General communication schemes for multiobjective decision making

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European Journal of Operational Research 26 (1986) 108-122 North-Holland

General communication schemes for multiobjective decision making Peter B O G E T O F T

Department of Management, Odense University, 5230 Odense M, Denmark

Abstract: In this paper four general communication schemes for multiobjective decision making are presented. The procedures are completely general and make no presumptions about linearity or convexity. Nevertheless, improving bounds on the optimal value are available in each iterative step. Hereby, the procedures model the essential judgements to make and the necessary information to exchange between the decision maker and an analyst in order to make the communication progress. The procedures provide valuable theoretical insight. They also supply a framework for synthesizing known and developing new procedures. Keywords: Multiple criteria decision making, interactive procedures, trade-off information, resource directive, price directive

I. Introduction The purpose of this paper is to present four general communication schemes for multiobjective decision making. Many interactive procedures have already been proposed for this kind of problems. For a recent overview, see Evans (1984). However, the present procedures are completely general and make no assumptions about linearity or convexity. Nevertheless, improving bounds on the optimal value are provided in each iterative step. Hereby, the communication is progressing. The procedures point out the essential judgements to make and the necessary information to submit to make the communication progress. The focal point is to balance approximate substitution wishes against approximate substitution possibilities. Thus, common economic concepts are applied and simple economic interpretations of the subproblems involved are made possible. Hereby, valuable theoretical insight is gained into the interaction between a decision maker and an analyst

Received July 1985

involved in multiobjective decision making. The procedures describe four different ways to structure the necessary investigations and comparisons of possibilities and wishes in a multiobjective decision context. These structures cover the ideas of a wide spectrum of known interactive procedures. Within each communication structure the precision of the investigations and comparisons is a design variable. We characterize the classes of investigations applicable in each structure, and we discuss the selection of convenient types in more concrete contexts. Hereby, the general procedures provide a starting point for developing new procedures. The outline of the paper is as follows. The decision context is defined in Section 2. In Section 3 the distinctive characteristics of the procedures developed in this paper are discussed. In Sections 4 and 5 the procedures are formalized and the mathematical foundation is developed. The selection of convenient substitution expressions is discussed in Section 6. Section 7 contains some final remarks. To simplify the exposition we assume all single-objective optimization programs to have an optimal solution. The procedures can be generalized to incorporate other cases as well.

0377-2217/86/$3.50 © 1986, ElsevierSciencePublishers B.V. (North-Holland)

P. Bogetoft / General communication schemes for MCDM

2. The decision context

Many decision contexts are characterized by incomplete a priori knowledge of the wishes, the possibilities, and the relations between wishes and possibilities. Several more or less conflicting objectives are persued and the appropriate compromise or aggregation of these is not known in advance. The decision maker does not know all decision alternatives and the consequences they generate. Or he finds it hard to comprehend this information. The purpose of multicriteria analysis is to help decision making in such cases. To formally represent the setting above we introduce the notion of a basic decision problem within a context of decentralized and not immediately available information. The basic decision problem considered is max F ( f t ( x ) . . . . . f~ ( x ) ) x~X

where X is the set of feasible decisions, fl . . . . . f , are real functions defining the criteria of relevance to the decision maker and F denotes his implicit, ordinal utility function. We assume without loss of generality that F is nondecreasing. Also, for case of simplicity, we assume F to be strictly increasing in fl. In value space, the decision problem may be stated as (P)

max r ( y ) y~Y

where Y = ((Yl . . . . . y,)} = ( ( f l ( x ) . . . . . f , ( x ) ) I x ~ X } is the set of feasible value vectors. In the setting considered we furthermore assume information to be decentralized and not immediately available. Thus, we assume that the decision maker (DM) 'knows' about F while an analyst (AN) 'knows' about Y. But we do not assume the DM and the AN to have explicit expressions of F

D M learns about Y Resource directive

Price directive

D M substitution decision J, T possibilities AN

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and Y, respectively. Rather, we simply assume that they are able to investigate F and Y, i.e. to answer certain questions about F and Y. The kind of questions to be answered depends on the case considered. One substantial assumption remains. We assume the DM to answer the sequence of questions consistently, i.e. we do not allow F to change during the communication process. The advantage of this is to make the notion of improving communication precise and provable. Thus, improving communication becomes equivalent to improving bounds on the optimal solution to the basic decision problem, c.f. the discussion in Section 3. Generally, questions are formulated as mathematical programs involving F and Y. Thus, for example, we may ask the DM to solve max ( F ( y ) [y ~ Y'). This simply means that the DM must select his most preferred point in Y'. One should note that this does not assume the existence of an explicit utility function. Formulating questions as mathematical programs is just an expositive convenience.

