Extension of Duality Results and a Dual Simplex Method for Linear Programming Problems With Intuitionistic Fuzzy Variables

The aim of this paper is to introduce a formulation of linear programming problems involving intuitionistic fuzzy variables. Here, we will focus on duality and a simplex-based algorithm for these problems. We classify these problems into two main different categories: linear programming with intuitionistic fuzzy numbers problems and linear programming with intuitionistic fuzzy variables problems. The linear programming with intuitionistic fuzzy numbers problem had been solved in the previous literature, based on this fact we offer a procedure for solving the linear programming with intuitionistic fuzzy variables problems. In methods based on the simplex algorithm, it is not easy to obtain a primal basic feasible solution to the minimization linear programming with intuitionistic fuzzy variables problem with equality constraints and nonnegative variables. Therefore, we propose a dual simplex algorithm to solve these problems. Some fundamental concepts and theoretical results such as basic solution, optimality condition and etc., for linear programming with intuitionistic fuzzy variables problems, are established so far. Moreover, the weak and strong duality theorems for linear programming with intuitionistic fuzzy variables problems are proved. In the end, the computational procedure of the suggested approach is shown by numerical examples.


Introduction
Linear Programming is a branch of science in operations research field which has many different applications. Parameters and values of an LP model should be accurate in a primal one. However, in the real world, this assumption does not coincide with the reality. Some sort of uncertainty about the parameters might exist in the problems that we ought to deal with in our daily lives. In these cases, parameters of LP problems would be presented in fuzzy numbers. Application of fuzzy numbers in mathematical programming has a profound history. Zadeh [1] was the first mathematician who proposed the Fuzzy Sets (FSs) theory for the first time. The notion of mathematical programming in the fuzzy environment was suggested by Tanaka et al. [2] in the fuzzy decision-making frame for the first time which had been presented by Bellman and Zadeh [3]. Noori-eskandari and CONTACT M. Goli mrtz.golii@gmail.com Ghaznavi [4] proposed an efficient algorithm for solving FLP problems. They consider some well-known approaches for solving FLP problems. They present some of the difficulties of these approaches and then, crisp LP problems are suggested for solving FLP problems. Linear programming problem in this environment, which is known as Fuzzy Linear Programming (FLP), was firstly investigated by Zimmerman [5]. Nasseri and Ebrahimnejad [6] proposed a novel approach to duality in FLP. Ghaznavi et al. [7] introduced parametric analysis in Fuzzy Number Linear Programming (FNLP) problems. They considered the problem variations by using a linear ranking function. They used the fuzzy primal simplex method, the fuzzy dual simplex method and the fuzzy primal -dual simplex method to find the new optimal basis. For solving FLP problem, Mahdavi-Amiri et al. [8] introduced the simplex algorithm of fuzzy primal. Also, other research has been done in this field, which is mostly based on comparing fuzzy numbers [9,10]. In other words, the ranking functions play a fundamental key role in the decision-making process [11][12][13][14].
Later scientists faced with some problems that the FS theory was unable to have an answer for these problems, among them, Atanassov [15] was the first scientist who presented a generalisation of FS theory to overcome to this obstacle which is known as Intuitionistic Fuzzy Set (IFS). The Degree of Membership (DM) and the Degree of Non-Membership (DNM) were applied to clarify the concept of the IFS theory. The only significant difference between the two theories is that in the FS theory, the summation of the DM of fuzzy numbers (calledμ) and Its complementary, the DNM 1 − μ,which are numbers in the interval of [0,1] is equal to 1 whereas about the IFS theory, in addition to those two mentioned concepts there exist another concept in the same interval which is called the degree of doubt in order to have the same result as summation of 1. Similar to