EFFICIENT WASTE COLLECTION BY MEANS OF ASSIGNMENT

PROBLEMS

Kostanca Katragjini, Federico Perea and Rub´en Ruiz

Grupo de Sistemas de Optimizaci´on Aplicada, Instituto Tecnol´ogico de Inform´atica

Universitat Polit`ecnica de Val`encia, Camino de Vera s/n, 46021 Valencia, Spain

Keywords:

Waste collection, Linear programming, Municipal solid waste management, Assignment.

Abstract:

This paper studies a municipal waste collection problem, whose ultimate goal is to ﬁnd an assignment of

containers to service days and posterior routing so that the total distance covered by the waste collection

trucks is minimized. Due to the complexity of this problem, we divide it into several subproblems. The focus

of this paper is on the ﬁrst problem, in which it must be decided which containers must be collected on each

service day so that the containers collected on the same day are as close to each other as possible. In a second

phase (out of the scope of this paper) the routes followed by the collection vehicles at each service day are

optimized. Such separation of the problem is needed as a result of the sheer size and complexity of the real

setting. Despite this separation into smaller subproblems, efﬁcient procedures must be designed in order to

solve real-sized instances. A computational experience over a number of instances shows the applicability of

our methods.

1 INTRODUCTION

This paper shows the initial results obtained when

working on a problem proposed by a municipality re-

garding their waste management. More speciﬁcally,

the municipality wants a support system able to de-

cide when and how to collect their waste containers

so that the associated costs are minimized. Other re-

quirements are that such support system should be

based on free software and that a solution should be

given in less than ﬁve minutes.

Waste collection (WC) is an essential problem in

many cities. Due to the increasing concerns about

pollution, and the ever raising fuel prices, an efﬁ-

cient planning is necessary for both saving money

and protecting the environment. WC can be seen as

part of the municipal solid waste (MSW) manage-

ment problem, in which several processes are consid-

ered: waste collection, transportation, treatment and

disposal, (Wilson, 1985). In the last decade, there has

been a trend to study the MSW management prob-

lem taking into account the uncertainty that some of

the involved parameters have, see for instance (Huang

et al., 2002; Li et al., 2008) and the references therein.

The WC problem can be deﬁned over a road net-

work of a given city. On such network, the different

elements of the problem are coded:

• one or more depots, where vehicles are stationed.

• landﬁlls and disposal plants, where waste has to

be brought.

• waste containers, which are usually specialized

for different types of waste.

• a ﬂeet of waste collection vehicles, potentially

heterogeneous.

• service frequency.

• and many other side constraints.

The actual problem faced asks for designing the

routes of the available vehicles, in order to collect all

speciﬁed containers, satisfying capacity and container

compatibility constraints with the objective of mini-

mizing the total travel distance and costs. In (Golden

et al., 2002), the waste collection problem is consid-

ered as an application of the vehicle routing prob-

lem. (Maniezzo, 2004) goes a bit further and formu-

lates the WC problem as a capacitated vehicle rout-

ing problem. In (Angelilli and Speranza, 2002) it is

discussed the importance of good estimates for the

operating costs in the corresponding vehicle routing

problem, which is applied to model two case studies.

The authors also consider work shifts and other im-

portant constraints. More complicated models arise

when other constraints are taken into account, such as

283

Katragjini K., Perea F. and Ruiz R..

EFFICIENT WASTE COLLECTION BY MEANS OF ASSIGNMENT PROBLEMS.

DOI: 10.5220/0003745102830292

In Proceedings of the 1st International Conference on Operations Research and Enterprise Systems (ICORES-2012), pages 283-292

ISBN: 978-989-8425-97-3

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

drivers’ lunch breaks, see e.g. (Kim et al., 2006; Ben-

jamin and Beasley, 2010) and many others. For exam-

ple, different types of waste might be present, some-

times collected with different trucks which employ

different technologies. Examples of different types

of waste are glass, plastic, containers, cardboard, dan-

gerous waste (such as ﬂuorescent light bulbs, used oil,

batteries, etc.), organic waste, etc. Moreover, WC

complicates further if we consider the fact that the

waste truck, once full, has to do a potentially long

trip to the treatment plant or landﬁll and then it has

to go back to a potentially different point in the route

to continue with the collection. Additionally, all con-

straints typical to labor scheduling problems also ap-

pear, as the maximum driving time and working time

of the waste collection operators must be observed.

As a result of all of the above, the WC can be con-

sidered as a very complicated and difﬁcult vehicle

routing problem, or better as a rich periodic vehi-

cle routing problem (PVRP). For early references on

PVRP and WC related problems the reader is referred

to (Beltrami and Bodin, 1974), who use the PVRP

to model a municipal waste collection problem, and

(Russell and Igo, 1979), who assign customer demand

points to days of the week so that a certain node rout-

ing problem is solved more efﬁciently. For a more

recent reference see for instance (Coene et al., 2010)

and the references therein.

