SUBJECTIVE TRADE-OFF RATE METHOD OF MULTIOBJECTIVE DECISION-MAKING

1996,16(4):432-441

SUBJECTIVE TRADE-OFF RATE METHOD OF MULTIOBJECTIVE DECISION-MAKING 1 Wang Xianjia (L7t \f ) Wuhan Unive1'sity of Hydraulic and Electric Enqineering, Wuhan 490072, China.

Wang Qi'Ut-ing (L,fti!.) Wuhan Yejin Unive1'sity of Science and Technology, Wuhan 480070, China..

Feng Shangyo'U (,~ ~.&.

)

Wuhlln Unive1'sity of Hydraulic and Electric Engin,:e1'ing, Wuhan 4/J0072, Ch·ina.

Abstract This paper proposes a mat.hod of subjective trade-off rate which describles decision-makers preferince in multio bjective decision-making. Decision-maker can arbitrarity determine his subjective trade-off rate ,but it is not sure to be effective.The paper finds an effective upper bound of subjective trade-off rate,which is the KuhnTucker multiplier of some mathematical programming.For the subjective trade-off rate not being larger than the upper bound, the solving method and properties of the optimal solution corresponding the trade-off rate are discussed.The paper lastly develops the process of solving multiobjective decision-making with the subjective trade-off rate method. Key words Multiobjective decision-making.Subjective trade-off rate, Kuhn-Tucker mul ti plier.

1 Introd uction According to [l],a decision problern is described as a pair (X, O),where X is a decisionrnaking possible set and 0 is a choice principle. To rnake a decision is to choose an element or a subset from the decision-making possible set. The choice principle n must be a criterion which can rank the alternatives in the making-decision possible set, so that the final decision can be selected. However, in practical rnaking-decision problerns, it is difficult directly to distinguish the good alternative frorn the alternatives themselvies. Generally, the alternatives which are either good or bad can be represented by the decision effects of the alternatives. One oiler uses a set of attributes or objectives to describe the decision effect.Mathematically,a vector-valued function f defined in X can represent the attributes. Therefore, a decision problern can be described as a triplet (X, t, 0). We call (X, f, 0) a multiobjective rnaking -decision problem.Traditional rnultiobjective rnaking-decision takes 1 Received Feb.a, 1995. Supported by the Nationel Natural Science Foundation of China and the Laboratory on Industrial Control of Zhejiang University of China.

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o as a natural partial order (i.e. if :1~, Y E Rn, x 2:: Y -¢=::::} Xi 2:: 1!i, i = 1"", n). Under the choice principle, the concept of Pareto, or noninferior solution was introduced at the turn of the century by Pareto, a prominent economist. [2] indicates that the set of Pareto solutions usually is large. The decision maker does not know how to choose the final decision, because the alternatives in the set of Pareto solutions can't be ranked by using the natural partial order. In order to make the final decision, it is necessary to change the choice principle, or decision rule. Hames et al. proposed a method called the surrogate worth trade-off method (SWT)[3].The method requires the decision rnader to give a scale of +10 to -10 to indicate one's preference based on Kuhn-Tucker multipliers of an [-constraint problem. In practice, it is difficult that the decision Blaker elaborates on the degree of preference. III addition, the method still requires that the noninferior solution, at which the trade is made, shoudl be generated by an £-constraint problem.In this paper, we proposes a mulbiobjecbive rnakingdecision rnethod, called subjective trade-off rate rnethod, for which the decision-rnader can arbitrarily give his trade-off rate between any two objectives to represent his aspiration at any noninferior solution, not necessarily generated by an £-constraint problem. Our method indicates that though the decision Blaker can arbitrarily give his aspiration by trade-off rate between two objectives, the aspiration rnay be ineffective.because it may be beyond the objective things. The method brings about an efficient interactive process of ruultiobjective making-decision.

2 TIle Concept of Subjective Trade-Off Rate Let X ~ R 1t be a set of a possible making-decision alternatives, called decision space, / : X

~

R n t be a function describing the rnaking-decision effects of the making-

decision alternatives, and f2 a choice principle. The rnultiobjective rnaking-decesion problem is formally described as follows:

(1) F

~ {y I 3x

E X, s.t.y = f(x)} is called the objective space (or criterion space). In fact, the

choice principle !1 is a binary relation on the objective space X. So, 11 can be represented as a subset of F x F,i.e.O ~ F x F,which can be used to be ranked alternatives through ranking corresponding objective vectors. Actually, 0 reflects the decision maker '8 preference. If the decision maker prefers J~ to y, or thinks that ~.D is at least as preferable as :'I, we say (~l;, y) E O. In our mind, a rnultiobjective making-decision problem can be represented as (1) unless f2 at least includes any pair (:1;, y) so long as .y1 ~ y2, y1, y2 E F, that is to say that if:'11 ~ y2, the decision maker prefers y1 to y2.

