Distinct Methods for Solving Fully Fuzzy Linear Programming Problems with Pentagonal Fuzzy Numbers

Distinct Methods for Solving Fully Fuzzy Linear Programming Problems with Pentagonal Fuzzy Numbers

D. Stephen Dinagar

and M. Mohamed Jeyavuthin

1

Associate Professor,

PG & Research Department of Mathematics, T.B.M.L. College, Porayar, Tamil Nadu, INDIA.

2

Research Scholar,

PG & Research Department of Mathematics, T.B.M.L. College, Porayar, Tamil Nadu, INDIA.

email: [email protected], [email protected]

(Received on: May 8, 2019) ABSTRACT

The focus of this paper is to find fuzzy optimal solution of fully fuzzy linear programming problems with pentagonal fuzzy numbers. New approaches for solving fully fuzzy linear programming problems with pentagonal fuzzy number have been proposed, based on distinct ranking functions. The proposed methods are very easy to understand.

Keywords: Linear programming problem, Fully fuzzy linear programming, Pentagonal fuzzy number, Robust ranking.

1. INTRODUCTION

The concept linear programming problem is to find out the best solution to the real- world problems where the available information is not exact or not precise. In that situation linear programming model helps lot.

First, the concept Fuzzy linear programming was proposed by Tanaka et al.

. It plays a vital role in Fuzzy modeling, which can formulate the uncertainty. Nasseri

has proposed a new method for solving the Fuzzy linear programming problems in which he has used the fuzzy ranking method for converting the fuzzy objective function into crisp objective function.

Fuzzy linear programming was studied by many researchers

. Liou and Wang

discussed ranking fuzzy numbers with interval values. Verdegay

have proposed three

methods for solving three models of fuzzy integer linear programming. Abbasbandy and

Asady

discussed a sign distance method for ranking fuzzy numbers. In 1998, Cheng

used a centroid based distance method. Kauffmann and Gupta

proposed Fuzzy Arithmetic and many researchers has studied in the relevant field

.

In 1984, Kolman and Hill

introduced fully fuzzy linear programming problem. A new method for solving fully fuzzy linear programming problems using ranking function is proposed by Allahviranloo et al.

. Kumar et al.

proposed a new method for solving fully fuzzy linear programming problems with inequality constraints. Apurba Panda and Madhumangal Pal

discussed a detailed study on pentagonal fuzzy number. Dhurai and Karpagam

proposed a new method for fully fuzzy linear programming problems with hexagonal fuzzy numbers by using ranking function. In

Stephen Dinagar and Mohamed Jeyavuthin discussed the concept of solving fully fuzzy integer linear programming problems under robust ranking technique.

The structure of this paper is, section 1 is introduction. In Section 2, some basic concepts have been reviewed. In Section 3, with the help of proposed ranking function algorithms of fully fuzzy linear programming problem have been presented. Section 4 is conclusion.

2 PRELIMINARIES

Definition 1 (Fuzzy set)

A Fuzzy set 𝐴̃ is defined by 𝐴̃ = {(𝑥, 𝜇

(𝑥)): 𝑥 ∈ 𝑋, 𝜇

(𝑥) ∈ [0,1] }. In the pair (𝑥, 𝜇

(𝑥)), the first element 𝑥 belong to the classical set 𝑋, the second element 𝜇

(𝑥) belong to the interval [0, 1], called Membership function.

Definition 2 (Support of Fuzzy set)

The support of fuzzy set 𝐴̃ is the set of all 𝑥 in 𝑋 such that 𝜇

(𝑥) > 0. That is 𝑆𝑢𝑝𝑝 (𝜇

) = {𝑥/ 𝜇

(𝑥) > 0}.

Definition 3 (𝜶-cut)

The𝛼-cut of fuzzy set 𝐴̃ is a set consisting of those elements of the universe 𝑋 whose membership values exceed the threshold level 𝛼.

That is 𝐴̃

= {𝑥 / 𝜇

(𝑥) ≥ 𝛼}.

Definition 4 (Convex Fuzzy set)

A fuzzy set 𝐴̃ is convex if 𝜇

(𝜆𝑥

+ (1 − 𝜆𝑥

)) ≥ min(𝜇

(𝑥

), 𝜇

(𝑥

)) , 𝑥

, 𝑥

∈ 𝑋 and 𝜆 ∈ [0,1]. Alternatively, a fuzzy set is convex, if all 𝛼- level sets are convex.