Additionally, the company wanted us to study a

speciﬁc generalizationof the WC problem. In this set-

ting, there are two types of waste that are collected si-

multaneously, typically cardboard and general waste,

as these two types of waste are the most frequent and

abundant. In order to do so, a special truck with two

compartments is employed. When the truck reaches a

location or node, it collects the cardboard container

(if present) and the general waste (usually always

present). From now on, following the usual nomen-

clature in WC problems, the two types of waste are

referred to as fractions.

Due to the high complexity of the WC problem,

a large variety of heuristics are present in the litera-

ture. In (Teixeira et al., 2004), a constructive heuris-

tic divided into three phases, namely deﬁnition of ge-

ographic zones, deﬁnition of the waste type to col-

lect each day, and deﬁnition of routes, is presented

to solve a urban recyclable waste collection problem.

A clustering-based waste collection algorithm is pre-

sented in (Kim et al., 2006). (Baustista et al., 2008)

model a WC problem as an capacitated arc routing

problem with some speciﬁc characteristics, which is

transformed into a node routing problem and solved

by means of an ant colony heuristic. Three meta-

heuristics are presented in (Benjamin and Beasley,

2010). The authors state that their algorithms perform

better in terms of quality of solutions than other algo-

rithms previously presented in the literature.

The company wanted this problem to be solved

(with around 1000 different collection sites) in less

than ﬁve minutes by means of free software. Time

constraints are due to the daily operations which

needed fast response times and free software was

needed due to budget limitations. Our experience

showed that it was impossible to do that (we could

solve to optimality instances consisting of 10 loca-

tions in around a minute by means of CPLEX, but

such computational time explodes as the number of

locations increases). Therefore, we had to propose

the following partition of the WC setting into two dif-

ferent problems:

1. Assignment: in a ﬁrst step we would ﬁnd an as-

signment of locations to service days so that all

locations collected on the same day are close to

each other and the amount of waste collected per

day is relatively constant.

2. Routing: the routing problem for each service day

would be solved on a second step. This problem

is not studied in this paper as it represents a very

complex vehicle routing problem. Additionally,

the problem may involve hundreds of nodes (up

to a thousand) and as much as 50 waste collection

trucks.

This paper shows the progress made on the ﬁrst

step. In Section 2, this assignment problem is mod-

eled as a mixed integer linear programming problem.

Section 3 introduces procedures aiming at solving our

assignment part of the WC problem efﬁciently. Such

procedures are tested and compared in Section 4.

Some additional techniques, yet to be tested, are pre-

sented in Section 5. The paper closes with some con-

clusions and pointers to future work.

2 ASSIGNING LOCATIONS TO

SERVICE DAYS

In this section we propose a mixed integer linear pro-

gramming (MILP) model that decides which loca-

tions should be collected on each service day, so that

no container overﬂows, the amount of waste collected

per service day is relatively constant, and the loca-

tions visited on each service day are as close to each

other as possible, over a weekly planning horizon.

This last objective has been treated by minimizing

the sum of the diameters of each day, as we will ex-

plain later. A service day is deﬁned as a day in which

some waste is collected. There are 7 days in a week

ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

284

but there does not need to be as many service days.

For example, busy streets with lots of population or

businesses nearby have their containers serviced ev-

ery day, potentially 7 days per week. Conversely,

sparsely populated areas might have containers ser-

viced twice or even just once per week. Note that the

problem also includes a given number of service days

per week.

Our problem has the following input data:

• Let i = 1,...,I be the locations where the con-

tainers are placed, j = 1,...,7 the possible ser-

vice days (Monday to Sunday), and k = 1, 2 waste

types, from now on denoted by fraction 1 and frac-

tion 2, respectively.

• Let (α

,β

) be the coordinates of location i on the

plane, i = 1,...,I.

• There are n

containers of fraction k at location

i, for all i = 1, ..., I, k = 1,2. In reality, there

are many more waste types (organic waste, pa-

per, plastic, hazardous, and others) but since the

collection technologies differ, the problems can

be solved separately in different runs. Therefore,

only two fractions are considered.

• Each container of fraction k receives g

kilograms

of waste every day, and has a maximum capacity

of c

, k = 1, 2.

• The company wants to collect fraction k s

days

per week, k = 1, 2, with s

∈ {1,...,7}. These are

the desired service days.

• Each fraction k container must be collected f

days per week. Note that f

≤ s

. This parameter

is the so called frequency of service.

• If location i is visited to collect fraction 2, then

fraction 1 on location i must be collected as well.

This implies that the actual number of service

days is s

The last point above is another speciﬁc constraint.

The company wants the “minor” fraction to be col-

lected each time the “major” fraction is collected. In

other words, when the general waste fraction is col-

lected by a truck in a given location, the cardboard

waste in the same location (if present) must be col-

lected as well.

