Geometric Programming Approach to an Interactive Fuzzy Inventory Problem

An interactive multiobjective fuzzy inventory problem with two resource constraints is presented in this paper. The cost parameters and index parameters, the storage space, the budgetary cost, and the objective and constraint goals are imprecise in nature. These parameters and objective goals are quantified by linear/nonlinear membership functions. A compromise solution is obtained by geometric programmingmethod. If the decision maker is not satisfied with this result, he/she may try to update the current solution to his/her satisfactory solution. In this way we implement manmachine interactive procedure to solve the problem through geometric programming method.


Introduction
In formulating an inventory problem, various parameters involve in the objective functions and constraints which are assigned by the decision maker DM from past experiences.But in real world situation, it is observed that the possible values of the parameters are often imprecise and ambiguous to the DM.In different situations, different circumstances, it takes different values.So, it is difficult to assign the precise values of the parameters.With this observation, it would be certainly more appropriate to interpret the DM's understanding of the parameters as fuzzy numerical data which can be represented by fuzzy numbers.In the conventional approaches the objective goals are taken as deterministic.The objective goals, however, may not be exactly known.The target may vary to some extent, which is then represented by the tolerance value.Due to inexactness of the objective goals, the objective functions may be characterized by different types of membership functions.
In a multiobjective nonlinear programming MONLP problem DM plays an important role to achieve his/her optimum goal.He/she has every right to choose or rechoose the suitable set of membership functions for different objective functions.He/she decides

Fuzzy Number and Its Membership Function
A fuzzy number A is a fuzzy set of the real line A whose membership function μ A x has the following characteristics with ∞ < a 1 ≤ a 2 ≤ a 3 < ∞: x , for a 2 ≤ x ≤ a 3 , 0, for otherwise,

2.1
where μ L A x : a 1 , a 2 → 0, 1 is continuous and strictly increasing; μ R A x : a 2 , a 3 → 0, 1 is continuous and strictly decreasing.
The general shape of a fuzzy number following the above definition is known as triangular-shaped fuzzy number TiFN Buckley and Eslami 16 .

α-Level Set
The α-level of a fuzzy number A is defined as a crisp set A α x : μ A x ≥ α, x ∈ X , where α ∈ 0, 1 .A α is a nonempty bounded closed interval contained in X and it can be denoted by A α A L α , A R α .A L α and A R α are the lower and upper bounds of the closed interval, respectively.

Multiobjective Nonlinear Programming (MONLP)
The MONLP problem is represented as the following vector minimization problem Sakawa 17 : Note.When m 1 problem 2.2 reduces to a single objective nonlinear programming problem.
In general, the parameters in objectives and constraints are considered as crisp numbers.But there is some ambiguity to express the parameters precisely.So, it will be more realistic, if the parameters are considered as fuzzy numbers.The multiobjective nonlinear programming with fuzzy parameters MONLP-FP is described as A j1 , A j2 , . . ., A jq j and B j represent, respectively, fuzzy parameters involved in the objective functions f r x, C r r 1, 2, . . ., m and the constraint functions g j x, A j j 1, 2, . . ., k .These fuzzy parameters, which reflect the expert's ambiguous understanding of the nature of the parameters in the problem formulation process are assumed to be characterized as fuzzy numbers.

2.4
The α-level sets have the following property:

2.5
A, B, C α are the nonempty bounded closed intervals contained in X and it can be defined A js , C rt are TiFNs with different types of left and right branch of the membership functions.They may be of linear, parabolic, exponential, and so forth, type membership functions Table 1 .
The constraint goals B j , j 1, 2, . . ., k may be more realistic if it can be taken a TiFN with only right membership functions called right TiFN such as B r 1 j B j1 , B j1 , B j2 .The corresponding membership function is

2.6
The right branch μ R B j x is monotone decreasing continuous function in x ∈ B j1 , B j2 which may be linear, parabolic, or exponential type membership functions.The corresponding α-level interval is Now suppose that the DM decides that the degree of all of the membership functions of the fuzzy numbers involved in the MONLP-FP should be greater than or equal to some value of α.Then for such a degree α, the α-MONLP-FP can be interpreted as the following crisp multiobjective linear programming problem which depends on the coefficient vector a, b, c ∈ A, B, C α :

2.7
Observe that there exists an infinite number of such problem 2.7 depending on the coefficient vector a, b, c ∈ A, B, C α , and the values of a, b, c are arbitrary for any a, b, c ∈ A, B, C α in the sense that the degree of all of the membership functions in the problem 2.7 exceeds the level α ∈ 0, 1 .However, if possible, it would be desirable for the DM to choose a, b, c ∈ A, B, C α in the problem 2.7 to minimize the objective functions under the constraints.From such a point of view, for a certain degree α, it seems to be quite natural to have the α-MONLP-FP as the following MONLP problem 2.7 :

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On the basis of the α-level sets of the fuzzy numbers, we can introduce the concept of an Pareto optimal solution to the α-MONLP.

