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Interactive decision making for multiobjective nonlinear programming problems with fuzzy parameters
Fuzzy Sets and Systems29 (1989) 31g-326 North-Halland
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INTERACTIVE DECISION MAKING FOR MULTIO~IIECTnrE NONLINEAR PROGRAMM|NG PROBLEMS WITH FUZZY PARAMETERS Masatoshi S A K A W A Deper~;ent of Co;uproarScience, Faculty of Engi~;eering, lwat¢ University iwme 020, Japan
Hitoshi ~,'ANO Dcpu~ment of Information Science, Collegeof Economics, KagawaUniversity, Kasawa 760, Japan Received October I98{~ Revised Febroa~ 1987 This paper presents an interactive decision making method for multlnbjective nonlln¢~t programming problems with fuzzy parameters. The fuzzy parameters in the objective luscious and the constraints are characterized by fuzzy numbers. The concept of ~-Parato opumality/s introduced in which the ordinary Pareto optimality is extended based on the ¢g-levelsets of fuzzy numbers. In our interactive method, if the decision maker (DM) specifiesthe dtgm© ~ of the ~-level sets and th~ referanee obiectiva valuas, a minimaxproblem is solved and the DM is supplied with the corresponding o~-Pareto optimal solution together with the trade-off rates among the values of the objective functions and the degree or. Then by considering the current values of the objective functions and ¢z as well as the trade-off rates, the DM responds by updating his reference objective values and/or the degree ~. In this way the satisf3fingsolution for the DM can be derived efficiently from among an ¢t-Pareto optimal solution set. A numerical example illustrates va~ous aspects of the results developed in this paper.
Keywords: Fuzzy multiobjective nonilnear programming problems, Fuzzy perameteFs, F~zzy numbers, ~-Pareto optimality, Interactive decision making.
1. I m H u c f i o n In the mnltiobjective nonlinear programming problem, L~ere is no optimal solution due to the inherent conflict between the objecfiws. Consequently, the aim is to find the satisficing solution of the decision m a k e r (DM) which is also Pareto optimal. However, when formulating the multinbjee~ve nonlinear prograntming problem which closely describes and represents the real decision s/tuation, various factors of the real systein ~hould be reflected in the description of the objective functions and the constraints. Na~ura|ly these objective functions and the constraints involve many parameters whose possible values may be assigned by the experts. In the conventional apploach, such parameters are fixed
at some values in an experimenta~ and/or s~bjecfive manner through the experts' understanding of the nature of the parameters. In most practical situations, he,never, it is nate~'ai to consider that the possible 0165-0114189/$3.50~1 1989, Elsevier Science Publishers B.V. (North-Holland)
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M. $akawa, H. Yano
values of these parameters are often only ambiguously known to the experts. In this case, it may be more appropriate to interpret the experts' understanding of the perameters as fuzzy numerical data which can be represented by means of ~ z ~ subsets of the real line known as fuzzy numbers [2, 3]. The resulting multiob]ectlve nonlinear programming problem involving f-tm~ parameters would be viewed as the more reanstic version of the conventional one. Recenfly, Tanaka and Asai [14,15] formulated multiobjective linear programnfing p~oblems with fuzzy parameters. Following the fuzzy decision or minimum operator proposed by Bellman and Zadeh [1] together with triangular membership fi~cfions for fuz~, parameters, they considered two types of fuzzy mulfiobjective linear programming problems; one is to determine the nonfezzy solution .and the other is to determine the ~ y solution. More recently, Orlovski [8,9] formulated general multiobjective nonlinear progra~lming problems with fuzzy parameters. He presented two approaches to the formulated problems by making systematic use of the extension princip|e of Zadeh [18] and demonstrated that there exist in some sense equivalent nonfi~zy formulations. In this paper, in order to deal with multiobjective nonlinear programming problen~ with fuzzy parameters characterized by fuzzy numbers, the concept of o~-Pareto optimality is introduced by extending the ordinary Pareto optimality on the basis of the ~-level sets of the fuzzy numbers. Then an interactive decision making method to derive the satisficing solution of the decision maker (DM) efficiently from among an ~r-Pareto optimal solution set is presented as a generaUy.ation of the rest~Rs obtained in Sakawa et aL [10-13].
In general, the multiobjecfive nonlinear programming (MONLP) problem is represented as the following vector-minimization problem:
~nin f ( x ) - (fi(x),A(x) . . . . . A(x)), subject to x ¢ X = {x e E ~ [ & ( x ) ~
0a) (lb)
where x is an n-dimensional vector of decision variables, f l ( x ) . . . . . f k ( x ) are k distinct objective functions of the decision vector x, gt(x) . . . . . gin(X) are r,~ inequality constraints, and X is the feasible set of constrained decisions. Fundamental to the MONLP is the Pareto optimality concept, also known as non/nferior solution. Qua~tatively, a Pareto optimal solution of the MONLP is one where any improvement of one objective function can be achieved only at the expense of annther. Mathematically, a formal definition of a Pareto optimal solution t o the MONLP is given below: D e ~ 1 (Pareto optimal solution), x * e X is said to be a Pareto optimal solution to the MONLP, if and only if there does not exist another x E X such that J~(x) ~<~(x*), i = 1. . . . . k, with strict inequality holding for at least one i.
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In practice, however, it would certainly be appropriate to consider that the possible values of the parameters in the description of the objective functions and the constraints muaHy involve the ambiguity of the expels' understanding of the real system. For this reason, in this paper, we consider the following fuzzy multiobjecfive nonlinear programming ( ~ I O N L P ) problem involving fuzzy parameters: rain f(x,~) A_(f~(x,~,),A(x, a~)..... A(x, ~,~)) subject to x ~ X ( f ) & - { x e E " l g j ( x , S l ) ~ O , j = l , " . . . . m},
(2a) (2b)
where al = (alt . . . . . a¢,,) and/;j = (G~l. . . . . 6jql) represent respectively a vector of fuzzy parameters involved in the objective function .~(x, a~) and the constraint
function gAx, 6j). Thea¢ fuzzy parameters are assumed to be characterized ~s the f ~ z y tlumbers introduced by Dubois and Prude [2, 3]. It is appropriate to recall here that a real fuzzy number ~ is a convex continuous fuzzy subset of the real fine whose membership function ~ ( p ) is defined 'by: (I) A continuous mapping from E ~ to the closed interval [0,1]. (2) ~ ( p ) ---0 for an p G (-~, ~].
(3) Strictly increasing on [Pl, P2]. (4) ~(p) = 1 for all p ¢ [P2, P3]. (5) Strictly decreasing on [P3, P4]. (6) ,~(p) = 0 for all p ¢ [P4, +~). Figure 1 illustrates the graph of the possible shape of a fuzzy number/~. We now assume that ~, and 6j~ in the FMONLP are fuzzy numbers whose membership functions are $a~(ai~) and ~;~(b~) respectively. For simplicity of notation, define the following vectors: ai -- (an . . . . . a~,,), b, = (bit . . . . . biq,), a = ( a t . . . . . a~), a f ( a t . . . . . a~,),
b=(b, ..... b,~), /~= (6,..... 6m).