It is also very important to note the difference be-

tween the service days of each fraction s

and the fre-

quency for each fraction f

. Let us advance an ex-

ample. The company wants to service an area with

maximum 4 service days in a week. This means that

= s

= 4. In other words, waste collection trucks

should operate only four days during the week. The

frequency for a different fraction is a different matter,

= 3 means that the containers of the ﬁrst fraction

have to be collected three times per week. Since not

all containers have to be collected in the same service

days, the problem is therefore to distribute contain-

ers in service days so that every container is collected

three times per week and that there are no more than

four days with service.

2.1 Preprocessing: Timetables

First, given the f

values, the set of feasible timeta-

bles is calculated for a given collection site. Each

timetable determines on which days a location is vis-

ited, and they will be assigned to collection sites.

From this information, and from the amount of waste

that arrives at each container per day (given by pa-

rameter g

), the amount of waste collected each day

can be easily calculated. A timetable is deﬁned as a

matrix p ∈ R

7×4

such that:

• p

j,k

= 1 if timetable p determines that on day j,

fraction k must be collected, j = 1,...,7, k = 1, 2,

zero otherwise.

• p

j,k+2

is the amount of fraction k collected on day

j if the timetable p is followed, j = 1, ..., 7,k =

1,2. Note that, if p

j,k

= 0, then p

j,k+2

= 0 too.

Note as well that p

j,k+2

only depends on g

and on

the last day before j that fraction k was collected.

Each feasible timetable p must satisfy:

1. p

j,2

≤ p

j,1

, j = 1, ...,7 (if fraction 2 is collected

on day j, so should fraction 1).

∑

j=1

j,k

= f

, k = 1,2 (for every collection site,

fraction k must be collected f

times per week).

3. p

j,k+2

≤ c

, j = 1,...,7,k = 1, 2 (containers can-

not overﬂow their capacity).

Note that the list of feasible timetables only depends

on f

, and on the ratio c

, for k = 1, 2. Besides,

the company may want to add some extra constraints,

such that no container must be collected on two con-

secutive days. We will consider that there are m fea-

sible timetables {p

,..., p

Example 1. As an example, consider a situation in

which s

= s

= 7, f

= 2, f

= 1, g

= g

= 10, c

40,c

= 80. Consider the following two timetables:







1 1 30 70

0 0 0 0

1 0 40 0

0 0 0 0







, p

′







1 1 20 70

0 0 0 0

1 0 50 0

0 0 0 0







Timetable p determines that the fraction 1 is to be col-

lected on Mondays and on Fridays, and fraction 2 on

EFFICIENT WASTE COLLECTION BY MEANS OF ASSIGNMENT PROBLEMS

285

Mondays. Timetable p

′

determines that fraction 1 is

to be collected on Mondays and on Saturdays, and

fraction 2 on Mondays. It is easy to see that p is

a feasible timetable, whereas p

′

is not, since a frac-

tion 1 container will overﬂow on Saturdays before the

collection (amount of garbage to be collected is 50,

whereas the container’s maximum capacity for frac-

tion 1 is 40).

Timetables are enumerated, note that in principle

there are







possible timetables. Nevertheless,

due to containers’ capacity and estimated amount of

waste received per day, many of these timetables are

unfeasible and therefore removed. Note that this pro-

cedure could be very well extended and generalized

so to work at each location, i.e., each location could

have a different capacity and a different generation of

garbage. However, the company did not have as much

precise and detailed data for such an extension.

2.2 Variables

Given the feasible timetables, all that is left to do is to

assign one timetable to each location so that the given

objective is optimized. For this assignment problem,

we employ the following variables:

• x

ℓ

= 1 if location i is assigned to timetable p

ℓ

, zero

otherwise.

• y

= 1 if on day j fraction k is collected, zero oth-

erwise.

• v is an internal variable, which is used to balance

the amount of waste collected per day.

• a

are the ﬁrst and second coordinates of the

center of day j, respectively.

• r

≥ 0 is the radius of day j.

2.3 Constraints

The constraints of our model are:

• Exactly one timetable for each location:

∑

ℓ=1

ℓ

= 1; i = 1, ..., I. (1)

• The following constraints force the number of ser-

vice days in which fraction k is collected to be s

∑

j=1

= s

; k = 1, 2. (2)

≤

∑

i=1

∑

ℓ=1

ℓ

j,k

; j = 1, ..., 7;k = 1, 2. (3)

ℓ

j,k

≤ y

; i = 1, ..., I; j = 1,...,7;

ℓ = 1,...,m;k = 1, 2. (4)

Note that Equations (3) and (4) force that y

= 1

if and only if at least one collection site is visited

to collect fraction k on day j.