3. The communication context

Obviously, some investigations of and some communication about F and Y is needed to solve the basic decision problem within the above context of decentralized and not immediately available information. In the remainder of this paper we develop four different procedures which may be used to define and structure the necessary investigations and communication. Many techniques from different disciplines are concerned with the issue of solving decision problems with decentralized and not immediately available information. Therefore, it must be em-

A N learns about F decision £

AN DM

substitution ? wishes

DM-R procedure

A N- R procedure

substitution D M wishes ,L 1"decision AN DM-P procedure

substitution AN possibillities ~' 1' decision DM A N-P procedure

Figure 1. General communication schemes

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/General communication schemes for M C D M

phasized that our starting point and prime interest is interactive multicriteria procedures. However, some ideas and terminology from multilevel planning and mathematical decomposition theory have been adopted. Useful references to these disciplines are Dirickx and Jennergren (1979), Geoffrion (1970), and Lasdon (1970). A first outline of the four general communication schemes are given in Figure 1. It may be useful to discuss the distinctive characteristics of these schemes and to make some comparisons with previous work on multiobjective decision making. First, consider the types of information exchanged. In this paper we only allow signals with resource or price interpretations. By resource signals we mean signals which inform about a given point or impose constraints on the set of points to be considered. By price signals we mean signals expressing a willingness or a possibility to substitute between different criteria values. We restrict attention to resource and price signals as they imply subproblems with simple economic interpretations. Also, they cover the ideas of a

wide spectrum of signals suggested in the multicriteria literature. The importance of substitution (or trade-off, price, dual) information has long been recognized. Several papers on the existence and characteristics of trade-offs in especially linear and convex cases have appeared. See for example Haimes and Chankong (1979), Hannan (1978), Kornbluth (1974), and Iserman (1976, 1977, 1978). Also, in concrete procedures it has been suggested to exchange information about the trade-off possibilities by means of ideal and nadir points, pay-off matrices, sets of non-dominated solutions, efficient trade-offs, Lagrange multipliers etc. and about trade-off wishes by means of the most preferred of several trial solutions or trade-off proposals, marginal rates of substitution etc., c.f. the references below. The effect of all these signals is to provide information about the possibilities and wishes around a given point. The rationale is the expectation that such more informative signals may assist the DM and the AN in their search for a compromise solution. In this paper we represent trade-off informa-

R1 ~ Yl

\ opt

F=F max

-,Yn ) Rn-i Figure 2. Basic concepts

P. Bogetoft / General communication schemes for MCDM

tion by introducing so called exaggerated substitution possibilities and wishes, and we explicitly model how this kind of information may assist the DM and the AN in their search for a compromise solution. Formal definitions are provided in Section 4 below. However, the idea is geometrically simple. Exaggerated substitution possibilities (ufunctions) are upper supports to the set of feasible value vectors Y and exaggerated substitution wishes (w-functions) are lower supports to indifference curves. These functions exaggerate as they give too optimistic pictures of the possibilities and the willingness to make trade-offs around the supporting point. See Figure 2. The use of this kind of dual information has two advantages. It allows us to do without convexity and concavity assumptions. Also, it allows us to use a wide spectrum of more or less informative signals in the general procedures. Next, consider the organization of the communication. In this paper we only discuss how the DM may learn about Y or how the AN may learn about F, and we only allow the pure resource and price signals described above. Clearly, many known procedures deviate from these pure arrangements. Sometimes, the DM and the AN simultaneously learn about Y and F. More often, mixed resource and price signals are implicitly or explicitly applied. Finally, there exist considerable variations in the substitution expressions suggested, c.f. the examples above. However, we believe the pure procedures to enlighten important aspects of these more complicated arrangements. Furthermore, it turns out that many known procedures are in fact surprisingly pure in the above sense. Hereby, the four general schemes of this paper cover an important spectrum of previous schemes for organizing the interaction between a DM and an AN involved in multicriteria decision making. Many of these procedures are reviewed in Kok (1984). The DM procedures describe how the DM may search the set of possibilities by (DM-R) imposing varying constraints (e.g. Benayoun e.a. (1971), Nakayama and Sawaragi (1984), Nijkamp and Spronk (1980) and Rietveld (1980, Ch. 9)), or by (DM-P) varying the weights of importance assigned to the different criteria (e.g. Geoffrion (1967), Soland (1979)), his priority-goal structure (e.g. Lee (1972)) or his reference point (e.g. Wierzbicki (1979), Lewandowski and Grauer

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(1982)). In several DM-R procedures some implicit price adjustment is involved as the metric applied depends on the constraints imposed. Explicit combinations of the DM-R and DM-P procedures are suggested by e.g. Kok and Lootsma (1984) and Soland (1979). In the former, the DM sets lower bounds on the different criteria and--through parwise comparisons--he indicates his reluctancy to different deviations from the ideal vector. Finally, the reference point approach could be considered a DM-R approach if it simple works as a steering instrument without any element of goal or aspiration interpretation. The AN-procedures describe how the AN may build up relevant information about the DM's preferences by (AN-R) asking for direct information about the DM's wishes around a given point (e.g. Geoffrion, Dyer and Feinberg (1972), Musselman and Talavage (1980), Oppenheimer (1978)), or by (AN-P) asking the DM to evaluate existing substitution possibilities (e.g. Chankong and Haimes (1983, Ch. 8), Korhonen, Wallenius and Zionts (1984), Steuer and Choo (1983), Steuer and Schuler (1978), White (1980), Zionts (1981), Zionts and Wallenius (1976, 1983)). Third, consider the quality of the communication. We assume a fixed implicit utility function and we focus on improving communication. Improving bounds on the optimal value are available in each iterative step. Our concept of improving communication is equivalent to the convergence concept applied in the AN-procedures above. In the DM-procedures convergence--if discussed--is usually equivalent to making the procedure stop. It has been widely criticized as being a pure mathematical concept forgetting that learning about the possibilities, changing the wishes etc. may represent improvements of its own. However, to the present author, even mathematically it is a poor concept as it forgets the quality of the final decision. Convergence should be against the optimal compromise solution. Of course, this latter concept is only defined if we assume a given utility function. Therefore, the method of this paper is to assume a fixed utility function and a DM capable of answering certain questions. Next, we look for improving bounds on the optimal value. In practice, the DM may not have a fixed utility function, he may not be able to answer the questions put to him, etc. We believe the procedures to be well-be-

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haved in such cases as well. However, this cannot be established nor even defined in a mathematical programming setting without such assumptions. So, we impose some behavioral assumptions on the D M and test the procedure in this setting.

