Inventory Example

Assume you own a sporting goods store in a small community whose local college basketball team has made it into the finals of the national championships. You are sure that if the local team wins you will be able to sell a significant number of T-shirts proclaiming the school as the national champion. Unfortunately, to have the shirts ready the day after the championship game, you will have to order the shirts at least a week in advance--before you know if the local team will win. You expect to sell between 2,000 and 10,000 shirts at $20 each. You can order the shirts for $7 each. Any shirts you do not sell you can sell as scrap for $2 each. In addition, you estimate that there is a 60% chance of the local team winning. You must decide today if you will order any shirts, and if so, how many.

In this problem you face two uncertain events, you don't know who will win the championship game, and you don't know how many shirts you can sell even if the local team does win.

Here is a decision tree constructed to solve the T-shirt problem:

In the first node you face a decision. You must decide how many shirts to order. This example has been simplified by assuming you can only order in quantities of 5,000. This means you must order 5,000, 10,000, or none at all.

Once you have made a decision about how many shirts to order, you next come to an event node--the local team either wins or loses the championship. If the team loses, you must sell all of the shirts as scrap losing either $25,000 if you order 5,000 shirts ($2*5,000-$7*5,000) or $50,000 if you order 10,000 shirts ($2*10,000-$7*10,000). If the team wins, you face another uncertainty--the demand for shirts.

The problem states that you expect demand to be between 2,000 and 10,000 shirts. A continuous, uncertain quantity, like forecast sales, can be modeled using several techniques. The simplest technique is to use three to five scenarios and assign a probability to each.

In the T-shirt example the range of expected demand was divided into four bands (2-4K, 4-6K, 6-8K, and 8-10K) and the midpoint in each band (3K,5K,7K,9K) is chosen as the scenario's demand.

Of course, you cannot sell more shirts than you have. Therefore, if you order only 5,000 shirts, you will make the same profit if demand is 5,000, 7,000 or 9,000. All shirts you have in excess of demand can be sold as scrap.

The solution indicated by this tree is that you should order 5,000 shirts because this option yields the highest expected value.