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International Journal of Systems Science
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Optimal policies for a finite-horizon batching inventory
model
a
a
b
Talal M. Al-Khamis , Lakdere Benkherouf & Mohamed Omar
a
Department of Statistics and Operations Research, College of Science, Kuwait University,
Safat, Kuwait
b
Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala
Lumpur, Malaysia
Published online: 05 Feb 2013.
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To cite this article: Talal M. Al-Khamis, Lakdere Benkherouf & Mohamed Omar (2014) Optimal policies for a finite-horizon
batching inventory model, International Journal of Systems Science, 45:10, 2196-2202, DOI: 10.1080/00207721.2013.765056
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International Journal of Systems Science, 2014
Vol. 45, No. 10, 2196–2202, http://dx.doi.org/10.1080/00207721.2013.765056
Optimal policies for a finite-horizon batching inventory model
Talal M. Al-Khamisa , Lakdere Benkheroufa,∗ and Mohamed Omarb
a
Department of Statistics and Operations Research, College of Science, Kuwait University, Safat, Kuwait; b Institute of Mathematical
Sciences, Faculty of Science, University of Malaya, Kuala Lumpur, Malaysia
Downloaded by [University of Malaya] at 21:17 04 August 2014
(Received 8 February 2012; final version received 12 December 2012)
This paper is concerned with finding an optimal inventory policy for the integrated replenishment-production batching model
of Omar and Smith (2002). Here, a company produces a single finished product which requires a single raw material and
the objective is to minimise the total inventory costs over a finite planning horizon. Earlier work in the literature considered
models with linear demand rate function of the finished product. This work proposes a general methodology for finding an
optimal inventory policy for general demand rate functions. The proposed methodology is adapted from the recent work of
Benkherouf and Gilding (2009).
Keywords: inventory model; optimal inventory policy; finite-horizon models
1. Introduction
Inventory management is concerned with the efficient control of the flow of goods in a supply chain. Typically this involves searching for optimal replenishment policies which
minimise some inventory costs.
This paper proposes a methodology for finding an
optimal inventory policy for a single item integratedreplenishment-production batching inventory model over a
finite planning horizon with the objective of minimising the
total inventory costs. Hill (1996) examined the issue of finding batching policies for linear increasing demand and fixed
production rate. Hill, Omar, and Smith (2000) proposed a
dynamic programming formulation for Hill’s earlier model
as well as a number of heuristics algorithms for finding the
optimal batching policy. Later on, Omar and Smith (2002)
extended Hill’s model to allow for the integration of the
costs related to raw materials with that of production in certain manufacturing systems. A number of heuristics were
also proposed for the optimal batching policy when the demand rate D for the finished product is linear and increasing
which were then compared numerically against the results
that were obtained from a dynamic programming formulation. Rau and Ou Yang (2007) made a significant progress
by solving the resulting optimisation problem completely
when D is linear (increasing and decreasing). Moreover, a
number of unanswered questions left in the work of Hill
(1996), Omar and Smith (2000) were addressed satisfactorily. Earlier work on economic lot sizing models for finite
horizon models was done in Donaldson (1997).
Recently, Benkherouf and Gilding (2009) presented
a general theoretical framework for finding the optimal
∗
Corresponding author. Email: [email protected]
C 2013 Taylor & Francis
replenishment schedule for finite horizon inventory problems which allow extension of Donaldson’s work. Related
work for finite horizon models may be found in Papachristos
and Skouri (2000) and Skouri and Papachristos (2002).
The objective of this paper is to show how the theory
developed in Benkherouf and Gilding (2009) applies to the
model of Omar and Smith (2002). This will enable us, as
we shall see, to extend the work of Rau and Ou Yang (2007)
to demand rate functions other than linear.
For completion, we present the derivation of the inventory costs for the model of Omar and Smith (2002) in
the next section. The main results of the paper are given
in Section 3. Section 4 contains numerical examples. The
conclusion is given in the last section.
2. The mathematical model
We assume that a company produces a single finished product which requires a single type of raw material. The planning horizon is assumed finite and made up of multiple
production runs: see Figure 1.
