Solving the optimal sequencing of skip collections and deliveries

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Solving the optimal sequencing of skip
collections and deliveries
F. Malucelli, M. Bruglieri (speaker), R.
Aringhieri, M. Nonato t D.E.I.
Politecnico di Milano, Italy D.T.I. Università
di Milano, Italy t D. I. Università di Ferrara,
Italy
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Collection of household recyclable waste is
operated

• by daily door to door pick up service
• by locating dedicated bins on the street network
  (periodic service)
• by establishing few collection centers (isole
  ecologiche) spread in the territory nearby urban
  areas where people bring recyclable waste of any
  kind (skips are emptied on demand)

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Problem description
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Left access container
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Problem description operations

• Users convey waste to their nearest collection
  center (CC) and dispose it into an appropriate
  container
• Once a container is full, a service request is
  issued
• bring the full container to a disposal plant to
  be emptied
• bring back to the CC an empty container of the
  same type
• The swap takes place when the CC is closed the
  removal and substitution of a container may take
  place in different moments and not necessarily in
  this order
• Each vehicle can carry a single container at the
  time

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Problem description optimization aspects

• The containers are owned by the company

A container, once emptied can be reused for
other materials
Several types of containers (left, right, with
compactor)
Maximum duration of a vehicle route
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A problem instance

• Given
• Locations of CCs, disposal plants, depot
• Travel times matrix,
• Additional empty skips at the depot,
• The set of full skips

• the problem consists of ROUTING the vehicles s.t.
• full skips are loaded and brought to a plant to
  be emptied,
• empty skips alike replace the full ones,
• without exceeding maximum route duration T

Goal minimize both the number of vehicles and
the global travelled time
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Differences with other VRP
Capacitated VRP
serve n customers with requests q1, , qn with a
fleet of k vehicles each with capacity Q located
at a depot.
It consists of two indipendent decision
problems
i) clustering (which vehicle to send to which
customer)
ii) routing (in what sequence each vehicle
should visit the customers
assigned to it)
The problem we consider is not a capacitated
VRP the request of each customer saturates
the capacity of the vehicle (1 container at a
time)
The actual capacity of the vehicle is the
maximum duration of routes
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Related problems in the literature
1 C. Archetti, M.G. Speranza, Collection
of waste with single load trucks a real case,
www.eco.unibs.it/dmq/speranza 2
L. Bodin, A. Mingozzi, R. Baldacci, M. Ball,
The Rollon-Rolloff Vehicle Routing Problem,
Transportation Science 34 (3) 271-288 (2000).
1 disposal plant 3 L. De Meulemeester,
G.Laporte, F.V.Louveaux, and F. Semet,
Optimal Sequencing of Skip Collections and
Deliveries, Journal of Operational
Research Society 48, 57-64 (1997). unbounded
number of spare containers
no container circulation
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Vehicle Routing graph construction
Nodes
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Vehicle Routing graph construction
Arcs
Physical graph
Vehicle
VR Graph
Cost of the arc
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Vehicle Routing graph construction
Arcs
Physical graph
Vehicle
VR Graph
Cost of the arc
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Vehicle routing tours and routes
Tour close path on the depot
Alternating sequence of loaded and unloaded arcs
Route sequence of tours
Feasible route route not exceeding maximum
duration time

Feasible solution set of feasible routes
covering all not dummy nodes
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From tours to routes
We heuristically solve the bin packing problem
associated to the assignment of tours to route
through the best fit algorithm
Performing the algorithm we take into account
of possible savings merging two tour
Case 2
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Constructing a feasible solution
Modified Clarke and Wright
1) Saving computation
for each pair (i,j) of compatible
nodes sijcij-ci0-c0j
2) Sort the savings in non increasing order
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3) Greedy phases
Phase 1
consider the savings in the order make the
shortcuts that decrease the infeasibilities
(i.e. decrease the use of spare containers)
i
j
0 spare container
1 spare container
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Lower bounds on the total travel time
Match the savings in the best possbile way 3
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MILP formulation
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LP based bound
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Improving the solution Local Search
12 different types of neighborhoods considering
move inter-tour, move intra-tour
some moves are customization of classical VRP
moves (Van Breedam 94), others have been tailored
for this specific network
String Exchange and Relocation
S1
S1, S2 both odd
r1
r2
S2
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Intra-tour 3-opt
It is a standard 3-opt move for VRP without arc
reversing operated on a single tour restricted to
unloaded arcs. For each triplet of unloaded
arcs a1, a2, a3 ? a single way of reconnecting
the four subsequences
a1
a2
a3
t
Scarr2carr
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Evaluation of a move
The move have impact on routes feasibility
If the move m generates an infeasible tour then
m is refused Else if the move generates an
infeasible route then call best fit obtaining
the route set R EndIF f(m) ?a?A ca - ?a? A-
ca a ?r?R (max (0, ?a?r ca-T))
infeasibility
saving
A is the set of arcs added and A- the set of
arcs removed from the original route set
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Local Search control algorithm
For all spare container ci not used in the
current solution For all tour tj which has a
request compatible with ci do scarr2carr(tj,
ci) While the solution improves do for
h1,,11 do Local Search with neighborhood Nh
The Local Search performs the exhaustive search
inside the neighborhood and selects the best
improvement
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Real instances
Regional area of about 4000 Km2 around Perugia

• 10 collection sites
• 10 types of material
• 6 types of containers
• 3 disposal plants
• 3 available vehicle at a single depot
• Up to 11 service requests

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Results on real cases
CPU times in seconds on a Pentium 2 GHz CPLEX
times in seconds on a biprocessor Xeon 2.8 GHz
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Results on randomly generated instances
Different numbers of available spare
containers T0 none T1 one for each type T2 ? T3
an intermediate number
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Conclusions and future work
Modified CW gives good results restart
procedur (randomization)
More sophisticated LS based procedures
Variable Neighborhood Search
Multidepot case
Multiperiod planning