Two methods for construction of suboptimistic and subpessimistic solutions of the interval problem of mixed-Boolean programming Текст научной статьи по специальности «Математика»

UDC 519.852.6

TWO METHODS FOR CONSTRUCTION OF SUBOPTIMISTIC AND SUBPESSIMISTIC SOLUTIONS OF THE INTERVAL PROBLEM OF MIXED-BOOLEAN PROGRAMMING

Mamedov K. Sh. - Dr. Sc., Professor of Baku State University and head of Department of the Institute of Control Systems of the National Academy of Sciences of Azerbaijan, Azerbaijan, Baku.

Mammadli N. O. - Doctorant, Institute of Control Systems, Azerbaijan National Academy of Sciences, Baku, Azerbaijan.

ABSTRACT

Context. The interval problem of mixed Boolean programming having numerous economic applications is considered. The object of the study was a model of the integer programming.

Objective. Development of methods for constructing suboptimistic and subpessimistic solutions of the mixed Boolean programming interval problem.

Two methods for constructing suboptimistic and subpessimistic solutions of mixed Boolean programming problems with interval initial data are introduced. These methods are based on some economic interpretation of the model considered.

Method. Two methods for constructing suboptimistic and subpessimistic solutions of mixed Boolean programming problems with interval initial data are introduced. These methods are based on some economic interpretation of the considered model. In the first method a criterion of selecting unknowns for assigning values, which is based on the principle of profit maximum for each unit of expenditure is introduced. Since the coefficients of the problem are intervals, two strategies are chosen: optimistic and pessimistic. In the optimistic strategy, the idea of choosing unknowns is used, which corresponds to the maximum ratio of the corresponding maximum profit to the minimum expenditure. And in the pessimistic strategy, the idea of maximum ratio of the minimum profit to the maximum expenditure is used. In the second method, the concept of a non-linearly increasing penalty (price) for using a unit of the remaining resources is introduced, that on the right side is bounded. Taking into account the principles of the above first and second methods, using this concept of penalty (price), methods for constructing suboptimistic and subpessimistic solutions have been developed.

Results. The algorithms for constructing suboptimistic and subpessimistic solutions to the interval problem of mixed Boolean programming are developed.

Conclusions. A software package was developed for constructing suboptimistic and subpessimistic solutions to the interval problem of mixed Boolean programming. A number of computational experiments have been carried out over random problems of various dimensions.

KEYWORDS: an interval problem of mixed Boolean programming, optimistic, pessimistic, sub-optimistic and sub-pessimistic

rt, ri - use of the i-th resource for the optimistic and pessimistic solutions, respectively;

Qj, Q . - the total penalty for using the remaining

^ —j

resources for the unknowns xj for an optimistic and pessimistic solutions, respectively;

fop, fp - upper bounds of the suboptimistic and sub-pessmistic values of the objective function, respectively;

fso, fso, fso.sht, fso.sht, fsp, fsp, fsp.sht, fsp.sht -

suboptimistic and subpessimistic values of the objective function obtained by the 1-st and 2-nd methods (non-linearly increasing penalty) corresponding to the 1-st and 2-nd approaches;

Xso - a suboptimistic solution; fso - a suboptimistic value; Xsp - a suppessimistic solution; fsp - a subpessimistic value;

^2 ^2 ^2 ^2 _

°so, °so, °so.sht, °so.sht, °sp, °sp, °sp.sht, °sp.sht - relative errors of the suboptimistic and subpessimistic values of

solutions, upper and lower bounds, errors, experiments.

NOMENCLATURE

N -the number of all variables, n - the number of boolean variables, m - number of bounders,

I = [1,..., n] - set of indexes of variables, taking boolean values;

R = [n +1, n + 2,..., N ] - set of indexes of variables, taking continuous values;

cj, cj, atj, aij, bi, bi - given positive integers; j* - fixed item; xj - j -th unknown; X - N - dimensijnal vector; X op - an optimistic solution; f op - an optimistic value; Xp - a pessimistic solution; fp - a pessimistic value;

qi, q - a penalty (price) for using the i-th

resource for the optimistic and pessimistic solutions, respectively;

the objective function from the optimistic and pessimistic values obtained by the 1-st and 2-nd methods (non-linearly increasing penalty) corresponding to the 1-st and 2-nd approaches;

kso, kso.sht, ksp, ksp.sht - the number of remaining

continuous variables after the application of the 1-st and 2-nd methods (nonlinearly increasing penalty) with the second approach for construction suboptimistic and subpessimistic solutions, respectively.

INTRODUCTION

At the begining of the mixed Boolean programming problems with interval data, we give some economic interpretation.

Let there are many objects. Some of these objects can be used or ignored, and the rest of the objects can be used to some extent. Suppose for the use of these objects, the resources belonging to a certain interval were distinguished.

If a fixed object is selected for use (or partial use), then the possible costs will be within the specified interval.

