# Solving Stochastic Fuzzy Transportation Problem with Mixed Constraints Using the Weibull Distribution

Authors: T.K. Buvaneshwari; D. Anuradha
Journal: Journal of Mathematics
 (2022)

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## Abstract

The development in the industries has necessitated the growth of transportation methods. Due to the variation in the transportation systems, many problems seem to arise in the present times. One such problem is the stochastic fuzzy transportation problem (SFTP). The SFTP is a chance-constrained programming (CCP) problem with probabilistic constraints where supply and demand are randomness, and the objective function is in fuzzy nature. In this paper, we have developed three models for the SFTP, where the constraints are mixed type following Weibull distribution (WD). The aim of the research work is to optimize the transportation cost in FTP under probabilistic mixed constraints. In order to achieve this, the cost coefficient of the fuzzy objective function is converted to alpha cut representation, and the probabilistic mixed constraints are converted to deterministic form by using the WD. The proposed models are illustrated by providing a numerical example, and the problem is solved using Lingo software. It is worth pointing out that these models are constructed from different points of view. The decision-maker’s (DM’s) preference has the final say in the usage of the models. A sensitivity analysis (SA) is performed to explore the sensitivity of the parameters in the proposed model.


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## Body

## 1. Introduction

Decision-making plays a vital role in many domains, including Economics, Psychology, Philosophy, Mathematics, and Statistics. The importance of transportation needs to be recognized as a necessary part of distribution networks. The primary objective of the transportation problem (TP) is to lower the cost of moving products and materials among customers and producers to facilitate the manufacturer in meeting the requirement of the customers. Cost, supply, and demand are the parameters of the TP. Although several modes of transportation are available for shipments of commodities in a transportation system, we may convey items from sources to destinations using distinct modes of transportation to save money or fulfill deadlines. The usual transportation system is transformed into an unequal transportation problem with mixed constraints (TPMC) if the capacity of a supply is appreciably expanded/reduced and the requirement of a demand is also appreciably expanded/decreased. The special type of transportation problem with mixed constraints has been meticulously studied first by Brigden [1]. Different approaches to solving different models of the TP with mixed constraints abound in the literature [2–7]. Gupta et al. [8] extended multiobjective capacitated TPMC in a fuzzy environment. In the real-world situation, the two main factors of uncertainties are randomness and fuzziness (or vagueness). The stochastic variability of all possible outcomes of a situation is referred as randomness, which may be completely and quantitatively characterized by using a random variable (RV) in probability theory. Fuzziness, on the other hand, arises from the imprecision of subjective human knowledge and manifests objectively in a range of contexts, including data gathering and processing, hazy parameter limits, expertise applications, and a lack of accurate information. Stochastic transportation problems (STP) and fuzzy transportation problems (FTP) are the names given to TP that are modeled in such conditions. It is observed that the randomness and fuzziness in information govern the decision-making problems in the real world. The information on weather forecasting, predicting the stock market, the study of the economic status of the industrial sector, etc., is not certain. The uncertainty in data is such that fuzzy programming problems and stochastic programming problems cannot be used in isolation to find the appropriate mathematical model. Hence, we consider both fuzziness and randomness while dealing with fuzzy stochastic programming problem. Since most of the mathematical programming models are governed by real-life decision-making problems having multiple numbers of conflicting functions, in our study, we have considered a fuzzy stochastic programming problem.Decision-making in a fuzzy environment is first developed by Zadeh. The fuzzy set theory has been used to handle a number of real-world problems, such as financial engineering and risk management. The reason seems to be that the strategy enables the DM to study and deal with the unknown elements that are present. Then, employing fuzzy quantities and/or fuzzy restrictions, the imperfect knowledge of asset returns, and the uncertainty associated with capital market behavior can be included. Fuzzy numbers, which are fuzzy subsets of the real line, can be used to represent fuzzy numerical data. Zimmermann [9] discussed fuzzy programming and linear programming with a number of different objective functions. Behera and Nayak [10] have solved the multiobjective mathematical programming in the fuzzy approach with the alpha cut. In stochastic models, the random variables are considered for each problem parameter.A stochastic programming (SP) problem arises when the probability is present in a mathematical programming problem. It is concerned with decision-making in which some or all parameters are traced as RVs to represent uncertainty. Researchers such as Kataoka, Williams, and Agarwal [5–7] have undertaken numerous studies on the topic of random transportation. Moreover, DMs are confronted with scenarios in which they must select a value from a group of options. In this case, mathematical programming comes to the DMs assistance. The chance-constrained technique is one of the most effective methods to handle optimization problems (OP) with numerous uncertainties. It is the formulation of an OP that ensures that the probability of satisfying a given constraint is greater than a certain benchmark. In other words, it restricts the feasible region so that the solution has a high level of confidence. Although the chance-constrained technique is a rather robust strategy, it can be challenging to implement. If the linear constraints of a TP are probabilistic, it becomes a CCP problem. This paper discusses the parameters used to address uncertainty using both probability theory and fuzzy set theory.


### 1.1. Literature Review

This section contains a survey of the literature on TP and FTP in SP. Charnes and Copper [11] established TP, and Kataoka [12] discussed the SP method for a TP. Spoerl and Wood [13] analysed the stochastic generalized assignment problem. Williams [14] defined a stochastic variable for demand using a joint cumulative distribution function and then investigated a stochastic TP with a supply and a demand. Agrawal and Ganesh [15] proposed a TP solution incorporating stochastic demand and nonlinear multichoice costs. Powell and Topaloglu [16] studied SP in transportation and logistics. Anholcer [17] proposed a stochastic generalized TP with a discrete distribution of demand. Ojha et al. [18] designed a stochastic discounted multiobjective STP with demand as a stochastic parameter, converting stochastic variables to deterministic ones using the expected value criteria. Holmberg and Tuy [19] demonstrated a branch-and-bound method for solving the TP when demand is stochastic and production costs are concave. However, in real-world situations, both supply and demand are stochastic. Mahapatra et al. [20] focused their efforts on inequality constraints in a multiobjective STP, where all supply and demand are log-normal RVs, and the objectives are incommensurable and contradictory. Agrawal [21] solved the STP by using an artificial intelligence approach in a stochastic environment. The following authors discussed articles that involve both fuzziness and randomness. A fuzzy goal programming method for STPs with restricted resources was discussed by Aruna Chalam [22]. Giri et al. [23] proposed fuzzy stochastic solid TP using a fuzzy goal programming approach. Acharya et al. [24] demonstrated how to compute a multiobjective fuzzy STP. Gessesse [25] developed a fuzzy programming approach based on a genetic algorithm for multiobjective linear fractional stochastic TP employing a four-parameter Burr distribution. Maity et al. [26]studied the optimal intervention in transportation networks using fuzzy stochastic multimodal systems.In this paper, the WD function could be employed to define the stochastic variable. We express supply and demand as stochastic variables. Weibull [27], a Swedish researcher, invented the WD in the 1950s, and it has since become a helpful and informative data treatment tool in reliability studies of components and electronic systems. The WD is both theoretically and practically pertinent to Fok’s [28] studies of future probability estimates under a given loading. Since the WD has flexibility in location, shape, and scale, it can design an extensive range of failure rates. The WD is defined mathematically as follows: the probability density function of an RV y is described by f=χ/λy−ξ/λχ−1e−y−ξ/λχ, where χ>0,λ​>0,y≥ξ andξ, χ and λ are the location, shape, and scale parameters, and then, random variable y is called WD. In dependability applications, the WD has been shown to be beneficial for characterizing cost times and lifetimes. Distribution has received the greatest attention in recent decades. Although the WD is a good choice for describing data on lifespan or strength, its performance falls below its competitors in several practical situations. Furthermore, the WD has applications in geosciences and transportation among other fields. Klakattawi [29] investigated the features and uses of the Weibull-gamma distribution. Mahapatra [30] investigated the characteristics and applications of the Weibull-gamma distribution.


