Geometric Programming Approach to an Interactive Fuzzy Inventory Problem

Abstract

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 programming method. 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 man-machine interactive procedure to solve the problem through geometric programming method.

1. 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 the types of fuzzy parameters and also free to assign the reference values of each objective functions/membership functions for the desired optimal solutions. In this way an interactive procedure can be established with the DM. Sakawa and Yano [1, 2] proposed a new technique to solve such type of problems.

Zadeh [3] first introduced the concept of fuzzy set theory. Later on, Bellman and Zadeh [4] used the fuzzy set theory to the decision-making problem. Fuzzy set theory now has made an entry into the inventory control systems. Park [5] examined the EOQ formula in the fuzzy set theoretic perspective associating the fuzziness with the cost data. Roy and Maiti [6] solved a single objective fuzzy EOQ model using GP technique. Ishibuchi and Tanaka [7] developed a concept for optimization of multiobjective-programming problem with interval objective function.

Such type of nonlinear programming (NLP) problem can be solved by geometric programming (GP) problem. It has a very popular and effective use to solve many real-life decision-making problem. Duffin et al. [8] first developed the idea on GP method. Kotchenberger [9] was the first to use it in an inventory problem. Later on, Worral and Hall [10] analysed a multi-item inventory problem with several constraints. Abou-El-Ata et al. [11] and Jung and Klein [12] developed single item inventory problems and solved by GP method. Mandal et al. [13] used GP technique in multi-item inventory problem and compared with the nonlinear programming (NLP) problem. Recently, inventory problems in fuzzy environment were formulated and solved by GP technique by Liu [14] and Sadjadi et al. [15].

In this paper, we formulate a multiobjective inventory problem. The manufacturer wants to minimize the total average cost which includes the production cost, set up cost and holding cost. He/she also wants to minimize the number of orders to supply the finished goods to different shops/wholesalers. The problem is formulated under total budgetary cost and total storage space capacity restrictions. The parameters involve in the model are taken as different types of fuzzy numbers. The fuzzy numbers are described by linear/nonlinear types of membership functions which will be selected by the DM. Interactive min-max procedure is followed up by two different ways. First, we solve a multiobjective interactive programming with crisp goal by GP method. Next, we proceed with fuzzy goal through GP method. The DM may update the solution until his/her satisfaction.

2. Definitions and Basic Concepts

2.1. 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 ∞<𝑎₁≤𝑎₂≤𝑎₃<∞:𝜇_𝐴(𝑥)=𝜇^𝐿_𝐴(𝑥),for𝑎₁≤𝑥≤𝑎₂,1,for𝑥=𝑎₂,𝜇^𝑅_𝐴(𝑥),for𝑎₂≤𝑥≤𝑎₃,0,forotherwise, (2.1) where 𝜇^𝐿_𝐴(𝑥)∶[𝑎₁,𝑎₂]→[0,1] is continuous and strictly increasing; 𝜇^𝑅_𝐴(𝑥)∶[𝑎₂,𝑎₃]→[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]).

2.2. 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.

2.3. Multiobjective Nonlinear Programming (MONLP)

The MONLP problem is represented as the following vector minimization problem (Sakawa [17]):Min𝑓(𝑥;𝐶)=𝑓₁𝑥;𝐶₁,𝑓₂𝑥;𝐶₂,…,𝑓_𝑚𝑥;𝐶_𝑚^𝑇, (2.2) subject to 𝑥∈𝑋(𝐴,𝐵)={𝑥∈𝑅^𝑛∣𝑔_𝑗(𝑥;𝐴_𝑗)≤𝐵_𝑗,𝑗=1,2,…,𝑘}, where 𝑚(≥2) objective functions 𝑓_𝑟∶𝑅^𝑛→𝑅 and 𝑥∈𝑅^𝑛.