In this case, the profit also belongs to a given other interval. It is required to choose for use (or partial use) such objects, the total costs of which do not exceed the allocated resources included in the corresponding intervals, and the total profit will be maximum. Taking the corresponding variables, we obtain a mathematical model of mixed-Boolean programming with the interval initial data. Here the aim is to develop methods for solving of the obtained problem, taking into account the basic properties of the model. In addition, carry out comparative computational experiments to identify the quality of the developed methods.

1 PROBLEM STATEMENT

The following problem is considered:

Z [cj, cJ ] xj + Z [ Cj, cj ] Xj ^ max (1)

j=1 j=n+1

Z[2j,aj] xj + Z [Oj,««] xj < [bj,bi],(i =1 m\ (2)

j = 1 j = n + 1

o < Xj < 1,(j = 1N), (3)

xj = 1 v 0,( j = \~n ),(n < N). (4)

Here it is assumed that

cj > 0, cj > 0, ay > 0, ajj > 0, bi > 0,

bi > 0 (i = 1, m, j = 1, N) are given integers.

We note the following natural conditions for the coefficients of the problem (1)-(4). First, for each

N _ _

conditions must be satisfied Z aij > bi ,(i = 1, m).

j=1

Conversely, if for all these conditions are not satisfied, then the solution X = (1,1,1,...,1) will satisfy the system (2)-

(4) and it will be the optimal solution. On the other hand, if

N

for some fixed i* the condition Z ai,j < b.u,( = 1 m) is

j=1 *

fulfilled then the inequality i* is not a restriction and it is excluded from the system (2). We assume that the above natural conditions are fulfilled for the problem (1)-(4).

This problem is called the problem of mixed-Boolean programming with interval data or simply the interval problem of mixed-Boolean programming. The considered problem (1)-(4) is a generalization of the Boolean programming problems, interval Boolean programming problems, and linear programming problems. In the case of n = 0 we obtain the linear programming problem with interval data, in the case of n = N an interval Boolean programming problem is obtained, in the case of Cj = cj, ajj = aij, bi = bi ,(i = 1, m, j = 1, N) the well-known

Boolean or mixed-Boolean programming problem is obtained.

In the beginning, for problems (1)-(4) we give some economic interpretation. Let there are N objects. From each object n(n < N) you can use or ignore, and for other objects N - n you can use to some extent. Assume that the resources included in the interval [bi,bi ](i = 1, m) are

allocated to use these objects. If the j-th object (j = 1, N) is selected for use (or partial use), then the possible costs enter the interval [aij-, aij ](i = 1, m; j = 1, N), while the profit

belongs to the interval [cj, cj ](j = 1, N).

It is required to choose for use (or partial use) such objects, which total costs did not exceed the allocated resources involved in the interval [bi,bi ](i = 1, m), and the total profit was maximum. Obviously, taking variables

[1, if j-th object is taken xj = I — and

[0, otherwise, (j = 1, n),

0 < xj < 1, (j = n +1, N), then the mathematical model of

the problem will be in the form (1)-(4).

To construct solutions for problem (1)-(4), we have introduced two criteria for choosing the number of unknowns and assigning specific values. Based on these criteria, two methods for constructing solutions have been developed.

2 LITERATURE REVIEW

It should be noted that since all the particular cases of problem (1)-(4) are in NP-complete class, this problem also belongs to the class NP-complete; difficult-solvable [1-2]. As far as we know, the interval problem of mixed Boolean programming has not yet been investigated. In spite of this, some classes of interval integer-programming problems were investigated in [3-6].

In this article, for the problem (1)-(4), the concepts of admissible, optimistic, pessimistic, suboptimistic and sub-

pessimistic solutions are introduced and methods for their solution are developed. These concepts are an extension of the concepts introduced in [7, 8]. It should be noted that a number of approximate and exact algorithms for solving the classical Boolean programming problem are presented in [9, 10]. And in [11] specific methods for construction of a suboptimal (or approximate) solution of Boolean programming problems were developed. The basic principles of interval calculus are presented in [12]. It should be noted that the concepts of a linearly-increasing penalty to construct an approximate Boolean programming solution were introduced in [13]. And in this paper a more powerful criterion is introduced, which we call a nonlinearly-increasing penalty for a more general class of problems.

3 MATERIALS AND METHODS

First we introduce an analog of the concepts introduced by the authors in [7, 8] for a more general class of mixed Boolean programming problems.

Definition 1. N -dimensional vector X = (x1,..., xN ) satisfying the system of conditions

(2)-(4) for VatJ e [atJ, a ] and Vb1 e [bi, b , ],

(i = 1, m; J = 1, N) is called an admissible solution of problem (1)-(4).

From this definition it immediately follows that the concepts of the optimal solution and the optimal value of the function (1) must have a different meaning, in contrast to the known ones. Because it is necessary to ensure that the sum of some intervals is not exceeded from a given specific interval [bi, bi ] and that the

maximum of some other intervals is reached. To this end, we introduce a few more definitions.

