Directory UMM :Data Elmu:jurnal:O:Operations Research Letters:Vol27.Issue4.2000:
Operations Research Letters 27 (2000) 175–184
www.elsevier.com/locate/dsw
An optimal replenishment policy for deteriorating items with
time-varying demand and partial–exponential
type – backlogging
S. Papachristosa; ∗ , K. Skourib
a Department
of Mathematics, Statistics and Operational Research Section, University of Ioannina, 45110 Ioannina, Greece
b Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
Received 1 July 1999; received in revised form 1 June 2000; accepted 1 June 2000
Abstract
We study a continuous review inventory model over a nite-planning horizon with deterministic varying demand
and constant deterioration rate. The model allows for shortages, which are partially backlogged at a rate which varies
c 2000 Elsevier Science B.V.
exponentially with time. For this model an optimal replenishment policy is established.
All rights reserved.
Keywords: Inventory; Lot sizing; Deterioration; Shortage; Partial backlogging
1. Introduction
The deterioration of many items during storage period is a real fact. Foods, pharmaceuticals, chemicals,
blood, drugs are a few examples of such items. Chare and Schrader [9], rst, proposed an inventory model
having a constant rate of deterioration and a constant rate of demand over a nite-planning horizon. Covert
and Phillip [4] extended Chare and Schrader’s model by considering variable rate of deterioration. Shah [15]
suggested a further generalization of all these models by allowing shortages and using a general distribution
for the deterioration rate. The common characteristic of these articles was that demand rate was taken as a
constant over the whole planning horizon.
The assumption of a constant demand rate is usually valid in the mature stage of the life cycle of the
product. In the growth and=or end stage of the product life cycle demand rate can be well approximated by
a linear function. Donaldson [7] developed an exact replenishment policy concerning this case.
Dave and Patel [6] studied an inventory model with deterministic but linearly changing demand rate and
constant deterioration rate over a nite planning horizon. In their formulation they assumed replenishment
Corresponding author. Fax: +30-651-98297.
E-mail address: spapachr@cc.uoi.gr (S. Papachristos).
∗
c 2000 Elsevier Science B.V. All rights reserved.
0167-6377/00/$ - see front matter
PII: S 0 1 6 7 - 6 3 7 7 ( 0 0 ) 0 0 0 4 4 - 4
176
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
intervals of equal lengths. Sachan [14] extended Dave and Pattel’s model to allow for shortages. In his
model, each cycle starts with a replenishment and the period of positive inventory is followed by a period
of shortages. Moreover, this replenishment policy requires an extra order to be placed at the end of the
planning horizon to clear all shortages accumulated during the nal cycle. Datta and Pall [5] developed
an EOQ model by introducing a variable deterioration rate and power demand pattern. They considered a
xed-length replenishment cycle and they determined the optimal shortage point and order quantity. Goswami
and Chaudhuri [10] attempted to solve a problem with exponentially deteriorating items, linearly time-varying
demand and shortages. They also assumed replenishment intervals of equal lengths. Their model starts with
replenishment and ends without replenishment (zero nal stock). Benkherouf [2] relaxed the assumption of
equal length for the replenishment cycles assuming a decreasing and logconcave demand rate. He proved
the existence of the optimal policy and presented an iterative procedure to obtain it. Hariga and Alyan [13],
who dealt with the same model, assuming a time-dependent demand rate and no shortages in the last cycle,
presented a heuristic procedure to obtain the replenishment policy. Hariga [12] extended already obtained
results by introducing a logconcave function to describe the demand rate. Chakrabarti and Chaudhurri [3]
studied an inventory model with linearly changing demand rate, constant deterioration rate and shortages over
a nite-planning horizon. In their model they assumed cycles of equal lengths (each cycle starts with shortages)
and the initial and nal inventory levels are both zero. Teng et al. [16] presented an inventory model under
a general-demand function constant deterioration rate and shortages over a nite-planning horizon. Moreover,
their model starts with shortages and ends with shortages. The characteristic of all the above articles is that
they allow shortages, while unsatis ed demand is completely backlogged.
For models where shortages are allowed, complete backlogging, or complete loss of unsatis ed demand are
two extreme cases. So researchers have turned their attention to models that allow partial backlogging, (e.g.
Wee [17,18]).
In this article, we develop an EOQ inventory model over a nite planning horizon, with constant deterioration rate, time-varying demand rate and time-dependent partial backlogging. More explicitly, we suppose
that the rate of backlogged demand increases exponentially as the waiting time for the next replenishment
decreases. We believe that this is a quite reasonable assumption since as the waiting time decreases, more
and more customers are willing to wait to get their orders as soon as the backlogged demand reaches the
system at the next replenishment.
The paper is organized as follows. In Section 2 we give the assumptions of the model and the notation
used. We continue with the mathematical formulation of the model in Section 3. In Section 4 we present
results, theorems and lemmas, which ensure the existence of the optimal plan and we give the algorithm that
can be used to nd the parameters of the optimal policy. The paper continues with Section 5 where we give
a numerical example, explaining the procedure proposed in Section 4. Some comments and conclusions are
given in Section 6. The paper closes with an appendix where we provide proofs for some of the results stated
in previous sections.
2. Assumptions and notation
The inventory model is a continuous review model developed under the following assumptions:
1. The planning horizon of the system is nite and is taken as H time units. The initial and the nal inventory
levels are both zero during this time horizon.
2. Replenishment is instantaneous (replenishment rate is in nite).
3. The lead time is zero.
4. The on hand inventory deteriorates at a constant rate (0 ¡ ¡ 1) per time unit. The deteriorated items
are withdrawn immediately from the warehouse and there is no provision for repair or replacement.
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
177
Fig. 1. Graphical representation of inventory.
5. The rate of demand, f(t); t ∈ [0; H ]; is a continuous, logconcave function of t; with f′ (t) 6= 0 ∀t.
6. The system allows for shortages in all cycles and each cycle starts with shortages.
7. Unsatis ed demand is backlogged at a rate exp(− x); where x is the time up to the next replenishment and
a parameter 0 ¡ ¡ (1=H ).
Nomenclature
C
C1
C2
C3
CIi
DIi
SIi
I (t)
n
si
ti
Ti
the replenishment cost per order.
holding cost per unit of stock carried per unit time.
shortage cost per unit of shortage per unit time.
deterioration cost per unit of deteriorated items.
the amount of inventory carried during the ith cycle.
the amount of deteriorated items during the ith cycle.
the amount of units in shortages during the ith cycle.
the inventory level at time t.
the number of replenishment cycles during the planning horizon.
time at which shortage starts during the ith cycle i = 1; : : : ; n − 1.
time at which the ith replenishment is made i = 1; : : : ; n.
length of the ith cycle
3. Mathematical formulation of the model
The
uctuation of the inventory level in the system is given in Fig. 1. The depletion of inventory during
the interval [ti ; si ]; of the ith replenishment cycle, is due to the joint e ect of the demand and the deterioration
of items. Hence the di erential equation, which describes the variation of inventory level, I (t); with respect
to time t; is
dI (t)
= −I (t) − f(t); ti 6 t ¡ si
dt
with boundary condition I (si ) = 0; i = 1; : : : ; n.
