Advanced Decision-Oriented Software for the Management of Hazardous Substances. Part VI: The Interactive Decision-Support Module

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

ADVANCED

DECISION-ORIENTED SOFlWAEE FOR THE W A G E J 6 E N T OF HAZARDOUS SUBSTANCES

PART VI:

The Interactive Decision-Support Module

Ch. Zhao L. Winkelbauer K. Fedra

December 1985 CP-85-50

Cotlaborative P a p e r s r e p o r t work which h a s not been performed solely at t h e International Institute for Applied Systems Analysis and which h a s received only limited review. Views o r opinions e x p r e s s e d herein do not necessarily r e p r e s e n t those of t h e Insti- tute, its National Member Organizations, or o t h e r organizations supporting t h e work.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

The r e s e a r c h described in this r e p o r t is sponsored by t h e Commission of t h e European Communities' (CEC) Joint Research Centre (JRC), I s p r a Establishment, under Study Contract No.2524-84-11 ED ISP A. I t i s c a r r i e d o u t by IIASA's Advanced Computer Applications (ACA) p r o j e c t , within t h e framework of t h e CEC/JRC Industrial Risk Programme, and in cooperation with t h e Centre's activities on t h e Management of Industrial Risk.

CONTENTS

1. Introduction: Model-based Decision Support 1.1 Background: Hazardous Substances Management

2. Risk-Cost Analysis f o r t h e Transportation of Hazardous Substances 2.1 Overall S t r u c t u r e of t h e Model

2.2 Modei Input

2.3 Evaluation of Alternatives 2.4 Model Output

3. Some Examples of Multiobjective Decision Analysis Software 3.1 E x p e r t Choice

3.2 MATS System 3.3 Arborist

4. The Methodology of Multi-Objective Decision Analysis 4.1 Selection of t h e Nondominated S e t of Alternatives 4.1.1 Problem Formulation

4.1.2 The Algorithm to Select t h e Nondominated S e t of Alternatives 4.2 The Reference Point Approach

4.2.1 General Concept

4.2.2 The Mathematical Description of t h e Approach 5. Implementation

6. References and Selected Bibliography

ADVANCED DECISION-ORIENTED SOFTWARE FOR THE

MANAGEMENT

OF HAZARDOUS SUBSTANCES

PART

VI:

The Interactive D e c i s i o n - S u p p o r t Module Ch. Zhao, L. Winkelbauer and

K.

Fedra

1. MTRODUCTIONI M ODEL-BASED DECISION SUPPORT

After t h e generally perceived failure of computer-based information systems t o provide t h e information needed by s t r a t e g i c decision makers, many r e s e a r c h e r s have recognized t h e potential of decision s u p p o r t sys- t e m s as a remedy f o r t h i s problem. A decision s u p p o r t system i s most com- monly d i r e c t e d toward providing s t ~ c t u r e d information t o managers faced with those ill-structured problems t h a t a r e typical of s t r a t e g i c planning and decision making.

From a decision s u p p o r t o r decision analysis point of view, t h e major components of a decision situation a r e :

a set of feasible alternatives, o r c o u r s e s of action open t o t h e decision maker, described in terms of decision-relevant c r i t e r i a and auxiliary d e s c r i p t o r s ;

a s e t of goals o r objectives t h a t t h e decision, i.e., t h e selection of any one alternative, has t o contribute to;

a value system, implicit o r explicit, t h a t describes t h e relative impor- tance of c r i t e r i a in r e s p e c t t o each o t h e r as w e l l a s t h e contribution of c e r t a i n c r i t e r i a values towards t h e respective goals o r objectives.

Depending on t h e level of detail, real-world alternatives in t h e domain of l a r g e and complex socio-technical systems such as t h e area of hazardous substances management addressed in t h e context of this study (Fedra, 1985;

Fedra and Otway, 1985) a r e usually very complicated, i.e., r i c h in detail, and complex, i.e., r i c h in s t r u c t u r e and relationships. Their extensive description, l e t alone t h e i r thorough evaluation, is a formidable task, f a r beyond t h e intellectual capabilities of any individual.

Modern information technology can certainly help to organize this weaith of information; d a t a bases a s w e i l a s models simulating t h e underlying processes and relationships a r e powerful tools in structuring and organiz- ing complex information. Simulation models can g e n e r a t e alternatives and estimate many of t h e c r i t e r i a necessary f o r t h e i r comparative evaluation.

The comparative evaluation itself and t h e eventual decision, however, r e q u i r e experience and judgement as w e l l as t h e information basis provided by t h e a p p r o p r i a t e information technology.

Thus, t o support policy and management decisions, i t is important t o provide substantive background information in t h e form of easily acces- sible data bases, a s well as models and tools f o r interactive decision sup- port Finally, t h e u s e r o r decision maker must b e allowed t o e x e r t a high level of control o v e r t h e software, and he must b e a b l e t o bring his experi- ence, judgement and discretion to b e a r in a substantial way. The system must be easy t o use, easy t o understand, and responsive. Clearly, tools t o m e e t t h e above requirements have t o b e tightly coupled, and integrated into one coherent decision support system. This would allow one t o iteratively gen- erate as well a s t o subsequently evaluate and select alternatives from t h e s e t generated, and described by a comprehensive list of c r i t e r i a .

In this p a p e r w e introduce a n interactive, display-oriented post- processor f o r multiobjective selection o r discrete optimization, which has been implemented within t h e framework of a p r o j e c t on Advanced Decision-

Oriented Software f o r t h e Management of Hazardous Substances (Fedra, 1985): The approach and software described h e r e is designed as a tool t o improve t h e usefulness and usability of decision support systems through t h e easy a c c e s s t o a r i c h set of powerful support functions and display options, and tight integration with substantive models and d a t a bases. A t t h e same time i t adds a new dimension of usefulness to t h e simulation models i t is connected t o as a n output post-processor, aiding in t h e comparative evaluation of complex modeling results.

1.1 Background: Hazardous Substances Management

Many industrial products and residuals such as hazardous and toxic substances are harmful t o t h e basic life support system of t h e environment.

In o r d e r t o e n s u r e a sustainable use of t h e biosphere f o r p r e s e n t and f u t u r e generations, i t i s imperative t h a t these substances a r e managed in a s u e a n d s y s t e m a t i c manner. The framework system (Fedra, 1985) is designed t o provide software tools which can b e used by those engaged in t h e management of t h e environment, industrial production, products, and waste streams, and hazardous substances and wastes in particular.

