Solution of Single and Multiobjective Stochastic Inventory Models with Fuzzy Cost Components by Intuitionistic Fuzzy Optimization Technique

Paknejad et al.’s model is considered in this paper. Itemwise multiobjective models for both exponential and uniform lead-time demand are taken and the results are compared numerically both in fuzzy optimization and intuitionistic fuzzy optimization techniques. Objective of this paper is to establish that intuitionistic fuzzy optimizaion method is better than usual fuzzy optimization technique as expected annual cost of this inventory model is more minimized in case of intuitionistic fuzzy optimization method. As a single objective stochastic inventory model where the lead-time demand follows normal distribution and with varying defective rate, expected annual cost is also measured. Finally the model considers for fuzzy cost components, which make the model more realistic, and numerical values for uniform, exponential, and normal leadtime demand are compared. Necessary graphical presentations are also given besides numerical illustrations.


Introduction
In conventional inventory models, uncertainties are treated as randomness and are handled by appealing to probability theory.However, in certain situations uncertainties are due to fuzziness and in these cases the fuzzy set theory, originally introduced by Zadeh 1 , is applicable.Today most of the real-world decision-making problems in economic, technical and environmental ones are multidimensional and multiobjective.It is significant to realize that multiple-objectives are often noncommensurable and conflict with each other in optimization problem.An objective within exact target value is termed as fuzzy goal.So a multiobjective model with fuzzy objectives is more realistic than deterministic of it.
In decision making process, first, Bellman and Zadeh 2 introduced fuzzy set theory; Tanaka et al. 3 applied concept of fuzzy sets to decision-making problems to consider the objectives as fuzzy goals over the α-cuts of a fuzzy constraints.Zimmermann 4, 5 showed Paknejad et al.'s 35 model is considered in this paper, as a single objective stochastic inventory model where the lead-time demand follows normal distribution and with varying defective rate, expected annual cost is measured.Itemwise multiobjective models for both exponential and uniform lead time demand are taken and the results are compared numerically both in fuzzy optimization and intuitionistic fuzzy optimization techniques.From our numerical as well as graphical presentations, it is clear that intuitionistic fuzzy optimization obtains better results than fuzzy optimization.Finally the model considers for several fuzzy costs and numerical values for uniform, exponential, and normal lead-time demand are compared.Necessary graphical presentations are also given besides numerical illustrations.

Mathematical Model
Paknejad et al. 35 presented a quality adjusted lot-sizing model with stochastic demand and constant lead time and studied the benefits of lower setup cost in the model.We note that the previous literature focuses on the issue of setup cost reduction in which information about lead-time demand, whether constant or stochastic, is assumed completely known.This paper considers Paknejad et al.'s model along with the notations and some assumptions that will be taken into account throughout the paper.Each lot contains a random number of defectives following binomial distribution.After the arrival purchaser examines the entire lot, an order of size Q is placed as soon as the inventory position reaches the reorder point s.The shortages are allowed and completely backordered.Lead-time is constant and probability distribution of lead-time demand is known.Now, we use the following notations: where f x is the density function of lead-time demand, EC Q, s : expected annual cost given that a lot size Q is ordered.

Single Objective Stochastic Inventory Model (SOSIM)
Thus a quality-adjusted lot-sizing model is formed as MinEC Q, s Setup cost non-defective item holding cost stockout cost defective item holding cost inspecting cost

3.1
It is the stochastic model, which minimizes the expected annual cost.

Multiobjective Stochastic Inventory Model (MOSIM)
In reality, a managerial problem of a responsible organization involves several conflicting objectives to be achieved simultaneously that refers to a situation on which the DM has no control.For this purpose a latest tool is linear or nonlinear programming problem with multiple conflicting objectives.So the following model may be considered.
To solve the problem in 3.1 as an MOSIM, it can be reformulated as 4.1

Multiitem Stochastic Model with Fuzzy Cost Components
Stochastic nonlinear programming problem with fuzzy cost components considers as Here K i , π i , h i , h i represent vector of fuzzy parameters involved in the objective function EC.We assume

h i , and h i h −
i , h 0 i , h i , all of which are triangular fuzzy numbers.

Fuzzy Nonlinear Programming (FNLP) Technique to Solve Multiobjective Nonlinear Programming Problem (MONLP)
A Multi-Objective Non-Linear Programming MONLP or Vector Minimization problem VMP may be taken in the following form: 6.1 Zimmermann 5 showed that fuzzy programming technique can be used to solve the multiobjective programming problem.
To solve MONLP problem, the following steps are used.
Step 1. Solve the MONLP of 6.1 as a single objective non-linear programming problem using only one objective at a time and ignoring the others; these solutions are known as ideal solution.
Step 2. From the result of Step 1, determine the corresponding values for every objective at each solution derived.With the values of all objectives at each ideal solution, pay-off matrix can be formulated as follows: Here x 1 , x 2 , . . ., x k are the ideal solutions of the objective functions L r and U r are lower and upper bounds of the rth objective functions f r x r 1, 2, . . ., k .
Step 3. Using aspiration level of each objective of the MONLP of 6.1 may be written as follows.
Find x so as to satisfy Here objective functions of 6.1 are considered as fuzzy constraints.These type of fuzzy constraints can be quantified by eliciting a corresponding membership function:

6.5
Having elicited the membership functions as in 6.5 μ r f r x for r 1, 2, . . ., k introduce a general aggregation function: So a fuzzy multi-objective decision making problem can be defined as Max μ D x subject to x ∈ X.

