Improved Route Planning And Scheduling Of Waste Collection And Transport

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Improved route planning and scheduling of waste
collection and transport
Teemu Nuortioa,
*, Jari Kytöjokib
, Harri Niskaa
, Olli Bräysyb
a
Department of Environmental Sciences, University of Kuopio, P.O. Box 1627, FI-70211 Kuopio, Finland
b
Agora Innoroad Laboratory, Agora Center, University of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland
Abstract
The collection of waste is a highly visible and important municipal service that involves large expenditures. Waste collection problems are,
however, one of the most difficult operational problems to solve. This paper describes the optimization of vehicle routes and schedules for
collecting municipal solid waste in Eastern Finland. The solutions are generated by a recently developed guided variable neighborhood
thresholding metaheuristic that is adapted to solve real-life waste collection problems. Several implementation approaches to speed up the
method and cut down the memory usage are discussed. A case study on the waste collection in two regions of Eastern Finland demonstrates
that significant cost reductions can be obtained compared with the current practice.
q 2005 Published by Elsevier Ltd.
1. Introduction
The collection of municipal solid waste is one of the most
difficult operational problems faced by local authorities in
any city. In recent years, due to a number of cost, health, and
environmental concerns, many municipalities, particularly
in industrialized nations, have been forced to assess their
solid waste management and examine its cost-effectiveness
and environmental impacts, e.g. in terms of designing
collection routes. During the past 15 years, there have been
numerous technological advances, new developments and
mergers and acquisitions in the waste industry. The result is
that both private and municipal haulers are giving serious
consideration to new technologies such as computerized
vehicle routing software.
This paper describes a study of planning vehicle routes
for the collection of municipal solid waste in two different
regions in Eastern Finland. In the past, the design of the
collection routes has been done manually. The real-life
waste collection problem under consideration can be
described as follows. Waste is located in containers along
the streets of a defined road network. The containers must be
collected by a fleet of compactor trucks whose capacity
cannot be exceeded. The considered collection areas
involve both heavily and thinly populated areas. There are
in total approximately 30,000 waste bins in the studied
region. Different types of municipal solid waste are
collected separately, and furthermore there are separate
containers for each waste type. Each vehicle can typically
collect refuse from several hundreds of waste bins before
going to the waste disposal site to unload. Only one type of
waste can be collected simultaneously by each vehicle.
There are various bin types and only deep collection
containers and pressing containers are collected separately.
Waste must be collected during workdays; collection on
Saturdays and Sundays is typically forbidden. The vehicle
leaves the depot at the start of the day and must return there
before the end of the day. At the end of the day, the vehicle
is unloaded at the waste disposal site before returning to the
depot in case more than 50% of the vehicle capacity is in
use. If the 8-h working day length is exceeded, an overtime
pay is incurred. A lunch break of 30 min splits the 8-h
working day into two equal halves. Thus, two different tours
can be operated for each collecting day; one in the morning
and one in the afternoon. In addition, the drivers have two
15-min coffee breaks during the workday. There are a
limited number of identical vehicles available with capacity
of 26 tons to collect the mixed waste in the considered two
regions, and each tour is served by one vehicle.
The waste bins and waste disposal site have given time
windows in which they must be visited. More precisely,
Expert Systems with Applications xx (xxxx) 1–10
www.elsevier.com/locate/eswa
0957-4174/$ - see front matter q 2005 Published by Elsevier Ltd.
doi:10.1016/j.eswa.2005.07.009
* Corresponding author.
E-mail address: teemu.nuortio@uku.fi (T. Nuortio).
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the waste disposal site is open from Monday 06:00 to Friday
19:00 and all collection activities must be done within 06:
00–22:00. The visits to the waste containers are assumed to
follow a given cyclic pattern such that the collections are
done similarly within different periods. Collection period
and container type are defined by the customer, but the
actual waste collection is scheduled within the specified
collection period by the waste management company. The
visits to same waste containers in consecutive days are
forbidden if waste must be collected twice per week or less
frequently.
Reliable information on the mass and volume of waste in
each container is not available. The amount of municipal
solid waste is highly variable and the accumulation of waste
depend on several factors such as the number of inhabitants
sharing a container, GDP per capita, lifestyle, time of the
year, etc. Therefore, the considered waste collection
problem is stochastic by nature. In this study, the average
accumulation rate of waste in each container type is
estimated separately using the historical weight and route.
