THEORETICAL ARTICLE

An inventory model for deteriorating itemswith stock-dependent demand, time-varyingholding cost and shortages

Karabi Dutta Choudhury & Biplab Karmakar &

Mantu Das & Tapan Kumar Datta

Accepted: 4 October 2013# Operational Research Society of India 2013

Abstract In this article, an inventory model for deteriorating items with two compo-nents demand and time-varying holding cost has been developed. Demand is assumed tobe stock-dependent up to the level of available stock. After which the demand isconsidered as constant i.e. during stock-out period. Shortages are allowed and are fullybacklogged. Profits are maximized in both the infinite and finite time horizon cases.Some special cases are also derived from the main models. Two numerical examples areprovided for both finite and infinite time horizon. Sensitivity analysis performed hasshown successful effects of various model parameters on net profit.

Keywords Inventory . Cost benefit analysis . Demand . Deterioration . EOQmodel .

Holding cost . Shortage

1 Introduction

Displayed stock (inventory) level plays an important role to attract consumers. Forcertain types of goods, particularly for consumer goods, the demand is proportional tothe stock level. Levin et al. [19] observed the motivational effect of the presence ofinventory on the customers. Silver and Peterson [29] noted that sales at the retail leveltend to be proportional to the inventory displayed. Encouraged by these, severalresearchers have made attempts to analyze inventory models assuming a functionaldependence between the demand rate and the on-hand inventory level.

OPSEARCHDOI 10.1007/s12597-013-0166-x

K. Dutta Choudhury (*) : B. Karmakar :M. DasDepartment of Mathematics, Assam University, Silchar, Silchar 788011, Assam, Indiae-mail: karabidc@gmail.com

T. K. DattaDepartment of Mathematics, BITS Pilani-Dubai Campus, Dubai, United Arab Emirates

Gupta and Vrat [17] developed an inventory model where demand rate is replen-ishment size (initial stock) dependent. They analysed the model through cost mini-mization. Baker and Urban [3] and Urban [30] developed inventory model withinstantaneous inventory level-dependent demand rate. Mandal and Phaujdar [20]analyzed two inventory models considering linearly stock-dependent demand rate,first with instantaneous replenishment without shortages and second with finitereplenishment rate allowing shortages. In the past two decades, many researchersacross the world have published several research articles with inventory level depen-dent demand, incorporating some other important factors—such as deterioration,pricing strategy, inflation, variability in holding cost, limited storage capacity etc.Some of such articles are by Pal et al. [24], Paul et al. [25], Giri et al. [15], Giri andChaudhuri [14], Datta and Pal [6], Datta and Paul [7, 8], Ouyang et al. [23], Alfares[2], Dutta Choudhury and Datta [11].

Maximum physical goods undergo decay or deterioration over time. Fruits, veg-etables etc. suffer from depletion by direct spoilage while stored. Highly volatileliquids such as gasoline, alcohol and turpentine undergo physical depletion over timethrough the process of evaporation. Electronic goods, radioactive substances, photo-graphic film, grain etc. deteriorate through a gradual loss of potential or utility withthe passage of time. So decay or deterioration of physical goods in stock is a veryrealistic feature. In this model we have considered this factor.

Whitin [33] considered deterioration of the fashion goods at the end of a prescribedshortage period. Ghare and Schrader [12] developed a model for exponentiallydecaying inventory. An order level inventory model for items deteriorating at aconstant rate was presented by Shah and Jaiswal [28], Aggarwal [1], Dave andPatel [9]. Inventory model with a time-dependent rate of deterioration were consid-ered by Deb and Chaudhuri [10]. Some of the recent work in this field has been doneby Chung and Ting [5], Giri and Chaudhuri [13], Jalan and Chaudhuri [18]. Burwellet al. [4] developed an economic lot size model for price-dependent demand underfreight discounts.

In most of the inventory models, holding cost is known and is considered asconstant. But in real life situation holding cost need not be a constant as becauseother parameters may govern the holding cost. In generalization of EOQ models,various functions describing holding cost were considered by several researcherslike Naddor [22], Van Der Veen [31], Muhlemann and Valtis-Spanopoulos [21],Weiss [32] and Goh [16]. Giri and Chaudhuri [14] treated the holding cost as anon-linear function of the length of the time for which the item is held in stockand as a functional form of the amount of the on-hand inventory. Roy [26]developed an EOQ model for deteriorating items where deterioration rate andholding cost are expressed as linearly increasing functions of time and demandrate is a function of selling price and shortages are allowed and completelybacklogged. Sahoo et al. [27] also considered time varying holding cost with constantdeterioration.

In any business, especially in retail business, goodwill of the retailer, good qualityof the items, prompt services etc. are very crucial factors. These factors have amotivational effect on the customers to purchase from a particular retail shop andto wait during stock-out period. These customers may not be influenced by thedisplayed stock-level.

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In the present work, attempt has been made to develop an EOQ model for deterio-rating items where holding cost is expressed as linearly increasing function of time,keeping deterioration under consideration and the demand rate is assumed to be a linearfunction of the on-hand inventory as considered in Mandal and Phaujdar [20] till thestock lasts and then allowing shortages with constant demand. When the shortage startsit is completely backlogged. Both infinite and finite time horizon were also consideredwhile developing the EOQmodel. Numerical examples were taken to discuss the modelas well as to describe different special cases. Attempt has also been made to validate thedeveloped model by sensitivity analysis.

