Lakdere Benkherouf and Dalal Boushehri

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Volume 2012, Article ID 768929,16pages doi:10.1155/2012/768929

Research Article

Optimal Policies for a Finite-Horizon Production Inventory Model

Lakdere Benkherouf and Dalal Boushehri

Department of Statistics and Operations Research, College of Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait

Correspondence should be addressed to Lakdere Benkherouf,[email protected] Received 26 October 2011; Revised 29 February 2012; Accepted 17 March 2012 Academic Editor: Imed Kacem

Copyrightq2012 L. Benkherouf and D. Boushehri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is concerned with the problem of finding the optimal production schedule for an inventory model with time-varying demand and deteriorating items over a finite planning horizon. This problem is formulated as a mixed-integer nonlinear program with one integer variable. The optimal schedule is shown to exist uniquely under some technical conditions. It is also shown that the objective function of the nonlinear obtained from fixing the integrality constraint is convex as a function of the integer variable. This in turn leads to a simple procedure for finding the optimal production plan.

1. Introduction

This paper is concerned with the optimality of a production schedule for a single-item inventory model with deteriorating items and for a finite planning horizon. The motivation for considering inventory models with time-varying demand and deteriorating items is well documented in the literature. Readers may consult Teng et al.1, Goyal and Giri2, and Sana et al.3and the references therein.

Earlier models on finding optimal replenishment schedule for a finite planning horizon may be categorized as economic lot sizeELSmodels dealing with replenishment only. The model treated in this paper is an extension of the economic production lot size EPLSto finite horizon models and time-varying demand. The model is close in spirit to that of1. However, in1, the possibility that products may experience deterioration while in stock was not considered. Deterioration was considered in 3 with the possibility of shortages. Nevertheless, their proposedEPLSschedule is not optimal.

Recently, Benkherouf and Gilding4suggested a general procedure for finding the optimal inventory policy for finite horizon models. The procedure is based on earlier work by Donaldson 5, Henery 6, and Benkherouf and Mahmoud 7. This procedure was

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motivated by applications toELSmodels. Nevertheless, it turned out that the applicability of the procedure goes beyond its original scope. The procedure has already been successful in finding the optimal inventory policy for an integrated single-vendor single buyer with time-varying demand rate: see Benkherouf and Omar8. The current paper presents another extension of the procedure toEPLS models. In our treatment, we have opted for a route of simplicity. In that, we selected a model with no shortages and where costs are fixed throughout the planning horizon. Various extensions of the model are discussed inSection 5.

The details of the model of the paper along with the statement of the problem to be discussed are presented in the next section. Section 3 contains some preliminaries on the procedure of Benkherouf and Gilding4. The main results are contained in Section 4.

Section 5is concerned with some general remarks and conclusion.

2. Mathematical Model

The model treated in this paper is based on the following assumptions:

1the planning horizon is finite;

2a single item is considered;

3products are assumed to experience deterioration while in stock;

4shortages are not permitted;

5initial inventory at the beginning of planning horizon is zero, also the inventory depletes to zero at the end of the planning horizon;

6the demand function is strictly positive;

We will initially look at a single periodiproduction cycle, say, starting at timet_i−1 and ending at timet_i, i1,2, . . . .Some of the notations used in the model are as follows:

H: the total planning horizon, p: the constant production rate,

Dt: the demand rate at timet, 0< Dt< p,

α: constant deteriorating rate of inventory items withα >0,

t^p_i: the time at which the inventory level reaches it is maximum in theith production cycle,

K: set up cost for the inventory model, c1: the cost of one unit of the item withc1 >0,

ch: carrying cost per inventory unit held in the model per unit time, TC: total system cost duringH.

Figure 1shows the changes of the level of stock for a typical production period.

LetItbe the level of stock at timet. The change in periodi, the level of inventory, may be described by the following diﬀerential equationi, i1,2, . . .,

It p−Dt−αIt, t_i−1 ≤t < t^p_i, It↓0 ast↓t_i−1. 2.1

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i−1 p

i i

Inventory level

Time t t t

Figure 1: The changes of inventory levels of various components of the model for a typical production batch.