!
Pl P2 P3 P4 Fig. 1. Membershipfunctionof f'uz~ number~.
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M. S~awa, H. Yano
Then we can introduce the following oMevel set or o~-cut [3] of the f~azy n ~ b e s ~ and ~j~. De~n 2 (~-level set). The ~-ievel set of the fuzzy numbers ~, (i = 1 . . . . . k; r = 1. . . . . Pl) and 6~ ( j = 1. . . . . m; s = 1. . . . . q/) is defined as the ordinary set L,~(~, •) for which the degree of their membership functions exceeds the level at: L,~(a, 6) -- {(a, b) I ~z,,(a~,) >~e~, i -- 1. . . . .
k; r = 1. . . . . Pi;
m~(bj~)~> ~,j = I ..... m ; s = 1 ..... q:).
(3)
It i~ clear that the level sets have the following property: or1 ~
rain f(x,a) ~ (/~(x,al),A(x, a2)..... A(x, ak))
(4a)
subject to x ~ X ( b ) & { x e E n l g ~ ( x , b / ) < ~ O , ] = l . . . . . m},
(4b)
(a,b) ~ L,(~, 6).
(4c)
It should be emphasized here that in the c~-MONLP the parameters (a, b) are treated as decision variables rather than constants. On the basis of the or-level sets of the fuzzy numbers, we introduce the concept of o~-Pareto optimal solutions to the ac-MONLP. Definition 3 (of-Pareto op,,,aal solution), x* ¢ X ( b ) is said to be an ec-Pareto optimal solution to the e~-MONLP (4), if and only ff there does not exist another x e X ( b ) , (a,b)¢L~(a, 6) such that f~(x, ai)<<-~(x*,a*), iffil . . . . . k, with strictly inequality holding for at least one i, where the corresponding values of parameters (a*, b*) are called et-level optimal parameters. It is significant to note here that from the property of the o~-level set, the following relation holds. 1Ptoi~sitlon 1. Let x t and x 2 be e~t- and ~-Pareto optimal solutions and (a t, b') and (a 2, b 2) be corresponding eft- and e~2-1eveloptimal parameters to the e~- and e~2-MONLP respeeavely. If e~t >~~2, then there exists x ~ and (a 2, b 2) such that
f ( x t, a~)>~f(x ~, a2) fer ~*nyx ~ ~nd (a t, b'). As can be immediately unde~tood from the definition, usually e~-Pareto optimal solutions consist of an infinite number of points, and the DM must select his compromise or satisficing solution from among e~-Pareto optimal solutions based on his su~ective v0.1ue-judgement. In order to generate a candidate for the safisficing solution which is also e~-Pareto optimal, the DM is asked to specify the degree e~of the e~-level set and the reference levels of achievement of the objective functions, called reference
lnter~cff~e decision rr~Idng
319
levels. Observe that the idea of the reference levels or the reference point was first appeared in Wie~bicki [16]. For the DM's degree ~ and reference levels f . i -- 1. . . . . k, the corresponding o~-Pareto optima| solution, which is in a sense close to his requirement or better than that if the reference levels are attainable, is obtained by solving the following minimax problcnh wbere it is assumed that the difference (~(x, a~) -f~) is of equal importance to the DM:
rain
JteX(b~ -
~ax (~(x, a~)-~),
(5)
l~i~k
or equivalently n~n v subject to ~(x, a,)-~<~v,
(*, b) E Lo(~, 6),
(6) (7) (8)
x ~ X(b).
(9)
i = 1 . . . . . k,
It should he emphasized here that the minimax problem is simply used as a means of generating an ~-Pareto optimal solm~tion,and if the DM is not satisfied with the current o~-Pareto optimal solution, it is possible for him to improve the solution by updating his reference levels. The relationships between the optimal solutions of the minimax problem and the a~-Pareto optimality concept of the ~-MONLP can be characterized by the following theorems. 'l'he~rem 1. If (x*, v*, a*, b*) is a unique optimal solution to the minimax problem for some f -- (~ . . . . . fk), then x* is an oc-Pareto optimal solution to the ¢~-MONLP. Proof. AssLn~ that x is not an ~-Pareto optimal solution to the ~r-MONLP. Then there exist x e X(b) and (a=b)¢ L~(& b) such that f(x~ a)<~f(x*, a*). It follows that
which contradicts the fact that (x*, v*, a*, b*) is a unique optimal solution to the minimax problem. Hence x* is an ~-Pareto optimal solution to the ¢-MONLP. Theorem 2. I f x* is an c¢-Pa:eto optimal solution and (u*, b*) is an ¢-levd optimal parameter to the ~-MONLP, then there exists f = (~ . . . . . fk) such that (x*, v*, a*, b*) is an optimal solution to the minimax problem. Proof. Assume that (x*, v*, a*, b*) is not an optimal solution to the minimax problem for any f satisfying
A(x*, a~) - ~ . . . . .
A(x*, at) -?~ -- v*.
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M, 8akawa, H. Yano
Then the~e exist x ~ X(b ) and (a, b) e Lo,(a,b) such that m a x ff,(x, u,) - ~) < ma~ ff,(x*, a~) - ~) = ~,*.
l~i~k
This implies that max
~(x, a~) -f~(x*, aD)<0.
Now if either any ~(x, a i ) - ~ ( x * , a~') is positive or any ~(x, a~)-fl(x*, a~), i -- 1 , . . . , k, is ~eto, this inequality would be violated. Hence
f(x, a)-f(x*, a*)< 0 must hold, which contradicts the fact that x* is an eL-Pareto optimal solution and (a*, b*) is an ~-level optimal parameter to the ~-MONLP, and the theorem is proved. If (x*, v*, a*, b*), an optimal solution to the minimax problem, is not unique, then we can test the o~-Pareto optimality for x* by solving the follow~ng problem: k
max
O0a)
~ ~,
snb]ectto ~(x,a,)+e,=~(x*,a*),
e,~O, i~1,2 ..... k,
(IOb)
Let (,~,a, l;) be an optimal solution to (10). if all ee =0, then x* is an e~-Pareto optimal solution. If at least one e~ >0, it can easily be shown that x is an o~-Pareto optimal solution. 3. Trade-off mtos Given the ~-Pareto optimal solution for the degree ~ and the reference levels spedfi~,d by the DM by solving the corresponding minimax problem, the DM must either be satisfied with the current e~-Pareto optimal solution and et, or update the reference levels and/or the degree ~. In order to help the DM express his degree of preference, trade-off informatio~-~ between a standins objective function a~d each of the other objective functions as well as between the degree and the objective functions is very useful. Fortunately, such a trade-off information is easily obtainable since it is closvly related to the strict positive Lagrange multipliers of the minimax problem!. To derive the trade-off information, we first define the Lagrangian function L for the minimax problem (6)-(9) as follows:
~=1
j=l rn qj
i = l r=l
j = l s=!
