36106 Managerial Decision Modeling Revenue Management
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1 36106 Managerial Decision Modeling Revenue Management Kipp Martin University of Chicago Booth School of Business October 5, 2015
2 Reading and Excel Files 2 Reading (Powell and Baker): Section 9.5 Appendix 9.1 Files used in this lecture: revenuemgmt.xlsx revenuemgmt key.xlsx revenuemgmt macro.xlsx revenuemgmt alt.xlsx
3 Lecture Outline 3 Revenue Management Revenue Management - Sensitivity Analysis Allowable Increase and Decrease Objective Function Coefficient Sensitivity
4 Learning Objectives 1. Learn about revenue management, a very important application of modeling. 2. Apply our dual price methodology to find the value of overbooking. 3. Learn to build a revenue management model in Solver.
5 Revenue Management Revenue Management: managing the short-term demand for perishable inventory in order to maximize the revenue potential for an organization. For example, seat 35C on United Flight 219 from ORD to HNL on October 30, 2015 is an item held in inventory. This item expires on October 30, 2015 and has zero value after this date. Started with the American Airlines Sabre system. Back then it was called yield management. It is a form of price discrimination. Was used to determine how many airline flight seats to sell at an early reservation discount fare and how many airline flight seats to sell at full fare. Used by all major airlines today. Generates about a billion in incremental revenue for American Airlines Revenue Management is offered this Spring.
6 Revenue Management 6 Revenue Management has been applied to: Airline seat allocation Hotel room booking (my Phoenix experience) Apartment rentals Car rentals Cruise lines Golf courses The Internet greatly facilitates revenue management.
7 Revenue Management Objective: Show how to formulate revenue management problems as a linear program in Solver. Understand how dual prices are used in revenue management. Revenue Management is driven by technology. 1. fast computers 2. the Internet
8 Revenue Management This presentation is based on Leisure Air. See Management Science by Anderson, Sweeney, Williams, and Camm. Two Boeing jets, each with 132 seat capacity. One jet originates in Pittsburgh, the other in Newark. The Pittsburgh jet flies to Orlando with a stop in Charlotte The Newark jet flies to Myrtle Beach with a stop in Charlotte There are two fare classes: Class Q discount fare Class Y full fare There are three destinations: Orlando, Myrtle Beach, and Charlotte.
9 Revenue Management Possible Itineraries: Pittsburgh-Orlando (stop in Charlotte) Pittsburgh-Myrtle Beach (stop in Charlotte) Pittsburgh-Charlotte Newark-Orlando (stop in Charlotte) Newark-Myrtle Beach (stop in Charlotte) Newark-Charlotte Charlotte-Orlando Charlotte-Myrtle Beach
10 10 Here is the network topology. Revenue Management
11 Revenue Management Key Concept: ODIF origin-destination-itinerary fare This is a valid combination of legs in a class. A leg is a flight between two cities without a stop. There are 16 possible ODIF fares. There are four distinct legs. Note: In the BlueSky case for homework an itinerary is called a route and a leg is called a flight.
12 Revenue Management 12 See revenuemgmt.xlsx.
13 Revenue Management Objective: Determine how many seats to make available for each ODIF in order to maximize revenue. This is called revenue management. How are costs treated in these models? Important: 1. We treat price as parameter! 2. We treat demand as a deterministic parameter! Variable Definition: PCQ = number of seats allocated to PCQ ODIF (Pittsburg-Charlotte Q class) POY = number of seats allocated to POY ODIF (Pittsburg-Orlando Y class) There is one such variable for each class. The optimal value of the variable is how many seats to allow in that ODIF.
14 Revenue Management Objective: Determine how many seats to make available for each ODIF in order to maximize revenue. Therefore maximize the sum product of the ODIF variables with the corresponding revenue, e.g. max 178PCQ + 268PMQ COQ + 582COY Sample Test Question: Write out the revenue for just the Charlotte-Orlando leg.
15 Revenue Management Plane Capacity Constraint: The number of seats allocated to a leg cannot exceed the plane capacity (in this case 132) of the leg. There is one such constraint for each leg for a total of four constraints. Consider Pittsburgh-Charlotte leg capacity constraint PCQ + PMQ + POQ + PCY + PMY + POY 132 Sample Test Question: Write out the Charlotte-Orlando leg capacity constraint.
