# An Uncertain Programming Model for Fixed Charge Transportation Problem with After-Sale Service

Authors: Kai Zhu; Yue Fan; Jiayu Shen; Yuyi Li; Mingfeng Yin
Journal: Journal of Mathematics
 (2022)

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## Abstract

This paper constructs an uncertain mathematical model for the fixed charge transportation problem in which several kinds of items are transported from different sources to various destinations via different methods. In addition, after-sale service such as product return and exchange that help to raise the customer’s satisfaction is considered during the trade to meet the reality. Meanwhile, some indeterministic factors may occur during the transportation process; probability theory is not the best option due to the lack of adequate historical data. In the light of this statement, some parameters like supplies, demands, return rate, and exchange rate are all defined as uncertain variables to formulate the uncertain programming model. After that, the equivalent deterministic models are derived with the help of uncertainty theory. Finally, some numerical experiments are implemented via three algorithms; the optimal values are displayed and compared to show the application of this problem. The study of this paper provides an applicable method for the decision maker.


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## Body

## 1. Introduction

Transportation has participated in human’s daily life for quite a long period, which establishes closer connections between different persons and locations, facilitating the globalization. Hitchcock [1] put forward a typical transportation problem (TP) for the first time, which was later discussed in [2], but only two constraints like availability and demand were put into consideration. After that, Arsham and Kahn [3] created a refined linear programming method for the transportation problems and Cooper and Cooper [4] designed a solution method for a series of concave transportation problems on the premise of separable objective function. However, the real transportation process is more complex; thus, more aspects such as transportation time, mode, capacity, and so on need to be discussed to make the problem closer to the reality. In the light of this statement, Das et al. [5] created a programming approach for a multiobjective TP from the perspective of decision maker, which was illustrated via three possible cases. Aneja and Nair [6] studied a double criterion TP with relevant objectives in practical situations and developed a nondominated solution to minimize the total cost. Sun [7] considered exclusionary side constraints for a TP and established a programming model that was settled by traditional algorithms to find feasible integer solutions.A special branch of the traditional TP is the fixed charge transportation problem (FCTP). Besides the direct transportation cost related to the product, an extra part called fixed charge based on the route independent of the transportation quantity is put into consideration. Moreover, in reality, many uncertain factors such as traffic flow or weather condition may occur during the transportation process, making the problems more complex. In this condition, this paper focuses on the fixed charge transportation problem under the uncertain environment, proposing the mathematical model and corresponding solution methods.


### 1.1. Literature Review

In the past several decades, many researchers have focused on the models and corresponding solution methods for various types of FCTPs. Since it was initiated in [8], many research studies began to study it with different considerations. Robers and Cooper [9] proposed a refined algorithm with adjacent extreme point technique to solve the FCTP. Farley and Richardson [10] studied a particular case of identical fixed charges during the transportation period. Sagratella et al. [11] introduced a noncooperative FCTP with several competing players and made discussions on three versions about it. A simple but effective heuristic procedure was designed in [12] to solve a series of small FCTPs, and the best solution was achieved via several steps. When coming to a two-stage FCTP that involved plants, customers, and depots, Calvete et al. [13] created an evolutionary algorithm-based algorithm to minimize the overall costs. A nonlinear factor was added for a FCTP with multiple transportation modes in crisp and interval environments [14], and a crossover and a mutation scheme was renewed for solving the numerical examples.The above researches about transportation problem are all within the deterministic framework, in which the parameters are deemed to be crisp values. However, indeterminacy tends to occur during the transportation process; thus, some parameters such as supplies, demands, and transportation time are not suitable to be discussed in indeterministic scenarios. Yang and Feng [15] studied a bicriteria solid TP in stochastic environment and constructed three mathematical models under different criteria for this problem. A stochastic multistage FCTP was designed in [16], where a time-limited demand was met by the producer. In 1965, Zadeh [17] created fuzzy set theory that was later applied in several aspects [18, 19], and many scholars made further investigations about FCTP in fuzzy environment. Yang and Liu [20] treated several parameters as fuzzy variables in a FCTP and studied the problem under three criteria with the help of credibility theory. A multiobjective FCTP with intuitionistic fuzzy data was studied in [21], which was then transformed into interval-valued problem. There exist more relevant research studies in indeterministic situations [22, 23].All these studies with stochastic or fuzzy variables make the problem more realistic, which are under the framework of probability theory and fuzzy theory. When some uncertain factors occur, the assumption of random or fuzzy distribution depends on the adequate sample size. However, there exist technical difficulties or economic reasons, which prevent sufficient experimental or observed data from being obtained. Under this situation, an alternative option is to rely on the belief degrees evaluated by domain experts. Nevertheless, there exists a big shortcoming in belief degree that some counterintuitive conclusions may occur in some occasions, which leads to improper assessments. In order to get rid of such problems, uncertainty theory [24] and its refined edition [25] came into being. Since then, it has gained its popularity in massive fields [26–30], in which transportation is an important section. A typical TP was discussed in [31] where truck times and transportation costs were supposed to follow uncertain distributions, and a multiobjective optimization model was created. For a typical Solid TP, two kinds of programming models under two criteria were constructed in [32] with the help of uncertainty theory. An uncertain multiobjective multi-item FCTP with budget constraint was studied in [33], and three different models under different criteria were designed. For other FCTPs, similar methods were applied in [34–38] with different applications.


