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# A matheuristic for the Multi-period Electric Vehicle Routing Problem

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## A matheuristic for the Multi-period Electric Vehicle Routing Problem

1

1

2

### , Emmanuel N´eron

1

1 Universit´e de Tours, LIFAT EA 6300, CNRS, ROOT ERL CNRS 7002 64 avenue Jean Portalis, Tours 37200, France

laura.echeverriguzman@etu.univ-tours.fr,{aurelien.froger,emmanuel.neron}@univ-tours.fr

2 HEC Montr´eal

3000 Chemin de la Cˆote-Sainte-Catherine, Montr´eal (Qu´ebec) H3T 2A7, Canada jorge.mendoza@hec.ca

Abstract

The Multi-period Electric Vehicle Routing Problem (MP-E-VRP) consists on designing routes to be performed by a fleet of electric vehicles (EVs) to serve a set of customers over a planning horizon of several periods. EVs are charged at the depot at any time, subject to the charging infrastructure ca- pacity constraints (e.g., number of available chargers, power grid constraints, duration of the charging operations). Due to the impact of charging and routing practices on EVs battery aging, degradation costs are associated with charging operations and routes. The MP-E-VRP integrates EV routing and depot charging scheduling, and has coupling constraints between days. These features make the MP-E-VRP a complex problem to solve. In this talk we present a two-phase matheuristic for the MP- E-VRP. In the first phase it builds a pool of routes via a set of randomized route-first cluster-second heuristics. Routes are then improved by solving a traveling salesman problem. In the second phase, the algorithm uses the routes stored in the pool to assemble a solution to the MP-E-VRP. We discuss computational experiments carried out on small-size instances.

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max

1}

i

0

i

ij

0

ij

0

p

p

p

p

[Ep

p]

k0

m

Pm∈M

m

m

max

m(∆)

m

(3)

i

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[Epi

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pi], 7) the power consumed by all chargers at any time

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### Algorithm 1 describes the general structure of our matheuristic. The procedure starts by entering the sampling phase (lines 1-1) for each period p ≤ |P| −

1. At each iteration

p

:

i =

### p). Then, the algorithm uses a tour splitting procedure (known as

split) to retrieve a set of routes added to the set of routes for period

0p

0p

p

p

custSeqs(Rp

0p

p

tsp

p

p

p

p

Ω.

mpevracsp

(4)

Algorithm 1

1: function

ATHEURISTIC

2: Ω

3:

1

4: for

= 0

=

1do

5: Ω0p

6: while

do

7: for

= 1

=

do

8:

l

9:

p

)

10: Ω0p

0p

split(I, tspp

11:

+ 1

12: end for

13: end while

14:

p

custSeqs(Ω0p

15: Ωp

tsp(CSp

16: Ω

p

17: end for

18:

19: return

20: end function

= 1000.

### [1] N. Christofides and J. E. Beasley. The period routing problem.

Networks, 14(2):237–256, 1984.

### [2] Sekyung Han, Soohee Han, and Hirohisa Aki. A practical battery wear model for electric vehicle charging applications.

Applied Energy, 113:1100–1108, 2014.

### [3] Jorge E Mendoza and Juan G Villegas. A multi-space sampling heuristic for the vehicle routing problem with stochastic demands.

Optimization Letters, 7(7):1503–1516, 2013.

### [4] Christian Prins, Philippe Lacomme, and Caroline Prodhon. Order-first split-second methods for ve- hicle routing problems: A review.

Transportation Research Part C: Emerging Technologies, 40:179–

### Cartagena, July 28-31, 2019

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