U.S. patent application number 10/426666 was filed with the patent office on 2003-11-06 for methods and apparatus for decision making.
Invention is credited to Xu, Dong-Ling, Yang, Jian-Bo.
Application Number | 20030208514 10/426666 |
Document ID | / |
Family ID | 29273094 |
Filed Date | 2003-11-06 |
United States Patent
Application |
20030208514 |
Kind Code |
A1 |
Yang, Jian-Bo ; et
al. |
November 6, 2003 |
Methods and apparatus for decision making
Abstract
There is disclosed a multiple criteria decision analysing method
in which a plurality L of basic criteria are assessed in order to a
general criterion, comprising the steps of: making an assessment
{(K.sub.m,l,.gamma..sub.m,l),m=1, . . . , M} of the l.sup.th basic
criteria under a set of grades {K.sub.m,l,m=1, . . . , M}; and
transforming the assessment {(K.sub.m,l,.gamma..sub.m,l),m=1, . . .
,M} to an assessment {(H.sub.n,.beta..sub.n,l),n=1, . . . ,N} of
the general criterion under a set of grades {H.sub.n,n=1, . . . N}
using the matrix equation: 1 [ 1 , l 2 , l N , l ] = [ 1 , 1 1 , 2
1 , M 2 , 1 2 , 2 2 , M N , 1 N , 2 N , M ] = [ 1 , l 2 , l M , l ]
wherein: H.sub.n is the n.sup.th grade for assessment of the
general criterion; K.sub.m,l is the m.sup.th grade for assessment
of the l.sup.th basic criterion; .alpha..sub.n,m is the m.sup.th
degree to which K.sub.m,l implies H.sub.n; .gamma..sub.m,l is the
degree to which the l.sup.th basic criterion is assessed to
K.sub.m,l; .beta..sub.n,l is the degree to which the th basic
criterion is assessed to H.sub.n; and 2 n = 1 N n , m = 1
Inventors: |
Yang, Jian-Bo; (Sale,
GB) ; Xu, Dong-Ling; (Sale, GB) |
Correspondence
Address: |
James E Bradley
Bracewell & Patterson, L.L.P.
P.O. Box 61389
Houston
TX
77208-1389
US
|
Family ID: |
29273094 |
Appl. No.: |
10/426666 |
Filed: |
April 30, 2003 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
60377350 |
Apr 30, 2002 |
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Current U.S.
Class: |
708/160 |
Current CPC
Class: |
G06F 3/0481 20130101;
G06N 7/005 20130101 |
Class at
Publication: |
708/160 |
International
Class: |
G06F 003/00 |
Claims
What is claimed is:
1. A multiple criteria decision analysis method in which a
plurality L of basic criteria are assessed in order to a general
criterion, comprising the steps of: making an assessment
{(K.sub.m,l,.gamma..sub.m,l),m=1, . . . ,M} of the l.sup.th basic
criteria under a set of grades {K.sub.m,l, Im=1, . . . ,M}; and
transforming the assessment {(K.sub.m,l.gamma..sub.m- ,l),m=1, . .
. , M} to an assessment {(H.sub.n,.beta..sub.n,l),n=1, . . . ,N} of
the general criterion undera set of grades {H.sub.n, n=1, . . . ,N}
using the matrix equation: 35 [ 1 , l 2 , l N , l ] = [ 1 , 1 1 , 2
1 , M 2 , 1 2 , 2 2 , M N , 1 N , 2 N , M ] = [ 1 , l 2 , l M , l ]
wherein: H.sub.n is the n.sup.th grade for assessment of the
general criterion; K.sub.m,l is the m.sup.th grade for assessment
of the l.sup.th basic criterion; .alpha..sub.n,m is the degree to
which K.sub.m,l implies H.sub.n; .gamma..sub.m,l is the degree to
which the l.sup.th basic criterion is assessed to K.sub.m,l;
.beta..sub.n,l is the degree to which the th basic criterion is
assessed to H.sub.n; and 36 n = 1 N n , m = 1.
2. A method according to claim 1 in which the values of
.alpha..sub.n,m(m=1, . . . , M and n=1, . . . ,N) are assigned by a
decision maker.
3. A method according to claim 2 in which a grade K.sub.m,l implies
a grade H.sub.n to a degree of .alpha..sub.n,m wherein m=1, . . .
,M.
4. A method according to claim 1 in which the values of
.alpha..sub.n,m(m=1, . . . ,M and n=1, . . . ,N) are determined
using the following equations: 37 n , m = u ( H n + 1 ) - u ( K m ,
l ) u ( H n + 1 ) - u ( H n ) , n + 1 , m = 1 - n , m
,.alpha..sub.i,m=0(i=1, . . . ,N,i.noteq.n,n+1) if
u(H.sub.n).ltoreq.u(K.sub.m,l).ltoreq.u(H.sub.n+1) for n=1, . . .
,N-1;m=1, . . . ,M where u(H.sub.n) and u(K.sub.m,l) are utilities
of H.sub.n and K.sub.m,l, respectively.
5. A method according to claim 4 in which at least one of
u(H.sub.n) and u(K.sub.m,l) are estimated by a decision maker.
6. A method according to claim 4 in which at least one of
u(H.sub.n) and u(K.sub.m,l) are determined using one or more of the
equations: 38 u ( H n ) = n - 1 N - 1 for n = 1 , , N if H n + 1 is
preferred to H.sub.n; and 39 u ( K m , l ) = m - 1 M - 1 for m = 1
, , M if K m + 1 , l is preferred to K.sub.m,l.
7. A method according to claim 1 in which: a basic criterion is
assessed using the set {(k.sub.j, p.sub.j),=1, . . . ,P}, where
k.sub.j is a number and p.sub.j is the probability of the basic
criterion taking the number k.sub.j; and the values of
.gamma..sub.m,l in the assessment {(K.sub.m,l,
.gamma..sub.m,l),m=1, . . . ,M} are calculated using the equation
40 m , l = j = 1 P S m , j p j for m = 1 , , M wherein : S m , j =
K m + 1 , l - k j K m + 1 , l - K m , l , S m + 1 , j = 1 - S m , j
, S i , j = 0 ; m=M, . . . ,M-1; j=1, . . . ,P; i=1, . . .
,M,i.noteq.m,m+1 if K.sub.m,i.ltoreq.k.ltoreq.K.sub.m+1,l
8. A multiple criteria decision analysis method in which a
plurality L of basic criteria are assessed in order to assess a
general criterion, comprising the steps of: assigning weights
W.sub.i(i=1, . . . L) to the L basic criteria; normalising the
weights using the equations 41 i = W i j = 1 L W j ( i = 1 , L )
;determining .beta..sub.n,l, wherein .beta..sub.n,i is the degree
to which the i.sup.th basic criterion is assessed to H.sub.n, and
H.sub.n is the n.sup.th grade for assessment of the general
criterion, the general criterion being assessed into N grades;
calculating weighted degrees of belief n i from the equation
m.sub.n,i=.omega..sub.i.beta..sub.n,i)=.omeg-
a..sub.i.beta..sub.n,i,(n=1, . . . , N; i=. . . , 1);and
calculating a remaining probability mass m.sub.H,i from the
equation 42 m H , i = 1 - n = 1 N m n , i ( i = 1 , , L ) .
9. A method according to claim 8 in which m.sub.H,i is decomposed
into {overscore (m)}.sub.H,i and {tilde over (m)}.sub.H,i, wherein:
m.sub.H,i={overscore (m)}.sub.H,i+{tilde over (m)}.sub.H,i;
{overscore (m)}.sub.H,i=1-.omega..sub.i;and 43 m ~ H , i = i ( 1 -
n = 1 N n , i ) for i = 1 , , L .
