Network Revenue Management

Network Revenue Management Page 1 Outline Network Management Problem Greedy Heuristic LP Approach Virtual Nesting Bid Prices Based on Phillips (2005) Chapter 8

Demand for Hotel Rooms Vary over a Week Page 2 Occupancy 100% Monday Tuesday Wednesday Thursday Friday Saturday Sunday Since the demand varies, so should the booking limits. Many customers book for several days in a row Which booking limits apply to a booking request for Tuesday night check-in and 3 night stay? Tue, Wed, or Thu? Since requests are coupled over days, so should the booking limits. Compute the booking limits not separately day by day but by jointly for seven days!

Hotel Booking Network is a Chain but Extensive Page 3 Sun -1 Sat -1 Fri -1 Thu -1 Wed -1 Tue -1 Mon -1 Sun -2 Time flow chronologically Mon Tue Wed Thu Fri Sat Sun Mo Tu We Th Fr Sa Su Mo+1 night night night night night night night O O D O O D Origin-Destination (check-in,check-out) day pairs with 7 nights: D D Mo-Tu, Mo-We, Mo-Th, Mo-Fr, Mo-Sa, Mo-Su, Mo-Mo+1 all check in on Monday but stay 1, 2,, 7 nights Tu-We, Tu-Th, Tu-Fr, Tu-Sa, Tu-Su, Tu-Mo+1 all check in on Tuesday but stay 1, 2,, 6 nights We-Th, We-Fr, We-Sa, We-Su, We-Mo+1 all check in on Wednesday but stay 1, 2,, 5 nights M T W Th F S Su MT Th-Fr, Th-Sa, Th-Su, Th-Mo+1 all check in on Thursday but stay 1, 2, 3, 4 nights W Fr-Sa, Fr-Su, Fr-Mo+1 all check in on Friday but stay 1, 2, 3 nights Th Sa-Su, Sa-Mo+1 all check in on Saturday but stay 1, 2 nights F S Su-Mo+1 checks in on Sunday and stays 1 night Su A total of 7+6+5+4+3+2+1=7*8/2=28 O-D pairs. Check-out Over 365 nights 365*366/2=66795 O-D pairs. Standard, deluxe, deluxe with a view, suit rooms, and suppose each has a fare class of its own. 66795*4=267180 ODFs. Check-in

Demand for Flights Vary over Legs Page 4 Occupancy 100% College Station Dallas Dallas Los Angeles Los Angeles Honolulu Many customers book for College Station Dallas Los Angeles Honolulu trip Which booking limits apply to this booking request? Since requests are coupled over legs, so should the booking limits.

Airline Network: Hub-and-spoke Las Vegas Honolulu Fresno Little Rock Page 5 LA Dallas Palm Springs Austin College Station Origin-Destination pairs with 10 cities: Houston College Station {Dallas, Houston, Austin, Little Rock, Las Vegas, LA, Fresno, Palm Springs, Honolulu} Fresno {LA, Honolulu, Palm Springs, Las Vegas, Dallas, Houston, Austin, Little Rock, College Station} Dallas {College Station, Houston, Austin, Little Rock, Las Vegas, LA, Fresno, Palm Springs, Honolulu} Number of pairs is 10*9=90, where Dallas Austin and Austin Dallas treated as different legs. Origin-Destination-Fare (ODF) class Combinations with 10 cities and 6 fares on every flight Number of ODFs is 90*6=540 Too many ODFs, so joint optimization is a big challenge!

1. Greedy (leg-by-leg decomposition) Heuristic Page 6 Since joint optimization is difficult, let us focus on each leg to develop a greedy heuristic. This is decomposition of the itineraries by force. Consider the origin destination pair of College Station Dallas LA. College Station Dallas leg: Full fare is $200 and discount fare is $120. Dallas LA leg: Full fare is $400 and discount fare is $350. Because of the high passenger traffic out of College Station, the discount fare class on College Station Dallas leg may be closed. Once this class is closed, we cannot accept a discount booking from College Station to LA which brings in a revenue of $470. This revenue of $470 is more than the full fare of $200 for a customer flying on College Station Dallas leg. Rather displace a full fare customer on College Station Dallas leg than reject a discount fare customer on College Station LA OD pair. By considering legs one by one, we never entertain the possibility of protecting seats for a discount fare passenger that travels multiple legs from a full fare customer that travels a single leg although multiple leg customer pays (much) more than single leg customer. Greedy heuristic does not work; it failed miserably when applied to hotel booking.