### 1.2. Research Gap, Motivation, and the Proposed Contribution

The incidence of mixed constraint in TPs is a common phenomenon that may be seen in reality. The mixed constraint paradox in a TP occurs when it is possible to ship more total goods for less (or equal) total cost, while transporting the same amount or more from each origin and to each destination and keeping all the shipment costs nonnegative. However, there may be situations where cost coefficients, availability, and demand quantities are uncertain due to some uncontrollable factors. Uncertainty can be mainly classified into two types, namely fuzzy TP and stochastic TP. The TP in which the transportation cost, supply, and demand are fuzzy quantities is called fuzzy TP. The objective of the FTP is to determine the shipping schedule that minimizes the total fuzzy transportation cost while satisfying fuzzy supply and fuzzy demand limits. The situation where the parameters are imprecise in a stochastic sense and described by random variables with known probability distributions is called stochastic TP. The main benefits of fuzziness and randomness techniques are that they do not require prior predictable regularities and can deal with imprecise input information incorporating feelings and emotions quantified based on the DM’s subjective evaluation. The STP and stochastic fuzzy transportation problems (SFTP) with equality constraints have recently received significant attention, with various models and solution methods applied to stochastic environments. The literature search (Table1) reveals no systematic model for addressing stochastic fuzzy transportation problem with mixed constraints (SFTPMC). But in reality, imprecise mixed constraints that supply at origins may be uncertain, since any trouble or delay may occur for various reasons and demand requirement uncertainty cannot be avoided often due to inexact forecasting or requirement volatility. This has motivated us to develop the model for FTP mixed constraint under a stochastic environment.Table 1
Comparison of the approach to the present models.
AuthorsProblem typeStochastic parametersDistributionMethodologyGessesse [31]Fractional TPDemand & supplyNormal distributionSimulation-based genetic algorithmJerbi [32]TPDemand & supplyThe power law distributionFuzzy programming approachAl Qahtani et al. [33]TPDemand & supplyExtreme value distributionGoal programming approachNasseri and Bavandi [34]TPDemand & supplyExpectation value modelFuzzy programming approachDutta et al. [35]Fuzzy TPDemand & supplyFuzzy lognormal distribution and the confidence levels are treated as fuzzy numbersGenetic algorithm approachMahapatra et al. [36]TPDemand & supplyLogistic distributionA transformation technique is presented for manipulating cost coefficients involving multichoice for binary variables with auxiliary constraints and solved by Lingo softwareAgrawal and Ganesh [15]TPDemandGalton distributionParameters are replaced by Newton’s divided difference interpolating polynomial and solved by Lingo softwareDas and Lee [37]Solid TPDemand, supply, and capacityWeibull distributionGlobal criterion method and fuzzy goal programming approachThis articleSFTPMCDemand & supplyWeibull distributionAlpha cut representation for cost function and WD for constraints and then solved by Lingo softwareBased on this motivation, the main contribution of this study is summarized as follows: the goal of this paper is as follows: (1) to propose a mathematical model that deals with fuzziness and randomness under one roof, (2) to consider the fuzzy objective value and probabilistic constraints, and (3) to present a simplified computation conversion of probabilistic constraints to their equivalent deterministic constraint using WD. In order to achieve this, the fuzzy objective is converted to alpha cut representation, and probabilistic constraints are converted to deterministic form by using the WD and then solved using Lingo software. To investigate the sensitivity of the parameters in the proposed model, a sensitivity analysis is performed.


### 1.3. Managerial Insights Based on the Research

The last decade has seen the growth in various fields such as economics, trade, health care, transportation, cultivation, army, engineering, and technology. It has increased the importance of stochastic fuzzy optimization in solving scientific managerial decision-making problems in every field. The uncertainty factors which make the problem more complex have not spared the transportation process. The objective of this paper is to optimize the total transportation cost under uncertain conditions by utilizing the SFTPMC model. Generally, uncertainty is described using fuzzy or probability theory. However, it is not always reasonable to use fuzzy theory or probability theory to illustrate all the indeterminacies, as it requires adequate information. So, we prefer to describe the uncertainty with both fuzzy and probability parameters. In fact, low-frequency events occur in our daily lives causing the uncertainty theory to take effect. Under these circumstances, scheduling a proper transportation plan to lower the total costs becomes a challenge. This study provides an applicable model for the DM to deal with some uncertain factors without losing reliability from customers. This research could also be extended to other aspects, in which a decision/plan needs to be made in uncertain conditions.The rest of this article is structured as follows. Section2 presents the preliminaries that are related to this article. Section 3 deals with the assumptions and notation used in this paper. Section 4 investigates and converts the mathematical programming models of SFTPMC and its variants to the comparable deterministic models. Section 5 presents the proposed solution method with an illustration. Section 6 presents and analyses the computational experiences of SA. Finally, Section 7 summarizes the research outcome and limitations and suggests future directions.


## 1.1. Literature Review

This section contains a survey of the literature on TP and FTP in SP. Charnes and Copper [11] established TP, and Kataoka [12] discussed the SP method for a TP. Spoerl and Wood [13] analysed the stochastic generalized assignment problem. Williams [14] defined a stochastic variable for demand using a joint cumulative distribution function and then investigated a stochastic TP with a supply and a demand. Agrawal and Ganesh [15] proposed a TP solution incorporating stochastic demand and nonlinear multichoice costs. Powell and Topaloglu [16] studied SP in transportation and logistics. Anholcer [17] proposed a stochastic generalized TP with a discrete distribution of demand. Ojha et al. [18] designed a stochastic discounted multiobjective STP with demand as a stochastic parameter, converting stochastic variables to deterministic ones using the expected value criteria. Holmberg and Tuy [19] demonstrated a branch-and-bound method for solving the TP when demand is stochastic and production costs are concave. However, in real-world situations, both supply and demand are stochastic. Mahapatra et al. [20] focused their efforts on inequality constraints in a multiobjective STP, where all supply and demand are log-normal RVs, and the objectives are incommensurable and contradictory. Agrawal [21] solved the STP by using an artificial intelligence approach in a stochastic environment. The following authors discussed articles that involve both fuzziness and randomness. A fuzzy goal programming method for STPs with restricted resources was discussed by Aruna Chalam [22]. Giri et al. [23] proposed fuzzy stochastic solid TP using a fuzzy goal programming approach. Acharya et al. [24] demonstrated how to compute a multiobjective fuzzy STP. Gessesse [25] developed a fuzzy programming approach based on a genetic algorithm for multiobjective linear fractional stochastic TP employing a four-parameter Burr distribution. Maity et al. [26]studied the optimal intervention in transportation networks using fuzzy stochastic multimodal systems.In this paper, the WD function could be employed to define the stochastic variable. We express supply and demand as stochastic variables. Weibull [27], a Swedish researcher, invented the WD in the 1950s, and it has since become a helpful and informative data treatment tool in reliability studies of components and electronic systems. The WD is both theoretically and practically pertinent to Fok’s [28] studies of future probability estimates under a given loading. Since the WD has flexibility in location, shape, and scale, it can design an extensive range of failure rates. The WD is defined mathematically as follows: the probability density function of an RV y is described by f=χ/λy−ξ/λχ−1e−y−ξ/λχ, where χ>0,λ​>0,y≥ξ andξ, χ and λ are the location, shape, and scale parameters, and then, random variable y is called WD. In dependability applications, the WD has been shown to be beneficial for characterizing cost times and lifetimes. Distribution has received the greatest attention in recent decades. Although the WD is a good choice for describing data on lifespan or strength, its performance falls below its competitors in several practical situations. Furthermore, the WD has applications in geosciences and transportation among other fields. Klakattawi [29] investigated the features and uses of the Weibull-gamma distribution. Mahapatra [30] investigated the characteristics and applications of the Weibull-gamma distribution.