10. A method according to claim 9 in which m.sub.n,i,{overscore
(m)}.sub.H,i and {tilde over (m)}.sub.H,i(i=1, . . . ,L) are
aggregated into combined probability masses I.sub.n,L,{overscore
(I)}.sub.H,L and .sub.H,L, respectively, using equations i) to ix)
in a recursive manner I.sub.n,1=m.sub.n,l(n=1,2, . . . ,N) i)
I.sub.H,l=m.sub.H,l ii) .sub.H,l={tilde over (m)}.sub.H,l iii)
{overscore (I)}.sub.H,l={overscore (m)}.sub.H,l iv) 44 K i + 1 = [
1 - t = 1 N j = 1 j t N I t , i m j , i + 1 ] - 1 v )
I.sub.n,i+1=K.sub.i+1[I.sub.n,im.sub.n,i+1+I.sub.H,im-
.sub.n,i+1+I.sub.n,im.sub.H,i+1](n=1,2 . . . , N) vi)
.sub.H,i+1=+K.sub.i+1[.sub.H,i{tilde over (m)}.sub.H,i+1+{overscore
(I)}.sub.H,i+1{tilde over (m)}.sub.H,i+1+.sub.H,i+1{overscore
(m)}.sub.H,i+1] vii) {overscore (I)}.sub.H,i+1=K.sub.i+1[{overscore
(H)}.sub.H,i+1{overscore (m)}.sub.H,i+1] viii)
I.sub.H,i+1={overscore (I)}.sub.H,i+1+{overscore (I)}.sub.H,i+1 ix)
i={1,2, . . . ,L-1}
11. A method according to claim 10 in which combined degrees of
belief .beta..sub.n and .beta..sub.H are generated using the
equations: 45 n = I n , L 1 - I _ H , L n = 1 , 2 , , N H = I ~ H ,
L 1 - I _ H , L wherein .beta..sub.n is a degree of belief to which
the general criterion is assessed to the n.sup.th grade H.sub.n and
.beta..sub.H is a remaining degree of belief which is not assigned
to any specific grade.
12. A method according to claim 11 wherein each grade H.sub.n+1 is
more favourable than H.sub.n and performance indicators of a
general criterion are generated using the equations: 46 u max = n =
1 N - 1 n u ( H n ) + ( N + H ) u ( H N ) u min = ( 1 + H ) u ( H 1
) + n = 2 N n u ( H n ) u avg = u max + u min 2 and wherein
u.sub.max, u.sub.min and u.sub.avg are the best possible, worst
possible and average performance indicators respectively, and
u(H.sub.n)(n=1, . . . N) is optionally defined by 47 u ( H n ) = n
- 1 N - 1
13. A method according to claim 8 in which the values of
.beta..sub.n,i are determined using a method according to claim
1.
14. A carrier medium storing a computer program, which computer
program performs a method according to claim 1.
15. A computer adapted to perform a method according to claim
1.
16. A carrier medium storing a computer program, which computer
program performs a method according to claim 8.
17. A computer adapted to perform a method according to claim 8.
Description
CROSS REFERENCE TO RELATED APPLICATIONS
[0001] This application claims priority from the provisional
application Serial No. 60/377,350, filed Apr. 30, 2002, entitled
"Methods and Apparatus for Decision Making".
FIELD OF THE INVENTION
[0002] The present invention relates to methods and apparatus for
decision making, including software therefor.
BACKGROUND OF THE INVENTION
[0003] Decision making is a most common human activity. Individuals
and organisations make all kinds of decisions in a variety of ways
on a regular basis. Most decision problems are associated with a
number of criteria, which may be measured in different ways, be in
conflict with one another, and comprise both a quantitative and
qualitative nature. In many situations, decision makers may have to
make decisions on the basis of incomplete or partial information.
For instance, buying a car may be an individual or a family
decision and a customer will not buy a car without taking into
account several criteria such as price, safety measures, size of
engine, and general quality. Similarly, a company often will not do
business with a supplier without assessing many criteria such as
financial stability, technical capability, quality and after sales
services.
[0004] There is a large literature associated with decision
sciences, in which techniques for aiding or actually making
decisions are proposed. Of most relevance to the present
application is Multiple Criteria Decision Analysis (MCDA), which is
an important area of decision sciences wherein scientific methods
are investigated and developed in order to support decision making
with multiple criteria.
[0005] A decision associated with multiple criteria is deemed to be
properly made if all criteria in conflict are properly balanced and
sufficiently satisfied. A MCDA problem can be generally modelled
using a decision matrix, where a column represents a criterion, a
row an alternative decision, and an element the outcome of a
decision on a criterion. The decision matrix for a car selection
problem, for example, may look like Table 1.
1TABLE 1 Example of Decision Matrix General Engine Size Fuel
Consumption Price . . . Quality Car 1 1400 cc 40 miles/gallon
.English Pound.8,000 . . . Good Car 2 1500 cc 45 miles/gallon
.English Pound.9,000 . . . Excellent . . . . . . . . . . . . . . .
. . . Car N 1100 47 miles/gallon .English Pound.7,000 . . .
Good
[0006] Several methods have been proposed to deal with MCDA
problems represented in the form of a decision matrix. Multiple
criteria utility (value) function (MCUF) methods are among the
simplest and most commonly used (see, for example, E.
Jacquet-Lagreze and J. Siskox, "Assessing a set of additive utility
functions for multicriteria decision making, the UTA method",
European Journal of Operational Research, Vol. 10, pp. 151-164,
1982, and R. L. Keeney and H. Raiffa, Decision with Multiple
Objectives: Preference and Value Tradeoffs, John Wiley and Sons,
New York, 1976).
[0007] The MCUF methods are based on the estimation of utility for
each outcome in a decision matrix. However, if a MCDA problem
involves a large number of criteria and alternative decisions,
estimating the utilities of all outcomes at every alternative on
each criterion will become a tedious procedure and as such the MCUF
methods will be difficult to apply in a satisfactory way (T. J.
Stewart, "A critical survey on the status of multiple criteria
decision making theory and practice", OMEGA International Journal
of Management Science, Vol. 20, No. 5-6, pp. 569-586, 1992).
[0008] Pairwise comparisons between pairs of criteria were
primarily used to estimate relative weights of criteria in several
methods including the eigenvector method (T. L. Saaty, The Analytic
Hierarchy Process, University of Pittsburgh, 1988), the geometric
least square method (G. Islei and A. G. Lockett, "Judgmental
modelling based on geometric least squares", European Journal of
Operational Research, Vol.36, No. 1, pp.27-35, 1988) and the
geometric mean method. Pairwise comparison matrices have also been
used to assess alterative decisions with respect to a particular
criterion such as in Analytical Hierarchy Process (AHP) (Saaty,
ibid) and in judgmental modelling based on the geometric least
square method (Islei and Locket, ibid). However, using pairwise
comparisons to assess alternatives may lead to problems such as
rank reversal as within the AHP framework (V Belton and T Gears "On
a short-coming of Saaty's method of analytic hierarchy", OMEGA,
vol. 11, No. 3, pp 228-230, 1981; Stewart, ibid). These
difficulties have lead to a long debate on how quantitative and
qualitative assessments should be modelled and aggregated.
Furthermore, both MCUF and AHP methods are incapable of properly
coping with decision problems with missing information. If
assessment information is missing for one criterion, one has to
either abandon this criterion altogether or make assumptions, ie to
use fabricated information. However, this may mislead the decision
making process.
[0009] Fuzzy sets based methods have been developed to deal with
MCDA problems with uncertainties. The main feature of such methods
is their capability of handling subjective judgements in a natural
manner. Therefore, they provide attractive frameworks to represent
qualitative criteria and model human judgements (R R Yager
"Decision-making under various types of uncertainties", Journal of
Intelligent and Fuzzy Systems, Vol.3, No. 4, pp 317-323, 1995).
However, fuzzy set methods suffer from two fundamental drawbacks.
Firstly, they use a simplistic approach and limited linguistic
variables to model a variety of information including both precise
numbers and imprecise judgements. The consequences of this
modelling strategy include the loss of precision in describing
precise data and the lack of flexibility in capturing the diversity
of information. The second drawback results from the use of fuzzy
operations for criteria aggregation. Traditional fuzzy operators
may lead to the loss of information in the process of aggregating a
large number of criteria (J Wang, J B Yang and P Sen "Safety
analysis and synthesis using fuzzy sets and evidential reasoning",
Reliability Engineering and Systems Safety, Vol. 47, No. 2, pp
103-118, 1995).
