Incorporating User Preferences in Many-Objective Optimization using

Relation Epsilon-Preferred

Nicole Drechsler

, Andr

e S

ulﬂow

and Rolf Drechsler

1,3

Institute of Computer Science, University of Bremen, 28359 Bremen, Germany

solvertec GmbH, 28359 Bremen, Germany

Cyber-Physical Systems, DFKI GmbH, 28359 Bremen, Germany

Keywords:

Many-Objective Optimization, Nurse Rostering Problem, Relation ε-Preferred, User Preferences.

Abstract:

During the last 10 years, many-objective optimization problems, i.e. optimization problems with more than

three objectives, are getting more and more important in the area of multi-objective optimization. Many real-

world optimization problems consist of more than three mutually dependent subproblems, that have to be

considered in parallel. Furthermore, the objectives have different levels of importance. For this, priorities

have to be assigned to the objectives. In this paper we present a new model for many-objective optimization

called Prio-ε-Preferred, where the objectives can have different levels of priorities or user preferences. This

relation is used for ranking a set of solutions such that an ordering of the solutions is determined. Prio-ε-

Preferred is controlled by a parameter ε, that is problem speciﬁc and has to be adjusted experimentally by the

designer. Therefore, we also present an extension called Adapted-ε-Preferred (AEP), that determines the ε

values automatically without any user interaction. To demonstrate the efﬁciency of our approach, experiments

are performed.

1 INTRODUCTION

Many real-world optimization problems consist of

several mutually dependent subproblems that have

to be optimized in parallel. The so called Multi-

Objective Optimization (MOO) problems and ap-

proaches for solving them have been intensively stud-

ied in the past. For this, in the area of Evolution-

ary Algorithms (EAs) many models and algorithms

for MOO have been presented (Fonseca and Flem-

ing, 1995; Zitzler and Thiele, 1999; Deb, 2001;

Bader and Zitzler, 2011). If more than three opti-

mization objectives are considered, the corresponding

MOO problems are called

Many-Objective Optimiza-

tion problems in literature. Especially, real-world op-

timization problems often have more than three objec-

tives (Drechsler et al., 2001; Hughes, 2007; Pizzuti,

2012). Furthermore, several MOO problems in indus-

trial applications consist of subproblems that have dif-

ferent levels of importance. The importance of a sub-

problem is speciﬁed by the user and different meth-

ods exist to model these user preferences or priori-

ties (Schmiedle et al., 2001; Wickramasinghe and Li,

2009; Wagner and Trautmann, 2012). Considering

both, many-objective optimization problems and user

preferences, there is a need for algorithms that can

combine these properties.

One classical approach to deal with multiple opti-

mization criteria is the weighted sum approach. It is

often used in industrial applications, since it is easy

to implement and on a ﬁrst view scales well (see

e.g. (Burke et al., 2004)). If only a small region of

the Pareto-front is of interest, weighted sums can be

used to control the optimization process. The ob-

jectives’ priorities can be set by the choice of the

weights. In the context of Many-Objective Opti-

mization this method reaches its limit, because it is

hard to determine the weights, such that the search is

guided in the desired direction (Drechsler et al., 2001;

Geiger, 2009). Additionally, the weighted sum ap-

proach is unable to ﬁnd compromise solutions of con-

cave Pareto fronts.

In evolutionary MOO one of the ﬁrst approaches

was the use of Pareto-optimal elements (Goldberg,

1989). Here, the goal is to explore the Pareto-set of

a given MOO problem, such that as many elements

as possible of the Pareto-set are calculated. To guide

the search, a basic Dominates relation is deﬁned, that

is used to compare the solution elements. Based on

this relation many approaches for MOO have been in-

Drechsler N., Sülﬂow A. and Drechsler R..

Incorporating User Preferences in Many-Objective Optimization using Relation Epsilon-Preferred.

DOI: 10.5220/0004496000670074

In Proceedings of the 5th International Joint Conference on Computational Intelligence (ECTA-2013), pages 67-74

ISBN: 978-989-8565-77-8

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

tensively studied (Fonseca and Fleming, 1995; Zit-

zler and Thiele, 1999; Deb, 2001). But often, these

methods only consider two or three optimization ob-

jectives (Deb, 2001). If many-objective optimization

problems are considered, these methods have several

drawbacks. For example in (Deb, 2001) it is reported,

that the number of individuals in the Pareto-set in-

creases with the number of optimization objectives.

Experiments have shown, that for 20 objectives the

ranking of solutions is nearly impossible. For this,

the ratio of solutions that cannot be distinguished us-

ing the Dominates relation is almost 100%. But, if

EAs are used, a ranking of the solutions is necessary

to guide the search.