2 Assumptions and notations

Proposed model has been developed with following assumptions:

1. Replenishment is instantaneous i.e. replenishment rate is infinite; replenishmentsize is finite.

2. The demand rate R(I) is a function of the on hand inventory level I(t) and is defined as

R Ið Þ ¼ αþ I tð Þ β;whenI tð Þ≥0¼ α;Otherwise;

α>0, 0<β<1, I(t) is the instantaneous stock level at time t.3. Holding cost C1(t) per unit time is time dependent and is assumed to be as

C1 tð Þ ¼ hþ bt;whereh > 0 and b > 0:

4. The deterioration function has been assumed to be θ0(t)=θH(t−t1); where t is thetime measured from the instant of arrival of a fresh replenishment indicating thatthe deterioration of the items begins after a time t1 from the instant of that arrival instock; θ is a constant (0<θ≪1) andH(t−t1) is the well-known Heaviside’s functiondefined as

H t−t1ð Þ ¼ 1; t > t1;¼ 0; otherwise:

5. At the beginning of each replenishment cycle, the replenishment size qraises the inventory level s after fulfilling the backorder quantity (q−s) ofthe previous cycle except the first cycle i.e. at time t=0, the stock level is raised tothe levels.

6. During time period (0, t1) the level of inventory is depleted due to demand andduring time period (t1, t2) it is depleted due to both deterioration and demand. Attime t2, the inventory became zero and shortages began. The level remains non-negative for the initial time duration t2 in every replenishment cycle.

7. Shortages, if any, are completely backlogged and are cleared as soon as a freshlot arrives. Shortage cost is C2 per unit per unit time.

8. The selling price of the item per unit item is p.9. The unit cost of the item is C per unit.10. The replenishment cost is C3 per replenishment.11. The system involves only one stocking point.

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3 Mathematical formulation and solution

3.1 Infinite time-horizon model

The on-hand inventory level s at time t=0 after fulfilling the backorder quantity (q−s)gradually falls to the level s1(<s)) at time t= t1 due to demand. After time t= t1, thestock level further decreases due to demand and deterioration and falls to zero at timet= t2. The shortages begin to accumulate after time t= t2 and continues up to time t=Ttill next lot arrives (Fig. 1).

Let I(t) be the inventory level at time t(0≤ t≤T) the differential equation for theinstantaneous state over (0, T) are given by

dI tð Þdt

¼ − αþ I tð Þ βð Þ; 0≤ t≤ t1 ð1Þ

With boundary condition I(t)=s, t=0; and boundary condition I(t)=s1, t= t1.

dI tð Þdt

þ I tð Þ θ ¼ − αþ I tð Þ βð Þ; t1≤ t≤ t2 ð2Þ

With boundary conditions I(t)=s1 at t= t1 and I(t)=0 at t= t2.

dI tð Þdt

¼ −α; t2≤ t≤T ð3Þ

With boundary condition I(t)=0 at t=t2.Thus the solution of the differential Eq. (1) is:

I tð Þ ¼ α=β þ sð Þe−tβ–α=β; 0≤ t≤ t1 ð4Þ

Using boundary condition, I(t)=0, t= t2; the solution of the differential Eq. (2) is:

I tð Þ ¼ α e θþβð Þ t2−tð Þ−1h i

= θþ βð Þ; t1≤ t≤ t2 ð5Þ

The solution of the differential Eq. (3) with the given boundary condition is:

I tð Þ ¼ −α t−t2ð Þ; t2≤ t≤T ð6Þ

Fig. 1 The system with infinite time-horizon

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Using the condition I(t)=s1 at t= t1 in Eqs. (4) and (5), we get respectively

s1 ¼ α=β þ sð Þe−t1β−α=βs1 ¼ α e θþβð Þ t2−t1ð Þ−1

h i= θþ βð Þ

Eliminating s1 from the above two equations, the following relation between s andt2 is obtained:

s ¼ αet1β e θþβð Þ t2−t1ð Þ−1h i

= θþ βð Þ þ α et1β−1� �

=β ð7Þ

The total holding cost in carrying the inventory for the entire cycle:

CH ¼Z t2

0

hþ btð ÞI tð Þdt; h > 0; b > 0

¼Z t1

0

hþ btð ÞI tð Þ dt þZ t2

t1

hþ btð ÞI tð Þ dt

¼ Aþ Bð Þ e θþβð Þ t2−t1ð Þ−1h i

= θþ βð Þ−Ge θþβð Þ t2−t1ð Þ= θþ βð Þ−Bt2−Dt22 þ E

ð8Þ

whereA ¼ α et1β−1

� �hβ þ bð Þ=β2

B ¼ α bþ h θþ βð Þ½ �= θþ βð Þ2

G ¼ αbθt1= β θþ βð Þð ÞD ¼ αb= 2 θþ βð Þð ÞE ¼ A−G−αbθt12= 2 θþ βð Þð Þ−αht1θ= θþ βð Þ� �

=β

Details are given in the Appendix.Total amount of inventory that has deteriorated during this cycle is:

Ds ¼ s1−Z t2

t1

αþ I tð Þ βð Þdt

¼ αθ e θþβð Þ t2−t1ð Þ−1h i

= θþ βð Þ2−αθ t2−t1ð Þ= θþ βð Þð9Þ

Total amount of shortage cost for the entire cycle is:

Cs ¼ −C2

Zt2

T

I tð Þ dt¼ C2α T−t2ð Þ2=2

ð10Þ

Total amount of backorder at the end of the cycle is α(T− t2)let

Bq ¼ α T−t2ð ÞHence,

q−s ¼ α T−t2ð Þ ð11Þ

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Average net profit per unit time for a replenishment cycle:

f t2;Tð Þ ¼ s−DSð Þp−sC−CH−Cs−C3 þ p−Cð ÞBq

� �=T

¼ Lþ Qð Þe θþβð Þ t2−t1ð Þ þMt2 þ Dt22−C2α T−t2ð Þ2=2þ α p−Cð Þ T−t2ð Þ þ N

h i=T

ð12Þ

where

L ¼ αet1βp= θþ βð Þ−αθp= θþ βð Þ2−αet1βC= θþ βð Þ− Aþ Bð Þ= θþ βð ÞM ¼ Bþ αθp= θþ βð ÞN ¼ −L−E þ α et1β−1

� �p−Cð Þ=β−αθpt1= θþ βð Þ−C3

Q ¼ G= θþ βð Þ

Thus the optimization problem is:

Maximize f (t2, T)Subject to: T> t2> t1.

In the above expression, t1 is a known non-negative parameter.The necessary conditions for optimum values are:

∂ f∂t2

¼ 0 and∂ f∂T

¼ 0 ð13Þ

The sufficient conditions for the existence of maxima are:

∂2 f∂t22

< 0;∂2 f∂T2 < 0 and

∂2 f∂t22

� ∂2 f

∂T2

66647775− ∂2 f

∂t2∂T

� �2> 0 ð14Þ

The equation ∂ f∂t2 ¼ 0 gives,

T ¼ X t2− θþ βð Þ Lþ Qð Þe θþβð Þ t2−t1ð Þ þ Yh i

= αc2ð Þ ð15Þ

whereX ¼ αC2−2D

Y ¼ α p−Cð Þ−MThe equation ∂ f

∂T ¼ 0 gives,

T2 ¼ X t22 þ 2Y t2−2 Lþ Qð Þe θþβð Þ t2−t1ð Þ−2N

h i= αc2ð Þ ð16Þ

where D, M, L, Q and N are defined earlier.Eliminating T from Eqs. (15) and (16), the following equation in t2 is obtained.

X t2− θþ βð Þ Lþ Qð Þe θþβð Þ t2−t1ð Þ þ Yh i2

¼ αc2 X t22 þ 2Y t2−2 Lþ Qð Þe θþβð Þ t2−t1ð Þ−2N

h ið17Þ

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The Eq. (17) is a non-linear equation in the variable t2. It will be either impossibleor a tedious job to get an analytical solution for this. However, for a given set ofvalues of the model parameters the solution of this equation can be obtained by usingiterative method or any equation solver. Then Eq. (15) can be used to find the mostsuitable value of T. The optimum solutions are obtained in the given numericalexample 4.1.

3.2 The finite time-horizon model

Sometimes, in some businesses, we need to control inventory over a finite period oftime. In this section, we have developed a finite time-horizon model. Let H be theprescribed time period during which the inventory will be controlled and (m+1) bethe number of replenishments to be made during this period H. A (T, Si) policy isadopted here in which all first m replenishment cycles have equal duration. The lastreplenishment is made at time t=H to clear up the backlog during the last (mth)replenishment cycle. The length of each of m replenishment cycles is H/m.

A pictorial representation of the system is given below (Fig. 2):The jth cycle starts at time t=(j−1)H/m=Tj−1, say and ends at time t=jH/m=Tj,

say. In this cycle, deterioration starts at time t=Tj−1+t1 and shortages start at time t=Tj−1+t2 ( j=1,2,3....m).

In the jth cycle:Holding cost for the jth cycle is

CHj ¼ Aj þ Bj

� �e θþβð Þ t2−t1ð Þ−1h i

= θþ βð Þ−Ge θþβð Þ t2−t1ð Þ= θþ βð Þ−Bjt2−Dt22 þ E j

ð18Þwhere

Aj ¼ α et1β−1� �

hþ bT j−1� �

β þ b� �

=β2

Bj ¼ α bþ hþ bT j−1� �

θþ βð Þ� �= θþ βð Þ2

E j ¼ Aj−G−αbθt12= θþ βð Þ−α hþ bT j−1� �

θt1= θþ βð Þ� �=β

and G, D are same as defined in the infinite time horizon case.The total amount deteriorated is

Dsj ¼ αθ e θþβð Þ t2−t1ð Þ−1

h i= β þ θð Þ2−αθ t2−t1ð Þ= θþ βð Þ ð19Þ

Fig. 2 The system with finite time-horizon

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Total amount of backorder quantity at the end of the cycle is

Bq ¼ α H=m−t2ð Þ ð20Þ

Therefore the net profit for the jth cycle is:

f j t2;mð Þ¼ Lj þ Q

� �e θþβð Þ t2−t1ð Þ þM jt2 þ Dt2

2−αC2 H=m−t2ð Þ2=2þ α p−Cð Þ H=m−t2ð Þ þ N j

ð21Þwhere

Lj ¼ αpet1β= θþ βð Þ−αθp= θþ βð Þ2−αet1βc= θþ βð Þ− Aj þ Bj

� �= θþ βð Þ

Q ¼ G= θþ βð ÞM j ¼ Bj þ αθp= θþ βð ÞN j ¼ −Lj−E j þ α et1β−1

� �p−Cð Þ=β−αθpt1= θþ βð Þ−C3

Hence, the optimum profit for the entire period is:

F t2;mð Þ ¼ −C3 þXj¼1

m

f j t2;mð Þ ð22Þ

Here the optimization problem is to Maximize F(t2, m)Subject to:

mt2 < H ;t2 > t1;t2 > 0 and m is apositive integer:

ð23Þ

For a given value of m, the value of t2 which will optimize F(t2, m) can be obtainedby solving the equation

dF

dt2¼ 0

This gives:

θþ βð Þe θþβð Þ t2−t1ð ÞXj¼1

m

L j þ Q� �þX

j¼1

m

M j þ 2Dt2 þ αC2 H=m−t2ð Þ−α p−Cð Þ ¼ 0

Or equivalently,

V 1eθþβð Þ t2−t1ð Þ þ V 2t2 þ V 3 ¼ 0

where

V 1 ¼ θþ βð ÞXj¼1

m

Lj þ Q� �

;V 2 ¼ 2D−αC2ð Þ and V 3 ¼Xj¼1

m

M j þ αC2H=m−α p−Cð Þ;

The sufficient condition d2 Fdt22

< 0 should be satisfied by the obtained value of t2.

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4 Numerical examples

4.1 Example (Infinite time-horizon)

α=20 units, β=0.2, θ=0.02, t1=0.2day, p= 20/unit, C= 18/unit, C2= 0.40/unit/day, C3= 80 per replenishment, h=0.4, b=0.01.

Solving Eqs. (16) and (17) by Newton-Raphson’s method for a positive t2,

t2 ¼ t2� ¼ 2:73days

Using this value in Eq. (17), we get

T ¼ T� ¼ 5:88 days:

The optimum values of s and q are s*=74.43 units, q*=137.56 units.Hence, the profit per day is f (t2

*, T*)= 14.75

4.2 Example (Finite time-horizon)

α=50 units, β=0.2, θ=0.02, t1=0.2day, p= 10/unit, C= 8/unit, C2= 0.40/unit/day,C3= 90 / replenishment, h=0.7, b=0.01, H=25 days.

For these parameter values, the value of t2 and F(t2, m) for m=1,2,3,.. are given inTable 1.

Clearly, m*=5, t2*=t2=2.37 days,

The optimum values of s and q are s*=155.056 units, q*=286.487 unitsThe maximum net profit during 25 days is F (t2

*, m*)= 1,159.81

5 Sensitivity analysis

Here we have studied the effects of changes in the values of the parameters α, β, θ,C2, C3, h, b, p and C on t2

*, T*, profit, stock-level and backlog level derived by theproposed method as in infinite time horizon. 4.1-example is used.

The effects of changes in the values of the parameters α, β, θ and h, b for finite timehorizon, 4.2-example is used. On the basis of the results shown in Tables 2 and 3, thefollowing observations can be made.

Table 1 Solution table ofexample 4.2 in finite timehorizon model

Data in bold represents themaximum profit obtained fromthe objective function of thegiven problem

m t2 (in days) F (t2, m) (in Rupees)

1 6.71057 −1942.282 3.46634 193.84

3 2.7096 833.74

4 2.46473 1084.65

5 2.37136 1159.81

6 2.33353 1138.03

7 2.31896 1056.74

8 2.315 936.25

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It is seen from Table 2 that the solution is less sensitive to changes in the parameterb, while it is quite sensitive to changes in the parameters α, C2, C3, p and C.

Table 2 Sensitivity analysis for infinite time horizon model

Parameter % change % changein t2

*% changein T*

% changein profit

% changein s*

% changein q*

α −50 22.306 27.3192 −787.903 −41.1857 −39.5583−25 10.1136 12.438 −86.9636 −15.4638 −14.879725 −8.80382 −10.8976 32.9914 10.6128 10.2333

50 −16.6869 −20.7109 50.0981 18.4789 17.8313

β −50 −22.387 −4.3224 −16.0532 −46.259 −15.4813−25 −11.6185 −2.43049 −8.48792 −23.3632 −8.8737325 12.8998 3.35151 9.77799 24.0743 12.4734

50 28.0406 8.60751 21.6162 49.3859 31.2256

θ −50 29.385 10.7348 18.8113 37.3533 20.667

−25 14.2548 4.49333 8.7375 18.2181 8.6645

25 −13.6374 −3.42774 −7.72868 −17.5418 −6.5556750 −26.7922 −6.14552 −14.6496 −34.5377 −11.6888

C2 −50 −16.1775 21.0526 24.6714 −21.599 15.7676

−25 −5.89756 8.82636 11.8566 −7.85461 6.14547

25 3.87515 −6.77384 −11.162 5.14612 −4.2638550 6.63378 −12.178 −21.7629 8.80161 −7.38936

C3 −50 −30.5207 −38.0616 34.8031 −40.8635 −42.8722−25 −11.5204 −14.2734 19.7278 −15.3649 −16.002825 7.95799 9.79413 −27.9718 10.5538 10.908