Interactiondecisionmaking
32[
In the following, for notational convenience we denote the decision variable in the minimax problem (6)-(9) by y = (x, v, a, b) and let us assume tha~ the minimax problem has a unique local optimal solution y* satisfying the following three assumptions. A~pfioe
1, y* is a regular point of the constraints of the minimax prob|em.
A~snmpflon 2. The second-order sufficiency conditions are satisfied ~t y*. A~mnpt]on 3. There are no degenerate constraints at y*. Then the following exls~.ence theo~¢ra, Which is based on the implicit function theorem [4], holds. T h e o ~ m 3. Let y* = (z*, v*, a*, b*) be a unique local solution o f the minimax problem (6)-{~) sallying the assumptions 1, 2 and 3. Let ~ * = ( ~r , M*, a b. a z') denote the L~grange multipliers corresponding to the constraints (7)-(9). Then there exist a continuously differentiable vector valued function y(.) and 3.(.) defined on some neighborhood N(~*) so that y(o~*) =y*, ~.(a~*)--~*, where y(a~) is a unique local solution o f the minimax problem (6)-(9) for any ~ E N ( c : ) satisfying the nssumpgons 1, 2 and 3, and ~(a) ~s the Lagrange multiplier corresponding to the constraints (7)-(9).
In Theorem 3, inf(v I,(x, a,) - ~ ~< v, i = 1. . . . .
k, (a, b) ~ L.(~, 6), x E X(b)},
can be viewed as the optimal value function of the minimax problem (6)-(9) for any ot E N(~*). Therefore, the following theorem holds under the same assumptions in Theorem 3. T h e o ~ m 4. If all the assumptions in Theorem 3 are satisfied, then the following relations hold on some neighborhood N( ~*) of ~*. aL
au
~
P'
"
~j
(12)
: - - - = - - £ £ ~,,+£ £ a~. _
0~'
_
0~0;"
a
b
l=l r=l
j=l s=l
If all the constraints (7) of the minimax problem are active, namely if u(0e*) =f~(x(~*), a~(~*)) - ~ , then the followin~ theorem holds. Theorem S. Let all the assumptions in Theorem 3 be sa~fed. Also assume that all the constraints (7) of the minimax problem are active. Then it hold* that k
~,~a[ O~'
I~'=w*
P~
m
=££~f+££~', i=l
r=l
qJ
j=l a=l
~=~ . . . . . ~.
o3)
Regarding a trade-off rate between f~(x) and f~(x) for each i = 2 . . . . . k, by
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M. Sakawa, tt. Fund
extend:rig the results in Ha/rues and Chankong [5], it can be proved that the following theorem holds [17]. Theorem 6. Lee all the assumption in Theorem 3 be satisfied. Also assume that the constraints (7) are active. Then it holds that
Of(x, a~)
a~(~,a,) ,,.,,.
=
~,{"
-~,
i=2 .....
t,.
(14)
It should be noted here that in order to obtain the trade-off rate information from (13) and (14), all the constraints (7) of the minimax problem must be active. Therefore, ff there are inactive constraints, it is necessary to replace ~ for inactive constraints by j~(x*, a*) and solve the corresponding minimax problem for obtaining the Lagrange multipliers.
4. I~erac~ve algn~thm and 8nmerlesi example Following the above discussions, we can now construct the interactive algorithm ha order to derive the satisficing solution for the D M from among the ot-Pareto optimal solution set. The steps marked with an asterisk involve interaction with the DM. Step 1. Calculate the incBvidual minimum and maximum of each objective function under the given constraints for o~= 0 and 0c = 1. Step 2". Ask the D M to select the initial value of of (0 < ~ < 1) and the initial reference levels ~, i = 1. . . . . . k. Step 3. For the degree a and the reference levels specified by the DI~, solve the minimax problem and perform the oc-Pareto optimality tesl. Step 4*. The DM is supplied with the corresponding ot-Pareto optimal solution and the trade-off rates between the objective f~nctions and the degree ~. If the D M is satisfied with the current values of the objective functions and oc of the ~-Pareto optima~ solution, stop. Otherwise, the DM must update the reference levels and/or the degree ~ by considering the current values of the objective functions and ot ft~gether with the trade-off rates between the objective functions and the degree ~ and returl~ to step 3. Here it should be stressed for the DM that (1) any improvement of one objective function can be achieved only at the expense of at least one of the other objective functions for some fixed degree 0t, and (2) the greater value of the degree 0~ gives worse values of the objective functions for some fixed reference levels. Based on the method described above, we have developed a new interactive computer program. Our program is composed of one main program a~d several
lnteract~eedecismn n',ak~g Table 1. ~
,~ta a~z a~l ~;',z
a31 ~3z ~.33
b.:: hi2
GI3
323
numbers for nnraedcal example
Pl
P2
P~
P4
|C~
n~lt
3.8 57.0 18.0 1.'/5
4.0 59.0 19.5 2.0
4.0 60.0 20.0 2.0
4.3 63.5 22.5
L E E
2.3
2.5
2.5
1.25 17,5 0.9 0.8 0.85
1.4 20.0
1.5
1.7
20.0
22.0
E L E L E L E ]E E L
1.0
0.95 1.0
1.0 1.0 1.0
2,23 2.75
1.1 1.2 1.15
E
L L L E E E
CONHAND: ?GO .....................
< ITERATION 1 >. . . . . . . . . . . . . . . . . . . . . .
INPUT YOUR REFERENCE VALUES F ( 1 ) ?5025 5650 6062,5
(I=1,3):
I ~ P U T THE DE~REE ~LFR OF THE RLFR LEVEL SETS FOR THE FUZZY PARAMETERS: ?0.9 ( KUHN-TUCKER COND|T]ONS S A T | S P I E D ) RLFA-PARETO OPTIMAL SOLUTION TO THE H I N I H A X PROBLEM FOR 1 N Z T I A L REFERENCE VALUES ............................. OBJECTIVE FUNCTION ............................. F(I) = 6438,9620 F(2) = 7063.9620 F(3) = 747&,4&20 .......................................................... X(I) = 6.3782 X(2) = X(3) = 4.407~ .......................................................... TRADE-OFF5 AHONG OBJECTIVE FUNCTIONS -DF(2)IDF(I) = !*Z52! -DF(3)/OF(I) = 1.3422 ........................................................... TRADE-OFFS BETWEEN ALFA AND OSaECTIVE FUNCTZONS DF/DALFA = 797.2460 .........................................................
6.5187
ARE YOU S A T I S F I E D N I T H THE CURRENT CoSJECT]VE VALUES OF THE ALFR-PARE~O OPTINAL SOLUTXON ? ?NO
Fig. 2. Interactive dcdsion maidng processes.