The system consists of a n integrated set of sojTware tools, building on existing models and computer-assisted procedures. This set of tools is designed f o r non-technical users.

To facilitate t h e access to complex computer models f o r t h e casual u s e r , ana f o r more experimental and explorative use, it also a p p e a r s neces- s a r y t o build much of t h e accumulated knowledge of t h e subject a r e a s into t h e u s e r interface f o r t h e models. Thus, t h e interface incorporates el*

ments of a knowledge-based e x p e r t system, t h a t is capable of assisting any non-expert u s e r t o seiect, set up, run, and i n t e r p r e t specialized software.

By providing a coherent u s e r interface, t h e interactions between different models, t h e i r data bases. and auxiliary software f o r display and analysis become t r a n s p a r e n t f o r t h e u s e r , and a more experimental, educational

.

T h i s s o f t w a r e s y s t e m f o r t h e management o f hazardous s u b s t a n c e s and industrial r i s k i s developed under c o n t r a c t t o t h e Commission o f t h e European Communities (CEC), J o i n t Research Centre (JHC), Ispra, Italy.

style of computer use can b e supported.

One important p a r t of t h e applications of t h e framework system is scenario analysis, i.e., within t h e context of one o r a group of linked simu- lation models, t h e u s e r defines a scenario, i.e., a set of assumptions, boun- d a r y conditions and control variables describing a specific problem situa- tion (e.g., t h e transportation of a c e r t a i n amount of a hazardous chemical substance from a supply point, t h e industrial plant o r chemical deposit, to a demand point) and then t r a c e s t h e consequences of this situation through modeling. In scenario anaiysis, t h e consequences of t h e settings of control variables and parameters, describing control o r policy options, as well as e x t e r n a l driving forces, each set defining one scenario, are estimated in the form of complex d a t a which r e p r e s e n t t h e answer t o t h e u s e r ' s question:

"What, if

...

^?".

Usually t h e consequences of each set of assumptions analyzed a r e quantifiable, t h a t is, they c a n be measured on some natural o r artificial, numerical o r descriptionai scaies. Quantified and, if necessary, aggregated a t t r i b u t e s become c r i t e r i a , which in most cases are incommensurable (e.g., cost and r i s k ) , d i s c r e t e and finite. They a r e d i s c r e t e and finite, because f o r many real world problems continuous variables a r e not meaningful (e.g., t r u c k s come only in a limited number of sizes, and they can have only one, two o r maybe t h r e e , d r i v e r s ) t h e values f o r some criteria come directly from e x p e r t s (e.g., c r i t e r i a of an aesthetic o r political n a t u r e and should b e expressed as a f e w classes r a t h e r than on a n a r b i t r a r i l y "precise"

scale), t h e set of feasible and meaningful control and policy options is usu- ally finite and small, and because scenario analysis is r e s t r i c t e d t o a finite number of simulation runs.

To evaluate t h e outcomes from different scenarios on control and pol- icy alternatives, t o p r e s e n t complex d a t a such t h a t d i r e c t comparison is supported, and finally t o select t h e alternative which "best" suits t h e client's preferences, it is necessary t o provide a tool f o r implicit optimiza- tion, i.e., multicriteria decision analysis.

2. RISK-COST ANALYSIS MODEL FOR THE TRANSPORTATION OF HAZARDOUS SUBSTANCES

A R~sK-COS~ 'Analysis Model f o r t h e Transportation of Hazardous Sub- stances (Kleindorfer and Vetschera, 1985) has been implemented as one of t h e simulation and decision support models within t h e overall framework system.

The model is based on

a geographical representation of a given region (e.g., of Europe) which specifies supply and demand points together with various routes connecting t h e s e points,

on regulatory policies such as risk minimization and on economic policies such as cost minimization.

The function of this model is to enable t h e u s e r t o solve t h e problem of choosing t h e "best" r o u t e and mode f o r t h e transportation of hazardous substances from a c e r t a i n supply point t o a c e r t a i n demand point, and in defining policies t h a t ensure t h e selection of t h e s e mode/route alternatives.

2.1 Overall Structure of the _Yodel

The moael is designed as a policy-oriented tool. I t s s t ~ c t u r e there- f o r e , has t o closely follow t h e s t r u c t u r e of decision variables open t o regu- lators. In general w e can distinguish two different levels a t which regula- tions might operate:

a micro Level, deaiing with individual t r a n s p o r t activities o r connec- tions,

a n aggregated level aiming at global regulations t h a t can b e applied t o a l a r g e class of shipments.

The model currently impiemented in t h e framework system concen- trates on t h e micro level decision problem. e.g., individual shipments of hazardous substances.

For analysis a t t h e micro level t h e model will g e n e r a t e and evaluate possible transportation alternatives f o r a given t r a n s p o r t objective. A

.

t r a n s p o r t objective is defined by t h e amount and type of hazardous sub- stance t o b e transported and the points between which t h e goods a r e t o b e transported.

A t r a n s p o r t alternative in t h e model is r e p r e s e n t e d by a geographical r o u t e along which t h e t r a n s p o r t i s t o o c c u r and t h e choice of a t r a n s p o r t mode, both associated with risk-cost c r i t e r i a . The possibility of m o d e changes along t h e r o u t e is also considered in t h e model.

A detailed cost and risk analysis f o r ail t h e alternatives generated is then performed and t h e r e s u l t s of this evaluation are presented t o t h e deci- sion maker f o r his final choice among t h e alternatives using t h e Interactive Data Post Processor.

From t h e perspective of software engineering t h e implementation of t h e model consists of t h r e e main modules (Figure 2.1).

The f i r s t module g e n e r a t e s candidate paths and consequently generates different route/mode combinations. To limit t h e amount of alternatives t o reasonable numbers, t h e s e a r c h area is r e s t r i c t e d .

The second module performs a risk-cost evaluation of t h e paths gen- e r a t e d in t h e first phase. The outcome of t h e second phase is a list of c r i t e r i a values of all t h e alternatives f o r f u r t h e r evaluation.

The third module selects the "best" transportation alternative with r e s p e c t t o t h e c r i t e r i a specified by t h e decision maker and t h e p r e f e r - ences expressed.

In most c a s e s t h e numoer of alternatives is l a r g e and t h e selection of a p r e f e r r e d alternative from t h e s e t of feasible alternatives generated will r e q u i r e computer-assisted information management and decision support.

