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

Authors: S. Sarkar; T. Chakrabarti
Journal: Advances in Operations Research
 (2012)

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## 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 usingα-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.


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## Body

## 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 usingα-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 subsetÃ of real number R with membership function μÃ:R→[0,1]is called a fuzzy number if(a)
Ãis normal, that is, there exists an element x0 such that μÃ(x0)=1 is normal, that is, there exists an element  x0 such that  μÃ(x0)=1;(b)
Ãis convex, that is, μÃ(λx1+(1-λ)x2)≥μÃ(x1)∧μÃ(x2)for all x1,x2∈R and λ∈[0,1];(c)
μÃ  is upper semicontinuous;(d)
supp(Ã) is bounded, here supp(Ã)=supp{x∈R:μÃ(x)>0}.Definition 2.2 (triangular fuzzy number).
A Triangular Fuzzy number (TFN)Ã is specified by the triplet (a1, a2, a3) and is defined by its continuous membership function μÃ(x), a continuous mapping μÃ:R→[0,1] as follows:
(2.1)μÃ(x)={x-a1a2-a1ifa1≤x<a2a3-xa3-a2ifa2≤x≤a30otherwise.Definition 2.3 (integration of a fuzzy function).
Letf̃(x) be a fuzzy function from  [a,b]⊆R to R such that f̃(x) is a fuzzy number, that is, a piecewise continuous normalized fuzzy set on R. Then integral of any continuous α-level curve of   f̃(x) over [a,b] always exists and the integral of   f̃(x)over [a,b] is then defined to be the fuzzy set
(2.2)Ĩ(a,b)={(∫abf-(α)(x)dx+∫abf+(α)(x)dx,α)}.
The determination of the integral  Ĩ(a,b) becomes somewhat easier if the fuzzy function is assumed to be LR type. We will therefore assume that  f̃(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
(2.3)Ĩ(a,b)=(∫abf1(x)dx,∫abf2(x)dx,∫abf3(x)dx)LR.Definition 2.4 (α-cut of fuzzy number).
Theα-cut of a fuzzy number is a crisp set which is defined as [Ã]α={x∈R:μÃ(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
(2.4)[Ã]α=[AL(α),AR(α)].AL(α) and AR(α) are the lower and upper bounds of the closed interval, where
(2.5)[Ã]L(α)=inf⁡⁡{x∈R:μÃ(x)≥α},[Ã]R(α)=sup⁡{x∈R:μÃ(x)≥α}.Definition 2.5 (Global Criteria method (Rao,nd)).
In the global criteria methodx* 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
(2.6)F(X)=∑i=1k{fi(Xi*)-fi(X)fi(Xi*)}psubjecttogj(X)≤0,j=1,2,…,m,
where p is a constant (as usual value of p is 2) and Xi* is the ideal solution for the ith  objective function. The  solution Xi*  is obtained by minimizing fi(X) subject to the constraints gj(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)
R̃(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 functionR̃(t)is taken to be fuzzy function of time, that is,R̃(t)=ãebt.(ii)
K1 and K2 are two demand depended production variables, that is, K1=β1R̃(t) and K2=β2R̃(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 timet>0 follows the two-parameter Weibull distribution: θ(t)=γβtβ-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 1Whenβ=1, z(t)=α (a constant) which is the case of exponential decay.Whenβ<1 rate of deterioration is decreasing with time.Whenβ>1, rate of deterioration is increasing with time.(vi)
Shortages are allowed and are fully backlogged.