For more information on estimating the waste mass and
volume, we refer to Nuortio, Niska, Hiltunen, and
Kolehmainen (2004). The time to unload a vehicle at the
waste disposal site as well as the time to empty each waste
bin is estimated based on historical data using regression
analysis. Historical data is also used in estimating the travel
times between the waste collection points. For more details,
see Nuortio et al. (2004). The objective is to schedule the
collection activities and to minimize the total distance
traveled by the collection vehicles within each day.
The above described problem can be viewed as a
Stochastic Periodic Vehicle Routing Problem with Time
Windows and a limited number of vehicles (SPVRPTW).
The basic Vehicle Routing Problem (VRP) is one of the
most widely studied problems in combinatorial optimiz-
ation. The objective is to route the vehicles (one route per
vehicle, starting and ending at the depot), so that all
customers are supplied with their demands and the total
travel distance is minimized. The Stochastic VRP (SVRP)
arises when some of the elements of the problem are
stochastic. The stochastic component could, for instance, be
uncertain travel times, unknown demands and the existence
of the customers. The general solution approach for SVRP is
to generate an a priori solution that has the least cost in the
expected sense. For more information on SVRP, we refer to
Dror, Laporte, and Trudeau (1989); Gendreau, Laporte, and
Sèguin (1996); Bräysy, Gendreau, Hasle, and Løkketangen
(2004). The periodic VRP (PVRP) can be defined as the
problem of finding routes for all days of a given T-day
period. It is often called allocation-routing problem because
of the need to assign customers to days in addition to
assigning customers each day to vehicles. Although, the
PVRP has been used in many applications, it has not been
extensively studied in related literature. For a recent
literature review, see Bräysy et al. (2004). The VRP with
Time Windows (VRPTW) has been studied extensively
during the past decade. In VRPTW, some or all customer
points have defined time windows in which they must be
visited, e.g. 08:00–10:00 in the morning. The time windows
can be also soft. In that case, the time windows can be
violated against a given penalty cost. For more information
on VRPTW, we refer to recent extensive two-part survey
paper by Bräysy and Gendreau (2005a,b). In the literature,
the size of the vehicle fleet is often part of the optimization
problem. However, in practice, a transportation company
usually has a given fixed number of vehicles, and the goal is
to use them as effectively as possible. This problem is called
in the scientific literature as the VRP with limited number of
vehicles. So far, there has been only a little research on this
topic. The interested reader is referred to Lau, Sim, and Teo
(2003).
In the vehicle routing problems discussed above, the
demand occurs at the nodes of the defined graph. However,
waste collection problems are typically modeled as Arc
Routing Problems (ARPs). In ARPs, the aim is to determine
a least cost traversal of defined edges or arcs of a graph,
subject to some side constraints. Compared to more
common node routing problems, customers are here
modelled as arcs or edges. For excellent surveys on ARPs,
see Eiselt, Gendreau, and Laporte (1995a,b), Assad and
Golden (1995); Dror (2000). The ARP is a natural way to
model waste collection problems in cases where most or all
bins along a given street segment must be collected at the
same time, and most of the street segments must be
traversed by the collection vehicle as is the case in densely
populated city areas. However, here we decided to use the
node routing approach. The rationale is that in our case the
waste management company must also consider large thinly
populated areas. Another argument is that a more detailed
modeling is possible with node routing. Through node
routing, it is possible to consider each bin separately and
provide a more detailed solution. As the waste management
company has detailed data on the waste accumulation levels
in each bin, exact stop points of the vehicles to collect each
bin, and there are several different types of waste with
different collection intervals, we considered node routing
more appropriate.
As this problem is computationally very hard, and cannot
be solved by optimal (exact) methods in practice, heuristics
are used for this purpose. The problem is solved by the
Guided Variable Neighborhood Thresholding (GVNT)
metaheuristic of Kytöjoki, Nuortio, and Bräysy (2004).
The obtained computational results were validated by
comparing the generated solutions with the current routes
of the waste management company. The main contribution
of this paper lies in presenting the conceptual solution
model and successful real-life application of the GVNT
metaheuristic, including a number of implementation
guidelines for designing efficient solution methods for
very large-scale practical routing problems. It is shown that
significant improvements can be obtained. It is expected that
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the practical implementation of the proposed model will
provide more information for its improvement.
The remainder of this paper is organized as follows.