To overcome these problems in many-objective

optimization several approaches have been presented

(see e.g. (Fleming et al., 2005; Corne and Knowles,

2007; Ishibuchi et al., 2008; Brockhoff and Zitzler,

2009; Bader and Zitzler, 2011)). A promising ap-

proach in evolutionary many-objective optimization

is objective reduction based on the Dominates relation

(Brockhoff and Zitzler, 2009). There, the considered

objectives are reduced during decision making while

preserving the dominance structure of the considered

optimization problem as much as possible. In (Brock-

hoff and Zitzler, 2009) the reduction concept has been

studied for the multi-objective knapsack problem, for

DTLZ test functions (Deb et al., 2005) with up to

25 objectives and for a radar waveform problem with

nine objectives (Hughes, 2007). The hypervolume

based MOO has also been extended to many-objective

optimization. A method that approximates the hyper-

volume indicator is presented in (Bader and Zitzler,

2011). In (Drechsler et al., 2001; di Pierro et al.,

2007; Li and Wong, 2009; S

ulﬂow et al., 2007) re-

lations are presented that distinguish between solu-

tions that are incomparable if the Dominates rela-

tion is considered. An overview and a comparison of

these methods is given in (Corne and Knowles, 2007;

Ishibuchi et al., 2008).

Furthermore, approaches are presented that con-

sider user preferences in many-objective optimization

(Wickramasinghe and Li, 2009; Auger et al., 2009;

Wagner and Trautmann, 2012). In (Wickramasinghe

and Li, 2009) a user-deﬁned distance metric is used to

guide the search on the basis of the Dominates rela-

tion. The incorporation of user preferences to the hy-

pervolume approach has been investigated in (Auger

et al., 2009; Wagner and Trautmann, 2012).

The approaches in (Drechsler et al., 2001;

Schmiedle et al., 2001; S

ulﬂow et al., 2007) are based

on a relation called Preferred. Relation Preferred is a

reﬁnement of relation Dominates, i.e. a ranking of so-

lution elements that are incomparable using the Dom-

inates relation is enabled. This results in a better

guided search if EAs are used. In (Drechsler et al.,

2001) the model has been applied to an optimization

problem from the area of computer-aided design of in-

tegrated circuits. There, ﬁve optimization objectives

have been considered in parallel. Experiments have

shown, that Preferred clearly outperforms the Domi-

nates relation and an approach based on a weighted

sum. In (Schmiedle et al., 2001) Preferred is ex-

tended, such that it can also handle different levels

of priorities. The model is applied to an approach that

makes use of Genetic Programming (Koza, 1992) in

computer-aided designed of integrated circuits.

An extension of Preferred, the so-called relation

ε-Preferred, has been introduced in (S

ulﬂow et al.,

2007). For this, Preferred is enriched by a parame-

ter ε, where ε deﬁnes a radius for each objective. If

an objective is outside this region, the corresponding

elements are “punished”. In these examinations expe-

riments are performed, where a complex scheduling

problem is considered, i.e. the Nurse Rostering Prob-

lem (NRP) is solved using an EA. The NRP is of high

practical relevance and consists of several constraints,

i.e. resource planning for employees in a hospital has

to be performed. In the experiments, an example from

a hospital that consists of 26 optimization objectives,

has been considered. It turns out, that two approaches

based on the Dominates relation and Nondominated

Sorting (NSGA-II) could be improved with respect to

quality signiﬁcantly. Additionally, ε-Preferred out-

performs Preferred with respect to quality and robust-

ness.

In this article relation ε-Preferred is considered

and further extended, such that it can handle differ-

ent levels of priorities. Optimization problems like

e.g. the NRP consist of several objectives with prob-

lem speciﬁc user preferences. For this, the new re-

lation model Prio-ε-Preferred is formally introduced.

This relation is used to guide the search of an EA,

where the parameter ε has to be set by the developer.

It is shown by experiments that the convergence be-

havior of the algorithm and the quality of the results

depend on the choice of ε. Thus, a new method is

presented, that allows to determine good choices of

ε automatically without user interaction. The result-

ing method Adapted-ε-Preferred (AEP) automatically

adapts parameter ε such that the same quality as the

“hand-crafted” results can be obtained without user

interaction.

