a comparison of two reference point methods in multiple objective mathematical programming

Continuous Optimization

A comparison of two reference point methods in multipleobjective mathematical programming

John Buchanan a,*, Lorraine Gardiner b

a Department of Management Systems, University of Waikato, Private bag 3105, Hamilton, New Zealandb Department of Accounting and Management Information Systems, College of Business, California State University,

Chico, CA 95929, USA

Received 8 March 2002; accepted 20 May 2002

Abstract

When making decisions with multiple criteria, a decision maker often thinks in terms of an aspiration point or levels

of achievement for the criteria. In multiple objective mathematical programming, solution methods based on aspiration

points can generate nondominated solutions using a variety of scalarizing functions. These reference point solution

methods commonly use a scalarizing function that reaches down from the ideal solution, in a direction specified by the

aspiration point. Conversely, a similar scalarizing function can push out from the nadir point toward a specified as-

piration point. These scalarizing functions are similar in structure but diametrically opposed in their reference points. In

this paper we examine how these approaches behave from a technical point of view and conduct an experiment to

understand better the human behaviour of users of these approaches. Insights we gain on the evenness of dispersion are

relevant when attempting to construct a representation of the nondominated set. Further, the technical characteristics

of the two formulations� solutions, combined with behavioural tendencies, allow us to comment on the implications fortheir use in interactive multiple objective methods.

� 2002 Elsevier Science B.V. All rights reserved.

Keywords: Multiple objective programming; Decision analysis

1. Introduction

Reference point methodology provides the

foundation for many interactive search procedures

in multiple objective programming (for exam-

ple, Wierzbicki, 1980; Steuer and Choo, 1983;

Korhonen and Wallenius, 1988; Steuer et al.,

1993). The concept of a reference point is consistentwith Simon�s (1957) description of satisficing deci-sion making where a decision maker tends to have

targets or goals in mind while proceeding toward a

decision. This satisficing approach is specifically

accommodated in the reference point methods of

Wierzbicki (1980) and Korhonen and Wallenius

(1988). However, reference point formulations have

also been used in ‘‘nonsatisficing’’ models such asSteuer and Choo (1983), where a reference point

formulation is used as a mechanism for implicitly

*Corresponding author. Tel.: +64-7-838-4470; fax: +64-7-

838-4270.

E-mail addresses: [email protected] (J. Buchanan),

[email protected] (L. Gardiner).

0377-2217/03/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0377-2217(02)00487-3

European Journal of Operational Research 149 (2003) 17–34

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mail to: [email protected]

optimizing a decision maker�s value function.Reference point methods may ask a decision

maker to specify either aspiration points (desirable

levels of achievement), reservation points (levels of

achievement that should be attained, if at all pos-

sible) or worst outcome points (levels of achieve-

ment to be avoided). In this study, we focus our

attention on aspiration points. We acknowledge,

however, that procedures which use reservationpoints (e.g., Reeves andMacLeod, 1999) and worst

outcome points (e.g., Michalowski and Szapiro,

1992) represent important classes of reference point

methods.

The use of reference points appeared in the

early development of multiple objective program-

ming as part of the work by Charnes and Cooper

(1961) on goal programming. Wierzbicki (1980)produced seminal research on reference point

methods, including an investigation of the char-

acteristics of various achievement functions for

allowing the search for attractive nondominated

solutions to be controlled by reference points.

These achievement functions were designed to have

a significant advantage over goal programming by

producing only nondominated, or Pareto-optimal,solutions. An important class of achievement

functions considered in Wierzbicki�s paper hasformulations similar to the weighted Tchebycheff

distance metric.

In addition to their desirable structural features,

reference point methods have also appeared useful

from a methodological or operational perspective.

They provide the ‘‘shell’’ for simple, decisionmaker controlled solution methods that do not

require the decision maker to conform to any

particular requirements or axioms. The Pareto

Race method of Korhonen and Wallenius (1988),

where a decision maker simply moves over the

efficient surface in a relatively unstructured search

for a most preferred solution, is one such example.

Buchanan (1997) has proposed an even sim-pler solution method, also using a Tchebycheff

achievement function. This ‘‘GUESS’’ method

comprises successive guesses by a decision maker

and has been successfully used in experiments by

Buchanan and Daellenbach (1987), Corner and

Buchanan (1997) and Downing and Ringuest

(1998).

In this paper we compare two formulations,both using Tchebycheff-based achievement func-

tions and aspiration points, but having quite dif-

ferent philosophies. One seeks to ‘‘push out’’ from

an undesirable solution while the other ‘‘reaches

down’’ from a desirable but unachievable solution.

It is known that these two formulations produce

different solutions, given identical input (that is,

the same aspiration point) from the same decisionmaker (Hwang et al., 1993; Martinson, 1993).

These two formulations have been incorporated in

the TOPSIS solution method of Lai et al. (1994).

TOPSIS endeavors to find a compromise between

the positive ideal solution (reaching down) and the

negative ideal solution (pushing out).

From a behavioral perspective, the choice of

reference point can considerably affect a decisionmaker�s perception of the resulting solution.

Proverbs 27:7 states the point succinctly, ‘‘He who

is full loathes honey, but to the hungry even what is

bitter tastes sweet’’. Thus, if the reference point is

the nadir or worst solution, then any resulting

solution is a significant improvement. Conversely,

if the reference point is the ideal or best solution,

any resulting solution must be worse. Even if twosolutions are the same, any comparison with these

two reference points is likely to result in different

‘‘values’’ being placed on them. The two opposing

reference points for the formulations (pushing out

from an undesirable solution or reaching down

from a desirable solution) is similar to the concept

of framing, which has been discussed in the be-

havioral decision making literature. Image theory,a descriptive decision making model proposed by

Beach and Mitchell (1990), provides a useful il-

lustration of framing. A decision maker has im-

ages or schemata which represent, in some way, his

view or values. Within image theory, the first stage

of any decision is a compatibility test where each

alternative in turn is assessed to determine its ‘‘fit’’

with a decision maker�s images. Usually these im-ages will be, in some sense, ideal. More specifically,

Tversky and Kahneman (1981) show that how a

problem is framed (in their example, whether the

same problem is stated negatively or positively)

significantly affects decision maker choice. The

pushing-out and reaching-down formulations can

be viewed as different ways of framing the decision

18 J. Buchanan, L. Gardiner / European Journal of Operational Research 149 (2003) 17–34

problem, for the same aspiration point. Lai et al.(1994) illustratively allegorize these two reference

points as heaven and hell. To examine the behav-

ioral effects of this framing, we also present the

results of an experiment on how the same human

decision makers respond to solutions from these

two formulations, given the same aspiration point.

