Stancu-Minasian, I. M.

Ţigan, Şt.

Institute of Mathematics AS CR, v. v. i.

KybernetikaIoan M. Stancu-Minasian; Ştefan Iulius Ţigan, Ştefan IuliusA stochastic approach to some linear fractional goal programming problemsKybernetika, Vol. 24 (1988), No. 2, 139--149Persistent URL: http://dml.cz/dmlcz/125124Terms of use:© Institute of Information Theory and Automation AS CR, 1988Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This paper has been digitized, optimized for electronic delivery and stamped withdigital signature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czK Y B E R N E T I K A - V O L U M E 24 (1988), N U M B E R 2 A STOCHASTIC APPROACH TO SOME LINEAR FRACTIONAL GOAL PROGRAMMING PROBLEMS* I. M. STANCU-MINASIAN, ST. TIGAN This paper deals with an extension of the goal programming with linear fractional criteria and linear constraints studied by Kornbluth and Steuer [9] (see also [1]). This extension covers the case in which the objective functions or target values are random variables. Three problems are considered. Using some arguments similar to those employed by Ben-Israel, Charnes and Kirby [2] and Stancu-Minasian [15] for the linear case, it is shown that the three stochastic problems can be reduced, under certain hypotheses, to deterministic linear fractional min-max problems with linear or convex constraints. The latter problems are solved by use of the para-metrical method presented in [18] and [19] or the procedures considered in [5] and [10] for the linear fractional case with linear constraints. 1. INTRODUCTION The classical goal programming problem (Charnes and Cooper [4]) is formulated as is shown below. Problem GP: Minimize g(x, y, z) subject to (1.1) Ft(x) - y, + z, = G; for iel = {1, 2 , . . . , r} , (1.2) x e S , y = (yu y2,..., yr) £ 0 , z = (z,, z2 z,) £ 0 , where: (i) Ft: S ~> R, (/' e /) are the objective functions; (ii) G, e R (/' e i ) are the levels (goals, target values) to be reached by the objective functions; (iii) v and z are unknown deviational vectors, whose components measure the deviation upwards and downwards, respectively, of Ft(x) values from the corresponding goal values G, (/' el); * Presented at the International Conference on Stochastic Programming, Prague, September, 15-19 , 1986. 139 (iv) S £ W is the set of feasible solutions to the problem; (v) g: R++ 2 r-> IR + represents a global deviation function depending on the deviational vectors y and z. Ordinarily, the objective function g assumes the form (1-3) g(x,y,z) = ^(aiyi + piZi), iel where <x; and /?,- are intragoal positive weights and specify the relative penalties to be applied for over- or underachievements in G;. However, in this paper we shall consider another form for g(x, y, z) (i.e. (2.9)). The goal programming problem has been investigated and generalized by several authors who gave many efficient solution methods. Thus, for instance, for the case when the objective functions Ft (iel) are linear, Lee [11] suggested a modified simplex algorithm, whereas Ijiri [8] proposed a generalized inverse technique. In [1], Agrawal, Swarup and Garg considered that the objective functions E; (i el) are linear fractional and proposed an algorithm for solving the goal programming problem. This method is based on Swarup's procedure [17] for solving linear frac-tional functional programming problems. Kornbluth and Steuer [9] considered the case when F ; (iel) are linear fractional and either (a) each linear fractional criterion is assigned to its own priority level or (b) all the criteria are at the same level. In case (a), a sequence of linear fractional programming problems has to be solved, whereas in case (b), one has to solve only one linearly constrained programming problem with a nonlinear objective