1. DOMinant workshop, Molde, September 20-22, 2009
Vehicle routing for propane delivery
Guy Desaulniers
Eric Prescott-Gagnon
Benoît Bélanger-Roy
Louis-Martin Rousseau
Ecole Polytechnique, Montréal
DOMinant workshop, Molde, September 20-22, 2009

2. DOMinant workshop, Molde, September 20-22, 2009
Overview
Context
Problem statement and literature
MIP-based heuristic
Large neighborhood search heuristics
Some computational results
Future work
DOMinant workshop, Molde, September 20-22, 2009

3. DOMinant workshop, Molde, September 20-22, 2009
Context (1)
Ongoing research with Info-Sys Solutions Inc.
Develops software solutions
Information systems
Different applications for propane distributors
Customer accounts
Automatic billing upon delivery
Inventory forecasting
Wants to develop a vehicle routing application
Not familiar with OR  no sophisticated tools such as column generation or Cplex
No real-life data yet!
DOMinant workshop, Molde, September 20-22, 2009

4. DOMinant workshop, Molde, September 20-22, 2009
Context (2)
Propane distributor to residential and commercial customers
One product (propane)
Vendor-managed inventory at customers
Distributor wants to determine vehicle delivery routes for the next day T
Mandatory customers (safety level reached on day T+1) must be serviced on day T
Optional customers (safety level reached on subsequent days T+2 or T+3) can be serviced on day T if feasible and profitable
For planning, we assume known quantities to deliver
DOMinant workshop, Molde, September 20-22, 2009

5. DOMinant workshop, Molde, September 20-22, 2009
Context (3)
Customer opening hours (time windows)
One or several depots
Predetermined driver shifts assigned to depots
Routes must fit into these shifts
Replenishment stations (certain depots and others)
Possible en-route replenishments
DOMinant workshop, Molde, September 20-22, 2009

6. DOMinant workshop, Molde, September 20-22, 2009
Problem statement (1)
Given
a day T
a set of mandatory customers, each with
known demand
time windows
service time
a set of optional customers, each with
an estimated cost saving if not serviced on day T+1 or T+2
DOMinant workshop, Molde, September 20-22, 2009

7. DOMinant workshop, Molde, September 20-22, 2009
Problem statement (2)
set of depots
a set of identical vehicles:
limited capacity
fixed cost per day
variable cost per mile
each assigned to a depot
a set of driver shifts, each
with fixed start and end times
with a fixed cost
assigned to a depot
with an available vehicle
DOMinant workshop, Molde, September 20-22, 2009

8. DOMinant workshop, Molde, September 20-22, 2009
Problem statement (3)
set of replenishment stations (certain depots and others)
replenishment time per visit
travel time matrix between each pair of locations
travel cost matrix between each pair of locations
DOMinant workshop, Molde, September 20-22, 2009

9. DOMinant workshop, Molde, September 20-22, 2009
Problem statement (4)
Find at most one vehicle route for each driver shift
such that
each route starts and ends at the depot associated with the driver
each route respects
time windows of visited customers
vehicle capacity including, if needed, visits to replenishment stations
DOMinant workshop, Molde, September 20-22, 2009

10. DOMinant workshop, Molde, September 20-22, 2009
Problem statement (5)
each mandatory customer is serviced exactly once
full delivery
each optional customer is serviced at most once
the objectives are
minimize the number of vehicles (fixed cost)
minimize the number of drivers (fixed cost)
minimize total travel costs
DOMinant workshop, Molde, September 20-22, 2009

11. DOMinant workshop, Molde, September 20-22, 2009
Literature (1)
Survey on propane delivery by Dror (2005)
Stochastic inventory routing problem
Multi-period (no optional customers)
8 to 10 customers per route
No replenishment stations, driver shifts, time windows
Stochastic model
Federgruen and Zipkin (1984), Dror and Ball (1987),
Dror and Trudeau (1988), Trudeau and Dror (1992)
Markov decision process model
Kleywegt et al. (2002, 2004)
Adelman (2003, 2004)
DOMinant workshop, Molde, September 20-22, 2009

12. DOMinant workshop, Molde, September 20-22, 2009
Literature (2)
Petrol stations replenishment
Multiple products
Multiple tank vehicles
1 to 3 customers per route
No replenishment stations, driver shifts, optional customers
Cornillier et al. (2008), Avella et al. (2004)
One period, exact and heuristic
Cornillier et al. (2008)
Multi-period (2 to 5 days), heuristic
Cornillier et al. (2009)
One period, time windows, heuristic
DOMinant workshop, Molde, September 20-22, 2009

13. DOMinant workshop, Molde, September 20-22, 2009
Overview
Context
Problem statement and literature
MIP-based heuristic
Large neighborhood search heuristics
Some computational results
Future work
DOMinant workshop, Molde, September 20-22, 2009 13 13

14. DOMinant workshop, Molde, September 20-22, 2009
Mathematical model (1)
D : set of all drivers
R : set of all routes feasible for driver d, must respect
driver shift
customer time windows
vehicle capacity (with replenishments if necessary)
start and end at the driver depot
c : cost of route r which includes
driver fixed cost
travel cost
a : equal to 1 if route r visits customer i, 0 otherwise d r ri
DOMinant workshop, Molde, September 20-22, 2009