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M. Sakawa, H. Yono
subroutines. The main program calls in an~ tuns the subprograms with commands indicated by the use (DM), Consider the following three-objective nonlinea~ programming problem with fuzzy parameters:
miu A(~, ~,) = ~l,x~ + (x~ + 5) ~ + 2 ( x ~ - ~,2Y, rain A(a, ~z) = (xl + 621)2 + ~2z(xz - 55) 2 + 3(x3 + 20) z, subject to
" -2.~ x e X(/~) -- {(xt, x2, x3) I gl(x, bl) - 6 n x l2 +/;l~c22 + G~3xa ~ 100,
0~&~
10, i = 1, 2, 3}.
The membership functions for the fuzzy number: ~ . . . . . ~ , / / , . . . . . G~3 in th/s ezample are explained in Table 1 where L and E re,present respectively linear and exponential memlmrship functions. In Figure 2, the interaction processes using a time-sharing computer program under TSS of a M E L C O M COSMO 700S digital computer ai th~ computer center of Kagawa University are explained especially for the first ;~era~ion through the aid of some of the computer outputs. 0~-Pareto optima| sobz~ions a~-e obtained by solving the minimax problem using the revised version of the genera~:ed reduced gradient (GRG) [6] program called G R G 2 [7]. Since the DM is not satisfied with the current objective values, the DM updates his reference :evels. In this example, at the 6th iteration, the satisflcing solution of the DM is derived. The complete interactive processes are sununadzed in Table 2. CPU time required in this interaction process was 312.815 seconds and the example session ~akes about 13 minutes.
Table 2. Interactive processesfor numericalexample Iteration f2 at ~(x, nO f2(x, a2) fa(x, a3) ~l a2
xa -OA/OA -Ofa/Of, a~/a¢~
1
2
3
4
5
6
5025.0 5650.0 6062.5 0.9 6438.96 7063.96 7476.46 6.3782 6.5187 4.4076 1.1521 1.3422 797.25
6860.0 6700.0 6800.0 0.9 7008.58 6908.58 7008.58 7.7803 6.1078 2.1969 0.9217 0.9205 835.g0
6500.0 6300.0 6700.0 0.7 6823.52 6623.52 7023.52 7.3317 6.8347 2.5747 0.9384 1.0323 719.52
6700.0 6:Yo0.0 6900.0 0.65 r 6797.90 6587.90 6987.90 7.3351 6.0771 2.6287 0.9448 1.0392 705.95
6300.0 6600.0 6~J00.O 0.65 6492.66 6792.66 7092.66 7.0352 6.5879 3.8383 !.0935 L 1903 o9.L09
65~0.0 6700.0 7000.0 0.65 6556.7I 6756.71 7056.71 7.1465
6.6174 3.5815 1.0629 1.1506 696.61
Interactive decisio~ making
3~
In this paper, we have proposed an interac¢ive decision mal~ng method in order to deal with the multiobjective nonlinear progra~tnnng problem with fe~,y pararaeters characterized by fuzzy numbers. Through the use of the ~oncept of the of-level sets of fuzzy numbers, a new solution concept caUed ~-Pateto optir~ali~y has been introduced. In our interactive scheme, the satisficiag solution of ~he D M can be ~efi,,,e~ from a m o n g an ~-Pareto optimal r~3lution s~t by updating the reference levels a n d / o r the degree ~ based on the current values of the membership functions and ~ together with the trade-off ~ates between the objective functions and the degree o~. A n illustrative numer/cal example clarified the various aspects of both the solution concept of ~-Pareto optimality and the proposed method. However, further applications must be carried out in cooperation w~th a person actually involved fin decision making.
References [1] R.E. Bellman and L.A. Zaduh, Decision making in a fuzzy environment, Management Science t7 (4) (1970) 141-164. [2] D. Dubois and H. Prade, Operations on fuzzy numbers, Internat. J. Systems SoL 9 (6) (1978) 613-626. [3] D. Dubois and H. Predc, Fuzzy Sets and Systems: Themy and Applications (Academic Pre~, New York, 1980). [4] A.V. Fleece, Introduction to Sensitivity and Stability Analysis in Nonlinear t~osr~mndng (Academic Press, New Yogk, 1983). [5] Y.Y. Haimes and V. Chankong, Kuhn-Tucher multipliers as trede-offs in multiobjecfive decisinn-makinganalysis, Aut~omaticat5 (1) (1979) 59--72. [61 L.S. Lasdon, R.L. Fox end M.W. Ruiner, Nonlinear optimizatinn winS the generalized reduced ~ndient method, Ray. F:au~aise Automat. Informer. Recherche Op6rationoelio 3 0974) 73-103. [7] L.S. Lasdan, A.D. Wasen and M.W. Ratner, GRG2 User's Guide, Teehglif~ Memorandum, University of Texas (1980). [8] S.A. Orlovski, Problen~ of decisinn-makingwith fuzzy information, Working Paper WP-83-28, International |nstitote for App:!~l Systems Analysis, Laxenburg, Austt'ia (1983). [9] S.A. Orlovski, Multlobjective programming problems with fuzzy parameters, Control and Cyhernet. 13 (3) (1984) 175-183. [10] M. Sakuwa, Interactive computer programs for fuzzy linear programming with multiple objectives, Interoat. J. Man-Machine Stud. lg (5) 0983) 489-503 [II] M. Sakawnand I". Yumine, Interactive fuzzy decislon-makingfor multinb~:ctivelincar fractional ptc,,gramndng problems, Large Scale Systems 5 (2) (1983) 105-114. [12] M Sakawa and H. Yano, An interactive fuzzy satisficing method using penalty ~'.a[arizing problems, Prec. Int. computer symposium,Tamkang Univ., Taiwan (1984) 1122-1129. [13] M. Sakawa and H. Yano, Interactive fuzzy decision making for multiobjective nonlincor programming using augmented minimaxprohlerss, Fuzzy Sets and Systems 20 (l) (1986) 31-43. [14] H. Tanaka and K. Asai, A formulation of linear programming ~obiems by fth~,y functinn, Systems and Control 25 (6) (198i) 351-357 (in Japanese). [15] H. Tanaka and K. Asai, Fuzzy linear programmingproblems wit~,fuzzy numbers, Fuzzy Sets and Systems 13 (1) (1984) 1-10. [16] A.P. Wie~bichi, The use of reference ob~ectiws i~ ~nulti~bje~ive optimization-~teotetical
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implications and practical experiences, Worlfing Paper WP-79-66, rmemation~ Institute for Applied $~temsAnalysis, L ~ e n b c r ~ Austria (1979). [17] H. Yeno and M. ~ a w a , Trade-off rates in the weighted Tchebychcff norm metho~, T t ~ . S.LC.E. 2I (3) (1985) 2 4 q - 2 ~ (in .Tapanes¢). [~18] L.A. Zadeh, The coneept of a |hlguhtlc variabJ~ and its application to a p p ~ x ~ ' e ~a.~nlvg-1, Inform. $ci. 8 (1975) 199-249.
