A Compensatory Approach to Multiobjective Linear Transportation Problem with Fuzzy Cost Coefficients

Abstract

This paper deals with the Multiobjective Linear Transportation Problem that has fuzzy cost coefficients. In the solution procedure, many objectives may conflict with each other; therefore decision-making process becomes complicated. And also due to the fuzziness in the costs, this problem has a nonlinear structure. In this paper, fuzziness in the objective functions is handled with a fuzzy programming technique in the sense of multiobjective approach. And then we present a compensatory approach to solve Multiobjective Linear Transportation Problem with fuzzy cost coefficients by using Werner's 𝜇_and operator. Our approach generates compromise solutions which are both compensatory and Pareto optimal. A numerical example has been provided to illustrate the problem.

1. Introduction

The classical transportation problem (TP) is a special type of linear programming problem, and it has wide practical applications in manpower planning, personnel allocation, inventory control, production planning, and so forth. TP aims to find the best way to fulfill the demand of 𝑛 demand points using the capacities of 𝑚 supply points. In a single objective TP, the cost coefficients of the objective express commonly the transportation costs. But in real-life situations, it is required to take into account more than one objective to reflect the problem more realistically, and thus multiobjective transportation problem (MOTP) becomes more useful. These objectives can be quantity of goods delivered, unfulfilled demand, average delivery time of the commodities, reliability of transportation, accessibility to the users, and product deterioration. Also in practice, the parameters of MOTP (supply&demand quantities and cost coefficients) are not always exactly known and stable. This imprecision may follow from the lack of exact information, changeable economic conditions, and so forth. A frequently used way of expressing the imprecision is to use the fuzzy numbers. It enables us to consider tolerances for the model parameters in a more natural and direct way. Therefore, MOTP with fuzzy parameters seems to be more realistic and reliable.

A lot of researches have been conducted on MOTP with fuzzy parameters. Hussein [1] dealt with the complete solutions of MOTP with possibilistic coefficients. Das et al. [2] focused on the solution procedure of the MOTP where all the parameters have been expressed as interval values by the decision maker. Ahlatcioglu et al. [3] proposed a model for solving the transportation problem whose supply and demand quantities are given as triangular fuzzy numbers bounded from below and above, respectively. Basing on extension principle, Liu and Kao [4] developed a procedure to derive the fuzzy objective value of the fuzzy transportation problem where the cost coefficients, supply and demand quantities are fuzzy numbers. Using signed distance ranking, defuzzification by signed distance, interval-valued fuzzy sets, and statistical data, Chiang [5] gets the transportation problem in the fuzzy sense. Ammar and Youness [6] examined the solution of MOTP which has fuzzy cost, source and destination parameters. They introduced the concepts of fuzzy efficient and 𝛼-parametric efficient solutions. Islam and Roy [7] dealt with a multiobjective entropy transportation problem with an additional delivery time constraint, and its transportation costs are generalized trapezoidal fuzzy numbers. Chanas and Kuchta [8] proposed a concept of the optimal solution of the transportation problem with fuzzy cost coefficients and an algorithm determining this solution. Pramanik and Roy [9] showed how the concept of Euclidean distance can be used for modeling MOTP with fuzzy parameters and solving them efficiently using priority-based fuzzy goal programming under a priority structure to arrive at the most satisfactory decision in the decision making environment, on the basis of the needs and desires of the decision making unit.

In this paper, we focus on the solution procedure of the multiobjective linear Transportation Problem (MOLTP) with fuzzy cost coefficients. We assume that the supply and demand quantities are precisely known. And the coefficients of the objectives are considered as trapezoidal fuzzy numbers. The fuzziness in the objectives is handled with a fuzzy programming technique in the sense of multiobjective approach [10]. And a compensatory approach is given by using Werner’s 𝜇_and operator.

This paper is organized as follows. Section 2 presents brief information about fuzzy numbers. Section 3 contains the MOLTP formulation with fuzzy cost coefficients and some basic definitions about multiobjective optimization. Section 4 introduces the compensatory fuzzy aggregation operators briefly. Section 5 explains our methodology using Werners’ compensatory “fuzzy and’’ operator. Section 6 gives two illustrative numerical examples. Finally, Section 7 includes some results.

2. Fuzzy Preliminaries

In this paper, we assumed that the fuzzy cost coefficients are trapezoidal fuzzy numbers. In this section, brief information about the fuzzy numbers especially trapezoidal fuzzy numbers is presented. For more detailed information, the reader should check [11, 12].

Definition 2.1. A fuzzy number 𝑀 is an upper semicontinuous normal and convex fuzzy subset of the real line 𝑅.

Definition 2.2. A fuzzy number 𝑀=(𝑚₁,𝑚₂,𝑚₃,𝑚₄) is said to be a trapezoidal fuzzy number (TFN) if its membership function is given by 𝜇_𝑀(𝑥)=⎧⎪⎪⎪⎨⎪⎪⎪⎩0,𝑥<𝑚¹𝑥−𝑚₁,𝑚₂−𝑚₁1,,𝑚𝑚₁₂≤𝑥<𝑚₂,𝑚₄≤𝑥≤𝑚₃,−𝑥𝑚₄−𝑚₃0,,𝑚𝑥₃><𝑚𝑥₄≤𝑚₄,,(2.1) where 𝑚₁,𝑚₂,𝑚₃,𝑚₄∈𝑅 and 𝑚₁≤𝑚₂≤𝑚₃≤𝑚₄. The figure of the fuzzy number 𝑀 is given in Figure 1. It should be noted that a triangular fuzzy number is a special case of a TFN with 𝑚₂=𝑚₃. In the literature, a TFN can also be represented with the ordered quadruplets (𝑚₂,𝑚₃,𝑚₂−𝑚₁,𝑚₄−𝑚₃). Here 𝑚₂−𝑚₁ is called as left spread of 𝑀, where 𝑚₄−𝑚₃ is right spread. In this paper, we use 𝑀=(𝑚₁,𝑚₂,𝑚₃,𝑚₄) fuzzy number notation and called these ordered elements as characteristic points of 𝑀.