2. Linear Programming Approach Suppose that the demands are known in advance. Think of assembling resources (legs, night-stays) to make up products (itineraries, hotel-stays). Index products by j=1,2,,n and resources by i=1,2,,m. n=540 and m=20 in the simple network. p j : price of product j; d j : demand of product j; c i : capacity of resource i; All resources 1,11: College Station, Dallas; 2,12: Little Rock, Dallas; 3,13: Austin, Dallas; 4,14: Houston, Dallas; 5,15: Dallas, LA; 6,16: Dallas, Las Vegas; 7,17: LA, Las Vegas; 8,18: LA, Fresno; 9,19: LA, Honolulu; 10,20: LA, Palm Springs; Some products Product 1: College Station LA discount fare, uses resources 1 and 5 or resource 1 and 5 are used to make up product 1: a 1,1 =1, a 5,1 =1. Product 2: Little Rock Honolulu full fare, uses resources 2, 5 and 9 or resources 2, 5, and 9 make up product 2: a 2,2 =1, a 5,2 =1, a 9,2 =1. Product 3: Las Vegas Dallas discount fare, uses only resource 16 or resource 16 is product 3: a 16,3 =1. Product 4: Las Vegas Little Rock discount fare, uses resources 12 and 16 or resources 12 and 16 make up product 4: a 12,4 =1, a 16,4 =1. Product 5: Las Vegas Little Rock full fare, uses resources 12 and 16 or resources 12 and 16 make up product 5: a 12,5 =1, a 16,45 =1. a ij =1 if resource i is used in product j; 0 otherwise. x j =Total seats allocated to product (ODF) j. max n x j j1 a ij 0 x n j1 x j p j c d j x st (subject to) j i j for all resources i for all products Page 7 j

Linear Programming Discussion Page 8 Example Solution: x 4 =5 for ODF Las Vegas Little Rock discount fare. 5 discount fare seats are allocated to this ODF on resource 12 and 16: Dallas Little Rock and Las Vegas Dallas legs. x 5 =10 for ODF Las Vegas Little Rock full fare. 10 full fare seats are allocated to this ODF on resource 12 and 16: Dallas Little Rock and Las Vegas Dallas legs. Observations: Different fare classes on the same OD pair consume identical resources. Consider two fare classes on the same pair: high-fare class and low-fare class. Optimization implies» If the high-fare class has no allocation, then the low-fare class will have no allocation.» If the low-fare class is allocated a capacity equal to all of its demand, then the high-fare class is allocated a capacity equal to all of its demand. In summary, a high-fare class OD pair has higher priority than the low-fare class OD pair when receiving resource capacities provided that both classes consume the same capacity. Linear programming is an efficient methodology; it is fast to find a solution. Linear programming may not be effective It assumes that the demands are known. It is an allotment (partitioned limits) method as it does not allow for nested classes. Question: Can we achieve some nesting, if not all?

3. Virtual Nesting Page 9 ODFs ODFs Bucket 1 Bucket 1 Bucket 1 Y-class Y-class ODFs Y-class Bucket 2 M-class, M-W M-class, M-T M-class, M-M B-class Bucket 2 M-class, M-W M-class, M-T M-class, M-M B-class Bucket 2 M-class, M-W M-class, M-T M-class, M-M B-class Monday Tuesday Wednesday Virtual Nesting starts by assigning every ODF to a bucket in a resource that is used in the ODF. A bucket works like a fare class. A bucket is a collection of ODF s. Think of higher buckets as higher customer fare classes where we price the resources higher. ODF-to-bucket assignment is called indexing. Given this assignment, find the booking limits of an ODF by using EMSR heuristics or another method.» Virtual Nesting is attributed to American Airlines and developed in 1983. The key issue here is indexing.

Indexing: Which ODF into which bucket? First attempt: Order ODF by prices and put highest priced ODFs into the highest bucket. Problem: High priced ODFs consume more resources» You pay more if you travel further or if you stay longer Classification by ODF price alone ignores the consumption of other sources. Discourage ourselves from giving priority to high priced ODFs that consume a lot of resources. Page 10 Inspired from profit margin=price-cost, consider net leg fare of an ODF based on opportunity cost of capacity: Net leg fare of ODF j on resource i p j sum of on all resources other than resource i. Example: An airline estimates the for a seat on College Station Dallas to be $80 a seat on Dallas LA to be $210 What are the net leg fares for M-fare class whose College Station LA itinerary is priced at $400? On College Station Dallas leg, net leg fare of this itinerary is 400-210=190, use this number when bucketing College Station LA itinerary M-fare class (product) on College Station Dallas leg (resource). On Dallas LA, net leg fare of this itinerary is 400-80=320, use this number when bucketing College Station LA itinerary M-fare class (product) on Dallas LA leg (resource).

