MODELS & METHODS FOR PROJECT SELECTION

Concepts from Management Science, Finance and Information Technology

INTERNATIONAL SERIES IN OPERATIONS RESEARCH & MANAGEMENT SCIENCE Frederick S. Hillier, Series Editor Stanford University

Weyant, J. / ENERGY AND ENVIRONMENTAL POLICY MODELING Shanthikumar, J.G. & Sumita, U. / APPLIED PROBABILITY AND STOCHASTIC PROCESSES Liu, B. & Esogbue, AO. / DECISION CRITERIA AND OPTIMAL INVENTORY PROCESSES Gal, T., Stewart, T.I, Hanne, T. / MULTICRITERIA DECISION MAKING: Advances in

MCDM Models, Algorithms, Theory, and Applications Fox, B.L. ! STRATEGIES FOR QUASI-MONTE CARLO Hall, R.W. / HANDBOOK OF TRANSPORTATION SCIENCE Grassman, W.K.! COMPUTATIONAL PROBABILITY Pomerol, J-e. & Barba-Romero, S. / MULTICRITERION DECISION IN MANAGEMENT Axsater, S. / INVENTORY CONTROL Wolkowicz, H., Saigal, R., & Vandenberghe, L. / HANDBOOK OF SEMI-DEFINITE

PROGRAMMING: Theory, Algorithms, and Applications Hobbs, B.F. & Meier, P. / ENERGY DECISIONS AND THE ENVIRONMENT A Guide

to the Use of Multicriteria Methods Dar-El, E. / HUMAN LEARNING: From Learning Curves to Learning Organizations Armstrong, IS. / PRINCIPLES OF FORECASTING: A Handbookfor Researchers and

Practitioners Balsamo, S., Persone, V., & Onvural, R.I ANALYSIS OF QUEUEING NETWORKS WITH

BLOCKING Bouyssou, D. et al. / EVALUATION AND DECISION MODELS: A Critical Perspective Hanne, T. / INTELLIGENT STRATEGIES FOR META MULTIPLE CRITERIA DECISION MAKING Saaty, T. & Vargas, L. / MODELS, METHODS, CONCEPTS and APPLICATIONS OFTHE

ANALYTIC HIERARCHY PROCESS Chatterjee, K. & Samuelson, W. / GAME THEORY AND BUSINESS APPLICATIONS Hobbs, B. et al. / THE NEXT GENERATION OF ELECTRIC POWER UNIT COMMITMENT

MODELS Vanderbei, R.I / LINEAR PROGRAMMING: Foundations and Extensions, 2nd Ed. Kimms, A / MATHEMATICAL PROGRAMMING AND FINANCIAL OBJECTIVES FOR

SCHEDULING PROJECTS Baptiste, P., Le Pape, C. & Nuijten, W. / CONSTRAINT-BASED SCHEDULING Feinberg, E. & Shwartz, A / HANDBOOK OF MARKOV DECISION PROCESSES: Methods

and Applications Ramfk, J. & Vlach, M. / GENERALIZED CONCAVITY IN FUZZY OPTIMIZATION

AND DECISION ANALYSIS Song, J. & Yao, D. / SUPPLY CHAIN STRUCTURES: Coordination, Information and

Optimization Kozan, E. & Ohuchi, A / OPERATIONS RESEARCH/ MANAGEMENT SCIENCE AT WORK Bouyssou et al. / AIDING DECISIONS WITH MULTIPLE CRITERIA: Essays in

Honor of Bernard Roy Cox, Louis Anthony, Jr. / RISK ANALYSIS: Foundations, Models and Methods Dror, M., L'Ecuyer, P. & Szidarovszky, F.! MODELING UNCERTAINTY: An Examination

of Stochastic Theory, Methods, and Applications Dokuchaev, N. / DYNAMIC PORTFOLIO STRATEGIES: Quantitative Methods and Empirical Rules

for Incomplete Information Sarker, R., Mohammadian, M. & Yao, X. / EVOLUTIONARY OPTIMIZATION Demeulemeester, R. & Herroelen, W. / PROJECT SCHEDULING: A Research Handbook Gazis, D.C. ! TRAFFIC THEORY Zhu, J. / QUANTITATIVE MODELS FOR PERFORMANCE EVALUATION AND BENCHMARKING Ehrgott, M. & Gandibleux, X. / MULTIPLE CRITERIA OPTIMIZATION: State of the Art Annotated