Section 2 gives a brief review of the recent research on
applying intelligent systems to waste collection. The
proposed solution model and its input requirements are
presented in Section 3, whereas the heuristic routing and
scheduling methodologies are explained in Section 4. Our
test data and computational results are described in
Section 5. The main conclusions of the study are provided
in Section 6.
2. Related work
The use of computers to solve waste collection problems
has been reported by various researchers during the last
15 years.
Ong, Goh, and Poh (1990) develop a heuristic route first-
cluster second approach for optimizing the refuse collection
in Singapore. Janssens (1993) describes a mathematical
model to determine the optimal vehicle fleet size for
collecting waste oil in Antwerp in Belgium. The model
consists of demand estimation and a collection model that
estimates the number of routes and required travel times.
Alvarez-Valdez et al. (1993) suggest a strategy where the
dead-heading of the vehicles is minimized by allowing some
containers to be moved by the workers to the nearest
junction.
Kulcar (1996) reports a case study on optimizing solid
waste collection in Brussels. A methodology is developed to
illustrate how waste transportation costs can be minimized
in an urban area. In addition, several means of transportation
including transportation by vehicle, rail and canal are
evaluated. Eisenstein and Iyer (1997) investigate the
scheduling of garbage trucks in the city of Chicago and
develop a flexible routing strategy with regard to the number
of visits to the waste disposal site. A dynamic solution
method based on Markov decision process is presented.
Chang, Lu, and Wei (1997) describe a mixed-integer
programming model for routing and scheduling of solid
waste collection trucks. The model is integrated with a GIS
environment and an interactive approach, and a case study
in Taiwan is presented. Smith Korfmacher (1997) presents a
case study on designing solid waste collection for urban
areas in South Africa and discusses a number of strategies
for arranging the collection operations.
Bommisetty, Dessouky, and Jacobs (1998) consider the
problem of collecting recyclable materials in a large
university campus. The problem is modeled as a periodic
VRP, and a heuristic two-phase solution method is
suggested. Tung and Pinnoi (2000) address the waste
collection activities in Hanoi, Vietnam. The underlying real-
life vehicle routing and scheduling problem is formulated as
a mixed integer program, and a hybrid of standard VRP
construction and improvement heuristics is proposed for its
solution. Mourão (2000) used a route first-cluster second
approach where a giant tour is generated first, and then
decomposed with a lower-bounding method into a set of
routes that are feasible with regard to the vehicle capacity.
Bodin, Mingozzi, Baldacci, and Ball (2000) study so called
rollon–rolloff VRP faced by a sanitation company. In the
rollon–rolloff VRP tractors move large trailers between
locations and a disposal facility, one at a time. The authors
present a mathematical programming model and four
heuristic algorithms for the problem. The same problem is
studied earlier in De Meulemeester et al. (1997).
Shih and Chang (2001) develop a two-phase approach for
routing and scheduling the collection of infectious medical
waste from a set of hospitals. In the first phase, a standard
VRP is solved by a dynamic programming method while the
second phase uses a mixed integer programming method to
assign routes to particular days of the week. Golden, Assad,
and Wasil (2001) give a short review and analysis of the
real-life applications. Baptista, Oliveira, and Zúquete
(2002) present a case study on collection of recycling
paper in Portugal. The problem is modeled as periodic VRP
and a heuristic approach is presented that consists of initial
assignment of collection tasks to days, and interchange
moves to improve the solution. Minciardi, Paolucci, and
Trasforini (2003) describe heuristic strategies to plan
routing and scheduling of vehicles for very large-scale
solid waste collection that takes place at district level
instead of municipal level. A case study at Geneva, Italy is
presented. Teixeira, Antunes, and de Sousa (2004) study the
planning of real-life vehicle routes for the collection of
different types of urban recyclable waste in Portugal.
Heuristic techniques were developed to define the geo-
graphic collection zones, waste types to be collected on each
day and vehicle routes. Koushki, Al-Duaij, and Al-Ghimlas
(2004) present a case study to evaluate the efficiency of
municipal solid waste collection in Kuwait. Several
indicators to measure the effectiveness are proposed and
discussed, and a comparative analysis of the collection costs
is reported. Amponsah and Salhi (2004) describe a
constructive look-ahead heuristic with tailored improve-
ment mechanisms that are specifically designed for
collecting garbage in developing countries. Aringhieri,
Bruglieri, Malucelli, and Nonato (2004) study the real-life
collection and disposal of special waste such as glass, metal
and food. The special waste is collected from containers at
collection centers instead of each household. Thus, the
problem can be modeled as the rollon–rolloff VRP.