Definition 1 Let

E X,

is said to be an optimal solution of the multiobjectiv« making-decision problem (1) if there is not x E X, so that f(J;) i= f(:.c*) and (/(:1;),/(:1;*)) E !l.The set of all optimal solutions of (1) is denoted as X n . :1;*

:1;*

x* E Xn can he interpreted as an optirnal choice (or decision) of rIlllltiohjective prohleul

(X, f) with regard to the decision rnaker's preference !l, because the decision maker dose

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not prefer any other alternatives in X to z". When

(2) the multiobjective making-decision problem (1) changes into the traditional multiobjective rnaking-decision problem finding its set X n p of noninferior solutions. However, the noninferior solution hardly seem to be thought corresponding a decision mader's preference. Our concept of the optimal solution of multiobjetive making-decision problem (1) includes the concept of the noninferior solution and makes the noninferior solution connected a decision maker's preference. There are many methods finding Xnp [4]. SinceXjj; is generally very large and FO F {y E F J 3x E Xnp,s.t.y I(x)} is also very large.It is difficult that the decision maker makes the final decision from the set of noninforior solutions. Futhermore, the decision maker may choose one between noninforior solutions through rnaking the trade-

=

=

off between any two objectives, and changing Op into a new choice principle O.Xn may still be large. The above process can be repeated. At any alternative level x O E X (or corresponing objective vector level yO I( xO)), the decision maker can make his trade-off between any two objectives to indicate his preference. The decision maker is asked, "how rnuch would you like to irnprove Ii for per one-unit increment of Ij(xO), while all other objectives remain fixed at Ik(XO), k =1= i~ j." The decision maker can subjectively give his respond. The more the degradation of li,the better for decision maker. There is the least degradation of Ii which can be accepted in decision maker's mind.The mount of least degradation of Ii is said to be the decision-maker's subjectiv.e trade-off rate of the objective Ii for the objective Ij. Its definition is as follows: Definitioin 2 Let yO E F, set

=

-Wj = min {-~ > 0

ify? - yt 2 A(Y] -

jJ),y~

=

y~, k E {1,··· ,m} \ {i,j} .u' =1= yO,

the decision maker prefers y1

to

yO .

A?i is said to be the decision maker's subjective trade-off rate of the objective objective

t;

Ii for

}

,

(3) the

Remark 1 Though the decision maker can give his subjective trade-off rate according to his subjective wishes, a rational decision maker should make his subjective trade-off rates among all objectives satisfy some consistency conditions: (a)

(b)

A?i>O, for any i,jE{l,···,m}, A?i = A?kAgi' i,j,k E {l,·.·,m}

(4a) (4b)

Relationships (4) rnay be seen as the text conditions of subjective trade-off rates which are whether reasonable or not. Remark 2 In fact, if A?i is given, a decision maker's new local preference is determined.The binary relation of the decision maker's new preference is as follows:

(5)

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We think that the decision maker's preference changes from np to n° = f2p U A?j' Of course, the decision maker can also made trade-off between the other two objectives. After

consturcted the decision maker's new preference, we want to find the optimal solution 3~1 with regard to the new preference n°.The decision rnaker rnay accept the decision 3~1 .or he must fur ther rnake a trade-off between two objectives. Thus, above process can be repeated. The process can actually be seen as an interactive process between the decision maker and the analyst. It can be tested that to find the optimal solution of (X, so that y*

= f(3~*) E Y!lp n ~~j,where

fnO)

is to seek some z" E X,

and ,1

Yi -

,0

Yi

> \(, 1 _ A Yj

-

Yj0) ,A\

> \ 0 ,1 _ Aij,!Jk

k E {I ... , m} \ fi, j} ,y

1

_

-

, 0

!Jk'

-I yO

}.