To demonstrate the efﬁciency of Prio-ε-Preferred

and AEP several experiments are performed. The

methods are used to guide the search of an EA for the

NRP, using benchmarks from (Benchmarks, 2012),

where the user preferences are given. Considering

IJCCI2013-InternationalJointConferenceonComputationalIntelligence

these benchmarks, the number of optimization objec-

tives ranges from 9 to 17

. In the experiments dif-

ferent choices of ε are examined such the potential of

Prio-ε-Preferred is shown. A comparison to relation

Dominates (Goldberg, 1989) and method NSGA-II

(Deb, 2001) shows that Prio-ε-Preferred clearly out-

performs these approaches

. Then the inﬂuence of pa-

rameter ε is studied in detail. The new method AEP

is applied to the considered benchmarks. It is shown,

that the automatic method AEP is as good as Prio-

ε-Preferred, where parameter ε is adjusted manually.

Notice, that for AEP no user interaction is required.

The paper is structured as follows: In Section 2

previous work is reviewed. Then, the considered ap-

plication, the NRP is explained in Section 3. The

new model Prio-ε-Preferred is described in Section 4.

Then in Section 5 experimental evaluations are per-

formed that demonstrate the properties of the Prio-

ε-Preferred relation. In Section 6 the method AEP

is introduced and experimental results are discussed.

A summary of the proposed relation and methods is

given in Section 7.

2 PRELIMINARIES

To make the article self-contained, we brieﬂy give an

overview on basic techniques in evolutionary multi-

objective optimization and on relations proposed in

this ﬁeld for comparison.

2.1 Multi-objective Optimization

In general, many optimization problems consist of

several mutually dependent and conﬂicting subprob-

lems, i.e. usually the improvement in one objective

leads to the deterioration of another objective.

A multi-objective optimization problem is deﬁned

as follows: Given a search space Ω, an evaluation

function f : Ω → R

is deﬁned to calculate the ﬁt-

ness vector F(A) : ∀A ∈ Ω of size m. Then we have to

minimize (or maximize) the elements of F(A). In the

following we assume, without loss of generality, that

F has to be minimized for all objectives. According

to (Goldberg, 1989) it holds:

In contrast to (S

ulﬂow et al., 2007) benchmarks from

(Benchmarks, 2012) are taken. This makes the results com-

parable to other approaches.

A comparison to a many-objective approach with user

preferences like e.g. performed in (Auger et al., 2009) can

not be directly given, because different modellings of user

preferences are used. But, the modelling of user preferences

as presented here in combination with the hypervolume in-

dicator is an interesting task for future work.

Deﬁnition 1. Let A, B ∈ Ω.

A ≺

dominates

B :⇔

∃ j : F

(A) < F

(B) ∧F

(A) 6 F

(B), 1 6 i 6 m.

Based on this, we can describe the

Pareto set

χ as

∀p ∈ χ : @ q ∈ Ω : q ≺

dominates

As can be seen from the deﬁnition above, if two

elements A, B ∈ Ω are compared with the Dominates

relation, then A dominates B only if it is less or equal

to B in all objectives and if it is better in at least one

objective. All elements A ∈ Ω in the Pareto-set are

equal or not comparable. Usually, all points in this

set are of interest for the decision maker or designer.

2.2 Relation Preferred

To perform a reﬁnement of the Dominates relation

Preferred is deﬁned. Using the Preferred relation a

set of solutions can be classiﬁed into so-called Satisﬁ-

ability Classes (SC) (Drechsler et al., 2001). Relation

Preferred respects the number of objectives in which

A differs from B:

Deﬁnition 2. Let A, B ∈ Ω. A ≺

pre f erred

B :⇔

|{i : F

(A) < F

(B), 1 6 i 6 m}| > |{ j : F

(B) <

(A), 1 6 j 6 m}|.

Using Deﬁnition 2 we are able to compare ele-

ments A, B ∈ Ω pairwise more precisely. A is pre-

ferred to B (A ≺

pre f erred

B) iff i (i ≤ n) components

of A are smaller than the corresponding components

of B and only j ( j < i) components of B are smaller

than the corresponding components of A. We use a

graph representation for the relation, where each ele-

ment is a node and “preferences” are given by edges.

Relation Preferred is not a partial order, because it is

not transitive, i.e. the relation graph can have cycles.

Solutions that are included in a cycle are denoted as

incomparable. They are computed by a linear time

graph algorithm (Drechsler et al., 2001).

2.3 Relation ε-Preferred

To improve the robustness of relation Preferred in

many-objective optimization ε-Preferred has been in-

troduced (S

ulﬂow et al., 2007). Hence, the idea is to

deﬁne ﬁtness limits ε

, 1 ≤ i ≤ m, for each dimension.

Deﬁnition 3. Let A, B ∈ Ω and ε

, 1 6 i 6 m

A ≺

ε−exceed

B ⇔

|{i : F

(A) < F

(B) ∧ |F

(A) − F

(B)| > ε

}| >

|{ j : F

(A) > F

(B) ∧ |F

(A) − F

(B)| > ε

}|.