fuzzy numbers, the ranking of Intuitionistic Fuzzy Numbers (IFNs) plays an essential role in decision process. Nagoorgani et al. [16] defined a ranking using score function based on (α, β) − cut method. Seikh et al. [17] introduced a method to approximate the IFNs of the triangular type, which we show here with TIFN. More recently, Atalik and Senturk [18] proposed a new approach using the gergonne point to rank Triangular Intuitionistic Fuzzy Numbers (TIFNs). Suresh et al. [19] introduced the ranking of TIFNs by means of magnitude and solved the intuitionistic FLP problems using this ranking. There are many other methods for ranking IFNs, that we refer to [20][21][22][23][24][25]. Angelov [26] studied the application of IFS to optimisation problems and proposed a solution approach to these problems. Sanny Kuriakose et al. [27] suggested a non-membership function for the IFLP problem. A new form of LP problems in the intuitionistic fuzzy environment can be seen in the research of Parvathi et al. [28,29]. Ejegwa et al. [30] presented a review paper on some definitions, basic operators, some algebras, etc., on intuitionistic FS. Dubey and Mehra [31] proposed an approach based on the value and ambiguity of the index to solve linear programming problems with TIFNs. Nagoorgani and Ponnalagu [32] studied the intuitionistic FLP problem methods using the intuitionistic fuzzy dual simplex method, in which the objective function can be maximise or minimise and also the constraints can be equal or unequal. Nasseri and Goli [33] presented a method for solving fully intuitionistic FLP problems. They use the sign distance between IFNs for their comparison and then proposed an algorithm for finding the optimal solution. Nagoorgani and Ponnalagu [34] used interval arithmetics to solve the intuitionistic FLP problems. Nachammai and Thangaraj [35] solved the intuitionistic FLP problem based on special indexes that convert any IFN to a set of real numbers. Hepzibah and Vidhya [36] and Sidhu [37] studied on symmetric trapezoidal intuitionistic fuzzy numbers (TrIFNs). After defining a ranking function and arithmetic operations on these numbers, they solved the intuitionistic FLP problems without converting these problems into the crisp linear programming problem. Nasseri et al. [38] proposed an approach for solving FLP problems based on comparison of IFNs by the help of linear accuracy function. They define an auxiliary problem, having only triangular intuitionistic fuzzy cost coefficients, and then study the relationships between these problems leading to a solution for the primary problem. Then, they develop intuitionistic fuzzy primal simplex algorithms for solving these problems. Prabakaran and Ganesan [39] introduced Duality Theory for Intuitionistic FLP Problems. They discuss about the solution procedure of primal and dual LP problems involving IFNs without changing in to classical LP problems and then by using new type of arithmetic operations between IFNs, they have proved the weak and strong duality theorems. Sidhu and Kumar [40] proposed mehar methods to solve intuitionistic FLP problems with trapezoidal intuitionistic fuzzy numbers. Ramik and Vlach [41] introduced the intuitionistic FLP problem and then expressed the concepts of duality and related theorems. Now, we consider a minimisation IFVLP problem with equality constraints and nonnegative variables. Here we establish duality results and complementary slackness conditions for IFVLP problems. Then, for solving these problems, we develop the dual simplex method for IFVLP problems that directly uses the primal simplex table. In this case, we utilise the ranking function that already introduced in [19].
In what follows, these topics would be described: Some of the basic concepts of IFS theory would be explained, in Section 2. In Section 3, we classify the IFLP problems into two main different categories: IFNLP problems and IFVLP problems. Then, some fundamental concepts and theoretical results related to IFVLP problem such as basic solution, optimality condition and, etc., are given. In Section 4, we will consider the dual of an IFVLP problem and then by ranking function that already introduced in Section 2, all the dual theorems and the results will be proved. In Section 5, we present numerical examples and finally we explain the result of our research in Section 6.

