Optimal Policies for a Finite-Horizon Production Inventory Model

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.

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. 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.

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 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,𝑐₁:the cost of one unit of the item with 𝑐₁>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.

Figure 1 

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

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=1,2,…,𝐼^′(𝑡)=𝑝−𝐷(𝑡)−𝛼𝐼(𝑡),𝑡_𝑖−1≤𝑡<𝑡^𝑝_𝑖,𝐼(𝑡)↓0as𝑡↓𝑡_𝑖−1. (2.1) The solution to (2.1) is given by𝐼(𝑡)=𝑒^{−𝛼𝑡𝑡}_𝑡𝑖−1𝑒^𝛼𝑢𝑝−𝐷(𝑢)𝑑𝑢,𝑡_𝑖−1≤𝑡<𝑡^𝑝_𝑖,𝐼^′(𝑡)=−𝐷(𝑡)−𝛼𝐼(𝑡),𝑡^𝑝_𝑖≤𝑡<𝑡_𝑖,𝐼(𝑡)↑0as𝑡↑𝑡_𝑖. (2.2)(2.3) The solution to (2.3) is given by𝐼(𝑡)=𝑒^{−𝛼𝑡𝑡𝑖}_𝑡𝑒^𝛼𝑢𝐷(𝑢)𝑑𝑢,𝑡^𝑝_𝑖≤𝑡<𝑡_𝑖. (2.4) The total costs, (excluding the setup cost) for period 𝑖, which consist of holding cost and deterioration cost are given by𝑐_ℎ^𝑡𝑝𝑖_𝑡𝑖−1𝑒^{−𝛼𝑡𝑡}_𝑡𝑖−1𝑒^𝛼𝑢𝑝−𝐷(𝑢)𝑑𝑢𝑑𝑡+𝑐_ℎ^𝑡𝑖_𝑡𝑝𝑖𝑒^{−𝛼𝑡𝑡𝑖}_𝑡𝑒^𝛼𝑢𝐷(𝑢)𝑑𝑢𝑑𝑡,+𝑐₁^𝑡𝑝𝑖_𝑡𝑖−1𝑝𝑑𝑢−^𝑡𝑖_𝑡𝑖−1𝐷(𝑢)𝑑𝑢. (2.5) We will call models with this cost OHD models. It is possible to consider instead of (2.5) the form𝑐_ℎ^𝑡𝑝𝑖_𝑡𝑖−1𝑒^{−𝛼𝑡𝑡}_𝑡𝑖−1𝑒^𝛼𝑢𝑝−𝐷(𝑢)𝑑𝑢𝑑𝑡+𝑐_ℎ^𝑡𝑖_𝑡𝑝𝑖𝑒^{−𝛼𝑡𝑡𝑖}_𝑡𝑒^𝛼𝑢𝐷(𝑢)𝑑𝑢𝑑𝑡+𝑐₁^𝑡𝑝𝑖_𝑡𝑖−1𝑝𝑑𝑢, (2.6) which considers only holding and purchasing costs, where the expression 𝑐₁∫^𝑡𝑝𝑖_𝑡𝑖−1𝑝𝑑𝑢 represents the purchasing cost. We call this OHP models.

Note that since the function 𝐼 is continuous at 𝑡^𝑝_𝑖, we have𝑒^{−𝛼𝑡𝑝𝑖𝑡𝑝𝑖}_𝑡𝑖−1𝑒^𝛼𝑢𝑝−𝐷(𝑢)𝑑𝑢=𝑒^{−𝛼𝑡𝑝𝑖𝑡𝑖}_𝑡𝑝𝑖𝑒^𝛼𝑢𝐷(𝑢)𝑑𝑢, (2.7) or^𝑡𝑝𝑖_𝑡𝑖−1𝑒^𝛼𝑢𝑝−𝐷(𝑢)𝑑𝑢=^𝑡𝑖_𝑡𝑝𝑖𝑒^𝛼𝑢𝐷(𝑢)𝑑𝑢, (2.8)