Definition 2. An admissible solution

Xop = ((,xOP,...,xNf), iscalled to be an optimistic

solution of problem (1)-(4) if that satisfies the ineN

qualities X -ijx'0P < bi, for Vbi e \b_i,bi ], J=1

(i = 1, m; J = 1, N ), and in this, the value of the function N _

f0p = XcjJ will be maximal.

J=1

Definition 3. An admissible solution Xp = (xp,x2p,...,xN) iscalled to be a pessimistic solution of problem (1)-(4) if that satisfies the inequali-

N _

ties X ayxp < bi

J=1

for

VbI e b,bi ]

(i = 1, m; J = 1, N), and in this, the value of the function

N

fp = X CJxp will be maximal.

J =1

From these definitions it is clear that in order to find the optimistic and pessimistic solutions of problem (1)-(4) it is necessary to solve many problems of mixed-Boolean programming, which is included in the class of NP-complete ones. And this requires unreal time to find the solution of large size problems. Therefore, we have introduced the following concepts of suboptimistic and subpessimistic i.e. approximate solutions of problem (1)-(4) and have developed algorithms for finding them.

Definition 4. An admissible solution

) is called to be a sub-optimistic

solution of the problem (1)-(4) if that satisfies the conditions N _ _ _

X jf < bi for Vb, £ [b1, b, ], (i = 1, m; j = 1, N) and the

j =1

N

value of the function fs0 = X cjxs° will take a large value.

J=1

Definition 5.

An

admissible

solution

Xsp = ( xs

- vsp xsp xsp

x2 N

) is called to be a sub-pessimistic

solution of the problem (1)-(4) if that satisfies the conditions

N

X ayxsp < bt for Vbt e [bi, bi ], (i = 1, m; J = 1, N) and in

J=1

N

this, the value of the function fsp = X CjXsp will take a

j=1

large value.

Theoretical justification of the 1st method.

Using the above economic interpretation of problem (1)-(4) introduced in paragraph 1, we derive the criterion of choosing unknowns for assigning specific values. Let the j -

th object (j = 1, N) be selected for use (or partial use). Then, the necessary expenses should be included in the interval [aij, a,j ](i = 1, m; j = 1, N). In this case, the obtained profit is

included in the given interval [cy-, cj ](j = 1, N). Obviously, the profit per unit of consumption included in the interval [a,j, aij ](i = 1, m; j = 1, N) will be at least

. [Cj, cJ] = [Cj, cj ] __

min

—- (J = 1, N).

i [ay, ay ] max[a;J, ay ]

From here it is directly visible that it is necessary to choose a number J, which is determined from the following conditions:

x = 1V 0,( J = 1, n),(n < N).

max

[C,, cJ]

c,, cJ.]

J max[ai/, aJ ] max^,, aJ,

(4)

(5)

Using the formula (5) and taking into account the above definitions 4 and 5, we obtain the following criteria for choosing the number j of unknowns Xj for

construction of suboptimistic and subpessimistic solutions, respectively:

j* = argmax-

j max a

,. —'J

(6)

Z cjxj + Z Cj xJ ^ max

J=1 J=n+1

n _ N _ _

£ j + £ ajXj < 1, (i = 1, m), J = 1 J = n+1

0 < Xj < 1,(j = 1, N),

(16)

(17)

(18)

j* = argmax-^ . (7)

j max aij v '

i

Thus, to construct a suboptimistic solution, one can use criterion (6), and for a sub-pessimistic solution, (7). In this case, it is necessary to take into account the case in what interval is j* i.e. j* e [1,...,n] = I or

j* e [n +1,n + 2,...,N] = R .

Theoretical justification of the 2nd method (nonlinearly-increasing penalty method). We write the problem (1)-(4) in the following equivalent form

for fixed bi, bi e [b_i, bi ], (i = 1, m):

n _ N _

Z |_Cj, cj J xj + Z [_Cj, cj J xj ^ max (8)

j=1 j =n+1

n r _ j N r _ J _

ZL«j,a>jJ xj + Z ,awJ xj - 1,(i=l,^ (9)

j=1 j=n+1

0 - Xj - 1,( j = 1N), (10)

xj = 1 v 0,( j = 1n),(n - N). (11)

Here aij = aij / bi, a;j- = aij / bi, bi := 1, (i = 1, m; j = 1, N). It is obvious, that cj > 0, cj > 0,

о - a -1, 0 - a^ -1,a ^ 0, aij > 0,(i=1m, j=1N).

Proceeding from problem (8)—(11), we construct the following problem (12)-(15) and (16)-(19) which we call optimistic and pessimistic, respectively. n _ N _

Z cjXj + Z cjx} ^ max, (12)

j=1 j=n+1

n N _

Zaj xj + Z aj xj - 1,(i = 1, m), (13)

j=1 j=n+1

0 - xj - 1,( j = 1N), (14)

x = 1 v 0,( j = 1n),(n - N). (15)

x. = 1 v 0,( j = 1, n),(n - N). (19)

For optimistic problems (12)—(15) below, a method for constructing suboptimistic solutions was developed. Similarly, it is possible to develop a method for constructing subpessimistic of solutions of problems (16)-(19).