The solution of (1) is
Z si
−t
eu f(u) du; ti 6 t ¡ si :
I (t) = e
t
(1)
(2)
178
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
From (2) the amount of inventory carried during the ith cycle is given by
Z
1 si (t−ti )
(e
− 1)f(t) dt:
CIi =
ti
(3)
The amount of deteriorated items during the ith cycle is
Z si
(e(t−ti ) − 1)f(t) dt:
DIi = CIi =
(4)
ti
The variation of inventory level, I (t); with respect to time, t; during the interval [si−1 ; ti ] can be described by
the following di erential equation:
dI (t)
= −e− (ti −t) f(t); si−1 6 t ¡ ti
dt
with boundary condition I (si−1 ) = 0; i = 1; : : : ; n.
The solution of (5) is
Z t
e− (ti −u) f(u) du; si−1 6 t ¡ ti :
I (t) = −
(5)
(6)
si−1
From (6) the amount of shortage during the ith cycle is given by
Z ti Z t
Z ti
− (ti −u)
e
f(u) du dt =
e− (ti −u) (ti − u)f(u) du:
SIi =
si−1
(7)
si−1
si−1
Now, we have all quantities needed to formulate the total inventory cost as the sum of ordering, holding,
deterioration and shortage cost, for any policy with n replenishments:
TC(n; si ; ti ) = nC + C1
n−1
X
CIi+1 + C2
i=0
n−1
X
SIi+1 + C3
i=0
n−1
C1 + C3 X
= nC +
i=0
Z
si+1
(e
(t−ti+1 )
n−1
X
DIi+1
i=0
− 1)f(t) dt + C2
ti+1
n−1 Z
X
i=0
ti+1
e− (ti+1 −t) (ti+1 − t)f(t) dt:
si
(8)
The goals of this paper are: (1) to present an algorithm which, for a given number of replenishments, n;
can be used to determine the optimal replenishment points, ti∗ , and the optimal shortage points, si∗ and (2) to
obtain the overall optimal policy, i.e. nd the optimal values for the parameters, n; ti and si .
4. The optimal replenishment procedure
In this section, we shall present all the results which will lead to the construction of the algorithm giving the
optimal ti ; si ; values, for any policy with n replenishments. The continuity of f(t) guarantees that TC(n; si ; ti )
is a continuous function of si ; ti and its rst- and second-order partial derivatives exist.
Taking rst-order derivatives of TC(n; si ; ti ) w.r.t. ti ; si ; and equating them to zero we obtain
Z si
Z ti
(t−ti )
e− (ti −t) [1 − (ti − t)]f(t) dt; i = 1; : : : ; n;
(9)
e
f(t) dt = C2
(C1 + C3 )
ti
si−1
C1 + C3 (si −ti )
− 1] = C2 (ti+1 − si )e− (ti+1 −si ) ;
[e
i = 1; : : : ; n − 1:
(10)
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
179
In Appendix A we prove that the solution, ti∗ ; si∗ ; i = 1; 2; : : : ; n; of the above system of equations, satis es
the second-order conditions for a minimum. Moreover, as we shall see, the system of equations (9), (10) has
a unique solution and so the above minimum is a global minimum.
We shall now present the methodology used to solve Eqs. (9) and (10), and after that, we shall prove
that the obtained solution is unique. It is easy to see that, once t1 is known s1 (t1 ) can be obtained from (9).
Then t2 (t1 ) can be obtained from (10) and following this alternate procedure we can nd s2 (t1 ); : : : ; sn (t1 ).
Similarly, if we start with tn ; as known, sn−1 (tn ) can be obtained from (9), tn−1 (tn ) can be obtained from
(10) and repeating this procedure we can determine sn−2 (tn ); tn−2 (tn ); : : : ; t1 (tn ) and s0 (tn ). It is obvious that
the optimal replenishment policy that minimizes the total inventory cost, for a given n; requires the selected
value of t1 to be such that sn (t1 ) = H or the value of tn to be such that s0 (tn ) = 0.
The theorem, which follows, ensures the existence of a unique optimal replenishment schedule for any
policy with n replenishments.
Theorem 1. For the given model there exists a unique solution t1∗ ∈ [0; H ] satisfying sn (t1∗ ) = H .
The proof of this theorem follows immediately using Lemmas 1 and 2, which are given subsequently.
Now we present the theorem, which combined with Theorem 1, guarantees the existence of a unique optimal
policy for the problem under consideration.
Theorem 2. The function TC(n; si ; ti ) is convex w.r.t. n.
Proof. The technique used in the proof of this theorem involves dynamic programming arguments and is
similar to that used by Teng et al. [16] and Friedman [8]. Let us set
TC(n; si ; ti ) = nC + T (n; si ; ti );
(11)
where T (n; si ; ti ) is the sum of inventory, deterioration and shortages costs incurred from 0 to H .
It is enough to prove that
T (n + 1; 0; H ) − T (n; 0; H ) ¿ T (n; 0; H ) − T (n − 1; 0; H ):
(12)
By Bellman’s principle of optimality [1], we obtain the minimum value of T (n; si ; ti ):
T ∗ (n; 0; H ) = Min {T ∗ (n − 1; 0; s) + T (1; s; H )}:
s∈[0; H ]
(13)
∗
(n; 0; H ));
Recursive application of (13) yields the optimal ith shortage point, si∗ (n; 0; H ) = si∗ (n − j; 0; sn−j
i = 1; : : : ; n − j − 1; when n orders are placed in the interval [0; H ].
Let s = H and hence T ∗ (n; 0; H ) ¡ T ∗ (n − 1; 0; H ). Thus, T ∗ (n; 0; H ) is strictly decreasing in n. Let us
choose H1 and H2 such that
∗
(n + 2; 0; H2 ) = H:
sn∗ (n + 1; 0; H1 ) = sn+1
(14)
Since sn∗ (n + 1; 0; H1 ) = H employing the principle of optimality, we have
T ∗ (n + 1; 0; H1 ) = Min {T ∗ (n; 0; s) + T (1; s; H1 )}
s∈[0; H ]
= T ∗ (n; 0; H ) + T (1; H; H1 ):
(15)
But this means that if t = H then
@[T ∗ (n; 0; t) + T (1; t; H1 )] @T ∗ (n; 0; t) @T (1; t; H1 )
=
+
= 0;
@t
@t
@t
(16)
180
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
So
@T ∗ (1; t; H1 )
@T ∗ (n; 0; t)
=
−
@t
@t
t=H
t=H
C1 + C3 (H −tn∗ (n;0; H ))
[e
=
− 1];
(17)
where tn∗ (n; 0; H ) is the last optimal replenishment time when n orders are placed during the interval [0; H ].
