A continuous review inventory model, with deteriorating items, time-varying demand, linear...

A continuous review inventory model, with deteriorating items,time-varying demand, linear replenishment cost, partially

time-varying backlogging

K. Skouri, S. Papachristos *

Department of Mathematics, Statistics and O.R. Section, University of Ioannina, 45110 Ioannina, Greece

Received 23 June 2000; received in revised form 31 May 2001; accepted 2 July 2001

Abstract

In this paper we study a continuous review inventory model. Five costs are considered as significant:deterioration, holding, shortage, opportunity cost due to the lost sales and the replenishment cost perreplenishment which is linear dependent on the lot size. The deterioration of items occurs at a fixed rateindependent of time. Demand rate is described by any logconcave function of time, which satisfies very mildconditions. The model allows for partial backlogging. The backlogging rate is an exponentially decreasing,time-dependent function specified by a parameter. For this model we derive results, which ensure the ex-istence of a unique optimal policy and we propose an algorithm to find it. Numerical examples are given toillustrate the application of the algorithm. This paper extends the model studied by Bhunia and Maiti[Appl. Math. Model. 23 (1999) 301] by considering a more general demand rate function, introducingpartial backlogging and relaxing the condition of replenishment cycles of equal lengths. � 2002 ElsevierScience Inc. All rights reserved.

Keywords: Inventory; Deterioration; Replenishment cost; Partial backlogging; Logconcave demand

1. Introduction

The deterioration of many items during storage period is a real fact. Ghare and Schrader [1]have studied an inventory model taking into account the effect of deterioration of items in storage.In their model they introduced a constant deterioration rate, while demand rate was also taken tobe constant. Covert and Phillip [2], Tadikamalla [3], and Shah [4] extended Ghare and Schrader’s

Applied Mathematical Modelling 26 (2002) 603–617

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*Corresponding author. Tel.: +30-651-98263; fax: +30-651-98207.

E-mail address: [email protected] (S. Papachristos).

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PII: S0307-904X(01)00071-3

model by relaxing the assumption of constant deterioration rate and introducing other modes ofdeterioration.

Later, Dave and Patel [5] developed an inventory model with deterministic but linearlychanging demand rate, constant deterioration rate and finite planning horizon. Sachan [6] ex-tended Dave and Patel’s model to allow for shortages. Datta and Pall [7] presented an EOQ modelconsidering variables deterioration and power demand pattern. Research on models with dete-riorating items, time-varying demand and shortages continues with Goswami and Glaudhuri [8],Benkherouf [9], Hariga [10], Chakrabarti and Chaudhri [11], Hariga and Alyan [12], Teng et al.[13]. The common characteristic of all the above papers is that they allow for shortages whileunsatisfied demand is completely backlogged.

Wee andMercan [14] relaxed this assumption. They considered an inventory model over a finite-planning horizon with constant demand and deterioration rates. Additionally they assumed thatonly a fraction b, 0 < b < 1, of demand during the stockout period is backlogged. Other re-searchers who used the idea of partial backlogging areWee [15,16], Chang and Dye [17] and others.

In a recent paper Bhunia and Maiti [18] developed a deterministic inventory model over a finiteplanning horizon with constant rate of deterioration, linearly increasing demand, and completebacklogging of the excess demand. The new contribution of their paper was the introduction of areplenishment cost function, which is linear with respect to the lot size. The authors consideredpolicies with replenishment cycles of equal length. Taking into account the assumption of linearlyincreasing demand, this seems to be a rather restrictive condition, because it narrows the set of alladmissible policies.

In this paper, we extend and modify the model of Bhunia and Maiti [18] in three directions.Firstly, we suppose that the demand rate is described by any logconcave function of time. Sec-ondly, we assume a partial backlogging rate, which is a time-dependent exponentially decreasingfunction, containing a parameter. Different values of this parameter can give models ranging fromcomplete backlogging up to almost complete loss of the unsatisfied demand. Thirdly, we relax therestrictive assumption of replenishment cycles of equal lengths and we allow shortages in the lastcycle, thus enriching the set of all admissible policies. We must mention that the cycles of themodel of Bhunia and Maiti [18] start with replenishment, here we adopt the formulation of Goyalet al. [19] where cycles start with shortage.