Note 2.3. 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 asMin𝑓𝑥;𝐶=𝑓₁𝑥;𝐶₁,𝑓₂𝑥;𝐶₂,…,𝑓_𝑚𝑥;𝐶_𝑚^𝑇, (2.3) subject to 𝑥∈𝑋(𝐴,𝐵)={𝑥∈𝑅^𝑛∣𝑔_𝑗(𝑥,𝐴_𝑗)≤𝐵_𝑗,𝑗=1,2,…,𝑘}, where 𝐶_𝑟=(𝐶_𝑟1,𝐶_𝑟2,…,𝐶_𝑟𝑝𝑟), 𝐴_𝑗=(𝐴_𝑗1,𝐴_𝑗2,…,𝐴_𝑗𝑞𝑗) 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.

We now assume that 𝐶_𝑟𝑡, 𝐴_𝑗𝑠, and 𝐵_𝑗 in the MONLP-FP (2.3) are fuzzy numbers whose membership functions are 𝜇𝐶_𝑟𝑡(𝑐_𝑟𝑡), 𝜇𝐴_𝑗𝑠(𝑎_𝑗𝑠), and 𝜇𝐵_𝑗(𝑏_𝑗)(𝑟=1,2,…,𝑚;𝑡=1,2,…,𝑝_𝑟;𝑗=1,2,…,𝑘;𝑠=1,2,…,𝑞_𝑗), respectively.

The 𝛼-level set of the fuzzy numbers 𝐶_𝑟𝑡,(𝑟=1,2,…,𝑚;𝑡=1,2,…,𝑝_𝑟) and 𝐴_𝑗𝑠,  (𝑗=1,2,…,𝑘; 𝑠=1,2,…,𝑞_𝑗)are defined as the ordinary set (𝐴,𝐵,𝐶)_𝛼 for which the degree of their membership functions exceeds the level 𝛼: 𝐴,𝐵,𝐶_𝛼=)𝐴(𝑎,𝑏,𝑐∣𝜇_𝑟𝑡𝑎_𝑟𝑡≥𝛼,𝑟=1,2,…,𝑚;𝑡=1,2,…,𝑝_𝑟;𝜇𝐵_𝑗𝑏_𝑗𝜇𝐶≥𝛼,𝑗=1,2,…,𝑘;_𝑗𝑠𝑐_𝑗𝑠≥𝛼,𝑗=1,2,…,𝑘,𝑠=1,2,…,𝑞_𝑗. (2.4)

The 𝛼-level sets have the following property:𝛼₁≤𝛼₂⇔𝐴,𝐵,𝐶_𝛼1⊇𝐴,𝐵,𝐶_𝛼2. (2.5)(𝐴,𝐵,𝐶)_𝛼 are the nonempty bounded closed intervals contained in 𝑋 and it can be defined as (𝐴,𝐵,𝐶)_𝛼=[(𝐴,𝐵,C)^𝐿_𝛼,(𝐴,𝐵,𝐶)^𝑅_𝛼], where (𝐴,𝐵,𝐶)^𝐿_𝛼 and (𝐴,𝐵,𝐶)^𝑅_𝛼 are the lower and upper bounds of (𝐴,𝐵,𝐶)_𝛼 and can be obtained from the left branch and right branch of the membership functions 𝜇𝐴_𝑗𝑠(𝑎_𝑗𝑠),𝜇𝐵_𝑗(𝑏_𝑗), and 𝜇𝐶_𝑟𝑡(𝑐_𝑟𝑡).

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).

Table 1 

𝛼-level interval of coefficient parameters.