## 1.2. Research Gap, Motivation, and the Proposed Contribution

The incidence of mixed constraint in TPs is a common phenomenon that may be seen in reality. The mixed constraint paradox in a TP occurs when it is possible to ship more total goods for less (or equal) total cost, while transporting the same amount or more from each origin and to each destination and keeping all the shipment costs nonnegative. However, there may be situations where cost coefficients, availability, and demand quantities are uncertain due to some uncontrollable factors. Uncertainty can be mainly classified into two types, namely fuzzy TP and stochastic TP. The TP in which the transportation cost, supply, and demand are fuzzy quantities is called fuzzy TP. The objective of the FTP is to determine the shipping schedule that minimizes the total fuzzy transportation cost while satisfying fuzzy supply and fuzzy demand limits. The situation where the parameters are imprecise in a stochastic sense and described by random variables with known probability distributions is called stochastic TP. The main benefits of fuzziness and randomness techniques are that they do not require prior predictable regularities and can deal with imprecise input information incorporating feelings and emotions quantified based on the DM’s subjective evaluation. The STP and stochastic fuzzy transportation problems (SFTP) with equality constraints have recently received significant attention, with various models and solution methods applied to stochastic environments. The literature search (Table1) reveals no systematic model for addressing stochastic fuzzy transportation problem with mixed constraints (SFTPMC). But in reality, imprecise mixed constraints that supply at origins may be uncertain, since any trouble or delay may occur for various reasons and demand requirement uncertainty cannot be avoided often due to inexact forecasting or requirement volatility. This has motivated us to develop the model for FTP mixed constraint under a stochastic environment.Table 1
Comparison of the approach to the present models.
AuthorsProblem typeStochastic parametersDistributionMethodologyGessesse [31]Fractional TPDemand & supplyNormal distributionSimulation-based genetic algorithmJerbi [32]TPDemand & supplyThe power law distributionFuzzy programming approachAl Qahtani et al. [33]TPDemand & supplyExtreme value distributionGoal programming approachNasseri and Bavandi [34]TPDemand & supplyExpectation value modelFuzzy programming approachDutta et al. [35]Fuzzy TPDemand & supplyFuzzy lognormal distribution and the confidence levels are treated as fuzzy numbersGenetic algorithm approachMahapatra et al. [36]TPDemand & supplyLogistic distributionA transformation technique is presented for manipulating cost coefficients involving multichoice for binary variables with auxiliary constraints and solved by Lingo softwareAgrawal and Ganesh [15]TPDemandGalton distributionParameters are replaced by Newton’s divided difference interpolating polynomial and solved by Lingo softwareDas and Lee [37]Solid TPDemand, supply, and capacityWeibull distributionGlobal criterion method and fuzzy goal programming approachThis articleSFTPMCDemand & supplyWeibull distributionAlpha cut representation for cost function and WD for constraints and then solved by Lingo softwareBased on this motivation, the main contribution of this study is summarized as follows: the goal of this paper is as follows: (1) to propose a mathematical model that deals with fuzziness and randomness under one roof, (2) to consider the fuzzy objective value and probabilistic constraints, and (3) to present a simplified computation conversion of probabilistic constraints to their equivalent deterministic constraint using WD. In order to achieve this, the fuzzy objective is converted to alpha cut representation, and probabilistic constraints are converted to deterministic form by using the WD and then solved using Lingo software. To investigate the sensitivity of the parameters in the proposed model, a sensitivity analysis is performed.


## 1.3. Managerial Insights Based on the Research

The last decade has seen the growth in various fields such as economics, trade, health care, transportation, cultivation, army, engineering, and technology. It has increased the importance of stochastic fuzzy optimization in solving scientific managerial decision-making problems in every field. The uncertainty factors which make the problem more complex have not spared the transportation process. The objective of this paper is to optimize the total transportation cost under uncertain conditions by utilizing the SFTPMC model. Generally, uncertainty is described using fuzzy or probability theory. However, it is not always reasonable to use fuzzy theory or probability theory to illustrate all the indeterminacies, as it requires adequate information. So, we prefer to describe the uncertainty with both fuzzy and probability parameters. In fact, low-frequency events occur in our daily lives causing the uncertainty theory to take effect. Under these circumstances, scheduling a proper transportation plan to lower the total costs becomes a challenge. This study provides an applicable model for the DM to deal with some uncertain factors without losing reliability from customers. This research could also be extended to other aspects, in which a decision/plan needs to be made in uncertain conditions.The rest of this article is structured as follows. Section2 presents the preliminaries that are related to this article. Section 3 deals with the assumptions and notation used in this paper. Section 4 investigates and converts the mathematical programming models of SFTPMC and its variants to the comparable deterministic models. Section 5 presents the proposed solution method with an illustration. Section 6 presents and analyses the computational experiences of SA. Finally, Section 7 summarizes the research outcome and limitations and suggests future directions.


## 2. Some Preliminaries

Definition 1.
(Fuzzy set, Zadeh., [38]). Let R be a collection of sets and μRx be a membership function from R to [0, 1]. A fuzzy set R˜ with the membership function μRx is defined by R˜=x,μRx/x∈RandμRx∈0,1.Definition 2.
(Fuzzy number, Zadeh., [38]). A real fuzzy number r˜=r1,r2,r3 is a fuzzy subset of the real line R with the membership function μr˜r satisfying the following conditions:(i)
μr˜r:R⟶0,1 is continuous(ii)
μr˜r=0for all−∞,r1∪r3,∞(iii)
μr˜r is strictly increasing on r1,r2 and strictly decreasing on r2,r3(iv)
μr˜r=1for allr∈r2wherer1≤r2≤r3Definition 3.
(Triangular fuzzy number, Zadeh., [38]).A fuzzy number R˜ is denoted as a triangular fuzzy number by r1,r2,r3 where a1, a2, and a3 are real numbers, and its membership function μA˜x is given as follows:(1)μR˜x=x−r1r2−r1,r1≤x≤r2,x−r3r2−r3,r2≤x≤r3,0,otherwise.Definition 4.
(Alpha cut, Zadeh., [38]). The alpha cut of a fuzzy number Rx is defined as Rα=x/μx≥α,α∈0,1.Definition 5.
(Linear membership function, [10]). A linear membership function can be defined as μRX=0ifxij<x¯ijx¯ij−xij/x¯ij−x¯ijifx¯ij<xij<x¯ij1ifxij>x¯ij.
In order to transform the fuzzy system to a deterministic set, the alpha cut representation using linear membership function isx¯ij−xij/x¯ij−x¯ij=α such that xij=1−αx¯ij+αx¯ij for all α ∈ [0, 1].Definition 6.
(Feasible solution). Any set of {xij ≥ 0,i = 1,2, ..., m; j = 1,2,…n} that satisfies all the constraints is called a feasible solution to the problem.Definition 7.
(Optimal solution). A feasible solution to the problem which minimizes the total shipping cost is called an optimal solution to the problem.


## 3. Assumptions and Notations

An SFTPMC is associated with costs, supply, and demand. The direct value is the cost of transportation per unit amount. The mixed constraint occurs for the transportation activity between a source and a destination. The following notations are introduced in order to develop the mathematical model for the SFTPMCs:ai: the amount of homogeneous product availability at the source ibj: the amount of homogeneous product demand at the destination jc˜ij: fuzzy transportation cost per unit for carrying one unit of goods from source i to destination jxij: the amount shipped from source i to destination jpai: probabilities for aipbj: probabilities for bjχai: shape parameter for aiχbj: shape parameter for bjλai: scale parameter for aiλbj: scale parameter for bjξai: location parameter for aiξbj: location parameter for bj


## 4. Problem Formulation and Solution Concepts

The objective value and constraints play an important role for SFTPMC. Minimizing the total transportation costs is our main objective. In different real-life situations, parameters (cost, supply, and demand) become uncertain in nature; then, DM faces difficulty to make the optimal decision. This situation can be handled with fuzzy and random variables. In this model, we consider cost as triangular fuzzy variables and constraints as random variables. The uncertainty in supply or demand constraints may or may not occur, depending on the situation of DM. Therefore, we formulate three SFTPMC models based on the uncertainty in constraints. The model explained below shows the formulation of an SFTPMC model withm sources and n destinations.


