A simple method for obtaining weakly efficient points in multiobjective linear fractional programming problems

European Journal of Operational Research 126 (2000) 386±390

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Theory and Methodology

A simple method for obtaining weakly ecient points in multiobjective linear fractional programming problems Boyan Metev *, Dessislava Gueorguieva Institute of Information Technologies, Bulgarian Academy of Sciences, Acad. G. Bonchev str., bl. 2, 1113 So®a, Bulgaria Received 7 July 1998; accepted 23 March 1999

Abstract The properties of linear fractional functions facilitate the usage of a well known scalar optimization problem that gives weakly ecient points. Separately, the weak eciency can be detected with another scalar optimization test based on the same properties. The numerical estimation of the nadir vector is considered as a possible application. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Fractional programming; Multiobjective decision making; Reference point method

1. Introduction The mathematical optimization problems with a goal function that is a ratio with a linear numerator and a linear denominator have many applications: in ®nance (corporate planning, bank balance sheet management), in marine transportation, water resources, health care and so forth (Steuer, 1986). In addition, if the constraints are linear we obtain the linear fractional programming (LFP) problem. It is de®ned as follows:   p…x† ; max f …x† ˆ q…x† …1† n s:t: x 2 S  R ; *

Corresponding author.

where p(x) and q(x) are linear functions and the set S is de®ned in the following way: S ˆ fx j Ax ˆ b; x P 0g: Here A is a real valued …m  n† matrix, b 2 Rm . We suppose that S is a nonempty bounded polyhedron. Many authors have proposed algorithms for solving problem (1), for example: Charnes and Cooper (1962), Martos (1975), Wolf (1985) and others. Comparative investigations of such algorithms can be found in Arsham and Kahn (1990) and Bhatt (1989). Additional information concerning especially the `bad points' is given in the paper of Verma et al. (1989). A point x1 2 S is called a bad point if f …x† ! 1 when x ! x1 . A complete simplex type algorithm for solving

0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 2 9 8 - 2

B. Metev, D. Gueorguieva / European Journal of Operational Research 126 (2000) 386±390

problem (1) is presented in Arsham and Kahn (1990). Bazaraa and Shetty (1979) have shown that the goal function in (1) has several important properties ± it is (simultaneously): pseudo convex, pseudo concave, quasi-convex, quasi-concave, strict quasi-convex and strict quasi-concave. This means that the point that satis®es the Kuhn± Tucker conditions for the maximization problem gives the global maximum on the feasible set. In addition, each local maximum is also a global maximum. This maximum is obtained at an extreme point of S. The multiobjective linear fractional programming (MOLFP) problem can be written as follows: max max

p1 …x† q1 …x† p2 …x† q2 …x†

.. . max s:t:

387

(1993) have shown that computationally some of these results can be improved for the case when the denominators are identical. Choo has shown that the weakly ecient set for problem (2) is not always a union of polyhedrons, it may contain some nonlinear parts (Steuer, 1986). The explicit description of the weakly ecient set can be very useful but it is often a hard problem to get such description. An advantage of the weakly ecient set of problem (2) is that it is always a closed set. (The ecient set may not be closed (Steuer, 1986)). A nonlinear programming technique and the reference point method (Wierzbicki, 1981, 1986) are proposed here for the analysis of problem (2).

2. Analysis of the MOLFP problem using an auxiliary nonlinear programming problem

…2† pk …x† qk …x† x 2 S:

Here S  Rn is a nonempty bounded polyhedron (as in problem (1)). All pi (x) and qi (x) are linear functions. We denote fi …x† ˆ pi …x†=qi …x† …8i† and suppose that qi …x† > 0 8x 2 S; i ˆ 1; 2; . . . ; k. The vector x1 2 S is said to be ecient if there does not exist another x2 2 S such that fi …x2 † P fi …x1 † 8i; and fi …x2 † 6ˆ fi …x1 † for some i . The vector x1 2 S is said to be weakly ecient if there does not exist another x2 2 S such that fi …x2 † > fi …x1 † 8i. The set of all ecient points is denoted by E  S and the set of all weakly ecient points is denoted by Ew  S. All criterion vectors f(x) corresponding to points x 2 E constitute the nondominated set. All criterion vectors corresponding to points x 2 Ew constitute the weakly nondominated set. A description of MOLFP problems, some basic information and many examples can be found in Steuer (1986). Nykowski and Zolkiewski (1985) have proposed a replacing multiobjective linear programming problem and a compromise procedure for its solving. Several years later Dutta et al.