50 13.8886 17.0589 −71.744 18.3783 18.9506

h −50 34.2961 12.4963 24.0286 44.9028 26.2636

−25 16.6286 5.02493 11.2018 21.9791 10.6032

25 −16.0697 −3.68811 −10.0434 −21.4547 −7.65644850 −31.7968 −6.54114 −19.1761 −42.5813 −13.4383

b −50 1.09212 0.365332 0.46622 1.45145 0.668093

−25 0.543011 0.180685 0.228684 0.721701 0.330404

25 −0.536645 −0.176978 −0.226909 −0.71382 −0.3242650 −1.06759 −0.350291 −0.451103 −1.4201 −0.64175

p −50 −300.97 −22.377 107.642 −409.077 −41.5686−25 −152.92 −17.5145 116.134 −206.987 −33.799325 −47.8129 −71.483 – – –

50 −160.481 −47.6659 – – –

C −50 −151.034 −47.9389 – – –

−25 27.2267 −71.865 – – –

25 −148.957 −17.4189 118.138 −201.587 −33.575150 −290.406 −22.2749 108.531 −394.646 −41.3657

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Moreover, when decreases the value of p, h by −50 % and increases the value of C by50 % the profit function do not satisfy the condition (14).

Therefore, the above sensitivity analysis indicates that sufficient care shouldbe taken to estimate the parameters α, β, θ, C2, C3, h, p, C in the marketstudies.

It is seen from Table 3 that the solution is quit sensitive to the changes in theparameters α, β, θ, h and b.

Therefore, the above sensitivity analysis indicates that sufficient care should betaken to estimate the parameters α, β, θ, h and b in the market studies.

6 Discussion

In the present work economic order quantity model for linearly stock-dependentdemand rate with deterioration and shortages has been examined. Deterioration ofitems has been assumed to begin after a fixed time period t1 from the instant of theirarrival in stock.

The traditional parameter of holding cost is assumed here to be time-varying insteadof taking constant. As the changes in the time value of money and in the price index,holding cost may not remain constant over time. It is assumed here that the holding costis linearly increasing function of time.

Let us now investigate some particular cases for both infinite and finite time horizonfrom the above discussions.

Table 3 Sensitivity analysis for finite time horizon model

Parameter % change % change in t2 % change in profit % change in s* % change in q*

α −50 .00018308 −274.248 −100.00 −99.9997−25 .00018308 −57.8282 −33.337 −33.333125 .00018308 26.8148 19.9998 20.0001

50 .00018308 42.2897 33.332 33.3334

β −50 −18.1734 −4.44021 −37.2027 −9.06376−25 −8.76683 −2.42611 −17.9565 −5.1557525 8.81828 2.82959 17.4428 6.91875

50 18.0859 6.10977 34.687 16.3927

θ −50 9.47258 1.92159 11.0325 2.32596

−25 4.71382 0.955496 5.49439 1.08761

25 −4.66982 −0.938401 −5.45235 −0.96073450 −9.29747 −1.85476 −10.8628 −1.8146

When h and b changes together

h and b −50 53.3719 4.62886 66.7715 38.0369

−25 25.8641 5.75563 32.9318 10.8234

25 −25.2734 −5.62218 −32.5797 −5.2079150 −50.2575 −10.5323 −64.9786 −8.07743

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Where

(1) Holding cost is constant.(2) Deterioration starts at the instant of its arrival in stock.(3) In absence of deterioration only.(4) Shortages and deterioration are not allowed.

6.1 Infinite time-horizon

6.1.1 Holding cost is constant

If the holding cost is constant i.e. C1(t)=h, then average profit per unit time is

f t2;Tð Þ ¼ Le θþβð Þ t2−t1ð Þ þMt2−C2α T−t2ð Þ2=2þ α p−Cð Þ T−t2ð Þ þ Nh i

=T ð23Þ

where

L ¼ αpet1β= θþ βð Þ−αθp= θþ βð Þ2−αCet1β= θþ βð Þ− α et1β−1� �

h θþ βð Þ þ hαβ� �

= β θþ βð Þ2

M ¼ αh= β þ θð Þ þ αθp= θþ βð ÞN ¼ −L−α et1β−1

� �h=β þ α et1β−1

� �p−Cð Þ=β−αθpt1= θþ βð Þ−C3

∂ f∂t2

¼ 0

T ¼ X t2− θþ βð Þ Lþ Qð Þe θþβð Þ t2−t1ð Þ þ Yh i

= αc2ð Þ

ð24Þ

∂ f∂T

¼ 0

T2 ¼hX t2

2 þ 2Y t2−2 Lþ Qð Þe θþβð Þ t2−t1ð Þ−2Ni= αc2ð Þ

ð25Þ

Eliminating T from Eqs. (24) and (25), t2 can be derived.

6.1.2 Deterioration starts at the instant of its arrival in stock

If the deterioration start from the instant of arrival of the stock (t1=0), the averageprofit per unit time is

f t2;Tð Þ ¼ Le βþθð Þt2 þMt2 þ Dt22−αc2 T−t2ð Þ2=2þ α p−Cð Þ T−t2ð Þ þ N

h i=T ð26Þ

where

L ¼ αpβ= β þ θð Þ2−αc= β þ θð Þ−α bþ h β þ θð Þ½ �= β þ θð Þ3

N ¼ −L−C3

M and D are same as in infinite time horizon.

