Computation Section
Subunit Inventory Game
 - Lost Sales

This model is similar to the Backorder case except whenever the inventory is exhausted, sales of subsequent demand is lost until the next delivery arrives. There is a specified cost for each lost sale. The goal is to place replenishment orders to minimize the total cost of orders, inventory and lost sales. Since revenues from sales and costs of purchasing the item is not included, the lost sales cost must reflect the lost profit plus any penalties for losing the customer.

The simulated model used for the game is slightly different than the situation pictured in that the inventory is observed weekly and orders can be placed only at the end of a week. This is called discrete review. The picture shows continuous review where orders can be placed at any time. In our model, at the end of each week the player must decide to order or not. If a nonzero order quantity is chosen, that quantity arrives L weeks later at the beginning of the week. The parameters of the model are shown to the left of the figure. Demand is a random variable governed by a normal distribution with given mean and standard deviation. All these parameters can be changed, but we use the parameters above for the example on this page.


To start playing the game press the Start button on the Start_LS worksheet.

A dialog opens to accept the inventory parameters and stores them on the game worksheet. The parameters become the default values for the next play of the game.


Clicking the OK button on the Initial Conditions dialog opens the Inv_LS worksheet where the game is played. The figure below shows the worksheet as it first appears. In addition to the control buttons, the dialog at the upper left holds the step number, a description of the decision to be made and the order quantity field. Step 1 requests the order quantity decision for at the end of the first week.

The display is similar to the backorder display with the exception of a new column for Lost Sales. Because shortages are not backordered, the ending inventory never goes below zero. The new column is used to record and evaluate the cost of lost sales.

The Next button leads to the next week. The Quit button temporary suspends time.


We figure below shows the results of 25 weeks of play. The first instance of lost sales occurs at week 13. The initial inventory is 21, there are no deliveries, and the demand is 30. The resulting final inventory is 0 and 9 sales are lost. The cost for the week is $1807. This includes $1800 for the lost sales ($200 each) and $7 for the average inventory during the week.


We continue the example until 52 weeks pass. The game ends with the message below.

An answer of No to retains the results. An answer of Yes resets the results to week 1. The summary below shows that the solution provided by the author lost over 10% of the demand due to shortages.

The complete history of the 52 week game is shown below. With practice, you can do much better. One interesting difference from the backorder case is the difference between the average delivery and the average demand. Since part of the demand is lost the total deliveries will usually be less than the total demand. For a finite game the initial and final inventories also affect this relationship.


Since this game is very similar to the Backorder game, we will not repeat the sections on simulation. The Simulation button, changes the seed for the random demand so that a specified policy can be evaluated for different demand streams.


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tree roots

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

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