The solution to2.1is given by

It e^−αt _t

t_i−1

e^αu

p−Du

du, t_i−1≤t < t^p_i, 2.2 It −Dt−αIt, t^p_i ≤t < ti, It↑0 ast↑ti. 2.3

The solution to2.3is given by

It e^−αt _t_i

t

e^αuDudu, t^p_i ≤t < ti. 2.4

The total costs, excluding the setup cost for period i, which consist of holding cost and deterioration cost are given by

ch

_t^p

i

ti−1

e^−αt _t

ti−1

e^αu

p−Du du

dt c_h

_t_i

t^p_i

e^−αt _t_i

t

e^αuDudu

dt, c1

t^p_i ti−1

pdu− _t_i

ti−1

Dudu

.

2.5

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We will call models with this cost OHD models. It is possible to consider instead of2.5the form

c_h _t^p

i

ti−1

e^−αt _t

ti−1

e^αu

p−Du du

dt c_h

_t_i

t^p_i

e^−αt _t_i

t

e^αuDudu

dtc₁ _t^p

i

ti−1

pdu, 2.6

which considers only holding and purchasing costs, where the expression c1 t^p_i t_i−1pdu represents the purchasing cost. We call this OHP models.

Note that since the functionIis continuous att^p_i, we have

e^−αt^pⁱ _t^p

i

ti−1

e^αu

p−Du

due^−αt^pⁱ _t_i

t^p_i

e^αuDudu, 2.7

or

_t^p

i

ti−1

e^αu

p−Du du

_t_i

t^p_i

e^αuDudu, 2.8

then

p _t^p

i

ti−1

e^αudu _t_i

ti−1

e^αuDudu, 2.9

or

p α

e^αt^pⁱ −e^αtⁱ⁻¹ p

αe^αtⁱ⁻¹

e^αt^pⁱ^−tⁱ⁻¹−1

_t_i

ti−1

e^αuDudu, 2.10

e^αt^pⁱ^−tⁱ⁻¹−1 α pe^−αtⁱ⁻¹

_t_i

ti−1

e^αuDudu. 2.11

Therefore,

t^p_i −t_i−1 1 αlog

1α

pe^−αtⁱ⁻¹ _t_i

ti−1

e^αuDudu

. 2.12

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Lemma 2.1. The expression of the cost in2.5is equal to

chαc₁ p

α² log

1α pe^−αtⁱ⁻¹

_t_i

ti−1

e^αuDudu

− 1 α

_t_i

ti−1

Dudu

. 2.13

Proof. Applying integration by parts, we get that2.5reduces to ch

α _t^p

i

ti−1

1−e^αu−t^pⁱ

p−Du du ch

α _t_i

t^p_i

e^αu−t^pⁱ−1

Dudu c1

_t^p

i

ti−1

pdu−c1

_t_i

ti−1

Dudu,

2.14

or

ch

α

p _t^p

i

ti−1

du− _t^p

i

ti−1

Dudu−p _t^p

i

ti−1

e^αu−t^pⁱdu _t^p

i

ti−1

e^αu−t^pⁱDudu

ch

α ti

t^p_i

e^αu−t^pⁱDudu− _t_i

t^p_i

Dudu

c1

_t^p

i

ti−1

pdu−c1

_t_i

ti−1

Dudu,

2.15

pch

α c₁

t^p_i −t_i−1

−ch

αe^−αt^pⁱ _t^p

i

ti−1

e^αu

p−Du du c_h

αe^−αt^pⁱ _t_i

t^p_i

e^αuDudu−c_h α

_t_i

t_i−1

Dudu−c₁ _t_i

t_i−1

Dudu.

2.16

This is equal, using2.7, to

pc_h α c₁

t^p_i −t_i−1

−c_h α c₁

ti

t_i−1

Dudu. 2.17

Now, the lemma follows from2.12and2.17.

