Lectu re Notes in Economics and Mathematical Systems Managing Editors: M. Beckmann and W. Krelle 209 Essays and Surveys on Multiple Criteria Decision Making Proceedings of the Fifth International Conference on Multiple Criteria Decision Making Mons, Belgium, August 9 -13,1982 Edited by Pierre Hansen Spri nger-Verlag Berlin Heidelberg New York 1983 Editorial Board H. Albach A. V. Balakrishnan M. Beckmann (Managing Editor) p. Ohrymes, J. Green W. Hildenbrand W. Krelle (Managing Editor) H. P. KOnzi K. Ritter R. Sato U. Schittko P. SchOnfeld R. Selten Managing Editors Prof. Dr. M. Beckmann Brown University Providence, RI 02912, USA Prof. Dr. W. Krelle Institut fOr Gesellschafts-und Wirtschaftswissenschaften der Universitat Bonn Adenauerallee 24-42, 0-5300 Bonn, FRG Editor Prof. Dr. Pierre Hansen Faculte Universitaire Catholique de Mons Faculte de Sciences Economiques Appliquees Chaussee de Binche, 151, B-7000 Mons, Belgium and Institut d'Economie Scientifique et de Gestion, Lille, France Sponsored by: Faculte Universitaire Catholique de Mons Fonds National de la Recherche Scientifique Ministere de l'Education Nationale et de la Culture Franc;:aise College Interuniversitaire d'Etudes Ooctorales en Management European Institute for Advanced Studies in Management European Research Office of the U.S. Army Municipalite de Mons ISBN-13: 978-3-540-11991-3 e-ISBN-13: 978-3-642-46473-7 001: 10.1007/978-3-642-46473-7 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to ·Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1983 Softcover reprint of the hardcover 1s t edition 1983 2142/3140-543210 PREFACE The Fifth International Conference on Multiple Criteria Decision Making, not suprisingly, had several objectives. First, it aimed at beinq a forum for exchange and intensive discussion of recent ideas on theory and practice of MCDM, following the now well-established tradition of the previous meetings in the series, organized by H. Thiriez and S. Zionts in Jouy-en-Josas (1975), S. Zionts in Buffalo (1977), G. Fandel and T. Gal in Hagen/Konigswinter (1979) and J. Morse in Newark (1980). Second, closer contacts Nere desired between participants in these meetings and other active groups in the field, prominent among which is the European Working Group on Multiple Criteria Decision Aid. Third, participation of senior or junior researchers who had recently developped important new methodolo0ies, such as the Analytical Hierarchy Process, was actively sought for. Fourth, a synthesis of the rapidly expanding field of MCDM was to be made through selective surveys by leading researchers in the various areas it comprises. Fifth, cross-fertilization and multidisciplinary research was to be encouraged through presentations on the connections between MCDM and mathematics, economics, game theory, computer science and other subjects. Sixth, much emphasis was to be given to real-world applications of MCDM, particularly large scale ones and/or pioneering work in new fields. The present volume reflects the general agreement observed among participants that these goals were largely attained. Fifty papers were presented at the meeting and many discussions were so animated that they continued during lunch time and evenings. The pleasant and stimulating atmosphere was not even hindered by the weather which chose to be bad during sessions and sunny during social events. The conference could not have been organized without the substantial help of our sponsors, to which warm thanks are due: the Faculte Universitaire Catholique de Mo~s, host institution, for providing facilities and rooms, the Fonds National BeIge de Ia Recherche Scientifique, the Ministere de l'Education Nationale et de La Culture Fran~aise, the College Interuniversitaire d'Etudes Doctorales en Management, the European Institute for Advanced Studies in Management, the European Research Office of the