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
1and I. Konstantaras
1, 21Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
2Hellenic 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 t1The time when the inventory level reaches zero
SThe maximum inventory level at each scheduling periodcycle c1 The inventory holding cost per unit per unit time
c2 The shortage cost per unit per unit time
c3 The cost incurred from the deterioration of one unit c4 The per unit opportunity cost due to the lost sales
μ The time point that increasing demand becomes steady
γ The time point, afterμ, until the demand is steady and then decreases It The inventory level at timet∈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 xβx ≤ 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 Tβx ≥ 0, where βx 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 isθt abtb−1 a >0, b > 0, t > 0. There is no replacement or repair of deteriorated units during the periodT. Forb 1,θtbecomes 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 ft is a positive, continuous, and increasing function of t, and gt is a positive, continuous and decreasing function oft.
3. The Mathematical Formulation of the Model Starting with No Shortages
The replenishment at the beginning of the cycle brings the inventory level up toS. Due to demand and deterioration, the inventory level gradually depletes during the period0, t1 and falls to zero att t1. 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≤t1, It1 0, 3.1
dIt
dt −DtβT−t, t1≤t≤T, It1 0. 3.2 The solutions of these differential equations are affected from the relation betweent1, μ,andγthrough the demand rate function. Since the demand has three components in three successive time periods, the following cases: it1 < μ < γ < T,iiμ < t1 < γ < T,and iiiμ < γ < t1 < T must be considered to determine the total cost and then the optimal replenishment policy.
Case 1t1< μ < γ < T. In this case,3.1becomes dIt
dt abtb−1It −ft, 0≤t≤t1, It1 0. 3.3 Equation3.2leads to the following three:
dIt
dt −ftβT−t, t1≤t≤μ, It1 0, 3.4 dIt
dt −f μ
βT−t, μ≤t≤γ, I μ−
I μ
, 3.5
dIt
dt −gtβT−t, γ≤t≤T. 3.6
The solutions of3.3,3.4,3.5, and3.6, are, respectively,
It e−atbt1
tfxeaxbdx, 0≤t≤t1, 3.7
It − t
t1
fxβT−xdx, t1≤t≤μ, 3.8
It −f μ
t μ
βT−xdx− μ
t1
fxβT−xdx, μ≤t≤γ, 3.9
It − t
γ
gxβT−xdx− μ
t1
fxβT−xdx−f μ
γ μ
βT−xdx, γ≤t≤T. 3.10
The total amount of deteriorated items during0, t1is
D t1
0
fteatbdt− t1
0
ftdt. 3.11
The cumulative inventory carried in the interval0, t1is found from3.7and is
I1 t1
0
Itdt t1
0
e−atb
t1
t
fxeaxbdx
dt. 3.12
Due to3.8,3.9, and3.10, the time-weighted backorders during the intervalt1, Tare
I2 T
t1
−Itdt
μ
t1
−Itdt γ
μ
−Itdt T
γ
−Itdt
μ
t1
μ−t
ftβT−tdtf μ
γ μ
t μ
βT−xdx
dt γ
μ μ t1
fxβT−xdx
dt
T
γ t γ
gxβT−xdx
dt T
γ μ t1
fxβT−xdx
dtf μ
T γ
γ μ
βT−xdx
dt.
3.13 The amount of lost sales duringt1, Tis
L μ
t1
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
TC1t1 c1I1c2I2c3Dc4L c1
t1
0
e−atb
t1
t
fxeaxbdx
dt
c3
t1
0
fteatbdt− t1
0
ftdt
c2
μ t1
μ−t
ftβT−tdtf μ
γ μ
t μ
βT−xdx
dt
γ
μ μ t1
fxβT−xdx
dt T
γ t γ
gxβT−xdx
dt
T
γ μ t1
fxβT−xdx
dtf μ
T γ
γ μ
βT−xdx
dt
c4 μ t1
1−βT−t
ftdtf μ
γ μ
1−βT −t dt T
γ
1−βT−t gtdt
. 3.15
Case 2μ < t1 < γ < T. In this case,3.1reduces to the following two:
dIt
dt abtb−1It −ft, 0≤t≤μ, I μ−
I μ
, dIt
dt abtb−1It −f μ
, μ≤t≤t1, It1 0.
