This document discusses modelling the periodic vehicle routing problem in an industrial context. It describes using a routing solution for delivery, pick-up, and field service routes over multiple time periods. The problem is modeled as a salesman tour problem that duplicates each point to serve according to its required number of visits. The model extends the vehicle routing problem to duplicate vehicles over the time period and ensure minimum and maximum lapse times between visits. A custom heuristic is described that generates initial routes by prioritizing shorter routes and points with more required visits, then duplicates routes over the time period.

1. Modelling the Periodic Vehicle
Routing Problem in an industrial
context
ISMP 2018 - Bordeaux
GROUPE

2. Introduction
1https://github.com/Mapotemp
Plan
1. Context
2. Specific constraints
3. Custom heuristic

3. Context
1
2
Routes Management Solution
● Delivery - Ecommerce
● Pick-up - Wastes
● Field service - Health care
A ready to use solution

4. Context
1
3
Open Source technological stack
● OpenStreetMap + OSRM
● Ruby + Rails
● OR-Tools
https://github.com/Mapotemp
o

5. Context
1
4
Problem : salesman tours
● Multiple visits to honor for each customer
● From 1 to 4 months of time horizon
● contractual or customer satisfaction based
visit frequency

6. Specific constraints
2
5
VRP model extension
● Duplication of each point, according to its
number of visits
K : set of Vehicles
P : set of days within
the period
N : set of points to
serve
Ti : set of visits of the
point i

7. Specific constraints
2
6
VRP model extension
● Duplicate every vehicle over the entire period
● Vehicles are ordered according to their day
index
K : set of Vehicles
P : set of days within
the period
N : set of points to
serve
Ti : set of visits of the
point i

8. Specific constraints
2
7
Variables
● Ensure that both visits are active
● Get the first vehicle with the right day index
K : set of Vehicles
P : set of days within
the period
N : set of points to
serve
Ti : set of visits of the
point i
Ei- : Minimum lapse
between two visits of i
Ei+ : Maximum lapse
between two visits of i

9. Specific constraints
2
8
Constraints
● Minimum and Maximum lapse between visits
K : set of Vehicles
P : set of days within
the period
N : set of points to
serve
Ti : set of visits of the
point i
Ei- : Minimum lapse
between two visits of i
Ei+ : Maximum lapse
between two visits of i

10. Specific constraints
2
9
Results
● Pretty good constraint propagation
● Few unvisited points
● But, poor local minimum due to inadequate
first solution strategy

11. Custom heuristic
3
10
● 70 customers
● Visits number per customer up to 8
● 167 visits
● 1 depot
● 24 business days

12. Custom heuristic
3
11

13. Custom heuristic
3
12
● Routes order preference
○ Priority for routes with a shorter work duration
● Point priority
○ Order of the point within the Giant tour
○ Priority based on visit number
8 - 4 - 2 - 1

14. Custom heuristic
3
13
Generation procedure
1. Filling first route
● Customers are ordered using an interest metric. As long as there is time left,
customers are inserted.
Interest : Function of giant tour diversion and visits number.
2. Duplicate route pattern
● The route built in the first step is duplicated all over the period regarding each
customer visit frequency

15. Custom heuristic
3
14
First day filled
● 8 visits
● Restricted area
● Currently no reorder phase

16. Custom heuristic
3
15
Customer id Visits Priority minimum lapse Linked days
1 2 12 12
2 8 3 3, 6, 9, 12, 15, 18, 21
3 8 3 3, 6, 9, 12, 15, 18, 21
4 8 3 3, 6, 9, 12, 15, 18, 21
5 2 12 12
6 2 12 12
7 2 12 12
8 2 12 12
9 2 12 12

17. Custom heuristic
3
16
● First day is full
● Every customer must be visited at least twice and have no restriction
● The first day route is duplicated on day 12.
● A subpart of the customers must be visited 8 times.
● A part of the first day route is duplicated on days 3, 6, 9, 15, 18 and 21 in
addition to day 12

18. Custom heuristic
3
17
Next
● Filled routes are processed following their working time left
● Empty days are processed following their working time available
● Generation continue until no more route left
Result
● 0 visit unaffected
● Between 4 and 9 visits a day
● A mean of 6 visits a day

19. Further work
18
● Perform improvement phase instead of only insertion
● Use this generation as starting point for local search (through OR-Tools)
● Handle impossible duplications (D : minimum lapse = 6)
Day 0 1 2 3 4 5 6 7 8
Customers A,B,C A,B,D A,B,C A,B A,B,C
Full? x x x

20. Merci
Gwénaël Rault
Operation Research Developer
&PhD Student
www.linkedin.com/in/rgwenael
www.mapotempo.com
Special thanks
Adeline Fonseca, Frédéric Rodrigo, Philippe Lacomme and Marc Sevaux