u.S. ATIny for their financial support, and the ~~nicipality of Mons for receiving the participants in the historic Town Hall. The Vice-Dean A. Bultez welcomed the participants and took a constant in terest in the meeting. The practical organization was done by the FUCAt1 staff; the gracious help of A. Segond, D. Raulier and Y. 14illame, and the cheerful readiness of P. Lietard allowed to manage smoothly all problems and are remembered gratefully. Finally, the success of the meeting was due to the participants, who carefully nre pared papers, presentations and discussions, chaired sessions, expressed their vieNS, IV criticism and responses and revised their papers for publication. They all deserve warm thanks for their so active participation. The list of participants is given on page 431, the program of the meeting on page 437. The table of contents is on the next page, papers being ordered alphabetically by name of first author. Pierre Hansen 1~ons, November 1982. TABLE OF CONTENTS BODILY. S.E. and C.C. WHITE: "Optimal Consumption and Portfolio Strategies in a Continuous-Time Model with Summary-Dependent Preferences" 1 BOGARDY. I. and A. BARDOSSY : "Application of MCDM to Geological Exploration Planning" 8 COHON.J.L. and T. W. EAGLES: "Analysis of Nuclear Reactor Siting Policy Using t·lultiobjecti \Ie 19 Programming" CRAMA. Y. and P. HANSEN "An Introduction to the ELECTRE Research Programme" 31 DUESING. E.C. : "Mul tiobjecti ve Li near Programming and the Theory of the Fi rm 1. Substitution and Sensitivity Analysis" 43 GERSHON. M. and L. DUCKSTEIN : "An Allgorithm for Choosing of a Multiobjective Technique" 53 FARQUHAR. P.H. : "Research Directions in Multiattribute Utility Analysis" 63 FRIESZ. T.L. and P.T. HARKER: "Multi-Objective Design of Transportation Networks: The Case of Spatial Price Equilibrium" 86 GAL. T : "nn Efficient Sets in Vector Maximum Problems - A Brief Survey" 94 GOLABI. K. : "A Markov Decision Modelling Approach to a Multi-Objective Maintenance Problem" 115 GRAUER. M. : "Reference Point Optimization - The Nonlinear Case" 126 HABENICHT. H. : "Quad Trees. A Datastructure for Discrete Vector Optimization Problems" 136 HINLOOPEN. E. • P. NIJKAMP and P. RIETVELD: "The Regime Method : A New Multicriteria Technique" 146 HO. J.K. : "Multiple Criteria Optimization Using Analytic Hierarchies and Holistic Preferences 156 HOLIN.S. and M. PREVOT : "An Application of the Multiobjective Programming to French Industry" 167 JAHN. J. : "Mathematical Applications of MCDM : Vector Approximation and Cooperative Differential Games" 177 KHAIRULLAH. Z. and D. KHAIRULLAH : "Importance of Characteristics of Graduating Seniors with Respect to Positions in Public Accounting" 187 VI KORHONEN, P. and J. WALLENIUS : "Principles of Solving Sequential Multiple Criteria Decision Problems" 195 KORNBLUTH, J.S.H. : "Max-Min Programming with Linear Fractional Functions : Algorithms and Examples" 204 KWON, Y.K. and P.L. YU : "Confl ict Dissolution by Reframing Game Payoffs: Introduction" 214 LEHERT, P. and A. DE WASCH : "Representation of Best Buys for a Heterogeneous population" 221 LAWRENCE, K.D., LAWRENCE, S.M. and R.A. MAROSE : "A Multiple Goal Portfolio Analysis Model for the Selection of MIS Projects" 229 LEE, S.M., SNYDER, C. and M. GEN : "The Microcomputer: Experience and Implications for the Future of Multiple Criteria Decision Making" 238 LOCKETT, A.G. and B. HETHERINGTON "Subjective Data and MCDM" 247 MICHALOWSKI, W. and Z. ZOLKIEWSKI "An Interactive Approach to the Solution of a Linear Production Planning Problem with Multiple Objectives" 260 MORSE, J.N. : "Banking in a Volatile World: Setting Country Lending Limits" 269 NAKAYAMA, H., TAKEGUCHI, T. and M. SANO : "Interactive Graphics for Portfolio Selection" 280 NARULA, S.C. and A.D. NWOSU : "Two-Level Hierarchical Programming Problem" 290 NYKOWSKI, I. and Z. ZOLKIEWSKI : "On Some Connectioris