[0010] The present inventors have developed a MCDA method which has
been termed evidential reasoning (ER) (see J. Wang, J. B. Yang and
P. Sen, "Safety analysis and synthesis using fuzzy sets and
evidential reasoning", Reliability Engineering and System Safety,
Vol. 47, No. 2, pp. 103-118, 1995, J. B. Yang and M. G. Singh, "An
evidential reasoning approach for multiple attribute decision
making with uncertainty", IEEE Transactions on Systems, Man and
Cybernetics, Vol. 24, No. 1, pp. 1-18, 1994; J. B. Yang and P. Sen,
"A general multi-level evaluation process for hybrid MADM with
uncertainty", IEEE Transactions on Systems, Man, and Cybernetics,
Vol. 24, No. 10, pp. 1458-1473, 1994; and Z. J. Zhang, J. B. Yang
and D. L. Xu, "A hierarchical analysis model for multiobjective
decision making", in Analysis, Design and Evaluation of Man-Machine
System 1989, Selected Papers from the 4th IFAC/IFIP/IFORS/IEA
Conference, Xian, P. R. China, September 1989, Pergamon, Oxford,
UK, pp.13-18, 1990).
[0011] In the ER approach, it is proposed to use the concept of
belief degrees in an assessment framework to model subjective
judgements and develop an evidential reasoning algorithm to
aggregate criteria in the assessment framework (Zhang, Yang and Xu;
ibid: Yang and Singh, ibid; Yang and Sen, ibid). Compared with
fuzzy sets methods, the ER approach provides a more flexible way of
modelling human judgements (Yang and Sen, ibid) and the ER criteria
aggregation process is also based on the rigorous Dempster-Shafer
theory of evidence (G. A. Shafer, Mathematical Theory of Evidence,
Princeton University Press, Princeton, USA, 1976, the contents of
which, together with the contents of the other publications cited
above, are hereby incorporated by reference). However, the prior
art ER technique as described in the above mentioned publications
is primarily of academic interest, since it is unable to properly
accommodate a variety of "real life" situations. For example, the
prior art technique is not capable of accommodating precise data or
properly handling incomplete information, which may be caused due
to a lack of information, the complexity of a decision problem and
the inability of humans to provide precise judgements. Also, the
old ER algorithm does not provide a rigorous process of aggregating
incomplete information.
[0012] Therefore, there is a need to provide an improved MCDA
technique which is capable of dealing with "real life" situations,
and of overcoming the above described problems associated with the
prior art.
SUMMARY OF THE INVENTION
[0013] The present invention addresses the aforesaid need, and
overcomes the above described problems. The present invention
provides a rigorous means to support in a practical way the
solution of MCDA problems. It is capable of dealing with
quantitative and qualitative information, and can handle imprecise
subjective information in a way that is consistent and
reliable.
[0014] According to a first aspect of the invention there is
provided a multiple criteria decision analysis method in which a
plurality L of basic criteria are assessed in order to assess a
general criterion, comprising the steps of:
[0015] making an assessment {(K.sub.m,l,.gamma..sub.m.l),m=1, . . .
,M} of the l.sup.th basic criteria under a set of grades
{K.sub.m,l, m=1, . . . , ml;
[0016] and transforming the assessment
{K.sub.m,l,.gamma..sub.m,l,m=1, . . . ,M} to an assessment
{(H.sub.n,.beta..sub.n,l),=n=1, . . . , N} of the general criterion
under a set of grades {H.sub.n,n=1, . . . ,N} using the matrix
equation: 3 [ 1 , l 2 , l N , l ] = [ 1 , 1 1 , 2 1 , M 2 , 1 2 , 2
2 , M N , 1 N , 2 N , M ] = [ 1 , l 1 , l M , l ]
[0017] wherein:
[0018] H.sub.n is the n.sup.th grade for assessment of the general
criterion;
[0019] K.sub.m,l is the m.sup.th grade for assessment of the
l.sup.th basic criterion;
[0020] .alpha..sub.n,m is the degree to which K.sub.m,l implies
H.sub.n;
[0021] .gamma..sub.m,l is the degree to which the l.sup.th basic
criterion is assessed to K.sub.m,l;
[0022] .beta..sub.n,l is the degree to which the l.sup.th basic
criterion is assessed to Hn; and 4 n = 1 N n , m = 1
[0023] This method provides a rule based transformation of
qualitative information. The values of .alpha..sub.n,m(m=1, . . .
,M and n=1, . . . ,N) maybe assigned by a decision maker. The
decision maker may be, for instance, the user of software embodying
the method. A grade K.sub.m,l may imply a grade H.sub.n to a degree
of .alpha..sub.n,m wherein m=1, . . . ,M.
[0024] The values of .alpha..sub.n,m (m=1, . . . , M and n=1, . . .
,N) may be determined using the following equations: 5 n , m = u (
H n + 1 ) - u ( K m , l ) u ( H n + 1 ) - u ( H n ) , n + 1 , m = 1
- n , m ,
[0025] .alpha..sub.i,m=0(i=1, . . . ,N,i.noteq.n,n+1)
[0026] if u(H.sub.n).ltoreq.u(K.sub.m,l).ltoreq.u(H.sub.n+1) for
n=I, . . . ,N-1;m=1, . . . ,M
[0027] where u(H.sub.n) and u(K.sub.m,l) are utilities of H.sub.n
and K.sub.m,l, respectively. This can be regarded as a utility
based information transformation. It can be used, for example, if
the user of the method does not want or is unable to enter values
for .alpha..sub.n,m.
[0028] At least one of u(H.sub.n) and u(K.sub.m,l) may be estimated
by a decision maker.
[0029] At least one of u(H.sub.n) and u(K.sub.m,l) may be
determined using one or more of the equations: 6 u ( H n ) = n - 1
N - 1 for n = 1 , , N if H n + 1
[0030] is preferred to H.sub.n; 7 u ( K m , l ) = m - 1 M - 1 for m
= 1 , , M if K m + 1 , l
[0031] is preferred to K.sub.m,l.
[0032] According further to the method:
[0033] a basic criterion may be assessed using the set
{(k.sub.j,p.sub.j), j=1, . . . ,P}
[0034] where k.sub.j is a number and p.sub.j is the probability of
the basic criterion taking the number k.sub.j and the values of
.gamma..sub.m,l in the assessment {(k.sub.m,l,.gamma..sub.m,l),m=1,
. . . ,M} are calculated using the equation 8 m , l = j = 1 P s m ,
j p j for m = 1 , , M
[0035] wherein: 9 s m , j = K m + 1 , l - k j K m + 1 , l - K m , l
, s m + 1 , j = 1 - s m , j , s i , j = 0 ;
(m=1, . . . ,M-1); j=1, . . . , P; i=1, . . . , M, i.noteq.m, m+1
if K.sub.m,l.ltoreq.k.sub.j.ltoreq.K.sub.m+1,l
[0036] In this way, .gamma..sub.m,l can be calculated for
quantitative criteria, and a quantitative data transformation can
be performed. .gamma..sub.m,l may be assigned directly by a
decision maker in the case of qualitative criteria.
[0037] The first aspect of the invention may be performed in
conjunction with the second aspect of the invention, ie, a MCDA
method may comprise both the first and second aspects of the
invention.
[0038] According to a second aspect of the invention there is
provided a multiple criteria decision analysis method in which a
plurality L of basic criteria are assessed in order to assess a
general criterion, comprising the steps of:
[0039] assigning weights W.sub.i(i=1, . . . L) to the L basic
criteria;
[0040] normalising the weights using the equations 10 i = W i j = 1
L W j ( i = 1 , L ) ;
[0041] determining .beta..sub.n,i, wherein .beta..sub.n,i is the
degree to which the i.sup.th basic criterion is assessed to
H.sub.n, and H.sub.n is the nth grade for assessment of the general
criterion, the general criterion being assessed into N grades;
[0042] calculating weighted degrees of belief m.sub.n,i from the
equation
m.sub.n,i=.omega..sub.i.beta..sub.n,i,(n=1, . . . , N;i=1, . . .
,L);
[0043] and
[0044] calculating a remaining probability mass m.sub.H,i from the
equation 11 m H , i = 1 - n = 1 N m n , i ( i = 1 , , L ) .
[0045] In this way, it is possible to deal with incomplete
information. The weights may be assigned directly by a decision
maker or estimated.
[0046] m.sub.H,i may be decomposed into {overscore (m)}.sub.H,i and
{tilde over (m)}.sub.H,i, wherein:
[0047] m.sub.H,i={overscore (m)}.sub.H,i+{tilde over
(m)}.sub.H,i;
[0048] {overscore (m)}.sub.H,i=1-.omega..sub.i; and 12 m ~ H , i =
i ( 1 - n = 1 N n , i ) for i = 1 , , L .
