=Paper=
{{Paper
|id=Vol-2362/paper30
|storemode=property
|title=The Method for Ranking Quasi-Optimal Alternatives in Interval Game Models Against Nature
|pdfUrl=https://ceur-ws.org/Vol-2362/paper30.pdf
|volume=Vol-2362
|authors=Yuri Romanenkov,Viktor Kosenko,Olena Lobach,Evgen Grinchenko,Marina Grinchenko
|dblpUrl=https://dblp.org/rec/conf/colins/RomanenkovKLGG19
}}
==The Method for Ranking Quasi-Optimal Alternatives in Interval Game Models Against Nature==
https://ceur-ws.org/Vol-2362/paper30.pdf
The Method for Ranking Quasi-Optimal Alternatives in
Interval Game Models Against Nature
Yuri Romanenkov1[0000-0002-3526-7237], Viktor Kosenko2[0000-0002-4905-8508],
Olena Lobach3[0000-0001-7494-9997], Evgen Grinchenko4[0000-0002-3973-9078],
Marina Grinchenko5[0000-0002-8383-2675]
1 N. Ye. Zhukovsky National Aerospace University "Kharkiv Aviation Institute", Chkalov. 17,
61070 Kharkov, Ukraine
2 State Enterprise "Kharkiv Research Institute of Mechanical Engineering", Krivokonevskaya.
30, 61016 Kharkiv, Ukraine
3National Technical University "Kharkiv Polytechnic Institute", Kyrpychova. 2, 61002,
Kharkiv, Ukraine
4Kharkiv National University of Internal Affairs, L. Landau avenue. 27, 61080 Kharkiv,
Ukraine
5National Technical University "Kharkiv Polytechnic Institute", Kyrpychova. 2, 61002,
Kharkiv, Ukraine
KhAI.management@ukr.net, kosv.v@ukr.ua,
e.v.lobach@gmail.com, gengrinchenko@gmail.com,
marinagrunchenko@gmail.com
Abstract. The task of selecting the optimal strategy in the interval game with
nature is considered; in particular, the situation when in the interactive dialogue
of an analyst and decision support system there are cases of objective ambiguity
caused, on the one hand, by interval uncertainty of data, and on the other hand –
by the chosen model of the task formalization. The method for ranking quasi-
optimal alternatives in interval game models against nature is proposed, which
enables comparing interval alternatives in cases of classical interval ambiguity.
In this case, the function of the analyst preferences is used with respect to the
values of the criterion that help determine the indicators for the quantitative
ranking of alternatives. By selecting a specific type of the preference function,
the researcher artificially converts the primary uncertainty of the data into the
uncertainty of the preference function form, which nevertheless enables avoid-
ing the ambiguity in the “fuzzy” areas of quasi-optimal alternatives.
Keywords: Playing Against Nature, Optimal Strategy, Interval Data.
1 Introduction
The formalization of the decision-making task is one of the key stages of the man-
agement cycle and the efficient control of organizational and technical systems large-
ly depends on the relevance and correctness of the management cycle. [1]. In the case
of an antagonistic situation, the section of applied mathematics known as “game theo-
ry” is used to solve such tasks. [2, 3]. Numerous methods of classical game theory are
successfully implemented in modern decision support system (DSS) [4], in particular,
decision-making techniques under complete ambiguity and risk. And classical and
derived criteria [5], as well as modified ones [6–8] are used for selecting alternatives.
The efficiency of these criteria is ensured when the initial data of the decision-
making task is absolutely correct. However, when there are various kinds of uncer-
tainties in the initial data, the problem of adapting the criteria arises as well as organ-
izing their final values for the pool of alternatives.
There are different approaches to solve similar tasks in the context of various data,
e.g. interval [9, 10], fuzzy [11], stochastic [12]. In this case, with some sets of initial
data, a situation may arise when alternatives are considered incomparable [13], that is,
there are “fuzzy” areas of quasi-optimal alternatives and the only best one cannot be
selected within them.
This leads to the situation when within the interactive dialogue between an analyst
and DSS there are cases of the objective ambiguity that is caused on the one hand, by
interval uncertainty of data, and on the other hand – by the chosen model of the task
formalization.
Thus, the topical scientific and practical task is to develop techniques and means
for avoiding the ambiguity in the “fuzzy” areas of quasi-optimal alternatives accord-
ing to the analyst request.
2 Problem statement
Consider the classical deterministic decision-making task under complete uncertainty
according to [8], which can be presented as the matrix whose lines correspond to de-
cision variants and columns – to factors. At the intersection of the columns and lines,
gains eij are located, they correspond to decisions Ei under appropriate conditions
F j (see Table 1).