Figure 1 

The membership function of 𝑀.

Some algebraic operations on TFNs that will be used in this paper are defined as follows.

Let 𝑎̃=(𝑎₁,𝑎₂,𝑎₃,𝑎₄) and 𝑏̃=(𝑏₁,𝑏₂,𝑏₃,𝑏₄) be TFNs.

(i) Addition:
𝑎̃+𝑏̃=𝑎₁+𝑏₁,𝑎₂+𝑏₂,𝑎₃+𝑏₃,𝑎₄+𝑏₄.(2.2)

(ii) Multiplication with a Positive Crisp Number (𝑘>0):
𝑘⋅𝑎̃=𝑘𝑎₁,𝑘𝑎₂,𝑘𝑎₃,𝑘𝑎₄.(2.3) One convenient approach for solving the fuzzy linear programming problems is based on the concept of comparison of fuzzy numbers by use of ranking functions.

Definition 2.3. The set of elements that belong to the fuzzy number 𝑎̃ at least to the degree 𝛼 is called the 𝛼-level set: 𝑎_𝛼̃={𝑥∈𝑋∣𝜇_̃𝑎(𝑥)≥𝛼}.

Definition 2.4. Let 𝑅∶𝐹(ℝ)→ℝ be a ranking function which maps each fuzzy number into the real line, where a natural order exists. We denote the set of all trapezoidal fuzzy numbers by 𝐹(𝑅). The orders could be defined on 𝐹(ℝ) by 𝑎𝑎̃̃≤≥_𝑅𝑅𝑏̃iff𝑅(𝑎)̃≥𝑅𝑏̃,𝑎̃=𝑏̃𝑏̃iffiff𝑅𝑅(𝑎(𝑎)̃)̃=≤𝑅𝑅𝑏̃𝑏̃,,(2.4) where 𝑎̃ and 𝑏̃ are in 𝐹(ℝ). Also we write 𝑎̃≤_𝑅𝑏̃ if and only if 𝑏̃≥_𝑅𝑎̃. We restrict our attention to linear ranking functions, that is, a ranking function 𝑅 such that 𝑅𝑘𝑎̃+𝑏̃=𝑘𝑅(𝑎)̃+𝑅𝑏̃.(2.5) In this paper, we used the linear ranking function which was first proposed by Yager [13]

𝑅(𝑎)̃=12¹⁰inf𝑎_𝜆̃+sup𝑎_𝜆̃𝑑𝜆,(2.6) which reduces to 𝑅(𝑎)̃=𝑎¹+𝑎₂+𝑎₃+𝑎₄4.(2.7) Then, for TFNs 𝑎̃ and 𝑏̃, we have 𝑎̃≤_𝑅𝑏̃iff𝑎₁+𝑎₂+𝑎₃+𝑎₄≤𝑏₁+𝑏₂+𝑏₃+𝑏₄.(2.8)

3. The MOLTP with Fuzzy Cost Coefficients

The MOLTP with fuzzy cost coefficients is formulated as follows: min𝑓^𝑘(𝐱)=^𝑚_𝑛𝑖=1_𝑗=1s.t.∶_𝑛𝑐̃^𝑘𝑖𝑗𝑥_𝑖𝑗,𝑘=1,2,…,𝐾,_𝑗=1𝑥_𝑖𝑗=𝑎_𝑖,𝑖=1,2,…,𝑚,_𝑚_𝑖=1𝑥𝑥_𝑖𝑗=𝑏_𝑗,𝑗=1,2,…,𝑛,_𝑖𝑗≥0,𝑖=1,2,…,𝑚;𝑗=1,2,…,𝑛.(3.1)𝑥_𝑖𝑗 is decision variable which refers to product quantity that is transported from supply point 𝑖 to demand point 𝑗. 𝑎₁,𝑎₂,…,𝑎_𝑚 and 𝑏₁,𝑏₂,…,𝑏_𝑛 are 𝑚 supply and 𝑛 demand quantities, respectively. We note that 𝑎_𝑖(𝑖=1,2,…,𝑚) and 𝑏_𝑗(𝑗=1,2,…,𝑛) are crisp numbers. 𝐾 is the number of the objective functions of MOLTP. 𝑐̃^𝑘𝑖𝑗 is fuzzy unit transportation cost from supply point 𝑖 to demand point 𝑗 for the objective 𝑘,(𝑘=1,2,…,𝐾). For our fuzzy transportation problem, the coefficients of the objectives 𝑐̃^𝑘𝑖𝑗 are considered as trapezoidal fuzzy numbers: 𝑐̃^𝑘𝑖𝑗=𝑐^{𝑘1𝑖𝑗},𝑐^{𝑘2𝑖𝑗},𝑐^{𝑘3𝑖𝑗},𝑐^{𝑘4𝑖𝑗}.(3.2) The membership function of the fuzzy number 𝑐̃^𝑘𝑖𝑗 and its figure are given in (3.3) and Figure 2, respectively,𝜇𝑐̃^𝑘𝑖𝑗(𝑥)=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩0,𝑥<𝑐^{𝑘1𝑖𝑗}𝑥−𝑐_{𝑘1𝑖𝑗1},𝑐^{𝑘2𝑖𝑗}−𝑐^{𝑘1𝑖𝑗}1,,𝑐𝑐^{𝑘1𝑖𝑗}_{𝑘2𝑖𝑗}≤𝑥<𝑐^{𝑘2𝑖𝑗},𝑐_{𝑘4𝑖𝑗}≤𝑥≤𝑐^{𝑘3𝑖𝑗},−𝑥𝑐^{𝑘4𝑖𝑗}−𝑐^{𝑘3𝑖𝑗}0,,𝑐𝑥^{𝑘3𝑖𝑗}><𝑐_{𝑘4𝑖𝑗}𝑥≤𝑐^{𝑘4𝑖𝑗},.(3.3) Now, in the context of multiobjective, let us give the definitions of efficient or nondominated or Pareto optimal solutions for MOLTP. These are used instead of the optimal solution concept in a single objective TP.

