An EPQ Model with Two-Component Demand under Fuzzy Environment and Weibull Distribution Deterioration with Shortages

Abstract

A single-item economic production model is developed in which inventory is depleted not only due to demand but also by deterioration. The rate of deterioration is taken to be time dependent, and the time to deterioration is assumed to follow a two-parameter Weibull distribution. The Weibull distribution, which is capable of representing constant, increasing, and decreasing rates of deterioration, is used to represent the distribution of the time to deterioration. In many real-life situations it is not possible to have a single rate of production throughout the production period. Items are produced at different rates during subperiods so as to meet various constraints that arise due to change in demand pattern, market fluctuations, and so forth. This paper models such a situation. Here it is assumed that demand rate is uncertain in fuzzy sense, that is, it is imprecise in nature and so demand rate is taken as triangular fuzzy number. Then by using 𝛼-cut for defuzzification the total variable cost per unit time is derived. Therefore the problem is reduced to crisp average costs. The multiobjective model is solved by Global Criteria method with the help of GRG (Generalized Reduced Gradient) Technique. In this model shortages are permitted and fully backordered. Numerical examples are given to illustrate the solution procedure of the two models.

1. Introduction

The classical EOQ (Economic Order Quantity) inventory models were developed under the assumption of constant demand. Later many researchers developed EOQ models taking linearly increasing or decreasing demand. Donaldson [1] discussed for the first time the classical no-shortage inventory policy for the case of a linear, positive trend in demand. Wagner and Whitin [2] developed a discrete version of the problem. Silver and Meal [3] formulated an approximate solution procedure as “Silver Meal heuristic” for a deterministic time-dependent demand pattern. Mitra et al. [4] extended the model to accommodate a demand pattern having increasing and decreasing linear trends. Deb and Chaudhuri [5] extended for the first time the inventory replenishment policy with linear trend to accommodate shortages. After some correction in the above model [5], Dave [6] applied Silver’s [7] heuristic to it incorporating shortages. Researchers have also worked on inventory models with time-dependent demand and deterioration. Models by Dave and Patel [8], Sachan [9], Bahari-Kashani [10], Goswami and Chaudhuri [11], and Hariga [12] all belong to this category. In addition to these demand patterns, some researchers use ramp type demand. Ramp type demand is one in which demand increases up to a certain time after which it stabilizes and becomes constant. Ramp type demand precisely depicts the demand of the items, such as newly launched fashion goods and cosmetics, garments, and automobiles for which demand increases as they are launched into the market and after some time it becomes constant. Today most of the real-world decision-making problems in economic, technical, and environmental ones are such that the inventory related demands are not deterministic but imprecise in nature. Furthermore in real-life problems the parameters of the stochastic inventory models are also fuzzy, that is, not deterministic in nature and this is the case of application of fuzzy probability in the inventory models.

In 1965, the first publication in fuzzy set theory by Zadeh [13] showed the intention to accommodate uncertainty in the nonstochastic sense rather than the presence of random variables. After that fuzzy set theory has been applied in many fields including production-related areas. In the 1990s, several scholars began to develop models for inventory problems under fuzzy environment. Park [14] and Ishii and Konno [15] discussed the case of fuzzy cost coefficients. Roy and Maiti [16] developed a fuzzy economic order quantity (EOQ) model with a constraint of fuzzy storage capacity. Chang and Yao [17] solved the economic reorder point with fuzzy backorders.

Some inventory problems with fuzzy shortage cost are analyzed by Katagiri and Ishii [18]. A unified approach to fuzzy random variables is considered by Krätschmer [19]; Kao and Hsu [20] discussed a single-period inventory model with fuzzy demand. Fergany and EI-Wakeel [21] considered the probabilistic single-item inventory problem with varying order cost under two linear constraints. Hala and EI-Saadani [22] analysed constrained single-period stochastic uniform inventory model with continuous distribution of demand and varying holding cost. Fuzzy models for single-period inventory problem were discussed by Li et al. [23]. Banerjee and Roy [24] considered application of the Intuitionistic Fuzzy Optimization in the constrained Multiobjective Stochastic Inventory Model. Banerjee and Roy [25] also discussed the single- and multiobjective stochastic inventory model in fuzzy environment. Lee and Yao [26] and Chang and Yao [17] investigated the economic production quantity model, and Lee and Yao [27] studied the EOQ model with fuzzy demands. A common characteristic of these studies is that shortages are backordered without extra costs.

Buckley [28] introduced a new approach and applications of fuzzy probability and after that Buckley and Eslami [29–31] contributed three remarkable articles about uncertain probabilities. Hwang and Yao [32] discussed the independent fuzzy random variables and their applications. Formalization of fuzzy random variables is considered by Colubi et al. [33]. Krätschmer [19] analyzed a unified approach to fuzzy random variables. Luhandjula [34] discussed a mathematical programming in the presence of fuzzy quantities and random variables.

Many researchers developed inventory models in which both the demand and deteriorating rate are constant. Although the constant demand assumption helps to simplify the problem, it is far from the actual situation where demand is always in change. In order to make research more practical, many researchers have studied other forms of demand. Among them time-dependant demand has attracted considerable attention. In this paper exponentially increasing demand has been considered which is a general form of linear and nonlinear time-dependent demand. Various types of uncertainties and imprecision are inherent in real inventory problems. They are classically modeled using the approaches from the probability theory. However there are uncertainties that cannot be appropriately treated by usual probabilistic models. The questions how to define inventory optimization task in such environment and how to interpret optimal solution arise. This paper considers the modification of EPQ formula in the presence of imprecisely estimated parameters with fuzzy demands where backorders are permitted, yet a shortage cost is incurred. The demand rate is taken as triangular fuzzy number. Since the demand is fuzzy the average cost associated with inventory is fuzzy in nature. So the average cost in fuzzy sense is derived. The fuzzy model is defuzzified by using 𝛼-cut of fuzzy number. This multiobjective problem is solved by Global Criteria method with the help of GRG (Generalized Reduced Gradient) technique, and it is illustrated with the help of numerical example.

2. Preliminaries

For developing the mathematical model we are to introduce certain preliminary definitions and results which will be used later on.

Definition 2.1 (fuzzy number). A fuzzy subset 𝐴 of real number 𝑅 with membership function 𝜇_𝐴∶𝑅→[0,1]is called a fuzzy number if(a)𝐴is normal, that is, there exists an element 𝑥₀ such that 𝜇_𝐴(𝑥₀)=1 is normal, that is, there exists an element 𝑥₀ such that 𝜇_𝐴(𝑥₀)=1;(b)𝐴is convex, that is, 𝜇_𝐴(𝜆𝑥₁+(1−𝜆)𝑥₂)≥𝜇_𝐴(𝑥₁)∧𝜇_𝐴(𝑥₂)for all 𝑥₁,𝑥₂∈𝑅 and 𝜆∈[0,1];(c)𝜇_𝐴 is upper semicontinuous; (d) supp(𝐴) is bounded, here supp(𝐴)=supp{𝑥∈𝑅∶𝜇_𝐴(𝑥)>0}.