Bibliographical Surveys Bienstock, D. / Potential Function Methodsfor Approx. Solving Linear Programming Problems Matsatsinis, N.F. & Siskos, Y. / INTELLIGENT SUPPORT SYSTEMS FOR MARKETING

DECISIONS Alpern, S. & Gal, S. / THE THEORY OF SEARCH GAMES AND RENDEZVOUS Hall, R.W./HANDBOOK OF TRANSPORTATION SCIENCE - 2nd Ed. Glover, F. & Kochenberger, G.A / HANDBOOK OF METAHEURISTICS

MODELS & METHODS FOR PROJECT SELECTION

Concepts from Management Science, Finance and Information Technology

by

Samuel B. Graves Boston College

Jeffrey L. Ringuest Boston College

with Andres L. Medaglia

SPRINGER SCIENCE+BUSINESS MEDIA, LLC

Library of Congress Cataloging-in-Publication Data

A C.I.P. Catalogue record for this book is available from the Library of Congress.

Graves, Samuel B. & Ringuest, Jeffrey L. / MODELS & METHODS FOR PROJECT SELECTION: Concepts /rom Management Science, Finance & Information Technology

ISBN 978-1-4613-5001-9 ISBN 978-1-4615-0280-7 (eBook) DOI 10.1007/978-1-4615-0280-7

Copyright © 2003 by Springer Science+Business Media New York Originally published by Kluwer Academic Publishers, New York in 2003 Softcover reprint of the hardcover 1 st edition 2003

All rights reserved. No part ofthis work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without the written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

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Printed on acid-free paper.

This book is dedicated to our families for their love and continuing support.

TABLE OF CONTENTS

DEDICATION v TABLE OF CONTENTS VB

PREFACE Xl

CHAPTER! THE LINEAR MUL TIOBJECTIVE PROJECT SELECTION PROBLEM

1.1 Introduction 1 1.2 An Example from the Literature 3 1.3 Towards a More General Multiobjective Formulation 7 1.4 A Second Example 9 1.5 Summary and Conclusions 11 References 15

CHAPTER 2 EVALUATING COMPETING INVESTMENTS

2.1 Introduction 19 2.2 Adjusting for Time Alone 19 2.3 Adjusting for Time and Risk 22 2.4 Conclusions 27 References 30

CHAPTER 3 THE LINEAR PROJECT SELECTION PROBLEM: AN ALTERNATIVE TO NET PRESENT VALUE

3.1 Introduction 31 3.2 An Example 32 3.3 The Behavioral Implications ofNPV 33 3.4 Multiple Objective Decision Methods 35 3.5 Conclusions 38 References 40

CHAPTER 4 CHOOSING THE BEST SOLUTION IN A PROJECT SELECTION PROBLEM WITH MULTIPLE OBJECTIVES

4.1 Introduction 41 4.2 Some Early Approaches 42 4.3 A Matching and Grouping Approach 46 4.4 A Stochastic Screening Approach 54 4.5 Conclusions 62 References 64

CHAPTERS EVALUATING A PORTFOLIO OF PROJECT INVESTMENTS

5.1 Introduction 65 5.2 Examples 67 5.3 Conclusions 74 References 76

CHAPTER 6 CONDITIONAL STOCHASTIC DOMINANCE IN PROJECT PORTFOLIO SELECTION

6.1 Introduction 77 6.2 The Model 78 6.3 Summary and Conclusions 89 References 93

CHAPTER 7 MEAN-GINI ANALYSIS IN PROJECT SELECTION

7.1 Introduction 95 7.2 The Model 101 7.3 Conclusions 114 References 117

CHAPTER 8 A SAMPLING-BASED METHOD FOR GENERATING NONDOMINATED SOLUTIONS IN STOCHASTIC MOMP PROBLEMS

8.1 Introduction 119 8.2 Stochastic, Nondominated Solutions 123 8.3 Sampling Approaches to Solving MOMP Problems 125 8.4 Computational Issues 126 8.5 Summary and Conclusions 134 Appendix 8.1