Definition 5 (Fuzzy Number)

A fuzzy number 𝐴̃ is a subset of real line R, with the membership function 𝜇

(𝑥) holds the following conditions:

(i) 𝜇

(𝑥) is piecewise continuous in its domain

(ii) 𝐴̃ is normal. That is, there is a 𝑥

∈ 𝐴̃ such that 𝜇

(𝑥

) = 1.

(iii) 𝐴̃ is convex. That is,𝜇

(𝜆𝑥

+ (1 − 𝜆𝑥

)) ≥ min(𝜇

(𝑥

), 𝜇

(𝑥

)) , 𝑥

, 𝑥

∈ 𝑋

Definition 6 (Pentagonal Fuzzy Number)

A fuzzy number 𝐴̃

is pentagonal fuzzy number denoted by 𝐴̃

= (𝑎

, 𝑎

, 𝑎

, 𝑎

, 𝑎

), where 𝑎

, 𝑎

, 𝑎

, 𝑎

, 𝑎

are real numbers and its membership function 𝜇

(𝑥) is given by

𝜇

(𝑥) =

{

0, 𝑥 < 𝑎

1

2 [ 𝑥 − 𝑎

𝑎

− 𝑎

], 𝑎

≤ 𝑥 ≤ 𝑎

1

2 + 1

2 [ 𝑥 − 𝑎

𝑎

− 𝑎

], 𝑎

≤ 𝑥 ≤ 𝑎

1, 𝑥 = 𝑎

1

2 + 1

2 [ 𝑎

− 𝑥

𝑎

− 𝑎

], 𝑎

≤ 𝑥 ≤ 𝑎

1

2 [ 𝑎

− 𝑥

𝑎

− 𝑎

], 𝑎

≤ 𝑥 ≤ 𝑎

0, 𝑥 > 𝑎

Definition 7

A pentagonal fuzzy number can be defined as 𝐴̃

= (𝑀

(𝑥),𝐽

(𝑥),𝐽

(𝑥),𝑀

(𝑥)) for 𝑥 ∈ [0,1]

where,

(i) 𝑀

(𝑥)is strictly increasing continuous function on [0,0.5]

(ii) 𝐽

(𝑥)is strictly increasing continuous function on [0.5,1]

(iii) 𝐽

(𝑥)is strictly decreasing continuous function on [1,0.5]

(iv) 𝑀

(𝑥) is strictly decreasing continuous function on [0.5,0]

Remark 8

The pentagonal fuzzy number 𝐴̃

becomes triangular fuzzy number if 𝑎

− 𝑎

= 𝑎

− 𝑎

.

Figure:1 Graph of a Pentagonal Fuzzy Number

3 FULLY FUZZY LINEAR PROGRAMMING PROBLEM (FFLPP) Consider the fully fuzzy linear programming problem

Maximize (or Minimize) 𝑍̃ = 𝐶̃𝑥̃, Subject to 𝐴̃𝑥̃ ≤, =, ≥ 𝑏̃ , 𝑥̃ ≥ 0

where 𝑥̃ is a non-negative pentagonal fuzzy number and 𝐶̃ = [𝑐̃

]

, 𝐴̃ = [𝑎̃

]

𝑥̃ = [𝑥̃

]

, 𝑏̃ = [𝑏̃

]

, 1 ≤ 𝑗 ≤ 𝑛 and 1 ≤ 𝑖 ≤ 𝑚.

3.1 Method I

In this method we are finding fuzzy optimal solution for fully fuzzy linear programming problem with pentagonal fuzzy number with the aid of average ranking function.

3.2 Algorithm

Step 1. Arrange the fully fuzzy linear programming problem in the following manner, Max(or)Min 𝑧̃ = 𝑐̃

𝑥̃

; Subject to𝐴̃

𝑥̃

≤ 𝑏̃

; 𝑥̃

≥ 0.

Step 2. Convert the problem into fuzzy variable linear programming problem by using the ranking function 𝔑(𝐴̃

) =

.

Step 3. Applying the simplex / Big-M procedure. to solve the fuzzy variable linear programming problem. Let the solution be 𝑥̃

. Hence the solution of the fully fuzzy linear programming problem 𝑥̃

.

