Approximation Methods for Pricing Problems under the Nested Logit Model with Price Bounds

Transcription

1 Approximation Methods for Pricing Problems under the Nested Logit Model with Price Bounds W. Zachary Rayfield School of ORIE, Cornell University Paat Rusmevichientong Marshall School of Business, University of Southern California Huseyin Topaloglu School of ORIE, Cornell University

2 Pricing under the Nested Logit Model $$ $$$$ $ $ $$$$ $ $$ $$$ $$

3 Pricing under the Nested Logit Model $$ $$$$ $ $ $$$$ $ $$ $$$ $

4 Pricing under the Nested Logit Model $$ $$$$$ $ $ $$$$ $ $$ $$$ $

5 Problem Formulation Store Competitor Space 1 2 j n

6 Problem Formulation Store Competitor Space 1 2 j n n products, N = {1,..., n}

7 Problem Formulation Store Competitor Space 1 2 j n n products, N = {1,..., n} Preference weight of product j: w j

8 Problem Formulation Store Competitor Space 1 2 j n n products, N = {1,..., n} Preference weight of product j: w j Total preference weight of competitor space normalized to 1

9 Problem Formulation Store Competitor Space w 1 w j w n w 0 =1 Price of product j:

10 Problem Formulation Store Competitor Space p 1 p j w 0 =1 p n Price of product j: p j

11 Problem Formulation Store Competitor Space p 1 p j w 0 =1 p n Price of product j: p j w j = e αj βjpj

12 Problem Formulation Store Competitor Space p 1 p j w 0 =1 p n Price of product j: p j w j = e αj βjpj Prices are constrained:

13 Problem Formulation Store Competitor Space p 1 p j w 0 =1 p n Price of product j: p j w j = e α j β j p j Prices are constrained: l j p j u j

14 Problem Formulation Store Competitor Space p 1 p j w 0 =1 p n Price of product j: p j w j = e α j β j p j p j (w j )= α j β j 1 β j log w j Prices are constrained: l j p j u j

15 Problem Formulation Store Competitor Space p 1 p j w 0 =1 p n Price of product j: p j w j = e α j β j p j p j (w j )= α j β j 1 β j log w j Prices are constrained: L j w j U j

16 Problem Formulation Store Competitor Space p 1 p j w 0 =1 p n Price of product j: p j w j = e α j β j p j p j (w j )= α j β j 1 β j log w j Prices are constrained: L j w j U j If customer buys from our firm, product j purchased with probability w j k N w k

17 Problem Formulation Store Competitor Space p 1 p j w 0 =1 p n Price of product j: p j w j = e α j β j p j p j (w j )= α j β j 1 β j log w j Prices are constrained: L j w j U j If customer buys from our firm, product j purchased with probability w j k N w k

18 Problem Formulation Store Competitor Space p 1 p j w 0 =1 p n Price of product j: p j w j = e α j β j p j p j (w j )= α j β j 1 β j log w j Prices are constrained: L j w j U j Expected revenue customer buys from our firm: R(w) = p j (w j ) j N w j k N w k

19 Problem Formulation Store Competitor Space p 1 p j w 0 =1 p n Price of product j: p j w j = e α j β j p j p j (w j )= α j β j 1 β j log w j Prices are constrained: L j w j U j Expected revenue customer buys from our firm: R(w) = j N p j (w j ) w j V (w)

20 Problem Formulation Store Competitor Space w 1 w j w n w 0 =1

21 Problem Formulation Store Competitor Space w 1 w j w n w 0 =1 V (w) = j N w j

22 Problem Formulation Store Competitor Space w 1 γ [0, 1] w j w n w 0 =1 V (w) = j N w j Customer buys from our firm with probability V (w) γ 1+V (w) γ

23 Problem Formulation Store Competitor Space w 1 γ [0, 1] w j w n w 0 =1 V (w) = j N w j Customer buys from our firm with probability Expected Revenue: Π(w) = V (w)γ 1+V (w) γ R(w) V (w) γ 1+V (w) γ

24 Problem Formulation Store Competitor Space w 1 γ [0, 1] w j w n w 0 =1 V (w) = j N w j Customer buys from our firm with probability Expected Revenue: Π(w) = V (w)γ 1+V (w) γ R(w) V (w) γ 1+V (w) γ Z = max Π(w) L w U