## 3.1. Notations

(i)
T
is the length of one cycle.(ii)
I(t)
is the inventory level at time t.(iii)
R̃(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 functionR̃(t)is taken to be fuzzy function of time, that is,R̃(t)=ãebt.(ii)
K1 and K2 are two demand depended production variables, that is, K1=β1R̃(t) and K2=β2R̃(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 timet>0 follows the two-parameter Weibull distribution: θ(t)=γβtβ-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 1Whenβ=1, z(t)=α (a constant) which is the case of exponential decay.Whenβ<1 rate of deterioration is decreasing with time.Whenβ>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 timet=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 Figure2.Figure 2The changes in the inventory level can be described by the following differential equations:(4.1)dĨ(t)dt+γβtβ-1Ĩ(t)=(β1-1)ãebt0≤t≤z,dĨ(t)dt+γβtβ-1Ĩ(t)=(β2-1)ãebtz≤t≤y,dĨ(t)dt+γβtβ-1Ĩ(t)=-ãebty≤t≤t1,dĨ(t)dt=-ãebtt1≤t≤t2,dĨ(t)dt=-(β1-1)ãebtt2≤t≤T
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
(4.2)dI1+(t)dt+γβtβ-1I1+(t)=(β1-1)a+(α)ebt,0≤t≤z,(4.3)dI1-(t)dt+γβtβ-1I1-(t)=(β1-1)a-(α)ebt,0≤t≤z,(4.4)dI2+(t)dt+γβtβ-1I2+(t)=(β2-1)a+(α)ebt,z≤t≤y,(4.5)dI2-(t)dt+γβtβ-1I2-(t)=(β2-1)a-(α)ebt,z≤t≤y,(4.6)dI3+(t)dt+γβtβ-1I3+(t)=-a-(α)ebt,y≤t≤t1,(4.7)dI3-(t)dt+γβtβ-1I3-(t)=-a+(α)ebt,y≤t≤t1,(4.8)dI4+(t)dt=-a-(α)ebt,t1≤t≤t2,(4.9)dI4-(t)dt=-a+(α)ebt,t1≤t≤t2,(4.10)dI5+(t)dt=-(β1-1)a-(α)ebt,t2≤t≤T,(4.11)dI5-(t)dt=-(β1-1)a+(α)ebt,t2≤t≤T,
where  (4.12)Ii+=sup⁡⁡{x∈R:μĨi(x)≥α},i=1,2,3,4,5,Ii-=inf⁡⁡{x∈R:μĨi(x)≥α},i=1,2,3,4,5.
Similarly a+(α), a-(α)  have usual meaning.The solutions of the differential equations (4.2) to (4.11) are, respectively, represented by (4.13)I1+(t)=((β1-1)b)(ebt-1){a3-α(a3-a2)}-γβtβ+1(β1-1)a3(1+β),0≤t≤z,I1-(t)=((β1-1)b)(ebt-1){a1+α(a2-a1)}-γβtβ+1(β1-1)a1(1+β),0≤t≤z,I2+(t)=P(1+γzβ-γtβ)+(β2-1){a3-α(a3-a2)}(ebt-ebz)b-γtβ(β2-1)a3b(ebt-ebz)+γ{a3-α(a3-a2)}(β2-1)btβ(ebt-1)-γ{a3-α(a3-a2)}(β2-1)zβ(ebz-1)b+{a3-α(a3-a2)}γβ(β2-1)(1+β)(zβ+1-tβ+1),z≤t≤y,I2-(t)=P(1+γzβ-γtβ)+(β2-1){a1+α(a2-a1)}(ebt-ebz)b-γtβ(β2-1)a1b(ebt-ebz)+{a1+α(a2-a1)}γ(β2-1)btβ(ebt-1)-{a1+α(a2-a1)}γ(β2-1)