Algorithms of discrete optimization
and their application to problems
with fuzzy coefficients
Petr Ya. Ekel
a,*, Fernando H. Schuffner Neto
ba
Graduate Program in Electrical Engineering, Pontifical Catholic University of Minas Gerais, Av. Dom Jose Gaspar, 500, 30535-610, Belo Horizonte, MG, Brazil
bDepartment of Electronics Engineering and Telecommunications, Pontifical Catholic University of Minas Gerais, Av. Dom Jose Gaspar, 500, 30535-610, Belo Horizonte, MG, Brazil
Received 2 November 2004; received in revised form 1 June 2005; accepted 3 June 2005
Abstract
An approach to solving optimization problems with fuzzy coefficients in objective functions and constraints is described. It consists in formulating and solving one and the same problem within the framework of mutually rel\ated models with constructing equivalent analogs with fuzzy coefficients in objective functions alone. It enables one to maximally cut off dominated alternatives ‘‘from below’’ as well as ‘‘from above’’. Since the approach is applied within the context of fuzzy discrete optimization prob-lems, several modified algorithms of discrete optimization are discussed. These algo-rithms are associated with the method of normalized functions, are based on a combination of formal and heuristic procedures, and allow one to obtain quasi-optimal solutions after a small number of steps, thus overcoming the computational complexity posed the NP-completeness of discrete optimization problems. The subsequent contrac-tion of the decision uncertainty regions is associated with reduccontrac-tion of the problem to multiobjective decision making in a fuzzy environment with using techniques based
0020-0255/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2005.06.001
* Corresponding author. Tel.: +55 31 3319 4305; fax: +55 31 3319 4225. E-mail address:[email protected](P.Ya. Ekel).
on fuzzy preference relations. The techniques are also directly applicable to situations in which the decision maker is required to choose alternatives from a set of explicitly avail-able alternatives. The results of the paper are of a universal character and can be applied to the design and control of systems and processes of different purposes as well as the enhancement of corresponding CAD/CAM systems and intelligent decision making sys-tems. The results of the paper are already being used to solve problems of power engineering.
2005 Elsevier Inc. All rights reserved.
Keywords: Discrete optimization; Fuzzy coefficients; Nonfuzzy analogs; Fuzzy number ranking indices; Multiobjective decision making; Fuzzy preference relations
1. Introduction
Discrete, integer, and Boolean (in the general case, discrete) problems have relevant applications in many fields [8,43,45]. Considering this, it should be stressed that direct determination of discrete solutions to problems of discrete nature is necessary. This is explained by the fact that even though at the cost of ignoring parameter discreteness, with smoothing of functions, it is possible to replace an actual objective function by a convex function defined on a convex region, with such an approach the danger exists that the objective function will be distorted (with a deviation from the optimum) or that the constraints will be violated. Besides, the transition from the discrete model to its convex analog can lead to considerable ‘‘coarsening’’ the model that often makes vapid its es-sence [17]. Hence the ability to solve discrete problems by discrete methods makes it possible in the course of the solution to consider detailed situations and reflect individual forms of initial data reliably to obtain solutions within the framework of more adequate models. Finally, with orientation to discrete methods it is possible to pose problems of combinatorial nature, which had previously not been considered[17].
Theoretical and experimental evaluations (for example, see[34,35]) have re-vealed essential drawbacks of exact methods of discrete programming. Fur-thermore, estimates of computational complexity [27] in solving discrete problems indicate that their NP-completeness does not permit one to develop general methods with polynomial dependence on the problem dimension
[35,40]. In this connection, the development and use of approximate methods are the main direction in the evolution of discrete programming.
Taking the above into account, the algorithms discussed in the paper are based on a combination of formal and heuristic procedures. They are associ-ated with the method of normalized functions[6]and use elements of the gree-dy heuristics [11–13,44], which basically provide the best heuristic among possible heuristics with a priori estimates and can be the basis for new effective
approximate approaches. The algorithms allow one to obtain quasi-optimal solutions after a small number of steps, thus overcoming the problem NP-com-pleteness. They do not require analytical specification of objective functions and constraints. This ensures the flexibility and the possibility to solve complex problems, for which adequate analytical descriptions are difficult.
In the process of posing and solving a wide range of problems related to the design and control of complex systems, one inevitably encounters different kinds of uncertainty[22]. Its consideration in shaping the mathematical models should be inherent to the practice of systems analysis. This serves as a means for increasing the adequacy of the models and, as a result, their credibility and the factual effectiveness of solutions based on their analysis.
At present, investigators have doubts about the validity or, at least, the expediency of including the uncertainty within the framework of models that are shaped by traditional approaches. Considering this, the application of the fuzziness concept to the systems to be studied may play a significant role in overcoming this situation. Besides, operation with a fuzzy parameter space allows one not only to be oriented toward the contextual or intuitive aspect of qualitative analysis as a fully substantiated process, but, by means of fuzzy set theory[41,48], also to use this approach as reliable source for obtaining quan-titative information.
Nevertheless, in solving problems under condition of uncertainty it is neces-sary to exert maximum efforts in seeking the possibilities for overcoming the uncertainty. This is done, for example, by using information of informal char-acter (based on experience, knowledge, and intuition of specialists) or, in the general case, by aggregating information arriving from various sources of both formal and informal nature[18]. Here we are essentially speaking of the fact that the characteristic of uncertain information (usually specified by intervals) may and should be supplemented by specifically adopted, well-founded assumptions as to differentiated reliability of different values of uncertain parameters. This represents a generalization of the interval specification of information and serves as a technique for removing the uncertainty, but re-quires the use of the corresponding apparatus. The apparatus of fuzzy set the-ory can serve as the latter. Its utilization in problems of complex system optimization offers advantages of both fundamental nature (based on the pos-sibility of validly obtaining the more effective, less ‘‘cautious solutions’’) and computational character[18,19].
