A Deterministic Inventory Model of Deteriorating Items with...

International Scholarly Research NetworkISRN Applied MathematicsVolume 2011, Article ID 657464, 16 pagesdoi:10.5402/2011/657464

Research ArticleA Deterministic Inventory Model of DeterioratingItems with Two Rates of Production, Shortages,and Variable Production Cycle

Jhuma Bhowmick1 and G. P. Samanta2

1 Department of Mathematics, Maharaja Manindra Chandra College, Kolkata 700003, India2 Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah 711103, India

Correspondence should be addressed to G. P. Samanta, g p [email protected]

Received 9 March 2011; Accepted 19 April 2011

Academic Editors: S. Cho and D. Spinello

Copyright q 2011 J. Bhowmick and G. P. Samanta. 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 Continuous production control inventory model is developed for a deteriorating item havingshortages and variable production cycle. It is assumed that the production rate is changed toanother at a time when the inventory level reaches a prefixed level Q1 and continued until theinventory level reaches the level S(> Q1). The demand rate is assumed to be constant, and theproduction cycle T is taken as variable. The production is started again at a time when the shortagelevel reaches a prefixed quantity Q2. For this model, the total cost per unit time as a function ofQ1, Q2, S, and T is derived. The optimal decision rules for Q1, Q2, S, and T are computed. Thesensitivity of the optimal solution towards changes in the values of different system parameters isalso studied. Results are illustrated by numerical examples.

1. Introduction

EOQ inventory models have long been attracting considerable amount of research attention.For the last fifteen years, researchers in this area have extended investigation into variousmodels with considerations of item shortage, item deterioration, demand patterns, item ordercycles, and their combinations. The control and maintenance of production inventories ofdeteriorating items with shortages have received much attention of several researchers inthe recent years because most physical goods deteriorate over time. In reality, some of theitems are either damaged or decayed or vaporized or affected by some other factors and arenot in a perfect condition to satisfy the demand. Food items, drugs, pharmaceuticals, andradioactive substances are examples of items in which sufficient deterioration can take placeduring the normal storage period of the units, and consequently this loss must be taken into

2 ISRN Applied Mathematics

account when analyzing the system. Research in this direction began with the work of Whitin[1] who considered fashion goods deteriorating at the end of a prescribed storage period.Ghare and Schrader [2] were the two earliest researchers to consider continuously decayinginventory for a constant demand. An order-level inventory model for items deteriorating at aconstant rate was discussed by Shah and Jaiswal [3]. Aggarwal [4] developed an order-levelinventory model by correcting and modifying the error in Shah and Jaiswal’s analysis [3]in calculating the average inventory holding cost. In all these models, the demand rate andthe deterioration rate were constants, the replenishment rate was infinite, and no shortage ininventory was allowed.

Researchers started to develop inventory systems allowing time variability in oneor more than one parameters. Dave and Patel [5] discussed an inventory model forreplenishment. This was followed by another model by Dave [6]with variable instantaneousdemand, discrete opportunities for replenishment, and shortages. Bahari-Kashani [7]discussed a heuristic model with time-proportional demand. An economic order quantity(EOQ)model for deteriorating items with shortages and linear trend in demand was studiedby Goswami and Chaudhuri [8]. It is a common belief that a large pile of goods attractsmore customers in the supermarket. This phenomenon is termed as stock-dependent demandrate. Baker and Urban [9] established an economic order quantity model for a power-forminventory-level-dependent demand pattern. Mandal and Phaujdar [10] then developed aneconomic production quantity model for deteriorating items with constant production rateand linearly stock-dependent demand. Later on, Datta and Pal [11] presented an EOQmodelin which the demand rate is dependent on the instantaneous stocks displayed until a givenlevel of inventory S0 is reached, after which the demand rate becomes constant. Other papersrelated to this area are Urban [12], Giri et al. [13], Padmanabhan and Vrat [14], Pal et al.[15], Ray and Chaudhuri [16], Urban and Baker [17], Ray et al. [18], Giri and Chaudhuri[19], Datta and Paul [20], Ouyang et al. [21], Teng et al. [22], Roy and Samanta [23],and others.

Another class of inventory models has been developed with time-dependentdeterioration rate. Covert and Philip [24] used a two-parameter Weibull distribution torepresent the distribution of the time to deterioration. This model was further developedby Philip [25] by taking a three-parameter Weibull distribution for the time to deterioration.Mishra [26] analyzed an inventory model with a variable rate of deterioration, finite rateof replenishment, and no shortage, but only a special case of the model was solved undervery restrictive assumptions. Deb and Chaudhuri [27] studied a model with a finite rate ofproduction and a time-proportional deterioration rate, allowing backlogging. Goswami andChaudhuri [8] assumed that the demand rate, production rate, and deterioration rate wereall time dependent. A detailed information regarding inventory modeling for deterioratingitems was given in the review papers of Nahmias [28] and Raafat [29]. An order-levelinventory model for deteriorating items without shortages has been developed by Jalan andChaudhuri [30]. Ouyang et al. [31] considered the continuous inventory system with partialbackorders. A production inventory model with two rates of production and backorders isanalyzed by Perumal and Arivarignan [32], Samanta and Roy [33].

