IND-NIMBUS Software for Multiobjective Optimization - wseas.us

Proceedings of the 6th WSEAS International Conference on Simulation, Modelling and Optimization, Lisbon, Portugal, September 22-24, 2006

IND-NIMBUS Software for Multiobjective Optimization

VESA OJALEHTO Department of Mathematical Information Technology University of Jyväskylä P.O. Box 35 (Agora), FI-40014 University of Jyväskylä FINLAND KAISA MIETTINEN∗ and MARKO M. MÄKELÄ† [email protected]., makela@utu., kaisa.miettinen@hse.

Abstract:

We describe the main features and principles of the IND-NIMBUSdmst system. This system is

an implementation of the interactive NIMBUS method for multiobjective optimization. The system can be connected with dierent types of software describing the mathematical model of the multiobjective optimization problem to be solved.

Keywords:

Multiple objective programming, interactive methods, NIMBUS, nonlinear programming,

decision support, optimization tool

1

Introduction

Practical

optimization

IND-NIMBUS is a portable desktop application tarapplications

typically

geted for commercial industrial application in addi-

in-

tion to academic use.

volve several conicting criteria to be optimized simultaneously.

This has been a fruitful starting

point for the development of various optimization

2

methods for multiple objectives (see, e.g. [11] and

Multiobjective optimization

In classical optimization we have only one objective

references therein). Despite of that the lack of soft-

function to be optimized. However, in real life ap-

ware for multiobjective optimization is, however,

plications it is typical to have several conicting ob-

evident. In order to ll this gap WWW-NIMBUS

jectives, that should be optimized simultaneously.

was developed at the University of Jyväskylä in

For this reason the classical single objective opti-

1995 [14]. The WWW-NIMBUS system is an im-

mization methods are not applicable as such, but

plementation of the interactive NIMBUS method

we need special multiobjective optimization meth-

[11, 12, 13, 14, 16] and it is operating on the Inter-

ods. For further information see [11].

net being freely available for academic use.

In

Another feature of practical optimization ap-

multiobjective

plications is that the functions characterizing the minimize

problem are not necessarily available in an explicit

subject to

form but they can be evaluated, for example, as a system of partial dierential equations.

This is

involving

a challenge for the optimization software, since it

Rn → R

should be able to connect with dierent simulation

The

codes in a exible way.

xn )T

consider

{f1 (x), f2 (x), . . . , fk (x)} x∈S

k (≥ 2) conicting objective

(1)

functions fi :

decision (variable) vectors x = (x1 , x2 , . . . , are supposed to belong to the nonempty com-

feasible region S ⊂ Rn . Objective vectors f (x) = (f1 (x), f2 (x), . . . , fk (x))T consist of objective (function) values and the image of the feasible

NIMBUS, a new implementation of the NIMBUS method for industrial optimization applications.

†

we

that we want to minimize simultaneously.

pact

The aim of this research was to develop IND-

∗

optimization

problems of the form

Helsinki School of Economics, P.O Box 1210, FI-00101 Helsinki, FINLAND University of Turku, Department of Mathematics, FI-20014 Turku, FINLAND

1

654


feasible objective region .

region is called a

655

and upper bounds. These subproblems are solved

In multiobjective optimization, objective vec-

to get one to four new Pareto optimal solutions.

tors are regarded as optimal if their components

(In the earlier versions of the NIMBUS method

cannot be improved without deterioration to at

[11, 12, 13, 14], only one subproblem was formed

least one of the other components. More precisely,

and solved.) However the same preference informa-

a decision vector

x0 ∈ S

is called

Pareto optimal

tion can be the basis of several dierent subprob-

if there does not exist another x ∈ S such that fi (x) ≤ fi (x0 ) for all i = 1, . . . , k and fj (x) < fj (x0 ) for at least one index j . Furthermore, an objective

lems (see [15].)

3.1

vector is Pareto optimal if the corresponding decision vector is Pareto optimal.

