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Generalized algorithms for solving problems of discrete, integer, and Boolean programming are discussed. These algorithms are associated with the method of normalized functions and are based on a combination of formal and heuristic procedures. This allows one to obtain quasi-optimal solutions after a small number of steps, overcoming the
*NP*-completeness of discrete optimization problems. Questions of constructing so-called “duplicate” algorithms are considered to improve the quality of discrete problem solutions. An approach to solving discrete problems with fuzzy coefficients in objective functions and constraints on the basis of modifying the generalized algorithms is considered. Questions of applying the generalized algorithms to solve multicriteria discrete problems are also discussed. The results of the paper are of a universal character and can be applied to the design, planning, operation, and control of systems and processes of different purposes. The results of the paper are already being used to solve power engineering problems.

Discrete, integer, and Boolean (in the general case, discrete) optimization problems have important applications in many fields [

Theoretical and experimental evaluations, discussed in [

The algorithms discussed in the present paper are based on a combination of formal (the method of normalized functions [

In the process of formulating and solving a wide range of problems related to the design, planning, operation, and control of complex systems, one inevitably encounters different types of uncertainty [

Investigations of recent years show the benefits of applying fuzzy set theory [

From the diversity of discrete optimization problems it is possible to distinguish two comprehensive classes. The first class is associated with the general problem of discrete programming, including integer, Boolean, and discrete programming problems proper. The problems with discrete variables may be reduced to integer or, in the general case, to Boolean models [

The second class of models is related to the problems of combinatorial type. In their solution, an extremum of the objective function is defined on a given finite discrete set A. The totality of objects obtained from A (for example, combinations or permutations) as well as objects obtained as a result of executing logical operations on elements of A [

The combinatorial problems are the most difficult from the computational viewpoint. Their solution is based, in the main, on finiteness of

In this connection, when solving discrete problems, it is important that their formulation and solution should exploit those properties and peculiarities of the problems, which promote their effective solution. Considering this, the desirability of allowing for constraints on the discreteness of in the form of discrete sequences

has been validated in [

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

is met while satisfying

The objective function (2) is interpreted as concave and the constraints (3) are interpreted as convex.

Given the maximization problem (1)-(3), one can formulate a minimization problem. In particular, from the discrete sequences of the type (1) it is necessary to choose parameters that the objective

is met while satisfying

The objective function (4) is interpreted as convex and the constraints (5) are interpreted as concave.

Let us consider the Boolean problem of maximization of

while satisfying

where

The idea of one of popular methods, associated with the class of heuristic methods, may be illustrated by analyzing (6) and (7) for

It allows one to try to maximize (6) on the basis of the largest

When analyzing (6) and (7) for d = 1, the maximization is reached by expending only one type of resources. If d > 1, the optimization process is stopped when a remaining amount of only one of resources is not sufficient for next incrementing any of x_{i},

where t is the optimization step number.

Applying (9), it is possible to transform the constraints (7) to equal conditions. For instance, before the first optimization step we have

Assume that at step t, variable

the respective levels

Then the algorithm of solving the maximization problem (1)-(3), taking into account the results of [

1) The components of the vector

where

For t = 1 we have

2) If

3) The components of the vector

4) If

5) The components of the vector

6) The index

7) We recalculate the current values of the following quantities:

8) If

Commenting the algorithm of solving the maximization problem (1)-(3), it is necessary to indicate that the execution of its operation 1 provides determination of the constraint with the most scarce type of the resource

a convolution:

which maximizes the increment of the objective function per unit normalized resource b. In this connection, the utilization of (14) and (15) is similar to (8). Besides, the execution of operations 2 and 4 is directed at excluding variables that lead to violation of the constraints (3) or to decreasing the objective function of (2).

The problem of minimization (1), (4), and (5) is more difficult than the problem (1)-(3). In particular, in the case of maximization we stop changing the variable x_{i} when at least one of the constraints (3) is violated. At the same time, in the minimization process the optimization is completed on any variable when all constraints (5) are obeyed. Therefore, there is in the case of maximization usually only one “deficient” constraint at each step requiring attention. At the same time, 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 are presented in the following form:

The algorithm of solving the minimization problem (1), (4), and (5), using the results of [

1) The components of the vector

In (19),

where

2) The components of the vector

3) The components of the vector

4) The index

5) We recalculate the current values of the quantities

6) If

7) If

Commenting the algorithm of solving the minimization problem (1), (4), and (5), it is necessary to indicate that the execution of its operation 1 provides 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 that

Numerous comparisons of solutions for diverse types of problems given, for instance, in [

for Boolean and

their convincing agreement. This is also confirmed by numerous results on “good” behavior of the greedy algorithms [

where

The utilization of the “duplicate” algorithms can be considered, in a certain measure, as a guarantee of obtaining optimal solutions. Furthermore, the analysis of one and the same problem on the basis of several algorithms allows obtaining a series of solutions of equal worth, which is important as well.

