#### Abstract

We define linear programming problems involving trapezoidal fuzzy numbers (LPTra) as the way of linear programming problems involving interval numbers (LPIn). We will discuss the solution concepts of primal and dual linear programming problems involving trapezoidal fuzzy numbers (LPTra) by converting them into two linear programming problems involving interval numbers (LPIn). By introducing new arithmetic operations between interval numbers and fuzzy numbers, we will check that both primal and dual problems have optimal solutions and the two optimal values are equal. Also, both optimal solutions obey the strong duality theorem and complementary slackness theorem. Furthermore, for illustration, some numerical examples are used to demonstrate the correctness and usefulness of the proposed method. The proposed algorithm is flexible, easy, and reasonable.

#### 1. Introduction

In the real-world environment, there are many problems which are a concern to the linear programming models, and sometimes it is necessary to formulate these models with parameters of uncertainty. Many numbers from these problems are linear programming problems with fuzzy variables. Interval analysis is an efficient and reliable tool that allows us to handle such problems effectively.

Linear programming problems with interval coefficients have been studied by several authors, such as Sengupta et al. [1, 2], Bitran , Chanas and Kuchta , Nakahara et al. , Steuer , and Shaocheng . Numerous methods for comparison of interval numbers can be found as in Sengupta et al. [1, 2], Ganesan and Veeramani [8, 9], etc. By taking maximum value range and minimum value range inequalities as constraint conditions, Shaocheng  reduced the interval linear programming problem into two classical linear programming problems and obtained an optimal interval solution to it. Ramesh and Ganesan  proposed a method for solving interval number linear programming problems without converting them to classical linear programming problems.

Bellman and Zadeh  introduced for the first time the concept of a fuzzy decision process as an intersection of the fuzzy objective function and resource constraints. From this idea, the authors of  introduced linear programming problems with fuzzy variables and semifully fuzzy linear programming problems. Some authors considered these problems and have developed various methods for solving these problems. Recently, some authors  considered linear programming problems with trapezoidal fuzzy data and/or variables and semifully fuzzy linear programming problems and stated a fuzzy simplex algorithm to solve these problems. Moreover, they developed the duality results in fuzzy environment and presented a dual simplex algorithm for solving linear programming problems with trapezoidal fuzzy variables and semifully fuzzy linear programming problems. Furthermore, the authors of  showed that the presented dual simplex algorithm directly using the primal simplex tableau algorithm tenders the capability for sensitivity (or postoptimality) analysis using primal simplex tableaus.

The contributions of the present study are summarized as follows. (1) In the LPTra under consideration, all of the parameters, such as the coefficients in the objective function, the right-hand side vector, and the decision variables, are a kind of trapezoidal fuzzy numbers, simultaneity. Furthermore, the coefficients in the left-hand side matrix are symmetric trapezoidal fuzzy numbers. (2) According to the proposed approach, the LPTra is converted into two linear programming problems involving interval numbers. The integration of the optimal solution of the two subproblems provides the optimal solution of the primary LPTra. (3) In contrast to most existing approaches, which provide a precise solution, the proposed method provides a fuzzy optimal solution. (4) Similarly, to the competing methods in the literature, the proposed method is applicable for all types of real problems. (5) The complexity of computation is greatly reduced compared with commonly used existing methods in the literature.

The rest of this paper is organized as follows. In Section 2, we recall the definitions of interval number linear programming, interval numbers, and existing method for solving linear programming problem involving interval numbers. In Section 3, a new method is proposed for obtaining the fuzzy optimal solution of the LPTra. The advantages of the proposed method are discussed in Section 4. Two application examples are provided to illustrate the effectiveness of the proposed method in Section 5. Comparative study is presented in Section 6. Finally, concluding remarks are presented in Section 7.

#### 2. Materials and Methods

In this section, some basic definitions and arithmetic operations for closed intervals’ numbers and of linear programming problems involving interval numbers are presented .

##### 2.1. Arithmetic Operations for Closed Intervals’ Numbers

In this section, some arithmetic operations for two intervals are presented [22, 24].

Let us denote by the class of all closed intervals in . If is closed interval, we also adopt the notation , where and mean the lower and upper bounds of , respectively, and with .

If , then . Any two intervals and and the arithmetic operations on and are defined as(i)Addition: , subtraction: Multiplication: Division: with

##### 2.2. A New Interval Arithmetic

In this section, some arithmetic operations for two intervals are presented [21, 23].

Let be the set of all proper intervals and be the set of all improper intervals on the real line . We shall use the terms “interval” and “interval number” interchangeably. The midpoint and width (or half-width) of an interval number are defined as and . The interval number can also be expressed in terms of its midpoint and width as .