Figure 2 

The membership function of the fuzzy cost coefficient 𝑐̃𝑘𝑖𝑗.

Definition 3.1 (Pareto optimal solution for MOLTP). Let 𝑆 be the feasible region of (3.1). 𝐱^∗∈𝑆 is said to be a Pareto optimal solution (strongly efficient or nondominated) if and only if there does not exist another 𝐱∈𝑆 such that 𝑅(𝑓^𝑘(𝐱))≤𝑅(𝑓^𝑘(𝐱^∗)) for all 𝑘=1,2,…,𝐾 and 𝑅(𝑓^𝑘(𝐱))≠𝑅(𝑓^𝑘(𝐱^∗)) for at least one 𝑘=1,2,…,𝐾 where 𝑅 is the ranking function defined in (2.7).

Definition 3.2 (Compromise solution for MOLTP). A feasible solution 𝐱^∗∈𝑆 is called a compromise solution of (3.1) if and only if 𝐱^∗∈𝐸 and 𝑅(𝑓^𝑘(𝐱^∗))≤∧_𝐱∈𝑆𝑅(𝑓(𝐱)) where 𝑅(𝑓(𝐱))=(𝑅(𝑓¹(𝐱)),𝑅(𝑓²(𝐱)),…,𝑅(𝑓^𝑘(𝐱))), ∧ stands for “min” operator and 𝐸 is the set of Pareto optimal solutions.

When the fuzzy cost coefficients are given as trapezoidal fuzzy number in the form 𝑐̃^𝑘𝑖𝑗=(𝑐^{𝑘1𝑖𝑗},𝑐^{𝑘2𝑖𝑗},𝑐^{𝑘3𝑖𝑗},𝑐^{𝑘4𝑖𝑗}) by means of (2.2) and (2.3), the objectives can be written as follows for each 𝑘=1,2,…,𝐾:𝑓^𝑘(𝐱)=^𝑚_𝑛𝑖=1_𝑗=1𝑐^{𝑘1𝑖𝑗},𝑐^{𝑘2𝑖𝑗},𝑐^{𝑘3𝑖𝑗},𝑐^{𝑘4𝑖𝑗}𝑥^𝑖𝑗=^𝑚_𝑛𝑖=1_𝑗=1𝑐^{𝑘1𝑖𝑗}𝑥_𝑖𝑗,^𝑚_𝑛𝑖=1_𝑗=1𝑐^{𝑘2𝑖𝑗}𝑥_𝑖𝑗,^𝑚_𝑛𝑖=1_𝑗=1𝑐^{𝑘3𝑖𝑗}𝑥_𝑖𝑗,^𝑚_𝑛𝑖=1_𝑗=1𝑐^{𝑘4𝑖𝑗}𝑥_𝑖𝑗.(3.4) Since 𝑓^𝑘(𝐱) is a trapezoidal fuzzy number for a given 𝐱∈𝑆, we need to define the minimum of fuzzy valued objective function. In this paper, the fuzziness in the objectives is handled with a fuzzy programming technique in the sense of multiobjective approach by means of (2.7). The ranking value of 𝑓^𝑘(𝐱) can be written as follows: 𝑅𝑓^𝑘𝐱=∑^𝑚𝑖=1∑^𝑛𝑗=1𝑐^{𝑘1𝑖𝑗}𝑥_𝑖𝑗+∑^𝑚𝑖=1∑^𝑛𝑗=1𝑐^{𝑘2𝑖𝑗}𝑥_𝑖𝑗+∑^𝑚𝑖=1∑^𝑛𝑗=1𝑐^{𝑘3𝑖𝑗}𝑥_𝑖𝑗+∑^𝑚𝑖=1∑^𝑛𝑗=1𝑐^{𝑘4𝑖𝑗}𝑥_𝑖𝑗4.(3.5) For obtaining a better ranking value, we want to find a feasible solution 𝐱∈𝑆 that minimizes all of the characteristic points of the fuzzy objective value 𝑓^𝑘(𝐱), simultaneously. It implies that the less the characteristic points of fuzzy objective value the better (preferable) the solution. Thus, the minimum of objective function can be handled asmin𝑓^𝑘(𝐱)=min^𝑚_𝑛𝑖=1_𝑗=1𝑐^{𝑘1𝑖𝑗}𝑥_𝑖𝑗,min^𝑚_𝑛𝑖=1_𝑗=1𝑐^{𝑘2𝑖𝑗}𝑥_𝑖𝑗,min^𝑚_𝑛𝑖=1_𝑗=1𝑐^{𝑘3𝑖𝑗}𝑥_𝑖𝑗,min^𝑚_𝑛𝑖=1_𝑗=1𝑐^{𝑘4𝑖𝑗}𝑥_𝑖𝑗.(3.6) And with (3.6), (3.1) can be reformulated to the following problem in the sense of multiobjective approach [10]: min𝑓^𝑘1(𝐱)=^𝑚_𝑛𝑖=1_𝑗=1min𝑓_𝑘2𝑐^{𝑘1𝑖𝑗}𝑥_𝑖𝑗,(3.7a)(𝐱)=^𝑚_𝑛𝑖=1_𝑗=1min𝑓_𝑘3𝑐^{𝑘2𝑖𝑗}𝑥_𝑖𝑗,(3.7b)(𝐱)=^𝑚_𝑛𝑖=1_𝑗=1min𝑓_𝑘4𝑐^{𝑘3𝑖𝑗}𝑥_𝑖𝑗,(3.7c)(𝐱)=^𝑚_𝑛𝑖=1_𝑗=1s.t.𝐱∈𝑆.𝑐^{𝑘4𝑖𝑗}𝑥_𝑖𝑗,((33..77de))Thus, by constructing four objectives for each 𝑘∈𝐾, fuzziness in (3.1) is eliminated. In other words, our aim is to find the Pareto optimal solutions of (3.1) in the manner of multiobjective linear programming problems. We note that since 𝑐^{𝑘1𝑖𝑗}≤𝑐^{𝑘2𝑖𝑗}≤𝑐^{𝑘3𝑖𝑗}≤𝑐^{𝑘4𝑖𝑗}, the same order is valid between 𝑓^𝑘1, 𝑓^𝑘2, 𝑓^𝑘3, 𝑓^𝑘4 for each 𝑘∈𝐾. So, we want to find a Pareto optimal solution that minimizes all of the objectives of (3.7a)–(3.7e) (all the characteristic points of all objectives). Certainly, the solution of (3.7a)–(3.7e) is not the same with the individual minima of objectives (3.7a)–(3.7d). In general, an optimal solution which simultaneously minimizes all objective functions in (3.7a)–(3.7e) does not always exist when the objective functions conflict with one another. When a certain Pareto optimal solution is selected, any improvement of one objective function can be achieved only at the expense of at least one of the other objective functions. A preferred compensatory compromise Pareto optimal solution is a solution which satisfies the decision maker’s preferences and is preferred to all other solutions, taking into consideration all objectives contained in (3.7a)–(3.7e).

