kuhn-tucker multipliers as trade-offs in multiobjective decision-making analysis

Automatica Vol. 15~ pp. 59 72 Pergamon Press Ltd. 1979. Printed in Great Britain © International Federation of Automatic Control

in Kuhn-Tucker Multipliers as Trade-Offs Multiobjective Decision-Making Analysis*

YACOV Y. HAIMESt and VIRA CHANKONG +

An extension of the theoretical basis to the Surrogate Worth Tradeoff (SWT) method provides useful relationships between optimal Kuhn Tucker multipliers and trade-offs in multiot~jective optimization problems.

Key Word Index Multiobjective optimization: Kuhn Tucker multipliers; trade-off analysis: Pareto optimum; efficient solution; decision-making; non-inferior solution: noncommensurable objectives.

Summary-Usefu l relationships between the optimal Kuhn Tucker multipliers and trade-offs in the multiobjective decision-making problems are developed based on the sensitivity interpretation of such multipliers. Practical and theoretical applications of these results are discussed. The results "provide a convenient way for obtaining necessary (trade-off) information for continuing into the analyst decision-maker interactive phase of the multiobjective decision-making process. This paper further extends the theoretical basis of the Surrogate Worth Trade-off (SWTI Method; a multiobjective optimization method which first appeared in the scientific literature in 1974.

1. INTRODUCTION

THIS paper focuses on a certain aspect of multiobjective decision-making in large and complex problems. In recent years, criticism over the use of a single criterion as a sole basis for decision- making in a large and complex system has brought about an enthusiastic search for effective techniques capable of handling multiple objective decision-making problems. It has become increas- ingly obvious that decisions that affect different segments of society generally need to be based on more than one decision criterion and that these criteria are often conflicting and noncommensurable in nature. In a planning for management and development of a river basin, for example, both economic efficiency and environmental im- pacts (e.g. water quality, land use, recreation, wildlife, etc.) have been identified as two equally important but conflicting and noncommensurable principal goals and they might be best treated as such. Correspondingly, most analyses of large and complex systems involving social decisions, whether it be for development planning or for operational management, usually results in multiobjective model formulations to which the application of traditional single criterion decision-

*Received February 13, 1978. The original version of this article was presented at the IFAC VII World Congress, Helsinki, Finland, June 12 18, 1978. This article was recom- mended for publication by associate editor K. J. ~,str6m.

#Professor of Systems and Civil Engineering, Case Institute of Technology, Case Western Reserve University, Cleveland, Ohio 44106, U.S.A.

+Faculty of Engineering, Khonkaen University, Khonkaen, Thailand.

59

making techniques (such as benefit-cost analysis etc.) has been found to be less than adequate; and so the search for multiple criteria techniques continues.

The general philosophy taken in this paper is that the multiobjective decision-making process should follow the following 3-step procedure.

(1) Generate noninferior solutions; (2) obtain meaningful information to interact

with the decision maker (DM); (3) use information obtained in step 2 to in-

teract with the DM and select the final solution based on the DM's preference response.

Step 1 which is normally performed by the analyst serves as a preliminary screening process designed to reduce the originally large set of feasible alternatives by eliminating inferior ones from further consideration. What remains is a set of noninferior (sometimes known as nondomi- nated, efficient or Pareto-optimal) alternatives. An alternative is said to be noni~ferior if there is no other feasible alternative which may improve one objective without degrading at least one of the others. The task of identifying noninferior alternatives can be made purely analytical once appropriate mathematical models have been for- mulated. There have been numerous theoretical results which can be used as a necessary theoretical machinery supporting this task. In particular, in Haimes, Hall, and Freedman (1975), Chankong (1977) or Chankong and Haimes (1976), different ways of generating noninferior solutions (Step 1) were discussed. For general nonlinear problems, it was pointed out that the e-constraint approach (Haimes, 1970, 1973, 1977; Haimes, Wismer and Lasdon, 1971) was considered to be superior to other approaches due to its ability to deal with any noninferior solutions even in the duality gaps which may exist in nonconvex problems. In view of this generality, we shall assume in the discussion of the multiobjective decision-making

60 ~' , \ ( ' ( ) \ 3'. H/klMt.IS and VIR,\ ( ' t lANKON(i

process hereafter that Step 1, i.e., generating noninferior solutions, has already been accomplished b~ means of the t:-constraint approach.

But as we have already mentioned, generating noninferior solutions serves only as the first screening process in the entire multiobjectivc decision making analysis. The next step is to select the "best' alternative from among noninferior ones. This task, itself, is not routine and, at times, requires careful and elaborated analysis and execution.

To select the 'best ' noninferior solution is to find some way of ordering those (noninferiorl solutions in a complete order. The natural ordering relation in an n-dimension Euclidean space, namely ~>', is only a partial order and, hence, inadequate for this purpose. By definition, two noninferior solutions are incomparable by the natural order ' > ' . Thus, to completely order the noninferior set we must call upon a new ordering relation, a logical choice of which is that of the prelerence relation, which reflects the preference structure of the decision maker. Steps 2 and 3 above represent a practical but indirect means for carrying out the construction of this preference relation.

Much of the effort in theoretical and metho- dological developments in multiobjective decision making has been devoted, directly or indirectly, to the construction of the preference relation. Since considerable subjectivity and value judgement are involved, common problems normally encountered in this construction are inaccuracy and inconsistency in the preference assessment process.

For the construction of a preference ordering relation to select the best alternative from the noninferior set, availability of additional information about the noninferior set itself would be helpful in making the task more consistent, accurate and perhaps easier to perform. In this paper, we develop analytical results which give information about the gradient or tangent of the noninferior surface. This, in turn, yields information about the trade-off rates (among at- tributes) within the noninferior set which are useful in constructing the preference relation as carried out in Steps 2 and 3 above.

Though the paper focuses particularly on Step 2 assuming the first step has been accomplished by using the r-constraint approach, a brief ac- count of relevant theoretical results required in Step 1 is presented in the next section. And before the main results are developed in later sections, we discuss the concepts of trade-off in multiobjective decision making and justify the use of trade-off as the information to be used in the DM-analyst interactive phase as set out in Steps

2 and 3. Finally the main results \~hicl~ vcla[,~ trade-ofl; to Kuhn Tucker multipliers of the ::- constraint problem are developed. Examplc~ il- lustrating various aspects of these resulis are also presented.

2. PRELIMINARIES

In a multiobjective decision-making problem (MDMP), we shall refer to an alternative x as being a member of X, a set of feasible alternatives and a subset of R s. Associated with alternative x, there correspond the levels of n quality measures or objectives (n>2} denoted by / l (x) ...... ]',(x) or for short, by n-dimensional vector f (x) . For a continuous MDMP, it will be assumed that X is a continuum (i.e., a union of "non-single point' connected subsets of R ") and that each J)(x}, j = 1 ..... n is a continuously differentiable real-valued function defined on X. Finally a feasible neighborhood of a feasible alternative x will be denoted by X c~ N(x} and a 6-neighborhood of x will be denoted by N(:\-,~i) where

Nix, 6 t Z" .Vl e", IIx- yll

Associated with an M D M P is a multiobjective optimization problem (MOP defined as min~ x(f, (x ) .. . . . . ]~ (x ) ).