As an example of the use of the “duplicate” algorithm associated with applying (23) we consider a problem of selecting locations and sizes of capacitors for a distribution network whose scheme and parameters are shown in

The discrete sequence of low voltage capacitor sizes is the following:

It is necessary to select sizes of capacitors which minimize the objective function

while satisfying the following constraints on lower voltage limits:

The solution of (24)-(26), based on applying of the “duplicate” algorithm associated with the use of (23), is the following:

The solution of the problem (25) and (26) with ignoring the constraints (24) on variable discreteness, based on applying the simplex method of linear programming, is the following: x_{1} = 0 kVAr, x_{2} = 34 kVAr, x_{3} = 104 kVAr, x_{4} = 12 kVAr and x_{5} = 54 kVAr. Rounding the obtained values to the nearest discrete ones does not permit one to obtain a feasible solution. Rounding the obtained values to the nearest higher discrete ones provides the following solution: x_{1} = 0 kVAr, x_{2} = 78 kVAr, x_{3} = 156 kVAr, x_{4} = 78 kVAr and x_{5} = 78 kVAr. The corresponding value of the objective function (25) is F = 390 kVAr.

Another “duplicate” algorithm is associated with the results of [

with recalculating

As an additional means for possible improving the efficiency of solutions may serve the formulation and analysis of one and the same problem within the framework of mutually interrelated models (1)-(3) and (1), (4), and (5). Applying this approach, if we have the increasing (decreasing) sequences (1) for the problem (1)-(3), the sequences (1) are to be decreasing (increasing) for the problem (1), (4), and (5). Thus, it is possible to solve one and the same problem “from above” as well as “from below” as well. This approach is fruitful and also serves for solving problems with fuzzy coefficients discussed below.

The described results have a high degree of generality and have been used in solving diverse power engineering problems discussed below.

Although there are diverse formulations of optimization problems with fuzziness (for instance, [

Generalizing (1)-(3), it is possible to construct the problem of choosing parameters from discrete sequences of the type (1) that the objective

is met while satisfying

The objective function of (29) and constraint (30) include fuzzy coefficients, as indicated by the ~ symbol.

Generalizing (1), (4), and (5), it is possible to construct the problem of choosing parameters from discrete sequence of the type (1) that the objective

is met while satisfying (30).

The models (1), (29), and (30) and (1), (31), and (30) generalize the models analyzed, for instance, in [

An approach [

To compare alternatives in accordance with (15) or (21) it is necessary to use the corresponding methods, considered, for instance, in [

If

is the degree of preference

is the degree of preference

The review of techniques which have been developed for ranking of fuzzy numbers can be found in [

The authors of [

There is another approach that is better validated and natural for the practice of decision making. It is associated with the transition to multiattribute choosing alternatives in a fuzzy environment because the application of additional criteria (including the criteria of qualitative character, such as “investment attractiveness”, “flexibility of development”, etc.) can serve as a convincing means to contract the decision uncertainty regions.

Before starting to discuss questions of multiattribute 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 interrelated models:

1) model of maximization (29) with the constraints (30) approximated by (3);

2) model of minimization (31) with the constraints (30) approximated by (5).

When using this approach, solutions dominated by the initial objective function are cut off from above as well as from below to the greatest degree [

Assume we are given a set X of alternatives (from the decision uncertainty region), which are to be examined by q criteria of quantitative and/or qualitative nature. This problem is presented by a pair

where

In (34), R_{p} is defined as a fuzzy set of all pairs of X × X, such that the membership function _{k} weakly dominates X_{l} (X_{k} is not worse than X_{l}) for the pth criterion.

A convincing and natural approach to obtaining matrices R_{p} is presented in [_{p},_{p} are discussed in [

At the same time, fuzzy preference relations are not a unique form of preference representation. For instance, the authors of [

The basic procedures to deal with multiple criteria, when preferences are modeled as a vector R of fuzzy preference relations, considered in [

The indicated procedures have served for developing other techniques: the analysis of alternatives with fuzzy ordering of criteria [

When analyzing multiobjective models (

where L is a feasible region in

The first step in solving the problem (32) is associated with determining a set of Pareto solutions

When analyzing multiobjective problems, it is necessary to solve some questions related to normalizing objective functions, selecting principles of optimality, and considering priorities of objectives. Their solution and, therefore, development of multiobjective methods is carried out in the following directions [

The lack of clarity in the concept of “optimal solution” is the basic methodological complexity in solving multiobjective problems. When applying the Bellman-Zadeh approach to decision making in a fuzzy environment [

