A Note on Integer Linear Programming Problems with LR Pentagonal Fuzzy Numbers

A Note on Integer Linear Programming Problems with LR 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: August 17, 2018) ABSTRACT

The objective of this paper is to deal with fuzzy integer linear programming problem in which all the parameters and variables are LR pentagonal fuzzy numbers.

In this paper we have introduced and studied the ranking function of LR pentagonal fuzzy numbers. A new approach for solving fuzzy integer linear programming problems with LR pentagonal fuzzy number is proposed, based on ranking function.

The proposed method is very easy to understand. This is illustrated with relevant numerical examples.

Keywords: Ranking function, LR Pentagonal fuzzy number, Fuzzy integer linear programming.

1. INTRODUCTION

The concept linear programming problem is to find out the best solution to the real-

world problems where the available informations are not exact or not precise. In that situation

linear programming model helps lot. Firstly, 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

. Sahaya Sudha et al.

proposed solving fuzzy linear programming

problem using pentagonal fuzzy numbers with robust ranking method. Nagoor Gani et al.

discussed Fuzzy linear programming problem using LR fuzzy number.

Herrera and Verdegay

have proposed three methods for solving three models of Fuzzy integer linear programming. Allahviranloo et al.

discussed a model of Fuzzy integer linear programming problem with fuzzy variable and proposed to solve a new method. Pandian and Jayalakshmi

have proposed a decomposition method for solving Fuzzy integer linear programming problem with fuzzy variables by using classical integer linear programming. In

Stephen Dinagar and Mohamed Jeyavuthin discussed the concept of solving integer linear programming problems with pentagonal fuzzy numbers.

In this paper, section 2 contains some basic definitions needed for this work. In section 3, parametric and LR type of pentagonal Fuzzy numbers are discussed. In section 4, Fuzzy integer linear programming with fuzzy variables are discussed and relevant numerical illustrations are given. Finally, conclusion is included in section 5.

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)

The 𝛼-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 3. PENTAGONAL FUZZY NUMBER (PFN)

3.1 Parametric Representation of Pentagonal Fuzzy Number

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 LR Pentagonal Fuzzy Number

3.2 LR Type Representation of Pentagonal Fuzzy Number (LR PFN) Definition 9 (LR Pentagonal Fuzzy Number)

A LR Pentagonal fuzzy number denoted by 𝑀 ̃

= (𝑚, 𝛼

, 𝛼

, 𝛽

, 𝛽

)

is a fuzzy number, where 𝑚, 𝛼

, 𝛼

, 𝛽

, 𝛽

are real numbers and its membership function 𝜇

(𝑥) is given by

𝜇

(𝑥) =

{

0, 𝑥 < 𝑚 − (𝛼

+ 𝛼

) 1

2 [ 𝑥 − (𝑚 − (𝛼

+ 𝛼

))

𝛼

] , 𝑚 − (𝛼

+ 𝛼

) ≤ 𝑥 ≤ 𝑚 − 𝛼

1

2 + 1

2 [ 𝑥 − (𝑚 − 𝛼

)

𝛼

] , 𝑚 − 𝛼

≤ 𝑥 ≤ 𝑚 1, 𝑥 = 𝑚

1

2 + 1

2 [ (𝑚 + 𝛽

) − 𝑥

𝛽

] , 𝑚 ≤ 𝑥 ≤ 𝑚 + 𝛽

1

2 [ (𝑚 + (𝛽

+ 𝛽

)) − 𝑥

𝛽

] , 𝑚 + 𝛽

≤ 𝑥 ≤ 𝑚 + (𝛽

+ 𝛽

) 0, 𝑥 > 𝑚 + (𝛽

+ 𝛽

)

Definition 10 (Equality of LR PFNs)

Two LR Pentagonal fuzzy number 𝐴̃

= (𝑚, 𝛼

, 𝛼

, 𝛽

, 𝛽

)

and 𝐵̃

= (𝑚

, 𝛼

, 𝛼

, 𝛽

, 𝛽

)

is said to be equal if and only if 𝑚 = 𝑚

, 𝛼

= 𝛼

, 𝛼

= 𝛼

, 𝛽

= 𝛽

, 𝛽

= 𝛽

.

Definition 11 (Symmetric LR PFNs)

A fuzzy number 𝐴̃

= (𝑚, 𝛼

, 𝛼

, 𝛽

, 𝛽

)

is said to be symmetric LR-Pentagonal fuzzy number if there exist real number such that 𝛼

+ 𝛼

= 𝛽

+ 𝛽

.