This paper is organized as follows. In Section 2 we give the assumptions of the model and thenotation used throughout the paper. We continue with Section 3 where we present the mathe-matical formulation of the model. In Section 4 we establish the theoretical results, theorems andlemmas, which guarantee the existence of a unique optimal plan, and we present the algorithm,which can be used to find the optimal policy. This paper continuous with Section 5 where theproposed algorithm is demonstrated by a numerical example. Comments and concluding remarksare given in Section 6. This paper closes with an Appendix A where we provide the proofs forsome of the results stated in previous sections.

2. Assumptions and notation

In this section we present the notation used throughout this paper and we state the assumptionsunder which our continuous review inventory model is developed.

604 K. Skouri, S. Papachristos / Appl. Math. Modelling 26 (2002) 603–617

2.1. Assumptions

1. The planning horizon of the system is finite and taken as H time units. The initial and finalinventory levels during the planning horizon are both set to zero.

2. Replenishment is instantaneous (replenishment rate is infinite).3. The lead-time is zero.4. The on-hand inventory deteriorates at a constant rate h (0 < h < 1) per time unit. The deteri-

orated items are withdrawn immediately from the warehouse and there is no provision forrepair or replacement.

5. The rate of demand at time, t 2 ½0;H �, is a continuous, logconcave function f ðtÞ with f 0ðtÞ 6¼ 0for all t.

6. The system allows for shortages in all cycles and each cycle starts with shortages.7. Unsatisfied demand is backlogged at a rate expðaxÞ, where x is the time up to the next replen-

ishment and a parameter with, 0 < a < C2=ðC þ C2H C4Þ.

2.2. Notation

3. The mathematical formulation of the model

A realization of the inventory level in the system is given in Fig. 1. The depletion of the in-ventory level during the interval ½ti; si�, of the ith replenishment cycle, is due to the joint effect ofdemand and deterioration. Hence the differential equation which describes the variation of in-ventory level, IðtÞ, with respect to time t is

dIðtÞdt

¼ hIðtÞ f ðtÞ; ti 6 t < si ð1Þ

with boundary condition IðsiÞ ¼ 0; i ¼ 1; . . . ; n.

n the number of replenishment cycles during the planning horizonsi time at which shortage starts during the ith cycles, i ¼ 1; . . . ; nti time at which the ith replenishment is made, i ¼ 1; . . . ; nqi variables replenishment lot size ordered at instant ti; i ¼ 1; . . . ; nAþ Cqi the replenishment cost for the ith cycle, where A is a fixed set up replenishment cost and

C is the additional replenishment cost paid per unit of ordered itemC1 holding cost per unit of stock carried per unit timeC2 shortage cost per unit of shortage per unit timeC3 deterioration cost per unit of deteriorated itemC4 opportunity cost due to lost sales per unit of lost salesCIi the amount of inventory carried during the ith cycleDIi the amount of deteriorated items during the ith cycleSIi the amount of units in shortages during the ith cycleLIi the amount of lost sales units during the ith cycleIðtÞ the inventory level at time t

K. Skouri, S. Papachristos / Appl. Math. Modelling 26 (2002) 603–617 605

The solution of (1) is

IðtÞ ¼ eht

Z si

tehuf ðuÞdu; ti 6 t < si: ð2Þ

From (2) we get that, the amount of inventory carried during the ith cycle is given by

CIi ¼1

h

Z si

ti

ðehðttiÞ 1Þf ðtÞdt: ð3Þ

The amount of deteriorated items during the ith cycle is

DIi ¼ hCIi ¼Z si

ti

ðehðttiÞ 1Þf ðtÞdt: ð4Þ

The variation of the inventory level, IðtÞ, at any time, t, in the interval ½si1; ti� is described bythe differential equation

dIðtÞdt

¼ eaðtitÞf ðtÞ; si1 6 t < ti ð5Þ

with boundary condition Iðsi1Þ ¼ 0; i ¼ 1; . . . ; n.The solution of (5) is

IðtÞ ¼ Z t

si1

eaðtiuÞf ðuÞdu; si1 6 t < ti: ð6Þ

From (6) we obtain that, the amount of shortage during the ith cycle is given by

SIi ¼Z ti

si1

Z t

si1

eaðtiuÞf ðuÞdudt ¼Z ti

si1

eaðtiuÞðti uÞf ðuÞdu: ð7Þ

The amount of lost sales during the ith cycle is

LIi ¼Z ti

si1

ð1 eaðtiuÞÞf ðuÞdu: ð8Þ

Fig. 1. Graphical representation of inventory.


