Guy Desaulniers

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Vehicle routing for propane delivery

• Guy Desaulniers
• Eric Prescott-Gagnon
• Benoît Bélanger-Roy
• Louis-Martin Rousseau
• Ecole Polytechnique, Montréal

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Overview

• Context
• Problem statement and literature
• MIP-based heuristic
• Large neighborhood search heuristics
• Some computational results
• Future work

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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!

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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
  T1) must be serviced on day T
• Optional customers (safety level reached on
  subsequent days T2 or T3) can be serviced on
  day T if feasible and profitable
• For planning, we assume known quantities to
  deliver

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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

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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
• known demand
• time windows
• service time
• an estimated cost saving if not serviced on day
  T1 or T2

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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

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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

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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

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Problem statement (5)

• 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

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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)

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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

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Overview

• Context
• Problem statement and literature
• MIP-based heuristic
• Large neighborhood search heuristics
• Some computational results
• Future work

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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
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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
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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
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Mathematical model (4)
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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

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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

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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

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Overview

• Context
• Problem statement and literature
• MIP-based heuristic
• Large neighborhood search heuristics
• Some computational results
• Future work

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Large neighborhood search (1)

• Iterative method

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Large neighborhood search (2)

• Iterative method
• Current solution

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Large neighborhood search (3)

• Iterative method
• Current solution
• Destruction

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Large neighborhood search (4)

• Iterative method
• Current solution
• Destruction

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Large neighborhood search (5)

• Iterative method
• Current solution
• Destruction
• Reconstruction

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Large neighborhood search (6)

• New solution

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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

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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

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Destruction

• 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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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

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• Select randomly customers, favoring those
  generating longer detours

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Time operator

• Select randomly a specific time
• Select customers whose possible visiting time is
  closest to selected time

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Random operator

• Select the customers to remove at random

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• Each operator i has an associated value pi
• If operator i finds a better solution pi pi1
• Probability of choosing operator i pi / Sjpj
• pi values are reset to 5 every 100 iterations

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Reconstruction with MIP

1. Fix parts of the current solution
2. Enumerate routes using the neighbor lists and
   respecting the fixed parts of the solution
3. Solve the resulting MIP using Cplex

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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

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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

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Overview

• Context
• Problem statement and literature
• MIP-based heuristic
• Large neighborhood search heuristics
• Some computational results
• Future work

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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

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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

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LNS-MIP vs LNS-TABU (2)
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LNS-MIP vs LNS-TABU (3)

• Small instances (45 customers)
• 10 customers removed per LNS iteration
• 2000 tabu search iterations per LNS iteration

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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

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LNS-Tabu No. of customers removed (2)
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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

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LNS-Tabu No. of LNS iterations (2)
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LNS-Tabu No. of customers (1)

• 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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LNS-Tabu No. of customers (2)
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Overview

• Context
• Problem statement
• MIP-based heuristic
• Large neighborhood search heuristics
• Some computational results
• Future work

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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

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Questions ?
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Tabu search heuristic alone
Medium-sized instances, 800,000 tabu search
iterations