# Optimal Policies for a Finite-Horizon Production Inventory Model

Authors: Lakdere Benkherouf; Dalal Boushehri
Journal: Advances in Operations Research
 (2012)

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## Abstract

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.


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## Body

## 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 Giri [2], and Sana et al. [3] and the references therein.Earlier models on finding optimal replenishment schedule for a finite planning horizon may be categorized as economic lot size (ELS) models dealing with replenishment only. The model treated in this paper is an extension of the economic production lot size (EPLS) to finite horizon models and time-varying demand. The model is close in spirit to that of [1]. However, in [1], 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 proposed (EPLS) schedule is not optimal.Recently, Benkherouf and Gilding [4] suggested 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 motivated by applications to (ELS) models. 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 Omar [8]. The current paper presents another extension of the procedure to (EPLS) 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 in Section 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. Section3 contains some preliminaries on the procedure of Benkherouf and Gilding [4]. The main results are contained in Section 4. Section 5 is concerned with some general remarks and conclusion.


## 2. Mathematical Model

The model treated in this paper is based on the following assumptions:(1)
the planning horizon is finite;(2)
a single item is considered;(3)
products are assumed to experience deterioration while in stock;(4)
shortages are not permitted;(5)
initial inventory at the beginning of planning horizon is zero, also the inventory depletes to zero at the end of the planning horizon;(6)
the demand function is strictly positive;We will initially look at a single periodi (production cycle), say, starting at time ti-1 and ending at time ti,  i=1,2,…. Some of the notations used in the model are as follows: H:
the total planning horizon,p:
the constant production rate,D(t):
the demand rate at timet, 0<D(t)<p,α:
constant deteriorating rate of inventory items withα>0,tip:
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.Figure1 shows the changes of the level of stock for a typical production period.Figure 1
The changes of inventory levels of various components of the model for a typical production batch.LetI(t) be the level of stock at time t. The change in period i, the level of inventory, may be described by the following differential equation i,i=1,2,…,(2.1)I′(t)=p-D(t)-αI(t),ti-1≤t<tip,I(t)↓0ast↓ti-1.
The solution to (2.1) is given by(2.2)I(t)=e-αt∫ti-1teαu{p-D(u)}du,ti-1≤t<tip,(2.3)I′(t)=-D(t)-αI(t),tip≤t<ti,I(t)↑0ast↑ti.
The solution to (2.3) is given by(2.4)I(t)=e-αt∫ttieαuD(u)du,tip≤t<ti.
The total costs, (excluding the setup cost) for period i, which consist of holding cost and deterioration cost are given by(2.5)ch∫ti-1tipe-αt[∫ti-1teαu{p-D(u)}du]dt+ch∫tiptie-αt{∫ttieαuD(u)du}dt,+c1{∫ti-1tippdu-∫ti-1tiD(u)du}.
We will call models with this cost OHD models. It is possible to consider instead of (2.5) the form(2.6)ch∫ti-1tipe-αt[∫ti-1teαu{p-D(u)}du]dt+ch∫tiptie-αt{∫ttieαuD(u)du}dt+c1∫ti-1tippdu,
which considers only holding and purchasing costs, where the expression c1∫ti-1tippdu represents the purchasing cost. We call this  OHP models.Note that since the functionI is continuous at tip, we have(2.7)e-αtip∫ti-1tipeαu{p-D(u)}du=e-αtip∫tiptieαuD(u)du,
or(2.8)∫ti-1tipeαu{p-D(u)}du=∫tiptieαuD(u)du,then(2.9)p∫ti-1tipeαudu=∫ti-1tieαuD(u)du,or(2.10)pα(eαtip-eαti-1)=pαeαti-1{eα(tip-ti-1)-1}=∫ti-1tieαuD(u)du,(2.11)eα(tip-ti-1)-1=αpe-αti-1∫ti-1tieαuD(u)du.Therefore,(2.12)tip-ti-1=1αlog⁡⁡{1+αpe-αti-1∫ti-1tieαuD(u)du}.Lemma 2.1.
The expression of the cost in (2.5) is equal to
(2.13)(ch+αc1)[pα2log⁡⁡{1+αpe-αti-1∫ti-1tieαuD(u)du}-1α∫ti-1tiD(u)du].Proof.
Applying integration by parts, we get that (2.5) reduces to
(2.14)chα∫ti-1tip{1-eα(u-tip)}{p-D(u)}du+chα∫tipti{eα(u-tip)-1}D(u)du+c1∫ti-1tippdu-c1∫ti-1tiD(u)du,
or
(2.15)chα{p∫ti-1tipdu-∫ti-1tipD(u)du-p∫ti-1tipeα(u-tip)du+∫ti-1tipeα(u-tip)D(u)du}+chα{∫tiptieα(u-tip)D(u)du-∫tiptiD(u)du}+c1∫ti-1tippdu-c1∫ti-1tiD(u)du,(2.16)=p(chα+c1)(tip-ti-1)-chαe-αtip∫ti-1tipeαu{p-D(u)}du+chαe-αtip∫tiptieαuD(u)du-chα∫ti-1tiD(u)du-c1∫ti-1tiD(u)du.
This is equal, using (2.7), to
(2.17)p(chα+c1)(tip-ti-1)-(chα+c1)∫ti-1tiD(u)du.
Now, the lemma follows from (2.12) and (2.17).Note that Lemma2.1 reduced the dependence of the inventory cost in period i from three variables to two variables. This reduction can be significant for an n-period model. Let(2.18)R(x,y):=(ch+αc1)[pα2log⁡⁡{1+αpe-αx∫xyeαtD(t)dt}-1α∫xyD(t)dt].Remark 2.2.
Letα→0, in (2.18), and recall that log(1+x) may be expanded as
(2.19)x-12x2+13x3-14x4+15x5+O(x6),
to get that as α→0,  R(x,y) is equivalent to
(2.20)(ch+αc1)[∫xy{eα(t-x)-1α}D(t)dt-12p{∫xyeα(t-x)D(t)dt}2],
which leads to the expression
(2.21)ch[∫xy(t-x)D(t)dt-12p{∫xyD(t)dt}2].
This expression may be found in Hill [9], Omar and Smith [10], and Rau and Ouyang [11]. 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 wheren ordered are made may be written as follows:(2.22)TC(t1,…,tn)=nK+∑i=1nR(ti-1,ti),
which is given by (2.18).The objective now is to findn and t1,…,tn which minimizes TC subject to t0=0<t1<⋯<tn=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 Gilding [4]. The next section contains the ingredients of the approach.