The lot size, qi, for the ith cycle, is made up by the sum of maximum and minimum stock levelsrecorded over the cycle. So by, using Eqs. (6) and (2), we obtain it as

qi ¼Z ti

si1

eaðtiuÞf ðuÞduþZ si

ti

ehðutiÞf ðuÞdu: ð9Þ

Now we have all necessary quantities to formulate the total inventory cost function for anypolicy, which has n replenishment cycles. This is the sum of the following costs: replenishment,holding, deterioration, shortage and opportunity, and is given by

TCðn; si; tiÞ ¼Xn1

i¼0

ðAþ Cqiþ1Þ þ C1

Xn1

i¼0

CIiþ1 þ C2

Xn1

i¼0

SIiþ1 þ C3

Xn1

i¼0

DIiþ1 þ C4

Xn1

i¼0

LIiþ1

¼ nAþ CXn1

i¼0

Z siþ1

tiþ1

ehðttiþ1Þf ðtÞdt þ C1 þ C3hh

Xn1

i¼0

Z siþ1

tiþ1

ðehðttiþ1Þ 1Þf ðtÞdt

þXn1

i¼0

Z tiþ1

si

ðC4 C4eaðtiþ1tÞ þ Ceaðtiþ1tÞ þ C2e

aðtiþ1tÞðtiþ1 tÞÞf ðtÞdt: ð10Þ

The minimization of this function w.r.t. n, ti, si, will give the optimal policy of the problem.

4. The optimal replenishment policy

In this section we derive all the results which ensure the existence of the optimal values ti, si, forany policy with n replenishments, we give the algorithm which can be used to find them and weprove the convexity of the total cost function, TCðn; si; tiÞ, with respect to n, a fact which guar-antees the existence of the overall optimal policy.

The continuity of f ðtÞ implies that TCðn; si; tiÞ is a continuous function of si, ti and its first- andsecond-order partial derivatives exist. Taking first-order derivatives of TCðn; si; tiÞ with respect toti, si and equating them to zero we obtain

ðCh þ C1 þ C3hÞZ si

ti

ehðttiÞf ðtÞdt ¼Z ti

si1

ðC2 þ aC4 aC aC2ðti tÞÞeaðtitÞf ðtÞdt;

i ¼ 1; . . . ; n; ð11Þ

CehðsitiÞ þ C1 þ C3hh

½ehðsitiÞ 1� ¼ Ceaðtiþ1siÞ þ C2ðtiþ1 siÞeaðtiþ1siÞ þ C4ð1 eaðtiþ1siÞÞ;

i ¼ 1; . . . ; n 1: ð12ÞIn Appendix A we prove that, if t�i , s

�i , i ¼ 1; 2; . . . ; n, is a solution of the above system of

equations, this satisfies the second-order conditions for a minimum. Moreover we shall prove thatthe system of equations (11), (12) has a unique solution in ti, si on the interval ½0;H � and so theabove minimum is a global minimum.

We shall now present the methodology used to solve the system of equations (11) and (12) andwe shall prove that the obtained solution is unique. It is easy to see that, once ti is known, s1ðt1Þcan be obtained from (11). Then t2 ðt1Þ can be obtained from (12) and following this alternateprocedure we can find s2ðt1Þ; . . . ; snðt1Þ. The choice of t1 plays a key role, as all subsequent


parameters become functions of this. It is obvious that the set of ti, si values so produced will bea solution of the above system only if the starting point t1 is such that snðt1Þ ¼ H . This gives riseto the problem of the existence of some t1, which satisfies this condition.

We shall now establish the results, which ensure that t1 can be chosen so as to satisfy thiscondition. First, we present two important lemmas and then we establish the main result.

Lemma 1. For t1 ¼ 0 or H we have snð0Þ < H and snðHÞ > H , respectively.

Proof. If t1 ¼ 0 from (11) we have s1 ¼ 0 and so (12) becomes

C ¼ Ceat2 þ C2eat2 t2 þ C4 C4e

at2 :

Let us set

hðt2Þ ¼ Ceat2 þ C2eat2 t2 þ C4 C4e

at2 C

and consider the equation hðt2Þ ¼ 0. Obviously hð0Þ ¼ 0 and so the value t12 ¼ 0 is a root of thisequation.