The paper is, firstly and principally, a sensitivity

analysis of the different solutions generated bythe two formulations, controlling for aspiration

points. Secondly, the paper provides recommen-

dations for decision support, particularly in the

context of interactive multiple criteria decision

methods. In this respect, the technical sensitivity

analysis provides a clear understanding of how

solution methods can be expected to behave (an

issue of controllability). Then the experiment addsto our understanding of how decision makers can

be expected to behave. A desired outcome is an

improved alignment of solution methods to deci-

sion makers. The paper is organized as follows. In

Section 2 we introduce necessary terminology and

present the two achievement function formula-

tions. Section 3 considers the discrepancy between

solutions generated by these two formulationsusing a q-dimensional hyperspherical approxima-

tion. Section 4 provides two illustrative examples

and Section 5 describes and summarizes results of

the behavioral experiment. A discussion of the

results and their implications for decision support

is presented in Section 6.

2. Terminology and definitions

The multiple objective mathematical program-

ming problem can be stated as:

MAX F ðxÞ ¼ ½f1ðxÞ; f2ðxÞ; . . . ; fqðxÞ�s:t: x 2 X

ð1Þ

where x is an n-dimensional vector of decision

variables; F is a vector-valued function comprising

q distinct objective functions of x; X � Rn is

the feasible set of constrained decisions; and Z ¼F ðxÞ � Rq is the image of X in objective function

(criterion) space. We further assume that set X

comprises m constraints of the form, X ¼ fx 2 Rn :gjðxÞ6 0; j ¼ 1; 2; . . . ;mg. The operator MAX

indicates that each objective function is to be

maximized over X.

The set of nondominated solutions N consists

of those criterion vectors z 2 Z which are not

dominated by another criterion vector in Z. A

criterion z1 is said to dominate solution z2 if

z2k 6 z1k for all k ¼ 1; 2; . . . ; qz2j < z1j for at least one j 2 f1; 2; . . . ; qg

ð2Þ

The set of weakly nondominated solutions consistsof those criterion vectors z 2 Z for which there isno other z0 2 Z such that z0 > z.In the remainder of the paper, two assumptions

are made about the preferences of the decision

maker. First, preferences are assumed to mono-

tonic; that is, given the MAX operator for each

objective, ‘‘more is always better than less.’’ Sec-

ond, a decision maker is presumed to want toconsider only nondominated solutions.

We define the maximum and minimum values

of each objective over the nondominated set N

as:

Maximum ¼ Uk ¼ maxz2Z

zk; k ¼ 1; 2; . . . ; q

Minimum ¼ Mk ¼ minz2N

zk; k ¼ 1; 2; . . . ; qð3Þ

The vector of Uk values, U, is usually referred to asthe ideal point while the vector of Mk values, M, isoften labelled the nadir point.

The Uk and Mk values are often used to rescale

the criterion vectors in order to provide commen-

surable measurement of all objectives over N(Steuer, 1986). The transformation is internal to

the procedure and in most solution methods the

decision maker is not asked to think in terms of

the transformed units. The function hðzÞ mapsany criterion vector z 2 N onto the range [0,1].

Using this transformation, hðMÞ ¼ ð0; 0; . . . ; 0Þand hðUÞ ¼ ð1; 1; . . . ; 1Þ and

hðzkÞ ¼zk �Mk

Uk �Mkð4Þ

The criterion space is therefore scaled and N is

contained within a q-dimensional unit hypercube

anchored at the origin of the nonnegative orthant.

The remaining discussion in this paper assumes the

transformation in (4).

J. Buchanan, L. Gardiner / European Journal of Operational Research 149 (2003) 17–34 19

2.1. PU and PM formulations

We now define the two different reference point

formulations which are compared in this paper.

They are hereafter referred to as Programs PU and

PM. PU uses a Tchebycheff-based achievement

function to minimize the maximum weighted de-

viation from the ideal solution U. Program PM, onthe other hand, seeks to maximize the minimum

weighted achievement from the nadir solution M.These two formulations use a simple form of the

Tchebycheff-based achievement function as op-

posed to an augmented form (Steuer, 1986). The

augmented form ensures that weakly nondomi-

nated solutions are not generated; however, for

our comparisons the simple form will suffice.For a given set of weights w, the PU and PM

formulations are:

PUðwÞ : min y

s:t: y Pwkð1� hðzkÞÞ; wk > 0;

k ¼ 1; 2; . . . ; qz 2 Z

ð5Þ

PMðwÞ : max y

s:t: y6wkhðzkÞ; wk > 0;

k ¼ 1; 2; . . . ; qz 2 Z

ð6Þ

The maxmin formulation (PM) was initially de-

veloped by Kaplan (1974) and subsequently dis-

cussed by Gupta and Arora (1977) and Posner andWu (1981). Corner and Buchanan (1995) used PM

in experimental work and the GUESS method

proposed by Buchanan (1997) is also based on the

PM formulation. The minmax (PU) formulation is

more commonly reported in the literature; varia-

tions of this formulation have been used in the

work of Benayoun et al. (1971), Nakayama and

Sawaragi (1984), and Steuer and Choo (1983), toname a few.

The PM formulation has M as its reference

point while for PU the reference point is U. Wedistinguish between a reference point such as either

M or U, which is an integral part of the problemformulation, and an aspiration point, which is an

input specified by the decision maker. The aspira-

tion point represents a target or goal and can beequivalently represented as a set of weights over the

criteria, contingent on the formulation being used.