[0049] This approach permits greatly advantageous treatment of
incomplete information.
[0050] m.sub.n,i, {overscore (m)}.sub.H,i, and {tilde over
(m)}.sub.H,i(i=1, . . . ,L) maybe aggregated into combined
probability masses I.sub.n,L, {overscore (I)}.sub.H,L and .sub.H,L
respectively, using equations i) to ix) in a recursive manner
I.sub.n,1=m.sub.n,l(n=1,2, . . . ,N) i)
I.sub.H,l=m.sub.H,l ii)
.sub.H,l={tilde over (m)}.sub.H,l iii)
{overscore (I)}.sub.H,l={overscore (m)}.sub.H,l iv) 13 K i + 1 = [
1 - t = 1 N j = j t 1 N I t , i m j , i + 1 ] - 1 v )
I.sub.n,i+1=K.sub.i+1[I.sub.n,im.sub.n,i+-
1+I.sub.H,im.sub.n,i+1+I.sub.n,im.sub.H,i+1](n=1,2 . . . , N)
vi)
.sub.H,i+1=+K.sub.i+1[.sub.H,i{tilde over (m)}.sub.H,i+1+{overscore
(I)}.sub.H,i+1{tilde over (m)}.sub.H,i+1+.sub.H,i+1{overscore
(m)}.sub.H,i+1] vii)
{overscore (I)}.sub.H,i+1=K.sub.i+1[{overscore
(H)}.sub.H,i+1{overscore (m)}.sub.H,i+1] Viii)
I.sub.H,i+1={overscore (I)}.sub.H,i+1+{overscore (I)}.sub.H,i+1
ix)
[0051] i={1,2, . . . ,L-1}
[0052] This permits the probability masses to be combined, and
allows upper and lower bounds of probability masses to be provided.
From this, ranges of combined assessments can be generated.
[0053] Combined degrees of belief .beta..sub.n, and .gamma..sub.H
may be generated using the equations: 14 n = I n , L 1 - I _ H , L
n = 1 , 2 , , N H = I ~ H , L 1 - I _ H , L
[0054] wherein .beta..sub.n is a degree of belief to which the
general criterion is assessed to the n.sup.th grade H.sub.n and
.beta..sub.H is a remaining degree of belief which is not assigned
to any specific grade.
[0055] Further according to the method wherein each grade H.sub.n+1
is more favourable than H.sub.n, performance indicators of a
general criterion may be generated using the equations: 15 u max =
N - 1 n = 1 n u ( H n ) + ( N + H ) u ( H N ) u min = ( 1 + H ) u (
H 1 ) + n = 2 N n u ( H n ) . u avg = u max + u min 2
[0056] wherein u.sub.max, u.sub.min and u.sub.avg are the best
possible, worst possible and average performance indicators
respectively, and u(H.sub.n) (n=1, . . . N) is optionally defined
by 16 u ( H n ) = n - 1 N - 1 .
[0057] Alternatively, u(H.sub.n) may be estimated by a decision
maker.
[0058] The values of .beta..sub.n,i may be determined using the
method of the first aspect of the invention.
[0059] According to a third aspect of the invention there is
provided a carrier medium storing a computer program, which
computer program performs the method of the first aspect of the
invention. In this instance, a decision maker may be a user of the
computer program.
[0060] According to a fourth aspect of the invention there is
provided a carrier medium storing a computer program, which
computer program performs the method of the second aspect of the
invention.
[0061] According to a fifth aspect of the invention there is
provided a computer adapted to perform the method of the first
aspect of the invention.
[0062] According to a sixth aspect of the invention there is
provided a computer adapted to perform the method of the second
aspect of the invention.
[0063] Methods, computer programs and carrier media therefor in
accordance with the invention will now be described with reference
to the accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
[0064] FIG. 1 is a schematic diagram of quality criteria for a
motor engine;
[0065] FIG. 2 is a schematic diagram of general and basic
criteria;
[0066] FIG. 3 shows the main window produced by an exemplary
computer program;
[0067] FIG. 4 shows a software driven interface for implementing a
rule based quantitative data transformation technique;
[0068] FIG. 5 shows a software driven interface for implementing a
rule based qualitative information transformation technique;
[0069] FIG. 6 shows a software driven interface which supports
utility estimation;
[0070] FIG. 7 shows a software driven interface which enables
random numerical data to be inputted;
[0071] FIG. 8 shows a software driven interface which permits a
user to assign degrees of belief;
[0072] FIG. 9 shows a graphical display of a distributed
assessment;
[0073] FIG. 10 shows a graphical display of utility intervals;
[0074] FIG. 11 shows a graphical display of attributes;
[0075] FIG. 12 shows a graphical display which portrays the ranking
of four motorcycles.
DETAILED DESCRIPTION OF THE INVENTION
Handling Oualitative and Quantitative Information
[0076] The present invention permits the assessment of both
quantitative and qualitative information which are subject to a
range of uncertainties. Instead of using a decision matrix, the
present invention describes a decision problem using a generalised
decision matrix, an example of which is shown in Table 2 for the
car selection problem which was described above in relation to
Table 1. The main difference between a decision matrix and a
generalised decision matrix is that the element of the latter can
be a value or a distribution in a belief structure to accommodate
uncertainties in human judgements.
2TABLE 2 Generalised Decision Matrix Fuel Engine Consumption
General Size (mile/gallon) Price . . . Quality Car 1 1400 cc 35
(50%) .English Pound.8,000 . . . Good (75%) 40 (50%) Excellent
(25%) Car 2 1500 cc 40 (33%), .English Pound.9,000 . . . Good (35%)
45 (33%) 50 (33%) Excellent (65%) . . . . . . . . . . . . . . . . .
. Car N 1100 45 (25%), .English Pound.7,000 . . . Average (15%) 46
(25%) 48 (25%), Good (70%) 49 (25%) Excellent (5%) Unknown
(10%)
[0077] Decision making with multiple criteria is based on the
assessment of criteria. For instance, the quality of a motor engine
may not be properly assessed without taking into account relevant
quality criteria such as quietness, responsiveness, fuel
consumption, vibration and starting, as shown in FIG. 1. Similar to
the motor engine example, any general (upper-level) criterion of an
object can be assessed through its basic (lower-level) criteria, as
shown in FIG. 2 or through a multi-level hierarchy of criteria.
[0078] Due to the subjective nature of the criterion, the quality
of a motor engine can be expressed in the present invention using
certain standards in terms of linguistic evaluation grades such as
poor, indifferent, average, good and excellent. For example, the
quality of an engine may be described using the following
distribution,
S(Quality)={(poor, 0.01)}, (indifferent, 0.1), (average, 0.15),
(good, 0.63), (excellent, 0.05)} (1)
[0079] which reads that the quality of the engine is 1% poor, 14%
indifferent, 15% average, 63% good, and 5% excellent. The
distribution provides a panoramic view of the engine's quality as
far as the quality criteria are concerned. The percentages in
equation (1) are referred to as the degrees of believe to which the
engine's quality is assessed to individual grades. For example, 63%
good means that the quality of the engine is assessed to the grade
"good" to a degree of 63%.
[0080] An assessment of quality is normally generated by
aggregating more than one quality criterion. The quality criteria
could be either quantitative or qualitative, and can be assessed in
different ways. For instance,fuel consumption is a quantitative
criterion and could be assessed using a quantity such as how many
miles a motor vehicles can travel per gallon of fuel (mpg). On the
other hand, it is more natural to assess a qualitative criterion
using a set of grades appropriate for this criterion but not
necessarily the same set as that used for assessing other criteria.
In terms of quietness, for example, it is natural to judge that an
engine is very quiet, quiet, normal, noisy or very noisy; in terms
of vibration, it is common to judge that an engine vibrates
heavily, normally or lightly.