4. Compensatory Operators

There are several fuzzy aggregation operators. The detailed information about them exists in Zimmermann [11] and Tiryaki [14]. The most important aspect in the fuzzy approach is the compensatory or noncompensatory nature of the aggregation operator. Several investigators [11, 15–17] have discussed this aspect.

Using the linear membership function, Zimmermann [18] proposed the “min” operator model to the multiobjective linear problem (MOLP). It is usually used due to its easy computation. Although the “min” operator method has been proven to have several nice properties [16], the solution generated by min operator does not guarantee compensatory and Pareto optimality [19–21]. The biggest disadvantage of the aggregation operator “min” is that it is noncompensatory. In other words, the results obtained by the “min” operator represent the worst situation and cannot be compensated by other members which may be very good. On the other hand, the decision modeled with average operator is called fully compensatory in the sense that it maximizes the arithmetic mean value of all membership functions.

Zimmermann and Zysno [22] show that most of the decisions taken in the real world are neither noncompensatory (min operator) nor fully compensatory and suggested a class of hybrid compensatory operators with 𝛾 compensation parameter.

Basing on the 𝛾-operator, Werners [23] introduced the compensatory “fuzzy and” operator which is the convex combinations of min and arithmetical mean𝜇_and=𝛾min_𝑖𝜇_𝑖+(1−𝛾)𝑚_𝑖𝜇_𝑖,(4.1) where 0≤𝜇_𝑖≤1,𝑖=1,…,𝑚, and the magnitude of 𝛾∈[0,1] represent the grade of compensation.

Although this operator is not inductive and associative, this is commutative, idempotent, strictly monotonic increasing in each component, continuous, and compensatory. Obviously, when 𝛾=1, this equation reduces to 𝜇_and=min(noncompensatory) operator. In literature, it is showed that the solution generated by Werners’ compensatory “fuzzy and” operator does guarantee compensatory and Pareto optimality for MOLP [14, 16, 17, 21–23]. Thus this operator is also suitable for our MOLTP. Therefore, due to its advantages, in this paper, we used Werners’ compensatory “fuzzy and” operator to aggregate the multiple objectives.

5. A Compensatory Approach to MOLTP with Fuzzy Cost Coefficients

Now, we have a multi objective programming problem ((3.7a)–(3.7e)). In this paper, we used a fuzzy programming technique to solve this problem. So, we need to define the membership functions of objectives firstly.