then𝑝^𝑡𝑝𝑖_𝑡𝑖−1𝑒^𝛼𝑢𝑑𝑢=^𝑡𝑖_𝑡𝑖−1𝑒^𝛼𝑢𝐷(𝑢)𝑑𝑢, (2.9)

or𝑝𝛼𝑒^{𝛼𝑡𝑝𝑖}−𝑒^{𝛼𝑡𝑖−1}=𝑝𝛼𝑒^{𝛼𝑡𝑖−1}𝑒^{𝛼(𝑡𝑝𝑖−𝑡𝑖−1)}−1=^𝑡𝑖_𝑡𝑖−1𝑒^𝛼𝑢𝐷(𝑢)𝑒𝑑𝑢,_{𝛼(𝑡𝑝𝑖−𝑡𝑖−1)}𝛼−1=𝑝𝑒^{−𝛼𝑡𝑖−1𝑡𝑖}_𝑡𝑖−1𝑒^𝛼𝑢𝐷(𝑢)𝑑𝑢. (2.10)(2.11)

Therefore,𝑡^𝑝_𝑖−𝑡_𝑖−1=1𝛼𝛼log1+𝑝𝑒^{−𝛼𝑡𝑖−1𝑡𝑖}_𝑡𝑖−1𝑒^𝛼𝑢𝐷(𝑢)𝑑𝑢. (2.12)

Lemma 2.1. The expression of the cost in (2.5) is equal to 𝑐_ℎ+𝛼𝑐₁𝑝𝛼²𝛼log1+𝑝𝑒^{−𝛼𝑡𝑖−1𝑡𝑖}_𝑡𝑖−1𝑒^𝛼𝑢𝐷(𝑢)−1𝑑𝑢𝛼^𝑡𝑖_𝑡𝑖−1𝐷(𝑢)𝑑𝑢. (2.13)

Proof. Applying integration by parts, we get that (2.5) reduces to 𝑐_ℎ𝛼^𝑡𝑝𝑖_𝑡𝑖−11−𝑒^{𝛼(𝑢−𝑡𝑝𝑖)}(𝑢+𝑐𝑝−𝐷)𝑑𝑢_ℎ𝛼^𝑡𝑖_𝑡^𝑝_𝑖𝑒^{𝛼(𝑢−𝑡𝑝𝑖)}−1𝐷(𝑢)𝑑𝑢+𝑐₁^𝑡𝑝𝑖_𝑡𝑖−1𝑝𝑑𝑢−𝑐₁^𝑡𝑖_𝑡𝑖−1𝐷(𝑢)𝑑𝑢, (2.14) or 𝑐_ℎ𝛼𝑝^𝑡𝑝𝑖_𝑡𝑖−1𝑑𝑢−^𝑡𝑝𝑖_𝑡𝑖−1𝐷(𝑢)𝑑𝑢−𝑝^𝑡𝑝𝑖_𝑡𝑖−1𝑒^{𝛼(𝑢−𝑡𝑝𝑖)}𝑑𝑢+^𝑡𝑝𝑖_𝑡𝑖−1𝑒^{𝛼(𝑢−𝑡𝑝𝑖)}𝐷(𝑢)+𝑐𝑑𝑢_ℎ𝛼^𝑡𝑖_𝑡^𝑝_𝑖𝑒^{𝛼(𝑢−𝑡𝑝𝑖)}𝐷(𝑢)𝑑𝑢−^𝑡𝑖_𝑡𝑝𝑖𝐷(𝑢)𝑑𝑢+𝑐₁^𝑡𝑝𝑖_𝑡𝑖−1𝑝𝑑𝑢−𝑐₁^𝑡𝑖_𝑡𝑖−1𝐷(𝑢)𝑐𝑑𝑢,=𝑝^ℎ𝛼+𝑐₁𝑡^𝑝_𝑖−𝑡_𝑖−1−𝑐_ℎ𝛼𝑒^{−𝛼𝑡𝑝𝑖𝑡𝑝𝑖}_𝑡𝑖−1𝑒^𝛼𝑢(𝑢+𝑐𝑝−𝐷)𝑑𝑢_ℎ𝛼𝑒^{−𝛼𝑡𝑝𝑖𝑡𝑖}_𝑡𝑝𝑖𝑒^𝛼𝑢𝐷(𝑢)𝑐𝑑𝑢−^ℎ𝛼^𝑡𝑖_𝑡𝑖−1𝐷(𝑢)𝑑𝑢−𝑐₁^𝑡𝑖_𝑡𝑖−1𝐷(𝑢)𝑑𝑢. (2.15)(2.16) This is equal, using (2.7), to 𝑝𝑐_ℎ𝛼+𝑐₁𝑡^𝑝_𝑖−𝑡_𝑖−1−𝑐_ℎ𝛼+𝑐₁^𝑡𝑖_𝑡𝑖−1𝐷(𝑢)𝑑𝑢. (2.17) 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. Let𝑅𝑥,𝑦∶=𝑐_ℎ+𝛼𝑐₁𝑝𝛼²𝛼log1+𝑝𝑒^{−𝛼𝑥𝑦}_𝑥𝑒^𝛼𝑡𝐷(𝑡)−1𝑑𝑡𝛼^𝑦_𝑥𝐷(𝑡)𝑑𝑡. (2.18)