To solve an M D M P is to find a feasible alternative x * e X so that x* is a noninferior (solution of MOP) and x* satisfies the DM most. To be more precise, we define:

Definition. x* is said to be a noninJerior alternative if there exists no other x e X so that f ( x ) _-< I (x*).

DeJTnition. x* is said to be a preferred alternative by the DM if x*Vx for all x e X where P is the preference ordering relation given by the DM and is to be read as ~is preferred to) (i.e., "x*Px" reads 'x* is preferred to x' and 'xpx*" reads "x is not preferred to x*').

We observe that to solve an M D M P is to use the composition of the ordering relations ' > ' and 'P ' to order the set F ~ , I ( x ) l x e X [ ( ' ) ~ R " ' SO

that an alternative with the highest rank can be selected as the final choice. From the practical veiwpoint both ordering relations should be used together. To use one without the other is generally inadequate. For example, to use 'P ' without considering noninferiority may lead to an i@erior final choice due to lack of detailed knowledge about the structure of the problem in constructing "P'. On the other hand, to consider only noninferiority without active participation of the DM, one usually lacks the necessary machinery to choose the preferred solution.

Kuhn Tucker multipliers as trade-offs in multiobjective decision-making analysis 61

One way of obtaining noninferior alternatives is to solve the e-constraint problem

where

Pk(e): min fk(X) x e X ~ X~lc)

and

x~(e) ~= {xl f,.(x)<ej,j # k}

It is known (Chankong and Haimes, 1976; Chankong, 1977) that:

Theorem 1. A unique solution of Pale), for any l<_k<n, is a noninferior solution of MDMP. Conversely any noninferior solution of M D M P solves (not necessarily uniquely) Pk(e) for some E ~ Yk and for all k = 1 ... . . n.

For practical purposes, since only local solutions are guaranteed in solving Pale) by any standard optimization technique, unless the problem is convex, we shall be working with local noninferior solutions (instead of noninferior solutions). The study hereafter will centre around local properties of M D M P and Pk(e). Also for the remainder of this paper, the set of local noninferior solutions of M O P will be denoted by X * .

Definition. x* is said to be a local noninferior solution of M O P (i.e., x*eX*) if there exists 6 > 0 so that x* is noninferior in X c~N(x*,6); i.e., there exists no other x e X ~ N(x*, 3) so that f (x) <-_f(x*).

The following two theorems are 'local' coun- terparts of results in Theorem 1.

Theorem 2. x* is a local noninferior solution of M O P if and only if x* is a local solution of P~ (e*) for all k = 1,..., n.

Proof The necessity part is proved by con- trapositive argument. If there is a k for which x* is not a local solution of Pk(e*), then for all 6 > 0 , there exists x ( 6 ) ~ X ~ N ( x * , 6 ) so that fk(X(6)) <J~(X*) and f j(x(6))<fj(x*) for all jsa k. Hence, x* C~ X*.

For the converse, if x* is a local solution of P~(e*) for all k = 1,...,n, then for each k, there exists 6k>0 so that x* is a minimizer of Pde*) within X c~ N(x*, 6k). By taking 6 =min{61 . . . . . 6,} >0, x* is then a minimizer of Pk(e*) in X c~N(x*,6) for all k = l ... . . n. Hence there is no x e X m N ( x * , 6 ) so that f ( x ) < f ( x * ) implying that x* ~ X*.

Theorem 3. x* is a local noninferior solution of M O P if x* is a strict local solution of Pk(e*) for some k.

Proofi Let x* be a local solution of Pk(e*) for some k. Then there exists 6 > 0 so that for all

xeXc~N(x* ,6 ) satisfying J)(x)<J)(x*) for all j-~k, J~(x)<L(x* ). Hence there is no xeXc~N(x* ,6 ) so that f ( x ) < f ( x * ) implying that x* e X*.

Corollary. If x* solves P~(~:*) with second- order sufficiency conditions being satisfied for some k, then x* is a local noninferior solution of MOP.

Upon imposing some convexity assumptions, as in the case of a scalar optimization problem, a local noninferior solution becomes a 'global' noninferior solution as the following theorem indicates.

Theorem 4. If X is a convex set and all fj are convex functions on X, then any local noninferior solution of M O P is also a global noninferior solution.

Proof. Let x* be a local noninferior solution of MOP. By Theorem 2, x* is a local solution of Pk(~*) for all k = l . . . . ,n. By convexity and Theorem 1, the results immediately follows.

3. T R A D E - O F F S IN MULTIOBJECTIVE DECISION

M A K I N G

3.1. Why trade-off? In the above 3-step procedure, given nonin-

ferior solutions from Step 1, the next question to be answered is "What kind of information should we use to interact with the DM?"

For obvious reasons, such information should be meaningful and easily comprehensible by the DM, easily obtainable by the analyst and contain sufficient information about the structure of the system. Among many used (e.g., trade-off, the pay-off matrix in the STEP method (Benayoun and co-workers, 1971), the concordance and dis- cordance in the ELECTRE method (Roy, 1971), etc.) trade-off is probably the most widely accep- ted and appears in most everyday decision-making problems. And for our purpose, it is also quite suitable to be used in conjunction with the e- constraint method of generating noninferior solutions for it satisfies all above three criteria. To the DM, the term 'trade-off ' is easy to under- stand and physically meaningful. Furthermore, with proper kinds of trade-offs, it has a potential of providing a systematic assessment, e.g., com- paring two objectives at a time. The task of assessing preferences thus becomes easier and is destined to be more accurate and more consistent. In a continuous multiobjective decision- making problem, from the point of view of the analyst who has the task of providing information to the DM, certain types of trade-offs are easily obtainable requiring, if any, only a little work in addition to already performed routine effort. Finally, as will be evident from the main

62 X(k(.()\ "I'. } tAIMl<, t tnd MIR\ ( ' I I . kNK( ) \ ( ;

results in the sections to follow, the kind of trade- off which we propose to use in conjunclion with the c-constraint approach contains significant information about the local behaviors of the noninferior set both in the decision and objective spaces.

3.2. What is trade-o[[? Trade-off, according to Webster's New World

Dictionary of the American Language, Second College Edition, is: "an example, especially a giving up of one benefit, advantage, etc. in order to gain another regarded as more desirable". The term has been used freely in various multiobjective decision-making problems. Yet it seems to take different forms under different contexts. Thus, quite often, ambiguity, vagueness and/or confusion usually arises when the term is used without some proper modifiers attached to it. This section attempts to identify some of those different forms of trade-offs according to the context in which the term is used.

7btal vs partial trade-ql, ls. When the number of objectives is mo,e than two, i.e., n > 2, it is useful to make a distinction between ' total ' and 'partial' trade-offs.

Given two feasible alternatives x ° and x*, the corresponding levels of objectives are f ( x ° ) = (.fl (X°) . . . . . . t',(x°)) and f ( x * ) = (fl (x*) ... . . f , (x* )) respectively. Denote the ratio of change o f f t and change in.L) by T~i(x °, x*) where

often in discrete MDM problems ~xilh a Ii;~it~ number of available alternatives. The types of trade-off used in Zionts and Wallenius's method for solving linear MDM problems (Zionts and Wallenius, 1976) also falls in this category. In that method the DM has the task of choosing whether or not to move from one extreme point. x* (of the feasible region) to another extreme point x. He is provided, of course, with values of the total trade-off "l~i(x~',.\ *) fo, all j = I ..... n and n 4-k, where k is a standing (primary) objective.