∗
Similarly, from sn+1
(n + 2; 0; H2 ) = H; we have
@T (1; t; H2 )
@T ∗ (n + 1; 0; t)
=−
@t
@t
t=H
t=H
∗
C1 + C3 (H −tn+1
(n+1;0; H ))
[e
=
− 1]:
(18)
Subtracting Eq. (18) from Eq. (17) we have
@[T ∗ (n; 0; H ) − T ∗ (n + 1; 0; H )]
¿0
@H
(19)
which implies that T ∗ (n; 0; H ) − T ∗ (n + 1; 0; H ) is a strictly increasing function of H . Again using (13) we
obtain
T ∗ (n; 0; H1 ) − T ∗ (n + 1; 0; H1 ) = min {T ∗ (n − 1; 0; s) + T (1; s; H1 )} − T ∗ (n; 0; H ) − T (1; H; H1 ): (20)
s∈[0; H1 ]
Taking s = H in Eq. (20) we have
T ∗ (n; 0; H1 ) − T ∗ (n + 1; 0; H1 ) ¡ T ∗ (n − 1; 0; H ) − T ∗ (n; 0; H ):
(21)
Since H ¡ H1 and T ∗ (n; 0; H ) − T ∗ (n + 1; 0; H ) is a strictly increasing function in H; we obtain relation (12).
This implies that TC(n; si ; ti ) is also convex in n.
Now, we present two lemmas. The results of these lemmas can be combined to give the proof of
Theorem 1.
Lemma 1. sn (0) ¡ H and sn (H ) ¿ H .
The proof of this lemma follows easily using (9) and (10).
Lemma 2. sn (t1 ) is an increasing function of the variable t1 .
Proof. Since f(t) is logconcave, f(t)=f′ (t) is strictly increasing in t; for ti 6 t 6 si ; and so we have f′ (t)6
(f′ (ti )=f(ti ))f(t). Then multiplying both sides of this inequality by e(t−ti ) , and adding e(t−ti ) f(t) ¿ 0 to
both sides we obtain the following inequality:
e(t−ti ) f′ (t) + e(t−ti ) f(t)6e(t−ti )
f′ (ti )
f(t) + e(t−ti ) f(t);
f(ti )
ti 6 t 6 si :
(22)
In (22) if we multiply both sides by C1 + C3 and integrate with respect to t in the interval [ti ; si ]; we have
Z si
′
f (ti )
(si −ti )
+
e(t−ti ) f(t) dt:
(23)
f(si ) − f(ti )]6(C1 + C3 )
(C1 + C3 )[e
f(ti )
ti
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
Due to (9) the above inequality becomes
Z
(C1 + C3 ) e(si −ti ) f(si ) − f(ti ) −
si
ti
′
181
Z
f′ (ti ) ti − (ti −t)
e(t−ti ) f(t) dt 6C2
e
f(t)[1 − (ti − t)] dt:
f(ti ) si−1
(24)
′
But f (ti )=f(ti ) ¡ f (t)=f(t) for si−1 6 t 6 ti ; and 1 − H ¿ 0. Taking these into account on the right-hand
side of (24) and then integrating by parts we obtain
Z si
(si −ti )
(t−ti )
f(si ) − f(ti ) −
e
f(t) dt
(C1 + C3 ) e
ti
"
#
Z
6C2 f(ti ) − e− (ti −si−1 ) (1 − (ti − si−1 ))f(si−1 ) −
ti
e− (ti −t) f(t)(2 − (ti − t)) dt :
(25)
si−1
Substituting Mi = si − ti and Ki = ti − si−1 into (9) and di erentiating (9) with respect to t1 (obviously Mi ; Ki
are functions of t1 ) we get for i = 1; : : : ; n;
Z si
dti Mi
dMi
e f(si ) − f(ti ) −
+ (C1 + C3 )
e(t−ti ) f(t) dt
(C1 + C3 )eMi f(si )
dt1
dt1
ti
"
dKi − Ki
dti
e
f(si−1 )(1 − Ki ) + C2
f(ti ) − e− Ki f(si−1 )(1 − Ki )
= C2
dti
dt1
#
Z
ti
e− (ti −t) f(t)(2 − (ti − t)) dt :
−
(26)
si−1
If i = 1; then dti =dt1 = 1 and dK1 =dt1 = 1 so
(C1 + C3 )eM1 f(s1 )
dM1
= C2 e− K1 f(s0 )(1 − K1 )
dt1
Z t1
− K1
− (t1 −t)
f(s0 )(1 − K1 ) −
e
f(t)(2 − (t1 − t)) dt
+C2 f(t1 ) − e
s0
Z s1
e(t−t1 ) f(t) dt :
(27)
−(C1 + C3 ) eM1 f(s1 ) − f(t1 ) −
t1
Using (25) and the fact that ¡ 1=H we see that on the right-hand side of (27) is a positive number, which
implies that dM1 =dt1 ¿ 0.
Next, di erentiating (10) with respect to t1 we have
C1 + C3 M1 dM1
dK2 − K2
= C2
e
(1 − K2 ):
e
dt1
dt1
The left-hand side of the above equation is positive so, dK2 =dt1 ¿ 0.
Continuing P
in the same way we can show that dKi =dt1 ¿ 0 and dMi =dt1 ¿ 0 for i = 1; : : : ; n. Moreover,
n
since sn (t1 ) = i=1 (Mi + Ki ), it is easily shown that dsn (t1 )=dt1 ¿ 0, which means that sn (t1 ) is an increasing
function of t1 .
Using induction and based on the proof of the above lemma we can prove the following.
Corollary 1. The functions ti (t1 ); si (t1 ); i = 1; 2; : : : ; n; are monotonically increasing w.r.t. t1 .
The next theorem relates the monotonicity of the demand rate, f, with ordering of the lengths of the
cycles Ti .
182
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
Table 1
Total cost (TC) for di erent values of n. The overall optimal replenishment policy
n
TC
1
19 057.5
n = 10
ti
0.56
si
0.9972
2
11 885.2
1.33028
1.65026
3
8945.57
1.88828
2.14093
4
7580.19
2.32634
2.53508
5
6268.46
2.68698
2.86484
6
5786.6
7
5418.34
2.9935
3.14844
8
5241.74
3.26003
3.39729
9
5160.99
3.49582
3.61901
10
5051.06
3.70722
3.81896
11
5101.5
3.8988
4
Theorem 3. (a) If f(t) is an increasing function of t; then T1 ¿ T2 ¿ · · · ¿ Tn .
(b) If f(t) is a decreasing function of t; then T1 ¡ T2 ¡ · · · ¡ Tn .
Proof. (a) Applying the mean value theorem to the integrals in Eq. (9), we obtain
C1 + C3
f(x1 )(e(si −ti ) − 1) = C2 (ti − si−1 )f(x2 )e− (ti −si−1 ) ;
where x1 ∈ [ti ; si ] and x2 ∈ [si−1 ; ti ].
Since f is increasing we have
(28)
C1 + C3 (si −ti )
(e
− 1) ¡ C2 (ti − si−1 )e− (ti −si−1 ) :
(29)
But from (10) follows that (C1 + C3 =)(e(si −ti ) − 1) = C2 (ti+1 − si )e− (ti+1 −si ) , so (29) can be written as
(ti+1 − si )e− (ti+1 −si ) ¡ (ti − si−1 )e− (ti −si−1 )
for any ti ; si :
(30)
Now if we set g(x) = xe− x , then g′ (x) = (1 − x)e− x ¿ 0 for x ¡ 1= and so g(x) is a strictly monotonic
function. Since (30) is valid it follows that ti+1 − si ¡ ti − si−1 or equivalently Ki+1 ¡ Ki .
If we multiply both the sides of (30) by C2 and take into account (10) we obtain
C1 + C3 (si −ti )
C1 + C3 (si−1 −ti−1 )
(e
(e
− 1) ¿
− 1):
This gives si−1 − ti−1 ¿ si − ti or Mi−1 ¿ Mi and since Ti = Ki + Mi we conclude that Ti ¿ Ti+1 .