Definition 2.2 (triangular fuzzy number). A Triangular Fuzzy number (TFN)𝐴 is specified by the triplet (𝑎₁, 𝑎₂, 𝑎₃) and is defined by its continuous membership function 𝜇_𝐴(𝑥), a continuous mapping 𝜇_𝐴∶𝑅→[0,1] as follows: 𝜇_𝐴(𝑥)=𝑥−𝑎₁𝑎₂−𝑎₁if𝑎₁≤𝑥<𝑎₂𝑎₃−𝑥𝑎₃−𝑎₂if𝑎₂≤𝑥≤𝑎₃0otherwise. (2.1)

Definition 2.3 (integration of a fuzzy function). Let 𝑓(𝑥) be a fuzzy function from [𝑎,𝑏]⊆𝑅 to 𝑅 such that 𝑓(𝑥) is a fuzzy number, that is, a piecewise continuous normalized fuzzy set on 𝑅. Then integral of any continuous 𝛼-level curve of 𝑓(𝑥) over [𝑎,𝑏] always exists and the integral of 𝑓(𝑥)over [𝑎,𝑏] is then defined to be the fuzzy set 𝐼̃(𝑎,𝑏)=^𝑏_𝑎𝑓⁻(𝛼)(𝑥)𝑑𝑥+^𝑏_𝑎𝑓⁺(𝛼)(𝑥)𝑑𝑥,𝛼. (2.2) The determination of the integral 𝐼(̃𝑎,𝑏) becomes somewhat easier if the fuzzy function is assumed to be 𝐿𝑅 type. We will therefore assume that 𝑓(𝑥)=(𝑓₁(𝑥),𝑓₂(𝑥),𝑓₃(𝑥))_𝐿𝑅 is a fuzzy number in 𝐿𝑅 representation for all 𝑥⊆[𝑎,𝑏], where 𝑓₁(𝑥), 𝑓₂(𝑥), and 𝑓₃(𝑥) are assumed to be positive integrable functions on [𝑎,𝑏]. Dubois and Prade have shown that under these conditions 𝐼̃(𝑎,𝑏)=^𝑏_𝑎𝑓₁(𝑥)𝑑𝑥,^𝑏_𝑎𝑓₂(𝑥)𝑑𝑥,^𝑏_𝑎𝑓₃(𝑥)𝑑𝑥_𝐿𝑅. (2.3)

Definition 2.4 (α-cut of fuzzy number). The 𝛼-cut of a fuzzy number is a crisp set which is defined as [𝐴]_𝛼={𝑥∈𝑅∶𝜇_𝐴(𝑥)≥𝛼}.According to the definition of fuzzy number it is seen that 𝛼-cut is a nonempty bounded closed interval; it can be denoted by 𝐴_𝛼=𝐴_𝐿(𝛼),𝐴_𝑅(𝛼). (2.4)𝐴_𝐿(𝛼) and 𝐴_𝑅(𝛼) are the lower and upper bounds of the closed interval, where 𝐴_𝐿(𝛼)=inf𝑥∈𝑅∶𝜇_𝐴(𝑥),𝐴≥𝛼_𝑅(𝛼)=sup𝑥∈𝑅∶𝜇_𝐴(𝑥)≥𝛼. (2.5)

Definition 2.5 (Global Criteria method (Rao,nd)). In the global criteria method 𝑥^∗ is found by minimizing a preselected global criteria, 𝐹(𝑋), such as the sum of the squares of the relative deviations of the individual objective functions from the feasible ideal solutions. The 𝑋^∗ is found by minimizing 𝐹(𝑋)=^𝑘_𝑖=1𝑓_𝑖𝑋^∗_𝑖−𝑓_𝑖(𝑋)𝑓_𝑖𝑋^∗_𝑖^𝑝subjectto𝑔_𝑗(𝑋)≤0,𝑗=1,2,…,𝑚, (2.6) where 𝑝 is a constant (as usual value of 𝑝 is 2) and 𝑋^∗_𝑖 is the ideal solution for the 𝑖th objective function. The solution 𝑋^∗_𝑖 is obtained by minimizing 𝑓_𝑖(𝑋) subject to the constraints 𝑔_𝑗(𝑋)≤0, 𝑗=1,2,…,𝑚.

3. Notations and Assumptions

3.1. Notations

(i)𝑇 is the length of one cycle.(ii)𝐼(𝑡) is the inventory level at time 𝑡.(iii)𝑅(𝑡) is the induced fuzzy demand around 𝑅 which is a function of time.(iv)𝐾₁ and 𝐾₂ are two production rates varies with demand.(v)𝐶₁ is the holding cost per unit time.(vi)𝐶₂ is the shortage cost per unit time.(vii)𝐶₃ is the unit purchase cost.(viii)𝐶₄ is the fixed ordering cost of inventory.(ix)𝑇 is the cycle time.(x)GC is the global criteria.(xi)𝜃  is the deterioration rate of finished items.

3.2. Assumptions

(i) The demand function 𝑅(𝑡)is taken to be fuzzy function of time, that is,𝑅(𝑡)=𝑎𝑒^𝑏𝑡.(ii)𝐾₁ and 𝐾₂ are two demand depended production variables, that is, 𝐾₁=𝛽₁𝑅(𝑡) and 𝐾₂=𝛽₂𝑅(𝑡), where 𝛽₁ and 𝛽₂ are constants.(iii)Replenishment is instantaneous.(iv)Lead time (i.e., the length between making of a decision to replenish an item and its actual addition to stock) is assumed to be zero. The assumption is made so that the period of shortage is not affected.(v) The rate of deterioration at any time 𝑡>0 follows the two-parameter Weibull distribution: 𝜃(𝑡)=𝛾𝛽𝑡^𝛽−1, where 𝛾(0<𝛾<1) is the scale parameter and 𝛽(>0)is the shape parameter. The implication of the Weibull rate (two parameter) is that the items in inventory start deteriorating the instant they are received into inventory. The rate of deterioration-time relationship is shown in Figure 1.

Figure 1 

When 𝛽=1, 𝑧(𝑡)=𝛼 (a constant) which is the case of exponential decay.

When 𝛽<1 rate of deterioration is decreasing with time.

When 𝛽>1, rate of deterioration is increasing with time.(vi)Shortages are allowed and are fully backlogged.

4. Mathematical Modeling and Analysis

Here we assume that the production starts at time 𝑡=0 at the rate 𝐾₁ and the stock attains a level 𝑃 at 𝑡=𝑧. Then at 𝑡=𝑧 the production rate changes to 𝐾₂ and continues up to 𝑡=𝑦, where the inventory level reaches the maximum level 𝑄. And the production stops at 𝑡=𝑦 and the inventory gradually depletes to zero at 𝑡=𝑡₁ mainly to meet the demands and partly for deterioration. Now shortages occur and accumulate to the level 𝑆 at time 𝑡=𝑡₂. The production starts again at a rate 𝐾₁ at 𝑡=𝑡₂ and the backlog is cleared at time 𝑡=𝑇 when the stock is again zero. The cycle then repeats itself after time 𝑇.

The model is represented by Figure 2.

Figure 2 

The changes in the inventory level can be described by the following differential equations:𝑑𝐼̃(𝑡)𝑑𝑡+𝛾𝛽𝑡^𝛽−1𝐼̃(𝑡)=𝛽₁−1𝑎𝑒^𝑏𝑡𝑑𝐼̃(𝑡)0≤𝑡≤𝑧,𝑑𝑡+𝛾𝛽𝑡^𝛽−1𝐼̃(𝑡)=𝛽₂−1𝑎𝑒^𝑏𝑡𝑑𝐼̃(𝑡)𝑧≤𝑡≤𝑦,𝑑𝑡+𝛾𝛽𝑡^𝛽−1𝐼̃(𝑡)=−𝑎𝑒^𝑏𝑡𝑦≤𝑡≤𝑡₁,𝑑𝐼̃(𝑡)𝑑𝑡=−𝑎𝑒^𝑏𝑡𝑡₁≤𝑡≤𝑡₂,𝑑𝐼̃(𝑡)𝛽𝑑𝑡=−₁−1𝑎𝑒^𝑏𝑡𝑡₂≤𝑡≤𝑇 (4.1) with initial conditions 𝐼(𝑧)=𝑃, 𝐼(𝑦)=𝑄, 𝐼(𝑡₁)=0, 𝐼(𝑡₂)=𝑆, and 𝐼(𝑇)=0.