The set U of actual exaggerated substitution possibilities u: R "-1 ---,R is defined by U : = ( u ~ U Ju(y 2 ..... Y,) >~Yl, V y ~ Y } =

{ue Ulu(~)>/¢(e),Ve~E}.

Also, to describe substitution wishes we introduce a class V of nonincreasing functions

4. Four interactive procedures

v: E ---, R. To formalize the general communication schemes we need a bit of notation. In general, if A is a set, A' denotes a subset of A, a denotes an element of A, and a---,A' is equivalent to the operation A' .'= A' U ( a } of expanding A' by a. Let the perturbation function ~b: R "-1 ~ R be given by • (e) := m a x ( y 1 l Y ~ Y, (Y2 . . . . . y . ) >/e}, eER

"-1.

G(v, y ) : = max F ( y l - v ( y 2 ..... y,) + v(e), e), e~E

v~V,

y~Y,

i.e. the maximum utility that may be obtained when substituting from y according to v. Now, the set W of actual exaggerated substitution wishes w: R " - ] ~ R becomes W : = { y , - v ( y 2 ..... y,) + v ( . ) l

We define m a x A a s - o o i f A = ~ a n d +~ ifA is unbounded from above. Consequently, • becomes a nonincreasing function. Also, let the sufficient e-values

E cc_R n-1 be defined as a set wide enough to ensure that {(~li(e), e ) l e e E } contains all efficient points in Y. For example, we may chose E as R "-1 or {(Y2 ..... y , ) l y ~ Y } . Now, the essential issue facing the DM in the D M - R and DM-P procedures becomes one of learning about • on E. To describe substitution possibilities we introduce a class U of nonincreasing functions u: E ~ R . D M learns a b o u t Y DM: max e~E

Resource directive

To determine substitution wishes in a given point it is useful to introduce an evaluation function

rain F(u(e), e) u~U"

y e Y, G ( y , v) ~ r ( y ) } . Note that w ( - ) = y l - v ( y 2 ..... y , ) + v(.) is a lower support function to the indifference curve through y. Finally, let

z:= ((u(e*), e*)lug V, e*~argmaxF(u(e),e),

i.e. a set of points which are all at least as good as any point in Y. The set Z is built up iteratively when needed. The general communication schemes may now be defined as in Figure 3 below. The outline in Figure 3 emphasizes the similari-

A N learns a b o u t F AN: max y~Y

z*-'= (u*(r*), ~*) z'J, t u*--' U' A N : rain_ u(z~,..., z,,)* - z 1. u~U

s.t.e~E},

m i n y l - w ( y 2 ..... y.) w~W"

y'J,

T w*-'W'

D M : m i n G(v, y*) o~V

w * ( ' ) := y? - v(y~ ..... y * ) + v ( ' ) DM: min vEV

Price directive

m a x G(v, y ) y~Y'

A N : rain u~U

m a x u(z 2. . . . . z , , ) - z 1 zEZ"

w*(-) := y? - v*(y~..... y,*)+ v*(.) w'J,

t Y* "-" Y'

A N : m a x y] - w * ( y 2 . . . . . y . ) y~y

u* $

T z* ~ Z '

D M : m a x F(u*(r), *) eEE

Z* := (U*(~*), e*) F i g u r e 3. F o u r interactive p r o c e d u r e s

P. Bogetoft /

General communication schemes

for MCDM

113

ties of the four schemes. Nevertheless, to aid interpreting and applying the procedures a few equivalent formulations may be useful. First, note that the DM-R subproblem is equivalent to

i.e. the problem of determining an upper support to the perturbation function at e*. Hence, the AN must solve the general (U-) dual program to the

rrfin(u(e*)lue UAu(y 2. . . . . Y.)>~Yl Vye Y},

(P~*) m a x { y 1 I Y E Y , ( Y 2 ..... y , ) ~ e * } .

e-constraint problem Hereby, the procedures are related to general dual-

2 z

1

u0 w2

Ie DM-R procedure

AN-R procedure

k ~ Y ~

~

F=Fmax w2

b w DM-P procedure Figure 4. The four procedures in a simple case

AN-P procedure

z

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P. Bogetoft / Generalcommunicationschemesfor MCDM

ity theory of mathematical programming, c.f. Section 6. Also, the DM-P subproblem is equivalent to (Pl-v*)