[0081] To aggregate both quantitative and qualitative criteria, the
relationships amongst various sets of grades have to be properly
interpreted. For instance, the performance of a motor engine is
said to be good if it is quiet, its responsiveness is good, its
fuel consumption is low (39 mpg for example), its vibration is
normal, and its starting is also good. In the above aggregation, it
is implied that a quiet engine means that the quality of the engine
is good as far as quietness is concerned. In other words, the grade
quiet in the quietness assessment is equivalent to the grade good
in quality assessment. Similarly, in the above aggregation if the
fuel consumption of an engine is 39 mpg then its quality is judged
to be good as far as fuel consumption is concerned.
[0082] In general, if both quantitative and qualitative criteria
are included in a decision making problem it is necessary to
transform various sets of assessment grades to a consistent
framework so that they can be compared and aggregated consistently.
In the following sections, techniques are exemplified which
facilitate the transformation.
Rule-Based Quantitative Information Transformation
[0083] As discussed in the previous section, different linguistic
evaluation grades may be used to describe the same standard. The
equivalence between an evaluation grade and its corresponding
standard can be established using equivalence rules to transform
various sets of grades to a unified set. To transform quietness
assessment to quality assessment, for example, the following simple
equivalence rule could be established.
[0084] Suppose an evaluation grade "very noisy" in a quietness
assessment is equivalent to a grade "poor" in a quality assessment,
"noisy" equivalent to "indifferent", "normal" to "average", "quiet"
to "good", and "very quiet" to "excellent". Then one could say that
the set of grades {very noisy, noisy, normal, quiet, very quiet} in
quietness assessment is equivalent to the set {poor, indifferent,
average, good, excellent} in quality assessment.
[0085] The above equivalence is based on the fact that individual
grades in the two sets are judged to be equivalent on the
one-to-one basis. In the case of transforming vibration assessment
(heavily, normally or lightly) to quality assessment, however, the
grade "heavily" for vibration criterion may imply a "poor" grade of
engine quality to a degree of 80% and an "indifferent" grade to
20%. In general, a grade for a basic criterion may imply several
grades for a general criterion to certain degrees. Suppose:
[0086] H.sub.n is the n.sup.th grade for assessment of a general
criterion
[0087] K.sub.m,i is the m.sup.th grade for assessment of the
l.sup.th basic criterion
[0088] .alpha..sub.n,m is the degree to which K.sub.m,i implies
H.sub.n
[0089] .gamma..sub.m,i is the degree to which the l.sup.th basic
criterion is assessed to K.sub.m,i
[0090] .beta..sub.n,l is the degree to which the l.sup.th basic
criterion is assessed to K.sub.m,i
[0091] Then, an assessment {(K.sub.m,l,.gamma..sub.m,l),m=1, . . .
M} ml under a set of grades {K.sub.m,im=1, . . . M,} can be
equivalently transformed to an assessment
{(H.sub.n,.beta..sub.n,l),n=1, . . . ,N} under another set of
grades {H.sub.n,n=1, . . . ,N} using the following matrix equation
17 [ 1 , l 2 , l N , l ] = [ 1 , 1 1 , 2 1 , M 2 , 1 2 , 2 2 , M N
, 1 N , 2 N , M ] [ 1 , l 2 , l M , l ] ( 2 )
[0092] The values of .alpha..sub.n,m(m=1, . . . ,M and n=1, . . .
,N) should satisfy the equation 18 n = 1 N n , m = 1
[0093] and are determined by the following rules extracted from
decision makers:
[0094] A grade K.sub.m,l implies
[0095] a grade H.sub.1 to a degree of .alpha..sub.1,m;
[0096] a grade H.sub.2 to a degree of .alpha..sub.2,m; . . . ;
and
[0097] a grade H.sub.N to a degree of .alpha..sub.N,m with m=1, . .
. , M.
[0098] Because the values of .alpha..sub.n,m(m=1, . . . ,M and n=1,
. . . ,N) are determined using rules, this approach can be termed a
rule-based information transformation technique.
Utility-Based Information Transformation
[0099] In the transformation technique described in the previous
section it was assumed that the original assessment is equivalent
to the transformed assessment in terms of value (also called
utility) to decision makers, though the utilities of both
assessments were not known explicitly. The utility of an assessment
is given by the weighted sum of the utilities of grades using the
degrees of belief as weights. The utility of a grade is a real
number that is normally between 0 (the value for the most
unfavourable grade) and 1 (the value for the most favourable
grade). The utility of a grade represents a value of the grade to
the decision maker. It is used to measure the decision maker's
preferences towards a grade. Therefore, there is an element of
subjectivity in utility estimation.
[0100] Suppose the utilities of all grades are already given by a
decision maker for both sets of grades {K.sub.m,l,m=1, . . . ,M}
and {H.sub.n,n=1, . . . ,N}, denoted by u(K.sub.m,l) and
u(H.sub.n). Then, an assessment {(K.sub.m,l.gamma..sub.m,l),m=1, .
. . ,M} under the set of grades {K.sup.m,i,m=1, . . . ,M} can be
equivalently transformed to another assessment
{(H.sub.n,.beta..sub.n,l),n=1, . . . ,N} under the set of grades
{H.sub.n,n=1, . . . ,N} using the following matrix equation 19 [ 1
, l 2 , l N , l ] = [ 1 , 1 1 , 2 1 , M 2 , 1 2 , 2 2 , M N , 1 N ,
2 N , M ] [ 1 , l 2 , l M , l ] where n , m = u ( H n + 1 ) - u ( K
m , l ) u ( H n + 1 ) - u ( H n ) , n + 1 , m = 1 - n , m ,
.alpha..sup.i,m=0 (i=1, . . . ,N,i.noteq.n,n+1) if
u(H.sub.n).ltoreq.u(K.sub.m,l).ltoreq.u(H.sub.n+1) for n=1, . . .
,N-1; m=1, . . . ,M (3)
[0101] If the utilities of both sets of grades are not given then
they can be determined using the following equal distance scaling
equations: 20 u ( H n ) = n - 1 N - 1 for n = 1 , , N if H n + 1 is
preferred to H n ( 4 ) u ( K m , l ) = m - 1 M - 1 for m = 1 , , M
if K m + 1 , l is preferred to K m , l ( 5 )
Quantitative Data Transformation
[0102] A quantitative criterion is assessed using numerical values
initially. To aggregate a quantitative criterion together with
other qualitative criteria, equivalence rules are extracted to
transform a value to an equivalent distribution using belief
degrees on the chosen set of grades. For instance, a fuel
consumption of 50 mpg of a motor engine may mean that the quality
of the engine is "excellent" as far as fuel consumption is
concerned. In other words, the 50 mpg fuel consumption is
equivalent to "excellent" engine quality as far as fuel consumption
is concerned. Similarly, fuel consumptions of 44, 38, 32 and 25 mpg
may be equivalent to "good" "average", "indifferent" and "poor",
respectively. Any other numbers between 25 and 50 mpg can be made
to be equivalent to a few grades with different degrees of belief.
For example,fuel consumption of 42 mpg might be held to be
equivalent to "good" to a degree of belief of 67% and "average" to
a degree of belief of 33%.
[0103] In general, to assess a quantitative criterion, for example,
the l.sup.th criterion, a set of grades K.sub.m,l(m=1, . . . , M)
can be chosen which suits the criterion best, and then the
assessment based on this set of grades transformed to an assessment
based on another set of grades, for example the set of grades used
to assess the general criterion associated with the quantitative
criterion.
[0104] For any set of grades K.sub.m,l(m=1, . . . ,M), a set of
corresponding values k.sub.m(m=1, . . . ,M) can be found that the
quantitative criterion may take so that k.sub.m is equivalent to
the grade K.sub.m,l, or
k.sub.m means K.sub.m(m=1, . . . , M). (6)
[0105] Without losing generality, one can even define a set of
numerical grades K.sub.m,l(m=1, . . . ,M) so that
K.sub.m,l=k.sub.m(m=1, . . . ,M), with K.sub.M,l being the most
favourable feasible value of the criterion and K.sub.1,l the least.