3.16
Equation3.2leads to the following two:
dIt dt −f
μ
βT−t, t1≤t≤γ, It1 0, dIt
dt −gtβT−t, γ ≤t≤T, I γ−
I γ
.
3.17
Their solutions are, respectively,
It e−atb
μ t
fxeaxbdxf μ
t1
μ
eaxbdx
, 0≤t≤μ,
It e−atbf μ
t1
t
eaxbdx, μ≤t≤t1, It −f
μ
t t1
βT−xdx, t1≤t≤γ,
It − t
γ
gxβT−xdx−f μ
γ t1
βT−xdx, γ≤t≤T.
3.18
The total amount of deteriorated items during0, t1is
DI0− t1
0
Dtdt μ
0
fteatbdtf μ
t1
μ
eatbdt− μ
0
ftdt−f μ
t1−μ
. 3.19
The total inventory carried during the interval0, t1is
I1 t1
0
Itdt μ
0
Itdt t1
μ
Itdt
μ
0
e−atb
μ t
fxeaxbdxf μ
t1
μ
eaxbdx
dtf μ
t1
μ
e−atb
t1
t
eaxbdx
dt.
3.20
The time-weighted backorders during the intervalt1, Tare
I2 T
t1
−Itdt γ
t1
−Itdt T
γ
−Itdt
γ
t1
f μ
t t1
βT−xdx dt T
γ t γ
gxβT−xdx dtf μ
T γ
γ t1
βT−xdx dt.
3.21
The lost sales in the intervalt1, Tare
Lf μ
γ t1
1−βT−t dt T
γ
1−βT−t
gtdt. 3.22
The inventory cost for this case is TC2t1 c1I1c2I2c3Dc4L
c1 μ 0
e−atb
μ t
fxeaxbdxf μ
t1
μ
eaxbdx
dtf μ
t1
μ
e−atb
t1
t
eaxbdx
dt
c2 γ t1
f μ
t t1
βT−xdx dt T
γ t γ
gxβT−xdx dt
f μ
T γ
γ t1
βT−xdx dt
c3 μ 0
fteatbdtf μ
t1
μ
eatbdt− μ
0
ftdt−f μ
t1−μ
c4
f
μ
γ t1
1−βT−t dt T
γ
1−βT−t gtdt
.
3.23
Case 3μ < γ < t1< T. In this case,3.1reduces to the following three:
dIt
dt abtb−1It −ft, 0≤t≤μ , I μ−
I μ
, dIt
dt abtb−1It −f μ
, μ≤t≤γ, I γ−
I γ
,
dIt
dt abtb−1It −gt, γ ≤t≤t1, It1 0.
3.24
Equation3.2leads to the following:
dIt
dt −gtβT−t, t1≤t≤T, It1 0. 3.25 Their solutions are, respectively,
It e−atb
μ t
fxeaxbdxf μ
γ μ
eaxbdx t1
γ
gxeaxbdx
, 0≤t≤μ, 3.26
It e−atb
f μ
γ t
eaxbdx t1
γ
gxeaxbdx
, μ≤t≤γ, 3.27
It e−atb
t1
t
gxeaxbdx, γ ≤t≤t1, 3.28
It − t
t1
gxβT−xdx, t1≤t≤T. 3.29
The total amount of deteriorated items during0, t1is
DI0− t1
0
Dtdt
μ
0
fxeaxbdxf μ
γ μ
eaxbdx t1
γ
gxeaxbdx
− μ
0
ftdt−f μ
γ−μ
− t1
γ
gtdt.