Step 4.Convert all the inequations to the constraints into equations by introducing the slack and/or surplus variables in the constrains.

Step 5.To compute the net evaluations using the relation 𝑍̃ = 𝐶

𝑌

− 𝐶

; 𝑗 = 1,2, … , 𝑛.

If, all 𝑍̃ ≥ 0for maximization problem,𝑍̃ ≤ 0 for minimization problem, then optimal solution is reached.

Step 6.Determine the basic variable 𝑥̃

, it will be replaced by the non-basic variable, where

𝑘 = arg min {𝔑(𝐵̃̃

)} , 𝑖 = 1,2, … , 𝑚 in maximization and 𝑘 = arg max {𝔑(𝐵̃̃

)}, 𝑖 = 1,2, … , 𝑚 for minimization problem.

Step 7. perform the pivot operation and return to step 5. Then repeat the procedure until a fuzzy optimal solution is obtained.

3.3 Method II

In this method we are finding fuzzy optimal solution for fully fuzzy linear programming problem with pentagonal fuzzy number. This method is used to convert the inequality constraints in to equality constraints.

3.4 Algorithm:

Step 1. To Check type of all the constraints, convert the inequality constraints into equality constraints. ∑

𝑎̃

× 𝑥̃

= 𝑏̃

𝑜𝑟 ∑

𝑎̃

× 𝑥̃

≠ 𝑏̃

∀ 𝑖 = 1,2, … , 𝑚

If ∑

𝑎̃

× 𝑥̃

≠ 𝑏̃

for some 𝑖, then the following cases arise:

Case (i) If ∑

𝑎̃

× 𝑥̃

≤ 𝑏̃

for some 𝑖, then convert such type of inequality constraints into

equality constraints by introducing non-negative variable 𝑆̃

to the left side of the constraints.

∑

𝑎̃

× 𝑥̃

+ 𝑆̃

= 𝑏̃

for some 𝑖, where 𝑆̃

is a non-negative pentagonal fuzzy number.

Case (ii) If ∑

𝑎̃

× 𝑥̃

≥ 𝑏̃

for some 𝑖, then convert such type of inequality constraints into equality constraints by introducing non-negative variable 𝑆̃

to the right side of the constraints. ∑

𝑎̃

× 𝑥̃

= 𝑏̃

+ 𝑆̃

for some 𝑖, where 𝑆̃

is a non-negative pentagonal fuzzy number.

Step 2. Let Maximize (or Minimize) 𝐶̃

× 𝑋̃ Subject to 𝐴̃𝑋̃ ≤, =, ≥ 𝑏̃ where 𝑋̃ is a non- negative pentagonal fuzzy number, be converted into Maximize (or Minimize) 𝐶̃

× 𝑋̃ Subject to 𝐴̃𝑋̃ = 𝑏̃ where 𝑋̃ is a non-negative pentagonal fuzzy number, and 𝐶̃ = [𝑐̃

]

, 𝐴̃ = [𝑎̃

]

, 𝑋̃ = [𝑥̃

]

, 𝑏̃ = [𝑏̃

]

, 1 ≤ 𝑗 ≤ 𝑛 and 1 ≤ 𝑖 ≤ 𝑚.

Step 3. Substituting 𝐶̃ = [𝑐̃

]

, 𝐴̃ = [𝑎̃

]

, 𝑋̃ = [𝑥̃

]

, 𝑏̃ = [𝑏̃

]

into the fully fuzzy linear programming problem, obtained in step 2, then the fully fuzzy linear programming problem may be written as, Maximize (or Minimize) ∑

(𝐶̃

× 𝑋̃) subject to

∑

𝑎̃

× 𝑥̃

= 𝑏̃

∀ 𝑖 = 1,2, … , 𝑚, 𝑥̃

is a non-negative pentagonal fuzzy number.