function which is the sum of the linear fractional objective functions. Stancu-Minasian [14] considers also that all the criteria are at the same priority level and a goal objective function of Chebyshev type must be minimized (see, e.g. (2.9)). Peteanu and Tigan [12] examined the case when all the objective functions are linear fractional and have the same priority level and, additionally, the goals G; (i e I) are extended to the interval goals [G;, G"] (i el). It is shown that the interval goal programming problem can be reduced to a min-max linear fractional problem with linear constraints. In [20], Tigan gives an extension of the interval goal programming [12] to include objective functions with inexact data. This extension is a natural link between goal programming, multiobjective programming and inexact programm-ing with set-inclusive constraints, as introduced by Soyster [13]. Stochastic approaches to goal programming are considered by Contini [7] and Chobot [6]. This paper extends goal programming with linear fractional criteria and linear constraints to include the case when the objective functions or target values are random variables. Three stochastic problems are considered and it is shown that these problems can be reduced, under certain hypotheses, to deterministic linear fractional min-max problems with linear or convex constraints. The latter problems can be solved by use of the parametrical methods presented in [18] and [19] or the procedures given in [5] and [10] for the linear fractional case with linear constraints. 140 Let us first replace Problem GP by the following equivalent Problem GPl: Minimize g(x, y, z) subject to (1.4) F;(x) - yt g G ; , iel, (1.5) F;(x) + z ; ^ G ; , i e / , (1.6) x e S , yt^Q, z, > 0 , i e / . This formulation of the goal programming problem will be used in the next sections for the extensions obtained in case when the functions F ; or target values G; (i e I) are assumed to be random variables. Problems amenable to this formulation are met with in many areas. For an application of a stochastic model dealing with production scheduling and investments in a firm, see [6]. It can be easily shown that problems GP and GPl have the same optimal solution set when the function g is of the form (1.3) with positive weights. Moreover, it can be proved that GP and GPl have the same optimal solution set when g(x, y, z) is a strictly increasing function with respect to y and z. 2. RANDOM TARGET VALUES In what follows, we shall assume that the objective functions F ; in Problem GPl are well determined, and the target values G; are independent random variables with known distributions T;(*) (i si). In this situation, we consider that constraints (1.4) —(1.5) are satisfied in a pro-portion of cases or, in other words, with certain given probabilities, rather than always satisfied. Like Ben-Israel, Charnes and Kirby [2] (see also Stancu-Minasian [15]), we con-sider the following stochastic goal. Problem SGP: Minimize g(x, y, z) subject to (2.1) ?{a> | F ;(x) ^ Gi(aj) - z,} ^ Pi, iel, (2.2) P{co | F;(x) < Gi(co) + yt} ^ g, , iel, (2.3) xeS, }>;^Q, Zi^O, iel, where p, and g ; e [0, 1] are lower bounds of these probabilities. The other notation is as in the foregoing section. Let us find the deterministic equivalent of Problem SGP. Let T-'\d) = inf {rj: Tt(t\) ^ 6) . O ^ ^ l , iel, and fr\e) = sup {n: Tin) < 9}, o ^ e ^ l , i el. 141 We rewrite (2.1) as: P{cu | E;(x) ^ G;(co) - z;} = P{« | Gi(co) ^ F,(x) + z;} = = T(E,(x) + z;) > ?