an epq model with two-component demand under fuzzy...

Hindawi Publishing CorporationAdvances in Operations ResearchVolume 2012, Article ID 264182, 22 pagesdoi:10.1155/2012/264182

Research ArticleAn EPQ Model with Two-Component Demandunder Fuzzy Environment and Weibull DistributionDeterioration with Shortages

S. Sarkar and T. Chakrabarti

Department of Applied Mathematics, University of Calcutta, 92 APC Road, Kolkata 700009, India

Correspondence should be addressed to S. Sarkar, [email protected]

Received 26 April 2011; Revised 28 June 2011; Accepted 15 July 2011

Academic Editor: Hsien-Chung Wu

Copyright q 2012 S. Sarkar and T. Chakrabarti. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

A single-item economic production model is developed in which inventory is depleted not onlydue 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. TheWeibull distribution, which is capable of representing constant, increasing, and decreasing ratesof 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 productionperiod. Items are produced at different rates during subperiods so as to meet various constraintsthat arise due to change in demand pattern, market fluctuations, and so forth. This paper modelssuch a situation. Here it is assumed that demand rate is uncertain in fuzzy sense, that is, it isimprecise in nature and so demand rate is taken as triangular fuzzy number. Then by using α-cutfor defuzzification the total variable cost per unit time is derived. Therefore the problem is reducedto crisp average costs. The multiobjective model is solved by Global Criteria method with the helpof GRG (Generalized Reduced Gradient) Technique. In this model shortages are permitted andfully backordered. Numerical examples are given to illustrate the solution procedure of the twomodels.

1. Introduction

The classical EOQ (Economic Order Quantity) inventory models were developed underthe assumption of constant demand. Later many researchers developed EOQ modelstaking linearly increasing or decreasing demand. Donaldson [1] discussed for the firsttime the classical no-shortage inventory policy for the case of a linear, positive trend indemand. Wagner and Whitin [2] developed a discrete version of the problem. Silver andMeal [3] formulated an approximate solution procedure as “Silver Meal heuristic” for

2 Advances in Operations Research

a deterministic time-dependent demand pattern. Mitra et al. [4] extended the model toaccommodate a demand pattern having increasing and decreasing linear trends. Deb andChaudhuri [5] extended for the first time the inventory replenishment policy with lineartrend 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 workedon inventory models with time-dependent demand and deterioration. Models by Daveand 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 researchersuse ramp type demand. Ramp type demand is one in which demand increases up to acertain time after which it stabilizes and becomes constant. Ramp type demand preciselydepicts 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 themarket 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 inventoryrelated demands are not deterministic but imprecise in nature. Furthermore in real-lifeproblems the parameters of the stochastic inventory models are also fuzzy, that is, notdeterministic in nature and this is the case of application of fuzzy probability in the inventorymodels.

In 1965, the first publication in fuzzy set theory by Zadeh [13] showed the intentionto accommodate uncertainty in the nonstochastic sense rather than the presence of randomvariables. 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 problemsunder fuzzy environment. Park [14] and Ishii and Konno [15] discussed the case of fuzzy costcoefficients. Roy and Maiti [16] developed a fuzzy economic order quantity (EOQ) modelwith a constraint of fuzzy storage capacity. Chang and Yao [17] solved the economic reorderpoint 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 Kratschmer [19]; KaoandHsu [20] discussed a single-period inventorymodel with fuzzy demand. Fergany and EI-Wakeel [21] considered the probabilistic single-item inventory problem with varying ordercost under two linear constraints. Hala and EI-Saadani [22] analysed constrained single-period stochastic uniform inventory model with continuous distribution of demand andvarying holding cost. Fuzzy models for single-period inventory problem were discussedby Li et al. [23]. Banerjee and Roy [24] considered application of the Intuitionistic FuzzyOptimization in the constrained Multiobjective Stochastic Inventory Model. Banerjee andRoy [25] also discussed the single- and multiobjective stochastic inventory model in fuzzyenvironment. Lee and Yao [26] andChang and Yao [17] investigated the economic productionquantity model, and Lee and Yao [27] studied the EOQ model with fuzzy demands. Acommon characteristic of these studies is that shortages are backordered without extracosts.

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

Advances in Operations Research 3

Many researchers developed inventory models in which both the demand anddeteriorating rate are constant. Although the constant demand assumption helps to simplifythe problem, it is far from the actual situation where demand is always in change. In orderto make research more practical, many researchers have studied other forms of demand.Among them time-dependant demand has attracted considerable attention. In this paperexponentially increasing demand has been considered which is a general form of linearand nonlinear time-dependent demand. Various types of uncertainties and imprecision areinherent in real inventory problems. They are classically modeled using the approaches fromthe probability theory. However there are uncertainties that cannot be appropriately treatedby usual probabilistic models. The questions how to define inventory optimization task insuch environment and how to interpret optimal solution arise. This paper considers themodification of EPQ formula in the presence of imprecisely estimated parameters with fuzzydemands where backorders are permitted, yet a shortage cost is incurred. The demand rateis taken as triangular fuzzy number. Since the demand is fuzzy the average cost associatedwith inventory is fuzzy in nature. So the average cost in fuzzy sense is derived. The fuzzymodel is defuzzified by using α-cut of fuzzy number. This multiobjective problem is solvedby 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 definitionsand results which will be used later on.

