Optimal Decision Making for Fractional Multi-commodity Network Flow Problem in a Multi-choice Fuzzy Stochastic Hybrid Environment

In this paper, a fractional multi-commodity network flow problem with multi-choice parameters is studied under hybrid fuzzy-stochastic conditions. In this problem, coefficients of the objective function in both the nominator and denominator take the form of multi-choice parameters, with the alternative choices for the nominator and denominator of the fraction being represented by fuzzy-stochastic and fuzzy variables, respectively. The arc capacities are also considered as fuzzy-stochastic variables. The main goal of the present research is to provide the decision-maker with a model by the help of which he/she can manage unknown factors across a multi-commodity network. Given that this problem is herein investigated in a hybrid fuzzy-stochastic environment and includes multi-choice parameters, we use the probability-possibility approach, Lagrange interpolating polynomial, and the Charnes–Cooper variable transformation technique to convert the problem into a deterministic one. Finally, efficiency of the proposed model is evaluated by presenting a couple of numerical instances.


Introduction
A group of optimization problems that have many applications in various fields of science and engineering are network problems. If we want to define a network from a mathematical point of view, we have to define it as a directed graph where one or more real numbers are attributed to its all or parts of nodes or arcs (indicating capacity, transmission cost and ...for arcs and capacity, stop cost, production rate and ...for nodes). For example, computer networks, power grids, telecommunications and internet networks, highway networks, rail and air networks, and finally production and distribution networks, all of which are tied to our lives. In all of these areas, the goal is to effectively transmit one or more commodity from one point to another point of a network so that better service and higher network efficiency are obtained. In the meantime, Multi-Commodity Networks Flow (MCNF) problems are a special category of these problems that due to their various applications on one hand and the theoretical attractiveness on the other hand have received much attention in recent years. In this type of problems, we want to transmit a certain number of different commodities from specific origins to specific destinations in a network, each arc of which has a certain capacity and cost, so that the demand for each destination is met according to the stock of the origin points.
Despite the presence of numerous deterministic models with thousands of variables and constraints in the industry, their results are often times simply ignored by managers and decision-makers. This can be linked to the fact that such results are usually outcomes of deterministic modeling studies and become simply invalid under highly variable conditions in reality. Indeed, such changes in the condition may make previously optimal results either infeasible or suboptimal. On the other hand, many real-world situations come 1 3 36 Page 2 of 17 with multiple alternatives. For instance, a shipping carrier considers different shipping costs for transporting the same goods depending on the actual carrier (i.e., truck, ship, train, or aircraft). In the meantime, various companies have implemented different price discrimination strategies [33] based on the price elasticity. For such companies, making optimal decisions when multiple choices are available is of paramount importance and imposes direct impacts on their profitability. When the decision criterion is singular, the problem at hand would be simple to address. With increasing the number of objectives to be addressed simultaneously, however, the problem becomes highly challenging to solve. In this situation, we encounter three challenges: 1. As a first challenge, one must present a model that ensures the decision-maker a high degree of isolation against real-world variations; a model that can generate feasible and optimal solutions even upon changes in the real environment of the problem. 2. The next challenge is associated with situations where the decision-maker deals with multiple alternatives amongst which he/she must choose for one. The fact that making the optimal decision affects the profitability directly doubles the importance of this challenge. 3. As a final challenge, one may refer to the necessity of considering multiple objectives for the problem, which is a common practice for decision-makers in the real world.
The first challenge is usually tackled through what we know as stochastic programming but as we know, the probability distribution of a random variable must be through statistical analysis and inference based on definite data of appropriate size. However, in many situations, data with adequate size are not available to determine the distribution, and as a result, expert opinion is inaccurately replaced. Since the random variable alone cannot justify this hybrid phenomenon, in such cases, the probability distribution faces both fuzzy and stochastic phenomena. On the other hand, in some cases, the collected data are not accurate themselves. In this situation, we encounter a fuzzy phenomenon between random values. In these cases, since the distribution of parameters includes both random and fuzzy elements, random variables are clearly not the best way to describe fuzzy stochastic hybrid uncertainty. Therefore, it makes more sense to use fuzzy-stochastic variables to represent these values. Focusing on the second challenge, we are dealing with problems where parameters can be set based on alternative choices and achieving an optimal solution is highly dependent on optimal selection of these parameters. In the literature, such problems have been referred to as Multi-Choice Programming (MCP). With the third challenge focusing on the multiplicity of the objectives for a given problem, one can utilize multi-objective optimization and fractional programming to tackle this challenge. Of course, when we have several objective functions for the problem, the complexity of the problem is much greater [22]. But, when only two objective functions are considered, the ratio of these objective functions can be optimized, for example, minimizing costs and maximizing profits or minimizing costs and maximizing reliability. In this research, we aim to combine these challenges into a comprehensive model to help managers and decisionmakers in making optimal decisions under close-to-reality conditions. In fact, this is a Fractional Multi-Commodity Network Flow (FMCNF) problem with multi-choice parameters. In this model, alternative choices in the nominator and denominator of the fraction are fuzzy-stochastic and fuzzy variables, respectively. Our final goal in this article is to present a model for managing unknown factors across a fractional multi-commodity network.

