Keywords: large-scale problem, dimension reduction, multicriteria programming, Pareto-set, dynamic programming.

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1 Advanced Mathematical Models & Applications Vol.3, No.3, 2018, pp REDUCTION OF ONE BLOCK LINEAR MULTICRITERIA DECISION-MAKING PROBLEM Rafael H. Hamidov 1, Mutallim M. Mutallimov 2, Khatin Y. Huseynova 1, Rufana R. Javadzada 1 1 Faculty of Applied Mathematics and Cybernetics, Baku State University, Baku, Azerbaijan 2 Institute of Applied Mathematics, Baku State University, Baku, Azerbaijan Abstract. This paper analyzes one large-scale multicriteria linear programming problem with block-triangular shape matrix A of constraints. The matrix A is taken as M-matrix. Its all diagonal blocks are also assumed to be M-matrix. We construct procedure to reduce the dimension of problem, preserving the initial structure as it was. Such property of procedure allows to use existence efficient procedures as an auxiliary ones in developing of decision making to have finally solution. Suggested reduction scheme allows one to treat information we deal with in part. We illustrate this process on a numerical example. It helps to master easily a reduction algorithm and to have an evident idea of its efficiency. We also consider linear dynamic multicriteria problem with a phase vector having large dimension and show applicability of reduction algorithm in this case. Keywords: large-scale problem, dimension reduction, multicriteria programming, Pareto-set, dynamic programming. AMS Subject Classification: 46A22, 90C26. Corresponding author: Mutallim M. Mutallimov, Institute of Applied Mathematics, Baku State University, Z.Khalilov 23, Baku, AZ1148, Azerbaijan, mmutallimov@bsu.edu.az Received: 08 October 2018; Revised: 26 October 2018; Accepted: 16 November 2018; Published: 28 December Introduction The following multicriteria problem of linear programming is considered: Ax b, x 0, Cx max, (1) where A = a ij is the matrix of dimension (n n), C = c ij is the matrix of dimension (k n), b is n dimensional vector column and x- unknown n-dimensional vector column. A has a property: a ii > 0, i = 1,..., n, a ij 0, i, j = 1,..., n where i j. All coordinates of b are non-negative. It is supposed that there is A 1 and her elements are non-negative. The matrix A specified properties is called a M-matrix (Gal et al., 2013). In addition we impose the following condition on (1): A has block-triangular shape and all diagonal blocks of A are M-matrix. A reduction algorithm of a problem (1) is developed. The purpose of the algorithm is to exclude at the same time a part of the x i variables and conditions (Ax) i b i from (1) and to receive a new problem of smaller dimension and with the same Pareto set as in (1). It is shown that the structure of the problem (1), after a reduction, remains as in (1). It allows us to use, if necessary, the effective algorithms developed in Belen kii (1968), Meerov (1986) when k = 1 for the reduced problem. Reduction process will be organized in part and begins with the last diagonal block. The explanation is offered to expediency of the organization of reduction process for this scheme. It allows to treat in part of information in reduction process and to arise a number of problems we can meet because of large dimension in process of the solution of 227