In the present paper, we have developed a continuous production control inventorymodel with variable production cycle for deteriorating items with shortages in which twodifferent rates of production are available, and it is possible that production started at one rateand after some time it may be switched over to another rate. Such a situation is desirable inthe sense that by starting at a low rate of production, a large quantum stock of manufactureditems at the initial stage is avoided, leading to reduction in the holding cost.

ISRN Applied Mathematics 3

2. Notations and Modeling Assumptions

The mathematical model in this paper is developed on the basis of the following notationsand assumptions:

(i) a is the constant demand rate;

(ii) p1 (>a) and p2 (> p1) are the constant production rates started at time t = 0 and attime t = t1 (>o), respectively;

(iii) C1 is the holding cost per unit per unit time;

(iv) C2 is the shortage cost per unit per unit time;

(v) C3 is the cost of a deteriorated unit. (C1, C2, and C3 are known constants);

(vi) C4 and C5 are the constant unit production costs when the production rates are p1and p2 respectively (C4 > C5);

(vii) Q(t) is the inventory level at time t(≥ 0);

(viii) A is the setup cost;

(ix) replenishment is instantaneous and lead time is zero;

(x) T is the variable duration of production cycle;

(xi) shortages are allowed and backlogged;

(xii) C is the average cost of the system;

(xiii) the distribution of the time to deterioration of an item follows the exponentialdistribution g(t), where

g(t) =

⎧⎨

⎩

θe−θt, for t > 0,

0, otherwise.(2.1)

θ is called the deterioration rate; a constant fraction θ (0 < θ �1) of the on-hand inventorydeteriorates per unit time. It is assumed that no repair or replacement of the deteriorateditems takes place during a given cycle.

In this paper, we have considered a single commodity deterministic continuousproduction inventory model with a constant demand rate a. The production of the item isstarted initially at t = 0 at a rate p1 (>a). Once the inventory level reaches Q1, the rate ofproduction is switched over to p2 (> p1), and the production is stopped when the level ofinventory reaches S(> Q1) and the inventory is depleted at a constant rate a. It is decidedto backlog demands up to Q2 which occur during stock-out time. Thus, the inventory levelreaches Q2 (backorder level is Q2), the production is started at a faster rate p2 so as to clearthe backlog, and when the inventory level reaches 0 (i.e., the backlog is cleared), the nextproduction cycle starts at the lower rare p1.

We denote by [0, t1], the duration of production at the rate p1, by [t1, t2], the durationof production at the rate p2, by [t2, t3], the duration when there is no production but onlyconsumption by demand at a rate a, by [t3, t4], the duration of shortage, and by [t4, T],the duration of time to backlog at the rate p2. The cycle then repeats itself after time T . Theduration of a production cycle T is taken as variable.

This model is represented by Figure 1.


Inve

ntory

S

Q1

t4 TO Time

t1 t2 t3 Q2

Figure 1

3. Model Formulation and Solution

Let Q(t) be the instantaneous state of the inventory level at any time t (0 ≤ t ≤ T),then thedifferential equations describing the instantaneous states of Q(t) in the interval [0, T] aregiven by the following:

dQ(t)dt

+ θQ(t) = p1 − a, 0 ≤ t ≤ t1,

dQ(t)dt

+ θQ(t) = p2 − a, t1 ≤ t ≤ t2,

dQ(t)dt

+ θQ(t) = −a, t2 ≤ t ≤ t3,

dQ(t)dt

= −a, t3 ≤ t ≤ t4,

dQ(t)dt

= p2 − a, t4 ≤ t ≤ T.

(3.1)

The boundary conditions are

Q(0) = 0, Q(t1) = Q1, Q(t2) = S, Q(t3) = 0, Q(t4) = −Q2, Q(T) = 0.(3.2)

The solutions of equations (3.1) are given by

Q(t) =1θ

(p1 − a

)(1 − e−θt

), 0 ≤ t ≤ t1 (3.3)

=1θ

(p2 − a

)+ e−θ(t−t1)

[

Q1 − 1θ

(p2 − a

)]

, t1 ≤ t ≤ t2 (3.4)

= −aθ+(

S +a

θ

)

e−θ (t−t2), t2 ≤ t ≤ t3 (3.5)

= −a(t − t3), t3 ≤ t ≤ t4 (3.6)

= −Q2 +(p2 − a

)(t − t4), t4 ≤ t ≤ T. (3.7)


From (3.2) and (3.3), we have

1θ

(p1 − a

)(1 − e−θt1

)= Q1 (3.8a)

∴ t1 =1θlog

[

1 +θQ1

(p1 − a

) +θ2Q2

1(p1 − a

)2

]

(neglecting higher powers of θ, 0 < θ � 1

)

=Q1

(p1 − a

) +θQ2

1

2(p1 − a

)2

(neglecting higher powers of θ, 0 < θ � 1

).

