MULTI-CHOICE GOAL PROGRAMMING APPROACH FOR A FUZZY TRANSPORTATION PROBLEM

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MULTI-CHOICE GOAL PROGRAMMING APPROACH FOR A FUZZY TRANSPORTATION PROBLEM

1 Debashis Dutta and ²A. Satyanarayana Murthy

1Department of Mathematics, National Institute of Technology Warangal , A.P, India.

2 Department of Mathematics, National Institute of Technology Warangal , A.P, India.

ABSTRACT

A transportation problem with fuzzy demand and supply values and objective function assuming multiple choices is considered. Two cases are dealt with where the goal function assumes a set of choices and a range. Multi-choice goal programming methodology is applied for both the cases. This results in a linear programming problem where some of the constraints are crisp and the remaining fuzzy. A method is proposed to solve this problem. The proposed method is illustrated through two numerical examples.

Keywords: Transportation problem, goal programming, linear programming, fuzzy constraints, multi-choice programming, non-linear programming, transformation technique.

1. INTRODUCTION

The classical transportation problem refers to a special class of linear programming problems. In a typical transportation problem, a product is to be transported from m sources to n destinations and their demand and supply values are a_1,a₂,…,am and b_1,b₂,…,bn respectively. In addition, there is a penalty c_ijassociated with transporting a unit of the product from source i to destination j. This penalty may be cost or delivery time or safety of delivery, etc. In practice, the parameters of transportation problem i.e demand and supply values are not always exactly known and stable. This paper deals with the case when the penalties are known exactly, but the estimate of the demand and supply values are only imprecise.

The main approaches to decision making under imprecision includes stochastic programming and fuzzy programming.

The concept of the fuzzy set theory, first introduced by Zadeh[16], is used for solving different types of linear programming(LP) problems. Zimmermann[17] first introduced fuzzy linear programming(FLP) as conventional LP.

He used linear membership functions and the min operator as an aggregator for these functions, and assigned an equivalent LP to fuzzy linear programming problem. Subsequently, Zimmermann’s fuzzy linear programming has developed into several fuzzy optimization methods for solving the transportation problems.

Chanas et.al[8] presented an FLP model for solving transportation problems with crisp cost coefficients and fuzzy supply and demand values. Also, Chanas and Kuchta[6] proposed the concept of the optimal solution for the transportation problem with fuzzy cost coefficients expressed as L-R fuzzy numbers, and developed an algorithm for obtaining the optimal solution. Additionally, Chanas and kuchta[7] designed an algorithm for solving integer fuzzy transportation problem with fuzzy demand and supply values in the sense of maximizing the joint satisfaction of the fuzzy goal and the constraints.

The term ‘Goal Programming’ was introduced by Charnes and Cooper[12] in 1961. Decision makers sometimes set such goals, even when they are unattainable within the available resources. Such problems are tackled with the help of the techniques of goal programming. Any constraint incorporated is called a goal. Whether the goals are attainable or not, the objective function is stated in such a way that it’s optimization means as ‘close as possible’

to the indicated goals.

Multi-choice linear programming problems exist in many managerial decision making problems. Hiller and Lieberman[13] and Ravindran et. al[15] have considered a mathematical model in which an appropriate constraint is to be chosen using binary variables.

A method for modeling the multi-choice goal programming problem, using the multiplicative terms of binary variables to handle the multiple aspiration levels was presented by Chang[9]. He has also given a method where the multiplicative terms of the binary variables are replaced y a continuous variable[11].

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In this paper, two goal programming models, where the multiple aspiration levels of the cost goal are handled by the use of multiplicative terms of the binary variables and by the use of a continuous variable for a transportation problem with fuzzy demand and supply values are formulated.

The organization of this paper is as follows. The two transportation models and their respective goal programming formulations are presented in section two. Section three explains the method of solution of the linear programming problem with fuzzy and crisp constraints. Two numerical examples on the proposed method are presented in section four. Section five gives some concluding remarks on the proposed method.

2. MATHEMATICAL MODELS

All fuzzy numbers considered in this paper are trapezoidal fuzzy numbers of the type A = (a₁,a₂,a₃,a₄) having the following membership function