For continuous problems, the following concepts are useful.

Delinition. T~j(x',x*) is said to be a local total (partial) trade-off involving .lj. and !) between x ° and x* if x ° ~ X c> N(x*) and Tkj(x°,x *) is a total (partial) trade-qff involving .[~ and .[i between .x" and x*.

Definition. Given a feasible alternative x* and a feasible direction d* emanating from x* (i.e., there exists % > 0 so that x*+z td*~X for 0 < < % ) , we define the ~li -.t}" total trade-off rate at x* along the direction d* as

tti(x*,d*)= lim 7~j(x* +~d*, x* ).

Note. Using continuous differentiability, it can easily be shown that

Tp, i (x" , x * ) = (.l;,(x °) /i,(.'c* ))/( l }(x (') - l } ( x * )). t~i(x*, d* )= Vti(x*)" d* /Vl}(x* )" d*.

Definitions. T~i(x°,x *) is said to be the 'partial" trade-off involving J~ and f i between x ° and x* if ji(x°)=.li(x *) for all 1=1 . . . . . it 1~1< and .j. On the other hand, i f .~(x°)¢j i (x *) for at least one /=1 . . . . . n, and l C k and j then Tki(x°,x *) is called the 'total" trade-off involving .lk and .t) between x ° and x*.

The terms 'partial ' and "total' are used here in an analogous manner as partial and total de- rivatives. The partial trade-off concept can generally be found in, for example, constructing multiattribute utility functions or constructing a family of indifference curves, etc. It is also used in the Surrogate Worth Trade-off (SWT) method (Haimes and Hall, 1974; Haimes, Hall and Freedman, 1975; Hall and Haimes, 1976), and thus it is one of the instrumental tools used in developing an interactive version of the SWT method (Chankong, 1977). The principal value of this concept is that it enables the DM to com- pare changes in two objectives at a time thus making the preference assessment task easier. The total trade-off concept is, perhaps, used more

Also if d~' is a feasible direction with the property that there exists ~ > 0 such that })(x*+~d~)= ,tilx*) for all l c k and j and for all 0<:~__<~2, then we shall call the corresponding t~j(x*,d*) the partial trade-oil rate.

To illustrate these concepts, consider a simplified water resources planning problem conceived by Reid-Vermuri (19711. A multipurpose dam is to be constructed at a certain site. After preliminary analysis, it was decided that three main objectives were suitable as the decision criteria. namely:

to minimize the cost of construction (.1~) to minimize water loss (.12 volume/year) to maximize the total storage capacity of the reservoir (.li~).

It was further found that these decision criteria depend mainly on two variables namely:

total man-hours devoted to building the dam (XI) mean radius of the lake impounded in some fashion (x2 in miles).

Moreover, relationships between the decision cri-

Kuhn Tucker multipliers as trade-offs in multiobjective decision-making analysis 63

TABLE 1. ALTERNATIVES APPROPRIATE FOR FINAL DECISON

Alternative Decision Variables

x I x 2

.70 22.36

116.19 44.72

172.95 44.72

Cost fl

500

7029

12500

Objective Functions

f2 [ Storage Capacity f3

l

Water Loss

I 250

i000

i000

500

3750

5000

teria and these decision variables were found to be as follows'

f l (X1, X2 ) = °0"0 l x 1 ,.0.02 v2 ~" -~" 1 ~'~ 2

.12 ( X I ' X2 ) = 0 ' 5 X 2

( [~. ~ ] _ _ , ~ 0 . 0 0 5 x 1 ~ 0 . 0 1 ~ 2 • J31."~ 1' "~2, ~ - ~ " "a'l "*~2

Due to physical and geological considerations, three alternatives listed in Table 1 were considered appropriate candidates for the final implementation. From Table 1, the corresponding trade-offs between the following pairs of alternatives A/B, A/C and B/C are given in Table 2.

TABLE 2. TRADE-OFF VALUES

fl/f2 fl/f3

A/B 8.7 2.0

A/C 16.0 2.7

B/C ~ 4 . 4

In choosing the final decision from among A, B and C, pairwise comparisons based on the use of subjective value judgement on trade-offs as well as the levels of objective functions are to be made. Given an explicit set of alternatives such as in the above example, trade-offs for each pair of those alternatives can be easily computed. The trade-offs between A & B and A & C are examples of what we defined as total trade-off since nonzero trade-offs occur for both pairs fl/f2 andfl/J3. Thus both numbers fill2 and fl/J~ must simul- taneously play important roles in the subjective value judgement for deciding between A& B and between A&C. On the other hand the trade-off, .]'l,(f3, between B & C is an example of partial trade-off, since f~,(f2 = ~ . Consequently in deciding between B & C apart from the levels of fl , J2 and j ; , only the value of fl/f2 figures in the preference assessment process.

When specification of alternatives is only in implicit form, e.g. when there is no information

available to limit the set of alternatives down to a small finite number such as in the above example, it is impractical if not impossible to compute trade-offs between each and every pair of alternatives as done previously. Decision- making in this case should also be carried out in an iterative manner. Starting from an initial trial point e.g. x°=(x° , x° )=(0 .7 ,22 .4 ) our objective is to find a new point x 1 which is better in terms of the DM's preference than x °. This is equivalent to moving in a certain direction in the xlx2 plane (or decision variable space). Lacking adequate analytical machinery, human judgement should be employed in choosing the direction to move. But before judgement can be used, it would be necessary to know how the levels of objective functions J~, ]2 and J~ are ellected by such moves. This is where the concept of trade-off rate is useful. The values of trade-off rates depend on the direction we move. For example, at x °, trade- off rates between fl and f2, and f~ and J; in the directions defined by d~=(0.98, 0.19), d2 = (0.99997, 0.00798) and d3=(1, 0), all emanating from x °, are given in Table 3 below.

TABLE 3. EXAMPLES OF TRADE-OFF RATES

T r a d e - o f f r a t e s

Direction fl/f2 fl/f3

d 1

d 2

d 3

5.31

81.00

1.46

1.96

2.01

The trade-off rates in directions d, and d2 illustrate the concept we defined as total trade-off rate whereas those in directions d3 are examples of partial trade-off rates, In this setting knowing the values of appropriate trade-off rates along any given trial point (alternative) will be helpful to continue into the DM-Analyst interactive phase of the multiobjective decision making process. The main results to be developed sub-

64 Y,xct)'~ "~ I I AIMts a n d \ : IRA ('t~\NK()N(I

sequcntly \sill provide nccessaly theoretical sup- port for computing these trade-off rates.

Before we proceed further, a few other remarks are noteworthy.

Remark 1. In normal decision-making si- tuations, the term 'trade-off' is involved in the situation where it is known that something has to be given up in order to obtain something else (which corresponds to the negativity of "~i as defined above). Although it may not be clearly specified, this is to imply the assumption that the decision-making problem at hand has, for the DM*s consideratiom nothing but a set of (local or global) noninferior alternatives (i.e., X is the set of (local or global) noninferior solutions). We shall, however, generalize the use of the term 'trade-off" to mean the ratio of changes of two objectives as defined above, i.e., we allow "I~/ to be positive for some pair k and j and tk)r some pair x ° and x*. The following observations are clear from definitions.