(3.8b)

Again, from (3.2) and (3.4), we have

1θ

(p2 − a

)+ eθ(t1−t2)

[

Q1 − 1θ

(p2 − a

)]

= S (3.9)

∴ eθ(t2−t1) =

[

1 − Sθ(p2 − a

)

]−1 [

1 − Q1θ(p2 − a

)

]

= 1 +(S −Q1)θ(p2 − a

) +θ2(S −Q1)S(p2 − a

)2

(neglecting higher powers of θ

),

(3.10a)

∴ t2 − t1 =1θlog

[

1 +(S −Q1)θ(p2 − a

) +θ2(S −Q1)S(p2 − a

)2

]

=(S −Q1)(p2 − a

) +θ(S2 −Q1

2)

2(p2 − a

)2


),

(3.10b)

t2 =Q1

(p1 − a

) +θQ2

1

2(p1 − a

)2 +(S −Q1)(p2 − a

) +θ(S2 −Q1

2)

2(p2 − a

)2

[by (3.8b)

]. (3.11)

From (3.5), we have

− 1θa +

(

S +1θa

)

eθ(t2− t3) = 0, since Q(t3) = 0 (3.12a)

=⇒ t3 − t2 =1θlog

(

1 +θS

a

)

=S

a− S2θ

2a2


)(3.12b)

∴ t3 =Q1

(p1 − a

) +θQ2

1

2(p1 − a

)2 +(S −Q1)(p2 − a

) +θ(S2 −Q1

2)

2(p2 − a

)2 +S

a− S2θ

2a2

[by (3.11)

]. (3.13)


Again, from (3.6), we have

a(t3 − t4) = −Q2, since Q(t4) = −Q2, (3.14)

∴ t4 = t3 +Q2

a. (3.15)

From (3.7) and Q(T) = 0, we have

− Q2 +(p2 − a

)(T − t4) = 0

∴ Q2 =(p2 − a

)(

T − t3 − Q2

a

)[by (3.15)

],

(3.16)

Q2 =a(p2 − a

)

p2

⎡

⎢⎣T − Q1

(p1 − a

) − θQ21

2(p1 − a

)2 − (S −Q1)(p2 − a

)

−θ(S2 −Q1

2)

2(p2 − a

)2 − S

a+S2θ

2a2

⎤

⎥⎦

[by (3.13)

],

(3.17)

D = Total number of deteriorated items in [0, T]

=[(p1 − a

)t1 +

(p2 − a

)(t2 − t1) − S

]+ [S − a(t3 − t2)]

=(p1 − a

)[

Q1(p1 − a

) +θQ2

1

2(p1 − a

)2

]

+(p2 − a

)

⎡

⎢⎣(S −Q1)(p2 − a

) +θ(S2 −Q1

2)

2(p2 − a

)2

⎤

⎥⎦

− a

(S

a− S2θ

2a2

)[using (3.8b), (3.10b), and (3.12b)

]

=θ

2

⎡

⎢⎣

Q12

(p1 − a

) +

(S2 −Q1

2)

(p2 − a

) +S2

a

⎤

⎥⎦,

(3.18)

S1 = Total Shortage over the period [0, T]

= −∫ t4

t3

a(t − t3)dt +∫T

t4

[−Q2 +(p2 − a

)(t − t4)

]dt

[by (3.6) and (3.7)

]

= − p2Q22

2a(p2 − a

)[by using (3.6) and (3.7)

],

(3.19)

IT = Total inventory carried over the period [0, T]

=∫ t1

0Q(t)dt +

∫ t2

t1

Q(t)dt +∫ t3

t2

Q(t)dt.(3.20)


Now,

∫ t1

0Q(t)dt =

1θ

(p1 − a

)∫ t1

0

(1 − e−θt

)dt

[by (3.3)

]

=1θ

(p1 − a

)[

t1 +1θ

(e−θt1 − 1

)]

=(p1 − a

)[t1θ− Q1

θ(p1 − a

)

][by (3.8a)

]

=(p1 − a

)[1θ2

log

{

1 +θQ1

(p1 − a

) +θ2Q2

1(p1 − a

)2 +θ3Q3

1(p1 − a

)3

}

− Q1

θ(p1 − a

)

]

[by (3.8a) and neglecting higher powers of θ

]

=(p1 − a

)[

Q12

(p1 − a

)2 +θQ3

1(p1 − a

)3 − Q21

2(p1 − a

)2 − θQ13

(p1 − a

)3 +θQ3

1

3(p1 − a

)3

]


)

=Q2

1

2(p1 − a

) +θQ3

1

3(p1 − a

)2 ,

(3.21)∫ t2

t1

Q(t)dt =∫ t2

t1

[1θ

(p2 − a

)+{

Q1 − 1θ

(p2 − a

)}

e−θ(t−t1)]

dt[by (3.4)

]

=1θ

(p2 − a

)(t2 − t1) − 1

θ

{

Q1 − 1θ

(p2 − a

)}{

e−θ(t2−t1) − 1}

=1θ2

(p2 − a

)log

[

1 +(S −Q1)θ(p2 − a

) +θ2(S −Q1)S(p2 − a

)2 +θ3S2(S −Q1)(p2 − a

)3

]

− (S −Q1)θ

[by (3.9) and (3.10a)

]

=(p2 − a

)[(S −Q1)θ(p2 − a

) +(S −Q1)S(p2 − a

)2 +S2(S −Q1)θ(p2 − a

)3 − (S −Q1)2

2(p2 − a

)2

− (S −Q1)2Sθ(p2 − a

)3 +(S −Q1)3θ

3(p2 − a

)3

]

− (S −Q1)θ


)

=

(S2 −Q2

1

)

2(p2 − a

) +θ(S3 −Q3

1

)

3(p2 − a

)2 ,

(3.22)


∫ t3

t2

Q(t)dt =∫ t3

t2

[

−aθ+(

S +a

θ

)

e−θ(t−t2)]

dt[by (3.5)

]

= −aθ(t3 − t2) − 1

θ

(

S +a

θ

){e−θ(t3−t2) − 1

}

= − a

θ2log

(

1 +Sθ

a

)

+S

θ

[by (3.12a)

]

= − a

θ2

[Sθ

a− S2θ2

2a2+S3θ3

3a3

]

+S

θ


)

=S2

2a− S3θ

3a2.