The derivative of hðt2Þ is

dhðt2Þdt2

¼ eat2ðC2 þ aC4 aC aC2t2Þ

and it is positive for t2 < d and negative for t2 > d, where

d ¼ 1

aþ C4

C2

CC2

:

This implies that the function hðt2Þ is increasing for t2 2 ½0; d� and decreasing for t2 2 ½d;þ1Þ. Soany other root of hðt2Þ ¼ 0, say t22, will be greater than d. But since a < C2=ðC þ C2H C4Þ wehave t22 > H . Therefore, the only root of the equation hðt2Þ ¼ 0 in the interval ½0;H � is t2 ¼ 0.Applying the same reasoning we find s2 ¼ 0; t3 ¼ 0; . . . ; sn ¼ 0.

If t1 ¼ H from Eq. (11) we have that s1ðHÞ > H . Eq. (12) then yields

Cehðs1HÞ þ C1 þ C3hh

½ehðs1HÞ 1� ¼ Ceaðt2s1Þ þ C2ðt2 s1Þeaðt2s1Þ þ C4ð1 eaðt2s1ÞÞ:

If we set

gðt2Þ ¼ Cehðs1HÞ þ C1 þC3hh

½ehðs1HÞ 1� Ceaðt2s1Þ C2ðt2 s1Þeaðt2s1Þ C4ð1 eaðt2s1ÞÞ;

then for t2 ¼ s1 we have

gðs1Þ ¼ Cðehðs1HÞ 1Þ þ C1 þ C3hh

½ehðs1HÞ 1� > 0:

The derivative of gðt2Þ isdgðt2Þdt2

¼ ðC2 þ aC4 aC aCðt2 s1ÞÞeaðt2s1Þ < 0

and so gðt2Þ is a strictly decreasing function of t2. Consequently t2 > s1 and so t2 > H . Continuingthis process we find that s2 > H ; t3 > H ; . . . ; sn > H , i.e. snðHÞ > H . �


Lemma 2. The function snðt1Þ is increasing with respect to the variables t1.

Proof. Since f ðtÞ is logconcave, f ðtÞ=f 0ðtÞ is strictly increasing in t, for ti 6 t6 si, and so we havef 0ðtÞ6 ðf 0ðtiÞ=f ðtiÞÞf ðtÞ. Multiplying this inequality by ehðttiÞ and adding the term hehðttiÞf ðtÞ > 0to both sides of the resulting inequality, we obtain

ehðttiÞf 0ðtÞ þ hehðttiÞf ðtÞ6 ehðttiÞ f0ðtiÞf ðtiÞ

f ðtÞ þ hehðttiÞf ðtÞ; ti 6 t6 si: ð13Þ

Now we multiply both sides of (13) by Ch þ C1 þ C3h and then we integrate with respect to t in theinterval ½ti; si�. After some elementary operations we obtain the inequality

ðCh þ C1 þ C3hÞ½ehðsitiÞf ðsiÞ f ðtiÞ�6 ðCh þ C1 þ C3hÞf 0ðtiÞf ðtiÞ

�þ h

� Z si

ti

ehðttiÞf ðtÞdt: ð14Þ

Due to (11) the above inequality becomes

ðCh þ C1 þ C3hÞ ehðsitiÞf ðsiÞ�

f ðtiÞ hZ si

ti

ehðttiÞf ðtÞdt�

6f 0ðtiÞf ðtiÞ

Z ti

si1

eaðtitÞf ðtÞ½C2 þ aC4 aC aC2ðti tÞ�dt

¼Z ti

si1

f 0ðtiÞf ðtiÞ

f ðtÞeaðtitÞ½C2 þ aC4 aC aC2ðti tÞ�dt: ð15Þ

We know that

a <C2

C þ C2H C4

andf 0ðtiÞf ðtiÞ

<f 0ðtÞf ðtÞ

for si1 6 t6 ti. Taking these inequalities into account in the right-hand side of (15) and thenintegrating by parts we obtain

ðCh þ C1 þ C3hÞ ehðsitiÞf ðsiÞ�

f ðtiÞ hZ si

ti

ehðttiÞf ðtÞdt�

6 ðC2 þ aC4 aCÞf ðtiÞ eaðtisi1ÞðC2 þ aC4 aC aC2ðti si1ÞÞf ðsi1Þ

Z ti

si1

aeaðtitÞf ðtÞð2C2 þ aC4 aC aC2ðti tÞÞdt: ð16Þ

Let us set Mi ¼ si ti and Ki ¼ ti si1. Obviously Mi, Ki are functions of ti. Substituting Mi, Ki to(11) and differentiating (11) with respect to ti for i ¼ 1; . . . ; n, we obtain the system of equations