Definition 2.1 (fuzzy number). A fuzzy subset Ã of real number Rwith membership functionμÃ : R → [0, 1] is called a fuzzy number if

(a) Ã is 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) Ã is convex, that is, μÃ(λx1 + (1 − λ)x2) ≥ μ

Ã(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 specifiedby the triplet (a1, a2, a3) and is defined by its continuous membership function μ

Ã(x), acontinuous mapping μ

Ã : R → [0, 1] as follows:

μÃ(x) =

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

x − a1

a2 − a1if a1 ≤ x < a2

a3 − x

a3 − a2if a2 ≤ x ≤ a3

0 otherwise.

(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 seton R. Then integral of any continuous α-level curve of ˜f(x) over [a, b] always exists and theintegral of ˜f(x) over [a, b] is then defined to be the fuzzy set

˜I(a, b) =

{(

∫b

a

f−(α)(x)dx +∫b

a

f+(α)(x)dx, α

)}

. (2.2)

The determination of the integral ˜I(a, b) becomes somewhat easier if the fuzzy function isassumed to be LR type. We will therefore assume that ˜f(x) = (f1(x), f2(x), f3(x))LR is a fuzzynumber in LR representation for all x ⊆ [a, b], where f1(x), f2(x), and f3(x) are assumedto be positive integrable functions on [a, b]. Dubois and Prade have shown that under theseconditions

˜I(a, b) =

(

∫b

a

f1(x)dx,∫b

a

f2(x)dx,∫b

a

f3(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 isdefined as [ Ã]α = {x ∈ R : μ

Ã(x) ≥ α}. According to the definition of fuzzy number it is seenthat α-cut is a nonempty bounded closed interval; it can be denoted by

[

Ã]

α= [AL(α), AR(α)]. (2.4)

AL(α) and AR(α) are the lower and upper bounds of the closed interval, where

[

Ã]

L(α) = inf

{

x ∈ R : μÃ(x) ≥ α

}

,

[

Ã]

R(α) = sup

{

x ∈ R : μÃ(x) ≥ α

}

.

(2.5)

Definition 2.5 (Global Criteria method (Rao,nd)). In the global criteria method x∗ is found byminimizing a preselected global criteria, F(X), such as the sum of the squares of the relativedeviations of the individual objective functions from the feasible ideal solutions. The X∗ isfound by minimizing

F(X) =k∑

i=1

{

fi(

X∗i

) − fi(X)

fi(

X∗i

)

}p

subject to gj(X) ≤ 0, j = 1, 2, . . . , m,

(2.6)


where p is a constant (as usual value of p is 2) andX∗i is the ideal solution for the ith objective

function. The solution X∗i 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) ˜R(t) is the induced fuzzy demand around Rwhich 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) = aebt.

(ii) K1 and K2 are two demand depended production variables, that is, K1 = β1 ˜R(t)and K2 = β2 ˜R(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 andits actual addition to stock) is assumed to be zero. The assumption is made so thatthe period of shortage is not affected.

(v) The rate of deterioration at any time t > 0 follows the two-parameter Weibulldistribution: θ(t) = γβtβ−1, where γ (0 < γ < 1) is the scale parameter and β(> 0) isthe shape parameter. The implication of theWeibull rate (two parameter) is that theitems in inventory start deteriorating the instant they are received into inventory.The rate of deterioration-time relationship is shown in Figure 1.


Time

β = 1

β < 1

Rateof

deter

orati

ion

1 < β < 2

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 attainsa level P at t = z. Then at t = z the production rate changes to K2 and continues up tot = y, where the inventory level reaches the maximum level Q. And the production stops att = y and the inventory gradually depletes to zero at t = t1 mainly to meet the demands andpartly for deterioration. Now shortages occur and accumulate to the level S at time t = t2.The production starts again at a rateK1 at t = t2 and the backlog is cleared at time t = T whenthe stock is again zero. The cycle then repeats itself after time T .

