Volume 2009, Article ID 679736,24pages doi:10.1155/2009/679736

*Research Article*

**Order Level Inventory Models for**

**Deteriorating Seasonable/Fashionable Products** **with Time Dependent Demand and Shortages**

**K. Skouri**

^{1}**and I. Konstantaras**

^{1, 2}*1**Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece*

*2**Hellenic Army Academy, Vari 16673, Attica, Greece*

Correspondence should be addressed to I. Konstantaras,ikonst@cc.uoi.gr Received 18 September 2008; Revised 3 July 2009; Accepted 20 July 2009 Recommended by Wei-Chiang Hong

An order level inventory model for seasonable/fashionable products subject to a period of increasing demand followed by a period of level demand and then by a period of decreasing demand ratethree branches ramp type demand rateis considered. The unsatisfied demand is partially backlogged with a time dependent backlogging rate. In addition, the product deteriorates with a time dependent, namely, Weibull, deterioration rate. The model is studied under the following diﬀerent replenishment policies:astarting with no shortages andbstarting with shortages. The optimal replenishment policy for the model is derived for both the above mentioned policies.

Copyrightq2009 K. Skouri and I. Konstantaras. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**1. Introduction**

It is observed that the life cycle of many seasonal products, over the entire time horizon, can be portrayed as a period of growth, followed by a period of relatively level demand and finishing with a period of decline. So researchers commonly use a time-varying demand pattern to reflect sales in diﬀerent phases of product life cycle. Resh et al.1and Donaldson 2 are the first researchers who considered an inventory model with a linear trend in demand. Thereafter, numerous research works have been carried out incorporating time- varying demand patterns into inventory models. The time dependent demand patterns, mainly, used in literature are, i linearly time dependent and, ii exponentially time dependentDave and Patel3, Goyal4, Hariga5, Hariga and Benkherouf6, Yang et al.

7. The time dependent demand patterns reported above are unidirectional, that is, increase continuously or decrease continuously. Hill8proposed a time dependent demand pattern by considering it as the combination of two diﬀerent types of demand in two successive

time periods over the entire time horizon and termed it as “ramp-type” time dependent demand pattern. Then, inventory models with ramp type demand rate also are studied by Mandal and Pal9, Wu et al.10and Wu and Ouyang11, Wu12, Giri et al.13, and Manna and Chaudhuri14. In these papers, the determination of the optimal replenishment policy requires the determination of the time point, when the inventory level falls to zero.

So the following two cases should be examined:1this time point occurs before the point, where the demand is stabilized, and 2 this time point occurs after the point, where the demand is stabilized. Almost all of the researchers examine only the first case. Deng et al.

15 reconsidered the inventory model of Mandal and Pal 9 and Wu and Ouyang 11 and studied it exploring these two cases. Skouri et al.16extend the work of Deng et al.

15by introducing a general ramp type demand rate and considering Weibull distributed deterioration rate.

The assumption that the goods in inventory always preserve their physical characteris- tics is not true in general. There are some items, which are subject to risks of breakage, deteri- oration, evaporation, obsolescence, and so forth. Food items, pharmaceuticals, photographic film, chemicals, and radioactivesubstances are few items in which appreciable deterioration can take place during the normal storage of the units. A model with exponentially decaying inventory was initially proposed by Ghare and Schrader 17. Covert and Phillip18and Tadikamalla 19developed an economic order quantity model with Weibull and Gamma distributed deterioration rates, respectively. Thereafter, a great deal of research eﬀorts have been devoted to inventory models of deteriorating items, the details can be found in the review articles by Raafat20, and Goyal and Giri21.

In most of the above-mentioned papers, the demand during stockout period is totally backlogged. In practice, there are customers who are willing to wait and receive their orders at the end of stockout period, while others are not. In the last few years, considerable attention has been paid to inventory models with partial backlogging. The backlogging rate can be modelled taking into account the customers’ behavior. The first paper in which customers’ impatience functions are proposed seems to be that by Abad 22. Chang and Dye23developed a finite horizon inventory model using Abad’s reciprocal backlogging rate. Skouri and Papachristos24studied a multiperiod inventory model using the negative exponential backlogging rate proposed by Abad 22. Teng et al. 25 extended Chang and Dye ’s 23 and Skouri and Papachristos’24 models, assuming as backlogging rate any decreasing function of the waiting time up to the next replenishment. Research on models with partial backlogging continues with Wang 26 and San Jose et al. 27 and 28.

Manna and Chaudhuri 14 noted that ramp type demand pattern is generally followed by new brand of consumer goods coming to the market. But for fashionable products as well as for seasonal products, the steady demand will never be continued indefinitely. Rather it would be followed by decrement with respect to time after a period of time and becomes asymptotic in nature. Thus the demand may be illustrated by three successive time periods that classified time dependent ramp-type function, in which in the first phase the demand increases with time and after that it becomes steady, and towards the end in the final phase it decreases and becomes asymptotic. Chen et al.29proposed a search procedure based on Nelder-Mead algorithm to find a solution for the case of inventory systems with shortage allowance and nonlinear demand pattern. Also, Chen et al. 30 proposed a net present value approach for the previous inventory system without shortages.

For both models, the demand rate is a revised version of the Beta distribution function and so is a diﬀerentiable with respect to time.

