Optimal Policies for a Finite-Horizon Production Inventory Model

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.


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 Advances in Operations Research 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.Section 3 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.

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 period i production cycle , say, starting at time t i−1 and ending at time t i , 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 time t, 0 < D t < p, α: constant deteriorating rate of inventory items with α > 0, t p i : the time at which the inventory level reaches it is maximum in the ith production cycle, K: set up cost for the inventory model, c 1 : the cost of one unit of the item with c 1 > 0, c h : carrying cost per inventory unit held in the model per unit time, TC: total system cost during H.
Figure 1 shows the changes of the level of stock for a typical production period.Let I 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 e αu D u du.

2.11
Therefore, e αu D u du .

2.12
Lemma 2.1.The expression of the cost in 2.5 is equal to Proof.Applying integration by parts, we get that 2.5 reduces to

2.16
This is equal, using 2.7 , to Now, the lemma follows from 2.12 and 2.17 .
Note that Lemma 2.1 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

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Remark 2.2.Let α → 0, in 2.18 , and recall that log 1 x may be expanded as to get that as α → 0, R x, y is equivalent to

2.21
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 where n ordered are made may be written as follows: which is given by 2.18 .
The objective now is to find n and t 1 , . . ., t n which minimizes TC subject to t 0 0 < t 1 < • • • < 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 Gilding 4 .The next section contains the ingredients of the approach.

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 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 t 0 0 and t n H and ignore the rest of the constraints 3.2 .Write Assuming that R i s are twice differentiable, then, for fixed n, the optimal solution in P subject to 3.2 reduces to minimizing S n .
Use the notation ∇ for the gradient, then setting ∇TC t 1 , . . ., t n ; n 0 gives Two sets of hypotheses were put forward in 4 .
Hypothesis 1.The functions R i satisfy, for i 1, . . ., n and y > x, Hypothesis 2. Define then there is a continuous function f such that L x R i ≥ 0, L y R i ≥ 0 for all i 1, . . ., n, and ∂R i y ∂R i 1 x 0 on the boundary of the feasible set.
The next theorem shows that under assumptions in Hypotheses 1 and 2, the function S n 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 S n is convex in n if 3.6 This is equivalent to

3.7
Theorem 3.2.If s n denotes the minimum objective value of 3.1 subject to 3.2 and R i x, y R x, y , then s n is convex in n.

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Based on the convexity property of s n , the optimal number of cycles n * is given by Now to solve 3.4 at i n − 1, Assume that t n−1 is known, according to 4 , t n−2 can be found uniquely as a function of t n−1 .Repeating this process for i n − 2, down to i 1, t n−3 ,. ..,t n are a function of t n−1 .So, the search for the optimal solution of 3.5 can be conducted using a univariate search method.

Optimal Production Plan
This section is concerned with the optimal inventory policy for the production inventory model.The model has been introduced in Section 2. This section will investigate the extent to which the function R given by 2.18 satisfies Hypotheses 1 and 2,

4.4
The following result indicates that R obtained in 4.2 satisfies Hypothesis 1. Proof.It is clear that for any x ∈ 0, H , R x, x 0.

4.5
Now, direct computations show that But p > D x , therefore ∂ x R x, y < 0 since G x, y > 0 for y > x.Also, it can be shown that We claim that ∂ y R x, y > 0. Indeed, the claim is equivalent to

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The function F x y is decreasing since with F x x 0. Hence, the claim is true.To complete the proof, we need to examine the sign of ∂ x ∂ y R. Again, some algebra leads to But ∂ y G x, y > 0, and p > D x .Therefore, ∂ x ∂ y R x, y < 0, for y > x, and the proof is complete.
Before we proceed further, we set We assume the following.

A1
The function Z x is nonincreasing.
Note that as α → 0, G x, u → 0, and consequently Z x 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. Let D u αe βu
, where α > 0 and is known and β > 0 and is known, then Z x u is nonincreasing.
Note that Z x is nonincreasing which is equivalent to 1/Z x non-decreasing.We have with g x u 2/β ∂ y G x, y > 0, which implies the result.
Example 4.3.Let D u a bu, where b > 0, then it is an easy exercise to check that assumption A1 is satisfied.Lemma 4.4.If Z x satisfies (A1) for all 0 ≤ x ≤ H, then L x R ≥ 0, where L x is defined in 3.5 .

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Proof.Tedious but direct algebra using the definition of L x R leads to 4.13

4.17
The left hand side of 4.16 may be written as This is equal by extended-mean value theorem to α/p F x ξ / H x ξ for some x < ξ ≤ y.However, The last inequality follows from assumption A1 .This completes the proof. 21 The next assumption is needed for L y R ≥ 0 of Hypothesis 2 to hold.
A2 V 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 as α → 0, A2 reduces to the condition that the function is non-decreasing.This property is satisfied by linear and exponential demand rate functions.In fact, assumption A1 is also, in this case, satisfied when D is linear or exponential.
Proof.Recall that

4.24
Direct and tedious computation leads to

4.28
Now, the extended-mean value theorem gives that W y ξ , for some x < ξ < y.

4.29
But assumption A2 implies that V y ξ /W y ξ ≤ V y y /W y y , where the right hand side of the above inequality is equal to , has a unique minimum, this minimum can be found using the iterative procedure mentioned in [4].
Let s n be the minimal value of S n , then the next theorem follows from Theorem 3.2.

Theorem 4.7. The function s n is convex in n.
As a consequence of Theorem 4.7, 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 s n .Then, the optimal value of n can be obtained using the following corollary.
Corollary 4.8.The optimal number of production period n * is such that

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 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 c h 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 2 sin 10t 2 cos 10t 4, c h 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 .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.
−x D t dt .
−x D t dt − e α y−x D y − D x y e α y−x 1 G x, y − 1 .

1 −D y 1 −
of the function f in 4.15 .Then L y R ≥ 0 is equivalent to ∂ y G x, y 1 G x, y e α y−x D y − D x − α D y /p e α y−x − 1 − G x, y , 4.26 or, by ∂ y G and 4.8 , we get that the requirement L y R ≥ 0 leads to α e α y−x e α y−x D y − D x − α y x e α t−x D t dt 1 G x, y e α y−x − 1 − G x, y ≤ D y /p .4.27 The left hand side of 4.27 is equal to V y y − V y x / W y y − W y x , where V y and W y are given by 4.21 , and 4.22 respectively.Computations show that V y u 2α 2 e α y−u e α y−u D y − D u − α y u e α t−u D t dt αe α y−u D y , W y u −∂ x G u, y e α y−u − 1 − G u, y − 1 G u, y −αe α y−u − ∂ x G u, y .

Theorem 4 . 6 .
right hand side of 4.27 .Hence, L y R ≥ 0. As a consequence of Lemmas 4.4 and 4.5 and Theorem 3.1, we have the following result.Under the requirements that assumptions (A1) and (A2) hold the function S n N

1 H
the optimization problem 2.22 with R given by 2.6 .It is clear that adding −c 1 D u du will have no effect on the optimization problem.Consequently, the results obtained for the OHP model apply to the OHD model.