2.2 Model Lnput

The d a t a s t r u c t u r e of t h e Risk-Cost Analysis Model f o r t h e Transporta- tion of Hazardous substances consists of f o u r main parts:

Pam h m Evaluation

Criteria

Sele3ion

of of

the

p

'

A l ~ t l v e s vrton

-

-best-

solution

R g u r e 2.1: Overall s t r u c t u r e of the t r a n s p o r t a t i o n model

a description of the transportation network, i . e . , the cit-ies and the links between them,

risk indicators, cost factors,

general information about the model.

The general i n f o r m a t i o n about the model i s represented by the follow- ing elements:

substances t o be transported, described by their specific gravity,

a d e s c r i p t i o n of t h e descriptors of t h e a r c s , a l i s t of r i s k g r o u p s : damages, i n j u r i e s a n d d e a t h s ,

a l i s t of land u s a g e classes: u r b a n , s u b u r b a n a n d a g r i c u l t u r a l ,

t h e v e h i c l e s (i.e. t r u c k s , c a r s , t r a i n s , e t c . ) , d e s c r i b e d by c a p a c i t i e s .

The t r a n s p o r t a t i o n network i s d e s c r i b e d as follows:

The nodes ciescribe t h e c i t i e s by t h e i r r e l a t i v e c o o r d i n a t e s .

The arcs d e s c r i b e t h e links between t h e c i t i e s , e.g., t h e r o a d o r r a i l system by t h e i r

length,

mode (e.g., r o a d , r a i l r o a d , e t c . ) d e s c r i p t o r s (e.g., tunnel, b r i d g e , e t c . ) t y p e (e.g., highway, minor r o a d , etc.)

s h a r e s of land u s a g e c l a s s , i.e., t h e kind of environment (e.g., u r b a n , s u b u r b a n , a g r i c u l t u r a l ) t h e r o a d o r r a i l p a s s e s t h r o u g h .

Based o n t h i s d a t a s t r u c t u r e initially all possible p a t h s (within a heu- r i s t i c a l l y defined "window") are g e n e r a t e d f o r e a c h v e h i c l e u n d e r con- s i d e r a t i o n f r o m t h e s p e c i f i e d s u p p o r t point t o t h e s p e c i f i e d demand point.

F o r t h e s e p a t h s r i s k a n d cost are estimated, a n d finally t h e y are com- p a r e d a n d e v a l u a t e d .

2.3 E v a l u a t i c n of A l t e r n a t i v e s

Alternatives are e v a l u a t e d in t e r m s of c o s t a n d r i s k . The c r i t e r i a of c o s t a n d r i s k are incommensurable; f o r i n s t a n c e , t h e c o s t of t r a n s p o r t a t i o n i s measured in monetary v a l u e a n d t h e r i s k of t r a n s p o r t a t i o n i s measured in t h e number of f a t a l i t i e s in t h e e v e n t of a n a c c i d e n t .

Sometimes c o s t a n d r i s k are c o n t r a d i c t o r y , f o r example t h e s h o r t e s t

-

a n d thus usually t h e most cost-effective

-

connection i s a highway t h a t p a s s e s c l o s e t o densely populated a r e a s , with a h i g h e r r i s k p o t e n t i a l t h a n m o r e r e m o t e , a n d t h e r e f o r e m o r e e x p e n s i v e r o u t e s .

In this model c o s t evaluation is based on f r e i g h t rate sampled from commercial t r a n s p o r t firms. The c o s t function i s simply described by t h e following formula.*)

where

c + fixed c o s t s

co: initial p a r t of t h e variable c o s t s function c,: slope of t h e v a r i a b l e costs function X: amount of substance t o b e shipped L: length of t h e path.

The r i s k analysis in t h e model c o v e r s both losses in t h e form of pro- p e r t y damage and losses in t h e form of 'injuries and fatalities. Considering t h e stochastic n a t u r e of t h e s e losses expected values and t h e variance of losses a r e taken as decision c r i t e r i a .

A simpiified lognormal distribution risk analysis submodel is employed t o evaluate t h e alternatives. A s outcomes of risk analysis, t h e c r i t e r i a of a l t e r n a t i v e s are described in terms of e x p e c t e d losses and variance of losses t o a given group along a r o u t e in t h e network. F u r t h e r on, t h e groups of o b j e c t s t h a t c a n b e affected by accidents (population, p r o p e r t y values e t c . ) will b e r e p r e s e n t e d by g.

The formulations of t h e s e c r i t e r i a are as follows. The expected loss E[Rg] of group g along r o u t e ( r l , r 2 , .

. .

,rl) i s :

3 This cost function i s only a very crude f i r s t approximation and strictly speaking only valid when the volume t o be shipped i s very large in relation t o the capacity of any vehi- cle t o be used. Also, the linear distance dependency only holds for relatively large ais- tances.

where

p,(rk): the probability of a n accident on arc k

q,: t h e probability of a n accident, which happens for type n land usage on arc r

p,, un2: p a r a m e t e r s of lognormal distribution of conditional density function f o r type n.

The variance of losses

to

a given group g along r o u t e ( r l , r2,...,rl) is :

var (Rg )

=

E [R:]

-

E [ R ] ~ where

and

Both t h e expected value and t h e variance of losses t o s e v e r a l groups are c h a r a c t e r i s t i c of a route/mode combination t h a t will b e used in evaluating t h e different alternatives. For t h r e e r i s k groups ( p r o p e r t y damage, fatal and non-fatal injuries) six risk-related objectives can be considered in t h e evaluation.

Combining t h e s e six objectives with cost, w e can g e t a well-defined mul- tiobjective decision problem with seven c r i t e r i a .

To simplify o u r description, f u r t h e r on t h e problem with only t h r e e c r i t e r i a (cost, expected loss i.e., p r o p e r t y damage, and expected number of fatalities) will b e considered as a n example.

2.4 Model Output

The output of t h e t r a n s p o r t a t i o n model consists of a List of c r i t e r i a f o r all t h e alternatives:

The

t i s k i n d i c a t o r s

are r e p r e s e n t e d as follows:

r i s k groups (e.g., damages, injuries, deaths);

possibilities of a c c i d e n t s ( a p r i o r i ) ;

consequences of a n accident, depending on t h e substance involved, land usage c l a s s and r i s k group.