max{yl-v*(y2 ..... y,)ly ~Y},

i.e. the AN most solve a (generalized) parametric approach problem. To get a more concrete impression of the procedures, it may be useful to consider a classical case with Y convex and F quasiconcave. Letting U and V be the set of nonincreasing affine functions from R n-] to R and by drawing up the signals submitted during a few iterations, the basic ideas should be clear. Illustrative examples are outlined in Figure 4, where { u ° } and {y0} denotes the a priori information available to the DM in the DM-R and DM-P procedures and yl and u 1 in the AN-R and AN-P procedures represent arbitra~T¢ initial signals submitted by the AN. 'in technical terms the procedures may be shortly ch; racterized as follows. In the DM-R (DM-P) procedure the DM produces decreasing outer (incre lsing inner) approximations of the set of feasible points. In the AN-R (AN-P) procedure the Ai~ produces decreasing outer (increasing inner) apj:broximations of the set of points above the op imal indifference curve. So, the approximation pri aciples in the resource and price directive procec ures are exactly opposite or dual. On the other ha,~d, the way the DM learns about the perturbatio a function is equivalent to the way the AN lea ms about the optimal indifference curve. In conceptual terms the procedures work as follows. A DM wanting to learn his possibilities Y may apply two approaches. He may demand (approximate) information about the substitution possibilities prevailing in a given direction. Comparing these possibilities with his wishes enables the DM to suggest a new and better informed direction of search in the next phase. This is the DM-R approach. Alternatively, the DM may inform the AN about his (approximate) substitution wishes around a given (feasible) point and instruct the AN to determine a decision balancing these wishes against the actual possibilities. The new proposal enables the DM to suggest a better substitution scheme in the next phase. This is the

DM-P approach. In both procedures the DM systematically builds up information about Y that is relevant in view of the actual F. Also, in both cases the DM directs the communication. However, he has an option between comparing the substitution wishes and possibilities himself (DMR) or letting the AN do so (DM-P). Similarly, the AN wanting to learn the important aspects of F has two obvious possibilities. He may suggest a decision to the DM and ask him to state his (approximate) substitution wishes at this point. This enables the AN to state a more relevant proposal in the next iteration. This is the AN-R procedure. Alternatively, the AN may suggest some (approximate) substitution possibilities and instruct the DM to select his most preferred point. The point selected enables the AN to suggest a more relevant scheme in the next iteration. This is the AN-P procedure, In both procedures, the AN systematically builds up information about F that is relevant in view of the actual Y. He does so directly by requesting substitution schemes, or indirectly, by asking the DM to select from a given scheme. Thus, the AN directs the communication. However, he has an option between comparing the possibilities and wishes himself (AN-R) or letting the DM take over (AN-P). Finally, if we think in production terms, we get more specific economic interpretations. One may think of the DM as a consumer and the AN as a producer in a two-person, n-commodity economy. The direct production function is ~li, the direct utility function is F and the first criterion constitutes the chosen measurement unit. Now, the subproblems involved are those of determining a price-offer (resource directive procedures) and those of determining a profit (or utility) maximizing production (consumption) plan given prices (price directive procedures). The set of (relative) prices considered are not simply a subset of R n-1 as we do not assume concavity of • nor F. Rather, full price schemes are applied. That is, the value of resources is defined for any given amount stead of simply per unit. Only linear value schemes can be expressed by simple prices, dual variable, Lagrange multipliers, shadow prices etc. The price functions, value schemes or dual functions allow one to express more specialized substitution conditions.

P. Bogetoft / General communication schemes f o r M C D M

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Assumption (VI). The set V is sufficiently wide to

5. Mathematical foundation

ensure Consider now the formal properties of the procedures suggested above. Two issues are essential. First, the equivalence between the basic decision problem and the different full master problems must be examined. Next, the question of improving communication must be discussed. Short-hand notations of the full master problems are convenient. Thus let

(PDM-R)

max min_ F(u(e),

(PDM-P)

min max G ( v , y ) ,

(PAN-,)

max m i n Yl -- w(y2 . . . . . Y,),

and (PAN-P)

min max u(z 2 . . . . . z . ) -- z 1.

e~E

vE V

y~ Y

uEU

uEU

e),

yE Y

w~ W

zEZ

In the procedures above the DM and the AN exchange information about substitution conditions by means of the u- and w-functions introduced. Obviously, for the procedures to work these signals must be able to describe the possibilities and wishes sufficiently precisely. Equivalently, the U and V classes must be sufficiently wide to make the procedures full master problems solve the original problem. The following assumptions prove relevant, c,f. Theorems 1 - 4 below.

Vy ~ Y 3v ~ V: G ( v , y ) ~ F ( y ) , i.e. at any point it must be possible to support the indifference curve from below by a w-function. []

Assumption 0/2). The set V is sufficiently wide to ensure

3v ~ V: max G ( v , y ) <~ max F ( y ) , y~Y

y~Y

i.e. a w ~ W running (weakly) between the perturbation function and the optimal indifference curve must exist. [] A fundamental issue is of course how many and diversified u- and v-functions are needed to ensure (U1), (U2), (V1) and (V2) above. We return to this issue in Section 6. The connections between the original problem P and the procedures full master problems PDM-R, PDM-P, PAN-R'and PAN-P may now be stated as follows.

Theorem 1 (DM-R). I f U fulfils (U1) we have max min_ F ( u ( e ) , e ) = max F ( y ) . e~E

u~U

y~Y

Furthermore, we get V(e*, u*) ~ argmax(PoM.R):

Assumption 0UI). The set U is sufficiently wide to ensure

u: u(,)

(1)

V y ' ~ argmax(P~.): y ' ~ argmax(P),

(2)

3y'~argmax(P~_v.): y'~argmax(P).

