BUS606: Operations and Supply Chain Management

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 Benkherouf and Gilding. 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

$-c_{1} \int_{t_{i-1}}^{t_{i}} D(u) d u$   (5.1)

Now, consider the optimization problem (2.22) with $R$ given by (2.6). It is clear that adding $-c_{1} \int_{0}^{H} D(u) d u=-c_{1} \sum_{i=1}^{n} \int_{t_{i-1}}^{t_{i}} D(u) d u$ 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 Teng et al. and note that the model in Teng et al. allows for the purchasing cost to vary with time, and therefore with fixed unit cost and no deterioration, the model in Teng et al. 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 Teng et al. is the standard approach for constrained nonlinear programming problem. The key result in Teng et al. is Theorem  1 which adapted to the model of this paper with $\theta=0$ requires that $c_{h}(1-(D(t) / p)) \text { > } 0$ to hold. This is satisfied since $D(t) \text { < } p$. Theorem  1 in Teng et al. 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 Teng et al. cannot be entirely correct in its present form. Indeed, for simplicity let $n=2$ then the problem treated in Teng et al. 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 $D(t)=2 \sin (10 t)+2 \cos (10 t)+4, c_{h}=1, \text { and } H \simeq 4.27$, 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 Teng et al also applies to part of Balkhi.

Figure 2 Behaviour of the objective function when $D(t)=2 \sin (10 t)+2 \cos (10 t)+4$.

It is worth noting that the keys to success in applying the approach in Benkherouf and Gilding 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.