between Bicriteria and Fractional Programming Prob 1e ms" 300 REEVES, G.R. and L.S. FRANZ: "A Simpl ified Approach to Interactive MOLP" 310 RIOS-GARCIA, S. and S. RIOS-INSUA : "Portfol io Selection Problem with Multiattributes and Multiple Criteria" 317 SAATY, T. L. : "Priority Setting in Complex Problems" 326 SARIN, R.K. : "Measurable Value Function Theory: Survey and Open Problems" 337 SERAFINI, P. : "Convergence of Dual Variables in Interactive Vector Optimization" 347 VII SPRONK, J. and F. VEENEKLAAS : "Scenarios for Economic Development: A Feasibility Study by Means of Interactive Multiple Goal Programming" 356 TELGEN, J. : "An MCDM Problem in Banking" 372 WEBER M.O. : "An Empirical Investigation on Multi-Attribute-Decision-Making" 379 WENDELL, R. E. : "Efficiency and Solution Approaches to Bi-Objective Mathematical Programs" l39 WHITE, C.C. and H.K. EL DEIB "Multi-Stage Decisionmaking with Imprecise Utilities" 400 WHITE, D.J. : "The Foundations of Multi-Objective Interacti ve Programming - Some Questions" 406 ZIONTS, S. : "A Report on a Project on Multiple Criteria Decision Making, 1982" 416 List of Participants 431 Conference program 437 OPTIMAL CONSUMPTION AND PORTFOLIO STRATEGIES IN A CONTINUOUS TIME MODEL WITH SUMMARY-DEPENDENT PREFERENCES Samuel E. Bodily Darden Graduate School of Business Administration and Chelsea C. White Department of Engineering Science and Systems University of Virginia ABSTRACT A preference model wherein current wealth and a summary description of past consumption condition preferences for future consumption is used to develop relationships between optimal consumption and portfolio strategies. Financial variables are described by a continuous-time, continuous-state, stochastic process. The results complement and extend results of previous models. I. INTRODUCTION Optimal consumption and portfolio mixture strategies are derived in this note for a continuous-time, continuous-state preference model. In this model current wealth and a summary descriptor of past consumption condition the investor's preferences for future consumption. Our model generalizes the models in previous work where the investor maximized the expectation of discounted utility of consumption [3,7J, the expectation of a utility of consumption that depends on time plus a bequest [2J, or the expectation of continuous-time analogs of an additive or a multiplicative multiperiod utility of consumption [4J. The investor's decisions are based on two separate considerations: 1. Investment dynamics - the investment opportunities and wealth to be invested now and their evolution in the future. 2. Consumption preferences - the relative desirability of alternative streams of consumption including the way in which they depend on the wealth path. This note begins by stating our models of these two aspects of the problem. Then results are given relating optimal investment and consumption. 2 II. INVESTMENT DYNAMICS At each time t in [O,t*] the investor selects a consumption rate c(t) and a portfolio mixture x(t) indicating the fraction (O~x(t)~l) of available wealth invested in a risky versus a risk-free investment. (The risk-free investment may be, for example, a treasury bill and the risky investment an index fund for Standard and Poors 500 stocks.) Let w(t) and s(t) respectively represent the level of wealth and the level of a summary descriptor of past consumption at time t. Assume these state variables are generated by the stochastic differential equations dw(t) = w(t) [x(t) (r-rf)+rf]dt-c(t)dt+w(t)x(t)de(t) (la) ds(t) = g[s(t),c(t)]dt (lb) where: g[.,.] = a (sufficiently smooth) function relating the effect of the present summary descriptor value and presently selected consumption rate to the rate of change of the summary descriptor value, r = expected rate of return on the risky asset, rf = rate of return on the sure asset, e(t) = a