Remark 2.2. Let 𝛼→0, in (2.18), and recall that log(1+𝑥) may be expanded as 1𝑥−2𝑥²+13𝑥³−14𝑥⁴+15𝑥⁵+𝑂𝑥⁶, (2.19) to get that as 𝛼→0, 𝑅(𝑥,𝑦) is equivalent to 𝑐_ℎ+𝛼𝑐₁^𝑦_𝑥𝑒^{𝛼(𝑡−𝑥)}−1𝛼𝐷(𝑡)1𝑑𝑡−2𝑝^𝑦_𝑥𝑒^{𝛼(𝑡−𝑥)}𝐷(𝑡)𝑑𝑡², (2.20) which leads to the expression 𝑐_ℎ^𝑦_𝑥()𝐷(𝑡)1𝑡−𝑥𝑑𝑡−2𝑝^𝑦_𝑥𝐷(𝑡)𝑑𝑡². (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:TC𝑡₁,…,𝑡_𝑛=𝑛𝐾+^𝑛_𝑖=1𝑅𝑡_𝑖−1,𝑡_𝑖, (2.22) which is given by (2.18).

The objective now is to find 𝑛 and 𝑡₁,…,𝑡_𝑛 which minimizes TC subject to 𝑡₀=0<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 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 problem𝐏∶TC𝑡₁,…,𝑡_𝑛;𝑛=𝑛𝐾+^𝑛_𝑖=1𝑅_𝑖𝑡_𝑖−1,𝑡_i, (3.1) subject to0=𝑡₀<𝑡₁<⋯<𝑡_𝑛=𝐻. (3.2) 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 𝑡_𝑛=𝐻 and ignore the rest of the constraints (3.2).

Write𝑆_𝑛∶=^𝑛_𝑖=1𝑅_𝑖𝑡_𝑖−1,𝑡_𝑖. (3.3) Assuming that 𝑅^′_𝑖𝑠 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(𝑡₁,…,𝑡_𝑛;𝑛)=0 gives(∇TC)_𝑖=𝜕𝑅_𝑖𝑦𝑡_𝑖−1,𝑡_𝑖+𝜕𝑅_𝑖+1𝑥𝑡_𝑖,𝑡_𝑖+1=0,𝑖=1,…,𝑛−1. (3.4) Two sets of hypotheses were put forward in [4].

Hypothesis 1. The functions 𝑅_𝑖 satisfy, for 𝑖=1,…,𝑛 and 𝑦>𝑥,(1)𝑅_𝑖(𝑥,𝑦)>0,(2)𝑅_𝑖(𝑥,𝑥)=0, (3)(𝜕𝑅_𝑖)_𝑥(𝑥,𝑦)<0<(𝜕𝑅_𝑖)_𝑦(𝑥,𝑦), (4)(𝜕_𝑥𝜕_𝑦𝑅)_𝑖(𝑥,𝑦)<0.