Next we describe shortly the current synchronous

The set of Pareto

NIMBUS method.

optimal objective vectors is called the Pareto opti-

set

Because all the Pareto optimal solutions are

maker

For more details, see [16].

A.

denote the set of saved solutions by

mal set. mathematically

Synchronous NIMBUS algorithm

equivalent,

we

need

a

decision

A = ∅.

We

At rst, we

The starting point of the solution pro-

cess can come from the decision maker or it can be some neutral compromise between the objectives.

(DM) to identify the most preferred one

among them. The decision maker is a person who 1. Get a Pareto optimal starting point.

can express preference information related to the conicting objectives and we assume that less is pre-

2. Ask the decision maker to classify the objec-

ferred to more in each objective for her/him.

tive functions at the current solution and to

3

specify the possible aspiration levels and up-

NIMBUS method

per bounds.

NIMBUS is an interactive multiobjective optimizaNIMBUS

3. Ask the decision maker to select the maxi-

has been used in solving several real-life problems.

mum number of dierent solutions to be gen-

Among others, it has been applied in a structural

erated (between one and four) and solve as

design problem [17], in the optimal control problem

many subproblems.

tion method (see [11, 12, 13, 14, 16]).

of the continuous casting of steel [18] and in the 4. Present the dierent new solutions obtained

paper machine headbox design [7].

to the decision maker.

The idea of NIMBUS is that the DM examines the values of the objective functions calculated at

5. If the decision maker wants to save one or

a current Pareto optimal decision vector and classies the objective functions into up to ve classes.

more

of

the

This means that the DM is asked to indicate (by

it/them to

new

solutions

to

A,

include

A.

the means of the classication) what kind of a so6. If the decision maker does not want to see

lution would be more satisfactory than the current

intermediate solutions between any two solu-

one. The classes are for functions whose values

tions, go to step 8.

Otherwise, ask the de-

cision maker to select the two solutions from

should be decreased,

among the new solutions or solutions in should be decreased to some aspiration level,

A.

Ask the number of new intermediate solutions from the decision maker.

are satisfactory at the moment,

7. Generate the desired number of Pareto opti-

are allowed to increase up till some upper bound,

mal intermediate solutions. Go to step 4. are allowed to change freely. 8. Ask the decision maker to choose the most preferred one among the new and/or interme-

In the so-called synchronous NIMBUS method

A.

from one to four dierent single objective sub-

diate solutions or solutions in

problems are formulated based on the classica-

cision maker wants to continue, go to step 2.

tion as well as the corresponding aspiration levels

Otherwise, stop.

2

If the de-


cally or graphically by indicating desirable changes

The algorithm is terminated if the DM does not

in a bar chart with a mouse.

want to decrease any objective value or is not willing to let any objective value increase. Otherwise,

In the WWW-NIMBUS system the user can

the search continues iteratively by moving around

save,

the Pareto optimal set.

load and modify problems.

The exible

In this way, the DM can

database management enables saving selected solu-

learn about the problem, adjust ones hopes and -

tions whenever required. In this way, the user can

nally identify the most desirable solution.

comfortably return to previous solutions if they turn out to be interesting, after all.

3.2

NIMBUS method implementation

The feasible region

S

The user can also

graphically illustrate a (sub)set of saved solutions and not only the latest ones, and generate interme-

may consist of nonlinear and

diate solutions between any two solutions. Because

linear equality and inequality constraints as well as

of the structure of the algorithm, all the solutions

lower and upper bounds for the variables. The ob-

handled are Pareto optimal.

jective functions can be either minimized or maximized. The

single

objective

subproblems

that

are

4

formed based on the classication information, can be solved with dierent underlying solvers.

IND-NIMBUS is a multi-platform desktop applica-

If the

tion implementing the NIMBUS algorithm. Unlike

DM wishes to use a local algorithm, it is possible to

WWW-NIMBUS, it does not contain any tools to

use the proximal bundle method [10]. This method

formulate the optimization problem. Instead, it is

can solve even nondierentiable problems and it as-

assumed that the problem is formulated in another

sumes the objective and the constraint functions to

program, which provides the necessary information

be locally Lipschitz continuous. Note that it needs

for the IND-NIMBUS software.