When using fuzzy set theory, certain fundamental problems arise in the comparison of alternatives on the basis of fuzzy values of objective functions, consideration of constraints with fuzzy coefficients, development of principles and concrete methods of solving associated optimization problems. Below we discuss some approaches to solving these problems and propose ways of imple-menting the approaches as applied to discrete programming models with fuzzy coefficients both in the objective function and constraints with the use of
modification of the algorithms of discrete optimization and contraction of aris-ing decision uncertainty regions on the basis of procedures of multiobjective choosing alternatives in a fuzzy environment.
2. Problem formulation
From the variety of discrete optimization models it is possible to distinguish two extensive classes. The first class is related to the general problem of discrete programming, including the problems of integer, Boolean, and discrete pro-gramming proper. The problems with discrete variables may be reduced to integer or, in the general case, to Boolean models [32,45]. However, such a reduction increases the problem dimensionality[17].
The second class of models is associated with the problems of combinatorial type. When solving them, an extremum of the objective function is defined on a given finite discrete setA. The totality of objects obtained fromA(for example, combinations or permutations) as well as objects obtained as a result of execu-tion of logical operaexecu-tions on elements ofA[17]may be considered as a combi-natorial spaceD. The problem is formulated as a search for X0¼ ðx0
1;. . .;x 0
nÞ
fromGDyielding an extremum of the objective function, i.e.,FðXÞ ! extr
x2GD.
The combinatorial problems are the most difficult from the computational standpoint. Their solution is based, in the main, on finiteness of GDand the specificity of the problem. Many of the combinatorial problems may be re-duced to problems of integer or Boolean programming. For instance, Ibaraki
[33] defines sufficient conditions of reducibility of combinatorial problems to models of integer programming but shows that there is no a general algorithm of such reducibility, even it is realizable. Besides, in many cases reducibility is reached by accepting considerable assumptions, sharp increasing model dimen-sion, and losing the possibility to effectively exploit combinatorial properties of the initial problem[17].
In this connection, when solving discrete problems, it is important that their formulation and solution algorithms should exploit those properties and pecu-liarities of the problems, which promote their effective solution. Taking this into account, the desirability of allowing for constraints on the discreteness of variables in the form of discrete sequences
xsi;asi;bsi;. . .; si¼1;. . .;ri ð1Þ has been validated in[49]; here asi;bsi;. . .are characteristics required for for-mation of objective functions, constraints, and their increments, which corre-spond to thesth discrete (integer, Boolean) value of the variablexi.
It is expedient to use discrete sequences of the type(1)because the charac-teristicsasi;bsi;. . .(technical, economic, etc. depending on values ofxi) cannot always be fitted closely to analytical relationships in terms ofxsi, but in discrete
sequences of the type(1)these characteristics may be taken as exact. Besides, a flexible formulation of combinatorial problems is possible on the basis of the discrete sequences because they can be different for different variables. Consid-ering this, a maximization problem may be formulated as follows.
Assume we are given discrete sequences of the type (1) (increasing or decreasing, depending on the problem formulation). From them it is necessary to choose parameters that the objective
maximizeFðxs1;as1;bs1;. . .;. . .;xsn;asn;bsn;. . .Þ ð2Þ is met while satisfying the constraints
gjðxs1;as1;bs1;. . .;. . .;xsn;asn;bsn;. . .Þ6bj; j¼1;. . .;m. ð3Þ The objective function(2)is interpreted as concave and the constraints(3)are interpreted as convex.
Given the maximization problem (1)–(3), we can formulate a problem of minimization.
From discrete sequences of the type(1)it is necessary to choose parameters that the objective
minimizeFðxs1;as1;bs1;. . .;. . .;xsn;asn;bsn;. . .Þ; ð4Þ subject to the constraints
gjðxs1;as1;bs1;. . .;. . .;xsn;asn;bsn;. . .ÞPbj; j¼1;. . .;m. ð5Þ The objective function(4) is interpreted as convex and the constraints(5)are interpreted as concave.
Generalizing the problem(1)–(3), it is possible to construct the problem of choosing parameters from discrete sequences of the type(1) that the objective maximizeF~ðxs1;as1;bs1;. . .;. . .;xsn;asn;bsn;. . .Þ ð6Þ is met while satisfying the constraints
~
gjðxs1;as1;bs1;. . .;. . .;xsn;asn;bsn;. . .Þ
~
bj; j¼1;. . .;p. ð7Þ
The objective function(6)and constraint(7)include fuzzy coefficients, as indi-cated by thesymbol.
Generalizing(1), (4), (5), it is possible to construct the problem of choosing parameters from discrete sequence of the type(1)that the objective
minimizeF~ðxs1;as1;bs1;. . .;. . .;xsn;asn;bsn;. . .Þ ð8Þ is met while satisfying the constraints (7).
The models(1), (6), (7)and(1), (8), (7)generalize the models analyzed, for instance, in[28–30]and their framework accommodates many practical prob-lems involving, for example, probprob-lems whose formulation with deterministic information have been considered in[17,49].
Initially we consider algorithms of solving the problems(1)–(3)and(1), (4), (5)whose modification permits one to solve the problems(1), (6), (7)and(1), (8), (7).
3. Solution algorithms
Let us consider the Boolean problem of maximization of FðxÞ ¼X
n
i¼1
cixi; ð9Þ
while satisfying the constraints
Xn
i¼1
ajixi6bj; j¼1;. . .;m; ð10Þ
where ci> 0, i= 1,. . .,n, aji> 0, j= 1,. . .,m, i= 1,. . .,n, and bj> 0,
j= 1,. . .,m.