### 1.2. Research Gap and the Proposed Contribution

As mentioned in Section1.1, there exist plenty of transportation problems that focus on several aspects such as transportation time, capacity, damage rate, and so on. However, in real-world scenarios, people will absolutely care about their benefits. For example, defective products may be sold to the customers due to defective rate, which would be unacceptable. At this moment, an exchange option will be made. Besides, impulsive purchasing always occurs around us, and product return policy must be available to guarantee the customer’s benefits according to relative regulations. Thus, under this circumstance, after-sale service such as item return and item exchange needs to been considered in order to facilitate the service quality. Based on the normal transportation problem, this paper studies a FCTP in which customer’s extra demand (return or exchange) is satisfied. As the manager has to afford the transportation cost, how to schedule a plan to minimize the cost becomes a challenge. Besides, in reality, their always exit some indeterministic factors during the transportation. To make this problem more realistic, this paper puts uncertain factors into this FCTP, establishing an uncertain programming model with the help of uncertainty theory, where after-sale service is particularly considered.Compared with the existing literature, the main contributions of this paper are organized as follows. This paper considers a FCTP with extra costs brought by after-sales services, and an uncertain model is built with some uncertain constraints. Then, the original programming model is extended under the expected value criterion and chance-constrained criterion, and corresponding deterministic forms are built with the help of uncertainty theory. After that, three kinds of algorithms are selected to solve the numerical examples, and their performances are compared. Moreover, we alter the original model by adding nonlinear elements to make further comparisons.


### 1.3. Managerial Insights Based on the Research

This paper considers the after-sale service in the transportation problem; the goal to minimize the total transportation cost is realized under the uncertain circumstance. On the one hand, after-sale service such as product return and exchange is essential, which helps to refine service quality and customer’s satisfaction. However, this part has not been considered yet. On the other hand, some indeterministic factors will occur everywhere, including the transportation process, causing the problem more complex. Generally, we prefer to describe the indeterminacy with probability. However, it is not always reasonable to use probability theory to illustrate all the indeterminacies as it requires adequate information. Actually, low frequency events will occur in our daily life; at this moment, uncertainty theory takes its effect. Under this circumstance, how to schedule a proper transportation plan to lower the total costs is a challenge. The study in this paper provides an applicable method for the decision maker so that he can deal with some uncertain factors without losing reliability from customers. This research could also be extended to other aspects in which decision/plan needs to be made within indeterministic condition.The structure of the remainder of this paper is organized as follows. Section2 illustrates some basic information about uncertainty theory, and Section 3 elaborates the transportation problem. In Section 4, expected value programming model and chance-constrained model are proposed, and their equivalent forms with determinacy are deducted in Section 5. Section 6 exhibits the applications of this uncertain model via several numerical experiments.


## 1.1. Literature Review

In the past several decades, many researchers have focused on the models and corresponding solution methods for various types of FCTPs. Since it was initiated in [8], many research studies began to study it with different considerations. Robers and Cooper [9] proposed a refined algorithm with adjacent extreme point technique to solve the FCTP. Farley and Richardson [10] studied a particular case of identical fixed charges during the transportation period. Sagratella et al. [11] introduced a noncooperative FCTP with several competing players and made discussions on three versions about it. A simple but effective heuristic procedure was designed in [12] to solve a series of small FCTPs, and the best solution was achieved via several steps. When coming to a two-stage FCTP that involved plants, customers, and depots, Calvete et al. [13] created an evolutionary algorithm-based algorithm to minimize the overall costs. A nonlinear factor was added for a FCTP with multiple transportation modes in crisp and interval environments [14], and a crossover and a mutation scheme was renewed for solving the numerical examples.The above researches about transportation problem are all within the deterministic framework, in which the parameters are deemed to be crisp values. However, indeterminacy tends to occur during the transportation process; thus, some parameters such as supplies, demands, and transportation time are not suitable to be discussed in indeterministic scenarios. Yang and Feng [15] studied a bicriteria solid TP in stochastic environment and constructed three mathematical models under different criteria for this problem. A stochastic multistage FCTP was designed in [16], where a time-limited demand was met by the producer. In 1965, Zadeh [17] created fuzzy set theory that was later applied in several aspects [18, 19], and many scholars made further investigations about FCTP in fuzzy environment. Yang and Liu [20] treated several parameters as fuzzy variables in a FCTP and studied the problem under three criteria with the help of credibility theory. A multiobjective FCTP with intuitionistic fuzzy data was studied in [21], which was then transformed into interval-valued problem. There exist more relevant research studies in indeterministic situations [22, 23].All these studies with stochastic or fuzzy variables make the problem more realistic, which are under the framework of probability theory and fuzzy theory. When some uncertain factors occur, the assumption of random or fuzzy distribution depends on the adequate sample size. However, there exist technical difficulties or economic reasons, which prevent sufficient experimental or observed data from being obtained. Under this situation, an alternative option is to rely on the belief degrees evaluated by domain experts. Nevertheless, there exists a big shortcoming in belief degree that some counterintuitive conclusions may occur in some occasions, which leads to improper assessments. In order to get rid of such problems, uncertainty theory [24] and its refined edition [25] came into being. Since then, it has gained its popularity in massive fields [26–30], in which transportation is an important section. A typical TP was discussed in [31] where truck times and transportation costs were supposed to follow uncertain distributions, and a multiobjective optimization model was created. For a typical Solid TP, two kinds of programming models under two criteria were constructed in [32] with the help of uncertainty theory. An uncertain multiobjective multi-item FCTP with budget constraint was studied in [33], and three different models under different criteria were designed. For other FCTPs, similar methods were applied in [34–38] with different applications.