5.1. Constructing the Membership Functions of Objectives

Now, the membership functions of 4𝐾objectives will be defined to apply our compensatory approach. Let 𝐿_𝑘𝑝 and 𝑈_𝑘𝑝 be the lower and upper bounds of the objective function 𝑓^𝑘𝑝(𝑘=1,2,…,𝐾;𝑝=1,2,3,4), respectively. In the literature, there are two common ways of determining these bounds. The first way: solve the MOLTP as a single objective TP using each time only one objective and ignoring all others. Determine the corresponding values for every objective at each solution derived. And find the best (𝐿_𝑘𝑝) and the worst (𝑈_𝑘𝑝) values corresponding to the set of solutions. And the second way: by solving 8𝐾 single-objective TP, the lower and upper bounds 𝐿_𝑘𝑝 and 𝑈_𝑘𝑝 can also be determined for each objective 𝑓^𝑘𝑝(𝐱),𝑘=1,2,…,𝐾;𝑝=1,2,3,4as follows: 𝐿_𝑘𝑝=min_𝐱∈𝑆𝑓^𝑘𝑝(𝐱),𝑈_𝑘𝑝=max_𝐱∈𝑆𝑓^𝑘𝑝(𝐱),(5.1) where 𝑆 is the feasible solution space, that is, satisfied supplydemand and non-negativity constraints. Here, we note that (5.1) will be used for determining the lower and upper bounds of objectives. There are several membership functions in the literature, for example, linear, hyperbolic and piecewise linear, and so forth. For simplicity, in this paper, we used a linear membership function 𝜇_𝑘𝑝𝑓^𝑘𝑝=⎧⎪⎪⎨⎪⎪⎩1,𝑓^𝑘𝑝𝑈_𝑘𝑝<𝐿_𝑘𝑝,−𝑓^𝑘𝑝𝑈_𝑘𝑝−𝐿_𝑘𝑝0,,𝐿𝑓_{𝑘𝑝𝑘𝑝}≤𝑓^𝑘𝑝≤𝑈_𝑘𝑝,>𝑈_𝑘𝑝.(5.2) Here, 𝐿_𝑘≠𝑈_𝑘,𝑘=1,2,…,𝐾 and in the case of 𝐿_𝑘=𝑈_𝑘,𝜇_𝑘𝑝(𝑓^𝑘𝑝(𝐱))=1. The membership function 𝜇_𝑘𝑝(𝑓^𝑘𝑝) is linear and strictly monotone decreasing for 𝑓^𝑘𝑝 in the interval [𝐿_𝑘𝑝,𝑈_𝑘𝑝].

Using Zimmermann’s minimum operator [18], MOLTP can be written as max_𝑥msi.n_𝑘𝑝t.𝐱𝜇∈_𝑘𝑝𝑆𝑓.^𝑘𝑝(𝐱),(5.3) By introducing an auxiliary variable 𝜆, min_𝑘𝑝𝜇_𝑘𝑝𝑓^𝑘𝑝(𝐱)=𝜆⟹𝜇_𝑘𝑝𝑓^𝑘𝑝(𝐱)≥𝜆.(5.4)

(5.3) can be transformed into the following equivalent conventional LP problem:msa.xt.𝜆𝜇,_𝑘𝑝𝐱𝜆∈∈𝑆[𝑓0,,^𝑘𝑝1(]𝐱.)≥𝜆,𝑘=1,…,𝐾;𝑝=1,2,3,4,(5.5) Here, we note that (5.5) is the “min” operator model for MOLTP. Its optimal objective value denotes the maximizing value of the least satisfaction level among all objectives. And it can also be interpreted as the “most basic satisfaction” that each objective in the transportation system can attain.

5.2. Werners’ Compensatory 𝜇_and Operator for MOLTP with Fuzzy Cost Coefficients

It is pointed out that Zimmermann’s min operator model is a noncompensatory model and does not always yield a Pareto optimal solution [19–21]. So we used Werners’ 𝜇_and operator to aggregate our objectives’ membership functions.

For every objective, after satisfying its most basic satisfaction level in the transportation system, to promote its satisfaction degree as high as possible, we can make the following arrangement:𝜇_𝑘𝑝𝑓^𝑘𝑝(𝐱)=𝜆+𝜆_𝑘𝑝,(5.6) where 𝜆=min𝜇_𝑘𝑝(𝑓_𝑘𝑝).

This arrangement is introduced to the constraints with the following expressions:(i)𝜇_𝑘𝑝(𝑓^𝑘𝑝(𝐱))=𝜆+𝜆_𝑘𝑝, (ii)𝜆+𝜆_𝑘𝑝≤1.

So, 𝜇_and used in this paper (4.1) can be formed as𝜇_and=𝛾min_𝑘𝑝𝜇_𝑘𝑝+(1−𝛾)4𝐾𝜇¹¹𝜇_and+𝜇₁₂+𝜇₁₃+𝜇₁₄+𝜇₂₁+⋯+𝜇_𝐾4,=𝛾𝜆+(1−𝛾)𝜇4𝐾𝜆+𝜆¹¹+𝜆+𝜆₁₂+𝜆+𝜆₁₃+𝜆+𝜆₁₄+𝜆+𝜆₂₁⋯+𝜆+𝜆_𝐾4_and,=𝜆+(1−𝛾)4𝐾𝜆¹¹+𝜆₁₂+𝜆₁₃+𝜆₁₄+𝜆₂₁+⋯+𝜆_𝐾4,(5.7) where the magnitude of 𝛾∈[0,1] represents the grade of compensation. Obviously, when 𝛾=1and0, 𝜇_and= “min” operator and 𝜇_and= “average” operator, respectively.