## 3. Technical Preliminaries

This section contains a summary of the work of [4] needed to tackle the problem of this paper. Proofs of the results are omitted. Interested readers may consult [4].Consider the problem(3.1)P:TC(t1,…,tn;n)=nK+∑i=1nRi(ti-1,ti),
subject to(3.2)0=t0<t1<⋯<tn=H.
It was shown in [4] that, under some technical conditions, the optimization problem (P) has a unique optimal solution which can be found from solving a system of nonlinear equations derived from the first-order optimality condition. To be precise, let t0=0 and tn=H and ignore the rest of the constraints (3.2).Write(3.3)Sn:=∑i=1nRi(ti-1,ti).
Assuming that Ri′s are twice differentiable, then, for fixed n, the optimal solution in (P) subject to (3.2) reduces to minimizing Sn.Use the notation∇ for the gradient, then setting ∇TC(t1,…,tn;n)=0 gives(3.4)(∇TC)i=(∂Ri)y(ti-1,ti)+(∂Ri+1)x(ti,ti+1)=0,i=1,…,n-1.
Two sets of hypotheses were put forward in [4].Hypothesis 1.
The functionsRi satisfy, for i=1,…,n and y>x,(1)
Ri(x,y)>0,(2)
Ri(x,x)=0,(3)
(∂Ri)x(x,y)<0<(∂Ri)y(x,y),(4)
(∂x∂yR)i(x,y)<0.Hypothesis 2.
Define(3.5)Lxz=∂x2z+∂x∂yz+f(x)∂xz,Lyz=∂y2z+∂x∂yz+f(y)∂yz,
then there is a continuous function f such that ℒxRi≥0,  ℒyRi≥0 for all i=1,…,n, and (∂Ri)y+(∂Ri+1)x=0 on the boundary of the feasible set.The next theorem shows that under assumptions in Hypotheses1 and 2, the function Sn has a unique minimum.Theorem 3.1.
The system (3.4) has a unique solution subject to (3.2). Furthermore, this solution is the solution of (3.1) subject to (3.2).  Recall that a function Sn is convex in n if
(3.6)Sn+2-Sn+1≥Sn+1-Sn.
This is equivalent to
(3.7)12(Sn+Sn+2)≥Sn+1.Theorem 3.2.
Ifsn denotes the minimum objective value of (3.1) subject to (3.2) and  Ri(x,y)=R(x,y), then sn is convex in n.Based on the convexity property ofsn, the optimal number of cycles n* is given by(3.8)n*=min⁡⁡{n≥1:sn+1-sn>0}.Now to solve (3.4) at  i=n-1,(3.9)∂Rn-1y(tn-2,tn-1)+(∂Rn)x(tn-1,H)=0.Assume thattn-1 is known, according to [4], tn-2 can be found uniquely as a function of tn-1. Repeating this process for i=n-2, down to i=1,tn-3,…,tn are a function of tn-1. So, the search for the optimal solution of (3.5) can 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 in Section2. This section will investigate the extent to which the function R given by (2.18) satisfies Hypotheses 1 and 2,(4.1)R(x,y)=(ch+αc1)[pα2log⁡⁡{1+αp∫xyeα(t-x)D(t)dt}-1α∫xyD(t)dt].Without loss of generality, we will setch+αc1 to 1. As this will have no effect on the solution of the optimization problem where K needs to be replaced by K/(ch+αc1), therefore, we set(4.2)R(x,y)=pα2log⁡⁡{1+αp∫xyeα(t-x)D(t)dt}-1α∫xyD(t)dt.
Write(4.3)G(x,y)=αp∫xyeα(t-x)D(t)dt.