The

cost factors

are d e s c r i b e d by t h e following variables:

t r a n s p o r t c o s t s , fixed and variable,

i n s u r a n c e costs, depending on t h e t y p e of a r c and t h e t r a n s p o r t a t i o n medium used.

3. SOME EXAMPLES OF II[ICROCOMPUTER-BASED DECISION

ANALYSIS SOFTWARE

-

S u p p o r t for t h e decision m W n g process

^is

t y p i c d l y present in three general fonns. R r s t , the

MS

[Decision S u p p o r t S y s t e m ] should provide accurate, timely i n f o r m a t i o n w h i c h s u p p o r t s t h e intelligence p h a ~ e of decision making. Second, ,the A S should assist in designing d t e r n a t i v e courses of action. The

LES

m a y develop d t e r n a t i v e s o n i t s own, (through a g o d seeking capability) a n d i t should be able to a n d y z e d i m r e n t d t e r n a t i v e s (through a what-qf capability). And f i n d l y , m a n y decision s u p p o r t systems recommend a specific course of a c t i o n to follow in order to s u p p o r t the choice phase of decision making.

(Hogue and Watson 1985)

Of c o u r s e , i t i s not n e c e s s a r y f o r a c e r t a i n decision s u p p o r t system t o h a v e a i l t h r e e supporting functions. They a r e important c r i t e r i a in describing decision s u p p o r t systems. Also, f o r such intrinsically

i n t e r a c t i v e a n d user-oriented software such as DSS, i t is interesting t o com- p a r e t h e u s e r i n t e r f a c e which is a n o t h e r c r i t i c a l c r i t e r i o n of p r a c t i c a l usa- bility.

Given below a r e brief descriptions and assessments for some microcomputer-based decision s u p p o r t systems in t h e market as compara- tive background material. These descriptions and assessments are based on t h e following simplified version of t h e t r a n s p o r t a t i o n problem introduced in c h a p t e r 2.

The s c e n a r i o under consideration i s t h e t r a n s p o r t a t i o n of a c e r t a i n amount of a chemical s u b s t a n c e f r o m A t o B. Five a l t e r n a t i v e pathways asso- c i a t e d with d i f f e r e n t t r a n s p o r t a t i o n modes are possible. A s c r i t e r i a f o r t h e multiobjective optimization only t h e cost of t r a n s p o r t a t i o n , t h e e x p e c t e d vaiue of losses of p r o p e r t y damage and t h e e x p e c t e d value of t h e number of fatalities are considered. Let u s suppose t h a t t h e decision maker wants

to

minimize all t h r e e criteria.

3.1 E x p e r t Choice

E x p e r t Choice i s a decision s u p p o r t system software package

.

developed by Decision S u p p o r t Software Inc., McLean, Virginia in 1983.

I t d o e s not p r o p o s e decisions, but i t h e l p s t h e u s e r t o make decisions based on his judgements. E x p e r t Choice d o e s not r e s t r i c t t h e judgment pro- cess to quantifiable a t t r i b u t e s . Both quantitative and qualitative judgments are a c c e p t e d .

With E x p e r t Choice the decision maker can organize a complex decision problem in a h i e r a r c h i c a l tree s t r u c t u r e . This makes i t possible to i n t e g r a t e judgements and measurements in t h e same h i e r a r c h i c a l s t r u c t u r e to a c h i e v e t h e "best" solution. The h i e r a r c h i c a l tree consists of nodes at d i f f e r e n t levels. Each of t h e s e nodes in t u r n c a n have at most seven b r a n c h nodes in e a c h of t h e six h i e r a r c h y levels. The goal node i s at level 0; t h e u s e r c a n define nodes at levels 1-5. Thus E x p e r t Choice is c a p a b l e of model- ing v e r y Large problems (thousands of nodes).

The decision t r e e for o u r sample t r a n s p o r t a t i o n problem is shown in Figure 3.1.

.

CURRENT NODE ( 0 ) GOAL LEVEL

=

0

LOCAL PRIORI'IT

=

1.000

ENTER

(?

FOR HELP)

SELECT THE. BEST TRANSPORTATION PATH

I

- I

PATH

1

PATH 2 PATH 3 PATH 3 PATH 3

PATH

4

PATH

4

PATH 5 ^PATH 5 ^PATH 5

Figure 3.1: E z p e r t Choice d e c i s i o n tree for sample t r a h s p o r t a t t o n problem

Once t h e E x p e r t Choice model is built, t h e u s e r can s t a r t t h e judgement process. First, E x p e r t Choice a s k s t h e u s e r to compare t h e main c r i t e r i a in p a i r s with r e s p e c t t o t h e goal in terms of importance, p r e f e r e n c e and likeli- hood. This i s done by asking t h e u s e r questions like "Do you think t h a t with r e s p e c t t o t h e goal COST i s extremely, v e r y s t r o n g , s t r o n g , moderate or equal to PROPERTY DAMAGE ?", or

-

in an a l t e r n a t i v e mode

-

by d i r e c t

input of a numerical specification t o e x p r e s s t h e importance of each cri- terion.

The a t t r i b u t e s of t h e alternatives a r e also determined by qualitative pairwise comparison. E x p e r t Choice derives priorities from these simple pairwise comparison judgements. It then synthesizes o r combines t h e s e priorities throygh weighting and obtains overall priorities f o r t h e alterna- tives at t h e bottom of t h e tree. T l ~ i s i s t h e final result and amounts to a ranking of t h e alternatives, which is shown in b a r c h a r t s (the alternative with t h e longest b a r is t h e "best" solution).

The technique employed in E x p e r t Choice is quite easy f o r t h e non- e x p e r t u s e r to understand. To run E x p e r t Choice, only t h e ability t o com- p a r e c r i t e r i a , and judgement, a r e required on the par% of t h e user.

Obviously t h e r e a r e some disadvantages t o E x p e r t Choice. Only a rough ranking of alternatives is provided t o t h e u s e r and t h e r e is no back- ground information available from o t h e r "hard" computer models in t h e sys- t e m .

E x p e r t Choice is most likely suitable f o r problems where t h e a t t r i b u t e s of t h e problems a r e difficult t o describe in terms of quantity. The decision recommended by E x p e r t Choice is t o a l a r g e d e g r e e based on t h e judgement of t h e decision maker.

E x p e r t Choice r e q u i r e s an IBM PC-XT o r similar PC.

3.2 MATS System

AUTS (Multi-Attribute Tradeoff System) is an interactive decision sup- p o r t system t o assist planners in t h e systematic evaluation of plans with impacts on many factors.