Therefore, depending on the compensation parameter 𝛾, (5.5) is converted to the following compensatory model that is solved for obtaining compromise Pareto optimal solutions for MOLTP:max𝜇_and=𝜆+(1−𝛾)4𝐾^𝐾_4𝑘=1_𝑝=1𝜆_𝑘𝑝s.t.∶𝐱𝜇∈𝑆_𝑘𝑝,𝜆+𝜆𝑓^𝑘𝑝_𝑘𝑝(𝐱)≥𝜆+𝜆_𝑘𝑝,∀𝑘=1,2,…,𝐾;∀𝑝=1,2,3,4,𝛾,𝜆,𝜆_𝑘𝑝≤1,∀𝑘=1,2,…,𝐾;∀𝑝=1,2,3,4,𝑥∈[0,1_𝑖𝑗],≥0,𝑖=1,2,…,𝑚;𝑗=1,2,…,𝑛.(5.8) It is noted that in order to avoid some possible computational errors in solution process, we added the condition (i) as 𝜇_𝑘𝑝≥𝜆+𝜆_𝑘𝑝 to the formulated problem (5.8). So, our compensatory model generates compensatory compromise Pareto optimal solutions for MOLTP.

We shall give this assertion in the following theorem.

Theorem 5.1. If (𝐱,𝝀^𝐱) is an optimal solution of problem (5.8), then 𝐱 is a Pareto optimal solution for MOLTP, where 𝝀^𝐱=(𝜆^𝐱,𝜆^𝐱11,𝜆^𝐱12,𝜆^𝐱13,𝜆^𝐱14,…,𝜆^𝐱𝐾1,𝜆^𝐱𝐾2,𝜆^𝐱𝐾3,𝜆^𝐱𝐾4).

If required, the proof of the theorem can be found in [23]. Also, Pareto optimality test [24] can be applied to the solutions of (5.8), and it could be seen that these solutions are Pareto optimal for MOLTP.

We also note here that our approach is valid for single objective TPs with fuzzy cost coefficients since our compensatory approach could be applied to four objectives which are constructed from the original single objective function of TP.

6. Illustrative Examples

In this section, two numerical examples are given to explain our approach. The first example contains only one objective. Second one dealt with the multiobjective version.

6.1. A Single Objective Transportation Problem with Fuzzy Cost Coefficients

In this paper, we studied MOLTP with crisp supply&demand parameters but fuzzy costs which are given as trapezoidal fuzzy numbers. To our knowledge this combination is not studied in the literature. But [8] dealt with the single objective version of TP which is studied in our paper. In [8], single objective TP is converted to a bicriterial TP, and an algorithm is given to solve this bicriterial problem by means of parametric programming. The numerical example that is given in [8] is as follows: Supplies: 𝑎₁=70; 𝑎₂=70; Demands: 𝑏₁=30; 𝑏₂=30; 𝑏₃=80;Penalties of objective: 𝑐^1𝑖𝑗 see Table 1.

Table 1 

The penalties of the single objective in Example 6.1.

Using (5.1), the lower and upper bounds of objectives are determined to construct the membership functions as mentioned in Table 2.

Table 2 

Bound values of objective.

Using (5.8), the compensatory model is constructed as follows:max𝜇_and=𝜆+(1−𝛾)4⁴_𝑝=1𝜆_1𝑝s.t.₃,_𝑗=1𝑥_1𝑗=70,³_𝑗=1𝑥_2𝑗=70,₂_𝑖=1𝑥_𝑖1=30,²_𝑖=1𝑥_𝑖2=30,²_𝑖=1𝑓𝑥_𝑖3=80,₁₁𝑓(𝐱)+0⋅𝜆+𝜆₁₁≤540,𝑓¹²(𝐱)+300𝜆+𝜆₁₂≤1030,₁₃𝜆+(𝜆𝐱)_1𝑝+780𝜆+𝜆₁₃≤2370,𝑓¹⁴(𝐱)+300𝜆+𝜆₁₄≤6700,𝜆,𝜆_1𝑝≤1,𝑝=1,2,3,4,𝑥,𝛾∈[0,1],𝑝=1,2,3,4,_𝑖𝑗≥0(𝑖=1,2;𝑗=1,2,3).(6.1) This compensatory model generates the following compensatory compromise Pareto optimal solutions for different 11 values of the compensation parameter 𝛾 with 0.1 increment.

For the value of 𝛾=0, we obtained three alternative solutions𝐗^1∗=⎧⎪⎨⎪⎩𝑥¹¹𝑥=30,𝑥₁₂=30,𝑥₁₃=10₂₁𝑓=0,𝑥₂₂=0,𝑥₂₃=70₁₁⎫⎪⎬⎪⎭,𝐗^1∗=540,𝑓𝐗¹²𝐗^1∗=730,𝑓¹³𝐗^1∗=1590,𝑓¹⁴𝐗^1∗=6700;_2∗=⎧⎪⎨⎪⎩𝑥¹¹𝑥=0,𝑥₁₂=30,𝑥₁₃=40₂₁𝑓=30,𝑥₂₂=0,𝑥₂₃=40₁₁⎫⎪⎬⎪⎭,𝐗^2∗=540,𝑓𝐗¹²𝐗^2∗=880,𝑓¹³𝐗^2∗=1980,𝑓¹⁴𝐗^2∗=6400;_3∗=⎧⎪⎨⎪⎩𝑥¹¹𝑥=10,𝑥₁₂=30,𝑥₁₃=30₂₁𝑓=20,𝑥₂₂=0,𝑥₂₃=50₁₁⎫⎪⎬⎪⎭,𝐗^3∗=540,𝑓¹²𝐗^3∗=830,𝑓¹³𝐗^3∗=1850,𝑓¹⁴𝐗^3∗=6500.(6.2) The solution 𝐗^3∗ is obtained for the value of 𝛾=0.1-𝛾=1.0.