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^𝑟1_𝑗=(𝐵_𝑗1,𝐵_𝑗1,𝐵_𝑗2). The corresponding membership function is𝜇_𝐵𝑗(𝑥)=1,for𝑥≤𝐵_𝑗1,𝜇^𝑅_𝐵𝑗(𝑥),for𝐵_𝑗1≤𝑥≤𝐵_𝑗2,0,for𝑥≥𝐵_𝑗2. (2.6)

The right branch 𝜇^𝑅_𝐵𝑗(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 L: [𝐵_𝑗1,𝐵_𝑗2−𝛼(𝐵_𝑗2−𝐵_𝑗1)],P: [𝐵_𝑗1,𝐵_𝑗1+√1−𝛼(𝐵_𝑗2−𝐵_𝑗1)],E: [𝐵_𝑗1,𝐵_𝑗2+((𝐵_𝑗2−𝐵_𝑗1)/𝛽₂)log(1−(𝛼/𝛼₂)].

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:Min𝑓₁𝑥,𝑐₁,𝑓₂𝑥,𝑐₂,…,𝑓_𝑚𝑥,𝑐_𝑚^𝑇,subjectto𝑥∈𝑋(𝑎,𝑏)=𝑥∈𝑅^𝑛∣𝑔_𝑗𝑥,𝑎_𝑗≤𝑏_𝑗,𝑗=1,2,…,𝑘;𝑥≥0. (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):Min𝑓₁𝑥,𝑐₁,𝑓₂𝑥,𝑐₂,…,𝑓_𝑚𝑥,𝑐_𝑚^𝑇,subjectto𝑥∈𝑋(𝑎,𝑏)=𝑥∈𝑅^𝑛∣𝑔_𝑗𝑥,𝑎_𝑗≤𝑏_𝑗,()∈𝐶,𝑗=1,2,…,𝑘;𝑥≥0𝑎,𝑏,𝑐𝐴,𝐵,_𝛼. (2.8)

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.

3. Interactive Min-Max Method

3.1. 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:Minmax_{𝑟=1,2,…,𝑚}𝑓_𝑟𝑥,𝑐_𝑟−𝑓_𝑟,subjectto(𝑎,𝑏,𝑐)∈(𝐴,𝐵,𝐶)_𝛼,𝑥∈𝑋(𝑎,𝑏), (3.1) or equivalentlyMin𝑣,subjectto𝑓_𝑟𝑥,𝑐_𝑟−𝑓_𝑟()∈()≤𝑣,𝑟=1,2,…,𝑚,𝑎,𝑏,𝑐𝐴,𝐵,𝐶_𝛼,𝑥∈𝑋(𝑎,𝑏). (3.2)

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_𝑟^min and maximum 𝑓_𝑟^max 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.

3.2.1. Parabolic Membership Function (Type 1)

For each of the objective functions, the corresponding quadratic membership functions are𝜇_𝑓𝑟𝑓_𝑟𝑥,𝑐_𝑟=1for𝑓_𝑟𝑥,𝑐_𝑟<𝑓¹_𝑟,𝑚_𝑟𝑥,𝑐_𝑟𝑓=1−^𝑟𝑥,𝑐_𝑟−𝑓¹_𝑟𝑝_𝑟²,for𝑓¹_𝑟≤𝑓_𝑟𝑥,𝑐_𝑟≤𝑓⁰_𝑟,0for𝑓_𝑟𝑥,𝑐_𝑟>𝑓⁰_𝑟,for𝑟=1,2,…,𝑚, (3.3) where 𝑓¹_𝑟 and 𝑓⁰_𝑟 are to be chosen such that 𝑓_𝑟^min≤𝑓¹_𝑟≤𝑓⁰_𝑟≤𝑓_𝑟^max and 𝑝_𝑟(=𝑓⁰_𝑟−𝑓¹_𝑟) is the tolerance of the 𝑟-th objective function 𝑓_𝑟(𝑥,𝑐_𝑟).