MATS

_{w a s} developed at t h e Environmental and Social Branch Division of t h e Planning Technical Services Engineering and Researcn Center, Denver, Colorado, in 1983. The MATS program w a s developed t o assist planners in analyzing tradeoffs between multiple objec- tives o r a t t r i b u t e s , in o r d e r t o a r r i v e at a judgment of t h e overall worth of a given mix of gains and losses f o r those attributes. The basic method employed in MATS is based on utility t h e o r y and t h e weighting coefficient method.

In o u r example t h e decision maker a t f i r s t is a s k e d t o e n t e r t h e c r i - teria and t h e i r r a n g e s (with specification of t h e b e s t and w o r s t level). Then MATS a s k s a s e r i e s of questions in t h e following form: 'Which change is more significant ?" followed by t w o change r a n g e s (e.g. 1000 to 2000 and 5000 to 4000) to select from and o n e possibility to e x p r e s s t h a t both changes are equal in t h e opinion of t h e decision maker. S o MATS obtains t h e subjective weightings f o r t h e c r i t e r i a from t h e decision maker.

A f t e r t h e elicitation of c r i t e r i a rankings MATS p r o d u c e s "subjective weighted" impacts f o r e a c h plan. These weighted impacts are on a common scale a n d c a n b e added t o a r r i v e at a t o t a l score f o r e a c h plan. According to t h e total score for e a c h plan t h e p r o c e d u r e of ranking a l t e r n a t i v e plans i s c a r r i e d out.

A f t e r t h e ranking p r o c e d u r e t h e utility functions are displayed in a simple g r a p h i c a l s t y l e a n d t h e n t h e a l t e r n a t i v e plans are listed on t h e s c r e e n in sequence of t h e i r p r i o r i t y , and for e a c h of them t h e i r total plan s c o r e a n d t h e i r objective values, subjective values (values of t h e utility) a n d subjective weighted values are displayed.

Only "quantifiable" a t t r i b u t e s c a n b e evaluated by t h i s software. The capability of t h e system i s limited to 40 plans which c a n b e evaluated and ranked. The system i s scrolling- and not screen-oriented, and only provides menus in e a c h i n t e r a c t i v e p h a s e which can not give t h e u s e r a visual impression of h i s problem, as for example, a graphics-based u s e r i n t e r f a c e could. The main disadvantage of MATS i s t h a t i t is difficult for a u s e r to specify h i s p r e f e r e n c e s in t e r m s of weighting coefficients.

An IBM PC-XT i s r e q u i r e d t o s u p p o r t t h e MATS software.

3.3 ARBORIST

ARBORIST

f e a t u r e s a g r a p h i c s u s e r i n t e r f a c e for decision-tree con- s t r u c t i o n , evaluation, a n d analysis. A s i s w e l l known, decision-tree metho- dology c a n help a ciecision maker t o s t r u c t u r e and formulate p r e f e r e n c e s and choices while analyzing a problem with a limited number of a l t e r n a t i v e s under uncertainty.

Unlike t h e systems discussed above, ARBORIST i s a single objective optimization system. Therefore i t is necessary f o r t h e decision maker t o transform t h e incommensurable c r i t e r i a into a unique unit using weighting coefficients to e x p r e s s his preferences.

The Arborist s c r e e n is divided into f o u r windows: Function Menu win- dow, Macro window, Focus window and Message window. The u s e r is guided through t h e whole system by t h e menus in t h e Function Menu window.

One of these menus helps t h e decision maker t o build up a decision tree which i s then shown in t h e Focus window. The t r e e consists of a r o o t node, decision nodes (i.e., nodes with b r a n c h e s which r e p r e s e n t alternatives).

chance nodes (i.8.. nodes at which one outcome of a chance event w i l l . o c c u r ) , end nodes (i.e., t h e final outcomes t h a t r e s u l t from t h e decisions made in conjunction with t h e c h a n c e events) and b r a n c h e s connecting t h e s e nodes. An ARBORIST s c r e e n showing a decision tree r e l a t e d to o u r tran- sportation problem i s shown in Figure 3.2.

The decision maker can assign descriptions (e.g., PATH1) and values (e.g., COST

=

1000) t o all nodes and formulas (e.g., a

*

COST

+

@

*

PROPERTYDAMAGE

+

⁷

*

INJURIES)' to end nodes.

A f t e r t h e s e specifications ARBORIST provides t h e following analysis functions:

calculate t h e expected value f o r t h e decision t r e e , and show t h e "best"

solution as a magenta colored path through t h e t r e e ;

display t h e probability distributions f o r t h e outcome at a selected chance node as histograms in t h e Macro window;

perform sensitivity analysis f o r one selected p a r a m e t e r at a selected node and display t h e r e s u l t s in t h e form of colored c u r v e s in t h e Focus window.

The main disadvantage of ARBORIST is t h a t i t is a single objective optimization package and t h a t t h e u s e r h a s t o p r e p a r e all t h e d a t a f o r his decision problem himself, i.e., t h e u s e r always h a s t o input all t h e d a t a of his problem description in a n interactive process, because t h e r e are no

1: a , fl and 7 i n t h e v a l u e s p e c i f i c a t i o n r e p r e s e n t w e i g h t i n g c o e f f i c i e n t s

f i g u r e 3.2: ARBORET decision tree for sampLe t r a n s p o r t a t i o n probLem.

Pam

1

path 2 ( . 6 )

Path

3

Path

4 (.4) Paul

5

data pre-processors or "hard" computer models in the system.

Despite the disadvantages mentioned, Arborist i s a useful tool f o r deci- sion analysis with uncertainty. Arborist was developed a t Texas Instruments Inc. in 1985 and requires a TI-PC or IBM-PC as hardware support.

Expected Value

Browse

R O ~ . W.

p i E Z q

Sensltivlty Other Menrr~s Fonnulas Quit Change Values

The problem mentioned in c h a p t e r 2 is a well known d i s c r e t e , multiob- jective decision problem, in which all feasible a l t e r n a t i v e s a r e explicitly listed in t h e finite set x0=fxl,x2.

...,

xnj, and t h e values of all c r i t e r i a of each a l t e r n a t i v e a r e known and listed in t h e set Q = If (xl),f ( x ~ ) , . .

.

,f (x,) j.

