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.

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. [

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 [

Recently, Benkherouf and Gilding [

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

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

the planning horizon is finite;

a single item is considered;

products are assumed to experience deterioration while in stock;

shortages are not permitted;

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

the demand function is strictly positive;

We will initially look at a single period

the total planning horizon,

the constant production rate,

the demand rate at time

constant deteriorating rate of inventory items with

the time at which the inventory level reaches it is maximum in the

set up cost for the inventory model,

the cost of one unit of the item with

carrying cost per inventory unit held in the model per unit time,

total system cost during

Figure

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

Let

Note that since the function

then

or

Therefore,

The expression of the cost in (

Applying integration by parts, we get that (

Note that Lemma

Let

The total inventory costs where

The objective now is to find

This section contains a summary of the work of [

Consider the problem

Write

Use the notation

The functions

Define

The next theorem shows that under assumptions in Hypotheses

The system (

If

Based on the convexity property of

Now to solve (

Assume that

This section is concerned with the optimal inventory policy for the production inventory model. The model has been introduced in Section

Without loss of generality, we will set

The function

It is clear that for any

Before we proceed further, we set

The function

Note that as

Let

Note that

with

Let

If

Tedious but direct algebra using the definition of

Now, set for

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

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

If assumption (A2) is satisfied, then

Recall that

Computations show that

As a consequence of Lemmas

Under the requirements that assumptions (A1) and (A2) hold the function

Let

The function

As a consequence of Theorem

The optimal number of production period

if

if there exists an

if there exists an

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 [

Now, consider the optimization problem (

Before we close, we revisit paper [

Behaviour of the objective function when

It is worth noting that the keys to success in applying the approach in [

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