Standard heuristic construction and improvement pro-
cedures as well as lower bounding procedures are presented.
3. Model
In this section, we describe the developed solution
model. First, the input requirements of the model are listed.
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Then, the conceptual model illustrating the considered
problem and the used solution approach is presented.
3.1. Input requirements
Inputs for the developed solution method are organized
into six data files. These are:
(a) Bins. This file contains information on the waste bins.
For each bin the ID, location (X- and Y-coordinates),
associated stop point ID of the vehicle, waste type,
mass and volume of the waste, service time, required
time interval for collection, previous collection date
and associated waste disposal site are listed.
(b) Facilities. This file contains information on the depot
and waste disposal sites. For each site, the file lists ID
(equals vehicle stop point ID), location (X- and Y-
coordinates), facility type (depot or waste disposal
site), service time and operating time windows.
(c) Vehicles. This file contains information on the
available vehicle fleet. For each vehicle, the ID number
and volume and weight capacities for different waste
types are given.
(d) Stoppoints. This file lists all registered stop points for
the vehicles based on historical data. A stop point is a
location where the vehicle is parked for loading or
unloading. Often several waste bins are collected in the
same stop point. For each stop point the file lists the ID
number and X- and Y-coordinates.
(e) Rules. This file contains information on the allowed
maximum daily and weekly working hours, working
days, start date of the current planning period and
collection intervals.
(f) Distance matrix. This file is created by calculating the
shortest paths between all pairs of stop points. It
describes the distance and drive time for paths
connecting the stop points.
3.2. Conceptual model
The used solution approach is based on the conceptual
model presented in Fig. 1. As can be seen from the figure,
the main function controls the InputParser, Optimizer and
Output classes. The InputParser is used to read in the six
data files described in Section 3.1. The optimizer is a class
of functions described in Section 4 to perform the required
scheduling and routing optimization. The user can
determine the optimization criteria using the SetObjective
function. The default objective is the total distance, but it
could be also, e.g. total time or cost. A solution is first
initialized and initial daily and period solutions are created
Fig. 1. The conceptual model of the waste collection scheduler and optimizer.
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with the InitializeSolution, ConstructDailySolution and
ConstructPeriodSolution functions, correspondingly. An
attempt is made to improve the daily solutions already
during the construction of the first solution, and also after
the initial feasible solution has been constructed using the
seven well-known improvement heuristics Heur2Opt,
Heur3Opt, HeurOrOpt, HeurRelocate, HeurExchange,
Heur2Opt* and HeurCROSS as described in Sections 4.1
and 4.2. The improvement heuristics are guided by the
GVNT metaheuristic of Kytöjoki et al. (2004) described in
Section 4.1. The optimization is done according to a number
of constraints. Some of these constraints are implicit, and
given by the problem, whereas others are set outside, e.g. by
work regulations. The latter type of constraints is handled by
the Rules class. The Rules class regulates, e.g. the allowed
work time and legislative collection intervals for different
types of waste.
As an outcome, the optimizer produces a PeriodSolution
that gives the allocation of tasks among the days within the
collection period (with given periodBegin and periodEnd
dates), i.e. the scheduling solution as well as detailed routes
for each collection truck in each day, i.e. the routing solution
(DailySolution). The PeriodSolution consists of a number of
DailySolutions. The solutions can be evaluated according to
total distance or total time of all routes (numRoutes). The
number of vehicles (numVehicles) is typically known in
advance, but it can also be an optimization criterion. Each
daily solution consists of a number of routes represented by
the Route class. A route has a given start and end locations
as well as associated start and end times, and it consists of an
optimally ordered set of stop points, i.e. locations where a
vehicle stops to, e.g. collect a waste from a container. The
location can be defined according to its x, y, and z (for
height) coordinates or according to the physical address and
zip code of the location.
A vehicle must be allocated to each route. Each vehicle
has its own code for identifying and a given capacity. The
capacity is defined separately for each type of waste in terms
of both volume and weight. Several capacities may be
associated with each vehicle. Each vehicle belongs to one of
the predefined vehicle types. The vehicle type is used to
model the compatibility of the vehicle with the stop points
and waste bins. In other words, a vehicle may or may not be
stopped at a given location or used to collect the waste from
a given waste bin, e.g. because of the size of the vehicle or
required special equipment. The accumulated load (acc-
Load) and the accumulated time (accTime) should not
exceed the vehicle capacity or violate the work time
regulations, respectively. Once the vehicle is full with
respect to its capacity, it is driven to the waste disposal site
for emptying. The waste disposal site is modeled here with
the Facility class. The Facility class can be used to model
also other positions (type) such as the depot where the
vehicle is refueled and repaired.