Example SAS Code 135 References 144

CHAPTER 9 AN INTERACTIVE MUL TIOBJECTIVE COMPLEX SEARCH FOR STOCHASTIC PROBLEMS

9.1 Introduction 147 9.2 Direct Search Methods 149 9.3 Applying Complex Search to Multiobjective Mathema-

tical Programming.Problems 152 9.4 An Example of Multi objective Complex Search 155 9.5 Conclusions 158 References 160

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CHAPTER 10 AN EVOLUTIONARY ALGORITHM FOR PROJECT SELECTION PROBLEMS BASED ON STOCHASTIC MULTIOBJECTIVE LINEARLY CONSTRAINED OPTIMIZATION

10.1 Introduction 163 10.2 Stochastic Multiobjective Linearly Constrained

Programs 164 10.3 Multiobjective Evolutionary-Based Algorithm 166 10.4 Computational Examples 174 10.5 Summary and Conclusions 183 Appendix 10.1

Input File for the Algorithm Parameters for the SMOLCP Example 185

Appendix 10.2 Java Program that Defines the First Objective Function in the SMOLCP Example 187

References 188

INDEX 191

IX

PREFACE

The project selection problem is one that has been given much atten­tion in the literature. In the project selection problem the decision maker is required to allocate limited resources across an available set of projects, for example, research and development (R&D) projects, information technology (IT) projects, or other capital spending projects. In choosing which projects to fund, the decision maker must have some concrete objective in mind, e.g., maximization of profit or market share or perhaps minimization of time to market. And in some cases the decision maker may wish to simultaneously satisfy more than one of these objectives. Most often, these multiple objec­tives will be in conflict, resulting in a more complicated decision making task.

The decision maker may be able to partially fund some projects, or conversely some projects may involve a binary decision of fully funding or not funding at all. The decision maker may also have to resolve issues of in­terdependency-that is that the value of funding an additional individual proj­ect may vary depending upon the success or failure of projects that are already in the portfolio. The decision maker then must take all these factors into ac­count in seeking an appropriate project selection model, choosing a method­ology which evaluates the appropriate objective(s), subject to relevant re­source constraints as well as constraints relating to projects with binary (full or none at all) funding restrictions.

There is a considerable body of literature describing an abundant va­riety of models designed for the project selection problem. For our purposes here, the literature can be broken down into two main streams: that which we will label the traditional management science stream and that which we will call the financial modeling stream. The first stream, the management science literature, derives largely from mathematical programming treatments along with some use of classical decision theory. In order to use these approaches it is usually assumed that the existing decision alternatives (projects) are rea­sonably well-known and that the necessary information for modeling these alternatives is at hand at the initiation of the planning process. The majority of the management science models treat the decision process of choosing a set of new projects to form a wholly new portfolio. But some of the models we will present also address the problem of adding one or more new projects to an already existing project portfolio. Most of the research in this body of lit­erature is confined to decisions which are made at one point in time, that is, the models are static in the sense that they represent a one-time decision to assemble or analyze a given portfolio.

An important junction in the decision making process occurs when the decision maker chooses the appropriate objective(s). If a single objective

(e.g., market share) is chosen, then the problem may be handled with ordinary mathematical programming techniques that have been used in project plan­ning models for some time now. If, however, the decision maker wishes to pursue several objectives simultaneously (e.g., maximization of revenue in each of several future time periods), some form of multiobjective program­ming will be needed. It is our belief that this multiobjective case is the more realistic one, thus, in this book, we will show several applications of multiob­jective programming to the project portfolio problem.

