International Journal of Mathematical Archive-4(3), 2013, 63-66
Available online through www.ijma.info ISSN 2229 – 5046
A NEW METHODOLOGY FOR SOLVING FUZZY
LINEAR PROGRAMMING PROBLEM IN TEA MANUFACTURING PROCESS
1
R. Karthika
Assistant Prof, Dept of Mathematics, SNMV College of Arts & Science, Coimbatore-21, India
2
A. SahayaSudha
Assistant Prof, Dept of Mathematics, Nirmala College for Women, Coimbatore-18, India
(Received on: 04-02-12; Revised & Accepted on: 27-02-13)
ABSTRACT
There are invariable complex theories associated to formulate a new strategy. In order to give a clear picture on the
complex doctrines a research paper is absolutely necessary. Hence this paper is prepared with the broad outlook to get
into the new decomposition method used for solving fuzzy linear programming problems. For this a classical linear
programming problem has been proposed using decomposition method. A new methodology has been adopted to
include the practical side of issues pertaining to Tea manufacturing.
Keywords: Fuzzy linear programming, Fuzzy number, Triangular Fuzzy, Linear Programming.
1. INTRODUCTION:
Fuzzy set theory is an extension of classical set theory where elements have degrees of membership. Fuzzy sets have
been introduced by Lotfi A Zadeh [16] (1965) and Dieter Klaua [8] (1965). Interval Arithmetic was first suggested by
Dwyer [5] in 1951. The usual Arithmetic operations on real numbers can be extended to the ones defined on fuzzy
numbers by means of Zadeh’s extension principle [16, 17].
In 1978 D. Dubois and H. Prade defined any of the fuzzy numbers as a fuzzy subset of the real line [4]. A fuzzy number
is a quantity whose values is imprecise, rather than exact as is the case with single valued numbers [6-7, 9-10]. H.M
.Lee and L. Lin [12] weighted Triangular fuzzy numbers to tackle the rate of aggregative risk in fuzzy circumstances.
Zimmerman [18] presented a fuzzy approach to multi objective linear programming problems. Fuzzy linear
programming problem with fuzzy coefficients was formulated by Negoita [13] and called robust programming. Tanaka
and Asai [14] also proposed a formulation of fuzzy linear programming with fuzzy constraints and gave a method for
its solution which bases on inequality relation between fuzzy numbers.
To have an effective and meaningful solution to the fuzzy linear programming problem we place a new approach to the
long standing unsolved problems on Decomposition method. There is a vast gap between theoretical analysis and
practical analysis, hence fuzzy linear programming (FLP) was introduced and many researchers are used to get for their
further research and specialization. Fuzzy set theory has been applied to various disciplines. The concept of FLP
problems suggested several approaches for solving these difficult problems.
This paper has been prepared with lots of facts and figures basing on the FLP and the concluding part of this reveals
that there are lots of chances to make use of this theory in various new problems. This Technique of FLP will solve
many difficult problems.
2. PRELIMINARIES: [1, 4, 12]
Definition: 2.1: A fuzzy number A is a convex normalized fuzzy set on the real line R such that:
1) There exists at least one x0 ∈ R with μA (x0) = 1.
2) μA (x) is piecewise continuous.
Definition: 2.2: A fuzzy number ã is a triangular fuzzy number denoted by (𝑎𝑎1 , 𝑎𝑎2 , 𝑎𝑎3 ) where 𝑎𝑎1 , 𝑎𝑎2 and 𝑎𝑎3 are real
numbers and its membership function is given below.
(𝑥𝑥−𝑎𝑎 1 )
𝑓𝑓𝑓𝑓𝑓𝑓 𝑎𝑎1 ≤ 𝑥𝑥 ≤ 𝑎𝑎2
⎧
μ ã (x) =
(𝑎𝑎 2 −𝑎𝑎 1 )
(𝑎𝑎 3 −𝑥𝑥)
⎨(𝑎𝑎 3 − 𝑎𝑎 2)
⎩ 0
𝑓𝑓𝑓𝑓𝑓𝑓 𝑎𝑎2 ≤ 𝑥𝑥 ≤ 𝑎𝑎3
𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
Corresponding author: 2A. SahayaSudha*
Assistant Prof, Dept of Mathematics, Nirmala College for Women, Coimbatore-18, India
International Journal of Mathematical Archive- 4(3), March – 2013
63
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R. Karthika & A. SahayaSudha*/ A NEW METHODOLOGY FOR SOLVING FUZZY LINEAR PROGRAMMING PROBLEM IN TEA
MANUFACTURING PROCESS/IJMA- 4(3), March.-2013.