Suppose this is the case. Then any value
k.sub.j(K.sub.m,l.ltoreq.k.sub.m+1,l,m=1, . . . ,M-1) of the
l.sup.th criterion can be expressed as: 21 k j = m = 1 M ( K m , l
s m , j ) ( 7 )
[0106] where 22 s m , j = K m + 1 , l - k j K m + 1 , l - K m , l ,
s m + 1 , j = 1 - s m , j if K m , l k j K m + 1 , l , m = 1 , , M
- 1 ( 8 )
[0107] s.sub.i,j=0 for i=1, . . . M; i.noteq.m,m+1
[0108] The assessment of k.sub.j in terms of the set of grades
K.sub.m,l(m=1, . . . , M) can be expressed as
{(K.sub.m,l,.gamma..sub.m,l), m=1, . . . M} with
.gamma..sub.m,l=s.sub.m,j(m=1, . . . , M) (9)
[0109] In many decision situations, a quantitative criterion may be
a random variable and take several values with different
probabilities. Such assessment information can be expressed using a
random number: {(k.sub.j,p.sub.j),j=1, . . . ,P} where k.sub.j(j=1,
. . . ,P) are possible values that the criterion may take, p.sub.j
is the probability that the criterion may take a particular value
k.sub.j and P is the number of possible values that the criterion
may take. Using equations (7) and (8), s.sub.m,j(m=1, . . . ,M) can
be calculated for each k.sub.j(j=1, . . . ,P). The random number
{(k.sub.j,p.sub.j),=1, . . . ,P} can then be transformed to an
assessment {(K.sub.m,l,.gamma..sub.m,l)- m=1, . . . ,,M} under the
set of grades {K.sub.m,l,m=1, . . . ,M} using the following matrix
equation 23 [ 1 , l 2 , l M , l ] = [ s 1 , 1 s 1 , 2 s 1 , P s 1 ,
1 s 1 , 2 s 1 , P s M , 1 s M , 2 s M , P ] [ p 1 p 2 p P ] ( 10
)
[0110] When the quantitative criterion takes a deterministic
number, such as k.sub.j, then p.sub.j=1 and p.sub.i=0 for i=1, . .
. ,M and i.noteq.j in equation (10). That is, for a deterministic
value, equation (10) becomes 24 [ 1 , l 2 , l M , l ] = [ s 1 , j s
2 , j s M , j ] ( 11 )
[0111] This is equivalent to equation (9). This is the special
feature of a deterministic criterion, and the analysis conforms to
the previous analysis.
[0112] In order to aggregate the basic quantitative criterion with
other basic criteria, it is necessary to transform the assessment
results {(K.sub.m,l,.gamma..sub.m,l)m=1, . . . ,M} under the set of
grades {K.sub.m,l, m=1, . . . , M to {(H.sub.n,.beta..sub.n,l),n=1,
. . . ,N} under the set of grades {H.sub.n,n=1, . . . ,N} of the
general criterion. We can use equation (2) to perform the
transformation. Combining equation (2) with equation (10), we can
transform a deterministic number k.sub.j or a random number
{(k.sub.j,p.sub.j),j=1, . . . ,P} to {(H.sub.n,.beta..sub.n,1),
n=1, . . . ,N} using the following equation: 25 [ 1 , l 2 , l N , l
] = [ 1 , 1 1 , 2 1 , M 2 , 1 2 , 2 2 , M N , 1 N , 2 N , M ] [ s 1
, 1 s 1 , 2 s 1 , P s 2 , 1 s 2 , 2 s 2 , P s M , 1 s M , 2 s M , P
] [ p 1 p 2 p P ] ( 12 )
Dealing with Incomplete Information
[0113] Following appropriate transformations, all criteria can be
described in the same framework. Using the techniques described in
the previous sections, it is possible to do so even if the criteria
comprise quantitative and qualitative criteria, and if the
quantitative criteria take random or precise numbers. An example of
such an instance is the assessment of quality criteria of a motor
engine using the following distributions under the same set of
grades.
[0114] S[quietness]={[good, 0.5], [excellent, 0.3]}
[0115] S[responsiveness]={[good, 1.0]}
[0116] S[fuel economy]=([indifferent, 0.5], [average, 0.5]}
[0117] S[vibration]={[good, 0.5], [excellent, 0.5]}
[0118] S[starting]={[good, 1.0]}
[0119] In an ideal situation, the quality of an engine will be
regarded as good if its responsiveness, fuel economy, quietness,
vibration and starting are all assessed to be exactly good.
However, such consensus assessments are rare, and criteria are
often assessed to different evaluation grades, as shown in the
above example. A further problem is that an assessment may not be
complete. For example, the assessment for quietness is not complete
as the total degree of belief in the assessment is 0.5+0.3=0.8. In
other words, 20% of the belief degrees in the assessment are
missing.
[0120] To judge the quality of an engine and compare it with other
engines, a question is how to generate a quality assessment for the
engine by aggregating the various assessments of the quality
criteria as given above, which could be incomplete. This question
is common to most MCDA problems. The present invention provides a
systematic and rational way of dealing with the aggregation
problem.
Generate Basic Probability Masses
[0121] In the engine quality assessment problem, each quality
criterion plays a part in the assessment but no single criterion
dominates the assessment. In other words, the quality criteria are
of relative importance. This is true of any MCDA problem.
[0122] Weights for each of the basic criteria W.sub.i(i=1, . . .
,L) should reflect the relative importance of each basic criterion
to the general criterion. They can be assigned using a few methods
such as a method based on pairwise comparisons or simply according
to the decision maker's judgement. For example, if the decision
maker thinks that basic criterion 1 is twice as important as basic
criterion 2, then he or she may assign {W.sub.1=10,W.sub.2=5} or
{W.sub.1=2,W.sub.2=1} as long as they are relatively correct.
[0123] The weights assigned by the decision maker need to be
normalised to arrive at a set of normalised weights
.omega..sub.i(i=1, . . . ,L) using the following equation 26 i = W
i j = 1 L W j ( i = 1 , , L ) ; ( 13 )
[0124] so that 0.ltoreq..omega..sub.i.ltoreq.1 and 27 i = 1 L i = 1
( 14 )
[0125] The present invention uses a new evidential reasoning
algorithm for criteria aggregation, which operates on probability
masses as described in the following sections. Since criteria are
of relative importance, the assessment of one criterion to a grade
to certain degree does not necessarily mean that all criteria would
be assessed to the grade to the same degree. For instance, if the
quietness of an engine is assessed to be good to a degree of 50%,
the quality of the engine would not necessarily be assessed to be
good to the same degree. This is because the engine quality is also
determined by the other four quality criteria.
[0126] In the present invention, the definition of a basic
probability mass takes into account the relative importance of
criteria. Let .beta..sub.n,i denote a degree of belief that the
i.sup.th basic criterion is assessed to a grade H.sub.n. Let
m.sub.n,i be a basic probability mass representing the degree to
which the i.sup.th basic criterion (quality criterion) supports the
hypothesis that the general criterion (quality) is assessed to the
n.sup.th grade H.sub.n. Let H denote the whole set of grades, or
H={H.sub.n,n=1, . . ., N}. m.sub.n,i is then defined as the
weighted degree of belief or probability mass as shown in equation
(15). Let m.sub.H,i be a remaining probability mass unassigned to
any individual grade after the i.sup.th basic criterion has been
assessed. Then, m.sub.n,i and m.sub.H,i are given by
m.sub.n,i=.omega..sub.i.beta..sub.n,in=1, . . ,N (15) 28 m H , i =
1 - n = 1 N m n , i = 1 - i n = 1 N n , i ( 16 )
[0127] m.sub.H,i is decomposed into two parts: {overscore
(m)}.sub.H,i and {tilde over (m)}.sub.H,i with m.sub.H,i={overscore
(m)}.sub.H,i+{tilde over (m)}.sub.H,i,
[0128] where 29 m _ H , i = 1 - i and m ~ H , i = i ( 1 - n = 1 N n
, i ) ( 17 )
[0129] {overscore (m)}.sub.H,i is the first part of the remaining
probability mass that is not yet assigned to individual grades due
to the fact that criterion i only plays one part in the assessment
relative to its weight. In other words, {overscore (m)}.sub.H,i
provides the scope where other criteria can play a role in the
assessment. {overscore (m)}.sub.H,i should eventually be assigned
to individual grades in a way that is dependent upon how all
criteria are assessed and weighted. {tilde over (m)}.sub.H,i is the
second part of the remaining probability mass that is not assigned
to individual grades due to the incompleteness in an assessment.
{tilde over (m)}.sub.H,i is proportional to .omega..sub.i and will
cause the subsequent assessments to be incomplete.