The differential equations (4.1) are fuzzy differential equations. To solve this differential equation at first we take the 𝛼-cut then the differential equations reduces to 𝑑𝐼⁺₁(𝑡)𝑑𝑡+𝛾𝛽𝑡^𝛽−1𝐼⁺₁(𝑡)=𝛽₁−1𝑎⁺(𝛼)𝑒^𝑏𝑡,0≤𝑡≤𝑧, (4.2)𝑑𝐼⁻₁(𝑡)𝑑𝑡+𝛾𝛽𝑡^𝛽−1𝐼⁻₁(𝑡)=𝛽₁−1𝑎⁻(𝛼)𝑒^𝑏𝑡,0≤𝑡≤𝑧, (4.3)𝑑𝐼⁺₂(𝑡)𝑑𝑡+𝛾𝛽𝑡^𝛽−1𝐼⁺₂(𝑡)=𝛽₂−1𝑎⁺(𝛼)𝑒^𝑏𝑡,𝑧≤𝑡≤𝑦, (4.4)𝑑𝐼⁻₂(𝑡)𝑑𝑡+𝛾𝛽𝑡^𝛽−1𝐼⁻₂(𝑡)=𝛽₂−1𝑎⁻(𝛼)𝑒^𝑏𝑡,𝑧≤𝑡≤𝑦, (4.5)𝑑𝐼⁺₃(𝑡)𝑑𝑡+𝛾𝛽𝑡^𝛽−1𝐼⁺₃(𝑡)=−𝑎⁻(𝛼)𝑒^𝑏𝑡,𝑦≤𝑡≤𝑡₁, (4.6)𝑑𝐼⁻₃(𝑡)𝑑𝑡+𝛾𝛽𝑡^𝛽−1𝐼⁻₃(𝑡)=−𝑎⁺(𝛼)𝑒^𝑏𝑡,𝑦≤𝑡≤𝑡₁, (4.7)𝑑𝐼⁺₄(𝑡)𝑑𝑡=−𝑎⁻(𝛼)𝑒^𝑏𝑡,𝑡₁≤𝑡≤𝑡₂, (4.8)𝑑𝐼⁻₄(𝑡)𝑑𝑡=−𝑎⁺(𝛼)𝑒^𝑏𝑡,𝑡₁≤𝑡≤𝑡₂, (4.9)𝑑𝐼⁺₅(𝑡)𝛽𝑑𝑡=−₁−1𝑎⁻(𝛼)𝑒^𝑏𝑡,𝑡₂≤𝑡≤𝑇, (4.10)𝑑𝐼⁻₅(𝑡)𝛽𝑑𝑡=−₁−1𝑎⁺(𝛼)𝑒^𝑏𝑡,𝑡₂≤𝑡≤𝑇, (4.11) where 𝐼⁺_𝑖=sup𝑥∈𝑅∶𝜇𝐼_𝑖̃(𝑥)𝐼≥𝛼,𝑖=1,2,3,4,5,_−𝑖=inf𝑥∈𝑅∶𝜇𝐼_𝑖̃(𝑥)≥𝛼,𝑖=1,2,3,4,5. (4.12) Similarly 𝑎⁺(𝛼), 𝑎⁻(𝛼) have usual meaning.

The solutions of the differential equations (4.2) to (4.11) are, respectively, represented by 𝐼⁺₁(𝑡)=𝛽₁−1𝑏𝑒^𝑏𝑡−1𝑎₃−𝛼𝑎₃−𝑎₂−𝛾𝛽𝑡^𝛽+1𝛽₁−1𝑎₃𝐼1+𝛽,0≤𝑡≤𝑧,₋₁(𝑡)=𝛽₁−1𝑏𝑒^𝑏𝑡−1𝑎₁+𝛼𝑎₂−𝑎₁−𝛾𝛽𝑡^𝛽+1𝛽₁−1𝑎₁𝐼1+𝛽,0≤𝑡≤𝑧,₊₂(𝑡)=𝑃1+𝛾𝑧^𝛽−𝛾𝑡^𝛽+𝛽₂𝑎−1³−𝛼𝑎₃−𝑎₂𝑒^𝑏𝑡−𝑒^𝑏𝑧𝑏−𝛾𝑡^𝛽𝛽₂−1𝑎₃𝑏𝑒^𝑏𝑡−𝑒^𝑏𝑧+𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝛽₂−1𝑏𝑡^𝛽𝑒^𝑏𝑡−1−𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝛽₂−1𝑧^𝛽𝑒^𝑏𝑧−1𝑏+𝑎₃−𝛼𝑎₃−𝑎₂𝛾𝛽𝛽₂−1𝑧1+𝛽^𝛽+1−𝑡^𝛽+1𝐼,𝑧≤𝑡≤𝑦,₋₂(𝑡)=𝑃1+𝛾𝑧^𝛽−𝛾𝑡^𝛽+𝛽₂𝑎−1¹+𝛼𝑎₂−𝑎₁𝑒^𝑏𝑡−𝑒^𝑏𝑧𝑏−𝛾𝑡^𝛽𝛽₂−1𝑎₁𝑏𝑒^𝑏𝑡−𝑒^𝑏𝑧+𝑎₁+𝛼𝑎₂−𝑎₁𝛾𝛽₂−1𝑏𝑡^𝛽𝑒^𝑏𝑡−1−𝑎₁+𝛼𝑎₂−𝑎₁𝛾𝛽₂−1𝑧^𝛽𝑒^𝑏𝑧−1𝑏+𝑎₁+𝛼𝑎₂−𝑎₁𝛽𝛾𝛽₂−1𝑧1+𝛽^𝛽+1−𝑡^𝛽+1𝐼,𝑧≤𝑡≤𝑦,₊₃(𝑡)=𝑎₁+𝛼𝑎₂−𝑎₁𝑏𝑒^𝑏𝑡1−𝑒^𝑏𝑡−𝑎₁𝛾𝑡^𝛽𝑏𝑒^𝑏𝑡1−𝑒^𝑏𝑡+𝑎₁+𝛼𝑎₂−𝑎₁𝑏𝛾𝑡^𝛽₁𝑒^𝑏𝑡1−1−𝑡^𝛽𝑒^𝑏𝑡−𝑎−1₁+𝛼𝑎₂−𝑎₁𝑡𝛾𝛽^1𝛽+1−𝑡^𝛽+11+𝛽,𝑦≤𝑡≤𝑡₁,𝐼⁻₃(𝑡)=𝑎₃−𝛼𝑎₃−𝑎₂𝑏𝑒^𝑏𝑡1−𝑒^𝑏𝑡−𝑎₃𝛾𝑡^𝛽𝑏𝑒^𝑏𝑡1−𝑒^𝑏𝑡+𝑎₃−𝛼𝑎₃−𝑎₂𝑏𝛾𝑡^𝛽₁𝑒^𝑏𝑡1−1−𝑡^𝛽𝑒^𝑏𝑡−𝑎−1₃−𝛼𝑎₃−𝑎₂𝑡𝛾𝛽^1𝛽+1−𝑡^𝛽+11+𝛽,𝑦≤𝑡≤𝑡₁,𝐼⁺₄(𝑡)𝑎=−¹+𝛼𝑎₂−𝑎₁𝑒^𝑏𝑡−𝑒^𝑏𝑡1𝑏,𝑡₁≤𝑡≤𝑡₂,𝐼⁻₄(𝑡)𝑎=−³−𝛼𝑎₃−𝑎₂𝑒^𝑏𝑡−𝑒^𝑏𝑡1𝑏,𝑡₁≤𝑡≤𝑡₂,𝐼⁺₅(𝑡)=𝛽₁−1𝑎₁+𝛼𝑎₂−𝑎₁𝑒^𝑏𝑇−𝑒^𝑏𝑡𝑏,𝑡₂𝐼≤𝑡≤𝑇,₋₅(𝑡)=𝛽₁−1𝑎₃−𝛼𝑎₃−𝑎₂𝑒^𝑏𝑇−𝑒^𝑏𝑡𝑏,𝑡₂≤𝑡≤𝑇. (4.13) Therefore the upper 𝛼-cut of fuzzy stock holding cost is given by𝐻𝐶⁺=𝐶₁𝑇^𝑧₀𝐼⁺₁(𝑡)𝑑𝑡+^𝛼1𝑡1_𝑧𝐼⁺₂(𝑡)𝑑𝑡+^𝑡1_𝛼1𝑡1𝐼⁺₃(𝑡)=C𝑑𝑡₁𝑇𝐴₁+𝐵₁𝑡₁+𝐶₁𝑡₁^𝛽+1+𝐷₁𝑡₁^𝛽+2+𝐸₁𝑡₁^𝛽+3+𝐹₁𝑒^𝑏𝑡1𝑡₁+𝐺₁𝑒^𝑏𝑡1𝑡₁^𝛽+1+𝐻₁𝑒^{𝑏𝛼1𝑡1}+𝐼₁𝑒^𝑏𝑡1. (4.14) Calculations are shown in Appendix A.