### 4.1. Chance-Constrained Programming Model for SFTPMC

To obtain the quantiles of a probability distribution function in a closed form, it is necessary to apply the constraints in the suggested SP model to the deterministic constraints. Another reason to adopt the WD is that it has the closed form of the quantiles. The CCP paradigm for SFTPMC is described in this paper as follows:(2)PMinimizez˜=∑i=1m∑j=1nc˜ijxij.Subject to(3)P∑j=1nxij≥ai≥pai,i∈q1,(4)P∑j=1nxij=ai≥pai,i∈q2,(5)P∑j=1nxij≤ai≥pai,i∈q3,(6)P∑i=1mxij≥bj≥pbj,j∈r1,(7)P∑i=1mxij=bj≥pbj,j∈r2,(8)P∑i=1mxij≤bj≥pbj,j∈r3,(9)xij≥0,i=1,2,…,m,j=1,2,…,n,where pai and pbj are probabilities given. It is observed that RVs ai and bj represent supply and demand, following the WD, respectively. The WD for ai has three parameters ξai, χai, and λai which serve as location, scale, and shape parameters. Similarly, the parameters for bj are defined as ξbj, χbj, andλbj. Constraints (3)–(5) are the probabilistic constraint for the quantity of supply at origin i, which ensures with a given probability pai, the sum of quantities of shipments from supply source i must distribute exactly ai units of supply or it must distribute at least ai units of supply or it must distribute at most ai units of supply which {q1, q2, and q3} are partitions of i = {1, 2, 3, …, m}. In a similar manner, constraints (6)–(8) can be construed for the demand at destination j, must receive exactly bj units, or it must receive at least bj units or it receives at most bj units of demand which {r1, r2, and r3} are partitions of j = {1, 2, 3, …, n}. c˜ij is the fuzzy transportation cost (1) per unit for carrying one unit of goods from sources i to destination j. The aim is to describe the quantity xij transported from origin i to destination j while minimizing the total transport cost when satisfying mixed type supply and demand constraints. For the general cases of TPMC type, in which a single random variable among ai and bj is uncertain, the transformation of the probabilistic parameter into a deterministic parameter is studied in Case I and II.


#### 4.1.1. Case (I) Onlyai is Uncertain

(i)
ForP∑j=1nxij≥ai≥pai,i∈q1 the proof is shown belowIt is considered thatai (i∈q1 =1, 2, m1) are independent RVs with the WD and three known parameters, ξai, χai, andλai. Equation (3) can therefore be transformed as follows:(10)Pai≤∑j=1nxij≥pai,i∈q1.Let us consider∑j=1nxij=δaiPai≤δai≥pai,i∈q1, by using WD for the probabilistic function,(11)∫−∞δaiχaiλaiai−ξaiλaiχai−1e−ai−ξai/λaiχaidai≥pai,i∈q1.Simultaneously,ai≥ξaii∈q1, and the integration of equation (11) gives the following form:(12)∫ξaiδaiχaiλaiai−ξaiλaiχai−1e−ai−ξai/λaiχaidai≥pai.After integrating equation (12), we get(13)1−e−δai−ai/λaiχai≥pai.Equation (13) can be further simplified by applying the logarithm as,δai≥ξai+λai−1n1−pai1/χai,i∈q1. Lastly, this can be stated as a deterministic constraint in the equivalent terms: ∑j=1nxij≥ξai+λai−1n1−pai1/χai,i∈q1.(ii)
ForP∑j=1nxij≤ai≥pai,i∈q3 (i = m2+1, m2+2,…, m) the proof of converting the probabilistic values into deterministic values by using WD is obtained in [37].


#### 4.1.2. Case (II) Onlybj is Uncertain

(i)
ForP∑i=1mxij≥bj≥pbj,j∈r1, (j = 1,2,…, n1) the proof is shown in [37].(ii)
ForP∑i=1mxij≤bj≥pbj,j∈r3 the proof is shown as follows.It is considered thatbj (j = n2+1, n2+2,…,n) are independent RVs with the WD and three known parameters ξbj, χbj, and λbj. Equation (8) can therefore be transformed as follows:(14)Pbj≥∑i=1mxij≥pbj,j∈r3.Let us consider∑i=1mxij=δbj,(15)Pbj≥δbj≥pbj,j∈r3,(16)∫δbj∞χbjλbjbj−ξbjλbjχbj−1e−bj−ξbj/λbjχbjdbj≥pbj,j∈r3.Simultaneously,bj≥ξbjj∈r3, and the integration of equation (16) gives the following form:(17)∫δbjξbjχbjλbjbj−ξbjλbjχbj−1e−bj−ξbj/λbjχbjdbj≥pbj.After integrating equation (17), we get(18)1−e−δbj−ξbj/λbjχbj≥pbj.Equation (18) can be further simplified by applying the logarithm as δbj≤ξbj+λbj−1npbj1/χbj,j∈r3. Lastly, this can be stated as a deterministic constraint in equivalent terms ∑i=1mxij≤ξbj+λbj−lnpbj1/χbj,j∈r3.


#### 4.1.3. Remarks

LetP∑j=1nxij=ai≥pai,i∈q2=m1+1,m1+2,…,m2, and P∑i=1mxij=bj≥pbj,j∈r2=n1+1,n1+2,…,n2 be the equality type supply and demand constraints. Then, change equality type to inequality type for the supply constraint as P∑j=1nxij≤ai≥pai and P∑j=1nxij≥ai≥pai for the demand constraint as P∑i=1mxij≤bj≥pbj and P∑i=1mxij≥bj≥pbj. Choosing the probability value at 50% level in the supply and demand inequality constraint by using WD, we obtain the same deterministic value for both inequality types for supply and demand constraints. So, the deterministic values of equality constraint for the supply as ∑j=1nxij=ξai+λai−ln1−pai1/χai,i∈q2 or ∑j=1nxij=ξai+λai−lnpai1/χai,i∈q2 and the demand as ∑i=1mxij=ξbj+λbj−1npbj1/χbj,j∈r2 or ∑i=1mxij=ξbj+λbj−1n1−pbj1/χbj,j∈r2.Deterministic mathematical programming models for SFTPMC are presented in this part, which use the quantiles of the WD as constraints and the fuzzy linear membership function for the cost function. In a real scenario, only some aspects of supply and demand may be uncertain, while others are certain. As a result, SFTPMC deterministic can be modified as needed according to the situation. Three models, in which any one of the parameters areai or bj are uncertain, are studied in Model 1 and 2, respectively. Model 3 presents a general model in which all RVs are uncertain. Commercially available solvers can be used to solve this deterministic mathematical programming for the optimal solution.Model 1.
For deterministic modeling of SFTPMC, a linear membership function [10] is used for the cost function, a quantile of the Weibull distribution is used for probabilistic supply constraint, and demand constraint remains certain.(19)D1Minimizez=∑i=1m∑j=1ncij1−αxij¯+αx¯ij.
Subject to(20)∑j=1nxij≥ξai+λai−1n1−pai1/χai,i∈q1,(21)∑j=1nxij=ξai+λai−1n1−pai1/χai,i∈q2,(22)∑j=1nxij≤ξai+λai−1npai1/χai,i∈q3,(23)∑i=1mxij≥bj​​​​ ​,j∈r1,(24)∑i=1mxij=bj​​​​ ​,j∈r2,(25)∑i=1mxij≤bj​​​​ ​,j∈r3,(26)xij≥0.​​  ​Model 2.
For deterministic modeling of SFTPMC, a linear membership function [10] is used for the cost function, a quantile of the Weibull distribution is used for demand constraint, and supply constraint remains certain.(27)D2Minimizez=∑i=1m∑j=1ncij1−αx¯ij+αx¯ij.
Subject to(28)∑j=1nxij≥ai,i∈q1,(29)∑j=1nxij=ai,i∈q2,(30)∑j=1nxij≤ai,i∈q3,(31)∑i=1mxij≥ξbj+λbj−1n1−pbj1/χbj,j∈r1,(32)∑i=1mxij=ξbj+λbj−1npbj1/χbj,j∈r2,(33)∑i=1mxij≤ξbj+λbj−1npbj1/χbj,j∈r3,(34)xij≥0.Model 3.
For deterministic modeling of SFTPMC, a linear membership function [10] is used for the cost function, and a quantile of the Weibull distribution is used for supply and demand constraints.(35)D3Minimizez=∑i=1m∑j=1ncij1−αx¯ij+αx¯ij.
Subject to(36)∑j=1nxij≥ξai+λai−1n1−pai1/χai,i∈q1,(37)∑j=1nxij=ξai+λai−1n1−pai1/χai,i∈q2,(38)∑j=1nxij≤ξai+λai−1npai1/χai,i∈q3,(39)∑i=1mxij≥ξbj+λbj−1n1−pbj1/χbj,j∈r1,(40)∑i=1mxij=ξbj+λbj−1npbj1/χbj,j∈r2,(41)∑i=1mxij≤ξbj+λbj−1npbj1/χbj,j∈r3,(42)xij≥0.