Literature Review
In this section, the literature of MCNF problems, FMCNF Problems, and FMCNF problems with multi-choice parameters, respectively.

MCNF Problems
During last decades, the MCNF problem has become popular in the academic literature and an increasing number of researchers have become interested in it. It is a powerful operations research method for solving many complex problems, especially in the fields of transportation and communications. The MCNF problem was introduced by Fords and Fulkerson [10] and Hu [15]. After that, many researchers tried to expand this model by adding different aspects to the problem. Even for a very simple MCNF problem with continuous flow and linear costs, it is very difficult to find an optimal solution [18]. These difficulties are due to the complex nature of the constraint structure and the large number of variables that require extensive computational time [21]. However, own to the developments in algorithms and computational technology, great achievements have been attained in solving the model in term of the solution time and optimal point [19]. For example, Yu et al. [35] studied the variable demand and multi-commodity extensions in Markovian network equilibrium. They also proposed efficient algorithms that outperform state-of-the-art commercial optimization software. A novel optimization approach for the assessment of different transportation systems has been introduced by Bevrani et al. [2]. Their multi-criteria multi-commodity flow model maximizes the flow of commodities and vehicles but also simultaneously minimizes the incurred cost of travel for those commodities. When a multi-commodity problem is presented in uncertainty conditions, the process of solving it is much more difficult than traditional methods [11]. Most research in this field has been done in fuzzy or stochastic conditions [3,16,29]. Among these, Shi et al. [32] considered a maximum flow problem under uncertainty conditions. The main purpose of their paper is to resolve the maximum flow in a non-deterministic network using the framework of uncertainty theory. Ding [6] presented a MCNF problem with non-deterministic costs and capacities. He developed a decomposition-based algorithm to solve a ( , )-minimum cost multi-commodity problem. A case of discrete cost multi-commodity flow with random demands has been investigated by Mejri et al. [20], where a penalty is considered for each un-routed demand. This problem requires finding a network topology that minimizes the total fixed start-up costs and expected penalties for not meeting the multi-commodity demand.

FMCNF Problems
When only two objective functions are considered, the ratio of these objective functions can be optimized, for example, minimizing costs and maximizing profits or minimizing costs and maximizing reliability. In fact, the problem of fractional programming can be solved by adopting some well-known approaches [5]. In addition, fractional programming occurs in many practical problems [30,31] that can be followed in the other network analyses. So far, good research has been done on deficit planning under uncertainty. Recently, Nasseri et al. [24] have investigated the fuzzy random linear fractional programming problem. In their proposed model, the coefficients and scalars in the objective function are fuzzy numbers and technical coefficients and the right-hand values are the fuzzy random variables with specified distribution. After converting the fuzzy random fractional problem to the equivalent multi-objective linear fractional programming problem, they reduced the problem to a single-objective linear programming problem and solved it using the fuzzy programming approach. In another study, they offered a proposed approach to stochastic interval-valued linear fractional programming problem [25].
In their proposed approach, the stochastic interval-valued linear fractional programming problem is converted to the optimization problem with an interval-valued objective function so that interval bounds are in the form of fractional functions. Xu et al. [34] a problem and a new algorithm proposed for the linear fractional minimal cost flow problem on network. Using a new check number and combining the characteristic of network to extend the traditional theories of minimum cost flow problem, discussed the relation between it and its dual problem. Also, optimality conditions derived and a Network Simplex Algorithm proposed that leads to optimal solution assuming certain properties. A fractional minimal cost flow problem under linear type belief degree based uncertainty is studied by Niroomand [26]. The problem is crisped using an uncertain chance-constrained programming approach and its non-linear objective function is linearized by a variable changing approach. Fakhri et al. [9] presented a deterministic model of fractional MCNF problems. They examined the conditions of optimality and the dual implications of this problem.