## 1.2. Research Gap and the Proposed Contribution

As mentioned in Section1.1, there exist plenty of transportation problems that focus on several aspects such as transportation time, capacity, damage rate, and so on. However, in real-world scenarios, people will absolutely care about their benefits. For example, defective products may be sold to the customers due to defective rate, which would be unacceptable. At this moment, an exchange option will be made. Besides, impulsive purchasing always occurs around us, and product return policy must be available to guarantee the customer’s benefits according to relative regulations. Thus, under this circumstance, after-sale service such as item return and item exchange needs to been considered in order to facilitate the service quality. Based on the normal transportation problem, this paper studies a FCTP in which customer’s extra demand (return or exchange) is satisfied. As the manager has to afford the transportation cost, how to schedule a plan to minimize the cost becomes a challenge. Besides, in reality, their always exit some indeterministic factors during the transportation. To make this problem more realistic, this paper puts uncertain factors into this FCTP, establishing an uncertain programming model with the help of uncertainty theory, where after-sale service is particularly considered.Compared with the existing literature, the main contributions of this paper are organized as follows. This paper considers a FCTP with extra costs brought by after-sales services, and an uncertain model is built with some uncertain constraints. Then, the original programming model is extended under the expected value criterion and chance-constrained criterion, and corresponding deterministic forms are built with the help of uncertainty theory. After that, three kinds of algorithms are selected to solve the numerical examples, and their performances are compared. Moreover, we alter the original model by adding nonlinear elements to make further comparisons.


## 1.3. Managerial Insights Based on the Research

This paper considers the after-sale service in the transportation problem; the goal to minimize the total transportation cost is realized under the uncertain circumstance. On the one hand, after-sale service such as product return and exchange is essential, which helps to refine service quality and customer’s satisfaction. However, this part has not been considered yet. On the other hand, some indeterministic factors will occur everywhere, including the transportation process, causing the problem more complex. Generally, we prefer to describe the indeterminacy with probability. However, it is not always reasonable to use probability theory to illustrate all the indeterminacies as it requires adequate information. Actually, low frequency events will occur in our daily life; at this moment, uncertainty theory takes its effect. Under this circumstance, how to schedule a proper transportation plan to lower the total costs is a challenge. The study in this paper provides an applicable method for the decision maker so that he can deal with some uncertain factors without losing reliability from customers. This research could also be extended to other aspects in which decision/plan needs to be made within indeterministic condition.The structure of the remainder of this paper is organized as follows. Section2 illustrates some basic information about uncertainty theory, and Section 3 elaborates the transportation problem. In Section 4, expected value programming model and chance-constrained model are proposed, and their equivalent forms with determinacy are deducted in Section 5. Section 6 exhibits the applications of this uncertain model via several numerical experiments.


## 2. Preliminary

As the uncertain theory is utilized in this paper, some basic concepts about it should be introduced before establishing the mathematical model.Supposeℒ is a σ-algebra over a nonempty set Γ. Each event depicted by element Λ∈ℒ with a value of ℳΛ labels the chance that uncertain event Λ occurs. A set function ℳ from ℒ to 0,1 is called an uncertain measure if it satisfies the axioms of normality, duality, and subadditivity [24].Then, if a variable is treated as an uncertain one, its uncertainty distribution is defined as follows [25]:(1)ℳξ≤x=Φx.Example 1.
An uncertain variableξ follows a normal uncertainty distribution:(2)Φx=1+expπe−x3σ−1,x∈ℜ,where e and σ are real numbers with σ>0. Then, the variable ξ is named as a normal uncertain variable expressed as ξ∼Ne,σ.Example 2.
A variable is called linear if it follows linear uncertain distribution:(3)Φx=0,ifx≤a,x−ab−a,ifa≤x≤b,1,ifx≥b,where 0<a<b, and it is denoted as ξ∼ℒa,b.Definition 1.
(see [24]). An uncertain distribution Φx is said to be regular if its inverse function Φ−1x exists and is unique for each α∈0,1. Then, the inverse function Φ−1 is called the inverse uncertainty distribution of ξ.
Then, for the normal distributionξ∼Ne,σ in Example 1, we get its inverse distribution function as(4)Φ−1α=e+σ3πlnα1−α.
For the linear distributionℒa,b in Example 2, the corresponding inverse distribution function can be expressed as(5)Φ−1α=1−αa+αb.Definition 2.
(see [24]). Let ξ be an uncertain variable. Then, the expected value of ξ is defined by(6)Eξ=∫0+∞ℳξ≥rdr−∫−∞0ℳξ≤rdr,on the premise that at least one of the two integrals is finite.
For simplicity, the expected value is defined as follows [25]:(7)Eξ=∫01Φ−1αdα.Theorem 1 (see [25]).
Letξ1,ξ2,…,ξn be independent regular uncertain variables with uncertainty distributions Φ1,Φ2,…,Φn, respectively. If the function is strictly increasing with respect to x1,x2,…,xm, ξ=fξ1,ξ2,…,ξn is an uncertain variable with inverse uncertainty distribution Ψ−1α=fΦ1−1α,…,Φn−1α.Then, based on equation (7), the expected value of uncertain variable ξ=fξ1,ξ2,…,ξn can be described as follows [25]:(8)Eξ=∫01fΦ1−1α,…,Φn−1αdα,as long as Eξ exists.Theorem 2 (see [25]).
Ifξ and η are independent uncertain variables with finite expected values, we have Eaξ+bη=aEξ+bEη for any real numbers a and b.Based on the above theorems, some results related to uncertain distribution can be achieved [39].(9)Eξk=∫01Φ−1αkdα,Eξ−Eξk=∫0∞1−Φxk−Eξdx−∫−∞0Φxk+Eξdx.