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INTERACTIVE DECISION MAKING FOR MULTIO~IIECTnrE NONLINEAR PROGRAMM|NG PROBLEMS WITH FUZZY PARAMETERS Masatoshi S A K A W A Deper~;ent of Co;uproarScience, Faculty of Engi~;eering, lwat¢ University iwme 020, Japan
Hitoshi ~,'ANO Dcpu~ment of Information Science, Collegeof Economics, KagawaUniversity, Kasawa 760, Japan Received October I98{~ Revised Febroa~ 1987 This paper presents an interactive decision making method for multlnbjective nonlln¢~t programming problems with fuzzy parameters. The fuzzy parameters in the objective luscious and the constraints are characterized by fuzzy numbers. The concept of ~-Parato opumality/s introduced in which the ordinary Pareto optimality is extended based on the ¢g-levelsets of fuzzy numbers. In our interactive method, if the decision maker (DM) specifiesthe dtgm© ~ of the ~-level sets and th~ referanee obiectiva valuas, a minimaxproblem is solved and the DM is supplied with the corresponding o~-Pareto optimal solution together with the trade-off rates among the values of the objective functions and the degree or. Then by considering the current values of the objective functions and ¢z as well as the trade-off rates, the DM responds by updating his reference objective values and/or the degree ~. In this way the satisf3fingsolution for the DM can be derived efficiently from among an ¢t-Pareto optimal solution set. A numerical example illustrates va~ous aspects of the results developed in this paper.
Keywords: Fuzzy multiobjective nonilnear programming problems, Fuzzy perameteFs, F~zzy numbers, ~-Pareto optimality, Interactive decision making.
1. I m H u c f i o n In the mnltiobjective nonlinear programming problem, L~ere is no optimal solution due to the inherent conflict between the objecfiws. Consequently, the aim is to find the satisficing solution of the decision m a k e r (DM) which is also Pareto optimal. However, when formulating the multinbjee~ve nonlinear prograntming problem which closely describes and represents the real decision s/tuation, various factors of the real systein ~hould be reflected in the description of the objective functions and the constraints. Na~ura|ly these objective functions and the constraints involve many parameters whose possible values may be assigned by the experts. In the conventional apploach, such parameters are fixed
at some values in an experimenta~ and/or s~bjecfive manner through the experts' understanding of the nature of the parameters. In most practical situations, he,never, it is nate~'ai to consider that the possible 0165-0114189/$3.50~1 1989, Elsevier Science Publishers B.V. (North-Holland)
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M. $akawa, H. Yano
values of these parameters are often only ambiguously known to the experts. In this case, it may be more appropriate to interpret the experts' understanding of the perameters as fuzzy numerical data which can be represented by means of ~ z ~ subsets of the real line known as fuzzy numbers [2, 3]. The resulting multiob]ectlve nonlinear programming problem involving f-tm~ parameters would be viewed as the more reanstic version of the conventional one. Recenfly, Tanaka and Asai [14,15] formulated multiobjective linear programnfing p~oblems with fuzzy parameters. Following the fuzzy decision or minimum operator proposed by Bellman and Zadeh [1] together with triangular membership fi~cfions for fuz~, parameters, they considered two types of fuzzy mulfiobjective linear programming problems; one is to determine the nonfezzy solution .and the other is to determine the ~ y solution. More recently, Orlovski [8,9] formulated general multiobjective nonlinear progra~lming problems with fuzzy parameters. He presented two approaches to the formulated problems by making systematic use of the extension princip|e of Zadeh [18] and demonstrated that there exist in some sense equivalent nonfi~zy formulations. In this paper, in order to deal with multiobjective nonlinear programming problen~ with fuzzy parameters characterized by fuzzy numbers, the concept of o~-Pareto optimality is introduced by extending the ordinary Pareto optimality on the basis of the ~-level sets of the fuzzy numbers. Then an interactive decision making method to derive the satisficing solution of the decision maker (DM) efficiently from among an ~r-Pareto optimal solution set is presented as a generaUy.ation of the rest~Rs obtained in Sakawa et aL [10-13].
In general, the multiobjecfive nonlinear programming (MONLP) problem is represented as the following vector-minimization problem:
~nin f ( x ) - (fi(x),A(x) . . . . . A(x)), subject to x ¢ X = {x e E ~ [ & ( x ) ~
0a) (lb)
where x is an n-dimensional vector of decision variables, f l ( x ) . . . . . f k ( x ) are k distinct objective functions of the decision vector x, gt(x) . . . . . gin(X) are r,~ inequality constraints, and X is the feasible set of constrained decisions. Fundamental to the MONLP is the Pareto optimality concept, also known as non/nferior solution. Qua~tatively, a Pareto optimal solution of the MONLP is one where any improvement of one objective function can be achieved only at the expense of annther. Mathematically, a formal definition of a Pareto optimal solution t o the MONLP is given below: D e ~ 1 (Pareto optimal solution), x * e X is said to be a Pareto optimal solution to the MONLP, if and only if there does not exist another x E X such that J~(x) ~<~(x*), i = 1. . . . . k, with strict inequality holding for at least one i.
/ncemcavedec/a~ making
317
In practice, however, it would certainly be appropriate to consider that the possible values of the parameters in the description of the objective functions and the constraints muaHy involve the ambiguity of the expels' understanding of the real system. For this reason, in this paper, we consider the following fuzzy multiobjecfive nonlinear programming ( ~ I O N L P ) problem involving fuzzy parameters: rain f(x,~) A_(f~(x,~,),A(x, a~)..... A(x, ~,~)) subject to x ~ X ( f ) & - { x e E " l g j ( x , S l ) ~ O , j = l , " . . . . m},
(2a) (2b)
where al = (alt . . . . . a¢,,) and/;j = (G~l. . . . . 6jql) represent respectively a vector of fuzzy parameters involved in the objective function .~(x, a~) and the constraint
function gAx, 6j). Thea¢ fuzzy parameters are assumed to be characterized ~s the f ~ z y tlumbers introduced by Dubois and Prude [2, 3]. It is appropriate to recall here that a real fuzzy number ~ is a convex continuous fuzzy subset of the real fine whose membership function ~ ( p ) is defined 'by: (I) A continuous mapping from E ~ to the closed interval [0,1]. (2) ~ ( p ) ---0 for an p G (-~, ~].
(3) Strictly increasing on [Pl, P2]. (4) ~(p) = 1 for all p ¢ [P2, P3]. (5) Strictly decreasing on [P3, P4]. (6) ,~(p) = 0 for all p ¢ [P4, +~). Figure 1 illustrates the graph of the possible shape of a fuzzy number/~. We now assume that ~, and 6j~ in the FMONLP are fuzzy numbers whose membership functions are $a~(ai~) and ~;~(b~) respectively. For simplicity of notation, define the following vectors: ai -- (an . . . . . a~,,), b, = (bit . . . . . biq,), a = ( a t . . . . . a~), a f ( a t . . . . . a~,),
b=(b, ..... b,~), /~= (6,..... 6m).
