithm i, and P j is the Pareto front resulted by algorithm j, then the quality index for algorithm i and j (Q(P i , P j )) can be calculated by Equations 29  developed by Deb et al. based on the genetic algorithm [53].Like other evolutionary algorithms, in the first step NSGA-II generates random solutions with the population of μ.In addition, each solution is evaluated by fitness functions, and, based on this evaluation, Pareto fronts are created by non-domination sorting.In the next step, each solution receives a rank equal to the level of the front that it belongs to.Then the crowding distance between the solutions on each front is measured.The selection procedure is the binary tournament method.The winner is the solution with the higher rank, and if the ranks are equal, the winner is the one with the higher crowding distance.The rest of the NSGA-II procedure is exactly the same as the genetic algorithm [54].Moore and Chapman were the first to present MOPSO [55].The procedure of MOP-SO is similar to PSO.The only difference is that since the problem is multi-objective, instead of determining global best solution (G-best), the Pareto front would be determined.So, an archive including all of non-dominated solutions found in each iteration are stored.The steps of MOPSO are: 1) Initialize random solutions with the population of n and the archive of non-dominated solutions.As presented in Table 5, the average of Q(MOP-SO,NSGA-II) is higher than Q(NSGA-II,MOPSO), so, based on the quality index, the first assumption is that MOPSO performs better than NSGA-II.Similarly, based on the quality index, we can assume that M-NR-GA performs better than both MOPSO and NSGA-II.Table 6 shows that, on average, NSGA-II has better performance in CPU time.In addition, the average number of solutions in the Pareto front is higher in M-NR-GA.The average diversity of the solutions in the Pareto front is higher in M-NRGA.It should be mentioned, since in the proposed M-NRGA P e present of solutions are selected based on elitism from the elitist solutions in the first front, by adjusting P e the number of solutions in the Pareto front can be adjusted to be at least equal to a fix number.In this paper, we consider 30 as the minimum number of solutions in the Pareto front.
To ensure that the presented algorithm can solve the network design problem efficiently and in reasonable time, statistical tests are implemented [57].The confidence level of 99% is considered for all of the tests.The first test is if NSGA-II performs better than the proposed M-NRGA in each index, and the second one is if MOPSO performs better than M-NRGA in each index at the confidence level of 99%.As presented in Considering Equations 29 and 30, it is clear that Q(P i , P j )+Q(P j , P i )=1.For the quality index, if Q(P i , P j )>Q(P j , P i ), then the Pareto front resulted by algorithm i is better than the Pareto front resulted by algorithm j [56].
The next index for evaluating the performance of a multi-objective algorithm is the diversity of the solutions in the Pareto front (D(P i )).The higher value of this index shows the better performance of the multi-objective algorithm.The diversity of solutions in a Pareto front is calculated as presented in Equation 31.
where Z j is the objective function j and X k is the solution in Pareto front P i [56].Two other indexes are CPU time and the number of solutions in the Pareto front (NSPF).The higher number of solutions in the Pareto front is more preferable because it can give the decision makers more options.We solve the presented case study 100 times.For the presented problem, Table 5 shows the quality index for presented algorithms, and other indexes are presented in Table 6.Based on Equations 29 and 30, Q(M-NRGA,NS-GA-II)=1 and Q(M-NRGA,MOPSO)=0.966.So, according to these indexes, M-NRGA performs better than the other two algorithms.However, in the CPU time index, NSGA-II preforms better.
To find the best solution in the Pareto front resulted from M-NRGA, as presented in the solution algorithm, PROMETHEE-II was applied.The best solution is also shown in Figure 7.For this solution, Objective function 2 -minimizing total cost -is: 1,237,447.6$, and Objective function 3 -minimizing fuel consumption -is: 13,233,837.6 liters.quality index of the Pareto front, number of solutions in the Pareto front, and Pareto front diversity, the performance of M-NRGA is better.
As mentioned before, the most important index for evaluating Pareto fronts is the quality index.The best Pareto fronts for the Fars healthcare supply network design problem, based on this index, are illustrated in Figure 5 and Table 8.As presented in Table 8, according to Equation 18, the diversity of the solutions in the Pareto front resulted from M-NRGA amounts to 9,235,474.32,from MOPSO to 8,868,654.35,and from NSGA-II to 6,119,401.00.20,000 19,000 18,000 17,000 16,000 15,000 14,000 13,000 12,000 11,000 10,000