(3.23)

Therefore, from (3.20), total inventory carried over the cycle [0,T]

=Q2

1

2(p1 − a

) +θQ3

1

3(p1 − a

)2 +

(S2 −Q2

1

)

2(p2 − a

) +θ(S3 −Q3

1

)

3(p2 − a

)2 +S2

2a− S3θ

3a2 , (3.24)

P = Production cost over the period [0, T]

= C4p1t1 + C5{p2(t2 − t1) + p2(T − t4)

}

=C4p1Q1(p1 − a

)

[

1 +θQ1

2(p1 − a

)

]

+C5p2

(p2 − a

)

[

S +Q2 −Q1 +θ(S2 −Q2

1

)

2(p2 − a

)

]

[by (3.8b), (3.10b), and (3.16)

].

(3.25)

Average cost of the system

= C(Q1, S, T)

=1T[C1IT − C2S1 + C3D + P +A]

=1T

[

A +C5p2

(p2 − a

)S +

{(C1 + C3θ)p22a

(p2 − a

) +C5p2θ

2(p2 − a

)2

}

S2 +C1θ

(2a − p2

)p2

3a2(p2 − a

)2 S3

+C5p2

(p2 − a

)Q2 +C2p2

2a(p2 − a

)Q22 +

{C4p1

(p1 − a

) − C5p2(p2 − a

)

}

Q1

+

{f

2(C1 + C3θ) +

θ

2

(C4p1

(p1 − a

)2 − C5p2(p2 − a

)2

)}

Q21 +

C1dfθ

3kQ3

1

]

[using (3.18), (3.19), (3.24), and (3.25)

],

(3.26)


where

d = p1 + p2 − 2a,

k =(p1 − a

)(p2 − a

),

f =

(p2 − p1

)

k,

(3.27)

and from (3.17),

Q2 =a(p2 − a

)

p2T − S − a

(p2 − p1

)

p2(p1 − a

)Q1 −θ(2a − p2

)

2a(p2 − a

)S2 − adfθ

2p2(p1 − a

)Q21. (3.28)

The necessary conditions for C(Q1, S, T) to be minimum are

∂C

∂Q1= 0,

∂C

∂S= 0,

∂C

∂T= 0, (3.29)

that is,

C1dfθ

kQ2

1 +

{

(C1 + C3θ)f + θ

(C4p1

(p1 − a

)2 − C5p2(p2 − a

)2

)}

Q1

− 1k(C5a + C2Q2)

(p2 − p1 + θdfQ1

)+

C4p1(p1 − a

) − C5p2(p2 − a

) = 0,

(3.30)

C1θp2(2a − p2

)S2 + ap2

{(C1 + C3θ)

(p2 − a

)+ C5aθ

}S

+ C5a2p2

(p2 − a

) − p2(aC5 + C2Q2){a(p2 − a

)+ θ

(2a − p2

)S}= 0,

(3.31)

T(C5a + C2Q2) −A −{

C4p1(p1 − a

) − C5p2(p2 − a

)

}

Q1

− p2(p2 − a

)

[

C5(Q2 + S) +C2

2aQ2

2 +

{(C1 + C3θ)

a+

C5θ(p2 − a

)

}

× S2

2+C1θ

(2a − p2

)

3a2(p2 − a

) S3

]

−{

(C1 + C3θ)f + θ

(C4p1

(p1 − a

)2 − C5p2(p2 − a

)2

)}Q2

1

2− C1dfθ

3kQ3

1 = 0.

(3.32)


Solving these and using (3.28), we get the optimal values Q1∗, Q2

∗, S∗, and T ∗ of Q1, Q2, S,and T , respectively, which minimizeC(Q1, S, T) provided they satisfy the following sufficientcondition:

H = The Hessian Matrix of C

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

∂2C

∂Q21

∂2C

∂Q1∂S

∂2C

∂Q1∂T

∂2C

∂S∂Q1

∂2C

∂S2

∂2C

∂S∂T

∂2C

∂T∂Q1

∂2C

∂T∂S

∂2C

∂T2

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(3.33)

is positive definite.If the solutions obtained from equations (3.30), (3.31), and (3.32) do not satisfy the

sufficient condition (3.33), we may conclude that no feasible solution will be optimal for theset of parameter values taken to solve equations (3.30), (3.31), and (3.32). Such a situationwillimply that the parameter values are inconsistent and there is some error in their estimation.

4. Numerical Examples

Example 4.1. Let A = 200, C1 = 1.5, C2 = 2, C3 = 18, C4 = 15, C5 = 13, a = 3, p1 = 4, p2 = 6, andθ = 0.002 in appropriate units.