Net Leg Fare Computation for Leg (1,2) With Two Opportunity Costs Page 11 Itinerary Price Opportunity cost Net leg fare Bucket 1 2 Itinerary Price 1,2 100 1,2,1 210 1,2,3 190 1,2,3,2 360 1,2,3,2,1 450 2,1 180 2,1,2 300 Resource Opportunity cost 1,2 100 2,1 100 2,3 100 3,2,1,2 420 100+100 220 1 3,2,1,2,3 510 100+100+100 210 1 2,1,2 300 100+100 200 1 1,2,3,2 360 100+100 160 2 1,2,3,2,1 450 100+100+100 150 2 1,2,1 210 100 110 3 1,2 100 0 100 3 1,2,3 190 100 90 3 3 2,3 220 2,3,2 300 3,2,3 210 3,2,1 190 3,2,1,2 420 3,2,1,2,3 510 Resource Opportunity cost 1,2 50 2,1 150 2,3 100 Bottom tables with altered, to see the effect of these alterations on the net leg fare Itinerary Price Opportunity cost Net leg fare Bucket 3,2,1,2 420 100+150 170 1 3,2,1,2,3 510 100+150+100 160 1 1,2,3,2 360 100+100 160 1 2,1,2 300 150 150 1 1,2,3,2,1 450 100+100+150 100 2 1,2 100 0 100 2 1,2,3 190 100 90 2 1,2,1 210 150 60 3

Indexing with Net leg fares on Houston Dallas leg (Resource) Page 12 Honolulu Fresno Little Rock Dallas Palm Springs LA Austin Houston Itinerary: Origin-Destination Net leg fare Bucket Houston Fresno 180 1 Houston Little Rock 120 1 Houston Austin 110 1 Houston Dallas 100 2 Houston Palm Springs 85 2 Houston Honolulu 40 3 Houston LA 30 3

Indexing with Net leg fares on Dallas LA leg Page 13 Fresno Little Rock Honolulu Palm Springs LA Dallas Houston Austin Itinerary: Origin-Destination Net leg fare Bucket Houston Fresno 160 1 Austin Honolulu 160 1 Houston Honolulu 140 1 Austin LA 110 1 Austin Fresno 90 2 Little Rock Fresno 85 2 Houston Palm Springs 80 2 Austin Palm Springs 60 2 Little Rock Honolulu 40 3 Dallas LA 30 3 Little Rock Palm Springs 25 3 Little Rock LA 25 3 Houston LA 20 3 Houston-Honolulu ODF is in bucket 1 on Dallas LA flight, but in bucket 3 on Houston Dallas flight. This is possible when DallasLA flight has a high opportunity cost. Recall that DallasLA opportunity cost is not deducted from the leg fare of DallasLA but is deducted from the leg fare of HoustonDallas. Inconsistency or not: As a result of these, if you are going from Houston to Honolulu, the system can find the DallasLA ticket but not the HoustonDallas ticket. The funny thing is that your friend can buy a HoustonDallas ticket after you!!