Note thatLemma 2.1reduced the dependence of the inventory cost in periodifrom three variables to two variables. This reduction can be significant for ann-period model. Let

R x, y

: c_hαc₁ p

α²log

1α pe^−αx

_y

x

e^αtDtdt

− 1 α

_y

x

Dtdt

. 2.18

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Remark 2.2. Letα → 0, in2.18, and recall that log1xmay be expanded as x−1

2x²1 3x³−1

4x⁴1

5x⁵O x⁶

, 2.19

to get that asα → 0,Rx, yis equivalent to chαc1

y x

e^αt−x−1 α

Dtdt− 1 2p

y x

e^αt−xDtdt 2

, 2.20

which leads to the expression

c_h _y

x

t−xDtdt− 1 2p

_y

x

Dtdt 2

. 2.21

This expression may be found in Hill9, Omar and Smith10, and Rau and Ouyang11.

However, their interest in finding the optimal inventory policy for their model centered around treating special cases for demand rate functions or devising heuristics.

The total inventory costs wherenordered are made may be written as follows:

TCt1, . . . , t_n nKⁿ

i1

Rti−1, t_i, 2.22

which is given by2.18.

The objective now is to findnandt1, . . . , tnwhich minimizes TC subject tot00< t1<

· · · < t_n H. The problem becomes a mixed integer programming problem. The approach that we will use to solve it is based on a procedure developed by Benkherouf and Gilding4.

The next section contains the ingredients of the approach.

3. Technical Preliminaries

This section contains a summary of the work of4needed to tackle the problem of this paper.

Proofs of the results are omitted. Interested readers may consult4.

Consider the problem

P : TCt1, . . . , tn;n nKⁿ

i1

Riti−1, ti, 3.1

subject to

0t₀< t₁<· · ·< t_nH. 3.2 It was shown in4that, under some technical conditions, the optimization problemPhas a unique optimal solution which can be found from solving a system of nonlinear equations

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derived from the first-order optimality condition. To be precise, lett0 0 andtn Hand ignore the rest of the constraints3.2.

Write

Sn:ⁿ

i1

Riti−1, ti. 3.3

Assuming thatR_isare twice diﬀerentiable, then, for fixedn, the optimal solution inPsubject to3.2reduces to minimizingS_n.

Use the notation∇for the gradient, then setting∇TCt1, . . . , t_n;n 0 gives

∇TC_i ∂Ri_yti−1, ti ∂Ri1_xti, t_i1 0, i1, . . . , n−1. 3.4 Two sets of hypotheses were put forward in4.

Hypothesis 1. The functionsRisatisfy, fori1, . . . , nandy > x, 1Rix, y>0,

2Rix, x 0,

3 ∂Ri_xx, y<0<∂Ri_yx, y, 4 ∂x∂yR_ix, y<0.

Hypothesis 2. Define

Lxz∂²_xz∂_x∂_yzfx∂xz, Lyz∂²_yz∂_x∂_yzf

y

∂_yz, 3.5

then there is a continuous functionf such thatLxR_i ≥ 0,LyR_i ≥ 0 for alli 1, . . . , n, and

∂Ri_y ∂Ri1_x0 on the boundary of the feasible set.

The next theorem shows that under assumptions in Hypotheses1and2, the function S_nhas a unique minimum.

Theorem 3.1. The system3.4has a unique solution subject to3.2. Furthermore, this solution is the solution of3.1subject to3.2. Recall that a functionSnis convex innif

S_n2−S_n1≥S_n1−S_n. 3.6

This is equivalent to

1

2SnS_n2≥S_n1. 3.7

Theorem 3.2. Ifsn denotes the minimum objective value of 3.1subject to3.2andRix, y Rx, y, thens_nis convex inn.

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Based on the convexity property ofsn, the optimal number of cyclesn^∗is given by

n^∗min{n≥1 :s_n1−s_n>0}. 3.8

Now to solve3.4atin−1,

∂R_n−1ytn−2, t_n−1 ∂Rn_xtn−1, H 0. 3.9

Assume thatt_n−1is known, according to4,t_n−2can be found uniquely as a function oft_n−1. Repeating this process fori n−2, down toi 1, t_n−3,. . .,t_n are a function oft_n−1. So, the search for the optimal solution of 3.5can be conducted using a univariate search method.

