Ann Oper Res (2006) 143: 305–315 DOI 10.1007/s10479-006-7390-1 Fuzzy linear programs with trapezoidal fuzzy numbers K. Ganesan · P. Veeramani C Springer Science + Business Media, Inc. 2006 Abstract The objective of this paper is to deal with a kind of fuzzy linear programming problem involving symmetric trapezoidal fuzzy numbers. Some important and interesting results are obtained which in turn lead to a solution of fuzzy linear programming problems without converting them to crisp linear programming problems. Keywords Fuzzy numbers . Ranking . Fuzzy linear programming AMS subject classification: 90C05 . 90C70 Bellman and Zadeh (1970) proposed the concept of decision making in fuzzy environments. After the pioneering work on fuzzy linear programming by Tanaka et al. (1974, 1984) and Zimmermann (1974), sever al kinds of fuzzy linear programming problems have appeared in the literature and diff erent methods hav e been proposed to solve such problems. Numerous meth ods for comp aris on of fuzz y numbe rs have been sugges ted in the literat ure. Male ki, Tata and Mashinchi (1996, 2000) us ed the Rouben’s me thod of comparison of fuzzy numbers and obtained an optimal solution. In this paper, we introduce a new type of fuzzy arithmetic for symmetric trapezoidal fuzzy numbers and propose a method for solving fuzzy linear programming problems without converting them to crisp linear programming problems. This paper is organized as follows: In section 1, we give the definitions of fuzzy linear programming, symmetric trapezoid al fuzzy numbers and some related results of fuzzy arith- metic on symmetric trapezoidal fuzzy numbers. In section 2, we prove fuzzy analogues ofsome imp ort ant the ore ms of lin ear pro grammi ng. A numeri cal example in vo lvi ng symmet ric trapezoidal fuzzy numbers is also given to illustrate the theory developed in this paper. K. Ganesan () Department of Mathematics, S. R. M Institute of Science and Technology, Deemed University, Chennai–603 203, India e-mail: gansan [email protected]P. Veeramani Department of Mathematics, Indian Institute of Technology, Madras, Chennai–600 036, India e-mail: [email protected]. in Springer
12
Embed
Fuzzy Linear Programs With Trapezoidal Fuzzy Numbers
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
8/3/2019 Fuzzy Linear Programs With Trapezoidal Fuzzy Numbers
Fuzzy linear programs with trapezoidal fuzzy numbers
K. Ganesan · P. Veeramani
C Springer Science + Business Media, Inc. 2006
Abstract The objective of this paper is to deal with a kind of fuzzy linear programming
problem involving symmetric trapezoidal fuzzy numbers. Some important and interesting
results are obtained which in turn lead to a solution of fuzzy linear programming problems
without converting them to crisp linear programming problems.
Keywords Fuzzy numbers . Ranking . Fuzzy linear programming
AMS subject classification: 90C05 . 90C70
Bellman and Zadeh (1970) proposed the concept of decision making in fuzzy environments.
After the pioneering work on fuzzy linear programming by Tanaka et al. (1974, 1984) and
Zimmermann (1974), several kinds of fuzzy linear programming problems have appeared in
the literature and different methods have been proposed to solve such problems. Numerous
methods for comparison of fuzzy numbers have been suggested in the literature. Maleki,
Tata and Mashinchi (1996, 2000) used the Rouben’s method of comparison of fuzzy numbers
and obtained an optimal solution. In this paper, we introduce a new type of fuzzy arithmeticfor symmetric trapezoidal fuzzy numbers and propose a method for solving fuzzy linear
programming problems without converting them to crisp linear programming problems.
This paper is organized as follows: In section 1, we give the definitions of fuzzy linear
programming, symmetric trapezoidal fuzzy numbers and some related results of fuzzy arith-
metic on symmetric trapezoidal fuzzy numbers. In section 2, we prove fuzzy analogues of
some important theorems of linear programming. A numerical example involving symmetric
trapezoidal fuzzy numbers is also given to illustrate the theory developed in this paper.
K. Ganesan ()Department of Mathematics, S. R. M Institute of Science and Technology, Deemed University,
In this paper we restrict ourself to symmetric trapezoidal fuzzy numbers due to the com-
plication involved in the multiplication of general fuzzy numbers. It is to be noted that if
a, b, c are intervals, then c(a + b) need not be even equivalent to (ca + cb), under the existing
interval arithmetic (and hence also for symmetric trapezoidal fuzzy numbers). But by using
the modified fuzzy arithmetic for symmetric trapezoidal fuzzy numbers we can establish thedistributive law (up to equivalence) and hence we are able to prove the main results of this
paper.
1. Preliminaries
The aim of this section is to present some notations, notions and results which are of useful
in our further consideration.