(6)

3 TIle Effectiveness of Subjective Trade-Off Rate We do not give any limit to how the decision rnaker makes his trade-off rate between two objectives corresponding to his subjective wishes. His ambition may is so great that his desire for the least amount of degradation of fi per one-unit increment of fj is beyond the extent of the objetively practical problem, that is to say that the alternative satisfying the decision maker's subjective trade-off rate cann't be found in the decision space.The subjective trade-off rates is said to he ineffective. What is the suitable upper bound of the subjective trade-off rate'? Definition 3 Let yO E F, A~j is a subjective trade-off rate of decision maker of the

objective fi with regard to the objective fj at the objective leves yO. If Yo p is said to be an effective subjective trade-off rate, where ~?j is given by (6).

n ~?j -I 1;, A~j

Remark 3 If only the subjective trade-off rate is effective, the decision maker's subjective wishes and the objective probability can be united.Obviously, the decision maker mainly concerns the least upper bound of the effective subjective trade-off rate. Lemma 1 Let h, Yi(i = 1,·,·,1) be real convex functions defined on R"', y* E RfI" Set (7)

If the single objective problem rnin h( x) exists the optimal solution for any :r.EX(e)

(where B(O) is a unit sphere of origin),then the optimal value function V(E) is a convex function, Proof Suppose

X(El),

X2

E

1,

E

2

E E

B(O)

~

R'

= :r.EX(e) ru!n h(:J;)

E B(O), 0 ~ a ~ 1. According to the conditions.there exists

E X(E2)' such that V(E 1 ) = f(3~1) ,and V(E 2 ) = f(:J~2). Let ?is = O:~L:1

+ (1 -

Xl

E

a)x2'

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because of the convexity of Yi, we have

Yi(aXl

+ (1 -

a)x2)

~

aYi(xl) + (1 - a)Yi(x2)

~ a(Yi

+ e;) + (1 -

a)(Yi

+ e;)

= Yi + act + (1 - a)e;. Therefore, x E X(ae l + (1- a)e 2 ) . Since for "'Ix E X(~t:el + (1- a)e 2 ) , V(ae l I(x) is right, we have

Lemma 2 Let C C R 1t be convex set and

+ (1- a)e 2 )

~

x E C,6(· 1 C) is the indicator function of

C, i.e.

6(x 1 C)

={

Then 86(x 1 C)

O

xEC

+00

x t/:. C,

= Nc(x),

(8)

where 81(x) indicates the subdifferential [5] of the convex function Nc(x) = {v E R"

Proof Let

x

E C and

'l'

1< v, x -

1

at x, and

(9)

X >~ 0, "'Ix E C}.

E 86(x I C), then for "'Ix E R"; we have

6(x I C) ~ 6(x I C)+ < "', X Because for "'Ix E C,6(x

I

-

X

>=<

C) = 0 is right, we have <

v, x - X

'l', X -

>.

X >~ 0, "'Ix E

C. Therefore

v E NcCx). Conversely,let v E Nc(x), then for "'Ix E C we have < v, x - X

6(x I C) = 0

~< v,

0, and we also have

x - X >= 6(x 1 C)+ < v, x - X > .

So we have v E 86(x 1 C). For x t/:. C, we have h(x I C) =

6(x I C)

>~

~

6(x 1 C)+ <

V,x

+00.

So we also have

-x >.

Lemma 3 Let h,Yi(i = 1"" ,p) be the real differentiable convex functions, B(O) be a unit sphere of origin in RP. Suppose the single objective problem rnin h(x) exists the xEX(e)

optimal solution and z" is an optirnal solution of the single objective problem rnin h( x) for e

= O.

Set Vee)

then

xEX(e)

= xEX(e) min hex). If {Vhj(x*),j E I} is a linear independent vector group, 8V

-Be, le=o= -Ai,

(10)

where 1= {j E {I"" ,p} I gj(x*) = yJ} and (AI,"', Ap) is the Kuhn-Tucker multipliers of the problern ruin hex) at (x*,O), i.e. (AI,'" ,Ap) satisfies that xEX(e)

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Wang et al: METHOD OF MULTIOBJECTIVE DECISION-MAKING

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(a)

(1))

(A1, .•. , Ap)

~

0,

(A1, ... , Ap

11

)

i- 0,

(1Ia)

= 0,

L A'i{Yi(X*) - Y;) i=l

(lIb)

p

\7h(x*)+ L\7gi(x*) = O. (lIe) i=l Proof According to Lemma 1, V (E) is a convex function, In the first place we give the subdifferential expression of V (E) at E = O. Let (c)

C

= {( e, z ) E RP x R

1l

lEE B (0), gi( x)