(a) x* is noninferior in X if and only if, for each x eX, there is at least a pair k and j (both depend on x) that 7~i(x,x* ) is negative.

(b) x* is locally noninferior if and only if there exists 6 > 0 so that for x e X ~ N ( x * , 6 ) , there exists at least a pair k and j so that Tki(x, x*) is negative.

Remark 2. Given two alternatives x ° and x*, if f ( x °) and f ( x* ) lie on the same indifference curve, the corresponding trade-off, whether it be total or partial, is usually known as the indifference trade-oil" or marginal rate ~?[" substitution; a term most widely used in decision- making today (see for example, MacCrimmon and Wehrung, 1975).

Remark 3. An interesting example of the total trade-off rate is the 'trade-off vector" as described by lsermann (1974) and as used by Zionts and Wallenius (1976) in their development of an interactive multiobjective algorithm for solving a linear MDMP. An Isermann's trade-off vector is a column vector obtained from the 'reduced costs' portion of the optimal multiple objective simplex tableau. Each component of this vector represents the change in each objective level per one unit change in the corresponding nonbasic variable. Thus lsermann's trade-off vector is the total trade-off rate along one of the edges of the feasible region emanating from the extreme point under study.

Objecti~,e ~s subjectire trade-@. The terms objective and subjective are used to reflect the degree of human factor involvement in evaluating trade-offs (Ackoff and Sasieni, 1968). If a trade-off

is calculated using purcl 5 thc internal >tructurc ~i the system under study, it is then called a~ objective trade-off. If, on the other hand, it i, determined solely by the DM who, mos~ of the time, has to ignore internal beha\.iors of thc system and concentrate on evaluating trade-offs to satisfy certain preference criteria {e.g., indifference criterion or preferred proportional criterion. see for example, MacCrimmon and Wehrung (1975)), it is then called a subjective trade-off. Subjective trade-offs are used most often in mo- dern decision-making that is based on lhe concepts of utility function or preference. Geoffrion's interactive procedure (Geoffrion, Dyer and Feinberg, 1972), for example, relies on subjective trade-off analysis, i.e., it requires subjective de- termination of the marginal rate of substitution or indifference trade-off.

Unlike the subjective trade-off analysis, i.e., instead or presetting some preference criteria and subjectively determining the trade-offs, preference is assigned to the already-determined trade-off in the objective trade-off analysis. The latter task is generally much easier than the former. In a large and complex decision-making problem in which the effect of the internal structure on decision- making is significant, and if objective trade-off values can be obtained in a systematic way, the objective trade-off analysis is thus preferred. The value of trade-off determined objectively is more realistic and, most importantly, guaranteed to be feasible, since it is determined from the system characteristics itself.

The major problem that seems to block the implementation of the objective trade-off analysis in problems with large set of feasible alternatives. e.g., when X is infinite, is the lack of an easy or systematic way for evaluating the required trade- off values. In the following development, by restricting the consideration to a set of locally noninferior choices, instead of the whole feasible region X, the task of obtaining objective trade-off values required for the DM to assign his preferences on, is as simple as obtaining positive Lagrange multipliers.

In the remainder of this paper, we shall care- fully set out precise relationships between some positive multipliers associated with an optimum solution of P~(c), i.e., a noninferior solution of MOP, and certain types of objective trade-offs.

4. TRADE-OFt" INTERPRETATION O t KUI tN

TUCKER MUL'YIPLIERS OF Pk{~:l

Let X~{x]xeRX gi(x)<O,i=l ..... m', where g~ is a continuously differentiable real-valued l'unc- lion defined o n R N. Hence Pk(e) is defined as

Kuhn Tucker multipliers as trade-offs

minf~(x) subject to ,[.)(x)<~j, j = l . . . . ,n, j # k , g/(x)__<O, i= 1 ... . . m, and x ~ R N.

Let x* solve Pk(e) with 2*j, j = 1 ... . . n, jg=k and it*, i= 1 ... . . m as the optimal Kuhn-Tucker multipliers associated with fSs and g~'s constraints respectively. (Note: 2k* and /t* are the optimal Kuhn Tucker multipliers of Pk(5) if

"~'k 1, ~'kk 1, A'kk + 1, ~n ~ ) • .., ..., P l , ' " ,Pm

satisfies the Kuhn-Tucker conditions for optimality for Pk(e)).

The key concept underlying relationships between Kuhn Tucker multipliers 2k* and trade-offs between .fk and Ji at x* is the sensitivity interpretation of the multipliers. For convenience we shall restate Luenberger's Sensitivity Theorem (Luenberger, 1973, p. 236) using notations suitable for our discussions. We shall also state some of its consequences in the form of corollaries so that they can be easily referred to (Haimes and Hall, 1974).

Sensitivity Theorem. Given ~°E Yk, let x* solve Pk(e °) with 2k*, j@k , being the corresponding Kuhn-Tucker multipliers associated with the constraints Ji(x ) < ~o • ~j, j =/= k. If

(i) x* is a regular point of the constraint of Pk(e°);

(ii) the second-order sufficiency conditions are satisfied at x*, and

(iii) there are no degenerate constraints at x*, then zkj'* -- - 63fk(x* )/~ej for all j :# k.

Corollary 1. If x* solves Pk(e, °) and satisfies (i), (ii), and (iii), then there exists a neighborhood N(e °) of s ° so that, for all EeN(e°), x(e) which uniquely solves Pk(e) locally exists, and is continuously differentiable function of e with x(e °) =X*.

Corollary 2. With all the hypotheses of the sensitivity theorem satisfied, there exists a neighborhood N(~, °) of ~,o so that for each j such that 2k*.>O, J j (x (e ) )=e ~ for all eeN(e° ) .

The results in the above two corollaries arise naturally from the proof of the sensitivity theorem which, in turn, relies heavily on the implicit function theorem. It should be stressed here that conditions (i), (ii) and (iii) required for the sensitivity interpretation of Kuhn-Tucker multipliers above are merely sufficient conditions. In the next section, we shall investigate another set of sufficient conditions applicable to the linear case. Also, as required by (ii), we shall assume throughout this paper that all functions ()'i and g~) are twice continuously differentiable.

For notational convenience in presenting the following results and without loss of generality,

in multiobjective decision-making analysis 65

we shall use P,(e) as a means for generating noninferior solutions.

Theorem 5. Let x* solve P,(e °) (for some given e°e Y,) with (i) x* being a regular point of the constraint of Pk(e°), (ii) the second-order sufficiency conditions satisfied at x*, and (iii) all binding constraints at x* being nondegenerate. Let 2,*. denote the optimal Kuhn-Tucker multipliers corresponding to the constraint J}(x)<e,j. Also assume without loss of generality that the first p, l < p < = n - 1 , of these multipliers are strictly positive and the remaining ( n - 1 - p ) multipliers are strictly zero (i.e., 2 * > 0 for all j = l , . . . , p and )~*j=0 for all j = p + l ..... n - l ) . Then:

(a) If e°=f~(x*) for all j = l .. . . . n - 1 (i.e., p = n -1 ) , there exists a neighborhood N(x* ) of x* and a continuously differentiable vector-valued function x ( ' ) defined on some neighborhood N(e, °) of c ° so that

X* (~ N (x* )~_x(N (e°) ) ~ X *

where x(N(e° ) ) is the image of N(~ °) under the function of x(-).