6.7
Here we adopt the fuzzy decision as follows.
Fuzzy decision is based on minimum operator like Zimmermann's approach 4 .In this case 6.7 is known as FNLP M .
Then the problem of 6.7 , using the membership function as in 6.5 , according to min-operator is reduced to

Stochastic Models with Fuzzy Cost Components
Stochastic non-linear programming problem with fuzzy objective coefficient considers as Min Z CX, X ≥ 0. 7.1

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Here C represents a vector of fuzzy parameters involved in the objective function Z.We assume C i c i − , c i 0 , c i , which is a triangular fuzzy number with membership function: where

7.4
According to Kaufman and Gupta 36 by combining three objectives into a single objective function, 7.3 can be reduced to an LPP by most likely criteria as

Formulation of Intuitionistic Fuzzy Optimization
When the degree of rejection nonmembership is defined simultaneously with degree of acceptance membership of the objectives and when both of these degrees are not complementary to each other, then IF sets can be used as a more general tool for describing uncertainty.

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Figure 1: Membership and nonmemebership functions of the objective goal.
To maximize the degree of acceptance of IF objectives and constraints and to minimize the degree of rejection of IF objectives and constraints, we can write max μ i X , XεR, i 1, 2, . . ., K n, where μ i X denotes the degree of membership function of X to the ith IF sets and ν i X denotes the degree of nonmembership rejection of X from the ith IF sets.

An Intuitionistic Fuzzy Approach for Solving MOIP with Linear Membership and Nonmembership Functions
To define the membership function of MOIM problem, let L acc k and U acc k be the lower and upper bounds of the kth objective function.These values are determined as follows.Calculate the individual minimum value of each objective function as a single objective IP subject to the given set of constraints.Let X * 1 , X * 2 , . . ., X * k be the respective optimal solution for the k different objective and evaluate each objective function at all these k optimal solutions.It is assumed here that at least two of these solutions are different for which the kth objective function has different bounded values.For each objective, find lower bound minimum value L acc k and the upper bound maximum value U acc k .But in intuitionistic fuzzy optimization IFO , the degree of rejection non-membership and degree of acceptance membership are considered so that the sum of both values is less than one.To define membership function of MOIM problem, let L The linear membership function for the objective Z k X is defined as 9.1

Lemma 9.1. In case of minimization problem, the lower bound for non-membership function (rejection) is always greater than that of the membership function (acceptance).
Now, we take new lower and upper bounds for the non-membership function as follows:

9.2
Following the fuzzy decision of Bellman-Zadeh 2 together with linear membership function and non-membership functions of 9.1 , an intuitionistic fuzzy optimization model of MOIM problem can be written as

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The problem of 9.3 can be reduced following Angelov 29 to the following form: Then the solution of the MOIM problem is summarized in the following steps.
Step 1. Pick the first objective function and solve it as a single objective IP subject to the constraint; continue the process K-times for K different objective functions.If all the solutions i.e., X * 1 . ., m; j 1, 2, . . ., n are the same, then one of them is the optimal compromise solution and go to Step 6. Otherwise go to Step 2. However, this rarely happens due to the conflicting objective functions.
Then the intuitionistic fuzzy goals take the form

9.5
Step 2. To build membership function, goals and tolerances should be determined at first.Using the ideal solutions, obtained in Step 1, we find the values of all the objective functions at each ideal solution and construct pay off matrix as follows: Step 3. From Step 2, we find the upper and lower bounds of each objective for the degree of acceptance and rejection corresponding to the set of solutions as follows: For linear membership functions,

9.8
Step 4. Construct the fuzzy programming problem of 9.3 and find its equivalent LP problem of 9.4 .
Step 5. Solve 9.4 by using appropriate mathematical programming algorithm to get an optimal solution and evaluate the K objective functions at these optimal compromise solutions.

Few Stochastic Models
Case 1 Demand follows Uniform distribution .We assume that lead time demand for the period for the ith item is a random variable which follows uniform distribution and if the decision maker feels that demand values for item I below a i or above b i are highly unlikely and values between a i and b i are equally likely, then the probability density function f i x is given by where b i s i are the expected number of shortages per cycle and all these values of b i s i affect all the desired models.
Case 2 Demand follows Exponential distribution .We assume that lead-time demand for the period for the ith item is a random variable that follows exponential distribution.Then the probability density function f i x is given by Case 3 Demand follows Normal distribution .We assume that lead-time demand for the period for the ith item is a random variable, which follows normal distribution.Then the probability density function f i x is given by where b i s i are the expected number of shortages per cycle and all these values of b i s i affects all the desired models and Φ x i represents area under standard normal curve from −∞ to x i .