Hypothesis 2. Define ℒ_𝑥𝑧=𝜕²_𝑥𝑧+𝜕_𝑥𝜕_𝑦𝑧+𝑓(𝑥)𝜕_𝑥ℒ𝑧,_𝑦𝑧=𝜕²_𝑦𝑧+𝜕_𝑥𝜕_𝑦𝑧+𝑓𝑦𝜕_𝑦𝑧, (3.5) then there is a continuous function 𝑓 such that ℒ_𝑥𝑅_𝑖≥0, ℒ_𝑦𝑅_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 𝑆_𝑛+2−𝑆_𝑛+1≥𝑆_𝑛+1−𝑆_𝑛. (3.6) This is equivalent to 12𝑆_𝑛+𝑆_𝑛+2≥𝑆_𝑛+1. (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.

Based on the convexity property of 𝑠_𝑛, the optimal number of cycles 𝑛^∗ is given by𝑛^∗=min𝑛≥1∶𝑠_𝑛+1−𝑠_𝑛>0. (3.8)

Now to solve (3.4) at 𝑖=𝑛−1,𝜕𝑅_𝑦𝑛−1𝑡_𝑛−2,𝑡_𝑛−1+𝜕𝑅_𝑛𝑥𝑡_𝑛−1,𝐻=0. (3.9)

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.

4. 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,𝑅𝑥,𝑦=𝑐_ℎ+𝛼𝑐₁𝑝𝛼²𝛼log1+𝑝^𝑦_𝑥𝑒^{𝛼(𝑡−𝑥)}𝐷(𝑡)−1𝑑𝑡𝛼^𝑦_𝑥𝐷(𝑡)𝑑𝑡. (4.1)

Without loss of generality, we will set 𝑐_ℎ+𝛼𝑐₁ to 1. As this will have no effect on the solution of the optimization problem where 𝐾 needs to be replaced by 𝐾/(𝑐_ℎ+𝛼𝑐₁), therefore, we set𝑅=𝑝𝑥,𝑦𝛼²𝛼log1+𝑝^𝑦_𝑥𝑒^{𝛼(𝑡−𝑥)}𝐷(𝑡)−1𝑑𝑡𝛼^𝑦_𝑥𝐷(t)𝑑𝑡. (4.2) Write𝐺=𝛼𝑥,𝑦𝑝^𝑦_𝑥𝑒^{𝛼(𝑡−𝑥)}𝐷(𝑡)𝑑𝑡. (4.3) Direct computations then lead to𝜕_𝑥𝑝𝑅=𝛼²𝜕_𝑥𝐺+11+𝐺𝛼𝐷(𝑥),𝜕_𝑦𝑝𝑅=𝛼²𝜕_𝑦𝐺−11+𝐺𝛼𝐷𝑦,𝜕_𝑥𝜕_𝑦𝑝𝑅=𝛼²𝜕_𝑥𝜕_𝑦𝐺(1+𝐺)−𝜕_𝑥𝐺𝜕_𝑦𝐺(1+𝐺)²,𝜕²_𝑥𝑝𝑅=𝛼²𝜕²_𝑥𝐺(1+𝐺)−𝜕_𝑥𝐺²(1+𝐺)²+1𝛼𝐷^′(𝑥),𝜕²_𝑦𝑝𝑅=𝛼²𝜕²_𝑦𝐺(1+𝐺)−𝜕_𝑥𝐺²(1+𝐺)²−1𝛼𝐷^′𝑦,𝜕_𝑥𝐷(𝑥)𝐺=−𝛼𝐺+𝑝,𝜕_𝑦𝛼𝐺=𝑝𝑒^{𝛼(𝑦−𝑥)}𝐷𝑦,𝜕_𝑥𝜕_𝑦𝐺=−𝛼𝜕_𝑦𝜕𝐺,_2𝑥𝐷(𝑥)𝐺=−𝛼−𝛼𝐺−𝛼𝑝+𝐷^′(𝑥)𝑝,𝜕²_𝑦𝐺=𝛼𝜕_𝑦𝛼𝐺+𝑝𝑒^{𝛼(𝑦−𝑥)}𝐷^′𝑦. (4.4) The following result indicates that 𝑅 obtained in (4.2) satisfies Hypothesis 1.