25 Transformation of Objective Z Π(w) = V (w)γ 1+V (w) γ R(w) L w U (tight at opt.)

26 Transformation of Objective Z Π(w) = V (w)γ 1+V (w) γ R(w) L w U (tight at opt.) Z (1 + V (w) γ ) V (w) γ R(w)

27 Transformation of Objective Z Π(w) = Z V (w) γ (R(w) Z ) V (w)γ 1+V (w) γ R(w) L w U (tight at opt.) Z (1 + V (w) γ ) V (w) γ R(w)

28 Transformation of Objective Z Π(w) = Z V (w) γ (R(w) Z ) V (w)γ 1+V (w) γ R(w) L w U (tight at opt.) Z (1 + V (w) γ ) V (w) γ R(w) Z = max V L w U (w)γ (R(w) Z )

29 Transformation of Objective Z Π(w) = Z V (w) γ (R(w) Z ) V (w)γ 1+V (w) γ R(w) L w U (tight at opt.) Z (1 + V (w) γ ) V (w) γ R(w) Z = max V L w U (w)γ (R(w) Z ) Use single decision variable to capture V (w)

30 Transformation of Objective Z Π(w) = V (w)γ 1+V (w) γ R(w) L w U (tight at opt.) Z (1 + V (w) γ ) V (w) γ R(w) Z V (w) γ (R(w) Z ) Z = max V L w U (w)γ (R(w) Z ) Use single decision variable to capture V (w) R(w) = j N p j(w j )w j V (w)

31 Transformation of Objective Z Π(w) = V (w)γ 1+V (w) γ R(w) L w U (tight at opt.) Z (1 + V (w) γ ) V (w) γ R(w) Z V (w) γ (R(w) Z ) Z = max V L w U (w)γ (R(w) Z ) Use single decision variable to capture V (w) R(w) = j N p j(w j )w j V (w)

32 Transformation of Objective Z Π(w) = V (w)γ 1+V (w) γ R(w) L w U (tight at opt.) Z (1 + V (w) γ ) V (w) γ R(w) Z V (w) γ (R(w) Z ) Z = max V L w U (w)γ (R(w) Z ) Use single decision variable to capture V (w) R(w) = j N p j(w j )w j V (w) Z = max y? {y γ ( )}? y Z

33 Transformation of Objective Z = max V L w U (w)γ (R(w) Z ) Use single decision variable to capture V (w) R(w) = j N p j(w j )w j V (w) Z = max y? {y γ ( )}? y Z

34 Transformation of Objective Capture numerator with g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N } Z = max V L w U (w)γ (R(w) Z ) Use single decision variable to capture V (w) R(w) = j N p j(w j )w j V (w) Z = max y? {y γ ( )}? y Z

35 Transformation of Objective Capture numerator with g(y) = max { j N p j (w j )w j : j N (Concave, non-linear knapsack problem) } w j y, w j [L j,u j ] j N Z = max V L w U (w)γ (R(w) Z ) Use single decision variable to capture V (w) R(w) = j N p j(w j )w j V (w) Z = max y? {y γ ( )}? y Z

36 Transformation of Objective Capture numerator with g(y) = max { j N p j (w j )w j : j N (Concave, non-linear knapsack problem) } w j y, w j [L j,u j ] j N Restrict attention to [ L, Ū] Z = max y? {y γ ( )}? y Z

37 Transformation of Objective Capture numerator with g(y) = max { j N p j (w j )w j : j N (Concave, non-linear knapsack problem) } w j y, w j [L j,u j ] j N Restrict attention to [ L, Ū] Fixed-point relationship: Z = max y [ L,Ū] {y γ ( g(y) y Z )}

38 Recovering Optimal Prices y g( ) If Z* known: Recover the optimal solution by solving max y [ L,Ū] { y γ g(y) y y γ Z } g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N }

39 Recovering Optimal Prices y g( ) If Z* known: Recover the optimal solution by solving max y [ L,Ū] { y γ g(y) y y γ Z } g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N }

40 Recovering Optimal Prices y g( ) w 1 (y)... w n (y) If Z* known: Recover the optimal solution by solving max y [ L,Ū] { y γ g(y) y y γ Z } g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N }

41 Recovering Optimal Prices y g( ) w 1 (y)... w n (y) If Z* known: Recover the optimal solution by solving max y [ L,Ū] { y γ g(y) y y γ Z } g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N }

42 Recovering Optimal Prices y g( ) w 1 (y)... w n (y) p 1 ((w 1 (y))... p n ((w n (y)) If Z* known: Recover the optimal solution by solving max y [ L,Ū] { y γ g(y) y y γ Z } g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N }