Booking with Virtual Nesting: Single Class Illustration Page 14 1 2 3 After virtual indexing is finished, List the O-D pairs on each resource from the highest indexed to the lowest O D Resource 1 2 Booking Limit 1 2 C(1 2) 1 3 b 1 3 O D Resource 2 1 Booking Limit 2 1 C(2 1) 3 1 b 3 1 O D Resource 3 2 Booking Limit 3 1 C(3 2) 3 2 b 3 2 O D Resource 2 3 Booking Limit 2 3 C(2 3) 1 3 b 1 3 1 2 2 1 3 2 2 3 1 2 1 3 2 1 3 1 3 1 3 2 2 3 1 3 C(1 2) b 1 3 C(2 1) b 3 1 C(3 2) b 3 2 C(2 3) b 1 3 280 120 280 210 170 60 170 140 Check availability on corresponding virtual classes on each involved resource Booking limit vector b = [280, 120; 280, 210; 170, 60; 170, 140] Booking vector B = [192, 118; 228, 123; 142, 59; 30, 16] A booking request for going from 3 to 1 x = [0, 0; 1, 1; 1, 0; 0, 0], b B + x accept B = [192, 118; 229, 124; 143, 59; 30, 16] A booking request for going from 3 to 2 x = [0, 0; 0, 0; 1, 1; 0, 0], b B + x accept B = [192, 118; 229, 124; 144, 60; 30, 16] A booking request for going from 3 to 2 x = [0, 0; 0, 0; 1, 1; 0, 0], b < B + x reject B = [192, 118; 229, 124; 144, 60; 30, 16] A booking request for going from 3 to 1 x = [0, 0; 1, 1; 1, 0; 0, 0], b B + x accept B = [192, 118; 230, 125; 145, 60; 30, 16] Puzzling for outsiders: (3 2) is unavailable but (3 1) is, despite (3 1)=(3 2)+ (2 1)

Dynamic Opportunity Cost of Capacity If we have a network with 8 legs in a network and b i denotes the capacity on leg (O-D) i while П(b) : Maximum Profit to be made from the network with capacity vector b=[b 1,b 2,b 3,b 4,b 5,b 6,b 7,b 8 ]. The marginal cost of accepting an ODF j on leg i is d aij ( b, t) aij ( ( b, t) ( b 1, t)) db Recall a ij =1 if leg (resource) i is used in ODF (product) j. d d d Before accepting an ODF 2 that uses legs 3,4 and 7, we consider ( b, t), ( b, t), ( b, t). db db db i Because only a 32 =a 42 =a 72 =1 for this ODF and others are 0. These computations are conceptually same as in single-leg. However, there are complexity and numerical issues when this type of computation is done in a network that includes hundreds and sometimes thousands of legs (resources), whose remaining capacity b changes dynamically over time t. If computation is not a concern, directly compute marginal revenue for accepting product j. Make up vector a j of the same size as b, the number of legs. The ith element in the vector a j is 1 if a ij = 1. Ex: a 2 = [0, 0, 1, 1, 0, 0, 1, 0] for product 2 that uses resources 3,4 and 7 If b a j 0, exact marginal opportunity cost of capacity Π b, t Π(b a j, t), there are n of these; n m. If concerned about computations, for each resource i used in product j, make up vector 1 ij of the same size as b, the number of legs. Only the ith element in the vector 1 ij has 1 all others are zero. 3 4 7 Page 15 Ex: 1 32 = [0, 0, 1, 0, 0, 0, 0, 0], 1 42 = [0, 0, 0, 1, 0, 0, 0, 0], 1 72 = [0, 0, 0, 0, 0, 0, 1, 0] so a 2 = 1 32 + 1 42 + 1 72 If b a j 0, approximate marginal opportunity cost of capacity a ij =1 Π b, t Π(b 1 ij, t) For approximation, compute each Π b, t Π(b 1 ij, t) in advance and keep in the memory, which requires a space of number of resources m. In real time, perform only the sum.

4. Additive Bid Prices Las Vegas Honolulu Fresno $210 $90 $290 $280 LA Palm Springs $110 $180 $120 Austin Dallas $170 $100 $150 Little Rock College Station Page 16 E.g., Dallas Houston = 100 = Π, b DH, Π, b DH 1, Would you close a fare class on Las Vegas Fresno itinerary if the price is $350? Little Rock Honolulu itinerary if the price is $800? Palm Springs Austin itinerary if the price is $400? Houston Skip the section on calculation of the bid prices in the book. Recall that bid prices are about. Just like, bid prices must be dynamic. Applications in Airlines (Scandinavian Airline System) and Rental Cars (Hertz).

Virtual Nesting or Bid Prices Page 17 Virtual nesting and bid prices both depend on the opportunity cost of the capacity. The opportunity cost needs to be updated dynamically as time passes and remaining capacity drops. Both virtual nesting and bid pricing methods perform similarly when are updated frequently. If the bid prices are not updated and demand is stronger than expected, bid price becomes lower than what it should be (underpricing), demand is weaker than expected, bid price become higher than what it should be (overpricing). Ignorance of dynamism is more costly when bid pricing methods are used. Historically, airlines had RM before hotels for single sources. Airlines used fare class based booking controls. When network RM management came out, the legacy systems at Airlines are modified to take legs of an OD into account. This led to virtual nesting. On the other hand, Hotels did not have any (legacy) RM systems so they started with bid pricing systems. Over time, more airlines adopted bid pricing.