!
Pl P2 P3 P4 Fig. 1. Membershipfunctionof f'uz~ number~.
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M. S~awa, H. Yano
Then we can introduce the following oMevel set or o~-cut [3] of the f~azy n ~ b e s ~ and ~j~. De~n 2 (~-level set). The ~-ievel set of the fuzzy numbers ~, (i = 1 . . . . . k; r = 1. . . . . Pl) and 6~ ( j = 1. . . . . m; s = 1. . . . . q/) is defined as the ordinary set L,~(~, •) for which the degree of their membership functions exceeds the level at: L,~(a, 6) -- {(a, b) I ~z,,(a~,) >~e~, i -- 1. . . . .
k; r = 1. . . . . Pi;
m~(bj~)~> ~,j = I ..... m ; s = 1 ..... q:).
(3)
It i~ clear that the level sets have the following property: or1 ~
rain f(x,a) ~ (/~(x,al),A(x, a2)..... A(x, ak))
(4a)
subject to x ~ X ( b ) & { x e E n l g ~ ( x , b / ) < ~ O , ] = l . . . . . m},
(4b)
(a,b) ~ L,(~, 6).
(4c)
It should be emphasized here that in the c~-MONLP the parameters (a, b) are treated as decision variables rather than constants. On the basis of the or-level sets of the fuzzy numbers, we introduce the concept of o~-Pareto optimal solutions to the ac-MONLP. Definition 3 (of-Pareto op,,,aal solution), x* ¢ X ( b ) is said to be an ec-Pareto optimal solution to the e~-MONLP (4), if and only ff there does not exist another x e X ( b ) , (a,b)¢L~(a, 6) such that f~(x, ai)<<-~(x*,a*), iffil . . . . . k, with strictly inequality holding for at least one i, where the corresponding values of parameters (a*, b*) are called et-level optimal parameters. It is significant to note here that from the property of the o~-level set, the following relation holds. 1Ptoi~sitlon 1. Let x t and x 2 be e~t- and ~-Pareto optimal solutions and (a t, b') and (a 2, b 2) be corresponding eft- and e~2-1eveloptimal parameters to the e~- and e~2-MONLP respeeavely. If e~t >~~2, then there exists x ~ and (a 2, b 2) such that
f ( x t, a~)>~f(x ~, a2) fer ~*nyx ~ ~nd (a t, b'). As can be immediately unde~tood from the definition, usually e~-Pareto optimal solutions consist of an infinite number of points, and the DM must select his compromise or satisficing solution from among e~-Pareto optimal solutions based on his su~ective v0.1ue-judgement. In order to generate a candidate for the safisficing solution which is also e~-Pareto optimal, the DM is asked to specify the degree e~of the e~-level set and the reference levels of achievement of the objective functions, called reference
lnter~cff~e decision rr~Idng
319
levels. Observe that the idea of the reference levels or the reference point was first appeared in Wie~bicki [16]. For the DM's degree ~ and reference levels f . i -- 1. . . . . k, the corresponding o~-Pareto optima| solution, which is in a sense close to his requirement or better than that if the reference levels are attainable, is obtained by solving the following minimax problcnh wbere it is assumed that the difference (~(x, a~) -f~) is of equal importance to the DM:
rain
JteX(b~ -
~ax (~(x, a~)-~),
(5)
l~i~k
or equivalently n~n v subject to ~(x, a,)-~<~v,
(*, b) E Lo(~, 6),
(6) (7) (8)
x ~ X(b).
(9)
i = 1 . . . . . k,
It should he emphasized here that the minimax problem is simply used as a means of generating an ~-Pareto optimal solm~tion,and if the DM is not satisfied with the current o~-Pareto optimal solution, it is possible for him to improve the solution by updating his reference levels. The relationships between the optimal solutions of the minimax problem and the a~-Pareto optimality concept of the ~-MONLP can be characterized by the following theorems. 'l'he~rem 1. If (x*, v*, a*, b*) is a unique optimal solution to the minimax problem for some f -- (~ . . . . . fk), then x* is an oc-Pareto optimal solution to the ¢~-MONLP. Proof. AssLn~ that x is not an ~-Pareto optimal solution to the ~r-MONLP. Then there exist x e X(b) and (a=b)¢ L~(& b) such that f(x~ a)<~f(x*, a*). It follows that
which contradicts the fact that (x*, v*, a*, b*) is a unique optimal solution to the minimax problem. Hence x* is an ~-Pareto optimal solution to the ¢-MONLP. Theorem 2. I f x* is an c¢-Pa:eto optimal solution and (u*, b*) is an ¢-levd optimal parameter to the ~-MONLP, then there exists f = (~ . . . . . fk) such that (x*, v*, a*, b*) is an optimal solution to the minimax problem. Proof. Assume that (x*, v*, a*, b*) is not an optimal solution to the minimax problem for any f satisfying
A(x*, a~) - ~ . . . . .
A(x*, at) -?~ -- v*.
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M, 8akawa, H. Yano
Then the~e exist x ~ X(b ) and (a, b) e Lo,(a,b) such that m a x ff,(x, u,) - ~) < ma~ ff,(x*, a~) - ~) = ~,*.
l~i~k
This implies that max
~(x, a~) -f~(x*, aD)<0.
Now if either any ~(x, a i ) - ~ ( x * , a~') is positive or any ~(x, a~)-fl(x*, a~), i -- 1 , . . . , k, is ~eto, this inequality would be violated. Hence
f(x, a)-f(x*, a*)< 0 must hold, which contradicts the fact that x* is an eL-Pareto optimal solution and (a*, b*) is an ~-level optimal parameter to the ~-MONLP, and the theorem is proved. If (x*, v*, a*, b*), an optimal solution to the minimax problem, is not unique, then we can test the o~-Pareto optimality for x* by solving the follow~ng problem: k
max
O0a)
~ ~,
snb]ectto ~(x,a,)+e,=~(x*,a*),
e,~O, i~1,2 ..... k,
(IOb)
Let (,~,a, l;) be an optimal solution to (10). if all ee =0, then x* is an e~-Pareto optimal solution. If at least one e~ >0, it can easily be shown that x is an o~-Pareto optimal solution. 3. Trade-off mtos Given the ~-Pareto optimal solution for the degree ~ and the reference levels spedfi~,d by the DM by solving the corresponding minimax problem, the DM must either be satisfied with the current e~-Pareto optimal solution and et, or update the reference levels and/or the degree ~. In order to help the DM express his degree of preference, trade-off informatio~-~ between a standins objective function a~d each of the other objective functions as well as between the degree and the objective functions is very useful. Fortunately, such a trade-off information is easily obtainable since it is closvly related to the strict positive Lagrange multipliers of the minimax problem!. To derive the trade-off information, we first define the Lagrangian function L for the minimax problem (6)-(9) as follows:
~=1
j=l rn qj
i = l r=l
j = l s=!