Fig. 1 shows, in two dimensions, the relation-

ships among these different terms. The set of

nondominated solutions is represented by a hy-

persphere. hða1Þ and hða2Þ are aspiration points,transformed from the original a1 and a2. The linesegment between ð0; 1Þ and ð1; 0Þ is the set of pointswhere h1 þ h2 ¼ c ¼ 1, and other values for c arealso shown. h M1 is the PM solution for hða1Þ whileh U1 is the PU solution for hða1Þ; similarly for hða2Þ.From Fig. 1, it can be seen that the discrepancy

between PM and PU solutions is a function of the

shape of the nondominated set, the location of the

aspiration point relative to the nondominated set,

and the distance of the aspiration point from theline segment L ¼ hðMÞhðUÞ. We take the closenessof the aspiration point to L as a measure of the‘‘centrality’’ of the aspiration point. Even in the

two-dimensional case, it is obvious that the dis-

crepancy between h M and h U can be substantial.Fig. 1 shows PM as ‘‘pushing out’’ from hðMÞ, andPU as ‘‘reaching down’’ from hðUÞ.The relationship between aspiration points and

weights is as follows. For a given aspiration point

a ¼ ða1; a2; . . . ; aqÞ, the weights w ¼ ðw1;w2; . . . ;wqÞare:

Fig. 1. Comparing PM and PU in two dimensions.


For PU : wk ¼1

1� hðakÞ; k ¼ 1; 2; . . . ; q

For PM : wk ¼1

hðakÞ; k ¼ 1; 2; . . . ; q

ð7Þ

These weights can be normalized so that they sum

to unity although it is not necessary to do so. In

this paper, however, our investigation is under-taken using aspiration points, not weights. The use

of aspiration points rather than weights better re-

flects the intent of most reference point methodo-

logies.

Using aspiration points, then, the intent of PU

and PM formulations can be stated. PU seeks to

minimize the maximum weighted deviation, but so

that the proportional deviations are equal; that is:

1� h U11� hða1Þ

¼ 1� h U21� hða2Þ

¼ � � � ¼1� h Uq1� hðaqÞ

;

ð1� hðakÞÞ > 0; k ¼ 1; 2; . . . ; q ð8Þ

PM seeks to maximize the minimum weighted

achievement so that, if possible, proportional

achievements are equal; that is:

h M1hða1Þ

¼ h M2hða2Þ

¼ � � � ¼h MqhðaqÞ

; hðakÞ > 0;

k ¼ 1; 2; . . . ; q ð9Þ

There is, of course, no guarantee that the equalities

of (8) and (9) will hold. Rather, these equations

reflect the intent of the different formulations. In

almost every case, for a given aspiration point,

a ¼ ða1; a2; . . . ; aqÞ, PU and PM result in different

solutions. Further analysis of the discrepancy be-

tween PM and PU solutions for the two-dimen-sional polyhedral case is provided in Buchanan

and Gardiner (2001).

2.2. Aspiration point location

We assume that all aspiration points have been

scaled according to (4) and are contained within

the unit hypercube. The location of aspiration

point hðaÞ can be described relative to N, the set ofnondominated solutions. First, hðaÞ is either on Nor not. If not, hðaÞ belongs to one of three mutu-ally exclusive categories: under N, over N or neither

as shown in Fig. 2 and defined in Appendix A. Thebold portion of the boundary represents N. The set

of aspiration points that are neither under nor over

N is contained in the interior of the rectangle la-

belled ‘‘Neither’’. The aspiration points over and

under N are shown relative to this rectangle and N.

Over, under and neither can be related to for-

mulations (5) and (6) of PU and PM respectively

in terms of the optimal solution y , as shown inTable 1.

Fig. 2. Location of aspiration points.

Table 1

Relating y to aspiration point location

PU PM

y < 1 All h Uk > hðakÞ; hðaÞ is under hðaÞ is not on or under; hðaÞ could be over or neithery ¼ 1 All h Uk P hðakÞ with at least one h Uk ¼ hðakÞ; either

h U ¼ hðaÞ or h U dominates hðaÞ; hðaÞ is either on orunder

All h Mk P hðakÞ with at least one h Mk ¼ hðakÞ; eitherh M ¼ hðaÞ or h M dominates hðaÞ; hðaÞ is either on or under

y > 1 hðaÞ is not on or under; hðaÞ could be over or neither All h Mk > hðakÞ; hðaÞ is under


For either formulation (PU or PM), if solutionh is a proportional solution as defined in (8) and(9), then the aspiration vector hðaÞ will be eitheron, over or under N. When h U and h Mð6¼ hðaÞ)are both proportional and hðaÞ is over N, h M will

be closer to L than h U . The reverse holds whenhðaÞ is under N. A proof is given in Appendix A.

3. Analysis of the hyperspherical case

We apply our analysis of discrepancy to the

case where the set of nondominated solutions, N,

is described by the boundary of the nonnegative

section of a unit hypersphere centered at the ori-

gin. In two dimensions, this is the intersection of

the boundary of a unit circle with the nonnegativequadrant, where the equation for N is h21 þ h22 ¼ 1.This simplification of representing N as the

boundary of the nonnegative portion of a q-

dimensional spheroid aids the ensuing analysis

without any loss of generality, especially in higher

dimensions. A hyperspherical approximation is

one (symmetric) limit of a general polyhedral set,

as the number of facets tends to infinity. Conse-quently, the analysis of differences between the PM

and PU formulations is less affected by the ‘‘noise’’

of the nondominated surface. In addition to being

smooth and symmetric, the hyperspherical ap-

proximation shares with the unit hypercube the q

extreme points which individually maximize each

objective function over the unit hypersphere. For

example in 3-dimensions, these extreme points are(1,0,0), (0,1,0) and (0,0,1). Also, since the dis-

crepancy associated with the hyperspherical ap-

proximation tends to be less than in the linear case,

it provides a conservative estimate of discrepancy.

We define dh as the Euclidean distance betweenthe PM and PU solutions for any given hðaÞ anduse it as the measure of discrepancy. In order to

explore how the amount of discrepancy dependson the location of hðaÞ, we examine how close hðaÞis to the line segment L ¼ hðMÞhðUÞ and charac-terize the hyperplane containing hðaÞ. We definedaL as the Euclidean distance from hðaÞ to theclosest point on L and let c ¼ h1 þ h2 þ � � � þ hq.Fig. 1 illustrates these measures in two dimensions.

