Last-mile delivery services design under stochastic user...

Last-mile delivery services design under stochastic userequilibrium∗

B. Tounsi1, Y. Hayel2, D. Quadri3, L. Brotcorne1

25 November 2016

1 INOCS, INRIA Lille Nord Europe

2 LIA, University of Avignon

3 LRI, University Paris-Sud

∗ Work supported by the ANR RESPET

COSMOS Day () Math. program with equilibrium constraint 25 November 2016 1 / 28

Plan

1 Introduction

2 Customers’choice process (lower level: follower)

3 Services pricing problem (upper level: leader)

4 Numerical example

5 Conclusion


Introduction

Plan

1 Introduction



4 Numerical example

5 Conclusion


Introduction

Study’s context

Merging two research fields (Game Theory and MathematicalProgramming)

Application : e-commerce delivery

Two objectives linked: a provider (who looks for determining servicespricing) and customers (who select services considering price andcongestion result of overall decisions). In other words, the leadersolves an optimization problem that includes an other optimizationproblem representing the decision of the follower→ the leader takes into account the response of the follower.

Study of the behavior of the users (stationary phase)

Proposition of a solution method to decision aiding


Introduction

The problem : a delivery services system

A last-mile delivery services system: a provider controls services’tariffs and customers who react by choosing their delivery serviceaccording to an utility function.

I Relay station service (RSS)I Delivery at home (DAH).

Each service has options (location, time window, ...).

DCM

Delivery Pick upServices

Options

Figure: The delivery systemCOSMOS Day () Math. program with equilibrium constraint 25 November 2016 5 / 28

Introduction

Methodologies employed

RSS: Erlang-B queuing model

DAH: M/G/1 queue (the arrival of demand follows a Poisson process)

The provider controls services’ tariffs [leader]

Each customer who react by choosing their delivery service accordingto an utility function [followers]

The behavior of the customers is modeled by the use of game theorytools

→ Congestion problem studied via adapted equilibrium: Nested Logit

Both provider and customers: bi-level programming


Customers’choice process (lower level: follower)

Plan

1 Introduction



4 Numerical example

5 Conclusion



Stochastic user equilibrium

Logit-based stochastic user equilibrium (LUE).

Error term:Cr = cr + εr

A random error εr , following a Gumbel distribution.

The probability for an option to be chosen :

pj =exp(−θcj)∑l

exp(−θcl)



Stochastic user equilibrium

Nested Logit based stochastic user equilibrium (NUE).The probability of option j in service n to be chosen by a customer:

∀n ∈ N , ∀j ∈ On, pnj = Pr(n)Pr(j |n) (1)

with

Pr(n) =

(∑

k∈On

e−θcnk/φn)φn∑m∈N

(∑

k∈Om

e−θcmk/φm)φmand Pr(j |n) =

e−θcnj/φn∑k∈On

e−θcnk/φn.



Resolution method for SUE

Let p∗ a solution of the following minimization problem:

[N-SUE] minp

Z (p) = Z1(p) + Z2(p) + Z3(p) (2a)

s. t. Z1(p) =∑n∈N

∑j∈On

∫ pnj

0cnj(s)ds (2b)

Z2(p) =∑n∈N

φnθ

∑j∈On

pnj ln(pnj) (2c)

Z3(p) =∑n∈N

1− φnθ

((∑j∈On

pnj) ln(∑j∈Jn

pnj)) (2d)

∑n∈N

∑j∈On

pnj = 1 (2e)

pnj ≥ 0 ∀n ∈ N ,∀j ∈ On (2f)



Resolution method for SUE

Then p∗ is a solution of the non-linear system (1) and then there exists aSUE considering a Nested Logit DCM with congestion costs functions.Moreover this solution is unique.



The method of successive averages

[Sheffi 1985, Chen et al. 1991]

Step 0: Initialization, i = 0. Find initial choice probabilities p0 andcompute initial costs C 0

nj = cnj(p0nj), ∀n ∈ N , j ∈ Sn.

Step 1: Direction finding. Apply equation (1) with fixed costs C inj in

order to get the auxiliary points yi .

Step 2: Move. Find the new solution pi+1 = (pi+1nj ) by:

∀n ∈ N , ∀j ∈ On, pi+1nj = pi

nj +y inj − pi

nj

i + 1.

Step 3: Convergence criterion. Compute the infinite norm differenceas ||pi+1 − pi ||∞ := max

n∈N ,j∈On

|pi+1nj − pi

nj |. If ||pi+1 − pi ||∞ < ε then

stop, else set i = i + 1 and go to step 1.



Sensitivity Analysis based heuristic

Tool for non linear optimisation problem [Fiacco 1983].

Estimate the solution under a perturbation.

P(ε) = min f (x , ε)

subject togi (x , ε) ≤ 0, i = 1, ..m

hi (x , ε) = 0, i = 1, ..n

At a point ε0

∇εy(ε0) = [Jy ]−1[−Jε]

Where y is the vector of variables of the Lagrangian of P(x , ε), Jy (respJε) are the Jacobians matrix with respect to y (resp ε) of equations systemformed by the KKT conditions of P(x , ε).


