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### Transcript

• 1.
• Lecture 4
• Non-Linear Models
• Optimal Pricing in Retail
• Modeling Pricing Competition
• 2. Nonlinear Programming
• So far we have analyzed problems with objective functions and constraints that are linear – what happens if these are non-linear ?
• Where does non-linearity come from in the first place?
• Economies or diseconomies of scale - e.g. Marginal costs/benefits that change as volume changes
• Interactions effects – e.g. price effects demand, yet revenue is the product of price times demand, leading to nonlinearity
• Threshold effects – e.g. the value of an option in the future is the maximum of i) the value if exercised, or ii) the value if not exercised.
• 3. Example: The optimal scale of business
• Consider a business with the choice of two technologies:
• The firm uses the lowest cost technology given its volume of output.
\$0 \$20,000 Tech. 2 \$2.00 \$0 Tech. 1 Variable Cost/Unit Annual Fixed Cost
• 4.
• 5.
• 6. What the problem looks like for small unit volumes …. Maximum profit?
• 7. “ Local” maximum – best solution within a limited “neighborhood”. “ Global” maximum – best solution overall But for a wider range of unit volumes things look different ….
• 8. Different starting points lead to different solutions
• Starting solver at v=1,000 we get
• x*=4,444 Local maximum
• Starting solver at v=10,000 we get
• x*=40,000 Global maximum
• Why? Solver checks for “local improvements” if marginal changes in volume do not improve profits, then it declares current value is “optimal”
• For linear problems, local optima are always global optima and this logic works. But for non-linear problems, this is not always true.
• 9. Some practical decision-making consequences
• Beware of the “small change” argument…
• Argument: “Every time we’ve tried lowering prices a little, our profits have declined because the increased revenue doesn’t offset our increased costs. We’re best off staying with our current price points.”
• Pitfall: If the price declines are large enough , the increased volume may justify a shift to a new technology that leads to MUCH higher profits (e.g. Wal-Mart).
• Why cost-benefit analysis can be misleading:
• Argument: If marginal benefits do not exceed marginal cost, the project is not worthwhile
• Pitfall: Making larger changes (e.g.volume in our example) or making several changes simultaneously may lead to an improvement even if marginal changes do not.
• 10. Nonlinear Programming: Key Points
• The solution returned by the optimizer may depend on the starting point: In general, optimizers are not guaranteed to give global optimal solutions to nonlinear programs. You may have to experiment with different starting points.
• This behavior is due to the inherent limitations of marginal analysis, which has broader implications for decision making.
• Also, nonlinear programs are less efficient numerically; they can take much more compute time to solve than linear programs.
• 11. Apparel Business Product Assortment Presentation Advertising Pricing Supply Chain Management
• 12. Pricing in the Apparel Business
• The price a customer is willing to pay for a particular product has little to do with how much it costs to produce it
• Products are seasonal, if a product is not sold-out at the end of the season, the remainder is sold at a deep loss
• There is a large number of products (styles, colors, sizes)
• For some companies, there are many stores in different regions
• Most product related decisions, including pricing, are made by “buyers”, whose main focus is selecting the “right” styles
• Buyers are eager to get rid of the slow-selling merchandise by offering deep markdowns (“I just want it gone!”)
• Recently, companies started realizing that a substantial amount is being left on the table by this mode of operation
• 13. Who does “Pricing Optimization”?
• 14. Example: Setting Retail Price
• A retail firm is considering a pricing decision for a fashion item
• According to marketing department estimates, if the price for the item is set at \$ P , the overall demand for the item over the 15-week selling season will be:
• D = 7,725 - 97.5 P
• What price should a firm set to maximize its revenues over the selling season
• 15. Demand-Price Curve
• Demand follows a linear pattern
D P 7,725 79.24 1,875 60 Example: for P 1 =\$60, demand is D =7,725-97.5x60=1,875 units
• 16.
• Revenues = R = P * D = P* (7,725 - 97.5 P ) = 7,725 P - 97.5 P 2
R, \$ P, \$ 79.24 Non-linear function of P 39.62 153,014 0
• Best price = \$39.62
• 17. What real-life factors are ignored in this model?
