Basic price optimization

Transcription

1 Basic price optimization Brian Kallehauge 4234 Advanced Topics in Operations Research Fall 29 Revenue Management Session 3

2 Outline The price-response unction Price response with competition Incremental costs The basic price optimization problem 2 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

3 Introduction to price optimization The basic pricing and revenue optimization problem can be ormulated as an optimization problem. The objective is to maximize contribution: total revenue minus total incremental cost rom sales. The ey elements o the optimization problem is: the price-response unction and the incremental cost o sales. In this lecture we will ormulate and solve the pricing and revenue optimization problem or a single product in a single maret without supply constraints. Furthermore, we will discuss some important optimality conditions. 3 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

4 The price-response unction A undamental input to any price and revenue optimization (PRO) analysis is the price-response unction (or curve) d(p). There is one price-response unction associated with each combination o product, maret-segment, and channel in the PRO cube. The price-response unction, d(p), speciies demand or the product o a single seller as a unction o the price, d, oered by that seller. This constrasts with the concept o a maret demand curve which speciies how an entire maret will respond to changing prices. Dierent irms competing in the same maret ace dierent priceresponse unctions. The price-response unctions may dier due to many actors, such as the eectiveness o their mareting campaigns, perceived customer dierences in quality, product dierences, location, etc. 4 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

5 Price-response unctions in a perectly competitive maret In a perectly competitive maret: The price-response aced by an individual seller is a vertical line at the maret price. For higher prices, the demand drops to. I he prices below the maret price, his demand equals the entire maret. For example a wheat armer: I he charges more than the maret price, he will sell nothing. I he charges below the maret price, the demand will be eectively ininite. Price-response curve in a perectly competitive maret. 5 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

6 Commodities in a perectly competitive maret Commodity producers, such as the wheat armer, have no pricing decision the price is set by the operation o the larger maret. I.e., in a competitive maret, each irm only has to worry about how much output it wants to produce. Whatever it produces can only be sold at one price: the going maret price. Thereore, sellers o true commodities in a perectly competitive maret have no need or pricing and revenue optimization (PRO). However, true comodities are surprisingly rare! 6 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

7 Price-response curves in non-competitive marets The price-response curves which ace most companies demonstrate some degree o smooth price response: As the price increases, the demand declines. Demand reaches zero at some satiating price P. Typical price-response curve. 7 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

8 Properties o the price-response unction The price-response unctions used in PRO analysis are time-dependent. We set prices that will be in place or some inite period o time. The period may be minutes or hours or longer. At the end o each period we have the opportunity to change prices. The demand we expect to see at a given price will depend on the length o the time period the price will be in place. I.e. there is no single price-response unction without an associated time interval. There are many dierent ways in which product demand might change in response to changing prices but all price-response unctions are assumed: nonnegative (p ), continuous (no gaps in maret response to prices), dierentiable (smooth and with well-deined slope at every point), and downward sloping (raising prices decreases demand). Implies imprecision since using derivatives rather than dierence equations. 8 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

9 Measures o price sensitivity The two most common measures o price sensitivity are the slope and the elasticity o the price-response unction. The slope measures how demand changes in response to a price change and equals the change in demand divided by the dierence in prices. The price elasticity is deined as the ratio o the percentage change in demand to the percentage change in price. 9 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

10 The slope o price-response unctions The slope equals the change in demand divided by the change in prices: Downward sloping: p > p 2 implies d(p ) d(p 2 ), i.e. δ(p,p 2 ). The slope at a single price, p, can be computed as the limit o the above equation as p 2 approaches p : where d (p ) is the derivative o the price-response unction at p. For small price changes we can write: I.e. a large slope means that demand is more responsive to prices than a smaller slope. DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

11 The price elasticity o price-response unctions The elasticity equals the percentage change in demand divided by the percentage change in prices: where ε(p,p 2 ) is the elasticity o a price change rom p to p 2. This equation can be reduced to: Since downward sloping priceresponse curve, ε(p,p2). EX: ε =.2 ε =.8 % price increase: % price decrease: 2 % demand decrease. 8 % demand increase. DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

12 Point elasticity The price elasticity at a single price, p, ( point elastiticy at p ) can be computed as the limit o the above price elasticity equation as p 2 approaches p : I.e. the points elasticity is equal to times the slope o the demand curve times the price divided by the demand. The point elasticity is useul as a local estimate o the change in demand resulting rom a small change in price. Note that, unlie the slope, the price elasticity is independent o the units in which the price and demand is measured. 2 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