Note that for c6 1, all corresponding aspiration

points are on or under N. As c approaches 0,the aspiration points approach hðMÞ. For cP2=

ffiffiffi2

pffi 1:414, all corresponding aspiration points

are on or over N. As c approaches 2, the aspiration

points approach hðUÞ.

3.1. Discrepancies in two dimensions

Figs. 3 and 4 show, for two dimensions, thediscrepancy between PU and PM solutions

for different aspiration points, characterized by

their c value, their distance from the line segment

L ¼ hðMÞhðUÞ, and whether the aspiration pointis over or under N.

Fig. 3 shows discrepancy curves for selected

values of c that correspond to aspiration points,

hðaÞ, over N. As c increases, the aspiration points

Fig. 3. Discrepancy between PU and PM solutions for aspi-

ration points on the line h1 þ h2 ¼ c, when the aspiration pointis over N.

Fig. 4. Discrepancy between PU and PM solutions for aspi-

ration points on the line h1 þ h2 ¼ c, when the aspiration pointis under N.


are closer to the ideal point and the amount ofdiscrepancy increases sharply with daL. For hðaÞunder N, as c increases and a is closer to thenondominated set, the discrepancy decreases. Note

that for aspiration points over N the discrepancies

are convex, while they are concave for aspiration

points under N. Choosing aspiration points very

near to the ideal point U will result in considerablediscrepancy, as also will aspiration points whichare near M.Let us put the discrepancy measure in context.

The maximum theoretically possible discrepancy

in two dimensions, with a hyperspherical N, is

0.7654. In this (unlikely) situation, for a given as-

piration point, PU would give solution (1,0) while

PM would give solution ð1=ffiffiffi2

p; 1=

ffiffiffi2

pÞ, or vice

versa. Consider a ‘‘middle of the road’’ discrep-ancy of 0.2. What does this mean? For the case of

c ¼ 1; that is, for all aspiration points on the hy-perplane h1 þ h2 ¼ 1, the median discrepancy is0.21, with a mean of 0.19. A discrepancy of 0.2 is,

therefore, quite realistic. An example of a 0.2 dis-

crepancy is shown in Table 2.

Thus, the solution is such that h U1 is 42% larger

than h M1 while h U2 is 89% of h M2 . Since the ob-jective value has been normalized over the range

[0,1], we have that h U1 is 17.3% further along the

range (i.e., 17.3% better achievement) than h M1while h M2 is 9.9% further along the range than

h U2 Clearly, very different solutions result from the

same aspiration point––depending entirely on

the framing; that is, whether it is deviation from

the ideal or achievement from the nadir.

3.2. Discrepancies in q dimensions

We now extend this analysis and demonstrate

that the results described visually in two dimen-

sions are no less applicable in higher dimensions.

In Fig. 5, we present the discrepancy for the case

where the aspiration points are on the hyperplaneh1 þ h2 þ � � � þ hq ¼ 1, and the number of dimen-

sions ranges from 2 to 10 million. Note that all

aspiration points on this hyperplane are either

under or on N and all resulting PU or PM solu-tions are proportional.

The discrepancy ðdhÞ is a function of the dis-tance of the aspiration point from L ðdaLÞ and q, asfollows

dh ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�

2qd2aL þ 2 1� d2aL � 1q

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq� 2þ d2aL þ 1

q � qd2aLq

q� 2þ d2aL þ 1q

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffid2aL þ 1

q

qvuuut

The discrepancy increases as q increases, but not

linearly, with limq!1 dh ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2daL

p.

The maximal distance from L for q objectives isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ð1=qÞ

pat which dh ¼ 0. It is interesting to

note that as q increases, the maximal discrepancy

shifts toward daL ¼ 0.

3.3. Volume of aspiration points under N

As the number of dimensions q increases, the

likelihood of a given aspiration point being above

N increases significantly. Let Vq be the volume of aq-dimensional hypersphere with radius r (see, e.g.,

Kendall (1961, p. 35)); Weisstein (1999, pp. 876–

878) and Vq-cube the volume of a q-dimensionalhypercube with side r. Since we are concerned onlywith the portion of Vq located in the nonnegativeorthant, the volume of interest is Vq=2q. The ratioVq=ð2qVq-cubeÞ describes the proportion of aspira-tion points under N, for any value of q. Table 3

Table 2

Discrepancy example where dh ¼ 0:2hða1Þ hða2Þ h U1 h U2 h M1 h M2

0.310 0.690 0.583 0.813 0.410 0.912

Fig. 5. Discrepancy for aspiration points on h1 þ h2 þ � � �þhq ¼ 1; q dimensions.


tabulates the proportion of aspiration points

under N for given values of q.

For a random allocation of aspiration points, at

least 83% of aspiration points will be over N forfive or more objectives, which affects the behavior

of the two reference point formulations, PU and

PM.

3.4. Dispersion of PU and PM solutions

Related to the discrepancy discussed above, the

behavior of these two formulations can also beexamined from the perspective of dispersion. That

is, for a given set of aspiration points, what does

the pattern of solutions produced by PU and by

PM look like? To examine this, we consider the

three-dimensional case with equally spaced aspi-

ration points within the unit cube. By projecting

the unit cube onto the plane, a visual picture of the

dispersion patterns for PU and PM can be ob-tained. Fig. 6 illustrates this projection; the per-

spective is that of looking straight down the ray L,from above the ideal point hðUÞ. The spherical,nondominated set is also shown.

Using a 0.05 grid, this represents 11� 11�11 ¼ 1331 aspiration points within the unit cube.Figs. 7 and 8 show the resulting dispersion of so-

lutions over the spherical nondominated set forPU and PM.

The dispersion of, and by implication the dis-

crepancy between, PU and PM solutions are dif-

ferent. Overall, PM provides a more even pattern

of solutions. PU solutions are either toward the

centre or to the absolute extremes. Although it

cannot be seen from Fig. 7, the three extreme point

solutions (1,0,0), (0,1,0) and (0,0,1) each comprises100 PU solutions in contrast to 10 PM solutions in

Fig. 8. For example, under PU there were 100

different aspiration points which all gave the same

final solution of (1,0,0). For the three-dimensional

sphere, 52% of all potential aspiration points are

under N. Figs. 9 and 10 show the patterns of dis-

persion depending on whether the aspiration point

is over N or under N.