The model is represented by Figure 2.The changes in the inventory level can be described by the following differential

equations:

d˜I(t)dt

+ γβtβ−1˜I(t) =(

β1 − 1)

aebt 0 ≤ t ≤ z,

d˜I(t)dt

+ γβtβ−1˜I(t) =(

β2 − 1)

aebt z ≤ t ≤ y,

d˜I(t)dt

+ γβtβ−1˜I(t) = −aebt y ≤ t ≤ t1,

d˜I(t)dt

= −aebt t1 ≤ t ≤ t2,

d˜I(t)dt

= −(β1 − 1)

aebt t2 ≤ t ≤ T

(4.1)

with initial conditions I(z) = P , I(y) = Q, I(t1) = 0, I(t2) = S, and I(T) = 0.


I(t)

P Q

z y t2 T

S

t1

Figure 2

The differential equations (4.1) are fuzzy differential equations. To solve this differen-tial equation at first we take the α-cut then the differential equations reduces to

dI+1 (t)dt

+ γβtβ−1I+1 (t) =(

β1 − 1)

a+(α)ebt, 0 ≤ t ≤ z, (4.2)

dI−1 (t)dt

+ γβtβ−1I−1 (t) =(

β1 − 1)

a−(α)ebt, 0 ≤ t ≤ z, (4.3)

dI+2 (t)dt

+ γβtβ−1I+2 (t) =(

β2 − 1)

a+(α)ebt, z ≤ t ≤ y, (4.4)

dI−2 (t)dt

+ γβtβ−1I−2 (t) =(

β2 − 1)

a−(α)ebt, z ≤ t ≤ y, (4.5)

dI+3 (t)dt

+ γβtβ−1I+3 (t) = −a−(α)ebt, y ≤ t ≤ t1, (4.6)

dI−3 (t)dt

+ γβtβ−1I−3 (t) = −a+(α)ebt, y ≤ t ≤ t1, (4.7)

dI+4 (t)dt

= −a−(α)ebt, t1 ≤ t ≤ t2, (4.8)

dI−4 (t)dt

= −a+(α)ebt, t1 ≤ t ≤ t2, (4.9)

dI+5 (t)dt

= −(β1 − 1)

a−(α)ebt, t2 ≤ t ≤ T, (4.10)

dI−5 (t)dt

= −(β1 − 1)

a+(α)ebt, t2 ≤ t ≤ T, (4.11)


where

I+i = sup{

x ∈ R : μ˜Ii(x) ≥ α

}

, i = 1, 2, 3, 4, 5,

I−i = inf{

x ∈ R : μ˜Ii(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 − 1)

b

)

(

ebt − 1)

{a3 − α(a3 − a2)} −γβtβ+1

(

β1 − 1)

a3(

1 + β) , 0 ≤ t ≤ z,

I−1 (t) =

((

β1 − 1)

b

)

(

ebt − 1)

{a1 + α(a2 − a1)} −γβtβ+1

(

β1 − 1)

a1(

1 + β) , 0 ≤ t ≤ z,

I+2 (t) = P(

1 + γzβ − γtβ)

+(

β2 − 1){a3 − α(a3 − a2)}

(

ebt − ebz)

b− γtβ

(

β2 − 1)

a3

b

(

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,

I−2 (t) = P(

1 + γzβ − γtβ)

+(

β2 − 1){a1 + α(a2 − a1)}

(

ebt − ebz)

b− γtβ

(

β2 − 1)

a1

b

(

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,

I+3 (t) =[{a1 + α(a2 − a1)}

b

]

(

ebt1 − ebt)

−(

a1γtβ

b

)

(

ebt1 − ebt)

+[{a1 + α(a2 − a1)}

b

]

γ{

tβ

1

(

ebt1 − 1)

− tβ(

ebt − 1)}

− {a1 + α(a2 − a1)}γβ

(

tβ+11 − tβ+1

)

(

1 + β) , y ≤ t ≤ t1,


I−3 (t) =[{a3 − α(a3 − a2)}

b

]

(

ebt1 − ebt)

−(

a3γtβ

b

)

(

ebt1 − ebt)

+[{a3 − α(a3 − a2)}

b

]

γ{

tβ

1

(

ebt1 − 1)

− tβ(

ebt − 1)}

− {a3 − α(a3 − a2)}γβ

(

tβ+11 − tβ+1

)

(

1 + β) , y ≤ t ≤ t1,

I+4 (t) = −{a1 + α(a2 − a1)}(

ebt − ebt1)

b, t1 ≤ t ≤ t2,

I−4 (t) = −{a3 − α(a3 − a2)}(

ebt − ebt1)

b, t1 ≤ t ≤ t2,

I+5 (t) =

(

β1 − 1){a1 + α(a2 − a1)}

(

ebT − ebt)

b, t2 ≤ t ≤ T,

I−5 (t) =

(

β1 − 1){a3 − α(a3 − a2)}

(

ebT − ebt)

b, t2 ≤ t ≤ T.