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In this case ∂ f∂t2 ¼ 0 , this implies

T ¼ X t2−L β þ θð Þe βþθð Þt2 þ Yh i

= αc2ð Þ ð27Þ

X and Y are same as in infinite time horizon.and ∂ f t2;Tð Þ

∂T ¼ 0 , this implies

T2 ¼ X t22 þ 2Y t2−2Le βþθð Þt2−2N

h i= αc2ð Þ ð28Þ

Solving (27) and (28), t2 can be determined.

6.1.3 In absence of deterioration only

If deterioration is absent altogether i.e. θ=0, t1=0; then the equation reduces to

f t2;Tð Þ ¼ Leβt2 þMt2 þ Dt22−αc2 T−t2ð Þ2=2þ α p−Cð Þ T−t2ð Þ þ N

h i=T ð29Þ

where

L ¼ α p−Cð Þ=β−α bþ hβð Þ=β2

D ¼ αb= 2βð ÞN ¼ −L−C3

M ¼ α bþ hβð Þ=β2

In this case ∂ f∂t2 ¼ 0 , this implies

T ¼ X t2−Lβeβt2 þ Y� �

= αc2ð Þ ð30Þ

where X and Y are same as defined in case 6.1.2and ∂ f

∂T ¼ 0 , this implies

T2 ¼ X t22 þ 2Y t2−2Leβt2−2N

� �= αc2ð Þ ð31Þ

Solving Eqs. (30) and (31), t2 can be determined.The same result could be obtained by assigning t1= t2, indicating that deterioration

does not occur for positive on-hand inventory.

6.1.4 Shortages and deterioration are not allowed

If shortages are not allowed i.e. θ=0; t1=0, t2=T, then the Eq. (12) reduces to

f Tð Þ ¼ LeβT þMT þ DT 2 þ N� �

=T ð32Þ

when t1=0, θ=0, t2=T, Eq. (7) reduces to

T ¼ 1=βð Þln 1þ sβ=αð Þ

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6.2 Finite time-horizon

6.2.1 Holding cost is constant

When the holding cost is constant i.e. C1(t)=h, expression for the profit during the jth

cycle in the Eq. (21) and is given by

fj t2;mð Þ ¼ Ljeθþβð Þ t2−t1ð Þ þM jt2 þ p−Cð Þα H=m−t2ð Þ− αC2=2ð Þ H=m−t2ð Þ2 þ Nj

where

Lj ¼ αpet1β= θþ βð Þ−αθp= θþ βð Þ2−αcet1β= θþ βð Þ− Aj þ Bj

� �= θþ βð Þ

Aj ¼ αh et1β−1� �

=β; Bj ¼ αh= θþ βð ÞM j ¼ Bj þ αθp= θþ βð Þ; E j ¼ Aj−αθht1= θþ βð Þ� �

=β

N j ¼ −Lj−E j þ α p−Cð Þ et1β−1� �

=β−αθpt1= θþ βð Þ−C3

F t2;mð Þ ¼ −C3 þXj¼1

m

fj t2;mð Þ ð33Þ

Now dFdt2

¼ 0

This gives, V 1e θþβð Þ t2−t1ð Þ þ V 2t2 þ V 3 ¼ 0where

V 1 ¼ θþ βð ÞXj¼1

m

Lj;V 2 ¼ αC2;V 3 ¼ αC2H=m−α p−Cð Þ þXj¼1

m

Mj

On simplification, t2 can be derived.

6.2.2 Deterioration starts at the instant of its arrival in stock

When deterioration starts from the instant of arrival of the item in stock, theexpression for the profit in the jth cycle is obtained by putting t1=0, in the Eq. (21)and is given by

fj t2;mð Þ ¼ Ljeθþβð Þt2 þ Dt2

2 þM jt2 þ p−Cð Þα H=m−t2ð Þ−αC2 H=m−t2ð Þ2=2þ Nj

Hence, the net profit for the entire period is:

F t2;mð Þ ¼ −C3 þXj¼1

m

fj t2;mð Þ ð34Þ

where

Lj ¼ αβp= θþ βð Þ2−αc= θþ βð Þ−Bj= θþ βð Þ

N j ¼ −Lj−C3

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D and Mj, Bj are same as in finite region.Now dF

dt2¼ 0 ;

This gives:

θþ βð Þe θþβð Þt2Xj¼1

m

Lj þXj¼1

m

M j þ 2Dt2 þ αC2 H=m−t2ð Þ−α p−Cð Þ ¼ 0

Or equivalently, V 1e θþβð Þt2 þ V 2t2 þ V 3 ¼ 0

where V 1 ¼ θþ βð Þ∑ j¼1

m

Lj ; V2, V3 are same as in finite region.

On simplification t2 can be derived.

6.2.3 In absence of deterioration only

In the deterioration is absent altogether, θ=0, t1=0, the Eq. (34) reduces to:

f j t2;mð Þ ¼ Ljeβt2 þ Dt2

2 þM jt2 þ p−Cð Þα H=m−t2ð Þ−αC2 H=m−t2ð Þ2=2þ N j

where

Lj ¼ αβp=β2−αc=β−Bj=β;Bj ¼ α bþ hþ bT j−1� �

β� �

=β2

M j ¼ Bj;N j ¼ −Lj−C3;D ¼ αb= 2βð Þ;Hence, the net profit for the entire period is:

F t2;mð Þ ¼ −C3 þΣmj¼1 f j t2;mð Þ ð35Þ

Now dFdt2

¼ 0 ;This gives, V 1e θþβð Þt2 þ V 2t2 þ V 3 ¼ 0

where V 1 ¼ β ∑j¼1

mLj ; V2, V3 are same as in finite region

On simplification t2 can be derived.

