1. 1 Retail Pricing using Dynamic Optimization October 2012 © 2012 Massachusetts Institute of Technology. All rights reserved. Sabanci University Fall 2012 Robert M. Freund

2. 2 A Retail Pricing Problem The Coop is preparing for the Fall sales season of Power-Pro laptop bags for MBA students. The “ back-to-school ” sales season runs from early August through late September, a total of 9 weeks.

3. 3 Retail Pricing Problem - Details 9-week sales period Set (or re-set) price each week Six different possible price levels Price cannot be increased from week to week The total stock level is 1,000 laptop bags Any bag remaining at the end of the season has a salvage value of $55

4. 4 Distribution of Demand Stock Level1,000 Salvage Value$55 Sales Season9 weeks Price cannot be increased from week to week. Weekly Demand Probability Price0.2 $70.00220225230235240 $75.00210215220225230 $80.00165170175180185 $103.0075859095105 $112.006065707580 $121.004045505560

5. 5 Pricing Strategy What decision do we need to make in week 1? Is week 1 different than any other week? What decision(s) do we need to make in weeks 2, 3, …, 9 ?

6. 6 Pricing Strategy What will influence the pricing decision in week 1? Starting StockOptimal 1 st Week Price 600? 500? 400? 300?

7. 7 Example of a Pricing Strategy “ Here is what you should do: In weeks 1, 2, 3 set price to $112 In weeks 4, 5set price to$103 In weeks 6, 7, 8set price to$80 In week 9set price to $75 ” Could this be a description of an optimal strategy?

8. 8 Another Type of Pricing Strategy “ Here is what you should do: Start with a price of $112 In every week t, do the following: Given last week’s price P t-1 and the average demand  at price P t-1 If: (current stock level)/(weeks left in season) > (0.87) , then decrease the price to the next lowest value.” Could this be a description of an optimal strategy?

9. 9 Yet Another Type of Pricing Strategy “ Here is what you should do: Start with a price of $112 When the stock level falls to: 525lower price to $103 390lower price to $80 265lower price to $75 ” Could this be a description of an optimal strategy?

10. 10 A “Dynamic” Pricing Strategy … If we are in week 5 and stock level is 420 and last week’s price was $103, then set this week’s price to $80 If we are in week 5 and stock level is 415 and last week’s price was $103, then set this week’s price to $103 … If we are in week t and stock level is U and last week’s price was P, then set this week’s price to ___ … Make these statements for all weeks, all stock levels, and all previous weeks’ prices.

11. 11 1-Period Model Suppose it is the last week of the selling season It is the start of week 9 1 pricing period Week 9 Starting Stock Level = U

12. 12 1-Period Model, Stock Level = 100 For price of $103, Total EMV = 103*(.2*75 +.2* 85 +.2*90 +.2*95 +.2*100) + 55 * (.2*(100–75) +.2*(100-85) +.2*(100-90) +.2*(100-95) +.2 * (100-100)) =9,167 + 605 = $9,772 For Stock Level = U = 100, and each price P, the expected total revenue is shown in the Total EMV column below: Units Sold Probability Current Week SalvageTotal Price0.2 Sales ValueEMV $70.00100 $7,000+$0$7,000 $75.00100 $7,500+$0$7,500 $80.00100 $8,000+$0$8,000 $103.0075859095100$9,167+$605$9,772 $112.006065707580$7,840+$1,650$9,490 $121.004045505560$6,050+$2,750$8,800

13. 13 1-Period Model, Stock Level = 300 For price of $75, Total EMV = 75 *(.2*210 +.2*215 +.2*220 +.2*225 +.2*230) + 55 * (.2*(300–210) +.2*(300-215) +.2*(300-220) +.2*(300-225) +.2 * (300-230)) =16,500 + 4,400 = $20,900 For Stock Level = U = 300, and each price P the expected total revenue is given in the Total EMV column below: Units Sold Probability Current Week SalvageTotal Price0.2 Sales ValueEMV $70.00220225230235240$16,100+$3,850$19,950 $75.00210215220225230$16,500+$4,400$20,900 $80.00165170175180185$14,000+$6,875$20,875 $103.0075859095105$9,270+$11,550$20,820 $112.006065707580$7,840+$12,650$20,490 $121.004045505560$6,050+$13,750$19,800

14. 14 Week-9 Table of Optimal EMV and Price Stock Level Last Week’s Price (U)$70$75$80$103$112$121 0 5 10 … 100$7,000 ($70)$7,500 ($75)$8,000 ($80)$9,772 ($103) … 300$19,950 ($70)$20,900 ($75) … 1,000

15. 15 Week-9 Model Interpretation Define J 9 (U, P) = Optimal EMV if we are in the last week (week 9 of a 9-week model) with a stock level of U units, and a price cap (i.e., previous week’s price) of P J 9 (U, P) is precisely what is portrayed in the Week-9 table