Interactive Nonlinear Programming with Fuzzy Parameter
To obtain the optimal solution, the DM is asked to specify the degree α of the α-level set and the reference levels of achievement of the objective functions.For the DM's degree α and reference levels f r , r 1, 2, . . ., m the corresponding optimal solution, which is, in the min-max sense, nearest to the requirement or better than that of the reference levels are attainable, is obtained by solving the following min-max problem: 3.2

Interactive Fuzzy Nonlinear Programming with Fuzzy Goals
Considering the imprecise nature of the DM's judgements, it is quite natural to assume that the DM may have imprecise or fuzzy goals for each of the objective functions in the α-MONLP.In a minimization problem, a fuzzy goal stated by the DM may have to achieve "substantially less than or equal to some value specified."This type of statement can be quantified by eliciting a corresponding membership function.
In order to elicit a membership function μ r f r x, c r from the DM for each of the objective functions f r x, c r in the α-MONLP, we first calculate the individual minimum f min r and maximum f max r of each objective function f r x, c r under the given constraints for α 0 and α 1.By taking into account the calculated individual minimum and maximum of each objective function for α 0 and α 1 together with the rate of increase of membership satisfaction, the DM may be able to determine a subjective membership function μ r f r x, c r which is a strictly monotone decreasing function with respect to f r x, c r .For example, two nonlinear membership functions are shown below.
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Parabolic Membership Function (Type 1)
For each of the objective functions, the corresponding quadratic membership functions are for r 1, 2, . . ., m,

3.3
where f 1 r and f 0 r are to be chosen such that r is the tolerance of the r-th objective function f r x, c r .

Exponential Membership Function (Type 2)
For each objective function, the corresponding exponential membership function is as follows: for r 1, 2, . . ., m,

3.4
The constants α r > 1, β r > 0 can be determined by asking the DM to specify the three points f 1 r , f 0.5 r , and and p r f 0 r − f 1 r are the tolerance of the r-th objective function f r x, c r .
In a minimization problem, DM has a target goal f 1 r x, c r with a flexibility p r .Having determined the membership functions for each of the objective functions, to generate a candidate for the satisficing solution which is also Pareto optimal, the DM is asked to specify the degree α of the α-level set and the reference levels of achievement of the membership functions called the reference membership values.Observe that the idea of the reference membership values e.g., Sakawa and Yano 1, 2 can be viewed as an obvious extension of the idea of the reference point.For the DM's degree α and the reference membership values μ r , r 1, 2, . . ., m the following min-max problem is solved to generate the Pareto optimal solution, which is, in the min-max sense, nearest to the requirement or better than that if the reference membership values are attainable Min max

3.6
The DM will select the membership functions for the corresponding objective functions from Type 1 and Type 2 membership functions.Then the above primal function can be solved by GP method as it has been expressed in signomial form and obtain optimal solution of ν says ν * : Now the DM selects his most important objective function from among the objective functions f j x j , j 1, 2, . . ., m .If it is the j-th objective, the following posynomial programming problem is solved by GP for ν ν * : Min f j x, c j , The problem is now solved by GP method and optimal solution is then examined by following Pareto optimality test by Wendell and Lee 18 .

Pareto Optimality Test
Let x * be the optimal decision vector which is obtained from 3.7 , solve the problem 3.8

Interactive Min-Max Method in Inventory Problem
The following notations and assumptions are used in developing a multiobjective multi-item inventory model.

Notations
For the i 1, 2, . . ., n th item, D i demand per unit item, Assumptions. 1 Production is instantaneous with zero lead-time, 2 when the demand of an item increases then the total purchasing cost spread all over the items and hence the demand of an item is inversely proportional to unit cost of production, that is, D i a i C −b i 0i since the purchasing cost and the demand of an item are nonnegative.We also require that the scaling constant a i > 0, and index parameter b i < 1 as C 0i and D i are inversely related to each other.