13 Price elasticity in practice The term price elasticity is oten used as a synonym or price sensitivity. High price elasticity items have very price sensitive demand, while low price elasticity items have much less price sensitive demand. Oten, a good with a price elasticity greater than is described as elastic, while one with an elasticity less than is described as inelastic. Elasticity is dependent on whether we measure the total maret response i all suppliers o a product change their prices or the price-response elasticity or an individual supplier within the maret. I all suppliers raise prices, the only alternative or customers is to purchase a substitute product or to go without. I a single supplier raises prices, customers can go to its competitor. Furthermore, as well as other aspects o price response, elasticity is dependent on the time period under consideration. There may be great dierence in price elasticity in the short run and in the long run... 3 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

14 Price elasticity or dierent goods For most products, short-run elasticity is lower than long-run elasticity since buyers have more lexibility to adjust to higher prices in the long run. For example, short-run elasticitiy or gasoline has been estimated to be.2, while the long-run elasticity has been estimated at.7. At irst, consumers still need to by gasoline, but in the long term, people will change habits, e.g. buying higher mile-per-gallon cars. On the other hand, or many durable goods, such as cars and washing machines, the long-run price elasticity is lower than the short-run elasticity. The reason is that customers initially respond to a price rise by postponing the purchase o a new item. However, they will still purchase at some time in the uture, so the long-run eect o the price change is less than the short-run eect. 4 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

15 Examples o price elasticity Salt has a low price elasticity as a respond to maret price changes (people will by salt even i prices go up) but or an individual seller, the price elasticity would be expected to be high due to competitiveness. Airline ticets have a large long-term price elasticity since passengers will change their tavel habits i prices stay high. Cars have a low long-tirm elasticity since initially posponed purchashes will be be realized later in time even though prices stay high. 5 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

16 Price response and willingness to pay In reality, the price-response unction is not simply given. Demand is the result o each potential customer observing the prices and deciding whether or not to buy a speciic product. The price-response unction speciies how many more o those potential customers would buy i we lowered our price and how many current buyers would not buy i we raised our price. I.e., the price-response unction is based on assumptions about customer behavior. The most important part o models o customer behavior is based on willingness to pay (w.t.p). The willingness-to-pay approach assumes that each potential customer has a maximum willingness to pay (also called a reservation price ) or a given product. A customer will purchase i and only i the price is less than his/her maximum w.t.p. 6 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

17 Willingness to pay The number o customers whose maximum willingness to pay (w.t.p.) is at least p is denoted d(p). I.e., d(p) is the number o customers who are willing to pay the price p or more or the product. Deine the unction w(x) as the w.t.p. distribution across the population. Then or any values p < p 2 : is the raction o the population that has w.t.p. between p and p 2. Note that w(x) or all nonnegative values o x. 7 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

18 The willingness to pay distribution Let D = d(), i.e. the number o customers willing to pay zero or more i.e. willing to buy the product at all, be the maximum demand achievable. Then we can derive d(p) rom the w.t.p distribution: Recall that d(p) is the number o customers who are willing to pay the price p. Note that the price-response unction is partitioned into two separate components: the total demand D and the w.t.p. distribution w(x). Next lecture considers examples o price-response unctions and the basic price optimization problem. 8 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

19 Simpliied airline are structures and marginal revenue transormation Brian Kallehauge 4234 Advanced Topics in Operations Research Fall 29 Revenue Management Session 3

20 The low cost carrier competition led to simpliied are structures in scheduled airlines Past Transition Future Strong maret segmentation Weaening o maret segmentation Increasing maret convergence Industry Monopoly Intense LCC competition Price transparency Consolidation o industry Fare structure First Business Economy First Business Business Economy Economy Traditional are restrictions (AP, RT, Less-restricted ares SA/SU, min/max) Lower prices High are ratios Stabilization o prices simpliied ares is the most important pricing development in the industry in the past 25 years Tretheway (24) 2 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

21 Without modiications o traditional RM systems are simpliication leads to spiraldown in revenues o 2-3% Fare simpliication Decrease in protection levels Decrease in sales o highpriced products Spiral-down in revenues 2-3% Decrease in orecast The root cause o spiral-down is the brea-down o the independent-demand assumption o RM systems 2 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