And the lower 𝛼-cut of fuzzy stock holding cost is given by𝐻𝐶⁻=𝐶₁𝑇^𝑧₀𝐼⁻₁(𝑡)𝑑𝑡+^𝛼1𝑡1_𝑧𝐼⁻₂(𝑡)𝑑𝑡+^𝑡1_𝛼1𝑡1𝐼⁻₃(𝑡)=𝐶𝑑𝑡₁𝑇𝐴₂+𝐵₂𝑡₁+𝐶₂𝑡₁^𝛽+1+𝐷₂𝑡₁^𝛽+2+𝐸₂𝑡₁^𝛽+3+𝐹₂𝑒^𝑏𝑡1𝑡₁+𝐺₂𝑒^𝑏𝑡1𝑡₁^𝛽+1+𝐻₂𝑒^{𝑏𝛼1𝑡1}+𝐼₂𝑒^𝑏𝑡1. (4.15) Calculations are shown in Appendix B.(1)As demand is fuzzy in nature shortage cost is also fuzzy in nature.

Therefore the upper 𝛼-cut of shortage cost is given by S.C⁺=𝐶₂𝑇_𝑡^𝜂𝑇₁𝐼⁺₄(𝑡)𝑑𝑡+^𝑇_𝜂𝑇𝐼⁺₅(𝑡)=𝐶𝑑𝑡₂𝑇𝑎₁+𝛼𝑎₂−𝑎₁𝑏−𝑒^𝑏𝜂𝑇𝑏+𝜂𝑇𝑒^𝑏𝑡1+𝑒^𝑏𝑡1𝑏−𝑡₁𝑒^𝑏𝑡1+C₂T𝑎₁+𝛼𝑎₂−𝑎₁𝛽₁−1𝑏²𝑒^𝑏𝑇𝑒𝑇−^𝑏𝑇𝑏−𝜂𝑇𝑒^𝑏𝑇+𝑒^𝑏𝜂𝑇𝑏. (4.16) Also the lower 𝛼-cut of shortage cost is given byS.C⁻=𝐶₂𝑇_𝑡^𝜂𝑇₁𝐼⁻₄(𝑡)𝑑𝑡+^𝑇_𝜂𝑇𝐼⁻₅(𝑡)=𝐶𝑑𝑡₂𝑇𝑎₃−𝛼𝑎₃−𝑎₂𝑏−𝑒^𝑏𝜂𝑇𝑏+𝜂𝑇𝑒^𝑏𝑡1+𝑒^𝑏𝑡1𝑏−𝑡₁𝑒^𝑏𝑡1+𝐶₂𝑇𝑎₃−𝛼𝑎₃−𝑎₂𝛽₁−1𝑏²𝑒^𝑏𝑇𝑒𝑇−^𝑏𝑇𝑏−𝜂𝑇𝑒^𝑏𝑇+𝑒^𝑏𝜂𝑇𝑏. (4.17)(2) Annual cost due to deteriorated unit is also fuzzy as demand is fuzzy quantity.

Therefore deterioration cost per year is given by𝐶D.C=³𝑇^𝑧₀𝐾₁−𝑅𝑑𝑡+^𝛼1𝑡1_𝑧𝐾₂−𝑅𝑑𝑡−^𝑡1_𝛼1𝑡1=𝐶𝑅𝑑𝑡₃𝑇^z₀𝑎𝛽₁−1𝑒^𝑏𝑡𝑑𝑡+^𝛼1𝑡1_𝑧𝑎𝛽₂−1𝑒^𝑏𝑡𝑑𝑡−^𝑡1_𝛼1𝑡1𝑎𝑒^𝑏𝑡=𝐶𝑑𝑡₃𝑇^𝑧₀𝑓₁(𝑡)𝑓₂(𝑡)𝑓₃(𝑡)𝑑𝑡+^𝛼1𝑡1_𝑧𝜙₁(𝑡)𝜙₂(𝑡)𝜙₃(𝑡)𝑑𝑡−^𝑡1_𝛼1𝑡1𝜑₁(𝑡)𝜑₂(𝑡)𝜑₃(𝑡)𝑑𝑡, (4.18) where𝑓₁(𝑡)=𝑎₁𝛽₁−1𝑒^𝑏𝑡,𝑓₂(𝑡)=𝑎₂𝛽₁−1𝑒^𝑏𝑡,𝑓₃(𝑡)=𝑎₃𝛽₁−1𝑒^𝑏𝑡,𝜙₁(𝑡)=𝑎₁𝛽₂−1𝑒^𝑏𝑡,𝜙₂(𝑡)=𝑎₂𝛽₂−1𝑒^𝑏𝑡,𝜙₃(𝑡)=𝑎₃𝛽₂−1𝑒^𝑏𝑡,𝜑₁(𝑡)=𝑎₃𝑒^𝑏𝑡,𝜑₂(𝑡)=𝑎₂𝑒^𝑏𝑡,𝜑₃(𝑡)=𝑎₁𝑒^𝑏𝑡. (4.19) Therefore upper 𝛼-cut of deterioration cost is given byD.C⁺=𝐶₃𝑇𝑎₃𝛽₁−1𝑒^𝑏𝑧−1𝑏−𝛼𝑎₃𝛽₁−1𝑒^𝑏𝑧−1𝑏+𝑎₂𝛼𝛽₁−1𝑒^𝑏𝑧−1𝑏+𝑎₃𝛽₂−1𝑏𝑒^{𝑏𝛼1𝑡1}−𝑒^𝑏𝑧−𝛼𝑎₃𝛽₂−1𝑏𝑒^{𝑏𝛼1𝑡1}−𝑒^𝑏𝑧+𝛼𝑎₂𝛽₂−1𝑏𝑒^{𝑏𝛼1𝑡1}−𝑒^𝑏𝑧−𝑎₁𝑏𝑒^𝑏𝑡1−𝑒^{𝑏𝛼1𝑡1}−𝑎₁𝛼𝑏𝑒^𝑏𝑡1−𝑒^{𝑏𝛼1𝑡1}+𝑎₂𝛼𝑏𝑒^𝑏𝑡1−𝑒^{𝑏𝛼1𝑡1}. (4.20) The lower 𝛼-cut of deterioration cost is given byD.C⁻=𝐶₃𝑇𝑎₁𝛽₁−1𝑒^𝑏𝑧−1𝑏+𝛼𝑎₂𝛽₁−1𝑒^𝑏𝑧−1𝑏−𝑎₁𝛼𝛽₁−1𝑒^𝑏𝑧−1𝑏+𝑎₁𝛽₂−1𝑏𝑒^{𝑏𝛼1𝑡1}−𝑒^𝑏𝑧+𝛼𝑎₂𝛽₂−1𝑏𝑒^{𝑏𝛼1𝑡1}−𝑒^𝑏𝑧−𝛼𝑎₁𝛽₂−1𝑏𝑒^{𝑏𝛼1𝑡1}−𝑒^𝑏𝑧−𝑎₃𝑏𝑒^𝑏𝑡1−𝑒^{𝑏𝛼1𝑡1}−𝑎₂𝛼𝑏𝑒^𝑏𝑡1−𝑒^{𝑏𝛼1𝑡1}+𝑎₃𝛼𝑏𝑒^𝑏𝑡1−𝑒^{𝑏𝛼1𝑡1}. (4.21)(3) The annual ordering cost =𝐶₄/𝑇.