These solutions imply following fuzzy objective values:𝑓𝑓¹𝐗^1∗=(540,730,1590,6700),₁𝑓𝐗^2∗=(540,880,1980,6400),₁𝐗^3∗=(540,830,1850,6500).(6.3) In [8], three alternative solution sets are obtained. Two of them are the same as our solution sets 𝐗^1∗, 𝐗^2∗. And the other solution set is𝐗^4∗=⎧⎪⎨⎪⎩𝑥¹¹𝑥=0,𝑥₁₂=0,𝑥₁₃=70₂₁𝑓=30,𝑥₂₂=30,𝑥₂₃=10₁₁⎫⎪⎬⎪⎭,𝐗^4∗=540,𝑓¹²𝐗^4∗=1030,𝑓¹³𝐗^3∗=2370,𝑓¹⁴𝐗^3∗=6400,(6.4) with the fuzzy objective value 𝑓¹(𝐗^4∗)=(540,1030,2370,6400).

To compare our results, the ranking function which is defined in (2.7) can be used. The ranking values of the solutions are as follows:𝑅𝑅𝑓𝑓¹𝐗^1∗=2390,𝑅𝑓¹𝐗^2∗=2450,₁𝐗^3∗=2430,𝑅𝑓¹𝐗^4∗=2585.(6.5) As it can be seen from the ranking values, our compensatory model generates better fuzzy objective values according to the ranking function (2.7).

6.2. Multiobjective Transportation Problem with Fuzzy Cost Coefficients

Let us consider a MOLTP with the following characteristics:supplies: 𝑎₁=24;𝑎₂=8;𝑎₃=18;demands: 𝑏₁=11;𝑏₂=9;𝑏₃=21;𝑏₄=9;penalties of the first objective: 𝑐^1𝑖𝑗 see Table 3; penalties of the second objective: 𝑐^2𝑖𝑗 see Table 4.

Table 3 

The penalties of the first objective in Example 6.2.

Table 4 

The penalties of the second objective in Example 6.2.

Using (5.1), the lower and upper bounds of the objectives are determined to construct the membership functions as mentioned in Table 5.

Table 5 

Bound values of objectives.

Using (5.8), the compensatory model is constructed as follows:max𝜇_and=𝜆+(1−𝛾)8²_4𝑘=1_𝑝=1𝜆_𝑘𝑝s.t.₄,_𝑗=1𝑥_1𝑗=24,⁴_𝑗=1𝑥_2𝑗=8,⁴_𝑗=1𝑥_3𝑗=18,₃_𝑖=1𝑥_𝑖1=11,³_𝑖=1𝑥_𝑖2=9,³_𝑖=1𝑥_𝑖3=21,³_𝑖=1𝑓𝑥_𝑖4=9,₁₁𝑓(𝐱)+183𝜆+𝜆₁₁≤513,𝑓¹²(𝐱)+225𝜆+𝜆₁₂≤697,₁₃𝑓(𝐱)+219𝜆+𝜆₁₃≤787,𝑓¹⁴(𝐱)+212𝜆+𝜆₁₄≤972,₂₁𝑓(𝐱)+167𝜆+𝜆₂₁≤452,𝑓²²(𝐱)+121𝜆+𝜆₂₂≤532,₂₃𝜆+(𝜆𝐱)_𝑘𝑝+128𝜆+𝜆₂₃≤603,𝑓²⁴(𝐱)+180𝜆+𝜆₂₄≤754,𝜆,𝜆_𝑘𝑝≤1,∀𝑘=1,2;𝑝=1,2,3,4,𝑥,𝛾∈[0,1],∀𝑘=1,2;𝑝=1,2,3,4,_𝑖𝑗≥0(𝑖=1,2,3;𝑗=1,2,3,4).(6.6) By solving (6.6), the results for different 11 values of the compensation parameter 𝛾 with 0.1 increment are obtained and given in Table 6. The results are the values of objective functions 𝑓^𝑘𝑝(𝑘=1,2;𝑝=1,2,3,4); the satisfactory levels of the objectives corresponding to solution 𝐱, (i.e. the values of membership functions) 𝜇_𝑘𝑝(𝑘=1,2;𝑝=1,2,3,4); the most basic satisfactory level 𝜆; the compensation satisfactory level 𝜇_and, respectively.

Table 6 

The results of our compensatory model.

So, our compensatory model generates the following compensatory compromise Pareto optimal solutions 𝐗^1∗, 𝐗^2∗, and 𝐗^3∗ for our MOLTP.

For the value of 𝛾=0,𝐗^1∗=⎧⎪⎪⎨⎪⎪⎩𝑥¹¹𝑥=0,𝑥₁₂=1,𝑥₁₃=14,𝑥₁₄=9₂₁𝑥=0,𝑥₂₂=8,𝑥₂₃=0,𝑥₂₄=0₃₁𝑓=11,𝑥₃₂=0,𝑥₃₃=7,𝑥₃₄=0₁₁⎫⎪⎪⎬⎪⎪⎭,𝑓𝐗^1∗=330,𝑓¹²𝐗^1∗=488,𝑓¹³𝐗^1∗=592,𝑓¹⁴𝐗^1∗=784,₂₁𝐗^1∗=323,𝑓²²𝐗^1∗=422,𝑓²³𝐗^1∗=475,𝑓²⁴𝐗^1∗=574.(6.7)