TIFNs and Their Arithmetic Operations
An IFSÃ I relates to each member of the universe set X, DM μÃ I (x) : X → [0, 1] and DNM vÃ I (x) : X → [0, 1] such that: is named the degree of hesitancy of x toÃ I . Definition 2.1: An IFNÃ I = (μÃ I , vÃ I ) in the set of real numbers R, is define as where 0 ≤ μÃ I + vÃ I ≤ 1 and a, b 1

Definition 2.2: A TIFNÃ
, is a special IFN with the following DM and DNM, respectively: where a v ≤ a μ ≤ a ≤ā μ ≤ā v . The value wÃ I represents the maximum DM and the value uÃ I represents the minimum DNM such that 0 ≤ wÃ I ≤ 1, 0 ≤ uÃ I ≤ 1 and 0 ≤ wÃ I + uÃ I ≤ 1 ( Figure 1).
In the following theorem, it is shown that in a special case, Mag is a linear ranking function.
In fact, if we assume that the considered TIFNs have the same maximum DM and the same DNM, then Mag becomes a linear ranking function. Proof: two TIFNs. Then for λ ≥ 0, we have: On the other hand, because wÃ I = wB I and uÃ I = uB I , so we have: The same is true for λ < 0.
In this paper, we consider TIFNs in such a way that Mag be a linear ranking function.

Intuitionistic Fuzzy Linear Programming Problems
A crisp LP problem is as: where the parameters c = (c 1 , . . . , c n ), b = (b 1 , . . . , b m ) T and A = [a ij ] m×n are given with crisp components and x ∈ R n is an unknown vector of variables to be found. Assume that some parameters are considered to be IFNs, then we obtain IFLP problem. In this section, we are going to consider them in details. Hence, based on their structures, we divide them into two main categories: (1) IFNLP problem, (2) IFVLP problems.
An IFNLP problem is defined as: where b ∈ R m ,c I ∈ (F I (R)) m , A ∈ R m×n are given and x ∈ R n is to be determined. Also, Mag is the ranking function defined by (1). An IFVLP problem is defined as: whereb I ∈ (F I (R)) m and A ∈ R m×n are given andx I ∈ (F I (R)) n is to be determined.

Intuitionistic Fuzzy Basic Feasible Solution
Here, we explain the notion of Intuitionistic Fuzzy Basic Feasible Solution (IFBFS) for IFVLP problem. Consider the following IFVLP problem: where its parameters are defined as in ( (c B 1 , . . . , c B m ). Now, corresponding to any index j, 1 ≤ j ≤ n,define Obviously, for every basic index (0, . . . , 1, 0, . . . , 0) T is the i th unit vector. From Be i = [a B 1 , . . . , a B i , . . . , a B m ]e i = a B i = a j it follows:

Theorem 3.1: (Optimality conditions). Suppose that the IFVLP problem (3) is non-degenerate and B is a feasible basis. An IFBFSx
Therefore, the corresponding IF objective value is On the other hand, for any IFBFSx I to (3), we havẽ Thus, from (9) we have: Hence, for each IFBFS to (3), we havẽ Thus, from (7) and (8), we havez So, if we have and thus we get that Therefore from (11) we seez hence,x I * is optimal. Now, assume thatx I * is an optimal IFBFS to (3). For j = B i , 1 ≤ i ≤ m, from (7), we have z j − c j = 0. From (11), it is obvious that if for some non-basic variablex I j we have z j > c j then this variable is a entering variable andz I * > Magz I , which is a contradiction to optimality ofz I * . So, we have z j ≤ c j , 1 ≤ j ≤ n.

Duality and the Main Results
In this part, we introduce the dual of an LP problem with IF variables (DIFVLP) and express the related dual results.
the DIFVLP problem is formulated as:

Proof:
The proof is an obvious conclusion of Theorem 4.1.

Corollary 4.2: Let one of the IFVLP or DIFVLP problems be unbounded, then the other one is infeasible.
Proof: The proof is an obvious conclusion from the Theorem 4.1.

Theorem 4.3: Consider a given IFLP problem and its dual. Only one of the statements (1) and
(2) is true: (1) One of the primal or dual problems is unbounded and the other one is infeasible.
(2) The primal problem and its dual have no feasible solution.