2.1. Obtaining weakly ecient points Let us consider problem (2). Using the reference point method we formulate the following nonlinear programming problem: min

D;

s:t: D P bi …ri ÿ fi …x††; x 2 S:

i ˆ 1; 2; . . . ; k;

…3†

Here bi > 0 …8i†, and ri 2 R1 …8i†; ri are the reference point components. The solution of problem (3) determines a weakly ecient point for problem (2). Really, suppose that x1 is a solution of problem (3), that gives the minimal value Dmin , but x1 is not a weakly ecient point. Then there exists another point x2 2 S, such that fi …x2 † > fi …x1 † …8i†. Therefore it is obvious that for the corresponding value D2 we get D2 < Dmin , and this is a contradiction. De®nition (Bazaraa and Shetty (1979)). Consider the function h: S ! R1 , where S is a nonempty convex set in Rn . The function h is called strict quasi-convex, if for each two points x1 , x2 2 S, such that h…x1 † 6ˆ h…x2 †, the following inequality holds:

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B. Metev, D. Gueorguieva / European Journal of Operational Research 126 (2000) 386±390

h…kx1 ‡ …1 ÿ k†x2 † < max fh…x1 †; h…x2 †g for all k 2 …0; 1†: The functions gi …x† ˆ bi …ri ÿ fi …x†† 8i are strict quasi-convex because fi (x) are linear fractional (Bazaraa and Shetty, 1979). It is obvious that in problem (3) the minimum of the following function is searched: u…x† ˆ max‰bi …ri ÿ fi …x††Š i

ˆ max‰gi …x†Š i

…i ˆ 1; 2; . . . ; k†:

Theorem. Let S  Rn be a nonempty convex set. Suppose that the functions gi …x† …i ˆ 1; 2; . . . ; k 8x 2 S† are strict quasi-convex. Then the function u…x† ˆ maxi ‰gi …x†Š is strict quasi-convex, too. Proof. Let 0 < k < 1; x1 ; x2 2 S. Then u…kx1 ‡ …1 ÿ k†x2 † ˆ max gi …kx1 ‡ …1 ÿ k†x2 † i   < max max…gi …x1 †; gi …x2 † i   ˆ max maxgi …x1 †; maxgi …x2 † i

i

  ˆ max u…x1 †; u…x2 † :



Therefore in problem (3) we have to minimize a strict quasi-convex function on the convex set S. Each local minimum of a strict quasi-convex function is also a global minimum of this function on the feasible set S (Bazaraa and Shetty, 1979). This means that we can solve problem (3) using nonlinear programming algorithms that give local minimum. The obtained solution will give a weakly ecient point for problem (2). The usage of reference points for the analysis of nonlinear multiobjective optimization problems is treated in Metev (1995). It is shown there that the obtained value of a given criterion can be improved by a correspondingly chosen reference point. More exactly, for the problems cosidered

here: if we have a weakly ecient point x1 obtained with a solution of problem (3) and we need to increase the value fq (x1 ), we have to increase the corresponding component of the reference point in problem (3). The same result gives a way for moving in the weakly nondominated set. The cited paper contains, in addition, a result about the attainability of the nondominated points. The sense is that if the reference point is close to a nondominated point then the obtained vector in criterion space is close to the same nondominated point. A similar result exists for the MOLFP problems: if the reference point in (3) is close to a given weakly nondominated point f a …x† then the solution of (3) determines a weakly nondominated point that is close to f a …x†. 2.2. Test for weak eciency Very often it is useful in MOLFP problems to check if x0 2 Ew where x0 2 S. For such purposes we can use the following test (with respect to MOLFP problem (2)): max

t

s:t: t 6 fi …x† ÿ fi …x0 †;

i ˆ 1; 2; . . . ; k;

