# Selecting Pareto Optimal Subsets From Multiattribute Alternatives

^{[ to cite ]:}

James B. Wiley (1978) ,"Selecting Pareto Optimal Subsets From Multiattribute Alternatives", in NA - Advances in Consumer Research Volume 05, eds. Kent Hunt, Ann Abor, MI : Association for Consumer Research, Pages: 171-174.

^{[ direct url ]:}

http://acrwebsite.org/volumes/9419/volumes/v05/NA-05

Previously, Green (1974) illustrated the use of fractional replication procedures for selecting multiattribute alternatives for conjoint measurement studies of consumer choice. This study suggests the information laden choices in conjoint measurement are among alternatives in pareto optimal subsets. Pareto optimal subsets are those in which no alternative "dominates" another on every attribute. Procedures for selecting pareto optimal subsets are illustrated.

INTRODUCTION

How do consumers choose among multiattribute alternatives? Recent studies by Green and Rao (1972), Green, Carmone, and Wind (1972), Fiedler (1972), and Bettman, Capon, and Lutz (1973, 1974) have investigated this question using the techniques of conjoint measurement (Krantz and Tversky, 1972), trade-off analysis (Johnson, 1974), or functional measurement (Anderson, 1972). At the core of the many similarities among these three techniques are the following four fundamental premises: a) individuals (or other decision-making units) assign "partworths" to attribute levels, b) "utility" is a function of these partworths, c) choice is proportional to utility, and d) although the individual may be unable to articulate the partworths, they may be inferred from numerous choices among alternatives having characteristics which are varied in systematic ways. In this paper we use "conjoint measurement" in a generic sense, to refer to the general problem of inferring "part-worth's" from choices.

Conjoint measurement is a recent development in mathematical psychology that is concerned with resolution of the Joint effect of a set of attributes (measured at the ordinal level) into separate, independent contributions, which are measured at the level (asymptotically) of interval scales with common unit. That is, it is concerned with the effect of two or more independent variables on the ordering of a dependent variable. The approach is designed to solve a fundamental scientific problem, that of determining the rules by which a set of independent variables can be combined and used to predict values for a dependent variable. In the behavioral sciences, however, only rarely can one measure the dependent variable at the interval level, let alone the independent variables. In order to bridge this discrepancy between technical feasibility and scientific objective, conjoint measurement is used to determine whether there exists any rescaling of the dependent and independent variables, that when combined according to a hypothesized composition rule, would yield results consistent with the observed ordinal data structure. Alternatively, conjoint measurement may be used as a scaling method. When used as a scaling method, one tries to solve the measurement and composition rule problems together by fitting scales that obey the stated composition rule to some suitable approximation (Green and Wind, 1973).

With the additive conjoint measurement algorithms, such as Kruskal's MONANOVA (1965), an additive composition rule is assumed. Then, the above four premises can be represented as follows. The utility U(x) of some multiattribute alternative x = (x_{1}, x_{2}, . . ., x_{n}), expressed as an __n__-component vector, is defined as,

where u_{j} is a real-valued function of x_{j}, and x_{j} is the "level" of alternative x on the j (th) attribute.

As Dawes and Corrigan (1973) point out, additive models typically are applied to problems in which it also is reasonable to assume that the real-valued function u_{j} (1) is conditionally monotonic over x_{j }(which also implies an __a priori__ ordering over x_{j}.) That is, given two alternatives, A = (x_{1}, x_{2}) and B = (x_{1}, x'_{2}), then

which implies utility is monotonic with x_{2} and, in general, with x_{i}. As a consequence, we can speak of an alternative, A, "dominating another alternative", B, whenever

Under such conditions, choice of A = (x_{i}, x_{j}) must obtain by virtue of the assumptions of the model and __a priori __information regarding the setting. Since an alternative that is equal to or greater than another alternative on every attribute must be chosen by assumption, it follows that valuable information for scaling purposes (as distinct from model testing) is gleaned from choices among sets in which no alternative "dominates" another. Alternatives in such sets are said to be Pareto optimal. This paper describes how to select Pareto optimal subsets of multiattribute alternatives constructed from p^{p} factorial series. This series includes three attributes at three levels designs, four attributes at four levels, five attributes at five levels and so forth. Pareto optimal subsets for the 33 through the 66 series are reported. A desirable property of the reported sets is that all subsets selected from the reported sets will also be Pareto optimal. As a consequence, Pareto optimal subsets can be organized according to balanced incomplete block designs (BIBD's), with the alternatives within blocks Pareto-optimal.