Some of the notation used for the model is given below:
H:
p:
D(t) :
p
ti :
K1 :
K2 :
c:
the total planning horizon,
the constant production rate,
the demand rate at time t, 0 < D(t) < p,
the time at which the inventory level reaches its maximum
in the i th production run,
ordering cost for a production run,
ordering cost for raw material,
the cost of one unit of the finished product with c > 0,
(cost/unit)
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International Journal of Systems Science
Figure 1.
c :
ch :
cr :
n:
ρ:
2197
a. Inventory level for finished product. b. Inventory level for raw material.
the cost of one unit of raw material with c > 0, (cost/unit)
carrying cost of the finished product, (cost/unit/unit time)
carrying cost of raw material, (cost/unit/unit time)
the number of production runs
the quantity of raw material required to produce one unit
of finished product
Let [ti−1 , ti ) be the time interval of the ith production
run and Ip (t) be the level of stock of the finished product.
The changes of the level of inventory in the interval
p
[ti−1 , ti ) is described by the differential equation (see
Figure 1a.):
Ip (t)
= p − D(t),
with boundary condition Ip (ti−1 ) = 0.
The solution of Equation (1) is given:
Ip (t) =
t
{p − D(u)} du.
(2)
ti−1
p
On the interval (ti , ti ] the change in the level of stock
may be described by
Ip (t) = −D(t),
(3)
(1)
p
where ti < t ≤ ti , with boundary condition Ip (ti ) = 0.
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T.M. Al-Khamis et al.
The solution of Equation (3) is given:
Now, expressions (7) and (10) imply that the total cost
during the production run i is equal to:
ti
Ip (t) =
D(u)du.
(4)
K1 + K2 + (c + ρc )
t
ti
D(t)dt + Hp (ti−1 , ti )
ti−1
+ Hr (ti−1 , ti ).
Furthermore, on [ti−1 , ti ), we must have
p
p ti − ti−1 =
ti
D(t)dt.
(5)
ti−1
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The holding cost in period i for the production run is
proportional to the area under Ip (t) in the interval [ti−1 , ti ],
which can be shown using (2)–(5), after some algebra, to
be equal to:
Hp (ti−1 , ti )
ti
2 ti
1
(t − ti−1 ) D(t)dt −
:= ch
D(t)dt
. (6)
2p ti−1
ti−1
It follows that the total production costs on [ti−1 , ti ) is equal
to
K1 + c
ti
D(t)dt + Hp (ti−1 , ti ).
H
Set K = K1 + K2 . Also, note that
nK +
where ti−1 < t ≤
with boundary condition
The solution of Equation (8) is given by:
Ir (t) = ρp
i = 1, . . . , n − 1; t0 = 0; tn = H },
ti−1
R(ti−1 , ti )
t ∈ ,
subject to
(13)
with
2
y
D(t)dt
x
y
y
(t − x)D(t)dt −
x
x
D(t)dt
2p
p
ti
D(t)dt
,
Note that since D(t) < p, for t in [0, H ], the expression
involving c2 in (14) is strictly positive for y > x. Indeed,
fix x and for y > x, define
2
ti
2 (14)
dt
y
G(y) :=
.
ti−1
D(t)dt + Hr (ti−1 , ti ).
y
(t − x)D(t)dt −
x
x
(9)
D(t)dt
2p
2
.
It follows that
It follows that the total raw material cost in a single run is
given by:
ti
n
i=1
+ c2
The holding cost can be shown using (5) to be
ρcr
2p
(12)
c1 = ρcr /2p, and c2 = ch . The problem of finding the optimal batching policy may be stated as that of solving the
mixed integer non-linear programme
= 0.
tp
= ρp ti − t .
{Hp (ti−1 , ti ) + Hr (ti−1 , ti )},
= {t = (t0 , . . . , tn ), with ti−1 < ti ,
R(x, y) = c1
p
Ir (ti )
D(t)dt =
where ti−1 < ti , i = 1, . . . , n − 1; t0 = 0; tn = H .
Let
(8)
p
ti ,
i=1 ti−1
i=1
min Cn (t0 , . . . , tn ) = nK +
Ir (t) = −ρp,
K2 + ρc
n
(7)
For raw material in the production run i, let Ir (t) be the
level of stock of raw material (see Figure 1b.)
The dynamics of Ir (t) is described by:
n ti
0 D(t)dt = constant. Therefore, finding the optimal
inventory (batching) policy reduces to finding n and
(t0 , . . . , tn ) which minimises the expression
ti−1
Hr (ti−1 , ti ) :=
(11)
(10)
y
G (y) = (y − x) −
x
D(t)dt
p
D(y),
where G is the derivative of the function G. Since D(t) <
p, and y > x, the expression with the integral term is strictly
International Journal of Systems Science
less than (y − x). The result is then immediate by noting
that G(x) = 0.