ðCh þ C1 þ C3hÞ ehMif ðsiÞdMi

dt1

�þ dtidt1

ehMif ðsiÞ�

f ðtiÞ hZ si

ti

ehðttiÞf ðtÞdt��

¼ ðC2 þ aC4 aCÞ dtidt1

f ðtiÞ dtidt1

� dKi

dti

�ðC2 þ aC4 aC aC2KiÞeaKif ðsi1Þ

adtidt1

Z ti

si1

eaðtitÞf ðtÞð2C2 þ aC4 aC aC2ðti tÞÞdt; i ¼ 1; . . . ; n: ð17Þ


For i ¼ 1 we have K1 ¼ t1 s0 ¼ t1 and since dt1=dt1 ¼ 1. By using (17) for i ¼ 1 and replacingthese derivatives we obtain

ðCh þ C1 þ C3hÞehM1f ðs1ÞdM1

dt1

¼ dK1

dt1eaK1f ðs0Þ þ ðC2 þ aC4 aCÞf ðt1Þ eaK1ðC2 þ aC4 aC aC2K1Þf ðs0Þ

aZ t1

s0

eaðt1tÞð2C2 þ aC4 aC aC2ðt1 tÞÞf ðtÞdt ðCh þ C1 þ C3hÞ

f ðs1ÞehM1

� f ðt1Þ h

Z s1

t1

ehðtt1Þf ðtÞdt�: ð18Þ

Using inequality (16) and the fact that a < C2=ðC þ C2H C4Þ we conclude that the right-handside of (18) is a positive number, which implies that dM1=dt1 > 0.

Next, differentiating (12) with respect to t1 we have

ðCh þ C1 þ C3hÞehM1dM1

dt1¼ ðC2 þ aC4 aC aC2K2Þ

dK2

dt1eaK2 :

The right-hand side of the above equation is positive and so, dK2=dt1 > 0.Using the same technique we can prove that dKi=dt1 > 0 and dMi=dt1 > 0 for i ¼ 1; . . . ; n. It is

obvious that snðt1Þ ¼Pn

i¼1ðMi þ NiÞ. The above results show that dsnðt1Þ=dt1 > 0, which impliesthat the function snðt1Þ is strictly increasing with respect to t1. �

Using induction on i we can prove the following

Corollary 1. The functions tiðt1Þ; siðt1Þ; i ¼ 1; 2; . . . ; n are monotonically increasing w.r.t. t1.

The above lemmas lead to the following theorem.

Theorem 1. For the given model there exists a unique value for t�1 2 ½0;H � satisfying the conditionsnðt�1Þ ¼ H .

This theorem guarantees the uniqueness of the optimal replenishment schedules for any policywith n replenishments.

Now we present the theorem, which ensures the existence of a unique optimal number of re-plenishments, n, and consequently the existence of a unique optimal policy for the problem underconsideration.

Theorem 2. The cost, TCðn; s�i ; t�i Þ, of the optimal policy with n replenishments, is a convex functionwith respect to n.

Proof. The proof of this theorem will be established by using dynamic programming. The tech-nique is similar to that used by Teng et al. [13] and Friedman [20]. A policy with n replenishmentcycles and si, ti shortages and ordering points, starts its first cycle at s0 ¼ 0 and ends its last cycle


at time sn ¼ H . The realization of the stock level in the system is monitored and as the ith cycle arecompleted we observe the value of the decision variables si. This can be any value in the intervalð0;HÞ. So, for this process the stage space (points at which we keep records of the process) isf1; 2; . . . ; ng, while the state space (points at which shortages start) at each stage is any value in theinterval ð0;HÞ. Let us set T ðn; 0;HÞ¼ the cost, except the fixed set up, of the optimal policy with nreplenishments with its first cycles starting at time zero and its last one finishing at time H .