Based on these input data, the computer outputs areQ1

∗ = 6.2681, S∗ = 13.9149, Q2∗ = 10.8676, T ∗ = 20.7353, C∗ = 60.7351.

These results satisfy the sufficient condition.

Example 4.2. The parameters are similar to those in Example 4.1, except that p2 is changed to8 units.


∗ = 8.2123, S∗ = 14.5851, Q2∗ = 11.3927, T ∗ = 20.4743, C∗ = 61.7855.


Example 4.3. The parameters are similar to those in Example 4.1, except that p2 is changed to10 units.


∗ = 8.9252, S∗ = 14.875, Q2∗ = 11.62, T ∗ = 20.3248, C∗ = 62.2401.


These examples reveal that a higher value of p2 causes higher values of Q1∗, S∗, Q2

∗,and C∗, but lower value of T ∗.

5. Sensitivity Analysis

Sensitivity analysis depicts the extent to which the optimal solution of the model is affectedby changes or errors in its input parameter values. In this section, we study the sensitivity of


the optimal inventory levelsQ1∗, S∗, backorder leverQ2

∗, cycle length T ∗, and average systemcost C∗ with respect to the changes in the values of the parameters C1, C2, C3, C4, C5, θ,A, a,p1, and p2. The results are shown in Tables 1–3.

Careful study of Table 1 reveals the following facts.

(i) It is seen that Q1∗ is insensitive to changes in the values of parameters C3 and θ,

moderately sensitive to changes in the values of parameters C1, C2, A, and a, andhighly sensitive to changes in the values of parameters C4, C5, p1, and p2.

(ii) It is observed that S∗ is insensitive to changes in the values of parameters C2,C3, and θ and moderately sensitive to changes in the values of parameters C1, C4,C5, A, a, p1, and p2.

(iii) It is seen that Q2∗ is insensitive to changes in the values of parameters C3, θ and

moderately sensitive to changes in the values of parameters C1, C2, C4, C5, A, a, p1and p2.

(iv) Table 1 reveals that T ∗ is insensitive to changes in the values of parameters C2, C3θ,and p2, moderately sensitive to changes in the values of parameters C1, C5, A, anda, and highly sensitive to changes in the values of parameters C4 and p1.

(v) It can be seen that the optimum total cost C∗ is insensitive to changes in the valuesof parameters C1, C2, C3, θ, A, and p2 and moderately sensitive to changes in thevalues of parameters C4, C5, a, and p1.



moderately sensitive to changes in the values of parameters C1, C2,A, a, p1, and p2,and highly sensitive to changes in the values of parameters C4 and C5.

(ii) It is observed that S∗ is insensitive to changes in the values of parameters C2, C3,θ,and p2, moderately sensitive to changes in the values of parameters C1, C5, A, a,and p1, and highly sensitive to changes in the values of parameters C4.

(iii) It is seen thatQ2∗ is insensitive to changes in the values of parameters C3, θ, and p2

moderately sensitive to changes in the values of parameters C1, C2, C5, A, a, andp1, and highly sensitive to changes in the value of parameter C4.

(iv) Table 2 reveals that T ∗ is insensitive to changes in the values of parameters C2, C3,θ, and p2, moderately sensitive to changes in the values of parameters C1, C4, C5,A, and a, and highly sensitive to changes in the value of parameter p1.




moderately sensitive to changes in the values of parameters C1, C2,A, a, p1, and p2,and highly sensitive to changes in the values of parameters C4 and C5.

(ii) It is observed that S∗ is insensitive to changes in the values of parameters C2, C3,and θ, p2,moderately sensitive to changes in the values of parameters C1,A, a, andp1, and highly sensitive to changes in the values of parameters C4 and C5.


Table 1: Sensitivity Analysis of Example 4.1.

Changingparameter Change (%)

Change (%) in

Q∗1 S∗ Q2

∗ T ∗ C∗

C1

+10 −3.786 −6.510 +2.467 −2.826 +0.883+5 −1.916 −3.376 +1.262 −1.462 +0.452− 5 +1.961 +3.649 −1.328 +1.571 −0.475−10 +3.958 +7.608 −2.726 +3.263 −0.975

C2

+10 +3.628 +1.624 −7.616 −1.191 +0.581+5 +1.881 +0.842 −3.960 −0.622 +0.301− 5 −2.028 −0.908 +4.306 +0.683 −0.325−10 −4.223 −1.891 +9.009 +1.437 −0.677

C3

+10 −0.089 −0.168 +0.062 −0.072 +0.022+ 5 −0.045 −0.084 +0.031 −0.036 +0.011−5 +0.045 + 0.085 −0.031 +0.036 −0.011−10 +0.091 +0.169 −0.063 +0.072 −0.022

C4

+10 −80.468 +5.337 + 5.337 −12.126 +1.910+5 −38.354 +3.478 +3.478 −5.062 +1.245−5 +35.143 −4.861 −4.861 +3.348 −1.740−10 +67.384 −10.973 −10.975 +5.140 −3.927

C5

+10 +58.906 −9.329 −9.179 +4.747 +3.137+ 5 +30.574 −4.196 −4.115 +2.965 +1.738−5 −33.032 +3.165 +3.079 −4.258 −2.109−10 −68.877 +5.148 +4.973 −9.992 −4.642