15. DOMinant workshop, Molde, September 20-22, 2009
Mathematical model (2)
M : set of all mandatory customers
O : set of all optional customers
s : estimated cost saving if servicing optional customer i, which is computed as
an average of the detours incurred for servicing it in between neighbor customers on the day it should become mandatory
H : set of time points (start and end times of shifts) where we count the number of vehicles used i
DOMinant workshop, Molde, September 20-22, 2009

16. DOMinant workshop, Molde, September 20-22, 2009
Mathematical model (3) d
Y : binary variable equal to 1 if route r is assigned to driver d and 0 otherwise
E : binary slack variable equal to 1 if optional customer i is not serviced and 0 otherwise
V : integer variable counting the number of vehicles used r i
DOMinant workshop, Molde, September 20-22, 2009

17. DOMinant workshop, Molde, September 20-22, 2009
Mathematical model (4)
DOMinant workshop, Molde, September 20-22, 2009

18. DOMinant workshop, Molde, September 20-22, 2009
MIP-based heuristic
Model (1)-(6) requires the enumeration of all feasible routes for all drivers
Impractical for real-life instances
We limit route enumeration and solve the restricted model (1)-(6) using the Cplex MIP solver
DOMinant workshop, Molde, September 20-22, 2009

19. DOMinant workshop, Molde, September 20-22, 2009
Route enumeration (1)
For each customer, find the nearest neighbor customers (2 or 3)
For each customer, find the nearest replenishment stations (1 or 2)
For each station, find the nearest customers (10 to 15)
For each customer, insert it into the neighbor list of its nearest stations
DOMinant workshop, Molde, September 20-22, 2009

20. DOMinant workshop, Molde, September 20-22, 2009
Route enumeration (2)
For each driver, enumerate all feasible routes that respect
location i can be visited after location j if i belongs to the neighbor list of j
a replenishment station cannot be visited unless the vehicle is less than half full
For all customers that belong to less than 10 routes, create additional routes by inserting the customer into existing routes
DOMinant workshop, Molde, September 20-22, 2009

21. DOMinant workshop, Molde, September 20-22, 2009
Overview
Context
Problem statement and literature
MIP-based heuristic
Large neighborhood search heuristics
Some computational results
Future work
DOMinant workshop, Molde, September 20-22, 2009 21 21

22. DOMinant workshop, Molde, September 20-22, 2009
Large neighborhood search (1)
Iterative method
DOMinant workshop, Molde, September 20-22, 2009

23. DOMinant workshop, Molde, September 20-22, 2009
Large neighborhood search (2)
Iterative method
Current solution
DOMinant workshop, Molde, September 20-22, 2009

24. DOMinant workshop, Molde, September 20-22, 2009
Large neighborhood search (3)
Iterative method
Current solution
Destruction
DOMinant workshop, Molde, September 20-22, 2009

25. DOMinant workshop, Molde, September 20-22, 2009
Large neighborhood search (4)
Iterative method
Current solution
Destruction
DOMinant workshop, Molde, September 20-22, 2009

26. DOMinant workshop, Molde, September 20-22, 2009
Large neighborhood search (5)
Iterative method
Current solution
Destruction
Reconstruction
DOMinant workshop, Molde, September 20-22, 2009

27. DOMinant workshop, Molde, September 20-22, 2009
Large neighborhood search (6)
New solution
DOMinant workshop, Molde, September 20-22, 2009

28. DOMinant workshop, Molde, September 20-22, 2009
LNS heuristics
Initial solution
Destruction
A roulette-wheel selection of four operators
Reconstruction
MIP-based heuristic (LNS-MIP) or
Tabu search heuristic (LNS-Tabu)
Stopping criterion: maximum number of iterations
DOMinant workshop, Molde, September 20-22, 2009

29. DOMinant workshop, Molde, September 20-22, 2009
Initial solution
Greedy algorithm
For each driver shift
Build a route starting from the beginning of the shift
Select the mandatory customer that can be reached at the earliest
Add this customer to the route if enough capacity and it is possible to return to the depot before the shift end time
Insert a replenishment to the nearest station when necessary
Return to the beginning of the loop (next customer)
No optional customers are serviced
DOMinant workshop, Molde, September 20-22, 2009

30. DOMinant workshop, Molde, September 20-22, 2009
Destruction
Fixed number of customers to remove
Neighborhood operators based on:
Proximity
Longest detour
Time
Random
Roulette-wheel selection based on performance
DOMinant workshop, Molde, September 20-22, 2009

31. Proximity operator (Shaw, 1998)
Select randomly a customer i
Order the remaining customers according to their proximity (in distance) to i
Select randomly a new customer i’ favoring those having a greater proximity
Select each subsequent customer according to its proximity to an already selected customer, which is chosen at random
DOMinant workshop, Molde, September 20-22, 2009

32. DOMinant workshop, Molde, September 20-22, 2009
Select randomly customers, favoring those generating longer detours
DOMinant workshop, Molde, September 20-22, 2009