3.2.2. Exponential Membership Function (Type 2)

For each objective function, the corresponding exponential membership function is as follows:𝜇_𝑓𝑟𝑓_𝑟𝑥,𝑐_𝑟=1for𝑓_𝑟𝑥,𝑐_𝑟<𝑓¹_𝑟,𝑚_𝑟𝑥,𝑐_𝑟=𝛼_𝑟1−𝑒^{−𝛽𝑟((𝑓𝑟(𝑥,𝑐𝑟)−𝑓1𝑟)/𝑝𝑟)},for𝑓¹_𝑟≤𝑓_𝑟𝑥,𝑐_𝑟≤𝑓⁰_𝑟,0for𝑓_𝑟𝑥,𝑐_𝑟>𝑓⁰_𝑟,for𝑟=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 𝑓¹_𝑟,𝑓_𝑟^0.5, and 𝑓⁰_𝑟 such that 𝑓_𝑟^min≤𝑓¹_𝑟≤𝑓⁰_𝑟≤𝑓_𝑟^max and 𝑝_𝑟(=𝑓⁰_𝑟−𝑓¹_𝑟) are the tolerance of the 𝑟-th objective function 𝑓_𝑟(𝑥,𝑐_𝑟).

In a minimization problem, DM has a target goal 𝑓¹_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 attainableMinmax_{𝑟=1,2,…,𝑚}𝜇_𝑟−𝜇_𝑟𝑓_𝑟𝑥,𝑐_𝑟,subjectto(𝑎,𝑏,𝑐)∈(𝐴,𝐵,𝐶)_𝛼,𝑥∈𝑋(𝑎,𝑏), (3.5) which is equivalent toMin𝜈,subjectto𝜇_𝑟−𝜇_𝑟𝑓_𝑟𝑥,𝑐_𝑟()∈()≤𝜈,𝑟=1,2,…,𝑚,𝑎,𝑏,𝑐𝐴,𝐵,𝐶_𝛼,𝑥∈𝑋(𝑎,𝑏). (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^∗:Min𝑓_𝑗𝑥,𝑐_𝑗,subjectto𝑓_𝑟𝑥,𝑐_𝑟≤𝑚_𝑟⁻¹𝜇_𝑟,()∈()−𝜈,𝑟,𝑗=1,2,…,𝑚𝑟≠𝑗𝑎,𝑏,𝑐𝐴,𝐵,𝐶_𝛼,𝑥∈𝑋(𝑎,𝑏), (3.7) 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 Min𝑉=^𝑚_𝑟=1𝑓_𝑟𝑥,𝑐_𝑟,subjectto𝑓_𝑟𝑥,𝑐_𝑟≤𝑓^∗_𝑟𝑥^∗,𝑐_𝑟()∈(),∀𝑟=1,2,…,𝑚,𝑎,𝑏,𝑐𝐴,𝐵,𝐶_𝛼,𝑥∈𝑋(𝑎,𝑏). (3.8)

4. 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,𝑄_𝑖= order quantity (decision variable), (𝑄≡(𝑄₁,𝑄₂,…,𝑄_𝑛)^𝑇),𝐶_0𝑖= unit cost of production (decision variable), (𝐶₀≡(𝐶₀₁,𝐶₀₂,…,𝐶_0𝑛)^𝑇),𝑐_1𝑖= inventory carrying cost per item,𝑐_3𝑖= set up cost per cycle,𝑤_𝑖= storage area per each item,𝐶= total budgetary cost,𝑊= total available storage area,TC(𝐶₀,𝑄)= total average inventory cost function, NO(𝐶₀,𝑄)= total number of order function,SS(𝑄)= storage space function,BC(𝐶₀,𝑄)= budgetary cost function.

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_𝑖𝐶^−𝑏𝑖_0𝑖 since the purchasing cost and the demand of an item are non-negative. 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.

4.1. 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 𝑇_𝑖=𝑄_𝑖/𝐷_𝑖 is purchasingcost=𝐶_0𝑖𝑄_𝑖,set-upcost=𝑐_3𝑖,holdingcost=𝑐_1𝑖(𝑄_𝑖/2)𝑇_𝑖.

