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AFIT/GOR/AA/81 0-1

MADAM:MULTIPLE-ATTRIBUTE DECISION ANALYSIS MODEL

VOLUME 11

T!IFSIS w T CAFIT/GOR/AA/81D-I Wayne A. Stimpson (J> CC' T

2Lt USAFR ~~FEB 1 9 1982

AFITj,0R/AA/81 D-1

Thes is

t", MADAM:

MULTIPLE-ATTRIBUTE DECISION ANALYSIS MODEL

Volume II

Wayne A.. Stimpson2Lt USAFR

Prepared in partialfulfillment of 1he

requirements fer aMasters Degr' e

December 1981

Accession ForNTIS Cr ' UDTt(, TkY

School of ErgineeringAir Force Institute of TechnologyWright-Patterson Air Force Base

Volume II

Table of Contents

List of Figures and Tables....... .... . ..... .... i

Abstract .. .. ..... ..... ..... ...... ... v

1. Introduction. .. .. .... ...... ..... ....

11. Data Structures .. .. ..... ..... ....... 2

III. Sensitivity Analysis .. .. ... ..... ........ 7

IV, Hierarchy Manipulation. .. .. .... ...... ... 9Subroutines NODIN and FIND .. .. ... ........ 9Subroutines PRETOT, PRENEX, and NEXT. .. .. .... 11Subroutine SPAN .. .. ..... ..... ..... 13Subroutine CALC .. .. ..... ..... ..... 13

V. Variable List. .. ... ...... ..... ..... 15Variables Used. .. .. .... ...... ...... 1Pseudo-variables. .. .. ..... ..... ... 18Others .. .. ... ..... ..... ....... 18C,,cs Reference Map. .. ... ...... ..... 19

VI Program Structure .. .. ..... ..... ...... 20

Program Source Code .. .. ... ..... ..... ...... 23

List of Figures and Tables

Table 1 Pseudo-array Variables ..... ............ 3

Figure I Data Fields for Node Record........... 4

Table 2 Record Content ...... ................ 4

Figure 2 The Search Process (FIND) .............. 10

Figure 3 Logic Structure for NEXT ..... .......... 12

Figure 4 Logic Flow of CALC ..... .............. 14

AFIT/GOR/AA/81 D-1

Abstract

The compldx multifaceted decision situations present today sugqest

the need for a timely, automated too. A~decision-maker is fof ed into

comparing alternative actions or systems .over an entire set of different

measures of merit. This effort is an on-line, real-time, compiter-based

decision aid designed to assist the decision-maker in clarifyiog prefer-

ences in a complex decision environment. It is applicable to problems

which may be represented by a hierarchy of objectives to be satisfied.

The program is MADAM: Multiple-Attribute Decision Analysis Model, and it

is written in FORTRAN V and is implemented on the CYBER 175 system. MADAM

is designed to aid the decision-maker as he or she progresses through prob-

lem formulation, parameterization, sensitvity analyses, and a decision,

including storage of all data and rationales. Deterministic problems are

analyzed through Multi-Attribute Utility Theory concepts and an additive

value function is utilized for sensitivity analysis. Pairwise preferential

independence is tested bqtween attrithutes. The sensitivity analyses in-)

cludc a cumulaive weight analysis, a re.lative weight analysis, and an

attribute level analysis. The analyses may be conducted by fixing an ob-

jective to he considered and conducting the analysis across the alternative

4 systems or actions, or conversely by fixing the alternative to be consid-

ered nd conducting the analysis across a desired set of objectives.

t he werk is divided into two volumes. Volume I is a theoretical pre-

sentation and includes a user's manual. It requires no programming ex-ipertise and may be used independently of Volume II, Vulume I is a pro-

igraiikming manual including the source code. t may not be used independently

of Volume 1.

I INTRODUCTION

This manual is designed to supplement the information contained in

Volume I of this thesis. The foundamental intention to this volume is to

provide decision analysts with an elucidation of the details of MADAM in

order to better understand the program and, hence, facilitate its imple-

mentation. Future analysts may also have the desire to modify or extend

the scope of problems suitable for analysis with MADAM, and the following

information will be very useful.

MADAM is written in FORTRAN V and implemented on the CYBER 175 system.

An effort was made to keep the code as machine independent as possible,

and for those situations where this was not possible, an explicit discus-

sion concerning Iodifications pre-requisite to transporting MADAM IS pro-

vided. All references to an example problem will refer to the 2xample

illustrated in Chapter V of Volume I.

