Capacity Allocation. Outline Two-class allocation Multiple-class allocation Demand dependence Bid prices. Based on Phillips (2005) Chapter 7.

Transcription

1 Caacity Allocation utallas.eu/~metin Page Outline Two-class allocation Multile-class allocation Deman eenence Bi rices Base on Phillis (2005) Chater 7

2 Booking Limits or 2 Fare Classes utallas.eu/~metin Page 2 Only one ecision variable: b= b 2 Secon class has booking limit b The irst class has rotection level o C-b Demans or Prices or Full-are class whose cumulative ensity is F ; Discount are class whose cumulative ensity is F. Full-are class ; Discount are class.

3 2-class Problem Increasing the Booking Limit utallas.eu/~metin Page 3 b u or same b u by b kee constant Execte marginal eect on revenue F (b): No secon class is rejecte with b or b+ Remaining caacity or the irst class is the same b iscount eman too low to be limite >b -F (b): Remaining caacity or the irst class is less by F [ F [ F ( b)0 [ F ( b)] ( b)] -F (C-b): First class rejecte because o b+ >C-b C-b F (C-b): Low irst class eman ( b)] [ F [ F ( C b)]( ( C b)] F ( C b) ( / ) ) F Marginal eect on revenue 0-0=0 - <0-0>0 0 ( C b)

4 2 Fare Classes Booking Limit an Protection Level utallas.eu/~metin Page 4 The critical term is F (C-b)-(- / ) F (C-b) ecreases as b increases because F is cumulative ensity an increases in its argument. I b=0 an still F (C-b) (- / ), set b=0. In this case, the iscount rice is very low so it is not worth oening u the iscount class. For any b, increase b as long as F (C-b)>(- / ). For a continuous ull are eman istribution, C-b*= F (- / ) an b* C. - / Otimal booking limit or the secon class C C-F (- / ) 0 F ( ) b y * * or max min / C max C min C F F C F ( P( ( C F / y * / ( ), C ). ),0 / ( / ),0 ),0 F 0 (- / ) Otimal rotection level or the irst class C Deman

5 2 Fare Classes Protection Level with Uniorm Deman utallas.eu/~metin Page 5 Suose that the irst class (ull-are) eman is uniorm between 20 an 40 seats on a light. Moreover, irst class assengers ay $000, while the iscount are is $300. What are booking limits or iscount are i the lane has C=00 or C=30 seats? F ( ) Protection level y irst: F (x)=(x-20)/20 or 20<x<40. F (u)=20+20u or 0<u<. F (- / )=F (7/0)=34. - / =-3/0 =7/0 I the caacity is 00, the booking limit is 00-34=66 or the iscount class. I the caacity is 30, the booking limit is (30-34) + =0 or the iscount class. 20 Otimal rotection level or the irst class F (- / ) =34 40 Deman

6 2 Fare Classes Protection Level with Normal Deman utallas.eu/~metin Page 6 Suose that the irst class (ull-are) class eman is normal with mean 30 seats an stanar eviation 5. Moreover, irst class assengers ay $000, while the iscount are is $300. What are booking limits or iscount are i the lane has C=00 or C=30 seats? Stanar cumulative normal istribution Φ(z)=robability. Φ (7/0)=norminv(0.7, mean=0, stev=) =0.52=z. - / =7/0 F ( ) Protection level mean+z*stev=30+(0.52)5=32.6. Or use norminv(0.7, mean=30, stev=5)=32.6. Let us roun u the rotection level to 33. I the caacity is 00, the booking limit is 00-33=67 or the iscount class. I the caacity is 30, the booking limit is (30-33) + =0 or the iscount class. 20 Deman Otimal norminv(0.7,30,5) rotection =32.6 level or the irst class

7 Multile Fare Classes Neste Protection Levels Fare classes inexe by j=,2,,n. j= is the highest class, while j=n is the lowest class. Protection level y i or class i rotects uture reservations or classes j={,2,, i}. Protection levels {,.,n} {,.,n} {,2,3} {,2} {} utallas.eu/~metin Page 7 Class rice Class 2 rice 2 Class 3 rice 3 Class n rice n Class n rice n Perios n n 3 2 Ineenent Demans n n 3 2 Monotone Prices n < n 3 < 2 < Time-wise Searate Bookings x n x n x 3 x 2 x Check-in Perio=0

8 Multile Fare Classes Increasing the Booking Limit utallas.eu/~metin Page 8 F 3 (b 3 ): No thir class is rejecte with b 3 or b 3 + Remaining caacity or irst/secon class is the same Marginal eect on revenue 0-0=0 b 3 u by 3 b 3 3 >b 3 First class rejecte because o b <0 b 3 u or same -F 3 (b 3 ): Remaining caacity or irst/secon class is less by Secon class rejecte because o b <0 b 3 limits 3 r class to oen sace or st an 2 n classes b 3 kee constant Low irst an secon class eman 3-0>0 0 Challenge: Comuting the robability that irst or secon class is rejecte by increasing the booking limit o the thir class.