## 4.1. Chance-Constrained Programming Model for SFTPMC

To obtain the quantiles of a probability distribution function in a closed form, it is necessary to apply the constraints in the suggested SP model to the deterministic constraints. Another reason to adopt the WD is that it has the closed form of the quantiles. The CCP paradigm for SFTPMC is described in this paper as follows:(2)PMinimizez˜=∑i=1m∑j=1nc˜ijxij.Subject to(3)P∑j=1nxij≥ai≥pai,i∈q1,(4)P∑j=1nxij=ai≥pai,i∈q2,(5)P∑j=1nxij≤ai≥pai,i∈q3,(6)P∑i=1mxij≥bj≥pbj,j∈r1,(7)P∑i=1mxij=bj≥pbj,j∈r2,(8)P∑i=1mxij≤bj≥pbj,j∈r3,(9)xij≥0,i=1,2,…,m,j=1,2,…,n,where pai and pbj are probabilities given. It is observed that RVs ai and bj represent supply and demand, following the WD, respectively. The WD for ai has three parameters ξai, χai, and λai which serve as location, scale, and shape parameters. Similarly, the parameters for bj are defined as ξbj, χbj, andλbj. Constraints (3)–(5) are the probabilistic constraint for the quantity of supply at origin i, which ensures with a given probability pai, the sum of quantities of shipments from supply source i must distribute exactly ai units of supply or it must distribute at least ai units of supply or it must distribute at most ai units of supply which {q1, q2, and q3} are partitions of i = {1, 2, 3, …, m}. In a similar manner, constraints (6)–(8) can be construed for the demand at destination j, must receive exactly bj units, or it must receive at least bj units or it receives at most bj units of demand which {r1, r2, and r3} are partitions of j = {1, 2, 3, …, n}. c˜ij is the fuzzy transportation cost (1) per unit for carrying one unit of goods from sources i to destination j. The aim is to describe the quantity xij transported from origin i to destination j while minimizing the total transport cost when satisfying mixed type supply and demand constraints. For the general cases of TPMC type, in which a single random variable among ai and bj is uncertain, the transformation of the probabilistic parameter into a deterministic parameter is studied in Case I and II.


### 4.1.1. Case (I) Onlyai is Uncertain

(i)
ForP∑j=1nxij≥ai≥pai,i∈q1 the proof is shown belowIt is considered thatai (i∈q1 =1, 2, m1) are independent RVs with the WD and three known parameters, ξai, χai, andλai. Equation (3) can therefore be transformed as follows:(10)Pai≤∑j=1nxij≥pai,i∈q1.Let us consider∑j=1nxij=δaiPai≤δai≥pai,i∈q1, by using WD for the probabilistic function,(11)∫−∞δaiχaiλaiai−ξaiλaiχai−1e−ai−ξai/λaiχaidai≥pai,i∈q1.Simultaneously,ai≥ξaii∈q1, and the integration of equation (11) gives the following form:(12)∫ξaiδaiχaiλaiai−ξaiλaiχai−1e−ai−ξai/λaiχaidai≥pai.After integrating equation (12), we get(13)1−e−δai−ai/λaiχai≥pai.Equation (13) can be further simplified by applying the logarithm as,δai≥ξai+λai−1n1−pai1/χai,i∈q1. Lastly, this can be stated as a deterministic constraint in the equivalent terms: ∑j=1nxij≥ξai+λai−1n1−pai1/χai,i∈q1.(ii)
ForP∑j=1nxij≤ai≥pai,i∈q3 (i = m2+1, m2+2,…, m) the proof of converting the probabilistic values into deterministic values by using WD is obtained in [37].


### 4.1.2. Case (II) Onlybj is Uncertain

(i)
ForP∑i=1mxij≥bj≥pbj,j∈r1, (j = 1,2,…, n1) the proof is shown in [37].(ii)
ForP∑i=1mxij≤bj≥pbj,j∈r3 the proof is shown as follows.It is considered thatbj (j = n2+1, n2+2,…,n) are independent RVs with the WD and three known parameters ξbj, χbj, and λbj. Equation (8) can therefore be transformed as follows:(14)Pbj≥∑i=1mxij≥pbj,j∈r3.Let us consider∑i=1mxij=δbj,(15)Pbj≥δbj≥pbj,j∈r3,(16)∫δbj∞χbjλbjbj−ξbjλbjχbj−1e−bj−ξbj/λbjχbjdbj≥pbj,j∈r3.Simultaneously,bj≥ξbjj∈r3, and the integration of equation (16) gives the following form:(17)∫δbjξbjχbjλbjbj−ξbjλbjχbj−1e−bj−ξbj/λbjχbjdbj≥pbj.After integrating equation (17), we get(18)1−e−δbj−ξbj/λbjχbj≥pbj.Equation (18) can be further simplified by applying the logarithm as δbj≤ξbj+λbj−1npbj1/χbj,j∈r3. Lastly, this can be stated as a deterministic constraint in equivalent terms ∑i=1mxij≤ξbj+λbj−lnpbj1/χbj,j∈r3.


### 4.1.3. Remarks

LetP∑j=1nxij=ai≥pai,i∈q2=m1+1,m1+2,…,m2, and P∑i=1mxij=bj≥pbj,j∈r2=n1+1,n1+2,…,n2 be the equality type supply and demand constraints. Then, change equality type to inequality type for the supply constraint as P∑j=1nxij≤ai≥pai and P∑j=1nxij≥ai≥pai for the demand constraint as P∑i=1mxij≤bj≥pbj and P∑i=1mxij≥bj≥pbj. Choosing the probability value at 50% level in the supply and demand inequality constraint by using WD, we obtain the same deterministic value for both inequality types for supply and demand constraints. So, the deterministic values of equality constraint for the supply as ∑j=1nxij=ξai+λai−ln1−pai1/χai,i∈q2 or ∑j=1nxij=ξai+λai−lnpai1/χai,i∈q2 and the demand as ∑i=1mxij=ξbj+λbj−1npbj1/χbj,j∈r2 or ∑i=1mxij=ξbj+λbj−1n1−pbj1/χbj,j∈r2.Deterministic mathematical programming models for SFTPMC are presented in this part, which use the quantiles of the WD as constraints and the fuzzy linear membership function for the cost function. In a real scenario, only some aspects of supply and demand may be uncertain, while others are certain. As a result, SFTPMC deterministic can be modified as needed according to the situation. Three models, in which any one of the parameters areai or bj are uncertain, are studied in Model 1 and 2, respectively. Model 3 presents a general model in which all RVs are uncertain. Commercially available solvers can be used to solve this deterministic mathematical programming for the optimal solution.Model 1.
For deterministic modeling of SFTPMC, a linear membership function [10] is used for the cost function, a quantile of the Weibull distribution is used for probabilistic supply constraint, and demand constraint remains certain.(19)D1Minimizez=∑i=1m∑j=1ncij1−αxij¯+αx¯ij.
Subject to(20)∑j=1nxij≥ξai+λai−1n1−pai1/χai,i∈q1,(21)∑j=1nxij=ξai+λai−1n1−pai1/χai,i∈q2,(22)∑j=1nxij≤ξai+λai−1npai1/χai,i∈q3,(23)∑i=1mxij≥bj​​​​ ​,j∈r1,(24)∑i=1mxij=bj​​​​ ​,j∈r2,(25)∑i=1mxij≤bj​​​​ ​,j∈r3,(26)xij≥0.​​  ​Model 2.
For deterministic modeling of SFTPMC, a linear membership function [10] is used for the cost function, a quantile of the Weibull distribution is used for demand constraint, and supply constraint remains certain.(27)D2Minimizez=∑i=1m∑j=1ncij1−αx¯ij+αx¯ij.
Subject to(28)∑j=1nxij≥ai,i∈q1,(29)∑j=1nxij=ai,i∈q2,(30)∑j=1nxij≤ai,i∈q3,(31)∑i=1mxij≥ξbj+λbj−1n1−pbj1/χbj,j∈r1,(32)∑i=1mxij=ξbj+λbj−1npbj1/χbj,j∈r2,(33)∑i=1mxij≤ξbj+λbj−1npbj1/χbj,j∈r3,(34)xij≥0.Model 3.
For deterministic modeling of SFTPMC, a linear membership function [10] is used for the cost function, and a quantile of the Weibull distribution is used for supply and demand constraints.(35)D3Minimizez=∑i=1m∑j=1ncij1−αx¯ij+αx¯ij.
Subject to(36)∑j=1nxij≥ξai+λai−1n1−pai1/χai,i∈q1,(37)∑j=1nxij=ξai+λai−1n1−pai1/χai,i∈q2,(38)∑j=1nxij≤ξai+λai−1npai1/χai,i∈q3,(39)∑i=1mxij≥ξbj+λbj−1n1−pbj1/χbj,j∈r1,(40)∑i=1mxij=ξbj+λbj−1npbj1/χbj,j∈r2,(41)∑i=1mxij≤ξbj+λbj−1npbj1/χbj,j∈r3,(42)xij≥0.