The idea of one of the most popular methods, related to the class of heuristic methods, may be illustrated by considering the problem(9), (10)form= 1 (the 0–1 knapsack problem[42]). It is possible to assume thatxi,i= 1,. . .,nare
ar-ranged as follows: c1 a1 Pc2 a2 P Pcn an . ð11Þ
It permits one to try to maximize(9)on the basis of the largestci/ai, taking
x1= 1, thenx2= 1, and so on until(10)is observed. Similar methods are called
greedy methods. In spite of their ‘‘naivety’’, in many cases they represent the best heuristic among other heuristics with a priory estimates. However, a range of problems is not restricted by the case ofm= 1. Considering this, we discuss below ways of constructing algorithms for the general case (m> 1) to solve problems (linear as well as nonlinear), which can include not only Boolean, but integer and discrete variables as well (some theoretical and experimental results in this field related to solving Boolean and integer linear problems are discussed in [11–13,44]).
When analyzing the models(9) and (10)form= 1, maximization is reached by expending only one resource type. If m> 1, the optimization process is stopped when a remaining amount of only one of resources is not sufficient for next incrementing any of xi, i= 1,. . .,n. It is possible to speak about
‘‘equivalence’’ of different types of resources from the standpoint of termina-tion of the process of maximizing(9). Thus, it is expedient to have a single mea-sure for different resources. This consideration leads to the idea of normalization [6]. For example, the constraints (10) are reduced to a single arbitrary resourcebas
aðtÞji ¼aji
b
bðtji 1Þ; j¼1;. . .;m; i¼1;. . .;n; ð12Þ wheretis the optimization step number.
Using(12), it is possible to convert the constraints(10)to equal conditions. For instance, before the first optimization step we have
Xn
i¼1
aðji0Þxi6b; j¼1;. . .;m. ð13Þ
The normalization (12)may also be useful for reducing the model dimen-sion. If in (13)
aðpi0Þ6að0Þ
qi; p6¼q; i¼1;. . .;n; ð14Þ
the qth constraint is disturbed earlier than the pth one. Thus, the pth con-straint, which is not active, can be eliminated from consideration (the principle of explicit domination).
The algorithm of solving the maximization problem(1)–(3), can be written in the form suitable for a subsequent modification on the basis of the results of
[17,49].
1. The components of the constraint increment vector fDGðtÞi g are
calculated: DGðtÞi ¼max
j Dg ðtÞ
ji; i2IðtÞ; j¼1;. . .;m; ð15Þ
wheretis the index number of the optimization step;I(t)is the set of vari-ables at thetth step which, at their present values, satisfy all constraints. In(15),DgðtÞji is the increment in thejth constraint whenxðtÞsi undergoes a change from the current level si to the level si+ 1 while all the other
xðtÞ
sk; k6¼iremain at current levels sk: DgðtÞji ¼ gj xðtÞs1;a ðtÞ s1;b ðtÞ s1;. . .;. . .;x ðtÞ siþ1;a ðtÞ siþ1;b ðtÞ siþ1;. . .;. . .;x ðtÞ sn;a ðtÞ sn;b ðtÞ sn;. . . h gj xðtÞs 1;a ðtÞ s1;b ðtÞ s1;. . .;. . .;x ðtÞ si;a ðtÞ si;b ðtÞ si;. . .;. . .;x ðtÞ sn;a ðtÞ sn;b ðtÞ sn;. . . i b bðtj 1Þ; j¼1;. . .;m; i2I ðtÞ. ð16Þ
In evaluatingDgðtÞji, for the first step (t= 1) we havei2In(Inis the initial
set of variables) and bðtj 1Þ¼bðj0Þ¼bj.
2. We refine the setI(t)of variables on which optimization is possible at the tth step:
3. We make a check for nonemptiness of the setI(t). IfI(t)5B, then go to
operation 4; otherwise go to operation 12.
4. The components of the increment vector of the objective functionfDFðtÞi g
are calculated as DFðtÞi ¼FðxðtÞs 1;a ðtÞ s1;b ðtÞ s1;. . .;. . .;x ðtÞ siþ1;a ðtÞ siþ1;b ðtÞ siþ1;. . .;. . .;x ðtÞ sn;a ðtÞ sn;b ðtÞ sn;. . .Þ FðxðtÞs 1;a ðtÞ s1;b ðtÞ s1;. . .;. . .;x ðtÞ si;a ðtÞ si;b ðtÞ si;. . .;. . .;x ðtÞ sn;a ðtÞ sn;b ðtÞ sn;. . .Þ; i2IðtÞ. ð18Þ
5. We refine the setI(t)of variables on which optimization is possible at the tth step:
IðtÞ¼ fijDFðtÞi >0; i2IðtÞg. ð19Þ
6. We make a check for nonemptiness of the setI(t). IfI(t)5B, then go to
operation 7; otherwise go to operation 12.
7. The components of the vectorfVðtÞi gare calculated as: VðtÞi ¼DF
ðtÞ i
DGðtÞi ; i2I
ðtÞ. ð20Þ
8. The indexi=ltof the most promising variable to be incremented is
deter-mined from VðtÞl
t ¼maxi V
ðtÞ
i ; i2IðtÞ. ð21Þ
9. We recalculate the current values of the quantities: xðtÞs i ¼ xðtÞ si if i6¼lt; i2I ðtÞ; xðtÞsiþ1 if i¼lt; ( bðtÞj ¼bðtj 1ÞDgðtÞjl t bðtj 1Þ b ; j¼1;. . .;m. ð22Þ
10. We refine the setI(t):
IðtÞ¼ fijsi<ri; i2IðtÞg; ð23Þ
taking into account that in (1)si= 1,. . .,ri.