Practice as of 2011 Dynamic Virtual Nesting or Bid Prices Page 18 Dynamic Virtual Nesting United Airlines Delta Airlines Bid Prices American Airlines Lufthansa/Swiss LAN Iberia British Airways Air France/KLM SAS Cathay Pacific Qatar Etihad Royal Jordanian Thai Some of these airlines are using what we refer to as hybrid controls, whereby leg authorization limits are used to manage local services, and bid prices are used for connecting ones. That particular variation of bid price controls retains aspects of leg authorizations levels (for RM analysts who are more familiar with these types of controls), but it enables bid prices for connecting services to avoid the huge increase in the total volume of controls required in an O&D implementation. Richard Ratliff, Senior Research Scientist, Sabre, Nov 2011 Lists above are based on his Nov 21 Demreman guest lecture.

Summary Page 19 Network Management Problem Greedy Heuristic LP Approach Virtual Nesting Bid Prices

Net Leg Fare Computation for Product (1,2) Deducting the opportunity cost of (1,2) as well 1 2 3 Itinerary Price 1,2 100 1,2,1 210 1,2,3 190 1,2,3,2 360 1,2,3,2,1 450 2,1 180 2,1,2 300 2,3 220 2,3,2 300 3,2,3 210 3,2,1 190 3,2,1,2 420 3,2,1,2,3 510 Resource Original opportunity cost 1,2 100 2,1 100 2,3 100 Resource Altered opportunity cost 1,2 50 2,1 150 2,3 100 Deduction of the opportunity cost of (1,2) does not change prioritization of itineraries or bucketing. Buckets with original 3,2,1,2; 3,2,1,2,3; 2,1,2 1,2,3,2; 1,2,3,2,1 Buckets with altered 3,2,1,2; 3,2,1,2,3; 2,1,2; 1,2,3,2 1,2,3,2,1; 1,2; 1,2,3 Itinerary Price Original Opportunity cost Net leg fare Bucket 3,2,1,2 420 100+100+100 120 1 3,2,1,2,3 510 100+100+100+100 110 1 2,1,2 300 100+100 100 1 1,2,3,2 360 100+100+100 60 2 1,2,3,2,1 450 100+100+100+100 50 2 1,2,1 210 100+100 10 3 1,2 100 100 0 3 1,2,3 190 100+100-10 * Itinerary Price Altered Opportunity cost Net leg fare Bucket 3,2,1,2 420 100+150+50 120 1 3,2,1,2,3 510 100+150+50+100 110 1 1,2,3,2 360 50+100+100 110 1 2,1,2 300 150+50 100 1 1,2,3,2,1 450 50+100+100+150 50 2 1,2 100 50 50 2 1,2,3 190 50+100 40 2 1,2,1 210 150+50 10 3 Page 20 1,2,1; 1,2; 1,2,3. 1,2,1.

Net Leg Fare Computation for Product (3,2) Deducting the opportunity cost of (3,2) as well 1 2 3 Itinerary Price 1,2 100 1,2,1 210 1,2,3 190 1,2,3,2 360 1,2,3,2,1 450 2,1 180 2,1,2 300 2,3 220 2,3,2 300 3,2,3 210 3,2,1 190 3,2,1,2 420 3,2,1,2,3 510 Buckets with original 3,2,1,2; 3,2,1,2,3; 2,3,2 1,2,3,2; 1,2,3,2,1 3,2,3; 3,2; 3,2,1. Resource Original opportunity cost 1,2 100 2,1 100 2,3 100 Resource Altered opportunity cost 1,2 50 2,1 150 2,3 100 Buckets with altered 3,2,1,2; 3,2,1,2,3; 1,2,3,2; 2,3,2 1,2,3,2,1; 3,2,3; 3,2; 3,2,1. Itinerary Price Net leg fare Bucket 3,2,1,2 420 120 1 3,2,1,2,3 510 110 1 2,3,2 300 100 1 1,2,3,2 360 60 2 1,2,3,2,1 450 50 2 3,2,3 210 10 3 0 3 3,2,1 190-10 * Itinerary Price Net leg fare Bucket 3,2,1,2 420 120 1 3,2,1,2,3 510 110 1 1,2,3,2 360 110 1 2,3,2 300 100 1 1,2,3,2,1 450 50 2 3,2,3 210 10 3 0 3 3,2,1 190-60 * Page 21