There is a temptation to write R in the compact form
R(x, y) = c1
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where c1
2
y
D(t)dt
x
c2
, and c2
2p
+ c2
y
(t − x)D(t)dt, (15)
2199
y
D(t)dt
(∂y R)(x, y) = 2c1 D(y)
x
1 y
+ c2 (y − x) −
D(t)dt D(y),
p x
(18)
x
c2 . Clearly c2
0 but c1
= c1 −
=
>
can
be negative. The form in (15) was adopted in Rau and Ou
Yang (2007). Therefore, with no restriction on the magnitude of c1 , it is possible to end up with R(x, y) strictly
negative, which is not possible in practice. In fact the form
may give misleading results. This possibility can be elimic
nated by requiring that c1 ≥ − 2p2 . In this paper, to eliminate
unwanted costs, we shall opt for the form (14).
A general procedure for the solution of (13), inspired from Donaldson (1977), goes through two stages.
In Stage 1, the variable n is fixed and the non-linear programme (13) is solved as an unconstrained optimisation
problem by setting t0 = 0, and tn = H and ignoring the
rest of the elements of which is defined in (12). The firstorder optimality condition leads to a system of non-linear
equation with unique solution. This solution can be found
using a simple univariate iterative scheme. Also, this solution turns out to be the optimal solution of the non-linear
programme. Moreover, the value of the objective function
at this optimal solution can be shown to be convex in n. This
justifies a simple search procedure for the optimal value of
n which is done in Stage 2. Benkherouf and Gilding (2009)
showed that Donaldson’s procedure may be applied to more
general inventory models than those considered earlier in
the literature. The next section shows that it could also be
applied to solve (13).
3. Optimal Procedure
This section contains the main results of the paper, namely
that the optimisation problem (13) can be solved using the
methodology proposed in Benkherouf and Gilding (2009).
The success of applying the result in Benkherouf and
Gilding depends on the extent to which hypotheses 1 and 2
are satisfied. Details of these will follow.
Recall the definition of R in (14), then direct computations show:
y
D(t)dt
(∂x R)(x, y) = −2c1 D(x)
x
y
D(x)
+ c2 −1 +
D(t)dt, (16)
p
x
y
2 c2
D (x)
D(t)dt
∂x R (x, y) = +2c1 D 2 (x) − 2c1 −
p
x
D(x)
D(x),
(17)
− c2 −1 +
p
(∂x ∂y R)(x, y) = −2c1 D(x)D(y)
D(x)
D(y), (19)
+ c2 −1 +
p
y
2 ∂y R (x, y) = 2c1 D 2 (y) + 2c1 D (y)
D(t)dt
x
D(y)
D(y)
+ c2 1 −
p
1 y
+ c2 (y − x) −
D(t)dt D (y).
p x
(20)
It is clear that since
satisfies for y > x
(iii)
D(x)
p
< 1, c1 > 0, and c2 > 0, R
(i) R(x, y) > 0,
(21)
(ii) R(x, x) = 0,
(22)
(∂x R)(x, y) < 0 < (∂y R)(x, y)
(23)
(iii) (∂x ∂y R)(x, y) < 0.
(24)
Therefore, Hypothesis 1, of Benkherouf and Gilding
(2009), holds.
Next, for some real-valued continuous function f on
[0, H ] define the operators Lx and Ly by:
Lx R = ∂x2 R + ∂x ∂y R + f (x)∂x R,
(25)
Ly R = ∂x ∂y R + ∂y2 R + f (y)∂y R.
(26)
and
If Lx R ≥ 0, and Ly R ≥ 0, then Hypothesis 2 holds: see
Benkherouf and Gilding (2009).
Now, assume that
(A1) The demand rate is log-concave and differentiable
on [0, H ].
Demand rate functions that are log-concave are common in the inventory control literature: see, for example,
Henery (1979). Log-concave functions have no multiple
maxima but can be flat on top. The linear, the exponential
together with the inverse logit functions are log-concave.
The linear and the exponential functions are suitable for
modelling demand rates that are monotonic (increasing or
decreasing). Demand rate function that initially increases
2200
T.M. Al-Khamis et al.
over time, then flattens up, and finally decreases are logconcave. Those demand functions are suitable for modelling
some new electronic equipments where a rise in demand is
seen immediately after their launch, followed by a period
of steady demand and finally a decrease in demand because
alternative new products are launched.
The next results show that log-concave functions play
an important role for Hypothesis 2 to hold.