Interactiondecisionmaking
32[
In the following, for notational convenience we denote the decision variable in the minimax problem (6)-(9) by y = (x, v, a, b) and let us assume tha~ the minimax problem has a unique local optimal solution y* satisfying the following three assumptions. A~pfioe
1, y* is a regular point of the constraints of the minimax prob|em.
A~snmpflon 2. The second-order sufficiency conditions are satisfied ~t y*. A~mnpt]on 3. There are no degenerate constraints at y*. Then the following exls~.ence theo~¢ra, Which is based on the implicit function theorem [4], holds. T h e o ~ m 3. Let y* = (z*, v*, a*, b*) be a unique local solution o f the minimax problem (6)-{~) sallying the assumptions 1, 2 and 3. Let ~ * = ( ~r , M*, a b. a z') denote the L~grange multipliers corresponding to the constraints (7)-(9). Then there exist a continuously differentiable vector valued function y(.) and 3.(.) defined on some neighborhood N(~*) so that y(o~*) =y*, ~.(a~*)--~*, where y(a~) is a unique local solution o f the minimax problem (6)-(9) for any ~ E N ( c : ) satisfying the nssumpgons 1, 2 and 3, and ~(a) ~s the Lagrange multiplier corresponding to the constraints (7)-(9).
In Theorem 3, inf(v I,(x, a,) - ~ ~< v, i = 1. . . . .
k, (a, b) ~ L.(~, 6), x E X(b)},
can be viewed as the optimal value function of the minimax problem (6)-(9) for any ot E N(~*). Therefore, the following theorem holds under the same assumptions in Theorem 3. T h e o ~ m 4. If all the assumptions in Theorem 3 are satisfied, then the following relations hold on some neighborhood N( ~*) of ~*. aL
au
~
P'
"
~j
(12)
: - - - = - - £ £ ~,,+£ £ a~. _
0~'
_
0~0;"
a
b
l=l r=l
j=l s=l
If all the constraints (7) of the minimax problem are active, namely if u(0e*) =f~(x(~*), a~(~*)) - ~ , then the followin~ theorem holds. Theorem S. Let all the assumptions in Theorem 3 be sa~fed. Also assume that all the constraints (7) of the minimax problem are active. Then it hold* that k
~,~a[ O~'
I~'=w*
P~
m
=££~f+££~', i=l
r=l
qJ
j=l a=l
~=~ . . . . . ~.
o3)
Regarding a trade-off rate between f~(x) and f~(x) for each i = 2 . . . . . k, by
322
M. Sakawa, tt. Fund
extend:rig the results in Ha/rues and Chankong [5], it can be proved that the following theorem holds [17]. Theorem 6. Lee all the assumption in Theorem 3 be satisfied. Also assume that the constraints (7) are active. Then it holds that
Of(x, a~)
a~(~,a,) ,,.,,.
=
~,{"
-~,
i=2 .....
t,.
(14)
It should be noted here that in order to obtain the trade-off rate information from (13) and (14), all the constraints (7) of the minimax problem must be active. Therefore, ff there are inactive constraints, it is necessary to replace ~ for inactive constraints by j~(x*, a*) and solve the corresponding minimax problem for obtaining the Lagrange multipliers.
4. I~erac~ve algn~thm and 8nmerlesi example Following the above discussions, we can now construct the interactive algorithm ha order to derive the satisficing solution for the D M from among the ot-Pareto optimal solution set. The steps marked with an asterisk involve interaction with the DM. Step 1. Calculate the incBvidual minimum and maximum of each objective function under the given constraints for o~= 0 and 0c = 1. Step 2". Ask the D M to select the initial value of of (0 < ~ < 1) and the initial reference levels ~, i = 1. . . . . . k. Step 3. For the degree a and the reference levels specified by the DI~, solve the minimax problem and perform the oc-Pareto optimality tesl. Step 4*. The DM is supplied with the corresponding ot-Pareto optimal solution and the trade-off rates between the objective f~nctions and the degree ~. If the D M is satisfied with the current values of the objective functions and oc of the ~-Pareto optima~ solution, stop. Otherwise, the DM must update the reference levels and/or the degree ~ by considering the current values of the objective functions and ot ft~gether with the trade-off rates between the objective functions and the degree ~ and returl~ to step 3. Here it should be stressed for the DM that (1) any improvement of one objective function can be achieved only at the expense of at least one of the other objective functions for some fixed degree 0t, and (2) the greater value of the degree 0~ gives worse values of the objective functions for some fixed reference levels. Based on the method described above, we have developed a new interactive computer program. Our program is composed of one main program a~d several
lnteract~eedecismn n',ak~g Table 1. ~
,~ta a~z a~l ~;',z
a31 ~3z ~.33
b.:: hi2
GI3
323
numbers for nnraedcal example
Pl
P2
P~
P4
|C~
n~lt
3.8 57.0 18.0 1.'/5
4.0 59.0 19.5 2.0
4.0 60.0 20.0 2.0
4.3 63.5 22.5
L E E
2.3
2.5
2.5
1.25 17,5 0.9 0.8 0.85
1.4 20.0
1.5
1.7
20.0
22.0
E L E L E L E ]E E L
1.0
0.95 1.0
1.0 1.0 1.0
2,23 2.75
1.1 1.2 1.15
E
L L L E E E
CONHAND: ?GO .....................
< ITERATION 1 >. . . . . . . . . . . . . . . . . . . . . .
INPUT YOUR REFERENCE VALUES F ( 1 ) ?5025 5650 6062,5
(I=1,3):
I ~ P U T THE DE~REE ~LFR OF THE RLFR LEVEL SETS FOR THE FUZZY PARAMETERS: ?0.9 ( KUHN-TUCKER COND|T]ONS S A T | S P I E D ) RLFA-PARETO OPTIMAL SOLUTION TO THE H I N I H A X PROBLEM FOR 1 N Z T I A L REFERENCE VALUES ............................. OBJECTIVE FUNCTION ............................. F(I) = 6438,9620 F(2) = 7063.9620 F(3) = 747&,4&20 .......................................................... X(I) = 6.3782 X(2) = X(3) = 4.407~ .......................................................... TRADE-OFF5 AHONG OBJECTIVE FUNCTIONS -DF(2)IDF(I) = !*Z52! -DF(3)/OF(I) = 1.3422 ........................................................... TRADE-OFFS BETWEEN ALFA AND OSaECTIVE FUNCTZONS DF/DALFA = 797.2460 .........................................................
6.5187
ARE YOU S A T I S F I E D N I T H THE CURRENT CoSJECT]VE VALUES OF THE ALFR-PARE~O OPTINAL SOLUTXON ? ?NO
Fig. 2. Interactive dcdsion maidng processes.