(x-a₁)/(a₂-a₁), a₁ ≤ x ≤ a₂

1, a₂ ≤ x ≤ a₃ µ_A(x) = (a₄-x)/(a₄-a₃) a₃ ≤ x ≤ a₄

0, otherwise Consider the transportation model

c(x) =



  m

i n

j

1 1

c_ijx_ij={ c₁,c₂,…,ck} Subject to



 n

j 1

x_ij  (a_i1, a_i2,a_i3,a_i4) (i=1,2,…,m) (2.1)



 m

i 1

xij  (bj1,bj2,bj3,bj4) (j=1,2,…,n)

xij ≥ 0 and are integers, i=1,2,…,m,j=1,2,…,n

The objective function can assume only one of the k choices (aspiration levels) b₁,b₂,…,bk. We illustrate the procedure for finding an optimal solution of the above problem for the case with four goals. Since the number of choices is 4(=2²) two binary variables are required to model the situation. We rewrite (2.1) for four choices case as follows

c(x) = z₁z₂c₁ + (1-z₁)z₂c₂ + z₁(1-z₂)c₃ + (1-z₁)(1-z₂)c₄ =  (say) Subject to



 n

j 1

x_ij (a_i1,a_i2,a_i3,a_i4) (i=1,2,…,m) (2.2)



 m

i 1

xij  (bj1,bj2,bj3,bj4) (j=1,2,…,n) x_ij ≥ 0 and are integers, i=1,2,…,m, j=1,2,…,n, z₁ = 0/1, z₂ = 0/1

In order to minimize , the flexible membership function goal with the aspired level 1(i.e the highest possible value of membership function) is used as follows

( gmax - )/(gmax – gmin) – di+

+ di -

=1

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where g_max and g_min are respectively upper and lower bunds of the aspiration levels of the cost goal and d_i⁺ and d_i^– are respectively, over and under achievements of the i^th goal.

Using the goal programming method presented by Chang[9] for a linear programming problem with minimization type objective function, we construct the following goal programming problem for problem (2.2).

Minimize d1+

+ d1-

+ d2+

+ d2-

Subject to



 n

j 1

x_ij (ai1,a_i2,a_i3,a_i4) (i=1,2,…,m) (2.3)



 m

i 1

x_ij  (b_j1,b_j2,b_j3,b_j4)(j=1,2,…,n)



  m

i n

j

1 1

c_ijx_ij –d₁⁺ + d₁^- = 

 = z1z2c1 + (1-z1)z2c2 + z1(1-z2)c3 + (1-z1)(1-z2)c4

/(g_max – g_min) + d₂⁺ - d₂^- = g_min/(g_max – g_min)

x_ij ≥ 0 and are integers, i=1,2,…,m, j=1,2,…,n, z₁ = 0/1, z₂ = 0/1

The nonlinear constraints of the above problem can be linearized by defining z₃ = z₁z₂ and adding the linear constraint z1 + z2 – 1 ≤ 2z3 ≤ z1 + z2, z3 = 0/1

Hence problem (2.3) can be written as

Minimize d₁⁺ + d₁^- + d₂⁺ + d₂^- Subject to



 n

j 1

x_ij (ai1,a_i2,a_i3,a_i4) (i=1,2,…,m) (2.4)



 m

i 1

x_ij  (b_j1,b_j2,b_j3,b_j4)(j=1,2,…,n)



  m

i n

j

1 1

c_ijx_ij –d₁⁺ + d₁^- = 

 = (c₂ – c₄)z₁ + (c₃ –c₄)z₂ + (c₁ – c₂ –c₃ +c₄)z₃ + c₄ /(gmax – gmin) + d2+

- d2 -

= gmin/(gmax – gmin)

x_ij ≥ 0 and are integers, i=1,2,…,m, j=1,2,…,n, z₁ = 0/1, z₂ = 0/1,z₃ = 0/1

We now consider the following transportation model where the cost goal can assume any value in a prescribed range

a ≤ c(x) ≤ b Subject to



 n

j 1

x_ij (a_i1,a_i2,a_i3,a_i4) (i=1,2,…,m) (2.5)



 m

i 1

x_ij  (b_j1,b_j2,b_j3,b_j4)(j=1,2,…,n)

x_ij ≥ 0 and are integers, i=1,2,…,m, j=1,2,…,n,

A penalty p is assigned for exceeding the cost goal and there is no penalty for achieving a value lesser than the aspiration levels. Using the goal programming method given by Chang[11] for a linear programming problem with minimization type objective function, we construct the following goal programming problem.

Minimize pd⁺ + e⁺ + e^– Subject to

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

  m

i n

j

1 1

c_ijx_ij–d⁺ + d^- = y

y – e⁺+ e^- = a (2.6) a ≤ y ≤ b

d⁺, d^- , e⁺, e^- ≥ 0

xij ≥ 0 and are integers, i=1,2,…,m, j=1,2,…,n, 3. SOLUTION OF THE PROBLEM

Problems (2.4) and (2.6) are linear programming problem with fuzzy and crisp constraints as Minimize f^Tx

Subject to Ax  B

Dx (≤ or = ) h (crisp constraints) (3.1) x ≥ 0

Where B = [b₁,b₂,b₃,b₄] is a column vector of trapezoidal fuzzy numbers. This problem can be solved by generalizing the method by. Let ^* be the maximum degree of satisfaction of the fuzzy constraints of problem (2.7).

The membership function of the objective function of problem (2.7) can be determined by solving the following two linear programming problems.