$(,), Theorem 2 (DM-P). If V fulfils (V2) we have

i.e. at any point it must be possible to support the perturbation function from above by a u-function. This is equivalent to U closing the duality gap between P~ and its U-dual program for all e ~ E. []

Assumption (U2). The set U is sufficiently wide to ensure

rain max G ( v, y ) = max F ( y ) . v~ V

y~ Y

y~ Y

Furthermore, we get 3(v*, y*) ~ argmax(PDM_e): (1)

y* ~ argmax(P),

(2)

y* ~ argmax(Pl_w. ).

[]

Theorem 3 (AN-R). If V fulfils (V1) we have 3u E U: max F ( u ( e ) , e) <~ max F ( y ) , e~E

y~ Y

i.e. an u ~ U running (weakly) between the perturbation function and the optimal indifference curve must exist. []

max min_ Yl - w ( y2 . . . . . y, ) = O. yEY

wEW

Furthermore, letting ~ ( e ) : = m a x w(e) w~W

Ve~E,

[]

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P, Bogetoft / General communication schemes for M C D M

we get

signal submitted by the restricted master problem is unique, we get a strict improvement in one of the bounds. []

3(y*, w*) ~ argmax(PAN.R): (1) y* ~ argmax(P), (2) y* ~ argmax(Pl_~), (3) y* ~ argmax(P1,_w,) in case of (V2).

[]

Theorem 4 (AN-P). I f U fulfils (U2) we have min

u~U

max zEZ

u(z 2.....

Zn) -- Z 1 = O.

Furthermore, we get

3(u*, z*) ~ argrnax(PAN_p): (1)

z* ~ argmax(P)

(2) 3 y ' ~ argmax(Pl._,,): y ' ~ argmax(P).

[]

The proofs of Theorems 1-4 are outlined in the appendix. Theorems 1-4 state conditions sufficient to ensure that the procedures full master problems solve the original problem. More precisely, they show that an optimal solution to the original problem may be obtained from an optimal solution to the procedures full master problems by direct, resource directive execution or by indirect, price directive execution. In cases of non-unique solutions to the full master problems or the execution problems an optimal solution to the original problem may not be uniquely identified by the procedures above. In such instances further optimization which is no part of the present methods may be undertaken to determine an optimal solution from the remaining--hopefully few--potential decisions. Now, consider the question of improving communication. The fundamental result may be stated as follows. Theorem 5. Assume the sets of substitution schemes to fulfil the appropriate assumption (U1), (U2), (V1) or (V2) stated in Theorems 1-4. For each procedure we get (1) Nondecreasing lower and nonincreasmg upper bounds on the optimal value of the full master problem are available in each iterative step. (2) I f a signal is repeated by the restricted master or subproblem, a global optimum has been reched. (3) I f optimality has not been reached and the

The proof of Theorem 5 is outlined in the appendix. Theorem 5 shows that the procedures do not cycle and that the communication is improving. It also shows that convergence is finite whenever the set of potential signals is finite. In other cases the question of convergence must be treated in more specific settings. In practice, different stopping rules may be applied. No detailed discussion is intended here. We simply note that optimality is reached if a signal is repeated, and that upper and lower bounds on the optimal value are determined in each iterative step. Also, different execution principles may be applied. The details may be developed along the lines of Theorems 1-4.

6. Substitution expressions The selection of convenient substitution expressions is of course a major problem in specific settings. No thorough discussion is intended nor indeed possible here. However, a few general remarks on the theoretical and practical issues involved may be useful. First, at the theoretical level, consider the assumptions (U1), (V1), (U2) and (V2). The assumption (U1) is simply the condition that U closes the duality gap between P, and its general U-dual program for all e ~ E. Consequently, general duality theory provides a genuine theoretical framework for discussing (U1). From this appropriate classes may be developed in many more concrete cases. For example, the set of affine functions will do when X is convex and fl . . . . . fn are concave, special classes of superadditive functions will do in integer cases and sets of step functions will do in bottleneck programs. A review of general duality theory is given by Tind and Wolsey (1981). General duality in multicriteria optimization is discussed in Bogetoft (1985a) and Bogetoft and Tind (1985). Finally, papers on vector optimization provide numerous results about the characterization of all efficient solutions by optimizing appropriate classes of value functions and distances, c.f. for example Gearhart (1983),

P. Bogetoft / General communication schemes for MCDM

Jahn (1984), Rasmussen (1985), Soland (1979), Yu (1974) and the references herein. Now, consider (V1). The author is not aware of a theoretical setting appropriate for this case. Nonetheless, two remarks are possible. First, the set of nonincreasing linear functions is sufficient whenever F is quasi-concave and nonsatiated in Yl- Since most utility functions are probably quasi-concave this provides a starting point for discussing (V1) in most cases. Second, note that general duality theory may still be useful. If F( ) is the value of a constraint optimization problem of its own