Lemma 3.1: Under Assumption (A1), there exists a continuous function f on [0, H ] such that Lx R ≥ 0.
Proof: Direct algebra using (16), (17), (19) and the definition of Lx in (25) we get
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Let
D (x)
c2
D (x)
+
− 2c1
f (x) := −
D(x)
p
2c1 D(x) + c2 1 −
The result is clearly true if r = 0.
Note that for y > x,
(y − x) −
D(x) p
.
It is not difficult to check that Lx R ≥ 0 is equivalent to
D (x)
D(y) − D(x)
y
.
≤
D(x)
x D(t)dt
The left hand-side of the inequality is equal to, by the ex
tended mean-value theorem, DD(ξ(ξ)) for some ξ ∈ (x, y). The
lemma then follows since D is log-concave. This finishes
the proof.
Now, we turn to the operator Ly R. The lemma below
shows that we need to consider two separate cases. Let
c2
,
p
D (x)
F (x) : =
.
rD(x) + c2
r : = 2c1 −
Lemma 3.2: If f is given by (27), then the requirement
Ly R(x, y) ≥ 0 holds, if either
1
p
y
D(t)dt > 0,
x
since D(t) < p, for t ∈ [x, y].
Let us consider the case (ii), that is r > 0 with F nondecreasing, in which case Ly R(x, y) ≥ 0 is equivalent to
2c1
y
≥
(27)
(i) r = 0,
or
(ii) r > 0, and F is non-increasing.
or
(iii) r < 0, and F is non-decreasing.
D (y)
2c1 D(y) + c2 1 − D(y)
p
y
1 y
D(t)dt + c2 (y − x) −
D(t)dt
≥ 0.
× 2c1
p x
x
r {D(y) − D(x)} − r
D(x)
Lx R = 2c1 D(x) {D(x) − D(y)} − c2 −1 +
p
y
c2
− 2c1 D (x)
× {D(x)−D(y)}+
D(t)dt
p
x
y
D(x)
D(t)dt.
+ f (x) −2c1 D(x)+c2 −1+
p
x
Proof: Recall the definition of the operator Ly in (26) with
(18)–(20) to get after some algebra that Ly R(x, y) ≥ 0 is
equivalent to
x
D(y) − D(x)
D(t)dt + c2 (y − x) −
D (y)
2c1 D(y) + c2 1 −
D(y) p
1
p
y
x
.
D(t)dt
(28)
The expression with the integral term in (28) reduces, by
the extended mean value theorem to
D (ξ )
2c1 D(ξ ) + c2 1 −
D(ξ ) p
,
which is equal to F (ξ ), whence part (ii) follows. The proof
of part (iii) is similar to that of part (ii) with the inequality
in (28) reversed.
Remark 1: Note that
(i) the linear and the exponential demand rate functions satisfy Assumption A1,
(ii) under Assumption A1, and r > 0, the function F
is non-increasing if D is concave. This is always
satisfied by a linear demand rate function, but is
not satisfied if D is exponential,
(iii) under Assumption A1, and r < 0, the function F
is non-decreasing if D is linear or exponential.
(iv) if r = 0, then the only requirement needed is Assumption A1. In fact this case reduces to the classical optimal EOQ problem for time-varying demand
and finite planning horizon.
Guided by Lemma 3.2, we assume that
(A2) The condition (i) or (ii) or (iii) of Lemma 3.2 is
satisfied.
Theorem 3.3: Under Assumptions A1 and A2, and for fixed
n, the non-linear programme (13) has a unique optimal
International Journal of Systems Science
solution. Moreover, if n denotes the value of the objective
function at this minimum, then n is convex in n.
gives
c2
2c1 D(ti ) + {p − D(ti )} (Qi − Qi+1 )
p
Qi
= 0.
+ c2 (ti − ti−1 ) D(ti ) −
ti − ti−1
Proof: Assumptions (A1) and (A2) imply by Lemmas 3.1
and 3.2 that Hypothesis 2 is satisfied. Also, (21)–(24) imply that Hypothesis 1 holds. Therefore, the existence and
uniqueness of the optimal solution for (13) follows from
Theorem 1 of Benkherouf and Gilding (2009) (p. 1000).
The convexity is a direct consequence of Theorem 2 in
Benkherouf and Gilding (p. 1002). This completes the
proof.
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(29)
We shall only show (i) as the proof of (ii) is similar and (iii) is
easy. Since D is increasing and p > D(ti ) then Qi+1 > Qi ,
otherwise we get a contradiction that (29) has a solution.