Definition: 2.3: Let A= (𝑎𝑎1 , 𝑎𝑎2 , 𝑎𝑎3 ) and B= ( 𝑏𝑏1 , 𝑏𝑏2 , 𝑏𝑏3 ) be two triangular fuzzy numbers. Then the following are the
basic operations that can be performed on triangular fuzzy numbers
(a) A +B = (𝑎𝑎1 , 𝑎𝑎2 , 𝑎𝑎3 ) + (𝑏𝑏1 , 𝑏𝑏2 , 𝑏𝑏3 ) = (𝑎𝑎1 + 𝑏𝑏1 , 𝑎𝑎2 + 𝑏𝑏2 , 𝑎𝑎3 + 𝑏𝑏3 )
(b) A –B =(𝑎𝑎1 , 𝑎𝑎2 , 𝑎𝑎3 ) - (𝑏𝑏1 , 𝑏𝑏2 , 𝑏𝑏3 ) = (𝑎𝑎1 − 𝑏𝑏1 , 𝑎𝑎2 − 𝑏𝑏2 , 𝑎𝑎3 − 𝑏𝑏3 )
(c) kA =k (𝑎𝑎1 , 𝑎𝑎2 , 𝑎𝑎3 ) = (𝑘𝑘𝑎𝑎1 , 𝑘𝑘𝑘𝑘2 , 𝑘𝑘𝑎𝑎3 ) for k ≥ 0.
(d) kA = k (𝑎𝑎1 , 𝑎𝑎2 , 𝑎𝑎3 ) = (𝑘𝑘𝑎𝑎3 , 𝑘𝑘𝑘𝑘2 , 𝑘𝑘𝑎𝑎1 ) for k < 0.
Definition: 2.4: Let 𝐴𝐴̃ = (𝑎𝑎1 , 𝑎𝑎2 , 𝑎𝑎3 ) be in F( R) then
(i) 𝐴𝐴̃ is said to be positive if 𝑎𝑎𝑖𝑖 ≥ 0, for all i= 1 to 3;
(ii) 𝐴𝐴̃ is said to be integer if 𝑎𝑎𝑖𝑖 ≥ 0, for all i = 1 to 3 are integers and
(iii) 𝐴𝐴̃ is said to be symmetric if 𝑎𝑎2 − 𝑎𝑎1 = 𝑎𝑎3 − 𝑎𝑎2 .
Definition: 2.5: A matrix A is called nonnegative and denoted by A ≥ 0 if each element of A be a nonnegative number.
Definition: 2.6:A fuzzy vector 𝑏𝑏� = (𝑏𝑏�𝑖𝑖 )mx1 is called non negative and denoted by 𝑏𝑏� ≥ 0, if each element of 𝑏𝑏� be a
nonnegative fuzzy number, that is 𝑏𝑏�𝑖𝑖 ≥ 0. where i = 1,2,3……
3. FUZZY LINEAR PROGRAMMING
Consider the following linear programming problem
Max 𝑧𝑧= c𝑥𝑥�
Subject to
A𝑥𝑥� ≤ 𝑏𝑏� ,
and 𝑥𝑥�𝑖𝑖 ≥ 0, i = 1,2,…..
where the coefficient matrix A = (𝑎𝑎𝑖𝑖𝑖𝑖 )m x n is a nonnegative real crisp matrix.
The cost vector c = (𝑐𝑐1 … … … . 𝑐𝑐𝑛𝑛 ) is nonnegative crisp vector and 𝑥𝑥� = (𝑥𝑥�𝑗𝑗 )nx1 and 𝑏𝑏� = (𝑏𝑏�𝑖𝑖 )mx1 are non negative real
fuzzy vectors such that 𝑥𝑥� j , 𝑏𝑏�i ∈ F(R) for all 1 ≤ j ≤ n and 1 ≤ i ≤ m.
4. CASE STUDY
The case study on Tea manufacturing Industry throws lot of new technical and meaningful derivations. The present
study illustrates the in and out of the tea manufacturing statistical points that has been neatly arranged with the
available facts.The gap between the theoretical and practical must be united and there by a new answer will give a
detailed discussion of the variables and real variables.
The process of the problem is as follows. Tea leaves plucked from the garden is sent to the factory for processing and
the profit made by the tea leaves of the two grades A and B are Rs 21 per Kg and Rs 36 per Kg. respectively.Each
grade leaves of A and B are processed by drying and fermenting .The availability of the machines per week is 3600
mins and 3000 mins in a week. The data are as follows,
Drying (mins)
Fermenting (mins)
Profit
Leaves of grade A
6
2
21
Leaves of grade B
6
3
36
Availability of machines in a week
3600
3000
Here, 𝑥𝑥1 denotes the leaves of grade A and 𝑥𝑥2 denotes the leaves of grade B
Also 𝑃𝑃� = (P1 , P2 , P3 ) , where the membership value reaches the highest point at P2 while P1 and P3 denotes the lower
bound and upper bound of the fuzzy set P.