Then

TCðn; s�i ; t�i Þ ¼ nAþ T ðn; 0;HÞ: ð19ÞTo prove the convexity it is enough to show that

T ðnþ 1; 0;HÞ T ðn; 0;HÞ > T ðn; 0;HÞ T ðn 1; 0;HÞ: ð20ÞApplying Bellman’s principles of optimality [21] we end up with the following forward dynamicprogramming equation

T ðn; 0;HÞ ¼ Mins2ð0;HÞ

fT ðn 1; 0; sÞ þ Gð1; s;HÞg; ð21Þ

where Gð1; s;H ) is the cost of the last cycle starting at any time s and finishing at H .Recursive application of (21) yields the optimal ith shortage point, s�njðn; 0;HÞ ¼ s�njðn jþ

1; 0; s�njþ1ðn; 0;HÞÞ, j ¼ 1; . . . ; n, for any policy with n orders placed in the interval ½0;H �.For s ¼ H (21) gives T ðn; 0;HÞ < T ðn 1; 0;HÞ. This proves that T ðn; 0;HÞ is strictly de-

creasing function with respect to n. Let us now choose H1;H2 > H so that

s�nðnþ 1; 0;H1Þ ¼ s�nþ1ðnþ 2; 0;H2Þ ¼ H : ð22ÞSince s�nðnþ 1; 0;H1Þ ¼ H , by employing the principles of optimality, we have

T ðnþ 1; 0;H1Þ ¼ Mins2ð0;H1Þ

fT ðn; 0; sÞ þ Gð1; s;H1Þg ¼ T ðn; 0;HÞ þ Gð1;H ;H1Þ: ð23Þ

But this implies that

o½T ðn; 0; tÞ þ Gð1; t;H1Þ�ot

¼ oT ðn; 0; tÞot

þ oGð1; t;H1Þot

t¼H

¼ 0: ð24Þ

So

oT ðn; 0; tÞot

t¼H

¼ oGð1; t;H1Þot

t¼H

¼ CehðHt�nðn;0;HÞÞ�

þ C1 þ C3hh

ðehðHt�nðn;0;HÞÞ 1Þ�f ðHÞ; ð25Þ

where t�nðn; 0;HÞ is the last optimal replenishment time when n orders are placed during the in-terval ½0;H �. Similarly, from s�nþ1ðnþ 2; 0;H2Þ ¼ H , we have

oT ðnþ 1; 0; tÞot

t¼H

¼ oGð1; t;H1Þot

t¼H

¼ CehðHt�nþ1ðnþ1;0;HÞÞ

�þ C1 þ C3h

hðehðHt�nþ1

ðnþ1;0;HÞÞ 1Þ�f ðHÞ: ð26Þ


Subtracting (26) from (25) we obtain

o½T ðn; 0;HÞ T ðnþ 1; 0;HÞ�oH

> 0; ð27Þ

which implies that T ðn; 0;HÞ T ðnþ 1; 0;HÞ is a strictly increasing function with respect to H .From (21) and (23) we obtain

T ðn;0;H1Þ T ðnþ 1;0;H1Þ ¼ Mins2ð0;H1Þ

fT ðn 1;0; sÞ þGð1; s;H1Þg fT ðn;0;HÞ þGð1;H ;H1Þg:

ð28ÞFor s ¼ H , (28) gives

T ðn; 0;H1Þ T ðnþ 1; 0;H1Þ < T ðn 1; 0;H1Þ T ðn; 0;HÞ: ð29ÞSince H < H1 and T ðn; 0;H1Þ T ðnþ 1; 0;HÞ is a strictly increasing function with respect to H , weobtain the relation (20). This implies that TCðn; s�i ; t�i Þ is also convex with respect to n. �

The methodology presented in this paper for the solution of the problem is based on the se-lection of the first replenishment point, t1, so that snðt1Þ ¼ H . One can proceed reversely, that isselect first, the last replenishment point, tn, so that s0ðtnÞ ¼ 0 and then proceed to obtain values forall other parameters involved. Applying a symmetric argument it can be proved that the reverseapproach leads exactly to the same results.

5. Numerical example

To illustrate the algorithm described above we consider the following example. f ðtÞ ¼ 20þ 2t,A ¼ 100, C ¼ 0:2, C1 ¼ 0:5, C2 ¼ 1:5, C3 ¼ 5, C4 ¼ 0:5, h ¼ 0:01, a ¼ 0:08;H ¼ 12. For a ¼ 0 thisexample yields the one used by Bhunia and Maiti [18].

