Single Period Stochastic Model

1. Single Period Stochastic Model Henry C. Co Technology and Operations Management, California Polytechnic and State University

2. Short product life cycles/Long lead times Computers Apparel Fresh food, newspapers Services Airline industry

3. These models have the objective of properly balancing the cost of Underage � having not ordered enough products vs. Overage � having ordered more than we can sell These models apply to problems like: Planning initial shipments of �High-Fashion� items Amount of perishable food products Item with short shelf life (like the daily newspaper) Because of this last problem type, this class of problems is typically called the �Newsboy� problem 3 Inventory Management Problems

4. The Newsboy Model At the start of each day, a newsboy must decide on the number of papers to purchase. Daily sales cannot be predicted exactly, and are represented by the random variable, D. The newsboy must carefully consider these costs: cU: underage cost (when D=S). This is the unit opportunity cost; for example, unit revenue r - unit cost c, i.e., (r-c) cO: overage cost (when D=S). This is the unit cost of overstocking; for example, unit cost c - unit salvage value u, i.e., (c-u). 4 Inventory Management Problems

5. The objective is to Minimize the expected cost: cu E[max{D-S, 0}] + co E[max{S-D, 0}] Solving for Q, the optimal order quantity S satisfies the following condition: Equation (4-36) can be rewritten as: Inventory Management Problems 5

6. Graphical Representation Inventory Management Problems 6

7. Uniform Demand Between [A,B] Inventory Management Problems 7

8. Try this on Excel! Emilio Tadini & Sons is a hand-made shirt retailer, located in Rome (Italy), close to Piazza di Spagna. This year Mr. Tadini faces the problem of ordering a new bright color shirt made by a Florentine firm. He assumes that the demand is uniformly distributed between 200 and 350 units. The purchasing cost is c = �18 while the selling price is r = �52 and the salvage value is u = �7. Thus co = 34 and cu = 11. Hence, Mr. Tadini should order S = 313 units. Inventory Management Problems 8

9. Inventory Management Problems 9

10. Another Example � The buyer for Needless Markup, a famous �high end� department store, must decide on the quantity of a high-priced women�s handbag to procure in Italy for the following Christmas season. The unit cost of the handbag to the store is $28.50 and the handbag will sell for $150.00. Any handbags not sold by the end of the season are purchased by a discount firm for $20.00. In addition, the store accountants estimate that there is a cost of $.40 for each dollar tied up in inventory, as this dollar invested elsewhere could have yielded a gross profit. Assume that this cost is attached to unsold bags only. Suppose that the sales of the bags are equally likely to be anywhere from 50 to 250 handbags during this season. Based on this, how many bags should the buyer purchase? Inventory Management Problems 10

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12. Distribution of Demand Is Normal 12 Inventory Management Problems

13. Example Every week, the owner of a newsstand purchases a number of copies of The Computer Journal. Weekly demand for the Journal is normally distributed with mean 10 and standard deviation 5. He pays 25 cents for each copy and sells each for 75 cents. Inventory Management Problems 13

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