The process of constructing of a suboptimistic solution

begins from an admissible solution Xso = (0,0,...,0). Then we accept ю=0 and ri :=0,(i = 1,m). Let some coordinate xj take the value of unit, for example xs° =1. Then on the right-hand side of the system (13) there are resources for further use 1 -aij1 (i = 1, m) .Obviously, these

resources are different. With a view of constructing a final solution containing a larger number of units, i.e. in order to uniform use of the remaining resources, it is necessary to assign a penalty (price) for the use of each resource. It is clear that the penalty (price) should have such property, that at reduction of the remaining resources, the penalty for their use should increase.

If j =1 is selected, then on the right side of the system (13) 1 -a -aij1 (i = 1,m) remains. In the general case, 1 - ri (i = 1, m) is on the right side, where L, = Za,j (i = 1m ffl= {j\xj = 1}.

j'effl

We note that in [13] a penalty is imposed, which increases linearly (proportional to) with decreasing right-hand sides, i.e. ti = ri (i = 1, m) is accepted . And in this

work as a penalty ti (i = 1, m), ti = 1/(1- ri) (i = 1,m) is

acciOljviously, with increasing used resources ri (i = 1, m),

the penalty for using the remaining resources increases nonlinearly, i.e. faster than linear. Therefore, this method will be called the method of non-linearly increasing fine. In

other words lim ti = да . This ti (i = 1, m) provides a high

Li "

price (penalty) for the use of scarce resources.

Note that the penalty in the form ti = —1— (i = 1, m)

1 - l,

was first introduced in the work for Boolean programming problems [11]. In this paper, these concepts were extended for a more general problem of partial Boolean programming

with interval data. It should be noted that in order to fewer use of the remaining smaller resources (right-hand parts of the system (13)), it is possible to increase the penalty t, (i = 1, m) as follows: 1

ti =-- (i = 1, m), here k is a fixed natural

(1 - L,)k

number. Computational experiments have shown that the best results are obtained mainly for k = 2 . Then the

total penalty xs,° = 1 for acceptance will be

q, =^La1JL1 (j = 1, N). i=1

At the same time, the profit per unit of the total penalty for constructing of a suboptimistic solution to take xj = 1 will be Q. = C, / q (j = 1N).

c

maxQ, = = -^ = Q, or

j J q-J q-J. J

c

J»

j* = argmax Q,

(20)

Hje

—j

j q,

1

q i = U (j = 1, N), t, = —^ (i = 1, m). i=1 1 - r'

r, = X aij (i = 1, m), ra = {jx, = 1}.

jera

(21)

In other words, if j* e I, then xs° can take the values

J*

either 0 or 1. If a,, < 1 - r, (i = 1, m), then xs° := 1

—u* j* '

Li := Li +aij„, (i = 1,m), I := I\{j*}, is accepted, and if at least for one i (i = 1, m), a,, > 1 - Li (i = 1, m), xs° := 0 ,

J*

I := I \{j*} is accepted. If j* e R , then the unknown x

J*

must take any values from the interval [0,1]. In this case, if

< 1 -Li

for all i (i = 1, m), then we accept xs° := 1.

Li := Li +aijt, (i = 1,m), R := R \{ j*}. And if at least for one i (i = 1, m) a,* > 1 - Li (i = 1, m), then we accept

xs° = min

J*

1 - r

ia

Obviously, it is necessary to choose xs° = 1, where the number j* is determined from the following criterion:

-1 R := R \ {j*}, L, := L, + a, xj . And for

iJ*

To construct a suboptimistic solution using criterion (20), it is necessary to take into account the circumstances j* e I or j* e R . The use of these circumstances in the construction of solutions are given below. To construct a subpessimistic solution, the process is carried out similarly as mentioned above, using the following criterion:

C, Cj max Q = max=^ = = Q .

the rest j it is accepted xsJ° := 0 , (j e I u R).

Obviously, in this case, at least for one i (i = 1, m), L, = 1 is obtained, the process of constructing a

suboptimistic solution is completed.

To continue the construction process of a suboptimistic

solution Xs° = (xS°,x2°,...,xsN), we find the next number j * from the criteria (6) or (20). Construction process of this solution is completed, if I = 0 and R = 0 .

Note that it is possible to construct a subpessimistic

solution Xsp = (xf, xSp,..., xNp )of problem (16)-(19) similarly to the above, only using criteria (7) or (21).

II approach: Here, in the case of j* e I , the first part

of the I approach still stands, and in the case of j* e R, i.e.

When the unknown xs° should take any values from the

J *

interval [0,1], we proceed as follows: if a,* < 1 - l, for all

(i = 1, m), then we accept xs° := 1.