33. DOMinant workshop, Molde, September 20-22, 2009
Time operator
Select randomly a specific time
Select customers whose possible visiting time is closest to selected time
DOMinant workshop, Molde, September 20-22, 2009

34. DOMinant workshop, Molde, September 20-22, 2009
Random operator
Select the customers to remove at random
DOMinant workshop, Molde, September 20-22, 2009

35. DOMinant workshop, Molde, September 20-22, 2009
Each operator i has an associated value πi
If operator i finds a better solution: πi= πi+1
Probability of choosing operator i = πi / Σjπj
πi values are reset to 5 every 100 iterations
DOMinant workshop, Molde, September 20-22, 2009

36. Reconstruction with MIP
Fix parts of the current solution
Enumerate routes using the neighbor lists and respecting the fixed parts of the solution
Solve the resulting MIP using Cplex
DOMinant workshop, Molde, September 20-22, 2009 36

37. Reconstruction with tabu (1)
Fix parts of the current solution
Fixed parts are treated as aggregated customers
Replenishments are always unfixed
Use a tabu search heuristic
Seven different move types
Tabu list
Infeasibility w.r.t. time windows and vehicle capacity allowed
DOMinant workshop, Molde, September 20-22, 2009 37

38. Reconstruction with tabu (2)
Move types
Move a customer from one route to another
Remove an optional customer from a route
Insert an optional customer into a route
Insert a visit to a replenishment station into a route
Remove a visit to a replenishment station from a route
Change the location of a replenishment
Exchange the routes of two drivers
All applied at each iteration
DOMinant workshop, Molde, September 20-22, 2009 38

39. DOMinant workshop, Molde, September 20-22, 2009
Overview
Context
Problem statement and literature
MIP-based heuristic
Large neighborhood search heuristics
Some computational results
Future work
DOMinant workshop, Molde, September 20-22, 2009 39 39

40. Randomly generated instances
1 depot, 8-hour driver shifts for all instances
Two different sizes (5 instances for each)
45 customers: 25 mandatory and 20 optional (small)
average of 3.3 drivers used
average of 8.9 optional customers serviced
average of 10.4 customers per route
250 customers: 150 mandatory and 100 optional (medium)
average of 5.5 drivers used
average of 22.9 optional customers serviced
average of 31.4 customers per route
5 runs for each instance
DOMinant workshop, Molde, September 20-22, 2009

41. LNS-MIP vs LNS-TABU (1) Small instances (45 customers)
10 customers removed per LNS iteration
2000 tabu search iterations per LNS iteration
Comparison of the performance of MIP vs Tabu for same neighborhoods
Percentage of improvement w.r.t. to current solution value

42. LNS-MIP vs LNS-TABU (2)

43. LNS-MIP vs LNS-TABU (3) Small instances (45 customers)
10 customers removed per LNS iteration
2000 tabu search iterations per LNS iteration

44. LNS-Tabu: No. of customers removed (1)
Medium-sized instances (250 customers)
400 LNS iterations
2000 tabu search iterations per LNS iteration
Tabu list length = 25% of no. of customers removed
Varying number of customers removed per LNS iteration
DOMinant workshop, Molde, September 20-22, 2009

45. LNS-Tabu: No. of customers removed (2)
DOMinant workshop, Molde, September 20-22, 2009

46. LNS-Tabu: No. of LNS iterations (1)
Medium-sized instances (250 customers)
80 customers removed per LNS iteration
Fixed total of tabu search iterations = 800,000
Tabu list length = 20
Varying number of LNS iterations
Approx.: 580 seconds
DOMinant workshop, Molde, September 20-22, 2009

47. LNS-Tabu: No. of LNS iterations (2)
DOMinant workshop, Molde, September 20-22, 2009

48. LNS-Tabu: No. of customers (1)
Instances with varying number of customers ( )
40% of optional customers
80 customers removed per LNS iteration
400 LNS iterations
2000 tabu search iterations per LNS iteration
Tabu list length = 20
DOMinant workshop, Molde, September 20-22, 2009

49. LNS-Tabu: No. of customers (2)

50. DOMinant workshop, Molde, September 20-22, 2009
Overview
Context
Problem statement
MIP-based heuristic
Large neighborhood search heuristics
Some computational results
Future work
DOMinant workshop, Molde, September 20-22, 2009 50 50

51. DOMinant workshop, Molde, September 20-22, 2009
Future work
Perform tests on real-life instances
Improve LNS-Tabu
Develop a two-phase method
Minimize number of vehicles and drivers in the first phase
Minimize total travel costs in the second phase
Develop LNS-column generation
Generalize to oil products
Multiple products
Heterogeneous fleet of vehicles
Vehicles with several compartments
DOMinant workshop, Molde, September 20-22, 2009

52. DOMinant workshop, Molde, September 20-22, 2009
Questions ?
DOMinant workshop, Molde, September 20-22, 2009

53. Tabu search heuristic alone
Medium-sized instances, 800,000 tabu search iterations
DOMinant workshop, Molde, September 20-22, 2009