The purpose of the present paper is to study an order level inventory model when the demand is described by a three successive time periods that classified time dependent ramp-type function. Any such function has points, at least one, where diﬀerentiation is not possible, and this introduces extra complexity in the analysis of the relevant models. The unsatisfied demand is partially backlogged with time dependent backlogging rate, and units in inventory are subject to deterioration with Weibull-distributed deterioration rate.

The rest of the paper is organized as follows. In the next section the assumptions and notations for the development of the model are provided. The model starting with no shortages is studied in Section3, and the corresponding one starting with shortages is studied in Section4. For each model the optimal policy is obtained. Numerical examples highlighting the results obtained are given in Section 5. The paper closes with concluding remarks in Section6.

**2. Notation and Assumptions**

The following notations and assumptions are used in developing the model:

*Notations*

*T* The constant scheduling periodcycle
*t*_{1}The time when the inventory level reaches zero

*S*The maximum inventory level at each scheduling periodcycle
*c*1 The inventory holding cost per unit per unit time

*c*_{2} The shortage cost per unit per unit time

*c*_{3} The cost incurred from the deterioration of one unit
*c*4 The per unit opportunity cost due to the lost sales

*μ* The time point that increasing demand becomes steady

*γ* The time point, after*μ, until the demand is steady and then decreases*
*It* The inventory level at time*t*∈0, T.

*Assumptions*

1The ordering quantity brings the inventory level up to the order level *S.*

Replenishment rate is infinite.

2Shortages are backlogged at a rate *βx, which is a nonincreasing function of*
*xβ*^{}x ≤ 0 with 0 ≤ *βx* ≤ 1, *β0 * 1, and *x* is the waiting time up to
the next replenishment. Moreover it is assumed that *βx* satisfies the relation
*βx Tβ*^{}x ≥ 0, where *β*^{}x is the derivate of *βx. The cases with* *βx * 1
or 0correspond to complete backloggingor complete lost salesmodels.

3The time to deterioration of the item is distributed as Weibull a, b; that is, the
deterioration rate is*θt * *abt** ^{b−1}* a >0, b > 0, t > 0. There is no replacement or
repair of deteriorated units during the period

*T. Forb*1,

*θt*becomes constant, which corresponds to exponentially decaying case.

4The demand rate *Dt* is a time dependent ramp-type function and is of the
following form:

*Dt *

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

*ft,* 0*< t < μ,*
*f*

*μ*
*g*

*γ*

*, μ*≤*t*≤*γ,*
*gt,* *γ < t,*

2.1

where *f*t is a positive, continuous, and increasing function of *t,* and *gt* is a positive,
continuous and decreasing function of*t.*

**3. The Mathematical Formulation of** **the Model Starting with No Shortages**

The replenishment at the beginning of the cycle brings the inventory level up to*S. Due to*
demand and deterioration, the inventory level gradually depletes during the period0, t1
and falls to zero at*t* *t*_{1}. Thereafter shortages occur during the periodt1*, T*, which are
partially backlogged. Consequently, the inventory level,*It, during the time interval 0*≤*t*≤
*T,*satisfies the following diﬀerential equations:

*dIt*

*dt* *θtIt *−Dt, 0≤*t*≤*t*1*, It*1 0*,* 3.1

*dIt*

*dt* −DtβT−*t,* *t*1≤*t*≤*T, It*1 0. 3.2
The solutions of these diﬀerential equations are aﬀected from the relation between*t*_{1},
*μ,*and*γ*through the demand rate function. Since the demand has three components in three
successive time periods, the following cases: i*t*1 *< μ < γ < T,*ii*μ < t*1 *< γ < T,*and
iii*μ < γ < t*_{1} *< T* must be considered to determine the total cost and then the optimal
replenishment policy.

*Case 1*t1*< μ < γ < T*. In this case,3.1becomes
*dIt*

*dt* *abt*^{b−1}*It *−ft, 0≤*t*≤*t*1*, It*1 0. 3.3
Equation3.2leads to the following three:

*dIt*

*dt* −ftβT−*t,* *t*_{1}≤*t*≤*μ, It*1 0, 3.4
*dIt*

*dt* −f
*μ*

*βT*−*t,* *μ*≤*t*≤*γ, I*
*μ*_{−}

*I*
*μ*_{}

*,* 3.5

*dI*t

*dt* −gtβT−*t,* *γ*≤*t*≤*T.* 3.6

The solutions of3.3,3.4,3.5, and3.6, are, respectively,

*It e*^{−at}^{b}_{t}_{1}

*t**fxe*^{ax}^{b}*dx,* 0≤*t*≤*t*_{1}*,* 3.7

*It *− ^{t}

*t*1

*fxβT*−*xdx,* *t*1≤*t*≤*μ,* 3.8

*It *−f
*μ*

*t*
*μ*

*βT*−*xdx*− ^{μ}

*t*1

*fxβT*−*xdx,* *μ*≤*t*≤*γ,* 3.9

*It *− ^{t}

*γ*

*gxβT*−*xdx*− ^{μ}

*t*1

*fxβT*−*xdx*−*f*
*μ*

*γ*
*μ*

*βT*−*xdx,* *γ*≤*t*≤*T.* 3.10

The total amount of deteriorated items during0, t1is

*D* ^{t}^{1}

0

*fte*^{at}^{b}*dt*− ^{t}^{1}

0

*ftdt.* 3.11

The cumulative inventory carried in the interval0, t1is found from3.7and is

*I*1 ^{t}^{1}

0

*Itdt* ^{t}^{1}

0

*e*^{−at}^{b}

*t*1

*t*

*fxe*^{ax}^{b}*dx*

*dt.* 3.12

Due to3.8,3.9, and3.10, the time-weighted backorders during the intervalt1*, T*are