F(y*) = m a x { a ( d ) I t ( d ) <~y*, d~D} and 7": R" + R is an optimal dual function closing the duality gap at y*, the equation 7(Y)= y*(y*) defines a lower support function to the indifference curve through y*. The conditions (U2) and (V2) are somewhat harder to fulfil since they involve functions supporting the perturbation function and the (optimal) indifference curve simultaneously. However,

and 1 ~ = { max w(-)lW'c_ W} WE W '

fulfil the assumptions in (U2) and (V2) whenever U and V fulfil (U1) and (V1). Thus, by combining signals fulfilling (U1) and (V1) applicable (U2) and (V2) classes may be developed. Next, let us turn to some more practical considerations. In all the procedures we have decomposed the total problem of investigating and comparing the wishes and possibilities into a sequence of--hopefully--simpler subproblems. Nevertheless, the DM and the AN still have to perform certain nontrivial evaluations and to store a great number of information to make the communication proceed as prescribed. This may not constitute a problem if the DM and the AN, respectively, are well equiped headquarters and division of a large organization. However, in most cases it remains a major issue to keep down the burdens imposed on the AN and--more importantly--the DM. To accomplish this, several instruments may be applied. First, the choice of overall communication

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structure is important. The four different structures imply different distributions of the burdens between the DM and the AN. However, the details depend on several contextual factors. Therefore, no adequate discussion is possible here, c.f. Bogetoft (1985b). Next, the choice of substitution functions within a given structure is important. The complexity of the subproblems may be reduced by diminishing the information value of the signals applied. Consider for example the DM-R procedure. The 'flatter' u-functions one applies, the better indication the DM receives of the possibilities to substitute in each interaction. However, the task of solving the AN's subproblem increases. Also the assessments in the DM subproblem must be more precise. A wide spectrum of possibilities exists between the one extreme where point estimates of the perturbation function are used, and the other extreme where full perturbation functions are submitted. So, one must seek an appropriate trade-off between the complexity of the subproblems and the number of iterations. Third, the possibility of constructing, evaluating and communicating about substitution conditions in a piecemeal fashion should be mentioned. For example, simple pairwise comparisons may suffice to construct w-functions in some cases, c.f. Korhonen, Wallenius and Zionts (1984) and Bogetoft (1985a). Again, at the possible cost of reduced information value and additional iterations, the subproblems may be extensively simplified. Finally, let us note that it may even be rational to ignore the conditions (U1), (V1), (U2) and (V2) when selecting U and V. The reduced burden of the decision process may outweigh the possible cost of a duality gap and theoretical suboptimality.

7. S o m e final remarks

In this paper four general communication schemes have been developed. They provide theoretical insight into the interaction between a decision maker and an analyst involved in multiobjective decision making. Also, they supply a framework for synthesizing the main ideas of a wide spectrum of known procedures. Finally, at the more practical level, they constitute a starting

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P. Bogetoft / General communication schemes for M C D M

point for the development of new procedures in more concrete settings. Numerous extensions are possible at all three levels. Let us just mention one study presently undertaken, eft. Bogetoft, Hallefjord and Kok (1985). In practice, methods based on reference points, moving targets etc. have proved attractive. Therefore, a more profound theoretical foundation of these procedures would be interesting. One approach would b e - - i n the vein of this paper--to establish conditions that are sufficient to ensure improving communication, Especially, it should be modelled how dual information may assist the DM's selection of the successive reference points.

i.e. the opportunity to use price directive execution when it is unique. It suffices to show y ' ~ argmax(P,.) ~

y ' ~ argmax(Pl._..)

(4)

as (3) has already been proven. To prove (4), first note that u* is an optimal solution to rain

u(e*),

s.t.

u(y2 ..... Y,)>~Yl V y ~ Y, ur= U ,

as F is strictly increasing in the first variable. So,

ya-u*(y2 .... y,)<~O V y ~ Y .

(5)

Also, from u* nonincreasing and y ' feasible in (P~.), equation (5), the definition of 4 and (U1) we get

Appendix

u*(e*) ~ u*(y~,..., y') ~ y ; = 4(e*) = u*(e*), Proo| of Theorem 1 (DM-R). First the equivalence of PDM-Rand P shall be proven. From U closing the duality gap (U1) and F nondecreasing we get the first equality below: ur~U

e~E

= max F(y).

(1)

y~Y

The last equality may be seen as follows. We have ( > ) as all efficient points are located at the perturbation function 4. We have ( ~ ) as all relevant points on 4 are weakly dominated by a point in Y, i.e. for any e ~ E satisfying 4 ( e ) > - o o we have

: l y r e : Yi>~(4(e), e),,

i = l . . . . . n,

(2)

c.f. the definition of 4. Next, let (e*, u*) be any optimal solution to PDM-R" Consider the statement V y ' ~ argmax(P~.): y ' ~ argmax(P)

y~ - u*(y~ . . . . . y ' ) = 0.

(6)

From (5) and (6) we get (4). This proves Theorem 1. []

max min F(u(e), e)= max F(4(e), e) e~E

i.e.

(3)

Proof of Theorem 2 (DM-P). First, consider the equality min max G (v, y ) -- max F ( y ) . v~g

y~Y

(7)

y~Y

The (~<) part is simply the assumption (V2). The (>~) part may be seen as follows. Let b be any scheme in V, and let ~ ~ argmax(Pl_ v). This implies •

-

.....