CONCLUSION AND FUTURE STUDY
In recent history, global warming has been one of the most important challenges for humanity.Understanding carbon footprint is vital in overcoming this challenge.In this paper, we provided a computational method for measuring TRCE in a healthcare supply network.In the first step, a mathematical model for designing a proper supply network under uncertainty was developed.To solve this model, we proposed a twophase algorithm based on NRGA and the PROMETH-EE-II method.The statistical tests showed that the proposed algorithm has better performance in solving the model in comparison to NSGA-II and MOPSO.In the proposed algorithm, the number of solutions in the Pareto front can be adjusted to be at least equal to a fixed number.This can provide more possibilities for decision-makers to obtain optimal solutions in real-world problems.In the next step, a computational method based on Monte Carlo was developed for measuring TRCE.The proposed approach was applied to the Fars healthcare supply network.After solving the model, the total cost is $1,237,447.6,and the fuel consumption is 13,233,837.6 liters in the planning time horizon.The probability distribution function of TRCE is normal with the mean of 32,007,127,597.94grams and variance of 6.8246E+19.The obtained results confirm the efficiency of the proposed approach as a practical tool for measuring TRCE.In this research, time windows for product delivery to distribution centers and TRCE vector is calculated by Equation 27.A vector with B=1,000,000 arrays for each probabilistic and fuzzy number based on their probability density function or fuzzy membership function is generated by MATLAB 2012a for measuring TRCE.Table 9 shows the descriptive statistics of TRCE in the Fars healthcare supply network.The average of TRCE is: 32,007,127,597.94grams, and the standard deviation of TRCE is: 8,261,131,178.88.
The histogram of TRCE is illustrated in Figure 6.According to this histogram, we expect that the probability distribution function of TRCE would be normal.The Kolmogorov-Smirnov test is applied to test the normality of TRCE.
Sets: I -Set of customers indexed by i=1,2,…,N; where N is the number of customers J -Set of candidate locations indexed by j=0,1,2,…,P; where P is the number of candidate locations.0 is the central depots index G -Set of all supply network graph nodes G=I,J={0,1,2,…,P,P+1,…,P+N} A -Set of arcs (i,j) T -Set of time periods indexed by t=1,2,…,K; where K is the number of time periods in the planning horizon M -Set of products indexed by m=1,2,…,M; where M is the number of products distributed in the supply network V -Set of vehicles indexed by v=1,2,…,V; where V is the number of available vehicles Parameters: dim t u -Customer i demand for product m in the period t j p -Fixed cost for opening a distribution center in the candidate location j v g -Fixed cost for using vehicle v ij c -The distance along the arc (i,j)!A ij v s -Traveling cost per distance unit for vehicle v in arc (i,j)!A j m a -The capacity of distribution center placed in node j for the product m.Capacity of central depot ( ) m 0 a is infinitive for all products.horizon.The customers' demand for each product is a trapezoidal fuzzy number ( d u =(d 1 ,d 2 ,d 3 ,d 4 ) where: d 1 ≤d 2 ≤d 3 ≤d 4 ) on R ).The membership function of the trapezoidal fuzzy number is presented in Equation 1

1 ũ 2 ũ
v m b -Capacity v-th vehicle for product m m j -Weight of each unit of product m v 0 ũ -Fuel consumption per distance unit for unloaded vehicle v in a road without slope v -Additional fuel consumption per distance unit for vehicle v associated with load unit v -Quotient of fuel consumption per distance unit for vehicle v associated with vehicle conditions

Figure 5 -
Figure 5 -The best Pareto fronts for the Fars healthcare supply network design problem, based on quality index

Figure 7
presents the normal Q-Q plot of TRCE.It proves that the probability distribution function of TRCE is normal.

Figure 6 -Figure 7 -
Figure 6 -Histogram of TRCE for Fars healthcare supply network 6ensures that customers are visited within vehicle capacity.Constraints 7 and 8 ensure that all customers and all distribution centers are visited once in each time period.Constraint 9 states that each vehicle can take only one tour in each time period.Constraints 10, 11, and 12 show the load of vehicles at each route.Sub-tour elimination in both graphs is assured in Constraints 13 and 14.The continuity of tours and returning the vehicle to the origin depot is ensured in Constraint 15.Constraint 16 states that customer can be allocated to a distribution center and distribution center can be allocated to central depot only if there is a route connected to them.The binary variables are defined inConstraints  17, 18, and 19.Finally, load and auxiliary variables taking positive values are declared in Constraints 20, 21, and 22.

Table 2 -
The centroid of demand for products in Fars healthcare supply networ

Table 3 -
Products classification

Table 7 ,
both MOPSO and NSGA-II have a better CPU time comparing to M-NRGA.However, in terms of the

Table 5 -
Quality index for presented algorithms

Table 6 -
The value of indexes for presented algorithms

Table 7 -
The statistical tests results for all hypotheses

Table 8 -
Best Pareto front for the presented problem based on quality index

Table 9 -
Descriptive statistics of TRCE