FMCNF Problems with Multi-choice Parameters
In some cases, we deal with multiple alternative choices for each parameter of the problem, amongst which one should opt for proper choices to optimize the problem. As mentioned earlier, we usually recognize such problems as multi-choice programming. The MCP was first introduced by Healy [14] in a study on a special case of mixed integer programming. Since then, the wide application of the MCP in real-world decision-making processes motivated numerous researchers toward incorporating it into goal programming, transportation problems, the game theory, etc. [4,8,27]. As a common aspect among most of relevant research works, one may refer to deterministic nature of the alternative choices for the multi-choice parameters.
Recently, however, Nasseri and Bavandi [23] for the first time investigated a multi-choice linear programming problem in fuzzy-stochastic environment, wherein the alternative choices were represented by fuzzy-stochastic variables. So far, no other research has been done on network flow problems with multi-choice parameters including uncertain alternative values.
In this paper, we focus on a MCNF problem with a fractional objective function in a hybrid fuzzy-stochastic environment for the first time, where the coefficients of the objective function in the nominator and denominator are multi-choice parameters represented by fuzzy-stochastic and fuzzy variables, respectively. This model can be used in various applications, especially in maritime transport, air transport and even the tourism industry. For example, suppose we are going to load a ship with a limited carrying shipping capacity and a certain number of goods. Also, suppose there is a certain number of ports through which these goods can be transported. The selling cost and transportation cost of the goods vary at different ports. Our goal is to choose one of these ports and the amount of each type of goods should be loaded in such a way that the profit gained per unit of transportation cost will be maximum. On the other hand, due to the fluctuations in the real situation, the problem data cannot be considered accurately during modeling. There is no existing mathematical model and methodology which can solve the above example directly, so we proposed a FMCNF problem in a multi-choice fuzzy stochastic hybrid environment.
We generalize the deterministic fractional MCNF problem to its non-deterministic form. The rest of this paper is organized as follows. In Sect. 2, we briefly discuss some of the basic concepts and theorems related to the topic. In Sect. 3, we first present the deterministic model of the FMCNF and then generalize it to its non-deterministic form. The deterministic equivalent form of the FMCNF problem in a multi-choice fuzzy stochastic environment is presented in Sect. 4. Finally, in Sect. 5, two examples are provided to illustrate the efficiency of the model and algorithm.

Preliminaries
In this section, required definitions and basic concepts if probability theory and fuzzy set theory as well as fuzzy random variables will be presented.

Definition 3.1 Let is a nonempty set of all possible events,
Γ is an -algebra over and P( ) is its power set. Therefore, for any A ∈ P( ) , the possibility measure Pos(A) as a function of Γ to interval [0, 1] have following properties [7]: is called a possibility space. P( ) elements are also called fuzzy events. Definition 3.2 A fuzzy interval of type LR is represented as Ã ∶ , m 1 , m 2 , LR where and are non-negative left and right spreads, respectively. Also, m 1 and m 2 are mean values of the fuzzy number Ã . The membership function of can be defined as following [7] where L and R are left and right continuous non-increasing functions from [0, 1] to [0, 1], respectively, so that then the LR fuzzy number is represented as following: and is referred as triangular fuzzy number. Therefore, we assume that functions L(.) and R(.) are as follows: Fuzzy random variables are generalizations of ordinary random variables. Therefore, before defining fuzzy random variables, it is appropriate to first recall the definition of ordinary random variables. Definition 3.3 Let X 1 , X 2 , … , X n are normal and independent random variables. So, if Y = C 0 + C 1 X 1 + C 2 X 2 + ⋯ + C n X n is a linear combination of these variables, then Y has also a normal distribution with mean and variance as follows: Definition 3.4 [17] The fuzzy random variable is a function from the possibility space (Ω, A, Pr) to a set of fuzzy variables such that for every Borel set B of R, Pos{ ( ), ∈ B} is a measurable function of . Definition 3.5 [17] Consider the fuzzy variable (a( ), , ) ( ∈ Ω) where a is a random variable with a normal distribution and indicated as a ∼ N , 2 . Then, is a fuzzy random variable with a membership function as follows: Theorem 3.6 [17] Let is a n-dimensional fuzzy random vector. If f ∶ R n → R is a measurable function, then f ( ) is a fuzzy random variable.