11. We make a check for nonemptiness of the setI(t). IfI(t)5B, then go to
operation 1, takingt=t+ 1; otherwise go to operation 12. 12. The calculations are completed because the solution is obtained.
The execution of operation 1 of the algorithm, in accordance with(15), pro-vides determination of the most ‘‘dangerous’’ constraint (the constraint with the most scarce type of the resource bðtÞj ; j¼1;. . .;m) for every variable at
the given step. In essence, the execution of operation 1 permits one to construct the following convolution:
X
i2IðtÞ
DGðtÞi 6b.
Thus, at each step of optimization we obtain an increment of thatltth variable
which maximizes the increment of the objective function per unit normalized resource b. In this connection, the use of(20) and (21)is similar to(11).
The refinements of the set I(t) (17) and (19) are associated with excluding such variables that lead to violation of the constraints (3) or to decreasing the objective function(2).
The problem of minimization(1), (4), (5)is more difficult than the problem of maximization (1)–(3). In the case of maximization we cease changing the variable xi when at least one of the constraints (3) is violated in accordance
with (17). In the case of minimization the optimization is completed on any variable when all constraints (5) are obeyed. On the other hand, there is in the case of maximization usually only one ‘‘deficient’’ constraint at each step requiring particular attention, while in minimization we have to pay attention to each constraint because the optimization process cannot be completed until all constraints(5)have been obeyed.
It is assumed that the initial constraints(5)are already normalized and have the following form:
gðj0Þðxs1;as1;bs1;. . .;. . .;xsn;asn;bsn;. . .Þ ¼gjðxs1;as1;bs1;. . .;. . .;xsn;asn;bsn;. . .Þ
b bj
Pb; j¼1;. . .;m.
The algorithm of solving the minimization problem(1), (4), (5)can be writ-ten in the form suitable for a subsequent modification as on the basis of the results of[17,49].
1. The components of the constraint increment vectorfDGðtÞi gare evaluated: DGðtÞi ¼ X j DgðtÞji ; i2IðtÞ; j2JðtÞ. ð24Þ In(24), DgðtÞji ¼ gj xðtÞs1;a ðtÞ s1;b ðtÞ s1;. . .;. . .;x ðtÞ siþ1;a ðtÞ siþ1;b ðtÞ siþ1;. . .;. . .;x ðtÞ sn;a ðtÞ sn;b ðtÞ sn;. . . h gj xðtÞs 1;a ðtÞ s1;b ðtÞ s1;. . .;. . .;x ðtÞ si;a ðtÞ si;b ðtÞ si;. . .;. . .;x ðtÞ sn;a ðtÞ sn;b ðtÞ sn;. . . i b ðt1Þ j b ; j2J ðtÞ; i2IðtÞ;
whereJ(t)is the set of the constraints(5)at thetth step. For the first step (t= 1) we havej2Jm(Jmis the initial set of constraints); i2In.
2. The components of the increment vector of the objective function fDFðtÞi g; i2IðtÞ are calculated in accordance with(18).
3. The components of the vectorfVðtÞi g; i2IðtÞare calculated in accordance with(20).
4. The indexi=ltof the most promising variable to be incremented is
deter-mined from VðtÞl
t ¼mini V
ðtÞ
i ; i2IðtÞ. ð25Þ
5. We recalculate the current values of the quantitiesxðtÞ si; i2I ðtÞ, using(22), and bðtÞj ¼bðtj 1ÞDgðtÞjl t b bðtj 1Þ; j2J ðtÞ.
6. We refine the set J(t):
JðtÞ¼ fjjbðtÞj >0; j2JðtÞg. ð26Þ
7. We make a check for nonemptiness of the setJ(t). IfJ(t)5B, then go to
operation 8; otherwise go to operation 11.
8. We refine the set I(t)in accordance with the condition(23).
9. We make a check for nonemptiness of the setI(t). IfI(t)5B, then go to
operation 1, takingt=t+ 1; otherwise go to operation 10.
10. The calculations are completed because the problem has no solution. 11. The calculations are completed because the solution is obtained.
The execution of operation 1 of the algorithm in accordance with(24) pro-vides the convolution of the set of constraints (5) at the given optimization step. In this connection, at each step of optimization we obtain an increment of thatltth variable which minimizes the increment of the objective function
per unit total expenditure of normalized resources.
The refinement of the setJ(t)(26)is associated with revealing and excluding such constraints(5) which are already satisfied.
The algorithm has no operation similar to operation 5 of the algorithm of solving the maximization problem. However, this does not narrow a field of its applications because prior to using the algorithm it is possible to carry out simple minimization of the objective function (4)without considering the constraints(5).
Our numerous comparisons of solutions for diverse types of discrete prob-lems given in[32,45], based on the paper results (with the upper bound of the number of operations N= (5mn+ 6m+ 11)n for Boolean and N¼ ð5mnþ 6mþ11ÞPniri for integer and discrete linear problems) and exact methods,
show their convincing agreement. This is also confirmed by numerous results on ‘‘good’’ behavior of the greedy algorithms [1] for wide classes of discrete problems. At the same time, the authors of[4]indicate that the greedy algo-rithms should be used with great care. Taking into account these opposite opinions, it is expedient to have not only one, but several algorithms realizing different strategies. Considering this, so-called ‘‘duplicate’’ algorithms have been developed on the basis of a qualitative analysis of the problem statement. One of them is based on evaluating the components of the vectorfDGðtÞi g
(oper-ation 1 of the algorithm of minimiz(oper-ation) as follows: DGðtÞi ¼min
j Dg ðtÞ
ji; i2IðtÞ; j2JðtÞ; ð27Þ
whereDgðtÞji; i2IðtÞ; j2JðtÞare calculated as (16).
