A Lagrangian relaxation method for solving choice-based mixed linear optimization models that integrate supply and demand interactions

Transcription

1 A Lagrangian relaxation method for solving choice-based mixed linear optimization models that integrate supply and demand interactions Meritxell Pacheco Shadi Sharif Azadeh, Michel Bierlaire, Bernard Gendron Transport and Mobility Laboratory (TRANSP-OR) École Polytechnique Fédérale de Lausanne September, 2017 MP, SSA, MB, BG OR Berlin 1 / 25

2 Outline 1 Introduction 2 Choice-based mixed linear optimization 3 Case study 4 Lagrangian relaxation 5 Future work MP, SSA, MB, BG OR Berlin 2 / 25

3 Introduction 1 Introduction 2 Choice-based mixed linear optimization 3 Case study 4 Lagrangian relaxation 5 Future work MP, SSA, MB, BG OR Berlin 3 / 25

4 Introduction Motivation Demand Choices of customers Discrete choice models Nonlinear and nonconvex formulations Supply Design and configuration of the system Mixed Integer Linear Problems (MILP) MP, SSA, MB, BG OR Berlin 4 / 25

5 Introduction Demand model Population of N customers (n) Choice set C (i) C n C: alternatives considered by customer n (N i = {n 1 i C n }) Behavioral assumption U in = V in + ε in V in = k β inkx e ink + qd (x d ) P n (i C n ) = Pr(U in U jn, j C n ) Simulation Distribution ε in R draws ξ in1,..., ξ inr U inr = V in + ξ inr MP, SSA, MB, BG OR Berlin 5 / 25

6 Introduction Supply model Operator selling services to a market Price p in (to be decided) Capacity c i Benefit (revenue cost) to be maximized Opt-out option (i = 0) Price characterization Lower and upper bound Discretization: price levels Binary representation (λ inl ) Capacity allocation Exogenous priority list of customers Here it is assumed as given Capacity as decision variable MP, SSA, MB, BG OR Berlin 6 / 25

7 Choice-based mixed linear optimization 1 Introduction 2 Choice-based mixed linear optimization 3 Case study 4 Lagrangian relaxation 5 Future work MP, SSA, MB, BG OR Berlin 7 / 25

8 Choice-based mixed linear optimization MILP model (in words) MILP max subject to benefit availability utility definition discounted utility choice capacity allocation price selection MP, SSA, MB, BG OR Berlin 8 / 25

9 Choice-based mixed linear optimization MILP model MILP max subject to benefit availability utility definition discounted utility choice capacity allocation price selection y i {0, 1} yin d {0, 1} y in {0, 1} y inr {0, 1} operator decision customer decision (data) product of decisions capacity restrictions Relations between availabilities y in = yiny d i i, n (1) y inr y in i, n, r (2) MP, SSA, MB, BG OR Berlin 9 / 25

10 Choice-based mixed linear optimization MILP model MILP Utility max subject to benefit availability utility definition discounted utility choice capacity allocation price selection U inr = V in {}}{ β in p in + q d (x d ) +ξ inr i, n, r (3) MP, SSA, MB, BG OR Berlin 9 / 25

11 Choice-based mixed linear optimization MILP model MILP max subject to benefit availability utility definition discounted utility choice capacity allocation price selection z inr = Discounted utility { Uinr if y inr = 1 l nr if y inr = 0 discounted utility l nr z inr i, n, r (4) z inr l nr + M inr y inr i, n, r (5) U inr M inr (1 y inr ) z inr i, n, r (6) z inr U inr i, n, r (7) MP, SSA, MB, BG OR Berlin 9 / 25

12 Choice-based mixed linear optimization MILP model MILP max subject to benefit availability utility definition discounted utility choice capacity allocation price selection U nr = max z inr i C { 1 if i = arg max{unr } w inr = choice 0 otherwise Choice z inr U nr i, n, r (8) U nr z inr + M nr (1 w inr ) i, n, r (9) w inr = 1 n, r (10) i w inr y inr i, n, r (11) MP, SSA, MB, BG OR Berlin 9 / 25

13 Choice-based mixed linear optimization MILP model MILP max subject to benefit availability utility definition discounted utility choice capacity allocation price selection Priority list y in r y inr i > 0, n < N, r (12) Capacity cannot be exceeded y inr = 1 n 1 w imr (c i 1)y inr + (n 1)(1 y inr ) i > 0, n > c i, r (13) m=1 Capacity has been reached y inr = 0 c i (y in y inr ) n 1 m=1 w imr i > 0, n, r (14) MP, SSA, MB, BG OR Berlin 9 / 25

14 Choice-based mixed linear optimization MILP model MILP max subject to benefit availability utility definition discounted utility choice capacity allocation price selection p in = 1 ( L in 1 ) 10 k l in + 2 l λ inl l=0 When calculating the benefit: λ inl w inr α inrl = λ inl w inr Linearization of α inrl + Price bounded from above λ inl + w inr 1 + α inrl i > 0, n, r, l (15) α inrl λ inl i > 0, n, r, l (16) α inrl w inr i > 0, n, r, l (17) L in 1 l in + 2 l λ inl m in i > 0, n (18) l=0 MP, SSA, MB, BG OR Berlin 9 / 25