L)

+

Ve~E. Hereby, G(~, P ) = max F(fv(e), e) eEE

>/max F ( 4 ( e ) , e ) = max F(y).

i.e. the possibility of using resource directive execution. If y ' is any optimal solution to P~. we get from (1), (U1), and (2) that

Consequently,

max F ( y ) = F(u*(e*), e*)

~'~ ~ V: max G(b, y) >1G(~, ~,) > max F(y).

y~Y

= r ( 4 ( e * ) , e*) <~r ( y ' ) . Therefore, since y ' ~ Y we get (3). Now, consider the statement : l y ' ~ argrnax(Pl,_..): y ' ~ argmax(P),

e~E

y~ Y

y~ y

yE Y

This proves the (>/) part of (7). Next, let (3, ~) be any optimal solution to PDM-P, and let y ' be any optimal solution to P. By F strictly increasing in the first variable we get

P. Bogetoft / General communication schemes for M C D M

:=Y, -

0(*)

b(y2 ..... L) +

Ve~E, since otherwise a better ) would have been chosen. Now, by the definition of • we get Yl - w(Y2 . . . . . y , ) <~0

Vy ~ Y.

max F ( ~ ( e ) , e ) = G(b, ~ ) = F ( y ' ) . eEE

As F is strictly increasing in y~ this implies Y,-

(Y2,..

Y')

Now, by F strictly increasing in the first variable and the definition of G we get from (lY) the existence of a fi ~ Y so that F ( y ' ) < F ( ~ ( y ~ , . . . , y~), y~ . . . . . y~) <~ max F ( ~ ( e ) , e) = G(b, fi) <~F ( ~ ) .

(8)

Also, by the definition of G(b, y) and (7) we get

0.

(9)

Finally, let (v*, y*) = (fi, y'). From (8) and (9) we get (v*, y*) = (~, y') ~ argmax(PDM.p), y* = y' ~ argmax(P), y* ~ argmax(P 1_ ~.). This proves the execution statements in Theorem 2. []

teE

This contradicts y ' being an optimal solution to P. Hereby (10) has been proven. Next, let y* = y ' and w* be optimal in (12). By (11) and (12) we get (y*, w*) E argmax(PAN.a ). Also, y* ~ argmax(P) by assumption and y* argmax(Pl_~) by (11) and (12). Finally, in case of (V2), v* could be chosen to run weakly above and y * ~ argmax(Pl_w, ) results. This prove the execution statements in Theorem 3. [] Proof of Theorem 4 (AN-P). First, note that by definition of Z

z = ((.(¢), * * ~ a r g m a x F ( u ( r ) , e),

Proof of Theorem 3 (AN-R). First the equality max rain_ Yl

-

w(y2 . . . . . y,) = 0

-

y~ Y w~ W

(10)

119

s.t. e ~ E }

we have Vz ~ Z: F ( z ) >1 max F ( y ) ,

as F is nondecreasing. So, Z is a set of points at least as good as any point in Y. Now, consider the equality

Vy ~ Y 3v ~ V: G ( y , v) <~F ( y ) . From the definition of W this implies vyEr3wEW:

y l - w ( y 2 . . . . . y,)<~O.

(11)

This proves the (~<) part of (10). The ( > ) part may be seen as follows. Let y' be any optimal solution to P. Then min y~ - w(y~ . . . . . y ' ) = 0 ,

(12)

wEW

(14)

y~Y

must be proven. From (Vl) we have

rain max u(z 2 . . . . . z , ) - z l = O . u~D zEZ

(15)

The (>~) part may be seen as follows. By definition of Z we have rUE

U 3z~:Z:

u(z 2 .....

Zn)--g 1

= 0 .

Hereby

as --

t

--

¢

3 ~ , ~ W: el - w ( y 2 . . . . . y~) < 0

(13)

would imply the existence of a lower support ~ to the indifference curve of a point ~ no better than y' so that ~ is running strictly above y', i.e. a contradiction. More precisely, to see (12), we note that (13) is equivalent to

3y

Y, b

v:

) =y, -

G(b, .~) ~
y ~ - ~(y~,..., y') < 0.

V u ~ U: max u(z 2 . . . . . z , ) - z 1 >10. zEZ

This proves the (>I) part of (15). The (~<) part runs as follows. Let u ' ~ U fulfil the assumption in (U2), i.e. max V(u'(e), e) <~ max r ( y ) . e~E

y~ Y

From this and (14) we get for any z ~ Z

..... y.) +

(13')

F ( u ( z 2. . . . . z , ) , z 2 . . . . . z , ) <~ max V(u'(e), e) eEE

P. Bogetoft / G e n e r a l communication schemes f o r M C D M

120

~< max F ( y ) <~F ( z ) .

utility. The upper bound is nonincreasing. A nondecreasing lower bound is

y~Y

Using F strictly increasing in the first variable this implies Y z ~ Z: u ' ( z 2 . . . . . z , ) - z 1 <~O.

(16)

This proves the (~<) part of (15). Next, let u* = u' and z* ~ argrnax(P). Hereby, to prove the execution statements, it suffices to show (u*, z*) ~ argmax(PAN_P),

(17)

z* ~ argmax(P1 _ ~.).

(18)

First, consider (17). By z* ~ Y and the definition of u* we get r ( z * ) <~ m a x F ( u * ( e ) , e)

In the DM-P procedure --s F;,Mp = G(v s, yS)

and --/:~M-P= min max G (v, y ) v ~ V y ~ yS

constitute an upper and a (nondecreasing) lower bound on the optimal utility. By m i n { F[)M_p -t It = 1 . . . . . s } a nonincreasing upper bound is available. In the AN-R procedure --s F~N_ R = max

e~E

Hence, as F is strictly increasing in the first variable, z* ~ Z,

z*)-z~ =0.