Part (b) of the theorem is proved using a similar reasoning.
The methodology presented previously for the solution of the problem is based on the fact that we have
chosen to select t1 , the rst replenishment point, so that sn (t1 ) = H . One can proceed in a reversed way, i.e.
select tn , and then proceed to obtain values for all the other parameters. It is obvious that similar results can
be obtained
5. Numerical example
In order to illustrate the preceding theory we consider the following example, which has been used by
Hariga and Alyan [13]:
f(t) = 10e0:98t ;
C = 250;
C1 = 40;
C2 = 80;
C3 = 200;
= 0:08;
= 0:2;
H = 4:
In Table 1, we present the total cost, TC, for di erent values of n and the overall optimal replenishment
policy. The results were obtained using Mathematica. The t1∗ value was obtained using the Bolzano method
for solving equations.
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
183
6. Concluding remarks
1. In this model if we set = 0, we obtain the model with complete backlogging developed by Hariga [10].
2. Moreover, taking = 0, = 0 and f(t) linear, we can obtain the model of Goyal et al. [11].
3. The total cost is a decreasing function of the parameter . This implies that the model with this type of
partial backlogging always has smaller total cost than that with complete backlogging.
Appendix A. Checking the conditions for a minimum of TC (n,si ,ti ).
For convenience let TC(n; si ; ti ) = TC. To ensure that the solution of Eqs. (9) and (10) gives a minimum,
it is enough to prove that the associated Hessian matrix has positive principal minors. The elements of this
Hessian matrix are
H2k+1; 2j+1 =
H2k; 2j =
@2 TC
;
@sk+1 sj+1
@2 TC
;
@tk @tj
k; j = 0; 1; : : : ; n − 1;
j; k = 1; 2; : : : ; n;
H2k+1; 2j =
@2 TC
;
@sk+1 @tj
k = 0; 1; : : : ; n − 1; j = 1; 2; : : : ; n;
H2k; 2j+1 =
@2 TC
;
@tk @sj+1
k = 1; 2; : : : ; n; j = 0; 1; : : : ; n − 1:
The nonzero entries of the Hessian matrix are
H2j+1; 2j+1 =
@2 TC
= [(C1 + C3 )e(sj+1 −tj+1 )
2
@sj+1
+C2 e− (tj+1 −sj+1 ) (1 − (tj+1 − sj+1 ))]f(sj+1 );
j = 0; 1; : : : ; n − 1;
Z sj
@2 TC
= (C1 + C3 )f(tj ) + (C1 + C3 )
H2j; 2j =
e(t−tj ) f(t) dt + C2 f(tj )
@tj2
tj
Z tj
(2 − (tj − sj−1 ))e− (tj −t) f(t) dt; j = 1; : : : ; n;
− C2
sj−1
H2k+1; 2j =
H2k; 2j+1 =
@2 TC
= −C2 e− (tj −sk+1 ) (1 − (tj − sk+1 ))f(sk+1 );
@sk+1 @tj
k = 0; 1; : : : ; n − 1; j = k + 1; k + 2;
@2 TC
= −(C1 + C3 )e(sj+1 −tk ) f(sj+1 );
@tk @sj+1
k = 1; 2; : : : ; n − 1;
We observe that H2j+1; 2j+1 ¿ 0; H2j; 2j ¿ 0; H2k+1; 2j ¡ 0 and H2k; 2j+1 ¡ 0.
Let Mk be the principal minor of order k, then
M1 = H1; 1 ¿ 0;
j = k − 1:
184
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
M2 =
@2 TC @2 TC
@2 TC @2 TC
−
¿0
@s1 @t1 @t1 @s1
@s12 @t12
(this follows using inequality (25)):
It is not dicult to verify that the principal minors of higher order satisfy the following recurrence relationships:
M2k+1 =
@2 TC
M2k ;
2
@s2k
M2k =
@2 TC @2 TC
−
2
2
@t2k−2
@s2k−2
k = 1; 2; : : : ; n − 1;
@2 TC
@s2k−2 @t2k−2
2 !
M2k−2 ;
k = 2; : : : ; n:
From these relationships we conclude that all principal minors of order k, are positive.
References
[1] R.E. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957.
[2] L. Benkherouf, On an inventory model with deteriorating items and decreasing time-varying demand and shortages, Eur. J. Oper.
Res. 86 (1995) 293–299.
[3] T. Chakrabarti, K.S. Chaudhuri, An EOQ model for deteriorating items with a linear trend in demand and shortages in all cycles,
Int. J. Prod. Econom. 49 (1997) 205–213.
[4] R.P. Covert, G.C. Philip, An EOQ model for items with Weibull distribution deterioration, Am. Inst. Ind. Eng. Trans. 5 (1973)
323–326.
[5] T.K. Datta, A.K. Pal, Order level inventory system with power demand pattern for items with variables rate of deterioration, Indian
J. Pure Appl. Math. 19 (1988) 1043–1053.
[6] U. Dave, L.K. Patel, (T, Si) policy inventory model for deteriorating items with time proportional demand, J. Oper. Res. Soc. 32
(1981) 137–142.
[7] W.A. Donaldson, Inventory replenishment policy for a linear trend in demand: an analytical solution, Oper. Res. Quart. 28 (1977)
663–670.
[8] M.F. Friedman, Inventory lot-size models with general time-dependent demand: an analytical solution, Oper. Res. Quart. 20 (1982)
157–167.
[9] P.M. Ghare, G.F. Shrader, A model for exponentially decaying inventories, J. Ind. Eng. 14 (1963) 238–243.
[10] A. Goswami, K.S. Glaudhuri, An EOQ model for deteriorating items with shortages and a linear trend in demand, J. Oper. Res.
Soc. 42 (1991) 1105–1110.
[11] S.K. Goyal, D. Morrin, F. Nebebe, The nite horizon trended inventory replenishment problem with shortages, J. Oper. Res. Soc.
43 (1992) 1173–1178.
[12] M. Hariga, Optimal EOQ models for deteriorating items with time-varying demand, J. Oper. Res. Soc. 47 (1996) 1228–1246.
[13] M. Hariga, A. Alyan, A lot sizing heuristic for deteriorating items with shortages in growing and declining markets, Comput. Oper.
Res. 24 (1997) 1075–1083.
[14] R.S. Sachan, On (T, Si ) inventory policy model for deteriorating items with time proportional demand, J. Oper. Res. Soc. 35 (1984)
1013–1019.
[15] Y.K. Shah, An order-level lot-size inventory for deteriorating items, Am. Inst. Ind. Eng. Trans. 9 (1977) 108–112.
[16] J.T. Teng, M.C. Chern, H.L. Yang, Y.J. Wang, Deterministic lot size inventory models with shortages and deterioration for
uctuating
demand, Oper. Res. Lett. 24 (1999) 65–72.
[17] H.M. Wee, Deterministic lot size inventory model for deteriorating items with shortages and a declining market, Comput. Oper.
Res. 22 (1995) 345–356.
