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DOMinant workshop, Molde, September 20-22, 2009
Guy Desaulniers Eric Prescott-GagnonBenoît Bélanger-Roy
Louis-Martin Rousseau
Ecole Polytechnique, Montréal
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Vehicle routing for propane delivery
DOMinant workshop, Molde, September 20-22, 2009
Context Problem statement and literature MIP-based heuristic Large neighborhood search heuristics Some computational results Future work
2
Overview
DOMinant workshop, Molde, September 20-22, 2009
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!
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Context (1)
DOMinant workshop, Molde, September 20-22, 2009
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
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
Given a day T a set of mandatory customers, each with ▪known demand ▪time windows▪service time
a set of optional customers, each with ▪known demand ▪time windows ▪service time▪an estimated cost saving if not serviced on day T+1 or T+2
DOMinant workshop, Molde, September 20-22, 2009
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
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
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
each mandatory customer is serviced exactly once▪ full delivery
each optional customer is serviced at most once▪ full delivery
the objectives are ▪minimize the number of vehicles (fixed cost)▪minimize the number of drivers (fixed cost)▪minimize total travel costs
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)
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DOMinant workshop, Molde, September 20-22, 2009
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
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DOMinant workshop, Molde, September 20-22, 2009
DOMinant workshop, Molde, September 20-22, 2009
Context Problem statement and literature MIP-based heuristic Large neighborhood search heuristics Some computational results Future work
13
Overview
DOMinant workshop, Molde, September 20-22, 2009
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
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
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
i
r
d
DOMinant workshop, Molde, September 20-22, 2009
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
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
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
Context Problem statement and literature MIP-based heuristic Large neighborhood search heuristics Some computational results Future work
21
Overview
DOMinant workshop, Molde, September 20-22, 2009
Iterative method▪Current solution
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Large neighborhood search (2)
DOMinant workshop, Molde, September 20-22, 2009
Iterative method▪Current solution▪Destruction
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Large neighborhood search (3)
DOMinant workshop, Molde, September 20-22, 2009
Iterative method▪Current solution▪Destruction
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Large neighborhood search (4)
DOMinant workshop, Molde, September 20-22, 2009
Iterative method▪Current solution▪Destruction▪Reconstruction
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Large neighborhood search (5)
DOMinant workshop, Molde, September 20-22, 2009
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
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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
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DOMinant workshop, Molde, September 20-22, 2009
DOMinant workshop, Molde, September 20-22, 2009
Fixed number of customers to remove
Neighborhood operators based on:▪Proximity▪Longest detour▪Time▪Random
Roulette-wheel selection based on performance
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DOMinant workshop, Molde, September 20-22, 2009
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
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DOMinant workshop, Molde, September 20-22, 2009
Select randomly customers, favoring those generating longer detours
ikjkij ccc
DOMinant workshop, Molde, September 20-22, 2009
Select randomly a specific time Select customers whose possible visiting time is closest
to selected time
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DOMinant workshop, Molde, September 20-22, 200935
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
1. Fix parts of the current solution2. Enumerate routes using the neighbor lists and
respecting the fixed parts of the solution3. Solve the resulting MIP using Cplex
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DOMinant workshop, Molde, September 20-22, 2009
1. Fix parts of the current solution◦ Fixed parts are treated as aggregated customers◦ Replenishments are always unfixed
2. Use a tabu search heuristic◦ Seven different move types◦ Tabu list◦ Infeasibility w.r.t. time windows and vehicle capacity allowed
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DOMinant workshop, Molde, September 20-22, 2009
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
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DOMinant workshop, Molde, September 20-22, 2009
Context Problem statement and literature MIP-based heuristic Large neighborhood search heuristics Some computational results Future work
39
Overview
DOMinant workshop, Molde, September 20-22, 2009
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
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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
Small instances (45 customers) 10 customers removed per LNS iteration 2000 tabu search iterations per LNS iteration
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
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DOMinant workshop, Molde, September 20-22, 2009
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
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DOMinant workshop, Molde, September 20-22, 2009
Instances with varying number of customers (150-450) 40% of optional customers 80 customers removed per LNS iteration 400 LNS iterations 2000 tabu search iterations per LNS iteration Tabu list length = 20
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DOMinant workshop, Molde, September 20-22, 2009
DOMinant workshop, Molde, September 20-22, 2009
Context Problem statement MIP-based heuristic Large neighborhood search heuristics Some computational results Future work
50
Overview
DOMinant workshop, Molde, September 20-22, 2009
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
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