Lemma 4.1. The function 𝑅 satisfies Hypothesis 1.

Proof. It is clear that for any 𝑥∈[0,𝐻], 𝑅(𝑥,𝑥)=0. (4.5) Now, direct computations show that 𝜕_𝑥𝑅=𝐺𝑥,𝑦𝑥,𝑦𝛼(𝑥1+𝐺𝑥,𝑦−𝑝+𝐷). (4.6) But 𝑝>𝐷(𝑥), therefore 𝜕_𝑥𝑅(𝑥,𝑦)<0 since 𝐺(𝑥,𝑦)>0 for 𝑦>𝑥. Also, it can be shown that 𝜕_𝑦𝑅=𝐷𝑦𝑥,𝑦𝛼𝑒^{𝛼(𝑦−𝑥)}.1+𝐺𝑥,𝑦−1 (4.7) We claim that 𝜕_𝑦𝑅(𝑥,𝑦)>0. Indeed, the claim is equivalent to 𝐹_𝑥𝑦=1+𝐺𝑥,𝑦−𝑒^{𝛼(𝑦−𝑥)}<0for𝑦>𝑥. (4.8) The function 𝐹_𝑥(𝑦) is decreasing since 𝐹^′_𝑥𝑦=−𝛼𝑒^{𝛼(𝑦−𝑥)}𝐷𝑦1−𝑝, (4.9) with 𝐹_𝑥(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 𝜕_𝑥𝜕_𝑦𝑅𝑥,𝑦=−𝛼𝜕_𝑦𝐺𝐷𝑦𝑥,𝑦1−𝑝. (4.10) 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 set𝑍_𝑥(𝑢)=𝐷^′(𝑢)𝐷(𝑢){1+2𝐺(𝑥,𝑢)},with0≤𝑥≤𝑢≤𝐻. (4.11) 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 have𝑔_𝑥(𝑢)1∶=𝑍_𝑥(𝑢)=𝐷(𝑢){1+2𝐺(𝑥,𝑢)}𝐷^′(𝑢)=1𝛽{1+2𝐺(𝑥,𝑢)}, (4.12)

with 𝑔^′_𝑥(𝑢)=(2/𝛽)𝜕_𝑦𝐺(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_𝑥𝑅≥0, where ℒ_𝑥 is defined in (3.5).