T h e r e are many tools which could be employed t o solve this problem (e.g., Korhonen, 1985, Majchrzak, 1984). We have drawn on t h e method developed by Majchrzak (1985).

Usually, t h e p r o c e d u r e of problem solving i s divided into two stages.

The f i r s t s t a g e i s t h e selection of elements of a nondominated set f r o m a l l t h e a l t e r n a t i v e s of set xO. In t h e second stage, t h e "best" solution is identi- fied as t h e decision maker's final solution t o t h e problem under considera- tion, in a c c o r d a n c e with his p r e f e r e n c e s , experience etc., as t h e basis f o r his decision.

In t h e d i s c r e t e , multicriteria optimization module of t h e o v e r a l l system, at t h e f i r s t s t a g e of problem solving, t h e dominated approximation method i s used to s e l e c t t h e elements of t h e p a r e t o set, because of i t s calculation effi- ciency and i t s ability t o solve relatively l a r g e scale p r o b l e m s . ' ~ o r instance, t h i s method c a n b e used to solve a problem with 15-20 c r i t e r i a and more than a thousand a l t e r n a t i v e s , which is sufficient f o r processing t h e d a t a arising from s c e n a r i o anaiysis in t h e framework system.

In t h e second s t a g e , a n i n t e r a c t i v e p r o c e d u r e based on t h e r e f e r e n c e point t h e o r y i s employed t o help t h e u s e r to find his final solution. This a p p r o a c h combines t h e analytical power of t h e "hard" computer model with t h e qualitative assessments of t h e decision maker in t h e decision process. I t makes t h e decision p r o c e s s more reasonable and c l o s e r to t h e human think- ing process. In t h e following, t h e methodology used in t h e s e two s t a g e s will b e described briefly.

T h i s s e c t i o n i s b a s e d o n t h e R e f e r e n c e P o i n t Approach d e v e l o p e d b y W i e r z b i c k i (1979, 1980) and d r a w s o n t h e DISCRET p a c k a g e d e v e l o p e d b y M a J c h r z a k (1984, 1985).

4.1 Selection of the Nondominated Set of Alternatives 4.1.1 Problem Formulation

W e may d e s c r i b e t h e problem considered as a minimizing ( o r maximizing o r mixed) problem of m c r i t e r i a with d i s c r e t e values of c r i t e r i a and a finite number of alternatives n.

L e t x0 b e t h e set of alternative admissible decisions. For each of t h e elements of xO, all c r i t e r i a under consideration have been evaluated. L e t Q be t h e c r i t e r i a values set f o r all feasible d i s c r e t e alternatives in t h e space of c r i t e r i a F. L e t a mapping f: x0 +'Q b e given.

Then the problem can b e formulated as follows:

min f ( z ) z e O

The partial pre-ordering relation in s p a c e Q is implied by t h e positive cone A

= R+?

f1,f2 E Q fl

<

f 2

<==>

f l E f 2

-

^A

This means f l dominates f 2 in t h e sense of partial pre-ordering.

Element f a E Q is nondominated in t h e s e t of feasible elements Q , if i t is not dominated by any o t h e r feasible element. Let N

=

N(Q)

c

Q denote t h e s e t of all nondominated elements in t h e c r i t e r i a s p a c e and let Nx

=

N(xO) C

x0 denote t h e set of t h e corresponding nondominated alternatives (deci- sions) in t h e decision space.

To solve this problem means t o delete all t h e dominated alternatives

-

t h a t is, alternatives f o r which a b e t t e r one can b e found in t h e sense of t h e natural partial ordering of t h e c r i t e r i a

-

o r t o find t h e set N of nondom- inated elements and t h e corresponding s e t N, of nondominated alternatives.

Eventually, a final solution should b e found from t h e set of nondominated alternatives.

4.1.2 The Algorithm t o S e l e c t t h e Nondominated S e t of Alternatives

The algorithm to s e l e c t t h e nondominated set of a l t e r n a t i v e s is quite simple. The method implemented in o u r system is of t h e explicit enumeration type. I t is called t h e method of dominated approximations and i s based on t h e following notion.

Def.

1: S e t A i s called a dominated approximation of N if, a n d only if N C A - A

i.e., if for each f i E N t h e r e e x i s t s f , E A such t h a t f i

<

f , in t h e s e n s e of p a r t i a l p r e + r d e r i n g induced by A .

Def.

2: The A2 approximation dominates t h e Al approximation of t h e nondominated set N if, and only if

Al C A2

+

A

The method of dominated approximations g e n e r a t e s a sequence of approximations Ak, k=0,1,2, ...I such t h a t

Q = A o > A , >

...

^{> A k}

>...

> A , = N

given Q and A select N

=

N (Q), and assuming t h a t all c r i t e r i a are t o b e minimized. Then t h e p r o c e d u r e of problem solving c a n b e d e s c r i b e d as fol- lows.

Step 0: l e t A.

=

Q, N

=

@,

K =

⁰

Step

1:

If Ak \ N

=

@ t h e n s t o p ,

else choose any index i E 1=11,2,

...

,mj and find f L E Q s u c h t h a t fLi

=

min f i

set N

=

N

u I

fs j and go to s t e p 2.

Step

2:

C r e a t e t h e new approximation A k + l by f s

A +

= 1

A + \ N

!(

f a + A n (Ak \ N ) l

u

N set

K = K +

1 and g o to s t e p 1.

A s a r e s u l t of t h e above p r o c e d u r e t h e nondominated set N of a l t e r n a - tives is found when t h e stopping condition Ak \ N

=

@ is satisfied. The selection of t h e p a r e t o set from all t h e a l t e r n a t i v e s in t h e c r i t e r i a s p a c e i s shown in Figure 4.1.

elements

of

Pareto

set

0

dominated elements

0

n g u r e 4.1: The pareto set from the a l t e r n a t i v e s i n the c r i t e r i a space

4.2 The Reference Point Approach 4.2.1 General Concept

After t h e system eliminates, by t h e method mentioned above, all t h e dominated alternatives, the s e t of remaining nondominated alternatives i s usually l a r g e and i t s elements are incomparable in t h e sense of natural p a r - tial ordering. To choose from among them, additional information must b e obtained from t h e decision maker. The main problem of multicriteria optimi- zation is how and in what form this additional information may b e obtained,

such t h a t i t s a t i s f a c t o r i l y r e f l e c t s t h e decision maker's p r e f e r e n c e s , e x p e r i e n c e and o t h e r subjective f a c t o r s .