2 ADVANCED MATH. MODELS & APPLICATIONS, V.3, N.3, 2018 the problem (1). One reduction algorithm without when A is not triangular shape is considered in Hamidov et al. (2017). At first on a numerical example we illustrate executions of separate steps of an algorithm. It helps to master easily a reduction algorithm and to have an evident idea of its efficiency. Then the general scheme of an algorithm is offered. After then, the linear dynamic multicriteria problem, with a phase vector, having large dimension is considered. Such problem arises, for example, by optimization of oil production in the elastic mode, maximizing full profit (Meerov, 1986). After finite-dimensional approximation according to the scheme from Meerov (1986) the problem becomes as problem (1). Then the offered reduction algorithm can be called reduction algorithm for considered class of problems of multicriteria programming in functional space. 2 Problem definition Problems in decision-making most commonly arise from the analysis of mathematical models of practical situations. A variety of practical situations of decision-making as a rule are considered in their mathematical analysis. It is the reason of lack of uniform reception of the decision to use him for each case. However, there is a number of fundamental properties that the seeking solutions possess, independently, in what way it is obtained. In many decision making problems one of such properties is a property of efficiency. It assumes that decision-making process should be organized in Pareto optimum set. However, for problems of large dimension, the organization of decision-making process considerably becomes complicated. Therefore if there is an opportunity, at first it is reasonable to reduce a problem and then to organize search of her decision. In this work such opportunity for following multicriteria linear programming with block triangular shape matrix is considered: i 1 x i A i x i H ik x k b i, x i 0, i = 1, 2,..., m, (2) k=1 c 1r x 1 + c 2r x c mr x m max, r = 1,..., l. (3) Here A i,h ik is a square (n n)- matrices with non-negative elements (A i 0, H ik 0), b i, x i is n dimensional column vectors and b i 0, i = 1, 2,..., m; c ir is n dimensional row vectors, i = 1, 2,..., m, r = 1,..., l Denote A i = E A i, i = 1, 2,..., m (E is n dimensional unit matrix) and problem (2), (3) is representable in block-matrix form:. Ax = Ā H 21 Ā H 31 H 32 Ā H m 11 H m 12 H m Ā m 1 0 H m1 H m2 H m3... H m m 1 Ā m x 1 0,..., x m 0, Cx = C 1 x 1 + C 2 x C m x m max. x 1 x 2 x 3... x m 1 x m b 1 b 2 b 3. b m 1 b m = b, Here C is (l nm) matrix, C i is (l n)-matrices, i = 1, 2,..., m. Note that an assumption of identical dimension of all vectors of x i, diagonal blocks A i and vectors of b i, i = 1, 2,..., m is not necessarily. For convenience of treatment it is assumed. We take that matrices A i, i = 1, 2,..., m are M- matrices, i.e. A i 1 exists, A i 1 0, i = 1, 2,..., m, and their non-diagonal elements aren t positive. The last condition is satisfied automatically as a result of the conditions A 0 and H ik 0. It is easy to check that, from A i 1 0, i = 1, 2,..., m follows also a condition of A 1 0, i.e. a matrix A becomes M-matrix. 228

3 R.H. HAMIDOV et al.: REDUCTION OF ONE BLOCK LINEAR MULTICRITERIA DECISION-MAKING... 3 Auxiliary facts Denote X = { x R n m Ax b, x 0 }. Let x 0 X and there exists j 0 such that c ij 0 0, i = 1, 2,..., n and (Ax 0 ) j 0 < b j 0. Then the following statement takes place. Statement 1. If l i=1 c ij 0 > 0, then x0 is non-efficient solution (dominated solution (Podinovsky & Nogin, 1982)) and if l i=1 c ij0 = 0, then there exist vector estimation which is not worse in Poreto sens, than Cx 0 and (Ax, ) j 0 = b j 0. Proof : The j 0 -th diagonal element of the matrix A are positive, and other elements of j th column of a matrix A are non-negative. Then it is possible to find number α > 0 and x Xx i = x 0 i, i j0, x j = x 0 j 0 + α, (Ax ) j 0 = b j. Then Cx = Cx 0 + αc j 0 Cx 0. If C j 0 0 we have Cx Cx 0, i.e x 0 can t be effective. Denote G = {j c j 0}. Let G and j 0 G. Statement 2. We eliminate variable x j 0 from variables involved in criterion functions (Cx) i max, i = 1,..., k expressing x j 0 trough the others of variables in the equation (Ax) j 0 = b j 0. Then if among other coefficients of a matrix C exist such which change its value, then such change can happen only by means of positive increments. The trustiness of statement can easily be checked. From statements 1,2 it follows that, if C j 0, j G, it is possible to exclude all variables from x j, j G(3), (4) by means of the system of the equations (Ax) j = b j, j G, preserving at the same time the set of all effective estimates. Thereby we eliminate all pare conditions (Ax) j = b j, x j, j G from further considiration. Elimination of the variables x i, i G from criterion function increase (don t reduce) coefficient of the other variables and we have an opportunity to make a new nonempty set G for again received new problem. Thus process we can continue until G. Below we will show one effective variant of a reduction of a problem (2), (3). At first, we illustrate this variant on the numerical example. 4 Numerical example We will consider problem (2), (3) with the following numerical data: 9x 1 x 2 x 3 3, x 1 + 9x 2 x 3 3, x 1 x 2 + 9x 3 3, x 1 x 2 x 3 + 6x 4 x 5 x 6 3, x 1 x 2 x 3 x 4 + 6x 5 x 6 3, x 1 x 2 x 3 x 4 x 5 + 6x 6 3, (4) x 1 x 2 x 3 x 4 x 5 x 6 + 3x 7 x 8 x 9 3, x 1 x 2 x 3 x 4 x 5 x 6 x 7 + 3x 8 x 9 3, x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 + 3x 9 3, x 1 0, x 2 0, x 3 0, x 4 0, x 5 0, x 6 0, x 7 0, x 8 0, x 9 0, 229