Step 4. If all the parameters 𝑐̃

, 𝑥̃

, 𝑎̃

and 𝑏̃

are represented by pentagonal fuzzy numbers (𝑝

, 𝑞

, 𝑟

, 𝑠

, 𝑡

), (𝑥

, 𝑦

, 𝑧

, 𝑢

, 𝑣

), (𝑎

, 𝑏

, 𝑐

, 𝑑

, 𝑒

) and (𝑏

, 𝑔

, ℎ

, 𝑘

, 𝑙

) respectively then the fully fuzzy linear programming problem written as,

Maximize (or Minimize) ∑

(𝑝

, 𝑞

, 𝑟

, 𝑠

, 𝑡

) × (𝑥

, 𝑦

, 𝑧

, 𝑢

, 𝑣

) subject to

∑

(𝑎

, 𝑏

, 𝑐

, 𝑑

, 𝑒

) × (𝑥

, 𝑦

, 𝑧

, 𝑢

, 𝑣

) = (𝑏

, 𝑔

, ℎ

, 𝑘

, 𝑙

) ∀ 𝑖 = 1,2, … , 𝑚, (𝑥

, 𝑦

, 𝑧

, 𝑢

, 𝑣

) is a non-negative pentagonal fuzzy number.

Step 5. Assuming (𝑎

, 𝑏

, 𝑐

, 𝑑

, 𝑒

) × (𝑥

, 𝑦

, 𝑧

, 𝑢

, 𝑣

) = (𝑎

, 𝑏

, 𝑐

, 𝑑

, 𝑒

) the fully fuzzy linear programming problem obtained in the previous step may be written as,

Maximize (or Minimize) ℛ (∑

(𝑝

, 𝑞

, 𝑟

, 𝑠

, 𝑡

) × (𝑥

, 𝑦

, 𝑧

, 𝑢

, 𝑣

)), where ℛ =

6

subject to ∑

(𝑎

, 𝑏

, 𝑐

, 𝑑

, 𝑒

) = (𝑏

, 𝑔

, ℎ

, 𝑘

, 𝑙

) ∀ 𝑖 = 1,2, … , 𝑚, (𝑥

, 𝑦

, 𝑧

, 𝑢

, 𝑣

) is a non-negative pentagonal fuzzy number.

Step 6. Find the optimal solution 𝑥

, 𝑦

, 𝑧

, 𝑢

and 𝑣

by solving the crisp linear programming problem obtained in step 5.

3.5 Method III

In this method we are finding fuzzy optimal solution for fully fuzzy linear programming problem with pentagonal fuzzy number with the aid of Robust ranking function.

where the Robust Ranking Technique satisfies the properties such as compensation,

linearity, additivity. It provides results which are consist human intuition. If 𝐴̃

is a pentagonal fuzzy number then the Robust ranking is defined by

R(𝐴̃

) = ∫(𝐴̃

, 𝐴̃

)(0.5)

1

0

𝑑𝛼

where (𝐴̃

, 𝐴̃

) is the 𝛼 level cut of the fuzzy number 𝐴̃

.

Figure 2

Pentagon is divided into four parts which includes two triangles and two trapezoids.

By using robust ranking technique for triangle fuzzy number and trapezoidal fuzzy number the ranking function is

R

(𝐴̃

) = ∫{(𝑎

− 𝑎

)𝛼 + 𝑎

, 𝑎

− (𝑎

− 𝑎

)𝛼}

1

0

(0.5) 𝑑𝛼

3.6 Algorithm

Step 1. Formulate the chosen problem into the following fuzzy linear programming problem as

Maximize 𝑧̃ = ∑

𝑐̃

𝑥̃

Subject to, ∑

𝑎̃

𝑥̃

≤, =, ≥ 𝑏̃

, i=1,2,3,… ,m, 𝑥̃

≥ 0, 𝑗 = 1,2,3, … , 𝑛

Step 2.Using the Ranking functions, the fully fuzzy integer linear programming problem (FFILPP) transformed into fuzzy variable integer linear programming problem (FVILPP).

Step 3. Optimal solution.

4. CONCLUSION

The proposed three methods provide solution to fully fuzzy linear programming problems with pentagonal fuzzy number. Some distinct ranking methods have been employed in each method for solving the optimal solutions. This notion can be extended to some other optimization problems, especially integer programming problems in future.

Figure 2 𝜇_𝐴̃𝑃(𝑥)

C 0.75

1

𝐽₂(𝑥) 𝐽₁(𝑥)

𝑀₂(𝑥) 𝑀₁(𝑥)

E D

A 𝑎

B 0.25

0.5

0 𝑎

𝑎

𝑎

𝑎

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