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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.16 No.4 pp.598-603
DOI : https://doi.org/10.7232/iems.2017.16.4.598

# A Possibility Approach to Solve Fuzzy Fixed Charge Transportation Problem

Houssine Tlig*, Adel Benhamed, Abdelwaheb Rebai*
Department of Quantitative Methods, Gabes University, Tunisia
Department of Quantitative Methods, Sfax University, Tunisia
Corresponding Author, Housni.tlig@gmail.com
20170111 20170526 20170731

## ABSTRACT

This paper studies the fixed charge transportation problem under uncertain environment, in which the capacities of sources, the direct costs, the fixed charges and the demands of destinations are not known with a precise manner. As a result, the transportation problem takes the form of fuzzy mixed-integer programming problem. In this paper, we propose a solution method based on the possibility approach. With this approach, the obtained transportation problem takes the form of a crisp mixed-integer linear programming problem and provides crisp values to different variables at different possibility levels. A numerical example with trapezoidal fuzzy parameters is given to demonstrate the effectiveness of the proposed method.

## 1.INTRODUCTION

The transportation problem is one of the most important areas of supply chain design that offers great potential to reduce costs. The fixed charge transportation problem (FCTP) is an extension of traditional transportation problem in which two kinds of constraints are taken into account (source and destination). The FCTP was introduced by Hirsch and Dantzi (1968). It consists to transport a homogeneous product from m sources to n destinations. The objective is to determinate the amounts of the product to be transported from all sources to all destinations such that the total transportation cost is minimized.

Several methods were developed in the literature to solve the FCTP under imprecise environment have been investigated. For example, Bit et al. (1993) applied fuzzy linear programming to the linear multiobjective solid transportation problem (STP). Li et al. (1997) proposed a neural network approach for multi-criteria (STP). Chanas and Machaj (1984) studied the transportation problems with fuzzy supplies and demands and solved them via the parametric programming approach. Chanas and Kuchta (1996) converted the fuzzy transportation problem a bicriterial problem with crisp objective function. Jimenez and Verdegay (1999) investigated the fuzzy solid transportation problem in which supplies, demands and conveyance capacities are represented by trapezoidal fuzzy numbers and applied a parametric approach for finding the fuzzy solution. Liu and Kao (2004) developed a method based on the extension principle to solve the FCTP when costs, supplies and demands are fuzzy numbers. Dinagar and Palanivel (2009) investigated the FTP when direct costs are trapezoidal fuzzy. Pandian and Natarajan (2010) proposed an algorithm called fuzzy zero point method for finding fuzzy optimal solution for FTP in which the transportation cost, supply and demand are represented by trapezoidal fuzzy numbers. Kumar and Kaur (2011) developed a technique based on fuzzy linear programming problem for finding the optimal solution of FTP. Gupta and Kumar (2012) proposed a method called Mehar’s method, to find the solution of the fully fuzzy multi objective transportation problems. Ebrahimnejad (2015) applied a fuzzy bounded dual algorithm for solving bounded transportation problems with fuzzy supplies and demands. Ebrahimnejad (2015) proposed a two-step method for solving FTP where all of the parameters are represented by non-negative triangular fuzzy numbers. Ebrahimnejad (2014) proposed a ranking function to solve FTP introduced by Kaur and Kumar (2012). Ojha et al. (2009) have studied solid transportation problem with general fuzzy costs and time. Molla-Alizadeh-Zavardehi and Tavakkoli- Moghaddam (2011) developed a technique based on artificial immune and genetic algorithms with a prufer number representation to solve a capacitated fixed-charge transportation problem. Lotfi and Tavakkoli-Moghaddam (2013) proposed genetic algorithm using priority-based encoding with new operators for fixed charge transportation problems.

In this paper, parallel to the chance constrained programming (CCP) approach to problems involving probabilistic uncertainty, a possibility approach is developed as a new way to treat fuzzy FCTP. This approach treats each constraint as a fuzzy event. Following this approach, the fuzzy FCTP is transformed into possibility mixed-integer programming problem. For the case of fuzzy parameters with trapezoidal membership functions, the possibility FCTP becomes a mixed-integer linear programming problem.

The rest of paper is organized as follows. Fuzzy FCTP problem is introduced in Section 2. Section 3 presents the possibility approach to solving fuzzy FCTP problem. In Section 4, by using the possibility approach, the fuzzy FSTP is transformed into possibility FCTP. Section 5 discusses a numerical experiment in determining the optimal transportation plan when trapezoidal fuzzy parameters are used in the possibility FCT. Finally, Section 6 concludes the paper, and discusses some future research directions.