Proof. Tedious but direct algebra using the definition of ℒ_𝑥𝑅 leads to ℒ_𝑥𝐷(𝑥)𝑅=1−/𝑝1+𝐺𝑥,𝑦²𝛼^𝑦_𝑥𝑒^{𝛼(𝑡−𝑥)}𝐷(𝑡)𝑑𝑡+𝐷(𝑥)−𝑒^{𝛼(𝑦−𝑥)}𝐷𝑦+1𝛼𝐷^′(𝑥)𝐺𝑥,𝑦−𝑝1+𝐺𝑥,𝑦𝛼𝐺𝑥,𝑦𝐷(𝑥)1+𝐺𝑥,𝑦1−𝑝𝑓(𝑥). (4.13)ℒ_𝑥𝑅≥0 is equivalent to 1−𝐷(𝑥)/𝑝𝛼1+𝐺𝑥,𝑦^𝑦_𝑥𝑒^{𝛼(𝑡−𝑥)}𝐷(𝑡)𝑑𝑡+𝐷(𝑥)−𝑒^{𝛼(𝑦−𝑥)}𝐷𝑦+1𝛼𝐷^′(𝑥)𝐺−𝑝𝑥,𝑦𝛼𝐺𝐷(𝑥)𝑥,𝑦1−𝑝𝑓(𝑥)≥0. (4.14) Let 𝑓(𝑥)𝐷=−^′(𝑥)𝐷^′(𝑥)+𝐷^′(𝑥)𝑝𝐷(𝑥).1−/𝑝 (4.15) It can be shown that (4.14) is true if 𝛼𝑝𝑒^{𝛼(𝑦−𝑥)}𝐷𝑦−𝐷(𝑥)−𝛼∫^𝑦_𝑥𝑒^{𝛼(𝑡−𝑥)}𝐷(𝑡)𝑑𝑡𝐺≤𝐷𝑥,𝑦1+𝐺𝑥,𝑦^′(𝑥)𝐷(𝑥). (4.16) Define, for 𝑢≥𝑥. 𝐹_𝑥(𝑢)=𝑒^{𝛼(𝑢−𝑥)}𝐷(𝑢)−𝛼^𝑢_𝑥𝑒^{𝛼(𝑡−𝑥)}𝐷(𝑡)𝐻𝑑𝑡,_𝑥(𝑢)=𝛼𝑝^𝑢_𝑥𝑒^{𝛼(𝑡−𝑥)}𝐷(𝑡)𝛼𝑑𝑡1+𝑝^𝑢_𝑥𝑒^{𝛼(𝑡−𝑥)}𝐷(𝑡)𝑑𝑡. (4.17) The left hand side of (4.16) may be written as 𝛼𝑝𝐹_𝑥𝑦−𝐹_𝑥(𝑥)𝐻_𝑥𝑦−𝐻_𝑥(𝑥). (4.18) This is equal by extended-mean value theorem to (𝛼/𝑝)((𝐹^′_𝑥(𝜉))/(𝐻^′_𝑥(𝜉))) for some 𝑥<𝜉≤𝑦. However, 𝐹^′_𝑥(𝑢)=𝛼𝑒^{𝛼(𝑢−𝑥)}𝐷^′(𝑢),𝐻^′_𝑥(𝑢)=𝜕_𝑦𝐺(𝑥,𝑢){1+2𝐺(𝑥,𝑢)}. (4.19) Therefore, 𝛼𝑝𝐹^′_𝑥(𝜉)𝐻^′_𝑥(𝜉)=𝐷^′(𝜉)𝐷(𝜉(≤𝐷){1+2𝐺𝑥,𝜉)}^′(𝑥)𝐷(𝑥(=𝐷){1+2𝐺𝑥,𝑥)}^′(𝑥)𝐷(𝑥). (4.20) The last inequality follows from assumption (A1). This completes the proof.

Now, set for 0≤𝑢≤𝑦≤𝐻,𝑉_𝑦(𝑢)=−𝛼𝑒^{𝛼(𝑦−𝑢)}𝑒^{𝛼(𝑦−𝑢)}𝐷𝑦−𝐷(𝑢)−𝛼∫^𝑦_𝑢𝑒^{𝛼(𝑡−𝑢)}𝐷(𝑡),𝑊𝑑𝑡_𝑦(𝑢)=−1+𝐺𝑢,𝑦𝑒^{𝛼(𝑦−𝑢)}−1−𝐺𝑢,𝑦. (4.21)(4.22) The next assumption is needed for ℒ_𝑦𝑅≥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 function𝐷ℱ=^′(𝑢)1−𝐷(𝑢)/𝑝 (4.23)

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.