• Prices can be changed during the selling season
• Supply constraints may limit the overall sales
• Demand curve can be controlled by advertising
• Retailers sell variety of products with inter-related demands
• Retailers typically face competition
• Etc. etc.
• 18. Ex: Multi-Period Revenue Management Problem
• A retail company stores are stocked with 2,000 units of a single fashion item
• The sales season consists of 15 weeks
• No chance for re-stocking the item of reallocating among stores
• Goal: maximize retailer’s profits over the selling season
• 19. Costs
• All production and distribution costs have already been paid: they are sunk costs
• Every unit kept in inventory at the end of each week incurs \$1 in inventory and maintenance costs
• All items in stores that are not sold at the end of 15 weeks are sold to discounters (“jobbers”) for \$25 per unit (salvage value)
• 20. Demand-Price Curve
• Demand in each week follows a linear pattern.
• Week 1:
D 1 P 1 655 100.8 265 60 Example: for P 1 =\$60, demand is D 1 =655-6.5x60=265 units
• 21. Time-Dependent Demand Functions
• We expect demand to decrease over time.
• Decreasing intercepts:
• Each week, intercept drops by 20 units
• 22. Time-Dependent Demand Functions
• Maximum demand in week 15 is about 60% lower than in week 1
Demand Price 655 375 100.8 57.7 Week 1 Week 15
• 23. Revenues
• Revenue earned in Week 8 (assuming enough inventory):
R 8 P 8 79.2 Non-linear function of P 8 39.6 10201.0
• 24. Decision Variables and Objective Function
• Goal: maximize retailer’s profits over the selling season
• Decision Variables:
• P t = Price in week t , t =1,…,15.
• S t = Sales in week t , t =1,…,15.
• Objective Function:
• Profits = Sales Revenue + Salvage Revenue – Inventory Cost
• Additional (definitional) variables: Ending inventory and demand for each week:
• I t = Inventory in week t , t =1,…,15
• D t = Demand in week t , t =1,…,15
• 25. Revenues and Costs
• Sales Revenue
• S 1 P 1 +…+ S 15 P 15 =
• Salvage Revenue
• 25 I 15
• Inventory Cost
• 1x I 1 +…+ 1x I 15 = 1x
• Objective Function:
• Profits =S 1 P 1 +…+ S 15 P 15 + 25 I 15 - (1x I 1 +…+ 1x I 15 )
• 26. Problem Constraints
• Constraints:
• For every week (t=1,…,15)
• Sales in week t
• S t ≤ D t
• S t ≤ I t- 1
• Inventory balance
• I t = I t- 1 - S t
• I 0 = 2000
• Demand in week t
• Non-negativity
• S t ,P t  0
I t -1 I t S t Week t
• 27. Multi-period Revenue Management Problem: Complete Formulation
• Non-linear model: objective function is quadratic in price decision variables
• Cannot check “Assume Linear Model” in Excel Solver options
• 28. Optimized Spreadsheet =C12-\$D\$4*B12 =F11-E12 =D6*F26 =F28+F29-F30 =D7*SUM(F12:F26)
• 29.
• The optimal solution is to gradually decrease the price.
• The total profit value is \$110,999.
• 30.
• Sales DECREASE over the course of the season as well. Why?
• 31.
• The LARGER is the inventory cost, the more FRONTLOADED sales are.
• 32. Modeling Pricing Competition
• Two retailers compete by setting different prices for similar items
• Demand for Company A depends on the prices of both Company A and Company B:
• What should Company A’s price be, if Company B sets a price of \$100?
• 3725+40*100 = 7,725
• Company A should charge \$39.62, if Company B charges \$100
• This is called the “Best Response” of Company A to Company B’s price
• 33. Company A’s Best Response Problem
• Given competitor’s price , select own price to maximize revenues
• Decision Variable:
• Objective Function to be maximized:
• Revenues =
• Constraints:
• Similar problem for Company B
• Can be solved using Solver (see Retail.xls)
• 34. Best Response
• If Company B charges \$100, the “Best Response” for Company A is to charge \$39.62
• What would Company B do in response?
