Integrating advanced discrete choice models in mixed integer linear optimization

Transcription

1 Integrating advanced discrete choice models in mixed integer linear optimization Meritxell Pacheco Shadi Sharif Azadeh, Michel Bierlaire, Bernard Gendron Transport and Mobility Laboratory (TRANSP-OR) École Polytechnique Fédérale de Lausanne June, 2017 MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

2 Outline 1 Introduction 2 General framework 3 Case study 4 Conclusions MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

3 Introduction 1 Introduction 2 General framework 3 Case study 4 Conclusions MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

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 Workshop on Discrete Choice Models / 23

5 Introduction Demand model Population of N customers (n) Choice set C (i) C n C: alternatives considered by customer 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 Workshop on Discrete Choice Models / 23

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 Workshop on Discrete Choice Models / 23

7 General framework 1 Introduction 2 General framework 3 Case study 4 Conclusions MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

8 General framework MILP (in words) MILP max subject to benefit utility definition availability discounted utility choice capacity allocation price selection MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

9 General framework Variables Availability y i {0, 1} y in {0, 1} y inr {0, 1} Utility and choice U inr z inr U nr w inr {0, 1} services proposed by the operator y i = 1 and services considered by customers capacity restrictions utility discounted utility maximum discounted utility choice Pricing λ inl {0, 1} binary representation of the price α inrl {0, 1} linearization of the product w inr λ inl MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

10 General framework MILP MILP Utility max subject to benefit utility definition availability discounted utility choice capacity allocation price selection U inr = p in V in {}}{ β in p in + q d (x d ) +ξ inr i, n, r (1) endogenous variable β in associated parameter (β 0n = 0) q d (x d ) exogenous demand variables MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

11 General framework MILP MILP max subject to benefit utility definition availability discounted utility choice capacity allocation price selection Product of decisions { yin d 1 if i Cn = 0 otherwise i, n y in = y d iny i i, n (2) Availability at operator and scenario level y inr y in i, n, r (3) MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

12 General framework MILP MILP max subject to benefit utility definition availability discounted utility choice capacity allocation price selection z inr = { Uinr if y inr = 1 l nr if y inr = 0 (l nr smallest lower bound) Discounted utility i, n, r 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 Workshop on Discrete Choice Models / 23

13 General framework MILP MILP max subject to benefit utility definition availability discounted utility choice capacity allocation price selection U nr = max z inr n, r i C { 1 if i = arg max{unr } w inr = i, n, r 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 Workshop on Discrete Choice Models / 23

14 General framework MILP MILP max subject to benefit utility definition availability discounted utility choice capacity allocation price selection Capacity allocation Priority list Two sets of constraints i > 0 Capacity cannot be exceeded ( y inr = 1) Capacity has been reached ( y inr = 0) Price selection p in = 1 10 k ( L in 1 l in + l=0 2 l λ inl ) When calculating the benefit: λ inl w inr α inrl = λ inl w inr + linearizing constraints MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

15 General framework MILP MILP max i>0(r i C i ) max subject to benefit utility definition availability 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 Workshop on Discrete Choice Models / 23

16 Case study 1 Introduction 2 General framework 3 Case study 4 Conclusions MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

17 Case study Parking choices N = 50 customers C = {PSP, PUP, FSP} C n = C n PSP: 0.50, 0.51,..., 0.65 (16 price levels) PUP: 0.70, 0.71,..., 0.85 (16 price levels) Capacity of 20 spots MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

18 Case study Choice model: mixtures of logit model 1 V FSP = β AT AT FSP + β TD TD FSP + β OriginINT FSP Origin INT FSP V PSP = ASC PSP + β AT AT PSP + β TD TD PSP + β FEE FEE PSP + β FEEPSP(LowInc) FEE PSP LowInc + β FEEPSP(Res) FEE PSP Res V PUP = ASC PUP + β AT AT PUP + β TD TD PUP + β FEE FEE PUP + β FEEPUP(LowInc) FEE PUP LowInc + β FEEPUP(Res) FEE PUP Res + β AgeVeh 3 AgeVeh 3 Parameters Circle: distributed parameters Rectangle: constant parameters Variables: all given but FEE (in bold) 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 Workshop on Discrete Choice Models / 23

19 Case study Experiment 1: uncapacitated vs capacitated case (1) Capacity constraints are ignored Unlimited capacity is assumed 20 spots for PSP and PUP Opt-out has unlimited capacity MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

20 Case study Experiment 1: uncapacitated vs capacitated case (2) Uncapacitated Log Solution time (s) Revenue 28 Log Solution time (s) Revenue Capacitated R Log Solution time (s) Revenue 28 Log Solution time (s) Revenue R MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

21 Case study Experiment 1: uncapacitated vs capacitated case (3) 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 Workshop on Discrete Choice Models / Demand

22 Case study Experiment 2: price differentiation by segmentation (1) Discount offered to residents Two scenarios (municipality) 1 Subsidy offered by the municipality 2 Operator obliged to offer reduced fees We expect the price to increase PSP: {0.60, 0.64,..., 1.20} PUP: {0.80, 0.84,..., 1.40} MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

23 Case study Experiment 2: price differentiation by segmentation (2) Scenario 1 PSP NR PSP R PUP NR PUP R Revenue Price Revenue Scenario 2 Discount (%) PSP NR PSP R PUP NR PUP R Revenue Price Revenue Discount (%) 20 MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

24 Case study Experiment 2: price differentiation by segmentation (3) Scenario 1 20 PSP NR PSP R PUP NR PUP R FSP NR FSP R 15 Demand 10 5 Scenario PSP NR PSP R Discount (%) PUP NR PUP R FSP NR FSP R 15 Demand Discount (%) MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

25 Case study Other experiments Impact of the priority list Priority list = order of the individuals in the data (i.e., random arrival) 100 different priority lists Aggregate indicators remain stable across random priority lists Benefit maximization through capacity allocation 4 different capacity levels for both PSP and PUP: 5, 10, 15 and 20 Optimal solution: PSP with 20 spots and PUP is not offered Both services have to be offered: PSP with 15 and PUP with 5 MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

26 Conclusions 1 Introduction 2 General framework 3 Case study 4 Conclusions MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

27 Conclusions Conclusions and ongoing research Conclusions Powerful tool to configure systems based on heterogenous behavior Computationally expensive, e.g., for N = 50 and R = 250 Uncapacitated: 2.5 h Capacitated: 1.7 days In practice, more individuals and a high number of draws is desirable Ongoing research Decomposition technique (Lagrangian relaxation) Faster subproblems that can be parallelized MP, SSA, MB, BG Workshop on Discrete Choice Models / 23

28 Conclusions Questions? MP, SSA, MB, BG Workshop on Discrete Choice Models / 23