As Figs. 9 and 10 show, PU solutions grouptowards the center for under aspiration points and,

similarly, PM solutions group towards the center

for over aspiration points. Thus, the position of

the aspiration point relative to N, combined with

the choice of reference point method, can result in

either greater or lesser dispersion of solutions. The

differences in dispersion are quantified in Table 4.

The PU solutions, however, are less uniformlydistributed than the PM solutions and have larger

gaps near the extremes in both cases. From the

perspective of controllability, this means that (in

some parts of the solution space) a minor change

in aspiration point location can result in a major

change in solution location––a result that may

prove less than intuitive for a decision maker.

We use the determinant of the variance–covariance matrix as a general measure of dis-

persion for each group of solutions. Using this

measure, for hðaÞ under N, the amount of disper-sion among the PM solutions is 28.5 times that of

Table 3

Tabulation of aspiration points under N

Number of objectives (q) Proportion of aspiration points

under N

2 0.785398

3 0.523599

4 0.308425

5 0.164493

10 0.002490

15 0.000012

Fig. 6. Two-dimensional projection of aspiration points in the

unit cube.


the PU solutions. Similarly the amount of PU

dispersion for solutions generated using hðaÞ overN is 22.2 times the dispersion among PM solu-

tions. It should be noted, however, that the dis-

persion measures in Table 4 reflect the large

number of extreme points for PU.

4. Illustrative examples

We now further examine the nature of the PM

PU discrepancy in the context of two examples.

Example 1. This example (modified from Zionts

and Wallenius, 1976) illustrates some results for a

simple MOLP problem. The problem formulation

is shown below, along with a two-dimensional

projection of the nondominated set.

MAX f1ðxÞ ¼ 3x1 þ x2 þ 2x3 þ x4f2ðxÞ ¼ x1 � 2x2 þ 2x3 þ 4x4f3ðxÞ ¼ �x1 þ 8x2 þ 1:6x3 þ 3x4

s:t: g1ðxÞ ¼ 2x1 þ x2 þ 4x3 þ 3x46 60g2ðxÞ ¼ 3x1 þ 4x2 þ x3 þ 2x46 60 xi P 0;

i ¼ 1; 2; 3; 4:

The nondominated set comprises facets ABDC,

DFB and the edge EC as shown in Fig. 11. At only

one point of the entire nondominated set (E) is oneobjective maximized at a value of 1 with all other

objectives at 0, unlike the hyperspherical non-

dominated set where each objective can attain a

value of 1 with all others at 0. While the PM and

PU formulations again produce different patternsof dispersion, they are similar to those already

shown for the hyperspherical nondominated set.

Here we consider all aspiration points inside the

unit cube, again using a 0.05 grid. The aspiration

points corresponding to hðMÞ and hðUÞ have,however, been excluded, leaving a total of 1329

aspiration points.

The patterns of dispersion for PU and PM areshown in Fig. 12. Both formulations give a good

coverage of solutions, except that PM provides

more solutions in the extreme areas of the non-

dominated set.

We now consider in the following tables how

the position of the aspiration point affects the re-

sulting solution. From Table 1, we use two cate-

gories of aspiration point position with respect toN-under/on and over/neither.

Fig. 8. Two-dimensional projection of PM dispersion.Fig. 7. Two-dimensional projection of PU dispersion.


Table 5 shows that about half the aspiration

points were dominated by or equal to the resulting

solution; that is, they were under/on N. Further-more about half of the solution points were such

that PMwas closer to the center (that is, ray L) thanPU. However, this depended almost entirely on theposition of the aspiration point. If hðaÞ was underN, then PU was almost always closer to the center.Consideration of proportional and nonpropor-

tional solutions is provided in Tables 6 and 7.

Table 4

Dispersion statistics

PU PM

hðaÞ is under NCentroid (0.55, 0.55, 0.55) (0.48, 0.48, 0.48)

Dispersion 8.490E�06 2.422E�04

hðaÞ is over NCentroid (0.43, 0.43, 0.43) (0.52, 0.52, 0.52)

Dispersion 1.079E�03 4.862E�05

Fig. 10. PU (left) and PM (right) dispersion for aspiration points over N.

Fig. 9. PU (left) and PM (right) dispersion for aspiration points under N.


Solution proportionality has been shown here

for two reasons. It is surmised that decision

makers would prefer a solution to be propor-tional to their guess in any interactive MCDM

solution method. Further, this tabulation gives

some indication of the relative magnitude of

proportional and nonproportional solutions, for

each formulation. For PM, only 38.83% of as-

piration points result in proportional solutions,

compared with 50.64% for PU. Considering only

the nonproportional solutions for PM, almost60% of them were under N; that is, the rayfrom hðMÞ through hðaÞ did not intersect

with the efficient set. For PU, almost 80% of

the nonproportional solutions resulted from

the case of over/neither; only 21.80% of non-

proportional PU solutions occurred when hðaÞwas under N.

Example 2. In this three-dimensional example

the nondominated set simply consists of a line

from (1,1,0) to (0,0,1). Again 1329 aspiration

points from within the unit cube are considered

and the resulting solutions are categorized in

Table 8.

Fig. 12. Example 1: Dispersion for PU (left) and PM (right).

Fig. 11. Two-dimensional projection of the simple MOLP

problem.

Table 5

Example 1: Categorization of aspiration points by closeness to center

Position of hðaÞ PM closer to center than PU PM ¼ PU PU closer to center than PM Total Percent

Under/on 10 33 632 675 50.79

Over/neither 602 36 16 654 49.21

Total 612 69 648 1329

Percent 46.05 5.19 48.76


With this nondominated set, a tidy result can be

seen. PM is closer to the center than PU 55.76% of

the time. Not surprisingly, only 9% of the solu-

tions are proportional; intuitively, this occurs only

when the ray through the aspiration point inter-sects with the nondominated line. In summary, if

hðaÞ is under or on, then PU is at least as close tothe center as PM. If hðaÞ is over or neither, thenPM is at least as close to the center as PU.