*I*_{2} ^{T}

*t*1

−Itdt

^{μ}

*t*1

−Itdt ^{γ}

*μ*

−Itdt ^{T}

*γ*

−Itdt

^{μ}

*t*1

*μ*−*t*

*ftβT*−*tdtf*
*μ*

*γ*
*μ*

*t*
*μ*

*βT*−*xdx*

*dt* ^{γ}

*μ*
*μ*
*t*1

*f*xβT−*xdx*

*dt*

^{T}

*γ*
*t*
*γ*

*g*xβT−*xdx*

*dt* ^{T}

*γ*
*μ*
*t*1

*fxβT*−*xdx*

*dtf*
*μ*

*T*
*γ*

*γ*
*μ*

*βT*−*xdx*

*dt.*

3.13
The amount of lost sales duringt1*, T*is

*L* ^{μ}

*t*1

1−*βT*−*t*

*ftdtf*
*μ*

*γ*
*μ*

1−*βT*−*t*
*dt* ^{T}

*γ*

1−*βT*−*t*

*gtdt.* 3.14

The total cost in the time interval0, Tis the sum of holding, shortage, deterioration, and opportunity costs and is given by

*TC*_{1}t1 *c*_{1}*I*_{1}*c*_{2}*I*_{2}*c*_{3}*Dc*_{4}*L*
*c*_{1}

*t*1

0

*e*^{−at}^{b}

*t*1

*t*

*fxe*^{ax}^{b}*dx*

*dt*

*c*_{3}

*t*1

0

*fte*^{at}^{b}*dt*− ^{t}^{1}

0

*ftdt*

*c*_{2}

*μ*
*t*1

*μ*−*t*

*ftβT*−*tdtf*
*μ*

*γ*
*μ*

*t*
*μ*

*βT*−*xdx*

*dt*

^{γ}

*μ*
*μ*
*t*1

*f*xβT−*xdx*

*dt* ^{T}

*γ*
*t*
*γ*

*g*xβT−*xdx*

*dt*

^{T}

*γ*
*μ*
*t*1

*fxβT*−*xdx*

*dtf*
*μ*

*T*
*γ*

*γ*
*μ*

*βT*−*xdx*

*dt*

*c*4
*μ*
*t*1

1−*βT*−*t*

*ftdtf*
*μ*

*γ*
*μ*

1−*βT* −*t*
*dt* ^{T}

*γ*

1−*βT*−*t*
*gtdt*

*.*
3.15

*Case 2*μ < t1 *< γ < T. In this case,*3.1reduces to the following two:

*dI*t

*dt* *abt*^{b−1}*It *−ft, 0≤*t*≤*μ, I*
*μ*^{−}

*I*
*μ*^{}

*,*
*dI*t

*dt* *abt*^{b−1}*It *−f
*μ*

*,* *μ*≤*t*≤*t*_{1}*, It*1 0.

3.16

Equation3.2leads to the following two:

*dIt*
*dt* −f

*μ*

*βT*−*t,* *t*1≤*t*≤*γ, It*1 0,
*dIt*

*dt* −gtβT−*t,* *γ* ≤*t*≤*T, I*
*γ*^{−}

*I*
*γ*^{}

*.*

3.17

Their solutions are, respectively,

*It e*^{−at}^{b}

*μ*
*t*

*fxe*^{ax}^{b}*dxf*
*μ*

*t*1

*μ*

*e*^{ax}^{b}*dx*

*,* 0≤*t*≤*μ,*

*It e*^{−at}^{b}*f*
*μ*

*t*1

*t*

*e*^{ax}^{b}*dx,* *μ*≤*t*≤*t*_{1}*,*
*It *−f

*μ*

*t*
*t*1

*βT*−*xdx,* *t*1≤*t*≤*γ,*

*It *− ^{t}

*γ*

*gxβT*−*xdx*−*f*
*μ*

*γ*
*t*1

*βT*−*xdx,* *γ*≤*t*≤*T.*

3.18

The total amount of deteriorated items during0, t1is

*DI0*− ^{t}^{1}

0

*Dtdt* ^{μ}

0

*fte*^{at}^{b}*dtf*
*μ*

*t*1

*μ*

*e*^{at}^{b}*dt*− ^{μ}

0

*ftdt*−*f*
*μ*

*t*1−*μ*

*.* 3.19

The total inventory carried during the interval0, t1is

*I*1 ^{t}^{1}

0

*Itdt* ^{μ}

0

*Itdt* ^{t}^{1}

*μ*

*Itdt*

^{μ}

0

*e*^{−at}^{b}

*μ*
*t*

*fxe*^{ax}^{b}*dxf*
*μ*

*t*1

*μ*

*e*^{ax}^{b}*dx*

*dtf*
*μ*

*t*1

*μ*

*e*^{−at}^{b}

*t*1

*t*

*e*^{ax}^{b}*dx*

*dt.*

3.20

The time-weighted backorders during the intervalt1*, T*are

*I*2 ^{T}

*t*1

−Itdt ^{γ}

*t*1

−Itdt ^{T}

*γ*

−Itdt

^{γ}

*t*1

*f*
*μ*

*t*
*t*1

*βT*−*xdx dt* ^{T}

*γ*
*t*
*γ*

*gxβT*−*xdx dtf*
*μ*

*T*
*γ*

*γ*
*t*1

*βT*−*xdx dt.*

3.21

The lost sales in the intervalt1*, T*are

*Lf*
*μ*

*γ*
*t*1

1−*βT*−*t*
*dt* ^{T}

*γ*

1−*βT*−*t*

*gtdt.* 3.22

The inventory cost for this case is
*TC*_{2}t1 *c*_{1}*I*_{1}*c*_{2}*I*_{2}*c*_{3}*Dc*_{4}*L*