ε-exceed additionally takes the distance of the so-

lutions components into account. Using ε-exceed the

extension ε-Preferred is deﬁned as follows:

IncorporatingUserPreferencesinMany-ObjectiveOptimizationusingRelationEpsilon-Preferred

Deﬁnition 4. Given two solutions A, B ∈ Ω.

A ≺

ε−pre f erred

B ⇔

A ≺

ε−exceed

B ∨ (B ⊀

ε−exceed

A ∧ A ≺

pre f erred

First it is counted how often a solution exceeds the

ε-limits and the better solution is determined. If both

solutions are in the given range Preferred is used for

comparison.

Example 1. Consider some solution vectors from R

i.e. the results of three objective functions:

(7,0,9) (8,7,1) (1,9,6)

Additionally, let ε

= 5, 1 ≤ i ≤ 3. (7,0,9)

≺

ε−pre f erred

(8,7,1), because for the second objective

it holds |0 − 7| > ε

, where solution (7,0,9) “wins”,

and for the third it holds |9 − 1| > ε

, where so-

lution (8,7,1) “wins”. Since each solution has an

ε-exceeding objective, Preferred is used for com-

parison. The same argumentation holds for (8,7,1)

≺

ε−pre f erred

(1,9,6) and (1,9,6) ≺

ε−pre f erred

(7,0,9).

3 THE APPLICATION:

UTILIZATION PLANNING

PROBLEM

The problem of utilization planning, i.e. the Nurse

Rostering Problem (NRP) (Burke et al., 2004; Burke

et al., 2012), is very complex and cannot fully be de-

scribed with all details. For this, we brieﬂy highlight

the main aspects to give an idea of the underlying op-

timization problem.

The problem is to determine a schedule for em-

ployees at a hospital. In our examinations schedules

for up to 16 persons for a planning period of 30 days

are computed. The computation of the ﬁtness can be

roughly categorized in three main areas:

1. Rules resulting from ergonomics, e.g. having reg-

ular shifts

2. Restrictions by law, e.g. maximal hours of work

per day or maximal working days per month

3. Rules of the nurse station, e.g. sufﬁcient nurses

per shift

Some of these constraints are “hard” in the sense

that they have to be fulﬁlled, while others are “soft”,

i.e. they improve the ﬁtness, but also without them

valid schedules result. All together up to 30 optimiza-

tion objectives are inﬂuencing the ﬁtness function.

Each one might have a different inﬂuence, e.g. some

are linear while others are exponential regarding their

inﬂuence.

In our application we make use of benchmarks

for the nurse rostering problem that are reported in

(Burke et al., 2012). The benchmarks are available

Name/Day

1 2 3 4 5 6 7 8 9 ... 30

C. Meyer

S S D D V V V - N

...

J. Smith

D D - - N N N - -

...

J. Doe

- - D D D D - - -

...

J. Blow

F F - L L L - - F

...

J. Bloggs

S V V - - N N N -

...

Figure 1: Example nurse rostering schedule.

from (Benchmarks, 2012). There, the rules of the

nurse station are modeled as hard constraints. The

remaining optimization objectives are considered as

soft constraints, where a weight for each rule is given

in the benchmark. Thus, a weighted sum can be

constructed in such a way that a single ﬁtness value

is calculated for each employee and schedule. Fol-

lowing this, we get one ﬁtness value for each em-

ployee that has to be optimized in parallel. The ﬁtness

values are stored in an m-dimensional vector, where

m = #employees +1. One dimension gives the objec-

tive function for the hard constraint and the remaining

dimensions the soft constraints for each employee.

Example 2. To give a better understanding of the ap-

proach in Figure 1 a sketch of a schedule is given. De-

pending on the grade of training the optimization al-

gorithm assigns exactly one shift to a person per day.

In this example all given shifts are marked with a let-

ter. These letters have the following meaning: Day

shift (D), Night shift (N), Late shift (L), Vacation (V),

Free shift (F) and Stand-by shift (S). For more details

see (Burke et al., 2004; Burke et al., 2012; Bench-

marks, 2012).

4 RELATION Prio-ε-Preferred

Most often in real-world applications the objectives

have different priorities, i.e. one objective is more im-

portant than another objective. For this it is of high

relevance to model the priorities during the optimiza-

tion process. For the NRP the user preferences have

been pointed out in the previous section. In the fol-

lowing we propose a new model that combines multi-

ple optimization criteria with user preferences.

4.1 Deﬁnition

To model priorities of optimization objectives in this

approach a new relation Prio-ε-Preferred is deﬁned.