…4†

x 2 S: If the solution of problem (4) gives t ˆ 0 then x0 2 Ew ; if t > 0 then x0 62 Ew . It is clear that here we need to maximize the function h…x† ˆ min…fi …x† ÿ fi …x0 ††: i

There is an analogy with the theorem proved above ± the function h(x) is strict quasi-concave when fi (x) are strict quasi-concave ± and this is the case here because fi (x) are linear fractional. Therefore each local maximum of h(x) is also its global maximum on S. It follows that for problem (4) we can use again nonlinear programming algorithms that give local maximums. 2.3. Numerical illustration The test (4) can have various applications. One possibility is to include it in procedures for ob-

B. Metev, D. Gueorguieva / European Journal of Operational Research 126 (2000) 386±390

taining estimates of the values ai ˆ minx2EW fi …x†. The vector a ˆ …a1 ; a2 ; . . . ; ak † with so de®ned components is called the nadir criterion vector. An approximation of this vector can be got from the payo€ table of the vector optimization problem. (The payo€ table is square and its ith row is the criterion vector obtained when we individually maximize the ith criterion.) Very useful information about the nadir vector in multiple objective linear programming problems and many references can be found in the paper of Korhonen et al. (1997) describing an heuristic approach for estimating this vector. Let us consider an example from the book of Steuer (1986, pp. 348 & 12.4). The example is: max max max s:t:

…ÿ2x1 ‡ x2 ˆ f1 †; …ÿ3x1 ÿ x2 ˆ f2 †; ……x1 ‡ x2 ÿ 2†=…ÿx1 ‡ 2x2 ‡ 5† ˆ f3 †; x1 P 0; x1 6 4; x2 P 0; x2 6 4:

max f1 ˆ 4 and the corresponding x ˆ …x1 ; x2 †

± for …x1 ; x2 † ˆ …4; 0†. The payo€ table gives the following estimate of the nadir vector: ()8, )12, )2/5). We ®nd that minx2S f1 ˆ ÿ8, thus the exact value of the ®rst component of the nadir vector is )8. However minx2S f2 ˆ ÿ16, so we could try to improve the value )12, obtained from the payo€ table. For instance we can set the question: is there a point xe 2 EW such that f2 …xe † ˆ ÿ15? The value f2 ˆ ÿ12 is obtained at point (4,0). At this point two of the constraints describing S are active: x2 ˆ 0 and x1 ˆ 4. Is there a point xa 2 S such that xa2 ˆ 0 and f2 …xa † ˆ ÿ15? We use the following problem: min s:t:

3y1 ‡ y2 ˆ 15;

min

ˆ …0; 4†;

s:t: 2

x1 6 4;

x2 ˆ 0:

The solution is: x1 ˆ 4:0; x2 ˆ 0; y2 ˆ 4:9; y2 ˆ 0:3 and the goal function is equal to 0.9. Therefore the searched point does not exist. Now we use the second active constraint. The corresponding problem is:

1

x2S

max f2 ˆ 0 and the corresponding x ˆ …x1 ; x2 †

  2 2 …x1 ÿ y1 † ‡ …x2 ÿ y2 † ; x1 P 0;

The constraints in this example describe here the set S. It is easy to ®nd the following data:

389

  …x1 ÿ y1 †2 ‡ …x2 ÿ y2 †2 3y1 ‡ y2 ˆ 15; x1 ˆ 4; x2 P 0; x2 6 4:

x2S

The solution is: x1 ˆ y1 ˆ 4; x2 ˆ y2 ˆ 3, the goal function is equal to 0. At the point (4,3) we have