*c*1
*μ*
0

*e*^{−at}^{b}

*μ*
*t*

*fxe*^{ax}^{b}*dxf*
*μ*

*t*1

*μ*

*e*^{ax}^{b}*dx*

*dtf*
*μ*

*t*1

*μ*

*e*^{−at}^{b}

*t*1

*t*

*e*^{ax}^{b}*dx*

*dt*

*c*2
*γ*
*t*1

*f*
*μ*

*t*
*t*1

*βT*−*xdx dt* ^{T}

*γ*
*t*
*γ*

*gxβT*−*xdx dt*

f
*μ*

*T*
*γ*

*γ*
*t*1

*βT*−*xdx dt*

*c*3
*μ*
0

*fte*^{at}^{b}*dtf*
*μ*

*t*1

*μ*

*e*^{at}^{b}*dt*− ^{μ}

0

*ftdt*−*f*
*μ*

*t*1−*μ*

*c*4

*f*

*μ*

*γ*
*t*1

1−*βT*−*t*
*dt* ^{T}

*γ*

1−*βT*−*t*
*gtdt*

*.*

3.23

*Case 3*μ < γ < t1*< T*. In this case,3.1reduces to the following three:

*dIt*

*dt* *abt*^{b−1}*It *−ft, 0≤*t*≤*μ , I*
*μ*^{−}

*I*
*μ*^{}

*,*
*dIt*

*dt* *abt*^{b−1}*It *−f
*μ*

*,* *μ*≤*t*≤*γ, I*
*γ*^{−}

*I*
*γ*^{}

*,*

*dIt*

*dt* *abt*^{b−1}*It *−gt, *γ* ≤*t*≤*t*_{1}*, It*1 0.

3.24

Equation3.2leads to the following:

*dIt*

*dt* −gtβT−*t,* *t*_{1}≤*t*≤*T, It*_{1} 0. 3.25
Their solutions are, respectively,

*It e*^{−at}^{b}

*μ*
*t*

*fxe*^{ax}^{b}*dxf*
*μ*

*γ*
*μ*

*e*^{ax}^{b}*dx* ^{t}^{1}

*γ*

*gxe*^{ax}^{b}*dx*

*,* 0≤*t*≤*μ,* 3.26

*It e*^{−at}^{b}

*f*
*μ*

*γ*
*t*

*e*^{ax}^{b}*dx* ^{t}^{1}

*γ*

*gxe*^{ax}^{b}*dx*

*,* *μ*≤*t*≤*γ,* 3.27

*It e*^{−at}^{b}

*t*1

*t*

*gxe*^{ax}^{b}*dx,* *γ* ≤*t*≤*t*_{1}*,* 3.28

*It *− ^{t}

*t*1

*g*xβT−*xdx,* *t*_{1}≤*t*≤*T.* 3.29

The total amount of deteriorated items during0, t1is

*DI0*− ^{t}^{1}

0

*Dtdt*

^{μ}

0

*fxe*^{ax}^{b}*dxf*
*μ*

*γ*
*μ*

*e*^{ax}^{b}*dx* ^{t}^{1}

*γ*

*gxe*^{ax}^{b}*dx*

− ^{μ}

0

*ftdt*−*f*
*μ*

*γ*−*μ*

− ^{t}^{1}

*γ*

*gtdt.*

3.30

The total inventory carried during the interval0, t1, using3.26,3.27, and3.28is

*I*1 ^{t}^{1}

0

*Itdt* ^{μ}

0

*Itdt* ^{γ}

*μ*

*Itdt* ^{t}^{1}

*γ*

*Itdt*

^{μ}

0

*e*^{−at}^{b}

*μ*
*t*

*fxe*^{ax}^{b}*dxf*
*μ*

*γ*
*μ*

*e*^{ax}^{b}*dx* ^{t}^{1}

*γ*

*gxe*^{ax}^{b}*dx*

*dt*

^{γ}

*μ*

*e*^{−at}^{b}

*f*
*μ*

*γ*
*t*

*e*^{ax}^{b}*dx* ^{t}^{1}

*γ*

*gxe*^{ax}^{b}*dx*

*dt* ^{t}^{1}

*γ*

*e*^{−at}^{b}

*t*1

*t*

*gxe*^{ax}^{b}*dx*

*dt.*

3.31

The time-weighted backorders during the intervalt1*, T*are

*I*2 ^{T}

*t*1

−Itdt ^{T}

*t*1

*t*
*t*1

*gxβT*−*xdx.* 3.32

The lost sales in the intervalt1*, T*are

*L* ^{T}

*t*1

1−*βT*−*t*

*gtdt.* 3.33

The inventory cost for this case is
*TC*_{3}t1 *c*_{1}*I*_{1}*c*_{2}*I*_{2}*c*_{3}*Dc*_{4}*L*