For the value of 𝛾=0.1-𝛾=0.4,𝐗^2∗=⎧⎪⎪⎨⎪⎪⎩𝑥¹¹=𝑥6.1809,𝑥₁₂=1,𝑥₁₃=7.8191,𝑥₁₄=9₂₁𝑥=0,𝑥₂₂=8,𝑥₂₃=0,𝑥₂₄=0₃₁𝑓=4.8191,𝑥₃₂=0,𝑥₃₃=13.1809,𝑥₃₄=0₁₁⎫⎪⎪⎬⎪⎪⎭,𝑓𝐗^2∗=330,𝑓¹²𝐗^2∗=506.5426,₁₃𝑓𝐗^2∗=598.1809,𝑓¹⁴𝐗^2∗=784,₂₁𝑓𝐗^2∗=310.6383,𝑓²²𝐗^2∗=415.8191,₂₃𝐗^2∗=475,𝑓²⁴𝐗^2∗=586.3617.(6.8)

For the value of 𝛾=0.5-𝛾=1.0,𝐗^3∗=⎧⎪⎪⎨⎪⎪⎩𝑥¹¹𝑥=5.0133,𝑥₁₂=2.2338,𝑥₁₃=7.7530,𝑥₁₄=9₂₁𝑥=1.2338,𝑥₂₂=6.7662,𝑥₂₃=0,𝑥₂₄=0₃₁𝑓=4.7530,𝑥₃₂=0,𝑥₃₃=13.2470,𝑥₃₄=0₁₁⎫⎪⎪⎬⎪⎪⎭,𝑓𝐗^3∗=333.7013,𝑓¹²𝐗^3∗=503.0398,₁₃𝑓𝐗^3∗=593.3119,𝑓¹⁴𝐗^3∗=780.2987,₂₁𝑓𝐗^3∗=308.0334,𝑓²²𝐗^3∗=421.9218,₂₃𝐗^3∗=481.1689,𝑓²⁴𝐗^3∗=598.8318.(6.9) These solutions imply following fuzzy objective values for our MOLTP: 𝑓𝑓¹𝐗^1∗=(488,592,158,192),₁𝑓𝐗^2∗=(506.5426,598.1809,176.5426,185.8191),₁𝑓𝐗^3∗=(503.0398,593.3119,169.3385,186.9868),₂𝑓𝐗^1∗=(422,475,99,99),₂𝑓𝐗^2∗=(415.8191,475,105.1808,111.3617),₂𝐗^3∗=(421.9218,481.1689,113.8834,117.6629).(6.10)

All of these solutions pointed out that for all possible values of 𝑐̃^𝑘𝑖𝑗(𝑖=1,2,3;𝑗=1,2,3,4;𝑘=1,2), the certainly transported amounts are⎧⎪⎪⎨⎪⎪⎩𝑥𝑥₁₄=9,𝑥₂₃=0,₂₄=𝑥0,𝑥₃₂=0,₃₄=0⎫⎪⎪⎬⎪⎪⎭.(6.11) And also, the least transported amounts are⎧⎪⎪⎨⎪⎪⎩𝑥₁₂≥𝑥1,𝑥₁₃≥7.7530,₂₂𝑥≥6.7662,₃₁≥4.7530,𝑥₃₃≥7⎫⎪⎪⎬⎪⎪⎭.(6.12)

For 𝛾=1,𝜇_and equals to min (noncompensatory) operator that is 𝜇_and=min_𝑘𝑝𝜇_𝑓𝑘𝑝=0.8620 and gives the solution 𝐗^3∗. This solution remains the same for 𝛾=[0.5,1].

For 𝛾=0,𝜇_and equals to average operator (full compensatory) operator, that is, 𝜇_and=min(1/4𝐾)∑𝜇_𝑓𝑘𝑝=0.9235 and gives the solution 𝐗^1∗.

These solutions and the values of all membership functions are offered to decision maker (DM). If DM is not satisfied with the proposed solution then he/she could assign the weights 𝑤_𝑘,(𝑤_𝑘>0,∑^𝐾𝑘=1𝑤_𝑘=1) on his/her objectives 𝑓^𝑘,𝑘=1,2. In this case, the weights 𝑤_𝑘 are inserted to the compensatory model as the following manner: 𝜇_𝑘𝑝𝑓^𝑘𝑝𝑤_𝑘𝑤≥𝜆+𝜆_𝑘𝑝,∀𝑘=1,2;𝑝=1,2,3,4,_𝑘𝜆+𝜆_𝑘𝑝≤1,∀𝑘=1,2;𝑝=1,2,3,4,(6.13) instead of the constraints𝜇_𝑘𝑝𝑓^𝑘𝑝(𝐱)≥𝜆+𝜆_𝑘𝑝,∀𝑘=1,2;𝑝=1,2,3,4.(6.14)

7. Conclusions

MOLTP which is a well-known problem in the literature has wide practical applications in manpower planning, personnel allocation, inventory control, production planning, and so forth. In this paper, we deal with MOLTP whose costs coefficients are given as trapezoidal fuzzy numbers. We assume that the supply and demand quantities are precisely known. The fuzziness in the objectives is handled with a fuzzy programming technique in the sense of multiobjective approach. And a compensatory approach is given by using Werner’s 𝜇_and operator. Our approach generates compromise solutions which are both compensatory and Pareto optimal for MOLTP. It is known that Zimmerman’s “min” operator is not compensatory and also does not guarantee to generate the Pareto optimal solutions. Werner’s 𝜇_and operator is useful about computational efficiency and always generates Pareto optimal solutions. The proposed approach also makes it possible to overcome the nonlinear nature owing to the fuzziness in the costs.

This paper discussed MOLTP with fuzzy cost coefficient. For further work, MOLTP with fuzzy supply&demand quantities and also multi-index form of this problem could be considered.