Mathematical Model
In this section, we first introduce the deterministic form of the MCNF problem and then generalize it into a hybrid fuzzy random environment.

Fractional Multi-commodity Network Flow Problem
Consider a directed network G(N, A) . In this network, the set of nodes is represented as N = {1, 2, … , n} and the set of arcs is indicated by A = {(i, j)|i, j ∈ N} . Suppose that K is a set of commodities K = {1, 2, … , |K|} . c ijk and d ijk are the coefficients for cost of arc (i, j) . Other variables and parameters are as follows: • x ijk : is the amount of commodity of type k which crosses the arc (i, j). • u ij : is the upper bound of the total flow for all commodities on the arc (i, j). • b k :is the amount of the commodity k that has to be sent from the origin s k to the destination t k .
Also assume that b k i is defined as the flow balance of commodity k at node i. In fact, b k i represent the supply or the demand of commodity k at the node i. p and q are given constants. Therefore, this problem can be formulated as follows: where the constraints (c 1 ) are the balance constraints which represent the conservation relations of network flow. Constraints (c 2 ) are called bundle constraints indicating that the total flow of commodities on every arc (i, j) are less than or equal to the given capacity. Constraints (c 3 ) are the nonnegativity constraints.
On the other hand, the condition It is worth noting that the objective function in Problem 1 is a pseudo-linear function and every local optimal solution in Problem 1 is a global optimal solution, too [28].
In the crisp multi-commodity problem, the coefficient of problem, are crisp non-negative values. In practice, however, these values are often not constant. In fact, flow boundaries are the flow values which pass through the arcs, and transfer costs cannot be accurately measured according to their nature. In this situation, if enough data are available, the probability distribution of these data can be created using statistical analysis and inference based on the crisp data with appropriate measure. Unfortunately, in many cases, proper amounts of data are not available to determine the distribution. The reason for this could be, for example, the congestion, events and weather conditions, repairs and even in some cases, the newness of the case under consideration, so that we do not have historical data with the appropriate size. For example, changes in the price of gasoline can affect transmission costs. On the other hand, in some cases the collected data are inaccurate. Therefore, the only hope in such circumstances is to use the inaccurate opinion of experts and professionals. In these cases, we face a fuzzy event among random variables.
Under these conditions, let the decision-maker be dealt with multiple choices amongst which he/she must opt for only one. For this purpose, the decision-maker must follow an MCP approach. The MCP is a mathematical programming model where each parameter can take multiple alternative values, among which one must select the one to optimize the problem. This indicates that we are herein investigating a multi-choice multi-commodity flow problem in which alternative choices are represented by fuzzy-stochastic variables. As a next step, the decision-makers often define the certainty level before proceeding to obtain the optimal solution based on the defined level. Accordingly, the use of chance constraints for ensuring a desirable level of management and formulating an uncertain programming model that represents the inherent uncertainty of the network is seemingly a suitable approach.

FMCNF Problem in a Multi-choice Fuzzy Stochastic Environment
To generalize the deterministic FMCNF problem to its uncertainty form, the parameters c ijk , p, d ijk and q are multichoice parameters. Alternative values of c ijk and p are represented by fuzzy-stochastic variables while those of d ijk and q are fuzzy variables. Moreover, u ij is herein a fuzzy-stochastic variable. As a result, an FMCNF problem can be rewritten as an FMCNF problem in a multi-choice fuzzy stochastic environment through the following expression: are alternative choices c ijk , p, d ijk and q respectively. Note that the changes in the fuzzy random variables are independent of the value of the crisp variable x, however, the value of the objective function f x;c,d changes as the values of these variables change. Given the existence of a fuzzy random variable in this objective function, we cannot calculate a specific value for every value of x. Therefore, its value cannot be determined by conventional methods. Also, the constraints in Problem 2, do not specify a specific area due to the constraint of c ′ 2 . In such cases, to deal with fuzzy random events, the probability-possibility approach with a predetermined confidence levels are utilized. Therefore, problem (2) can be rewritten as a Probability-Possibility Constrained Programming (PPCP) model as follows: Problem 3: In this model, the decision-maker hopes to obtain the minimum value of f 0 , so that f x,c,d is equal to or less than Min f 0