4. Optimal Production Plan

This section is concerned with the optimal inventory policy for the production inventory model. The model has been introduced inSection 2. This section will investigate the extent to which the functionRgiven by2.18satisfies Hypotheses1and2,

R x, y

chαc1 p

α²log

1α p

_y

x

e^αt−xDtdt

− 1 α

_y

x

Dtdt

. 4.1

Without loss of generality, we will setc_hαc₁to 1. As this will have no eﬀect on the solution of the optimization problem whereKneeds to be replaced byK/chαc1, therefore, we set

R x, y

p α²log

1α

p _y

x

e^αt−xDtdt

− 1 α

_y

x

Dtdt. 4.2

Write

G x, y

α p

_y

x

e^αt−xDtdt. 4.3

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Direct computations then lead to

∂xR p α²

∂xG 1G 1

αDx,

∂_yR p α²

∂_yG 1G − 1

αD y

,

∂_x∂_yR p α²

∂_x∂_yG1G−∂xG

∂_yG 1G² ,

∂²_xR p α²

∂²_xG1G−∂xG² 1G² 1

αDx,

∂²_yR p α²

∂²_yG1G−∂xG² 1G² −1

αD y

,

∂_xG−α

GDx

p

,

∂yG α

pe^αy−xD y

,

∂_x∂_yG−α∂yG,

∂²_xG−α

−αG−αDx

p Dx p

,

∂²_yGα∂yGα

pe^αy−xD y

.

4.4

The following result indicates thatRobtained in4.2satisfies Hypothesis1.

Lemma 4.1. The functionRsatisfiesHypothesis 1.

Proof. It is clear that for anyx∈0, H,

Rx, x 0. 4.5

Now, direct computations show that

∂xR x, y

G x, y α

1G x, y

−pDx

. 4.6

Butp > Dx, therefore∂xRx, y<0 sinceGx, y>0 fory > x. Also, it can be shown that

∂yR x, y

D y α

e^αy−x 1G

x, y−1

. 4.7

We claim that∂yRx, y>0. Indeed, the claim is equivalent to Fx

y

1G x, y

−e^αy−x<0 fory > x. 4.8

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The functionFxyis decreasing since

F_x y

−αe^αy−x

1−D y p

, 4.9

withF_xx 0. Hence, the claim is true. To complete the proof, we need to examine the sign of∂x∂yR. Again, some algebra leads to

∂_x∂_yR x, y

−α∂yG x, y

1−D y p

. 4.10

But∂yGx, y > 0, andp > Dx. Therefore,∂x∂yRx, y < 0, fory > x, and the proof is complete.

Before we proceed further, we set

Z_xu Du

Du{12Gx, u}, with 0≤x≤u≤H. 4.11

We assume the following.

A1The functionZ_xis nonincreasing.

Note that asα → 0,Gx, u → 0, and consequentlyZ_xureduces toDu/Du. In other words, assumptionA1implies thatDis logconcave. This property of the demand rate function may be found in4,6, when considering models with infinite production rates. As a matter of fact, this property ofDcan also be obtained if we letp → ∞.

Example 4.2. LetDu αe^βu, whereα >0 and is known andβ >0 and is known, thenZxu is nonincreasing.

Note thatZ_xis nonincreasing which is equivalent to 1/Z_xnon-decreasing. We have

gxu: 1

Z_xu Du{12Gx, u}

Du 1

β{12Gx, u}, 4.12

withg_xu 2/β∂yGx, y>0, which implies the result.

Example 4.3. LetDu abu, whereb >0, then it is an easy exercise to check that assumption A1is satisfied.

Lemma 4.4. IfZ_xsatisfies (A1) for all 0≤x≤H, thenLxR≥0, whereLxis defined in3.5.

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Proof. Tedious but direct algebra using the definition ofLxRleads to

LxR 1−

Dx/p

1G x, y₂

α

_y

x

e^αt−xDtdtDx−e^αy−xD y 1

α

DxG x, y 1G

x, y − p α

G x, y 1G

x, y

1−Dx p

fx.

4.13

LxR≥0 is equivalent to 1−

Dx/p

1G x, y

α

_y

x

e^αt−xDtdtDx−e^αy−xD y 1

αDxG x, y

−p αG

x, y

1−Dx p

fx≥0.