*c*_{1}

*μ*
0

*e*^{−at}^{b}

*μ*
*t*

*fxe*^{ax}^{b}*dxf*
*μ*

*γ*
*μ*

*e*^{ax}^{b}*dx* ^{t}^{1}

*γ*

*gxe*^{ax}^{b}*dx*

*dt*

^{γ}

*μ*

*e*^{−at}^{b}

*f*
*μ*

*γ*
*t*

*e*^{ax}^{b}*dx* ^{t}^{1}

*γ*

*gxe*^{ax}^{b}*dx*

*dt* ^{t}^{1}

*γ*

*e*^{−at}^{b}

*t*1

*t*

*gxe*^{ax}^{b}*dx*

*dt*

*c*2
*T*
*t*1

*t*
*t*1

*gxβT*−*tdx*

*c*4
*T*
*t*1

1−*βT*−*t*
*g*tdt

*c*3
*μ*
0

*fxe*^{ax}^{b}*dxf*
*μ*

*γ*
*μ*

*e*^{ax}^{b}*dx* ^{t}^{1}

*γ*

*gxe*^{ax}^{b}*dx*

− ^{μ}

0

*f*tdt−*f*
*μ*

*γ*−*μ*

− ^{t}^{1}

*γ*

*gtdt*

*.*

3.34 Finally the total cost function of the system over0, Ttakes the following form:

*TCt*1

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

*TC*_{1}t1, if*t*_{1}≤*μ,*
*TC*_{2}t1, if*μ < t*_{1}*< γ,*
*TC*_{3}t1, if*γ*≤*t*_{1}*.*

3.35

It is easy to check that this function is continuous at *μ* and *γ. The problem now is the*
minimization of this, three branches, function*TCt*1. This requires, separately, studying each
of these branches and then combining the results to state the algorithm giving the optimal
policy.

**3.1. The Optimal Replenishment Policy**

In this subsection we present the results, which ensure the existence of a unique optimal value
for *t*1, say *t*^{∗}_{1}, which minimizes the total cost function. Although the optimality procedure
requires the constrained optimization of the functions*TC*_{1}t1,TC2t1, and*TC*_{3}t1, we will,
firstly, search for their unconstrained minimum. The first- and second-order derivatives of
*TC*1t1,*TC*2t1,and*TC*3t1are, respectively,

*dTC*_{1}t1

*dt*1 *f*t1ht1,
*d*^{2}*TC*_{1}t1

*dt*_{1}^{2} *dft*1

*dt*_{1} *ht*1 *f*t1*dht*1
*dt*_{1} *,*
*dTC*2t1

*dt*_{1} *f*
*μ*

*ht*1,
*d*^{2}*TC*_{2}t1

*dt*12 *f*

*μdht*1
*dt*1 *,*
*dTC*_{3}t1

*dt*_{1} *g*t1ht1,
*d*^{2}*TC*3t1

*dt*12 *dgt*1

*dt*_{1} *ht*1 *g*t1*dht*1
*dt*_{1} *,*

3.36

where

*ht*1 *c*_{1}*e*^{at}^{1}^{b}

*t*1

0

*e*^{−at}^{b}*dtc*_{3}

*e*^{at}^{1}* ^{b}*−1

−*c*_{2}T−*t*_{1}βT−*t*_{1}−*c*_{4}

1−*βT*−*t*_{1}

*.* 3.37

Equation 3.37 is the same as 16 of the paper of Skouri et al. 16. So, following the methodology proposed by Skouri et al. 16, the algorithm, which gives the optimal replenishment policy, is as follows.

*Step 1. Computet*^{∗}_{1}from*ht*1 0.

*Step 2. Ift*^{∗}_{1}≤*μ, then the optimal order quantity is given by*

*Q*^{∗} ^{t}

∗1

0

*f*te^{at}^{b}*dt* ^{μ}

*t*^{∗}_{1}

*βT*−*tftdtf*
*μ*

*γ*
*μ*

*βT*−*tdt* ^{T}

*γ*

*βT*−*tgtdt* 3.38

and the total cost is given by*TC*_{1}t^{∗}_{1}.

If*μ < t*^{∗}_{1} *< γ,*then the optimal order quantity is given by

*Q*^{∗} ^{μ}

0

*fte*^{at}^{b}*dtf*
*μ*

*t*^{∗}_{1}
*μ*

*e*^{at}^{b}*dtf*
*μ*

*γ*
*t*^{∗}_{1}

*βT*−*tdt* ^{T}

*γ*

*gtβT*−*tdt* 3.39

and the total cost is given by*TC*2t^{∗}_{1}.

If*γ < t*^{∗}_{1} *< T,*then the optimal order quantity is given by

*Q*^{∗} ^{μ}

0

*f*te^{at}^{b}*dtf*
*μ*

*γ*
*μ*

*e*^{at}^{b}*dt* ^{t}

∗1

*γ*

*gte*^{at}^{b}*dt* ^{T}

*t*^{∗}_{1}

*gtβT*−*tdt* 3.40

and the total cost is given by*TC*_{3}t^{∗}_{1}.

*Remark 3.1. The previous analysis shows thatt*^{∗}_{1}is independent from the demand rate*Dt.*

This very interesting result agrees with the classical result, in many order level inventory
systems, that the point*t*^{∗}_{1}is independent from the demand rateNaddor31, page 67.