[18] H.M. Wee, Deteriorating inventory model with quantity discount, pricing and partial backordering, Int. J. Prod. Econom. 59 (1999)
511–518.
www.elsevier.com/locate/dsw
An optimal replenishment policy for deteriorating items with
time-varying demand and partial–exponential
type – backlogging
S. Papachristosa; ∗ , K. Skourib
a Department
of Mathematics, Statistics and Operational Research Section, University of Ioannina, 45110 Ioannina, Greece
b Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
Received 1 July 1999; received in revised form 1 June 2000; accepted 1 June 2000
Abstract
We study a continuous review inventory model over a nite-planning horizon with deterministic varying demand
and constant deterioration rate. The model allows for shortages, which are partially backlogged at a rate which varies
c 2000 Elsevier Science B.V.
exponentially with time. For this model an optimal replenishment policy is established.
All rights reserved.
Keywords: Inventory; Lot sizing; Deterioration; Shortage; Partial backlogging
1. Introduction
The deterioration of many items during storage period is a real fact. Foods, pharmaceuticals, chemicals,
blood, drugs are a few examples of such items. Chare and Schrader [9], rst, proposed an inventory model
having a constant rate of deterioration and a constant rate of demand over a nite-planning horizon. Covert
and Phillip [4] extended Chare and Schrader’s model by considering variable rate of deterioration. Shah [15]
suggested a further generalization of all these models by allowing shortages and using a general distribution
for the deterioration rate. The common characteristic of these articles was that demand rate was taken as a
constant over the whole planning horizon.
The assumption of a constant demand rate is usually valid in the mature stage of the life cycle of the
product. In the growth and=or end stage of the product life cycle demand rate can be well approximated by
a linear function. Donaldson [7] developed an exact replenishment policy concerning this case.
Dave and Patel [6] studied an inventory model with deterministic but linearly changing demand rate and
constant deterioration rate over a nite planning horizon. In their formulation they assumed replenishment
Corresponding author. Fax: +30-651-98297.
E-mail address: spapachr@cc.uoi.gr (S. Papachristos).
∗
c 2000 Elsevier Science B.V. All rights reserved.
0167-6377/00/$ - see front matter
PII: S 0 1 6 7 - 6 3 7 7 ( 0 0 ) 0 0 0 4 4 - 4
176
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
intervals of equal lengths. Sachan [14] extended Dave and Pattel’s model to allow for shortages. In his
model, each cycle starts with a replenishment and the period of positive inventory is followed by a period
of shortages. Moreover, this replenishment policy requires an extra order to be placed at the end of the
planning horizon to clear all shortages accumulated during the nal cycle. Datta and Pall [5] developed
an EOQ model by introducing a variable deterioration rate and power demand pattern. They considered a
xed-length replenishment cycle and they determined the optimal shortage point and order quantity. Goswami
and Chaudhuri [10] attempted to solve a problem with exponentially deteriorating items, linearly time-varying
demand and shortages. They also assumed replenishment intervals of equal lengths. Their model starts with
replenishment and ends without replenishment (zero nal stock). Benkherouf [2] relaxed the assumption of
equal length for the replenishment cycles assuming a decreasing and logconcave demand rate. He proved
the existence of the optimal policy and presented an iterative procedure to obtain it. Hariga and Alyan [13],
who dealt with the same model, assuming a time-dependent demand rate and no shortages in the last cycle,
presented a heuristic procedure to obtain the replenishment policy. Hariga [12] extended already obtained
results by introducing a logconcave function to describe the demand rate. Chakrabarti and Chaudhurri [3]
studied an inventory model with linearly changing demand rate, constant deterioration rate and shortages over
a nite-planning horizon. In their model they assumed cycles of equal lengths (each cycle starts with shortages)
and the initial and nal inventory levels are both zero. Teng et al. [16] presented an inventory model under
a general-demand function constant deterioration rate and shortages over a nite-planning horizon. Moreover,
their model starts with shortages and ends with shortages. The characteristic of all the above articles is that
they allow shortages, while unsatis ed demand is completely backlogged.
For models where shortages are allowed, complete backlogging, or complete loss of unsatis ed demand are
two extreme cases. So researchers have turned their attention to models that allow partial backlogging, (e.g.
Wee [17,18]).
In this article, we develop an EOQ inventory model over a nite planning horizon, with constant deterioration rate, time-varying demand rate and time-dependent partial backlogging. More explicitly, we suppose
that the rate of backlogged demand increases exponentially as the waiting time for the next replenishment
decreases. We believe that this is a quite reasonable assumption since as the waiting time decreases, more
and more customers are willing to wait to get their orders as soon as the backlogged demand reaches the
system at the next replenishment.
The paper is organized as follows. In Section 2 we give the assumptions of the model and the notation
used. We continue with the mathematical formulation of the model in Section 3. In Section 4 we present
results, theorems and lemmas, which ensure the existence of the optimal plan and we give the algorithm that
can be used to nd the parameters of the optimal policy. The paper continues with Section 5 where we give
a numerical example, explaining the procedure proposed in Section 4. Some comments and conclusions are
given in Section 6. The paper closes with an appendix where we provide proofs for some of the results stated
in previous sections.
2. Assumptions and notation
The inventory model is a continuous review model developed under the following assumptions:
1. The planning horizon of the system is nite and is taken as H time units. The initial and the nal inventory
levels are both zero during this time horizon.
2. Replenishment is instantaneous (replenishment rate is in nite).
3. The lead time is zero.
4. The on hand inventory deteriorates at a constant rate (0 ¡ ¡ 1) per time unit. The deteriorated items
are withdrawn immediately from the warehouse and there is no provision for repair or replacement.
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
177
Fig. 1. Graphical representation of inventory.
5. The rate of demand, f(t); t ∈ [0; H ]; is a continuous, logconcave function of t; with f′ (t) 6= 0 ∀t.
6. The system allows for shortages in all cycles and each cycle starts with shortages.
7. Unsatis ed demand is backlogged at a rate exp(− x); where x is the time up to the next replenishment and
a parameter 0 ¡ ¡ (1=H ).
Nomenclature
C
C1
C2
C3
CIi
DIi
SIi
I (t)
n
si
ti
Ti
the replenishment cost per order.
holding cost per unit of stock carried per unit time.
shortage cost per unit of shortage per unit time.
deterioration cost per unit of deteriorated items.
the amount of inventory carried during the ith cycle.
the amount of deteriorated items during the ith cycle.
the amount of units in shortages during the ith cycle.
the inventory level at time t.
the number of replenishment cycles during the planning horizon.
time at which shortage starts during the ith cycle i = 1; : : : ; n − 1.
time at which the ith replenishment is made i = 1; : : : ; n.
length of the ith cycle
3. Mathematical formulation of the model
The
uctuation of the inventory level in the system is given in Fig. 1. The depletion of inventory during
the interval [ti ; si ]; of the ith replenishment cycle, is due to the joint e ect of the demand and the deterioration
of items. Hence the di erential equation, which describes the variation of inventory level, I (t); with respect
to time t; is
dI (t)
= −I (t) − f(t); ti 6 t ¡ si
dt
with boundary condition I (si ) = 0; i = 1; : : : ; n.