*1 w • • • • •

II DATA STRUCTURES

MADAM utilizes a random access file for storing and manipulating data.

Throughout the program are pairs ofsubroutines and functi'ons which are re-

sponsible for setting and retrieving data, This is possible because FORTRAN

V will not recognize an array variable until it is determined that no func-

tion by that name exists. Thus, for example, MADAM uses the function ARAY

(I, J) to represent a pseudo-array variable. A complete l.ist of the sub-

routine function pairs which represent psuedo-array variables is given in

Table 1. Each record of the random access file is 100 words in length.

This 100 word record is divided up in to data fields as shown in Figure 1

for the first 500 records. The remaining records have data fields as shown

in Table 2.

r NOTE***The current ability to access both real and

integer (character) data from a record is dependent

on the fact that the CYBER 175 system uses one word

for both types of information. In order to imple-

ment MADAM on a system which uses two words for real

data and one word fur integer data, it would be nec-Iessary to modify the psuedo-array variables so thatthe desired words will be accessed. This will re-

quire manipulation of the record length and storage

: (retrieval) subscripts.

Each data file (tree) constructed will require about 300 PRU for per-

manent storage.

A§ illustrated in Figure 1, each node record contains information

about the position of the node in the objective hierarchy plus any node-

4_) . ..

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Node Data Word(s) Variable~s)

Node Reference Number Digit 1 IRAY (I,')

First Descendant 2 IRAY (1,2)First Crosslink 3 RAY (1,3)

Backlink 4 RAY (1,4)

Relative Weight 5 ARAY (I)Cumulative Weight 6 ARAY (1,2)

Associated Attribute 7 LABEL (1)System Values 8 to (NSYS+7) VRAY (1,J)

Comments (Rationale) (NSYS+7) to 64 -none-

Objective 65 to 80 OBJCTV (1)

to OBjCTV (16

Figure 1 Data Fields for Node Record

Information Record No,

Node Record 1 to 500

Number of Nodes (NNODES) 501

Number of Attributes (NATT) 502

Number of Systems (NSYS) 503

Null Record 504

Regression Intercepts 505

Regression Slopes 506

Sum of Squared Errors 507

Regression Form Indicators 508

System Names 509

Attribute Names 510

Null Record 511

Worst Attribute Levels 512

Best Attribute Levels 513Table 2 Record Content

specific data. rhe positional information is provided by tile first four

words in the record. These words contain the node reference number digit,

the first descendant, the first crosslink, and tile backlink respectively.

The tode reference number digit indicates where this node is on its span.

For instance, in the sample problem, the node reference number digit of

the node with the objective: to maximize aircraft survivability, is two.

The first descendant pointer contains a number of the first child of the

given node (if any). Absence of a first child (IRAY (1,2) <.0) implies

that the given node is a data node. The first crosslink pointer contains

a number of the immediate siblings of the given node (if any). The back-

link pointer contains the number of the node which has this given node as

a first descendant or a first crosslink.

The remaining words in a node record provide the data for identifying

the node and incorporating it into the decision analysis. The fifth word

of the record (ARRAY (1,1)) contains the weight of the given node relative

to its siblings. These relative weights are normalized to a unity sum

across the span. The sixth word of a node record (ARRAY (1,2)) contains

I the cumulative weight of a given node. It is automatically calculated by

MADAM during the "folding-back" of the objective hierarchy (see subroutine

4 CALC), and it is a function of the node's relative weight and the cumulative

weight of its parent. The seventh word in a node record (LABEL (I)) con-

tains the attribute label to be associated with that node, it is relevant

only for data nodes, and is left a null field for all noJes interior to the

INOTE*I*All character data is designed around theCYBER's 10 letters/word fcrinat. In order to

implement MADAM on a system with a differert

1storage format, it will be necessary to scale

all input data or data structures to incorporate

a different number of characters per word.

The next set of words in a node record (VRAY (1,J)) hold the values

of the alternative systems relative to a given node. These values are

either calculated by MADAM (for an interior node) or indirectly input by

the user (for a data node). Only as many words as needed are used for

this information baseo on the number of systems. Those words between

the system vaiues and the 64th word are used to hold comments or ration-

ale concerning either the relative weighting of the children of a given

node or the input attribute levels of the alternative systems. Thus, the

amount of space for these comments is inversely proportional to the number

of alternative systems. Also, because of this structure, the number of

alternative systems is limited to 57. Finally, the objective associated

with the given node is contained in the 64th to the 80th words.