Solution of the Model of 3.1
The lead-time demand follows normal distribution and thus b i s i , the expected demand short at the end of the cycle, takes up the value according to 10.2 .Thus for single item model, we consider the following data: D 2750; K 10; h 0.25; v 0.02; π 1; h 0.15 11.1 all the cost-related parameters are measured in "$" .Here, the lead-time demand follows Normal distribution with mean μ 20 and standard deviation σ 2.
The lead-time demand follows normal distribution and thus b i s i , the expected demand short at the end of the cycle, takes up the value according to 10.2 .
In Table 1, a study of expected annual cost EC Q, s with lot size Q and reorder point s is given for different defective rate θ.We conclude from Table 1 as well asFigure 2 that the order quantity as well as the expected annual cost increases as θ increases.

Solution of the Model of 4.1
In case of MOSIM of 4.1 , we use the methods described in Section 6, to solve it by fuzzy optimization technique, and Sections 8 and 9, to solve it by intuitionistic fuzzy optimization technique and the following data are considered.In this case our objective is to analyze the expected annual cost of different stochastic models when their cost components are not deterministic but several triangular fuzzy numbers and thus the model becomes more practical and realistic.

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To solve the model of 5.1 we use the method described in Section 7.
The lead time demand follows uniform, exponential, and normal distribution, respectively, and thus the expected demand short at the end of the cycle, takes up the value according to 10.2 , 10.4 , and 10.5 , respectively.
In case of Uniform demand, all the cost-related parameters are measured in "$" .

Conclusion
Paknejad et al.'s model is considered in this paper, as a single objective stochastic inventory model where the lead-time demand follows normal distribution, and with varying defective rate, expected annual cost is measured.Our objective is to minimize the expected annual cost.Itemwise multiobjective models for both exponential and uniform lead-time demand are taken and the results are compared numerically both in fuzzy optimization and in intuitionistic fuzzy optimization techniques.From our numerical as well as graphical presentations, it is clear that intuitionistic fuzzy optimization obtains better results than fuzzy optimization.Thus expected annual cost is more minimized in case of intuitionistic fuzzy optimization than the usual fuzzy optimization technique.Finally the model considers for several fuzzy costs and numerical values for uniform, exponential, and normal lead-time demand are compared.Necessary graphical presentations are also given besides numerical illustrations.This model can also be extended taking lead-time demand as fuzzy random variables.

D:
expected demand per year, Q: lot size, s: reorder point, K: setup cost, θ: defective rate in a lot of size Q, 0 ≤ θ ≤ 1, h: nondefective holding cost per unit per year, h : defective holding cost per unit per year, π: shortage cost per unit short, ν: cost of inspecting a single item in each lot, μ: expected demand during lead time, b s : the expected demand short at the end of the cycle rej k and U rej k be the lower and upper bounds of the objective function Z k X where L acc k ≤ L rej k ≤ U rej k ≤ U acc k .These values are defined as follows.

Figure 4 11. 3 .
Figure 4 are the expected number of shortages per cycle and all these values of b i s i affect all the desired models.

Table 1 :
Variation of "EC" and "s" with "θ".The lead-time demand follows uniform distribution and thus b i s i , the expected demand short at the end of the cycle, takes up the value according to 10.2 .

Table 2 :
Comparison of solutions of FO and IFO UNIFORM .Then from Table2andFigure 3 we conclude that Intuitionistic fuzzy optimization IFO obtains more optimized values of EC 1 and EC 2 than fuzzy optimization FO .Also solution obtained by IFO 516.8335, 563.7091 is closer to the ideal solution 506.0453, 562.4362 than the solution obtained by FO.The lead-time demand follows exponential distribution and thus b i s i , the expected demand short at the end of the cycle, takes up the value according to 10.4 .

Table 3 :
Comparison of solutions of FO and IFO EXPONENTIAL .

Table 3 and
Figure 4we conclude that Intuitionistic fuzzy optimization IFO obtains more optimized values of EC 1 and EC 2 than fuzzy optimization FO .Also solution obtained by IFO 377.5540, 412.6827 is closer to the ideal solution 378.0060, 382.4234 than the solution obtained by FO.Expected annual costs EC 1 and EC 2 are more minimized in case of IFO than FO because, according to Section 6, only the membership functions μ EC 1 and μ EC 2 are maximized when FO technique is applied and the degree of acceptance of the IF objectives is measured by α * .But, when IFO technique is applied, according to Sections 8 and 9 not only are the membership functions μ EC 1 and μ EC 2 maximized but also the non-membership functions ν EC 1 and ν EC 2 are minimized and degree of acceptance of the IF objectives is measured by α * as well as degree of rejection of the IF objectives is measured by β * .As a result of that, IFO obtains more optimized values than FO.

Table 4 :
Expected Annual costs of different stochastic models.