( , iel, or, equivalently, (2.1') E;(x) + z ; > Tf1 ^ , - ) , /<=/. Similarly, (2.2) is re-written as (2.2') E;(x) - y, ^ t f x(l - q,), iel . Thus, we have Theorem 1. If G; (i el) are independent random variables with distributions T;( •) (iel), then Problem SGP is equivalent to the deterministic Problem PGD: Minimize g(x, y, z) subject to (2.4) E;(x) + z ; ^ Tf1^,), / £ / , (2.5) Ffa)-y,£ tr\\ -q,), iel, (2.6) x e S , yt£0, z ; > 0 , i e l , Further on, we need supplementary conditions on E; and g and shall show that the stochastic goal problem SGP reduces to a linear fractional min-max problem with linear constraints. Thus, we assume that: (HI) The objective functions are linear fractional: (2.7) F,(x) = SfLLSli, for all xeS, dtx + di0 where c; e R", dt e W, ci0, d,0 e R. (H2) For each i e I: (2.8) dtx + d,o > 0 , for all xeS . (H3) The global deviation function g is of Chebyshev type, i.e. (2.9) g(x, y, z) = max (a ;r ; + jS;z;), (a;, /?; > 0 , iel). iel Such functions were considered, for example, in [12], [14] and [20]. (H4) The set S is polyhedral nonempty and bounded subset of R", defined by (2.10) S = { x e R " | A x = b, x > 0] , where A is a given m x n real matrix and b e Rm. 142 Using these statements, problem PGD becomes Problem PGD1: Find min max (a^ + /?;z;) iel subject to (2.11) C'X + C'°- + z, _: Tr] (p,), / e / , d;x + d;o (2.12) 5_+_J2._ j , , .gf--(i _ , , . ) , i e / f a,x + d;o (2.13) xeS, yt>0, zt = 0 , i e J . Using assumption (H2), we transform constraints (2.H) and (2.12) into the follow-ing constraints: (2.11') (c. - Trj(p.) dt) x + z,.(_ ;x + di0) = - c;o + di0Tr >(p.) , / e / , (2.12') (c, - TrX1 - _) -*i) * ~ * ( -> + tl.'o) _ -c , 0 + di0tr\\ - q.), iel . Therefore, problem PGD1 becomes Problem PGD2: Find min max (a,y; + /?;z;) iel subject to constraints (2.1T), (2.12') and (2.13). The constraints of Problem PGD2 are nonlinear, but we can linearize them by the Kornbluth-Steuer transformation [10]: (2.14) u( = z;(_;x + di0) ; p, = y,(d,x + _ „ ) , i e / . Employing this change of variable, problem PGD2 becomes a deterministic linear fractional min-max problem with linear constraints: Problem PM: Find a..y; + R-M-. min max ;s; d,x + d)0 subject to (c; - TT1^,) _,) x + it, _: - c I 0 + di0Tr\p) , iel, (c; - f f *(l - «,) </,) x - V-, <, - c ; 0 + d.-cffTl ~ 1i). ' ' e l , x e S , M — (MJ, M2, ..., Mr) — 0 , i ' = (i'j, u2, •••, fr) _: 0 . Under assumptions (HI) —(H4), the following theorem is immediate: Theorem 2. If G; are independent random variables and if (x*, _*, u*) is an optimal 143 solution of Problem PM, then (x*, y*, z*) with rf- Pl—, z* = - Ut - , iel, dtx* + di0 d,x* + di0 is an optimal solution to Problem SGP and conversely. Proof. The proof follows from the fact that Problem SGP is equivalent to Problem PGD (Theorem 1), Problem PGD is equivalent to Problem PGD1 or Problem PGD2 by relations (2.7)-(2.9) and Problem PGD2 reduces to Problem PM by the variable transformation (2.14). • 3. RANDOM OBJECTIVE FUNCTIONS In this section, we assume that the target values G; (i e I) in Problem GP1 are well determined, but the objective functions F, are random. We adopt the following assumption: (H5) The objective functions Ft (i e 7) are of the form: (3.1) F i ( x , w ) _ £ _ _ ) _ L ± _ _ H ) d,x + di0 where the components of the vectors (c;, c;o) e W+1 (i e I) are normal independ-dent random variables with expectations c y and variances Oy (i e I,j = 0, 1 , . . . ...,«). We also suppose that assumptions (H2) —(H4) in Section 2 are true. With the notations adopted in Section 2, the stochastic goal programming problem is Problem GPF: Find min max (a,y; + /?,z;) iel subject to (3.2) ?{w | Fi(x, co) = G,- z;} _ p,, iel, (3.3) P{w | F,(x, co) < G, + y j _ q,, iel, (3.4) xeS, yi = 0, zi = 0, iel. Having in view (2.8) and (3.1), constraints (3.2) and (3.3) can be written as (3.5) ?