L, := L, +a,j *,

At the beginning of the constructing process of a sub-pessimistic solution ra = {0} and

L, := 0, (i = 1m), i.e.Xsp = (0,0,...,0) are accepted. Using the criteria (20) or (21) to construct a suboptimistic or subpessimistic solution two approaches were developed, respectively. These approaches to the construct of a suboptimistic solution was presented as follows.

I approach: In the case when for the first time it is

impossible to assign to an unknown xsJ° , (j e R) a

unit , then for this unknown we take the possible fractional values, and for the remaining variables we assign zero.

(i = 1, m) R := R \ {j* }. And if it is impossible to assign a

unit to an unknown xs° , i.e. at least for one i (i = 1, m)

J *

the condition a,, > 1 -r, (i = 1,m) is fulfilled, then for

j e I we accept x, := 0. And for the rest xs° (j e R) we

construct a linear programming problem and solve it by some well-known method. Obviously, the dimension of the obtained problem will be much smaller. These circumstances are confirmed once again in computational experiments.

Finally, we will write an algorithm for constructing of a suboptimistic solution by the nonlinearly increasing penalty method (The algorithm for constructing of a subpessimistic solution is compiled similarly).

Algorithm of the non-linearly increasing penalty method (I approach)

Step 1. Input

N, n, Oy, a,j, Cj, Cj, bi, bi, (i = 1, m; j = 1, N).

- aj — atj

Step 2. Accept b, := bi,a,-,- =-L-, an =-L-,

1 ~'J b, b,

b, := 1,(1 = 1m; J = 1N).

Step 3. Accept

xf,(J = 1N), ra = {0}, l, := 0, (i = and sets I := {1,2,..., n},R := {n +1,n + 2,...N}.

Step 4. Compute t_. = 1/(1 - r_i )(i = 1, m),

m

=^Lafii, J 61 u R. 1=1

Step 5. Compute Q = cn / q (j 61 <u R) and find

J —j

j. from relation j; = arg max Q ..

j j

Step 6. If j. 61 and for all i (i = 1, m) the relation aj; < 1 - r_i is fulfilled, then accept xj := 1, Li := Li +ai7., I := I \{ j.} and pass to step 4.

Step 7. If j. 61 and at least for one i (i = 1,m) the relation aij. > 1 - l, is fulfilled, then accept xj := 0, I := I \ {j.} and pass to step 4.

Step 8. . If j. 6 R and at least for any i (i = 1, m) the relation a j. < 1 - L, is fulfilled, then accept xso := 1, L' := L' +M'i , R := R \{j.} and pass to step

4.

Step 9. If j. 6 R and at least for one i (i = 1, m), relation a j. > 1 - L, is fulfilled, then accept

1 - r,

x = min-

j.

ia

L, := L, +Oj. j, R := R \{j.} and

xso := 0, j 61 u R .

Step 10. Compute fso := ^

CjX.

in the case of j. 61 and at least for one i (i = 1, m), the

relation a 17 > b, is satisfied, then we take for x = 0, and

—(/. j

for j 61 but for all other non-fixed variables j, j 6 R we compose and solve a linear programming problem of smaller dimension. Then we add the obtained solution to the fixed coordinates of the solution.

To estimate errors of the obtained suboptimistic and suboptimistic values from the optimistic and pessimistic values, the original problem is solved as a linear programming problem and corresponding values f p and

f are obtained, respectively. Then the relative errors are estimated as follows:

«1 < f op f s

0 so < -

op

f

02 < f op f 2 0so < -

f

op

0 so. sht < '

f op fs

1

so. sht

f

so. sht < ~

f op fs

so. sht

op

f

op

«ip <

f p fs

sp

fp

0% <

f p f s

sp

fr

«1 <

sp.sht -

f p fsp.sht «2 f p fsp.sht

fp

, «sp.sht < '

fp

j=1

Step 11. Print fso, xso = (x;°, x2o,..., xNo).

Step12. Stop.

Note that, a suboptimistic solution of the problem (1)-(4)is found by the application of the above algorithm . And to construct sub-pessimistic solution, you can use the same algorithm completely, but instead of using the criterion (20), you need to use criterion (21).

It is important to note that the algorithm for constructing suboptimistic and sub-pessimistic solutions by the second method, one can use this algorithm, but

It must be noted, that in development of methods for solving problems (1)-(4), the ideas of work [9-12] were used.

4 EXPERIMENTS

To identify the quality of the developed algorithms in this paper, the programs of these algorithms are compiled and a number of computational experiments were carried out on problems of large dimension. Using the work [11], the coefficients of these problems are chosen as randomly two-digit or three-digit numbers as follows:

I.0 < ay < 99,1 < a-j < 99,1 < cj < 99, 1 < Cj < 99, (i = 1m; j = 1N).

II. 0 < a.. < 999,1 < ajj < 999,1 < c. < 999,

—'J 7 J ' —J '

1 < Cj < 999, (i = i^m; j = 1N).

1 N

3 Z aij

3 j=1

1 N -

3 Z a'j

3 j=1

,(' = 1, m).