## 2.MATHEMATICAL FORMULATION OF FUZZY FCTP

Generally, in balance FCTP, the sum of supplies and the sum of demands are supposed to be equal to each other. But in the real systems, the balance condition does not always holds. It suffices to suppose that there are enough products in the sources to satisfy the demand of each destination. In order to construct the mathematical model for the non-balance FCTP, we firstly introduce the following notations:

• $a ˜ i$ is the capacity at source i;

• $b ˜ j$ is the minimal demand of products in destination j;

• $c ˜ i j$ is the direct cost of unit product transported from source i to destination j.

• $w ˜ i j$ is the fixed charge with respect to transportation activity from source i to destination j.

• xij is the quantity transported from source i to destination j.

• yij is a binary variable and is defined as

$y i j = { 1 i f x i j > 0 0 f x i j = 0$

Mathematically a transportation problem can be stated as follows:

$min ∑ i = 1 m ∑ j = 1 n ( c ˜ i j x i j + w ˜ i j y i j ) S . T ∑ j = 1 n x i j ≤ a ˜ i , i = 1 , ⋯ m , ∑ = 1 m x i j ≥ b ˜ j , j = 1 , ⋯ n , x i j ≥ 0 a n d y i j = { 1 i f x i j > 0 0 i f x i j = 0$
(1)

## 3.POSSIBILITY APPROACH

Possibility theory was formulated in terms of fuzzy set theory by Zadeh (1978) and has been developed by many researchers. Zadeh suggested that fuzzy sets can be used as a basis for the theory of possibility similar to the way that measure theory provides the basis for the theory of probability. He introduced the “fuzzy variable”, which is associated with a possibility distribution in the same manner that a random variable is associated with a probability distribution. In the fuzzy linear programming model, each fuzzy coefficient can be viewed as a fuzzy variable and each constraint can be considered to be a fuzzy event. Using possibility theory, possibilities of fuzzy events (i.e., fuzzy constraints) can be determined.

Fuzzy event via possibility measure (Dubois and Prade, 1988).

Let $( Ω i , P ( Ω i ) , π i )$ , for each $i = 1 , 2 , ⋯ , n$, be a possibility space with Ωi being the nonempty set of interest, P(Ωi) the collection of all subsets of Ωi , and i π the possibility measure from P(Ωi) to [0,1] . Given a possibility space $( Ω i , P ( Ω i ) , π i )$ with $π ( ∅ ) = 0 , π ( Ω i ) = 1$ and $π ( ∪ i A i ) = sup i { π ( A i ) }$ with each $A i ∈ P ( Ω i )$ Zadeh defined a fuzzy variable, $r ˜$, as a real-valued function defined over Ωi with the membership function:

$μ r ˜ ( s ) = π ( { θ i ∈ Ω i / r ˜ ( θ i ) = s } ) = sup θ i ∈ Ω i { π ( { θ i } ) / r ˜ ( θ i ) = s } , s ∈ R$

Let $( Ω i , P ( Ω i ) , π i )$ be a product possibility space such that $Ω = Ω 1 × Ω 2 × ⋯ × Ω n$ and from possibility theory, $π ( A ) = min i = 1 , 2 , ⋯ , n { π ( A i ) / A = A 1 × A 2 × ⋯ × A n , A i ∈ P ( Θ i ) }$ Suppose $a ˜$ and $b ˜$ are two fuzzy variables on the possibility spaces $( Ω 1 , P ( Ω 1 ) , π 1 )$ and $( Ω 2 , P ( Ω 2 ) , π 2 )$, respectively. Then $m ˜ ≤ n ˜$ is a fuzzy event defined on the product possibility space $( Ω = Ω 1 × Ω 2 , P ( Ω ) , π )$, with:

$π ( m ˜ ≤ n ˜ ) = sup θ 1 , θ 2 { π { ( θ 1 , θ 2 ) / m ˜ ( θ 1 ) ≤ n ˜ ( θ 2 ) } , θ 1 ∈ Ω 1 , θ 2 ∈ Ω 2 } = sup θ 1 , θ 2 { min { π ( θ 1 ) , π ( θ 2 ) } / m ˜ ( θ 1 ) ≤ n ˜ ( θ 2 ) , θ 1 ∈ Ω 1 , θ 2 ∈ Ω 2 }$

Furthermore, from the definition of fuzzy variables, we have:

$π ( m ˜ ≤ n ˜ ) = sup s , t ∈ R { min { μ a ˜ ( s ) , μ b ˜ ( t ) } / s ≤ t }$

Similarly, possibilities of the fuzzy events $m ˜$ < $n ˜$ and $m ˜$ = $n ˜$ defined on the product possibility space (Ω, Ρ(Ω), π) are given as:

$π ( m ˜ < n ˜ ) = sup s , t ∈ R { min { μ m ˜ ( s ) , μ n ˜ ( t ) } / s < t } π ( m ˜ = n ˜ ) = sup s , t ∈ R { min { μ m ˜ ( s ) , μ n ˜ ( t ) } / s = t }$

When right hand side is crisp, the possibilities of the corresponding fuzzy events are given as:

$π ( m ˜ ≤ n ) = sup s ∈ R { μ m ˜ ( s ) / s ≤ n } π ( m ˜ < n ) = sup s ∈ R { μ m ˜ ( s ) / s < n } π ( m ˜ = n ) = μ m ˜ ( n )$

Let $m ˜ 1 , m ˜ 2 , ⋯ , m ˜ n$ be fuzzy variables and : $f j : R n → R$ be a real-valued function, for j = 1, …, m. The possibility of the fuzzy event $≪ f j ( m ˜ 1 , m ˜ 2 , ⋯ , m ˜ n ) ≤ 0 , j = 1 , ⋯ , m ≫$ is given by

$π ( f j ( m ˜ 1 , m ˜ 2 , ⋯ , m ˜ n ) ≤ 0 , j = 1 , ⋯ , m ) = sup s 1 , ⋯ , s n ∈ R { min 1 ≤ i ≤ n { μ m ˜ i ( s i ) } / f j ( s 1 , s 2 , ⋯ , s n ) ≤ 0 , j = 1 , ⋯ , m }$

Using the above concept, fuzzy transportation model can be transformed into possibility integer linear programming model.

## 4.THE PROPOSED POSSIBILITY FCTP

In this section, a method based on both possibility theory and chance-constrained programming (CCP) introduced by Charnes and Cooper (1959) is proposed to solve the fuzzy fixed charge transportation model in which constraints are considered as fuzzy events. This method can be summarized as follows.

• The CCP deals with uncertainty by specifying the desired levels of confidence with which the constraints in fuzzy FCTP hold.

• The possibility theory provides possibilities of fuzzy events (i.e., fuzzy constraints)

• Depending upon the membership of fuzzy capacities, fuzzy demands and fuzzy costs, the obtained possibilistic FCTP may be a linear mixed-integer programming model or a nonlinear mixed-integer programming model.

Using the concepts of CCP and possibility of fuzzy events, the fuzzy transportation model becomes the following possibility model:

$min x i j k , y i j k , f ¯ f s . t π ( ∑ i = 1 m ∑ j = 1 n ( c ˜ i j x i j + w ˜ y i j ) ≤ f ) ≥ β , π ( ∑ j = 1 n x i j ≤ a ˜ i ) ≥ α i , i = 1 , 2 , ⋯ , m π ( ∑ i = 1 m x i j ≥ b ˜ j ) ≥ λ j , j = 1 , 2 , ⋯ , n x i j k ≥ 0 , y i j k = { 1 i f x i j > 0 0 i f x i j = 0$
(2)

Where β, αi and λi are predefined acceptable levels of possibility for constraints (1) and (2) and (3), respectively. The interpretation of the fuzzy FCTP is that the objective value f should be the minimum Value that the return function $∑ i = 1 m ∑ j = 1 n ( c ˜ i j x i j + w ˜ i j y i j )$ can achieve with “possibility” level β or higher, subject to the possibility levels of constraints (2) and (3) being at least αi and λj respectively. In other words, at the optimal solution, we obtain the value of $∑ i = 1 m ∑ j = 1 n ( c ˜ i j x i j + w ˜ i j y i j )$ at least equal to f with possibility level β while at the same time all constraints are satisfied at the predefined possibility levels.

In this paper we used normal and convex fuzzy variables. These variables were defined by Zimmermann (1996) as follows.

Definition 1: (Normal fuzzy variables). Given a fuzzy variable $m ˜$ on a possibility space $( Ω , P ( Ω ) , π )$ the fuzzy variable $m ˜$ is normal if $sup μ m ˜ ( s ) = 1$

Definition 2: (Convex fuzzy variables). A fuzzy variable is convex $m ˜$ if:

$μ m ˜ ( λ s 1 + ( 1 − λ ) s 2 ) ≥ min ( μ m ˜ ( s 1 ) , μ m ˜ ( s 2 ) ) , s 1 , s 2 ∈ R , λ ∈ [ 0 , 1 ]$

Alternatively, the fuzzy variable $m ˜$ is convex if all α -level sets are convex.