Minimize f^Tx Subject to Ax ≤ d – (d-c)^*

Ax ≥ a + (b-a)* (3.2) Dx (≤ or = ) h

x ≥ 0

yielding the optimal value f₁ and Minimize f^Tx

Subject to Ax ≤ d

Ax ≥ a (3.3) Dx (≤ or = ) h

x ≥ 0

yielding the optimal value f0.

The membership function of the objective function of problem (2.7) is therefore 1, if f^Tx ≤ f0

f(x) = (f^Tx – f1)/(f0 – f1), if f0 ≤ fTx ≤ f1

0, if f^Tx ≥ f1

By applying fuzzy programming technique, we get Maximize 

Subject to (f₁ – f₀) + f^Tx ≥ f1

(d-c) + Ax ≤ d

-(b-a) + Ax ≥ a (3.4) Dx (≤ or = ) h

x ≥ 0

The optimal solution of the above LP gives the optimal solution to the considered fuzzy transportation problem.

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4. NUMERICAL EXAMPLES:

Example 1: ( Case with discrete choices)

c(x) = 2x₁₁ + 3x₁₂ + 4x₂₁ + 2x₂₂ = {9,10,11,12}

Subject to

x₁₁ + x₁₂ [1,2,3,4]

x₂₁ + x₂₂  [2,3,5,6] (4.1) x11 + x21  [1,2,3,5]

x₁₂ + x₂₂  [2,4,5,6]

x_ij ≥ 0 for i=1,2,j=1,2 and are integers.

The goal programming formulation using (2.4) is

Min d₁⁺ + d₁^- + d₂⁺ + d₂^- Subject to

x₁₁ + x₁₂ [1,2,3,4]

x₂₁ + x₂₂  [2,3,5,6]

x₁₁ + x₂₁  [1,2,3,5] (4.2) x₁₂ + x₂₂  [2,4,5,6]

2x₁₁ + 3x₁₂ + 4x₂₁ + 2x₂₂ - d₁⁺ + d₁^- -  = 0  + 2z₁ + z₂ = 12

 + 3d₂⁺-3d₂^- = 9

We solve the following two linear programming problems from (3.2) and (3.3)

x₁₁ + x₁₂≤ 3 x₁₁ + x₁₂≥ 2 x₂₁ + x₂₂ ≤ 5 x₂₁ + x₂₂ ≥ 3

x₁₁ + x₂₁ ≤ 3 (4.3) x₁₁ + x₂₁ ≥ 2

x₁₂ + x₂₂ ≤ 5 x₁₂ + x₂₂ ≥ 4

2x₁₁ + 3x₁₂ + 4x₂₁ + 2x₂₂ - d₁⁺ + d₁^- -  = 0  + 2z₁ + z₂ = 12

 + 3d2+

-3d2-

= 9

, d₁⁺,d₁^-,d₂⁺,d₂^- ≥ 0, z₁ = 0/1, z₂ = 0/1

The optimal solution of this linear programming problem is x₁₁ = 2, x₂₂ = 4, d₂^- = 4,  = 12 with the optimal value of the objective function f₁ = 1

x₁₁ + x₁₂≤ 4 x₁₁ + x₁₂≥ 1 x₂₁ + x₂₂ ≤ 6

x₂₁ + x₂₂ ≥ 2 (4.4) x₁₁ + x₂₁ ≤ 5

x₁₁ + x₂₁ ≥ 1 x₁₂ + x₂₂ ≤ 6 x₁₂ + x₂₂ ≥ 2

2x₁₁ + 3x₁₂ + 4x₂₁ + 2x₂₂ - d₁⁺ + d₁^- -  = 0  + 2z₁ + z₂ = 12

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137  + 3d2+

-3d2-

= 9

, d₁⁺,d₁^-,d₂⁺,d₂^- ≥ 0, z₁ = 0/1, z₂ = 0/1

The optimal solution of this linear programming problem is x12 = 1, x21 = 1, x22 = 1, d2^- = 4,  = 12 with the optimal value of the objective function f₀ = 0. Hence we have, from (3.4)

Maximize  Subject to

 + d₁⁺ + d₁^- + d₂⁺ + d₂^- ≥ 1 x₁₁ + x₁₂+ ≤ 4

x11 + x12 - ≥ 1 x₂₁ + x₂₂ +  ≤ 6

x₂₁ + x₂₂ -  ≥ 2 (4.5) x₁₁ + x₂₁ + 2 ≤ 5

x₁₁ + x₂₁ -  ≥ 1 x₁₂ + x₂₂ +  ≤ 6 x₁₂ + x₂₂ -2≥ 2

2x₁₁ + 3x₁₂ + 4x₂₁ + 2x₂₂ - d₁⁺ + d₁^- -  = 0  + 2z₁ + z₂ = 12

 + 3d₂⁺-3d₂^- = 9

, d₁⁺,d₁^-,d₂⁺,d₂^- ≥ 0, z₁ = 0/1, z₂ = 0/1

Hence the optimal solution of problem (4.1) is x₁₁ = 2, x₂₂ = 4 with the total transportation cost being 12 and with degree of satisfaction  = 1.