6.2.4 Shortages and deterioration are not allowed

In the absence of shortage, i.e. t2=H/m , the expression for profit in the jth cycle is:

f j H=mð Þ ¼ LjeβHm þ D H=mð Þ2 þM jH=mþ N j;

where Lj; D; Mj; Nj are same as in 6.2.3Hence, the net profit for the entire period is:

F H=mð Þ ¼ −C3 þXj¼1

m

f j H=mð Þ ð36Þ

The total profit F(H/m) is a function of the discrete variable m only. For maximumprofit, the following inequality has to be satisfied:

Δ F m−1ð Þ > 0 > ΔF mð Þ

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where

Δ F mð Þ ¼ F mþ 1ð Þ−F mð ÞHence, the optimal m (=m*) satisfies the inequality

F m�−1ð Þ < N

Lþ Q< F m�ð Þ

F m�ð Þ ¼ −C3 þXj¼1

m

f j H=mð Þð37Þ

The proposed model has also been considered under different situations withnumerical examples. Using 4.1-example (infinite time-horizon) and 4.2-example(finite time-horizon) under these special situations Table 4 has been prepared.

Where

(1) Holding cost is constant (HCC)(2) Deterioration starts at the instant of its arrival in stock (DSAS)(3) In absence of deterioration only (DAA)(4) Shortages and deterioration are not allowed (SNA).

From the table, it is clear that though the net profit is maximum when holding costis constant, but in real life holding cost may not be a constant. When the deteriorationstarts from the time t=0, then in infinite time horizon holding cost will be less as it istime dependent and hence, profit will be maximum. If the deterioration is absentaltogether and shortages are not also allowed, then also the profit will be maximum.

7 Conclusion

In the present work, a deterministic inventory model has been developed for deteri-orating items with two- component demand rate and time variable holding costallowing shortages. The principal features of the model are as follows:

1) The deterministic demand rate is linear and is assumed to be a function of on handinventory level with deterioration and shortages for infinite and finite time-horizons.

2) The deterioration has been assumed to begin after a fixed time from the instant oftheir arrival in stock (like fruits, vegetables etc).

3) The traditional parameters of holding cost is assumed here to be time varying. Asthe changes in the time value of money and in the price index, holding cost maynot remain constant over time. It is assumed here that the holding cost is linearlyincreasing function of time.

The comparative studies between the results of the following cases have been prepared:

1. Holding cost is constant and holding cost is linear increasing function of time.2. Deterioration starts from the instant of arrival of stock and deterioration starts

after a fixed time.3. Deterioration is absent and along with deterioration.4. Without shortage and with shortages have been discussed in details.

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Tab

le4

Com

parativ

estudiesof

infinite

andfinite

timehorizonmodelsunderdifferentcases

Infinite

timeho

rizon

Finite

timeho

rizon

t 2days

Tdays

Profit(per

day)

inRupees()

sun

itsqun

itsm

t 2Days

Profitdu

ring

Hdays

inRupees()

sun

itsqun

its

Ex1

&2

2.73

5.88

14.75

74.429

137.55

52.37

1159

.81

155.05

628

6.49

HCC

2.79

5.93

14.89

76.681

139.46

52.86

1175

.27

198.16

930

5.26

DSAS

2.60

5.86

13.91

70.189

135.43

52.32

1144

.56

151.56

528

5.44

DAA

8.11

9.63

27.81

405.91

436.38

52.93

1206.08

199.41

430

2.79

SNA

–2.78

10.94

74.429

–5

596

.08

429.57

–

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Though the result obtained is more profitable when the deterioration starts fromthe beginning in the finite horizon as we cannot ignore that the holding cost will beless in such situation. The model may also be analysed further by taking the holdingcost to be stock-dependent one.

Acknowledgments We would like to be thankful to the Department of Mathematics, Assam Univer-sity, Silchar for giving us chance to use the UGC funded computer laboratory. The authors would liketo thank the referees for their constructive suggestions in the conference ORSI- Kolkata, where thepaper was presented. At last not the least, special thanks to ORSI- Kolkata for giving us a chance topresent the paper in the International Conference at Calcutta Business School from 6th–8th January2012.

The authors are also thankful to the referees for their valuable suggestions to modify the present paperfor publication.

Appendix

Total holding cost in carrying the inventory for the entire cycle is

CH ¼Z t2

0hþ btð ÞI tð Þdt; h > 0; b > 0

¼Z t1

0hþ btð ÞI tð Þdt þ

Z t2

t1

hþ btð ÞI tð Þdt

¼ I1 þ I2

For 0≤ t≤ t1I1 ¼ ∫t10 hþ btð Þ α=β þ sð Þe−tβ−α=β� �

dt [from Eq. (4)]