## 4.1.1. Case (I) Onlyai is Uncertain

(i)
ForP∑j=1nxij≥ai≥pai,i∈q1 the proof is shown belowIt is considered thatai (i∈q1 =1, 2, m1) are independent RVs with the WD and three known parameters, ξai, χai, andλai. Equation (3) can therefore be transformed as follows:(10)Pai≤∑j=1nxij≥pai,i∈q1.Let us consider∑j=1nxij=δaiPai≤δai≥pai,i∈q1, by using WD for the probabilistic function,(11)∫−∞δaiχaiλaiai−ξaiλaiχai−1e−ai−ξai/λaiχaidai≥pai,i∈q1.Simultaneously,ai≥ξaii∈q1, and the integration of equation (11) gives the following form:(12)∫ξaiδaiχaiλaiai−ξaiλaiχai−1e−ai−ξai/λaiχaidai≥pai.After integrating equation (12), we get(13)1−e−δai−ai/λaiχai≥pai.Equation (13) can be further simplified by applying the logarithm as,δai≥ξai+λai−1n1−pai1/χai,i∈q1. Lastly, this can be stated as a deterministic constraint in the equivalent terms: ∑j=1nxij≥ξai+λai−1n1−pai1/χai,i∈q1.(ii)
ForP∑j=1nxij≤ai≥pai,i∈q3 (i = m2+1, m2+2,…, m) the proof of converting the probabilistic values into deterministic values by using WD is obtained in [37].


## 4.1.2. Case (II) Onlybj is Uncertain

(i)
ForP∑i=1mxij≥bj≥pbj,j∈r1, (j = 1,2,…, n1) the proof is shown in [37].(ii)
ForP∑i=1mxij≤bj≥pbj,j∈r3 the proof is shown as follows.It is considered thatbj (j = n2+1, n2+2,…,n) are independent RVs with the WD and three known parameters ξbj, χbj, and λbj. Equation (8) can therefore be transformed as follows:(14)Pbj≥∑i=1mxij≥pbj,j∈r3.Let us consider∑i=1mxij=δbj,(15)Pbj≥δbj≥pbj,j∈r3,(16)∫δbj∞χbjλbjbj−ξbjλbjχbj−1e−bj−ξbj/λbjχbjdbj≥pbj,j∈r3.Simultaneously,bj≥ξbjj∈r3, and the integration of equation (16) gives the following form:(17)∫δbjξbjχbjλbjbj−ξbjλbjχbj−1e−bj−ξbj/λbjχbjdbj≥pbj.After integrating equation (17), we get(18)1−e−δbj−ξbj/λbjχbj≥pbj.Equation (18) can be further simplified by applying the logarithm as δbj≤ξbj+λbj−1npbj1/χbj,j∈r3. Lastly, this can be stated as a deterministic constraint in equivalent terms ∑i=1mxij≤ξbj+λbj−lnpbj1/χbj,j∈r3.


## 4.1.3. Remarks

LetP∑j=1nxij=ai≥pai,i∈q2=m1+1,m1+2,…,m2, and P∑i=1mxij=bj≥pbj,j∈r2=n1+1,n1+2,…,n2 be the equality type supply and demand constraints. Then, change equality type to inequality type for the supply constraint as P∑j=1nxij≤ai≥pai and P∑j=1nxij≥ai≥pai for the demand constraint as P∑i=1mxij≤bj≥pbj and P∑i=1mxij≥bj≥pbj. Choosing the probability value at 50% level in the supply and demand inequality constraint by using WD, we obtain the same deterministic value for both inequality types for supply and demand constraints. So, the deterministic values of equality constraint for the supply as ∑j=1nxij=ξai+λai−ln1−pai1/χai,i∈q2 or ∑j=1nxij=ξai+λai−lnpai1/χai,i∈q2 and the demand as ∑i=1mxij=ξbj+λbj−1npbj1/χbj,j∈r2 or ∑i=1mxij=ξbj+λbj−1n1−pbj1/χbj,j∈r2.Deterministic mathematical programming models for SFTPMC are presented in this part, which use the quantiles of the WD as constraints and the fuzzy linear membership function for the cost function. In a real scenario, only some aspects of supply and demand may be uncertain, while others are certain. As a result, SFTPMC deterministic can be modified as needed according to the situation. Three models, in which any one of the parameters areai or bj are uncertain, are studied in Model 1 and 2, respectively. Model 3 presents a general model in which all RVs are uncertain. Commercially available solvers can be used to solve this deterministic mathematical programming for the optimal solution.Model 1.
For deterministic modeling of SFTPMC, a linear membership function [10] is used for the cost function, a quantile of the Weibull distribution is used for probabilistic supply constraint, and demand constraint remains certain.(19)D1Minimizez=∑i=1m∑j=1ncij1−αxij¯+αx¯ij.
Subject to(20)∑j=1nxij≥ξai+λai−1n1−pai1/χai,i∈q1,(21)∑j=1nxij=ξai+λai−1n1−pai1/χai,i∈q2,(22)∑j=1nxij≤ξai+λai−1npai1/χai,i∈q3,(23)∑i=1mxij≥bj​​​​ ​,j∈r1,(24)∑i=1mxij=bj​​​​ ​,j∈r2,(25)∑i=1mxij≤bj​​​​ ​,j∈r3,(26)xij≥0.​​  ​Model 2.
For deterministic modeling of SFTPMC, a linear membership function [10] is used for the cost function, a quantile of the Weibull distribution is used for demand constraint, and supply constraint remains certain.(27)D2Minimizez=∑i=1m∑j=1ncij1−αx¯ij+αx¯ij.
Subject to(28)∑j=1nxij≥ai,i∈q1,(29)∑j=1nxij=ai,i∈q2,(30)∑j=1nxij≤ai,i∈q3,(31)∑i=1mxij≥ξbj+λbj−1n1−pbj1/χbj,j∈r1,(32)∑i=1mxij=ξbj+λbj−1npbj1/χbj,j∈r2,(33)∑i=1mxij≤ξbj+λbj−1npbj1/χbj,j∈r3,(34)xij≥0.Model 3.
For deterministic modeling of SFTPMC, a linear membership function [10] is used for the cost function, and a quantile of the Weibull distribution is used for supply and demand constraints.(35)D3Minimizez=∑i=1m∑j=1ncij1−αx¯ij+αx¯ij.
Subject to(36)∑j=1nxij≥ξai+λai−1n1−pai1/χai,i∈q1,(37)∑j=1nxij=ξai+λai−1n1−pai1/χai,i∈q2,(38)∑j=1nxij≤ξai+λai−1npai1/χai,i∈q3,(39)∑i=1mxij≥ξbj+λbj−1n1−pbj1/χbj,j∈r1,(40)∑i=1mxij=ξbj+λbj−1npbj1/χbj,j∈r2,(41)∑i=1mxij≤ξbj+λbj−1npbj1/χbj,j∈r3,(42)xij≥0.