3.30
The total inventory carried during the interval0, t1, using3.26,3.27, and3.28is
I1 t1
0
Itdt μ
0
Itdt γ
μ
Itdt t1
γ
Itdt
μ
0
e−atb
μ t
fxeaxbdxf μ
γ μ
eaxbdx t1
γ
gxeaxbdx
dt
γ
μ
e−atb
f μ
γ t
eaxbdx t1
γ
gxeaxbdx
dt t1
γ
e−atb
t1
t
gxeaxbdx
dt.
3.31
The time-weighted backorders during the intervalt1, Tare
I2 T
t1
−Itdt T
t1
t t1
gxβT−xdx. 3.32
The lost sales in the intervalt1, Tare
L T
t1
1−βT−t
gtdt. 3.33
The inventory cost for this case is TC3t1 c1I1c2I2c3Dc4L
c1
μ 0
e−atb
μ t
fxeaxbdxf μ
γ μ
eaxbdx t1
γ
gxeaxbdx
dt
γ
μ
e−atb
f μ
γ t
eaxbdx t1
γ
gxeaxbdx
dt t1
γ
e−atb
t1
t
gxeaxbdx
dt
c2 T t1
t t1
gxβT−tdx
c4 T t1
1−βT−t gtdt
c3 μ 0
fxeaxbdxf μ
γ μ
eaxbdx t1
γ
gxeaxbdx
− μ
0
ftdt−f μ
γ−μ
− t1
γ
gtdt
.
3.34 Finally the total cost function of the system over0, Ttakes the following form:
TCt1
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
TC1t1, ift1≤μ, TC2t1, ifμ < t1< γ, TC3t1, ifγ≤t1.
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, functionTCt1. 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 t1, say t∗1, which minimizes the total cost function. Although the optimality procedure requires the constrained optimization of the functionsTC1t1,TC2t1, andTC3t1, we will, firstly, search for their unconstrained minimum. The first- and second-order derivatives of TC1t1,TC2t1,andTC3t1are, respectively,
dTC1t1
dt1 ft1ht1, d2TC1t1
dt12 dft1
dt1 ht1 ft1dht1 dt1 , dTC2t1
dt1 f μ
ht1, d2TC2t1
dt12 f
μdht1 dt1 , dTC3t1
dt1 gt1ht1, d2TC3t1
dt12 dgt1
dt1 ht1 gt1dht1 dt1 ,
3.36
where
ht1 c1eat1b
t1
0
e−atbdtc3
eat1b−1
−c2T−t1βT−t1−c4
1−βT−t1
. 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∗1fromht1 0.
Step 2. Ift∗1≤μ, then the optimal order quantity is given by
Q∗ t
∗1
0
fteatbdt μ
t∗1
βT−tftdtf μ
γ μ
βT−tdt T
γ
βT−tgtdt 3.38
and the total cost is given byTC1t∗1.
Ifμ < t∗1 < γ,then the optimal order quantity is given by
Q∗ μ
0
fteatbdtf μ
t∗1 μ
eatbdtf μ
γ t∗1
βT−tdt T
γ
gtβT−tdt 3.39
and the total cost is given byTC2t∗1.
Ifγ < t∗1 < T,then the optimal order quantity is given by
Q∗ μ
0
fteatbdtf μ
γ μ
eatbdt t
∗1
γ
gteatbdt T
t∗1
gtβT−tdt 3.40
and the total cost is given byTC3t∗1.
Remark 3.1. The previous analysis shows thatt∗1is independent from the demand rateDt.
This very interesting result agrees with the classical result, in many order level inventory systems, that the pointt∗1is independent from the demand rateNaddor31, page 67.
3.2. The Special Case βx 1anda0
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:
TCt1
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩ c1
t1
0 t1
t
fxdx dtc2
μ t1
t t1
fxdx dtc2f μ
γ μ
t−μ dtc2
γ μ
μ t1
fxdx dt
c2 T γ
t γ
gxdx dt dtc2f μ
γ−μ T−γ
c2
T γ
μ t1
fxdx dt,
t1< μ < γ < T,
c1 μ 0
μ t
fxdx dtc1 μ 0
t1
μ
f μ
dx dtc1 t1
μ t1
t
f μ
dx dtc2 γ t1
t t1
f μ
dx dt
c2 T γ
t γ
gxdxdtc2
T γ
γ t1
f μ
dx dt, μ < t1< γ < T,
c1 μ 0
μ t
fxdx dtc1 μ 0
γ μ
f μ
dxdtc1 μ 0
t1
γ
gxdxdtc1 γ μ
γ t
f μ
dx dt
c1 γ μ
t1
γ
gxdx dtc1
t1
γ t1
t
gxdx dtc2
T t1
t t1
gxdx dt,
μ < γ < t1< T.