Net Leg Fare Computation for Product (2,1) Deducting the opportunity cost of (2,1) as well 1 2 3 Itinerary Price 1,2 100 1,2,1 210 1,2,3 190 1,2,3,2 360 1,2,3,2,1 450 2,1 180 2,1,2 300 2,3 220 2,3,2 300 3,2,3 210 3,2,1 190 3,2,1,2 420 3,2,1,2,3 510 Buckets with original 3,2,1,2; 3,2,1,2,3; 2,1,2; 2,1 1,2,3,2,1 1,2,1; 3,2,1. Resource Original opportunity cost 1,2 100 2,1 100 2,3 100 Resource Altered opportunity cost 1,2 50 2,1 150 2,3 100 Buckets with altered 3,2,1,2; 3,2,1,2,3; 2,1,2 1,2,3,2,1; 2,1 1,2,1; 3,2,1. Itinerary Price Net leg fare Bucket 3,2,1,2 420 120 1 3,2,1,2,3 510 110 1 2,1,2 300 100 1 2,1 180 80 1 1,2,3,2,1 450 50 2 1,2,1 210 10 3 3,2,1 190-10 * Itinerary Price Net leg fare Bucket 3,2,1,2 420 120 1 3,2,1,2,3 510 110 1 2,1,2 300 100 1 1,2,3,2,1 450 50 2 2,1 180 30 2 1,2,1 210 10 3 3,2,1 190-60 * Page 22

Net Leg Fare Computation for Product (2,3) Deducting the opportunity cost of (2,3) as well 1 2 3 Itinerary Price 1,2 100 1,2,1 210 1,2,3 190 1,2,3,2 360 1,2,3,2,1 450 2,1 180 2,1,2 300 2,3 220 2,3,2 300 3,2,3 210 3,2,1 190 3,2,1,2 420 3,2,1,2,3 510 Buckets with original 2,3; 3,2,1,2,3; 2,3,2; 1,2,3,2; 1,2,3,2,1; 3,2,3; 3,2,1. Resource Original opportunity cost 1,2 100 2,1 100 2,3 100 Resource Altered opportunity cost 1,2 50 2,1 150 2,3 100 Buckets with altered 2,3; 3,2,1,2,3; 1,2,3,2; 2,3,2; 1,2,3,2,1; 1,2,3; 3,2,3 Itinerary Price Net leg fare Bucket 2,3 220 120 1 3,2,1,2,3 510 110 1 2,3,2 300 100 1 1,2,3,2 360 60 2 1,2,3,2,1 450 50 2 3,2,3 210 10 3 1,2,3 190-10 * Itinerary Price Net leg fare Bucket 2,3 220 120 1 3,2,1,2,3 510 110 1 1,2,3,2 360 110 1 2,3,2 300 100 1 1,2,3,2,1 450 50 2 1,2,3 190 40 2 3,2,3 210 10 3 Page 23

Net Leg Fare Computation Results Page 24 1 2 3 2 1 Buckets with original Buckets with original Buckets with original Buckets with original 3,2,1,2; 3,2,1,2,3; 2,1,2 2,3; 3,2,1,2,3; 2,3,2; 3,2,1,2; 3,2,1,2,3; 2,3,2 3,2,1,2; 3,2,1,2,3; 2,1,2; 2,1 1,2,3,2; 1,2,3,2,1 1,2,3,2; 1,2,3,2,1; 1,2,3,2; 1,2,3,2,1 1,2,3,2,1 1,2,1; 1,2; 1,2,3. 3,2,3; 3,2,1. 3,2,3; 3,2; 3,2,1. 1,2,1; 3,2,1. Buckets with altered opportunity costs Buckets with altered Buckets with altered opportunity costs Buckets with altered 3,2,1,2; 3,2,1,2,3; 2,1,2; 1,2,3,2 2,3; 3,2,1,2,3; 1,2,3,2; 2,3,2; 3,2,1,2; 3,2,1,2,3; 1,2,3,2; 2,3,2 3,2,1,2; 3,2,1,2,3; 2,1,2 1,2,3,2,1; 1,2; 1,2,3 1,2,3,2,1; 1,2,3; 1,2,3,2,1; 1,2,3,2,1; 2,1 1,2,1. 3,2,3 3,2,3; 3,2; 3,2,1. 1,2,1; 3,2,1.