In (Schmiedle et al., 2001) relation Priority Preferred

has been deﬁned that combines relation Preferred

IJCCI2013-InternationalJointConferenceonComputationalIntelligence

with priorities. Following this, the model proposed

in this approach combines relation ε-Preferred with a

lexicographic ordering of the objectives.

Let us assume that priorities 1, 2, . . . , k are as-

signed to the m objectives in an ascending ordering,

i.e. the lower the index i, 1 ≤ i ≤ k, the higher the pri-

ority.

Deﬁnition 5. Let p = (p

, . . . , p

) be a priority vec-

tor. p

determines the number of objectives that have

priority i. The priority of an objective is calculated by

the function pr : {1, . . . , m} → { 1, . . . , k}. The sub-

vector of objectives c|

of priority i is deﬁned as

∈ R

, c|

= (c

, . . . , c

where

r =

i−1

∑

j=1

+ 1 ∧ s =

∑

j=1

For A, B ∈ Ω relation ≺

ε−priopre f

(Prio-ε-

Preferred) is deﬁned by:

A ≺

ε−priopre f

B :⇔

∃ j ∈ {1, . . . , k} : A|

≺

ε−pre f erred

∧

(∀h < j : A|

⊀

ε−pre f erred

∧ B|

⊀

ε−pre f erred

)

More informally, Prio-ε-Preferred considers the

objectives that have the same priority and then per-

forms comparisons using the ε-Preferred relation on

these objectives. The objectives are considered in de-

scending order of their priority. A solution A is better

than B with respect to relation Prio-ε-Preferred if the

objectives with highest priority of A are ε-Preferred to

B. If they are not comparable or equal, the objectives

with the next priority in descending ordering are con-

sidered and compared using the ε-Preferred relation.

This is done until the better solution, A or B, is found

or all priorities are considered. If no better solution is

found, A and B are denoted as incomparable.

Example 3. Let us consider a problem with 5

objectives and 3 different priorities. Let c =

, c

) a solution vector and p = (1, 3, 1)

a priority vector, i.e. one objective has priority 1

(i.e. p

= 1), three objectives have priority 2 (p

= 3)

and one objective has priority 3 (p

= 1). This leads to

the function pr with pr(1) = 1, pr(2) = 2, pr(3) = 2,

pr(4) = 2, and pr(5) = 3 what means that the ﬁrst

objective has priority 1, the second objective prior-

ity 2, and so on. For priority 2 the projection is

∈ R

, c|

= (c

, c

), since r = 1 + 1 = 2 and

s = 1 + 3 = 4.

Now, let us consider two solution vectors, A =

(2, 7, 0, 9, 15) and B = (2, 1, 9, 6, 5). Then it holds,

that B ≺

ε−priopre f

A. For this, ﬁrst the objectives with

priority 1 are compared. Since they are equal, next

the objectives with priority 2 are compared with re-

lation ≺

ε−pre f erred

, i.e. (1, 9, 6) ≺

ε−pre f erred

(7, 0, 9)

(see Example 1) which leads to B ≺

ε−priopre f

A. The

last objective does not have to be considered anymore,

because it has lowest priority and the decision, which

solution is better with respect to relation ≺

ε−priopre f

has already been made.

Then, after a pairwise comparison using Prio-

ε-Preferred analogously to (S

ulﬂow et al., 2007) a

relation graph is constructed. Then the strongly-

connected components are computed to perform a ﬁ-

nal ranking as described in (S

ulﬂow et al., 2007).

4.2 Methods

To test and compare the methods proposed in this pa-

per, a framework of an Evolutionary Algorithm (EA)

that optimizes schedules of the NRP has been used.

Details of the representation of the individuals and the

genetic operators are left out due to page limitation.

The objective function measures the ﬁtness of each

individual/schedule, i.e. the rules given by the bench-

marks are evaluated and the hard constraint for the

schedule and soft constraints for each employee are

calculated. These optimization rules are given in the

benchmarks. The ﬁrst objective of the m-dimensional

ﬁtness vector gives the hard constraint and the re-

maining objectives the soft constraints. The priorities

for the optimization objectives are set such that the

ﬁrst objective, the hard constraint, is of highest prior-

ity 1 and the soft constraints for the employees have

priority 2. Using the notation from Deﬁnition 5, for

the two priorities we have p = (1, m − 1), pr(1) = 1

and pr(i) = 2, 2 ≤ i ≤ m.

5 EXPERIMENTAL RESULTS

In this section we give an insight into the behavior

of the many-objective optimization methods for the

nurse rostering problem presented in this paper. For

the experiments, the algorithms are applied to bench-

marks that are taken from (Benchmarks, 2012).