• Since the company wants the amount of waste col-

lected per day to be relativelyconstant, we have to

add:

−(1− y

)M + (1− ε)v ≤

∑

i=1

∑

ℓ=1

∑

k=1

ℓ

j,k+2

j = 1,...,7, (5)

∑

i=1

∑

ℓ=1

∑

k=1

ℓ

j,k+2

≤ (1+ ε)v+ (1− y

)M,

j = 1,...,7, (6)

where ε ∈ [0,1] is a small tolerance, and M is

a large enough positive number. These two sets

of constraints force that the amount of collected

waste on a service day (when y

= 1) is in the in-

terval [(1− ε)v,(1+ ε)v].

• The radius of a given day j is deﬁned as:

min

)∈R

max

i collected on day j

d((a

),i),

where d((a

),i) is the distance from (a

) to

location i. The point (a

) deﬁning the radius

of a given day j is called the center of day j. In

order to keep the linearity of the problem we will

use the Manhattan distance. The following con-

straints force that all locations collected on day j

are less than r

far from the center (a

) of that

day, using the Manhattan distance:

−M(1−

∑

ℓ=1

ℓ

j,1

) + |α

− a

| + |β

− b

| ≤ r

i = 1,...,I, j = 1, ..., 7.

These constraints can easily be made linear

by dividing them into the four possible cases.

−M(1−

∑

ℓ=1

ℓ

j,1

) + (α

− a

) + (β

− b

) ≤ r

, (7)

−M(1−

∑

ℓ=1

ℓ

j,1

) + (α

− a

) − (β

− b

) ≤ r

, (8)

−M(1−

∑

ℓ=1

ℓ

j,1

) − (α

− a

) + (β

− b

) ≤ r

, (9)

−M(1−

∑

ℓ=1

ℓ

j,1

) − (α

− a

) − (β

− b

) ≤ r

, (10)

i = 1,..., I, j = 1,..., 7.

2.4 Objective

The objective is to minimize the average radius of

each day

∑

j=1

, or equivalently:

ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

286

min z =

∑

j=1

. (11)

The presented MILP model that assigns collection

sites to service days so that the collection sites visited

on a given day are as close to each other as possible,

consists of minimizing (11) subject to (1)-(10), and

will be denoted by ASD (“Assigning Sites to Days”).

3 IMPROVEMENT PROCEDURES

Despite the considerable reduction in the amount

of variables and constraints of the previous assign-

ment models with respect to other waste collection

problems found in the literature, the proposed MILP

model was only capable of solving instances of mod-

erate size, very far away from the required 1000 loca-

tions of the company. Additionally, the fact that the

company wanted to solve this part of the problem in

less than ﬁve minutes by means of free software calls

for the application of some heuristic procedures.

3.1 Reducing the Number of Variables

In our efforts to reduce the complexity of the prob-

lem, a ﬁrst attempt to solve ASD more efﬁciently was

to set variable v to be equal to the total amount of

waste to be collected per week divided by the num-

ber of service days (the arithmetic mean), that is,

v = 7(

∑

i=1

∑

k=1

)/s

. This way ASD has one

variable less, and therefore a reduction in computa-

tional time was hoped for. The ﬁrst we noted is that,

despite the intuitive idea that both problems should be

equivalent (v free and v equal to the mean), they are

not. Consider the following example.

Example 2. Consider a WC instance in which the to-

tal amount of waste to be collected is 4, over 3 ser-

vice days. Assume that the amounts collected per day

are w

= w

= 1,w

= 2. Take ε = 1/3. It is easy

to see that, for v = 3/2, w

∈ [(1 − ε)v,(1 + ε)v] =

[1,2]. Nevertheless, if we ﬁx v = 4/3, we have that

[(1−ε)v,(1+ε)v] = [8/9, 16/9] and therefore w

falls

out of the allowed interval. Therefore, with ε = 1/3,

the previous instance would be infeasible if v is ﬁxed

to be the mean, and it would be feasible if v is free.

In order to see whether or not the difference be-

tween the performance of both models (v free and v

ﬁxed) is signiﬁcant, we tested both models over 12

different-sized instances. The values of I and the ob-

jective values obtained with CPLEX after 5 minutes

for each instance and each model are shown in Ta-

ble 1. The values of s

, f

are the same as in

Example 1.

Table 1: Tests to ﬁx variable v.

I 16 25 36 49 64 81

v free 9.5 15 17.5 22.5 29.5 35

v = mean 9.5 13 18 24.5 31 35

I 100 121 144 169 196 225

v free 47 50 61 68 73 83.5

v = mean 45.5 49 60 69.5 75 81

From these data, a hypothesis test with 95% con-

ﬁdence interval does not reject that the objective val-

ues obtained with both models are equal (p-value =

0.93). In other words, ﬁxing the value of v does not

seem to improve the objective function value with the

allowed computation time. This, together with Exam-

ple 2, made us keep the ASD formulation described

before without ﬁxing variable v.

