APPLICATION OF ORDINAL INFORMATION IN DECISION MODELS
HELENA BROŽOVÁ
CZECH UNIVERSITY OF AFRICULTURE PRAGUE, CZECH REPUBLIC
Abstract
This paper deals with special ways of data setting in decision models and with the
possibility of their solving.
Payoffs and probabilities of states of nature are a quantitative crisp estimation in
typical decision model applications. The quality of model results depends on quality of these
estimations. Suppose now, that both estimations needn’t be particular numbers and the
decision-maker knows only preference information either for payoffs or for probabilities. That
means, ordinal information for both data sets can be known.
ORESTE method for multiple attribute model solving can be used for decision models
solving according to certain parallelism between these two types of models.
1. Introduction
The main problems of mathematical methods application can be explained by two
questions: How to choose the proper model and solving method and how to set all data for
model quantification. In this paper we would like to answer the second question particularly
for decision models.
The decision theory and decision models enable the decision-maker to make rational
decisions. There are many principles for decision model solving, but all of them need crisp
data assessing. This may often lead to unsatisfactory results because of the accuracy of data
estimation.
For this reason we suggest using methods for multiple attribute decision making for
decision model solving whenever the crisp quantification is difficult or impossible. This idea
is based on the facts of some similarity of both models.
We can see in the literature, that some simple methods as maximin, maximax or
dominance are used in decision theory and multiple attribute decision making too. We want
to show that more sophisticated methods as ORESTE are also applicable. These methods can
be used in the case of special type of decision model input data. The first part of this paper
describes the different kinds of decision model input information and special type of decision
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situations and possibility of their solving. The practical application of this idea is in the
second part.
2. Input information types and solving possibility of decision models
Decision model can be easily conformable to multiple attribute model. List of
alternative is in both model definitions, attributes correspond with states of nature and payoffs
can be considered as attribute evaluation. Both models have the same matrix form so the
payoff matrix can be taken as a criterion matrix of multiple attribute model with unique
criterion the values of which differ according to the state of nature. Table 1 shows the
similarity of elements of both models.
Alternatives
Probabilities/Preferences
States of nature/Criteria
s1/K1 s2/K2 .....
sn/Kn
v11
v12
.....
v1n
v21
v22
.....
v2n
.....
.....
.....
.....
vm1
vm2
.....
vm3
p1
p2
.....
pn
a1
a2
.....
am
Table 1: Correspondence of decision model and multiple attribute model elements
The input information for multiple attribute models can have a different character and
the solving methods differ according to type of preference information.
It is not necessary to know the quantitative estimation of input data. Qualitative or soft
preference information can be used in these models and this information type is less
demanding for the decision-maker to assess than the quantitative information. Preference
information sometimes may not be indicated at all. The preference information can be
expressed in following ways:
ƒ
no information – importance of attributes in not known
ƒ
nominal information - importance of attributes is expressed by standard level of
each attribute,
ƒ
ordinal information - relative importance of attributes or alternatives is set by
ordinal data,
ƒ
cardinal information - relative importance of attributes or alternatives is assessed
by cardinal data.
Standard form of decision model and its solving methods needs cardinal data for
payoffs. Cardinal data for probabilities of states of nature have to be assessed in the case of
decision under risk or can be missing in the case of decision under uncertainty. Quality of
results depends on quality of input data, but this data is often an expert estimation and
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depends on the degree of judgement skill. Therefore it is necessary to make sensitivity
analysis for model results.
Solving methods split into two main parts – the principles for decision under risk and
the principles for decision under uncertainty.
Suppose now, that both data estimations needn’t be crisp numbers. Generally there are
six possible types of decision model definitions according to the available type of
information. The possibilities are ordinal or cardinal information for payoffs and no
information, ordinal or cardinal information for possibilities of states of nature.
Following table shows different characters of decision model in relation with type of
Information about
probabilities of
states of nature
input information.
No
information
Ordinal
Cardinal
Information about payoffs
Ordinal
Cardinal
Decision model with qualitative estimation of
Standard decision model under
payoffs under uncertainty
uncertainty
Decision model with qualitative estimation of
Decision model with qualitative
payoffs and states of nature probabilities
estimation of states of nature
probabilities
Decision model with qualitative estimation of
Standard decision model under risk
payoffs under risk
Table 2: Type of input data and decision model character
From this view decision under uncertainty can be characterised as no information
about the state of nature and cardinal information about payoffs. Decision under risk includes
cardinal information about states of nature as well as about payoffs. These types of decision
model are widely used and there are many methods for their solution.