T'nere are many methods f o r obtaining t h a t additional information and to t h e n find t h e final o r t h e "best" solution according to t h e decision maker's p r e f e r e n c e . The most common method is t h e weighting coefficients method, which plays a c e n t r a l r o l e in t h e basic classical t h e o r y of multiob- jective decision analysis. I t r e p r e s e n t s a traditional method of multicri- teria optimization.

However, c e r t a i n difficulties often arise when applying t h e weighting coefficients method to real-world decision processes: Decision makers usu- ally d o not know how t o specify t h e i r p r e f e r e n c e s in t e r m s of weighting coefficients. Before running a multiobjective model, some of them d o not even h a v e a n idea a b o u t t h e i r weighting coefficients.

Most of them a r e not willing t o t a k e p a r t in psychometric experiments in o r d e r t o l e a r n a b o u t t h e i r own p r e f e r e n c e s . Sometimes t h e decision maker h a s v a r i a b l e p r e f e r e n c e s as time, and t h e information available t o him changes. The applicability of t h e weighting coefficients method to real world problems is s e v e r e l y r e s t r i c t e d by t h e s e factors.

I t i s obvious t h a t decision makers need a n a l t e r n a t i v e a p p r o a c h for multicriteria optimization problems. Since 1980 many versions of software tools based on r e f e r e n c e point t h e o r y have been developed at IIASA, such as DIDASS/N, DIDASS/L, MM, MZ, Micro DIDASS etc. These tools c a n d e a l with nonlinear problems, l i n e a r problems, dynamic t r a j e c t o r y problems, and committee decision problems. Recently many application experiments have been r e p o r t e d by numerous scientific p a p e r s and r e p o r t s (e.g., G r a u e r , et a l . 1982, Kaden, 1985,).

The r e f e r e n c e point a p p r o a c h is based on t h e hypothesis t h a t in every- day decisions individuals think r a t h e r in t e r m s of goals and a s p i r a t i o n lev- e l s than in terms of weighting coefficients o r maximizing utility. This hypothesis is q u i t e close to t h e real-world decision-making process.

Using t h e r e f e r e n c e point a p p r o a c h , t h e decision maker works with a computer interactively. T h e r e a r e two distinct p h a s e s in t h e a p p r o a c h :

In t h e f i r s t s t a g e , t h e e x p l o r a t o r y s t a g e , t h e decision maker may a c q u i r e information about t h e r a n g e and t h e frequency distribution of t h e a l t e r n a t i v e s thus giving him a n overview of t h e problem t o b e solved. The decision maker may also set some bounds f o r t h e c r i t e r i a values of t h e a l t e r n a t i v e s set t o focus his i n t e r e s t s on a c e r t a i n area.

In t h e second s t a g e , t h e s e a r c h s t a g e , at f i r s t t h e decision maker is r e q u i r e d t o specify his p r e f e r e n c e s in t e r m s of a r e f e r e n c e point in t h e c r i - t e r i a space. The values of t h e c r i t e r i a r e p r e s e n t e d by t h e r e f e r e n c e point in t h e c r i t e r i a s p a c e are t h e values t h e decision maker wants t o obtain, i.e., t h e goal of t h e decision maker, which reflects his e x p e r i e n c e and p r e f e r - ences.

Next, t h e system identifies an efficient point, which is one of t h e alter- natives closest to t h e r e f e r e n c e point. The efficient point i s t h e "best"

solution of t h e problem under t h e c o n s t r a i n t s of t h e model and with r e s p e c t to t h e r e f e r e n c e point specified by t h e decision maker.

If t h e decision maker i s satisfied by t h i s solution, h e can t a k e i t as a basis f o r his final decision. If t h e decision maker is not satisfied by t h i s solution, h e may modify h i s goal, i.e., change t h e r e f e r e n c e point or change t h e c o n s t r a i n t s , i.e., change t h e bounds h e had set b e f o r e , o r both, o r create some additional a l t e r n a t i v e s in o r d e r to obtain a new efficient point.

In t h e case of continuous variables problems, i.e., t h e problems described by continuous m o d e l s (linear o r nonlinear programming m o d e l s o r dynamic c o n t r o l models), t h e r e f e r e n c e point method is a b l e t o g e n e r a t e new alter- natives by running t h e model again.

4.2.2 The Mathematical Description of t h e Approach

The a p p r o a c h c u r r e n t l y implemented in t h e framework system i s as fol- lows: f o r t h e s a k e of computability, i t is n e c e s s a r y t o define a n achievement scalarizing function which transforms t h e multiobjective optimization prob- l e m into a single objective optimization problem. A f t e r having specified h i s p r e f e r e n c e s in t e r m s of a r e f e r e n c e point, which need not b e attainable, t h e decision maker obtains a n efficient point which i s t h e nondominated point n e a r e s t t o t h e r e f e r e n c e point in t h e s e n s e of t h e scalarizing function.

elements of Pareto set

0

dorninateb

0~~ a

elements

O

Set of Alternatives

O O 0 0

Reference

⁹¹

/'?

O o O , , o 0 o o

:

^{O o o O}

p i n t

Aspiration level

4

Figur- 4.2: The i n t e r a c t i v e procedure of the reference point a p p r o a c h

In o u r d a t a post-processor t h e Euclidean-norm scalarizing function i s used. Let q b e t h e r e f e r e n c e point specified by t h e user. Then assuming t h a t t h e optimization problem under consideration i s a minimization problem f o r all criteria (for maximizing problems one may easily transform i t into a minimizing problem by changing t h e sign of t h e r e l a t e d c r i t e r i a ) , t h e follow- ing scalarizing function i s minimized:

where ( f q ) , denotes t h e v e c t o r with components max(O,fq),

II.((

denotes t h e Euclidean norm and p >1 i s a penalty scalarizing coefficient.

The solution f e f o r minimizing t h e scalarizing function S i s an efficient point of t h e problem with r e s p e c t t o t h e specified r e f e r e n c e point.

If n e c e s s a r y , this p r o c e d u r e c a n b e r e p e a t e d until t h e decision maker i s satisfied by a n efficient point.

Figure 4.2 shows t h a t a f t e r changing t h e r e f e r e n c e point twice, finally t h e decision maker obtains a s a t i s f a c t o r y efficient point fe3 corresponding t o r e f e r e n c e point q3.

