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Inventory Theory
- Deterministic/Lost Sales/Infinite Replenishment

Here we consider the situation when the customer will not wait when the on-hand inventory is exhausted. Rather the customer will obtain the product elsewhere and the sale is lost.


Infinite Replenishment Rate with Lost Sales


The inventory position is shown in the figure below for several cycles. The minimum position is 0 and the maximum position is the lot size q. After a replenishment the inventory decreases at the constant demand rate until it reaches the 0 level. In the remainder of the cycle all demand results in lost sales.


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. Again we choose the lot size, q, and the fill rate, v. The fill rate is the proportion of the demand that is satisfied. The proportion that results in lost sales is 1-v. The model requires one new parameter, the cost of a lost sale (). 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. After this point in time, the lost sales cost is expended at the rate equal to the demand rate. The red area represents holding cost and the purple area represents the penalty due to lost sales. The fixed setup cost A and the product cost Cq are incurred at the beginning of each cycle. Note that in this case the entire net profit (revenue less product cost) for the entire demand is not received. With lost sales, the lot size does not satisfy the entire demand unless the fill rate is 1.

A single inventory cycle is shown below.

Since the demand is not entirely satisfied, the fill rate is the lot size divided by the demand during a cycle. The cycle time is a function of q and v. The time when the inventory is exhausted is t'.

The table below shows the various revenue and cost components and their respective rates. 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

Lost Sales Cost
Revenue from product Sales

For this model there are two decision variables q and v. We use the profit rate as a measure because the entire demand is not met and the profit is reduced when there are lost sales

Additional quantities associated with the inventory policy are below.

When we compute the mean residence time for items held in inventory, we use Little's Law with the flow rate through the inventory. Backorder quantities are 0 since there are no backorders.

The measures are computed below for two examples using a lot size of 400 units and a fill rate of 90%. The figure shows a single cycle of the inventory pattern. The example assumes cost and revenue for the product are both zero. The lost sales cost includes both the lost profit and any additional charges associated with the lost sale. The case in column N has the lost sales cost of 2, while column O shows the case with a lower lost sales cost of 1.2. The results are not markedly different. We will see considerable difference when the optimum solutions are presented.


Optimum Policy


For determination of the optimum lot size q* and fill rate v*, the unit revenue and unit purchase cost are included explicitly since a lost sale results in a lost profit. The objective is to maximize the profit rate. The profit is a concave function of the variables. Here we neglect the lower and upper bound on lot size.

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

The term on the left of the inequality is the profit for operating the inventory and the term on the right is the profit associated with meeting no demand. A fill rate of 0 is equivalent to not operating the inventory since there is no replenishment if no demand is met. Combining these results we find the optimum policy for the inventory.

The analysis could have been done without considering the product revenue and cost by defining an effective lost sales cost that is equal to the sum of the lost profit and other charges due to a lost sale.


The table below shows the parameters and instance results as well as the results for the optimum lot size and fill rate for two cases. The critical lost sales cost with this data is 1.414. The case in column N where the lost sale cost is 2 (>1.414) has the optimum values when there are no lost sales. For the case in column O with lost sale cost of 1.2 (<1.414) there is no inventory. We show q* as 0, but the lot size is irrelevant since no product passes through the inventory and all demand is lost.


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

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