Combine Probability Masses
[0130] In equations (15) to (17), the contribution of the i.sup.th
basic criterion to the assessment of the general criterion (see
FIG. 2) is represented as the basic probability masses. A new
algorithm is developed to aggregate the basic probability masses.
An important feature of the algorithm is its capacity of handling
incomplete assessments by providing the upper and lower bounds of
probability masses, based upon which the range of combined
assessments can be generated, as discussed in the following
sections.
[0131] Let I.sub.n,i(n=1,2, . . . ,N), {overscore (I)}.sub.H,i and
{overscore (I)}.sub.H,i denote the combined probability masses
generated by aggregating the first i criteria. The following
algorithm can be used to combine the first i criteria with the
(i+1).sup.th criterion in a recursive manner.
I.sub.n,l=m.sub.n,l (n=1,2, . . . ,N) (18a)
I.sub.H,l=m.sub.H,l (18b)
.sub.H,l={tilde over (m)}.sub.H,l (18c)
{overscore (I)}.sub.H,l={overscore (m)}.sub.H,l (18d) 30 K i + 1 =
[ 1 - t = 1 N j = 1 j t N I t , i m j , i + 1 ] - 1 (18e)
I.sub.n,i+1=K.sub.i+1[I.sub.n,im.sub.n,i+1+I.sub.H,im.sub.n,i+1+m.sub.H,i+-
1](n=1, . . . ,N) (18i)
.sub.H,i+1K.sub.i+1[.sub.H,i{tilde over (m)}.sub.H,i+1+{overscore
(I)}.sub.H,i i{tilde over (m)}.sub.H,i+1+.sub.H,i{overscore
(m)}.sub.H,i+1] (18g)
{overscore (I)}.sub.H,i+1=K.sub.i+1[{overscore
(I)}.sub.H,i+{overscore (m)}.sub.H,i+1] (18h)
I.sub.H,i+1={overscore (I)}.sub.H,i+1.sub.H,i+1 (18i)
[0132] i={1,2, . . . ,L-1}
[0133] wherein I.sub.n,L is the combined probability mass assigned
to the n.sup.th grade (n=1, . . . N),{overscore (I)}.sub.H,L the
combined probability that needs to be redistributed over the N
grades, and .sub.H,L the remaining combined probability mass that
is unable to be distributed to any specific grade due to
insufficient information, and I.sub.H,L={overscore
(I)}.sub.H,L+.sub.H,L.
Generate Combined Degrees of Belief
[0134] After all L basic criteria have been aggregated, the overall
combined probability masses are given by I.sub.n,L(n=1,2, . . .
,N), .sub.H,L and {overscore (I)}.sub.H,L. {overscore (I)}.sub.H,L
denotes the remaining probability mass that can be assigned to
individual grades. In the present invention, it is assigned to all
individual grades proportionally using the following normalisation
process so as to generate the combined degrees of belief to the
grade H.sub.n. 31 n = I n , L 1 - I _ H , L n = 1 , 2 , , N ( 19a
)
[0135] The degree of belief that is not assigned to any individual
grades is assigned to the whole set H by 32 H = I ~ H , L 1 - I _ H
, L ( 19b )
[0136] It has been proven that the combined degrees of belief
generated using the above normalisation process satisfy the common
sense synthesis rules (CSSR).sup. in MCDA whilst incompleteness in
original assessments is preserved and represented by .beta..sub.H.
The generated assessment for a general criterion can be represented
by a distribution {(H.sub.n,.beta..sub.n),n=1, . . . ,N}, which
reads that the general criterion is assessed to the grade H.sub.n
with the degree of belief .beta..sub.n(n=1, . . . ,N). .sup. CSSR
1: If no basic criterion is assessed to an evaluation grade at all
then the general criterion should not be assessed to the same grade
either. CSSR 2: If all basic criteria are precisely assessed to an
individual grade, then the general criterion should also be
precisely assessed to the same grade. CSSR 3: If all basic criteria
are completely assessed to a subset of grades, then the general
criterion should be completely assessed to the same subset as well.
CSSR 4: If basic assessments are incomplete, then a general
assessment obtained by aggregating the incomplete basic assessments
should also be incomplete with the degree of incompleteness
properly expressed.
Generate a Utility Interval
[0137] There may be occasions where distributed descriptions are
not directly comparable to show the difference between two
assessments. In such circumstances, it is desirable to generate
numerical values equivalent to the distributed assessments in some
sense. The present invention introduces the concept of expected
utility to define such a value. Suppose u(H.sub.n) is the utility
of the grade H with u(H.sub.n+1)>u(H.sub.n) if H.sub.n+1 is
preferred to H.sub.n. If all assessments are complete and precise,
then .beta..sub.H=0 and the expected utility which is calculated by
33 u = n = 1 N n u ( H n )
[0138] can be used for ranking alternatives.
[0139] Note that .beta..sub.H given in Equation (19b) is the
unassigned degree of belief representing the extent of the
incompleteness (ignorance) in the overall assessment. Within the
evaluation framework discussed in the previous sections,
.beta..sub.n provides the lower bound of the likelihood to which
H.sub.n is assessed to. The upper bound of the likelihood is given
by a belief degree of (.beta..sub.n+.beta..sub.H). Thus the belief
interval [.beta..sub.n(.beta..sub.n+.beta..sub.H)] provides the
range of the likelihood that H.sub.n may be assessed to. The
interval will reduce to a point .beta..sub.n if all basic
assessments are complete (or .beta..sub.H=0).
[0140] If any basic assessment is incomplete, the likelihood to
which H.sub.n may be assessed to is not unique and can be anything
in the interval [.beta..sub.n, (.beta..sub.n+.beta..sub.H)]. In
such circumstances, three values are defined to characterise a
distributed assessment, namely the minimum, maximum and average
utilities. Without loss of generality, suppose H.sub.1 is the least
preferred grade having the lowest utility and H.sub.N the most
preferred grade having the highest utility. Then the maximum,
minimum and average utilities are given by 34 u max = n = 1 N - 1 n
u ( H n ) + ( N + H ) u ( H N ) ( 20 ) u min = ( 1 + H ) u ( H 1 )
+ n = 2 N n u ( H n ) ( 21 ) u avg = u max + u min 2 ( 22 )
[0141] wherein u.sub.max, u.sub.min and u.sub.avg are the best
possible, worst possible and average performance indicators in
terms of utility values respectively, and u(H.sub.n) (n=1, . . . N)
are the utility values of the grade H.sub.n(n=1, . . . N) as
mentioned previously.
[0142] The present invention includes within its scope computer
programs which perform the above described methods, carrier media
storing said computer programs, and computers which are adapted to
perform the above described methods. Typically, a computer would be
adapted to perform the methods of the invention by virtue of
running computer programs of the present invention. Suitable
carrier media include, but are not limited to, hard discs, floppy
discs, compact discs, tapes, DVD and memory devices such as PROMs
and EEPROMs. Computer programs, such as an embodiment which is
exemplified below, can allow users to enter the transformation
rules, to define assessment grades, to conduct evidence mapping
processes and to aggregate multiple criteria using the ER
algorithm. Additionally, the computer program can provide a
graphical display of the results of an assessment. Computer
programs can be provided which interface with commercially
available operating systems or specific programs. The skilled
reader will readily appreciate how such interfacing can be
achieved.
EXAMPLE
[0143] Assessment Criteria
[0144] In this example a motorcycle assessment problem is examined
using both complete and incomplete (imprecise) data of both a
quantitative and qualitative nature. The belief structure will be
used to facilitate continuous and imprecise assessments for
qualitative criteria. For quantitative criteria, both certain and
random numbers are taken into account. The transformation
techniques are used to transform the various types of information
into a unified framework. Software is used to support the analysis.
The main window of the display produced by the software is shown in
FIG. 3 for the motorcycle selection problem.
[0145] The assessment problem has seven main criteria: Price,
Displacement, Range, Top speed, Engine quality; Operation system
and General finish. The first four criteria are quantitative and
are measured using the following different units: poundsterling,
cc, miles and mph, respectively.