(b) Again as in (a), if p = n - 1 and with N(x* ) so obtained as in (a), let

. . . . , f .

j = l , . . . , n , x ~ X * m N ( x * ) }

and let

F : ----A {(fl . . . . . fn- 1 )] fj =f j (X),

j = l .. . . . n - l , x ~ X * n N ( x * ) } .

There exists a continuously differentiable function f , defined on F* so that for each

( f l . . . . . L ) e F * , L = L ( f l . . . . . L 1).

Moreover for each 1 < j < n - 1,

Ofn ( f'~ .. . . . f 'n-1 ) = -- ).,*j

where f* =fj(x* ). °>.fj(x*) for some l < j (c) If l < p < n - 1 (i.e., sj

< n - 1 ) , let

F: = { ( f , . . . . . f . ) l / i =JS( ),

.] = 1 . . . . . n, x a x ( N ( e ° ) ) } .

There exist continuously differentiable functions f , ( ' ) , f p + l ( ' ) . . . . . f , 1( ' ) defined on N(e °) so that, for each (.f, . . . . . f , ) e F * , f j = f j ( . f l . . . . . . fp, e°+~ ... . .

66 } \ ( ( ) \ ~, lt,,~,l\ll,:S and VIR,\ ( II'~NIK~N(;

, ) ,,,, ~ ) for all j = p + 1 . . . . . n. Moreover , for each I ~ /

~5 P.

,7; I . . . . . " = - )"3 = ( V l ; (x* ) . d,* ),'(Vt;(x* ). d;' )

where d* is the direction of ?x0;°)..?~:v Also for each p + 1 < j < n - 1

'~L = v . / ) ( x * ) . -~. •

Before we prove these results, it is appropr ia te to discuss their practical implications. Fur thermore , this theorem provides a b roader theoretical basis to some earlier results associated with the deve lopment of the SWT method (Haimes and Hall, 19741. In general, Theo rem 5 enables one to make a comprehens ive study of local propert ies of the local noninferior set both in the decision and objective spaces. First, in simple words, part (a) indicates that under the specified condit ions and hypotheses, the locally noninferior surface in the ne ighborhood of x* can be complete ly specified parametr ical ly in terms of the s's. Tha t is, there is a one- to-one cor respondence between every local noninferior solution in the ne ighborhood of x* and ~; in the ne ighborhood NO: °) of s °. Although the explicit form of such paramet r ic description is not generally known, the knowledge that it exists can be very useful in some theoretical or a lgori thmic developments . For example, suppose we want to find a local noninferior solution which maximizes a utility function U(.I] ...... l,,l. On the locally noninferior surface in a ne ighborhood of x*, since all ./~} are functions of x which is in turn a function of c, U I ' ) is then a (implicitl function of c. Hence, instead of searching over X* n N L v * ) , we need only to search over N0:") over which we have complete control.

Par t (by gives similar results to part (a) but in the objective space. It says that, under the specified condit ions and hypotheses, there are exactly n - 1 degrees of freedom to specify a point on the locally noninferior surface in a ne ighborhood of

. t '(x*), in the objective space. In fact. one can choose any values o f / ; ...... Ii, t from N(r° l and compute./i , from ~, t l l ...... t,, 1), the resulting point 11'~ ..... f~) will lie on the locally noninferior surface in the objecuve space. But most impor tant ly . since " * - '-= ~ '* * .... - / , , j - - (q l , , , ~ : l j ) l . l l . . . . . . ],, 1) (for any 1 < i < n - 1 ) which represents the rate of change of tl, per one unit change in ,~} when all o ther objectives remain unchanged. -"~.s'* represents the par-

tial t rade-q[] r m e bet~ecn Ii, and l ) a n d ," i the concept which was discusscd in the Ia~i ~cction L

Finally, part (el extends the result in part (hi a little further by relaxing one assumption. Among others, one of the assumpt ions required in tbj i~ that all constraints .l)(x)=~,~, j l . . . . n .... i arc binding at x* tand nondegenerate} or z, , i . -0 lor all . j= 1 . . . . . n 1. If this assumpt ion is violated, i.e., if there is only p, where l < p < n .... 1. nondegenerate binding constraints, then the degrees of freedom to specify' a locally noninferior point in a ne ighborhood of IL'c*I in the objective space arc exactly the number of nondegenerate binding constraints, which is p in this case. In particular, one can choose any \.alues of./'~ . . . . . . 11, iin some ne ighborhood of l'~ (x*) ...... lp(x* )). then determine lp ~ 1 ...... Ii, ~ by somc specific rules (to bc denoted by Av+l . . . . . . 4,, 1) defined in terms o f . l l . . . . . Ii, and finally compute Ii, from a specified function of./'1 . . . . . . l,, ~-The resulting points (.t'1 . . . . . . ll,) will be a local noninferior solution in the objective space• These specific rules .4r,~ 1 . . . . . . 4, ~ are generally a point- to-set mapp ing defined on NO: °) and each./}, p + I < j < n I. is chosen from A it . I i.e..

t ) ~ .,4 s( 1~ . . . . . . /;,, ,,o ,, . . . . ,,p ~ . . . . , ;~ ~ t. Further investi- gat ion will show that A i will take on one of the following three forms. (il the set of points (on the real line in a ne ighborhood of . / ){x*))less than or

, O (I equal to .~(.11 . . . . . . 1),, ,,p~, . . . . . e,, ~1, or (it) the single point f i(.l: . . . . ii,, ,o o . , L p ~ 1 . . . . . E n 11" or (iii) the set of points (in a ne ighborhood of./)(x*)l greater than or equal to ~ ( l l . . . . iv, .o • '~I ' ~ 1 , " ' "" ;n I ) ; s e e

example in Section 6. ] 'he criterion for deciding which of these forrns A i should take is a little complicated and not worth report ing at this point. In part (c), we consider only those local noninferior solutions which cor respond to the case where , . t i - I ~ and I,,=.)7,(jl . . . . . .11, ~). f o remain within this special subset of local noninferior solution, a small change of (!li in the level of the binding objective .Ii(1=1 . . . . . p), with all other bind ing objectives remaining unchanged. will induce the (approximate t change in /i, by --).*t~ll as well as the tapproximate) change in each f). p + 1 ~j__<_ n - 1. by, V/}lx* ) (~:.\'1~:*),'(cl )dli. In this case -2 ,* represents the to ta l t rade -o i l rate between f , and l; at v* in the direction of i:x (c*) "~:l.

