1. Stochastic Modeling & Simulation Lecture 16 : Probabilistic Inventory Models

2. Administrative Problem Set 6 Good news: NOT DUE Friday. Due next Friday Bad news: I’ll add a couple of problems tonight and repost it.

3. Projects Comments on projects – offline / in my office. Literature review. Just because this is a project you’re constructing doesn’t mean you shouldn’t look at what others have done. It’ll be very helpful. Check out winter sim conference papers & proceedings. http://wintersim.org/ Also check out INFORMS.

4. Last time EOQ model

5. Uncertain Demand Back to our calendar ordering problem from the beginning of the semester: Suppose you need to order 2016 calendars to sell next year Each one you order will cost you $7 Demand is uncertain but suppose that you have historical data and you average around 200 when you price at $10 But it varies and actually looks somewhat symmetric and Normal-shaped with a standard deviation of 15 Let’s suppose you price at $10 How many do you order? This is a one period inventory problem.

6. Calendar Ordering The calendar ordering problem is often referred to as a newsvendor problem. Order too much  you have leftover inventory that you can’t sell Order too little  you could have made more money. Insightful problem not just about calendars or newspapers. Before we played around with different order amounts. How do we find the optimal solution? What would it entail? Balancing the overstocking cost and understocking cost.

7. Calendar Ordering Solution Suppose we ordered Q=200 calendars. What would our overstocking cost be if we actually sold 199? We had to pay $7 for the unsold calendar, so c over = $7. What if demand was X < 200? Overstocking cost would be $7(200-X) What would our understocking cost be if demand was actually 201? That’s a sale we could have had for a profit of $10 - $7 = $3. Right now we’ll just assume that c under = lost profit from that one sale = $3. What if demand was Y>200? Understocking cost would be $3(Y-200)

8. Calendar Ordering Solution Let f() be the density function of the demand for calendars, with distribution function F() Recall that we’ve assumed F() to be Normal(200,15 2 ) What is the expected cost of ordering q calendars less the ordering cost of the calendars? Expected overstocking cost + Expected understocking cost Expected overstocking cost = c over E(max[0,Q-D]) Expected overstocking cost = c under E(max[0,D-Q])

9. Calendar Ordering Solution What is the expected cost of ordering q calendars less the ordering cost of the calendars? Expected overstocking cost = c over E(max[0,Q-D]) Expected overstocking cost = c under E(max[0,D-Q])

10. Calendar Ordering Solution using Leibniz’s Rule, we can take the derivative of the above and set it equal to zero. Note the second order condition: second derivative >0.

11. Calendar Ordering Solution Rearranging terms we get: In our example from before c over = $7 and c under = $3 Implying F(Q*) = 3/(7+3) = 0.3 Using @Risk or norm.inv  optimal order quantity is 192.

12. Example: Airline Overbooking The ticket price for a Doha-Dubai flight is $200. Each plane can hold up to 100 passengers. Some passengers who buy tickets don’t show up To protect against no-shows the airline might try to sell more than 100 tickets. Suppose that if a ticketed passenger shows up but can’t board because it’s full gets a $100 voucher and is put on the next flight. Assume all passengers purchase refundable tickets such that they can get the $200 back if they cancel by the flight time (for ease = don’t show). If the number of no-shows is ~Normal(20,5 2 ), how many tickets should the airline sell? What if no-shows were distributed Triangle(0,20,35)?