(4.13)

Therefore the upper α-cut of fuzzy stock holding cost is given by

HC+ =(

C1

T

)

[

∫z

0I+1 (t)dt +

∫α1t1

z

I+2 (t)dt +∫ t1

α1t1

I+3 (t)dt

]

=(

C1

T

)

[

A1 + B1t1 + C1tβ+11 +D1t

β+21 + E1t

β+31 + F1e

bt1t1 +G1ebt1t

β+11 +H1e

bα1t1 + I1ebt1]

.

(4.14)

Calculations are shown in Appendix A.And the lower α-cut of fuzzy stock holding cost is given by

HC− =(

C1

T

)

[

∫z

0I−1 (t)dt +

∫α1t1

z

I−2 (t)dt +∫ t1

α1t1

I−3 (t)dt

]

=(

C1

T

)

[

A2 + B2t1 + C2tβ+11 +D2t

β+21 + E2t

β+31 + F2e

bt1t1 +G2ebt1t

β+11 +H2e

bα1t1 + I2ebt1]

.

(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+ =C2

T

{

∫ηT

t1

I+4 (t)dt +∫T

ηT

I+5 (t)dt

}

=C2

T

{a1 + α(a2 − a1)}b

{

−ebηTb

+ ηTebt1 +ebt1

b− t1e

bt1

}

+C2

T{a1 + α(a2 − a1)}

(

β1 − 1)

b2

{

ebTT − ebT

b− ηTebT +

ebηT

b

}

.

(4.16)

Also the lower α-cut of shortage cost is given by

S.C− =C2

T

{

∫ηT

t1

I−4 (t)dt +∫T

ηT

I−5 (t)dt

}

=C2

T

{a3 − α(a3 − a2)}b

{

−ebηTb

+ ηTebt1 +ebt1

b− t1e

bt1

}

+C2

T

{a3 − α(a3 − a2)}(

β1 − 1)

b2

{

ebTT − ebT

b− ηTebT +

ebηT

b

}

.

(4.17)

(2) Annual cost due to deteriorated unit is also fuzzy as demand is fuzzy quantity.

Therefore deterioration cost per year is given by

D.C =C3

T

{

∫z

0(K1 − R)dt +

∫α1t1

z

(K2 − R)dt −∫ t1

α1t1

Rdt

}

=C3

T

{

∫z

0a(

β1 − 1)

ebtdt +∫α1t1

z

a(

β2 − 1)

ebtdt −∫ t1

α1t1

aebtdt

}

=C3

T

{

∫z

0f1(t)f2(t)f3(t)dt +

∫α1t1

z

φ1(t)φ2(t)φ3(t)dt −∫ t1

α1t1

ϕ1(t)ϕ2(t)ϕ3(t)dt

}

,

(4.18)

where

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) = a2e

bt, ϕ3(t) = a1ebt.

(4.19)


Therefore upper α-cut of deterioration cost is given by

D.C+ =C3

T

{

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)

−a1

b

(

ebt1 − ebα1t1)

− a1α

b

(

ebt1 − ebα1t1)

+a2α

b

(

ebt1 − ebα1t1)

}

.

(4.20)

The lower α-cut of deterioration cost is given by

D.C− =C3

T

{

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)

− a3

b

(

ebt1 − ebα1t1)

− a2α

b

(

ebt1 − ebα1t1)

+a3α

b

(

ebt1 − ebα1t1)

}

.

(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+ = sup{

x ∈ R : μ˜TVC(x) ≥ α

}

,

TVC− = inf{

x ∈ R : μ˜TVC(x) ≥ α

}

.

(4.23)

The upper α-cut of total variable cost per unit time is

TVC+ =(

C1

T

)

[

A1 + B1t1 + C1tβ+11 +D1t

β+21 + E1t

β+31 + F1e

bt1t1 +G1ebt1t

β+11 +H1e

bα1t1 + I1ebt1]

+C2

T

{a1 + α(a2 − a1)}b

{

−ebηTb

+ ηTebt1 +ebt1

b− t1e

bt1

}


+C2

T

{a1 + α(a2 − a1)}(

β1 − 1)

b2

{

ebTT − ebT

b− ηTebT +

ebηT

b

}

+C3

T

{

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)

− a1

b

(

ebt1 − ebα1t1)

− a1α

b

(

ebt1 − ebα1t1)

+a2α

b

(

ebt1 − ebα1t1)

}

+C4

T.