A key assumption of the mathematical programming models above is that all relevant information about the projects is known. However, this may not always be true. Some allocation decisions must be made in the presence of uncertainty. Uncertainty may exist concerning the ultimate result of a proj­ect (e.g., the amount of revenue) or the success or failure likelihood may be known only as a probability distribution. Uncertainty may be represented by probability distributions around the coefficients in the objective function or in the constraints. In this book we will illustrate treatments for each of the above forms of uncertainty. We will, however, assume that adequate information is available to represent these projects in the model. The required information may be in the form, for example, of a probability of project success or a prob­ability distribution around a coefficient in the objective function (e.g. project return).

When we are dealing with uncertainty and multiple objectives, we may need to resort to the use of stochastic dominance criteria to screen a set of solutions. Stochastic dominance is appropriate for all probability distribu­tions and is minimally restrictive with respect to thedecision maker's utility function. In this monograph there are several forms of stochastic dominance, which are of interest. First order stochastic dominance simply compares the cumulative distribution functions for two projects and makes the choice on this basis alone. The first order criterion is applicable to all decision makers with monotone utility functions; that is, decision makers who prefer higher returns to lower ones and/or those who prefer less risk to more risk. In some instances, the first order criterion does not yield an unambiguous choice. In these cases it may be necessary to resort to second order stochastic dominance. The decision calculus here is based on the area between the two cumulative distribution curves. This second-order criterion is appropriate for a narrower class of decision makers, those who are risk-averse. We will also in some cases apply a conditional stochastic dominance criterion. Conditional stochastic dominance analysis identifies dominant and nondominant projects conditioned on the projects, which make up the current portfolio. This criterion requires no explicit knowledge of the decision maker's utility function and is applicable to all risk averse decision makers. Finally, in some cases we will apply a stochastic dominance criteria which compares

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alternatives based on the expected value and the probability of achieving desired levels of one or more measures.

The second main stream of literature that we wish to consider here is that which derives from financial portfolio research, some of which can be applied directly to the project selection model. The earliest such methodology is the mean-variance model, which compares two projects according to their mean and variance. Any project which has a higher mean return for a given variance or a lower variance for a given mean will be preferred. The mean­variance approach is limited in its practical applicability because it involves rather strict assumptions about the decision maker's risk orientation, and be­cause it may require a large number of pairwise comparisons if there is a large number of projects under consideration.

Another source of financial literature is the traditional financial opti­mization models in which the variance in the portfolio returns is minimized subject to a constraint on expected return. This approach, however, requires solution of a non-linear optimization problem (The portfolio variance is non­linear.) which may be impractical when there are large numbers of projects to consider. A more recent treatment of the project selection problem deriving from the financial literature is the mean-Gini approach. This, like the mean­variance criterion, is a two-parameter method. That is, only two parameters must be estimated for each R&D project. (Mean and variance for the mean­variance approach; mean and Gini coefficient for the mean-Gini approach). The Gini coefficient, like the variance, is a measure of dispersion in outcomes or investment risk. However, when the Gini is used, as opposed to the vari­ance, preferred portfolios may be designed based on a simple heuristic. In the mean-Gini analysis--as will be shown later in this work--there is one impor­tant difference between the financial application and the project portfolio ap­plication. In the financial application the necessary probability distributions for each security are unknown and are estimated from sample (i.e., market) data. For project portfolio applications the probability distributions (describ­ing various levels of success) tend to be simple discrete distributions, permit­ting (in principle) complete enumeration of all possible outcomes in the port­folio.

Largely in parallel to (and distinct from) the R&D project portfolio selection literature is a body of work describing Information Technology (IT) portfolio selection. The objectives of this body of work have much in com­mon with the R&D portfolio modeling work and we will assume in most of this work that our models apply equally well to R&D or IT project selection problems. Essentially the problem here is to find the optimal set (portfolio) of IT projects when resources are constrained. The greatest difference between the IT models and the R&D models is the heightened importance of project interdependencies in the IT models. In IT project applications, as opposed to R&D project applications, there is, due to the very nature of the projects, an

Xlll

increased incidence of interdependency. For example, two IT projects may share some identical sections of computer code. They may share as well hardware such as workstations and networks. And whereas R&D interaction modeling is typically pairwise, realistic IT modeling requires that higher-or­der interdependencies (among three or more projects) be represented.