An alternative ‘‘duplicate’’ algorithm is associated with the results of [13]
and is based on calculatingDgðtÞji; i2IðtÞ; j2JðtÞ(operation 1 of the algorithm
of minimization) as DgðtÞji ¼min gj xðtÞs 1;a ðtÞ s1;b ðtÞ s1;. . .;. . .;x ðtÞ siþ1;a ðtÞ siþ1;b ðtÞ siþ1;. . .;. . .; h n xðtÞs n;a ðtÞ sn;b ðtÞ sn;. . . gj xðtÞs 1;a ðtÞ s1;b ðtÞ s1;. . .;. . .;x ðtÞ si ;a ðtÞ si ;b ðtÞ si ;. . .;. . .; xðtÞsn;aðtÞ sn;b ðtÞ sn;. . . ;b ðt1Þ j io ; i2IðtÞ; j2JðtÞ ð28Þ with recalculatingbðtÞj ; j2JðtÞ (operation 5 of the algorithm of minimization) as bðtÞj ¼bðtj 1Þ gj xðtÞs 1;a ðtÞ s1;b ðtÞ s1;. . .;. . .;x ðtÞ sltþ1;a ðtÞ sltþ1;b ðtÞ sltþ1;. . .;. . .; h xðtÞs n;a ðtÞ sn;b ðtÞ sn;. . . gj xðtÞs 1;a ðtÞ s1;b ðtÞ s1;. . .;. . .;x ðtÞ slt;a ðtÞ slt;b ðtÞ slt;. . .;. . .; xðtÞsn;aðtÞ sn;b ðtÞ sn;. . . i ; j2JðtÞ. ð29Þ
The availability and use of the ‘‘duplicate’’ algorithms can be considered, in a certain measure, as an assurance of obtaining optimal solutions. Besides, the analysis of one and the same problem on the basis of several algorithms per-mits one to reveal a series of solutions of equal worth, which is important as well.
The described results have a high degree of generality and have been used in solving power engineering problems: optimization in the design and develop-ment (selecting eledevelop-ments of power systems, allocating reactive power sources, selecting means for increasing reliability, etc.), load management, and reactive power control.
As an additional means for possible improving the solution performance may serve formulating and solving one and the same problem within the frame-work of so-called mutually related models(1)–(3) and(1), (4), (5). Using this
approach, if we have the increasing (decreasing) sequences(1) for(1)–(3), the sequences(1)must be decreasing (increasing) for(1), (4), (5). Thus, it is possi-ble to solve one and the same propossi-blem ‘‘from above’’ and ‘‘from below’’ as well. This approach is fruitful and also serves for solving problems with fuzzy (or interval) coefficients.
4. Constraints with fuzzy coefficients
The basic question, which arises when solving optimization problems under conditions of uncertainty is how to account for the different nature of the straints and primarily the functional constraints. For simplicity we shall con-sider one constraint of the following form:
Xn
i¼1
~
AixiB~; ð30Þ
whereA~i; i¼1;. . .;nandB~ are fuzzy coefficients with membership functions
lAiðaiÞ; i¼1;. . .;nandlBðbÞ, respectively.
An approach to analyzing constraints of the form(30)is proposed in[37]. It involves approximate replacement of each of the constraints of the form(30)
by a finite set of nonfuzzy (deterministic) constraints, represented in the form of inequalities; these can be formulated readily, but with a considerable in-crease in the dimension of the problem being solved. However, when using the algorithms given above, this fact does not give rise to any difficulties, since the finite set of deterministic constraints at each step of the optimization is ‘‘rolled up’’ into a single inequality. Moreover, the principle of explicit domi-nation described above substantially reduces the dimension of the resulting equivalent nonfuzzy analog.
When a number of conditions are satisfied (in particular, with regard to the convexity of the fuzzy coefficients A~i; i¼1;. . .;n and B~ [41,48]), and we
as-sume the possibility of ordering
06r1< <rk < <rK 6min min
16i6nsuplAiðaiÞ; suplBðbÞ
;
then the constraint(30)can be changed approximately to the following system of constraints Xn i¼1 SrkA ixiS rk B; k¼1;. . .;K; ð31Þ where SrkA i and S rk
B; k¼1;. . .;K are sets of the rk-level, respectively of ~
Ai; i¼1;. . .;n and B. For example, the set~ SrkB of the rk-level of the fuzzy
Naturally, the accuracy of approximation(31)is readily adjusted by varying the value ofK.
Taking into account the definition of the set of the rk-level, we can write
from(31): Xn i¼1 ark i1;a rk i2 xi b rk 1 ;b rk 2 ½ ; k¼1;. . .;K; whence we obtain Xn i¼1 ark i2xi6b rk 2; k¼1;. . .;K ð32Þ and Xn i¼1 ark i1xiPb rk 1; k¼1;. . .;K.
This allows us to solve concrete problems in which we may encounter con-straints with inequalities in either direction.
Using the principle of explicit domination, we can reduce the dimension of the set of inequalities, for example, (32). As a result of normalization
hrki ¼ark i2 b brki 2 ; k¼1;. . .;K; i¼1;. . .;n;
we can change to the set of constraints
Xn
i¼1
hrk
i xi6b; k¼1;. . .;K;
wherebis an arbitrary positive number.