Ordinal information of states of nature means that their probabilities could not be set
exactly but we know which state of nature is more probable and which one is less probable.
Corresponding decision situation with ordinal information about states of nature and cardinal
information about payoffs lies between decision under risk and decision under uncertainty.
We think, that this character of decision situation is very frequent in practice. These problems
can be solved by lexicographic method, for instance, or ordinal information about states of
nature can be transformed to cardinal information using some method for assessing weights.
Last three types of decision model are remarkably different. Decision situations with
ordinal information about payoffs and no information, ordinal or cardinal information about
states of nature could not be solved using classical methods for decision models solving. If
payoffs could be in ordinal form, decision-maker could set only preference order or
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preference for all pair of alternatives and state of nature. This would be the possibility, how to
deals with data, which are very imperfect or really soft.
Multiple attribute decision-making methods can be used for selection of the best
alternative in these cases. For instance ORESTE method can be used for models with ordinal
information about states of nature and payoffs.
3. Application of ORESTE method for decision model solving
Management of DOAGRA s.r.o. wants to select the optimal further organisation of
machine repairing, because costs of own repair shop seem to be very high.
There are three main strategies - alternatives, the last one can be split into five
alternatives:
ƒ
to keep the repair shop with no organisational changes and offer services for
external clients;
ƒ
to expand the repair shop and offer services for external clients;
ƒ
to inactivate the repair shop (to sell it) and use external services – there are five
companies, which offer repair shop services (Fronk, s. r. o., Agro Domažlice, a.s.,
Karpem, s. r. o., Bodas, a. s. and ZD Draženov.
The consequence of each alternative depends on the number of own repaired machines
and on the expectant demand of external clients. While the estimation of the number of own
repaired machines and its probability is known, there is no quantitative estimation of the
demand of external clients and its probability.
Four states of nature on the first level have been identified – less than 40, 40 – 60, 60 –
80 and more than 80 reparations of own machine. Their probabilities have been calculated
from the frequency of repairs made in previous years.
Four states of nature on the second level have been assumed – less than 40, 40 – 50, 50
– 60 and more than 60 reparations of external client machine. Probabilities of these states of
nature could not be estimated, because there is no experience of the previous period.
However, it is possible to set an ordinal information. Preferences of these states of nature
have been set by simple method for pairwise comparison.
Table 3: Pairwise comparison
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Altogether there are sixteen states of nature on the main level. Their probabilities are
unknown, only ordinal information about them is known, because global preference of states
of nature can been calculated as a multiple of probabilities of the first level states of nature
and results of pairwise comparison of the second level states of nature.
Table 4: Original decision table
Payoffs - costs or revenue for each alternative depend on the number of reparations of
own machine and also on the demand of the external client. Therefore, payoffs have been
estimated for combination of intervals, which correspond with number of internal and
external reparations. This estimation can be used as ordinal information rather then cardinal,
because the real payoff values may differ around this estimation and we can only suppose that
the order of these values does not change.
Decision model with described estimation of payoffs and states of nature probabilities
have been constructed for this problem. Table 4 shows the decision table of this situation.
Because this estimation has been taken only as ordinal information, this model data
have been transformed, so payoffs are ranked in the order of their expected values (Table 5).
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Table 5: Ordinal model definition
The model has been solved by ORESTE method. Table 6 shows results of model
solving. The best alternative is to inactivate the repair shop and use external services from
Agro Domažlice, a.s.. Additional matrix of preference relations between pairs of alternatives
enables further analysis of this result.
Table 6: Result of model solving
4. Conclusion
This paper examined a possibility of different type of input information for decision
model and possibility of its solving. Decision models can be solved when their data
estimation is either ordinal or cardinal.
Generally there are six possible types of decision situations according to the type of
information available - ordinal or cardinal information for payoffs and no information, ordinal
or cardinal information for possibilities of states of nature. The standard models describe two
of them - decision under risk and under uncertainty.
Because data estimation is often difficult, decision model with ordinal information as
for payoffs as for probabilities of states of nature can be very useful in real situations. The
solving methods of multiple attribute model as ORESTE can be used for its solving.
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RNDr. Helena Brožová, CSc.
Dept. of Operations Research and System Analysis, Faculty of Economics and Management,
Czech University of Agriculture, 165 21 Praha 6 – Suchdol, e-mail: [email protected]
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