• Company B would charge \$21.10, if Company A charges \$39.62
• If Company A knows how Company B chooses prices, would Company A set a price of \$39.62?
• 35. Nash Equilibrium
• If there is a set of prices such that nobody has an incentive to deviate unilaterally, that set is called a Nash Equilibrium
• The Nash Equilibrium is a central concept in Game Theory, which analyzes competition
• The Nash Equilibrium for our pricing model is:
• 36. Equilibrium: Prices, Sales, Revenues
• How would these change, if Company A acquires Company B?
\$ 26,384 \$ 50,314 Revenues 1,498 2,215 Sales \$ 17.62 \$ 22.72 Price Company B Company A
• 37. Joint Optimization Problem
• We can formulate this as a joint optimization problem: Select and to maximize total revenues.
• Decision Variables:
• Objective Function to be maximized:
• Total Revenues =
• Constraints:
• See Retail.xls for spreadsheet implementation
• 38. Jointly Optimal Prices, Sales, Revenues \$ 26,443 \$ 55,880 Revenues 1,028 1,927 Sales \$ 25.74 \$ 29.00 Price Company B Company A
• 39. Competition vs. Centralized Management \$ 82,323 2,955 \$ 27.87 Centralized 7 % -20 % 35 % Percentage Difference \$ 76,698 Total Revenues 3,713 Total Sales \$ 20.66 Average Price Competition
• 40. Revenue Management Case Study: American Airlines & People Express
• Airline Deregulation Act 1978
• Price controls lifted
• Free entry and exit from markets
• Rise of low-cost carriers
• PeopleExpress started 1981
• 1984 Results: \$1B Rev., \$60M profit
• Major airlines like American were significantly affected, especially by loss of discretionary (leisure) travelers.
• 41. Crandall’s solution
• Bob Crandall (VP Marketing at the time) recognized some essential facts:
• Many AA flights departed with empty seats
• The marginal cost of using these seat was very small
• AA could in fact compete on cost with the new entrants using these “surplus seats”
• But how?
• Created new restricted, discounted fares (“Super Saver” and “Ultimate Super Saver” fares)
• Capacity-controlled the availability of these fares
• DINAMO – D ynamic I nventory A llocation and M aintenance O ptimizer: optimized the number of discount seats to sell on each and every flight departure (5000+ flights/day).
• 42. Results of the new strategy & capability
• AMR shares initially plunged on first announcement of “Ultimate Super Saver” fares Jan. 1985
• Analysts thought it was the start of a price war
• “ AMR cannot operate profitably at these fares.”
• But RM systems proved very effective
• AA revenues rose
• Competitors suffered: e.g. PeopleExpress
• 1984 \$60M profit (all-time high)
• 1985 \$160M loss
• 1986 Bankruptcy
• Sold to Continental
DINAMO launched
• 43. “ We were a vibrant, profitable company from 1981 to 1985, and then we tipped right over into losing \$50 million a month. We were still the same company. What changed was American’s ability to do widespread Yield Management in every one of our markets. … That was the end of our run because they were able to under-price us at will and surreptitiously.” “ Obviously PeopleExpress failed . . .We did a lot of things right. But we didn’t get our hands around Yield Management and automation issues. [If I were to do it again . . . ] the number one priority on my list every day would be to see that my people got the best information technology tools. In my view, that’s what drives airline revenues today more than any other factor–more than service, more than planes, more than routes .” Donald Burr CEO PeopleExpress
• 44. Pricing Optimization Software Companies DemandTec www.demandtec.com KhiMetrics www.khimetrics.com ProfitLogic www.profitlogic.com i2 www.i2.com Zilliant www.zilliant.com
• 45. Summary
• Non-linear optimization
• Price-dependent demands result in non-linear problems
• Beware of marginal analysis!
• Modeling pricing competition
• Best Response
• Nash Equilibrium
• Modeling inter-temporal dependencies
• Flow balance equations
• Surplus cash in the cash-flow matching problem vs. inventories in the revenue management problem
• Reference reading: P&B p. 200, p. 203-209