Definition 1.1. A fuzzy set a on R is said to be a symmetric trapezoidal fuzzy number if there exist real numbers a1, a2, a1 ≤ a2 and h > 0 such that
a( x) =
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
x
h+
h − a1
h, for x ∈ [a1 − h, a1]
1, for x ∈ [a1, a2]
− x
h+
a2 + h
h, for x ∈ [a2, a2 + h]
0, otherwise
We denote it by a = [a1, a2, h, h]. When h = 0; a = [a1, a2]. We use F (S) to denote the set
of all symmetric trapezoidal fuzzy numbers.
Let a = [a1, a2, h, h] and b = [b1, b2, k , k ] be two symmetric trapezoidal fuzzy numbers.
Then the arithmetic operations on a and b are given by:
(i) Addition: a + b = [a1, a2, h, h] + [b1, b2, k , k ] = [a1 + b1, a2 + b2, h + k , h + k ].
(ii) Subtraction: a − b = [a1, a2, h, h] − [b1, b2, k , k ] = [a1 − b2, a2 − b1, h + k , h + k ].
(iii) Multiplication: ab = [a1, a2, h, h][b1, b2, k , k ]
Definition 1.3. For any symmetric trapezoidal fuzzy number ˜ x , let us define ˜ x 0 if there
exist a ≥ 0 and h ≥ 0 such that ˜ x [−a, a, h, h]. We also denote [−a, a, h, h] by 0. Note
that 0 is equivalent to [0, 0, 0, 0] = 0. It is easy to see that if ˜ x ˜ y, then ( ˜ x − ˜ y) 0.
Remark 1.3. If ˜ x ≈ 0, then ˜ x is said to be a zero symmetric trapezoidal fuzzy number. Itis to be noted that if ˜ x = 0, then ˜ x ≈ 0, but the converse need not be true. If ˜ x ≈ 0 (that
is ˜ x is not equivalent to 0), then ˜ x is said to be a non-zero symmetric trapezoidal fuzzy
number. It is to be noted that if ˜ x ≈ 0, then ˜ x = 0, but the converse need not be true. If ˜ x 0
and ˜ x ≈ 0, then ˜ x is said to be a positive symmetric trapezoidal fuzzy number and is denoted
by ˜ x 0.
Definition 1.4. Let F (S) be the set of all symmetric trapezoidal fuzzy numbers.
The model
max ˜ z ≈n
j=1
c j ˜ x j
subject to
n j=1
aij
˜ x j bi , i = 1, 2, 3, · · · , m0,
n j =1
aij
˜ x j bi , i = m0 + 1, m0 + 2, m0 + 3, · · · , m (1)
and ˜ x j 0 for all j = 1, 2, 3, · · · , n,
where aij
∈ R, c j , ˜ x j , bi ∈ F (S), i = 1, 2, 3, · · · , m, j = 1, 2, 3, · · · , n
is called a fuzzy linear programming problem.
Definition 1.5. Any x = ( ˜ x1, ˜ x2, ˜ x3, · · · , ˜ xn ) ∈ F n (S), where each ˜ x i ∈ F (S), which satisfies
the constraints and non-negativity restrictions of (1) is said to be a fuzzy feasible solution to (1).
Definition 1.6. Let Q be the set of all fuzzy feasible solutions of (1). A fuzzy feasible
solution x0 ∈ Q is said to be a fuzzy optimum solution to (1) if cx0 cx for all x ∈ Q,
where c = (c1, c2, · · · , cn ) and cx = c1 ˜ x1 + c2 ˜ x2 + · · · + cn ˜ xn .
Definition 1.7. Let x = ( ˜ x1, ˜ x2, ˜ x3, · · · , ˜ xn ). Suppose x solves Ax ≈ b. If all
˜ x j ≈ [−α j , α j , h j , h j ] for some α j ≥ 0and h j ≥ 0, then x issaidtobea fuzzy basic solution.
If ˜ x j ≈ [−α j , α j , h j , h j ] for α j ≥ 0 and h j ≥ 0, then x has some non-zero components, say
˜ x1, ˜ x2, ˜ x3, · · · , ˜ xk , 1 ≤ k ≤ n. Then Ax ≈ b can be written as:
a1 ˜ x1 + a2 ˜ x2 + a3 ˜ x 3 + · · · + ak ˜ xk + ak +1[−βk +1, βk +1, hk +1, hk +1]
+ ak +2[−βk +2, βk +2, hk +2, hk +2] + · · · + an [−βn , βn , hn , hn ] ≈ b.
If the columns a1, a2, a3, · · · , ak corresponding to these non-zero components ˜ x1, ˜ x2, · · · , ˜ xk
are linearly independent, then x is said to be a fuzzy basic solution.