The Deterministic Equivalence of Model
In this section, we intend to convert Problem 3 to its crisp form. In fact, one way to solve this problem is to convert it to a crisp equivalent form. The model presented in the previous section has a number of multi-choice parameters whose alternative values take the form of fuzzy-stochastic and fuzzy variables. This implies that the problem cannot be solved directly. In order to solve the problem, we begin by transforming the multi-choice parameters using a predefined interpolation polynomial. An interpolating polynomial introduces an integer variable for each multi-choice parameter. If the selected multi-choice parameter comes with k alternative values, the corresponding integer variable should include exactly k nodal points, with each node referring to exactly one functional value of the selected multi-choice parameter. Herein, the functional value of each node takes the form of    Table 4 Data for multi-choice parameter q z 0 a fuzzy stochastic and a fuzzy variable, and a multi-choice parameter is substituted by an interpolating polynomial using the Lagrange formula. For the multi-choice parameters c ijk , p, d ijk and q, we herein introduce the integer variables u ijk , w, v ijk and z which take l ijk , k, p ijk , and l values, respectively. Then, we formulate the Lagrangian interpolating polynomials f (c ijk ) (u ijk ) , f̃̄p(w) , f (d ijk ) (v ijk ) , and f̃q(z) that pass through all l ijk , k, p ijk , and l values given in Tables 1, 2, 3 and 4, respectively.
According to the values given in Tables 1, 2, 3 and 4 and the Lagrange formula, interpolation polynomials for multichoice parameters of the problem are obtained as follows: For convenience, in the continuation of the paper, instead of the phrase on the left in the above relations, we use f̃̄c ijk u ijk , f̃̄p(w) , fd ijk v ijk and f̃q(z) , respectively.

Problem 4:
As mentioned earlier, a solution method for Problem 4 is to transform its constraints to their deterministic forms. Since t h e a l t e r n a t i ve ch o i c e s c (r) ijk (r = 1, … , l ijk ) a n d p (s) (s = 1, … , k) as well as ũ ij are fuzzy stochastic parameters, one can write them in the following forms, respectively: where the first and third elements represent the left and right tails, respectively, with the second elements being the center value. The second elements are normally distributed random variables with known mean f̃̄p w,p (1) ,p (2) , … ,p (k) = (w−1)(w−2)…(w−k+1) values and variances. On the other hand, the variables d (t) ijk (t = 1, … , p ijk ) and q (u) (u = 1, … , l) are fuzzy numbers. This implies that they can be expressed as d (t) ijk = (d ijk , d ijk , d ijk ) LR and q (u) = (q , q, q ) , respectively. Now, considering these parameters in Eqs. 8 , the following inequality holds: According to the Definition 3.7, we have: or equivalent, In (13), nominator of the fraction still has some stochastic parameters. Therefore, set: Clearly, h also is a random variable with following mean and variance: Therefore, according to (13) and constraint ( c 0 ), we have: where Φ −1 is the inverse function of the distribution function Φ . Therefore, the expression (13) is equivalent to: Similarly, the deterministic form of constraint ( c ′′ ) will be as follows: Finally, according to (14) and (15) and since the minimization of f 0 is equivalent to the minimization of the left side of (14), The crisp form of Problem 4 will be as follows:

Problem 5:
Problem 5 is a deterministic multi-choice fuzzy stochastic fractional multi-commodity network flow problem. Since the numerator of the fraction is convex and the denominator of the fraction is greater than or equal to zero, then the objective function of Problem 5 is clearly quasi-convex. However, using the variable transformation technique introduced by Charness and Coope [5] and Gupta and Jain [12], Problem 5 converted to the following equivalent form [13]: Problem 6: The objective function of this problem is convex according to Φ −1 ( ) ≥ 0 for every ∈ [0.5, 1) and its constraints are linear. Therefore, Problem 6 is a convex problem which by solving it, we will be able to provide a solution to Problem 3.