III SENSITIVITY ANALYSIS

The sensitivity analyses are controlled by the routines in the pri-

mary overlay (4,0). The types of analysis handled by the program are:

relative weight analysis (RSA), cumulative weight analysis (CSA), attri-

bute level analysis (LSA), and a system analysis (SSA) based on one of

the first three types. The first three types of analysis involve deter-

mining a fixed node and conducting the appropriate analysis for all sys-

tems at that node. The last type of analysis (actually three subtypes

SSA/RSA, SSA/CSA, SSA/LSA) involves fixing the alternative system and then

conducting the appropriate analysis over a set of nodes,

The entry point of the overlay, routine SENSTV is used to determine

the main type of analysis (CSA, RSA, LSA, or SSA). For any of the first

three types or sensitivity analysis, control is shifted to routine DETNOD

which is used to fix the analysis to the desired node. Routine DETNOD also

establishes the range over which the factor (cumulative weight, relative

weight, or attribute level) is to be considered. For the last type of

analysis (SSA), control shifts to routine SYSSEN which in turn calls upon

the routines, SENVAL, SENTYP, and NODSET to establish the fixed system,

the factor to be considered, and the set of nodes over which the analysis

is to be conducted.

Regardless of the type of analysis to be performed, once the initial-

ization process (as described in the preceding paragraph), has been com-

pleted, control shifts to the routine COMPUT. This routine utilizes the

equations presented in Chapter IV (of Volume I) to generate a matrix of

overall values. Each row of this matrix corresponds to a different level

of the factor under consideration. For the CSA, RSA, and LSA, each column

corresponds to a different system. For the SSA, each column represents a

diffe-ent node.

IAfter the overall values have been calculated, the routines GRAPH

and TABDIS are used to display the information in a graphial or tabular

manner respectively (see Figures 4.1 through 4.4 in Volume I). At the

conclusions of the desired type of sensitivity analysis, control is turned

over to the main option selection.

IV HIERARCHY MANIPULATION

Subroutines NODIN and FIND

These subroutines are used to allow the user to enter the tree struc-

ture at a desired node. If the desired node is not an element of the tree,

FIND will identify that node which is closest to the input node. All nodes

are referenced by their node reference number (NRN). Except for option

***MOD)***, the input of a non-validated NRN will be interpreted as an in-

dication that the user desires to exit from the option. Program control

will be return to the calling portion of MADAM. For option ***MOD*** an

invalide NRN is interpreted as a desire to introduce a new node having

the input NRN, so the appropriate branches (and dummy connecting nodes)

are constructed so as to create a vaiid environment for the input node.

Subroutine NODII. This routine allows the user to input a desired

NRN. If anything but a null string (space followed by <CR>) is entered,

this routine will call on FIND to establish the existence of a node cor-

responding to the input NRN.

Subroutine RIND. This routine implements a modified breadth-first

search of the tree to search for a node corresponding to the input NRN.

With each digit of the NRN representing a level, FIND starts with the

first digit and attempts to locate a node with the desired NRN digit by

crossing the span at that level. This process is repeated for all re-

ipaining levels (digits) of the input NRN. If a match cannot be estab-

lished at a given level, this indicates that the desired node would be

4 an added crosslink at that level. If a match is established, the next

digit iF brought up For c,'nsideration. if no more downlink path is

available (a dat node has been reached) the routine is terminated.

Figure 2 1lustrates an example of this search process.

Input NRN: 1.3.2

Iteration Result Rehadks

1 Fi'rst digit is satisfted

2 1 No Match

3 1 No Match

t 4 Second digit is satisfied

I I 13

No Match

I6 All digits satisfied soexit

Figure 2 The Search Process (FIND)

Subroutines PRENEX, PRETOT, and NEXT

These routines drive a depth-first search of the tree structure. SUbp

routines PRENEX and PRETOT are used to initialize the search process at a'

desired node ort at the root node respectively. Once all variables have

been initialized, subroutine NEXT performs a depth-first traversal, advant-

ing one node each time it is called.

Subroutine PRENEX. This routine elicits a NRN from the user by in-

voking NODIN. If a valid NRN is entered, the traversal will begin at the

node corresponding to the input NRN. All descendants of this input node

will be encountered in the resulting transversal.