(sub)gradient information.

In IND-NIMBUS, the DM has access to all so-

Alternatively, if the DM wishes to use a global optimization

solver,

he

can

solve

with an genetic algorithm [3].

the

lutions generated during the process. Any solution

problem

can be selected as a starting solution for a new clas-

NIMBUS contains

sication, or as an end point for generating interme-

two variants of genetic algorithms with dierent

diate solutions. In this way, the DM can generate a

constraint-handling techniques, based on adaptive

representation of the Pareto optimal set according

penalties [4] and method of parameter free penal-

to her/his preferences for further study

ties [2]. See also [19]

In order to organize these solutions, the DM can

It is also possible to use other global solvesr

create dierent lters, for example to show only

like Controlled Random Search (CRS) [20] or Dif-

those solutions where the rst objective function

ferential Evolution [22] global solvers, but at this time

these

are

available

only

for

internal

IND-NIMBUS

has values above zero.

use.

There is also a possibility

to view only those solutions obtained from the last

These solvers have two dierent variants, due their

classication or alternative generation.

constraint-handling techniques.

The DM

can also store the best solution candidates to the Best Candidates panel at any point during the

3.3

WWW-NIMBUS

solution process. The undesirable solutions can be

WWW-NIMBUS is an freely available implemen-

deleted from the solution list.

tation of the NIMBUS algorithm operating on the

The graphics used in IND-NIMBUS are gener-

Internet [14]. WWW-NIMBUS has been developed

ated using a standalone package, which can also be

for solving nonlinear and mathematically challeng-

used to generate graphics for other NIMBUS imple-

ing multiobjective optimization problems. WWW-

mentations. One of the main duties of the graph-

NIMBUS can be accessed at

http://nimbus.mit.

ics package is to generate the visualization graph-

jyu.fi/ and can be used, for example, for teaching

ics for comparing dierent Pareto optimal solutions.

purposes.

These visualization graphics include two-dimension

The problem to be solved in WWW-NIMBUS

and three-dimension bar-chart, spider web chart,

can be specied for the system either by lling a

value paths, whisker plot, petal diagram and multi-

web form or as a subroutine. The classication of

way dot plot. The same ltering tools are available

the objective functions can be carried out symboli-

for visualizations as for other action windows.

3

656


Figure 1: IND-NIMBUS classication window

r

BALAS

Numerical variable and objective function val-

[1, 5, 6] is a process simulator de-

ues of each solution can also be viewed, and ex-

veloped at the VTT Technical Research Center

ported, from the IND-NIMBUS software.

of Finland.

There

It operates on the Windows operat-

also exists views for changing the default parameter

ing system, consisting of a separate program Flow-

values of optimization solvers used, and for viewing

sheet for the creation of the process owsheet and

possible warning messages these solvers have pro-

the BALAS software, which formulates the process

duced during the optimization process.

simulation problem to be optimized.

The IND-NIMBUS classication window can be

Numerrin 2.0 is a Linux based software for phe-

seen in Figure 1. Here the classication information

nomenon based physical modeling of real-world sys-

is given by clicking the objective function bars on

tems.

the left. The DM can give the preferred aspiration

tran 95 programming language, using the nite el-

level or the upper bound to the edit box next to each

ement analysis library included [8].

function bar. Alternatives between any two Pareto

Numerrin models are created by the For-

The virtual paper making simulation and opti-

optimal solutions can be generated in a secondary

mization system (MOP) is designed to provide sup-

window, not shown here.

port for solving complex paper making problems in

It should be noted that the IND-NIMBUS user

a paper machine.

interface is concurrent. In other words, the user can

It contains several unit process

models that are combined as one large multidisci-

move between dierent windows, even if some op-

plinary simulation model [9].

eration is still running. Nevertheless, at this time

The MOP software

operates on the Windows platform.

there can be only one concurrent computational op-

MATLAB

eration running at a time.

r

is a interactive environment that

can be used to solve a wide range of problems. MATLAB is used with IND-NIMBUS system to de-

4.1

Using

IND-NIMBUS

with

third

velop a multicriteria model for intensity modulated

party applications

radiotherapy [21].