Therefore total variable cost per unit time is a fuzzy quantity and is defined by TVC=TVC⁺TVC⁻, (4.22) where TVC⁺=sup𝑥∈𝑅∶𝜇_TVC(𝑥)≥𝛼,TVC⁻=inf𝑥∈𝑅∶𝜇_TVC(𝑥)≥𝛼. (4.23) The upper 𝛼-cut of total variable cost per unit time isTVC⁺=𝐶₁𝑇𝐴₁+𝐵₁𝑡₁+𝐶₁𝑡₁^𝛽+1+𝐷₁𝑡₁^𝛽+2+𝐸₁𝑡₁^𝛽+3+𝐹₁𝑒^𝑏𝑡1𝑡₁+𝐺₁𝑒^𝑏𝑡1𝑡₁^𝛽+1+𝐻₁𝑒^{𝑏𝛼1𝑡1}+𝐼₁𝑒^𝑏𝑡1+𝐶₂𝑇𝑎₁+𝛼𝑎₂−𝑎₁𝑏−𝑒^𝑏𝜂𝑇𝑏+𝜂𝑇𝑒^𝑏𝑡1+𝑒^𝑏𝑡1𝑏−𝑡₁𝑒^𝑏𝑡1+𝐶₂𝑇𝑎₁+𝛼𝑎₂−𝑎₁𝛽₁−1𝑏²𝑒^𝑏𝑇𝑒𝑇−^𝑏𝑇𝑏−𝜂𝑇𝑒^𝑏𝑇+𝑒^𝑏𝜂𝑇𝑏+𝐶₃𝑇𝑎₃𝛽₁−1𝑒^𝑏𝑧−1𝑏−𝛼𝑎₃𝛽₁−1𝑒^𝑏𝑧−1𝑏+𝑎₂𝛼𝛽₁−1𝑒^𝑏𝑧−1𝑏+𝑎₃𝛽₂−1𝑏𝑒^{𝑏𝛼1𝑡1}−𝑒^𝑏𝑧−𝛼𝑎₃𝛽₂−1𝑏𝑒^{𝑏𝛼1𝑡1}−𝑒^𝑏𝑧+𝛼𝑎₂𝛽₂−1𝑏𝑒^{𝑏𝛼1𝑡1}−𝑒^𝑏𝑧−𝑎₁𝑏𝑒^𝑏𝑡1−𝑒^{𝑏𝛼1𝑡1}−𝑎₁𝛼𝑏𝑒^𝑏𝑡1−𝑒^{𝑏𝛼1𝑡1}+𝑎₂𝛼𝑏𝑒^𝑏𝑡1−𝑒^{𝑏𝛼1𝑡1}+𝐶₄𝑇. (4.24) The lower 𝛼-cut of total variable cost per unit time isTVC⁻=𝐶₁𝑇𝐴₂+𝐵₂𝑡₁+𝐶₂𝑡₁^𝛽+1+𝐷₂𝑡₁^𝛽+2+𝐸₂𝑡₁^𝛽+3+𝐹₂𝑒^𝑏𝑡1𝑡₁+𝐺₂𝑒^𝑏𝑡1𝑡₁^𝛽+1+𝐻₂𝑒^{𝑏𝛼1𝑡1}+𝐼₂𝑒^𝑏𝑡1+𝐶₂𝑇𝑎₃−𝛼𝑎₃−𝑎₂𝑏−𝑒^𝑏𝜂𝑇𝑏+𝜂𝑇𝑒^𝑏𝑡1+𝑒^𝑏𝑡1𝑏−𝑡₁𝑒^𝑏𝑡1+𝐶₂𝑇𝑎₃−𝛼𝑎₃−𝑎₂𝛽₁−1𝑏²𝑒^𝑏𝑇𝑒𝑇−^𝑏𝑇𝑏−𝜂𝑇𝑒^𝑏𝑇+𝑒^𝑏𝜂𝑇𝑏+𝐶₃𝑇𝑎₁𝛽₁−1𝑒^𝑏𝑧−1𝑏+𝛼𝑎₂𝛽₁−1𝑒^𝑏𝑧−1𝑏−𝑎₁𝛼𝛽₁−1𝑒^𝑏𝑧−1𝑏+𝑎₁𝛽₂−1𝑏𝑒^{𝑏𝛼1𝑡1}−𝑒^𝑏𝑧+𝛼𝑎₂𝛽₂−1𝑏𝑒^{𝑏𝛼1𝑡1}−𝑒^𝑏𝑧−𝛼𝑎₁𝛽₂−1𝑏𝑒^{𝑏𝛼1𝑡1}−𝑒^𝑏𝑧−𝑎₃𝑏𝑒^𝑏𝑡1−𝑒^{𝑏𝛼1𝑡1}−𝑎₂𝛼𝑏𝑒^𝑏𝑡1−𝑒^{𝑏𝛼1𝑡1}+𝑎₃𝛼𝑏𝑒^𝑏𝑡1−𝑒^{𝑏𝛼1𝑡1}+𝐶₄𝑇. (4.25) The objective in this paper is to find an optimal cycle time to minimize the total variable cost per unit time.

Therefore this model mathematically can be written as MinimizeTVC⁺,TVC⁻Subjectto0≤𝛼≤1. (4.26) Therefore the problem is a multiobjective optimization problem. To convert it as a single-objective optimization problem we use global criteria (GC) method.

Then the above problem reduces toMinimizeGCSubjectto0≤𝛼≤1. (4.27)

5. Global Criteria Method

The model presented by (4.26) is a multiobjective model which is solved by Global Criteria (GC) Method with the help of Generalized Reduced Gradient Technique.

The Multiobjective Nonlinear Integer Programming (MONLIP) problems are solved by Global Criteria method converting it to a single-objective optimization problem. The solution procedure is as follows.

Step 1. Solve the multiobjective programming problem (4.26) as a single-objective problem using only one objective at a time ignoring others.

Step 2. From the results of Step 1, determine the ideal objective vector, say (TVC^+min, TVC^−min) and the corresponding values of (TVC^+max, TVC^−max). Here, the ideal objective vector is used as a reference point. The problem is then to solve the following auxiliary problem: Min(GC)=MinimizeTVC⁺−TVC^+minTVC^+max−TVC^+min𝑄+TVC⁻−TVC^−minTVC^−max−TVC^{−min𝑄1/𝑄}, (5.1) where 1≤𝑄<∞. This method is also sometimes called Compromise Programming.

6. Numerical Example

We now consider a numerical example showing the utility of the model from practical point of view. According to the developed solution procedure of the proposed inventory system, the optimal solution has been obtained with the help of well-known generalized reduced gradient method (GRG). To illustrate the developed model, an example with the following data has been considered.

Let 𝑎₁=100 units/month, 𝑎₂=200 units/month, 𝑎₃=100 units/month, 𝐶₁=$.1 per unit, 𝐶₂=$.5 per unit, 𝐶₃=$.2 per unit, 𝐶₄=$100 per order, 𝑏=.2, 𝛼=.25,𝛽=1.9, 𝛽₁=0.9,𝛽₂=1.9, Υ=.001𝑇=7hrs.

Substituting the above parameters, Global Criteria (GC) is obtained as GC=0.0926. (6.1) The compromise solutions are TVC⁺=$548.62, TVC⁻=$503.74.

7. Conclusion

In this paper the multiobjective problem is solved by Global Criteria method. In reality, in different systems, there are some parameters which are imprecise in nature. The present paper proposes a solution procedure to develop an EPQ inventory model with variable production rate and fuzzy demand. In most of the real life problem demand of a newly launched product is not known in advance. This justifies the introduction of fuzzy demand. The technique for multiobjective optimization may be applied to the areas like environmental analysis, transportation, and so forth. In this paper we have taken two rates of production but this can be extended to 𝑛 number of production rates 𝑝_𝑖 during the time when the inventory level goes from(𝑖−1)𝑄 to 𝑖𝑄(𝑖=1,2,…,𝑛), where 𝑄 is prefixed level.