4 ADVANCED MATH. MODELS & APPLICATIONS, V.3, N.3, , 65x 1 1, 5x 2 2, 7x 3 6, 5x 4 + x 5 1, 5x 6 4x 7 x 8 + 6x 9 max, (5) x 1 x 2 x 3 x 4 7x 5 x 6 2x 7 + x 8 + x 9 max. The problem (4), (5) has three diagonal blocks. We will reduce it on blocks beginning from the last one. First step. We take the last three columns of the matrix of coefficients of the criterion function Cx and check whether there is among them a column with non-negative coordinates. The last column of a matrix of C, i.e. coefficients of variable x 9 has such property. Define = {9} =. We eliminate variable x 9 by using the equation (Ax) 9 = b 9. As a result, new coefficients of variable x 8 in criterion functions become non-negative, i.e. the set G extends and we have G = {8, 9}. Now at the same time we eliminate the variables x 8, x 9 from the variables in criterion functions. However this time there is no new non-negative column from coefficients of the remained variables, corresponding to the third blocks. Such variable in our case is only x 7. Then we exclude the variables x 8, x 9 from a condition (Ax) 7 b 7 and remember it. After elimination x 8 x 9 we come to the following problem with two diagonal blocks A 1, A 2 : 9x 1 x 2 x 3 3, x 1 + 9x 2 x 3 3, x 1 x 2 + 9x 3 3, x 1 x 2 x 3 + 6x 4 x 5 x 6 3, (6) x 1 x 2 x 3 x 4 + 6x 5 x 6 3, x 1 x 2 x 3 x 4 x 5 + 6x 6 3, x 1 0, x 2 0, x 3 0, x 4 0, x 5 0, x 6 0, 0, 15x 1 + x 2 0, 2x 3 4x 4 + 3, 5x 5 + x 6 1, 5x max, (7) 0 x x x x 4 6x x 6 + x max. New problem (6), (7) preserves initial structure of problem (4), (5). Second step. Now we carry out reductions concerning the second block A 2 observing at the same time to the rule of the first step. We have G = {6}. We eliminate variable x 6 in (7) by means of the equation (Ax) 6 = b 6. However after elimination x 6 we don t have new non-negative columns from the coefficients x 4, x 5. Therefore we exclude the variable x 6 from the conditions (Ax) 4 b 4, (Ax) 5 b 5 and remember them. We come to the problem: 9x 1 x 2 x 3 3, x 1 + 9x 2 x 3 3, x 1 x 2 + 9x 3 3, (8) x 1 0, x 2 0, x 3 0, 1 60 x x x x x x max, (9) 4 0 x x x x 4 6x 5 + x max. 230

5 R.H. HAMIDOV et al.: REDUCTION OF ONE BLOCK LINEAR MULTICRITERIA DECISION-MAKING... Third step. G = {2}. We eliminate x 2 from (9) by using the equation x 1 + 9x 2 x 3 = 3 and after then the column from new coefficients of x 3 becomes non-negative. Then at the same time we eliminate x 2 and x 3 from criterion functions. As a result the first column of coefficients x 1 in criterion functions becomes non-negative. Now we eliminate variables x 1, x 2, x 3 from criterion functions by setting x 1 = x 2 = x 3 = 3 7. In this case there is no condition to be remembered. After three steps problem (4), (5) gets a form: 33x 4 7x 5 7x 7 5, 7x x 5 7x 7 5, 2x 4 2x 5 + 2x , x 4 0, x 5 0, x 7 0, x x x max, 0 x 4 6x 5 + x max. The condition of new the problem is formed by remembered conditions of (3), (4) after elimination all of variables x 1, x 2, x 3, x 6, x 8, x 9. Remark: If to begin reduction process not from the last diagonal block, and other such block, then we wouldn t gain that effect as a result of reduction of the problem (4), (5) as it took place. All variables which to be subject to elimination when we begin with the last block, aren t present in the higher blocks. Therefore we have an opportunity to process initial information in part to reduce it. It is very important when reducing of the large scale problems. 5 Description of an problem reduction Step 1. We take the last diagonal block of problem (2), (3) and set the problem: A m x m b m, x m 0, (10) C n x n m. (11) We reduce problem (10), (11) according to the scheme from Hamidov et al. (2017): a) define a set: G = {i C m i 0, i = 1,..., n}, b) we eliminate all variables x n i, i G from the criterion functions and then check whether appears among again received columns in criterion functions non-negative. In positive answer, we eliminate all variables corresponding to these columns. If answer is negative, we eliminate from all conditions (Ax) i b i of a problem (2), (3) the conditions by which the variables x i are eliminated from criterion functions.we also remember other constraints of the problem (2), (3) containing rows of a matrix A m after elimination of variables. As a result we have a problem like (2), (3). All the blocks constraints of a new problem remain as in initial case. Only the matrix of criterion functions changes. If elements of matrix of new criteria differ from the corresponding element of a former elements of criteria matrix, following a statement 2, we can say that the new value will be strict larger than old values. Thus the chance of emergence of new non-negative columns in a matrix of criterion functions increases. However it can t be said if we begin process of a reduction with the first block A 1. Step 2. We repeat step 1 for again received problem. Step 3. Process comes to the end when G =. 231