3.2 Clustering

Solving a 1000 location problem is a real challenge.

However, a practical observation brings an important

venue for problem size reduction. Often, a relatively

short street has two container locations, spaced less

than a few meters away. Additionally, and specially in

road crossways, containers are located in the corners

of several roads. All these different locations can be

easily clustered into a single location with very little

expected degradation in the ﬁnal solutions. Such sce-

narios are very likely in large cities which have thou-

sands of containers but these are always tightly placed

in streets. By following simple clustering approaches

(like K-nearest neighbor) we can reduce the size of

the original problem by a factor of K in polynomial

time.

These simple clustering approaches were fast and

easy to implement but we applied a different ap-

proach. Instead of applying a K-nearest neighbor we

employ effective Traveling Salesman Problem (TSP)

heuristics. The Lin-Kernighan heuristic (Lin and

Kernighan, 1973) is one of the most well known and

powerful heuristics. More speciﬁcally, an effective

implementation (Helsgaun, 2000) is employed. This

Helsgaun heuristic is regarded as the most effective

method for solving the TSP which can yield close to

optimal solutions in very short computational times

with problems with hundreds or even thousands of

nodes. The procedure is the following: We construct

a TSP problem with all the locations and run the Hels-

gaun heuristic. The result is the shortest TSP tour that

traverses all locations, starting and ﬁnishing at a given

one. Then we apply a K clustering procedure. We

start from the ﬁrst location in the tour and create a

new virtual location with this one and the following

K − 1 locations. Location K + 1 is clustered with lo-

EFFICIENT WASTE COLLECTION BY MEANS OF ASSIGNMENT PROBLEMS

287

Table 2: Instances tested and their characteristics.

Instance Locations (I) s

Feasible time tables ε value

1 40 6 2 6 2 10 45 5 25 7 0.2

2 40 6 3 6 2 10 35 5 25 35 0.05

3 70 6 2 6 2 10 45 5 25 7 0.2

4 70 6 3 6 2 10 35 5 25 35 0.05

5 100 6 2 6 2 10 45 5 25 7 0.2

6 100 6 3 6 2 10 35 5 25 35 0.05

7 130 6 2 6 2 10 45 5 25 7 0.2

8 130 6 3 6 2 10 35 5 25 35 0.05

9 260 6 2 6 2 10 45 5 25 7 0.2

10 260 6 3 6 2 10 35 5 25 35 0.05

11 520 6 2 6 2 10 45 5 25 7 0.2

12 520 6 3 6 2 10 35 5 25 35 0.05

cations K + 2,...,2K and so on. This procedure con-

tinues until the last location of the tour is considered.

The result is a I/K location problem in which each

virtual location contains K original locations.

We expect the deterioration in the solution to be

very small. Note that after applying the Helsgaun

heuristic, the locations at each clustered virtual loca-

tion are most of the time very close to each other.

Note that many other clustering methods and algo-

rithms are possible.

4 INITIAL COMPUTATIONAL

RESULTS

In the following we present the results of the simple

ASD method when run with real instances. Some re-

sults with the application of the clustering procedure

will be given as well.

We employ a set of 12 real instances with data ob-

tained from the company. These range from 40 to 520

locations in size. While still deemed as “medium”,

they serve as a good benchmark of what can be done

with the ASD method alone. All instances contain

two fractions, whose full parameters are given in Ta-

ble 2.

As we can see, the mix of conﬁgurations yield two

cases with either 7 possible timetables per location or

35. Note that in the largest case of 520 locations and

35 possible timetables, the total number of x

ℓ

vari-

ables is 520× 35 = 18, 200 which is a really big num-

ber of binary variables considering the maximum al-

lowed computational time (5 minutes) with free soft-

ware (in all tested instances this time limit is reached,

i.e., no solution found is guaranteed to be optimal).

All tests are carried out in a high performance

computing cluster with 30 blades, each one contain-

ing 16 GBytes of RAM memory and two Intel XEON

E5420 processors running at 2.5 GHz. Note that

each processor has 4 physical computing cores (8

per blade). There is no parallel computing. The

30 blade servers are just used in order to divide the

workload and experiments. Experiments are carried

out in virtualized Windows XP machines, each one

with one virtualized processor and 2 GB of RAM

memory. We solve all 8 instances ﬁrst using IBM

ILOG CPLEX 12.2 with a single thread with a max-

imum CPU time limit of 5 minutes. Later, results in

the same conditions with the freely available 2.7 ver-

sion of COIN-OR Branch-and-cut MIP solver, also

referred to as CBC (https://projects.coin-or.org/Cbc)

will be given. We also employed LP-Solve 5.5

(http://sourceforge.net/projects/lpsolve/). However,

this free solver has a much inferior performance and

in these initial experiments, results are reported with

just CPLEX 12.2 and CBC 2.7.