The solution of (1) is
Z si
−t
eu f(u) du; ti 6 t ¡ si :
I (t) = e
t
(1)
(2)
178
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
From (2) the amount of inventory carried during the ith cycle is given by
Z
1 si (t−ti )
(e
− 1)f(t) dt:
CIi =
ti
(3)
The amount of deteriorated items during the ith cycle is
Z si
(e(t−ti ) − 1)f(t) dt:
DIi = CIi =
(4)
ti
The variation of inventory level, I (t); with respect to time, t; during the interval [si−1 ; ti ] can be described by
the following di erential equation:
dI (t)
= −e− (ti −t) f(t); si−1 6 t ¡ ti
dt
with boundary condition I (si−1 ) = 0; i = 1; : : : ; n.
The solution of (5) is
Z t
e− (ti −u) f(u) du; si−1 6 t ¡ ti :
I (t) = −
(5)
(6)
si−1
From (6) the amount of shortage during the ith cycle is given by
Z ti Z t
Z ti
− (ti −u)
e
f(u) du dt =
e− (ti −u) (ti − u)f(u) du:
SIi =
si−1
(7)
si−1
si−1
Now, we have all quantities needed to formulate the total inventory cost as the sum of ordering, holding,
deterioration and shortage cost, for any policy with n replenishments:
TC(n; si ; ti ) = nC + C1
n−1
X
CIi+1 + C2
i=0
n−1
X
SIi+1 + C3
i=0
n−1
C1 + C3 X
= nC +
i=0
Z
si+1
(e
(t−ti+1 )
n−1
X
DIi+1
i=0
− 1)f(t) dt + C2
ti+1
n−1 Z
X
i=0
ti+1
e− (ti+1 −t) (ti+1 − t)f(t) dt:
si
(8)
The goals of this paper are: (1) to present an algorithm which, for a given number of replenishments, n;
can be used to determine the optimal replenishment points, ti∗ , and the optimal shortage points, si∗ and (2) to
obtain the overall optimal policy, i.e. nd the optimal values for the parameters, n; ti and si .
4. The optimal replenishment procedure
In this section, we shall present all the results which will lead to the construction of the algorithm giving the
optimal ti ; si ; values, for any policy with n replenishments. The continuity of f(t) guarantees that TC(n; si ; ti )
is a continuous function of si ; ti and its rst- and second-order partial derivatives exist.
Taking rst-order derivatives of TC(n; si ; ti ) w.r.t. ti ; si ; and equating them to zero we obtain
Z si
Z ti
(t−ti )
e− (ti −t) [1 − (ti − t)]f(t) dt; i = 1; : : : ; n;
(9)
e
f(t) dt = C2
(C1 + C3 )
ti
si−1
C1 + C3 (si −ti )
− 1] = C2 (ti+1 − si )e− (ti+1 −si ) ;
[e
i = 1; : : : ; n − 1:
(10)
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
179
In Appendix A we prove that the solution, ti∗ ; si∗ ; i = 1; 2; : : : ; n; of the above system of equations, satis es
the second-order conditions for a minimum. Moreover, as we shall see, the system of equations (9), (10) has
a unique solution and so the above minimum is a global minimum.
We shall now present the methodology used to solve Eqs. (9) and (10), and after that, we shall prove
that the obtained solution is unique. It is easy to see that, once t1 is known s1 (t1 ) can be obtained from (9).
Then t2 (t1 ) can be obtained from (10) and following this alternate procedure we can nd s2 (t1 ); : : : ; sn (t1 ).
Similarly, if we start with tn ; as known, sn−1 (tn ) can be obtained from (9), tn−1 (tn ) can be obtained from
(10) and repeating this procedure we can determine sn−2 (tn ); tn−2 (tn ); : : : ; t1 (tn ) and s0 (tn ). It is obvious that
the optimal replenishment policy that minimizes the total inventory cost, for a given n; requires the selected
value of t1 to be such that sn (t1 ) = H or the value of tn to be such that s0 (tn ) = 0.
The theorem, which follows, ensures the existence of a unique optimal replenishment schedule for any
policy with n replenishments.
Theorem 1. For the given model there exists a unique solution t1∗ ∈ [0; H ] satisfying sn (t1∗ ) = H .
The proof of this theorem follows immediately using Lemmas 1 and 2, which are given subsequently.
Now we present the theorem, which combined with Theorem 1, guarantees the existence of a unique optimal
policy for the problem under consideration.
Theorem 2. The function TC(n; si ; ti ) is convex w.r.t. n.
Proof. The technique used in the proof of this theorem involves dynamic programming arguments and is
similar to that used by Teng et al. [16] and Friedman [8]. Let us set
TC(n; si ; ti ) = nC + T (n; si ; ti );
(11)
where T (n; si ; ti ) is the sum of inventory, deterioration and shortages costs incurred from 0 to H .
It is enough to prove that
T (n + 1; 0; H ) − T (n; 0; H ) ¿ T (n; 0; H ) − T (n − 1; 0; H ):
(12)
By Bellman’s principle of optimality [1], we obtain the minimum value of T (n; si ; ti ):
T ∗ (n; 0; H ) = Min {T ∗ (n − 1; 0; s) + T (1; s; H )}:
s∈[0; H ]
(13)
∗
(n; 0; H ));
Recursive application of (13) yields the optimal ith shortage point, si∗ (n; 0; H ) = si∗ (n − j; 0; sn−j
i = 1; : : : ; n − j − 1; when n orders are placed in the interval [0; H ].
Let s = H and hence T ∗ (n; 0; H ) ¡ T ∗ (n − 1; 0; H ). Thus, T ∗ (n; 0; H ) is strictly decreasing in n. Let us
choose H1 and H2 such that
∗
(n + 2; 0; H2 ) = H:
sn∗ (n + 1; 0; H1 ) = sn+1
(14)
Since sn∗ (n + 1; 0; H1 ) = H employing the principle of optimality, we have
T ∗ (n + 1; 0; H1 ) = Min {T ∗ (n; 0; s) + T (1; s; H1 )}
s∈[0; H ]
= T ∗ (n; 0; H ) + T (1; H; H1 ):
(15)
But this means that if t = H then
@[T ∗ (n; 0; t) + T (1; t; H1 )] @T ∗ (n; 0; t) @T (1; t; H1 )
=
+
= 0;
@t
@t
@t
(16)
180
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
So
@T ∗ (1; t; H1 )
@T ∗ (n; 0; t)
=
−
@t
@t
t=H
t=H
C1 + C3 (H −tn∗ (n;0; H ))
[e
=
− 1];
(17)
where tn∗ (n; 0; H ) is the last optimal replenishment time when n orders are placed during the interval [0; H ].
∗
Similarly, from sn+1
(n + 2; 0; H2 ) = H; we have
@T (1; t; H2 )
@T ∗ (n + 1; 0; t)
=−
@t
@t
t=H
t=H
∗
C1 + C3 (H −tn+1
(n+1;0; H ))
[e
=
− 1]:
(18)
Subtracting Eq. (18) from Eq. (17) we have
@[T ∗ (n; 0; H ) − T ∗ (n + 1; 0; H )]
¿0
@H
(19)
which implies that T ∗ (n; 0; H ) − T ∗ (n + 1; 0; H ) is a strictly increasing function of H . Again using (13) we
obtain
T ∗ (n; 0; H1 ) − T ∗ (n + 1; 0; H1 ) = min {T ∗ (n − 1; 0; s) + T (1; s; H1 )} − T ∗ (n; 0; H ) − T (1; H; H1 ): (20)
s∈[0; H1 ]
Taking s = H in Eq. (20) we have
T ∗ (n; 0; H1 ) − T ∗ (n + 1; 0; H1 ) ¡ T ∗ (n − 1; 0; H ) − T ∗ (n; 0; H ):
(21)
Since H ¡ H1 and T ∗ (n; 0; H ) − T ∗ (n + 1; 0; H ) is a strictly increasing function in H; we obtain relation (12).