zβ(ebz-1)b+{a1+α(a2-a1)}βγ(β2-1)(1+β)(zβ+1-tβ+1),z≤t≤y,I3+(t)=[{a1+α(a2-a1)}b](ebt1-ebt)-(a1γtβb)(ebt1-ebt)+[{a1+α(a2-a1)}b]γ{t1β(ebt1-1)-tβ(ebt-1)}-{a1+α(a2-a1)}γβ(t1β+1-tβ+1)(1+β),y≤t≤t1,I3-(t)=[{a3-α(a3-a2)}b](ebt1-ebt)-(a3γtβb)(ebt1-ebt)+[{a3-α(a3-a2)}b]γ{t1β(ebt1-1)-tβ(ebt-1)}-{a3-α(a3-a2)}γβ(t1β+1-tβ+1)(1+β),y≤t≤t1,I4+(t)=-{a1+α(a2-a1)}(ebt-ebt1)b,t1≤t≤t2,I4-(t)=-{a3-α(a3-a2)}(ebt-ebt1)b,t1≤t≤t2,I5+(t)=(β1-1){a1+α(a2-a1)}(ebT-ebt)b,t2≤t≤T,I5-(t)=(β1-1){a3-α(a3-a2)}(ebT-ebt)b,t2≤t≤T.
Therefore the upper α-cut of fuzzy stock holding cost is given by(4.14)HC+=(C1T)[∫0zI1+(t)dt+∫zα1t1I2+(t)dt+∫α1t1t1I3+(t)dt]=(C1T)[A1+B1t1+C1t1β+1+D1t1β+2+E1t1β+3+F1ebt1t1+G1ebt1t1β+1+H1ebα1t1+I1ebt1].
Calculations are shown in Appendix A.And the lowerα-cut of fuzzy stock holding cost is given by(4.15)HC-=(C1T)[∫0zI1-(t)dt+∫zα1t1I2-(t)dt+∫α1t1t1I3-(t)dt]=(C1T)[A2+B2t1+C2t1β+1+D2t1β+2+E2t1β+3+F2ebt1t1+G2ebt1t1β+1+H2ebα1t1+I2ebt1].
Calculations are shown in Appendix B.(1)
As demand is fuzzy in nature shortage cost is also fuzzy in nature.Therefore the upperα-cut of shortage cost is given by
(4.16)S.C+=C2T{∫t1ηTI4+(t)dt+∫ηTTI5+(t)dt}=C2T{a1+α(a2-a1)}b{-ebηTb+ηTebt1+ebt1b-t1ebt1}+C2T{a1+α(a2-a1)}(β1-1)b2{ebTT-ebTb-ηTebT+ebηTb}.
Also the lower α-cut of shortage cost is given by(4.17)S.C-=C2T{∫t1ηTI4-(t)dt+∫ηTTI5-(t)dt}=C2T{a3-α(a3-a2)}b{-ebηTb+ηTebt1+ebt1b-t1ebt1}+C2T{a3-α(a3-a2)}(β1-1)b2{ebTT-ebTb-ηTebT+ebηTb}.(2)
Annual cost due to deteriorated unit is also fuzzy as demand is fuzzy quantity.Therefore deterioration cost per year is given by(4.18)D.C=C3T{∫0z(K1-R)dt+∫zα1t1(K2-R)dt-∫α1t1t1Rdt}=C3T{∫0zã(β1-1)ebtdt+∫zα1t1ã(β2-1)ebtdt-∫α1t1t1ãebtdt}=C3T{∫0zf1(t)f2(t)f3(t)dt+∫zα1t1ϕ1(t)ϕ2(t)ϕ3(t)dt-∫α1t1t1φ1(t)φ2(t)φ3(t)dt},
where(4.19)f1(t)=a1(β1-1)ebt,f2(t)=a2(β1-1)ebt,f3(t)=a3(β1-1)ebt,ϕ1(t)=a1(β2-1)ebt,ϕ2(t)=a2(β2-1)ebt,ϕ3(t)=a3(β2-1)ebt,φ1(t)=a3ebt,φ2(t)=a2ebt,φ3(t)=a1ebt.
Therefore upper α-cut of deterioration cost is given by(4.20)D.C+=C3T{a3(β1-1)(ebz-1)b-αa3(β1-1)(ebz-1)b+a2α(β1-1)(ebz-1)b+a3(β2-1)b(ebα1t1-ebz)-αa3(β2-1)b(ebα1t1-ebz)+αa2(β2-1)b(ebα1t1-ebz)-a1b(ebt1-ebα1t1)-a1αb(ebt1-ebα1t1)+a2αb(ebt1-ebα1t1)(β1-1)(ebz-1)b}.