~

y; + < e, e, > ,i = 1, ... , p} ,

where e, is IJ-dirnensioll unit vector whose 'ith elernent is 1 and the others is Since for VE E B(O), the problem

rr~n :r.EX(g)

V (E) = ruin h( x) = inf {h( x) :l:EX(g)

o.

h(x) exists the optimal solution, we have

+ s(( E, 3;) I C) I 3; E R l- . 1L

Since {\7 Yj (:1;*), j E I} is a linear independent vector group, there exists dO E RJl. such tha.t

< \7Yj (a;*), dO >< 0,.7 EI. Thus there also exists :1; E Rn such that Yj (J;) < Yj,.i = 1,· .. ,1). By the continuity of Yj, we can obtain (0,:1;) EintC (inner of C), and we can also have

(0, ;/;) Eillt(doIIH5(. I C)) (darn f represents the effective domain of f).Set
t

.1l

intC; -#
Db ( (0, J; *)

I C)

=

L {Aj {- e j, \7Yj (

J; * ))

I A.i 2:

O} .

.lEI

Also,8
= (0, \7h(;c)) + L

{Aj(-ej, \7Yj{a;*)) I Aj

2: O} .By [6] , we

have

JEE

8V ( 0) = DV(O)

=L

lEI

{~ E

R 1U

I (~, 0) E D (0, :D *)} ,

{-Aje I Aj 2: 0, Vh{x*) + L(A.i VfI;{:I;*)) = O} . j

lEI

p

For

.7 E {I,··· ,p} \ I, let Aj = 0, 111,

we have (Al,···, Al' )

2: 0,

L Ai(Yi(:I;*) -

yi) = 0, and

i=l

Ai\7fJi(:I;*) = O. If (Al,···,A p ) = 0, for.i E I, we have NC.i(O,x*) = {OJ, i=l this leads to a contradictioll.SO,(Al'···' Ap ) is the Kuhn-Tucker multipliers satisfying(lla)-\7h i(:c*)

+L

(lIe). Again.since the vector group {\7 gj (:1;*),.i E I} is linear independent and \7 h( x*) +

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ACTA MATHEMATICA SCIENTIA

E AjVgi(X*)

Vol.16

= 0, by the unique theorem of linear expression,(Al,···, Ap ) satisfying the

iEI

p

above conditions is unique. Thus aV(O) is singleton.and the elernent is

E -Ajej .By

[5],

j=l

V(E) is differential at E = O,and aV(O) = VV(O). So, g~.3 le=o= -Aj. Theorem 1 Let X = {x E R" Igj(X) ~ O,j = 1,··· ,p}, I(x) = (/l(X),···, Iru(x)), Ii, gj(i = 1,· .. ,rn; j = 1,·· ·,1)) be real differentiable convex functions, xO be a noninferior solution of Op -rnin f( x) and yO = l(xO).Suppose {V li(XO), V gj(xO), i = 1,... , m, j = 1, ... ,p} xEX

is a linear independent vector group. Consider the following

E-

constraint problern PP(Ej) :

(12a) (12b) then (a)

(b) (c)

If Ej = 0, then Vi(O)

-t:

= yp,

d~t;) le;=o= (13) i, Let A?j be the decision maker's subjective trade-off rate with regard to the

-t:

t:

objective Ii for the objective Ij at the level yO. If f:1?j :I ¢J, then A?j ~ j , where j is the Kuhn-Tucker multiplier of E- constraint problem PP(Ej) at xO and Ej = 0 and f:1~j is given by (6). Proof (a) If Ej = 0, obviously xO E Xi(Ej). So we have Vi(O) ~ fi(XO) = yp. If there

yr.

is xl E Xi(O), so that Vi(O) = fi(X l ) < Since xl E Xi(O) we have Ij (xl) ~ yJ, fk(X l ) :s y~, k E {1, . · . ,m} \ {i, j} .So f(x l ) ~ f(xO) and I(x l ) :I f( xO).This leads to a contradiction with XO being a noninferior solution of the problem Op - lrlll! f(x).So, for any x E Xi(O), xEX

then Vi(O) ~ Vi(x), and Vi(O) = yp. (b) By Lemma 3,it can immediately be proved. \0 > ~ . dVi(ej) I - -/\ij' ~ we Ilave TT.(.) ( c0)" If /\ij /\ij' SInce ~ ej=OVi E J

where o( Ej) represents the higher infinitely small of

~

Since >..?; - Ai; > 0, there is lj > 0, so that when

Ej

-

TT·(O) ~. Vi -- -/\ijE J

+ 0 (.) EJ ,

.Also,

°< E < lj

we have

(14) Again since f:1?j

:I ¢J,let yl

E f:1?;, there is xl E X ,so that f(x l

)