Typically, each route starts and ends at the depot. A visit
to a facility has a certain begin and end time as well as
duration (stoptime). The total distance of the daily solution
is calculated using the accumulated drive distances (acc-
Distance) of each route. The distances and times between
each stop point are calculated beforehand as described in
Section 4.3 and stored in the CostMatrix that is read in with
the input parser. Each stop point consists of one or more
MacroBins and each MacroBin consists of one or more
waste bins. The MacroBin is an abstract class that is used to
aggregate waste containers containing same type of waste
and having same location and collection interval. These
containers are emptied at the same time so it is reasonable to
consider them as one to reduce the problem size and
facilitate the optimization. Because of different waste types,
several Macrobins may be associated with each StopPoint.
Each waste container (Bin) has its own identifier (id) and it
contains only one predefined type of waste. The visits to the
container must occur within given time windows, defined by
begin and end times. The visits have certain duration, i.e. the
time to empty the container that depends e.g. on the size of
the container and the distance of the container from the stop
point. Each waste container has a specific capacity
measured in terms of both volume and weight. There may
be restrictions regarding to compatibility of the waste
container and the vehicle. These are modeled using the
cartype attribute that lists the vehicle types that may be used
to handle the bin in question. In addition to the collection
intervals set by regulations and defined in the Rules class,
one may set preferred collection intervals separately for
each bin using the interval attribute. One may also define a
target facility (waste disposal site) for the collected waste by
specifying the facility identifier. The same attributes apply
also to MacroBins, although the durations, volumes,
weights and time windows must first be aggregated.
4. Methodology
In this section, we describe the applied solution
approach. First, the developed routing algorithm, the guided
variable neighborhood thresholding of Kytöjoki et al.
(2004) is shortly presented. Second, the heuristic scheduling
methodology is detailed. Then, the shortest path and travel
time calculation are briefly explained. Finally, some
implementation guidelines are given.
4.1. Routing algorithm
The routing algorithm of Kytöjoki et al. (2004)
consists of two phases. First, a feasible solution is
created with a hybrid insertion heuristic. In the second
phase, an attempt is made to improve the initial solution
with the Guided Variable Neighborhood Thresholding
(GVNT) metaheuristic of Kytöjoki et al. (2004). As
the name implies, the GVNT is based on three well-
known metaheuristic principles: guided local search
(Voudouris & Tsang, 1998), variable neighborhood
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search (Mladenovic & Hansen, 1997) and threshold
accepting (Dueck & Scheurer, 1990), which are applied
simultaneously.
The insertion heuristic constructs routes sequentially one
at a time until all waste containers have been routed. A route
is first initialized with a ‘seed’ container. The seed
containers are selected by finding the geographically closest
unrouted container in relation to the depot. The remaining
unrouted containers are then added into this route according
to distance minimization criterion until it is full with respect
to the scheduling horizon and/or capacity constraint. If
unrouted containers remain, the initializations and insertion
procedures are repeated until all waste containers are
serviced. Each time after a user-defined number of contain-
ers (64) have been inserted in the route currently under
construction, an attempt is made to reorder the route with
the well-known 2-opt (Flood, 1956), Or-opt (Or, 1976) and
3-opt (Lin, 1965) improvement heuristics. These three
operators are applied in turn until no more improvements
can be found.
Once the incumbent route is full with respect to the
scheduling horizon and/or capacity constraint an attempt is
made to improve the current partial solution by applying a
set of four well-known inter-route improvement heuristics
to all two-route combinations with the just finished route
according to the variable neighborhood search strategy of
Mladenovic and Hansen (1997). The applied inter-route
heuristics are: relocate, exchange (Savelsbergh, 1992),
2-opt* (Potvin & Rousseau, 1995) and CROSS (Taillard,
Badeau, Gendreau, Guertin, & Potvin, 1997). If an
improvement can be found, the above mentioned three
intra-route operators are launched as well. In case all
containers have been routed and no more improvements can
be found, the second phase is launched. In the second phase,
inter-route exchanges are tested with all two-route
combinations, and in case of improvements, intra-route
operators are also applied. In addition, a guided local search
based objective function is used in conjunction with a
threshold accepting type limit for the allowed worsening of
the objective value. The second phase is repeated until no
more improvements can be found. For more details, we refer
to Kytöjoki et al. (2004).