Services pricing problem (upper level: leader)

Plan

1 Introduction



4 Numerical example

5 Conclusion



The BiLevel Program

Denoting by t the leader variables, t = (tnj), and by p the userequilibrium, p = (pi ). The problem can be formulated as follows

BLP (U) maxt,p

F (t,p) (3a)

s. t. Lnj ≤ tnj ≤ Unj ∀n ∈ N , j ∈ On (3b)

(L) minp

Z (t,p) (3c)

s. t.∑n,j

pnj = 1 (3d)

pnj ≥ 0 ∀n ∈ N , j ∈ On (3e)

F (t,p) =∑n∈N

∑j∈On

λtnjpnj .



Gradient descent algorithm

∀n ∈ N , j ∈ On,∂F (t, p̄)

∂tnj= p̄nj +

∑m∈N

∑k∈Om

tmk∂p̄mk

∂tnj.

Leader ProblemCompute descent direction

Follower ProblemCompute new SUE

Set new Tariffs

Compute Derivatives

(SA)

Figure: Gradient descent algorithmCOSMOS Day () Math. program with equilibrium constraint 25 November 2016 16 / 28


Sensitivity Analysis based heuristic

Step 0: Initialization. Determine an initial set of leader variablest0 = (t0

nj)n∈N ,j∈On . Set counter i = 0.

Step 1: Lower-level. Solve the lower-level problem for t i and obtaincorresponding SUE p̄i+1(ti ).

Step 2: Sensitivty analysis. Compute the derivatives∂p̄i+1

nj

∂t imk

,

∀n ∈ N ,m ∈ N , j ∈ On, k ∈ Om at t i using the sensitivity analysismethod.

Step 3: Upper-level. Compute descent direction ∀n ∈ N , j ∈ On

d i+1nj := p̄i+1

nj +∑

m∈N

∑k∈Om

tmk∂p̄i+1

mk

∂t inj.

Step 4: Move. Compute ∀n ∈ N , j ∈ On, ti+1nj = t inj + d i+1

nj .

Step 5: Convergence. A stopping criterion when|F (ti+1, p̄i+1)− F (ti , p̄i )| < ε then stop, else go to step 1 andi = i + 1.



Bi-level Local search Heuristic (BLS)

1 2

3

σ1

t 1

t 2

σ2

σ 3

Figure: Local search progress.



Bi-level Local search Heuristic (BLS)

Step 0: Initialization. Determine an initial set of leader variablest0 = (t0

nj)n∈N ,j∈On . Set counter i = 0.

Step 1: Neighborhood. Build the neighborhood set V i of thesolution ti .

Step 2: Evaluation. At each neighbor v, get the corresponding SUEp̄(v) and evaluate leader objective function F (v, p̄(v)).

Step 3: Selection. Select the best neighborti+1 = argmaxv∈V i F (v, p̄(v)).

Step 4: Convergence. A stop criterion when|F (ti+1, p̄i+1)− F (ti , p̄i )| < ε then stop, else go to step 1 andi = i + 1.



SA based Local search (SLS)

Same neighberhood than BLS.

Evaluation of leader objectif for a neighbour, use the approximatedformulae:

∀n ∈ N , j ∈ On, p̄nj = p̄inj +

∑m∈N

∑k∈Om

∂p̄nj

∂tmk(vmk − t imk). (4)


Numerical example

Plan

1 Introduction



4 Numerical example

5 Conclusion


Numerical example

Numerical example

DCM

Delivery Pick upServices

Options

Figure: Example 1


Numerical example

Numerical example

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

1 2 3 4 5 6 7 8 9 10Tariff of service 21

Figure: Impact of nesting coefficient on the SUE


Numerical example

Numerical example

t 11

t 21

Revenue

Figure: Leader revenue depending on tariffs t11 and t21.


Numerical example

Numerical example

Ex 1: 3 options

GDA SLS BLS

Revenue 89.534 89.571 89.571Tariffs 3.928 5.575 3.862 5.646 3.872 5.661Nb iter 69 34 30

Time (s) 0.636 0.376 1.656

Table: Heuristics comparison

Ex 1: 6 options

GDA SLS BLS

Revenue 78.964 79.311 83.142Nb iter 527 32 24

Time (s) 14 1.3 41

Table: Heuristics comparison with 6 options.


Conclusion

Plan

1 Introduction



4 Numerical example

5 Conclusion


Conclusion

Conclusion

Nested DCM model for customers behavior.

Bi-level model for services design.

Multi-class customer consideration.

Extend leader variables. Discret variables .

F (t,p) =∑n∈N

∑j∈On

λtnjpnj − anjKnj .


Conclusion

[1] Y. Hayel and D. Quadri and T. Jimenez and L. Brotcorne, Decentralizedoptimization of last-mile delivery services with non-cooperative boundedrational customers, Annals of Operations Research, 2014.

[2] Y. SHEFFI, Urban transportation networks: Equilibrium analysis withmathematical programming methods, European Journal of OperationalResearch, 1985

[3] M. Chen and A. S. Alfa, Algorithms for solving fisk’s stochastic trafficassignment model, Transportation Research, 1991.

[4] S. Bekhor and L. Reznikova and T. Toledo, Application of cross-nested logitroute choice model in stochastic user equilibrium traffic assignment,Transportation Research Record, 2003

[5] A. V. Fiacco, Introduction to sensitivity and stability analysis in nonlinearprogramming, New York Academic Press, 1983

[6] H. Yang and S. Yagar and Y. Iida and Y. Asakura, An algorithm for the

inflow control problem on urban freeway networks with user-optimal flows,

Transportation Research, 1994


Last-mile delivery services design under stochastic user...

Documents

Transcript of Last-mile delivery services design under stochastic user...