We include the two examples in this section to

illustrate how PU and PM solutions compare in

nonhyperspherical cases. The patterns of disper-

sion are similar to the hyperspherical results.Further, these examples have significant num-

bers of nonproportional solutions. The centrality

properties proven for proportional solutions in

Appendix A are also generally the case for the

mixture of proportional and nonproportional so-lutions found in these examples.

5. Experiment

The previous sections provide a sensitivity

analysis of PU and PM solutions, controlling for

aspiration point location, and demonstrate howthe differences between PM and PU solutions are

largely a function of the location of the aspiration

point. Clearly, in any practical use of reference

point methods, decision makers do not randomly

choose aspiration points evenly throughout the

unit hypercube. Perhaps in practice with actual

aspiration points, these differences are not as sig-

nificant. We therefore designed a simple experi-ment to examine this question and to also see

which of the two types of solutions (PM or PU)

decision makers preferred.

A production scheduling decision problem was

used for the experiment. This simple problem has

been used successfully in other experimental con-

texts by one of the authors (e.g., Corner and

Buchanan, 1997). The problem concerned themanufacture of electrical components for lamps in

a medium size New Zealand company where most

of the company�s problems were attributable topoor production scheduling. The goal was to de-

velop a production schedule which minimized the

three conflicting objectives of operating costs,

stockouts, and labor temporarily laid off. A more

complete problem description can be found inCorner and Buchanan (1997). The nondominated

set of the three objective production scheduling

problem was chosen to be spherical.

A naive solution method, similar to the GUESS

method (Buchanan, 1997), was used to solve the

Table 8

Example 2: Categorization of aspiration points by closeness to center

Position of hðaÞ PM closer to center than PU PM ¼ PU PU closer to center than PM Total Percent

Under/on 0 144 361 505 38.00

Over/neither 741 83 0 824 62.00

Total 741 227 361 1329

Percent 55.76 17.08 27.16 100.00

Table 6

Example 1: Categorization of aspiration points by propor-

tionality of PM solution

Position of hðaÞ PM solution is not

proportional

PM solution is

proportional

Under/on 473 58.18% 202 39.15%

Over/neither 340 41.82% 314 60.85%

Total 813 516

61.17% 38.83%

Table 7

Example 1: Categorization of aspiration points by propor-

tionality of PU solution

Position of hðaÞ PU solution is not

proportional

PU solution is

proportional

Under/on 143 21.80% 532 79.05%

Over/neither 513 78.20% 141 20.95%

Total 656 673

49.36% 50.64%


problem whereby the decision maker guesses asolution (the aspiration point) between M and Uand the method finds a solution using both the PM

and PU formulations. The method presents the

PM and PU solutions in random order to elimi-

nate any order effects. The decision maker chooses

between the PM and PU solutions (or indicates

indifference) and the method proceeds until a sat-

isfactory solution is found. Although it was thegoal of the participants to derive a good produc-

tion schedule, we were only interested in the choice

between the PM and PU solutions at each itera-

tion. The solution method was programmed in

Cþþ to run in a Windows environment. Fifty-

eight students participated in the experiment which

was treated as a case study and became part of

their assessable work for a course; 38 were MBAstudents at Auburn University (from three differ-

ent classes), while the remaining 20 were students

from the University of Waikato (18 from an

undergraduate course in Operations, 2 from a

graduate course in decision making).

The hypothesis was that participants would

prefer PM to PU because, we assumed, decision

makers are more ‘‘achievement-oriented’’ than‘‘deviation-oriented’’. Because the production

scheduling problem was formulated with a spher-

ical nondominated set, PM will always produce a

proportional solution. This is not the case for PU.

We therefore eliminated all iterations where the

PU solution was not proportional. The data was

further reduced by eliminating all responses of

indifference. The final data set contained aspira-

tion points which resulted in a proportional PM orPU solution, between which participants expressed

a clear preference.

5.1. Results

The essential results are presented here, with a

full discussion reserved for Section 6. Aspiration

points were distributed such that almost 75% wereover N. This is not surprising––participants simplywanted greater achievement than was often feasi-

ble. If aspiration points were randomly distrib-

uted, then only 47.64% of aspiration points would

be expected to be over N. The distribution ofguesses, therefore, is not random ðp ¼ 0:0000Þ.With the raw data (before nonproportional PU

solutions and indifference were excluded), 82% ofaspiration points were over N.Table 9 provides a contingency table showing

the location of hðaÞ and solution preference.

Overall, PM is preferred at 57.0%. However, since

aspiration point location and solution prefer-

ence are not independent ðp ¼ 0:0006Þ, this overallpreference for PM cannot be interpreted in isola-

tion. If hðaÞ is over N, PM is clearly preferred at61.0% ðp ¼ 0:0000Þ. If hðaÞ is under N, PU is

preferred in this sample at 55.2%, but the result is

not statistically significant ðp ¼ 0:2131Þ.Recall from Fig. 1 and Appendix A that gen-

erally, if the aspiration point is over N, then PM is

closer to the center than PU; the converse is true

for an aspiration point under N. The data abovesuggest that decision makers prefer solutions

Table 9

Categorization of aspiration points and solution choices

Prefer PM solution Prefer PU solution Total

hðaÞ under 19.3% 31.5% 24.5%

65 80 145

44.8% 55.2% 100.0%

hðaÞ over 80.7% 68.5% 75.5%

272 174 446

61.0% 39.0% 100.0%

Total 100.0% 100.0% 100.0%

337 254 591

57.0% 43.0% 100.0%


which are closer to the center, which representmore of a compromise. Certainly the achievement

hypothesis, where decision makers are assumed to

prefer an achievement-oriented approach, is not

confirmed. Rather the position of the aspiration

point (under N or over N) and the apparent desireof decision makers for ‘‘centered’’ solutions would

seem to explain the choice of solution.

The dispersion of PM and PU solutions pre-sents no surprises. PU shows greater dispersion

than PM (a factor of 2) because 75% of the aspi-

ration points are over N, a result consistent withTable 4.

We now turn to a discussion of these results and

consider the implications for decision support.