Project selection models that capture the various characteristics de­scribed above (interdependencies, uncertainty and the ability to partially fund or the requirement to either not fund or fully fund projects) may be quite complicated mathematically. These characteristics can lead to complicated mathematical programs that include one or more objectives that may be linear or nonlinear, deterministic or stochastic and with variables that are real, inte­ger, or binary. Appropriate solution procedures for these complex mathe­matical programs are also needed. We will present these as well.

This monograph is intended to pull together in a single publication the latest work in this field. It is not intended as a survey, but rather as a vehicle for establishing some unity in the field of project selection modeling. The models presented here rely heavily on mathematical programming but also draw from decision theory and finance. Our intention is to present models that are broadly applicable in the project selection context, to describe the as­sumptions and limitations of these models, and to provide solution method­ologies appropriate for solving these models. The chapter outline below traces out the main themes of the book.

CHAPTERS 1-3: CRITERIA FOR CHOICE

Chapters 1-3 investigate the effect of the choice of optimization crite­ria on the results of the portfolio optimization problem. Chapter 1 lays out the multiobjective linear programming approach to the project selection problem. The multiobjective approach is contrasted with the goal program­ming approach, which had been used in earlier applications. This chapter shows that the multiobjective formulation of the problem is superior to the earlier goal programming approach in that the multiobjective technique yields several nondominated solutions to the problem, in contrast to the single solu­tion revealed by the goal programming approach. The multiobjective ap­proach is recommended here as a more general approach that will reveal all nondominated solutions.

Chapter 2 diverges from the discussion of optimization models for project selection to introduce a discussion of appropriate methods for adjust­ing for time and risk in the project selection problem. Projects and their as­sociated revenue streams typically last for a number of years and all projects involve some level of risk. In order to compare two projects with different time profiles, we need methods for adjusting for time and for risk. This

xiv

chapter shows some weaknesses of the traditional net present value (NPV) calculations (using the discount rate and the risk-adjusted discount rate) and shows how to use generalized NPV to avoid some of these weaknesses.

Chapter 3 continues the investigation of NPV as a decision criterion and returns to the discussion of optimization models by showing how mul­tiobjective linear programming can be used as an improvement over tradi­tional NPV. The chapter shows that the NPV formulation of the project se­lection problem is a special case of optimizing a multi-attribute value func­tion, and that the NPV formulation may impose undesirably strict assumptions about the decision maker's preferences over time. Multiobjective linear pro­gramming is then used in place ofNPV to solve the project selection problem.

CHAPTERS 4-7: RISK AND UNCERTAINTY

Chapters 4-7 deal with uncertainty in the project selection problem. Most of the models developed in this section are based on the assumption that a probability distribution is known or can be estimated to deal with uncer­tainty in some parameter of the project selection model. The multiobjective models used in Chapters 1 and 3 result in a set of nondominated solutions from which the decision maker must choose a single preferred solution. The process of making this choice may not be an easy task.

In Chapter 4 several methods are shown for screening nondominated solutions. One of these which introduces the problem of uncertainty is cov­ered in detail. That method uses a multiobjective model as in the earlier chapters and adds the additional complications associated with uncertainty. We solve the multiobjective portfolio problem (with profit and market share as objectives), yielding a list of nondominated solutions from which the deci­sion maker must choose. Next we establish goals, or desired levels of achievement for each objective. Then, assuming we can estimate the prob­ability distributions describing each objective, we calculate the probability of attaining the goals for each of the nondominated solutions.