Direct computations then lead to(4.4)∂xR=pα2∂xG1+G+1αD(x),∂yR=pα2∂yG1+G-1αD(y),∂x∂yR=pα2∂x∂yG(1+G)-(∂xG)(∂yG)(1+G)2,∂x2R=pα2∂x2G(1+G)-(∂xG)2(1+G)2+1αD′(x),∂y2R=pα2∂y2G(1+G)-(∂xG)2(1+G)2-1αD′(y),∂xG=-α{G+D(x)p},∂yG=αpeα(y-x)D(y),∂x∂yG=-α∂yG,∂x2G=-α{-αG-αD(x)p+D′(x)p},∂y2G=α∂yG+αpeα(y-x)D′(y).
The following result indicates that R obtained in (4.2) satisfies Hypothesis 1.Lemma 4.1.
The functionR satisfies Hypothesis 1.Proof.
It is clear that for anyx∈[0,H],
(4.5)R(x,x)=0.
Now, direct computations show that
(4.6)∂xR(x,y)=G(x,y)α{1+G(x,y)}{-p+D(x)}.
But  p>D(x), therefore ∂xR(x,y)<0 since  G(x,y)>0 for  y>x. Also, it can be shown that
(4.7)∂yR(x,y)=D(y)α{eα(y-x)1+G(x,y)-1}.
We claim that  ∂yR(x,y)>0. Indeed, the claim is equivalent to
(4.8)Fx(y)=1+G(x,y)-eα(y-x)<0fory>x.
The function Fx(y) is decreasing since
(4.9)Fx′(y)=-αeα(y-x){1-D(y)p},
with  Fx(x)=0. Hence, the claim is true. To complete the proof, we need to examine the sign of  ∂x∂yR. Again, some algebra leads to
(4.10)∂x∂yR(x,y)=-α∂yG(x,y){1-D(y)p}.
But  ∂yG(x,y)>0, and p>D(x). Therefore, ∂x∂yR(x,y)<0, for y>x, and the proof is complete.Before we proceed further, we set(4.11)Zx(u)=D′(u)D(u){1+2G(x,u)},with0≤x≤u≤H.
We assume the following.(A1)
The functionZx is nonincreasing.Note that asα→0, G(x,u)→0, and consequently Zx(u) reduces to D′(u)/D(u). In other words, assumption (A1) implies that D is logconcave. This property of the demand rate function may be found in [4, 6], when considering models with infinite production rates. As a matter of fact, this property of D can also be obtained if we let  p→∞.Example 4.2.
LetD(u)=αeβu, where α>0 and is known and β>0 and is known, then Zx(u) is nonincreasing.Note thatZx is nonincreasing which is equivalent to 1/Zx non-decreasing. We have(4.12)gx(u):=1Zx(u)=D(u){1+2G(x,u)}D′(u)=1β{1+2G(x,u)},withgx′(u)=(2/β)∂yG(x,y)>0, which implies the result.Example 4.3.
LetD(u)=a+bu, where  b>0, then it is an easy exercise to check that assumption (A1) is satisfied.Lemma 4.4.
IfZx satisfies (A1) for all 0≤x≤H, then ℒxR≥0, where ℒx is defined in (3.5).Proof.
Tedious but direct algebra using the definition ofℒxR leads to
(4.13)LxR={1-(D(x)/p)}{1+G(x,y)}2{α∫xyeα(t-x)D(t)dt+D(x)-eα(y-x)D(y)}+1αD′(x)G(x,y)1+G(x,y)-pαG(x,y)1+G(x,y){1-D(x)p}f(x).ℒxR≥0 is equivalent to
(4.14){1-(D(x)/p)}{1+G(x,y)}{α∫xyeα(t-x)D(t)dt+D(x)-eα(y-x)D(y)}+1αD′(x)G(x,y)-pαG(x,y){1-D(x)p}f(x)≥0.
Let
(4.15)f(x)=-D′(x)D′(x)+D′(x)p{1-(D(x)/p)}.
It can be shown that (4.14) is true if
(4.16)αpeα(y-x)D(y)-D(x)-α∫xyeα(t-x)D(t)dtG(x,y){1+G(x,y)}≤D′(x)D(x).
Define, for u≥x. 							(4.17)Fx(u)=eα(u-x)D(u)-α∫xueα(t-x)D(t)dt,Hx(u)=αp∫xueα(t-x)D(t)dt{1+αp∫xueα(t-x)D(t)dt}.
The left hand side of (4.16) may be written as
(4.18)αpFx(y)-Fx(x)Hx(y)-Hx(x).
This is equal by extended-mean value theorem to (α/p)((Fx′(ξ))/(Hx′(ξ))) for some x<ξ≤y. However,
(4.19)Fx′(u)=αeα(u-x)D′(u),Hx′(u)=∂yG(x,u){1+2G(x,u)}.
Therefore,