## 5. Solution Method

The solution method for the model SFTPMC is given as follows:Step 1.
Convert the given triangular fuzzy cost of problem (P) into an equivalent deterministic cost (17) by using alpha cut.Step 2.
Then, problem (P) is transformed into 3 models depending on its constraint.Step 3.
(i) For model 1,ai is an uncertain probabilistic supply constraint, and bj remains precise values. For ai, we use WD to reduce the probabilistic supply constraint into an equivalent deterministic supply constraint (20)–(22). Then, construct the problem (D1) with the reduced deterministic cost (17), deterministic supply constraint (20)–(22), and precise demand constraint (23)–(25). (ii) At alpha level 0, solve the problem (D1) to obtain the optimal transportation cost and unit flow, using the Lingo software.Step 4.
(i) For model 2, onlybj is uncertain, and ai remains precise values. For bj, we use WD to reduce the probabilistic demand constraint into an equivalent deterministic demand constraint (31)–(33). Then, construct the problem (D2) with the reduced deterministic cost (17), deterministic demand constraint (31)–(33), and precise supply constraint (28)–(30). (ii) At alpha level 0, solve the problem (D2) to obtain the optimal transportation cost and unit flow, using the Lingo software.Step 5.
(i) For model 3,ai and bj are uncertain values, and we use WD to reduce the probabilistic supply and demand constraint into an equivalent deterministic supply (36)–(41). Then, construct the problem (D3) with the reduced deterministic cost (17) and deterministic constraint (36)–(41). (ii) At alpha level 0, solve the problem (D3) to obtain the optimal transportation cost and unit flow, using the Lingo software.Step 6.
Repeat the above steps (3–5) to obtain the optimal transportation cost and unit flow for different alpha levels.


### 5.1. Numerical Experiment and Discussions

This segment provides an illustration to demonstrate the efficacy and applicability of SFTPMC and its variants. The coal plant generates a homogeneous product, and there are three plants and four repositories. Coal plant A has a manufacturing capacity of exactlya1 units, coal plant B has a production capacity of at least a2 units, and coal plant C has a manufacturing capacity of the most a3 units. Likewise, repository 1 has a capacity of demands at least b1 units; repository 2 has a capacity of demands at the most b2 units; repository 3 has a capacity of demands of at least b3 units. Repository 4 has a capacity of demands exactly b4 units. If transporting per unit cost from each coal plant to each repository is c˜ij, it is given in triangular fuzzy cost data as shown in Table 2.Table 2
Stochastic fuzzy transportation problem mixed constraint (P).
Repositories1234aiCoal plantA(0,0.5,1)(2,4,6)(1.5,2,3)(2,4,5)=a1B(3,5,7)(1.5,2,3)(0,0.5,1)(4,5.8,6)≥a2C(7,8.5,9)(2.5,3,4)(3,4,5)(2,3,4)≤a3bj≥b1≤b2≥b3=b4In the numerical experiment, nominal values of certain constants are supplied asa1=20, a2=16a3=25, and demand as b1=11, b2=13, b3=17, b4=14 in the following sections. Furthermore, the following arbitrary probabilities are given as Pa1=0.50, Pa2=0.96, Pa3=0.95, Pb1=0.26, Pb2=0.29, Pb3=0.25, Pb4=0.28. Since ai and bj are presumed to follow WD, the distinct values for the parameters are ξa1=19, ξa2=13, ξa3=24, ξb1=10, ξb2=11, ξb3=16, ξb4=13, χai=χbj=2, λai=λbj=2. The probabilistic constraints can simply be transformed to deterministic forms by utilizing equations (36)–(41). By using Step 1, we have to convert the given triangular fuzzy cost of the problem (P) into deterministic cost using the linear membership function. Table 3 shows the alpha cut representation for the cost function of a given problem (P).Table 3
Alpha cut representation of cost function for problem (P).
1234aiA1−α1x¯11+0αx¯111−α6x¯12+2αx¯121−α3x¯13+1.5αx¯131−α5x¯14+2αx¯14=a1B1−α7x¯21+3αx¯211−α3x¯22+1.5αx¯221−α1x¯23+0αx¯231−α6x¯24+4αx¯24≥a2C1−α9x¯31+7αx¯311−α4x¯32+2.5αx¯321−α5x¯33+3αx¯331−α4x¯34+2αx¯34≤a3bj≥b1≤b2≥b3=b4Now, using Step2, problem (P) is transformed into 3 models depending on its constraint. By Step 3, only supply constraints are uncertain, and demand remains certain for model 1. At alpha level 0 on the cost function, WD is used for the probabilistic supply constraint. Here, supply is a1=20.67, a2=16.59, and a3=24.45, and demand is b1=11, b2=13, b3=17, and b4=14. By using the Lingo software, we obtain the optimal transportation cost as 93.67 and x11 = 11, x14 = 9.67, x23 = 17, x34 = 4.33 and unit flow as 42.By Step4, only demand constraints are uncertain, and supply remains at certain values for model 2. At alpha level 0 on the cost function, WD is used for the demand constraint. Here, supply is a1=20, a2=16, a3=25, and demand is b1=11.1, b2=13.23, b3=17.07, b4=14.67, and we obtain the optimal transportation cost as 95.75 and x11 = 11.10, x14 = 8.9, x23 = 17.07, x34 = 5.77 and unit flow as 42.84.By Step5, both supply and demand constraints are uncertain for model 3. Here, supply is a1=20.67, a2=16.59, a3=24.45, and demand is b1=11.1, b2=13.23b3=17.07, b4=14.67. Similarly, we obtain the optimal transportation cost as 96.42 and x11 = 11.10, x14 = 9.57, x23 = 17.07, x34 = 5.1 and unit flow as 42.84. In the same manner, we have to solve different alpha levels for the 3 cases of constraints. Computational results for 3 models are shown in Table 4. Using Step 6, we can solve for different alpha levels.Table 4
Computational results for 3 models.
Optimization method (Lingo software)Model 1Model 2Model 3Optimal transportation cost93.6795.7596.42Units in flow4242.8442.84Three new models have been constructed for SFTPMC. While model 1 presents supply as a probabilistic value and demand as a certain value, model 2 presents demand as a probabilistic value and supply as a certain value, and model 3 presents supply and demand as probabilistic values. The cost coefficient of the objective function is transformed to alpha cut representation, and the stochastic mixed constraints are transformed to deterministic form by using the WD for solving the models. The fact that these models are constructed from different points of view is worth noting. Since the usage of the models is dependent on the DM’s preference, we cannot conclude which model is the best in the process of decision-making.