(19)

Now, by (15), (16), and (19) we get that (u*, z*) solves P A N - P ' i.e. (17) has been proved. Next, consider (18). By u* ~ U we have Y y ~ Y: u * ( y 2 . . . . . y , ) - y , >10.

w~ W s

and

y~Y

(20)

F£N_R =y~ - wS( y~ . . . . . y~ ) = 0 constitute nonincreasing upper and nondecreasing lower bounds on the optimal value (0) of the full master problem. In the AN-P procedure --s

=

u

s

Proof of Theorem 5. Consider iteration number s. Let ~s ___~, y~ ___ y, W 5 ___~ and Z s _c Z, respectively, be the information available in the (restricted) masterproblems from previous iterations. Also, let (z s, uS), (w s, yS), (yS, wS), and (u s, z s) be the information exchanged during the present iteration, respectively. Now, in the DM-R procedure _F~.....-s ,,~,_~- max rain F ( u ( e ) , e) u~U s

and _F~M_R = F(uS(z~ . . . . . z,~), z~ . . . . . z~) constitute upper and lower bounds on the optimal

s

..... < )

-

z,

s

=

0

and s _ min max u ( z 2 , . . . ,z,, ) - z 1 <~0 F~N.pu~U

Using (19) and (20) we get (18). Hereby, the execution statements of Theorem 4 have been proved. []

e~-E

m i n Yl -- w(Y2 . . . . . y , ) >~0

y~ Y

~< max F( y ) = r ( z* ).

u* ~ (z 2 . . . . .

max(_F~M_R [ / = 1, 2 . . . . . S).

z~Z*

provide nonincreasing upper and nondecreasing lower bounds on the optimal value (0) of the full masterproblem. Next, consider the statement that optimality is reached when a signal is repeated. By not too hard thinking we get the following implications proving the statement. In the DM-R procedure ~s=Fs+l

VuS=uS+I

~

~s+l _ K,s+l ~t D M - R - - ~ D M - R ,

as u s+~ does not change the evaluation of es+l, i.e. es + 1 has already been evaluated by an appropriate scheme u s. Equivalently, d +I is evaluated similarly in the (restricted) masterproblem and the subproblem. In the DM-P procedure wS = w s + l V yS = y s + l

P. Bogetoft / General communication schemes for MCDM ~-s+l __ l T s + l a D M - P - - ~-- D M - P '

::~

as y~+~ does not change the evaluation of w ~+~, i.e. w s+~ has already been evaluated at an appropriate point. So, w ~÷1 is evaluated similarly in the (restricted) masterproblem and the subproblem. In the AN-R procedure y,

y~+l

V W s = W s+l

~s+l = 0 -- itTs+l " AN-R - - ~--AN-R

~

as w ~+1 does not change the evaluation of y,+l, i.e. y~+l has already been evaluated (to zero) by an appropriate scheme. So, y s + l is evaluated similarly in the (restricted) masterproblem and the subproblem. In the AN-P procedure uS~uS+l

=~

VzSzS+1

~s+l ~0__ ~7s+1 AN-P - - a---AN - P

as z "÷1 does not change the evaluation of u ~÷1, i.e. u s ÷ a has already been evaluated (to zero) at an appropriate point. So, u ~+~ is evaluated similarly in the (restricted) masterproblem and the subproblem. Finally, consider the statement about strictly improving bounds. Assume that optimality has not yet been reached, i.e. ff*>_F ~. Also, by assumption, e~, v~, y~ and u s are unique. Now, if es + l = es, vs + l = v', y s + l = yS or us + l = u s, it means that optimality is reached in the next iteration and consequently one of the bounds must be strictly improved. Otherwise, if d + 1 4= d, Os + l ~ v s, y S + l 4=yS and u ~+ ~ 4= u s, respectively, we get --s

FDM_R = m i n

UE U s

~ I~M-P

r(u(e), e)

> lgEE rain ~s

F(u(d

~

G(v', y )

max y ~ ys

< max

y ~ y~

c(o

--$

F2N_R = min Y lS W~ ~s > t~N_p-

min

w~ ~s max z~Z s

< max

+

,

w

1),

Es+ 1) ~.

y) s

(Y2 .....

~s+l DM-R'

l~'s + 1

s

Yn)

~s+l y~+l _ w(y~+l . . . . . yZ+l) >/"AN-R,

u'(z

.....

z.) - zl

_< / T s + l u s + I ( z 2 , " " " , Z n ) - - 21 "~ ~ A N - P "

z~Z s

The strict inequalities follow from uniqueness and the weak inequalities follow from the optimi-

121

zation over lager sets at the r.h.s. So, a strict improvement in one of the bounds is obtained. [] Remark. The basic argument above is tightly related to the theory of two general principles for solving maxrnin and minmax problems developed by Burkard, Hamacher and Tind (1984). The main difference is that the subproblem considered here is not simply the inner optimization problem for fixed value of the outer variable. However, by the assumption of monotony the subproblems of the procedures above effectively do the s a m e - - t h e y propose new evaluations whenever the present evaluations are inappropriate. []

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