AN EXAMPLE

Suppose we wish to determine the relative importance of transportation characteristics. Conversations with experts and traffic managers as well as a review of the literature suggests three dominant service criteria; reliability, rate, and transit time. Further conversation with experts suggest the following levels are realistic for a given origin/destination/commodity shipment:

For example, one alternative may cost $1.13 per hundred weight, take six days normal transit time, with an expectancy that 95% of the time it will arrive on or before the sixth day and a 3% chance it will arrive on the seventh day. There are 273 such alternatives that can be constructed from this 3 design.

Aside from the so-called "alphabet routes" which it is said once were favored for some commodity shipments, it is reasonable to assume that a decision-maker would prefer a cheaper, quicker, more reliable alternative to a more expensive, slower, less reliable alternative. Informative choices, therefore, are among alternatives that require the decision-maker to forego performance according to one criteria in order to gain performance according to another criteria. Requiring the decision-maker to rank order the following alternatives requires such choices.

While the above list is not exhaustive (for example, the alternative B(2), B(2), B(2) could be added, the alternatives do comprise a balanced, pareto optimal set which can be constructed following a simple heuristic.

HOW TO CONSTRUCT BALANCED PARETO OPTIMAL SUBSETS

A simple procedure for constructing a balanced set of Pareto optimal alternatives is based on cyclic hyper- Greco-Latin square designs. This somewhat awesome term describes the straightforward generalization of Latin square designs with multiple factors. For example, a cyclic Greco-Latin square of a 33 design can be represented as:

The position in a given cell refers to attribute and number in that position refers to attribute level. Three alternatives are represented in a given row or column and these three alternatives are Pareto optimal. To return to the above example, the first row represents alternatives 1, 4 and 5 respectively of Table 1.

The above design can be represented as a composite of the following three Latin squares:

Square (1) comprises the elements in the first position, square two comprises the elements in the second position, and square three comprises the elements in the third position. The tables are constructed using a one-step cyclic permutation of the numbers in the design. That is, the next design in the sequence is constructed by moving the first number of the current design to the extreme right, simultaneously moving all other numbers to the left (Winer, 1962). A composite design constructed from cyclic elements is balanced in the sense that each attribute level appears once and only once in a cell __and__ once and only once in each position in a row or column. Thus, a row or column of a composite design represents a balanced Pareto optimal subset; i.e., every attribute and attribute level is represented and no alternative dominates another.

The remaining alternatives of Table 1 can be constructed by interchanging the second and third arrays of Table 2 to yield a new composite design:

The first row of the above design represents alternatives 2, 3, and 6 respectively of Table 1. With the addition of the alternative comprised of the second level of each attribute, the set will remain Pareto optimal, but it will no longer be balanced. The set of seven alternatives will be exhaustive, however, in the sense that each of the remaining 20 alternatives that can be constructed from the 33 design will be dominated, or will dominate at least one of the seven alternatives in the Pareto optimal set. In this sense the seven alternatives represent the largest Pareto optimal subset that can be constructed from a 33 design.

CONSTRUCTING SETS FOR LARGER DESIGNS

While Pareto optimal subsets can be constructed for larger designs following generalizations of the procedures discussed above, the exercise is cumbersome and can become quite laborious. However, if we note that each attribute level will occur once and only once in each alternative generated by such procedures, a simple, easily programmable rule can be explicated which facilitates identification of desired alternatives.

This is the rule. For p^{p} designs select alternatives whose attribute levels sum to (^{p+1}/2). Since each level 2 occurs once in an alternative, the sum of the levels will be the same for all alternatives. For example, in Table 2, if the letter (A) is replaced with the value one, (B) with the value two, and (C) with the value three, it is clear that for each of the six alternatives in Table 1, the sum of the attribute levels is six. Likewise, the desired alternatives from a 4^{4} design will have attribute levels which sum to ten, the desired alternatives from a 5^{5} design will have attribute levels which sum to 15, and so forth. The Pareto optimal alternatives based on cyclic hyper- Greco-Latin squares will form a subset of the alternatives generated following the above rule. The remaining alternatives identified are those in which a given attribute level appears more than once. For example, the alternative characterized by the second level of each attribute (2,2,2) of a 3- design is such an alternative. The second level appears three times, but the sum of these levels is equal to six, as required by the rule.

Table 3 presents Pareto optimal subsets for the 3^{3}, 4^{4}, and 5^{5} factorial series. The sets are complete for the 3^{3} and 4^{4} series. As indicated in the above discussion, the 3^{3} series contains seven alternatives, six of which are balanced in the sense that each level appears once and only once in an alternative. The 4^{4} series contains 44 alternatives, 24 of which are balanced. The 5^{5} series contains 381 elements. Only the 120 balanced elements are reported in order to conserve space. The 6^{6} series is presented in Table 4. There are 4332 alternatives in this series, 720 of which are balanced. Only the seed balanced alternatives are listed.