If, as a result of analyzing the last set of constraints withhrki P0, it turns out that
hrpi 6hrq
i ; p6¼q; i¼1;. . .;n; ð33Þ
theqth constraint, for a purposeful increase in the variablesxi,i= 1,. . .,n, is
disturbed earlier than thepth constraint. Thus, thepth constraint can be elim-inated from consideration. An example illustrating the application of the prin-ciple of explicit domination(33)is given in[26].
Thus, as regards a problem with constraints containing fuzzy coefficients, one can obtain an equivalent nonfuzzy analog whose dimension is reduced by using the principle of explicit domination (33). The solution of problems containing fuzzy coefficients in the objective functions alone is possible by a modification of traditional mathematical programming methods.
5. Problems with fuzzy coefficients in objective functions
Taking the above into account, it is possible to change from the problem(1), (6), (7)or(1), (8), (7)to the problem(1), (6), (3)or(1), (8), (5)with fuzzy coef-ficients in the objective function alone. The solution of the problem(1), (6), (3)
is possible on the basis of modifying the algorithm of solving the maximization problem (1)–(3). The solution of the problem (1), (8), (5) is associated with modifying the algorithm of solving the minimization problem (1), (4), (5) or corresponding ‘‘duplicate’’ algorithms. In particular, the execution of algebraic operations on fuzzy numbers by means of the expressions (18) and (20) is accomplished on the basis of algorithms given in[14], which in turn take into account the results of[16].
To compare alternatives on the basis of(21)or(25)(in essence, the ranking of fuzzy numbersVeðtÞi ; i2IðtÞin order to choose the largest or the smallest) it is necessary to use the corresponding methods, which are considered and ana-lyzed in [9,36,46,47]. In particular, the authors of [9] classify four groups of methods related to the ordering of fuzzy quantities. Among these groups, the authors of[31]consider the construction of fuzzy preference relations for pair-wise comparisons as the most practical and justified way. Taking this into ac-count, it is necessary to distinguish the fuzzy number ranking index introduced by Orlovsky[39]based on the conception of a membership function of a gen-eralized preference relation.
If Ve1 and Ve2 have the membership functionsl(v1) andl(v2), the quantity
g{l(v1),l(v2)} is the degree of preference l(v1)¤l(v2), while g{l(v2),l(v1)}
is the degree of preference l(v2)¤l(v1). Then, the membership functions of
the generalized preference relationsg{l(v1),l(v2)} andg{l(v2),l(v1)} take the
following form: gflðv1Þ;lðv2Þg ¼ sup v1;v22V minflðv1Þ;lðv2Þ;lRðv1;v2Þg; ð34Þ gflðv2Þ;lðv1Þg ¼ sup v1;v22V minflðv1Þ;lðv2Þ;lRðv2;v1Þg; ð35Þ
wherelR(v1,v2) andlR(v2,v1) are the membership functions of the
correspond-ing fuzzy preference relations.
IfF is the numerical axis on which the values of the maximized objective function, for example, are plotted, andRis the natural order (P) alongF, then
(34) and (35)reduce to: gflðv1Þ;lðv2Þg ¼ sup v1;v22V v1Pv2 minflðv1Þ;lðv2Þg; ð36Þ gflðv2Þ;lðv1Þg ¼ sup v1;v22V v2Pv1 minflðv1Þ;lðv2Þg. ð37Þ
These agree with the Baas–Kwakernaak[2], Baldwin–Guild[3], and one of the Dubois–Prade [15]fuzzy number ranking indices.
On the basis of the relations between(36) and (37), it is possible to judge the degree of preference of any of the alternatives compared. Utilization of this ap-proach is well founded. However, experience shows that in many cases the membership functions of the alternatives l(v1) and l(v2) compared form flat
apices (for example, [18,23]), i.e., they are so-called flat or trapezoidal fuzzy numbers [41,48]. In view of this, we can say that for the situation shown in
Fig. 1the alternatives Ve1and Ve2are indistinguishable if
gflðv1Þ;lðv2Þg ¼gflðv2Þ;lðv1Þg ¼r. ð38Þ
In such situations the modified algorithms of discrete optimization do not al-low one to obtain a unique solution because they ‘‘stop’’ when conditions like
(38)arise. This occurs also with other modifications of traditional mathemat-ical programming methods (this is illustrated in[26]by a simple example) be-cause combination of the uncertainty and the relative stability of optimal solutions can produce these so-called decision uncertainty regions. In this con-nection, other indices may be used as additional means for the ranking of fuzzy numbers.
Wang and Kerre [46] count more than 35 existing fuzzy number ranking indices, indicating that unlike in the case of real numbers, fuzzy quantities have no natural order. The idea with the ordering of fuzzy quantities is to convert a fuzzy quantity into a real number and base the comparison of fuzzy quantities on that of real numbers. Each individual conversion way, however, pays atten-tion to a special aspect of fuzzy quantity. As a consequence, each approach suf-fers from some defects if only one real number is associated with each fuzzy quantity. The authors of[10,36]share this opinion as well. Cheng[10]also indi-cates that many of indices produce different rankings for the same problem. The authors of [10,22,36] underline that fuzzy number ranking indices occa-sionally result in choices which appear inconsistent with intuition. Finally,
μ(v)
σ
µ(v1)
µ (v2)
v
the majority of indices for the ranking of fuzzy quantities have been proposed with the aspiration for obligatory distinguishing the alternatives. This is not natural because the uncertainty of information creates the decision uncertainty regions.
There actually is another approach that is better validated and natural for the decision making practice. This approach is associated with transition to multiobjective choosing alternatives in a fuzzy environment because the appli-cation of additional criteria (including the criteria of qualitative character, such as ‘‘comfort of operation’’, ‘‘flexibility of development’’, etc.) can serve as a convincing means to contract the decision uncertainty regions.