We show the ﬁnal results using CPLEX after the

assignment carried out by ASD both in total radii and

radii per day of the week (both measured as the Eu-

clidean distance between latitude-longitude like coor-

dinates) in Table 3 as well as in total collected waste

(in kilograms) in Table 4.

The results with CBC are presented in Tables 5

and 6.

It is observed that CPLEX provides smaller radii

than CBC for all instances except 6 and 8.

As regards the total collected waste we can see

that it is fairly constant throughout the days for all in-

stances. The observed ﬂuctuations are often seen at

days before and/or after the “break”. Note that all in-

stances have 6 service days in the week and therefore,

after the day without service, a larger amount of waste

needs to be collected. When comparing CPLEX and

CBC we have added a column with the standard de-

viation of the collected waste through the days. It

appears that CBC has the upper hand. Although the

small number of tested instances precludes us from

drawing strong conclusions, it seems that CPLEX is

not performing alarmingly better than CBC. There-

fore, our objective of employing a freely available

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288

Table 3: CPLEX Radii results for the instances (

∗

no feasible solution obtained for instance 12 in 5 minutes, the reported

solution was obtained after 14 minutes and 37 seconds).

Radii

Instance Monday Tuesday Wednesday Thursday Friday Saturday Sunday Radii sum

1 69.5 96.5 31.5 69.5 96.5 28.5 392

2 91 89 84.5 89 80 96,5 530

3 82.5 88 95 82.5 88 88.5 524.5

4 77 162 109 179.5 78 179.5 785

5 68 66.5 179.5 57 63 179.5 613.5

6 150 179.5 126 174.5 139 175.5 944.5

7 69 70.5 202 78 112 177 708.5

8 158.5 196 172 158.5 186 202 1073

9 101.5 294 187.5 90.5 241.5 240.5 1155.5

10 286.5 268.5 286.5 285.5 250 278.5 1656

11 274 294 148.5 244 294 155.5 1410.5

12* 280 298 285.5 298.5 298.5 276 1736.5

Table 4: CPLEX total collected waste at each day for the instances (

∗

no feasible solution obtained for instance 12 in 5 minutes,

the reported solution was obtained after 14 minutes and 37 seconds).

Total collected waste

Instance Monday Tuesday Wednesday Thursday Friday Saturday Sunday St. Dev

1 1560 1605 1545 2080 2275 1820 304.59

2 1720 1875 1875 1885 1760 1770 72.35

3 3760 2670 2655 2820 3965 2820 586.89

4 3065 3240 3015 3010 3050 3310 127.59

5 3625 5385 5259 3740 3600 4290 819.10

6 4200 4575 4490 4250 4215 4205 165.79

7 4275 4260 6375 6150 6320 5940 1007.50

8 5770 5740 5280 5305 5410 5815 274.65

9 7245 9590 8700 6460 6405 6575 1342.33

10 7865 7875 7855 7125 7125 7130 404.46

11 19315 19135 12900 12900 15360 12895 3100.07

12* 14795 14790 15035 16315 16320 15250 717.72

Table 5: CBC Radii results for the instances.

Radii

Instance Monday Tuesday Wednesday Thursday Friday Saturday Sunday Radii sum

1 83.5 42 89 83.5 50.5 100.5 449

2 112.5 103 89 98 139.5 105.5 647.5

3 136.5 158.5 61 136.5 158.5 73.5 724.5

4 174.5 113 110.5 179.5 84.5 179.5 841.5

5 119.5 99.5 179.5 119.5 97 179.5 794.5

6 162 175.5 162 69.5 187 170 926

7 196 148.5 92.5 196 148.5 92.5 874

8 202 107.5 170.5 166 172 194 1012

9 185.3 154.8 294.2 185.3 158.3 294.2 1272.2

10 285.6 256.4 269.4 275.0 294.2 286.5 1667.2

11 229.6 283.9 289.1 229.6 283.9 289.1 1605.1

12 298.3 263.7 290.6 283.9 298.3 280.1 1714.9

solver is satisﬁed.

We now take the four largest instances: 7-12, and

apply the clustering procedure prior to the resolution

of the ASD model by CBC. K = 2, meaning that the

clustering process reduces the original locations by a

factor of two. The results are given in Tables 7 and 8.

As can be seen, in almost all cases both the to-

tal radii as well as the standard deviation of col-

EFFICIENT WASTE COLLECTION BY MEANS OF ASSIGNMENT PROBLEMS

289

Table 6: CBC total collected waste at each day for the instances.