16 Revenue Management ODIF Demand Constraints: The number of seats allocated to an ODIF cannot exceed the demand on the ODIF. PCQ 33 PMQ 44 Sample Test Question: What is the Charlotte-Orlando discount demand constraint?
17 Revenue Management 17 Here is the optimal allocation.
18 Revenue Management 18
19 Revenue Management - Sensitivity Analysis 19 Okay, now let s see what might happen in practice after the initial model solution. We are going to tweak the demands. Remember, we assumed that demand was deterministic, but in fact the demand numbers come from forecasts and are stochastic. We are going to tweak the fares. Excel and Solver are really good tools for this. Open revenuemgmt macro.xlsm.
20 Revenue Management - Sensitivity Analysis The scenario: The perishable inventory consists of the seats in each ODIF (origin-destination-itinerary-fare) on Leisure s two jets for January 23, On January 19, 2015 a user comes to Leisure s Web Site and wants to buy a PCY ODIF the full fare from Pittsburgh to Charlotte. However, all 16 seats allocated in this PCY ODIF are sold. As of January 19, 2015 Leisure has not sold all of the 22 seats allocated to the POQ ODIF. The number seats allocated to the COQ ODIF for January 23, 2015 is 41 and the forecasted demand is 46. What should the revenue management system do?
21 21 Revenue Management - Sensitivity Analysis What is the effect of over booking PCY by 1? Each color represents an itinerary. P +1 N 1 C 1 O +1 M
22 Revenue Management - Sensitivity Analysis 22 What is the effect of over booking PCY by 1?
23 Revenue Management - Sensitivity Analysis Revenue Change ODIF Change Revenue PCY $380 POQ $228 COQ $224 + $376 Sell the ticket! And adjust the system so that: Add one more seat between Pittsburgh and Charlotte in class Y Allocate one less seat between Pittsburgh and Orlando in the Q class Allocate one more seat between Charlotte and Orlando in the Q class
24 Revenue Management - Sensitivity Analysis The scenario: The perishable inventory consists of the seats in each ODIF (origin-destination-itinerary-fare) on Leisure s two jets for January 23, On January 19, 2015 a user comes to Leisure s Web Site and wants to buy a PMQ ODIF the cheap Pittsburgh to Myrtle Beach. However, all 44 seats allocated in the PMQ ODIF are sold. As of January 19, 2015 Leisure has not sold all of the 22 seats allocated to the POQ ODIF. As of January 19, 2015 Leisure has sold not all of the 31 seats allocated to the CMQ ODIF. The number seats allocated to the COQ ODIF for January 23, 2015 is 41 and the forecasted demand is 46. What should the revenue management system do?
25 Revenue Management - Sensitivity Analysis 25 What is the effect of over booking PMQ by 1? P +1 N 1 1 C +1 O +1 1 M Each color represents a distinct ODIF.
26 Revenue Management - Sensitivity Analysis 26 What is the effect of over booking PMQ by 1? Each color represents a distinct ODIF.
27 Revenue Management - Sensitivity Analysis 27 Revenue Change ODIF Change Revenue PMQ $268 POQ $228 CMQ $179 COQ $224 + $85 Sell the ticket! And adjust the system so that: Add one more seat between Pittsburgh and Myrtle Beach in class Q Allocate one less seat between Pittsburgh and Orlando in the Q class Allocate one less seat between Charlotte and Myrtle Beach in the Q class Allocate one more seat between Charlotte and Orlando in the Q class
28 Revenue Management - Sensitivity Analysis 28 The dual prices give us this information immediately!
29 Revenue Management - Sensitivity Analysis The dual price on the PMQ ODIF is 85. What happens if you set the right-hand-side on the PMQ ODIF to 45 and solve again? The dual price on the PCY ODIF is 376. What happens if you set the right-hand-side on the PCY ODIF to 17 and solve again? See the Workbook revenuemgmt macro.xlsx. Key Idea: use the dual prices to adjust the number of seats made available in each ODIF as reservations come in.
