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" ^ AD-766 096 STATISTICAL ESTIMATION OF THE COEFFI CIENT OF PAIRWISE COMPATIBILITY FOR

Balintfy, et al Massachusetts University Pre pare d for: Office of Naval Research

1 " ^ AD STATISTICAL ESTIMATION OF THE COEFFI CIENT OF PAIRWISE COMPATIBILITY FOR MENU ITEMS Joseph L. Balintfy, et al Massachusetts University Pre pare d for: Office of Naval Research August 1973 DISTRIBUTED BY: mh National Technical Information Service U. S. DEPARTMENT OF COMMERCE 5285 Port Royal Road. Springfield Va _MMM i - i <

2 mi lamn in i -, I I o STATISTICAL ESTIMATION OF THE COEFFICIENT OF PAIRWISE COMPATIBILITY FOR MENU ITEMS by I ^ 1 I *; Joseph L. Balintfy and Ram C. Dahiya R«procJu{«d bp NATIONAL TECHNICAL INFORMATION SERVICE U S D^pa'l^»"«r.l Con.mtft, Spnngli.M VA 2J:3I n ' '.. 1 i-zb-'fjl^

1 ' - "F'" "'" i i. itimiwmmm*^ " m*mm*mm*m*im^*rtrmm*mmrmmm MUM - -. 1.. STATISTICAL ESTIMATION OF THE COEFFICIENT OF PAIRWISE COMPATIBILITY FOR MENU ITEMS by Joseph L.

10 August, 1973 I i; Prepared under Contract N00014-67-A-0230-0006 (NR 047-100) for the Office of Naval Research Reproduction In Whole or In Part Is Permitted for any Purpose of the United

3 1 ' - "F'" "'" i i. itimiwmmm*^ " m*mm*mm*m*im^*rtrmm*mmrmmm MUM STATISTICAL ESTIMATION OF THE COEFFICIENT OF PAIRWISE COMPATIBILITY FOR MENU ITEMS by Joseph L. Ballntfy and Ram C, Dahlya.. Technical Report No. 10 August, 1973 I i; Prepared under Contract N A (NR ) for the Office of Naval Research Reproduction In Whole or In Part Is Permitted for any Purpose of the United States Government This document has been approved for public release and sale; its distribution is unlimited i I I i School of Business Administration Department of General Business and Finance and Schi-ol of Natural Sciences and Mathematics Department of Mathematics and Statistics University of Massachusetts Amtierst, Massachusetts Il D D C nrpftts E. SEP io m - B ^ i

i I. ( I».,1. i. «I Mj.. Unclassified ib onoum Not Applicable Statistical Estlnmtlon of the Coefficient of Palrwlse Compatibility for Menu Items. * OKlCHiBri^fNorKi (Typ» "I rtpoit *n<j iiuluovv.(..i.-u Technical Report No.

4 I ( ^' "'" " ' ' wwnmimphwllii mi» UnclaHBlfled Srt'inlt i l.i..ill,.in n DOCUMENT CONTROL DATA R&D *>*< unit i la* iihi attan! tiiiu, i... u ut ab*fiin i itttti ihür*<",' n K'H! hr 0nt*'**t»hi»/! '' tivttjll 'i-fnitt f i ( I.* ili*,it I QHiiiIHA 1I*4Q AC rivitv t.ifp'ifjlm atlthitf) School of BualncuH Administration University of Massachusetts at Amherst I M. MO" ' Tl TLf i». «l 'IM I M. ul.i I. ( I».,1. i. «I Mj.. Unclassified ib onoum Not Applicable Statistical Estlnmtlon of the Coefficient of Palrwlse Compatibility for Menu Items. * OKlCHiBri^fNorKi (Typ» "I rtpoit *n<j iiuluovv.(..i.-u Technical Report No. 10. August % *u rmo^dl (FlfH nmm0. innljim inntmt. Imti nttm») Joseph L. Ballntfy, Ram C. Dahlya S MCPORT QA11 August, 1973 tm. TOTAL NO or PACK* 21 7b. NO OF Ml F» («. CONTHACT O«GRANT NO N A b. PBOJCC T NO NR «A. ORIOINATOn** HCPORT NUMBERHI Technical Report No. 10 tb. OTHEK «t "OR T NOISI (Any other numban r>i»l may b* mtblgntd Ihtt npotl) Not Applicable 10 OltTRISUTION ST ATCMCN T This document has been approved for public release and sale; its distribution is unlimited. II tupplcmcntarv NOTES \t SPONSORING WII.I T ARV ACTIVITY Not Applicable Office of Naval Research, Washington, D.C, II ABSTRACT The coefficient of compatibility between two menu items is defined as the difference between the conditional and marginal probabilities of selecting one after the other. Maximum likelihood estimates from selection statistics are given, and the asymptotic distribution of the estimator is determined so that test of null hypothesis and confidence intervals regarding the coefficients can be obtained. A method Is described for building data base for large joint probability matrices from menu selection data. DD. F N 0 O R :J473 S/N OI01-e07-6BO1,PAGE,, I*. UnclaaaiUett Secunly CUssi(ica lion.jimk I

5 I 4 Unclassified Srcuuty Clas slhc at o n KEY WOAOS - LtN K 4 LIN I( 8,_,,.. AO I.. WT A Ol. WT RO I.. E Food Preference ~1enu Planning Mathemati cal Statistics Food Service Data Banks D 0,F.,OoRv 14 u (BACK) (PAGE 2) tb -~- -. Unclassified ~cu rity Classifi ca t io n

diet problems by mathematical programming techniques [ 1 ], difficult questions emerge concerning the role and : measurability of compatibility between menu items.