6 ADVANCED MATH. MODELS & APPLICATIONS, V.3, N.3, Reduction of a linear multicriteria problem in dynamics Let L n 2 [0, T ] be the space of measurable of n-dimensional vector-functions, integratable with a square. C i (t), i = 1,..., l, b (t), x (t) L n 2 [0, T ]. Ci (t), i = 1,..., l, b(t) are given, and x (t) is unknown. A (t), H(t, τ) are the matrix functions of dimension (n n) and all components of a matrix A (t), H(t, τ) and a vector of b (t) aren t negative. Let F [x(t)] = A (t) x (t) + t 0 H (t, τ) x (τ) dτ and the operator F maps space of L n 2 [0, T ] into itself. Let X be the set of all feasible solution of system: x (t) F [x(t)] b (t), x (t) 0, 0 t T. (12) All equalities and inequalities are carried out for almost all t [0, T ]. The following problem is considered: T T J = (J 1,..., J k ) = C 1 (t) x (t) dt,..., C k (t) x (t) dt max, (13) 0 0 when x (t) X. The problem such as (13) arises, for example, in problem of optimization of oil production in the elastic mode (Meerov, 1986), maximizing full profit (Meerov, 1986). Following Meerov (1986) we approximate the problem (13) and we have: n C j i xi max, j = 1,..., k, i=1 i 1 x i A i x i h H ij x j b i, x i 0, i = 1,..., N, j=1 N = 2 m, h = T/N and b i = min { b i (t) } {, C j i = min } C j i (y). H ij = min {H (t, τ)} when (i 1) h t ih, (j 1) h τ jh, A i = min {A(t)}, when (i 1) h t ih. For the measurable function f(t) the minimum is understood as an exact top side of all numbers µ, for which µ µ (t) at almost all t. In (Meerov, 1986) it is proved that the optimal solution (14) when j = 1 tends to an optimal solution of (12), (13) in the following sense. Let x m = (x m 1,..., xm N ) an optimal solution of a (14) and xm (t) step-like function with value of x m i when (i 1) h t ih. Then x m (t) monotoncially tends to an optimal solution of (12), (13) in the norm L 2. A sufficient condition for existence of an optimal solution of (12), (13) when l = 1 is existence (E A) 1 0 (Meerov, 1986). We will assume that this condition takes place. Then the set of Y R k of all vector estimates of (12), (13) will be limited, closed and convex set. Then justice of the following statement follows from S. Karlin s theorem (see (Karlin & Gillis, 1960)) Statement 3.The set of effective points of (14) approximates the set of all effective points of (12), (13) and the accuracy of approximation depends on number N. It follows from statement 3 that the suggested reduction algorithm for (2), (3) becomes also the reduction algorithm for (12), (13). 7 Conclusion In this paper we considered one large- scale multicriteria linear programming with block-triangular shape matrix of constrains. We developed dimension reduction procedure by using the special structure of constraints. As a result we have the problem with the same structure as it was but with less variables and conditions. We also show the applicability of reduction algorithm to one linear dynamic multicriteria decision making problem. Illustration of the algorithm is given on a numerical example. (14) 232

7 R.H. HAMIDOV et al.: REDUCTION OF ONE BLOCK LINEAR MULTICRITERIA DECISION-MAKING... References Belen kii, V.Z.E. (1968). Problems of mathematical programming which have a minimal point. Doklady Akademii Nauk, 183(1), (in Russian). Gal, T., Stewart, T. & Hanne, T. (Eds.). (2013). Multicriteria decision making: advances in MCDM models, algorithms, theory, and applications (Vol. 21). Springer Science & Business Media. Hamidov, R., Mutallimov, M., Amirova, L. (2017). Application of reduction in one problem of decision-making, NAUPRI, 7, 6-12 (in Russian). Karlin, S. & Gillis, J. (1960). Mathematical methods and theory in games, programming, and economics. Physics Today, 13, 54p. Meerov, M.V. (1986). Research and optimization of multiply connected control systems, Nauka, Moscow, 1986 (in Russian). Podinovsky, V.V, Nogin, V.D. (1982). Poreto-optimal solutions of multicriteria problems. Moscow, Nauka (in Russian). 233

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