43 Fixed Point Problem To find Z*, solve z = max y [ L,Ū] { y γ g(y) y y γ z }

44 Fixed Point Problem To find Z*, solve z = max y [ L,Ū] { y γ g(y) y y γ z } Problem: RHS function not necessarily concave in y

45 Fixed Point Problem To find Z*, solve z = max y [ L,Ū] { y γ g(y) y y γ z } Problem: RHS function not necessarily concave in y Ex: 2.5 y γ g(y) y y γ z y

46 Approximation Framework Construct a set of grid points {ỹ t : t =1,..., T }, solve z = max y {ỹ t :t=1,...,t } { y γ g(y) y y γ z }

47 Approximation Framework Construct a set of grid points {ỹ t : t =1,..., T }, solve z = max y {ỹ t :t=1,...,t } { y γ g(y) y y γ z Objective : given ρ > 0, obtain an approximate solution ẑ (w/corresponding prices): }

48 Approximation Framework Construct a set of grid points {ỹ t : t =1,..., T }, solve z = max y {ỹ t :t=1,...,t } { y γ g(y) y y γ z Objective : given ρ > 0, obtain an approximate solution ẑ (w/corresponding prices): Want ẑ 1 1+ρ Z }

49 Approximation Framework Construct a set of grid points {ỹ t : t =1,..., T }, solve z = max y {ỹ t :t=1,...,t } { y γ g(y) y y γ z Objective : given ρ > 0, obtain an approximate solution ẑ (w/corresponding prices): Want ẑ 1 1+ρ Z Keep grid size small }

50 Approximation Framework Construct a set of grid points {ỹ t : t =1,..., T }, solve z = max y {ỹ t :t=1,...,t } { y γ g(y) y y γ z Objective : given ρ > 0, obtain an approximate solution ẑ (w/corresponding prices): Want ẑ 1 1+ρ Z Keep grid size small } Thm: A grid {ỹ t : t =1,..., T } with g(ỹ t+1 ) (1 + ρ)g(ỹ t ) t =1,..., T 1 guarantees a worst-case expected revenue of 1 1+ρ Z

51 Knapsack Properties g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N } g( ) concave, increasing on [ L, Ū] Optimal decision variable values monotonically increasing with y

52 Knapsack Properties g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N } g( ) concave, increasing on [ L, Ū] Optimal decision variable values monotonically increasing with y U j FREE L j w 1 w n L

53 Knapsack Properties g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N } g( ) concave, increasing on [ L, Ū] Optimal decision variable values monotonically increasing with y U j FREE L j w 1 w n L

54 Knapsack Properties g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N } g( ) concave, increasing on [ L, Ū] Optimal decision variable values monotonically increasing with y U j FREE L j w 1 w n L

55 Knapsack Properties g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N } g( ) concave, increasing on [ L, Ū] Optimal decision variable values monotonically increasing with y U j FREE L j w 1 w n L

56 Knapsack Properties g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N } g( ) concave, increasing on [ L, Ū] Optimal decision variable values monotonically increasing with y U j FREE L j w 1 w n L

57 Knapsack Properties g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N } g( ) concave, increasing on [ L, Ū] Optimal decision variable values monotonically increasing with y U j FREE L j w 1 w n L

58 Knapsack Properties g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N } g( ) concave, increasing on [ L, Ū] Optimal decision variable values monotonically increasing with y U j FREE L j w 1 w n L Ū

59 Knapsack Properties g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N } g( ) concave, increasing on [ L, Ū] Optimal decision variable values monotonically increasing with y U j FREE L j w 1 w n L Ū We can compute these points explicitly

60 Knapsack Properties g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N } g( ) concave, increasing on [ L, Ū] Optimal decision variable values monotonically increasing with y U j FREE L j w 1 w n L Ū We can compute these points explicitly # of points = O(n)

61 Grid Points g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N } U k F k L k L Ū

62 Grid Points g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N } U k F k L k L Ū

63 Grid Points g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N } U k F k L k L Ū Intermediate points Ỹ kq = j L k L j + j U k U j + (1 + ρ) q,q Z

64 Grid Points g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N } U k F k L k L Ū Intermediate points Ỹ kq = L j + U j + (1 + ρ) q,q Z j L k j U k Free product capacity increments exponentially

65 Grid Points g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N } U k F k L k L Ū Intermediate points Free product capacity increments exponentially g(ỹ k,q+1 ) (1 + ρ)g(ỹ kq ) q Z Ỹ kq = L j + U j + (1 + ρ) q,q Z j L k j U k