For any two intervals and , the arithmetic operations on and are defined as(i)Addition: Subtraction: Multiplication: Division: with

##### 2.3. Ramesh and Ganesan’s Method for Solving Linear Programming Problem Involving Interval Numbers

In this section, to overcome all the limitations of the method presented in , Ramesh and Ganesan’s method  is presented to find the exact optimal solution of linear programming problems involving interval numbers in which all the parameters are represented by intervals numbers.

###### 2.3.1. Formulation of a Linear Programming Problem Involving Interval Numbers (LPIn)

Now, we are in a position to prove interval analogue of some important relationships between the primal and dual linear programming problems.

We consider the primal (P) and dual (D) linear programming problems involving interval numbers (LPIn) as follows :

For the rest of this paper, we will consider the following notations:where , , , , , , , and are real numbers ().

For the rest of this paper, we will consider the following primal linear programming problem involving interval numbers (LPIn14):is equivalent toand primal linear programming problem involving interval numbers (LPIn23)is equivalent to

###### 2.3.2. Optimal Solution for Linear Programming Problem Involving Interval Numbers (LPIn)

In this section, we will describe how to determine the values of the primal (P) and dual (D) variables.

Calculation of the values of the primal variables (LPIn): .

Assume that is optimal; then, the current basis is . Moreover, the current nonbasic variables is and the corresponding solution is . Hence, the optimal solution to the problem (LPIn) can be written as for with the associated value of the objective function: or .

Calculation of the values of the dual (D) variables in .

We have and with the associated value of the objective function: or Min.

#### 3. Main Results

In this section, we will describe our method of solving.

##### 3.1. A New Interval Arithmetic for Trapezoidal Fuzzy Numbers via Intervals’ Numbers

The aim of this section is to present some notations, notions, and results which are useful for our further consideration.

A number (where ) is said to be a trapezoidal fuzzy number if its membership function is given by [12, 24, 25]

Assume that and are two trapezoidal fuzzy numbers. For any two trapezoidal fuzzy numbers and , the arithmetic operations on and are defined as follows.Addition: Subtraction: Multiplication:

For the rest of this paper, we will consider the following notations.

Assume that , , and are trapezoidal fuzzy numbers:where , , , , , , , and are real numbers ().

##### 3.2. Formulation of Linear Programming Problems Involving Trapezoidal Fuzzy Numbers

In this section, we introduce a kind of linear programming problems where the coefficients in objective function, the right-hand-side vector, and the decision variables are a type of trapezoidal fuzzy numbers, simultaneity, and the left-hand-side matrix having symmetric trapezoidal fuzzy numbers coefficients. We name such problems as linear programming problems involving trapezoidal fuzzy numbers (LPTra).

We consider the primal (P) and dual (D) linear programming problems involving trapezoidal fuzzy numbers (LPTra) as follows [12, 25]:

For the rest of this paper, we will consider the following primal linear programming problems involving trapezoidal fuzzy numbers (LPTra):

##### 3.3. Our Method for Solving the Linear Programming Problems Involving Trapezoidal Fuzzy Numbers (LPTra)

In this section, a method to find a fuzzy optimal solution of linear programming problems involving trapezoidal fuzzy numbers (LPTra) is presented.

###### 3.3.1. Formulation of a Linear Programming Problem Involving Midpoint (LPMi14)

Thanks to the new interval arithmetic and (LPIn14), we can write the following linear programming problem involving midpoint (LPMi14) :

###### 3.3.2. Formulation of a Linear Programming Problem Involving Midpoint (LPMi23)

Thanks to the new interval arithmetic and (LPIn23), we can write the following linear programming problem involving midpoint (LPMi23) :

Thanks to the new interval arithmetic, we can write the following lemma .

Lemma 1. is an optimal solution to (LPMi14) if and only if is an optimal solution to (LPIn14).

Proof (see ). If is an optimal solution to (LPMi14) and for , then is an optimal solution to (LPIn14).

Lemma 2. is an optimal solution to (LPMi23) if and only if is an optimal solution to (LPIn23).

Proof (see ). If is an optimal solution to (LPMi23) and for , then is an optimal solution to (LPIn23).
Thanks to the lemma above, we can write the following corollary .

Corollary 1. If is an optimal solution to (LPIn14) and is an optimal solution to (LPIn23), then is an optimal solution to (LPTra) with .