F rom the practical viewpoint, parts Ib) and {el provide a convenient way for est imating 'objec- t ive trade-off information necessary for continuing into the ' analys t -DM" interactive phase of the objective trade-off analysis. To see how' this infor- mat ion may be used, suppose we have generated a noninferior solution .-<" by means of solving P,,(~:') and with it we have -A,, i'° for all j - 1 . . . . . ~7 1. If case (bY applies, lhcn for each and e,,cr) i:-:1 . . . . . , 1. x~c ma'~ ask the DM the questiolY

Kuhn-Tucker multipliers as trade-offs in multiobjective decision-making analysis 67

Given the current levels of objectives (i.e., fj =J)(x °) for all j = l , . . . , n ) , how would you like to decrease f , by 2, ° units per one unit increase in f~ when all other objective levels remain unchanged ?

Or when case (c) applies (i.e., o 2,j > 0, j = l ..... p and 2,°=0, j = p + l ..... n - l ) then, for each and every j = l .. . . . p we may ask the DM the question"

Given the current levels of objectives, how would you like to decrease f , by 2, ° units and change each f;, j = p + l ... . . n - 1 by Vf~(x °) ×Ox(e°)/O~ units per one unit increase in fi

with all other objectives f~, k = l . . . . ,p and k ~ l remaining unchanged?

The DM should respond by expressing his preference on the prescribed trade-off in some given ordinal scale. Although there are p ques- tions (p = n - 1 if (b) applies) to be asked, the DM will only need to consider one question at a time.

How to process the DM's response to arrive at the final solution is treated elsewhere (Haimes and Hall, 1974; Haimes, Hall and Freedman, 1975; Hall and Haimes, 1976; Haimes, 1977). The 'Interactive Surrogate Worth Trade-off Method' (Chankong, 1977) utilizes the results developed in this section and proceeds to complete the multiobjective decision-making process with an efficient 'analyst-DM' interactive scheme.

Proof of Theorem 5. (a) From Corollary 1 of the sensitivity theorem, there exists a neighborhood N(e, °) of e ° so that for each e~N(~°), P,(e) has a strict local solution x(e) and that x(e) is a continuously differentiable function of defined on N (co). Thus the statement 'x(N(e°))~X *' easily follows from Theorem 3. To show the first part of (a) is to show that there exists a neighborhood N(x*) of x* so that for each R e X * ~ N(x*), there exists esN(e °) so that R =x(~).

We can construct such N(x*) in the following manner. Let

N(e°) ~ {ele ~ R"-1, [l~-~°]] ~ <6~}

where 5~ > 0 and I lll is the /n-norm, 1~ p ~ 0% in R"-1. Due to the continuity of fj, for each l_<__j < n - 1, in x, there exists 6j > 0 so that

A .

Let 6 =mln 1 ~=j~=n- 1 3£ The set

{xlx e ,ll x - x*llx

is therefore the required neighborhood of x*. Now to complete the proof, consider any

R~X*c~N(x*). Define ~=f j (x) for all j = l . . . . n °=fj(x*) for all j - 1 . Since ReN(x*) and e;

= l , . . . , n - l , it follows that gEN(e°). Hence, by definition of N(e°), P,(g) has the unique local solution x(g). Also since REX*, by Theorem 2, R is also a local solution of P,(e), It immediately follows that ,'~=x(g) which completes the proof for part (a).

(b) From (a), for any R~X*c~N(x*), there exists ~eN(e °) so that 2=x(g). Hence for all ReX* raN(x*), f,(2)=f,(x(g)) A=f,(~), where f , ( ' ) is a function of ~ defined on N(E°). The continuous differentiability o f f , ( . ) follows from that of f , ( - ) and x( . ) , Also by Corollary 2 of the sensitivity theorem and by the definition of x( . ) , we have for each ReX* c~N(x*),fj(x)=gj for all j = l , . . . , n - 1 . Hence L ( R ) = f . ( L I R ) ... . . L I(R)) for each 2eX*c~N(x*). Consequently for each (L ... . . L ) e F * , L = f . ( L .. . . . L - ,) as required.

Moreover for each j = 1,..., n - 1

~f" ,c* r* ~f.(~o) ~ r 1 , J 1 ~ " " " , J n -- 1 ) ~ - - - ~'nj~ *

where the equality is given by the sensitivity theorem.

(c) We again prove the first part of (c) by constructing the required functions. By definition of x ( ' ) and x(N(e°)), for each R~x(N(e°)) there exists g~N(e °) so that R=x(g). Hence

fA~) =fAx (~)) -fj(~)

for all j = p + 1 ... . . n - 1, n. And by Corollary 2 of the sensitivity theorem, f i (2)=g z for all l= 1 ... . . p. Hence for each j=p+ 1 ..... n,

fAR)=fAL (R) ... . . f~(R),en+, ..... ~ . - , )

for all R~x(N(g)) which shows that ~ ( . ) , j=p + 1,..., n are the required functions.

To prove the remainder of (c) is now simple. First observe that, for any 1 < l__< p

~Tfn E=g° -~fn E=EO

I.[)(x)-f;(x*)l < 6j(n - l) ~/p.

where again the equality is given by the sensitivity theorem.

Moreover, since for each l<=l<p, fi(x(e))=e~

68 YACOV Y. HAIMES and VIRA CHANKONG

for all eeN(~:°), by the chain rule Furthermore, if

Also

Hence

~x (e (') '7, = v J ; i x * ) "--~- . 1 =i'~" " cg,

I Ig=£ ~

~gt !')j~ ~:=~ ° = V fn (x* ) " ~-x(~°)

~x(~. °) VL(x*) . . . .

(?g t - z . , - O x ( e ° )

V~(x*) . . . . ~g,t

VL(x* )"d?'

V~x*)'dff

Finally, for each 1 < l ~ p and p + 1 ~ j =< n - 1, by the chain rule,

which completes the proof.

5. LINEAR CASE

So far we have insisted on using the regularity, second-order sufficiency and nondegeneracy conditions as the underlying assumptions on which our results have been developed. It has been pointed out, however, that those conditions are merely sufficient, but not necessary. For the linear case for example, the second-order sufficiency condition (which requires that the Hessian of the Lagrangian of Pk(e), be positive definite on the subspace corresponding to the supporting hyper- plane to the nondegenerate binding constraints surface) is never satisfied since the Hessian matrix of a linear Lagrangian is always a zero matrix. Yet similar 'shadow price' interpretations of Lagrange multipliers for LP problems have long been known (Dantzig, 1963).

In solving linear Pk(r°), a special technique like the simplex method, or any of its variants, is normally used. As a result, the simplex multipliers

n o = ( _ o _20 ~ k l , • " "~ k~k 1 ,

)o o _ o~,~+, . . . . . _ & . , _ ~ o . . . . . _ ~ o )

are also generated. Suppose x ° solves a linear Pk(e °) having x ° as the optimal basic variables and no as the corresponding simplex multipliers. Let B(e °) be the coefficient matrix of the optimal basic variables and credo ) be the cost coefficients of the optimal basic variables. It is known (see Dantzig, 1963, for example) that B(r, °) is nonsingular and n o = cs(~%B(e ° ) 1

/.) (g,O) = (g,? . . . . . ~.~l 1, ':[)+ 1 . . . . . ~Jl~, 0 . . . . . O ) | ,

then

and

x ° = B(e,°) - l b(e, ° )

= cn(~o)B (e.°) - l b (e ° ) = n°b(e ° ).

