Graduate Program in Business Information Systems

1. Graduate Program in Business Information Systems Inventory Decisions with Uncertain Factors Aslı Sencer

2. Uncertainties in real life • Demand is usually uncertain. • Probability distributions are used to represent uncertain factors. Ex: Demand is normally distributed OR Demand is either 20,30,40 with respective probabilities 0.2, 0.5, 0.3. BIS 517- Aslı Sencer

3. Stochastic versus Deterministic Models • Mathematical models involving probability are referred to as stochastic models. • Deterministic models are limited in scope since they do not involve uncertain factors. But they are used to develop insight! • Stochastic models are based on “expected values”, i.e. the long run average of all possible outcomes! BIS 517- Aslı Sencer

4. Example: Drugstore • A drugstore stocks Fortunes.They sell each for $3 and unit cost is $2.10. Unsold copies are returned for $.70 credit. There are four levels of demand possible. How many copies of Fortune should be stocked in October? Payoff Table: BIS 517- Aslı Sencer

5. Solution: • The expected payoffs are computed for each possible order quantity: Q = 20 Q = 21 Q = 22 Q = 23 $18.00 $18.44 $17.90 $16.79 Optimal stocking level, Q*=21 at an optimal expected profit of $18.44 • If the probabilities were long-run frequencies, then doing so would maximize long-run profit. BIS 517- Aslı Sencer

6. Example: Drugstore Payoff Table(Figure 16-1) BIS 517- Aslı Sencer

7. Single-Period Inventory Decision:The Newsvendor Problem • Single period problem (periodic review) • Demand is uncertain (stochastic) • No fixed ordering cost • Instead of h ($/$/period) we have hE($/unit/period=ch) • Instead of p ($/unit) we have pSand pR-c • Q: Order Quantity (decision variable) D: Demand Quantity • Costs: c = Unit procurement cost hE= Additional cost of each item held at end of inventory cycle = unit inventory holding cost-salvage value to the supplier pS= Penalty for each item short (loss of customer goodwill) pR= Selling price BIS 517- Aslı Sencer

8. Modeling the Newsvendor Problem The objective is to minimize total expected cost, which can be simplified as: where m is the expected demand. BIS 517- Aslı Sencer

9. Optimal order quantity of the Newsboy Problem Q* is the smallest possible demand such that BIS 517- Aslı Sencer

10. Example: Newsboy Problem • A newsvendor sells Wall Street Journals. She loses pS= $.02 in future profits each time a customer wants to buy a paper when out of stock. They sell for pR = $.23 and cost c = $.20. Unsold copies cost hE = $.01 to dispose. Demands between 45 and 55 are equally likely. How many should she stock? BIS 517- Aslı Sencer

11. Example: Solution • Discrete Uniform Distribution Demand is either 45,46,47,..., 55 each with a probability of 1/11. P(D<=Q*)=0.2 Q*=47 units. BIS 517- Aslı Sencer

12. Newsvendor Problem (Figure 16-3) BIS 517- Aslı Sencer

13. Multiperiod Inventory Policies • When demand is uncertain, multiperiod inventory might look like this over time. BIS 517- Aslı Sencer

14. Multiperiod Inventory Policies • The multiperiod decisions involve two variables: • Order quantity Q • Reorder point r • The following parameters apply: • A = mean annual demand rate • k = ordering cost • c = unit procurement cost • ps= cost of short item (no matter how long) • h = annual holding cost per dollar value • = mean lead-time demand BIS 517- Aslı Sencer

15. Multiperiod Inventory Policies: Discrete Lead-Time Demand • The following is used to compute the expected shortage per inventory cycle: • The following is used to compute the total annual expected cost: BIS 517- Aslı Sencer

16. Multiperiod Inventory Policies: Discrete Lead-Time Demand • Solution Algorithm. • Calculate the starting order quantity: • Determine the reorder point r*: • Determine optimal order quantity: • This procedure continues –using the last Q to obtain r and r to obtain the next Q- until no values change. BIS 517- Aslı Sencer

17. Example: Annual demand for printer cartridges costing c = $1.50 is A = 1,500. Ordering cost is k = $5 and holding cost is $.12 per dollar per year. Shortage cost is pS= $.12, no matter how long. Lead-time demand has the following distribution.Find the optimal inventory policy. BIS 517- Aslı Sencer

18. Example: Solution • The starting order quantity is: r* = 7 cartridges. • B(7) = (8–7)(.03) + (9–7)(.01) + (10–7)(.01)=.08 and the optimal order quantity is: BIS 517- Aslı Sencer

19. Example: Solution (cont’d.) • Q=290 leads to r=7, so the solution is optimal. The optimal inventory policy is: r* = 7 Q* = 290 • Optimal annual expected cost is: BIS 517- Aslı Sencer

20. Multiperiod Discrete BackorderingIteration 1 BIS 517- Aslı Sencer

21. Multiperiod Discrete BackorderingIteration 10 BIS 517- Aslı Sencer

22. Multiperiod Discrete BackorderingSummary BIS 517- Aslı Sencer