Subroutine PRETOT. This routine initialized the variables required

by NEXT so as to run a transversal of the entire tree structure, It is

usually invoked just prior to a potential application of PRENEX, thus al-

lowing the, user to gracefully opt a tour of the entire hierarchical struc-

Subroutine NEXT. This routine tours the hierarchical structure begin-

ning at the node specified by either PRETOT or PRENEX. Since a check is

made as to the validity of the input node before any processing is done,

it 'is pcssible to exit the current option by entering a null string during

option '?RENEX. The logic implemented in NEXT 'is illustrated in Figure 3i

START)

KICONT=O)

'A ~~one ~ TR~

I a Downik i n DoTNeRTR

ISet DownlinkTravelled GetFlag and Add Crossi ink

new Downl ink

-T{ ~Qonti nue(ICONT=l )

RETURN

Figure 3 Logic Flow for N'EXT

~Subroutine SPAN

This routine allows for the input of the objective hierarchy stru;-

ture by sequentially providing nodes in a depth-first fashion. The user

inputs immediatedescendants of the current node. Subroutine CHECK is

used to establish a satisfactory set of children for the current node.

Once a satisfactory set is established the first descendant of the

current node becomes the new parent node. When a data node is reached

(no descendants are input)i the, crosslink to the current node becomes

the new parent node. If no more siblings exist, the routine will back

up one level and establish the first crosslink as the new ,parent node

(see Figure in the main thesis text).

Subroutine CALC

This routine calculates intermediate values (collapses the tree)

by implementing a modified depth-first search. The tree is examined by'

p, each path from the root node to a data rode, and the cumulative weightsare calculated. System values based on the attributes are then computed

for each node. The logic flow of CALC is provided in Figure 4. The

cumulative weight of each node is given by:

CUMWT (NODE) = CUMWT(PARENT)* RELWT(NODE)

The system value at a particular node are the products of the systemIt value at a data node and the cumulative weight of the upper nodes for

each system.

€+ 13

Calculate CUMWT ofnode

his a NoClear Obt& 1nData Node System -0'next

Values node

For each level abovethe current node foreach alternative system,

add the product of theCUMWT at this node and-the value at this node

tthe value at theuprlevels.]

Figure 4 Logic Flow of CALC

V. VariablesLi st

Variables Used

The following i's a list of variables used in the program. This list

includes only important variables and does not include scratch variables,

do-loop indices, and dummy arguments. Psuedo-array variables are in the

next section.

AIN - vector of relative weights

ARRAY - used to read and write to the data file. It holds onerecord worth of information

ATTAX -maximum level oT attribute

ATTMIN - minimum level of attribute

ATTNAM name of attribute being perturbed during sensitjyityanalysis

ATTRI - name of first attribute

ATTR2 - name of second attribute

BO - intercept of linear regression (linear form)

I BOEXP - intercept of linear regression (exponential form)

BOLN - intercept of linear regression (logarithmic form)

I BOROOT intercept of linear regression (square-root form)

BOSQR - intercept of linear regression (squared form)

Bi - slope coefficient of linear regression (linear form)

81ExP - slope coefficient of linear regression (exponential form)

BILN - >lope coefficient of linear regression (logarithmic form)

BIROOT slope coefficient of linear regression (square-root form)

BISQR - slope coefficient of linear regression (squared form)

-' BAND - boundaries of indifference curve window

CMD - main option selection

DELl - change in first attribute,

DEL2 - change in second attribute

ERRFLG - counts the number of consecutive errors in user entryduring main option selection

FACLVL - level of attribute

FINDST - set of IFIND for SSA

ICHECK - flag if using between-node check0 - not using checkI - using check

ICONT - flags whether or not to continue processing0 - stop1 - continue

IDATA -flags whether or not node is data node0 - interior node

1 - ddta node

FADD -flags relationship of current node to input node.~Determines how to add th input branch to the existing

f 'I branch." . ) . 2 - IFIND is parent to input node

3 - IFIND is sibling to input node

IFIND - points to toe last node of the branch contained in INNRN

INDERR - flag for PPI0- PPI holdI - PPI fails

INDX - NRN digit for IFIND

I INDXST - set of INDX for SSA

t IQUIT - flag to continue processingI1 0 - continue

1 - stop

IRRAY used to read and write to the data file. It holds onej record worth of information.

