No Shortages

Shortages Backordered

Lost Sales


Inventory Theory
- Deterministic/Shortages Backordered/Infinite Replenishment

We consider again an independent inventory with infinite 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.


Infinite Replenishment Rate with Shortages Backordered


The deterministic model considered in this section allows shortages to be backordered. This means that when the inventory is empty and additional demand occurs, customers will wait for delivery until the next inventory replenishment. During that time, a charge is incurred proportional to the time the customer must wait until delivery. The situation is illustrated in the figure below. The figure shows inventory position. When the inventory position is positive, the inventory position is the same as the on-hand inventory level (inventory available for immediate delivery). When the inventory position is negative the shortage amount is the negative of the inventory position. The maximum inventory position occurs when the inventory replenished. Continuous demand reduces the inventory linearly, until it is 0. This interval is indicated by the blue area. Additional demand results in a shortage indicated by the red area. The replenishment amount is q and is the difference between the maximum and minimum inventory position. The inventory position repeats with a cycle time of.


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 before, one of the variables is the lot size, q. We choose as the second variable the 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. Other quantities could have been selected for the second variable such as maximum or minimum inventory position, but fill rate characterizes most of the situations involving shortages on these pages. The model requires one new parameter, the cost of a backordered order per unit time (). 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 a mathematical model describing the economic costs or profits associated with the inventory system, we show the cash flows below. This figure is a mixed representation of discrete as well as continuous cash flows. The arrows represent amounts paid or received at points in time. The areas represent continuous cash flows given by rates. Positive amounts are revenues, while negative amounts are expenditures or costs. The blue area indicates revenues from sales from the inventory. The revenue arrives continuously during the early part of the cycle until the level reaches 0. The remainder of the revenue is received at the end of the cycle when the replenishment arrives. The red area represents holding cost and the purple area represents backorder cost. The fixed setup cost A and the product cost Cq are 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.

When we compute the mean residence time for items held in inventory, we use Little's Law but only use the flow rate for items that are actually held in on-hand inventory. Similarly, when we compute the mean backorder time, we use as the flow rate only those items backordered. We feel that these measures are more interesting than the corresponding quantities computed with the entire flow rate through the system.

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 find the optimum values by setting the first partial derivatives to zero and solving. The solution is a global optimum since the functions in the inventory cost expression are all convex.

We first take the partial derivative with respect to the fill rate.

The optimal fill rate is a ratio involving the unit holding cost and backorder cost. It does not depend on the lot size.

We next take the partial derivative with respect to the lot size.

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.

Note that the lot size is larger than the lot size when no shortages are allowed by the ratio


The optimum inventory cost is reduced by the inverse of the ratio.

The table below shows the parameters and instance results as well as the results for the optimum lot size and fill rate (q*= 876, v*= 86.96%).


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. We will see that the results of this case lead to an unrealistic solution.

There are two expressions for the inventory cost depending on whether a shortage is allowed or not.

If we specify a specific value of q and require that v < 1, the minimum inventory cost occurs when v = 0, and all demand is satisfied through backorders. The figure below shows the inventory position when demand is entirely satisfied from backorders.

The optimum policy for a given q is found by comparing the cost for the two cases: v = 1 and v = 0.

When q is allowed to vary, it is clear that the optimum lot size is unbounded. It is best to increase the lot size as much as possible with v = 0.

The model is unrealistic in this case, because demand is satisfied only after an infinite wait. The revenue as well as product cost is not received until after a very long time, but the model does not penalize this because it does not consider the time value of money.The analysis assumes that lot sizes are unrestricted. When the lot size is restricted by , the inventory cost for the two cases v = 1 and v = 0.must be compared to find the optimum.


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 inventory cost rate has three terms: setup cost, holding cost and shortage cost. We set the partial of the cost function with respect to v to zero.

The value of the fill rate can be no more than one so we find two cases for the optimum.

When the unit shortage cost, (), is less than the cost to hold an item for the full cycle (), it is optimal to allow a shortage, otherwise no shortage should be allowed.

To find the optimum lot size we must compare the inventory cost with and without shortages. When a shortage is allowed it is optimum to have a fill rate of 0 and an unbounded lot size. When a shortage is not allowed, the optimum lot size is given by the EOQ formula.

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.

The analysis assumes that lot sizes are unrestricted. If the upper or lower bound on the lot size is violated by a solution, the inventory cost must be evaluated it the limit values to determine the optimum values.


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

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