Lemma 1: Let $m ˜ 1 , m ˜ 2 , ⋯ , m ˜ n$ be fuzzy variables with normal and convex membership functions. Let $( ⋅ ) α i L$ and $( ⋅ ) α i U$ denote the lower and upper bounds of the α -level set of , $m ˜ i , i = 1 , ⋯ , n$ . Then, for any given possibility levels $α 1 , α 2 , α 3$ with $0 ≤ α 1 , α 2 , α 3 ≤ 1$,

Proof.

First $π ( m ˜ 1 , m ˜ 2 + ⋯ + m ˜ n ≤ n ) ≥ α 1$ then

$sup s 1 , ⋯ , s n ∈ R { min 1 ≤ i ≤ n { μ m ˜ i ( s i ) } / ∑ i = 1 n s i ≤ n } ≥ α 1$

Assume that $s i * = arg sup { min 1 ≤ i ≤ n { μ m ˜ i ( s i ) } / ∑ i = 1 n s i ≤ n }$

It follows that $min { μ m ˜ i ( s i * ) } ≥ α 1 and ∑ i = 1 n s i * ≤ n$

Since $min { μ m ˜ i ( s i * ) } ≥ α 1$ implies that $μ m ˜ i ( s i * ) ≥ α 1$, for normal and convex fuzzy sets, we have

$s i * ∈ { ( m ˜ i ) α 1 L , ( m ˜ i ) α 1 U }$

Hence, with $∑ i = 1 n s i * ≤$ implies $∑ i = 1 n ( m ˜ i ) α 1 L ≤ n$.

Conversely, if $∑ i = 1 n ( m ˜ i ) α 1 L ≤ n$, there exists $α ′ 1$ with $α 1 ≤ α ′ 1 ≤ 1$ such that $∑ i = 1 n ( m ˜ i ) α ′ 1 L ≤ n$ and $μ m ˜ i ( m ˜ i ) α ′ 1 L ≥ α 1$.

This is equivalent to $min { μ m ˜ i ( m ˜ i ) α ′ 1 L } ≥ α 1$.

so $min { μ m ˜ i ( m ˜ i ) α ′ 1 L / ∑ i = 1 n ( m ˜ i ) α ′ 1 L ≤ n } ≥ α 1$.

Consequently,

$π ( ∑ i = 1 n m ˜ i ≤ n ) = sup s 1 , ⋯ , s n ∈ R { min 1 ≤ i ≤ n { μ m ˜ i ( s i ) } / ∑ i = 1 n s i ≤ n } ≥ α 1$

this proves case (i). cases (ii) and (iii) can be proved in the same way.

Given that fuzzy parameters in the FCTP are normal and convex, it follows from Lemma 1 that this transportation model can be solved by considering:

$min x i j , y i j , f f s . t ( c ˜ i j x i j + w ˜ i j y i j ) β U ≤ f , ∑ j = 1 n x i j ≤ ( a ˜ i ) α i U , i = 1 , 2 , ⋯ , m ∑ i = 1 m x i j ≤ ( b ˜ j ) λ j L , j = 1 , 2 , ⋯ , n x i j ≥ 0 and ​ y i j = { 1 i f x i j > 0 0 f x i j = 0$
(3)

Depending upon the membership functions of fuzzy parameters in the problem, the transportation model may take the form of an integer linear programming problem or an integer nonlinear programming model.

A special case: fuzzy FCTP with trapezoidal fuzzy parameters

From Lemma 1, for a trapezoidal fuzzy number $m ˜ i = ( ( m ˜ i ) 0 L , ( m ˜ i ) 1 L , ( m ˜ i ) 1 U ) , i = 1 , … , n$ and any given possibility levelα , 0 ≤α ≤ 1, the following are true:

$π ( m ˜ 1 + m ˜ 2 + ⋯ + m ˜ n ≤ n ) ≥ α if and only if ( 1 − α ) ( ( m ˜ 1 ) 0 L , ⋯ ⋯ , ( m ˜ n ) 0 L ) + α ( ( m ˜ 1 ) 1 L , ⋯ ⋯ , ( m n ) 1 L ) ≤ n , π ( m ˜ 1 + m ˜ 2 + ⋯ + m ˜ n ≤ n ) ≥ α if and only if ( 1 − α ) ( ( m ˜ 1 ) 0 U , ⋯ ⋯ , ( m ˜ n ) 0 U ) + α ( ( m ˜ 1 ) 1 U , ⋯ ⋯ , ( m ˜ n ) 1 U ) ≥ n , π ( m ˜ 1 + m ˜ 2 + ⋯ + m ˜ n = n ) ≥ α if and only if ( 1 − α ) ( ( m ˜ 1 ) 0 L , ⋯ ⋯ , ( m ˜ n ) 0 L ) + α ( ( m ˜ 1 ) 1 L , ⋯ ⋯ , ( m n ) 1 L ) ≤ n and ( 1 − α ) ( ( m ˜ 1 ) 0 U , ⋯ ⋯ , ( m ˜ n ) 0 U ) + α ( ( m ˜ 1 ) 1 U , ⋯ ⋯ , ( m n ) 1 U ) ≥ n$

Therefore, when all parameters of the problem are trapezoidal fuzzy numbers, the PFCP becomes the following linear integer programming model.(4)

$min x i j , y i j , f f ( 1 − β ) ( c ˜ i j k x i j k + w ˜ i j k y i j k ) 0 U + β ( c ˜ i j k x i j k + w ˜ i j k y i j k ) 1 U ≤ f ∑ j = 1 n x i j ≤ ( 1 − α i ) ( α ˜ i ) 0 U + α i ( α ˜ i ) 1 U , i = 1 , 2 , ⋯ , m ∑ i = 1 m x i j ≥ ( 1 − λ j ) ( b ˜ j ) 0 L + λ j ( b ˜ j ) 1 L , j = 1 , 2 , ⋯ , n x i j ≥ 0 and y i j = { 1 s i x i j > 0 0 s i x i j = 0$
(4)

## 5.NUMERICAL EXAMPLE

In this section, an example of phosphate transportation is given to demonstrate the possibility approach. In Tunisia, phosphate represents the main export commodity and the main source of state income. The quantities extracted from four mines (Metlaoui(S1), Moulareas (S2), Redeyef (S3) and Mdhilla (S4)) are transported to four production centers (Sfax (D1), Mdilla2 (D2), Gannouch (D3) and Skira (D4)).

Therefore, how to transport the phosphate from mines to the different production units economically is also an important issue in the phosphate industry. In this example, our goal is to make the transportation plan for the next month. At the beginning of this task, we need to obtain the basic data, such as supply capacity, demand, transportation direct and fixed cost of unit product, and so on. In fact, since the transportation plan is made in advance, we generally cannot get these data with precise manner.

So we assume that all parameters are trapezoidal fuzzy numbers as follows.

• fuzzy demands $b ˜ j$ and fuzzy capacities $a ˜ i$ (in million tons):

• Fuzzy direct costs and fuzzy fixed costs (in thousands of Tunisian dinars) per ton are listed in Tables 1, 2, 3 and 4.

In this example, all fuzzy constraints should be satisfied with same possibility level, i.e., $β = α i = λ j = h$. To obtain the optimal solution we used LINGO (14.0) software. We solved a mixed-integer linear programming at each possibility level (0, 0.25, 0.5, 0.75, 1).

The results for different possibility levels are provided in Table 5.

In the case of large problem, the decision maker can use general mixed-integer programming solution methods (e.g., the branch-and-bound and cutting plane methods).

The possibility approach provides the flexibility to decision makers to set their own acceptable (possibility) levels to obtain his optimal transportation plan. In a competitive, vague and uncertain environment, this approach should enhance the capability of decision makers to improve their operations.

## 6.CONCLUSION

This paper mainly investigated a fixed charge transportation problem in fuzzy environment. As a result, a decision method based on possibility theory is presented. This method permits to transform the fuzzy fixed charge solid transportation problem into a crisp integer programming problem. It makes the decision maker flexible when choosing the possibility level. In addition, when the parameters are trapezoidal fuzzy numbers, the crisp equivalent problem takes the form of a linear programming model and can be solved by standard integer liner programming algorithms.

Generally, fuzzy numbers are used to deal with the imprecision in transportation problems. We see that in more complex environments, intuitionistic fuzzy sets may be useful to minimize the imprecision in transportation problems.

## Table

fuzzy direct costs (from Si to D1 and D2)

fuzzy direct costs (from Si to D3 and D4)

fuzzy fixed costs (from Si to D1 and D2)

fuzzy fixed costs (from Si to D3 and D4)

Values of decision variables (in million tons) at different possibility levels

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