15 Choice-based mixed linear optimization MILP model MILP max i>0(r i C i ) max subject to benefit availability utility definition discounted utility choice capacity allocation price selection Revenue R i = 1 R 1 10 k [ n r ( l in w inr + l 2 l α inrl )] Cost C i = (f i + v i c i )y i MP, SSA, MB, BG OR Berlin 9 / 25

16 Case study 1 Introduction 2 Choice-based mixed linear optimization 3 Case study 4 Lagrangian relaxation 5 Future work MP, SSA, MB, BG OR Berlin 10 / 25

17 Case study Parking choices 1 PSP PUP FSP (opt-out) N = 50 customers C = {PSP, PUP, FSP} C n = C n p in = p i n Mixtures of a logit model 1 A. Ibeas, L. dellolio, M. Bordagaray, et al., Modelling parking choices considering user heterogeneity, Transportation Research Part A: Policy and Practice, vol. 70, pp , MP, SSA, MB, BG OR Berlin 11 / 25

18 Case study General experiments Uncapacitated vs Capacitated case Maximization of revenue Unlimited capacity Capacity of 20 spots for PSP and PUP Price differentiation by population segmentation Reduced price for residents Two scenarios 1 Subsidy offered by the municipality 2 Operator is obliged to offer a reduced price MP, SSA, MB, BG OR Berlin 12 / 25

19 Case study Uncapacitated vs Capacitated case Uncapacitated Capacitated Price Price PSP Price PUP Demand PSP Demand PUP R Demand FSP Demand Price Price PSP Price PUP Demand PSP Demand PUP R Demand FSP MP, SSA, MB, BG OR Berlin 13 / Demand

20 Case study Price differentiation by population segmentation Subsidy offered by the municipality Price PSP NR PSP R PUP NR PUP R Revenue Discount (%) Operator is obliged to offer a reduced price Price PSP NR PSP R PUP NR PUP R Revenue Discount (%) MP, SSA, MB, BG OR Berlin 14 / 25 Revenue Revenue

21 Case study But... MP, SSA, MB, BG OR Berlin 15 / 25

22 Case study Computational time Uncapacitated case Capacitated case R Sol time PSP PUP Rev Sol time PSP PUP Rev s s s s s min min min min h h days MP, SSA, MB, BG OR Berlin 16 / 25

23 Lagrangian relaxation 1 Introduction 2 Choice-based mixed linear optimization 3 Case study 4 Lagrangian relaxation 5 Future work MP, SSA, MB, BG OR Berlin 17 / 25

24 Lagrangian relaxation General idea 1 Relax complicating constraints Lagrangian subproblem 2 Define 2 separable subproblems: Identify common variable Create a copy Relax associated constraints 3 Solve the subproblems independently 4 Solve the Lagrangian dual to provide an upper bound (subgradient method) MP, SSA, MB, BG OR Berlin 18 / 25

25 Lagrangian relaxation First attempt: subproblems Relaxed constraints Common variable: w inr (copy: v inr ) Transferred to the objective function: Copy constraints (γ) Utility definition: involved in the choice + contains price variables (θ) Choice subproblem Variables: U inr and w inr Decomposes by n and r Choice subproblem: Z c nr (θ, γ) Price subproblem Variables: λ inl, α inrl and v inr Decomposes by n Price subproblem: Zn p (θ, γ) MP, SSA, MB, BG OR Berlin 19 / 25

26 Lagrangian relaxation First attempt: drawbacks Variables reaching the bounds: Utility Price poor upper bound High importance placed on the subgradient method MP, SSA, MB, BG OR Berlin 20 / 25

27 Lagrangian relaxation Current approach: sketch Relaxed constraints Relation between availability at operator and customer level Copy constraints for the choice variables also introduced 2 subproblems: Operator subproblem Customer subproblem MP, SSA, MB, BG OR Berlin 21 / 25

28 Lagrangian relaxation Current approach: subproblems Operator subproblem Capacitated Facility Location Problem Customer supbroblem Assumption: utility decreases as a function of the price Iterate over customers (priority list) and over scenarios Highest price such that the customer does not change the choice MP, SSA, MB, BG OR Berlin 22 / 25

29 Future work 1 Introduction 2 Choice-based mixed linear optimization 3 Case study 4 Lagrangian relaxation 5 Future work MP, SSA, MB, BG OR Berlin 23 / 25

30 Future work Ongoing research and future work Ongoing research Implementation of the 2 subproblems Subgradient method to solve the Lagrangian dual Future work Provide a lower bound on the original problem If the gap between bounds is significant column generation MP, SSA, MB, BG OR Berlin 24 / 25

31 Future work Questions? MP, SSA, MB, BG OR Berlin 25 / 25