66 Grid Points g(y) = max { j N p j (w j )w j : j N w j y, w j [L j,u j ] j N } U k F k L k L Ū Intermediate points Free product capacity increments exponentially g(ỹ k,q+1 ) (1 + ρ)g(ỹ kq ) q Z (Grid satisfies theorem conditions) Ỹ kq = L j + U j + (1 + ρ) q,q Z j L k j U k

67 Multiple Product Categories 11 w i1 Product Categories 22 γ i [0, 1] jj w ij nn w in Competitor Space w 0 =1

68 Multiple Product Categories 11 w i1 Product Categories 22 γ i [0, 1] jj w ij nn w in Competitor Space w 0 =1 m categories, M = {1,..., m}

69 Multiple Product Categories 11 w i1 Product Categories 22 γ i [0, 1] jj w ij nn w in Competitor Space w 0 =1 m categories, M = {1,..., m} Probability that customer buys product j customer chooses category i: w ij V i (w i )

70 Multiple Product Categories 11 w i1 Product Categories 22 γ i [0, 1] jj w ij nn w in Competitor Space w 0 =1 m categories, M = {1,..., m} Probability that customer buys product j customer chooses category i: w ij V i (w i ) Expected revenue customer buys from category i: w ij R i (w i )= p ij (w ij ) V i (w i ) j N

71 Multiple Product Categories 11 w i1 Product Categories 22 γ i [0, 1] jj w ij nn w in Competitor Space w 0 =1 Customer buys from category i with probability V i (w i ) γ i 1+ l M V l(w l ) γ l Expected Revenue: Π(w 1,..., w m )= i M V i (w i ) γ i 1+ l M V l(w l ) γ l R i (w i ) Z = max Π(w 1,..., w m ) (w 1,...,w m ):L i w i U i,i M

72 General Approximate Problem z = i M max y i {ỹ t i : t =1,...,T i} { y γ i i g i (y i ) y i y γ i i z } min s.t. z z i M max y i {ỹ t i :t=1,...,t i} { y γ i i g i (y i ) y i y γ i i z }

73 General Approximate Problem z = i M max y i {ỹ t i : t =1,...,T i} { y γ i i g i (y i ) y i y γ i i z } min s.t. z z i M max y i {ỹ t i :t=1,...,t i} { y γ i i g i (y i ) y i y γ i i z } x i (ỹ t i) γ i g i(ỹ t i ) ỹ t i (ỹ t i) γ i z, t {1,..., T i }, i M

74 General Approximate Problem z = i M max y i {ỹ t i : t =1,...,T i} { y γ i i g i (y i ) y i y γ i i z } min s.t. z z i M max y i {ỹ t i :t=1,...,t i} { y γ i i g i (y i ) y i y γ i i z } x i (ỹ t i) γ i g i(ỹ t i ) ỹ t i (ỹ t i) γ i z, t {1,..., T i }, i M

75 General Approximate Problem z = i M max y i {ỹ t i : t =1,...,T i} { y γ i i g i (y i ) y i y γ i i z }

76 General Approximate Problem z = i M max y i {ỹ t i : t =1,...,T i} { y γ i i g i (y i ) y i y γ i i z } The approximate problem is a linear program min (x,z) z s.t. z i M x i x i (ỹ t i) γ i g i(ỹ t i ) ỹ t i (ỹ t i) γ i z, t {1,..., T i }, i M.

77 General Approximate Problem z = i M max y i {ỹ t i : t =1,...,T i} { y γ i i g i (y i ) y i y γ i i z } The approximate problem is a linear program min (x,z) z s.t. z i M x i x i (ỹ t i) γ i g i(ỹ t i ) ỹ t i (ỹ t i) γ i z, t {1,..., T i }, i M. We apply the same method of grid construction LP has m+1 variables, # constraints m (grid size)

78 Upper Bound LP Given any set of grid points for each i M, solve {ȳ t i : t =1,..., τ i } min (x,z) z s.t. z i M x i x i (ȳ t i) γ i g i(ȳ t+1 i ) ȳ t i (ȳ t i) γ i z, t {1,..., τ i 1}, i M. Allows comparison of the performance of our algorithm, other heuristics, etc.

79 Algorithm Performance ρ =.005 APP - Avg. Gap with Upper Bound Rounding Heuristic - Avg. Gap with Upper Bound 10% 8.75% 7.5% 6.25% 5% 3.75% 2.5% 1.25% 0% S M L S M L S M L 25 total products 100 total products 225 total products S,M,L - small/medium/large degree of constraint violation

80 Sample Computation Times CPU Seconds total products / Small violation 225 total products / Large violation ρ

81 Thank you!