This implies that TC(n; si ; ti ) is also convex in n.
Now, we present two lemmas. The results of these lemmas can be combined to give the proof of
Theorem 1.
Lemma 1. sn (0) ¡ H and sn (H ) ¿ H .
The proof of this lemma follows easily using (9) and (10).
Lemma 2. sn (t1 ) is an increasing function of the variable t1 .
Proof. Since f(t) is logconcave, f(t)=f′ (t) is strictly increasing in t; for ti 6 t 6 si ; and so we have f′ (t)6
(f′ (ti )=f(ti ))f(t). Then multiplying both sides of this inequality by e(t−ti ) , and adding e(t−ti ) f(t) ¿ 0 to
both sides we obtain the following inequality:
e(t−ti ) f′ (t) + e(t−ti ) f(t)6e(t−ti )
f′ (ti )
f(t) + e(t−ti ) f(t);
f(ti )
ti 6 t 6 si :
(22)
In (22) if we multiply both sides by C1 + C3 and integrate with respect to t in the interval [ti ; si ]; we have
Z si
′
f (ti )
(si −ti )
+
e(t−ti ) f(t) dt:
(23)
f(si ) − f(ti )]6(C1 + C3 )
(C1 + C3 )[e
f(ti )
ti
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
Due to (9) the above inequality becomes
Z
(C1 + C3 ) e(si −ti ) f(si ) − f(ti ) −
si
ti
′
181
Z
f′ (ti ) ti − (ti −t)
e(t−ti ) f(t) dt 6C2
e
f(t)[1 − (ti − t)] dt:
f(ti ) si−1
(24)
′
But f (ti )=f(ti ) ¡ f (t)=f(t) for si−1 6 t 6 ti ; and 1 − H ¿ 0. Taking these into account on the right-hand
side of (24) and then integrating by parts we obtain
Z si
(si −ti )
(t−ti )
f(si ) − f(ti ) −
e
f(t) dt
(C1 + C3 ) e
ti
"
#
Z
6C2 f(ti ) − e− (ti −si−1 ) (1 − (ti − si−1 ))f(si−1 ) −
ti
e− (ti −t) f(t)(2 − (ti − t)) dt :
(25)
si−1
Substituting Mi = si − ti and Ki = ti − si−1 into (9) and di erentiating (9) with respect to t1 (obviously Mi ; Ki
are functions of t1 ) we get for i = 1; : : : ; n;
Z si
dti Mi
dMi
e f(si ) − f(ti ) −
+ (C1 + C3 )
e(t−ti ) f(t) dt
(C1 + C3 )eMi f(si )
dt1
dt1
ti
"
dKi − Ki
dti
e
f(si−1 )(1 − Ki ) + C2
f(ti ) − e− Ki f(si−1 )(1 − Ki )
= C2
dti
dt1
#
Z
ti
e− (ti −t) f(t)(2 − (ti − t)) dt :
−
(26)
si−1
If i = 1; then dti =dt1 = 1 and dK1 =dt1 = 1 so
(C1 + C3 )eM1 f(s1 )
dM1
= C2 e− K1 f(s0 )(1 − K1 )
dt1
Z t1
− K1
− (t1 −t)
f(s0 )(1 − K1 ) −
e
f(t)(2 − (t1 − t)) dt
+C2 f(t1 ) − e
s0
Z s1
e(t−t1 ) f(t) dt :
(27)
−(C1 + C3 ) eM1 f(s1 ) − f(t1 ) −
t1
Using (25) and the fact that ¡ 1=H we see that on the right-hand side of (27) is a positive number, which
implies that dM1 =dt1 ¿ 0.
Next, di erentiating (10) with respect to t1 we have
C1 + C3 M1 dM1
dK2 − K2
= C2
e
(1 − K2 ):
e
dt1
dt1
The left-hand side of the above equation is positive so, dK2 =dt1 ¿ 0.
Continuing P
in the same way we can show that dKi =dt1 ¿ 0 and dMi =dt1 ¿ 0 for i = 1; : : : ; n. Moreover,
n
since sn (t1 ) = i=1 (Mi + Ki ), it is easily shown that dsn (t1 )=dt1 ¿ 0, which means that sn (t1 ) is an increasing
function of t1 .
Using induction and based on the proof of the above lemma we can prove the following.
Corollary 1. The functions ti (t1 ); si (t1 ); i = 1; 2; : : : ; n; are monotonically increasing w.r.t. t1 .
The next theorem relates the monotonicity of the demand rate, f, with ordering of the lengths of the
cycles Ti .
182
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
Table 1
Total cost (TC) for di erent values of n. The overall optimal replenishment policy
n
TC
1
19 057.5
n = 10
ti
0.56
si
0.9972
2
11 885.2
1.33028
1.65026
3
8945.57
1.88828
2.14093
4
7580.19
2.32634
2.53508
5
6268.46
2.68698
2.86484
6
5786.6
7
5418.34
2.9935
3.14844
8
5241.74
3.26003
3.39729
9
5160.99
3.49582
3.61901
10
5051.06
3.70722
3.81896
11
5101.5
3.8988
4
Theorem 3. (a) If f(t) is an increasing function of t; then T1 ¿ T2 ¿ · · · ¿ Tn .
(b) If f(t) is a decreasing function of t; then T1 ¡ T2 ¡ · · · ¡ Tn .
Proof. (a) Applying the mean value theorem to the integrals in Eq. (9), we obtain
C1 + C3
f(x1 )(e(si −ti ) − 1) = C2 (ti − si−1 )f(x2 )e− (ti −si−1 ) ;
where x1 ∈ [ti ; si ] and x2 ∈ [si−1 ; ti ].
Since f is increasing we have
(28)
C1 + C3 (si −ti )
(e
− 1) ¡ C2 (ti − si−1 )e− (ti −si−1 ) :
(29)
But from (10) follows that (C1 + C3 =)(e(si −ti ) − 1) = C2 (ti+1 − si )e− (ti+1 −si ) , so (29) can be written as
(ti+1 − si )e− (ti+1 −si ) ¡ (ti − si−1 )e− (ti −si−1 )
for any ti ; si :
(30)
Now if we set g(x) = xe− x , then g′ (x) = (1 − x)e− x ¿ 0 for x ¡ 1= and so g(x) is a strictly monotonic
function. Since (30) is valid it follows that ti+1 − si ¡ ti − si−1 or equivalently Ki+1 ¡ Ki .
If we multiply both the sides of (30) by C2 and take into account (10) we obtain
C1 + C3 (si −ti )
C1 + C3 (si−1 −ti−1 )
(e
(e
− 1) ¿
− 1):
This gives si−1 − ti−1 ¿ si − ti or Mi−1 ¿ Mi and since Ti = Ki + Mi we conclude that Ti ¿ Ti+1 .