Total average inventory cost TC(𝐶₀,𝑄)=averagepurchasingcost  +  averageset-upcost  +  averageholdingcost=^𝑛_𝑖=1𝐶_0𝑖𝑄_𝑖+𝑐_3𝑖+𝑐_1𝑖𝑄_𝑖/2𝑇_𝑖𝑇_𝑖=^𝑛_𝑖=1𝑎_𝑖𝐶^1−𝑏𝑖_0𝑖+𝑎_𝑖𝑐_3𝑖𝐶^−𝑏𝑖_0𝑖𝑄_𝑖+𝑐_1𝑖𝑄_𝑖2. (4.1)

Total number of orders NO(𝐶₀,𝑄) = sum of orders of all items=^𝑛_𝑖=1𝐷_𝑖𝑄_𝑖=^𝑛_𝑖=1𝑎_𝑖𝐶^−𝑏𝑖_0𝑖𝑄_𝑖. (4.2)

Total budgetary cost BC(𝐶₀,𝑄) = sum of purchasing cost of all items=^𝑛_𝑖=1𝐶_0𝑖𝑄_𝑖. (4.3)

Total storage space SS(𝑄) = sum of storage space of all items=^𝑛_𝑖=1𝑤_𝑖𝑄_𝑖. (4.4)

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 MinTC𝐶₀,𝑄=^𝑛_𝑖=1𝑎_𝑖𝐶^1−𝑏𝑖_0𝑖+𝑎_𝑖𝑐_3𝑖𝐶^−𝑏𝑖_0𝑖𝑄_𝑖+𝑐_1𝑖𝑄_𝑖2,MinNO𝐶₀,𝑄=^𝑛_𝑖=1𝑎_𝑖𝐶^−𝑏𝑖_0𝑖𝑄_𝑖,subjecttoBC𝐶₀(𝑄),𝑄≤𝐶,SS≤𝑊,andboundaryconditions𝐶^𝑙_0𝑖≤𝐶_0𝑖≤𝐶^𝑢_0𝑖,𝑄^𝑙_𝑖≤𝑄_𝑖≤𝑄^𝑢_𝑖,𝑖=1,2,…,𝑛. (4.5)