3.3 Arithmetic Operations on LR Pentagonal Fuzzy Numbers

Let us consider 𝐴̃

= (𝑚, 𝛼

, 𝛼

, 𝛽

, 𝛽

)

and 𝐵̃

= (𝑚

, 𝛼

, 𝛼

, 𝛽

, 𝛽

)

be two pentagonal fuzzy numbers then,

(i) Addition

𝐴̃

(+)𝐵̃

= (𝑚 + 𝑚

, 𝛼

+ 𝛼

, 𝛼

+ 𝛼

, 𝛽

+ 𝛽

, 𝛽

+ 𝛽

)

(ii) Subtraction

𝐴̃

(−)𝐵̃

= (𝑚 − 𝑚

, 𝛼

− 𝛼

, 𝛼

− 𝛼

, 𝛽

− 𝛽

, 𝛽

− 𝛽

)

(iii) Multiplication

𝐴̃

(×)𝐵̃

= (

5

𝜎

,

5

𝜎

,

5

𝜎

,

5

𝜎

,

5

𝜎

)

𝐿𝑅

. Where 𝜎

= 𝑚

+ 𝛼

+ 𝛼

+ 𝛽

+ 𝛽

(or) 𝐴̃

(×)𝐵̃

= (𝑚𝑅̌(𝑏), 𝛼

𝑅̌(𝑏), 𝛼

𝑅̌(𝑏), 𝛽

𝑅̌(𝑏), 𝛽

𝑅̌(𝑏))

𝐿𝑅

. Where 𝑅̌(𝐵̃

) =

(or) 𝑅̌(𝑏) =

(iv) Division

𝐴̃

(/)𝐵̃

= (

𝑏

,

𝑏

,

𝑏

,

𝑏

,

𝑏

)

𝐿𝑅

. Where 𝜎

= 𝑚

+ 𝛼

+ 𝛼

+ 𝛽

+ 𝛽

, (or)

𝐴̃

(/)𝐵̃

= (

𝑅̌(𝑏)

,

𝑅̌(𝑏)

,

𝑅̌(𝑏)

,

𝑅̌(𝑏)

,

𝑅̌(𝑏)

)

𝐿𝑅

. Where 𝑅̌(𝐵̃

) =

5

(or) 𝑅̌(𝑏) =

5

.

Definition 12 (Ranking Function)

A ranking function is a map from 𝐹(𝔑) into real line. Now, we define the orders 𝐹(𝔑) as follows;

(i) 𝐴̃

≥ 𝐵̃

if and only if 𝔑(𝐴̃

) ≥ 𝔑(𝐵̃

) (ii) 𝐴̃

≤ 𝐵̃

if and only if 𝔑(𝐴̃

) ≤ 𝔑(𝐵̃

) (iii) 𝐴̃

= 𝐵̃

if and only if 𝔑(𝐴̃

) = 𝔑(𝐵̃

)

Where 𝐴̃

, 𝐵̃

are elements of 𝐹(𝔑). Let 𝐴̃

= (𝑚, 𝛼

, 𝛼

, 𝛽

, 𝛽

)

be pentagonal fuzzy number, the ranking function is

𝔑(𝐴̃

) = 𝑎

+ 2(𝑎

− 𝑎

) + 2(𝑎

− 𝑎

) − (𝑎

− 𝑎

) − (𝑎

− 𝑎

) 2

𝔑(𝐴̃

) =

2

, where 𝑚 = 𝑎

, 𝛼

= (𝑎

− 𝑎

), 𝛼

= (𝑎

− 𝑎

), 𝛽

= (𝑎

− 𝑎

), 𝛽

= (𝑎

− 𝑎

)

𝔑(𝐴̃

) = 𝑚 + 2(𝛼

+ 𝛼

) − (𝛽

+ 𝛽

) 2

4. FUZZY INTEGER LINEAR PROGRAMMING (FILP)

Consider the following fully fuzzy integer linear programming problems Maximize (or Minimize) 𝑧̃ = 𝑐̃

𝑥̃ Subject to, 𝐴̃ 𝑥̃{≤, =, ≥}𝑏̃, 𝑥̃ ≥ 0 and are integers, where the cost vectors 𝑐̃

= (𝑐̃

)

, 𝐴̃ = (𝑎̃

)

, 𝑥̃ = (𝑥̃

)

and 𝑏̃ = (𝑏̃

)

and 𝑎̃

, 𝑥̃

, 𝑏̃

, 𝑐̃

∈ 𝐹(𝑅), for all 1 ≤ 𝑗 ≤ 𝑛 and 1 ≤ 𝑖 ≤ 𝑚.