{co | Ci(co) x + cio(co) _ Gt(d,x + di0) - Zi(dtx + dt0)} _ p,, iel, (3.6) P{co | c;(w) x + c,0(co) < G,(d,x + di0) + y,(d,x + d,0)} _ q,, iel, or, equivalently, as (3.7) gfel d»\- 4^±1A^1^Z1^ < rt(i _ pi)i ieI> d°fjxj + 4>yn 144 (3.8) G ' ( d ' * + rf'o) + -V '^'X + dio) - ° V ~ 5 '° _ 0"*(«i) . iel, ( i ^ 2 + 4 r J = I where </>(•) is the probability distribution function of the standard normal variable N(0, 1). Thus, problem GPF is Problem GPF1: Find min max (a^,- + /J.z.) subject to constraints (3.7), (3.8) and (3.4). Employing the change of variable (2.14), problem GPF1 becomes a deterministic linear fractional min-max problem with nonlinear constraints. Problem GPF2: Find Cl:V: + BlUi mm max iel dtx + di0 subject to (3.9) G i ^ ± j y _ Z L M i . r ctx-cjo^ ^ - 1 ( 1 _ pt) f i e I t {iofjX} + of0y» J = I (3.10) - f e j l ^ o ) + p ' - ^ - ^ o ^ ^ - i ( g < ) , iel (£*? ,* 2 + *2o)1/a y = i (3.11) x e S , M , _ 0 , t ' i _ 0 , ; ' e / . It is reasonable to choose pt > 0-5 and qt > 0-5 such that the constraints of Problem GPF2 define a convex set. Hence, we have Theorem 3. If the objective functions E, (iel) are random of the form (3.1) and assumptions (H2)-(H4) are true and if (x*, u*, v*) is an optimal solution of Problem GPF2, then (x*, y*, z*), with ,,* .,* , iel, dfx* + diQ dtx* + di0 is an optimal solution to Problem GPF and conversely. The proof is similar to that of Theorem 2 and so, it will be omitted. 145 4. SIMPLE RANDOMIZATION OF OBJECTIVE FUNCTIONS Throughout this section, we shall adopt the following assumptions for Problem GPF: (H6) The objective functions Ft (iel), given by (3.1), are simply randomized, i.e., (4.1) c;(a>) = c\ + ti(co) c'l, ci0(co) = c'i0 + t_(co) c"i0 , i el, where c'_, c'l e W ( i e / ) are constant vectors, c'i0, c"i0 e 03 ( i e / ) are scalar constants and /,(•) (i el) are random variables on a probability space (Q, K, P) with continuous and strictly increasing distribution functions T(-) (i el). We note that the stochastic programming problems with simple randomization were examined by Bereanu [3] in the linear case and Stancu-Minasian [15], Stancu-Minasian and Tigan [16], [21] for some special classes of nonlinear minimum-risk problems. (H7) The global function g assumes the form (4.2) g(x, y, z) = max (a,->>, + /J,z,) iel (H8) d.x + di0 > 0, for all i el and for all x e S. (H9) c"x + c'_'0 > 0, for all i e / and for all x e S. Hence, problem GPF becomes Prob lem GPSR: Find min max (u._y_ + /?;Z,) iel subject to (c\ + tjto) c'l) x + (c'i0 + ti(co) c"i0) (4.3) P L (4.4) P L d_x + di0 [c\ + tjm) c'l) x + (c'i0 + t_(aj) c"i0) è G_ - zЛ _> Pì, iel, < G_ + yX ^ qt, iєl, dtx + di0 (4.5) xeS, J , _ : 0 , z_ _: 0 , iel. Let us find the deterministic equivalent of Problem GPSR. Under assumptions (H6), (H7), (H8) and (H9), we transform constraints (4.3) and (4.4) into the following constraints: (4.6) P L J tjm) _> G ^ X + d^ ^ f e + ^ ~ C'X ~ C'<°\ * p, , iel, [ c'!x + c"i0 j (4.7) P L | ti(co) < °^iX ± d'^-+ yJ&*± d ^ - C''X ~ CH £ q,, iel, { c'[x + c"i0 J 146 (4. Gj(dtx + di0) - Zj(djX + di0) - c'tx - c'i0 < j . - ! ^ _ -.•) ( e / (4.9) g j f o + ^ l i ^ i i J l M z i ^ ~ C'Q ?> Tr\qi), iel. c'[x + c"i0 According to assumption (H9), constraints (4.8) and (4.9) become (4.10) (Gtdt - c \ - rr\\ - Pt) c",) x - z,(dtx + dt0) £ ^ • o + 4 7 7 1 ( l - - P . O - ^ o G , , ie/, (4.11) (Gtdt - c't - Trl(qt) c'!) x + yt(dtx + dt0) L> £ -Jo + c J o T f 1 ^ . ) - d,oG,-, r e / . Therefore, problem GPSR can be formulated as follows: Problem GPSR1: Find min max (atyt + (j(zt) iel subject to constraints (4.5), (4.10) and (4.11). By the change of variable (2.14), Problem GPSR1 becomes: Problem GPSR2: Find atv, + B,u, nun max --iel dtX + d;0 subject to (4.12) (Gtd, - c't - rr\l - Pi)c';)x - ut g ISCJO + ct0Tr\\ - Pi)-dt0Gt, iel, (4.13) (G,dt - c't - Tr'(qt) c"t) x + vtZ c't0 + c"i0Tr\q) - di0Gt, iel, (4.14) xeS, w , ^ 0 , vt^0, iel. We have Theorem 4. If assumptions (H4) —(H9) are true, and if (x*, w*, p*) is an optimal solution of the min-max problem GPSR2, then (x*, y*, z*), where v, = , 2* = , ' e / , d,x* + di0 dtx* + di0 is an optimal solution of the stochastic goal programming problem GPSR, and conversely. 