###### 3.3.3. The Steps of Our Computational Method

The steps of our method for solving the linear programming problems involving trapezoidal fuzzy numbers (LPTra) as follows:Step 1: consider primal and dual linear programming problems involving trapezoidal fuzzy numbers (LPTra).Step 2: identify (LPIn14) and (LPIn23).Step 3 (Ramesh and Ganesan’s method, see ): solve the primal and dual (LPIn14) via (LPMi14).Apply the simplex method to (LPMi14) to determine the primal variables (LPMi14): . Assume that is optimal, then the current basis is . Moreover, the current nonbasic variables is and the corresponding solution is. Hence, the optimal solution to problem (LPIn14) can be written as for with the associated value of the objective function: or .Then, the corresponding dual variables is given by in . We have and with the associated value of the objective function: or Min.Step 4 (Ramesh and Ganesan’s method, see ): solve the primal and dual (LPIn23) via (LPMi23).Apply the simplex method to (LPMi23) to determine the primal variables (LPMi23): . Assume that is optimal, then the current basis is . Moreover, the current nonbasic variables is , and the corresponding solution is . Hence, the optimal solution to the problem (LPIn23) can be written as for with the associated value of the objective function: or .Then, the corresponding dual variables is given by in . We have and with the associated value of the objective function Min: or . Step 5: fuzzy optimal solution of linear programming problems involving trapezoidal fuzzy numbers (LPTra):(i)Primal optimal solution: , , with the associated value of the objective function Max.(ii)Then, the corresponding dual optimal solution problem is given by , with the associated value of the objective function Min.(iii)Comparisons using ranking function [12, 24, 25]: and .NB: let and ; then, iff , iff , and iff .

#### 4. Advantages of the Proposed Method

In this section, the advantages of the proposed method over the existing methods for solving LPTras are discussed:(1)The proposed approach can be applied for solving fully fuzzy linear programming problems where all the parameters are represented as fuzzy numbers.(2)The ranking functions of the optimal solutions are nonnegative real numbers, i.e., the midpoint values are no negative.(3)In contrast to the existing method [12, 25], the proposed method provides fuzzy optimal solutions that indicate possible outcomes with a certain degree of membership to the decision maker. This is especially useful for strategic decisions in cases more uncertainty exists.(4)The proposed method obeys the strong duality theorem and complementary slackness theorem  (i.e., Ramesh and Ganesan  have proved the weak and strong duality theorems. Complementary slackness theorem is also proved for the LPIns).(5)The main advantage of the proposed method is that utilizing problems LPIn14 and LPIn23 for solving LPTra is highly economical compared with problem LPTra from a computational viewpoint, regarding the number of constraints and variables. There is a direct relationship between the computational complexity of LPMi14 and LPMi23 problems and the number of their constraints and variables. Because the memory size needed for maintaining the basis (or its inverse) in the simplex algorithm is given by the square of the number of constraints, reducing the number of constraints in LPMi14 and LPMi23 models is crucial for increasing the computational efficiency.(6)In contrast to existing methods [12, 25], the proposed approach does not utilize fuzzy ranking functions for modelling the objective and constraint functions.(7)The proposed method is easy to apply for finding the fuzzy optimal solution of LPTra in real-world applications compared with the existing methods.(8)It is highlighted in [12, 25] that the computational efforts required to solve an LPTra problem can be reduced if trapezoidal fuzzy numbers are used to convey the subjective evaluations of decision makers. Using such fuzzy numbers allows us to compare the proposed method with most of the existing approaches in the literature. Because the proposed scheme considers the trapezoidal fuzzy numbers as intervals, the approximated multiplication does not impact the results, i.e., exact formulas yield the same results as the presented approach.

#### 5. Numerical Examples

Numerical examples are provided to illustrate the theory developed in this paper.

Example 1.  Step 1: consider the primal (P) and dual (LPTra).Primal (P):, , and are trapezoidal fuzzy numbers.The corresponding dual problem is given by :where , , and are trapezoidal fuzzy numbers.Step 2: Identify (LPIn14) and (LPIn23):Then, the corresponding dual problem is given byStep 3 (Ramesh and Ganesan’s method, see ): solve the primal and dual (LPIn14) via (LPMi14).Apply the simplex method to (LPMi14): Min subject to , , . (Table 1)(1)Optimal solution primal to LPIn14:(i)Optimal solution to LPMi14:, , , , , and with Min .(ii)Optimal solution to LPIn14: .We have . We get , , , , , and .(iii)Value of the objective function:Min or Min. We get Min .(2)Then, the corresponding dual problem (D) LPIn14 is given by(i)Optimal solution dual to LPMi14 with and . We get , , , , , and with Max .(ii)Optimal solution dual to LPIn14. We have