¼Z t1

0h α=β þ sð Þe−tβ−α=β� �

dt þZ t1

0bt α=β þ sð Þe−tβ−α=β� �

dt

¼ −h α=β þ sð Þe−tβ=β−hαt=β� �t10 þ b α=β þ sð Þ −te−tβ=β−e−tβ=β2

� �t10 − αbt2= 2βð Þ� �t1

0

¼ h=βð Þ α=β þ sð Þ 1−e−t1β� �

−hαt1=β þ b α=β þ sð Þh−t1e−t1β=β −e−t1β=β2 þ 1=β2

i−αbt21= 2βð Þ

¼ h=βð Þ α= θþ βð Þð Þ e θþβð Þ t2−t1ð Þ−1h i

et1β þ αet1β=βn o

1−e−t1β� �

−hαt1=β

þ b α= θþ βð Þð Þ e θþβð Þ t2−t1ð Þ−1h i

et1β þ αet1β=βn o

1−e−t1β� �

=β2−t1e−t1β=β� �

−αbt21= 2βð Þ

[From Eq. (7) & putting α=β þ sð Þ ¼ α= θþ βð Þð Þ e θþβð Þ t2−t1ð Þ−1� �

et1β þ αet1β=β�

¼ e θþβð Þ t2−t1ð Þ−1h i

et1β−1� �

αh= β θþ βð Þð Þ þ αh et1β−1ð Þ=β2−αht1=βþe θþβð Þ t2−t1ð Þ−1h i

et1β−1� �

αb = β2 θþ βð Þ� �

þ bα et1β−1ð Þ = β3 −

e θþβð Þ t2−t1ð Þ−1h i

αbt1= β θþ βð Þð Þ − αbt1= β2 − αbt2 1= 2βð Þ

For t1≤ t≤ t2

I2 ¼Z t2

t1

hþ btð ÞI tð Þdt

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¼ ∫t2t1 hþ btð Þ α= θþ βð Þð Þ e θþβð Þ t2−tð Þ−1h i

dt [from Eq. (5)]

¼ αh= θþ βð Þð ÞZ t2

t1

e θþβð Þ t2−tð Þ−1h i

dt þ bα= θþ βð Þð ÞZ t2

t1

t e θþβð Þ t2−tð Þ−1h i

dt

¼ αh= θþ βð Þð Þ −e θþβð Þ t2−tð Þ= θþ βð Þ−th it2

t1

þ bα= β þ θð Þð Þ −e θþβð Þ t2−tð Þ= θþ βð Þ−e θþβð Þ t2−tð Þ= θþ βð Þ2−t2=2h it2

t1

¼ αh= θþ βð Þð Þh−1= θþ βð Þ þ e θþβð Þ t2−t1ð Þ= θþ βð Þ−t2 þ t1

iþ bα= θþ βð Þð Þ

h−t2= θþ βð Þ

þ t1 eθþβð Þ t2−t1ð Þ= θþ βð Þ−1= θþ βð Þ2 þ e θþβð Þ t2−t1ð Þ= θþ βð Þ2−t22=2þ t1

2=2i

¼ αh e θþβð Þ t2−t1ð Þ−1h i

= β þ θð Þ2−αht2= θþ βð Þ þ αht1= θþ βð Þ−bαt2= θþ βð Þ2 þ

bαt1eθþβð Þ t2−t1ð Þ= θþ βð Þ2 þ bα e θþβð Þ t2−t1ð Þ−1

h i= θþ βð Þ3−bαt22= 2 θþ βð Þð Þ þ bαt21= 2 θþ βð Þð Þ

CH ¼ I2 þ I2

¼ 1= θþ βð Þð Þ e θþβð Þ t2−t1ð Þ−1h i

hα et1β−1� �

=β þ αb et1β−1� �

=β2 þ hα= θþ βð Þ þ bα= θþ βð Þ2n o

þhe θþβð Þ t2−t1ð Þ= θþ βð Þ

i−αbt1=β þ αbt1= θþ βð Þf g−t2α

hh θþ βð Þ þ b

i= θþ βð Þ2−bαt22= 2 θþ βð Þð Þ

þ αh et1β−1� �

=β2−hαt1=β þ αbt1= β θþ βð Þð Þ−αbt1=β2 þ αb et1β−1ð Þ=β3−αbt12= 2βð Þ

þ hαt1= θþ βð Þ þ αbt21= 2 θþ βð Þð Þ

¼ 1= θþ βð Þð Þ e θþβð Þ t2−t1ð Þ−1h i

α et1β−1� �

hβ þ bð Þ=β2 þ α h θþ βð Þ þ b½ �= θþ βð Þ2n o

−αbθt1e θþβð Þ t2−t1ð Þ= β θþ βð Þ2

−t2αhh θþ βð Þ þ b

i= θþ βð Þ2−bαt22= 2 θþ βð Þð Þ

þ hα et1β−1� �

β þ αb et1β−1� �� �

=β3 þ hαt1= θþ βð Þ−hαt1=β−αbθt1= β2 θþ βð Þ� �

þ αbt12= 2 θþ βð Þð Þ

−αbt12= 2βð Þ ¼ e θþβð Þ t2−t1ð Þ−1h i

Aþ Bð Þ= θþ βð Þ−Ge θþβð Þ t2−t1ð Þ= θþ βð Þ−t2B−t22Dþ E

where

A ¼ α et1β−1� �

hβ þ bð Þ=β2

B ¼ α bþ h θþ βð Þ½ �= θþ βð Þ2G ¼ αbθt1= β θþ βð Þð ÞD ¼ αb= 2 θþ βð Þð ÞE ¼ A−G−αbθt12= 2 θþ βð Þð Þ−αht1θ= θþ βð Þ� �

=β

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