In t h e o v e r a l l software system, t h e multi-criteria optimizer o r post- p r o c e s s o r is implemented as a n independent module as well as a n optional function of s e v e r a l o t h e r modules, notably t h e t r a n s p o r t a t i o n risk-cost analysis model. The only difference is in terms of access

-

e i t h e r from t h e system's master menu level, o r from t h e a p p r o p r i a t e level of o t h e r m o d e l s . If used as a stand-alone module, t h e program f i r s t examines i t s data direc- tory and lists all d a t a sets by a one-line identification in a sequence depend- ing on modification d a t e s , i.e., t h e data set g e n e r a t e d last is o f f e r e d as t h e f i r s t choice. The u s e r t h e n simply points at t h e desired d a t a s e t , which i s then loaded f o r f u r t h e r analysis.

Wherever t h e multi-criteria optimization package i s used as a n inter- g r a t e d post-processor, t h i s s t e p i s not n e c e s s a r y , since only one d a t a s e t , namely t h e one g e n e r a t e d with t h e c u r r e n t model, will b e examined.

In c a s e of t h e t r a n s p o r t a t i o n risk-cost analysis m o d e l , this d a t a s e t , one r e c o r d f o r e a c n feasible a l t e r n a t i v e generated, consists of:

a n a l t e r n a t i v e identification;

an a r r a y of c r i t e r i a f o r each feasible t r a n s p o r t a t i o n a l t e r n a t i v e ; additional m o d e l output f o r e a c h alternative, e.g., t h e node-arc sequence of t h e path;

a n a r r a y of c o n t r o l a n d policy v a r i a b l e s c o r r e s p o n d i n g t o e a c h a l t e r - native.

All i n t e r a c t i o n with t h e system i s menu-driven. A t t h e t o p level, summary information on t h e set of a l t e r n a t i v e s loaded i s provided (Figure 5.1).

f i g u r e 5.1% Top LeveL m e n u : s e l e c t i o n of t h e p o s t - p r o c e s s o r

BI-

Demonstrabon Prototype: Hazardous Substances Rlsk Yanaqem-t $+

eJ

L b r s l e u d D W l n - t V s s a a lo/=.

M s d km t m I S -1- 3. rm l m m t n r l Int>w* 6- -1 #r SF- h i q r .

-

t t l Y . A - Z l l .L- k w 1 . .

-

^u

This information includes:

r - 1 ~.brrrcsl ~ . t -

1-id ei.' F@nwu

w s l a t i a d -lati- -4d W m i c mr- M i a d Tr.IC w i a r-CAI -1 A u l y s i s r l i u l P l a t A v l y s i r

T m o l r md Disposal m r r r r m i d Wute S r r a . r Trmrqaeazia Rirt/Cat Amlysis bvir-tml 1-t A l s u a r a t

t h e number of a l t e r n a t i v e s ;

-

^{-n..*m d tm t.i,t*-. a,}

I .

*

-_{A1 1}

-

_Rights

^-.

Resawd. ^{1-8 -11-. IUI,.}

w

._ e

I

I

I

U r k ?- ^k

-.I

I-!

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t h e number of c r i t e r i a c o n s i d e r e d ;

a listing of c r i t e r i a , t o g e t h e r with t h e i r s t a t u s information (default s e t t i n g s f o r t h e t h r e e possible s t a t u s i n d i c a t o r s m i n i m i z e , mcrzimize, i g n o r e ) , a n d b a s i c s t a t i s t i c a l information ( a v e r a g e , minimum, maximum) f o r t h e individual c r i t e r i a .

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A t t h a t level, t h e menu o f f e r s t h e following choices:

d i s p l a y d a t a s e t s a v a i l a b l e f o r a n a l y s i s : (Figure 5 . 2 ) ;

select c r i t e r i a - this allows t h e u s e r to modify t h e s t a t u s c h a r a c t e r i z a - tion, i.e., change t h e dimensionality of t h e problem by ignoring o r including additional c r i t e r i a from t h e list (Figure 5.3);

s t a t i s t i c a l a n a l y s i s : h e r e statistical information on t h e d a t a s e t o t h e r t h a n t h e minima, maxima, and a v e r a g e values displayed by default can b e generated and displayed. In p a r t i c u l a r , t h i s includes s t a n d a r d devi- ations and median values as well as pairwise and multiple c o r r e l a t i o n coefficients, indicating relationships of indicators. Also, a c l u s t e r analysis option is f o r e s e e n , allowing a similarity ranking of a l t e r n a - tives and s u b s e t s of alternatives.

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r a n k i n g b y i n d i v i d u a i c r i t e r i a : h e r e t h e alternatives are ranked according t o t h e individual c r i t e r i a , resulting in a table of color-coded relationships.

d i s p l a y d a t a set: this invokes t h e second level menu f o r t h e display options, discussed below;

c o n s t r a i n c r i t e r i a : h e r e upper and lower bounds f o r t h e individual c r i t e r i a can b e defined, based on a graphical representation of t h e r a n g e and distribution of t h e c r i t e r i a values (Figures 5.3 and 5.4); set- ting t h e s e constraints results in t h e reduction of t h e set of alterna- tives considered; t h e bounds a r e defined by dragging, with t h e mouse graphical input device, a vertical b a r within t h e r a n g e of c r i t e r i a values, and cutting off alternatives left o r r i g h t of t h e b a r . The system displays t h e c u r r e n t value of t h e constraint, and indicates how many alternatives will b e deleted whenever t h e u s e r sets a constraint. If t h e constraint setting is verified by t h e u s e r , t h e alternatives excluded are deleted from t h e data set and new values f o r t h e descriptive statis- tics are computed.

f i n d pareto set: this option identifies t h e set of nondominated alterna- tives (see section 4.1), and indicates how many nondominated alterna- tives have been identified;

a n o t h e r f e a t u r e at this, as well as any o t h e r , level in t h e system is a n explain function t h a t provides a more detailed explanation of t h e menu options c u r r e n t i y available.

The option: d i s p l a v d a t a set generates a new menu of options. The display options are:

default scattergrams: t h e default scattergrams provide 2D projections of t h e d a t a s e t , using painrise combinations of t h e relevant c r i t e r i a (Figure 5.5). The f i r s t t h r e e combinations a r e displayed in t h r e e graphics windows. If t h e set of nondominated alternatives has already been identified, t h e pareto-optimal points will b e displayed in yellow and will be l a r g e r than t h e small, red, normal (dominated) alternatives;