θ

+10 −0.108 −0.294 +0.102 −0.101 +0.036+ 5 −0.054 −0.147 +0.052 −0.051 +0.018−5 +0.056 +0.148 −0.052 +0.051 −0.018−10 +0.110 +0.296 −0.102 +0.103 −0.036

A

+10 +9.654 +4.321 +4.320 +5.422 +1.546+ 5 +4.890 +2.189 +2.189 +2.746 +0.783−5 −5.027 −2.250 −2.252 −2.823 −0.805−10 −10.204 −4.568 −4.570 −5.729 −1.635

a

+10 +2.527 −4.392 −4.400 +10.160 +4.847+ 5 +2.127 −1.809 −1.814 +4.189 +2.562−5 −3.529 +1.179 +1.812 −2.983 −2.787−10 −8.259 +1.820 +1.182 −5.084 −5.767

p1

+10 −36.526 +4.298 +4.296 -9.557 +1.538+ 5 −14.757 +2.565 +2.564 -5.313 +0.918− 5 + 8.642 −3.632 −3.634 +7.214 −1.300−10 +11.062 −8.804 −8.804 +18.052 −3.150

p2

+10 +13.856 +2.009 +2.016 −0.440 +0.721+ 5 +7.752 +1.098 +1.100 −0.223 +0.394− 5 −10.207 −1.355 −1.360 +0.210 −0.486− 10 −24.350 −3.081 −3.090 +0.358 −1.106



Parameter Change %Change % in

Q1∗ S∗ Q2

∗ T ∗ C∗

C1

10 −4.5943 −6.4038 2.5824 −3.1601 0.95225 −2.3550 −3.3212 1.3201 −1.6357 0.4865−5 2.4780 3.5920 −1.3798 1.7607 −0.5090−10 5.0899 7.4926 −2.8281 3.6622 −1.0426

C2

10 2.4512 1.3747 −7.8410 −1.0047 0.50695 1.2725 0.7137 −4.0824 −0.5250 0.2632−5 −1.3784 −0.7727 4.4502 0.5773 −0.2850−10 −2.8762 −1.6119 9.3209 1.2152 −0.5946

C3

10 −0.1120 −0.1659 0.0641 −0.0806 0.02385 −0.0560 −0.0830 0.0325 −0.0401 0.0120−5 0.0560 0.0836 −0.0316 0.0405 −0.0118−10 0.1108 0.1673 −0.0641 0.0806 −0.0238

C4

10 −43.0659 8.6979 8.6994 −8.1600 3.20805 −20.6130 4.8550 4.8575 −3.4526 1.7907−5 18.9557 −5.7662 −5.7669 2.3200 −2.1267−10 36.3674 −12.3825 −12.3851 3.5786 −4.5673

C5

10 31.7463 −10.6369 −10.4892 3.2802 2.44395 16.4668 −5.0024 −4.9233 2.0421 1.3403−5 −17.7368 4.3277 4.2413 −2.8973 −1.5920−10 −36.8618 7.9142 7.7348 −6.7284 −3.4595

θ

10 −0.1705 −0.2935 0.1053 −0.1114 0.03875 −0.0852 −0.1467 0.0527 −0.0557 0.0194−5 0.0852 0.1481 −0.0527 0.0562 −0.0194−10 0.1717 0.2955 −0.1053 0.1123 −0.0388

A

10 7.4498 4.1769 4.1781 5.2598 1.54055 3.7724 2.1152 2.1163 2.6633 0.7801−5 −3.8759 −2.1728 −2.1733 −2.7351 −0.8015−10 −7.8638 −4.4086 −4.3686 −5.5479 −1.6261

a

10 −3.0734 −4.3647 −4.3695 10.4272 4.70055 −0.7781 −1.7559 −1.7573 4.2653 2.5075−5 −0.4420 1.0710 1.0735 −2.9881 −2.7602−10 −1.9093 1.5646 1.5721 −5.0468 −5.7329

p1

10 −7.0102 5.7998 5.8002 −9.1725 2.13925 −2.4147 3.2567 3.2591 −5.2099 1.2011−5 −0.1413 −4.2530 −4.2536 7.2799 −1.5687−10 −3.6117 −9.9986 −9.9994 18.3850 −3.6878

p2

10 4.2741 0.9565 0.9611 −0.3409 0.35405 2.3160 0.5129 0.5161 −0.1802 0.1899−5 −2.7861 −0.5999 −0.6021 0.2032 −0.2222−10 −6.2029 −1.3130 −1.3184 0.4322 −0.4860



Parameter Change %Change % in

Q1∗ S∗ Q2

∗ T ∗ C∗

C1

10 −4.7428 −6.3395 2.6515 −3.2960 0.99005 −2.4358 −3.2881 1.3537 −1.7063 0.5053−5 2.5770 3.5570 −1.4139 1.8377 −0.5278−10 5.3086 7.4192 −2.8924 3.8229 −1.0800

C2

10 2.1053 1.2585 −7.9466 −0.9186 0.47005 1.0935 0.6541 −4.1394 −0.4802 0.2442−5 −1.1865 −0.7092 4.5172 0.5284 −0.2649−10 −2.4784 −1.4810 9.4647 1.1119 −0.2521

