We consider again an independent inventory with finite replenishment rate. In this section we allow shortages. That is, we allow the inventory to run out and customers who arrive and find the inventory empty do not receive immediate satisfaction. The unsatisfied customers will respond in one of four ways.

• The customer will wait for delivery until the next replenishment but there is a cost associated with waiting that is proportional to the waiting time. This is called the backorder case.
• There is a fixed charge that occurs whenever the event of shortage occurs during a cycle. The charge is independent of the number of shortages that occur.
• The customer will wait for delivery until the next replenishment but there is a cost associated with dissatisfaction that is a constant, independent of the waiting time. This is called the fixed shortage cost case.
• The customer will not accept delivery at any future time and the sale is lost. This is called the lost sales case.

The first three cases involve backordered shortages, while the last involves lost sales. We consider the backorder cases on this page.

The finite model is often used when the inventory is being replenished by a production process. The backorder case is not realistic when the material being drawn from the inventory is another production process. A process cannot wait for backordered items. The model only has relevance when customers can wait for delivery.

Finite Replenishment Rate with Shortages Backordered

This situation is similar to the infinite replenishment case except that inventory is replenished at a finite rate P which is greater than the demand rate D. As shown in the figure below, the inventory position does not reach the extremes of the infinite case.

Formulas for Instance Results

To construct the model describing the costs or profits associated with the inventory system, we show the cash flows below. The blue area indicates revenues from sales from the inventory. The revenue has two components. The rate of revenue is RP during the interval when the shortage is being satisfied and RD after the shortage is satisfied and demand is drawing the inventory down. When the on-hand inventory reaches 0, there is no revenue. The expenditure for product cost is CP during production. The red area represents holding cost and the purple area represents backorder cost. The fixed setup cost A is incurred at the beginning of each cycle.

A single inventory cycle is shown below.

The table below shows the various revenue and cost components and their respective cost rates. We compute the holding and backorder costs during a cycle with expressions for the associated triangular areas. The rates are the cycle costs or revenues divided by the cycle time.

Item	Amount during a cycle	Cost or revenue rate
Setup Cost	A
Product Cost	Cq
Holding Cost
Backorder Cost
Revenue from product Sales	Rq	RD

For this model there are two decision variables q and v.

Additional quantities associated with the inventory policy are derived below.

All realization results are the same as those computed for the infinite replenishment case except they are multiplied by the factor (P - D)/P. This factor goes to 1 as the production rate increases without bound.

The measures are computed below for the example using a lot size of 400 units and a fill rate of 90%. The figure shows a single cycle of the inventory pattern.

Optimum Policy

For determination of the optimum lot size q* and fill rate v*, the unit revenue and unit purchase cost can be neglected and we write the expression for the cost rate for operating the inventory.

We first take the partial derivative with respect to the fill rate. The optimal fill rate is is the same as for the infinite replenishment rate case. We next take the partial derivative with respect to the lot size. The optimum lot size is increased over the infinite case by the square root of the factor P/(P - D). Given the fill rate, the optimum lot size for that fill rate can be determined from the formula above. Using the optimum fill rate, the optimum lot size is determined. The results below assume the lot size is unrestricted. Compared to the model without shortages and with an infinite replenishment rate, the optimum lot size is larger by the ratio . The optimum inventory cost is reduced by the inverse of the ratio.

Fixed Charge in the Event of Shortage

The two remaining cases considered on page involve backordered shortages with different cost models. Here we consider a situation where the only charge associated with a shortage is a fixed charge that is incurred in any cycle that a shortage occurs. The charge is independent of the number of customers that are affected.

Fixed Charge for Each Unit of Shortage

Here we consider a situation where there is a fixed charge that is incurred for each unit of shortage. The charge is independent of the wait for the backordered item.