(4.20)αpFx′(ξ)Hx′(ξ)=D′(ξ)D(ξ){1+2G(x,ξ)}≤D′(x)D(x){1+2G(x,x)}=D′(x)D(x).
The last inequality follows from assumption (A1). This completes the proof.Now, set for0≤u≤y≤H,(4.21)Vy(u)=-αeα(y-u){∫eα(y-u)D(y)-D(u)-α∫uy∫eα(t-u)D(t)dt},(4.22)Wy(u)=-{1+G(u,y)}{eα(y-u)-1-G(u,y)}.
The next assumption is needed for ℒyR≥0 of Hypothesis 2 to hold.(A2)
Vy′/Wy′ 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, (A2) reduces to the condition that the function(4.23)F=D′(u)1-(D(u)/p)is non-decreasing. This property is satisfied by linear and exponential demand rate functions. In fact, assumption (A1) is also, in this case, satisfied whenD is linear or exponential.Lemma 4.5.
If assumption (A2) is satisfied, thenℒyR≥0.Proof.
Recall that(4.24)LyR=∂y2R+∂x∂yR+f(y)∂yR.
Direct and tedious computation leads to
(4.25)LyR(x,y)=1α∂yG(x,y){1+G(x,y)}2×[α∫xyeα(t-x)D(t)dt-{eα(y-x)D(y)-D(x)}]-1αD′(y)+1αeα(y-x)D′(y){1+G(x,y)}+1αf(y)D(y){eα(y-x)1+G(x,y)-1}.
Recall the definition of the function f in (4.15). Then ℒyR≥0 is equivalent to
(4.26)∂yG(x,y)1+G(x,y){eα(y-x)D(y)-D(x)-α∫xyeα(t-x)D(t)dt}≤D′(y)D(y)p{1-(D(y)/p)}{eα(y-x)-1-G(x,y)},
or, by (∂yG) and (4.8), we get that the requirement ℒyR≥0 leads to
(4.27)αeα(y-x){eα(y-x)D(y)-D(x)-α∫xyeα(t-x)D(t)dt}{1+G(x,y)}{eα(y-x)-1-G(x,y)}≤D′(y){1-(D(y)/p)}.
The left hand side of (4.27) is equal to  (Vy(y)-Vy(x))/(Wy(y)-Wy(x)),  where Vy and Wy are given by (4.21), and (4.22) respectively.
Computations show that(4.28)Vy′(u)=2α2eα(y-u){eα(y-u)D(y)-D(u)-α∫uyeα(t-u)D(t)dt}+αeα(y-u)D′(y),Wy′(u)=-∂xG(u,y){eα(y-u)-1-G(u,y)}-{1+G(u,y)}{-αeα(y-u)-∂xG(u,y)}.
Now, the extended-mean value theorem gives that
(4.29)Vy(y)-Vy(x)Wy(y)-Wy(x)=Vy′(ξ)Wy′(ξ),forsomex<ξ<y.
But assumption (A2) implies that (Vy′(ξ)/Wy′(ξ))≤(Vy′(y)/Wy′(y)), where the right hand side of the above inequality is equal to
(4.30)αD′(y)-{-α+α(D(y)/p)}=D′(y){1-α(D(y)/p)}.
This is the right hand side of (4.27). 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 functionSn=∑i=1NR(ti-1,ti), with t0=0<t1<⋯<tn, has a unique minimum, this minimum can be found using the iterative procedure mentioned in [4].Letsn be the minimal value of Sn, then the next theorem follows from Theorem 3.2.Theorem 4.7.
The functionsn is convex in  n.As a consequence of Theorem4.7, the search for the optimal inventory policy can be conducted in two grids: the integer grid and ℝ+n. That is, for fixed integer n, the corresponding optimal times are found from the solution of the system of nonlinear equations (3.4) with corresponding objective value sn. Then, the optimal value of n can be obtained using the following corollary.Corollary 4.8.
The optimal number of production periodn* is such that(1)
ifK>s1-s2, then n*=1,(2)
if there exists anN≥2 such that sN-1-sN>K>sN-sN+1, then n*=N,(3)
if there exists anN≥1 such that K=sN-sN+1, then n*=N and n*=N+1.