Example 2: (Case with continuous choice)

c(x) = 2x₁₁ + 3x₁₂ + 4x₂₁ + 2x₂₂ ≥ 8 and ≤ 10 Subject to

x₁₁ + x₁₂ [1,2,3,4]

x₂₁ + x₂₂  [2,3,5,6] (4.6) x₁₁ + x₂₁  [1,2,3,5]

x₁₂ + x₂₂  [2,4,5,6]

penalty 2 is assigned for exceeding the cost goal.

The goal programming formulation is

Min 2d⁺ + e⁺ + e^- Subject to

x₁₁ + x₁₂ [1,2,3,4]

x21 + x22  [2,3,5,6]

x₁₁ + x₂₁  [1,2,3,5] (4.7) x₁₂ + x₂₂  [2,4,5,6]

2x₁₁ + 3x₁₂ + 4x₂₁ + 2x₂₂ – d⁺ + d ^- - y = 0 y – e⁺ + e^- = 8

8 ≤ y ≤ 10

xij ≥ 0 for i=1,2,j=1,2 and are integers.

d⁺, d^-,e⁺, e^- ≥ 0

We solve the following two linear programming problems from (3.2) and (3.3) Min 2d⁺ + e⁺ + e^-

Subject to x₁₁ + x₁₂≤ 3

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138 x₁₁ + x₁₂≥ 2

x₂₁ + x₂₂ ≤ 5 x₂₁ + x₂₂ ≥ 3

x₁₁ + x₂₁ ≤ 3 (4.8) x₁₁ + x₂₁ ≥ 2

x12 + x22 ≤ 5 x₁₂ + x₂₂ ≥ 4

2x₁₁ + 3x₁₂ + 4x₂₁ + 2x₂₂ – d⁺ + d ^- - y = 0 y – e⁺ + e^- = 8

8 ≤ y ≤ 10

d⁺, d^-,e⁺, e^- ≥ 0

The optimal value of the objective function is f₁ = 6. Then we have Min 2d⁺ + e⁺ + e^-

Subject to x₁₁ + x₁₂≤ 4 x₁₁ + x₁₂≥ 1 x21 + x22 ≤ 6 x₂₁ + x₂₂ ≥ 2

x₁₁ + x₂₁ ≤ 5 (4.9) x₁₁ + x₂₁ ≥ 1

x12 + x22 ≤ 6 x₁₂ + x₂₂ ≥ 2

2x₁₁ + 3x₁₂ + 4x₂₁ + 2x₂₂ – d⁺ + d ^- - y = 0 y – e⁺ + e^- = 8

8 ≤ y ≤ 10

d⁺, d^-,e⁺, e^- ≥ 0

The optimal value of the objective function is f₀ = 0. Hence we have, from (3.4) Maximize 

Subject to

6 + 2d⁺ + e⁺ + e^- ≥ 6 x₁₁ + x₁₂+ ≤ 4 x₁₁ + x₁₂- ≥ 1 x₂₁ + x₂₂ +  ≤ 6 x₂₁ + x₂₂ -  ≥ 2

x₁₁ + x₂₁ + 2 ≤ 5 (4.10) x₁₁ + x₂₁ -  ≥ 1

x₁₂ + x₂₂ +  ≤ 6 x12 + x22 -2≥ 2

2x₁₁ + 3x₁₂ + 4x₂₁ + 2x₂₂ – d⁺ + d ^- - y = 0 y – e⁺ + e^- = 8

8 ≤ y ≤ 10

d⁺, d^-,e⁺, e^- ≥ 0

The optimal solution of problem (4.6) is x₁₁ = 2, x₂₂ = 4 with the total transportation cost being 12 and degree of satisfaction  = 1.

5. CONCLUSIONS

In the present paper, goal programming models are constructed for two transportation problems with fuzzy demand and supply values, the cost goal assuming a set of discrete values and a range of values respectively. A method is proposed to find the optimal solution of the resultant linear programming problem with crisp and fuzzy constraints and hence that of the transportation problem under consideration. Two numerical examples with the same numerical data were considered in both the cases. The considered example gives the same solution for both the cases. However, it is noticed that the goal is exceeded in the second case which has also happened for Chang[11].

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139 6. ACKNOWLEDGEMENT

The second author expresses his sincere thanks to the Council of Scientific and Industrial Research(CSIR), India for providing financial support for this research in the form of a senior research fellowship(Grant No.09/922(0001)/2006-EMR-1)

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