To compare the results of the high dimensional op-

timization, we use a weighted sum approach to trans-

form the ﬁtness vector back to one dimension. The

justiﬁcation of the weights results from the experi-

ence of experts and they are given in the benchmarks.

To measure the inﬂuence of random seeds on the re-

sults, the random number generator has been initial-

ized with 10 different values. The results in the fol-

lowing give the average value for these 10 runs. The

average values of the results are used for comparison

of the methods in the following. In all experiments

the population size is set to 50 and the EA runs for

5000 generations.

IncorporatingUserPreferencesinMany-ObjectiveOptimizationusingRelationEpsilon-Preferred

Table 1: Fitness for generation 5000.

Benchmark #Objectives Method AV G Quality

NSGA-II

36993 0%

GPost 9

PrioPref

10512 71%

Prio

Pref

-5000 10485 72%

Weighted Sum 7159 80%

NSGA-II

2140 0%

Millar-2Shift-DATA1 9

PrioPref

2670 -24%

Prio

Pref

-5000 2720 -27%

Weighted Sum 1310 39%

NSGA-II

92022 0%

ORTEC01 17

PrioPref

12976 86%

Prio

Pref

-5000 15448 83%

Weighted Sum 9132 90%

NSGA-II

114604 0%

Valouxis-1 17

PrioPref

21048 82%

Prio

Pref

-5000 19872 83%

Weighted Sum 13986 88%

Table 2: Comparison of different epsilon values after 5000 generations.

Benchmark

Prio

Pref

-5000

Prio

Pref

-1000

Prio

Pref

-500

Prio

Pref

-10

AVG Quality AVG Quality AVG Quality AVG Quality

GPost 10485 0% 9014 14% 7925 24% 8502 19%

Millar-2Shift 2720 0% 1970 28% 1540 43% 2340 14%

ORTEC01 15448 0% 11181 28% 11275 27% 14496 6%

Valouxis-1 19872 0% 17508 12% 17018 14% 17954 10%

First, the presented approaches

PrioPref

and

Prio

Pref

are compared to the well-known method

NSGA-II

. The results are summarized in Table 1. The

epsilon value in

Prio

Pref

is set to 5000 by the user.

Experiments have shown that this setting is a good

starting point for our investigations. It can be seen

that both methods based on relation Preferred outper-

form

NSGA-II

enormously in most cases. Only for

benchmark Millar-2Shift-DATA1

NSGA-II

has a bet-

ter performance, but for the other cases it is improved

by more than 70%. In row Weighted Sum additionally

the results for the weighted sum approach, i.e. a sin-

gle objective optimization approach, are given. The

results obtained by the weighted sum are better than

the results calculated by the multi-objective optimiza-

tion methods. This can be explained by the fact that

the benchmarks are designed such that the weights are

directly given, which is an advantage of the weighted

sum approach.

For the experiments above a relatively high ep-

silon value has been used. In a next series of expe-

riments we brieﬂy discuss the inﬂuence of alternative

choices, the results are summarized in Table 2. By

this, directions for improvements are pointed out (see

also Section 6).

As reference for the experiments we use the ep-

silon value of 5000 from the previous section. The

results are given in column

Prio

Pref

-5000. In a

ﬁrst run denoted in column

Prio

Pref

-1000 the ep-

silon values of the optimization objectives were set

to 1000. Even by these ﬁrst experiments, the quality

could be improved by more than 10% to nearly 30%.

The trend holds, if epsilon value 500 is considered.

Especially for benchmark Millar-2Shift-DATA1

im-

provements of more than 40% can be observed. If the

value of ε was too low, e.g. 10 as given in Table 2 in

column

Prio

Pref

-10, the quality of the results is de-

creasing. For all considered benchmarks the average

value of the ﬁtness function is worse than the quality

for epsilon values 1000 and 500, respectively. The ex-

periments also show that the best choice of the epsilon

value depends on the considered benchmark. For OR-

TEC01 the best average results are determined with

epsilon value 1000, while the best results for GPost,

Millar-2Shift-DATA1 and Valouxis-1 are obtained for

epsilon value 500. Thus, there is a need for meth-

ods to determine a good epsilon value for the bench-

In Table 2 and Table 3 Millar-2Shift is the abbreviation

for benchmark Millar-2Shift-DATA1.

IJCCI2013-InternationalJointConferenceonComputationalIntelligence

Table 3: Comparison of methods for adapted epsilon values after 5000 generations.