Here [z] denotes the integer part of the number z. The results of the computational experiments are presented in the following tables, where for each dimension, 5 different problems were calculated.

5 RESULTS

Table 1 - Experiments with two-digit coefficients (N = 500; n = 300; m = 10)

№ 1 2 3 4 5

f J op 22948.176 22737.032 22307.446 22490.986 21982.270

f1 l—so 22458.800 22234.909 21877.868 21908.750 21598.278

f 2o 22499.599 22244.156 21885.093 21916.256 21607.766

f1 —so.sht 22867.500 22646.917 22175.650 22419.818 21853.667

f2 —so.sht 22875.682 22700.419 22223.520 22438.152 21878.973

0.021 0.022 0.019 0.026 0.017

sL 0.020 0.022 0.019 0.026 0.017

81 so.sht 0.004 0.004 0.006 0.003 0.006

s2 ° so.sht 0.003 0.002 0.004 0.002 0.005

ko 119 109 97 102 103

kso.sht 108 97 93 102 100

fP 14039.384 14183.660 13947.478 13755.646 13584.626

f1 —sp 13949.091 14082.000 13809.471 13609.842 13466.579

f —sp 13949.091 14103.551 13824.456 13611.735 13487.778

f1 _sp.sht 13973.145 14121.324 13877.356 13696.894 13502.867

f2 _sp.sht 13980.549 14133.362 13887.259 13716.408 13507.457

Ssp 0.006 0.007 0.010 0.011 0.009

s% 0.006 0.006 0.009 0.010 0.007

s1 ° sp.sht 0.005 0.004 0.005 0.004 0.006

s2 ° sp.sht 0.004 0.004 0.004 0.003 0.006

k 140 139 128 136 139

ksp.sht 143 138 129 136 135

Table 2 - Experiments with two-digit coefficients (N = 1000; n = 600; m = 10)

№ 1 2 3 4 5

f J op 45911.804 45296.379 44437.319 45092.610 44435.775

f1 —so 44627.593 44136.731 43596.684 44301.667 43647.305

f2 —so 44679.811 44198.527 43610.339 44358.495 43675.640

f1 —so.sht 45828.458 45178.727 44385.097 45017.759 44376.333

f2 —so.sht 45896.308 45217.426 44394.583 45021.415 44397.888

s1so 0.028 0.026 0.019 0.018 0.018

s2o 0.027 0.024 0.019 0.016 0.017

s1 so.sht 0.002 0.003 0.001 0.002 0.001

s2 so.sht 0.000 0.002 0.001 0.002 0.001

kso 199 225 211 213 220

K.sht 183 211 193 196 25

Table 2 continuation

fp 27827.451 28181.955 28069.358 27822.487 27432.328

f1 —sp 27642.257 27889.179 27762.000 27613.937 27139.276

f2 —sp 27642.720 27903.243 27762.000 27630.092 27153.737

f1 —sp.sht 27775.275 28094.640 28007.694 27751.538 27359.500

f2 _sp.sht 27780.474 28127.185 28019.607 27768.076 27362.182

sSp 0.007 0.010 0.011 0.007 0.011

s2p 0.007 0.010 0.011 0.007 0.010

s1 ° sp.sht 0.002 0.003 0.002 0.003 0.003

s2 ° sp.sht 0.002 0.002 0.002 0.002 0.003

K 266 271 269 276 280

Ksp.sht 264 269 260 277 275

Table 3 - Experiments with three-digit coefficients (N = 500; n = 300; m = 10)

№ 1 2 3 4 5

f J op 207813.440 204799.686 201112.681 203689.588 199601.713

f1 _so 203492.135 198601.718 196161.118 197544.935 193740.212

f2 _so 204080.969 198679.476 196336.007 197629.799 194101.881

f1 —so. sht 207116.555 204110.132 200646.886 202920.037 198132.469

f2 —so. sht 207142.991 204348.405 200734.420 202993.009 198240.447

Sso 0.021 0.030 0.025 0.030 0.029

sL 0.018 0.030 0.024 0.030 0.028

ft1 so.sht 0.003 0.003 0.002 0.004 0.007

s2 so.sht 0.003 0.002 0.002 0.003 0.007

kso 120 115 105 110 112

Koh 112 106 95 108 105

fp 141571.166 142834.086 139843.917 138310.900 136465.802

f1 ±-sp 140092.684 141415.095 138470.466 136071.083 134879.129

f —sp 140104.789 141629.340 138705.641 136149.312 134982.791

f1 —sp.sht 140808.383 142269.891 139219.236 137639.519 135628.630

f2 _sp.sht 140849.800 142344.442 139375.527 137692.414 135778.228

Ssp 0.010 0.010 0.010 0.016 0.012

s2sp 0.010 0.008 0.008 0.016 0.011

s1 sp.sht 0.005 0.004 0.004 0.005 0.006

s2 s sp.sht 0.005 0.003 0.003 0.004 0.005

ksp 139 136 133 135 139

Ksp.sht 141 135 131 133 137

Table 4 - Experiments with three-digit coefficients ( N = 1000; n = 600; m = 10)