Moreover, if each component of x ° is positive (or nonzero), i.e., nondegenerate case, then there exists a neighborhood of ~o, N(Eo), so that for all ~;eN(e°), the optimal set of basic variables of Pk(g) does not change, i.e., B(e)=B(~ °) and cB(~) =cR(~o ). Hence n(e)=n(~,°)A=n ° for all ~N(~° ) . Also x~(~)=B(e °) lb(e) which is a continuously differentiable (linear) function of ~;. From this, two conclusions follow: (i) since .l~(x(e))=n°b(e,) for all e,~N(e °) then -)~°~=(?ft(x°)/(?ej for all . j~k, and (ii) since the set of optimal basic variables does not change with ~ in N(e°), we conclude that any constraints that are binding at x ° will also stay binding at x(~), i.e., j)(x(t:)}=~:~ for a l l j so that o 2kj>O and for all eeN(e°).

With all these observations, we have estab- lished equivalent statements of the previously stated sensitivity theorem and its three corollaries by using the existence o.f B(e °) 1 and nondegeneracy as the underlying assumptions. From here we can proceed the same way as before to arrive at the same relationships between trade- otis and Lagrange multipliers (or simplex multipliers) as in nonlinear cases.

Again we summarize these results with a theorem.

7heorem 6 (linear case). Let j), j = 1 ..... n and g~, i=1 ..... m be all linear. For some given s:OeE, l, let x* be a nondegenerate solution of Pdc °) and let -),~'~ denote the optimal simplex multipliers corresponding to the constraint I)(x) <~j, j f :k . Then (a), (b} and (c) of Theorem 5 hold true at x* for the linear case.

6. AN EXAMPLE

We shall end this paper with a simple numeri- cal example that will illustrate operational aspects of the main results discussed in this paper.

Consider a convex multiobjective optimization problem

min {Ji (x),.12(x),.li~lx))... (MOP,) x e X


where

and

f l (X) = (X1 --3) 2 + (X2 --2) 2,

f2(X)= X~ +X2,

f3(X)=Xl +2X2

X ~- {xIx@E 2, gl ( x ) = - x 1 =~0 and

g2(x)= -x2<O}.

Taking fx as the primary objective, the e- constraint problem which will be used to generate noninferior solutions of MOP~ is minf~ (x) subject to

f2(X)=XI-'}"X2 ~-~-e2 } f3(X)=XI +2X2 ~e 3 gl(x) = --Xl=<0 P1 (e2, e3 )"

g2(x)= --X2=<0

It is not difficult to show, using Theorems 3 and 4, that those and only those points that lie within and on the boundary of the triangle PAB in Fig. 1 are nonsingular solutions of MOP1.

Before we go on, it is worthwhile to note that irrespective of what x, /~12, /~13, ]A1 and /./2 may be, the Hessian matrix H(x,J.12,Jq3,kt1,#2 ) of the Lagrangian function of MOP ~ is 21 and I is the 2 x2 identity matrix. Thus, irrespective of its argument, H ( . ) is positive definite in E 2 or any of its subsets, including X and any subset of X. We have therefore justified that the second-order sufficiency conditions for strict optimality will always prevail.

Now, we consider the point x°=(2,0.5) and the corresponding problem Pl(e°,e °) where (e °, e°)= (2.5, 3), as shown in Fig. 1. At this point we have V f l ( x ° ) = ( - 2 , - 3 ) r, Vf2(x°)=(1,1) v and Vf3(x°)=(1,2) x. Since x ° is the (unique) solution of Plte ° e, °~ 2, ,3, as well as a regular point,

x I

P 4i3,21

/ / / / /

~ , ° = 3 / / / / / o / /

~ 2 =2.5 / 0 / ",~. _ i d 3 i I

.

I 2 3

x I FI6. 1. Graphic solution of P~ (2.5.3}.

since the gradients of binding constraints Vf2(x °) and Vf3(x °) are linearly independent, the Kuhn- Tucker conditions must be satisfied. In particular we must have

v L (x ° ) + ,~° 2 vf2 (x ° ) + x° 3vf3 (x ° ) = 0

which implies /]'02 = 1 and 2o3 = 1. Also, because of the simplicity of the problem, we can obtain the following quantities and expressions:

N(e°): a neighborhood of e ° so that there exists a continuously differentiable function x:N(g°)~E N defined by the rule: 'for each e~N(e°), x(e) solves Pl(e).' There are many such neighborhoods, e.g.,

N l ( e ° } = {(e2,e3)]/~ 2 = 2 . 5 , 2 .833 <e, 3 < 3.250}

o r

N2(e °) = {(e2, e3) ] 2.333 < ~2 < 2.60, e 3 = 3}

o r

N3 (e°) = { (e2, a3)[2.417 <g2

<2.552, 2.92<% <3.125}

etc.

The simplicity of the problem permits us to find analytical expression for x(e) (defined above).

For any esN(e°), the unique solution of Pl(e) is always the point of intersection of the two constraints; fz(X)=<g2 and f3(x)~e3 as shown in Fig. 2. Thus for any eeN(e°), x(e), the solution of Pl(e) is given by

f 2 ( X ) = XI -}- X 2 = 8 2

f 3 ( x ) = x l + 2 X 2 = g 3

Hence,

and

, , fxl(8)'~ (282-e3~

~e 2 e.=e °

~3 ° = ~ C/; 3 e=eo =

The directions of d o and d o are shown in Fig. 1. Observe that for all eeN(a°), the constraints

f2(x)<% and f3(x)<% are always binding at the optimal solution x(8) of Pl(e,), i.e.,Jz(X(8))=e2 and

70 YACOV Y. HAIMES a n d VIRA C H A N K O N G

21 /4 (3,21

/ //i/I x ~ i I ---..~:3 ~=z'6"/" //

^ / /

, v ~ - - ~ l ~ -, .¢ \ 0 I ^ ~ ^2 3

-vf2(x) # -vf3(x) vf~(~') X I

2~- ./(3 2) /t / /

~= El3:3 0 / / / //

/

/ / _. , v./" , vl % ~ " . ~ 1 0 I 2 3

X 1

FIG. 3. Graphic solution of P1 (2.51, 3).

FI(;. 2. Graphic solution of P~ /2.6, 3t.

. l ; (x (g) )=g 3. Clearly, for each ~¢eN(c°),

,ll (X(g)} = ((2g 2 - - 83 ) - -3 ) 2 -I- ((g3 - - r 2 ) - - 2) 2

= (2A - A - 3 )2 + (.f i _ j ~ _ 2 )~-. Hence

and

~Ji (x ° ) - 1 = - ~ °

,?J_; i . , a ) = _ 1 : - , ~ L . ~2)`;

An operational meaning of all these results goes as follows: The DM may want to find a new alternative, rather than stay with x °, and he may want to make the resolution that for a new alternative it will be best to let the level off3 be at most as high as before and to let the level of J~ increase by at most O~ 2 where 0 <6g.. 2 ~ 1 . If so, then according to this resolution, there is no other alternative better than x ~ =X(~C;2-l-(]gE, te,,3) a t which

j~ ( x l ) = g0 _t_ (~g2, ' 1 , fa(X ) = ~:o

and

.l; (xl) = (2f2(x')-f3(x')- 3) 2

+ (,13(X t ) - f ( x I ) - -2 ) 2

or approximately given by .11 ( x°)+)~°26t:2. Moreover, in moving from x ° to x 1, we have moved in the direction d o in the (xl,x2) decision space. For example, if &:2 =0.0t , the new noninferior alternative x 1, as shown in Fig. 3, is (2.02, 0.49) at which .12 (xl) = 2.51, ./~(x~)= 3 and 2"1 ( x l ) = 3.2405 (exact). Thus the change in f~, which is exactly -0 .0095 units, is approximately given by -- ,:~°26t:2( = - - 0.01 units) thereby justifying the use of -2o2 as an approximation of the noninferior

partial trade-off rate at x °. Also, --2(]2 c a n be interpreted as the total trade-off rate at x* along a ° '

Similar illustrations involving 2°3 can also be made. These will be omitted however.