## 3. Problem Description

### 3.1. Assumptions

A FCTP with after-sale service under uncertain environment is analyzed. The products are available at several sources, which need to be transported via various conveyances to different destinations. To make it closer to actuality, return and exchange policies are available for customers. The manager’s duty is to schedule a proper transportation plan that minimizes the total cost. To make the problem more explicit, we make the following assumptions:(i)
Items are carried from sources to destinations via different conveyances without any loss, which means no damageability will occur during the transportation.(ii)
Some customers plan to return or exchange the item after the purchase due to different reasons, which is fully supported by the policy. And the customer will receive the same amount of same item for exchange option.(iii)
All the returned items or exchanged items must be transported back to the manufacture at the retailer’s expense before sold for the second time.


### 3.2. Notations

The research is to schedule an optimal transportation plan under some constraints to make the total transportation cost lowest. The following notations are described from three aspects as indices, parameters, and decision variable.Indices:(10)i:index for sources where products are availablei=1,2,…,M,j:index for destinations where products are conveyedj=1,2,…,N,k:index for conveyances through which the products are transportedk=1,2,…,K.Parameters:(11)Ai:the maximum amount at sourcei,Bj:the minimum demand at destinationj,Vk:the maximum transportation capacity of conveyancek,λj:return rate at destinationjηj:exchange rate at destinationj,cijk:direct transportation cost from sourceito destinationjvia conveyancek,ej:direct transportation cost from destinationjback to the manufacture,fijk:the fixed charge from sourceito destinationjvia conveyancek.Decision variables:(12)xijk:the product qunantity carried from source it o destinationjvia conveyancek.


### 3.3. Model Formulation

In this problem, the manager needs to afford two kinds of costs. The first one happens from sources to destinations, which includes direct cost∑i=1M∑j=1N∑k=1Kcijkxijk and the fixed charge ∑i=1M∑j=1N∑k=1Kyijkfijk, where(13)yijk=1,ifxijk>0,0,otherwise,indicates that the manager has to pay for the fixed charge when conveyance k from source i to destination j is occupied. The second one is the cost from destinations back to the manufacturer.Based on the above assumptions and notations, the total transportation cost described as the objective function is(14)Z=∑i=1M∑j=1N∑k=1Kcijkxijk+∑i=1M∑j=1N∑k=1Kyijkfijk+∑j=1Nejλj+ηj∑i=1M∑k=1Kxijk=∑i=1M∑j=1N∑k=1Kcijkxijk+yijkfijk+ejλj+ηjxijk,subjected to(15)∑j=1n∑k=1lxijk≤Ai,(16)∑i=1m∑k=1lxijk≥1+ηjBj,(17)∑i=1m∑j=1n∑p=1qxijk≤Vk,(18)xijk≥0;yijk∈0,1,where i=1,2,…,M; j=1,2,…,N; k=1,2,…,K; and r=1,2,…,R, respectively.The total cost is defined in the objective function equation (14) when after-sale service is considered. Both returned items and exchanged items increase the cost for transportation back to the manufacture. Besides, constraint equation (15) means that the inventory Ai is sufficient to satisfy the total product amount carried from source i.As we have made the assumption that due to some different reasons, some customers will return or change the items that were purchase previously. For the returned item, there is no need to increase the product amount, but for the customer who wants to make an exchange, an extra demand is required. As shown in constraint equation (16), an additional amount over its least demand Bj should be put into consideration based on the exchange rate.Constraint equation (17) implies that the capacity Vk of conveyance k should be no less than the total transportation amount.Formula (18) denotes the variables definition constraints.


## 3.1. Assumptions

A FCTP with after-sale service under uncertain environment is analyzed. The products are available at several sources, which need to be transported via various conveyances to different destinations. To make it closer to actuality, return and exchange policies are available for customers. The manager’s duty is to schedule a proper transportation plan that minimizes the total cost. To make the problem more explicit, we make the following assumptions:(i)
Items are carried from sources to destinations via different conveyances without any loss, which means no damageability will occur during the transportation.(ii)
Some customers plan to return or exchange the item after the purchase due to different reasons, which is fully supported by the policy. And the customer will receive the same amount of same item for exchange option.(iii)
All the returned items or exchanged items must be transported back to the manufacture at the retailer’s expense before sold for the second time.