Appendices

A. The Upper 𝜶-Cut of Fuzzy Stock Holding Cost

The upper 𝛼-cut of fuzzy stock holding cost is given by: 𝐻𝐶⁺=𝐶₁𝑇^𝑧₀𝐼⁺₁(𝑡)𝑑𝑡+^𝛼1𝑡1_𝑧𝐼⁺₂(𝑡)𝑑𝑡+^𝑡1_𝛼1𝑡1𝐼⁺₃(𝑡)=𝐶𝑑𝑡₁𝑇𝛽₁−1𝑎₃−𝛼𝑎₃−𝑎₂𝑏𝑒^𝑏𝑧𝑏1−𝑧−𝑏−𝛾𝛽𝛽₁−1𝑎₃𝑧^𝛽+2𝛼1+𝛽2+𝛽+𝑃₁𝑡₁+𝛾𝛼₁𝑧^𝛽𝑡₁−𝛾𝛼₁^𝛽+1𝑡₁^𝛽+1𝛽+1−𝑧−𝛾𝑧^𝛽+1+𝛾𝑧^𝛽+1+𝛽𝛽+1₂−1𝑏𝑎₃−𝛼𝑎₃−𝑎₂𝑒^{𝑏𝛼1𝑡1}𝑏−𝛼₁𝑡₁𝑒^𝑏𝑧−𝑒^𝑏𝑧𝑏+𝑧𝑒^𝑏𝑧+𝛾𝑎₃𝛽₂−1𝛽𝛼₁^𝛽+1𝑡₁^𝛽+1𝑏−1+𝛽𝛾𝑎₃𝛽₂−1𝛽𝑧^𝛽+1𝑏−1+𝛽𝛾𝑎₃𝛽₂−1𝑏𝛼₁𝑡₁^𝛽+1−𝑧^𝛽+1+𝑏2𝛼₁𝑡₁^𝛽+2−𝑧^𝛽+2+𝑏²6𝛼₁𝑡₁^𝛽+3−𝑧^𝛽+3+𝛾𝑎₃𝛽₂−1𝑒^𝑏𝑧𝑏𝛼1+𝛽¹𝑡₁^𝛽+1−𝑧^𝛽+1−𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝛽₂−1𝛽𝛼₁^𝛽+1𝑡₁^𝛽+1𝑏+𝛾𝑎𝛽+1³−𝛼𝑎₃−𝑎₂𝛽₂−1𝛽𝑧^𝛽+1𝑏+𝛾𝑎1+𝛽₃−𝛼𝑎₃−𝑎₂𝛽₂−1𝑏𝛼₁𝑡₁^𝛽+1−𝑧^𝛽+1+𝑏2𝛼₁𝑡₁^𝛽+2−𝑧^𝛽+2+𝑏²6𝛼₁𝑡₁^𝛽+3−𝑧^𝛽+3−𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝛽₂−1𝑏𝛼𝛽+1¹𝑡₁^𝛽+1−𝑧^𝛽+1−𝛾𝑎₃+𝛼𝑎₃−𝑎₂𝛽₂−1𝑧^𝛽𝑒^𝑏𝑧𝛼₁𝑡₁−𝑧𝑏+𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝛽₂−1𝑧^𝛽𝛼₁𝑡₁−𝑧𝑏+𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝛽𝛽₂−1𝑧1+𝛽2+𝛽𝛽+2^𝛽+1𝛼₁𝑡₁−𝛼₁^𝛽+2𝑡₁^𝛽+2−𝑧^𝛽+2𝛽+2+𝑧^𝛽+2+𝑎₁+𝛼𝑎₂−𝑎₁𝑏𝑒^𝑏𝑡1𝑡₁−𝑒^𝑏𝑡1𝑏−𝛼₁𝑡₁𝑒^𝑏𝑡1+𝑒^{𝑏𝛼1𝑡1}𝑏−𝑎₁𝛾𝑒^𝑏𝑡1𝑡₁^𝛽+1−𝛼₁^𝛽+1𝑡₁^𝛽+1𝑏−𝑎𝛽+1₁𝛾𝛽𝑏𝑡1+𝛽^1𝛽+1−𝛼₁^𝛽+1𝑡₁^𝛽+1+𝑎₁𝛾𝑏𝑡₁^𝛽+1−𝛼₁^𝛽+1𝑡₁^𝛽+1+𝑏2𝑡₁^𝛽+2−𝛼₁^𝛽+2𝑡₁^𝛽+2+𝑏²6𝑡₁^𝛽+3−𝛼₁^𝛽+3𝑡₁^𝛽+3−𝑎₁+𝛼𝑎₂−𝑎₁₂𝛾𝑏𝑡₁^𝛽+1𝑒^𝑏𝑡1−𝑡₁^𝛽+1−𝛼₁𝑡₁^𝛽+1𝑒^𝑏𝑡1+𝛼₁𝑡₁^𝛽+1+𝛽𝛾𝑎₁+𝛼𝑎₂−𝑎₁₂𝑏𝑡1+𝛽^1𝛽+1−𝛼₁^𝛽+1𝑡₁^𝛽+1−𝛾𝑎₁+𝛼𝑎₂−𝑎₁𝑏𝑡₁^𝛽+1−𝛼₁^𝛽+1𝑡₁^𝛽+1+𝑏2𝑡₁^𝛽+2−𝛼₁^𝛽+2𝑡₁^𝛽+2+𝑏²6𝑡₁^𝛽+3−𝛼₁^𝛽+3𝑡₁^𝛽+3+𝛾𝑎₁+𝛼𝑎₂−𝑎₁𝑡₁^𝛽+1−𝛼₁^𝛽+1𝑡₁^𝛽+1𝑏−𝑎1+𝛽₁+𝛼𝑎₂−𝑎₁𝛾𝛽𝑡1+𝛽^1𝛽+2−𝑡₁^𝛽+2𝛽+2−𝛼₁𝑡₁^𝛽+2+𝛼₁^𝛽+2𝑡₁^𝛽+2𝛽+2𝐻𝐶⁺=𝐶₁𝑇^𝑧₀𝐼⁺₁(𝑡)𝑑𝑡+^𝛼1𝑡1_𝑧𝐼⁺₂(𝑡)𝑑𝑡+^𝑡1_𝛼1𝑡1𝐼⁺₃(𝑡)=𝐶𝑑𝑡₁𝑇𝐴₁+𝐵₁𝑡₁+𝐶₁𝑡₁^𝛽+1+𝐷₁𝑡₁^𝛽+2+𝐸₁𝑡₁^𝛽+3+𝐹₁𝑒^𝑏𝑡1𝑡₁+𝐺₁𝑒^𝑏𝑡1𝑡₁^𝛽+1+𝐻₁𝑒^{𝑏𝛼1𝑡1}+𝐼₁𝑒^𝑏𝑡1, (A.1) where 𝐴₁=𝛽₁−1𝑎₃−𝛼𝑎₃−𝑎₂𝑏𝑒^𝑏𝑧𝑏1−𝑧−𝑏−𝛾𝛽𝛽₁−1𝑎₃𝑧^𝛽+21+𝛽2+𝛽−𝑃𝑧−𝑃𝛾𝑧^𝛽+1+𝑃𝛾𝑧^𝛽+1+𝛽𝛽+1₂−1𝑏𝑎₃−𝛼𝑎₃−𝑎₂−𝑒^𝑏𝑧𝑏+𝑧𝑒^𝑏𝑧−𝛾𝑎₃𝛽₂−1𝛽𝑧^𝛽+1𝑏+1+𝛽𝛾𝑎₃𝛽₂−1𝑏𝑧^𝛽+1+𝑏2𝑧^𝛽+2+𝑏²6𝑧^𝛽+3−𝛾𝑎₃𝛽₂−1𝑒^𝑏𝑧𝑏𝑧1+𝛽^𝛽+1+𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝛽₂−1𝛽𝑧^𝛽+1𝑏−𝛾𝑎1+𝛽³−𝛼𝑎₃−𝑎₂𝛽₂−1𝑏𝑧^𝛽+1+𝑏2𝑧^𝛽+2+𝑏²6𝑧^𝛽+3+𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝛽₂−1𝑏𝑧𝛽+1^𝛽+1+𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝛽₂−1𝑧^𝛽+1𝑒^𝑏𝑧𝑏−𝑎₃−𝛼𝑎₃−𝑎₂𝛾𝛽₂−1𝑧^𝛽+1𝑏−𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝛽𝛽₂−1𝑧2+𝛽^𝛽+2,𝐵₁=𝑃𝛾+𝑃𝛾𝛼₁𝑧^𝛽−𝛽₂−1𝑏𝑎₃−𝛼𝑎₃−𝑎₂𝛼₁𝑒^𝑏𝑧−𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝛽₂−1𝑧^𝛽𝑒^𝑏𝑧𝛼₁𝑏+𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝛽₂−1𝑧^𝛽𝛼₁𝑏+𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝛽𝛽₂−1𝑧1+𝛽^𝛽+1𝛼₁,𝐶₁=−𝑃𝛾𝛼₁^𝛽+1+𝛽+1𝛾𝑎₃𝛽₂−1𝛽𝛼₁^𝛽+1𝑏−1+𝛽𝛾𝑎₃𝛽₂−1𝑏𝛼₁^𝛽+1+𝛾𝑎₃𝛽₂−1𝑒^𝑏𝑧𝑏𝛼1+𝛽^1𝛽+1−𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝛽₂−1𝛽𝛼₁^𝛽+1𝑏+𝛾𝑎𝛽+1³−𝛼𝑎₃−𝑎₂𝛽₂−1𝑏𝛼₁^𝛽+1−𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝛽₂−1𝑏𝛼𝛽+1^1𝛽+1−𝑎₁𝛾𝑒^𝑏𝑡11−𝛼₁^𝛽+1𝑏−𝑎𝛽+1¹𝛾𝛽𝑏1+𝛽1−𝛼₁^𝛽+1+𝑎₁𝛾𝑏1−𝛼₁^𝛽+1−𝑎₁+𝛼𝑎₂−𝑎₁𝛾𝑏𝛼₁+𝑎−1𝛽𝛾₁+𝛼𝑎₂−𝑎₁𝑏1+𝛽1−𝛼₁^𝛽+1−𝛾𝑎₁+𝛼𝑎₂−𝑎₁𝑏1−𝛼₁^𝛽+1+𝛾𝑎₁+𝛼𝑎₂−𝑎₁1−𝛼₁^𝛽+1𝑏,𝐷1+𝛽₁=−𝛾𝑎₃𝛽₂−1𝛼₁^𝛽+22+𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝛽₂−1𝛼₁^𝛽+22−𝑎₃−𝛼𝑎₃−𝑎₂𝛾𝛽𝛽₂−1𝛼1+𝛽2+𝛽^1𝛽+2−𝑎₁𝛾1−𝛼₁^𝛽+22−𝛾𝑎₁+𝛼𝑎₂−𝑎₁1−𝛼₁^𝛽+22−𝑎₁+𝛼𝑎₂−𝑎₁𝛾𝛽+𝑎2+𝛽₁+𝛼𝑎₂−𝑎₁𝛾𝛼₁𝛽−𝛼1+𝛽^1𝛽+2𝛾𝛽𝑎𝛽+1𝛽+2¹+𝛼𝑎₂−𝑎₁,𝐸₁=−𝛾𝑎₃𝛽₂−1𝑏𝛼₁^𝛽+36+𝑎₃−𝛼𝑎₃−𝑎₂𝛽₂−1𝛾𝑏𝛼₁^𝛽+36+𝑎₁𝛾𝑏61−𝛼₁^𝛽+3−𝛾𝑏𝑎₁+𝛼𝑎₂−𝑎₁1−𝛼₁^𝛽+36,𝐹₁=𝑎₁+𝛼𝑎₂−𝑎₁𝑏1−𝛼₁,𝐺₁=𝑎₁𝛾𝛼₁^𝛽+1−1𝑏+𝑎𝛽+1¹+𝛼𝑎₂−𝑎₁𝛾𝑏𝛼₁,𝐻−1₁=𝑎₁+𝛼𝑎₂−𝑎₁𝑏²+𝛽₂−1𝑏²𝑎₃−𝛼𝑎₃−𝑎₂,𝐼₁𝑎=−¹+𝛼𝑎₂−𝑎₁𝑏². (A.2)