(4.24)

The lower α-cut of total variable cost per unit time is

TVC− =(

C1

T

)

[

A2 + B2t1 + C2tβ+11 +D2t

β+21 + E2t

β+31 + F2e

bt1t1 +G2ebt1t

β+11 +H2e

bα1t1 + I2ebt1]

+(

C2

T

){a3 − α(a3 − a2)}b

{

−ebηTb

+ ηTebt1 +ebt1

b− t1e

bt1

}

+C2

T

{a3 − α(a3 − a2)}(

β1 − 1)

b2

{

ebTT − ebT

b− ηTebT +

ebηT

b

}

+C3

T

{

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)

− a3

b

(

ebt1 − ebα1t1)

− a2α

b

(

ebt1 − ebα1t1)

+a3α

b

(

ebt1 − ebα1t1)

}

+(

C4

T

)

.

(4.25)

The objective in this paper is to find an optimal cycle time to minimize the total variable costper unit time.

Therefore this model mathematically can be written as

Minimize{

TVC+,TVC−}

Subject to 0 ≤ 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 to

Minimize GC

Subject to 0 ≤ 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 solvedby Global Criteria method converting it to a single-objective optimization problem. Thesolution procedure is as follows.

Step 1. Solve the multiobjective programming problem (4.26) as a single-objective problemusing 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 objectivevector is used as a reference point. The problem is then to solve the following auxiliaryproblem:

Min(GC) = Minimize

⎧

⎨

⎩

(

TVC+ − TVC+min

TVC+max − TVC+min

)Q

+

(

TVC− − TVC−min

TVC−max − TVC−min

)Q⎫

⎬

⎭

1/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 pointof 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 reducedgradient method (GRG). To illustrate the developed model, an example with the followingdata has been considered.

Let a1 = 100 units/month, a2 = 200 units/month, a3 = 100 units/month, C1 = $.1 perunit, 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 = 7 hrs.

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, indifferent systems, there are some parameters which are imprecise in nature. The presentpaper proposes a solution procedure to develop an EPQ inventory model with variableproduction rate and fuzzy demand. In most of the real life problem demand of a newlylaunched 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 environmentalanalysis, transportation, and so forth. In this paper we have taken two rates of production butthis can be extended to n number of production rates pi during the time when the inventorylevel 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+ =(

C1

T

)

[

∫z

0I+1 (t)dt +

∫α1t1

z

I+2 (t)dt +∫ t1

α1t1

I+3 (t)dt

]

=(

C1

T

)

⎡

⎣

(

β1 − 1){a3 − α(a3 − a2)}

b

(

ebz

b− z − 1

b

)

− γβ(

β1 − 1)

a3zβ+2

(

1 + β)(

2 + β)

+ P

⎧

⎨

⎩

α1t1 + γα1zβt1 −

γαβ+11 t

β+11

(

β + 1) − z − γzβ+1 +

γzβ+1

β + 1

⎫

⎬

⎭

+

(

β2 − 1)

b{a3 − α(a3 − a2)}

{

ebα1t1

b− α1t1e

bz − ebz

b+ zebz

}

+γa3(

β2 − 1)

βαβ+11 t

β+11

b(

1 + β) − γa3

(

β2 − 1)

βzβ+1

b(

1 + β)

− γa3(

β2 − 1)

b

[

{

(α1t1)β+1 − zβ+1}

+b

2

{

(α1t1)β+2 − zβ+2}

+b2

6

{

(α1t1)β+3 − zβ+3}

]

+γa3(

β2 − 1)

ebz

b(

1 + β)

{

(α1t1)β+1 − zβ+1}

− γ{a3 − α(a3 − a2)}(

β2 − 1)

βαβ+11 t

β+11

b(

β + 1) +

γ{a3 − α(a3 − a2)}(

β2 − 1)

βzβ+1

b(

1 + β)

+γ{a3 − α(a3 − a2)}

(

β2 − 1)

b

[

{

(α1t1)β+1 − zβ+1}

+b

2

{

(α1t1)β+2 − zβ+2}

+b2

6

{

(α1t1)β+3 − zβ+3}

]


− γ{a3 − α(a3 − a2)}(

β2 − 1)

b(

β + 1)

{

(α1t1)β+1 − zβ+1}

− γ{a3 + α(a3 − a2)}(

β2 − 1)

zβebz(α1t1 − z)b

+γ{a3 − α(a3 − a2)}

(

β2 − 1)

zβ(α1t1 − z)b

+γ{a3 − α(a3 − a2)}β

(

β2 − 1)

(

1 + β)(

2 + β)

{

(

β + 2)

zβ+1α1t1 − αβ+21 t

β+21 − zβ+2

(

β + 2)

+ zβ+2}

+{a1 + α(a2 − a1)}

b

{

ebt1t1 − ebt1

b− α1t1e

bt1 +ebα1t1

b

}

−a1γe

bt1(

tβ+11 − α

β+11 t

β+11

)

b(

β + 1)

− a1γβ

b(

1 + β)

{

tβ+11 − α

β+11 t

β+11

}

+a1γ

b

{

(

tβ+11 − α

β+11 t

β+11

)