Theorem 4.4: (Complementary slackness theorem)
Assume thatx I * and w * are feasible solutions to IFVLP problem and its DIFVLP problem, respectively. Thenx I * and w * are optimal iff Proof: Suppose thatx I * and w * are feasible solutions for IFVLP problem and DIFVLP problem, respectively. Then and If we multiply w * ≤ 0 by the inequality Ax I * ≤ Magb I we obtain If we multiplyx I * ≥ Mag0 I by the inequality w * A ≤ c we obtain Thus, we have From optimality ofx I * and w * for the primal and dual problems, we conclude w * b I = Mag cx I * and using the relationship (20) we have From (21) we will have To prove the converse part of this theorem, we utilise the facts (w * A − c)x I * = Mag0 I and Thus, Corollary 4.1, results optimality ofx I * and w * .

The Dual Simplex Method
Here, we first introduce the dual simplex method to solve an IFVLP problem and then describe its algorithm.
Consider the following IFVLP problem as:

Main steps of the dual simplex algorithm
Step 1. If the IFVLP problem is of maximisation type, convert the given IFVLP problem into minimisation problem.
Step 2. Convert the problem into a standard form. ii) If at least one of Mag(ỹ I 0 ) < 0 then go to the next step.
Step 4. Choose the most negative value, if there was more than one Mag(ỹ I 0 ) value less than zero. Now, i) If all y rj ≥ 0 for j = 1, . . . , n then stop. The IFVLP problem is infeasible. ii) If at least one y rj < 0, j = 1, . . . , n then consider the pivot column l.
Step 5. Let y 0l y rl be minimum value of the ratio test. Then y r will be the leaving variable.
Step 6. Update the table and obtain the new value of the objective function by applying the relationȳ 00 = Magỹ I 00 − y 0l y rl y r0 .
Step 7. Go to Step 3 and proceed with the procedures until obtain an optimal solution.

Numerical Examples
Example 5.1: A senior's centre wants to change a menu-planning system. As the first step, its staff tries to change the dinner program. Vegetables, meat and dessert are in the dinner menu. Each serving must contain at least one of these three categories. Table 1 shows the cost per serving of some suggested items as well as their contents.  Hence:x   [16], the results would be: x 1 = 0.6050000, x 2 = 1.197500 and z = 2.407500 in comparison, based on the achieve results, it is obvious that our method is better.

Example 5.3:
In Example 5.2, we just consider the fuzzy part of the numbers and solve it with the ranking function used in [47]. Then x 1 = 0.6100000, x 2 = 1.195000 and z = 2.415000. Obviously, a better answer will be obtained when the numbers are considered intuitionistic fuzzy. Example 5.4: Ekbatan oil refinery is able to extract three types of crude oil from its oil wells in three different cities. These three types of crude oil must be combined with two other types of additives to obtain gasoline. Table 3 shows the amount of sulphur and other additives, including lead and phosphorus used in crude oil.
Each gallon of crude oil makes up only a certain percentage of a gallon of gasoline due to by-products and non-consumable waste from crude oil. Thus, each gallon of city one crude oil is converted to 0.35 gallons, each gallon of city two crude oil is converted to 0.40 gallons, and each gallon of city three crude oil is converted to 0.30 gallons of gasoline. The refinery instructions for the amount of sulphur, lead and phosphorus in each gallon of gasoline are as follows: The question is, what combination of crude oil should we consider in order to minimise the cost of producing gasoline?
Because of uncertainty in resources, the problem can be modeled as an IFVLP problem by using TIFNs. So we will have minz I = 0.55x I 1 + 0.

Remark 5.1:
We emphasis that the above numerical discussion is given to explain our suggested theoretical results as well as the extension of the duality theorems and results in fuzzy environment. We saw that some of these numerical examples are compared with the other works which was used some ranking functions.

Conclusion
In this paper, we formulated a kind of linear programming problems involving intuitionistic fuzzy variables. We investigated the dual of this problem with intuitionistic fuzzy parameters. Also, we developed some duality results, including weak and strong duality and also complementary slackness, for the intuitionistic fuzzy problems. Using these results, we proposed a solution approach and a dual simplex algorithm for the IFVLP problem. The approach offered here is useful for sensitivity analysis. However, considering post analysis results in IFLP problems is a worthwhile area of research that will be investigated in our future works.