Next we consider the nondegeneracy and non- binding cases. Suppose we set % = 3 and G 2 to be 2.6 or higher as illustrated in Fig. 4, the unique solution of Pl(e2,3) is always at ~ = (2.2, 0.4). In particular, when % > 2.6, the constraint .12 (x) < e2 will always be non-binding (and hence 212 =0). If we attempt to make this binding by setting ~:2 =2.6, we run into another problem, i.e., the constraint J2(x)__<2.6 now, although binding, is degenerate since the corresponding optimal Kuhn Tucker multipliers ~12 is still zero. Here it can be shown that there exists no neighborhood N(g) where ~.=(2.6,3) on which a continuously differentiable function x could be defined with the rule 'x(c) solves Pl(g) for each teN(g)' .

Here 2~2 (which is equal to 1.6) represents the total trade-off rate (at .i) between fl and.)'2 along d as indicated in Fig. 4. However ,~3:=0 does not represent any kind of trade-off rate (at #) between fl and.J3.

To illustrate the result in Theorem 5c, suppose that ~ and ,~12 are obtained by solving P1(%,3) where c2>2.6 (i.e., the non-binding constraint

O

/(3,2) ///

~ . ~ , , . ~ = 3 E2= 2.6 / / i i ~\\ /// ii

, J~L~._ ~ ,N ~"--~__ I 2 3

Z I Fl(; 4 Graphic solution of Pt 0;2, 31, c2 ~ r 6


case). Now there exists a neighborhood N(e2,3 ) around (e2,3) so that for every e~N(e2,3) there exists x(~) which solves P~(~) and that x(e) is a continuously differentiable function of e. In fact, x(e) will move along the line OB. Observe that we deliberately make the constraint fz(x)<e. 2 non-binding throughout so that it would not have an active and direct role in determining the optimal solution of P~(e). Observe also that each time there is a change in optimal solution from to x(e) due to change in ~, the levels o f f l and f2 also change. In particular, if f3 changes by &3 (say) and e2 remains constant then the change in fl is approximately -7~26g3 and the corresponding change in f2 is

f2 (X (g2, g3 -{- (~'g3)) - - f2 (x (g2, g3 ))

where

ax(g) - - = (2/5, 1 /5) x,

c~g 3

respectively. Finally we consider the case of 'irregular

point.' Here we reconsider the point 2=(1.5,0) and the problem P~(gz, g3), where g2=1.5 and g3 =1.5. As discussed earlier 2 is the unique solution but an irregular point of P1 (g2,~3)- Let us try to find the optimal Kuhn-Tucker multipliers for this particular case. At ~, Vfl (2)= ( - 3 , - 4 ) T, Vf2(~)=(1,1) T, Vf3(2)=(1,2) T and Vg2(2)=(0, -1)T. Thus, if we attempt to solve

VTI (x) + ~.12vf2 (2) + ~ 13vf3 (-~) + #2Vg2 ( 2 ) =0. . . (*)

which is approximately given by

&(g)

Observe that the above discussion refers to only those noninferior solutions, in a neighborhood of if, which can be represented by x(e), i.e., all points along the line OB. It can be easily shown that to specify a noninferior solution in this special subset in the objective space one need only to specify the level f3 and calculate fl and f2

from the expressions (J3-7)2/5 and (3f3+4)/5, respectively.

It is also easy to show that to specify any noninferior solution in a neighborhood of ~, one need only to specify the level of f3, in a neighborhood of 3, and determine f2 from :

.['2 ~ { f2 i f2eR, (2f3 + 1 )/3 --<f2 =< (3f3 + 4)/5}

and finally compute fl from

J] = (2f2 - f - 3)2 + (f3 - f - 2) 2.

It is clear that the degrees of freedom in specifying a noninferior solution in a neighborhood of ~ in the objective space is exactly one. With a little more work, the following useful and more complete information concerning trade- otis at :e can be obtained. If f3 changes slightly from f3(~) (=3) by 6f3, then to stay in the noninferior surface in a neighborhood of ~ in the objective space, the corresponding changes in fl and./2 will be at m o s t -~-12c~J3 and

&(g) vL ( ~ ) - - - ,

we have, as its solution, 0_--</,/2__<2 , 0~_~.12 <2 and 1=<213--<3, with 2 1 2 = 2 - - f l 2 and 2 1 3 = 1 + # 2 . The non-uniqueness of the solution of (*) arises due to the fact that 2 is not a regular point of the constraint of P1(3/2,3/2). Here again, trade-off interpretations of 212 and 213 obtained by solving (*) fail to apply.

oa X

\ ~ E2>1,5

\ \

" ~ / /

~ . ~ = 1.5 / / ' I \ ,

0 l x 2 3

X i

FIG. 5. Graphic solution of P1 (e.2, 1.5), e. 2 > 1.5.

,/(3,2) / /

/ / / / /

/ / / / /

/ / / /

To overcome this difficulty, we relax the constraint f2(x)< 1.5 by choosing e2 to be any number greater than 1.5, as shown in Fig. 5, and let e3 remain at 1.5, 2 is still a unique solution of the new problem. Moreover, the regularity condition is satisfied and the new solution of (*) setting 212 =0 is ~13 =3 and fi2 =2. Now ~ 1 3 ( = 3 ) represents the total trade-off (along direction ( - 1,0)) rate at

involving f l and f3- Previous discussion regard- ing the trade-off information of J2 also applies here.

8. CONCLUDING REMARKS

Through the solutions of the e-constraint problem and through the sensitivity interpretation of the optimal Kuhn-Tucker multipliers associated

72 YAC()V Y. HAIMES and VIRA ( 'HANKON( i

with those solutions, we were able to study local properties of the inferior surface (frontier) of an MOP both in the decision and objective space. An important outcome of this study is that the gradient of the noninferior surface at a given point x* (on the surface) can be determined from the values of the corresponding optimal Kuhn- Tucker multipliers associated with the e- constraint problem to which x* is the strict local solution. This leads to the trade-off interpretation of such multipliers thereby providing a convenient way of obtaining necessary information for continuing into the 'analyst-DM' interactive mode of the M D M process.

Acknowledgements- This research was supported in part by the National Science Foundation, Research Applied to National Needs Program under grant no. AEN75015820 and by the Mekong Secretariate of the Economic and Social Commission to Asia and the Pacific, United Nations.

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kuhn-tucker multipliers as trade-offs in multiobjective decision-making analysis

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