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

Proof. Recall that ℒ_𝑦𝑅=𝜕²_𝑦𝑅+𝜕_𝑥𝜕_𝑦𝑅+𝑓𝑦𝜕_𝑦𝑅. (4.24) Direct and tedious computation leads to ℒ_𝑦𝑅=1𝑥,𝑦𝛼𝜕_𝑦𝐺𝑥,𝑦1+𝐺𝑥,𝑦²×𝛼^𝑦_𝑥𝑒^{𝛼(𝑡−𝑥)}𝐷(𝑡)𝑑𝑡−𝑒^{𝛼(𝑦−𝑥)}𝐷𝑦(𝑥−1−𝐷)𝛼𝐷^′𝑦+1𝛼𝑒^{𝛼(𝑦−𝑥)}𝐷^′𝑦+11+𝐺𝑥,𝑦𝛼𝑓𝑦𝐷𝑦𝑒^{𝛼(𝑦−𝑥)}.1+𝐺𝑥,𝑦−1 (4.25) Recall the definition of the function 𝑓 in (4.15). Then ℒ_𝑦𝑅≥0 is equivalent to 𝜕_𝑦𝐺𝑥,𝑦𝑒1+𝐺𝑥,𝑦^{𝛼(𝑦−𝑥)}𝐷𝑦−𝐷(𝑥)−𝛼^𝑦_𝑥𝑒^{𝛼(𝑡−𝑥)}𝐷(𝑡)≤𝐷𝑑𝑡_′𝑦𝐷𝑦𝑝𝐷𝑦𝑒1−/𝑝^{𝛼(𝑦−𝑥)}−1−𝐺𝑥,𝑦, (4.26) or, by (𝜕_𝑦𝐺) and (4.8), we get that the requirement ℒ_𝑦𝑅≥0 leads to 𝛼𝑒^{𝛼(𝑦−𝑥)}𝑒^{𝛼(𝑦−𝑥)}𝐷𝑦−𝐷(𝑥)−𝛼∫^𝑦_𝑥𝑒^{𝛼(𝑡−𝑥)}𝐷(𝑡)𝑑𝑡1+𝐺𝑥,𝑦𝑒^{𝛼(𝑦−𝑥)}≤𝐷−1−𝐺𝑥,𝑦^′𝑦𝐷𝑦.1−/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)=2𝛼²𝑒^{𝛼(𝑦−𝑢)}𝑒^{𝛼(𝑦−𝑢)}𝐷𝑦−𝐷(𝑢)−𝛼^𝑦_𝑢𝑒^{𝛼(𝑡−𝑢)}𝐷(𝑡)𝑑𝑡+𝛼𝑒^{𝛼(𝑦−𝑢)}𝐷^′𝑦,𝑊^′_𝑦(𝑢)=−𝜕_𝑥𝐺𝑢,𝑦𝑒^{𝛼(𝑦−𝑢)}−−1−𝐺𝑢,𝑦1+𝐺𝑢,𝑦−𝛼𝑒^{𝛼(𝑦−𝑢)}−𝜕_𝑥𝐺𝑢,𝑦. (4.28) Now, the extended-mean value theorem gives that 𝑉_𝑦𝑦−𝑉_𝑦(𝑥)𝑊_𝑦𝑦−𝑊_𝑦(𝑥)=𝑉^′_𝑦(𝜉)𝑊^′_𝑦(𝜉),forsome𝑥<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 𝛼𝐷^′𝑦−𝐷𝑦=𝐷−𝛼+𝛼/𝑝^′𝑦𝐷𝑦.1−𝛼/𝑝 (4.30) This is the right hand side of (4.27). Hence, ℒ_𝑦𝑅≥0.

As a consequence of Lemmas 4.4 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 𝑆_𝑛=∑^𝑁_𝑖=1𝑅(𝑡_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].

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(1)if 𝐾>𝑠₁−𝑠₂, then 𝑛^∗=1,(2)if there exists an 𝑁≥2 such that 𝑠_𝑁−1−𝑠_𝑁>𝐾>𝑠_𝑁−𝑠_𝑁+1, then 𝑛^∗=𝑁,(3)if there exists an 𝑁≥1 such that 𝐾=𝑠_𝑁−𝑠_𝑁+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−𝑐₁^𝑡𝑖_𝑡𝑖−1𝐷(𝑢)𝑑𝑢. (5.1)

Now, consider the optimization problem (2.22) with 𝑅 given by (2.6). It is clear that adding −𝑐₁∫^𝐻₀𝐷(𝑢)𝑑𝑢=−𝑐₁∑^𝑛_𝑖=1∫^𝑡𝑖_𝑡𝑖−1𝐷(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 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)=2sin(10𝑡)+2cos(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].

Figure 2 

Behaviour of the objective function when 𝐷(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.

Acknowledgments

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