An Efficient Large Neighborhood
Search based Matheuristic for Rich
Vehicle Routing Problems
S im ona M ancini
DIST, Politecnico di Torino, Torino, Italy
Outlines
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VRP and its variants
Real life VRPs
Problem description
Mixed Integer Programming formulation
Large Neighborhood based matheuristic
Computational results
Conclusions and future perspectives
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The Vehicle Routing
Problem
Further constraints:
• Limited number of vehicles
• Capacity
• Route length
• Route duration
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VRP variants
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VRP with Time Windows
VRP with Pick-up and Delivery
VRP with backhauls
VRP with heterogeneus fleet
Multi Depot VRP
Multi-Period VRP
Periodic VRP
VRP with Stochastic Demands
Dynamic VRP
….
Multi Echelon VRP (Two-Echelon, Intermediate Facilities, Location routing,
Truck and Trailer..)
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A complex real-life VRP
VRP
Heterogeneous
Fleet
Multi-Depot
MDMPVRPHF
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Multi-Period
Heterogeneus VRP
Real-life problems
Studied in literature
• Capacity
• Usage cost
• Kilometric cost
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vehicle characteristics and
compatibility with customers
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Refrigerated
Freezer
Armoured
• Autonomy
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smaller for electric vehicles
• Fuel consumption rate
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Larger for big trucks
Multi-depot VRP
Routes start from and end at the
same depot
Can be split into an assignment
problem plus several VRPs which
can be solved separately
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Real life Multi-Depot
Routes may end at a different
depot form the starting one if it is
convenient
More complex to model
Cannot be split into
assignment+routing
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Multi-period VRP
• Works with a planning horizon of T days
• Each customer has frequency of visit requirements (k out of T
days)
• Visits to customers must occor on allowed k-days
combinations
• Decision made for one period impact outcomes in other
periods
Assign visits schedule to each customer
Solve a vehicle routing problem for each day
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Real life multi-period I
Distribution
• Grocery and soft drink distribution
• Fuel oil and industrial gas delivery
• Pharmacies supply
• Automobile parts distribution
Waste Collection
Maintenance services
Surveillance patrols
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Real life multi-period II
• What does matter is not only the period in which a customer
is served, but also the moment within that period, in which
the delivery is performed
• In a multi-depot multi-period problem the fleet available at
each depot may change across the periods, if vehicles do not
come back to their starting depot
Visits Scheduling and Routing cannot be treated separately anymore,
since the input data for a VRP related to a given period,
depends on the routing plan of the previous ones
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MDMPVRPHF I
• No multiple visits
• Customers may be served in one of the periods in which they
are available, taking into account
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Days off
Good available only after a given period
• Multi depot where routes may start from and end at different
depots
• Depending on products availabilty, not all the customer may
be served by all the depots (may be bypassed adding fictitious
huge distance between customers and not available depots)
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MDMPVRPHF II
• Heterogeneous fleet
 Vehicles initially located at a depot
 Variable capacity, usage cost, cost/km
 Characteristics (refrigerated, freezer..) and compatibility
with customers
 Product request (perishable food, ice cream..)
 Customer location limitation (large trucks cannot enter city
center..)
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Problem definition I
• The problem consist of serving a set of customers, I, at the
minimum cost.
• Each customer i requires a quantity of goods qi, which can be
delivered from one of more depots d belonging to the set of depots
D, depending on the availability of the requested products at the
depots.
• Delivery may be carried out by vehicles which are compatible with
customers request, during a time-slot (day) in which the customer is
available.
• Further temporal constraint may be given by
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strictly deadlines imposing that customer i must receive his order
before day s
goods availability (in some cases goods may be available only after
day s).
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Problem definition II
• A set of vehicles V is located at the depots.
• For each vehicle v is known the capacity Cv,
the cost per min of usage μv, and the depot d
where the vehicle is located.
• A set of possible routes K is given, where for
each route k is known
 the vehicle v to which it is associated
 the day s in which it is scheduled.
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Problem definition III
• A customer may be assigned to a route only if it is
performed
• A vehicle v is supposed to be located at depot d
on day s (for s > 1) if d is the arrival depot of the
routes performed by v on day s - 1
• If v has not been used in day s - 1 he is supposed
to be located at the arrival depot of its last
performed route, or if it has not been used yet, at
the depot where it was located at the beginning
of the time-horizon.
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Input Data I
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Input data II
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Input Data III
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Variables
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Objective function
Cost minimization
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Assignment constraints I
A customer can be
visited by a route only
if it has been assigned
to it
Each customer must be
assigned to one and only one
route
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Assignment constraints II
Customer-depot
compatibility
Each route, if
performed must start
and end at a depot
Customers may be
assigned to a route
only if is performed
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Constraints I
Subtours elimination
Capacity not exceeded
Route duration
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Constraints II
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Constraints allowing to determine the starting depot for each routes, depending
on the vehicle and the routes performed by it on the previous days.