Benchmark

NSGA-II Prio

Pref

-1000

Prio

Pref

-500 SEP AEP

AVG Quality AVG Quality AVG Quality AVG Quality AVG Quality

GPost 36993 0% 9014 75% 7925 79% 15755 57% 7557 80%

Millar-2Shift 2140 0% 1970 8% 1540 28% 2400 -12% 1590 26%

ORTEC01 92022 0% 11181 88% 11275 88% 14058 85% 11672 87%

Valouxis-1 114604 0% 17508 85% 17018 85% 22594 80% 16692 85%

mark under consideration automatically. In summary,

based on relation Prio-ε-Preferred the quality mea-

sured by the ﬁtness value could be signiﬁcantly im-

proved. Even for example Millar-2Shift-DATA1 bet-

ter results have been calculated. But as can be seen

by the experiments, there is a need to determine good

problem speciﬁc epsilon values. For this, in the next

section an approach is presented that sets the epsilon

values automatically, i.e. no user interaction is needed

anymore.

6 ADAPTATION OF EPSILON

VALUES

In this section ﬁrst a description of the automatic

adaptation of the epsilon values is given. Then ex-

perimental results are carried out to demonstrate the

efﬁciency of the approach.

6.1 The Idea

In the last section we gave insight into the inﬂuence

of the epsilon values during the optimization process.

Altogether, the choice of the epsilon values has a large

inﬂuence on the quality of the results. Now, the prob-

lem is to ﬁnd a good epsilon value, such that the algo-

rithm has its best performance. For this, two methods

Separated ε-Preferred (SEP) and Adapted ε-Preferred

(AEP) have been developed that are introduced in the

following. Starting from an initial point, both meth-

ods reduce the epsilon values used throughout the al-

gorithm automatically.

For method SEP for each objective one separated

epsilon value is provided. It is determined such that

for each objective the average ﬁtness value over the

whole population is calculated:

|P|

∑

i=1

Ind

i, j

|P|

, 1 ≤ j ≤ m,

where m is the number of objectives, |P| is the size of

the population and Ind

i, j

is the j-th objective of the

i-th individual in population P. The epsilon values

, 1 ≤ j ≤ m, are updated in each generation.

In contrast, for method AEP one epsilon value for

all objectives is determined. Therefore, one individ-

ual out of the best Satisﬁability Class (SC) derived

Prio

Pref

is randomly chosen. The new epsilon

value is determined by the average value of all objec-

tives of that individual:

ε =

∑

j=1

Ind

best, j

where Ind

best, j

is the j-th objective of a randomly cho-

sen individual out of the best SC. The epsilon value

is updated in each generation. The idea behind this

method is that individuals can be distinguished by re-

lation Prio-ε-Preferred, if the difference of the indi-

viduals exceeds the calculated average range.

6.2 Experimental Evaluation

The results of methods SEP and AEP are summa-

rized in Table 3. For comparison, the results that

are derived from

NSGA-II

and

Prio

Pref

with ep-

silon values 1000 and 500 are shown in the table.

First, we take a closer look to column SEP where

the results of separated epsilon values are summa-

rized. The average is better than the average values

derived by

NSGA-II

, but worse than the results de-

rived by

Prio

Pref

-500 and

Prio

Pref

-1000. In

column AEP the method for automatic setting of ep-

silon values shows the same performance than the al-

gorithms that resulted from setting the epsilon values

manually. If doing so, many experiments have to be

performed to ﬁnd a good choice of epsilon. In con-

trast, if using AEP epsilon is set automatically and

no user interaction is necessary. Even in two cases

(GPost, Valouxis-1) the best average quality over all

considered epsilon values could be improved. For this

it is recommended to use method AEP.

7 CONCLUSIONS

Many-objective optimization is becoming more im-

portant in real-world applications. In this article an

industrial scheduling problem has been examined. It

consists of up to 17 optimization objectives that have

IncorporatingUserPreferencesinMany-ObjectiveOptimizationusingRelationEpsilon-Preferred

to be optimized in parallel. The objectives have differ-

ent levels of importance that are deﬁned by the user.

A new relation model called Prio-ε-Preferred for

incorporating user preferences in many-objective op-

timization has been presented and experimentally

evaluated. It was shown, that high quality results

could be obtained, i.e. the standard method NSGA-

II was improved by more than 80%.

Furthermore, a new automatic technique for deter-

mining the parameters for relation Prio-ε-Preferred

automatically has been developed. In this context, the

epsilon values have been reduced dynamically during

the optimization run. Experiments showed that best

quality results could be obtained while adjusting the

parameters automatically.

REFERENCES

Auger, A., Bader, J., Brockhoff, D., and Zitzler, E. (2009).

Articulating user preferences in many-objective prob-

lems by sampling the weighted hypervolume. In

Genetic and Evolutionary Computation Conference,

pages 555–562.

Bader, J. and Zitzler, E. (2011). HypE: An algorithm for

fast hypervolume-based many-objective optimization.