**3.2. The Special Case***βx *1* anda*0

If we are considering the case that there is no deterioration of the product a 0 and unsatisfied demand is complete backloggedβx 1, then the total cost function of the model starting with no shortages over0, Ttakes the following form:

*TCt*1

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩
*c*_{1}

*t*1

0
*t*1

*t*

*fxdx dtc*_{2}

*μ*
*t*1

*t*
*t*1

*fxdx dtc*_{2}*f*
*μ*

*γ*
*μ*

*t*−*μ*
*dtc*_{2}

*γ*
*μ*

*μ*
*t*1

*fxdx dt*

c2
*T*
*γ*

*t*
*γ*

*gxdx dt dtc*_{2}*f*
*μ*

*γ*−*μ*
*T*−*γ*

*c*_{2}

*T*
*γ*

*μ*
*t*1

*fxdx dt,*

*t*_{1}*< μ < γ < T,*

*c*1
*μ*
0

*μ*
*t*

*fxdx dtc*1
*μ*
0

*t*1

*μ*

*f*
*μ*

*dx dtc*1
*t*1

*μ*
*t*1

*t*

*f*
*μ*

*dx dtc*2
*γ*
*t*1

*t*
*t*1

*f*
*μ*

*dx dt*

c2
*T*
*γ*

*t*
*γ*

*gxdxdtc*_{2}

*T*
*γ*

*γ*
*t*1

*f*
*μ*

*dx dt,* *μ < t*_{1}*< γ < T,*

*c*1
*μ*
0

*μ*
*t*

*fxdx dtc*1
*μ*
0

*γ*
*μ*

*f*
*μ*

*dxdtc*1
*μ*
0

*t*1

*γ*

*gxdxdtc*1
*γ*
*μ*

*γ*
*t*

*f*
*μ*

*dx dt*

c1
*γ*
*μ*

*t*1

*γ*

*gxdx dtc*_{1}

*t*1

*γ*
*t*1

*t*

*gxdx dtc*_{2}

*T*
*t*1

*t*
*t*1

*gxdx dt,*

*μ < γ < t*1*< T.*

3.41

Following the previous procedure for the optimal replenishment policy, the optimal value of
*t*_{1}, say*t*^{∗}_{1}, is given by the very simple and known, in classical order level inventory system
Naddor31, equation:

*t*^{∗}_{1} *c*_{2}*T*
*c*1*c*2

*.* 3.42

**4. The Mathematical Formulation of** **the Model Starting with Shortages**

In this section the inventory model starting with shortages is studied. The cycle now starts
with shortages, which occur during the period0, t1,and are partially backlogged. At time
*t*1a replenishment brings the inventory level up to*S. Demand and deterioration of the items*
deplete the inventory level during the periodt1*, T*until this falls to zero at*tT*. Again the
three cases*t*_{1}*< μ < γ < T*,*μ < t*_{1}*< γ < T,*and*μ < γ < t*_{1}*< T* must be examined.

*Case 4*t1*< μ < γ < T. The inventory level,I*t, 0≤*t*≤*T* satisfies the following diﬀerential
equations:

*dIt*

*dt* −ftβt1−*t,* 0≤*t*≤*t*_{1}*, I0 *0,
*dIt*

*dt* *abt*^{b−1}*It *−ft, *t*1≤*t*≤*μ, I*
*μ*_{−}

*I*
*μ*_{}

*,*
*dIt*

*dt* *abt*^{b−1}*It *−f
*μ*

*,* *μ*≤*t*≤*γ, I*
*γ*_{−}

*I*
*γ*_{}

*,*
*dIt*

*dt* *abt*^{b−1}*It *−gt, *γ*≤*t*≤*T, I*T 0.

4.1

The solutions of4.1, are, respectively,

*It *− ^{t}

0

*fxβt*1−*xdx,* 0≤*t*≤*t*_{1}*,* 4.2

*It e*^{−αt}^{b}

*μ*
*t*

*e*^{αx}^{b}*fxdxf*
*μ*

*γ*
*μ*

*e*^{αx}^{b}*dx* ^{T}

*γ*

*e*^{αx}^{b}*gxdx*

*,* *t*1≤*t*≤*μ,* 4.3
*It e*^{−αt}^{b}

*f*

*μ*

*γ*
*t*

*e*^{αx}^{b}*dx* ^{T}

*γ*

*e*^{αx}^{b}*gxdx*

*,* *μ*≤*t*≤*γ,* 4.4

*It e*^{−at}^{b}

*T*
*t*

*e*^{ax}^{b}*gxdx,* *γ*≤*t*≤*T.* 4.5

The total amount of deteriorated units duringt1*, T*is

*De*^{−at}^{b}^{1}

*μ*
*t*1

*f*xe^{ax}^{b}*dxf*
*μ*

*γ*
*μ*

*e*^{αx}^{b}*dx* ^{T}

*γ*

*e*^{αx}^{b}*gxdx*

− ^{μ}

*t*1

*fxdx*−
*γ*−*μ*

*f*
*μ*

− ^{T}

*γ*

*gxdx.*

4.6

The total inventory carried during the intervalt1*, T*is found using4.3,4.4, and4.5and
is