B. The Lower 𝜶-Cut of Fuzzy Stock Holding Cost

The lower 𝛼-cut of fuzzy stock holding cost is given by: 𝐻𝐶⁻=𝐶₁𝑇^𝑧₀𝐼⁻₁(𝑡)𝑑𝑡+^𝛼1𝑡1_𝑧𝐼⁻₂(𝑡)𝑑𝑡+^𝑡1_𝛼1𝑡1𝐼⁻₃(𝑡)=𝐶𝑑𝑡₁𝑇𝛽₁−1𝑎₁+𝛼𝑎₂−𝑎₁𝑏𝑒^𝑏𝑧𝑏1−𝑧−𝑏−𝛾𝛽𝛽₁−1𝑎₁𝑧^𝛽+2𝛼1+𝛽2+𝛽+𝑃₁𝑡₁+𝛾𝛼₁𝑧^𝛽𝑡₁−𝛾𝛼₁^𝛽+1𝑡₁^𝛽+1𝛽+1−𝑧−𝛾𝑧^𝛽+1+𝛾𝑧^𝛽+1+𝛽𝛽+1₂−1𝑏𝑎₁+𝛼𝑎₂−𝑎₁𝑒^{𝑏𝛼1𝑡1}𝑏−𝛼₁𝑡₁𝑒^𝑏𝑧−𝑒^𝑏𝑧𝑏+𝑧𝑒^𝑏𝑧+𝛾𝑎₁𝛽₂−1𝛽𝛼₁^𝛽+1𝑡₁^𝛽+1𝑏−1+𝛽𝛾𝑎₁𝛽₂−1𝛽𝑧^𝛽+1𝑏−1+𝛽𝛾𝑎₁𝛽₂−1𝑏𝛼₁𝑡₁^𝛽+1−𝑧^𝛽+1+𝑏2𝛼₁𝑡₁^𝛽+2−𝑧^𝛽+2+𝑏²6𝛼₁𝑡₁^𝛽+3−𝑧^𝛽+3+𝛾a₁𝛽₂−1𝑒^𝑏𝑧𝑏𝛼1+𝛽¹𝑡₁^𝛽+1−𝑧^𝛽+1−𝛾𝑎₁+𝛼𝑎₂−𝑎₁𝛽₂−1𝛽𝛼₁^𝛽+1𝑡₁^𝛽+1−𝑧^𝛽+1𝑏+𝛾𝑎𝛽+1₁+𝛼𝑎₂−𝑎₁𝛽₂−1𝑏𝛼₁𝑡₁^𝛽+1−𝑧^𝛽+1+𝑏2𝛼₁𝑡₁^𝛽+2−𝑧^𝛽+2+𝑏²6𝛼₁𝑡₁^𝛽+3−𝑧^𝛽+3−𝛾𝑎₁+𝛼𝑎₂−𝑎₁𝛽₂−1𝑏𝛼𝛽+1¹𝑡₁^𝛽+1−𝑧^𝛽+1+𝛾𝑎₁+𝛼𝑎₂−𝑎₁𝛽₂−1𝑧^𝛽1−𝑒^𝑏𝑧𝛼₁𝑡₁−𝑧𝑏+𝑎₁+𝛼𝑎₂−𝑎₁𝛽𝛽₂−1𝑧1+𝛽2+𝛽𝛽+2^𝛽+1𝛼₁𝑡₁−𝛼₁^𝛽+2𝑡₁^𝛽+2−𝑧^𝛽+2𝛽+2+𝑧^𝛽+2+𝑎₃−𝛼𝑎₃−𝑎₂𝑏𝑒^𝑏𝑡1𝑡₁−𝑒^𝑏𝑡1𝑏−𝛼₁𝑡₁𝑒^𝑏𝑡1+𝑒^{𝑏𝛼1𝑡1}𝑏−𝑎₃𝛾𝑒^𝑏𝑡1𝑡₁^𝛽+1−𝛼₁^𝛽+1𝑡₁^𝛽+1𝑏−𝑎𝛽+1³𝛾𝛽𝑏𝑡1+𝛽^1𝛽+1−𝛼₁^𝛽+1𝑡₁^𝛽+1+𝑎₃𝛾𝑏𝑡₁^𝛽+1−𝛼₁^𝛽+1𝑡₁^𝛽+1+𝑏2𝑡₁^𝛽+2−𝛼₁^𝛽+2𝑡₁^𝛽+2+𝑏²6𝑡₁^𝛽+3−𝛼₁^𝛽+3𝑡₁^𝛽+3−𝑎₃−𝛼𝑎₃−𝑎₂𝛾𝑏𝑡₁^𝛽+1𝑒^𝑏𝑡1−𝑡₁^𝛽+1−𝛼₁𝑡₁^𝛽+1𝑒^𝑏𝑡1+𝛼₁𝑡₁^𝛽+1+𝛽𝛾𝑎₃−𝛼𝑎₃−𝑎₂b𝑡1+𝛽^1𝛽+1−𝛼₁^𝛽+1𝑡₁^𝛽+1−𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝑏𝑡₁^𝛽+1−𝛼₁^𝛽+1𝑡₁^𝛽+1+𝑏2𝑡₁^𝛽+2−𝛼₁^𝛽+2𝑡₁^𝛽+2+𝑏²6𝑡₁^𝛽+3−𝛼₁^𝛽+3𝑡₁^𝛽+3+𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝑡₁^𝛽+1−𝛼₁^𝛽+1𝑡₁^𝛽+1𝑏−𝑎1+𝛽₃−𝛼𝑎₃−𝑎₂𝛾𝛽𝑡1+𝛽^1𝛽+2−𝑡₁^𝛽+2𝛽+2−𝛼₁𝑡₁^𝛽+2+𝛼₁^𝛽+2𝑡₁^𝛽+2=𝐶𝛽+2₁𝑇𝐴₂+𝐵₂𝑡₁+𝐶₂𝑡₁^𝛽+1+𝐷₂𝑡₁^𝛽+2+𝐸₂𝑡₁^𝛽+3+𝐹₂𝑒^𝑏𝑡1𝑡₁+𝐺₂𝑒^𝑏𝑡1𝑡₁^𝛽+1+𝐻₂𝑒^{𝑏𝛼1𝑡1}+𝐼₂𝑒^𝑏𝑡1, (B.1) where 