Definition 13

A linear Programming problem is called fuzzy variable linear programming problem (FVLPP), if some of the parameters are crisp, and variables and right-hand sides are fuzzy numbers.

General form of FVLPP is as follows:

Maximize (or Minimize) 𝑧̃ = 𝑐̃ 𝑥̃

Subject to 𝐴̃ 𝑥̃{≤, =, ≥}𝑏̃, 𝑥̃ ≥ 0 and are integers, Where 𝑐̃ ∈ 𝑅

, 𝐴̃ ∈ (𝑅)

, 𝑅

∈ (𝐹(𝑅))

, and 𝑥̃ ∈ (𝐹(𝑅))

. 4.1 Algorithm:

Step 1. Arrange the fully fuzzy linear programming problem in the following manner,

Max(or)Min 𝑧̃ = 𝑐̃

𝑥̃

; Subject to 𝐴̃

𝑥̃

≤ 𝑏̃

; 𝑥̃

≥ 0.

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

) =

.

Step 3. Using an integer linear programming procedure finally we get an Optimum solution.

4.2 Numerical Examples

Example: 1

Consider the following fuzzy integer linear programing problem with LR type of pentagonal fuzzy numbers

Maximize 𝑍̃ = (6,2,2,2,2) 𝑥̃

+ (8,2,2,2,2) 𝑥̃

Subject to

(4,2,2,2,2) 𝑥̃

+ (12,4,4,4,4) 𝑥̃

≤ (10,2,2,2,2);

(8,2,2,2,2) 𝑥̃

+ (14,2,2,2,2) 𝑥̃

≤ (16,4,4,4,4);

𝑥̃

, 𝑥̃

≥ 0 and are integers.

Now we calculate, 𝔑(6,2,2,2,2) =

2

= 5.

Similarly,

𝔑 ( 8,2,2,2,2 ) = 6, 𝔑(4,2,2,2,2 ) = 4, 𝔑(12,4,4,4,4) = 10, 𝔑(10,2,2,2,2 ) = 7, 𝔑(8,2,2,2,2 ) = 6, 𝔑 (14,2,2,2,2) = 9, 𝔑(16,4,4,4,4) = 12,

Using the ranking function, the problem becomes:

Maximize 𝑍̃ = 5 𝑥̃

+ 6 𝑥̃

Subject to 4 𝑥̃

+ 10 𝑥̃

≤ 7;

6 𝑥̃

+ 9 𝑥̃

≤ 12;

𝑥̃

, 𝑥̃

≥ 0 and are integers.

Using an algorithm for integer linear programming problem, The solution of the problem is 𝑥̃

= 1, 𝑥̃

= 0 and 𝑍̃ = 5.

Example: 2

Consider the following fuzzy integer linear programing problem with LR type of pentagonal fuzzy numbers

Maximize 𝑍̃ = (7,2,4,6,2) 𝑥̃

+ (6,2,2,3,4) 𝑥̃

Subject to

(16,7,4,4,5) 𝑥̃

+ (14,6,2,3,7) 𝑥̃

≤ (15,5,5,7,3);

(12,5,3,4,3) 𝑥̃

+ (15,8,4,5,3) 𝑥̃

≤ (25,9,8,5,8);

𝑥̃

, 𝑥̃

≥ 0 and are integers.

Using the ranking function, the problem becomes:

Maximize 𝑍̃ = 5.5 𝑥̃

+ 3.5 𝑥̃

Subject to 14.5 𝑥̃

+ 10 𝑥̃

≤ 12.5;

10.5 𝑥̃

+ 15.5 𝑥̃

≤ 23;

𝑥̃

, 𝑥̃

≥ 0 and are integers.

Again, by using algorithm for integer linear programming problem, 𝑥̃

= 0, 𝑥̃

= 1 and 𝑍̃ = 3.5

5. CONCLUSION

This paper deal with fuzzy integer linear programming problem where all the parameters and variables are LR pentagonal fuzzy numbers. We have introduced and studied the ranking function of LR pentagonal fuzzy numbers. A new approach for solving fuzzy integer linear programming problems with LR pentagonal fuzzy number is proposed based on ranking function. This notion can be extended to some other optimization problems in future.

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