6. Discussion and implications

The motivation for this paper came from an

examination of reference point solution methods;

specifically, what is the effect of the choice of ref-

erence point? We limited our study to a simple

Tchebycheff formulation and considered both

ideal and nadir reference points. These two for-mulations (PU––reaching down and PM––pushing

up) typically give different solutions for the same

aspiration point (and decision maker). The choice

of reference point influences the structure of any

interactive solution model, resulting solutions and,

perhaps, the behavior of a decision maker.

To summarize, the two formulations have dif-

ferent philosophies. PU seeks to minimize themaximum deviation from the ideal vector; i.e., to

make all weighted deviations equal, if possible.

PM seeks to maximize the minimum achievement;

i.e., to make all weighted achievements equal, if

possible. Our study shows that different solutions

result from the same aspiration point, and that this

difference can be significant. The magnitude of this

difference depends on the location of the aspira-tion point, the number of objectives and the shape

of the nondominated surface. Particularly, if as-

piration points are located near the nadir or ideal

reference points and they are somewhat away from

the center, then the two solutions are often sub-

stantially different. We also compare the non-

dominated set coverage patterns for the two

formulations. The PM formulation generally pro-duces more evenly spaced solutions with smaller

gaps near the boundaries of N.We develop a classification of aspiration points

as under N, over N or neither. In general, if an

aspiration point is under N then the PU solution

will be closer to the center than the PM solution.

Conversely, if the aspiration point is over N then

the PM solution will be closer to the center thanthe PU solution. We also demonstrate that as the

number of objectives increases, the proportion of

potential aspiration points located over N tends toincrease dramatically. Further, evidence from our

experiment and other experiences with the refer-

ence point methods suggests that decision makers

typically want more than they can get, specifying

more aspiration points over N than would be ex-pected given the number of objectives.

These results enable us to comment further. If

the intent is to obtain relatively evenly spaced

samples of solutions from N, the PM formulation

appears preferable. The sensitivity of solutions is

much greater under the PU formulation when the

aspiration points approach the boundary of the

hypercube. Thus, the congruence between the as-piration point and the resulting solution becomes

an important consideration, particularly in an in-

teractive reference point solution method. An even

coverage of solutions is also important when

generating a representation of the nondominated

set. Steuer (1986, pp. 328–330) addresses this issue

through a choice of weights derived from uniform

and Weibull distributions. In contrast, we haveused a uniform distribution of weights and con-

sidered the effect of the different reference point

methods. Thus, if the aspiration point is over Nand near the ideal point, there is likely to be a

considerable difference between the aspiration

point and the resulting solution when PU is used.

The difference will also occur with PM, if the as-

piration point is under N and near the nadir point;however this appears less likely to happen because

of decision makers� persistent and understandablechoice of aspiration points near the ideal point. If

some congruence is desired, this suggests that PM

should be used so that the resulting solution will

generally be similar to the aspiration point. This

congruence, or local controllability (Wierzbicki,


1986), is a desirable feature of a reference pointmethod.

We hypothesized that decision makers may be

more comfortable with achievement, rather than

deviation, oriented solution methods. We hypoth-

esized that decision makers would prefer PM so-

lutions for this reason. While the experiment

showed evidence of preference for one type of so-

lution, this is not simply a preference for the resultsof one formulation over another. The preference

appears, rather, to be related to the centrality of

the resulting solution, suggesting that decision

makers tend to seek compromise solutions, when

given the opportunity. Practically speaking, then,

what should a decision analyst do? Recommend a

method that will give a centralising tendency (as-

suming most aspiration points are over N) andthereby support a decision maker in this direction?

If so, then PM should be recommended, given that

most aspiration points are over N. Or should ananalyst recommend a method which counteracts

this tendency of decision maker toward compro-

mise in the hope that new and better solutions will

be found? In this case, then, PU should be con-

sidered, again since most aspiration points are overN. However, as shown, the divergence obtainedfrom using PU may be extreme.

The sensitivity analysis and the behavioral

study underscore the importance and influence of

the choice of aspiration point. If a decision maker

is aware of how a particular reference point for-

mulation ‘‘works’’, he can choose aspiration points

accordingly. We recommend that a decision makermove away from the ideal point and choose aspi-

ration points more widely. Such a strategy should

generate a greater spread/variety of solutions, re-

gardless of the formulation.

Both the PM and PU methods have their ad-

vantages, although the above discussion suggests

that the PM formulation would appear to be a

better choice. However, since both methods exist,we should encourage decision makers to make use

of both methods; as Russo and Schoemaker (1989)

suggest, it can be advantageous to use more than

one frame of reference. The value comes not from

choosing the ‘‘best’’ or most appropriate frame of

reference, but from considering more than one

perspective or frame.

Appendix A

Location of aspiration point and solution cen-

trality.

Let domination cone D ¼ Rqþ n f0g ¼ fh 2 Rq 3

hk P08 k¼1;. . . ; q; 9 k¼1; . . . ; q 3 hk > 0g whereRq

þ is the nonnegative orthant of q-dimensionalspace. Then we define:

under N: Aspiration point hðaÞ is under N if

fhðaÞ þ Dg \ N 6¼ £. Thus hðaÞ is consideredto be under N if there is at least one h0 2 Nwhich dominates it. When hðaÞ is under N,h M and h U will both dominate hðaÞ. If not,the optimality of y M and y U is contradicted.

over N: Aspiration point hðaÞ is over N if there

exists some h0 2 N such that hðaÞ 2 fh0 þ Dg.Therefore hðaÞ is defined to be over N if it dom-inates at least one h0 2 N . An aspiration vectorhðaÞ over N may or may not dominate the re-sulting h M and h U solutions.

neither: Aspiration points may also be neitherunder nor over N. When this is the case, h M

and h U will neither dominate nor be dominatedby hðaÞ.

For proportional solutions, the location of theaspiration vector a will be either on, over or underN. If hðaÞ is on N, h U ¼ h M ¼ hðaÞ. Let us ex-amine the case where aspiration vector hðaÞ is noton N and both h U and h Mð6¼ hðaÞÞ are propor-tional. In this case, hðaÞ is either over or under N.We first consider hðaÞ over N and show that h M ismore central (closer to L) than h U . From this it

can be seen that when hðaÞ is under N, h U is themore central solution.