𝐴₂=𝛽₁−1𝑎₁+𝛼𝑎₂−𝑎₁𝑏𝑒^𝑏𝑧𝑏1−𝑧−𝑏−𝛾𝛽𝛽₁−1𝑎₁𝑧^𝛽+21+𝛽2+𝛽−𝑃𝑧−𝑃𝛾𝑧^𝛽+1+𝑃𝛾𝑧^𝛽+1+𝛽𝛽+1²−1𝑏𝑎₁+𝛼𝑎₂−𝑎₁−𝑒^𝑏𝑧𝑏+𝑧𝑒^𝑏𝑧−𝛾𝑎₁𝛽₂−1𝛽𝑧^𝛽+1𝑏+1+𝛽𝛾𝑎₁𝛽₂−1𝑏𝑧^𝛽+1+𝑏2𝑧^𝛽+2+𝑏²6𝑧^𝛽+3−𝛾𝑎₁𝛽₂−1𝑒^𝑏𝑧𝑏𝑧1+𝛽^𝛽+1+𝛾𝑎₁+𝛼𝑎₂−𝑎₁𝛽₂−1𝛽𝑧^𝛽+1𝑏−𝑎1+𝛽₁+𝛼𝑎₂−𝑎₁𝛾𝛽₂−1𝑏𝑧^𝛽+1+𝑏2𝑧^𝛽+2+𝑏²6𝑧^𝛽+3+𝑎₁+𝛼𝑎₂−𝑎₁𝛽₂−1𝛾𝑏𝑧𝛽+1^𝛽+1+𝑎₁+𝛼𝑎₂−𝑎₁𝛾𝛽₂−1𝑧^𝛽+1𝑒^𝑏𝑧𝑏−𝛾𝑎₁+𝛼𝑎₂−𝑎₁𝛽₂−1𝑧^𝛽+1𝑏−𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝛽𝛽₂−1𝑧2+𝛽^𝛽+2,𝐵₂=𝑃𝛾+𝑃𝛾𝛼₁𝑧^𝛽−𝛽₂−1𝑏𝑎₁+𝛼𝑎₂−𝑎₁𝛼₁𝑒^𝑏𝑧−𝑎₁+𝛼𝑎₂−𝑎₁𝛽₂−1𝛾𝑧^𝛽𝑒^𝑏𝑧𝛼₁𝑏+𝛾𝑎₁+𝛼𝑎₂−𝑎₁𝛽₂−1𝑧^𝛽𝛼₁𝑏+𝛾𝑎₁+𝛼𝑎₂−𝑎₁𝛽𝛽₂−1𝑧1+𝛽^𝛽+1𝛼₁,𝐶₂=−𝑃𝛾𝛼₁^𝛽+1+𝛽+1𝛾𝑎₁𝛽₂−1𝛽𝛼₁^𝛽+1𝑏−1+𝛽𝛾𝑎₁𝛽₂−1𝑏𝛼₁^𝛽+1+𝛼𝑎₁𝛽₂−1𝑒^𝑏𝑧𝑏𝛼1+𝛽^1𝛽+1−𝛾𝑎₁+𝛼𝑎₂−𝑎₁𝛽₂−1𝛽𝛼₁^𝛽+1𝑏+𝛾𝑎𝛽+1₁+𝛼𝑎₂−𝑎₁𝛽₂−1𝑏𝛼₁^𝛽+1−𝛾𝑎₁+𝛼𝑎₂−𝑎₁𝛽₂−1𝑏𝛼𝛽+1^1𝛽+1−𝑎₃𝛾𝑒^𝑏𝑡11−𝛼₁^𝛽+1𝑏−𝑎𝛽+1³𝛾𝛽𝑏1+𝛽1−𝛼₁^𝛽+1+𝑎₃𝛾𝑏1−𝛼₁^𝛽+1−𝑎₃−𝛼𝑎₃−𝑎₂𝛾𝑏𝛼₁+𝑎−1𝛽𝛾₃−𝛼𝑎₃−𝑎₂𝑏1+𝛽1−𝛼₁^𝛽+1−𝛾𝑎₃−𝛼𝑎₃−𝑎₂𝑏1−𝛼₁^𝛽+1+𝛾𝑎₃−𝛼𝑎₃−𝑎₂1−𝛼₁^𝛽+1𝑏,𝐷1+𝛽₂=−𝛾𝑎₁𝛽₂−1𝛼₁^𝛽+22+𝛾𝑎₁+𝛼𝑎₂−𝑎₁𝛽₂−1𝛼₁^𝛽+22−𝑎₁+𝛼𝑎₂−𝑎₁𝛽𝛾𝛽₂−1𝛼1+𝛽2+𝛽^1𝛽+2−𝑎₃𝛾1−𝛼₁^𝛽+22−𝛾𝑎₃−𝛼𝑎₃−𝑎₂1−𝛼₁^𝛽+22−𝑎₃−𝛼𝑎₃−𝑎₂𝛾𝛽+𝑎2+𝛽³−𝛼𝑎₃−𝑎₂𝛾𝛼₁𝛽−𝛼1+𝛽_1𝛽+2𝛾𝛽𝑎𝛽+1𝛽+2³−𝛼𝑎₃−𝑎₂,𝐸₂=−𝛾𝑎₁𝛽₂−1𝑏𝛼₁^𝛽+36+𝛾𝑎₁+𝛼𝑎₂−𝑎₁𝛽₂−1𝑏𝛼₁^𝛽+36+𝑎₃𝛾𝑏61−𝛼₁^𝛽+3−𝛾𝑏𝑎₃−𝛼𝑎₃−𝑎₂1−𝛼₁^𝛽+36,𝐹₂=𝑎₃−𝛼𝑎₃−𝑎₂𝑏1−𝛼₁,𝐺₂=𝑎₃𝛾𝛼₁^𝛽+1−1𝑏+𝑎𝛽+1³−𝛼𝑎₃−𝑎₂𝛾𝑏𝛼₁,𝐻−1₂=𝑎₃−𝛼𝑎₃−𝑎₂𝑏²+𝛽₂−1𝑏²𝑎₁+𝛼𝑎₂−𝑎₁,𝐼₂𝑎=−³−𝛼𝑎₃−𝑎₂𝑏². (B.2)

Acknowledgments

The author is thankful to the honourable referee and the editor-in-chief for their constructive suggestions for preparing this final version of the paper.