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 MinTC𝐶₀,𝑄=^𝑛_𝑖=1𝑎_𝑖𝐶¹⁻_𝑖0𝑏_𝑖̃+𝑎_𝑖𝑐̃_3𝑖𝐶⁻_𝑖0𝑏_𝑖̃𝑄_𝑖+𝑐̃_1𝑖𝑄_𝑖2,MinNO𝐶₀,𝑄=^𝑛_𝑖=1𝑎_𝑖𝐶⁻_𝑖0𝑏_𝑖̃𝑄_𝑖,subjecttoBC𝐶₀,𝑄=^𝑛_𝑖=1𝐶_0𝑖𝑄_𝑖≤(𝑄)=𝐶,SS^𝑛_𝑖=1𝑤_𝑖𝑄_𝑖≤𝑊,andboundaryconditions𝐶^𝑙_0𝑖≤𝐶_0𝑖≤𝐶^𝑢_0𝑖,𝑄^𝑙_𝑖≤𝑄_𝑖≤𝑄^𝑢_𝑖,𝑖=1,2,…,𝑛, (4.6) where 𝑐̃_1𝑖=(𝑐_11𝑖,𝑐_12𝑖,𝑐_13𝑖), 𝑐̃_3𝑖=(𝑐_31𝑖,𝑐_32𝑖,𝑐_33𝑖),  𝑏̃_𝑖=(𝑏_1𝑖,𝑏_2𝑖,𝑏_3𝑖), 𝐶^𝑟1=(𝐶₁,𝐶₁,𝐶₂), 𝑤_𝑖=(𝑤_1𝑖,𝑤_2𝑖,𝑤_3𝑖), 𝑊^𝑟1=(𝑊₁,𝑊₁,𝑊₂).
The 𝛼-level interval of these fuzzy numbers are represented by
(𝑐̃_1𝑖)_𝛼=[𝑐^𝐿_1𝑖𝛼,𝑐^𝑅_1𝑖𝛼], (𝑐̃_3𝑖)_𝛼=[𝑐^𝐿_3𝑖𝛼,𝑐^𝑅_3𝑖𝛼], (𝑏̃_𝑖)_𝛼=[𝑏^𝐿_𝑖𝛼,𝑏^𝑅_𝑖𝛼], (𝐶^𝑟1)_𝛼=[𝐶^𝐿_𝛼,𝐶^𝑅_𝛼],(𝑤_𝑖)_𝛼=[𝑤^𝐿_𝑖𝛼,𝑤^𝑅_𝑖𝛼], and (𝑊^𝑟1)_𝛼=[𝑊^𝐿_𝛼,𝑊^𝑅_𝛼].
For any given 𝛼∈[0,1], the problem (4.6) is then reduced to MinTC^𝐿_𝛼𝐶₀,𝑄=^𝑛_𝑖=1𝑎_𝑖𝐶^{1−𝑏𝑅𝑖𝛼}_0𝑖+𝑎_𝑖𝑐^𝐿_3𝑖𝛼𝐶^{−𝑏𝑅𝑖𝛼}_0𝑖𝑄_𝑖+𝑐^𝐿_1𝑖𝛼𝑄_𝑖2,MinNO^𝐿_𝛼𝐶₀,𝑄=^𝑛_𝑖=1𝑎_𝑖𝐶^{−𝑏𝑅𝑖𝛼}_0𝑖𝑄_𝑖,subjectto^𝑛_𝑖=1𝐶_0𝑖𝑄_𝑖≤𝐶^𝑟1𝑅_𝛼,_𝑛𝑖=1𝑤^𝐿_i𝛼𝑄_𝑖≤𝑊^𝑟1𝑅_𝛼,𝐶^𝑙_0𝑖≤𝐶_0𝑖≤𝐶^𝑢_0𝑖,𝑄^𝑙_𝑖≤𝑄_𝑖≤𝑄^𝑢_𝑖,(for𝑖=1,2,…,𝑛). (4.7)

4.2. Interactive Geometric Programming (IGP) Technique with Fuzzy Parameters

Following Section 3, the problem (3.2) can be written as Min𝑣,subjecttoTC^𝐿_𝛼𝐶₀,𝑄−TC≤𝑣,NO^𝐿_𝛼𝐶₀,𝑄−NO≤𝑣,_𝑛𝑖=1𝐶_0𝑖𝑄_𝑖≤𝐶^𝑟1𝑅_𝛼,_𝑛𝑖=1𝑤^𝐿_𝑖𝛼𝑄_𝑖≤𝑊^𝑟1𝑅_𝛼,𝐶^𝑙_0𝑖≤𝐶_0𝑖≤𝐶^𝑢_0𝑖,𝑄^𝑙_𝑖≤𝑄_𝑖≤𝑄^𝑢_𝑖,𝑖=1,2,…,𝑛,𝑣>0. (4.8) where TC and NO are the reference values of TC^𝐿_𝛼(𝐶₀,𝑄) and NO^𝐿_𝛼(𝐶₀,𝑄), respectively.

For any given 𝛼∈[0,1], the problem (4.8) is equivalent to the standard form of primal GP problem1Min𝑣,subjecttoTC^𝑛_𝑖=1𝑎_𝑖𝐶^{1−𝑏𝑅𝑖𝛼}_0𝑖+𝑎_𝑖𝑐^𝐿_3𝑖𝛼𝐶^{−𝑏𝑅𝑖𝛼}_0𝑖𝑄_𝑖+𝑐^𝐿_1𝑖𝛼𝑄_𝑖21−𝑣≤1,NO^𝑛_𝑖=1𝑎_𝑖𝐶^{−𝑏𝑅𝑖𝛼}_0𝑖𝑄_𝑖1−𝑣≤1,𝐶^𝑟1𝑅_𝛼^𝑛_𝑖=1𝐶_0𝑖𝑄_𝑖1≤1,𝑊^𝑟1𝑅_𝛼^𝑛_𝑖=1𝑤^𝐿_𝑖𝛼𝑄_𝑖𝐶≤1,_𝑙0𝑖≤𝐶_0𝑖≤𝐶^𝑢_0𝑖,𝑄^𝑙_𝑖≤𝑄_𝑖≤𝑄^𝑢_𝑖,(for𝑖=1,2,…,𝑛),𝑣>0. (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 Pareto-optimal solutions for different choices of 𝛼 and membership functions of fuzzy parameters by DM.

