No Shortages

Shortages Backordered

Lost Sales


Inventory Theory
- Deterministic/Shortages Backordered/Finite Replenishment

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



Click buttons to see the notation

Here we derive the formulas for the results for an instance. In addition to the inventory parameters, two variables are necessary to describe the pattern in the figure. As for the finite replenishment case, we use the variables lot size, q and fill rate, v. The fill rate is the proportion of the demand that is satisfied immediately from inventory. The proportion that is not satisfied from inventory, but is backordered is 1-v. This model assumes that the cost of a backordered item is proportional to the waiting time. The backorder cost per unit time is . Click the Parameters button on the left to see the definitions of the parameters and variables used for the analysis. We will derive formulas for the quantities defined on the page reached by the Results button. These formulas are implemented in the Inventory add-in.


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.

Amount during a cycle
Cost or revenue rate

Setup Cost

Product Cost
Holding Cost

Backorder Cost
Revenue from product Sales

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.


The table shows the results for the optimum lot size and fill rate

(q*= 1238, v*= 86.96%).

The instance results shown previously are hidden in rows 15 through 26.


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.

The analysis for this case has the same result as for the infinite replenishment case with a fill rate of 0 and an infinite lot size. The production rate is irrelevant since in the limit the demand is not fulfilled until an infinite time has passed.


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.

The results are similar to the infinite replenishment rate case except the breakpoint on the fixed charge is reduced.

The optimum solution is the policy with the smallest cost. Note that the first case suggests an infinite lot size with all items backordered and the second case suggests no shortages.


Return to Top

tree roots

Operations Management / Industrial Engineering
by Paul A. Jensen
Copyright 2004 - All rights reserved

Next Page