When using the Bellman-Zadeh approach, objective functions

solution d is defined as

The use of (36) allows one to get the solution

Therefore, the problem (35) is reduced to search for

To obtain (38), one needs to build

by

for minimized objective functions or

for maximized ones. In (39) and (40),

The construction of (39) or (40) demands the solution of the following problems:

Thus, the solution of the problem (35) on the basis of the Bellman-Zadeh approach demands analysis of

Since

where

Procedures for solving the problem (37) discussed in [

Finally, the existence of additional conditions (indices, criteria or constraints) of qualitative character, defined by linguistic variables [

where

Taking the above into account, the solution of the multicriteria discrete problems is reduced to modifying the generalized algorithms of solving discrete optimization problems discussed above to solve the maxmin problem (38).

Although the results of the present Section do not take into account the uncertainty of initial information, they can be used within the framework of a general scheme of multicriteria analysis under information uncertainty [

The described results have found wide applications in the analysis of diverse problems of power engineering, related to improving reliability, quality, and economical feasibility of power supply. These problems are characterized by an extremely high number of variables and, often, by the impossibility of the adequate analytical description of objective functions and constraints.

The following classes of problems of power systems and subsystems have been solved with the use of the results of the present paper:

・ choice and allocation of means for increasing reliability of power supply in distribution systems in different settings;

・ reinforcement (allocation of reactive power sources, allocation of voltage regulators, and reconduction) of distribution systems in different settings;

・ real-time active power control in power systems and subsystems;

・ real-time voltage and reactive power control in distribution systems in different settings.

As an example, it possible to present the solution of the problem of allocating reactive power sources in distribution systems [

Traditionally, problems of reactive power compensation in distribution systems are associated with the selection of locations, sizes, and types of capacitors to minimize the objective function of an economical nature, while the constraints on upper and lower voltage limits at different load levels are satisfied. However, our experience in solving the problems of reactive power compensation shows that the necessity to simultaneously observe constraints on upper and lower voltage limits at different buses creates essential obstacles. It is not uncommon to face situations when these constraints induce empty feasible regions. These obstacles can be avoided by minimizing the objective function of an economical nature as well as the objective function which reflects a volume of poor quality energy consumption. Besides, if the problem is associated with the determination of capacitor types (fixed or switched), the quantity of objectives should be increased.

Taking this into account, the developed computing platform EPODIAN [

Solution | Economic objective function (BR$∙10^{3}) | Poor quality energy consumption (MWh/year) | Total installed reactive power (kVAr) |
---|---|---|---|

I | 708 | 826 | 600 |

A | 543 | 472 | 630 |

B | 661 | 384 | 1215 |

C | 568 | 396 | 900 |

This network includes 2 feeders with more than 5000 consumers connected to them. The total length of feeders is 729 km . They are modeled by 9660 electrical nodes. In

In this paper, the generalized algorithms for solving problems of discrete, integer, and Boolean programming are discussed. These algorithms are based on a combination of formal procedures (associated with the method of normalized functions) and informal procedures (related to elements of greedy heuristics). The application of the algorithms allows one to obtain quasi-optimal solutions after a small number of steps, thus overcoming the NP- completeness of discrete optimization problems. Questions of constructing so-called “duplicate” algorithms are considered. Their use as well as the formulation and solution of one and the same discrete optimization problem within the framework of mutually related models serve as means for possible improving the quality of discrete problem solutions.

The approach to solving optimization problems, formalized within the framework of “soft” models containing fuzzy coefficients in objective functions and constraints, has been discussed. This approach is associated with modifying traditional mathematical programming methods and, in particular, the generalized algorithms presented in the paper. It is based on solving one and the same problem within the framework of mutually related models to maximally cut off dominated alternatives from above as well as from below. The subsequent contraction of the decision uncertainty regions is associated with reducing the problem to multiattribute choosing alternatives in a fuzzy environment.

It has shown the possibility of rational solving multiobjective discrete problems on the basis of applying the Bellman-Zadeh approach to decision making in a fuzzy environment and modifying the generalized algorithms presented in the paper.

The results of the paper have found wide applications in the analysis of power engineering problems, related to improving the reliability, quality, and economic efficiency of power supply.

This research was supported by the National Council for Scientific and Technological Development of Brazil (CNPq)―grants 305036/2011-4, 307466/2011-6, and the Energy Company of Minas Gerais (CEMIG)―R&D ANEEL Program projects GT480 and D535.

RobertoBerredo,PetrEkel,HelderFerreira,ReinaldoPalhares,DouglasPenaforte, (2015) Generalized Algorithms of Discrete Optimization and Their Power Engineering Applications. Engineering,07,530-543. doi: 10.4236/eng.2015.78049