Part (b) of the theorem is proved using a similar reasoning.
The methodology presented previously for the solution of the problem is based on the fact that we have
chosen to select t1 , the rst replenishment point, so that sn (t1 ) = H . One can proceed in a reversed way, i.e.
select tn , and then proceed to obtain values for all the other parameters. It is obvious that similar results can
be obtained
5. Numerical example
In order to illustrate the preceding theory we consider the following example, which has been used by
Hariga and Alyan [13]:
f(t) = 10e0:98t ;
C = 250;
C1 = 40;
C2 = 80;
C3 = 200;
= 0:08;
= 0:2;
H = 4:
In Table 1, we present the total cost, TC, for di erent values of n and the overall optimal replenishment
policy. The results were obtained using Mathematica. The t1∗ value was obtained using the Bolzano method
for solving equations.
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
183
6. Concluding remarks
1. In this model if we set = 0, we obtain the model with complete backlogging developed by Hariga [10].
2. Moreover, taking = 0, = 0 and f(t) linear, we can obtain the model of Goyal et al. [11].
3. The total cost is a decreasing function of the parameter . This implies that the model with this type of
partial backlogging always has smaller total cost than that with complete backlogging.
Appendix A. Checking the conditions for a minimum of TC (n,si ,ti ).
For convenience let TC(n; si ; ti ) = TC. To ensure that the solution of Eqs. (9) and (10) gives a minimum,
it is enough to prove that the associated Hessian matrix has positive principal minors. The elements of this
Hessian matrix are
H2k+1; 2j+1 =
H2k; 2j =
@2 TC
;
@sk+1 sj+1
@2 TC
;
@tk @tj
k; j = 0; 1; : : : ; n − 1;
j; k = 1; 2; : : : ; n;
H2k+1; 2j =
@2 TC
;
@sk+1 @tj
k = 0; 1; : : : ; n − 1; j = 1; 2; : : : ; n;
H2k; 2j+1 =
@2 TC
;
@tk @sj+1
k = 1; 2; : : : ; n; j = 0; 1; : : : ; n − 1:
The nonzero entries of the Hessian matrix are
H2j+1; 2j+1 =
@2 TC
= [(C1 + C3 )e(sj+1 −tj+1 )
2
@sj+1
+C2 e− (tj+1 −sj+1 ) (1 − (tj+1 − sj+1 ))]f(sj+1 );
j = 0; 1; : : : ; n − 1;
Z sj
@2 TC
= (C1 + C3 )f(tj ) + (C1 + C3 )
H2j; 2j =
e(t−tj ) f(t) dt + C2 f(tj )
@tj2
tj
Z tj
(2 − (tj − sj−1 ))e− (tj −t) f(t) dt; j = 1; : : : ; n;
− C2
sj−1
H2k+1; 2j =
H2k; 2j+1 =
@2 TC
= −C2 e− (tj −sk+1 ) (1 − (tj − sk+1 ))f(sk+1 );
@sk+1 @tj
k = 0; 1; : : : ; n − 1; j = k + 1; k + 2;
@2 TC
= −(C1 + C3 )e(sj+1 −tk ) f(sj+1 );
@tk @sj+1
k = 1; 2; : : : ; n − 1;
We observe that H2j+1; 2j+1 ¿ 0; H2j; 2j ¿ 0; H2k+1; 2j ¡ 0 and H2k; 2j+1 ¡ 0.
Let Mk be the principal minor of order k, then
M1 = H1; 1 ¿ 0;
j = k − 1:
184
S. Papachristos, K. Skouri / Operations Research Letters 27 (2000) 175–184
M2 =
@2 TC @2 TC
@2 TC @2 TC
−
¿0
@s1 @t1 @t1 @s1
@s12 @t12
(this follows using inequality (25)):
It is not dicult to verify that the principal minors of higher order satisfy the following recurrence relationships:
M2k+1 =
@2 TC
M2k ;
2
@s2k
M2k =
@2 TC @2 TC
−
2
2
@t2k−2
@s2k−2
k = 1; 2; : : : ; n − 1;
@2 TC
@s2k−2 @t2k−2
2 !
M2k−2 ;
k = 2; : : : ; n:
From these relationships we conclude that all principal minors of order k, are positive.
References
[1] R.E. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957.
[2] L. Benkherouf, On an inventory model with deteriorating items and decreasing time-varying demand and shortages, Eur. J. Oper.
Res. 86 (1995) 293–299.
[3] T. Chakrabarti, K.S. Chaudhuri, An EOQ model for deteriorating items with a linear trend in demand and shortages in all cycles,
Int. J. Prod. Econom. 49 (1997) 205–213.
[4] R.P. Covert, G.C. Philip, An EOQ model for items with Weibull distribution deterioration, Am. Inst. Ind. Eng. Trans. 5 (1973)
323–326.
[5] T.K. Datta, A.K. Pal, Order level inventory system with power demand pattern for items with variables rate of deterioration, Indian
J. Pure Appl. Math. 19 (1988) 1043–1053.
[6] U. Dave, L.K. Patel, (T, Si) policy inventory model for deteriorating items with time proportional demand, J. Oper. Res. Soc. 32
(1981) 137–142.
[7] W.A. Donaldson, Inventory replenishment policy for a linear trend in demand: an analytical solution, Oper. Res. Quart. 28 (1977)
663–670.
[8] M.F. Friedman, Inventory lot-size models with general time-dependent demand: an analytical solution, Oper. Res. Quart. 20 (1982)
157–167.
[9] P.M. Ghare, G.F. Shrader, A model for exponentially decaying inventories, J. Ind. Eng. 14 (1963) 238–243.
[10] A. Goswami, K.S. Glaudhuri, An EOQ model for deteriorating items with shortages and a linear trend in demand, J. Oper. Res.
Soc. 42 (1991) 1105–1110.
[11] S.K. Goyal, D. Morrin, F. Nebebe, The nite horizon trended inventory replenishment problem with shortages, J. Oper. Res. Soc.
43 (1992) 1173–1178.
[12] M. Hariga, Optimal EOQ models for deteriorating items with time-varying demand, J. Oper. Res. Soc. 47 (1996) 1228–1246.
[13] M. Hariga, A. Alyan, A lot sizing heuristic for deteriorating items with shortages in growing and declining markets, Comput. Oper.
Res. 24 (1997) 1075–1083.
[14] R.S. Sachan, On (T, Si ) inventory policy model for deteriorating items with time proportional demand, J. Oper. Res. Soc. 35 (1984)
1013–1019.
[15] Y.K. Shah, An order-level lot-size inventory for deteriorating items, Am. Inst. Ind. Eng. Trans. 9 (1977) 108–112.
[16] J.T. Teng, M.C. Chern, H.L. Yang, Y.J. Wang, Deterministic lot size inventory models with shortages and deterioration for
uctuating
demand, Oper. Res. Lett. 24 (1999) 65–72.
[17] H.M. Wee, Deterministic lot size inventory model for deteriorating items with shortages and a declining market, Comput. Oper.
Res. 22 (1995) 345–356.
[18] H.M. Wee, Deteriorating inventory model with quantity discount, pricing and partial backordering, Int. J. Prod. Econom. 59 (1999)
511–518.