№ 1 2 3 4 5

f J op 416772.431 407262.286 400559.019 410320.331 402729.978

f1 _so 403111.858 396721.293 390388.890 401217.913 392177.833

f —so 403492.387 396912.010 390687.146 401814.400 392573.481

f1 —so.sht 416005.141 406270.986 399687.804 409494.837 401871.992

f2 —so.sht 416238.934 406405.856 399782.798 410041.511 402047.775

§1so 0.033 0.026 0.025 0.022 0.026

§2o 0.032 0.025 0.025 0.021 0.025

81 ° so.sht 0.002 0.002 0.002 0.002 0.002

82 ° so.sht 0.001 0.002 0.002 0.001 0.002

ko 218 237 214 226 232

195 223 204 214 210

fp 280754.495 284249.634 282822.257 280536.958 277027.700

f1 —sp 278290.868 280818.009 279785.821 278534.584 274033.651

/ —sp 278305.366 280972.875 279895.739 278678.161 274132.534

f1 —sp.sht 279728.741 283343.339 282041.765 279638.956 276478.038

f2 —sp.sht 280001.797 283362.542 282256.459 279670.043 276552.316

0.009 0.012 0.011 0.007 0.011

c2 0.009 0.012 0.010 0.007 0.010

°.sp..sht 0.004 0.003 0.003 0.003 0.002

82 Osp.sht 0.003 0.003 0.002 0.003 0.002

k sp 265 274 265 279 279

k'sp.sht 263 269 261 278 273

6 DISCUSSION As will be seen from the above tables it is clear that the suboptimistic and subpessimistic values obtained by 1 and 2 methods of the objective function differ from each other (non-linearly increasing penalty). Taking into account that in the second approach the apparatus of the linear programming method is being used, which gives the best result both for the 1-st and the 2-nd methods. The more practical method can be considered the 2-nd method corresponding to the 2nd approach. Because this algorithm works faster than the application of linear programming apparatus. The above experiments of the 1-st method show that the relative errors of the suboptimistic and sub-pessimistic values of the objective function from the upper and lower bounds of the suboptimistic and pessimistic values for the 1st method vary within the limits of 0.016-0.033 and 0.006-0.016, and for the 2-nd method 0.000-0.007 and

0.002-0.006 respectively. And this means that using the methods developed in this article, the relative errors are not greater than 3.3%. On the other hand, in order to apply the 2-nd approach for constructing of suboptimistic and sub-pessimistic solutions for problems with two-digit coefficients, for the 1-st method on the average remains 106 and 136 variables out of 500 respectively, 214 and 272 out of 1000 variables respectively, and to construct suboptimistic and subpessimistic solutions for the 2nd method, the remaining number of variables is 100 and 136 of 500, 198 and 269 of 1000 variables. The above experiments once again confirm the efficiency and practicality of the developed methods in this work.

CONCLUSIONS

Proceeding from the above, the following conclusions may be drawn. In this article effective methods for solving problems of mixed Boolean programming with interval data have been developed. As far as we know, the problem of

mixed Boolean programming with interval data has not yet been studied in detail.

To this end, the concepts of optimistic, pessimistic, suboptimistic and subpessimistic solutions were introduced. Using these concepts, two types of methods were proposed. Computational experiments have showed that the method of nonlinearly increasing penalty in most cases exceeds the first method. Therefore, to solve practical problems, it is necessary to solve both methods and choose the best.

ACKNOWLEDGEMENTS

The work was carried out within the framework of the state budget research work of the Institute of Control Systems of ANAS "Development of solutions methods for solving algorithms and software for solving various classes of mixed integer programming problems" (State Registration No. 0101 Az 00736). We note that a particular case of this paper was considered by the authors in [7].

REFERENCES

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Journal Vicislitelnoy Matematiki i Matematiceskoy Fiziki, 1990, Vol. 30, No. 5, pp. 786-791.

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6. Emelichev V. A., Podkopaev D. P. Quantitative stability analysis for vector problems of 0-1 programming, Discrete Optimi-tation, 2010, No. 7, pp. 48-63.

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Received 05.03.2018.

Accepted 18.06.2018.

УДК 519. 852.6

ДВА МЕТОДА ДЛЯ ПОСТРОЕНИЯ СУБОПТИМИСТИЧЕСКОГО И СУБПЕССИМИСТИЧЕСКОГО РЕШЕНИЙ ИНТЕРВАЛЬНОЙ ЗАДАЧИ ЧАСТИЧНО-БУЛЕВОГО ПРОГРАММИРОВАНИЯ

Мамедов К. Ш. - д-р физ.-мат. наук, профессор Бакинского Государственного Университета и зав. отделом Института Систем Управления НАН Азербайджана, Азербайджан, Баку.