## 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 in [4]. The extension to OHD models should not pose any difficulty. Indeed, note that by comparing (2.5) and (2.6), the OHP and OHD models differ in the expression(5.1)-c1∫ti-1tiD(u)du.Now, consider the optimization problem (2.22) with R given by (2.6). It is clear that adding -c1∫0HD(u)du=-c1∑i=1n∫ti-1tiD(u)du will 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 paper [1] 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 in [1] is a special case of the model of the present paper. The reduction (2.18) in the present paper allows a direct approach as though the problem on hand is an unconstrained optimization problem. The approach adopted in [1] is 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 with θ=0 requires that ch(1-(D(t)/p))>0 to hold. This is satisfied since D(t)<p. Theorem  1 in [1] states, with no conditions imposed on D, 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 to Theorem 3.2 related to convexity of the corresponding objective value with respect to n is 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 (equivalently) to minimizing (2.22) with R given by (2.21). The objective function in this case is a function of a single variable. Take D(t)=2sin(10t)+2cos(10t)+4, ch=1, and  H≃4.27, and ignore the setup cost. Figure 2 shows the plot of the objective function. It is clear that multiple critical points can be observed as well as multiple optima. The remark on [1] also applies to part of Balkhi [12].Figure 2
Behaviour of the objective function whenD(t)=2sin(10t)+2cos(10t)+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 Hypotheses 1 and 2. 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.



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