147 5. CONCLUSIONS Three classes of stochastic goal programming problems were considered. Determin-istic linear fractional min-max problems with linear or convex constraints (as in the case of Problem GPF2) were obtained for each of these problems. These deterministic problems can be solved by use of the parametrical method presented in [18], [19] or the procedures given in [5], [10] for the linear fractional case with linear constraints. ACKNOWLEDGEMENTS The authors are grateful to the referees for their valuable comments which improved the quality of the presentation of the paper. (Received January 3, 1987.) R E F E R E N C E S [1] Usha Agrawal, K. Swarup and K. C. Garg: Goal programming problem with linear frac-tional objective function. Cahiers Centre Etudes Rech. Oper. 26 (1984), 1—2, 33—41. [2] A. Ben-Israel, A. Charnes and M. J. L. Kirby: On stochastic linear approximation problems. Oper. Res. 18 (1970), 3, 555-558. [3] B. Bereanu: On stochastic linear programming, I. Distribution problems: A single random variable. Rev. Roumaine Math. Pures Appl. 8 (1963), 4, 683 — 697. [4] A. Charnes and W. W. Cooper: Management Models and Industrial Applications of Linear Programming, Vol. 1, John Wiley, New York 1961. [5] A. Charnes and W. W. Cooper: Goal programming and multiple objective optimizations, Part I. European J. Oper. Res. 1 (1977), 39 -54 . [6] M. Chobot: A stochastic approach to goal programming (in Slovak). Ekonom.-mat. obzor 9 (1973), 305-319. [7] B. Contini: A stochastic approach to goal programming. Oper. Res. 16 (1968), 3, 576—586. [8] Y. Ijiri: Management Goals and Accounting for Control. North Holland, Amsterdam 1965. [9] J. S. H. Kombluth and R. E. Steuer: Goal programming with linear fractional criteria. European J. Oper. Res. 1 (1981), 58 -65 . [10] J. S. H. Kornbluth: Max-min programming with linear fractional functions. Algorithms and examples. In: Essays and Surveys on Multiple Criteria Decision Making (Mons, 1982 P. Hansen, ed.). (Lecture Notes in Econom. and Math. Systems 209.) 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The distribution function of the optimal value. Anal. Numer. Theor. Approx. 13 (1984), 2, 175-187. [17] K. Swarup: Linear fractional functional programming. Oper. Res. 13 (1965), 6, 1029— 1036. [18] S. Tigan: Sur une methode pour la resolution d'un probleme d'optimisation fractionnaire par segments. Anal. Numer. Theor. Approx. 4 (1975), 1, 87—97. [19] S. Tigan: A parametrical method for max-min nonlinear fractional problems. Itinerant Seminar on Functional Equations, Approximation and Convexity, Cluj —Napoca Univ., 1983,175-194. [20] S. Tig a n : Goal programming with inexact data. Seminar on Optimization Theory, Cluj — Napoca Univ., Report no. 5, 1985, 95-106. [21] S. Tigan and I. M. Stancu-Minasian: The stochastic max-min problem. Operations Research Verfahren5/(1984), 119-126. Dr. I. M. Stancu-Minasian, Laboratories of the Department of Economic Cybernetics — The Academy of Economic Studies, Calea Dorobanti 15—17, 71131 Bucharest. Romania. Dr. S. Tigan, Territorial Computing Center, Sir. Republicii 107, 3400 Cluj— Napoca. Romania. 149 

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A stochastic approach to some linear fractional goal programming problems

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A stochastic approach to some linear fractional goal programming problems

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Institute of Mathematics AS CR, v. v. i.

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