## 3.2. Notations

The research is to schedule an optimal transportation plan under some constraints to make the total transportation cost lowest. The following notations are described from three aspects as indices, parameters, and decision variable.Indices:(10)i:index for sources where products are availablei=1,2,…,M,j:index for destinations where products are conveyedj=1,2,…,N,k:index for conveyances through which the products are transportedk=1,2,…,K.Parameters:(11)Ai:the maximum amount at sourcei,Bj:the minimum demand at destinationj,Vk:the maximum transportation capacity of conveyancek,λj:return rate at destinationjηj:exchange rate at destinationj,cijk:direct transportation cost from sourceito destinationjvia conveyancek,ej:direct transportation cost from destinationjback to the manufacture,fijk:the fixed charge from sourceito destinationjvia conveyancek.Decision variables:(12)xijk:the product qunantity carried from source it o destinationjvia conveyancek.


## 3.3. Model Formulation

In this problem, the manager needs to afford two kinds of costs. The first one happens from sources to destinations, which includes direct cost∑i=1M∑j=1N∑k=1Kcijkxijk and the fixed charge ∑i=1M∑j=1N∑k=1Kyijkfijk, where(13)yijk=1,ifxijk>0,0,otherwise,indicates that the manager has to pay for the fixed charge when conveyance k from source i to destination j is occupied. The second one is the cost from destinations back to the manufacturer.Based on the above assumptions and notations, the total transportation cost described as the objective function is(14)Z=∑i=1M∑j=1N∑k=1Kcijkxijk+∑i=1M∑j=1N∑k=1Kyijkfijk+∑j=1Nejλj+ηj∑i=1M∑k=1Kxijk=∑i=1M∑j=1N∑k=1Kcijkxijk+yijkfijk+ejλj+ηjxijk,subjected to(15)∑j=1n∑k=1lxijk≤Ai,(16)∑i=1m∑k=1lxijk≥1+ηjBj,(17)∑i=1m∑j=1n∑p=1qxijk≤Vk,(18)xijk≥0;yijk∈0,1,where i=1,2,…,M; j=1,2,…,N; k=1,2,…,K; and r=1,2,…,R, respectively.The total cost is defined in the objective function equation (14) when after-sale service is considered. Both returned items and exchanged items increase the cost for transportation back to the manufacture. Besides, constraint equation (15) means that the inventory Ai is sufficient to satisfy the total product amount carried from source i.As we have made the assumption that due to some different reasons, some customers will return or change the items that were purchase previously. For the returned item, there is no need to increase the product amount, but for the customer who wants to make an exchange, an extra demand is required. As shown in constraint equation (16), an additional amount over its least demand Bj should be put into consideration based on the exchange rate.Constraint equation (17) implies that the capacity Vk of conveyance k should be no less than the total transportation amount.Formula (18) denotes the variables definition constraints.


## 4. Uncertain Mathematical Models

As mentioned above, when sufficient historical data are accessible, we can easily calculate the distributions of the different parameters. However, the actuality is not always as what we expect. In this problem, we regard availabilityAi, demand Bj, transportation capacity Vk, return rate λj, exchange rate ηj, unit transportation cost cijk, and fixed charge fijk as uncertain variables and the rest as deterministic ones. To make the problem more clear, the uncertain parameters are rewritten in the uncertain forms as A˜i, B˜j, V˜k, λ˜j, η˜j, c˜ijk, and f˜ijk. Then, the problem description could be altered to(19)minZs.t.∑j=1N∑k=1Kxijk≤A˜i∑i=1M∑k=1Kxijk≥1+η˜jB˜j∑i=1M∑j=1Nxijk≤V˜k,where(20)Z=∑i=1M∑j=1N∑k=1Kc˜ijkxijk+yijkf˜ijk+ejλ˜j+η˜jxijk.Definition 3.
A feasible solutionx∗ is called the expected optimal solution if(21)Efx∗≥Efx,holds for any feasible solution x, where f represents the objective function.
This criterion aims to minimize the expected value of objective function within several constraints. Thus, the corresponding model of equation (19) under this criterion could be described as follows:(22)minEZs.t.E∑j=1N∑k=1Kxijk−A˜i≤0,i=1,2,…,ME∑i=1M∑k=1Kxijk−1+η˜jB˜j≥0,j=1,2,…,NE∑i=1M∑j=1Nxijk−V˜k≤0,k=1,2,…,K.
From another point of view, in reality, the manager will consider the risk when making transportation plan, which means the manager has to schedule an optimal transportation plan on the premise of a predetermined upper bound. Under this circumstance, when a confidence levelα∈0,1 is given, the manager is willing to achieve a smallest value f¯ such that the uncertain variable Efx∗ is not more than f¯, which leads to the following chance-constrained criterion.Definition 4.
A feasible solutionx∗ is called α-optimal solution if(23)minf¯|ℳfx∗≤f¯≥α≤maxf¯|ℳfx≤f¯≥α,holds for any feasible solution x, where α is a confidence level.
Thus, with the different confidence levelsα, the model of equation (19) under this case could be altered to(24)minZ¯s.t.ℳZ≤Z¯≥αℳ∑j=1N∑k=1Kxijk≤A˜i≥αi,i=1,2,…,Mℳ1+η˜jB˜j≤∑i=1M∑k=1Kxijk≥βj,j=1,2,…,Nℳ∑i=1M∑j=1Nxijk≤V˜k≥γk,k=1,2,…,K,where α, αii=1,2,…,M, βjj=1,2,…,N, and γkk=1,2,…,K are prespecified confidence levels.