4.2. Scheduling algorithm
In this section, we shortly describe some of the basic
functionalities of the scheduling algorithm. In this case
study, the period length was set to four weeks and most of
the containers have 7 or 14 days as their schedule interval,
i.e. the containers must be collected once a week or every
two weeks.
Before constructing a new daily route, all waste
containers are divided in two groups. The first group
represents waste containers that require urgent service
whereas the less urgent containers are in the second group.
The filtering happens through checking the last collection
date. In many cases, the previous collection date is not
known, thus all containers must be treated equally. If it is
known that a container has been emptied already earlier
during the period the next collection time will be defined by
the last known collection day and a container specific
schedule interval. In the next phase, the containers are
sorted in ascending order according to their pickup interval.
That is because the containers that require more frequent
collection must be assigned first and the rest can be left over
for the next day if there is no space for them on the current
day.
After all daily routes have been constructed for a specific
day, and no more improvements to the daily solution can be
found with the GVNT metaheuristic, an attempt to
reschedule the containers between the days is made. A
move to another day is allowed only if the container must be
collected only once during the period and the last collection
date is not assigned or known. One must also check that the
new collection day is within the collection interval. For
example, if the interval is 7 days the first collection must be
done within the first 7 days from the start of the period.
The candidates for rescheduling are found by the relocate
and exchange inter-route heuristics guided by the GVNT
metaheuristic. The other two inter-route heuristics, 2-opt*
and CROSS, are not used in the rescheduling context
because they would make the implementation too complex
and in many cases the improvements would not be possible.
Obtaining improvements with the 2-opt* and CROSS would
require that collection intervals of the containers in the
exchanged route parts are not too strict and none of the
containers is collected more than once. After the initial
scheduling of the first period is done and all daily solutions
have been constructed, the rescheduling between the days is
disabled to speed up the route construction and scheduling
for the following periods. That is because now the last
collection date is known for each container and the
collection interval predefines the next collection time.
4.3. Shortest path calculation
The routing and scheduling algorithm requires infor-
mation on the shortest paths (in terms of travel time or
distance) between pairs of waste containers and between
waste containers and depot(s) and waste disposal site(s). In
this study, the shortest paths are calculated with the well-
known Dijkstra algorithm (Dijkstra, 1959) that finds the
shortest paths from the ‘source’ node to all other nodes of
the defined network. The shortest path source and
destination matrix is computed by repeating the application
of Dijkstra algorithm iteratively setting each node as source
node and calculating the shortest paths to every other node
of the network. The distance is used as the only shortest path
criterion. That is because distances between the waste
containers can be calculated quite reliably from digital road
information whereas travel times have a highly stochastic
nature. The other reason for using distances is that according
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to the haulers the waste collection costs are highly
correlated with the total traveling distance.
After calculating the shortest distances between each pair
of nodes the traveling times are estimated from the historical
GPS-data. Regression analysis is used to estimate traveling
speed in each road class, and the time spent on crossroads.
An alternative approach for calculating the shortest paths
could be using some Floyd–Warshall based variants.
Another improvement could be merging neural network
prediction model to shortest path algorithm in order to get
better estimates for travel times.
4.4. Implementation details
In this section, we shortly describe some ideas that we
used in implementing our solution algorithms to speed-up
the computations and to reduce the memory usage at the
same time.
One of the most innovative parts of our algorithm is a
new and very efficient coding of the distance and time
matrices that significantly reduces the memory require-
ments. Usually, both distance and time between each pair of
stop points is represented using separate floating point
numbers. Any floating point number mantissa has so called
least significant bits. By wisely controlling these least
significant bits, it is possible to significantly cut down the
memory usage. We were able to present the distance values
with just 15–20 bits in the mantissa without causing any
bigger impairment of the results, which can be seen here as
small noise in the overall computing precision. This way a
number of ‘saved’ bits can be used for coding the speed
information of the associated shortest path. With standard
IEEE float types the computing precision, i.e. the machine
epsilon is 1.192!10K7
. In case of double types it is 2.220!