324
M. Sakawa, H. Yono
subroutines. The main program calls in an~ tuns the subprograms with commands indicated by the use (DM), Consider the following three-objective nonlinea~ programming problem with fuzzy parameters:
miu A(~, ~,) = ~l,x~ + (x~ + 5) ~ + 2 ( x ~ - ~,2Y, rain A(a, ~z) = (xl + 621)2 + ~2z(xz - 55) 2 + 3(x3 + 20) z, subject to
" -2.~ x e X(/~) -- {(xt, x2, x3) I gl(x, bl) - 6 n x l2 +/;l~c22 + G~3xa ~ 100,
0~&~
10, i = 1, 2, 3}.
The membership functions for the fuzzy number: ~ . . . . . ~ , / / , . . . . . G~3 in th/s ezample are explained in Table 1 where L and E re,present respectively linear and exponential memlmrship functions. In Figure 2, the interaction processes using a time-sharing computer program under TSS of a M E L C O M COSMO 700S digital computer ai th~ computer center of Kagawa University are explained especially for the first ;~era~ion through the aid of some of the computer outputs. 0~-Pareto optima| sobz~ions a~-e obtained by solving the minimax problem using the revised version of the genera~:ed reduced gradient (GRG) [6] program called G R G 2 [7]. Since the DM is not satisfied with the current objective values, the DM updates his reference :evels. In this example, at the 6th iteration, the satisflcing solution of the DM is derived. The complete interactive processes are sununadzed in Table 2. CPU time required in this interaction process was 312.815 seconds and the example session ~akes about 13 minutes.
Table 2. Interactive processesfor numericalexample Iteration f2 at ~(x, nO f2(x, a2) fa(x, a3) ~l a2
xa -OA/OA -Ofa/Of, a~/a¢~
1
2
3
4
5
6
5025.0 5650.0 6062.5 0.9 6438.96 7063.96 7476.46 6.3782 6.5187 4.4076 1.1521 1.3422 797.25
6860.0 6700.0 6800.0 0.9 7008.58 6908.58 7008.58 7.7803 6.1078 2.1969 0.9217 0.9205 835.g0
6500.0 6300.0 6700.0 0.7 6823.52 6623.52 7023.52 7.3317 6.8347 2.5747 0.9384 1.0323 719.52
6700.0 6:Yo0.0 6900.0 0.65 r 6797.90 6587.90 6987.90 7.3351 6.0771 2.6287 0.9448 1.0392 705.95
6300.0 6600.0 6~J00.O 0.65 6492.66 6792.66 7092.66 7.0352 6.5879 3.8383 !.0935 L 1903 o9.L09
65~0.0 6700.0 7000.0 0.65 6556.7I 6756.71 7056.71 7.1465
6.6174 3.5815 1.0629 1.1506 696.61
Interactive decisio~ making
3~
In this paper, we have proposed an interac¢ive decision mal~ng method in order to deal with the multiobjective nonlinear progra~tnnng problem with fe~,y pararaeters characterized by fuzzy numbers. Through the use of the ~oncept of the of-level sets of fuzzy numbers, a new solution concept caUed ~-Pateto optir~ali~y has been introduced. In our interactive scheme, the satisficiag solution of ~he D M can be ~efi,,,e~ from a m o n g an ~-Pareto optimal r~3lution s~t by updating the reference levels a n d / o r the degree ~ based on the current values of the membership functions and ~ together with the trade-off ~ates between the objective functions and the degree o~. A n illustrative numer/cal example clarified the various aspects of both the solution concept of ~-Pareto optimality and the proposed method. However, further applications must be carried out in cooperation w~th a person actually involved fin decision making.
References [1] R.E. Bellman and L.A. Zaduh, Decision making in a fuzzy environment, Management Science t7 (4) (1970) 141-164. [2] D. Dubois and H. Prade, Operations on fuzzy numbers, Internat. J. Systems SoL 9 (6) (1978) 613-626. [3] D. Dubois and H. Predc, Fuzzy Sets and Systems: Themy and Applications (Academic Pre~, New York, 1980). [4] A.V. Fleece, Introduction to Sensitivity and Stability Analysis in Nonlinear t~osr~mndng (Academic Press, New Yogk, 1983). [5] Y.Y. Haimes and V. Chankong, Kuhn-Tucher multipliers as trede-offs in multiobjecfive decisinn-makinganalysis, Aut~omaticat5 (1) (1979) 59--72. [61 L.S. Lasdon, R.L. Fox end M.W. Ruiner, Nonlinear optimizatinn winS the generalized reduced ~ndient method, Ray. F:au~aise Automat. Informer. Recherche Op6rationoelio 3 0974) 73-103. [7] L.S. Lasdan, A.D. Wasen and M.W. Ratner, GRG2 User's Guide, Teehglif~ Memorandum, University of Texas (1980). [8] S.A. Orlovski, Problen~ of decisinn-makingwith fuzzy information, Working Paper WP-83-28, International |nstitote for App:!~l Systems Analysis, Laxenburg, Austt'ia (1983). [9] S.A. Orlovski, Multlobjective programming problems with fuzzy parameters, Control and Cyhernet. 13 (3) (1984) 175-183. [10] M. Sakuwa, Interactive computer programs for fuzzy linear programming with multiple objectives, Interoat. J. Man-Machine Stud. lg (5) 0983) 489-503 [II] M. Sakawnand I". Yumine, Interactive fuzzy decislon-makingfor multinb~:ctivelincar fractional ptc,,gramndng problems, Large Scale Systems 5 (2) (1983) 105-114. [12] M Sakawa and H. Yano, An interactive fuzzy satisficing method using penalty ~'.a[arizing problems, Prec. Int. computer symposium,Tamkang Univ., Taiwan (1984) 1122-1129. [13] M. Sakawa and H. Yano, Interactive fuzzy decision making for multiobjective nonlincor programming using augmented minimaxprohlerss, Fuzzy Sets and Systems 20 (l) (1986) 31-43. [14] H. Tanaka and K. Asai, A formulation of linear programming ~obiems by fth~,y functinn, Systems and Control 25 (6) (198i) 351-357 (in Japanese). [15] H. Tanaka and K. Asai, Fuzzy linear programmingproblems wit~,fuzzy numbers, Fuzzy Sets and Systems 13 (1) (1984) 1-10. [16] A.P. Wie~bichi, The use of reference ob~ectiws i~ ~nulti~bje~ive optimization-~teotetical
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implications and practical experiences, Worlfing Paper WP-79-66, rmemation~ Institute for Applied $~temsAnalysis, L ~ e n b c r ~ Austria (1979). [17] H. Yeno and M. ~ a w a , Trade-off rates in the weighted Tchebychcff norm metho~, T t ~ . S.LC.E. 2I (3) (1985) 2 4 q - 2 ~ (in .Tapanes¢). [~18] L.A. Zadeh, The coneept of a |hlguhtlc variabJ~ and its application to a p p ~ x ~ ' e ~a.~nlvg-1, Inform. $ci. 8 (1975) 199-249.