= yl, y? - y; = A?j(yJ - y])

and y~ ~ y~ fork E {1,···, m} \ {i, j}. Without lossing generality, let

°

(1

0)

t

Yl - yJ > 0, then

there is A > 0, so that < Yj~Yj < lj. Set x = + A~lxO, and y2 = f(x 2 ) . Since gj(j = 1,··· ,p) are convex functions ,X is a convex set.We have x 2 E X and y2 E Y = f(X). Again since f is a convex function,we have 2

xl

Wang et al: METHOD OF MULTIOB.TECTIVE DECISION-MAKING

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2 and 'fl .1

-

'yO •

<

l('yl - 'uO).Therefore A' ,J ,

-

o ') 0 A?i ('II'1 - 'y.) 0 1 0 1 A''I).(.,y-: - 'II.) < .:»: .) .I) A.-I) ) < - -(yo A 'I - 'fl') .1'1 2

1 (.

0

1

0)

Yk - 11,-: ~. ~ Yk - Yk ~ (1

0)

In the problem PP(£j) , take £j = Yi~Yi, then 0

Vi(E j) ~ Ii ( X

2

439

)

= y;, an d

0

< ('fl'0 _.11

?

- '1/";') .'It'

(15)

.

< £j < [j, and

3.;2

E Xi(£j). So we have (16)

Cornhining (15) with (16),we can get

This leads to a contradiction with

(14).So,A~~i ~ \~.

Definition 4 \~ given by (13) is said to he the objective trade-off rate of multiobjective problem (X, I) with regard to the objective I. for the objective lj at -the level yO (or ;cO). Remark 4 Theorern 1 gives an important result: at some objective level yO of the noninferior solution Xo, though decision maker can arbitrarily give his subjective trade-off rate between two objectives, it will be ineffective when it is greater than the objective tradeoff rate. This shows the restrain of the objective problem for the subjective wishes.The objective trade-off rate \~ given by (13) is an upper bound of the effective subjective trade-

off rates. Obviously, any number which is greater than >!;j can be taken a.s the upper bound of the effective subjective trade-off rates. The most important one of all upper bounds of the effective subjective trade-off rates is the least upper bound.

Theorem 2

Let the conditions of Theorcrll 1 hold, \~i he the decision maker's

subjective trade-off rate with regard to the objective Ii for the objective

/i

at the level

yO,\~ be given by (13), and \~ < A~j. If £- constraint problem pi(£j) exists the optimal solution for any £j > 0, then there is f > 0, so that if 3~* is an optimal solution of £constraint problern PP(£j) for 0 < £j < r., then f(3~*) E ~~j, where P/)(£.i) and ~~j are given by (12) and (()) .respectively.

Proof

By Theorem l(b ),let lfi(£j) =

dVi(£.1) d£j

Vi(£j) - Vi(O) = -t:.i£j

rf!in

:1:EXi(£:i)

I£i=O-- -

+ o(£j)

.

Ii (:l;),where Xi(£j) is given by (12b),then

s:. 1.)'

= -(>::'i - A?j)£j - A~~Ej

+ O(Ej),

< \~, there is fj .> 0,80 that when o· < Cj < Ej,- Aij - Aij)£j + o(£j < O. So,Vi(£j) - 'Ui(O) < -AijE.i for 0 < £j < fj. Let x is an optimal solution of PP(£j),then Vi(£j) = fi(x).Set y = I(x). Since x E Xi(£j),we where o( Ej) is the higher infinitely small of Ej .Since A?j (.)

-

(~)

0

)

.

.

have Yl~ ~ yO for k: E {I",· ,Tn} \ {-i"i} , and Yj ~

:'Ij + £j'

By Theorcrn l(a),we have

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ACTA MATHEMATICA SCIENTIA

y?

= 'lJi(O).Thus,Vi(cj) - Vi(O) = Yi Y E .x?j and f{x) E .x?j.

y? < -.x?jcj,and y? - f!/

Vol.16

> .x?jCj ~ .x?j(Yj -

Theorem 3 Let the conditions of Theorem 2 hold. If there is an cJ, 0

yJ).