10-16
, and when encoding the speed the overall precision
worsens as much as number of bits used for presenting the
speed. With this technique both distance and time can be
represented with a single floating point number, requiring,
e.g. just four bytes memory. Time can be computed in a
simple way from the stored distance and encoded speed. To
further reduce the memory usage, we calculated so-called
twisty-road-curve factors of the actual distances with
respect to the Euclidean distances and coded both speeds
and curve factors as fixed point numbers using either 16 or 8
bits depending how close the real cost matrix data is to the
respective Euclidian values. This makes it possible to
calculate actual distances and times based on the Euclidean
distances that can be calculated using the x- and y-
coordinates. The rationale is that modern processors execute
simple calculations faster than they do certain memory
operations. Especially, all kinds of cache memory misses
should be avoided as much as possible because they
generate a lot of penalty clock cycles. As a result, much
larger problems can be solved within reasonable amount of
memory and computation times are only a bit longer. For
more information and methods how to reduce the memory
usage even more with drive time, Euclidean distances and
twisty-road-coefficients for huge size cost matrices, we refer
to Kytöjoki (2004).
To limit the computational effort in the initial construction
heuristic, the waste containers considered for insertion to the
current route are limited to the containers with the smallest
insertion cost. More precisely, the 1000 smallest insertion
costs and the corresponding container for each potential
insertion position between two adjacent containers are kept in
memory in an ordered one-dimensional table for each
container node on the currently constructed route. Thus,
containerswhose insertion costisnot among the smallest1000
are ignored. Every time a container is inserted in the route, two
ordered lists of 1000 containers and their insertion cost have to
be updated (in the beginning an ordered list is initialized for
each container) corresponding to the two new potential
insertion positions. With this strategy it is possible to quickly
determine the cheapest insertion and its cost if we keep a
simple heap-priority queue structure.
To speed up the computations of seed containers, all
containers are ordered in the beginning primarily according
to their distance to the depot and secondarily according to
their collection schedule interval using the well-known
Quicksort algorithm. In case of equal distance to the depot,
the containers with a more frequent collection interval are
considered first. The next seed is then determined by finding
the first unrouted container in this ordered list.
To further speed up the computations, we use a one-way
linked list data-structure in an array to represent the solutions.
This structure includes the information of the following
container for each container and the array size is simply the
number of the containers. In practice, the array just represents
the list-nextvaluesfor eachofthe containersandanycontainer
is found quickly with simple index reference. Using this
structure, the changes to the route solution can be done very
quickly and in most of the cases in a constant time. For
example an addition of a new container u between two
adjacent containers i and j is done simply by changing the
‘next-values’ of u to j and i to u. Similarly, one can remove a
container orreverse a partofthe routevery quickly.Compared
to the typical one-dimensional table representation of
solutions, our strategy is much quicker. For example to add
a new container in case of table presentation, one must change
the position index of all containers behindthe current insertion
position, resulting in O(n) complexity.
5. Computational experiments
The suggested solution method has been implemented in
Visual CCC6.0, and it has been tested with a Pentium III
1.0 GHz (512 MB RAM) computer. This case study is based
on a real-life waste collection problem in Eastern Finland.
Only the collection of mixed waste is included in the study.
In this section, we describe the problem data and the
computational results.
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5.1. Problem data and the experimental setting
The case area consist of the operation area of Jätekukko
Ltd which is responsible for organizing waste management
in 18 municipalities (City of Kuopio and the surrounding
areas) serving approximately 180,000 people in Eastern
Finland. The total area consists of more than 150 collection
routes that are mainly handled by two haulers. Two different
regions were selected for closer examination in order to
re-optimize and schedule the collection routes using the
proposed solution methodology. In general, a lot of effort
was put on the validation of the historical data in order to
provide reliable inputs for the optimization. For more
detailed description of the data validation we refer to
Nuortio et al. (2004).
The first case region, Pieksämäki, consists of three
communes: Pieksämäki, Pieksämäki province and
Virtasalmi. Mixed waste is collected and transported to
the transfer station located in Pieksämäki town area from
which it is separately transported to the waste disposal
site. The waste is mainly collected by a single truck
which is daily starting from and ending the journey at
the transfer station. First, collection order of single routes
was optimized. Second, an attempt to optimize both
routing and scheduling was made. The planning period
was set to four weeks, due to the fact that a large
majority of the waste containers must be emptied at least
once in 4 weeks.