3.41
Following the previous procedure for the optimal replenishment policy, the optimal value of t1, sayt∗1, is given by the very simple and known, in classical order level inventory system Naddor31, equation:
t∗1 c2T c1c2
. 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 t1a replenishment brings the inventory level up toS. Demand and deterioration of the items deplete the inventory level during the periodt1, Tuntil this falls to zero attT. Again the three casest1< μ < γ < T,μ < t1< γ < T,andμ < γ < t1< T must be examined.
Case 4t1< μ < γ < T. The inventory level,It, 0≤t≤T satisfies the following differential equations:
dIt
dt −ftβt1−t, 0≤t≤t1, I0 0, dIt
dt abtb−1It −ft, t1≤t≤μ, I μ−
I μ
, dIt
dt abtb−1It −f μ
, μ≤t≤γ, I γ−
I γ
, dIt
dt abtb−1It −gt, γ≤t≤T, IT 0.
4.1
The solutions of4.1, are, respectively,
It − t
0
fxβt1−xdx, 0≤t≤t1, 4.2
It e−αtb
μ t
eαxbfxdxf μ
γ μ
eαxbdx T
γ
eαxbgxdx
, t1≤t≤μ, 4.3 It e−αtb
f
μ
γ t
eαxbdx T
γ
eαxbgxdx
, μ≤t≤γ, 4.4
It e−atb
T t
eaxbgxdx, γ≤t≤T. 4.5
The total amount of deteriorated units duringt1, Tis
De−atb1
μ t1
fxeaxbdxf μ
γ μ
eαxbdx T
γ
eαxbgxdx
− μ
t1
fxdx− γ−μ
f μ
− T
γ
gxdx.
4.6
The total inventory carried during the intervalt1, Tis found using4.3,4.4, and4.5and is
I1 μ
t1
e−atb
μ t
eαxbfxdxf μ
γ μ
eαxbdx T
γ
eαxbgxdx
dt
γ
μ
e−αtb
f μ
γ t
eαxbdx T
γ
eαxbgxdx
dt T
γ
e−atb
T t
eaxbgxdx
dt.
4.7
Due to4.2the time-weighted backorders during the time interval0, t1are
I2 t1
0 t 0
fxβt1−xdx dt. 4.8
The amount of lost sales during0, t1is
L t1
0
1−βt1−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
TC1t1 c1I1c2I2c3Dc4L c1
μ t1
e−atb
μ t
eαxbfxdxf μ
γ μ
eαxbdx T
γ
eαxbgxdx
dt
c1
γ μ
e−αtb
f μ
γ t
eαxbdx T
γ
eαxbgxdx
dt T
γ
e−atb
T t
eaxbgxdx
dt
c2
t1
0 t 0
fxβt1−xdx dtc4
t1
0
1−βt1−t ftdt
c3
e−atb1
μ t1
fxeaxbdxf μ
γ μ
eαxbdx T
γ
eαxbgxdx
− μ
t1
fxdx− γ−μ
f μ
− T
γ
gxdx
.
4.10
Case 5μ < t1< γ < T. The inventory level,It, 0≤t≤T satisfies the following differential equations:
dIt
dt −ftβt1−t, 0≤t≤μ, I0 0, dIt
dt −f μ
βt1−t, μ≤t≤t1, I μ−
I μ
, dIt
dt abtb−1It −f μ
, t1≤t≤γ, I γ−
I γ
, dIt
dt abtb−1It −gt, γ≤t≤T, IT 0.