ORGANIZING ACCORDING TO BIBD'S

Even restricted to Pareto optimal subsets, collecting choice data vis-a-vis the selected alternatives can become a formidable task. Twenty-one pairwise comparisons are required for the 7 alternatives in the 3^{3} series. Two hundred and seventy-six pairwise presentations would be required for the 24 balanced alternatives in the 4^{4} series. However, the 24 alternatives can be conveniently organized into six blocks of four alternatives each, such that the incidence of attribute levels is balanced within blocks and within the entire design. The following blocks illustrate this property.

Note that the blocks are constructed by cyclic permutation using the first six balanced elements of the 4 series (listed in Table 3). The balanced alternatives in the 5 series can be organized into 246 blocks of size five following similar procedures. The 6 series required 120 blocks of size six. Only a small fraction should be assigned a single subject and these should be replicated. Most subjects will find ranking six Pareto optimal 6 attribute alternatives a challenge.

In each of the above cases, the subject is required to rank the alternatives within blocks in order of preference. The resulting data can be analyzed "non-metrically," or following procedures suggested by Gulliksen and Tucker (1961), McKeon (1961), Wiley, MacLachlan, and Moinpour (1976), or Green and Carmone (1977).

BALANCED INCOMPLETE BLOCK DESIGNS

REFERENCES

Anderson, N. H., __Information Integration Theory: A Brief Survey__, Center for Human Information Processing, University of California, La Jolla, California, April 1972.

Bettman, J. R., N. Capon, and R. J. Lutz, "Cognitive Algebra in Multiattribute Models," __Journal of Marketing Research__, 12 (May 1975), pp. 151-64.

Bettman, J. R., "Multiattribute Measurement Models and Multi-attribute Theory: A Test of Construct Validity," __Journal of Consumer Research__, (March 1975), pp. 1-85.

Dawes, R. M., and B. Corrigan, "Linear Models in Decision Making," __Psychological Bulletin__, 1974, pp. 95-106.

Fiedler, J. A., "Optimizing Product Design Through Analysis of Consumer Trade Offs," M. Venkateson (ed.), __Proceedings 3rd Annual Conference Association for Consumer Research__, 1972, pp. 279-93.

Green, P. E., "On the Design of Choice Experiments Involving Large Numbers of Factors or Factor Levels." In S. Ward and P. Wright (eds.), __Proceedings of the 4th Annual Conference of the Association for Consumer Research__, 1974, pp. 228-41.

Green, P. E., and F. J. Carmone, "A BIBD/Logit Approach to Conjoint Measurement," University of Pennsylvania Working Paper, April 1977.

Green, P. E., F. J. Carmone, and Y. Wind, "Subjective Evaluation Models and Conjoint Measurement," __Behavioral Science__, Vol. 17 (May 1972), pp. 288-99.

Green, P. E. and V. R. Rao, "Conjoint Measurement Method of Quantifying Judgmental Data," __Journal of Marketing Research__, Vol. III (August 1971), pp. 355-62.

Green, P. E., and Y. Wind, __Multiattribute Decisions in Marketing: A Measurement Approach__. Hinsdale, Illinois: The Dryden Press, 1973.

Gulliksen, H. and L. R. Tucker, "A General Procedure for Obtaining Paired Comparisons from Multiple Rank Orders," __Psychometrika__, 26 (June 1961), pp. 173-83.

Johnson, R. M., "Trade-off Analysis of Consumer Values," __Journal of Marketing Research__, Vol. XI (May 1974), pp. 121-27.

Krantz, D. and A. Tversky, "Conjoint-measurement Analysis of Composition Rules in Psychology," __Psychological Review__ (March 1971), pp. 151-69.

Kruskal, J. R., "Analysis of Factorial Experiments by Estimating Monotone Transformations of the Data," __Journal of the Royal Statistical Society__, __B__. 27, 1965, pp. 251-63.

McKeon, J. J., 1961. "Measurement Procedures Based on Comparative Judgement," unpublished doctoral dissertation, University of North Carolina, Chapel Hill.

Wiley, J. B., D. L. MacLachlan, R. Moinpour, "Comparison of Stated and Inferred Parameter Values in Additive Models: An Illustration of a Paradigm," William Perrault, Jr. (ed.), __Advances in Consumer Research__, Vol. IV, pp. 98-105.

Winer, B. J., __Statistical Principles in Experimental Design__. New York: McGraw-Hill, 1962.

----------------------------------------

Tweet
window.twttr = (function (d, s, id) { var js, fjs = d.getElementsByTagName(s)[0], t = window.twttr || {}; if (d.getElementById(id)) return; js = d.createElement(s); js.id = id; js.src = "https://platform.twitter.com/widgets.js"; fjs.parentNode.insertBefore(js, fjs); t._e = []; t.ready = function (f) { t._e.push(f); }; return t; } (document, "script", "twitter-wjs"));