## 5. Deterministic Transformations

Obviously, there exist some uncertain variables that will bring difficulty for the computing and simulation. To simplify the calculation process, a natural option is to transform the uncertain models into deterministic ones, which is fully supported by the uncertain theory. The transformation of the equivalent deterministic forms is explicated in this section.Theorem 3.
We assume that the independent uncertain variablesA˜i, B˜j, V˜k, λ˜j, η˜j, c˜ijk, and f˜ijk have corresponding uncertainty distributions as ΦA˜i,ΦB˜j , ΦV˜k, Φλ˜j, Φη˜j, Φc˜ijk, and Φf˜ijk, respectively, and then the model in equation (22) will have the following deterministic description:(25)min∑i=1M∑j=1N∑k=1Kxijk∫01Φc˜ijk−1αdα+yijk∫01Φf˜ijk−1αdα+ej∫01Φλ˜j−1αdα+∫01Φη˜j−1αdαxijk,(26)s.t.∑j=1N∑k=1Kxijk−∫01ΦA˜i−11−αdα≤0,(27)∫011+Φη˜j−1αΦB˜j−1αdα−∑i=1M∑k=1Kxijk≤0,(28)∑i=1M∑j=1Nxijk−∫01ΦV˜k−11−αdα≤0,(29)i=1,2,…,M;j=1,2,…,N;k=1,2,…,K,where ΦA˜i−1,ΦB˜j−1 , ΦV˜k−1, Φλ˜j−1, Φη˜j−1, Φc˜ijk−1, and Φf˜ijk−1 are the inverse distributions of uncertain variables A˜i, B˜j, V˜k, λ˜j, η˜j, c˜ijk, and f˜ijk, respectively.Proof.
According to equation (8) and Theorem 2, in the objective function equation (22), we have(30)E∑i=1M∑j=1N∑k=1Kc˜ijkxijk=∑i=1M∑j=1N∑k=1Kxijk∫01Φc˜ijk−1αdα,and similarly, for the second part, we can get(31)E∑i=1M∑j=1N∑k=1Kyijkf˜ijk=∑i=1M∑j=1N∑k=1Kyijk∫01Φf˜ijk−1αdα,(32)E∑i=1M∑j=1N∑k=1Kejλ˜j+η˜jxijk=∑i=1M∑j=1N∑k=1K∫01Φλ˜j−1αdα+∫01Φη˜j−1αdαxijk.
With the combination of equations (30)–(32), the transformation of objective function in expect value case is completed.
Also, we can demonstrate that the constraints can be converted as equations (26)–(28) in the same way. Then, Theorem 3 is proved.Lemma 1.
Letξ1,ξ2,…,ξn be independent uncertain variables with regular uncertainty distributions Φ1,Φ2,…,Φn, respectively. Assume fξ1,ξ2,…,ξn is strictly increasing, then(33)ℳfξ1,ξ2,…,ξn≤f¯≥α,which is equivalent to(34)fΦ1−1α,…,Φn−1α≤f¯,where α is a given confidence level.Theorem 4.
Based on the assumptions in Theorem3, the equivalent expression of the model in equation (24) is(35)minΦZ−1α,(36)s.t.∑j=1N∑k=1Kxijk−ΦA˜i−11−αi≤0,(37)1+Φη˜i−1βjΦB˜j−1βj−∑i=1M∑k=1Kxijk≤0,(38)∑i=1M∑j=1Nxijk−ΦV˜k−11−γk≤0,(39)i=1,2,…,M;j=1,2,…,N;k=1,2,…,K,where(40)ΦZ−1α=∑i=1M∑j=1N∑k=1KΦc˜ijk−1αxijk+yijkΦf˜ijk−1α+ejΦλ˜j−1α+Φη˜j−1αxijk,is the inverse distribution of ΦZ and α, αi, βj, and γk are preset confidence levels.Proof.
Based on Lemma1, ℳZ≤Z¯≥α is equivalent to(41)ΦZ−1α≤Z¯,and the objective function in equation (24) can be altered to(42)minZ¯s.t.ΦZ−1α≤Z¯,which can also be converted to(43)minΦZ−1α.
Then, due to Theorem1 and the independence of all the uncertain variables, the expression of ΦZ−1α can be easily inferred as equation (40).
Similarly, the remaining constraints in equation (24) can be converted to equations (36)–(38). Then, the proof of Theorem 4 is eventually finished.