4.3. 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 asMin𝜈,subjectto𝜇_TC−𝜇_TCTC^𝐿_𝛼𝐶₀,𝑄≤𝜈,𝜇_NO−𝜇_NONO^𝐿_𝛼𝐶₀,𝑄≤𝜈,_𝑛𝑖=1𝐶_0𝑖𝑄_𝑖≤𝐶^𝑟1𝑅_𝛼,_𝑛𝑖=1𝑤^𝐿_i𝛼𝑄_𝑖≤𝑊^𝑟1𝑅_𝛼,𝐶^𝑙_0𝑖≤𝐶_0𝑖≤𝐶^𝑢_0𝑖,𝑄^𝑙_𝑖≤𝑄_𝑖≤𝑄^𝑢_𝑖,𝜈>0. (4.10) For any given 𝛼∈[0,1] the problem (4.10) is equivalent to the standard form of primal GP problem1Min𝜈,subjectto𝜈𝜇_TC−1𝜈𝜇_TCTC^𝐿_𝛼𝐶₀1,𝑄≤1,𝜈𝜇_NO−1𝜈𝜇_NONO^𝐿_𝛼𝐶₀1,𝑄≤1,𝐶^𝑟1𝑅_𝛼^𝑛_𝑖=1𝐶_0𝑖𝑄_𝑖1≤1,𝑊^𝑟1𝑅_𝛼^𝑛_𝑖=1𝑤^𝐿_𝑖𝛼𝑄_𝑖𝐶≤1,_𝑙0𝑖≤𝐶_0𝑖≤𝐶^𝑢_0𝑖,𝑄^𝑙_𝑖≤𝑄_𝑖≤𝑄^𝑢_𝑖,𝜈>0. (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.

5. Numerical Example

A contractor undertakes to supply two types of goods to different distributors. The minimum storage space requirement for the goods are 600 m². He can also arrange up to 640 m² 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 $2 (i.e., 𝑐̃₁₁≡$(1.2,1.5,2)). Similarly, holding cost of item-II is 𝑐̃₁₂≡$(1.5,1.8,2.3). The set-up cost and the index parameter of purchasing cost of each item are 𝑐̃₃₁≡$(100,115,130), 𝑐̃₃₂≡$(130,145,160); 𝑏̃₁≡(0.2,0.3,0.45) and 𝑏̃₂≡(0.5,0.65,0.9), respectively. The storage spaces of each item are 𝑤₁≡(1.4,1.8,2.2)m² and 𝑤₂≡(2.6,3,3.5)m², respectively. It is also recorded from past that the scaling constant of the purchasing cost of each item are 1000(=𝑎₁) and 1120(=𝑎₂), 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.INPUT THE VALUE OF 𝛼=0.7DO YOU WANT LIST OF MEMBERSHIP FUNCTIONS FOR FUZZY PARAMETERS? = YES(1) LINEAR (L) (2) PARABOLIC (P) (3) EXPONENTIAL (E)INPUT THE LIST OF MEMBERSHIP FUNCTIONS FOR LEFT BRANCH AND RIGHT BRANCH OF THE FUZZY PARAMETERS:

Table 2 

Boundary level of decision variables (𝐶₀,𝑄).