4.14

Let

fx −Dx

Dx Dx

p 1−

Dx/p. 4.15

It can be shown that4.14is true if α

p

e^αy−xD y

−Dx−α _x^ye^αt−xDtdt G

x, y 1G

x, y ≤ Dx

Dx. 4.16

Define, foru≥x.

Fxu e^αu−xDu−α _u

x

e^αt−xDtdt, H_xu α

p _u

x

e^αt−xDtdt

1α p

_u

x

e^αt−xDtdt

.

4.17

The left hand side of4.16may be written as α p

F_x y

−F_xx H_x

y

−H_xx. 4.18

This is equal by extended-mean value theorem toα/pF_xξ/H_xξfor somex < ξ≤y.

However,

F_xu αe^αu−xDu,

H_xu ∂yGx, u{12Gx, u}. 4.19

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Therefore, α p

F_xξ

H_xξ Dξ

Dξ{12Gx, ξ} ≤ Dx

Dx{12Gx, x} Dx

Dx. 4.20

The last inequality follows from assumptionA1. This completes the proof.

Now, set for 0≤u≤y≤H, Vyu −αe^αy−u

e^αy−uD y

−Du−α _u^ye^αt−uDtdt

, 4.21

W_yu − 1G

u, y

e^αy−u−1−G u, y

. 4.22

The next assumption is needed forLyR≥0 of Hypothesis2to hold.

A2V_y/W_y is non-decreasing.

Assumption A2 is technical and is needed to complete the result of the paper.

This assumption may seem complicated but, it is not diﬃcult to check it numerically using MATLAB or Mathematica,say, once the demand rate function is known. Moreover, it can be shown that asα → 0,A2reduces to the condition that the function

F Du 1−

Du/p 4.23

is non-decreasing. This property is satisfied by linear and exponential demand rate functions.

In fact, assumptionA1is also, in this case, satisfied whenDis linear or exponential.

Lemma 4.5. If assumption (A2) is satisfied, thenLyR≥0.

Proof. Recall that

LyR∂²_yR∂x∂yRf y

∂yR. 4.24

Direct and tedious computation leads to

LyR x, y

1 α

∂_yG x, y 1G

x, y₂

×

α _y

x

e^αt−xDtdt−

e^αy−xD y

−Dx

−1 αD

y 1

α

e^αy−xD y 1G

x, y 1

αf y

D y

e^αy−x 1G

x, y −1

.

4.25

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Recall the definition of the functionfin4.15. ThenLyR≥0 is equivalent to

∂yG x, y 1G

x, y

e^αy−xD y

−Dx−α _y

x

e^αt−xDtdt

≤ D y

D y p

1− D

y /p

e^αy−x−1−G x, y

,

4.26

or, by∂yGand4.8, we get that the requirementLyR≥0 leads to

αe^αy−x

e^αy−xD y

−Dx−α _x^ye^αt−xDtdt 1G

x, y

e^αy−x−1−G

x, y ≤ D

y 1−

D y

/p. 4.27

The left hand side of4.27is equal toVyy−V_yx/Wyy−W_yx, whereV_yandW_y are given by4.21, and4.22respectively.

Computations show that

V_yu 2α²e^αy−u

e^αy−uD y

−Du−α _y

u

e^αt−uDtdt αe^αy−uD

y , W_yu −∂xG

u, y

e^αy−u−1−G u, y

− 1G

u, y

−αe^αy−u−∂xG u, y

.

4.28

Now, the extended-mean value theorem gives that V_y

y

−V_yx W_y

y

−W_yx V_yξ

W_yξ, for somex < ξ < y. 4.29 But assumptionA2 implies thatV_yξ/W_yξ ≤ V_yy/W_yy, where the right hand side of the above inequality is equal to

αD y

−

−αα D

y

/p D

y 1−α

D y

/p. 4.30

This is the right hand side of4.27. Hence,LyR≥0.

As a consequence of Lemmas4.4 and 4.5 and Theorem 3.1, we have the following result.