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Route Compatibility Constraints
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Variables domain
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Classical Local Search heuristics
• Small neighborhood search often remain
trapped into local minima
• Only very limited regions of the solution space
are reached
• Multi-start methods allow to explore different
regions but we have no guarantees that we
did not discard promising areas
• Too sensitive to the diversification process
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Large Neighborhood Search I
• Large neighborhoods allow to explore wider
regions of the solution space
• Exponential number of solutions
• Analyze the neighborhood is strongly time
consuming
• Only a subset of the solutions is actually
analyzed
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Large Neighborhood Search II
• Neighborhoods are implicitly defined
• Destroy and repair operators are applied
 Destroy operators
 Route removal
 Depot closure
 Remove k% of the customers form the solution
 Remove k% longest arcs
 Random selection vs quality criteria based
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Large Neighborhood Search III
• Repair operators:
 Optimal
 Shortest path
 Network-flow based improvement algorithms
 Explicitly enumeration (time-consuming)
 Heuristic
Best insertion
 Auction insertion
 Truncated Branch and Bound
Further considerations
• Too Small destroy operators yields to small
neighborhoods
• Too Large destroy operators may yield a loss of
good information and the reconstructed
solution may be worse than the previous one
• Simply repair operators may yield to poor
quality solutions
• Complex repair operators may be very time
consuming
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A LNS based Matheuristic I
• General idea
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Exahustively exploit the whole neighborhood in a short
computational time
High quality solution
Add at each iteration a set of constraints which force the largest part
of the variables to assume the same values they assumed in the
current solution
Let the model solve the ultra-constrained problem and to determine
the optimal value for the other variables
This correspond to solve to the optimality an exponential
neighborhood
The ultra-constrained problem may be solved to the optimality in
short computational times
At each iteration a different subset of the variables is kept fixed
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A LNS based Matheuristic II
• Working on the customer-to-route assignment
variables Yik
• A initial feasible solution is computed
• At each iteration p customers are randomly
selected
• All the others N-p customers are assigned to the
same route to which they were assigned in the
initial solution
• The p selected ones assignment is let free
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A LNS based Matheuristic III
• This can be exploited adding the following set
of constraints to the model
• The ultra-constrained model is run and solved
to the optimality
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A LNS based Matheuristic IV
• The new obtained solution is the best one in
the exponential neighborhood
• This solution is kept as initial solution for the
following iteration
• The algorithm terminates after
 a maximum number of iterations (MAXITER) is
reached
 a maximum number of iterations without
improvements (MAXNOIMPROVE) is reached
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Further considerations I
• The parameter p can be aritbrarly chosen
keeping in mind that small values of p allow a
limited perturbation with the risk to remain
trapped into local minima, while very big
values generate such a large neighborhood
search which cannot be easily exhaustively
explored in a short computational
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Further considerations II
• This procedure need an initial feasible solution form
which to start which is computed letting the model run
for TIMELIMIT seconds
 enough time to find a feasible solution
• The initial solution quality is not a crucial issue,
because, due to the strong diversication inserted in the
algorithm, the matheuristic is capable to explore
regions potentially far, in the solutions space, from the
starting point and to converge to a very good solution
even starting from a poor quality one.
This is a strong good point!
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Computational tests I
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Number of customers: 30
Number of depots: 3
Number of vehicles: 6
Number of days: 5
Number of routes: number of vehicles *
number of days = 30
• Maximum route duration: 11 hours
• Average speed 80 Km/h
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Instances Description I
• 9 instances with different combination of customers-vehicles
compatibility and customers availability levels
• Food delivery is generally characterized by a low compatibility
and a high availability
• Electronic commerce (like amazon..) is characterized by
complete compatibility and low availability
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Instances Description II
• Three customers-vehicles compatibility levels:
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High compatibility: around 90% of customer-vehicle assignments are
feasible
Medium compatibility: around 80% of customer-vehicle
assignments are feasible
Low compatibility: around 60% of customer-vehicle assignments are
feasible
• Three customers availability levels:
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High availability: around 95% of customer-day assignments are
feasible
Medium availability: around 70% of customer-day assignments are
feasible
Low availability: around 30% of customer-day assignments are
feasible
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Model results
Time-limit 3600 sec – corei7 at 1.8 Ghz and 8 Gb of RAM
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Comparison LNS vs MODEL
Average results are very similar to best results
Robustness of the solution
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Comparison LNS vs INITIAL SOL
Averaged computational times for the MH are around 200 seconds
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Conclusions I
• The MH is robust
• It is able to strongly improve the initial solution
(25%)
• It is able to sensibly improve (5-6%) the MODEL
results strongly reducing computational times
• The MODEL performs better on highly
constrained problems, therefore the
improvement with MH is smaller because we
compare its results with an high quality solution
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Future Developments I
• Extremely portable method
• Application to other rich VRPs
 Multi-Depot
 Two-Echelon VRP
• Can be easily embedded in a more complex
metaheuristic framework
 VNS
 ALNS
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Future Developments II
• The same approach can be used for other kind
of neighborhoods
 Arcs variables fixing
 Route destruction
 Depote closure
• And for other kind of problems
 Scheduling
 Packing
 Other problems working with binary variables
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Thank you for the
kind attention
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