9 Execte Marginal Seat Revenue (EMSR-a) Version-a Heuristic: Protection Level Decomosition utallas.eu/~metin Page 9 In erio 3, when acceting bookings or class 3, rotect class an class 2. Let y,3 an y 2,3 be the rotection levels o irst an secon classes against the thir class. We can set y F ( 3 / ) an y2,3 F2 ( 3 / 2),3 The rotection level or both irst an secon classes while booking or class 3 is y y 2,3 y2,3 In general, y i y y... y, i 2, i i, i where y j,i+ is the rotection level o class j rom class i+. That is, y F i j y j, i ( i / j ) or i / j P( j y j, i ) j, i j Protect class j more against class i+ i Class j eman is larger Class j rice is larger

10 Execte Marginal Seat Revenue (EMSR-b) Version-b Heuristic: Remaining Deman Aggregation utallas.eu/~metin Page 0 In erio 3, when acceting bookings or class 3, rotect class an class 2. Aggregate classes an 2 into a new eman class {,2}. Set eman an rice or the class {,2}: an {,2} 2 {,2} E E 2E E The rotection level or the aggregate class {,2} while booking or class 3 is P( ) 3 / {,2} {,2} y2 2 2 In general, {,2,., i} 2... i an {,2,., i} / {,2,., i} P( {,2,., i} i i y ) i j j E i k j E k Protect class {,2,.,i} more against class i+ i Class {,2,.,i} eman is larger Class {,2,.,i} rice is larger

11 Version-b Heuristic with Normal Demans utallas.eu/~metin Page With normal ineenent emans mean μ i an stanar eviation δ i i {,2,., i}..., 2 i {,2,., i} i j 2 j an {,2,., i} j i i {,2,., i} The rotection level or the aggregate class {,2,,i} while booking or class i+ is y {,2,.,i} =Norminv(- i+ / {,2,.,i}, μ {,2,.,i}, δ {,2,.,i} ) Examle: Suose that irst an secon class emans are normal with the same arameters 2 20; 2 5 What is the istribution o the eman o aggregate class {,2}? The aggregate istribution is also Normal with arameters: 2 2 {,2} 40; {,2}

12 Comarison o Version-a an Version-b Heuristics uner Normal Demans utallas.eu/~metin Page 2 Examle: Suose that irst an secon class emans are normal with the same arameters 2 20; 2 5. Then the istribution o the eman o aggregate class {,2} is also Normal with arameters: 40; {,2} {,2} Furthermore, suose that the rices or classes are =000; 2 =700; 3 =400. Fin the rotection level or classes an 2 by using EMSR-a an EMSR-b heuristics. EMSR-a heuristic: The rotection level or class rom class 3: y,3 = norminv(-400/000, 20, 5)=2.27 The rotection level or class 2 rom class 3: y 2,3 = norminv(-400/700, 20, 5)=9.0 The rotection level or classes an 2 rom class 3: y 2 = =40.37 EMSR-b heuristic: Price or the aggregate class {,2}=850. The rotection level or the aggregate class {,2} while booking or class 3 is y {,2} =norminv(-400/850, 40, 7.)=40.52 The rotection level or classes an 2 rom class 3: y 2 =40.52 The rotection levels above are almost the same.

13 2-class Problem Deman Deenence By Buying u F (b): No secon class is rejecte with b or b+ Remaining caacity or irst class is the same Marginal eect on revenue 0-0=0 utallas.eu/~metin Page 3 b u by b u or same b kee constant b >b -F (b): Remaining caacity or irst class is less by Execte marginal eect on revenue F [ F [ F -F (C-b): First class rejecte because o b+ >C-b C-b F (C-b): Low irst class eman ( b)0 [ F ( b)] ( b)] ( ) ( b)] [ F [ F [ ( C b)]( ( C b)]( ) -α α: Rejecte class 2 buys u class ]/[( ) ) F - <0-0>0 - <0 0 ] [ F ( C b)( Increasing b, we lose the oortunity o selling at to low-aying customers who buy u when their class is close ( C b)] )