4.11
The solutions of4.11, are, respectively,
It − t
0
fxβt1−xdx, 0≤t≤μ,
It − μ
0
fxβt1−xdx−f μ
t μ
βt1−xdx, μ≤t≤t1,
It e−αtb
f μ
γ t
eαxbdx T
γ
eαxbgxdx
, t1≤t≤γ,
It e−atb
T t
eaxbgxdx, γ ≤t≤T.
4.12
The total cost of this case is obtained with a similar way of the previous cases and is,
TC2t1 c1
γ t1
e−αtb
f μ
γ t
eαxbdx T
γ
eαxbgxdx
dt
T
γ
e−atb
T t
eaxbgxdx
dt
c3
e−atb1
γ t1
f μ
eaxbdx T
γ
eaxbgxdx
−f μ
γ−t1
− T
γ
gxdx
c2 μ 0
t 0
fxβt1−xdx
dt
t1
μ μ 0
fxβt1−xdxf μ
t μ
βt1−xdx
dt
c4 μ 0
1−βt1−t
ftdtf μ
t1
μ
1−βt1−t dt
.
4.13
Case 6μ < γ < t1< T. The inventory level,It, 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
βt1−t, μ≤t≤γ, I μ−
I μ
, 4.15
dIt
dt −gtβt1−t, γ≤t≤t1, I γ−
I γ
, 4.16
dIt
dt abtb−1It −gt, t1≤t≤T, IT 0. 4.17
The solutions of4.14,4.15,4.16, and4.17, are, respectively,
It − t
0
fxβt1−xdx, 0≤t≤μ,
It − μ
0
fxβt1−xdx−f μ
t μ
βt1−xdx, μ≤t≤γ,
It − t
γ
gxβt1−xdx− μ
0
fxβt1−xdx−f μ
γ μ
βt1−xdx, γ≤t≤t1,
It e−atb
T t
eaxbgxdx, t1≤t≤T.
4.18
The total cost of this case is obtained with a similar way of the previous cases and is,
TC3t1 c1 T t1
e−αtb
T t
eaxbgxdx dt
c3
e−atb1
T t1
eaxbgxdx− T
t1
gxdx
c2 μ 0
t 0
fxβt1−xdx
dt γ
μ μ 0
fxβt1−xdxf μ
t μ
βt1−xdx
dt
c2 t1
γ t γ
gxβt1−xdx μ
0
fxβt1−xdxf μ
γ μ
βt1−xdx
dt
c4 μ 0
1−βt1−t
ftdtf μ
γ μ
1−βt1−t dt t1
γ
1−βt1−t gtdt
. 4.19
Finally the total cost function of the system over0, Ttakes the following form:
TCt1
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
TC1t1, ift1≤μ, TC2t1, ifμ < t1< γ, TC3t1, ifγ≤t1.
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, functionTCt1. 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, sayt∗1, which minimizes the total cost function. Taking the first-order derivative ofTC1t1, sayK1t1, and equating it to zero gives:
K1t1 −
c1c3αbtb−11 e−αtb1
μ t1
eαxbfxdxf μ
γ μ
eαxbdx T
γ
eαxbgxdx
t1
0
c2βt1−t c2t1−tβt1−t−c4βt1−t
ftdt0.
4.21
Ift∗1is a root of4.21, for this root the second-order condition for minimum is
c1c3αb t∗1b−1
−c3b−1 t∗1−1
αb t∗1b−1
e−αt∗b1
× μ
t∗
1
eαxbfxdxf μ
γ μ
eαxbdx T
γ
eαxbgxdx
c1c3αb t∗1b−1
f t∗1
c2−c4β0 f
t∗1 t
∗1
0
2c2β t∗1−t
c2
t∗1−t β
t∗1−t
−c4β t∗1−t
ftdt >0.
4.22
So, if4.22holds andt∗1 ≤μ, then the value of order level,S, is
S∗I t∗1
e−αt∗1b
μ t∗1
eαxbfxdxf μ
γ μ
eαxbdx T
γ
eαxbgxdx
, 4.23