## 5.1. Numerical Experiment and Discussions

This segment provides an illustration to demonstrate the efficacy and applicability of SFTPMC and its variants. The coal plant generates a homogeneous product, and there are three plants and four repositories. Coal plant A has a manufacturing capacity of exactlya1 units, coal plant B has a production capacity of at least a2 units, and coal plant C has a manufacturing capacity of the most a3 units. Likewise, repository 1 has a capacity of demands at least b1 units; repository 2 has a capacity of demands at the most b2 units; repository 3 has a capacity of demands of at least b3 units. Repository 4 has a capacity of demands exactly b4 units. If transporting per unit cost from each coal plant to each repository is c˜ij, it is given in triangular fuzzy cost data as shown in Table 2.Table 2
Stochastic fuzzy transportation problem mixed constraint (P).
Repositories1234aiCoal plantA(0,0.5,1)(2,4,6)(1.5,2,3)(2,4,5)=a1B(3,5,7)(1.5,2,3)(0,0.5,1)(4,5.8,6)≥a2C(7,8.5,9)(2.5,3,4)(3,4,5)(2,3,4)≤a3bj≥b1≤b2≥b3=b4In the numerical experiment, nominal values of certain constants are supplied asa1=20, a2=16a3=25, and demand as b1=11, b2=13, b3=17, b4=14 in the following sections. Furthermore, the following arbitrary probabilities are given as Pa1=0.50, Pa2=0.96, Pa3=0.95, Pb1=0.26, Pb2=0.29, Pb3=0.25, Pb4=0.28. Since ai and bj are presumed to follow WD, the distinct values for the parameters are ξa1=19, ξa2=13, ξa3=24, ξb1=10, ξb2=11, ξb3=16, ξb4=13, χai=χbj=2, λai=λbj=2. The probabilistic constraints can simply be transformed to deterministic forms by utilizing equations (36)–(41). By using Step 1, we have to convert the given triangular fuzzy cost of the problem (P) into deterministic cost using the linear membership function. Table 3 shows the alpha cut representation for the cost function of a given problem (P).Table 3
Alpha cut representation of cost function for problem (P).
1234aiA1−α1x¯11+0αx¯111−α6x¯12+2αx¯121−α3x¯13+1.5αx¯131−α5x¯14+2αx¯14=a1B1−α7x¯21+3αx¯211−α3x¯22+1.5αx¯221−α1x¯23+0αx¯231−α6x¯24+4αx¯24≥a2C1−α9x¯31+7αx¯311−α4x¯32+2.5αx¯321−α5x¯33+3αx¯331−α4x¯34+2αx¯34≤a3bj≥b1≤b2≥b3=b4Now, using Step2, problem (P) is transformed into 3 models depending on its constraint. By Step 3, only supply constraints are uncertain, and demand remains certain for model 1. At alpha level 0 on the cost function, WD is used for the probabilistic supply constraint. Here, supply is a1=20.67, a2=16.59, and a3=24.45, and demand is b1=11, b2=13, b3=17, and b4=14. By using the Lingo software, we obtain the optimal transportation cost as 93.67 and x11 = 11, x14 = 9.67, x23 = 17, x34 = 4.33 and unit flow as 42.By Step4, only demand constraints are uncertain, and supply remains at certain values for model 2. At alpha level 0 on the cost function, WD is used for the demand constraint. Here, supply is a1=20, a2=16, a3=25, and demand is b1=11.1, b2=13.23, b3=17.07, b4=14.67, and we obtain the optimal transportation cost as 95.75 and x11 = 11.10, x14 = 8.9, x23 = 17.07, x34 = 5.77 and unit flow as 42.84.By Step5, both supply and demand constraints are uncertain for model 3. Here, supply is a1=20.67, a2=16.59, a3=24.45, and demand is b1=11.1, b2=13.23b3=17.07, b4=14.67. Similarly, we obtain the optimal transportation cost as 96.42 and x11 = 11.10, x14 = 9.57, x23 = 17.07, x34 = 5.1 and unit flow as 42.84. In the same manner, we have to solve different alpha levels for the 3 cases of constraints. Computational results for 3 models are shown in Table 4. Using Step 6, we can solve for different alpha levels.Table 4
Computational results for 3 models.
Optimization method (Lingo software)Model 1Model 2Model 3Optimal transportation cost93.6795.7596.42Units in flow4242.8442.84Three new models have been constructed for SFTPMC. While model 1 presents supply as a probabilistic value and demand as a certain value, model 2 presents demand as a probabilistic value and supply as a certain value, and model 3 presents supply and demand as probabilistic values. The cost coefficient of the objective function is transformed to alpha cut representation, and the stochastic mixed constraints are transformed to deterministic form by using the WD for solving the models. The fact that these models are constructed from different points of view is worth noting. Since the usage of the models is dependent on the DM’s preference, we cannot conclude which model is the best in the process of decision-making.


## 6. Sensitivity Analysis and Discussion

In this section, SFTPMC performed a SA of optimality in terms of variations in probabilities on uncertain parameters such as source and demand. For the SA, we used the model 3 problem and changed the probability from (0≤P≤1), where P is the probability on ai or bj. We analysed the problem by holding any one parameter of probabilities (pai or pbj) as constant at 0.5 and changed the probabilities of the other parameter of probabilities (pai or pbj). Using the remark (4.1.3), the probability of the particular parameter remains 0.5 for equality constraint. In optimal solutions for model 3 in each stochastic parameter and fuzzy cost, both transportation cost and total shipping units (Flow) were obtained and listed. To handle several objective functions in this SA, the Lingo software was employed to solve this optimization problem. The SA findings for the probability for demand bj are shown in Table 5. These analysis’s graphical representations for transportation cost and unit flow in relation to the probability for bj are shown in Figures 1 and 2. The transportation cost gradually increases with the probability for bj, as shown in Figure 1. It is worth noting that transportation costs are sensitive to variations in demand requirements probability. In Figure 2, the unit flow decreases gradually depending on variations in demand requirements probability. Similarly, we can find the SA for ai. The SA of the probability reveals some intriguing patterns. DM’s can get knowledge and capacity to develop the transportation system by gaining an understanding of the sensitivity patterns of probability for uncertain parameters.Table 5
Sensitivity analysis of SFTPMC by the varying probability ofpbj.
Probabilities forbjPbjProbabilitiesaipaiOptimal transportation costUnit flow0.990.599.6449.250.9598.8147.590.998.3846.730.8598.146.170.897.8945.740.7597.745.050.797.5445.050.6597.444.770.697.2644.370.5597.1444.250.597.0244.010.4596.943.770.496.7843.530.3596.6643.290.396.5443.050.2596.4243.410.296.2942.550.1596.1642.290.19641.970.0595.841.57Figure 1
SA of the optimal transportation cost.Figure 2
SA of the unit’s flow.The SA of SFTPMC is obtained by varying the probability of demand constraint in Table5, and it is also depicted using the graph, and the outcomes of optimal transportation cost and unit flow are shown in Figures 1 and 2.In this study, the proposed SFTPMC models obtain the best solutions in an unpredictable situation, according to the results. It is easy to contemplate more conservative decisions in extremely uncertain conditions. In the results, it is noticed that the varying probabilities forbj are different values of transportation cost and unit flow for this problem. It is common to move toward more conservative solutions when there is greater uncertainty in the optimization problem. It is observed that when modeling SFTPMC with the uncertainty introduced by the probability for ai and bj, more conservative solutions are selected as the optimal. This analysis, on the other hand, demonstrates the need of grasping the sensitivity of mixed constraints in the face of rising uncertainty. It helps a DM choose the right level of uncertainty for uncertain parameters.


## 7. Conclusions and Future Scope

This article describes a model for solving an SFTPMC that includes the cost coefficient of the objective function as a fuzzy number and probabilistic constraints accompanying the WD. The fuzzy objective value is converted into an equivalent deterministic objective function using alpha cut representation, and all stochastic constraints are converted into an equivalent deterministic constraint using WD. Three models, models 1, 2, and 3, of SFTPMC are also established, and the optimal solution for each model has been obtained by using Lingo software. A numerical example is given to show the performance of the models. The SA results are presented for model 3 supply and demand parameters. A pictorial representation (Figures1 and 2) of model 3 on SA is to illustrate the effect of the change in demand parameters on total transportation costs and unit flow. Likewise, perform SA for the supply parameters. In addition, this analysis demonstrates the need of grasping the sensitivity of mixed constraints in the face of rising uncertainty. It assists a DM to choose the right level of uncertainty for uncertain parameters. The computed results clearly indicate that the designed model is robust with respect to the different parameters. As this is the problem, SFTPMC plays a vital role in many cases of managerial decision-making situations such as the planning of many complex resource allocation problems in the areas of industrial production, in which demand and supply are random variables in nature. In such situations, this model will serve as an efficient tool for optimal planning.The limitations of the current study are presented. This research has considered an FTP with mixed constraints under a stochastic environment, where the single objective that is transportation cost is considered. In reality, the decision-making process handling with the complex organizational situation cannot depend exclusively on a single condition. So, we must understand the presence of numerous criteria that can improve multicriteria decision-making. In view of this, in our future study, we plan to investigate SFTPMC scenarios by considering multiobjective with multiitem parameters. Using the approach presented in this work, supply and demand variations can be considered in the economic order quantity model. In our forthcoming research, we plan to get real-world data from proper authorities and employ statistical regularity criteria to derive its probability distribution. In this case, the WD has a broad range of applications.



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