A Fuzzy E . O . Q Model With Unit Production Cost , Time Depended Holding Cost , WithOut Shortages Under A Space Constraint : A Fuzzy Geometric Programing ( FGP ) Approach

In this paper, an Economic Order quantity (E.O.Q) model with unit production cost, time depended holding cost, with-out shortages is formulated and solved. In most real world situation, the objective and constraint function of the decision makers are uncertainty in nature so the coefficients, indices the objective function and constraint goals are imposed here in fuzzy environment The problem is then solved using both Fuzzy Max-Min Geometric-Programming technique and Fuzzy parametric GeometricProgramming. Sensitivity analysis is also presented here.


Introduction
An inventory deals with decision that minimize the cost function or maximize the profit function.For this purpose the task is to construct a suitable mathematical model of the real life Inventory system, such a mathematical model is based on various assumption and approximation.
Production and hosting by ISPACS GmbH.http://www.ispacs.com/journals/ojids/2017/ojids-00009/International Scientific Publications and Consulting Services d) Demand rate is constant.e) The unit production cost is continuous function of demand and Set-up cost and take the following form: p= θ −  −1 , θ,x∈ ℝ (>0).f) Holding cost time depended as at.The differential equation describing I(t) as follows

Fuzzy model
When the objective constraint goals and coefficients become fuzzy sets and fuzzy numbers, then the crisp model (2.6) written to be a fuzzy model, as (3.11) Similarly, we can determine the δ-cut of  ̃ (0≤i≤n; 1≤k≤Ti; 1≤i≤m).Proposition 3.1.When the coefficients and indexes of the fuzzy geometric programming problem are taken as fuzzy numbers, then ) xj > 0, using δ-cut of fuzzy numbers coefficients and indexes, the above problem is reduces to the following form where Definition 3.6.For any x ∈ ℝ m and feasible index di ∈ ℝ (ℝ is the real number set), if ) ≤ 1 (1≤i≤n), then the linear membership function are given by Based on Zimmerman, first finding -cut of the fuzzy numbers in coefficients and indexes then we built membership functions of both objective and constraints goals and using max-min operator the above problem (3.13) reduced to a fuzzy Non-Linear Programming (FNLP) problem Max x > 0, ,  ∈ [0,1], which is equivalent to a geometric programming problem with parameters , Proof.Pls.see reference [13] S.Islam, T.K. Roy (2006).)  00 ∏ ∏ ( and subject to
Applying geometric programming GP technique the dual programming of the problem (4.24) is  Putting the values in (4.24) we get the optimal solution of dual problem.The values of D, S, q is obtained by using the primal dual relation as follows; From the primal dual relation we have, The optimal solution of the given model through the parametric approach is given by and )) .

Numerical example and solution:
A manufacturing company produces a item.It is given that the inventory holding cost of the item is $15 per unit per year.The production cost of the item varies inversely with the demand and set-up cost.From the past experience, the production cost of the item is 120 −3  −1 where D is the demand rate and S is setup cost.Storage space area per unit item ( 0 ) and total storage space area (W) are 100 sq.ft. and 2000 sq.ft.respectively.Determine the demand rate (D), set-up cost (S), production quantity (q), and optimum total average cost (TAC) of the production system.

Outcome of sensitivity analysis
Effect, for increment parameters-1) Fig. 3. shows that as α changes increasingly the total average cost of the given problem decreases.
2) Fig. 4. shows that as α changes increasingly the total average cost of the given problem increases.

Conclusion
In this paper, we have proposed a real life inventory problem in crisp and fuzzy environment and presented solution along with sensitivity analysis approach.The inventory model is developed with unit production cost, time depended holding cost, with-out shortages.This model has been developed for a single item.In this paper, we first create a model then it transformed as a fuzzy model.At last we give a real example and solved it various methods.In fuzzy we have considered triangular fuzzy number (T.F.

2 ( 2 . 3 )
Total inventory related cost per cycle = set-up cost + holding cost + production cost ., total average cost per cycle is given by TAC(D,S,q) =

Figure 3 :Figure 4 :
Figure 3: Change of the value of objective function for change of α by Fuzzy Max-Min Geometric Programming Technique.
N) and solved by fuzzy Max-Min Geometric-Programming and fuzzy Parametric Geometric-Programming Technique.In future, the other type of membership functions such as piecewise linear hyperbolic, L-R fuzzy number, http://www.ispacs.com/journals/ojids/2017/ojids-00009/International Scientific Publications and Consulting Services Trapezoidal Fuzzy Number (TrFN), Parabolic flat Fuzzy Number (PfFN), Parabolic Fuzzy Number (pFN), pentagonal fuzzy number etc can be considered to construct the membership function and then model can be easily solved.

3 Some basic concept & definition 3.1. Pre-requisite mathematics
[18]y sets first introduced by Zadeh[18]in 1965 as a mathematical way of representing vagueness in every life.

Table 3
We now examine to sensitivity analysis of the optimal solution of the given problem for changes of α, keeping the other parameters unchanged.The initial data is given from the above numerical example.://www.ispacs.com/journals/ojids/2017/ojids-00009/ 6.1.Sensitivity test of fuzzy E.O.Q problemHere we have given a rough graph, which shown how change the value of TAC * (S * ,D * ,q * ) for different values of α. http