C3

10 −0.1165 −0.1654 0.0680 −0.0836 0.02465 −0.0583 −0.0827 0.0336 −0.0418 0.0124−5 0.0583 0.0834 −0.0327 0.0423 −0.0124−10 0.1154 0.1661 −0.0654 0.0841 −0.0247

C4

10 −33.843 9.8266 9.8287 −6.8744 3.66985 −16.218 5.3284 5.3305 −2.9216 1.9900−5 14.9308 −6.0887 −6.0912 1.9744 −2.2741−10 28.6470 −12.895 −12.897 3.0524 −4.8160

C5

10 24.9866 −11.109 −10.962 2.7853 2.17245 12.9599 −5.2894 −5.2117 1.7319 1.1870−5 −14.171 4.7287 4.6429 −2.4468 −1.3999−10 −28.957 8.8491 8.6695 −5.6635 −3.0291

θ

10 −0.1860 −0.2931 0.1084 −0.1151 0.04005 −0.0930 −0.1466 0.0542 −0.0576 0.0199−5 0.0930 0.1472 −0.0525 0.0581 −0.0201−10 0.1871 0.2951 −0.1084 0.1161 −0.0402

A

10 6.9018 4.1264 4.1274 5.2060 1.54085 3.4946 2.0894 2.0904 2.6357 0.7802−5 −3.5898 −2.1452 −2.1446 −2.7056 −0.8014−10 −7.2805 −4.3523 −4.3537 −5.4879 −1.6256

a

10 −4.5355 −4.4437 −4.4484 10.6200 4.60495 −1.5260 −1.7768 −1.7780 4.3356 2.4685−5 0.3339 1.0629 1.0654 −3.0219 −2.7352−10 −0.3328 1.5267 1.5318 −5.0893 −5.6941

p1

10 −1.4812 6.2783 6.2806 −9.2306 2.34465 0.0560 3.4931 3.4923 −5.2601 1.3045−5 −2.0862 −4.4867 −4.4880 7.3826 −1.6758−10 −6.9959 −10.463 −10.463 18.6688 −3.9079

p2

10 2.3775 0.6158 0.6170 −0.2480 0.23065 1.2739 0.3281 0.3305 −0.1309 0.1228−5 −1.4902 −0.3765 −0.3778 0.1486 −0.1414−10 −3.2537 −0.8148 3.1885 0.3178 −0.1334


(iii) It is seen that Q2∗ is insensitive to changes in the values of parameters C3 and θ,

moderately sensitive to changes in the values of parameters C1, C2, C5, A, a, p1,and p2, and highly sensitive to changes in the value of parameter C4.

(iv) Table 3 reveals that T ∗ is insensitive to changes in the values of parameters C2, C3,and θ, p2, moderately sensitive to changes in the values of parameters C1, C4, C5,A, and a, and highly sensitive to changes in the value of parameter p1.


6. Concluding Remarks

In the present paper, we have dealt with a continuous production inventory model fordeteriorating items with shortages in which two different rates of production are available,and it is possible that production started at one rate and after some time it may be switchedover to another rate. It is assumed that the demand and production rates are constant and thedistribution of the time to deterioration of an item follows the exponential distribution. Such asituation is desirable in the sense that by starting at a low rate of production, a large quantumstock of manufactured item, at the initial stage is avoided, leading to reduction in the holdingcost. The variation in production rate provides a way resulting consumer satisfaction andearning potential profit. For this model, we have derived the average system cost and theoptimal decision rules forQ1,Q2, S, and T when the deterioration rate θ is very small. Resultsare illustrated by numerical examples.

However, success depends on the correctness of the estimation of the inputparameters. In reality, however, management is most likely to be uncertain of the true valuesof these parameters. Moreover, their values may be changed over time due to their complexstructures. Therefore, it is more reasonable to assume that these parameters are known onlywithin some given ranges.

References

[1] T. M. Whitin, Theory of Inventory Management, Princeton University Press, Princeton, NJ, USA, 1957.[2] P. M. Ghare and G. P. Schrader, “Amodel for exponentially decaying inventories,” Journal of Industrial

Engineering, vol. 14, pp. 238–243, 1963.[3] Y. K. Shah and M. C. Jaiswal, “An order-level inventory model for a system with constant rate of

deterioration,” Opsearch, vol. 14, no. 3, pp. 174–184, 1977.[4] S. P. Aggarwal, “A note on an order level inventory model for a system with constant rate of

deterioration,” Opsearch, vol. 15, no. 4, pp. 184–187, 1978.[5] U. Dave and L. K. Patel, “Policy inventory model for deteriorating items with time proportional

demand,” Journal of the Operational Research Society, vol. 32, no. 2, pp. 137–142, 1981.[6] U. Dave, “An order level inventory model for deteriorating items with variable instantaneous

demand and discrete opportunities for replenishment,” Opsearch, vol. 23, no. 4, pp. 244–249, 1986.[7] H. Bahari-Kashani, “Replenishment schedule for deteriorating items with time-proportional

demand,” Journal of the Operational Research Society, vol. 40, pp. 75–81, 1989.[8] A. Goswami and K. S. Chaudhuri, “An EOQmodel for deteriorating itemswith shortages and a linear

trend in demand,” Journal of the Operational Research Society, vol. 42, no. 12, pp. 1105–1110, 1991.[9] R. C. Baker and T. L. Urban, “A deterministic inventory system with an inventory level dependent

demand rate,” Journal of the Operational Research Society, vol. 39, pp. 823–831, 1988.[10] B. N. Mandal and S. Phaujdar, “An inventory model for deteriorating items and stock-dependent

consumption rate,” Journal of the Operational Research Society, vol. 40, pp. 483–488, 1989.