+b

2

(

tβ+21 − α

β+21 t

β+21

)

+b2

6

(

tβ+31 − α

β+31 t

β+31

)

}

− {a1 + α(a2 − a12)}γb

{

tβ+11 ebt1 − t

β+11 − α1t

β+11 ebt1 + α1t

β+11

}

+βγ{a1 + α(a2 − a12)}

b(

1 + β)

(

tβ+11 − α

β+11 t

β+11

)

− γ{a1 + α(a2 − a1)}b

{

(

tβ+11 − α

β+11 t

β+11

)

+b

2

(

tβ+21 − α

β+21 t

β+21

)

+b2

6

(

tβ+31 − α

β+31 t

β+31

)

}

+γ{a1 + α(a2 − a1)}

(

tβ+11 − α

β+11 t

β+11

)

b(

1 + β)

−{a1 + α(a2 − a1)}γβ

(

1 + β)

⎧

⎨

⎩

tβ+21 − t

β+21

(

β + 2) − α1t

β+21 +

αβ+21 t

β+21

(

β + 2)

⎫

⎬

⎭

⎤

⎦

HC+ =(

C1

T

)

[

∫z

0I+1 (t)dt +

∫α1t1

z

I+2 (t)dt +∫ t1

α1t1

I+3 (t)dt

]

=(

C1

T

)

[

A1 + B1t1 + C1tβ+11 +D1t

β+21 + E1t

β+31 + F1e

bt1t1 +G1ebt1t

β+11 +H1e

bα1t1 + I1ebt1]

,

(A.1)


where

A1 =

(

β1 − 1){a3 − α(a3 − a2)}

b

(

ebz

b− z − 1

b

)

− γβ(

β1 − 1)

a3zβ+2

(

1 + β)(

2 + β) − Pz − Pγzβ+1 + P

γzβ+1

β + 1

+

(

β2 − 1)

b{a3 − α(a3 − a2)}

{

−ebz

b+ zebz

}

− γa3(

β2 − 1)

βzβ+1

b(

1 + β)

+γa3(

β2 − 1)

b

{

zβ+1 +b

2zβ+2 +

b2

6zβ+3}

− γa3(

β2 − 1)

ebz

b(

1 + β) zβ+1

+γ{a3 − α(a3 − a2)}

(

β2 − 1)

βzβ+1

b(

1 + β) − γ{a3 − α(a3 − a2)}

(

β2 − 1)

b

{

zβ+1 +b

2zβ+2 +

b2

6zβ+3}

+γ{a3 − α(a3 − a2)}

(

β2 − 1)

b(

β + 1) zβ+1 +

γ{a3 − α(a3 − a2)}(

β2 − 1)

zβ+1ebz

b

− {a3 − α(a3 − a2)}γ(

β2 − 1)

zβ+1

b− γ{a3 − α(a3 − a2)}β

(

β2 − 1)

(

2 + β) zβ+2,

B1 = Pγ + Pγα1zβ −(

β2 − 1)

b{a3 − α(a3 − a2)}α1e

bz − γ{a3 − α(a3 − a2)}(

β2 − 1)

zβebzα1

b

+γ{a3 − α(a3 − a2)}

(

β2 − 1)

zβα1

b+γ{a3 − α(a3 − a2)}β

(

β2 − 1)

(

1 + β) zβ+1α1,

C1 =

⎡

⎣

−Pγαβ+11

(

β + 1) +

γa3(

β2 − 1)

βαβ+11

b(

1 + β) − γa3

(

β2 − 1)

bαβ+11 +

γa3(

β2 − 1)

ebz

b(

1 + β) α

β+11

− γ{a3 − α(a3 − a2)}(

β2 − 1)

βαβ+11

b(

β + 1) +

γ{a3 − α(a3 − a2)}(

β2 − 1)

bαβ+11

− γ{a3 − α(a3 − a2)}(

β2 − 1)

b(

β + 1) α

β+11 −

a1γebt1(

1 − αβ+11

)

b(

β + 1) − a1γβ

b(

1 + β)

(

1 − αβ+11

)

+a1γ

b

(

1 − αβ+11

)

− {a1 + α(a2 − a1)}γb

(α1 − 1) +βγ{a1 + α(a2 − a1)}

b(

1 + β)

(

1 − αβ+11

)

−γ{a1 + α(a2 − a1)}b

(

1 − αβ+11

)

+γ{a1 + α(a2 − a1)}

(

1 − αβ+11

)

b(

1 + β)

⎤

⎥

⎦,


D1 = −γa3(

β2 − 1)

αβ+21

2+γ{a3 − α(a3 − a2)}

(

β2 − 1)

αβ+21

2− {a3 − α(a3 − a2)}γβ

(

β2 − 1)