14 2 Fare Classes Booking Limit an Protection Level utallas.eu/~metin Page 4 The critical term is F (C-b) +[ - α ]/[(- α) ] Or equivalently F (C-b) -[ - ]/[(- α) ] F (C-b) ecreases as b increases. I b=0 an still F (C-b) [ - ]/[(- α) ], set b=0. In this case, either the iscount rice is very low or the buy u robability α is high so it is not worth oening u the iscount class. For any b, increase b as long as F (C-b)>[ - ]/[(- α) ]. For a continuous ull are eman istribution, C-b*= F ([ - ]/[(- α) ]) an b* C. [ - ]/[(- α) ] Otimal booking limit or the secon class C-F ([ - ]/[(- α) ] ) C 0 F ( ) b y * * max min or [ C F F ([ ([ ]/[( ) ]/[( ) ]/[( ) ] P( ]), C y * ]),0 ). 0 C F ([ - ]/[(- α) ]) Otimal rotection level or the irst class Deman

15 2 Fare Classes Protection Level with Uniorm Deman utallas.eu/~metin Page 5 Suose that the irst class (ull-are) class eman is uniorm between 20 an 40 seats on a light. Moreover, irst class assengers ay $000, while the iscount are is $300. Buy u robability is 0.2. What are booking limits or iscount are i the lane has C=00 or C=30 seats? Protection level y irst: F (x)=(x-20)/20 or 20<x<40. F (u)=20+20u or 0<u<. F ([ - ]/[(- α) ])=F (700/[(0.8)000])=37.5. Let us roun u the rotection level to 38. [ - ]/[(- α) ] =7/8 F ( ) I the caacity is 00, the booking limit is 00-38=62 or the iscount class. I the caacity is 30, the booking limit is (30-38) + =0 or the iscount class. Booking limit with buy-u robability 0 is 67, it ecreases to 62 with buy-u robability o Otimal rotection level or the irst class Deman

16 2 Fare Classes Protection Level with Normal Deman utallas.eu/~metin Page 6 Suose that the irst class (ull-are) class eman is normal with mean 30 seats an stanar eviation 5. Moreover, irst class assengers ay $000, while the iscount are is $300. Buy u robability is 0.2. What are booking limits or iscount are i the lane has C=00 or C=30 seats? Stanar cumulative normal istribution Φ(z)=robability. Φ (7/8)=norminv(7/8, mean=0, stev=) =.5=z. [ - ]/[(- α) ] =7/8 F ( ) Protection level mean+z*stev=30+(.5)5= Or use norminv(7/8, mean=30, stev=5)= Let us roun u the rotection level to 36. I the caacity is 00, the booking limit is 00-36=64 or the iscount class. I the caacity is 30, the booking limit is (30-36) + =0 or the iscount class. 20 Otimal rotection level or the irst class Deman norminv(7/8,30,5) =35.75

17 Bi (Dislacement) Prices A bi rice or a light is the threshol rice that eens on the remaining caacity an the time until earture, such that a booking request is accete i its are is more rejecte i its are is less than the threshol rice. The bi rice is about the marginal (oortunity) cost o caacity. The latter is known as islacement cost. Bi rice can be equal to islacement cost. I the booking is accete, the remaining light caacity ros. The marginal cost o acceting a booking is about the marginal cost o the caacity b, which in a static setting woul be b Π b Π b Π b In the ynamic setting, the marginal cost o caacity when t units o time let is b Π b, t Π b, t Π b, t Dislacement cost Marginal cost o caacity an bi rice ecrease with large caacity (suly) an with less amount o remaining time (eman) Price went u by 2 bucks While you have been eciing! Oortunity cost o caacity eens on the caacity b. I b is very large, the oortunity cost is low: Not enough customers to sell the large caacity ater enying the current. Oortunity cost o caacity also eens on the time t that remains to sell the caacity. I t is very small, the oortunity cost is low: Not enough time to sell ater enying the current. utallas.eu/~metin Page 7

18 Bi Price Path utallas.eu/~metin Page 8 Oortunity cost curves with ierent caacities over time or a leg b-2 b Erratic bi rices right beore earture b b+ Bi rice ath t t Sale Sale Sale Sale Sale Sale Sale Sale Bi rice is not actually a rice, rather it is a cost. A roit margin on the bi rice to in the actual rice to quote to the customer. How much roit margin? In comarison to are classes, booking by bi rices Is simler as there is a single bi rice or a single light while there are many are classes Is more comlex as the bi rices must be comute ynamically while are class booking limits can be static. No ree lunch by booking via are classes or bi rices.