Table 1. Decision efficiency matrix eij
F1 F2 F3 … Fm
E1 e11 e12 e13 … e1m
E2 e21 e22 e23 … e2m
E3 e31 e32 e33 … e3m
… … … … … …
En en1 en 2 en3 … enm
A set of optimal variants E0 consists of the variants Ei 0 , which belongs to the set of
all variants E and the value of the Z i 0 criterion which is maximal among all its val-
ues Z i :
E0 Ei 0 | Ei 0 E Z i 0 max Z i .
i
(1)
Let one of the classical or derived criteria be used as Z i 0 criterion [8]:
maximin (Wald)
Z MM max min eij
i j (2)
Bayes-Laplace
m
Z BL max eij q j ; (3)
i
j 1
Savage
i
Z S min max max eij eij
j i
; (4)
extended maximin
n m
Z ME max min eij pi q j ; (5)
p q
i 1 j 1
gambler
Z AG max max eij ;
i j (6)
Hurwitz
i
Z HW max min eij 1 max eij , 0,1 ;
j j (7)
Hodge-Lehman
n
Z HL max v eij q j 1 v min eij , 0,1 ; (8)
i
j 1 j
Germier
ZG max min eij q j ;
i j (9)
BL(MM)-criterion
I1 : i i 1,...,m & ei 0 j 0 min eij 0 доп ,
j
I : i i 1,...,m & max e max e e min e ,
2 ij i0 j i0 j0 ij 0
(10)
j j j
m
Z BL MM max eij q j ;
I1 I 2
j 1
product-criterion
m
Z P max eij q j . (11)
i
j 1
When gains are presented as an interval
eij eij , eij , (12)
the values of the selected criterion for each alternative can be calculated according to
(2)(12) as intervals
Z i Z i , Z i . (13)
The basis for considering the estimates in the interval form is formed by the following
circumstances [14]:
1. in the process of short-term prediction, estimates in the interval form can be syn-
thesized in a natural way, that is, as a result of fulfilling a prediction task;
2. results of measuring the parameters of the system, direct or indirect, performed
with errors (strictly speaking, results of all measurements), can be represented in
the interval form;
3. if there is at least one model parameter in the interval form in a model, all parame-
ters of the model must be reduced to the interval form as the least complex form of
description of parametric uncertainty in order to observe data homogeneity;
4. interval models are more preferable than the probabilistic-statistical ones in the
case of making one-moment single decisions;
5. the apparatus of interval analysis proved its effectiveness in solving different scien-
tific and practical tasks;
6. interval algorithms typically do not require specialized tools for software imple-
mentation.
We imply by interval z z, z a closed limited subset R of the form
a, a x R | a x a , which can be described by the following characteristics:
z, in f [z] is the left end of interval [z]; z, sup[z] is the right end of interval [z];
zz
mid z is the middle (median) of interval [z]; wid z z z is the width of
2
interval [z].
For the two intervals z z, z and y y, y in classical interval arithmetic
z , y IR , the following operations were assigned:
z y z y, z y ; (14)
z y z y, z y ; (15)
z y min z y,z y,z y,z y , max z y,z y,z y,z y ; (16)
z y z 1 y , 1 y , 0 y . (17)
Interval arithmetic operations have the following properties:
z y x z y x ; (18)
z y x z y x ; (19)
z y z y ; (20)
z y z y; (21)
z y x z x y x . (22)
The distance between two intervals z , y IR is determined by magnitude
dist z , y max z y , z y z , y (23)
and have the following properties:
dist z , y 0; (24)
dist z , y 0, when z y ; (25)
dist z , y dist z , y ; (26)
dist z , x dist z , y dist y , x . (27)
The key difference between classical interval arithmetic and interval analysis is in the
following. In classic interval arithmetic, the distribution law is not observed, there are
no inverse elements, similar terms cannot be reduced within its frameworks. This
leads to that the technique of symbol transformations is lost during formalization of
operations with intervals.
The main objective of interval analysis, by contrast, is not automation of compu-
ting, but rather finding the region of possible result values, taking into consideration
structures of functions and data, assigned in symbolic form.
Within this approach, interval magnitudes are considered at the intermediate stages
of calculations and analysis. Only at the last stage of decision-making, if necessary,
they are transformed into pointwise solutions. It will make it possible to give the pos-
sibility to save completeness of information on the set of possible solutions up to the
last moment.
The specific algorithmic implementation of operations with interval values eij
does not play a decisive role in this case, although it can be the subject of the specific
studies to narrow final intervals artificially.
According to the rules of classical interval analysis [15], a set (13) can be unam-
biguously ranked only when intervals Z i do not intersect
Z k Zl Z k Z k Zl Zl Z k Z k . (28)
Otherwise, there is a “weak” inequality:
Z k Zl Z k Z k Z l Z l Z k Z k , (29)
that is, the intervals are considered incomparable in the context of the classical para-
digm of interval analysis.
The formulation of the research task. In the situation described by formula (29), i.e.
when a group of intersecting interval values appears, among which it is impossible to
choose a larger value, it is necessary to develop a method for overcoming the uncer-
tainty that can be used by a direct request from the analyst.