Total collected waste

Instance Monday Tuesday Wednesday Thursday Friday Saturday Sunday St. Dev

1 2010 2025 2000 1395 1415 2040 317.29

2 1800 1795 1780 1810 1810 1890 38.78

3 3885 3775 2780 2745 2760 2745 555.08

4 3265 3195 2955 3050 2980 3245 137.00

5 4700 4805 4920 3300 3340 4870 780.11

6 4460 4195 4140 4110 4500 4530 194.00

7 7040 6570 5000 4965 4835 4910 982.44

8 5870 5335 5400 5885 5385 5445 253.59

9 8625 8410 8600 6390 6380 6570 1153.69

10 7840 7845 7160 7160 7275 7695 332.98

11 17535 17640 17500 13050 13095 13685 2356.30

12 16010 15980 14595 14495 15535 15890 697.43

Table 7: CBC radii results for the instances 7 - 12 after applying the clustering procedure.

Radii

Instance Monday Tuesday Wednesday Thursday Friday Saturday Sunday Radii sum

7 151 134 124.5 151 124 124.5 809

8 188.5 136 187 136.5 188.5 136 972.5

9 120.5 234.6 219.2 120.5 234.6 219.2 1148.57

10 217.9 294.2 161.6 294.2 150.8 294.2 1412.95

11 241.7 243.9 208.6 241.7 243.9 208.6 1388.49

12 271.2 280.1 205.1 298.3 183.3 298.3 1536.41

Table 8: CBC total collected waste at each day for the instances 7 - 12 after applying the clustering procedure.

Total collected waste

Instance Monday Tuesday Wednesday Thursday Friday Saturday Sunday St. Dev

7 6595 5840 6220 4575 4545 5545 846.83

8 5790 5490 5765 5245 5725 5305 241.26

9 8620 8500 8580 6465 6375 6435 1174.03

10 7940 7225 7305 7485 7235 7785 302.93

11 17520 17700 17640 13140 13275 13230 2413.81

12 15870 16020 14600 15120 14850 16045 638.81

lected waste are signiﬁcantly smaller once the clus-

tering procedure is applied, i.e., applying clustering

ﬁrst, optimization with CBC second, and declustering

third, performs better than directly optimizing with

CPLEX or CBC without clustering/declustering. The

explanation is the following: in ﬁve minutes solving

time, larger models have very little time to converge.

Once the clustering procedure is applied, the size of

the model is largely reduced and the ﬁnal results are

better. This means that far from deteriorating results,

the clustering procedure improves performance.

All instances and results were checked by the

company and results have exceeded their expecta-

tions. Considering that the results are only prelimi-

nary, this is a good outcome. We have achieved, using

CBC and for instances of up to 130 locations, to form

packed groups of locations that satisfy all constraints:

the number of service days is not exceeded, all col-

lection frequencies for both waste fractions are ob-

served, radii for each service day are minimized and

the amount of waste collected at each day is much

more stable than currently observed by the company.

5 FURTHER IMPROVEMENTS:

FEASIBLE SOLUTION FIRST,

GREEDY LOCAL SEARCH

SECOND

The procedure we propose in this section is based on

the following idea: Once a feasible solution has been

found by means of the previous integer linear pro-

gramming problem, we later try to improve the ob-

jective function by moving the collection day of the

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290

sites located furthest away from their day’s center to

either the previous day or the following day, provided

that such a movement does not make the containers

in this site to overﬂow. An interchange with another

collection site will be done in such a way that none

of the containers overﬂow, the amount collected each

day remains within the allowed interval, and the sum

of radii decreases. This is justiﬁed by the fact that

ASD can ﬁnd a feasible solution relatively quickly,

but the improvements in the objective function are

found very slowly. For instance, a 400-collection-site

example found a feasible solution in around 1 sec-

ond, with objective value equal to 127, but after 5

minutes the value of the objective function had de-

creased eight units only. A 625-collection-site exam-

ple, found a feasible solution in 11 seconds, with ob-

jective function equal to 163.5, and after 5 minutes the

improvement in the objective function was also quite

marginal.

Therefore, we propose to run the ASD until a fea-

sible solution has been found and then to run a local

search procedure. The pseudocode of this procedure

is as follows:

For a given day j, let F

, ¯r

, ¯w

be the collection

sites visited, the radius, the center, and the amount of

collected waste, respectively. For the sake of notation,

we let u(i, j

, j

) be the amount of waste collected at

site i on day j

if its collection day has been moved

from j

to j

and the rest of its schedule remains un-

changed. Let e

be the amount collected at site i if no

changes in its schedule have been done.

1. Set E =

0. For each j, ﬁnd i

: max

i∈F

d(i,

) =

d(i

). Let t

′

and r

′

be the center and the radius

of the new set F

′

= F

\ {i

}. Let q

= ¯r

− r

′

2. Let j

∗

: q

∗

= max

j/∈E

. If i

∗

cannot be moved

neither to the previous day nor to the following,

go to step 3. Otherwise, set h

= h

= −1 and go

to step 4.

3. Set E = E ∪{ j

∗

}. If E is the complete set, STOP.