In Table 1 we give the optimal total cost (TC) for different values of n and a. In Table 2 wepresent the optimal replenishment policy, i.e. the optimal values for n; ti; si in case a ¼ 0:08. In

Table 1

Optimal total cost (TC) for policies with given numbers of cycles, n, and different values of a ðC4 6¼ 0Þa n

1 2 3 4

0.08 1064.48 743.831 686.028 711.116

0.05 1098.99 744.852 687.834 713.044

0.02 1129.05 752.365 691.787 714.399

Table 2

The overall optimal replenishment policy for a ¼ 0:08 ðC4 6¼ 0Þn ¼ 3

ti 1.518 5.77374 9.53789

si 4.54621 8.46272 12


Tables 3 and 4 we present respective results in the case C4 ¼ 0, which corresponds to the modelwithout opportunity cost.

6. Concluding remarks

1. In the model studied in this paper if we set the parameter a, of the backlogging rate functionequal to zero, i.e. a ¼ 0, we revert to the model with complete backlogging of the unsatisfieddemand. If we additionally take the demand function, f ðtÞ, to be linear, we obtain the modelstudied by Bhunia and Maiti [18]. However we must mention that the optimal policy found bythese authors belongs to the class of policies with replenishment cycles of equal lengths, while inour model we have relaxed this condition by allowing policies with replenishment cycles of anylength. The optimal cost given by Bhunia and Maiti [18] for their model is 1394 units. The cor-responding cost resulting from our model is 711.475 units. This indicates that the condition ofreplenishment cycles of equal lengths is really very restrictive. The reduction in the total cost issignificant which favors the application of our model.

2. If we set C ¼ 0, a ¼ 0 and h ¼ 0 we obtain the model developed by Hariga [10].3. If we set C ¼ 0, a ¼ 0, h ¼ 0 and f ðtÞ linear, our model yields the one studied by Goyal et al.

[19]. By applying the technique proposed in this paper in the examples used by these authors weobtain the same optimal cost and policy.

4. If, in our model, we set C4 ¼ 0 (the case of the model without opportunity cost) the optimalcost is 711.475 for a ¼ 0, and 683.498 for a ¼ 0:08, indicating that the total optimal cost func-tion is decreasing with respect to the parameter a. This indication is a valid result and can beproved using the relation (10). This implies that in this model with this type of partial backlog-ging, the optimal total cost is always smaller than that with complete backlogging. If C4 6¼ 0, wecannot give a general result concerning the behavior of the total cost in relation to a. Indeed,taking the first derivative in (10) with respect to a its dependence on the values of C;C2 and C4

is revealed. For instance, if we consider the previous examples with C2 ¼ 1:5 and C4 ¼ 4:5, then

Table 3

Optimal total cost (TC) for policies with given numbers of cycles, n, and different values of a ðC4 ¼ 0Þa n

1 2 3 4

0.08 1052.15 739.619 683.498 707.037

0.05 1092.28 742.841 686.988 710.211

0.02 1126.69 751.267 691.731 711.359

Table 4

The overall optimal policy for a ¼ 0:08 ðC4 ¼ 0Þn ¼ 3

ti 1.55 5.80218 9.56305

si 4.54948 8.46617 12


the overall optimal total cost is 695.393 for a ¼ 0 and 711.343 for a ¼ 0:08. Taking C2 ¼ 1:5,and C4 ¼ 0:5 the overall optimal total cost is 692.888 for a ¼ 0 and 686.028 for a ¼ 0:08.

5. The parameter a, which is present in the backlogging rate function, gives the decision makersome flexibility to include in his theoretical model backlogging rate curve, which fits betterto his personal estimation for the true backlogging rate. By choosing a suitable value of thisparameter he can obtain models ranging from complete backlogging, a ¼ 0, up to almostany desirable rate of partial backlogging. Examples of this are given in Table 3.

Acknowledgements

The authors would like to thank the referees for their constructive suggestion to improve thepaper.