Problem Formulation
A wholesaling organisation purchase and stocks some commodities in his/her godown.He/she then supplies that commodities to some retailers.In such environment, the wholesaler always tries to minimize the total average cost which includes the purchasing cost, set-up and cost, and holding cost.His/her aim is also to minimize the total numbers of order supply to the retailer.
For the i 1, 2, . . ., n th item, the inventory costs over the time cycle Total average inventory cost TC C 0 , Q average purchasing cost average set-up cost average holding cost

4.1
Total number of orders NO C 0 , Q sum of orders of all items

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Total budgetary cost BC C 0 , Q sum of purchasing cost of all items Total storage space SS Q sum of storage space of all items In formulating the inventory models, the effect of constraints like total budgetary cost and total storage space cannot be unlimited, they must have restrictions.

Crisp Model
Under these circumstances the multiobjective inventory problem is then written as and boundary conditions

Fuzzy Model
In reality, the inventory costs such as carrying cost c 1i , set-up cost c 3i , the index parameter b i , storage area per item w i , total budgetary cost C , and total available storage area W are not exactly known previously.They may fluctuate within some range and can be expressed as a fuzzy number and boundary conditions where The α-level interval of these fuzzy numbers are represented by For any given α ∈ 0, 1 , the problem 4.6 is then reduced to 4.7

Interactive Geometric Programming (IGP) Technique with Fuzzy Parameters
Following Section 3, the problem 3.2 can be written as Min v,

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4.8 where TC and NO are the reference values of TC L α C 0 , Q and NO L α C 0 , Q , respectively.For any given α ∈ 0, 1 , the problem 4.8 is equivalent to the standard form of primal GP problem Min v, 4.9 Problem 4.9 is a constrained signomial problem with 6n 3 − 2n 1 4n 2 degree of difficulty.Following Kuester and Mize 19 , the problem is solved to obtain the Paretooptimal solutions for different choices of α and membership functions of fuzzy parameters by DM.

Interactive Fuzzy Geometric Programming (IFGP) Technique with Fuzzy Parameters and Fuzzy Goals
After determining the different linear/nonlinear membership functions for each of the objective functions proposed by Bellman and Zadeh 4 and following Zimmermann 20 the given problem can be formulated as Min ν, For any given α ∈ 0, 1 the problem 4.10 is equivalent to the standard form of primal GP problem Min ν,

4.11
Primal GP 4.11 may be solved by Fortran 77 with software code Kuester and Mize 19 .Following 3.7 and 3.8 , we get the Pareto optimal solution.

Numerical Example
A contractor undertakes to supply two types of goods to different distributors.The minimum storage space requirement for the goods are 600 m 2 .He can also arrange up to 640 m 2 storage space for the goods if necessary.The contractor invests $220 for his business with a maximum limit up to $250.From the past experience it was found that the holding cost of item-I is near about $1.5 but never less than $1.2 and above $2 i.e., c 11 ≡ $ 1.2, 1.5, 2 .Similarly, holding cost of item-II is c 12 ≡ $ 1.5, 1.8, 2.3 .The set-up cost and the index parameter of purchasing cost of each item are c 31 ≡ $ 100, 115, 130 , c 32 ≡ $ 130, 145, 160 ; b 1 ≡ 0.2, 0.3, 0.45 and b 2 ≡ 0.5, 0.65, 0.9 , respectively.The storage spaces of each item are w 1 ≡ 1.4, 1.8, 2.2 m 2 and w 2 ≡ 2.6, 3, 3.5 m 2 , respectively.It is also recorded from past that the scaling constant of the purchasing cost of each item are 1000 a 1 and 1120 a 2 , respectively.
The contractor wants to find the purchasing cost and inventory level of each item so as to minimize the total average cost and total number of order supply to the distributors.
The boundary level of purchasing cost and inventory level are given in Table 2.

Conclusion
In a real-life problem, it is not always possible to achieve the optimum goal set by a DM.Depending upon the constraints and unavoidable, unthinkable and compelling conditions prevailed at that particular time, a DM has to comprise and to be satisfied with a near optimum Pareto-optimal solution for the decision.This phenomenon is more prevalent when there is more than one objective goal for a DM.But, the usual mathematical programming methods in both crisp and fuzzy environments evaluate only one best possible solution against a problem.Moreover, GP method is the most appropriate method applied to engineering design problems.Nowadays, it is also applied to solve the inventory control problems.
In this paper, for the first time GP methods in an imprecise environment have been used to obtain a Pareto-optimal solution for most suitable choice of the DM.In this connection, we introduce here a man-machine interaction for the DM.This may be easily applied to other inventory models with dynamic demand, quantity discount, and so forth.The method can be easily expanded to stochastic, fuzzy-stochastic environments of inventory models.
lower and upper bounds of A, B, C α and can be obtained from the left branch and right branch of the membership functions μ A js a js , μ B j b j , and μ C rt c rt .

Table 2 :
Boundary level of decision variables C 0 , Q .