Indeed, D increasing implies that
As a result of Theorem 3.3, the procedure described in
Rau and Ou Yang (2007) or Benkherouf and Gilding (2009)
may be applied to find the optimal solution of (13), as the
following corollary shows.
D(ti ) −
Corollary 3.4: The optimal number of replenishment
schedule is such that
Qi
> 0.
ti − ti−1
This leads to the required result since ti > ti−1 .
(i) If K > 1 − 2 , then the optimal number of replenishment schedule is n = 1.
(ii) If there exists an N ≥ 2 such that N−1 − N >
K > N − N+1 , then the optimal number of replenishment schedule is N.
(iii) If there exists an N ≥ 1 such that K = N −
N+1 , then there are two optimal numbers of replenishment schedule. These are N and N + 1.
Corollary 3.6: If the demand rate D is linear then hypotheses 1 and 2 are satisfied, and consequently the optimal replenishment policy exists and is unique. Moreover,
Theorem 3.5 applies.
Note that Corollary 3.6 enables us to recover the result
in Rau and Ou Yang (2007).
The next theorem shows that if the demand rates are
monotonic, then so are the lengths of the optimal consecutive replenishment schedules.
4. Examples
The methodology proposed by Benkherouf and Gilding
(2009) was applied to the linear demand rate examples
found in Rau and Ou Yang (2007). Also, cases with exponential demand rate functions were considered. For the
linear demand rate functions the results in Rau and Ou Yang
(2007) were confirmed. We shall only present the results for
the exponential demand rate function. A case which was not
treated previously in the literature (Table 1).
Theorem 3.5: If t1 , t2 , . . . , tn−1 is the optimal solution
for the optimisation problem given by (13) then for i =
1, . . . , n − 1, we have
(i) If D is increasing, then ti+1 − ti < ti − ti−1 .
(ii) If D is decreasing, then ti+1 − ti > ti − ti−1 .
(iii) If D is constant, then ti+1 − ti = ti − ti−1 .
Proof: Let t1 , t2 , . . . , tn−1 be the optimal solution for the
optimisation problem given by (13). For i = 1, . . . , n − 1,
let
ti
Qi =
D(t)dt.
Example 4.1: Let D(t) = a exp(bt), a = 1, b = ±0.1,
H = 20, K = 50, c1 = 2, c2 = 90, p = 20. The value of
r = −0.5 < 0, and therefore by part (iii) of Remark 1,
hypotheses 1 and 2 are satisfied. Hence, the results of the
previous section apply. The optimal inventory policies are
given in Table 1 below. Column 2 shows the optimal values
of n. The last column contains the optimal inventory costs.
Note that the numerical results are in line with
Theorem 3.5, which is to be expected.
ti−1
Direct computations show that the first-order condition
(∂y R)(ti−1 , ti ) + (∂x R)(ti , ti+1 ) = 0,
Table 1.
2201
Optimal batching policies for the exponential demand rate function.
b
n
t1
t2
0.1
−0.1
8
3
3.801
4.552
7.024
10.631
t3
9.821
20
t4
t5
t6
t7
t8
Cn
12.289
−
14.498
−
16.497
−
18.321
−
20
–
2340.9874
809.4046
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2202
T.M. Al-Khamis et al.
5. Conclusion
In this paper, we have proposed a methodology for finding
the optimal solution for an integrated-production batching
inventory model which was considered in Omar and Smith
(2002). The methodology is based on an earlier work by
Benkherouf and Gilding (2009) carried out in the context
of finding an optimal inventory policy for finite horizon
classical inventory models. This procedure has already been
successful in tackling integrated vendor-buyer inventory
models: see Benkherouf and Omar (2010). Furthermore, in
order to succeed in solving the optimisation problems of the
type treated in Benkherouf and Gilding (2009) expression
(24) and Hypothesis 2 are critical. The rest of the elements
of Hypothesis 1 are not crucial. The justification of this will
lead to arguments beyond the objective of this paper. It is
hoped that this procedure could be extended to other models
since its scope seems to go beyond its motivating models.
A direct extension of the present model is to allow for
the inclusion of time-value of money in the cost function.
Inflation is one example as treated in Chern, Yang, Teng,
and Papachristos (2008) or Hou and Lin (2006). Another
possibility is to consider permissible delay in payments with
inflation as discussed in Chang, Wu and Chen (2009).
Acknowledgements
The authors would like to thank three anonymous referees for their
valuable comments on an earlier version of the paper.