## 6. Numerical Experiment

In order to explicate this kind of FCTP, a numerical experiment is designed in this section. We make an assumption that there exist three origins, three destinations, and two conveyances (truck and train), and then the uncertain model in equation (19) could be detailed as(44)min∑i=13∑j=13∑k=12c˜ijkxijk+yijkf˜ijk+ejλ˜j+η˜jxijks.t.∑j=13∑k=12xijk≤A˜i∑i=13∑k=12xijk≥1+η˜jB˜j∑i=13∑j=13xijk≤V˜k.The parameters are detailed in Tables1–5.Table 1
Supplies and demands.
Supplies at originsDemands at destinationsA˜1∼N80,3B˜1∼N52,1A˜2∼N95,2B˜2∼N45,2A˜3∼N69,2B˜3∼N40,1Table 2
Return rate and exchange rate.
Return rateExchange rateλ˜1∼ℒ0.05,0.07η˜1∼ℒ0.03,0.05λ˜2∼ℒ0.06,0.11η˜2∼ℒ0.02,0.06λ˜3∼ℒ0.04,0.09η˜3∼ℒ0.04,0.08Table 3
The maximum capacity.
V˜1∼N120,1V˜2∼N113,2Table 4
Direct transportation costc˜ijk and fixed charge f˜ijk.
k12i/j123123c˜ijk1ℒ0.32,0.41ℒ0.26,0.31ℒ0.36,0.52ℒ0.26,0.48ℒ0.31,0.35ℒ0.22,0.282ℒ0.40,0.46ℒ0.39,0.55ℒ0.22,0.30ℒ0.40,0.55ℒ0.33,0.40ℒ0.32,0.373ℒ0.30,0.40ℒ0.30,0.55ℒ0.42,0.45ℒ0.27,0.31ℒ0.23,0.25ℒ0.40,0.52f˜ijk1ℒ3,5ℒ6,9ℒ7,9ℒ4,8ℒ2,7ℒ4,82ℒ5,6ℒ8,10ℒ6,10ℒ6,10ℒ3,9ℒ5,63ℒ5,7ℒ6,7ℒ7,10ℒ5,8ℒ5,6ℒ7,9Table 5
Direct transportation costej back to the manufacture.
Transportation coste1=0.25e2=0.31e3=0.42As the transportation problem constructed in this paper belongs to a NP-hard problem, intelligent algorithms need to be utilized to implement the simulations. For simplicity, three strategies are adopted and tested in Matlab environment. The first one is the traditional interior point method (IPM), and the two other intelligent algorithms are genetic algorithm (GA) and particle swarm optimization algorithm (PSO). The corresponding flow charts of GA and PSO are shown in Figures1 and 2. Besides, the parameters are set as follows. GA: maxgeneration = 500, Popsize = 50, crossover probability Pc = 0.8, and mutation probability Pm = 0.2; PSO: Popsize = 40, maxgeneration = 1000, and c1=c2=2.Figure 1
Flow chart of GA.Figure 2
Flow chart of PSO.Under the expected value criterion, the optimal results obtained by GA and PSO are displayed in Table6. Here, “AL” (Algorithm) represents the GA or PSO algorithm, “Optimum solution” explicates the optimal decision variables xijk that lead to the total minimum transportation cost “OV” (Objective value), and “TA” (Total amount) indicates the whole transportation amount.Table 6
Optimal results of the expected value model.
ALOVTAOptimum solutionIPM144.7967146.498110.4891, 10.3917, 13.7854, 9.2263, 0, 16.6921, 5.1146, 0.4321, 06.7833, 17.1350, 8.7592, 11.6155, 17.0715, 1.1317, 17.8708, 0, 0GA165.3065146.88949.0120, 9.5263, 7.8053, 7.8889, 8.0551, 13.8736, 9.5404, 9.0114, 16.00300, 0, 6.6735, 9.0118, 9.0116, 9.1580, 8.3339, 6.6843, 7.3004PSO173.6376146.49869.0461, 13.5694, 26.2959, 10.0123, 4.5613, 7.7789, 1.0765, 8.1923, 0.62760, 10.7862, 9.8593, 2.8752, 20.3554, 6.4939, 5.3677, 2.4219, 7.1787The corresponding evolution process of GA and PSO is revealed in Figures3 and 4, respectively.Figure 3
The evolution process of GA.Figure 4
The evolution process of PSO.We can learn from Table6 that under the expected value criterion, IPM performs best while PSO loses its advantage. To reveal their effectiveness further, the three algorithms are tested for the chance-constrained case with different confidence levels. For simplicity, α, αi, βj, and γk are deemed to be identical, and the optimal results for the three methods are elaborated in Table 7.Table 7
Optimal results of the chance-constrained model.
αALOVTAOptimum solution0.95IPM173.9528152.165911.8470, 6.0939, 14.2020, 10.9980, 0, 18.2653, 6.2078, 0, 0,5.9004, 16.0804, 9.1164, 11.8861, 20.2160, 0, 19.9447, 1.4078, 0GA203.3171152.262910.2387, 0, 8.2735, 8.2736, 7.3241, 7.3596, 10.0567, 13.4528, 8.27359.1262, 7.3240, 7.3770, 8.9023, 13.6982, 8.8244, 8.2733, 7.8315, 7.6536PSO205.1389152.21006.7949, 8.4686, 17.6868, 8.7530, 3.0638, 9.5009, 6.0860, 8.9347, 010.4751, 10.5855, 8.0048, 9.7952, 16.1715, 0.0316, 14.0991, 7.1646, 6.59390.9IPM169.1728150.187511.5477, 6.3658, 13.2551, 11.2050, 0, 15.9906, 6.3572, 0, 07.4859, 15.4963, 9.7381, 11.9720, 19.5227, 0, 18.1324, 3.1183, 0GA197.7742150.18518.9181, 0, 7.5977, 7.5987, 7.2601, 6.9119, 8.9181, 9.1070, 8.84697.5987, 6.9122, 8.3678, 8.9181, 19.9036, 8.0825, 10.3531, 7.3877, 7.5028PSO198.7735150.18758.5525, 6.8108, 17.2381, 10.1274, 6.5618, 14.0327, 5.7079, 8.7489, 07.6798, 17.1644, 6.2950, 9.8664, 16.0792, 0, 15.0332 0.2170, 0.07250.85IPM162.0722148.878311.0557, 6.8205, 13.7784, 10.6551, 0, 15.7933, 7.0420, 1.1576, 07.4030, 15.1529, 9.2647, 11.7167, 17.6529, 0, 17.6095, 3.7762, 0GA193.0846148.87618.2291, 9.5366, 7.7029, 8.5107, 7.0891, 7.6889, 10.6834, 8.9365, 09.3323, 7.7720, 7.2546, 9.8298, 8.2291, 15.4059, 8.4937, 6.7206, 7.4610PSO196.3841148.878411.1494, 8.2094, 8.6612, 7.7997, 5.8796, 16.0841, 8.8765, 6.3582, 07.3664, 17.0024, 4.3000, 10.1840, 10.6677, 7.1591, 18.4595, 0.7094, 0.0116Table7 reveals that IPM is prior in this situation, which is in accordance with the result for the expected value model case. Besides, with the decrement of α, the total transportation cost shows a decreasing trend as well. The reason is that as the confidence level decreases, the corresponding uncertain variable’s value lowers too, leading to less total transportation cost. More simulations under different confidence levels are implemented for the chance-constrained case, with the sensitivity analysis explicated in Figure 5.Figure 5
Sensitivity analysis of the chance-constrained model.Figure5 illustrates that the objective value reveals increasing trend with the increment of confidence level α. For the chance-constrained model, its feasible domain will alter according the confidence level and the higher confidence level means more requirements should be satisfied, leading to higher transportation cost. This trend also obeys the common sense because when the constraints become more elastic, less cost is required, which is also reflected in the simulation. Besides, the optimal results gained via IPM are always most competitive, which is similar to the previous conclusion. To verify their effectiveness in a relatively complex situation, we add a nonlinear factor to the model in equation (44) where the objective function is modified to(45)min∑i=13∑j=13∑k=12c˜ijkxijk2+yijkf˜ijk+ejλ˜j+η˜jxijk,while other parameters or settings keep unchanged. For the chance-constrained case, the optimal values gained via the three methods are explicated in Figure 6.Figure 6
Sensitivity analysis of the chance-constrained model with nonlinear factor.The simulation results displayed in 6 demonstrate that IPM still performs best in the nonlinear situation. The above various simulation results indicate the effectiveness of the different solving algorithms in searching for the optimal solutions, making this problem applicable in the decision-making scenario.