As stated previously, the IND-NIMBUS software

In practice, there exist two dierent ways of

does not include tools to formulate optimization

connecting the optimization problem to the IND-

problem to be solved.

NIMBUS system. In a simpler approach, the prob-

Therefore, IND-NIMBUS

software must be used in conjunction with a exter-

lem can be given as an executable le.

nal simulation software. The problem dimensions,

cutable contains or dynamically links to the IND-

objective functions, variable names, and other in-

NIMBUS software for the actual optimization. Fur-

formation dening the problem in question must be

thermore, the executable must be able to access all

specied in a external software. For this purpose,

tools needed to formulate and solve the problem

the following applications have been used as IND-

in question.

NIMBUS test cases.

executable is run for each of the NIMBUS opera-

4

This exe-

During the optimization process this

657


tions as a separate process, and the communication

In addition, several parts of the NIMBUS algo-

between the executable and the IND-NIMBUS soft-

rithm could be run in parallel. Using suitable tools,

ware is done by services provided by the underlying

simple parallelization of the initial step and classi-

operating system.

cation operations should not be an overwhelming

However, it is likely that the simulation pro-

task. As the current CPU development trends are

cesses generated by a simulation software cannot be

heading towards multiprocessor systems, the paral-

separated to a single executable. In this case, the

lelization could in the future oer clear advantages

simulation process and the IND-NIMBUS software

regarding the time used for the optimization pro-

are controlled with a separate application, which

cess, even in standard desktop environments. Fur-

contains the NIMBUS implementation.

thermore, the industrial applications are often computationally very demanding, so with suitable par-

In both approaches the communication between

allelization implementation the optimization pro-

IND-NIMBUS and the problem entity is done via

cess could be distributed to several dierent com-

standard les. There exist some basic facilities for

puters.

socket based communication, but these have not yet

It could also be of some interest to store the

been used in any of the test cases used in the IND-

dierent objective function values obtained during

NIMBUS development.

the optimization process.

As mentioned earlier, the IND-NIMBUS soft-

This information could

be used to examine the simulation model after the

ware has some data visualization tools but it is also

optimization, and with these values it would be pos-

possible to use problem domain specic tools re-

sible to eliminate duplicate objective function cal-

lated to each industrial application to examine dif-

culations for the same, or similar, decision variable

ferent solutions generated during the optimization

values.

process. For this purpose, the problem designer can

The Pareto optimal solution set obtained during

specify a set of les that the problem generates dur-

the optimization process could be better examined

ing the objective function calculations. These les

with more advanced ltering tools.

are stored with the objective function values, and

Furthermore,

as the solutions are generated in a treelike struc-

they can be exported to the industrial application

ture, this structure could be shown to the DM to

for further referencing.

give him/her a better view of the whole solution process.

5

Conclusions and Future Plans

The IND-NIMBUS software is a desktop application

Acknowledgments

that has been successfully connected with several

The IND-NIMBUS development has been a part of

dierent applications to solve multiobjective opti-

the NIMBUS and MASIT01 projects supported by

mization problems. During this process, user feed-

TEKES Finnish Funding Agency for Techonoly

back from dierent test cases has given new ideas

and Innovation.

how to improve the IND-NIMBUS software. Some

MSc.

Paavo Nieminen, Ph.Lic.

Jussi Hakanen and Ph.Lic.

of these features and ideas are discussed here.

Elina Madetoja have

given valuable help in connecting IND-NIMBUS

During the IND-NIMBUS development it was

with dierent industrial applications. MSc. Tommi

apparent that the NIMBUS Fortran77 implementa-

Ronkainen has been the principal developer of IND-

tion cannot be easily extended to meet with chang-

NIMBUS software.

ing requirements. One of the most problematic aspect of the current implementation is the lack of

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