Appendix A. Checking the conditions for a minimum of TCðn; si; tiÞ

For convenience let us set TCðn; si; tiÞ ¼ TC. To ensure that the solution of equations (11) and(12) gives a minimum it is sufficient to prove that the associated Hessian matrix has all itsprincipal minors positive. The elements of the Hessian matrix are

H2kþ1;2jþ1 ¼o2TC

otkþ1 otjþ1

; j; k ¼ 0; 1; . . . ; n 1;

H2k;2j ¼o2TC

osk osj; k; j ¼ 0; 1; . . . ; n 1;

H2kþ1;2j ¼o2TC

otkþ1 osj; k ¼ 0; 2; . . . ; n 1; j ¼ 1; . . . ; n;

H2k;2jþ1 ¼o2TC

osk otjþ1

; k ¼ 1; . . . ; n; j ¼ 1; 2; . . . ; n 1:

The non-zero entries of the Hessian matrix are

H2jþ1;2jþ1 ¼o2TC

ot2jþ1

¼ ðCh þ C1 þ C3hÞf ðtjþ1Þ

þ hðCh þ C1 þ C3hÞZ sjþ1

tjþ1

ehðttjþ1Þf ðtÞdt þ ðC2 þ aC4 aCÞf ðtjþ1Þ

aZ tjþ1

sj

eaðtjþ1tÞð2C2 þ aC4 aC aC2Þðtjþ1 tÞf ðtÞdt;

j ¼ 0; . . . ; n 1;


H2j;2j ¼o2TC

os2j

¼ ½ðCh þ C1 þ C3hÞehðsjtjÞ þ ðC2 þ aC4 aC aC2ðtjþ1 sjÞÞeaðtjþ1sjÞ�f ðsjÞ;j ¼ 1; 2; . . . ; n;

H2kþ1;2j ¼ H2j;2kþ1 ¼o2TC

otkþ1osj¼ ðCh þ C1 þ C3hÞehðsjtkþ1Þf ðsjÞ;

k ¼ 0; 2; . . . ; n 1; j ¼ k þ 1;

H2jþ1;2k ¼ H2k;2jþ1 ¼o2TC

oskotjþ1

¼ eaðtjþ1skÞðC2 þ aC4 aC aC2ðtjþ1 skÞÞf ðskÞ;

k ¼ 1; . . . ; n 1; j ¼ k:

We observe that H2jþ1;2jþ1 > 0, H2j;2j > 0;H2kþ1;2j < 0 and H2k;2jþ1 < 0.Let Mk be the principal minor of order k. Then

M1 ¼ H1;1 > ðCh þ C1 þ C3hÞehðsjtjÞf ðsjÞ þ ðC2 þ aC4 aC aC2ðtj sj1ÞÞ eaðtjsj1Þf ðsj1Þ > 0:

This follows using inequality (16).

M2 ¼o2TC

os21

o2TC

ot21 o2TC

os1 ot1

o2TC

ot1 os1> M1e

aðt1s0ÞðC2 þ aC4 aC aC2ðt1 s0ÞÞ½f ðs1Þ�þ ðCh þ C1 þ C3hÞehðs1t1Þeaðt1s0ÞðC2 þ aC4 aC aC2ðt1 s0ÞÞf ðs1Þf ðs0Þ > 0

or, equivalently

M2 þo2TC

ot2 os1M1 > ðCh þ C1 þ C3hÞehðs1t1Þeaðt1s0ÞðC2 þ aC4 aC aC2ðt1 s0ÞÞf ðs1Þf ðs0Þ

> 0:

So we have shown that

M1 > 0; M2 > 0 and M2 þo2TC

ot2 os1M1 > 0:

It is not difficult to verify that the principal minors of higher order satisfy the following re-currence relationships

M2jþ1 ¼o2TC

ot2jþ1

M2j o2TC

otjþ1 osj

� �2

M2j1; j ¼ 1; 2; . . . ; n 1; ðA:1Þ

M2j ¼o2TC

os2jM2j1

o2TC

otj osj

� �2

M2j2; j ¼ 2; . . . ; n: ðA:2Þ


Also we observe that

o2TC

ot2jþ1

> o2TC

otjþ1 osj o2TC

otjþ1 osjþ1

ando2TC

os2jþ1

¼ o2TC

otjþ1 osj o2TC

otjþ1 osjþ1

:

From (A.1) and (A.2) we obtain

M2jþ1 þo2TC

otjþ1 osjþ1

M2j > o2TC

otjþ1 osjM2j

�þ o2TC

otjþ1osjM2j1

�; ðA:3Þ

M2j þo2TC

otjþ1 osjM2j1 ¼ o2TC

otj osjM2j1

�þ o2TC

otj osjM2j2

�: ðA:4Þ

Now, by applying inequality (A.3) for j ¼ 1, we getM3 > 0. This result, accompanied by Eq. (A.4)for j ¼ 2 results to M4 > 0. Using induction we can prove that the principal minors of any orderare positive.

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