6. Procedures of multiobjective choosing alternatives in a fuzzy environment
Before starting to discuss questions of multiobjective decision making in a fuzzy environment, it is necessary to note that considerable contraction of the decision uncertainty regions may be obtained by formulating and solving one and the same problem within the framework of mutually related models: (a) the model of maximization(6)with the constraints(7)approximated by
(3);
(b) the model of minimization(8)with the constraints (7)approximated by
(5).
It is natural that the number of the constraints(3)or(5)is equal to or great-er than the numbgreat-er of the constraints(7).
When using this approach, solutions dominated by the initial objective func-tion are cut off from above as well as from below to the greatest degree[22]if we solve, for example, the problem(6), (3)with the increasing (decreasing) dis-crete sequences(1)and the problem(8), (5)with the decreasing (increasing) dis-crete sequences (1). It should be stressed that this approach is of a universal character and may also be used in solving continuous problems as well.
Assume we are given a setXof alternatives (from the decision uncertainty region), which are to be examined byqcriteria of quantitative and/or qualita-tive nature. The problem of decision-making is presented by a pair hX,Ri where R= {R1,. . .,Rq} is a vector fuzzy preference relation [25,39]. In this
case, we have
Rp¼ ½XX;lRpðXk;XlÞ; p¼1;. . .;q; Xk;Xl2X; ð39Þ wherelRpðXk;XlÞis a membership function of fuzzy preference relation.
In (39), Rp is defined as a fuzzy set of all pairs of X·X, such that the
dominatesXl, i.e., the degree to which Xkis at least as good asXl (Xk is not
worse thanXl) for thepth criterion.
It is supposed in[25,39], that the matrices,Rp,p= 1,. . .,qare directly given
as expertÕs estimates (from the interval [0, 1]) denoting the degree of preference of one alternative over the other. However, there is another, more convincing and natural, approach to obtaining these matrices. In particular, the availabil-ity of fuzzy or linguistic estimates of alternativesF~pðXkÞ; p¼1;. . .;q; Xk 2X
(constructed on the basis of expert estimation or on the basis of aggregating information arriving from different sources of both formal and informal char-acter [18]) with the membership functions l[fp(Xk)], p= 1,. . .,q, Xk2X
per-mits one to construct the matrices Rp, p= 1,. . .,q as follows, using the
expressions (36) and (37): lRpðXk;XlÞ ¼ sup Xk;Xl2X fpðXkÞPfpðXlÞ minfl½fpðXkÞ;l½fpðXlÞg; ð40Þ lRpðXl;XkÞ ¼ sup Xk;Xl2X fpðXlÞPfpðXkÞ minfl½fpðXkÞ;l½fpðXlÞg. ð41Þ
If thepth criterion is associated with minimization, then(40) and (41)are writ-ten for regionsfp(Xk)6fp(Xl) andfp(Xl)6fp(Xk), respectively.
Let us consider the situation of setting up a single preference relationR. It can be represented by the strictRSand indifferentRIfuzzy preference relations
[25,39]. It is possible to use the inverse relationR1((Xk,Xl)2R1is equivalent
to (Xl,Xk)2R) to obtain
RS¼RnR1. ð42Þ
If (Xk,Xl)2R
S
, thenXkdominatesXl, i.e.,XkXl. The alternative Xk2Xis
nondominated inhX,Riif (Xk,Xl)2R
S
for anyXl2X.
If we havelR(Xk,Xl) as a nonstrict fuzzy preference relation, then the value
lR(Xk,Xl) is the degree of preferenceXk¤Xlfor anyXk,Xl2X. The
member-ship function, which corresponds to(42)is the following:
lSRðXk;XlÞ ¼maxflRðXk;XlÞ lRðXl;XkÞ;0g. ð43Þ
The expression(43)serves as the basis for the choice procedure introduced by Orlovsky[38]. Many authors have studied this procedure. For instance, it was shown in [5] that this choice procedure possesses many interesting desirable properties. Its axiomatic characterization is given, for example, in[7].
The use of(43)permits one to carry out the choice of alternatives. In par-ticular,lS
RðXl;XkÞis the membership function of the fuzzy set of allXk, which
are strictly dominated byXl. Its complement by 1lSRðXl;XkÞgives the fuzzy
set of alternatives, which are not dominated by other alternatives fromX. To choice the set of all alternatives, which are not dominated by other alternatives fromX, it is necessary to find the intersection of all 1lS
allXl2X[38,39]. This intersection is the set of nondominated alternatives with
the membership function lNDR ðXkÞ ¼ inf Xl2X 1lSRðXl;XkÞ ¼1sup Xl2X lSRðXl;XkÞ. ð44Þ BecauselND
R ðXkÞis the degree of nondominance, it is natural to obtain
alter-natives providing XND¼ XNDk jXNDk 2X;lND R ðX ND k Þ ¼ sup Xk2X lND R ðXkÞ . ð45Þ
WhenRis a vector fuzzy preference relation, the expressions (43)–(45)can be applied if we takeR¼Tqp¼1Rp, i.e.,
lRðXk;XlÞ ¼ min
16p6qlRpðXk;XlÞ; Xk;Xl2X.
When using this intersection, the setXNDfulfils the role of a Pareto set[39]. Its contraction is possible on the basis of differentiating the importance of Rp,
p= 1,. . .,qwith the use of the following convolution (aggregation of monob-jective fuzzy preference relations)[39]:
lTðXk;XlÞ ¼
Xq
p¼1
kplRpðXk;XlÞ; Xk;Xl2X;
wherekpP0,p= 1,. . .,qare weights (importance factors) for the
correspond-ing criteria normalized asPqp¼1kp¼1.