(

1 + β)(

2 + β) α

β+21

−a1γ(

1 − αβ+21

)

2−γ{a1 + α(a2 − a1)}

(

1 − αβ+21

)

2− {a1 + α(a2 − a1)}

γβ(

2 + β)

+ {a1 + α(a2 − a1)}γα1β(

1 + β) − α

β+21 γβ

(

β + 1)(

β + 2){a1 + α(a2 − a1)},

E1 = −γa3(

β2 − 1)

bαβ+31

6+{a3 − α(a3 − a2)}

(

β2 − 1)

γbαβ+31

6+a1γb

6

(

1 − αβ+31

)

−γb{a1 + α(a2 − a1)}

(

1 − αβ+31

)

6,

F1 ={a1 + α(a2 − a1)}

b(1 − α1),

G1 =a1γ(

αβ+11 − 1

)

b(

β + 1) +

{a1 + α(a2 − a1)}γb

(α1 − 1),

H1 ={a1 + α(a2 − a1)}

b2+

(

β2 − 1)

b2{a3 − α(a3 − a2)},

I1 = −{a1 + α(a2 − a1)}b2

.

(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− =(

C1

T

)

[

∫z

0I−1 (t)dt +

∫α1t1

z

I−2 (t)dt +∫ t1

α1t1

I−3 (t)dt

]

=(

C1

T

)

[(

β1 − 1){a1 + α(a2 − a1) }

b

(

ebz

b− z − 1

b

)

− γβ(

β1 − 1)

a1zβ+2

(

1 + β)(

2 + β)

+ P

⎧

⎨

⎩

α1t1 + γα1zβt1 −

γαβ+11 t

β+11

(

β + 1) − z − γzβ+1 +

γzβ+1

β + 1

⎫

⎬

⎭

+

(

β2 − 1)

b{a1 + α(a2 − a1)}

{

ebα1t1

b− α1t1e

bz − ebz

b+ zebz

}


+γa1(

β2 − 1)

βαβ+11 t

β+11

b(

1 + β) − γa1

(

β2 − 1)

βzβ+1

b(

1 + β)

− γa1(

β2 − 1)

b

[

{

(α1t1)β+1 − zβ+1}

+b

2

{

(α1t1)β+2 − zβ+2}

+b2

6

{

(α1t1)β+3 − zβ+3}

]

+γa1(

β2 − 1)

ebz

b(

1 + β)

{

(α1t1)β+1 − zβ+1}

−γ{a1 + α(a2 − a1)}

(

β2 − 1)

β(

αβ+11 t

β+11 − zβ+1

)

b(

β + 1)

+γ{a1 + α(a2 − a1)}

(

β2 − 1)

b

[

{

(α1t1)β+1 − zβ+1}

+b

2

{

(α1t1)β+2 − zβ+2}

+b2

6

{

(α1t1)β+3 − zβ+3}

]

− γ{a1 + α(a2 − a1)}(

β2 − 1)

b(

β + 1)

{

(α1t1)β+1 − zβ+1}

+ γ{a1 + α(a2 − a1)}(

β2 − 1)

zβ(

1 − ebz)

(α1t1 − z)b

+{a1 + α(a2 − a1)}β

(

β2 − 1)

(

1 + β)(

2 + β)

{

(

β + 2)

zβ+1α1t1 − αβ+21 t

β+21 − zβ+2

(

β + 2)

+ zβ+2}

+{a3 − α(a3 − a2)}

b

{

ebt1t1 − ebt1

b− α1t1e

bt1 +ebα1t1

b

}

−a3γe

bt1(

tβ+11 − α

β+11 t

β+11

)

b(

β + 1) − a3γβ

b(

1 + β)

{

tβ+11 − α

β+11 t

β+11

}

+a3γ

b

{

(

tβ+11 − α

β+11 t

β+11

)

+b

2

(

tβ+21 − α

β+21 t

β+21

)

+b2

6

(

tβ+31 − α

β+31 t

β+31

)

}

− {a3 − α(a3 − a2)}γb

{

tβ+11 ebt1 − t

β+11 − α1t

β+11 ebt1 + α1t

β+11

}

+βγ{a3 − α(a3 − a2)}

b(

1 + β)

(

tβ+11 − α

β+11 t

β+11

)

− γ{a3 − α(a3 − a2)}b

{

(

tβ+11 − α

β+11 t

β+11

)

+b

2

(

tβ+21 − α

β+21 t

β+21

)

+b2

6

(

tβ+31 − α

β+31 t

β+31

)

}


+γ{a3 − α(a3 − a2)}

(

tβ+11 − α

β+11 t

β+11

)

b(

1 + β)

−{a3 − α(a3 − a2)}γβ

(

1 + β)

⎧

⎨

⎩

tβ+21 − t

β+21

(

β + 2) − α1t

β+21 +

αβ+21 t

β+21

(

β + 2)