16. 16 2-Period Model Stock level at beginning of week 8 = U 2 weeks, set price in each week Now suppose it is the start of week 8 of the sales season:

17. 17 2-Period Model, Stock Level = 200 For price of $103, Total EMV = 103*(.2*75 +.2*85 +.2*90 +.2*95 +.2*105) +.2*J 9 (200–75, 103) +.2*J 9 (200-85, 103) +.2*J 9 (200-90,103) +.2*J 9 (200-95, 103) +.2*J 9 (200-105, 103) =9,270 + 10,351 = $19,621 For Stock Level = U 8 = 200, and each price P, the expected total revenue is given in the Total EMV column below: Units Sold ProbabilityWeek 8 EMV afterTotal Price0.2 Sales Week 8EMV $70.00200 $14,000+$0$14,000 $75.00200 $15,000+$0$15,000 $80.00165170175180185$14,000+$2,000$16,000 $103.0075859095105$9,270+$10,351$19,621 $112.006065707580$7,840+$11,470$19,310 $121.004045505560$6,050+$12,570$18,620

18. 18 2-Period Model I can do this for all stock levels U = 0, 5, 10, … 1,000 and all prices P = $70, $75, $80, $103, $112, $121

19. 19 Week-8 Table of Optimal EMV and Price Stock Level Last Week’s Price (U)$70$75$80$103$112$121 0 5 10 … 200$14,000 ($70)$15,000 ($75)$16,000 ($80)$19,621 ($103) … … … 1,000

20. 20 Week-8 Model Interpretation Define J 8 (U, P) = Optimal EMV if we are in the next-to- last week (week 8 of a 9-week model) with a stock level of U units, and a price cap of $P J 8 (U, P) is precisely what is portrayed in the 2- period table

21. 21 9-Period Model 9 weeks, 9 prices to determine What is the optimal pricing strategy in Week 1? Now suppose that we are in the first week of the full 9-week selling season.

22. 22 Week-1 Table of Optimal EMV and Price Stock Level Last Week’s Price (U)$70$75$80$103$112$121 0 5 10 … 200 … … … 1,000$96,730 ($103)

23. 23 9-Period Model Interpretation Define J 1 (U, P) = Optimal EMV if we are in week 1 of a 9-week model with a stock level of U units, and a price cap of $P J 1 (U, P) is precisely what is portrayed in the week-1 table We wish to know J 1 (1000, $121) (for starters)

24. 24 The General Model Let P denote the price decision for period t. Let D P (i) denote the demand at price P, for i = 1, …, 5 Let J t (U t, P t-1 ) denote the value function for period t, where U t is the stock level at the beginning of period t P t-1 is the price level of the previous period (t-1) J t (U t, P t-1 ) is the optimal expected value of revenues from week t through the end of the season, given that it is now the beginning of week t, there are U t bags in stock and we cannot set the price higher than P t-1.

25. 25 The General Model The value function for period t is essentially: However, when the stock level falls to 0, there are no bags left to sell regardless of the demand. Therefore, the equation is actually:

26. 26 Dynamic Optimization “Dynamic Optimization,” “Dynamic Programming,” “DP” State of the System Bellman Equation Backward Recursion

27. 27 Dynamic Optimization Dynamic optimization models capture complex interactions over time Dynamic optimization is a great conceptual tool, and very often is a good practical tool as well Dynamic optimization models are used in power system pricing, revenue management for airlines, hotels, car rental companies, internet retail pricing, military wargame models, etc.

28. 28 State of the System In this model, the state of the system is the stock level and price cap from the previous period: (U t, P t-1 ) In other models, the state of the system might be completely different State of the system in period t

29. 29 State of the System, continued The “state of the system” in each period t is comprised of values of quantities that capture what is needed to determine the current status of the situation. In pricing, this might include: Current stock level U t Last week’s price P t-1 Last week’s demand or average recent demand (indication of non-independent demand) D t-1 Number of markdowns already taken N t-1 others….

30. 30 Modeling Correlated (“learned”) Demand Demand Level this Week Low: 1234High: 5 Low: 10.500.300.2000 20.250.500.150.100 3 0.150.500.150.10 40 0.150.500.25 High: 5000.200.300.50 Demand Probabilities after week 1: Demand Level last Week This models demand being correlated from week to week

31. 31 Value Function For this model, the value function is J t (U t, P t-1 ) The value function is the optimal value of the system in period t for each possible state that the system might be in

32. 32 Backward Recursion We construct the value function recursively, starting with its value in the last time period, and working backwards to the first period. This is similar in concept to the way we construct EMV values at points in a decision tree, working from the end of the tree and working to the start of the tree. For our model, the backward recursion is: However, the exact nature of the recursion will differ from model to model

33. 33 Final Comments Dynamic optimization models compute the optimal decisions and associated EMV values Dynamic optimization models are very powerful Dynamic optimization models can be very computationally intensive When the models/systems become too complex, one needs to use “approximate dynamic optimization” methods to solve for very good (but necessarily optimal) pricing decisions