22 The are simpliication groups ares with similar restrictions into are amilies Fare classes Independent demand model Fare amilies Lowest-open-are demand model 2 2 Family 3 Strong ence 4 Fare simpliication 3 Strong ence 4 How do we optimize the revenue o the are amilies? Family 2 Fence -6 Price-points 22 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

23 What we need to solve The are amily networ revenue management problem with dependent demand Marginal revenue transormation rom original are structure to independentdemand model 2 Decomposition approximation The single-leg revenue management problem with independent demand vs. what we can solve 23 DTU Management Engineering, Technical University o Denmar Revenue Management Session 3 8//29

24 Can Existing RM Systems be Saved?* Marginal revenue transormation (Fiig et al. 29) The authors present a marginal revenue transormation that transorms any are structure (with any set o restrictions) into an independent demand model. This allows all the traditional RM methods (that was invented assuming independent demand) to be used unchanged. The standard availability control methods can be used unchanged provided that the eicient rontier is nested (or approximately nested). Previous wor has discussed methods to avoid spiral down and optimize simpliied ares. Sell-up models in Leg based EMSR, Belobaba and Weatherord (996) Hybrid Forecasting o Price vs. Product Demand, Boyd, Kallesen (25) DAVN-MR (Networ optimization, mix o ully un-restricted and ully restricted), Fiig et al (25), Isler et al (25). Fare Adjustment Methods with Hybrid orecasting, PROS, PODS research. Revenue Management with customer choice models, Talluri and van Ryzin (24), Gallego et al. (27). *Source: Thomas Fiig, Chie Scientist, Scandinavian Airlines. 24

25 Overview Fully dierentiated are structure Independent demand by class Class based RMsystem 25

26 Overview Fully dierentiated are structure Fully un-dierentiat are structure Independent demand by class Dependent demand Marginal Revenue Transormation Class based RMsystem 26

27 Fully dierentiated are structure Fully un-dierentiat are structure Overview Any are structure Independent demand by class Dependent demand Dependent demand Marginal Revenue Transormation Marginal Revenue Transormation Independent demand in policy space Map policies to classes Yes Nested policies No Class based RMsystem Policy based RM 27

28 Fare TR Optimization: Fully dierentiated - Deterministic Demand - Single Leg Fully dierentiated i di i TRi MRi $.2 3,2 3,2 $ $.2 $.,9 42,2 $48.45 $. $8 4,8 56,9 $6.27 $8 $6 9,9 76,8 $72.65 $6 $4 26,9 3,7 $82.98 $4 $2 36,3 4, $9.75 $2 d j j TR j j d j CAP

29 Marginal Revenue (Intuitive derivation) -Fully un-dierentiated, - Single Leg d j j Fully un-dierentiated i di i TRi MRi $.2 3,2 3,2 $ $.2 $.,9 42,2 $42.67 $428 $8 4,8 56,9 $ $228 $6 9,9 76,8 $46. $28 $4 26,9 3,7 $ $72 $2 36,3 4, $28. -$372 TR MR 2 $428 TR $42,67 42,2 2 2 TR $37, d 2 Revenue recieved.9 * $= $,9 2 = d 2 $,9 -$6,24 = $4,66 MR 2 MR 2 $4,66.9 $428 Loss due to buy-down 3.2* ($-$2)= - $6,24 Net revenue 29

30 Fare TR Optimization: Fully un-dierentiated - Deterministic Demand - Single Leg Fully un-dierentiated i di i TRi MRi $.2 3,2 3,2 $ $.2 $.,9 42,2 $42.67 $428 $8 4,8 56,9 $ $228 $6 9,9 76,8 $46. $28 $4 26,9 3,7 $ $72 $2 36,3 4, $28. -$372 d j j TR CAP MR() P() di. un-di. CAP Max 5 5 3

31 Deinition o policies. C D J I R Y S B E M H W U K L T G X N Policies: the set o are products S that the airline chooses to have open. n classes gives potentially 2 n policies. Examples could be: All classes closed {}, All classes in economy open {E,,T}, Only classes E,H, and K open: {E,H,K}. Fare amilies: {Y,S; E,M,H,} Nested policies: S Sl, Examples l Nested in economy: {},{E},{E,M},...,{E,M,...,L} Non-nested in economy: {},{E},{E,H},...,{E,M,...,L} 3