Notes on contributors
Talal M. Al-Khamis is a professor at the
Department of Statistics and Operations
Research (Kuwait University). He is also
the vice-dean for student affairs at the
Faculty of Science (Kuwait University). He
received his PhD in operations research
from the Florida Institute of Technology
(1989). His research interests are in stochastic modelling simulation and optimisation.
He has contributed papers to the Journal of Applied Probability,
European Journal of Operational Research, International Journal
of Production Economics, Computers and Operations Research,
Computers and Industrial Engineering.
Lakdere Benkherouf is a professor at the
Department of Statistics and Operations Research (Kuwait University). He is also the
Vice-Dean for Research and Laboratories
Affairs at the Faculty of Science (Kuwait
University). He received his PhD in statistics from Imperial College (London). His
research interests are in stochastic modelling and inventory control. He has contributed papers to the Journal of Applied Probability, Journal of
The Royal Statistical Society, Statistics and Probability Letters,
Communications in Statistics-Simulation and Computation,
Sankhya, Journal of The Operational Research Society, European Journal of Operational Research, Operations Research, Operations Research Letters, International Journal of Production
Economics, Computers and Operations Research, Computers
and Industrial Engineering, SIAM Journal of Control and
Optimization, etc.
Mohd Omar is a professor and head in the
Institute of Mathematical Sciences, University of Malaya, Malaysia. He received his
PhD in Operational Research from University of Exeter, United Kingdom. His current
research interests include production and inventory control and multi-objective optimisation. He has contributed articles to the
European Journal of Operational Research,
Computers and Industrial Engineering, International Journal of
Production Economics and International Journal of Production
Research.
References
Benkherouf, L., and Gilding, B.H. (2009), ‘On a Class of Optimization Problems for Finite Time Horizon Inventory Models’, SIAM Journal on Control and Optimization, 48, 993–
1030.
Benkherouf, L., and Omar, M. (2010), ‘Optimal Integrated Policies for a Single Vendor Single Buyer Time-Varying Demand
Model’, Computers and Mathematics with Applications, 60,
2066–2077.
Chang, C.-L., Wu, S.-J., and Chen L.-C., (2009), ‘Optimal Payment Time with Deteriorating Items Under Inflation and Permissible Delay in Payments’, International Journal of Systems
Science, 40, 985–993.
Chern, M.-S., Yang, H.-L., Teng, J.T., and Papachristos, S. (2008),
‘Partial Backlogging Inventory Lot-size Models for Deteriorating Items With Fluctuating Demand Under Inflation’,
European Journal of Operational Research, 191, 127–141.
Donaldson, W.A. (1977), ‘Inventory Replenishment Policy for a
Linear Trend in Demand—An Analytical Solution’, Operational Research Quarterly, 28, 663–670.
Henery, R.J. (1979), ‘Inventory Replenishment Policy for Increasing Demand’, Journal of the Operational Research Society,
30, 611–617.
Hill, R.M. (1996), ‘Batching Policies for Linearly Increasing Demand With a Finite Input Rate’, International Journal of Production Economics, 43, 149–154.
Hill, R.M., Omar, M., and Smith, D.K. (2000), ‘Stock Replenishment Policy for Deterministic Linearly Increasing Demand
With a Finite Input Rate’, Journal Sains, 8, 977–986.
Hou, K.-L., and Lin, L.-C. (2006), ‘An EOQ Model for Deteriorating Items With Price- and Stock-dependent Selling Rates
Under Inflation and Time Value of Money’, International
Journal of Systems Science, 37, 1131–1139.
Omar, M., and Smith, D.K. (2002), ‘An Optimal Batch Size for a
Production System Under Linearly Increasing-Time Varying
Demand Process’, Computers and Industrial Engineering, 42,
35–42.
Papachristos, S., and Skouri, K. (2000), ‘An Optimal Replenishment Policy for Deteriorating Items With Time-varying Demand and Partial-exponential Type Backlogging’, Operations
Research letters, 27, 175–184.
Rau, H., and Ou Yang, B.C. (2007), ‘A General and Optimal
Approach for Three Inventory Models With a Linear Trend in
Demand’, Computers and Industrial Engineering, 52, 521–
532.
Skouri, K., and Papachristos, S. (2002), ‘A Continuous Review Inventory Model, With Deteriorating Items, Time-varying Demand, Linear Replenishment Cost, Partially Time-varying
Backlogging’, Applied Mathematical Modelling, 26, 603–
617.