30 Allowable Increase and Decrease 30
31 Allowable Increase and Decrease Consider the constraint on the PMQ ODIF. The current right-hand-side demand limit is 44, and this implies that we cannot allocate more than 44 seats in this ODIF. We also know that if we sell one more seat, revenue goes up by $85. However, the allowable increase is 5. If we observe an increase in demand of six seats, and allocate six seats in this fare class, the sixth seat may not generate $85 dollars. At the margin, it can never generate more than $85. Run the model with demands of 49 and 50. What do you observe? For a maximization problem, the value of the dual price will decrease when the right-hand-side is increased by more than the allowable increase. Okay, so why is the allowable increase 5?
32 Allowable Increase and Decrease Important: See revenuemgmt alt.xlsx. We formulate the leg capacity and demand limit constraints differently in this workbook. We do not put a constraint on slack. Instead we say: The number of seats allocated to a leg cannot exceed the the capacity of the plane on the leg. The seats allocated for an ODIF cannot exceed the demand for that ODIF. The sensitivity report is a bit different. We discussed this last week. The dual price information on the ODIF constraints appears in the reduced cost information.
33 Allowable Increase and Decrease 33
34 Allowable Increase and Decrease 34 We lose the allowable increase and allowable decrease information on the ODIF demand constraints. The allowable increase and decrease information in the Variable Cells section is on the objective function coefficient, not on the reduced cost.
35 Allowable Increase and Decrease 35 The allowable decrease on the PMQ ODIF is 23. Why 23? Hint: The model wants to fill both planes to capacity. If we decrease the number of seats on PMQ, we want to increase seats on some other leg(s). What restricts this? What if we decrease the demand by more than 23? Will the new dual price be smaller or larger than $85?
36 Allowable Increase and Decrease Remember our two important rules: Rule 1: Helping helps less and less! Rule 2: Hurting hurts more and more! These two rules will always tell you what happens as your scarce resource levels increase or decrease.
37 Allowable Increase and Decrease 37 Summary: what happens to the optimal objective function value under the following scenarios? Constraint maximum minimum Type objective function objective function Increase Decrease Increase Decrease Increase Decrease Increase Decrease
38 38 Objective Function Coefficient Sensitivity Now, for the rest of the story! In addition to information about the right-hand-sides, Solver provides useful economic information about objective function coefficients.
39 Objective Function Coefficient Sensitivity The allowable increase and allowable decrease on an objective function coefficient is the amount the objective function coefficient can increase or decrease without changing the optimal solution. Consider the POQ ODIF. The associated adjustable cell is I7 (workbook revenuemgmt macro.xlsx). The current objective coefficient is 228. Let s say we raise the fare on this POQ by 70 dollars. That is, increase the objective function coefficient from 228 to = 298. The optimal solution does not change. Now exceed the allowable increase of 85. Increase the objective function coefficient from 228 to = 318. Not surprisingly, we increase the seat allocation from 22 to 45.
40 Objective Function Coefficient Sensitivity Now consider reducing an objective function coefficient. This makes the ODIF less profitable. The allowable decrease on the objective function coefficient on the NMQ ODIF is 70. Decrease the value of the coefficient by 75, that reduce the current 249 to = 174. The number seats allocated in the NMQ ODIF is reduced from 36 to 3.
41 Objective Function Coefficient Sensitivity Reduced Cost: two ways to define a reduced cost. the dual price on the nonnegativity constraint for a maximization, the allowable increase on an objective function coefficient if the variable is currently in the solution at 0 For a maximization problem, if a variable is currently at zero in the optimal solution, the reduced cost is the amount by which the objective function coefficient must increase in order for there to be an optimal solution where the variable is positive. For a minimization problem, if a variable is currently at zero in the optimal solution, the reduced cost is the amount by which the objective function coefficient must be reduced in order for there to be an optimal solution where the variable is positive.
42 42 Objective Function Coefficient Sensitivity Below is the sensitivity report when we change the demand on the POY ODIF from 11 to 50. In this optimal solution, the POQ ODIF is allocated zero seats. It has a reduced cost of 85 (actually, -85, we only worry about the absolute value).
43 Objective Function Coefficient Sensitivity In this optimal solution, the POQ ODIF is allocated zero seats. It has a reduced cost of 85 (actually, -85, we only worry about the absolute value). The objective coefficient on the POQ ODIF is $228. We would need to raise the fare on the POQ ODIF to $313 = $228 + $85 in order for there to be an optimal solution with a strictly positive number of seats allocated to the POQ ODIF.
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