Remark 1.4. Given a system of m simultaneous fuzzy linear equations involving symmetric
trapezoidal fuzzy numbers in n unknowns (m ≤ n) Ax ≈ b ; b ∈ F m (S), where A i s a (m × n)
real matrix and rank of A is m.Let B beany(m × m) matrix formed by m linearly independent
Springer
8/3/2019 Fuzzy Linear Programs With Trapezoidal Fuzzy Numbers
We have seen that for a column vector a j of A which is not in B, for which (˜ z j − c j ) ≺ 0 and
yij
> 0, for some i , is alone considered for inserting into the basis. Let us now discuss the sit-
uation when there exists an a j such that (˜ z j − c j ) ≺ 0 and yij ≤ 0, for all i = 1, 2, 3, · · · , m.If ˜ x = [ x1, x2, h, h] 0 and λ > 0, then λ ˜ x = [λ x1, λ x2, λh, λh] 0. Now λ can be
made sufficiently large so that λ ˜ x ˜ y for any symmetric trapezoidal fuzzy number ˜ y. If
λ > 0, (˜ z j − c j ) ≺ 0, then λ(˜ z j − c j ) ≺ 0. Now the proof of the following theorem follows
easily.
Theorem 2.3. Let x B = B−1b be a fuzzy basic feasible solution of (2). If there exist an a j of
A which is not in B such that (˜ z j − c j ) ≺ 0 and yij
≤ 0 , for all i, i ∈ {1, 2, 3, · · · , m} then
the fuzzy linear programming problem (2) has an unbounded solution.
2.3. Conditions of optimality
As in the classical linear programming problems, we can prove that the process of inserting
and removing vectors from the basis matrix will lead to any one of the following situations:
(i). there exist j such that (˜ z j − c j ) ≺ 0, yij
≤ 0, i = 1, 2, 3, · · · , m or
(ii). for all j , (˜ z j − c j ) 0.
In the first case, we get an unbounded solution and if the second case occurs , it is easy to
show that the fuzzy linear programming problem has a fuzzy optimal solution.
Theorem 2.4. If x B = B−1b is a fuzzy basic feasible solution of (2) and if (˜ z j − c j ) 0 for
every column a j of A, then x B is a fuzzy optimal solution to (2).
2.4. A numerical example
A company produces three products P1, P2 and P3. These products are processed on three
different machines M 1, M 2 and M 3. The time required to manufacture one unit of each
product and the daily capacity of the machines are given below:
Time per unit (minutes) Machine capacity
Machines P1 P2 P3 (min/day)
M 1 12 13 12 490
M 2 14 − 13 470
M 3 12 15 − 480
Note that the time availability can vary from day to day due to break down of machines,
overtime work etc. Finally the profit for each product can also vary due to variations in price.
At the same time the company wants to keep the profit somewhat close to Rs .14 for P1,
Rs .13 for P2 and Rs .16 for P3. The company wants to determine the range of each product
to be produced per day to maximize its profit. It is assumed that all the amounts produced
are consumed in the market.
Springer
8/3/2019 Fuzzy Linear Programs With Trapezoidal Fuzzy Numbers
Since the profit from each product and the time availability on each machine are uncertain,
the number of units to be produced on each product will also be uncertain. So we will model
the problem as a fuzzy linear programming problem. We use symmetric trapezoidal fuzzy
numbers for each uncertain value.
Profit for P1 which is close to 14 is modelled as [13, 15, 2, 2]. Similarly the other parame-ters are also modelled as symmetric trapezoidal fuzzy numbers taking into account the nature
of the problem and other requirements. So we formulate the given fuzzy linear programming
Here (˜ z j − c j ) 0, for all j . Hence by theorem (2.4), the current fuzzy basic feasible
solution is a fuzzy optimal solution.
Acknowledgement The authors would like to thank the referees for their critical comments and valuable
suggestions which helped the authors to improve the presentation of this paper. Also the authors would like tothank Professor Dan Butnariu, Department of Mathematics, University of Haifa, Israel, for the critical reading
of the preliminary version of this paper and for many valuable suggestions for the improvement of this paper.
The first author would like to thank his supervisor Professor S.Elumalai for his valuable suggestions.
References
Bellman, R.E. and L.A. Zadeh. (1970). Decision Making in fuzzy environment. Management science 17,
141–164.
Maleki, H.R., M. Tata and M. Mashinchi. (1996). Fuzzy number Linear Programming. Proc Internat Conf on
intelligent and Cognitive Systems FSS Sponsored by IEE and ISRF, Tehran, Iran 145–148.Maleki, H.R., M. Tata and M. Mashinchi. (2000). Linear Programming with Fuzzy Variables. Fuzzy Sets and
Systems 109, 21–33.
Tanaka, H., T. Okuda, and K. Asai. (1974). On Fuzzy Mathematical Programming. Journal of Cybernetics 13,
37–46.
Tanaka, H. and K. Asai. (1984). Fuzzy Linear Programming Problems with Fuzzy numbers. Fuzzy Sets and
Systems 13, 1–10.
Zimmermann, H.J. (1974). Optimization in fuzzy environment. presented at XXI Int. TIMES and 46th ORSA