ISAVD flags whether current information is saved on data filein case of abort0 -not savedI saved

- - ISIN - record number of record currently in memory (in ARRAY orIRRAY)

7£" i ISYS - number of systems being considered during sensiti.vityanalysis.

ITOL, - tolerance during PPI testing

ITOTL - flags typeof tree traversal,0 - not a total tree traversal1 - traverse ,all °descendants

KIDSET - set of NCHILD -for nodes during SSA

LEVEL - vector containing branch to node, IFIND(*,1) - location of record containing node *

(*,2) - flags whether or not visited by PRETOT-PRENEX-NEXT(*,3) - used in copying

LINE - one line of data for plot

LVL - length of LEVEL vector

MATRIX - pairwise.-comparison matrix

MAXI - maximum level of first attribute

MAX2 - maximum level of second attribute

MINI - minimum level of first attribute

MIN2 - minimum level of second attribute

MTYPE - type of SSA to be done

NATT - nuiroer of attributes

NCHILD - number of children for a node

NCROSS - number of siblings on a span

NDEEP - deoth of tree (maximum number of levels)

NDIFF - number of digits in INNRN not matched by routine,,:IND

NLOUD - number of allowed nodes plus four

NLVLS - length of INNRN

NNODES - number in tree plus one (master)

NSYS - number of alternative systems

NTAPE - defines which data files is in use

tNTREE - maximum number of simultaneous data files

(9OBiCT-V.- objective assdc'i'ated ,with- 'pdrt-iAcular node

SENS - type of sensitivity, analysis;

SSE - sumrof squared error (linear form)

SSEEXP - sum of squared error ('exponential-fo )' .

SSELN - sum :of squared error (logarithmic form)

SSERT - sum of-squared~error (square-root form)

SSESQR - sum of squared error (squared form)

TOLER - tolerance for PPI testing

TOP - objective of parent node during between-node check

USER - user identifier

VAL - attribute levels which define data points for theindividual value function

!I WHOLD - matrix of overall value of systems during sensitivityanalysis

WMAX - maximum level of factor during sensitivity analysis

WMIN - minimum level of factor during sensitivity analysis

Psuedo-arry Variables

The following are functions which are treated as if they were array

variables (se.e Table 1).

ARAY(I,J) relative and cumulative weight of node

ATT(I,J) best/worst levels of attributes

ATTI(I) attribute names

CRAY(I) comments or rationale (subroutine)

I LRAY(I,J) node position information

) oLABEL(I) name of attribute associated with node I

' ° OBJECT(I,OBJCTV) objective at node I

PARAM(I,J) regression parameters

r n • •18

(j. SYSLBL(I) alternative system names

VRAY(I,J) system values at node I

Cross Reference Map

The listing contains a partial load map, and combined with the com-

ments within each program unit, a corss reference map of where each varY-'

able is used or modified may be generated.

, , VI. Proqrani Structure

MADAM is divided up into five overlays (one main, four primary)

which reduce the required field length for loading to about 60K. These

overlays are machine dependent, but they may be removed by deleting the

overlay statements, changing the program statement following each over-

lay statement to a subroutine statement (except for PROGRAM MADAM), and

changing all overlay calls to subroutine calls to the subroutine making

entry into the described overlay. Each primary overlay is designed to

operate independently of the other primary overlays. Only the main over-

lay and one primary overlay are in the user's field length at one time,

thus considerably reducing the required core memory allocation. The

following is a list of routines in MADAM in the order in which they ap-

pear in the source code.

OVERLAY (XFILE,O,O) - main overlayPROGRA14 MADAM

FUNCTION ARAY(I,J)SUBROUTINE ASET(I,J,V)FUNCTION ATT(I,,3)FUNCTION ATTI(1)SUBROUTINE ATTLBL(I,ATTRIB)SUBROUTINE ATTSET(I,ATTRIB)SUBROUTINE ATTVAL(I,J,V)SUBROUTINE COPYRSUBROUTINE CRAY()SUBROUTINE CSET(1)SUBROUTINE FINDSUBROUTINE HELPSUBROUTINE INTROFUNCTION IRAY(I,J)SUBROUTINE ISET(I ,S,M)

FUNCTION LABEL(I)i' , iSUBROUTINE LSET(I,LABELT)

V SUBROUTIC: NEXTSUBROUTINE NODINSUBROUTINE OBJECT(I,OBJCTV)

