^{1}

^{2}

^{1}

^{2}

Disruption management has recently become an active area of research. In this study, an extension is made to consider the fact that some products may deteriorate during storage. A production-inventory model for deteriorating items with production disruptions is developed. Then the optimal production and inventory plans are provided, so that the manufacturer can reduce the loss caused by disruptions. Finally, a numerical example is used to illustrate the model.

In real life, the effect of decay and deterioration is very important in many inventory systems. In general, deterioration is defined as decay, damage, spoilage, evaporation, obsolescence, pilferage, loss of utility, or loss of marginal value of a commodity that results in decreasing usefulness [

At the beginning of each cycle, the manufacturer should decide the optimal production time, so that the production quantity should satisfy the following two requirements: one, it should meet demand and deterioration; second, all products should be sold out in each cycle, that is, at the end of each cycle, the inventory level should decrease to zero. Some researchers have studied such production model for deteriorating items under different condition. For example, Yang and Wee [

However, after the plan was implemented, the production run is often disrupted by some emergent events, such as supply disruptions, machine breakdowns, earthquake, H1N1 epidemic, financial crisis, political event and policy change. For example, the Swedish mining company Boliden AB suffered the production disruptions at its Tara zinc mine in Ireland due to an electric motor breakdown at one of the grinding mills. As a result of the breakdown, the production of zinc and lead concentrates is expected to fall by some 40% over the next six weeks [

In most of the existing literature, products are assumed to be no deterioration when the production disruptions are considered. But, in real situation the deterioration is popular in many kinds of products. Hence, if the deterioration rate is not small enough, the deterioration factor cannot be ignored when the production system is disrupted.

Therefore, in this paper, we develop a production-inventory model for deteriorating items with production disruptions. Once the production rate is disrupted, the following questions are considered in this paper.

Whether to replenish from spot markets or not?

How to adjust the production plan if the new production system can still satisfy the demand?

How to replenish from spot markets if the new production system no longer satisfies the demand?

The paper is organized as follows. Section

Suppose a manufacturer produces a certain product and sells it in a market. All items are produced and sold in each cycle. The following assumptions are used to formulate the problem.

A single product and a single manufacturer are assumed.

Demand rate is deterministic and constant.

Normal Production rate is greater than demand rate.

Lead time is assumed to be negligible.

Deterioration rate is deterministic and constant.

Shortages are not allowed.

Time horizon is finite.

There is only one chance to order the products from spot markets during the planning horizon.

Let the basic parameters be as follows:

normal production rate,

demand rate,

constant deterioration rate of finished products,

planning horizon,

the normal production period without disruptions,

the production disruptions time,

the new production period with disruptions,

the replenishment time from spot markets once shortage appears,

the order quantity from spot markets once shortage appears,

inventory level in the

At first, the manufacturer makes decisions about the optimal production time

Inventory system without disruptions.

The instantaneous inventory level at any time

Assuming that

Corollary

In the above model, the production rate is assumed to be deterministic and known. In practice, the production system is often disrupted by various unplanned and unanticipated events. Here, we assume the production disruptions time is

If

Without considering the stop time of production or replenishment, the inventory system with production disruptions can be depicted as Figure

From Section

But if

Inventory system with production disruptions.

From Proposition

If

From Proposition

So if

The proposition is proved.

Inventory system (

Since

If

If

Expanding the exponential functions and neglecting second and higher power of

Assuming that

If

Inventory system (

If

First, we need to determine the order time point

The proposition is proved.

If

If

Our objective in this section is to gain further insights based on a numerical example. We use the following numbers as the base values of the parameters:

From Figure

From Figure

In this paper, we propose a production-inventory model for a deteriorating item with production disruptions. Here, we analyze this inventory system under different situations. We have showed that our method helps the manufacturer reduce the loss caused by production disruptions.

In this study, the proposed model considers the deterioration rate as constant. In real life, we may consider the deterioration rate as a function of time, stock, and so on. This will be done in our future research.

The authors thank the valuable comments of the referees for an earlier version of this paper. Their comments have significantly improved the paper. This research is supported by a grant from the Ph.D. Programs Foundation of Ministry of Education of China (no. 200802861030). Also, this research is partly supported by the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, the National Natural Science Foundation of China, the China Postdoctoral Science Foundation (no. 20070411043), and the Postdoctoral Foundation of Jiangsu Province of China (no. 0701045C).