The other case region was selected from the Kuopio city
area and consists of 82 collection routes in total. In Kuopio
city, area population density is much higher than in
Pieksämäki area. The purpose was to find an area which is
as independent as possible from other collection areas (in
the sense that there are not overlapping routes), and two
historical time periods in which the most of the containers in
that area are emptied. The first period, Kuopio 1, was
collected using 6 different vehicles ending up total 58
routes. The other period, Kuopio 2, consists of 61 routes in
total. The daily journey starts and ends at the depot located
in Southern part of Kuopio city. All waste is finally
transported to waste disposal site which is approximate
16 km south-west from Kuopio center.
5.2. The results of single route optimization
The Matlab Matlog toolbox was used to estimate the
possible improvements obtained by optimizing the collec-
tion order individually for existing single routes, i.e. tackle
the traveling salesman problem. The Matlab optimization is
based on the well-known Clarke and Wright (1964) savings
algorithm and 2-opt improvement heuristic available in
the toolbox. All routes are extracted from the Pieksämäki
area. The study period (October 2003) consists of 3386
route points (waste containers) and 180 routes. Both remote
and dense areas are included. The total distance was used
as the optimization criterion. The results are presented
in Figs. 2 and 3.
As can be seen from the figures, significant savings can
be achieved by optimizing just the collection order of the
existing single routes. The difference (in distance) between
current route and optimized route varies from 0 to 70 km,
averaging to 16 km. The average route improvement in
terms of distance is about 12%.
1 25 49 73 97 121 145 169
Route
0
5
10
15
20
25
30
35
40
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improvement
(%)
Fig. 3. The difference between current routes and optimized routes in
percents.
0
10
20
30
40
50
60
70
80
1 25 49 73 97 121 145 169
Route
Route
improvement
(km)
Fig. 2. The difference between current routes and optimized routes in
kilometers.
Table 1
Computational results of routing and scheduling optimization for real-life waste collection problems in Eastern Finland
Problem n Collection
events
Period start Period end Estimated total
distance (km)
Optimized total
distance (km)
Savings
(km)
Savings
(%)
CPU
Pieksämäki 3225 5709 29.3.04 25.4.04 2693 2587 106 3.9 813
Kuopio 1 4718 14687 6.10.03 2.11.03 5487 3080 2591 43.9 697
Kuopio 2 4353 13738 3.5.04 30.5.04 5483 3073 2410 44.0 1040
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5.3. The results of routing and scheduling optimization
The results of routing and scheduling optimization are
described in Table 1. The first two columns refer to problem
name and size. The next three columns give the period start
and end dates and the number of collection events within the
period. The estimated total distance, presented in the next
column, is defined from the historical route data based on
the collection order of the containers. Optimized total
distance has been calculated with the model described in
Section 3. The last columns show the obtained savings in
kilometer and in percents compared with the current
practice and the CPU time in seconds.
According to Table 1, significant route improvements are
quickly achieved also as a result of combined routing and
scheduling optimization. Average saving in Kuopio area is
about 2500 km and on the average the routes were improved
by nearly 46%. Exceptionally, high improvement perform-
ance can be partially explained by the operational limitations.
The current operation is strictly bounded to the existing
schedules. Thus, by allowing rescheduling it is possible to
significantly increase the improvement rate. The used
geographical road data has also some inconsistencies and
there is not any data available on the shift exchanges. Finally,
according to the GPS data on the driven routes several
incomplete loads were emptied and in some cases even three
dump-visits were done during the workday. In the Pieksämäki
area, the data on the visits to disposal site (transfer station) is
also missing. The above problems hamper the comparison of
the actual and the optimized schedules.
6. Conclusions
In this paper, we have described a case study on a real-
life waste collection problem in Eastern Finland. We have
presented a conceptual model of the problem that can easily
be generalized also to other related problems, and an
efficient heuristic solution method based on the guided
variable neighborhood thresholding metaheuristic of Kytö-
joki et al. (2004). The applied solution methodology
integrates efficiently seven well-known improvement
heuristics, and the variable neighborhood search, guided
local search and threshold accepting metaheuristics as well
as a number of new speedup and memory reduction
techniques. The experimental results demonstrate that
significant savings compared to the current practice can be
obtained in both studied levels of optimization: optimization
of single routes only, and optimization of both routing and
scheduling for the whole collection period.
Acknowledgements
This work was partially supported by the Foundation for
Economic Education and Volvo Foundation. This support is
gratefully acknowledged. We would like to thank also the
iWaste project financers and partners, especially the
Environmental Informatics research unit at the University
of Kuopio, National Technology Agency of Finland
(Tekes), Jätekukko Ltd and Ecomond Ltd.
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