In Chapters 5 and 6 we again diverge from the discussion of optimi­zation models to examine the importance of decision context on the appropri­ate analysis of risk. In Chapter 5 we use concepts from decision theory to directly address the treatment of risk and uncertainty. We show that the tra­ditional methods for treating risk tend to introduce a bias into the project se­lection decision. This bias results from the common practice of analyzing each project in isolation, rather than considering the risk-reducing effects which result from aggregation of diverse projects into the same portfolio. This chapter demonstrates that managers who separately analyze only the next project on the horizon without considering the risk-mitigating effects of ag­gregation will tend toward excess timidity in decision making.

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Chapter 6 provides a more sophisticated treatment of risk that consid­ers the decision context and that is applicable to the project selection problem. Using an approach taken from financial modeling, this chapter shows a prac­tical method of developing a nondominated portfolio of risky projects based on the criterion of conditional stochastic dominance. The method is simple and highly intuitive, requiring only the estimation of two parameters, the ex­pected return and the Gini coefficient. The chapter demonstrates a successful application of this technique to a real-world R&D portfolio and shows that it is a practical method for screening large numbers of candidate portfolios to discover those which are nondominated.

Chapter 7 continues the use of the Gini. In this chapter, however, we set up a branch and bound heuristic, which is based on the mean return and the Gini coefficient of each project portfolio. This heuristic produces a set of solutions (portfolios) which are nondominated in the mean-Gini sense. The results of this branch and bound heuristic are then plotted with the return on the vertical axis and the associated Gini value on the horizontal axis. The points on this graph are a mean-Gini efficient frontier. We then screen the points to find those which are stochastically nondominated.

CHAPTERS 8-10: NON-LINEARITY AND INTERDEPENDENCE

These chapters deal with problems of non-linearity and interdepend­ence as they arise in the project selection problem. The ability to handle non­linear problems allows the application of the methodology to a far wider range of problems. Similarly, the ability to model interdependence between projects, as noted in the discussion ofIT models above, is an important step in generalization. Chapters 8, 9 and 10 present solution methodologies, which can be used to solve these most general project selection models.

Chapter 8 presents a method for generating nondominated solutions for stochastic multiobjective mathematical models of the project selection problem, which is applicable to both continuous and zero-one variables. The method is based on the assumption that the objective coefficients are random variables with probability distributions that are known or can be estimated. The method shown in this chapter generates solutions that are nondominated in terms of the expected value of each objective and the probability that each objective meets or exceeds a specified target value. The method in Chapter 8 is most applicable to integer 0,1 problems and is limited in the real variable case to problems with relatively few variables. This limitation is addressed in Chapter 9.

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Chapter 9 shows another approach to the stochastic non-linear prob­lem, this one having non-linearity in both the objective function and the con­straints. The method presented in Chapter 8 uses random sampling to identify a set of feasible solutions for the project selection problem. It is shown that this random sampling can be computationally burdensome so that the method presented in Chapter 8 is limited to a small number of real valued variables. In Chapter 9 complex search is applied to the project selection problem. In complex search only the initial feasible solution is generated randomly. Then subsequent solutions are found using a systematic search. In this way the computational burden is greatly reduced. The example problem shown in Chapter 9 illustrates a model with a non-linear objective function and several non-linear constraints, which are solved by relying on the progressive defini­tion of the decision maker's preferences.

Chapter 10 concludes our treatment of non-linearity and interdependence in the project selection problem. This chapter presents a new algorithm that treats the project selection problem in cases of uncertain objectives, partial funding, and interdependencies in the objectives. The method shown here is based on a multiobjective evolutionary algorithm and on concepts from linear programming and presents the decision maker with a very good approximation of the true efficient frontier. The algorithm is able to solve project selection problems modeled as multiobjective linear programs and multiobjective non-linear programs with linear constraints.

We would like to acknowledge here our indebtedness to individuals who have contributed to the research in this volume. In particular we want to acknowledge Randy Case, with whom we have co-authored work in this area and whose data is used in several places here. We are also indebted to the many editors and referees who have helped to sharpen and clarify the research we have performed over the past ten years. Finally, we want to thank Suzanne Proulx for technical assistance in producing this manuscript.

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