6 ' I"" 'I --ii". -. -,..» imemflhta..^1..,.. r STATISTICAL ESTIMATION OF THE COEFFICIENT OF FAIRWISE COMPATIBILITY FOR MENU ITEMS by Joseph L. Ballntfy and Ram C. Dahiya I INTRODUCTION With recent advancements In the art of representing food and meal preferences by mathematical models [ 2 ] along with the possibilities of defining and solving optimum human diet problems by mathematical programming techniques [ 1 ], difficult questions emerge concerning the role and : measurability of compatibility between menu items. A meal selected by, or planned for an Individual usually consists of a set of menu items which are mutually complementary in the sense that each Item represents one of the courses of the meal. It is tempting to consider the utility of a meal in terms of the utilities of its components. Recent results in multlattrlbute utility theory open the way for a variety of representations, and this is where the issue of compatibility comes in. The'simplest additive utility model would Imply that the utility of a meal is equal to some weighted sum of the unlvarlate utilities of the menu items in the meal. Addltivity, therefore, means that the utility of the meal is completely explained by the utilities of the items irrespective -1-

If the set of menu Items under consideration Is preference and therefore utility Independent, a measure of compatablllty can be derived from the multlattribute utility model of Keeney [6 ].

7 " ' ' - - " T 1 "- - - ^ of their relative combination. whelming that this Is not so. Empirical evidence { 5 ] Is, however, over- People find some combinations of Items more or less compatible than others, meaning that compatibility Is also a factor In the utility of a meal. If the set of menu Items under consideration Is preference and therefore utility Independent, a measure of compatablllty can be derived from the multlattribute utility model of Keeney [6 ]. This model expresses the utility of a meal as a weighted sum of additive utilities plus a weighted sun of all possible crossproducts of unlvarlate utilities. Depending upon the sign of the coefficients, this second sum may Increase or decrease the value of additive utilities, and hence would represent the effect of compat- ibility In a given combination of items. Unfortunately, again, utility independence can not be assumed to be true for any set of menu items, and not even for any set of pairwlse combinations of items. Consequently, re- search is still in progress to find the appropriate expression of compatibility in utility terms. In a recent report by Ballntfy and Sinha [ 3 ] a meal planning model is presented which does not require the representation of compatibility of menu items in the utility measure of the meals. The idea is advanced that a mathematical programming model which maximizes an additive utility function could determine which items would be most liked by a given population In a given time period, and from this fixed set of items a separate scheduling activity could combine the items into a sequence oi' neals on the criterion of compatibility alone. This way the concepts of the utility and compati- bility of a meal ate conveniently separated; the first being included only -2- mmnm --- '

8 PWWWWWW1iW* ll^",^^^* lw^^l^^w"^^^^''^ ' " ' " in the objective function addlllvely, while the second appears among the scheduling constraints, and hence Is no longer linked directly to utility measures. Consequently, it can be represented by any measure for which data collection is feasible. This paper suggests the introduction of a statistical measure for compatibility of menu items in the form of a coefficient of pairwise compatibility of menu items. This coefficient is defined on a cartesian product set of two sets of menu items where the sets under consideration are identified with two nonidentical courses of the meal., such as entrees and vegetables. Limiting the notion of compatibility to pairwise compatibility alone is arbitrary, but it is the first step of the investigation which may lead Co further extensions later. The first part of the paper provides the deifinitlon, some properties and examples of the coefficient of pairwise compatibility. The second part describes the method of estimation and the probability distribution of the estimator. The last part is devoted to the techniques of estimating I I I I I coefficients for the whole cartesian product set from subsets of data as it is usually available through a sequence of selective menu schedules. The application of this coefficient to menu scheduling algorithms and experience with data collection will be the subject of later reports. -3-

W Hw «-»»"«"-r. AD A NOTE ON EXCHANGES IN MATROID BASES Harold Gabow, et al Texas University

W Hw «-»»«-r. AD A NOTE ON EXCHANGES IN MATROID BASES Harold Gabow, et al Texas University -»-»-» -r - i mm.^ I»I «i i]11 m,»mj i BIII. 11 i ^ ^uiimn.iiiijp^i W Hw «-»»"«"-r AD-784 0 19 A NOTE ON EXCHANGES IN MATROID BASES Harold Gabow, et al Texas University Prep are d for: Naval Personnel