32 . Optimization: General Formulation - Arbitrary are structure - Deterministic Demand - Single Leg Fare products Policy Z (any set o open classes) Demand N j, j,..., d j {},{},{,3},... (Z ) n Total Revenue TR i TR i- Marginal revenue: i S i- Eicient Frontier S i Optimum S m Accumulated Dem. Total Revenue ( Z) d j ( Z) j Z TR ( Z) d ( Z) j Z j j S S d i All policies Z Objective maxtr( Z) s. t. ( Z) cap i- i CAP Demand 32

33 Marginal Revenue Transormation Policies on the convex hull Independent demand Policy Dem. TR S S S m m TR TR 2 TR m Partition Dem. Adj. Fare d d2 2 2 TR2 TR d d m m m m TR m TR m d m Marginal Revenue Transormation Theorem The transormed policies are independent. Optimization using the original are structure and the marginal revenue transormed in policy space gives identical results. 33

34 Mapping nested policies to Classes Many choice models have the desirable property that the policies are nested on the eicient rontier. For nested policies we can assign demand and adj. ares bac to the original classes and continue reusing class based RM-systems. Mapping rom policies to classes Newly added classes Partitioned demand S \ S d Split demand any way between newly added classes Total Revenue Marginal revenue: S - S Eicient Frontier S m Adj are Assign the adjusted are to all newly added classes. S d S i- i Demand 34

35 Applications: Fully dierentiated demand Assume are class independence. (the are products are adequately dierentiated, such that demand or a particular are product will only purchase that are product) Acc. demand d j Total Revenue TR j j,..., j,..., d j Partitioned demand Adjusted are d d j,..., j j,..., d j d TR d j,..., j TR j j d,..., d j j Thus demand and ares are unchanged by the MR transormation. Denote the unadjusted are: prod 35

36 Applications: Fully un-dierentiated demand Passengers will only buy the lowest available are (demand or all other are products except the lowest becomes zero) Acc. demand n psup Total Revenue TR Partitioned demand d n psup psup Adjusted are TR psup TR psup psup psup price Denote the partitioned d demand: Denote the adjusted are: price 36

37 Applications: Fully un-dierentiated demand (exponential sell-up, equal spaced are grid) Passengers will only buy the lowest available are Acc. demand n exp( ( n )) Total Revenue TR Partitioned demand d n exp( exp( n ) ) exp( ) Adjusted are M M, exp( exp( where ) ) M is called the are modiier 37

38 Applications: Hybrid demand The are class demands are decomposed into contributions rom both dierentiated (product-oriented) and un-dierentiated (price-oriented) demand Acc. demand Partitioned demand d j,..., d d prod d prod j price d price Total Revenue Adjusted are where j,..., The adjusted are in the hybrid case equals a demand-weighted average o: the unadjusted are or the product-oriented demand the adjusted are or the price-oriented demand. r d TR TR r d prod prod prod d d TR price prod j ( j r d ) price price. 38

39 Extension to Stochastic models - Derivation using DP Fare products j, j,..., n Policy Z N (any set o open classes) {},{},{,3},..., {2,4},{,2,4},... Arrival rate Prob. o booing Accumulated Dem. Total Revenue p j (Z) ( Z) p j ( Z) j Z TR ( Z) p ( Z) j Z j j Objective (Bellman recursion ormula) J t ( x) max Z N TR( Z) ( Z) ( Z) J t J ( x) t ( x ) 39

40 Extension to Stochastic models - Derivation using DP Bidprice vector Bellman recursion eq. BP ( x) J ( x) J ( x J t t ( x) J ( x) max TR( Z) Z N t t t BP( x) Independent demand model t ) ( Z) Total Revenue TR i TR i- Marginal revenue: i S i- S i max Eicient Frontier S m Partitioned demand p ( S ) ( S ) S p i BP* Adj. are. TR( S ( S ) ) TR( S ( S ) ) S i- i Demand Recover the Marginal Revenue Transormation: Using the transormed choice model (primed demand and ares) in an independent demand DP instead o the original choice model DP, the Bellman equation will produce the same bid-prices. 4

41 Applications: EMSRb-MR Coo boo constructing EMSRb-MR (How to construct XXX-MR). Determine the policies on the eicient rontier 2. Apply the marginal revenue transormation to both demands and ares. 3. Map policies bac to classes 4. Apply EMSRb in the normal ashion using the transormed demands and ares. Partitioned demand Adjusted are Protection Level Booing Limit d ~ N(, 2 ),,, BL cap Fare Fare Mean Standard EMSRb Adjusted EMSRb-MR Product Value Demand Deviation Limits Fares (MR) Limits $, $,2 2 $, $ $ $ $ $ $ $ (72) 6 $ $ (372) EMSRb-MR applied to the un-restricted are structure example. 4