[11] T. K. Datta and A. K. Pal, “A note on an inventory model with inventory-level-dependent demandrate,” Journal of the Operational Research Society, vol. 41, no. 10, pp. 971–975, 1990.

[12] T. L. Urban, “An inventory model with an inventory-level-dependent demand rate and relatedterminal conditions,” Journal of the Operational Research Society, vol. 43, no. 7, pp. 721–724, 1992.

[13] B. C. Giri, S. Pal, A. Goswami, and K. S. Chaudhuri, “An inventory model for deteriorating itemswith stock-dependent demand rate,” European Journal of Operational Research, vol. 95, no. 3, pp. 604–610, 1996.

[14] G. Padmanabhan and P. Vrat, “EOQmodels for perishable items under stock dependent selling rate,”European Journal of Operational Research, vol. 86, no. 2, pp. 281–292, 1995.

[15] S. Pal, A. Goswami, and K. S. Chaudhuri, “A deterministic inventory model for deteriorating itemswith stock-dependent demand rate,” International Journal of Production Economics, vol. 32, no. 3, pp.291–299, 1993.

[16] J. Ray and K. S. Chaudhuri, “An EOQ model with stock-dependent demand, shortage, inflation andtime discounting,” International Journal of Production Economics, vol. 53, no. 2, pp. 171–180, 1997.

[17] T. L. Urban and R. C. Baker, “Optimal ordering and pricing policies in a single-period environmentwith multivariate demand and markdowns,” European Journal of Operational Research, vol. 103, no. 3,pp. 573–583, 1997.

[18] J. Ray, A. Goswami, and K. S. Chaudhuri, “On an inventory model with two levels of storage andstock-dependent demand rate,” International Journal of Systems Science, vol. 29, no. 3, pp. 249–254,1998.

[19] B. C. Giri and K. S. Chaudhuri, “Deterministic models of perishable inventory with stock-dependentdemand rate and nonlinear holding cost,” European Journal of Operational Research, vol. 105, no. 3, pp.467–474, 1998.

[20] T. K. Datta and K. Paul, “An inventory system with stock-dependent, price-sensitive demand rate,”Production Planning and Control, vol. 12, no. 1, pp. 13–20, 2001.

[21] L. Y. Ouyang, T. P. Hsieh, C. Y. Dye, and H. C. Chang, “An inventory model for deterioratingitems with stock-dependent demand under the conditions of inflation and time-value of money,”The Engineering Economist, vol. 48, no. 1, pp. 52–68, 2003.

[22] J. T. Teng, L. Y. Ouyang, and M. C. Cheng, “An EOQ model for deteriorating items with power-formstock-dependent demand,” International Journal of Information and Management Sciences, vol. 16, no. 1,pp. 1–16, 2005.

[23] A. Roy andG. P. Samanta, “An EOQmodel for deteriorating itemswith stock-dependent time varyingdemand,” Tamsui Oxford Journal of Mathematical Sciences, vol. 20, no. 2, pp. 37–46, 2004.

[24] R. P. Covert and G. C. Philip, “An EOQ model for items with Weibull distribution deterioration,”American Institute of Industrial Engineers Transactions, vol. 5, no. 4, pp. 323–326, 1973.

[25] G. C. Philip, “A generalized EOQ model for items with Weibull distribution deterioration,” AmericanInstitute of Industrial Engineers Transactions, vol. 6, no. 2, pp. 159–162, 1974.

[26] R. B. Misra, “Optimum production lot-size model for a system with deteriorating inventory,”International Journal of Production Research, vol. 13, no. 5, pp. 495–505, 1975.

[27] M. Deb and K. S. Chaudhuri, “An EOQ model for items with finite rate of production and variablerate of deterioration,” Opsearch, vol. 23, pp. 175–181, 1986.

[28] S. Nahmias, “Perishable inventory theory : a review,” Operations Research, vol. 30, no. 4, pp. 680–708,1982.

[29] F. Raafat, “Survey of literature on continuously deteriorating inventory model,” Journal of theOperational Research Society, vol. 42, no. 1, pp. 27–37, 1991.

[30] A. K. Jalan and K. S. Chaudhuri, “Structural properties of an inventory systemwith deterioration andtrended demand,” International Journal of Systems Science, vol. 30, no. 6, pp. 627–633, 1999.

[31] L. Y. Ouyang, C. K. Chen, and H. C. Chang, “Lead time and ordering cost reductions in continuousreview inventory systems with partial backorders,” Journal of the Operational Research Society, vol. 50,no. 12, pp. 1272–1279, 1999.

[32] V. Perumal and G. Arivarignan, “A production inventory model with two rates of production andbackorders,” International Journal of Management and Systems, vol. 18, pp. 109–119, 2002.

[33] G. P. Samanta and A. Roy, “A deterministic inventory model of deteriorating items with two rates ofproduction and shortages,” Tamsui Oxford Journal of Mathematical Sciences, vol. 20, no. 2, pp. 205–218,2004.

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