Evolutionary Computation, 19(1):45–76.

Benchmarks (2012). Employee scheduling benchmark data

set: http://www.cs.nott.ac.uk/ tec/nrp/. Technical re-

port, ASAP, School of Computer Science, The Uni-

versity of Nottingham, UK.

Brockhoff, D. and Zitzler, E. (2009). Objective reduction in

evolutionary multiobjective optimization: Theorie and

applications. Evolutionary Computation, 17(2):135–

166.

Burke, E., Causmaecker, P. D., Berghe, G., and Landeghem,

H. (2004). The state of the art of nurse rostering. Jour-

nal of Scheduling, 7:441–499.

Burke, E., Curtois, T., Qu, R., and Vanden-Berghe, G.

(2012). Problem model for nurse rostering benchmark

instances. Technical report, ASAP, School of Com-

puter Science, University of Nottingham, UK.

Corne, D. and Knowles, J. (2007). Techniques for highly

multiobjective optimization: Theorie and applica-

tions. In Genetic and Evolutionary Computation Con-

ference, pages 773–780.

Deb, K. (2001). Multi-objective Optimization using Evolu-

tionary Algorithms. John Wiley and Sons, New York.

Deb, K., Thiele, L., Laumanns, M., and Zitzler, E.

(2005). Scalable test problems for evolutionary multi-

objective optimization. In In Evolutionary Multiob-

jective Optimization: Theoretical Advances and Ap-

plications, pages 105–145.

di Pierro, F., Khu, S., and Savic, D. (2007). An investi-

gation on preference order ranking scheme for multi-

objective optimization. IEEE Trans. on Evolutionary

Comp., 11(1):17–45.

Drechsler, N., Drechsler, R., and Becker, B. (2001). Multi-

objective optimisation based on relation favour. In

Int’l Conference on Evolutionary Multi-Criterion Op-

timization, pages 154–166.

Fleming, P., Purshouse, R., and Lygoe, R. (2005). Many-

objective optimization: An engineering design per-

spective. In International Conference on Evolutionary

Multi-Criterion Optimization, pages 14–32.

Fonseca, C. and Fleming, P. (1995). An overview of evo-

lutionary algorithms in multiobjective optimization.

Evolutionary Computation, 3(1):1–16.

Geiger, M. (2009). Multi-criteria curriculum-based course

timetabling - a comparison of a weighted sum and a

reference point based approach. In International Con-

ference on Evolutionary Multi-Criterion Optimiza-

tion, pages 290–304.

Goldberg, D. (1989). Genetic Algorithms in Search, Opti-

mization & Machine Learning. Addison-Wesley Pub-

lisher Company, Inc.

Hughes, E. (2007). Radar waveform optimization as a

many-objective application benchmark. In Interna-

tional Conference on Evolutionary Multi-Criterion

Optimization, pages 700–714.

Ishibuchi, H., Tsukamoto, N., and Nojima, Y. (2008). Evo-

lutionary many-objective optimization: A short re-

view. In IEEE Congress on Evolutionary Computa-

tion, pages 2424–2431.

Koza, J. (1992). Genetic Programming - On the Program-

ming of Computers by means of Natural Selection.

MIT Press.

Li, X. and Wong, H. (2009). Logic optimality for multi-

objective optimization. Applied Mathematics and

Computation, 215:3045–3056.

Pizzuti, C. (2012). A multiobjective genetic algorithm to

ﬁnd communities in complex networks. IEEE Trans.

on Evolutionary Comp., 16(3):418–430.

Schmiedle, F., Drechsler, N., Große, D., and Drechsler, R.

(2001). Priorities in multi-objective optimization for

genetic programming. In Genetic and Evolutionary

Computation Conference, pages 129–136.

ulﬂow, A., Drechsler, N., and Drechsler, R. (2007). Ro-

bust multi-objective optimization in high-dimensional

spaces. In International Conference on Evolutionary

Multi-Criterion Optimization, pages 715–726.

Wagner, T. and Trautmann, H. (2012). Integration of pref-

erences in hypervolume-based multiobjective evolu-

tionary algorithms by means of desirability functions.

IEEE Trans. on Evolutionary Comp., 14(5):688–701.

Wickramasinghe, U. and Li, X. (2009). A distance met-

ric for evolutionary many-objective optimization algo-

rithms using user-preferences. In 22nd Australasian

Joint Conference on Advances in Artiﬁcial Intelli-

gence (AI’09), pages 443–453.

Zitzler, E. and Thiele, L. (1999). Multiobjective evolu-

tionary algorithms: A comparative case study and the

strength pareto approach. IEEE Trans. on Evolution-

ary Comp., 3(4):257–271.

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