42 Applications: DAVN-MR - Follow Coo Boo EWR TOS DAVN-MR constructed to handled a mix o ully dierentiated and undierentiated are structures. Assuming exponential sell-up and equally spaced ares or simplicity. The are modiier is calculated individually by path. OSL Adj are adj DC M DC CPH Dierentiated are products M The are modiier since path are not aected by ris o buy-down. AMS Un-dierentiated are products M Mapped to lower bucets since Thus ares M are closed regardless o remaining capacity. Thus avoiding spiral down. Dierentiated Undierentiated 42

43 PODS Simulations PODS networ D 2 airlines. ALand AL2 2 cities east/west. 2 hubs 26 legs in 3 bans 482 marets. 446 paths. Sell-up parameters Input Frat5 sell-up. Forecasting Standard path/are class orecasting Hybrid path/are class orecasting Fare structure 6 are classes Unrestricted & Semi-restricted RM methods Standard DAVN (std. orecast, no are adj.) (Baseline) Hybrid DAVN (hybrid orecast. No are adj.) Full DAVN-MR (hybrid orecasting and are adj.) Competitive Scenarios Monopoly and Competition H2(42) 9 Traic Flows H(4)

44 PODS Simulations - Fare structure A un-dierentiated structure A semi-dierentiated structure UNDIFFERENTIATED SEMI-DIFFERENTIATED FARE CLASS AP Min Stay Cancel Fee Non Reund FARE CLASS AP Min Stay Cancel Fee Non Reund 2 2 YES 3 3 YES YES 4 4 YES YES 5 5 YES YES 6 6 YES YES 44

45 Revenue Index Load Factor PODS Simulations -Monopoly Un-dierentiated Monopoly: Un-dierentiated are-structure Monopoly: Un-dierentiated are-structure , 5,8 34, ,4 85, 74, 25 2 Standard Hybrid DAVN-MR Standard Hybrid DAVN-MR Hybrid orecasting leads to 6% gain compared to standard due to reduced spiral down. Full DAVN-MR (hybrid orecasting + are adjustment) adds an additional 8% gain. The eect comes rom closing lower ineicient classes, which leads to lower LF

46 Revenue Index Load Factor PODS Simulations -Monopoly Semi-dierentiated Monopoly: Semi-dierentiated are-structure Monopoly: Semi-dierentiated are-structure ,7 7,5 35,9 8 85,3 85, 72, Standard Hybrid DAVN-MR Standard Hybrid DAVN-MR Same overall trend compared to un-dierentiated. Slightly less eect due to restrictions

47 Revenue Index Load Factor PODS Simulations -Competition Un-dierentiated Competition: Un-dierentiated are-structure Competition: Un-dierentiated are-structure 5 25,,, 2, 2,3 4, 75 85,5 84,4 84,8 85,5 68,5 93, Standard Hybrid DAVN-MR AL AL2 Standard Hybrid DAVN-MR AL AL2 Hybrid orecasting leads to % gain compared to standard. Less than monopoly due to competition. Full DAVN-MR (hybrid orecasting + are adjustment) adds an additional % gain. The eect comes rom closing lower ineicient classes, which leads to lower LF

48 Revenue Index Load Factor PODS Simulations -Competition Semi-dierentiated Competition: Semi-dierentiated are-structure Competition: Semi-dierentiated are-structure ,7 2,9 99,7,9 28,9 2, ,5 84,4 84,6 85,7 64,6 93, Standard Hybrid DAVN-MR AL AL2 Standard Hybrid DAVN-MR AL AL2 48

49 Conclusion Marginal revenue transormation transorms a general discrete choice model to an equivalent independent demand model. The marginal revenue transormation allows traditional RM systems (that assumed demand independence) to be used continuously. The marginal transormation is valid or: Static optimization Dynamic optimization Networ optimization (provided the networ problem is separable into independent path choice probability). I the eicient rontier is nested (or approximately nested), the policies can be remapped bac to the original classes allowing the class based control mechanism to be used in the standard way. DAVN-MR was tested using PODS or both un-dierentiated and semidierentiated networs. Revenue gains are signiicant, -2 pct point better that hybrid orecasting. 49