3 Solution
The variant of formalizing interval comparison proposed in [15], which is reduced to
determining the reliability of hypotheses about the actual location of real numbers
within the corresponding intervals, cannot be used as a quantitative measure of the
ratio between these numbers [14]. The other way is proposed in [13] and is linked to
the correction of the interval logic which, however, fails in some particular cases [14].
Another option for lax formalization of the problem of comparing interval numbers
is to use the magnitudes of the distance between interval numbers as a comparison
measure (23). In this case, it becomes fundamentally possible to construct and analyze
the graph with interval numbers in vertices, however lax compliance with distribution
logic makes practical application of this approach difficult.
Let us use the method of formalizing the interval comparison proposed in [14], in
particular, let us introduce a monotonically increasing function that is not negative on
the whole real axis (see Fig. 1).
u
uZ
Z
0
Fig. 1. An example of the function of the decision maker's preferences regarding the values of
the criterion
When a specific type of function u Z is selected, the indicator can be calculated for
each alternative as follows:
Z
1
u* u Z dZ ,
Z Z Z
(30)
it is numerically equal to the height of a rectangle equivalent in area to a certain inte-
gral of the function u Z within the interval of the criterion value (see Fig. 2).
u
uZ
u*
Z
0 Z Z
Fig. 2. Graphical interpretation of the characteristic indicator of the criterion interval value
Using indicators (30) calculated for every alternative, a set (13) can be ranked quanti-
tatively.
The shape of the function u Z (for example u Z Z , u Z Z 2 , u Z Z 3 )
determines the analyst's preferences for the interval value of the criterion. For exam-
ple, when u Z Z , interval alternatives with equal midpoints are interpreted as
equivalent, while u Z Z 2 preference will be given to a wider interval alternative.
By selecting a specific type of function u Z , the researcher artificially converts
the initial ambiguity of data into the ambiguity of the preference function form,
which, nevertheless, enables avoiding the ambiguity in “fuzzy” arear of quasi-optimal
alternatives.
3.1 Example
The Table 2 represents an interval matrix of the game with nature.
Table 2. The decision efficiency interval matrix example
F1 F2 F3 F4
E1 5 ,5 . 5 15,16.5 1,1.1 5 ,5 . 5
E2 6 , 6 .6 12 ,13.2 19,20.9 2 , 2 .2
E3 10,11 14 ,15.4 0 , 0 .5 6 , 6 .6
E4 1,1.1 15,16.5 4 , 4 .4 6 , 6 .6
E5 12 ,13.2 1.9 ,2.3 5 ,5 . 5 16 ,17.6
The alternatives estimates obtained according to the maximin criterion are presented
in Table 3.
Table 3. Simulation results
Z1 u1* u*2 u*3
uZ Z uZ Z 2 uZ Z3
E1 1,1.1 1,05 1,103 1,1603
E2 2 , 2 .2 2,1 4,413 9,282
E3 0 , 0 .5 0,25 0,083 0,0313
E4 1,1.1 1,05 1,103 1,1603
E5 1.9 ,2.3 2,1 4,423 9,345
Obviously, by the maximin criterion, two alternatives are quasioptimal – E2 and E5 ,
whose estimates are incomparable in the paradigm of classical interval analysis. The
calculation of the indicator u* according to (30) for different forms of the preference
function allows to make an unequivocal reasonable choice of the only optimal alterna-
tive – E5 .
3.2 The critical analysis of results
The proposed technique for ranking alternatives has the following features.
1. The developed method cannot and should not be considered as the only or “best”
one within the given task. However, the fact that this technique is rather subjective
(while selecting the function of preference) does not violate the logic of the deci-
sion-making process. The analyst can work with the uncertainty until he makes
sure that the only optimal solution according to the selected criterion cannot be ob-
tained.
2. The proposed technique is algebraically simple and does not contain operations
that can lead to the artificial broadening of intervals. However, the researcher, that
is formalizing the decision-making task and selecting nontrivial criterion, should
take into consideration the fact that operations with interval data (especially with
intervals containing zero) can dramatically extend the criterion final interval crite-
rion. That is why the proposed technique (as the interval analysis as a whole) is ef-
ficient only for interval data of small width or for sparse interval matrices.
3. The interactive mode of operation of an analyst and DSS should remain dominant
with respect to automatic modes in the context of decision-making tasks; the pro-
posed technique should be used according to the analyst request.
4 Conclusions
1. The method is proposed for ranking quasi-optimal alternatives in interval game
models against nature, which enables comparing interval alternatives in cases of
classical interval ambiguity.
2. Recommendations on the practical implementation of the proposed method were
compiled. Specifically, recommendations for parametric setting of preference func-
tions depending on the location of interval estimates were formulated.
3. The algorithm that implements the proposed method is simple and its result is
clear, which is important in the process of making managerial decisions.
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