So,

< cJ < Ej, so

that the optirnal solution of c- constraint problem PP(cJ) is unique, then .x?j is an effective subjective trade-off rate and the optimal solution z" of c- constraint problem Pl{cJ) is in ~?j,where PP(cJ) and ~?j are given by (12) and (6), respectively, and Ej is given in the process of Theorem 2'8 proof. Proof By the above Theorem 2 and Theorem 4.30 in [8],it is easily proved. Remark 5 Theorem 2 and Theorern 3 indicate that the objective trade-off rate is the least upper bound of the effective subjective trade-off rates, and the optirnal solution of multiobjective making-decision problem (X, t, 0), where choice principle gives an effective subjective trade-off rate, can be solved by a nonlinear programming problem, c- constraint

problem PP{Cj).

4 Solving Process of the Subjective Trade-Off Rate Method According to above analysis, the subjective trade-off rate method of multiobjective decision-making can be designed as an interactive decision-making process between the decision maker and the analyst. The process is as follows: yO

=

Step 1 Finding a noninferior solution xO of the problem Op - nllnf(x) and setting xEX f{xO);

Step 2 Asking the decision maker to make his subjective trade-off rate .x?j between two objectives fi and fjat the level yO; Step 3 Solving c- constraint problem PP(Cj) and the objective trade-off rate

>::j =

- dv~til lej=o, where PiO(Cj) and >:;j are given by (12) and (13),respeetively; Step 4 Comparing the subjective trade-off rate .x?j with the objective trade-off rate >:;j, and judging the effectiveness of the subjective trade-off rate. If A?j ~ >:;j, we will return to step 2, or we will turn to further step; Step 5 Choosing r, > 0, so that Vi(cj) - V(O) < -A?jcj when 0 < Cj < Ej, where Vi(cj) is the optimal value function of c- constraint problem PP(Cj) given by (12); Step 6 Finding an optimal solution z" of E>: constraint problem PP{Cj), for () < Cj < Ej; Step 7 Testing whether z" is the noninferior solution of the problem Op - lllinxEx f( x). If No, return to step 6, or turn to further step; Step 8 Telling decision maker that z" is better than x O under his given subjective teade-off rate and asking him whether to accept z" as the final decision. If his respond is Yes, then the process is completed and x* will be taken as the final decision, or return to step 2 at the level y* = j(x*).

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441

5 Conclusion In.thispaper..a multiobjective decision-making problem is described as a triplet (X, t, 0) . . - ,. ,

.'

and formally' as (1). "'~'raditional rnultiobjective problem (only finding noninferior solution) is a special form of (X, t, 0). Since the noninferior solution set is generally very large, the final decision canu't be conveniently made. So we must change the choice principle 0 because the other elements X and

f cannt be changed. Actually, the

choice. principle reflects the

decision Blaker's preference. This paper proposes a niothed called the subjective trade-off rate to describe the decision maker's preference.The rnethod asks t'he decision maker to make the trade-off 'between gain and loss of two objectives. The decision maker can arbitrarily give his subjective trade-off rate, but our analysis will tell him which subjective trade-off rate is ineffective. We give the least upper bound of the effective subjective trade-off rates, and the solving rnethod when the subjective trade-off rate is effective. We also design the interactive decision-making process of multiobjective decision-making with the subjective trade-off method.

References Wa.ng Xianjia.. A Stucly on Theory and Application of two-level System Opbirniwation. Ph. D.

Thesi~,

Wuha.ll University of Hych'oulic & Electric Engineering, 1992. 2 Clune I S, Yu P L, Zhang D. Indefinite Preference Structures and Decision Analysis . .T. of Opti. The. and Appl., 1990,64: 71-85. 3 Hairnes Y Y, Hall W A. Multiobjectives in Water Resources Systems Analysis: the Surr'ogat.e Worth Tra.de-off Method. Water Resources Research, 1974,10: 614-624. 4 Chankong V, Haimes Y Y. Multiobjective Decision Making, 'I'heory and Methodology. North-Holland, New York, Amsterdam, Oxford, 1983. 5 Rockafellar R T. Convex Analysis, Princeton Univ. Press, 1970. 6 Rockafellar R T. The Theory of Subgradicnts and Its Applications to PrnblcIllS of Optirnizubion of Convex and Non Convex Functions. Helderrnann Verlag, Berlin,1981. 7 Rockafellar R T. Conjugate Duality and Optimisation. SIAM Publications, 1974. 8 Haimes Y Y, Chandong V. Kuhn- Tucker Multipliers as Trade-offs in Multinhjeetive Decision-Making Analysis, AutolIlfl.ties, 1979, 15: 59-72.

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