⎫

⎬

⎭

⎤

⎦

=(

C1

T

)

[

A2 + B2t1 + C2tβ+11 +D2t

β+21 + E2t

β+31 + F2e

bt1t1 +G2ebt1t

β+11 +H2e

bα1t1 + I2ebt1]

,

(B.1)

where

A2 =

(

β1 − 1){a1 + α(a2 − a1)}

b

(

ebz

b− z − 1

b

)

− γβ(

β1 − 1)

a1zβ+2

(

1 + β)(

2 + β) − Pz − Pγzβ+1

+ Pγzβ+1

β + 1+

(

β2 − 1)

b{a1 + α(a2 − a1)}

{

−ebz

b+ zebz

}

− γa1(

β2 − 1)

βzβ+1

b(

1 + β) +

γa1(

β2 − 1)

b

{

zβ+1 +b

2zβ+2 +

b2

6zβ+3}

− γa1(

β2 − 1)

ebz

b(

1 + β) zβ+1 +

γ{a1 + α(a2 − a1)}(

β2 − 1)

βzβ+1

b(

1 + β)

− {a1 + α(a2 − a1)}γ(

β2 − 1)

b

{

zβ+1 +b

2zβ+2 +

b2

6zβ+3}

+{a1 + α(a2 − a1)}

(

β2 − 1)

γ

b(

β + 1) zβ+1 +

{a1 + α(a2 − a1)}γ(

β2 − 1)

zβ+1ebz

b

− γ{a1 + α(a2 − a1)}(

β2 − 1)

zβ+1

b− γ{a3 − α(a3 − a2)}β

(

β2 − 1)

(

2 + β) zβ+2,

B2 = Pγ + Pγα1zβ −(

β2 − 1)

b{a1 + α(a2 − a1)}α1e

bz

− {a1 + α(a2 − a1)}(

β2 − 1)

γzβebzα1

b+γ{a1 + α(a2 − a1)}

(

β2 − 1)

zβα1

b

+γ{a1 + α(a2 − a1)}β

(

β2 − 1)

(

1 + β) zβ+1α1,

C2 =

⎡

⎣

−Pγαβ+11

(

β + 1) +

γa1(

β2 − 1)

βαβ+11

b(

1 + β) − γa1

(

β2 − 1)

bα1

β+1

+αa1(

β2 − 1)

ebz

b(

1 + β) α1

β+1 − γ{a1 + α(a2 − a1)}(

β2 − 1)

βαβ+11

b(

β + 1)


+γ{a1 + α(a2 − a1)}

(

β2 − 1)

bα1

β+1 − γ{a1 + α(a2 − a1)}(

β2 − 1)

b(

β + 1) α1

β+1

−a3γe

bt1(

1 − αβ+11

)

b(

β + 1) − a3γβ

b(

1 + β)

(

1 − αβ+11

)

+a3γ

b

(

1 − αβ+11

)

− {a3 − α(a3 − a2)}γb

(α1 − 1) +βγ{a3 − α(a3 − a2)}

b(

1 + β)

(

1 − αβ+11

)

−γ{a3 − α(a3 − a2)}b

(

1 − αβ+11

)

+γ{a3 − α(a3 − a2)}

(

1 − αβ+11

)

b(

1 + β)

⎤

⎥

⎦,

D2 = −γa1(

β2 − 1)

αβ+21

2+γ{a1 + α(a2 − a1)}

(

β2 − 1)

αβ+21

2

− {a1 + α(a2 − a1)}βγ(

β2 − 1)

(

1 + β)(

2 + β) α

β+21 −

a3γ(

1 − αβ+21

)

2−γ{a3 − α(a3 − a2)}

(

1 − αβ+21

)

2

− {a3 − α(a3 − a2)}γβ

(

2 + β) + {a3 − α(a3 − a2)}

γα1β(

1 + β)

− αβ+21 γβ

(

β + 1)(

β + 2){a3 − α(a3 − a2)},

E2 = −γa1(

β2 − 1)

bαβ+31

6+γ{a1 + α(a2 − a1)}

(

β2 − 1)

bαβ+31

6

+a3γb

6

(

1 − αβ+31

)

−γb{a3 − α(a3 − a2)}

(

1 − αβ+31

)

6,

F2 ={a3 − α(a3 − a2)}

b(1 − α1),

G2 =a3γ(

αβ+11 − 1

)

b(

β + 1) +

{a3 − α(a3 − a2)}γb

(α1 − 1),

H2 ={a3 − α(a3 − a2)}

b2+

(

β2 − 1)

b2{a1 + α(a2 − a1)},

I2 = −{a3 − α(a3 − a2)}b2

.

(B.2)

Acknowledgments

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


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