4 -SUBROUTINE OSET(IOBJCTV)FUNCTION DPRAM(IJ)

K. SUBROUTINE PRENEX

SUBROUTINE PRETOTSUBROUTINE PSET(I ,J,V)FUNCTION SYSLBL(I)SUBROUTINE SYSSET(I,SYSLBL)SUBROU-INE TLOADSUBROUTINE TSAVEFUNCTION VALU(X,LABEL)FUNCTION VRAY(I,J)SUBROUTINE VSET(I,K',V)FUNCTION XLEVEL(X,LABEL)

OVERLAY (XFILE,l,O) - primary overlay

PROGRAM DUMMYSUBROUTINE ADOSYSSUBROUTINE ASK(I, X1,X2,XMID)SUBROUTINE CHECKSUBROUTINE DELSYSSUBROUTINE GRAFIXSUBROUTINE INITTSUBROUTINE M4ASTERSUBROUTINE MODIFY

SUBROUTINE NEWSUBROUTINE PICTUR(INDEX)SUBROUTINE PHISUBROUTINE PRUNESUBROUTINE RDATTSUBROUTINE RDSYSLSUBROUTINE RDTTLSUBROUTINE REGRS(INDEX)SUBROUTINE SPANSUBROU2TINE STATSUBROUTINE TAPERSUBROUTIN't VALUE

OVERLAY (XFIL1,2,0) - primary overlay

PROfGRtAM WV LOADSU6ROUrINE CALLSIJIROUT INF DVIDE(NAJ))

11,1101TINE GMVEC (NCROSS , I FI ND , AI1N)SUL"ROUT INF RrivS1HBROIJT It'll RDWTFUNCTION SUM(N,A)SUBRCIITINE NVLODI(OPT)

OVERLAY (Xf-ILV,3,O) - primary overlay

PROGRAM NUMSUBROUTINE DISPLASUBROUTINE DSPLOfSUBROUTINE NEVIPGSUBROUTTIE jNUMREV

OVERLAY (XFILE,4,O) - primary overlay

PROGRAM SENSTV- SUBROUTINE COMPUT

SUBROUTINE DETNOD3: SUBROUTINE HEADER(I ,OBJCTV,ANSWER)jSUBROUTINE SENVALSUBROUTINE TABDIS

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t'1VED FOR PULI- RELEASE' AFR 190.17.FMRIC C. LYNC1 ,Major USAFDirector of Publc ALI;?.

19. KEY WORDS (Continue on reveree aid* If neceeary md Identtf) 6y bfack numbr)

Multi ple-Attri bute 7r r/ce h stiuo of Thhnd?~, (ATO}Decision Analysis " 'a- ...... ,, i, O 5i " "Computer AidPreferenceValue20. ABSTRACT (Continue on reveree side it neceeeary and Identify by Wlock numbet)

The complex multifaceted decision situations present today suggest the needfor a timely, automated tool. A decision-maker is forced into comparing alterna-tive actions or systems over an entire set of different measures of merit, Thiseffort is an on-line, real-time, computer-based decision aid designed to assistthe decision-maker in clarifying preferences in a cooplex decision environment.It is applicable to problems which may be represented by a hierarchy of objec-tives to be satisfied. The program is MADAM: Multiple-Attribute Decision Anal-ysls Model, and it is written in FORTRAN V and is implemented on the CYBER 175

DD I jAN 73 1473 EDITION OF I NOV 6s IS OBSOLETE UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE (ihen Date Entered)

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system. MADAM is designed to aid the decision-maker as he or she progressesthrough problem formulation, parameterization, sensitivity analyses, and adecision, including storage of all data and rationales. Deterministic problemsare analyzed through Multi-Attribute Utility Theory concepts and an dditivevalue function is utilized for sensitivity analysis. Pairwise prefe'Aenti'alindependence is tested between attributes. The sensitivity analysers include acumulative weight analysis, a relative weight analysis, and an attribute levelanalysis. The analyses may be conducted by fixing an objective to be consideredand conducting the analysis across the alternative systems or actions, or con-versely by fixing the alternative to be considered and conducting the analysisacross a desired set of objectives.

The work is divided into two volumes. Volume I is a theoretical presenta-tion and includes a user's manual. It requires no programming expertise andmay be used independently of Volume II. Volume II is a programming manual in-cluding the source code. It may not be used indepeadently of Volume I.

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