The lower α-cut of deterioration cost is given by(4.21)D.C-=C3T{a1(β1-1)(ebz-1)b+αa2(β1-1)(ebz-1)b-a1α(β1-1)(ebz-1)b+a1(β2-1)b(ebα1t1-ebz)+αa2(β2-1)b(ebα1t1-ebz)-αa1(β2-1)b(ebα1t1-ebz)-a3b(ebt1-ebα1t1)-a2αb(ebt1-ebα1t1)+a3αb(ebt1-ebα1t1)(β1-1)(ebz-1)b}.(3)
The annual ordering cost=C4/T.Therefore total variable cost per unit time is a fuzzy quantity and is defined by(4.22)TVC̃={TVC+TVC-},
where
(4.23)TVC+=sup⁡⁡{x∈R:μTVC̃(x)≥α},TVC-=inf⁡⁡{x∈R:μTVC̃(x)≥α}.
The upper α-cut of total variable cost per unit time is(4.24)TVC+=(C1T)[A1+B1t1+C1t1β+1+D1t1β+2+E1t1β+3+F1ebt1t1+G1ebt1t1β+1+H1ebα1t1+I1ebt1]+C2T{a1+α(a2-a1)}b{-ebηTb+ηTebt1+ebt1b-t1ebt1}+C2T{a1+α(a2-a1)}(β1-1)b2{ebTT-ebTb-ηTebT+ebηTb}+C3T{a3(β1-1)(ebz-1)b-αa3(β1-1)(ebz-1)b+a2α(β1-1)(ebz-1)b+a3(β2-1)b(ebα1t1-ebz)-αa3(β2-1)b(ebα1t1-ebz)+αa2(β2-1)b(ebα1t1-ebz)-a1b(ebt1-ebα1t1)-a1αb(ebt1-ebα1t1)+a2αb(ebt1-ebα1t1)(β1-1)(ebz-1)b}+C4T.
The lower α-cut of total variable cost per unit time is(4.25)TVC-=(C1T)[A2+B2t1+C2t1β+1+D2t1β+2+E2t1β+3+F2ebt1t1+G2ebt1t1β+1+H2ebα1t1+I2ebt1]+(C2T){a3-α(a3-a2)}b{-ebηTb+ηTebt1+ebt1b-t1ebt1}+C2T{a3-α(a3-a2)}(β1-1)b2{ebTT-ebTb-ηTebT+ebηTb}+C3T{a1(β1-1)(ebz-1)b+αa2(β1-1)(ebz-1)b-a1α(β1-1)(ebz-1)b+a1(β2-1)b(ebα1t1-ebz)+αa2(β2-1)b(ebα1t1-ebz)-αa1(β2-1)b(ebα1t1-ebz)-a3b(ebt1-ebα1t1)-a2αb(ebt1-ebα1t1)+a3αb(ebt1-ebα1t1)(β1-1)(ebz-1)b}+(C4T).
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(4.26)Minimize{TVC+,TVC-}Subjectto0≤α≤1.
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 to(4.27)MinimizeGCSubjectto0≤α≤1.


## 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 Step1, 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:
(5.1)Min⁡(GC)=Minimize{(TVC+-TVC+min⁡TVC+max⁡-TVC+min⁡)Q+(TVC--TVC-min⁡TVC-max⁡-TVC-min⁡)Q}1/Q,
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.Leta1=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(6.1)GC=0.0926.
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 ton number of production rates pi during the time when the inventory level goes from(i-1)Q to  iQ(i=1,2,…,n), where Q is prefixed level.



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*Source: 264182-2011-10-26.xml*