*I*_{1} ^{μ}

*t*1

*e*^{−at}^{b}

*μ*
*t*

*e*^{αx}^{b}*fxdxf*
*μ*

*γ*
*μ*

*e*^{αx}^{b}*dx* ^{T}

*γ*

*e*^{αx}^{b}*g*xdx

*dt*

^{γ}

*μ*

*e*^{−αt}^{b}

*f*
*μ*

*γ*
*t*

*e*^{αx}^{b}*dx* ^{T}

*γ*

*e*^{αx}^{b}*gxdx*

*dt* ^{T}

*γ*

*e*^{−at}^{b}

*T*
*t*

*e*^{ax}^{b}*gxdx*

*dt.*

4.7

Due to4.2the time-weighted backorders during the time interval0, t1are

*I*2 ^{t}^{1}

0
*t*
0

*fxβt*1−*xdx dt.* 4.8

The amount of lost sales during0, t1is

*L* ^{t}^{1}

0

1−*βt*1−*t*

*ftdt.* 4.9

The inventory cost during the time interval 0, T is the sum of holding, shortage, deterioration, and opportunity costs and is given by

*TC*_{1}t1 *c*_{1}*I*_{1}*c*_{2}*I*_{2}*c*_{3}*Dc*_{4}*L*
*c*_{1}

*μ*
*t*1

*e*^{−at}^{b}

*μ*
*t*

*e*^{αx}^{b}*f*xdx*f*
*μ*

*γ*
*μ*

*e*^{αx}^{b}*dx* ^{T}

*γ*

*e*^{αx}^{b}*gxdx*

*dt*

*c*_{1}

*γ*
*μ*

*e*^{−αt}^{b}

*f*
*μ*

*γ*
*t*

*e*^{αx}^{b}*dx* ^{T}

*γ*

*e*^{αx}^{b}*gxdx*

*dt* ^{T}

*γ*

*e*^{−at}^{b}

*T*
*t*

*e*^{ax}^{b}*gxdx*

*dt*

*c*_{2}

*t*1

0
*t*
0

*fxβt*1−*xdx dtc*_{4}

*t*1

0

1−*βt*1−*t*
*ftdt*

*c*3

*e*^{−at}^{b}^{1}

*μ*
*t*1

*fxe*^{ax}^{b}*dxf*
*μ*

*γ*
*μ*

*e*^{αx}^{b}*dx* ^{T}

*γ*

*e*^{αx}^{b}*gxdx*

− ^{μ}

*t*1

*fxdx*−
*γ*−*μ*

*f*
*μ*

− ^{T}

*γ*

*gxdx*

*.*

4.10

*Case 5*μ < t1*< γ < T. The inventory level,I*t, 0≤*t*≤*T* satisfies the following diﬀerential
equations:

*dIt*

*dt* −ftβt1−*t,* 0≤*t*≤*μ, I0 *0,
*dIt*

*dt* −f
*μ*

*βt*1−*t,* *μ*≤*t*≤*t*1*, I*
*μ*^{−}

*I*
*μ*^{}

*,*
*dIt*

*dt* *abt*^{b−1}*It *−f
*μ*

*,* *t*1≤*t*≤*γ, I*
*γ*_{−}

*I*
*γ*_{}

*,*
*dIt*

*dt* *abt*^{b−1}*It *−gt, *γ*≤*t*≤*T, IT* 0.

4.11

The solutions of4.11, are, respectively,

*It *− ^{t}

0

*fxβt*1−*xdx,* 0≤*t*≤*μ,*

*It *− ^{μ}

0

*fxβt*1−*xdx*−*f*
*μ*

*t*
*μ*

*βt*1−*xdx,* *μ*≤*t*≤*t*1*,*

*It e*^{−αt}^{b}

*f*
*μ*

*γ*
*t*

*e*^{αx}^{b}*dx* ^{T}

*γ*

*e*^{αx}^{b}*g*xdx

*,* *t*_{1}≤*t*≤*γ,*

*It e*^{−at}^{b}

*T*
*t*

*e*^{ax}^{b}*gxdx,* *γ* ≤*t*≤*T.*

4.12

The total cost of this case is obtained with a similar way of the previous cases and is,

*TC*_{2}t1 *c*_{1}

*γ*
*t*1

*e*^{−αt}^{b}

*f*
*μ*

*γ*
*t*

*e*^{αx}^{b}*dx* ^{T}

*γ*

*e*^{αx}^{b}*gxdx*

*dt*

^{T}

*γ*

*e*^{−at}^{b}

*T*
*t*

*e*^{ax}^{b}*gxdx*

*dt*

*c*3

*e*^{−at}^{b}^{1}

*γ*
*t*1

*f*
*μ*

*e*^{ax}^{b}*dx* ^{T}

*γ*

*e*^{ax}^{b}*gxdx*

−*f*
*μ*

*γ*−*t*1

− ^{T}

*γ*

*g*xdx

*c*2
*μ*
0

*t*
0

*fxβt*1−*xdx*

*dt*

^{t}^{1}

*μ*
*μ*
0

*fxβt*1−*xdxf*
*μ*

*t*
*μ*

*βt*1−*xdx*

*dt*

*c*4
*μ*
0

1−*βt*1−*t*

*ftdtf*
*μ*

*t*1

*μ*

1−*βt*1−*t*
*dt*

*.*

4.13

*Case 6*μ < γ < t**1***< T*. The inventory level,*I*t, 0≤*t*≤*T* for this case satisfies the following
diﬀerential equations:

*dIt*

*dt* −ftβt1−*t,* 0≤*t*≤*μ, I0 *0, 4.14
*dIt*

*dt* −f
*μ*

*βt*1−*t,* *μ*≤*t*≤*γ, I*
*μ*^{−}

*I*
*μ*^{}

*,* 4.15

*dIt*

*dt* −gtβt1−*t,* *γ*≤*t*≤*t*1*, I*
*γ*_{−}

*I*
*γ*_{}

*,* 4.16

*dIt*

*dt* *abt*^{b−1}*It *−gt, *t*1≤*t*≤*T, IT* 0. 4.17

The solutions of4.14,4.15,4.16, and4.17, are, respectively,

*It *− ^{t}

0

*fxβt*1−*xdx,* 0≤*t*≤*μ,*

*It *− ^{μ}

0

*fxβt*1−*xdx*−*f*
*μ*

*t*
*μ*

*βt*1−*xdx,* *μ*≤*t*≤*γ,*

*It *− ^{t}

*γ*

*gxβt*1−*xdx*− ^{μ}

0

*fxβt*1−*xdx*−*f*
*μ*

*γ*
*μ*

*βt*1−*xdx,* *γ*≤*t*≤*t*_{1}*,*

*It e*^{−at}^{b}

*T*
*t*

*e*^{ax}^{b}*gxdx,* *t*1≤*t*≤*T.*

4.18

The total cost of this case is obtained with a similar way of the previous cases and is,

*TC*3t1 *c*1
*T*
*t*1

*e*^{−αt}^{b}

*T*
*t*

*e*^{ax}^{b}*gxdx dt*

*c*3

*e*^{−at}^{b}^{1}

*T*
*t*1

*e*^{ax}^{b}*gxdx*− ^{T}

*t*1

*gxdx*

*c*2
*μ*
0

*t*
0

*fxβt*1−*xdx*

*dt* ^{γ}

*μ*
*μ*
0

*fxβt*1−*xdxf*
*μ*

*t*
*μ*

*βt*1−*xdx*

*dt*

*c*2
*t*1

*γ*
*t*
*γ*

*gxβt*1−*xdx* ^{μ}

0

*f*xβt1−*xdxf*
*μ*

*γ*
*μ*

*βt*1−*xdx*

*dt*

*c*4
*μ*
0

1−*βt*1−*t*

*f*tdt*f*
*μ*

*γ*
*μ*

1−*βt*1−*t*
*dt* ^{t}^{1}

*γ*

1−*βt*1−*t*
*g*tdt

*.*
4.19

Finally the total cost function of the system over0, Ttakes the following form:

*TCt*1

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

*TC*_{1}t1, if*t*_{1}≤*μ,*
*TC*_{2}t1, if*μ < t*_{1}*< γ,*
*TC*_{3}t1, if*γ*≤*t*_{1}*.*

4.20

It is easy to check that this function is continuous at *μ* and *γ. The problem now is the*
minimization of this, three branches, function*TCt*1. This requires, separately, studying each
of these branches and then combining the results to state the algorithm giving the optimal
policy.

**4.1. The Optimal Replenishment Policy**

In this subsection we derive the optimal replenishment policy, that is, we calculate the value,
say*t*^{∗}_{1}, which minimizes the total cost function. Taking the first-order derivative of*TC*1t1,
say*K*_{1}t1, and equating it to zero gives:

*K*_{1}t1 −

*c*_{1}*c*_{3}*αbt*^{b−1}_{1}
*e*^{−αt}^{b}^{1}

*μ*
*t*1

*e*^{αx}^{b}*f*xdx*f*
*μ*

*γ*
*μ*

*e*^{αx}^{b}*dx* ^{T}

*γ*

*e*^{αx}^{b}*gxdx*

^{t}^{1}

0

*c*2*βt*1−*t c*2t1−*tβ*^{}t1−*t*−*c*4*β*^{}t1−*t*

*ftdt*0.

4.21

If*t*^{∗}_{1}is a root of4.21, for this root the second-order condition for minimum is

*c*1*c*3*αb*
*t*^{∗}_{1}_{b−1}

−*c*3b−1
*t*^{∗}_{1}_{−1}

*αb*
*t*^{∗}_{1}_{b−1}

*e*^{−αt}^{∗b}^{1}

× ^{μ}

*t*^{∗}

1

*e*^{αx}^{b}*fxdxf*
*μ*

*γ*
*μ*

*e*^{αx}^{b}*dx* ^{T}

*γ*

*e*^{αx}^{b}*gxdx*

*c*_{1}*c*_{3}*αb*
*t*^{∗}_{1}_{b−1}

*f*
*t*^{∗}_{1}

*c*_{2}−*c*_{4}*β*^{}0
*f*

*t*^{∗}_{1}
^{t}

∗1

0

2c2*β*^{}
*t*^{∗}_{1}−*t*

*c*2

*t*^{∗}_{1}−*t*
*β*^{}

*t*^{∗}_{1}−*t*

−*c*4*β*^{}
*t*^{∗}_{1}−*t*

*f*tdt >0.

4.22

So, if4.22holds and*t*^{∗}_{1} ≤*μ, then the value of order level,S, is*

*S*^{∗}*I*
*t*^{∗}_{1}

*e*^{−αt}^{∗}^{1}^{b}

*μ*
*t*^{∗}_{1}

*e*^{αx}^{b}*fxdxf*
*μ*

*γ*
*μ*

*e*^{αx}^{b}*dx* ^{T}

*γ*

*e*^{αx}^{b}*gxdx*

*,* 4.23