Мамедли Н. О. - докторант Института Системных Управлений НАН Азербайджана, Азербайджан, Баку.

АННОТАЦИЯ

Актуальность. Рассмотрена интервальная задача частично-Булевого программирования, имеющая многочисленные экономические применения. Объектом исследования являлась модель целочисленного программирования.

Цель работы. Разработка методов построения субоптимистического и субпессимистического решений интервальной задачи частично-Булевого программирования.

Метод. Введены два метода для построения субоптимистического и субпессимистического решений задач частично-Булевого программирования с интервальными исходными данными. Эти методы основаны на некоторой экономической интерпретации рассмотренной модели.

В первом методе введен критерий выбора неизвестных для присвоения значений, который основан по принципу максимальности прибыли на каждую единицу расхода. Поскольку коэффициенты задачи являются интервалами, выбраны две стратегии: оптимистическое и пессимистическое. В оптимистической стратегии используется идея выбора неизвестных, которая соответствует максимальности отношения соответствующей максимальной прибыли на минимальный расход. А в пессимистической стратегии использована идея максимальности отношения минимальной прибыли на максимальный расход.

Во втором методе введено понятие нелинейно-возрастающего штрафа (цены) за использование единицы оставшихся ресурсов т.е. в правой части ограниченный.

Учитывая принципы вышеуказанных первого и второго методов с использованием этого понятия штрафа (цены), разработаны методы построения субоптимистического и субпессимистического решений.

Результаты. Разработаны алгоритмы построения субоптимистического и субпессимистического решений интервальной задачи частично-Булевого программирования.

Выводы. Составлен программный комплекс для построения субоптимистического и субпессимистического решений интервальной задачи частично-Булевого программирования. Проведен ряд вычислительных экспериментов над случайными задачами различной размерности.

КЛЮЧЕВЫЕ СЛОВА: интервальная задача частично-Булевого программирования, оптимистическое, пессимистическое, субоптимистическое и субпессимистическое решения, верхняя и нижняя границы, погрешности, вычислительный эксперимент.

УДК 519. 852.6

ДВА МЕТОДУ ДЛЯ ПОБУДОВИ СУБОПТ1МСТ1ЧЕСКОГО I СУБПЕСС1М1СТ1ЧЕСКОГО Р1ШЕНЬ ШТЕРВАЛЬНОГО ЗАВДАННЯ ЧАСТКОВО-БУЛЕВОГО ПРОГРАМУВАННЯ

Мамедов К. Ш. - д-р ф1з.-мат. наук, професор Бакинська Державного Ушверситету та зав. вщ-дшом 1нституту Систем Управлшня НАН Азербайджану, Азербайджан, Баку.

Мамедлi Н. О. - докторант 1нституту системного Управлшь НАН Азербайджану, Азербайджан, Баку.

АНОТАЦ1Я

Актуальшсть. Розглянута интервальна задача частково-Булевого програмування, що мае численш економ1чн1 застосу-вання. Об'ектом дослщження була модель цшочисельного програмування.

Мета роботи. Розробка методгв побудови субоптимютичного i субпессимгстичного ршень штервального завдання час-тково-Булевого програмування.

Метод. Введено два методи для побудови субоптимiстичного i субпессимютичного ршень задач частково-Булевого програмування з штервальними вихщними даними. Цi методи засноваш на деякш економiчнiй штерпретацп розглянуто! моделi.

У першому методi введений критерiй вибору невiдомих для присвоення значень, який заснований за принципом макси-мальностi прибутку на кожну одиницю витрат. Оскшьки коефщенти завдання е штервалами, обранi двi стратеги: оптимгс-тичний i песимютичний. В оптимiстичнiй стратеги використовуеться вдея вибору невiдомих, яка вдаоввдае максимальном! вщносини вщповщно! максимального прибутку на шшмальну витрату. А в песимютичнш стратеги використана iдея макси-мальносп вщносини мiнiмального прибутку на максимальны витратi.

У другому методi введено поняття нелшшно-зростаючого штрафу (цiни) за використання одинищ ресурс1в, що залиши-лися тобто в правiй частит обмежений.

З огляду на принципи вищевказаних першого i другого метод1в з використанням цього поняття штрафу (цши), розроб-ленi методи побудови субоптимютичного i субпессимiстичного ршень.

Результати. Розроблено алгоритми побудови субоптимiстичного i субпессимiстiчного ршень штервального завдання частково-Булевого програмування.

Висновки. Складено програмний комплекс для побудови субоптимютичного i субпессимютичного ршень штервально-го завдання частково-Булевого програмування. Проведено ряд обчислювальних експерименпв над випадковими завданнями рiзноl розмiрностi.

КЛЮЧОВ1 СЛОВА: штервальна задача частково-Булевого програмування, оптимютичне, песимiстичне, субоптимгс-тичне i субпессимiстичне ршення, верхня i нижня межi, похибки, обчислювальний експеримент.

Л1ТЕРАТУРА/ЛИТЕРАТУРА

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