## 7. Conclusions

This paper studies an uncertain FCTP in which after-sale service as return and exchange policies are guaranteed to meet the customer’s demand and raise the service quality. To make the problem more realistic, some indeterministic factors such as supplies, demands, and capacities are considered. When the sample size is large enough, it is possible for us to believe that the estimated probability distribution is close enough to the long-run cumulative frequency. However, when sufficient historical data are unavailable, uncertainty theory is the best option. Based on uncertainty theory, some parameters are supposed to follow uncertain distributions. Then, the basic model is evolved to two other uncertain models in the expected value criterion and chance-constrained criterion, respectively, and their deterministic forms are achieved via uncertainty inverse distribution method. After that, three algorithms such as IPM, GA, and PSO are utilized to solve the numerical examples, and the corresponding results are displayed in figures. Besides, a modified objective function is proposed for further tests of the three solving methods, and corresponding sensitivity analyses are made. The comparison results reveal that the IPM method provides best performance while the others are less competitive. Moreover, in chance-constrained case, the objective function is nondecreasing with increment of confidence levels, which is in accordance with the theoretical analysis.Compared with the fixed charge transportation problem in this paper, more aspects could be considered to make it more realistic. For example, product damage may occur during the transportation and it varies a lot according to the product variety. To meet the demand, more products should be conveyed. Also, more constraints such as time limit, time window, and price limitation can be supplemented, which are common in daily transportation and trade. From another point of view, profit-oriented problem is a nice topic. For example, for a particular product, there are always different brands with several different packaging amounts, and the prices are relative to the quality. For the customers, their special demands and purchasing powers will lead to different options, so how to make a proper schedule to maximize the profit is a challenge. Simultaneously, customers are generally price-sensitive, and a transportation problem with two decision variables can be studied.Based on the study of this paper, more applications could be considered. In task allocation, the same method can be utilized to minimize the total cost. When disassembling some equipment, how to make a proper disassembly process to save time is a challenge. When indeterminacy is put into consideration, all the problems related to schedule or plan can be dealt in a similar way. Also, multiobjective function with more constraints can be studied to make the problem more applicable, providing a more effective way for the decision maker.



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