ˆ …0; 0†; max f3 ˆ 2 and the corresponding x ˆ …x1 ; x2 †3 x2S

ˆ …4; 0†: All points …x1 ; x2 †i must be weakly ecient and the test (4) gives a con®rmation. Thus we have the following payo€ table for the example f1

f2

f3

4 0 )8

)4 0 )12

2/13 )2/5 2

The ®rst row is obtained for …x1 ; x2 † ˆ …0; 4†, the second row ± for …x1 ; x2 † ˆ …0; 0†, and the third row

f1 …4; 3† ˆ ÿ5;

f2 …4; 3† ˆ ÿ15;

f3 …4; 3† ˆ 5=7:

Test (4) con®rms that point ()5, )15, 5/7) is a weakly nondominated point, so (4, 3) is a weakly ecient point (this can be seen in the Steuer's book, too). Thus we obtain a better estimate of the second component of the nadir vector (in comparison with the data from the payo€ table). The third component of this vector cannot be improved because minx2S f3 ˆ ÿ0:4. The reasons used here for an analysis of this example are not sucient for creating an exact algorithm for ®nding the nadir vector in MOLFP problems, of course. We have shown only that test (4) can be included in such algorithm.

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B. Metev, D. Gueorguieva / European Journal of Operational Research 126 (2000) 386±390

3. Conclusion This paper shows that the property of strict quasi-convexity allows to use successfully the reference point method for the analysis of MOLFP problems ± it is easy to get weakly ecient points and then to improve the obtained value of chosen criterion. The testing for weak eciency is similar to the analogous operation in MOLP problems. It can be expected that some ideas about the approximate estimating of the nadir vector in MOLP problems (Korhonen et al., 1997) can be extended for MOLFP problems. Acknowledgements This research was supported by National Scienti®c Research Fund under the contract No b-616/1996. References Arsham, H., Kahn, A.B., 1990. A complete algorithm for linear fractional programs. Computers & Mathematics with Applications 20 (7), 11±23. Bhatt, S.K., 1989. Equivalence of various linearization algorithms for linear fractional programming. ZOR ± Methods and Models of Operations Research 33, 39±43. Bazaraa, M.S., Shetty, C.M., 1979. Nonlinear Programming. Theory and Algorithms. Wiley, New York.

Charnes, A., Cooper, W.W., 1962. Programming with linear fractional functionals. Naval Research Logistics Q. 9, 181± 186. Dutta, D., Rao, J.P., Tiwari, R.N., 1993. A restricted class of multiobjective linear fractional programming problems. European Journal of Operational Research 68 (3), 352± 356. Korhonen, P., Salo, S., Steuer, R., 1997. A heuristic for estimating nadir criterion values in multiple objective linear programming. Operations Research 45 (5), 751±757. Martos, B., 1975. Nonlinear Programming. Theory and Methods. North-Holland, Amsterdam. Metev, B., 1995. Use of reference points for solving MONLP problems. European Journal of Operational Research 80, 193±203. Nykowski, I., Zolkiewski, Z., 1985. A compromise procedure for the multiple objective linear fractional programming problem. European Journal of Operational Research 19, 91±97. Steuer, R., 1986. Multiple criteria optimization ± theory, computation, and application. Wiley, New York, Chichester. Verma, V., Khanna, S., Puri, M.C., 1989. On Martos' and Charnes-Cooper's approach vis- a-vis singular-points. Optimization 20 (4), 415±420. Wierzbicki, A., 1981. A mathematical basis for satis®cing decision making. In: Morse, J.N. (Ed.), Organizations: Multiple Agents with Multiple Criteria, Proceedings, University of Delaware, Newark, 1980; LNEMS, vol. 190, Springer, Berlin, pp. 465±485. Wierzbicki, A., 1986. On the completeness and constructiveness of parametric characterization to vector optimization problems. OR Spektrum 8, 73±87. Wolf, H., 1985. A parametric method for solving the linear fractional programming problems. Operations Research 33, 835±841.