Wiener process with de(t)-N(0,u2dt). Definitions, references, and related discussions concerned with Ito and Weiner processes ((la) generates an Ito process) can be found in [2]. Note that all of the randomness in the wealth process is represented by the process [e(t), t~O]. An admissible strategy is a decision rule which selects the pair [c(t),x(t)] on the basis of the triple [t,w(t),s(t)] for all t in the planning horizon [O,t*]; that is, it is assumed that throughout the planning horizon the investor/decision maker selects the rate of consumption and fraction of risky investment at time t knowing the current values of the level of wealth and summary descriptor. The investment/consumption decisions would ideally be based on the future needs and the likelihoods of future levels of wealth. The investor evaluates any sequence of decisions, however, ultimately through enjoyment of the consumption stream, as represented by his subjective utility of consumption. III. SUMMARY-DEPENDENT PREFERENCES FOR CONSUMPTION We assume that preferences are expressed by a von Neumann-Morgenstern utility function u[t,t(t)] where t(t) = [c(e): e>t]. Hence, uncertain consumption streams are ranked cardinally by their expected utility. Previous continuous-time models have employed simplistic forms of u[t,~(t)] such as the following: t* 1. continuously discounted utility, J" t e-peu[c(e)]de, [3,7] 3 I t* 2. time-dependent utility, t u[e,c(e)]de [2] 3. continuous-time analogs of additive or multiplicative utility [4], t* a) It ae u[c(e)]de (additive) b) -exp -8 {I!*u[c(e)]de} (multiplicative). where the rate for discounting utility, at represents the weight on utility p= at time t and 8 is a parameter analagous to the interaction terms in a multiplicative model (see [4] for an interpretation of these last two parameters). Each of these utility forms ignores fundamental behavioral properties of most individuals. They require, for example, that the investor has a steady enjoyment of consumption which is unaffected by fortuitous surges in wealth or by the level of past consumption. The model of preferences we adopt, however, allows the investor's attitude about future consumption to depend on current wealth and past consumption. The investor will generally not need to know the full detail of past consumption in order to adjust his consumption to the "life-style to which he has become accustomed"; the summary descriptor s(t) defined in the previous section will suffice. The function g[s(t),c(t)] used to update this summary descriptor will insure that if preferences regarding future consumption are equivalent for two different histories of consumption prior to t, then the summary descriptor of these two histories will be equal. (See [5] for models of summary-dependent preferences where there is a summary descriptor of both the past and the future.) We denote the instantaneous utility gain from consumption at time t by a[c(t)lw(t),s(t)] which expresses the conditioning of consumption preferences on the state of wealth and summary descriptor. A desirable property of the utility function is that decisions made at t are consistent with decisions made at t+6t; then u[t,t(t)] must be a positive linear transformation of U[t+6t,C(t+6t)], i.e. u[t,t(t)] = a[c(t)lw(t),S(t)]6t + (1+b[c(t)lw(t),S(t)]6t)-lu[t+6t,~(t+6t)] (2) where (1+b[c(t)lw(t),S(t)]6t}-1~O. Here b6t is the rate of discount which relates utility for future consumption beginning at time t + 6t to utility for future consumption beginning at t. Note that the consumption stream t(t) is longer tnan th~2~onsumption stream t(t+6t); hence u[t,t(t)]>U[t+6t,t(t+6t)]. KeWrltlng \ } glVes - U[t+6t,t(t+6t)] - u[t,c(t)] = u[t,t(t)]b[c(t)lw(t),S(t)]6t-a[c(t)lw(t),S(t)]6t - a[c(t)lw(t),s(t)]b[c(t)lw(t),S(t)]6t2• Dividing through by 6t and taking the limit as results in the following 6t~O differential equation Ii = bu - a. (3)