A STUDY OF INTUITIONSTIC FUZZY OPTIMIZATION TECHNIQUE FOR THE SOLUTION OF SINGLE AND MULTI OBJECTIVE COST INVENTORY SYSTEM IN MANUFACTURING
Meena p Dr.N.Srinivasan
Research Scholar, Professor & Head Of Mathematics,
Department Of Mathematics Department Of Mathematics,
St.Peters Institute of Higher St.Peters Institute of Higher
Education and Research, Education and Research,
Avadi, Chennai,India, Avadi, Chennai,India,
Meena11mathi11@gmail.com. Sri24455@yahoo.com
ABSTRACT
This paper is present to establish that intuitionistic optimization method is better than usual fuzzy optimization technique as expected annual cost of this inventory system in manufacturing is more minimized in case of intuitionistic fuzzy optimization method.
KEYWORDS:Inventory,fuzzy set, fuzzy optimization, multi objective system.
Introduction
Today most of the real-world decision making problems in economic, technical and environmental ones or multidimensional and multiobjective .It is significant to realize that multiobjectives are often noncommensurable and conflict with each other in optimization problem. The objective exact target value is referred as fuzzy goal.
DEFINITIONS
Intuitionistic Fuzzy set
An IFS A in X is given by where the functions. : X ‘ [0,1] define respectively, the degree of membership and degree of Non-membership of the element x ‘ X to the set A, which is a subset of X, and for every x ‘ X, 0 ‘ ‘(x) + ‘(x) ‘ 1.
Cost Components
This approach is used for the compilation of output construction price indices. It regards construction output as bundles of standardized homogeneous components. These components corresponds to the supply of standard operations.
Stochastic Process
A statistical process involving a number of random variables depending a variable parameter . A stochastic process is a collection of random variables indexed by time. It will be useful to consider separately the cases of discrete time and continuous time. a process of which the outcome appears to be unpredictable.
Intutionistic fuzzy programming technique to solve multi-objective linear programming problem
We first find the lower bound as Lk (least value) and upper bound was Uk(worst value) for the k-th objective function of the problem, k = 1,2,3, ‘ K where Uk is the highest acceptable level of achievement for kth object and Lk the aspired level of achievement for the objective k. when the aspiration levels for each objectives and constraints in both of membership and non-membership function have been specified, then we formed a intuitionistic fuzzy model and then convert the intuitionistic fuzzy model onto a crisp model.
Algorithm
Step 1
Solve the multi objective programming as a single objective transportation problem K times for each problem by taking one of the objective at a time.
Step 2
From the result of Step 1, determine the corresponding values for every objective at each solutions derived and construct a payoff matrix as:
Step 3
From step-2. We find the worst(Uk) and the best (Lk) values of each objectives for the degree of acceptance and rejection corresponding to the set of solutions as follows:
1 ‘ r ‘ k 1 ‘ r ‘ k
s = {1,2 } s = {1,2 }
for degree of acceptance of objectives.
We presents a new upper bound for the degree of rejection of objectives as follows:
Step 4
The initial intuitionistic fuzzy model becomes (in term of aspiration levels with each objectives)
Find {xij, i = 1,2,3, ‘.., m ; j = 1,2,3, ‘.,n} (2.10.1)
where = t with 0 < t < 1
where = t with 0 < t < 1
Step 5
Define the membership (acceptance) and non-membership(rejection) functions of IF objectives and constraint (or part of them) as follows:
For the k-th (k = 1,2,3, ‘.,K) objectives functions,
a hyperbolic membership function (‘k (Zk(xij)) ) is define by
= 1 ,
= 0 ,
Parabolic non-membership function (‘k(Zk(xij)) ) is defined as
= 0,
= 1,
For the j-th (j = 1,2,3, ‘.., n) constraint,
the hyperbolic membership function is defined by
= 1,
= 0,
here mj = (2bj – )/2
Parabolic non-membership function is defined as
= 0,
= 1,
For the i-th (i = 1,2,3, ‘.., m) constraint,
the hyperbolic membership function is defined by
= 1,
= 0,
here mj = (2ai + )/2
Parabolic non-membership function is defined as
= 0,
= 1,
Step 6
Find an equivalent crisp model by using the membership and non-membership functions of objectives, constraints by IF as follows:
xij
subject to
‘s(xij) + ‘s (xij) ‘ 1
‘s(xij) ‘ ‘s (xij)
‘s (xij) ‘ 0
xij ‘ 0, for all i, j, s = 1,2,3, ‘..m + n + K
Step 7
Solve the above crisp model by an appropriate mathematical programming algorithm.
Intuitionistic fuzzy optimization (IFO) problem such as fuzzy optimization problem can be represented as a two-stage process, which includes aggregation of objectives and constraints and defuzzification (maximization of aggregation function).
Conjunction of IF set is defined as
where denotes an IF objective (gain) and denotes an IF constraint.
Appling the above to the IFO problem (2.10.2), we have the following:
‘ ‘ ‘s (xij), s = { k, i, j / k = 1,2,3, ‘.K; i = 1,2,3, ‘, m; j = 1,2,3, ‘,m}
‘ ‘ ‘s (xij), s = { k, i, j / k = 1,2,3, ‘.K; i = 1,2,3, ‘, m; j = 1,2,3, ‘,m}
‘ + ‘ ‘ 1
‘ ‘ ‘, ‘ ‘ 0, xij ‘ 0, for all i, j
where ‘ denotes the minimal degree of acceptance of objective(s) and constraint(s) and ‘ denotes the maximal degree of rejection of objective(s) and constraint(s).
Now the IFO problem can be transformed to the following crisp optimization problem:
max (‘ – ‘)
subject to
‘ ‘ ‘s (xij), s = { k, i, j / k = 1,2,3, ‘.K; i = 1,2,3, ‘, m; j = 1,2,3, ‘,m}
‘ ‘ ‘s (xij), s = { k, i, j / k = 1,2,3, ‘.K; i = 1,2,3, ‘, m; j = 1,2,3, ‘,m}
‘ + ‘ ‘ 1
‘ ‘ ‘, ‘ ‘ 0, xij ‘ 0, for all i, j
It is equivalent to
max (‘ – ‘)
subject to
tanh(mk ‘ Zk(xij)) ‘ 2’-1,
,
,
,
‘ + ‘ ‘ 1
‘ ‘ ‘, ‘ ‘ 0, xij ‘ 0, for all i, j
In the above formulation (2.10.3), however all the membership and non-membership function are non-linear functions and hence we can not directly apply the linear programming method. To circumvent such difficulty, we have transferred the problem in the following way:
we define, tanh-1 (2′-1) = ” and = ”
such that ‘ = tanh(”) + and ‘ = ”2
since tanh-1(x) is strictly increasing function with respect to x them maximization of ‘ is equivalent to the maximization of ”. Also since is a strictly increasing function as than of ‘, the minimization of ‘ is equivalent to the minimization of ”. Hence the above problem can be transformed to the following ordinary linear programming problem:
max (” – ”)
subject to
Zk(xij) + ” ‘ mj
Step – 8
Determine if the decision maker is satisfied with the solution identified in step 7.If the decision maker is satisfied, STOP. If the decision maker is not satisfied, continue with step 3b, again. And define a new upper bound for the degree of rejection of objectives and constraints.This iteration will continue until and unless the decision maker is satisfied with the solutions.
Conclusion
The great advantage is that provides relatively good results compared by numerically and graphically both in fuzzy optimization and intuitionistic fuzzy optimization techniques , and.its provide to that intuitionistic fuzzy optimization obtains better results than fuzzy optimization
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