[0146] The last three criteria are qualitative and difficult to
measure directly. Therefore they are assessed through detailed
sub-criteria. For example, engine quality is assessed through
responsiveness, fuel consumption, quietness, vibration and
starting; general finish through quality of finish, seat comfort,
headlight, mirrors and horns. Operation system can be assessed
through handling, transmission and brakes, which however are still
difficult to assess directly and therefore are evaluated through
more detailed sub-sub criteria. For example, handling is assessed
through steering, bumpy bends, manoeuvrability and top speed
stability; transmission through clutch operation and gearbox
operation; and brakes through stopping power, braking stability and
feel at control.
[0147] Input Information
[0148] Table 4 describes the motorcycle assessment problem, which
involves four candidate motorcycles for assessment based on 29
criteria of a hierarchy as described in the previous section. The
input information includes the relative weights among groups of
criteria and the assessment outcome of each motorcycle on every
criterion. The relative weights of the same group of criteria are
shown in the brackets. Outcomes include precise numbers, random
numbers and subjective assessments.
[0149] Price, Displacement, Range and Top speed are all assessed
using precise numbers. For examples, the price, displacement, range
and top speed of Honda are given by .English Pound.6199, 998 cc,
170 miles and 160 mph, respectively. Fuel consumption varies in
different weather and road conditions. For example,fuel consumption
is assessed on four conditions: (1) winter & urban, (2) winter
& suburb, (3) summer & urban and (4) summer & suburb as
well as the frequencies that a motorcycle is used in these
conditions. For example, the fuel consumption of Honda is 31 mpg,
35 mpg, 39 mpg and 43 mpg under these four conditions with the
equal frequency of 25% recorded by {[31, 0.25], [35,0.25], [39,
0.25], [43, 0.25]}. Quantitative numbers can be transformed to
qualitative assessments using the techniques described previously.
FIG. 4 shows an interface for implementing the rule-based data
transformation technique which is supported by the software.
[0150] For simplicity, the qualitative criteria in this example are
all assessed using the same five evaluation grades, which are
defined as Poor (H.sub.1), Indifferent (H.sub.2), Average
(H.sub.3), Good (H.sub.4) and Excellent (H.sub.5) and abbreviated
by P, I, A, G and E respectively. The overall assessment of a
motorcycle is also based on this set of grades. For example, the
responsiveness of Yamaha is assessed to 30% good, and 60%
excellent, denoted by {[G, 0.3], [E, 0.6]}. If different sets of
grades are used for lower-level criteria, the rule or utility-based
techniques described in the previous sections can be used to
transform them to the same set of grades. FIG. 5 shows an interface
for implementing the rule-based qualitative information
transformation technique. The utility-based information
transformation techniques are implemented in the software by
estimating the utilities of grades. FIG. 6 shows an interface to
support utility estimation.
[0151] Imprecise assessments are lightly shaded in Table 4 and data
absence is also assumed, as shown by the shaded blank boxes. Some
judgements and random numbers are incomplete in the sense that the
total degree of belief in an assessment is not summed to unity. For
example, the assessment of the responsiveness of Yamaha is {[G,
0.3], [E, 0.6]} where the total belief degree is (0.3+0.6)<1 (or
30%+60%<100%). The assessment for the fuel consumption of Yamaha
is {[28, 0.25], [34, 0.25], [38, 0.25]} with the total belief
degree of 0.75[or 75%], since thefuel consumption data in urban
areas in winter are not available. All input information, either
quantitative or qualitative, can be fed into software using its
input dialogue windows such as those shown in FIGS. 7 and 8.
3TABLE 4 Assessment Data for Motorcycle Selection Problem 1
[0152] If traditional MCDA methods were applied to the above
problem, then at best one would have to make efforts to try to find
the missing information and eliminate the imprecision. This is
assuming that such efforts are practical and cost effective.
Otherwise, additional assumptions need to be made about these
missing and imprecise assessments, or certain criteria have to be
abandoned for further analysis. In either event, the outcome is
less than satisfactory. In contrast, the present invention is well
suited to solving the problem using the very information of Table
4. The software may be used to support the following analysis.
[0153] Ranking and Results
[0154] The present invention can operate on degrees of belief. To
generate utility intervals, it is necessary to estimate the
utilities of values and grades. The certain monetary equivalent
(CME) approach can be used to estimate the utilities of
quantitative criteria. Take price for example. Suppose for this
range of motorcycles the highest acceptable price is ".English
Pound.9,000" and the lowest possible price is ".English
Pound.5,000". Note that the price is a cost criteria and therefore
low price is preferred. First of all, the utility of price is
normalised by assigning u(9000)=0 and u(5000)=1.
[0155] Following the procedure of the CME approach, a price value
having the average utility of .English Pound.9,000 and .English
Pound.5,000 is identified first. Suppose the price value is
.+-.7,500. Thus u(7500)=(u(9000)+u(5000))/2=0.5. Furthermore,
suppose .English Pound.6,500 has the average utility of .English
Pound.7,500 and .+-.5,000, or u(6500)=(u(7500)+u(5000))/2=0.75, and
.English Pound.8,500 has the average utility of .English
Pound.9,000 and .English Pound.7,500, or
u(8500)=(u(9000)+u(7500))/2=0.25.Let K.sub.1,1=9000,
K.sub.2,1=8500, K.sub.3,1=7500, K.sub.4, 1=6500, K.sub.5,1=5000.
Note that if no preference information is available then a linear
marginal utility function could be assumed for price. Under this
assumption there would be {overscore (K)}.sub.1,1=9000, {overscore
(K)}.sub.2,1=8000, {overscore (K)}.sub.3,1=7000, {overscore
(K)}.sub.4,1=6000, and {overscore (K)}.sub.5,1=5000.
[0156] The probability assignment approach could be used to
estimate the utilities of the five evaluation grades for the
qualitative attributes. To illustrate the process and simplify
discussion, suppose the utility of the five evaluation grades are
equidistantly distributed in the normalised utility space, or
u(P)=0, u(I)=0.25, u(A)=0.5 u(G)=0.75, u(E)=1.
[0157] In Table 4, the criteria are of a three-level hierarchy. In
the present example, each group of the bottom level criteria
associated with the same upper-level criterion are first aggregated
to generate an assessment for the upper-level criterion. Once the
assessments for a group of upper-level criteria associated with the
same higher-level criterion are all generated, these assessments
can be further aggregated in the same fashion to generate an
assessment for the higher-level criterion. This hierarchical
aggregation process is based on the techniques previously described
herein, and implemented in the software. The assessment of each
motorcycle on any criterion can be reported graphically in the
software, as shown in FIGS. 9 and 10, which display data concerning
the quality of the Honda engine. Table 5 shows the final
assessments generated using the software for the four motorcycles
by aggregating all the criteria shown in Table 4. The comparison
and ranking of the four motorcycles on the overall criterion and
other selected criteria can be reported graphically as shown in
FIGS. 11 and 12.
[0158] The above results show that Honda is clearly the most
recommended motorcycle as its minimum utility is larger than the
maximum utilities of the other motorcycles. This is logical as it
has the best engine quality, excellent general finish and
relatively low price. Yamaha is ranked the second due to its low
price followed by Kawasaki. BMW is ranked the last due to its high
price and below average transmission and handling system. The above
ranking is conclusive for the weights provided despite the
imprecision and absence of some data. This shows that decision
could be made on the basis of incomplete information. Note,
however, that the above ranking is the personal choice of the
decision maker who provided the weights of all the criteria and
also estimated their marginal utilities. This means that given the
same assessment data shown in Table 4, another decision make may
achieve a different ranking.
4TABLE 5 Overall Assessment of Motorcycles Assessment Kawasaki
Yamaha Honda BMW Distributed {[P, 0.0515], {[P, 0.0455], {[P,
0.003], {[P, 0.0888], assessment [I, 0.0858], [I, 0.1153], [I,
0.0782], [I, 0.2499], [A, 0.2147], [A, 0.2106], [A, 0.1646], [A,
0.1904]. [G, 0.5693], [G, 0.3232], [G, 0.5374], [G, 0.1155], [E,
0.0572]} [F, O.2745]} [E, 0.2010]} [E, 0.3288]} Maximum 0.6345
0.6820 0.7217 0.5996 utility Minimum 0.6130 0.6511 0.7059 0.5864
utility Average 0.6237 0.6665 0.7138 0.5732 utility Ranking 3 2 1
4
[0159] It will be appreciated that this assessment is for exemplary
purposes only, and that the invention is not limited in its scope
by the specific disclosures of the example.
* * * * *