The construction oflT(Xk,Xl),Xk,Xl2Xallows one to obtain the
member-ship functionlND
T ðXkÞof the set of nondominated alternatives according to an
expression similar to(44). The intersection oflND
R ðXkÞandlNDT ðXkÞdefined as lNDðX kÞ ¼minflNDR ðXkÞ;lNDT ðXkÞg; Xk2X provides us with XND¼ XNDk jXNDk 2X;lNDðXNDk Þ ¼ sup Xk2X lNDðXkÞ .
The expressions(44) and (45)can also serve as the basis for building another procedure, which is of a lexicographic character. It is associated with step-by-step introduction of criteria for comparing alternatives. The procedure permits one to construct a sequenceX1,X2,. . .,Xqso thatX X1 X2 Xqwith the use of the following expressions:
lND Rp ðXkÞ ¼ inf Xl2Xp1 ½1lS RpðXl;XkÞ ¼1 sup Xl2Xp1 lS RpðXl;XkÞ; p¼1;. . .;q; Xp¼ XND;p k jX ND;p k 2X p1; lNDRp ðXNDk ;pÞ ¼ sup Xl2Xp1 lNDRp ðXkÞ ( ) .
It should be noted that ifRpis transitive, we can bypass the pairwise
compar-ison of alternatives at the pth step. In this situation, the comparison can be done on a serial basis (the direct use of(40) and (41)) with memorizing the best alternatives.
The described choice procedures have found applications in solving power engineering problems [24]. However, it is possible to propose the third proce-dure to contract the decision uncertainty region. In particular, the use of
(44)represented in the form lND
Rp ðXkÞ ¼1sup
Xl2X lS
RpðXl;XkÞ; p¼1;. . .;q ð46Þ permits one to construct the membership functions of the set of nondominated alternatives for each fuzzy preference relation.
The membership functions lND
Rp ðXkÞ; p¼1;. . .;q play a role identical to membership functions replacing objective functionsFp(X),p= 1,. . .,qin
solv-ing traditional multiobjective optimization problems [20] on the basis of the Bellman–Zadeh approach to decision making in a fuzzy environment[41,48]. Therefore, it is possible to construct
lNDðX kÞ ¼ min 16p6ql ND Rp ðXkÞ to obtain XND.
If necessary to differentiate the importance of different preferences, it is pos-sible to use lNDðX kÞ ¼ min 16p6q½l ND Rp ðXkÞ kp. ð47Þ
The utilization of(47)does not require the normalization ofkp,p= 1,. . .,q.
It is natural that the use of the second procedure may lead to solutions dif-ferent from results obtained on the basis of the first procedure. However, solu-tions based on the first and third procedures, which have a single generic basis, may at time also be different. At the same time, the third procedure is more preferential from the substantial point of view. In particular, the use of the first procedure can lead to choosing alternatives with the degree of nondominance equal to one, though these alternatives are not the best ones from the point of view of all preference relations. The third procedure can give this result only for alternatives that are the best solutions from the point of view of all fuzzy pref-erence relations.
Taking the above into account, it should be stressed that the fact of the pos-sibility to obtain different solutions on the basis of different approaches is nat-ural, and the choice of the approach is a prerogative of the decision maker. Considering this, all procedures have been implemented within the framework of an interactive decision making system MDMS (developed in C++ in the Builder–Borland environment)[21]. Its flexibility and user-friendly interaction
with a decision maker makes it possible to use MDMS for convincing solving problems of multiobjective choosing alternatives with the use of criteria of quantitative as well as qualitative nature.
It is natural that there are possibilities for constructing other types of pro-cedures for contracting the decision uncertainty regions based on results of sev-eral authors related to processing preference relations, including diverse preference structures. However, the main goal of the paper is to show the pos-sibility and rational ways of solving general problems[41,48]with fuzziness in objective functions and constraints on the basis of modifying traditional opti-mization methods and subsequent contraction of the decision uncertainty re-gions by reducing problems to multiobjective decision making in a fuzzy environment. In particular, it can be considered as a convincing way for devel-oping interactive decision support systems in fuzzy integer programming prob-lems, which would allow intelligent decisions according to the actuation preferences of the decision makers. The necessity of developing these systems was stated by the authors of[28].
7. Conclusions
In this paper, the approach to solving optimization problems formalized within the framework of ‘‘soft’’ models containing fuzzy coefficients in objec-tive functions and constraints has been considered. This approach is associated with modifying traditional mathematical programming methods and consists in formulating and solving one and the same problem within the framework of mutually related models that allows one to maximally cut off dominated alternatives from above and from below as well. The subsequent contraction of the decision uncertainty regions is associated with reduction of the problem to multiobjective choosing alternative in a fuzzy environment that is natural and acceptable in the decision making practice.
The approach has been realized in accordance with the analysis of fuzzy dis-crete optimization models based on modification of the algorithms of disdis-crete optimization. The algorithms are associated with the method of normalized functions, are based on a combination of formal and heuristic procedures, and allow one to obtain quasi-optimal solutions after a small number of steps. Prior to application of these algorithms there is a transition from the models with fuzzy coefficients in objective functions and constraints to equivalent ana-logs containing fuzzy coefficients in objective functions alone.
The results of the paper are of a universal character and can be applied to the design and control of systems and processes of different purposes as well as the enhancement of corresponding CAD/CAM systems and intelligent deci-sion making systems. In practical aspect, the results of the paper are already being used to solve problems of power engineering.
Acknowledgment
This research is supported by the National Council for Scientific and Tech-nological Development of Brazil (Grant No. 550408-2002/9).
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