(LEFT AND RIGHT BRANCH OF FUZZY PARAMETERS)

BRANCH	𝑐11	𝑐12	𝑐31	𝑐32	𝑏1	𝑏2	𝑤1	𝑤2	𝐶	𝑊
LEFT	P	E	E	P	L	L	L	E	P	E
RIGHT	P	P	L	E	P	L	E	E	L	P

INPUT THE VALUES OF (𝛼₁,𝛽₁) FOR 𝑐₁₂,𝑐₃₁,𝑤₂,𝑊

	𝑐12	𝑐31	𝑤2	𝑊
(𝛼1,𝛽1)	(1.4, 0.6)	(1.2, 1.6)	(2.2, 1.2)	(1.3, 0.6)

INPUT THE VALUES OF (𝛼₂,𝛽₂) FOR 𝑐₃₂,𝑤₁,𝑤₂

	𝑐32	𝑤1	𝑤2
(𝛼2,𝛽2)	(1.5, 0.2)	(1.9, 0.7)	(2.4, 1.3)

Solution with IGP
CALCULATION OF MAXIMUM AND MINIMUM VALUES OF OBJECTIVE FUNCTIONS

(INDIVIDUAL MINIMUM AND MAXIMUM)

OBJECTIVE FUNCTIONS	MINIMUM	MAXIMUM
TC($)	3861.849	6316.976
NO	15.17298	43.39745

CHOICE THE REFERENCE VALUE OF 𝑇𝐶=3860CHOICE THE REFERENCE VALUE OF 𝑁𝑂=13CALCULATION OF OPTIMAL SOLUTION

𝑖	𝐶∗0𝑖($)	𝑄∗𝑖	TC∗($)	NO∗	𝑣∗
1	0.2436070	208.4311	3862.605	15.60456	2.604558
2	1.788949	99.62540

ARE YOU SATISFIED WITH THE CURRENT OPTIMAL SOLUTION (OTHERWISE RECHOICE THE VALUE 𝛼 OR MEMBERSHIP FUNCTIONS OF FUZZY PARAMETERS)?= YES SATISFIED.

Solution with IFGP
DO YOU WANT LIST OF MEMBERSHIP FUNCTIONS? = YES LIST OF MEMBERSHIP FUNCTIONS (1) PARABOLIC (2) EXPONENTIAL INPUT MEMBERSHIP FUNCTION TYPE FOR 1ST OBJECTIVE:=2INPUT TWO POINTS TC¹,TC⁰=3861.849,6316.976INPUT MEMBERSHIP FUNCTION TYPE FOR 2ND OBJECTIVE:=1INPUT TWO POINTS NO¹,NO⁰=15.17298,43.39745INPUT 𝜇_TC𝜇_NO=0.99,0.988𝜈-MAX CALCULATION:OPTIMAL VALUE OF 𝜈=0.004781238CHOICE MOST IMPORTANT OBJECTIVE FUNCTION TC(𝐶₀,𝑄),NO(𝐶₀,𝑄)= TC(𝐶₀,𝑄)MINIMIZE TC(𝐶₀,𝑄)

𝑖	𝐶∗0𝑖($)	𝑄∗𝑖	TC∗($)	NO∗
1	0.2350434	205.9959	3861.849	15.71666
2	1.785726	101.1253

PARETO OPTIMALITY TEST

𝑖	𝐶∗0𝑖($)	𝑄∗𝑖	TC∗($)	NO∗
1	0.2351641	206.0291	3861.849	15.71500
2	1.7857650	101.1048

ARE YOU SATISFIED WITH THE CURRENT PARETO OPTIMAL SOLUTION (OTHERWISE RECHOICE THE VALUE OF 𝛼 OR THE MEMBERSHIP FUNCTIONS OF FUZZY PARAMETERS OR OBJECTIVE FUNCTIONS AND PROCEED AS BEFORE)?= YES SATISFIED.

6. 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.