An Illustrative Example
To evaluate the performance of the proposed model, two examples are solved in this section. Attempts have been made to extract samples from the subject literature so that, in addition to better analysis and comparison, the benefits of the proposed model in uncertainty environments are also shown.
Example 6.1 [1 ] Consider a network consisting of 6 nodes and 10 edges that is shown by a directed network in Fig. 1. Data related to the problem are presented in Tables 5 and 6 where p = 0 and = q (1) = (0.5, 0.3),q (2) = (1, 0.4),q (3) = (1.5, 0.5) . In this example, the coefficients of the nominator of the fraction and the edge capacity are fuzzy random coefficients and the coefficients of the denominator of the fraction are assumed to be fuzzy numbers. Each of the fuzzy and fuzzy random coefficients has a symmetrical triangular membership function. Hence, we show them as pairs of (a( ), ) and (m, ).
Using the data of Tables 5 and 6, the deterministic model of the problem is obtained in the form of Problem 6. This model is coded in Mathematica 10 software for different and values and the obtained solution is presented in both numerical and schematic forms in Table 7 and Fig. 2 respectively. Example 6.2 [9] Consider a network consisting of 12 nodes and 16 edges which is shown by a directed   Table 6 The related data of the Example 1 Edge Using the data of Tables 8 and 9, the deterministic model of the problem is obtained in the form of Problem 6. This model is coded in Mathematica 10 software for different and values and the solution is numerically and schematically presented in Table 11 and Fig. 4.
The results obtained from these two examples show that the proposed model can be very efficient if it has suitable inputs. As you can see in these examples, for different probability levels, different but close solutions are obtained. In fact, there are different situations before the decision maker who can make the best decision considering the circumstances. Figure 4 clearly shows that the presented model is appropriate for all three groups of risk-averse people (People who have a conservative approach and prefer to have a reliable return and have a high probability of success), risk seeking people (People who have a bold approach and accept high risk in order to receive a higher profit) and ultimately people who are neutral to risk (So-called riskneutral people). Indeed, by choosing a probability level of  0.00 0.00 0.00 (3,2) 0.00 0.00 0.00 (2, 4) 1.00 0.00 1.64 (2,5) 0.00 3.00 0.36 (3,4) 0.00 0.00 0.00 (3,5) 0.00 0.00 0.00 (4,5) 0.00 0.00 0.00 (4,6) 0.00 0.00 1.64 (5,6) 0.00 0.00 0.36     Table 9 (continued)  Edge 0.9, the risk is lower due to the lower objective value, and at a probability level of 0.1, the risk is higher due to the higher objective value. If the problem were solved without considering the uncertainty of the alternative values of the multi-choice parameters, we would eventually have the optimal value 2.4545, while the optimal value for the problem was obtained under uncertainty equal to 0.7155851 for = = 0.1 and 1.115010 for = = 0.9 . This difference will be very significant for real problems with large scale and will cause great losses to the decision-maker. On the other hand, in this model, we make decisions that ensure its feasibility as much as possible. In fact, constraints are not violated except in emergencies and unpredictability. It is worth noting that here only an idea was presented for the selection of and but in practice, this value is determined by the decision maker in a way that meets the safety and security needs.

Conclusion
Optimal decision making often requires exposure to unknown or non-deterministic parameters. No matter how accurately the model is designed, the data used in it determine the reliability of the model. In this paper, we tried to analyze several natures simultaneously in a model. A fractional multi-commodity network flow problem with multichoice parameters was studied in a hybrid fuzzy-stochastic environment. The aim was to present a model for managing unknown factors in a fractional multi-commodity network. In this problem, coefficients of the objective function in the nominator and denominator were multi-choice parameters. In this respect, the alternative choices in the nominator as well as the arc capacities were represented by fuzzy-stochastic variables while the alternative choices in the denominator were expressed as fuzzy variables. To transform the model into a deterministic problem, we used the probabilitypossibility approach, Lagrangian interpolating polynomial, and the Charnes-Cooper variable transformation technique.  Finally, performance of the presented methodology was evaluated on a few numerical instances. A major advantage of modeling this kind of problem in a fuzzy-stochastic environment emerges when we are dealing with multiple choices with no historical data and would rather rely on experts' opinions-a situation where a major challenge is how to quantify an expert's opinion in a consistent and robust way. This implies that the problem cannot be barely seen from a probabilistic point of view but rather the fuzzy nature of the problem may also play a pivotal role. In the present research, we tried to carefully analyze a particular category of multicommodity network flow problems. It is worth noting that we are looking for a sequential method to solve this problem which is postponed to our future work.
Author Contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Funding Authors received no specific funding for this study.

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Data provided in the paper would be enough for the reader to overview the numerical analysis steps.

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