Theorem 4.6. Under the requirements that assumptions (A1) and (A2) hold the function Sn _N

i1Rt_i−1, t_i, with t₀ 0 < t₁ < · · · < t_n, has a unique minimum, this minimum can be found using the iterative procedure mentioned in [4].

Lets_nbe the minimal value ofS_n, then the next theorem follows fromTheorem 3.2.

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Theorem 4.7. The functionsnis convex inn.

As a consequence ofTheorem 4.7, the search for the optimal inventory policy can be conducted in two grids: the integer grid andRⁿ. That is, for fixed integern, the corresponding optimal times are found from the solution of the system of nonlinear equations3.4with corresponding objective value s_n. Then, the optimal value of ncan be obtained using the following corollary.

Corollary 4.8. The optimal number of production periodn^∗is such that 1ifK > s1−s2, thenn^∗1,

2if there exists anN≥2 such thats_N−1−s_N> K > s_N−s_N1, thenn^∗N, 3if there exists anN≥1 such thatKs_N−s_N1, thenn^∗Nandn^∗N1.

5. Conclusion

This paper was concerned with finding the economic-production-lot-size policy for an inventory model with deteriorating items. An optimal inventory policy was proposed for a class of cost functions named OHD models. The proposed optimality approach was based on an earlier work in4. The extension to OHD models should not pose any diﬃculty. Indeed, note that by comparing2.5and2.6, the OHP and OHD models diﬀer in the expression

−c1

_t_i

t_i−1

Dudu. 5.1

Now, consider the optimization problem2.22withRgiven by2.6. It is clear that adding−c1 H

0 Dudu−c1_n

i1 ti

ti−1Duduwill have no eﬀect on the optimization problem.

Consequently, the results obtained for the OHP model apply to the OHD model.

Before we close, we revisit paper1 and note that the model in 1 allows for the purchasing cost to vary with time, and therefore with fixed unit cost and no deterioration, the model in1is a special case of the model of the present paper. The reduction2.18in the present paper allows a direct approach as though the problem on hand is an unconstrained optimization problem. The approach adopted in1is the standard approach for constrained nonlinear programming problem. The key result in 1 is Theorem 1 Page 993 which adapted to the model of this paper withθ 0 requires that c_h1−Dt/p > 0 to hold.

This is satisfied sinceDt < p. Theorem 1 in1states, with no conditions imposed onD, that for fixed n the optimal inventory policy is uniquely determined as a solution of the first order condition of the optimization problem on hand. A result similar toTheorem 3.2 related to convexity of the corresponding objective value with respect tonis also presented.

The following counterexample shows that Theorem 1 in 1 cannot be entirely correct in its present form. Indeed, for simplicity let n 2, then the problem treated in 1 reduces equivalentlyto minimizing2.22withRgiven by2.21. The objective function in this case is a function of a single variable. TakeDt 2 sin10t 2 cos10t 4,c_h1, andH4.27, and ignore the setup cost.Figure 2shows the plot of the objective function. It is clear that multiple critical points can be observed as well as multiple optima. The remark on1also applies to part of Balkhi12.

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1 2 3 4 6

7 8 9 10 11

Figure 2: Behaviour of the objective function whenDt 2 sin10t 2 cos10t 4.

It is worth noting that the keys to success in applying the approach in 4 are the separability of the cost functions between periods and Hypotheses1and2. With this in mind, we believe that the approach of this paper to models with shortages and possibly with costs that are a function of time are possible. The technical requirement needed to generalize the results will be slightly more involved but essentially similar.

Acknowledgments

The authors would like to thank three anonymous referees for helpful comments on an earlier version of the paper.

References

1 J. T. Teng, L. Y. Ouyang, and C. T. Chang, “Deterministic economic production quantity models with time-varying demand cost,” Applied Mathematical Modelling, vol. 29, pp. 987–1003, 2005.

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Lakdere Benkherouf and Dalal Boushehri

Research Article

Optimal Policies for a Finite-Horizon Production Inventory Model

Lakdere Benkherouf and Dalal Boushehri

1. Introduction

2. Mathematical Model

3. Technical Preliminaries

4. Optimal Production Plan

5. Conclusion

Acknowledgments

References

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Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific World Journal

Discrete Mathematics

Stochastic Analysis