Hence the above problem can be as formulated as linear programming as follows
Max Z = 21𝑥𝑥1 + 36 𝑥𝑥2
Sub to
6𝑥𝑥1 + 6𝑥𝑥2 ≤ 3600
2𝑥𝑥1 + 3 𝑥𝑥2 ≤ 3000
and 𝑥𝑥1 , 𝑥𝑥2 ≥ 0
© 2013, IJMA. All Rights Reserved
64
1
2
R. Karthika & A. SahayaSudha*/ A NEW METHODOLOGY FOR SOLVING FUZZY LINEAR PROGRAMMING PROBLEM IN TEA
MANUFACTURING PROCESS/IJMA- 4(3), March.-2013.
THE ALGORITHM
Following are the steps to illustrate the above case study problem.
Step 1: Solve the problem as linear programming problem.
Step 2: From Step-1, determine the corresponding values of the Objective function
Step 3: The above problem is converted into crisp linear programming problem.
Step 4: Solve the problem until we get an Optimum Solution
Step 5: Determine the final Optimal solution as z1, z2 , z3
Based on triangular fuzzy numbers the above formulation can be illustrated as fuzzy linear programming as follows,
Consider
Max Z = 21𝑥𝑥1 ⨁36 𝑥𝑥2
Sub to
6𝑥𝑥1 ⨁ 6𝑥𝑥2 ≤ (3200, 3600, 4000)
2𝑥𝑥1 ⨁ 3 𝑥𝑥2 ≤ (2800, 3000, 3200)
𝑥𝑥1 , 𝑥𝑥2 ≥ 0.
𝑏𝑏2 = (2800, 3000, 3200)
where 𝑏𝑏�1 = (3200, 3600, 4000) and �
Solution: Now the problem for (P2 ) is given below
(P2 ) Max Z = 21𝑥𝑥1 + 36 𝑥𝑥2
Sub to
6𝑥𝑥1 + 6𝑥𝑥2 ≤ 3600
2𝑥𝑥1 + 3 𝑥𝑥2 ≤ 3000
and 𝑥𝑥1 , 𝑥𝑥2 ≥ 0
Solving the above problem using simplex method,
The solution of (P2) is x1 = 0 and 𝑥𝑥2 = 600 and Max Z2 = 21600
Now the problem for (P1) is given below:
(P1) Max Z = 21𝑥𝑥1 + 36 𝑥𝑥2
Sub to
6𝑥𝑥1 + 6 𝑥𝑥2 ≤ 3200
2𝑥𝑥1 + 3 𝑥𝑥2 ≤ 2800
and 𝑥𝑥1 , 𝑥𝑥2 ≥ 0
Similarly using simplex method, we obtain the optimal solution of (P1).
We obtain 𝑥𝑥1 = 0 and 𝑥𝑥2 = 1600/3 Max Z1 = 19200
Now the problem for (P3) is given below:
(P3) Max Z = 21𝑥𝑥1 + 36 𝑥𝑥2
Sub to
6𝑥𝑥1 + 6𝑥𝑥2 ≤ 4000
2𝑥𝑥1 + 3 𝑥𝑥2 ≤ 3200
and 𝑥𝑥1 , 𝑥𝑥2 ≥ 0
Similarly using simplex method, we obtain the optimal solution of (P3)
We obtain 𝑥𝑥1 = 0 and 𝑥𝑥2 = 2000/3 Max Z3 = 24000
Therefore the solution for the given fuzzy linear programming problem is
𝑃𝑃�
𝑥𝑥1
P2
0
P1
P3
© 2013, IJMA. All Rights Reserved
0
0
𝑥𝑥2
1600
3
600
2000
3
Z
19200
21600
24000
65
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R. Karthika & A. SahayaSudha*/ A NEW METHODOLOGY FOR SOLVING FUZZY LINEAR PROGRAMMING PROBLEM IN TEA
MANUFACTURING PROCESS/IJMA- 4(3), March.-2013.
5. CONCLUSION
Based on the analysis of the results of a practical problem in Tea manufacturing Industry, the following conclusions are
drawn.
• Fuzzy Linear Programming is simple and suitable tool when compared to other methods.
• This model can be extended to any number of objectives.
• This model can be extended to any situation not only to Tea Manufacturing but in any field such as Engineering,
Agriculture with little or more modifications.
• Analysis of results indicated that 𝑧𝑧1 = 19200, 𝑧𝑧2 = 21600, 𝑧𝑧3 = 24000
• On analyzing the result, the company can plan their production, profit and other parameters related in a particular
product.
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Source of support: Nil, Conflict of interest: None Declared
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