19 Towars Bi Price Comutation utallas.eu/~metin Page 9 Ex: Suose that = i Fin total margin maximizing rice when the cost is c = 0 an the caacity constraint is b = 20. b = 20 From max = , the unconstraine rice 0 = 0. But 0 = 00 > 20 = b. To sell all o the caacity 20 = b = = or = 8. Π b = 20 = 360. Ater selling, b = 9 From max = , the unconstraine rice 0 = 0. But 0 = 00 > 9 = b. To sell all o the caacity 9 = b = = or = 8.. Π b = 9 = Π 20 Π 9 = = 6.. This rice is ierent rom 8 because o iscreteness, see the next age. Suose we set = 8 slightly higher than 6.. When caacity is bining, bi rice otimal rice, subject to iscreteness o sellable caacity units When caacity is not iscrete, but continuous at small ε, bi rice is» When the caacity is bining, Π b = b an = () so In the iscrete examle o the revious age Π b Π b Π b Π b ε Π b b ϵ, which is Π b b. = b b = =. = 6. while = 8.

20 Bi Prices or Otimal Prices? utallas.eu/~metin Page 20 When caacity is non-bining, bi rice = 0 <otimal rice Value-base ricing: Otimal rice (Oortunity) Cost-base ricing: Bi rice» Oortunity cost o one ewer unit o caacity is nothing i caacity > eman Assume that uture rices are equal to the current rice an solve: Exemliie on the last an next ages In reality, uture rices can reson to current eman, so they shoul be ierent rom the current one Future rices are anticiatory with resect to current eman To allow or uture rices to reson, we nee a recursive revenue comutation. Π(b, t) is otimal roit obtaine by selling b caacity to h an l tye customers. h an l customers are high-tye (ull are) an low-tye (iscount are). WTP h an WTP l are willingness to ay o h an l customers q h an q l are robability o h an l customers showing u, q h + q l h an l are current rices or h an l customers Π b, t = max h, l q h P WTP h > h h + q l P WTP l > l l Current revenue rom selling to h or l customer + q h P WTP h > h + q l P(WTP l > l ) Π b, t Caacity becomes b with a sale + q h P(WTP h > h ) q l P(WTP l > l ) Π b, t Caacity remains at b without a sale Π 0, t = Π b, 0 = 0 Above ormulation is an examle o obtaining roits over t erios but beyon the scoe o this course.

21 Prices over Time utallas.eu/~metin Page 2 With t = 20 ays until light earture/room check-in/car ick-u, ecie on the rice:, t = 20 = an b t = 20 = 20 yiels t = 8. Deman or the remaining t ays can be given as, t = 0t 0. Suose erson buys on ay 20, so b t = 9 = 9. On ay 9, b(9) = 9 an, 9 = For b(9) = 9,» max, 9 = = 9.5. But 9.5,9 = 95 > 9.» To sell all o the caacity 9 = 90 0 or = 7.. Π b = 9, t = 9 = For b(9) = 8,» From max = = 9.5. But 9.5,9 = 95 > 8.» To sell all o the caacity 8 = 90 0 or = 7.2. Π b = 8, t = 9 = Π 9,9 Π 8,9 = = 5.3. Suose we set = 7 slightly higher than 5.3. Suose noboy buys at =7. On ay 8, b(8) = 9 an, 8 = For b(8) = 9,» From max = = 9. But 9,8 = 90 > 9 = b(8).» To sell all o the caacity 9 = 80 0 or = 6.. Π b = 9, t = 8 = For b(8) = 8,» From max = = 9. But 9,8 = 90 > 8 = b(8).» To sell all o the caacity 8 = 80 0 or = 6.2. Π b = 8, t = 8 = Π 8,8 Π 7,8 = = 4.3. Suose we set = 6 slightly higher than 4.3. Three buy at 6 on ay 8, an so on

22 Summary utallas.eu/~metin Page 22 Two-class allocation Multile-class allocation Deman eenence Bi rices

23 Normal Density Function utallas.eu/~metin Page 23 requency normist(x,.,.,) normist(x,.,.,0) Prob Mean 95.44% 99.74% Excel statistical unctions : Density unction ()at x : normist( x, mean, st _ ev,0) Cumulative unction (c) at x : normist( x, mean, st _ ev,) x

24 Cumulative Normal Density utallas.eu/~metin Page 24 rob normist(x,mean,st_ev,) 0 x norminv(rob,mean,st_ev) Excel statistical unctions : Cumulative unction (c) at x : normist( x, mean, st _ ev,) Inverse unction o c at "rob": norminv( rob, mean, st _ ev)