A.1. h(a) over N

When hðaÞ is over N, h M is on the line seg-

ment from hðMÞ to hðaÞ. In other words, h M ¼hðMÞ þ ahðaÞ for a < 1. With the transformationin (4), hðMÞ is at the origin, so h M ¼ ahðaÞ fora < 1. Similarly, h U is the end point of a line seg-ment extending from hðUÞ through hðaÞ. So h U ¼hðUÞ þ bðhðaÞ � hðUÞÞ for b > 1. Since hðUÞ is aunit vector, h U ¼ 1þ bhðaÞ � b, for b > 1. For


any hðzÞ, the point on L closest to hðzÞ using theEuclidean metric is

e ¼Pq

i¼1 hðziÞq

; . . . ;

Pqi¼1 hðziÞq

�

Consider kh M ; e Mk2 and kh U ; e Uk2, the mini-mal Euclidean distances between the PM and

PU solutions, respectively, and L. Let Q M ¼ðkh M ; e Mk2Þ

2and Q U ¼ ðkh U ; e Uk2Þ

2.

Q M ¼Xq

k¼1h Mk

�Pq

i¼1 h Mi

q

�2

Since h M ¼ ahðaÞ,

Q M ¼Xq

k¼1ahðakÞ

� a

Pqi¼1 hðaiÞq

�2

¼ a2Xq

k¼1hðakÞ

�Pq

i¼1 hðaiÞq

�2

¼ a2ðkhðaÞ; eak2Þ2

where ea is the point on L closest to hðaÞ.Similarly,

Q U ¼Xq

k¼1h Uk

�Pq

j¼1 h Ui

q

�2

Since h U ¼ 1þ bhðaÞ � b,

Q U ¼Xq

k¼1ð1

þ bhðakÞ � bÞ

�Pq

i¼1 ð1þ bhðaiÞ � bÞq

�2

¼ b2Xq

k¼1hðakÞ

�Pq

i¼1 hðaiÞq

�2

¼ b2ðkhðaÞ; eak2Þ2

Thus, kh M ; e Mk2 ¼ akhðaÞ; eak2 and kh U ; e Uk2 ¼

bkhðaÞ; eak2. Since a < 1 and b > 1, kh M ; e Mk2 <kh U ; e Uk2.

A.2. h(a) under N

Using the logic above, it can be shown that h U

is closer to L when hðaÞ is under N.

Appendix B

Terminology and definitions

a aspiration vector; a ¼ fa1; a2; . . . ; aqg;Mk < ak < Uk 8 k ¼ 1; 2; . . . ; q

daL Euclidean distance between given hðaÞand the point closest to it on L, ea

dh Euclidean distance or discrepancy between

the h M and h U solutions for any given hðaÞD domination cone; D¼Rq

þnf0g¼fh2 Rq3hkP08k¼1;2;...;q;9k¼1;...; q 3 hk > 0gwhere Rq

þ is the nonnegative orthant of q-dimensional space

e point on L closest to any given hðzÞ;e ¼ ð

Pqi¼1 hðziÞ=q; . . . ;

Pqi¼1 hðziÞ=qÞ

ea point on L closest to hðaÞ; ea ¼ðPq

i¼1 hiðaÞ=q; . . . ;Pq

i¼1 hiðaÞ=qÞe M point on L closest to h M ; e M ¼

ðPq

i¼1 h Mi =q; . . . ;

Pqi¼1 h

Mi =qÞ

e U point on L closest to h U ;e U ¼ ð

Pqi¼1 h

Ui =q; . . . ;

Pqi¼1 h

Ui =qÞ

fkðxÞ kth objective function of x 2 X � Rn,

k ¼ 1; 2; . . . ; qFðxÞ F ðxÞ ¼ ff1ðxÞ; f2ðxÞ; . . . ; fqðxÞg � Rq

gjðxÞ jth constraint on set of alternatives,

j ¼ 1; 2; . . . ;mhðzÞ transformation function; hðzkÞ¼ðzk�MkÞ=

ðUk �MkÞ; k ¼ 1; 2; . . . ; qhk transformed criterion values; hk ¼ hðzkÞ;

k ¼ 1; 2; . . . ; qh vector of transformed criterion values; h ¼

fh1; h2; . . . ; hqgh M an optimal solution to PM(w); h M ¼

fh M ;1 ; h M ;

2 ; . . . ; h M ;q Þ

h U an optimal solution to PU(w); h U ¼fh U ;

1 ; h U ;2 ; . . . ; h U ;

q ÞhðaÞ transformed aspiration vector; 0< hðakÞ<

1 8 k ¼ 1;2; . . . ;qhðMÞ transformed nadir vector; hðMÞ ¼ f0;

0; . . . ; 0ghðUÞ transformed ideal vector; hðUÞ ¼ f1;

1; . . . ; 1gL line segment hðMÞhðUÞMk nadir value for kth criterion; Mk ¼

minz2Nfzkg; k ¼ 1; 2; . . . ; qM nadir vector; M ¼ fM1;M2; . . . ;MqgN set of nondominated criterion vectors;

N ¼ fz : 9= z0 2 Z where z0 P z; z0 6¼ zg


PMðwÞ multiple objective mathematical program

with weighted achievement function

based on MPUðwÞ multiple objective mathematical program

with weighted deviation function based

on Uq number of objective functions

Q M squared Euclidean distance between h M

and the point on L closest to it, e M

Q U squared Euclidean distance between h U

and the point on L closest to it, e U

Uk ideal value for kth criterion; Uk ¼maxz2Zfzkg; k ¼ 1; 2; . . . ; q

U ideal vector; U ¼ fU1;U2; . . . ;UqgVq hypervolume of a q-dimensional hyper-

sphere with radius rVq-cube hypervolume of a q-dimensional hyper-

cube with side rw vector of weights; w ¼ fw1;w2; . . . ;wqgX set of feasible alternatives; X ¼ fx 2

Rn : gjðxÞ6 0; j ¼ 1; 2; . . . ;mgy minimax or maximin variable in the PU

and PM formulations

zk criterion values; zk ¼ fkðxÞ; k ¼ 1; 2; . . . ; qz vector of criterion values; z¼fz1;z2;...;zqgZ image of X in criterion space; Z ¼

F ðxÞ � Rq

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a comparison of two reference point methods in multiple objective mathematical programming

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