Otherwise, go to Step 2.

4. If i

∗

could be moved to the previous day, ﬁnd

a collection site i

∈ F

∗

−1

that can be moved

to F

∗

such that r

∗

−1

+ r

∗

< ¯r

∗

−1

+ ¯r

∗

, and

∗

= w

∗

− e

∗

+ u(i

, j

∗

− 1, j

∗

) and w

∗

−1

∗

−1

− e

+ u(i

∗

, j

∗

, j

∗

− 1) (the amount of col-

lected waste) are in the allowed intervals. r

∗

−1

and t

∗

−1

are the radius and the center of F

∗

−1

∗

−1

\ {i

}) ∪ {i

∗

}, r

∗

and t

∗

are the radius

and the center of F

∗

= (F

∗

\ { i

∗

}) ∪ {i

}. Set

= (¯r

∗

−1

+ ¯r

∗

) − (r

∗

−1

+ r

∗

5. If i

∗

could be moved to the next day, ﬁnd a col-

lection site i

∈ F

∗

that can be moved to F

∗

such that r

∗

+ r

∗

< ¯r

∗

+ ¯r

∗

, and w

∗

− e

∗

+ u(i

, j

∗

+ 1, j

∗

) and w

∗

= w

∗

−

+ u(i

∗

, j

∗

, j

∗

+ 1) (the amount of collected

waste) are in the allowed intervals. r

∗

and

∗

are the radius and the center of F

∗

\ {i

}) ∪ {i

∗

}, r

∗

and t

∗

are the radius

and the center of F

∗

= (F

∗

\ { i

∗

}) ∪ {i

}. Set

= (¯r

∗

+ ¯r

∗

) − (r

∗

+ r

∗

6. If h

≥ h

, move i

∗

to day j

∗

− 1, and i

day j

∗

. Set F

∗

= F

∗

, F

∗

−1

= F

∗

−1

, ¯r

∗

= r

∗

¯r

∗

−1

= r

∗

−1

∗

= t

∗

−1

= t

∗

−1

, ¯w

∗

= w

∗

and ¯w

∗

−1

= w

∗

−1

. Go to Step 1. The objective

function has improved h

7. If h

> h

, move i

∗

to day j

∗

+ 1, and i

to day

∗

. Set F

∗

= F

∗

, F

∗

, ¯r

∗

= r

∗

, ¯r

∗

= r

∗

= t

∗

= t

∗

, ¯w

∗

= w

∗

and ¯w

∗

. Go to Step 1. The objective function has

improved h

We note that the complexity of this algorithm is

given by the calculation of the radii and centers of a

set of O(I) points, for at most O(I) times. From the

algorithms in (Megiddo, 1983), who showed that the

minimum enclosing circle could be found in O(I), we

derive that this step can be done in O(I

), and there-

fore the complexity of one movement of the previous

local search algorithm is O(I

) as well.

An important aspect of the local search procedure

is that it can be safely applied after a de-clustering

procedure, i.e., the ASD is applied to a clustered prob-

lem. We carry out a de-clustering after an initial feasi-

ble solution has been found and later we can apply the

local search. In this way, much of the precision lost

in the clustering is later regained in the local search

stage.

With the proposed local search procedure there

are four possible algorithms: Simple ASD (no local

search, no clustering), ASD with clustering, ASD un-

til feasibility with local search, and ASD until feasi-

bility with clustering, de-clustering and local search.

We are currently testing this proposed local search

and other variants and will present the results at the

conference.

6 CONCLUSIONS AND FUTURE

WORK

In this short paper we have studied a realistic waste

collection problem that we have divided into two sub-

EFFICIENT WASTE COLLECTION BY MEANS OF ASSIGNMENT PROBLEMS

291

problems. In the ﬁrst subproblem, the only one stud-

ied in this initial research, collection locations are as-

signed to service days in a complex assignment mixed

integer linear programming model. In a foreseen sec-

ond subproblem, each service day is solved separately

as a complex vehicle routing problem. This is needed

in order to solve the real problem, which involves the

collection of up to a thousand nodes each day during

the whole week with a ﬂeet of more than 50 waste

collection trucks. Although not fully tested, we have

proposed several techniques aimed at speeding up the

solution of the ﬁrst subproblem. More speciﬁcally, we

have presented some variable reduction techniques,

a clustering approach and a local search technique.

Initial results using commercial and freely available

solvers indicate that the solution for the ﬁrst sub-

problem is near and viable within short computational

times. Initial assessments by the company also indi-

cate that our results are already better than the manual

solutions currently employed.

In the next weeks we intend to test the local search

procedure. These results will be compared with the

previous ones. The desired goal is to reduce com-

putational time and at the same time to improve the

quality of the solutions. We also intend to test more

real instances and not just 12. Finally, a full ﬂedged

comparison with the current status quo at the com-

pany will be provided.

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