A research project was undertaken for the U.S. ArmyCorps of Engineers to determine the relative utility and effectiveness of four well-known multicriteria decision making (MCDM) models for applications in realistic water resources planning settings. A series of experiments was devised to examine the impact of rating and ranking procedures on the decision making behavior of users (e.g., planners, managers, analysts, etc.) when faced with situations involving multiple evaluation criteria and numerous alternative planning projects. The four MCDM models tested were MATS-PC, EXPERT CHOICE, ARIADNE, and ELECTRE.Two groups of analysts and decision makers were tested. One group consisted of experienced U.S. Army Corps planners, while the other was comprised of graduate students. Based on a series of nonparametric statistical tests, the results identified EXPERT CHOICE as the preferred MCDM model by both groups based largely on ease of use and understandability. ARIADNE fostered the largest degree of agreement within and among the two groups of individuals tested. The tests also lend support to the claim that rankings are not affected significantly by the choice of decision maker (i.e., who uses any of these MCDM models) or which of these four models is used. (KEY TERMS: multiple criteria decision making; decision support systems; water resources planning; experiments in decision making; design of computer-based decision support systems.)
A multiple-objective approach to decision making in watershed management is developed and demonstrated within the framework of nonlinear programing by means of the case study of the Charleston watershed in southern Arizona. The effects of various land treatments and management practices on water runoff, sediment, recreation, wildlife levels, and commercial potential of a study area are investigated, constraints on the available land and capital being observed. This leads to a model with five objective functions to be maximized under 18 constraints. In an iterative manner the decision maker proceeds from one noninferior solution to another, comparing sets of land management activities for reaching specified goals and evaluating trade-offs between individual objective functions. This technique, labeled Trade, involves the formulation of a surrogate objective function and the use of the cutting plane method to solve the general nonlinear problem; it hopefully provides a compromise between computationally intractable and excessively simplified approaches to truly multiobjective watershed management. sertation, Univ. of Ariz., Tucson, 1973. Roy, B., Problems and methods with multiple objective functions,Math. Progr., •(2), 240-266, 1970. U.S. Congress, National Forests, multiple-use and sustained yield, hearings before the Subcommittee on
This paper presents an input-output model where tlze technology coefficients and the demand variables are random variables, rather than fixed quantities. The model is able to relate data acquisition costs to forecasted production levels and uncertainty, as the error in the estimate of distribution parameters is reduced. The nonlinearity of the resulting systern of equations remains tractable and techniques such as the Newton-Raphson method can be applied advantageously. A numerical example is included.
A methodology for incorporating risk and uncertainty considerations in the benefit-cost analysis of water resources projects is presented and illustrated with a case study. Estimates of benefits and costs are allowed to vary over ranges subjectively assessed, modeled as random variables with known probability density functions, and, finally aggregated analytically to yield a distribution of the B/C ratio. The advantages and limitations of this approach are discussed against a background of current guidelines by the U.S. Water Resources Council and existing U.S. Army Corps of Engineers procedures. Lemma 2. Let X•, X2, ß ß ß, Xn be statistically independent gamma random variables with parameters (ai, 13i) for i = 1, 2, ß ß ß , n. Then the new random variable Y = X• + X2 + ß ß ß + Xn has the moment generating function of form M(t) = [(1 -/3•t)"•(1 -•t) "2... (1 -lgnt)"n] -• (13) GOICOECHEA ET AL.' RISK/UNCERTAINTY IN BENEFIT-COST ANALYSIS 795 for all real t such that i3it < 1, where/3i = max {/3•, •, ß ß ß , The above results for lemmas 1 and 2 can be verified easily using the definition and unique properties of the moment generating function [Hogg and Craig, 1972]. Lemma 3. Let XB and Xc be two statistically independent normal random variables with parameters (/•i, ai 2) for i = B, C. Then, the new random variable Y = X•/Xc has the distribution g•) = y [4H: a•:ac:]ß ff• exp -• • + ac dz (14) for -• < Y < •. The proof is given in the appendix. Lemma 4. Let X• and Xc be two statistically independent gamma random variables with parameters (hi, •i) for i = B, C. Then the new random variable Y = X•c has the distribution
A decision making method labeled Protrade has been developed to account for both multiple objectives and uncertainty aspects. This method is applied to the case study of the Black Mesa Region in Northern Arizona which is being strip mined for coal. One important concern in large-scale surface mining is the reclamation of mine spoils to bring about beneficial land uses while observing economic, social, environmental, and legal constraints. Protrade considers a set of linear objective functions with random parameters, subject to a set of physical constraints; the preferences of the decision maker are articulated in a progressive manner, and alternative solutions are generated. Nonlinear deterministic equivalents are introduced to account for uncertainty in some of the parameters involved. While nq•fi•al policy recommendations are made, this illustration of Protrade is carried out to investigate the p6te'•ial of the method for trading off multiple objectives and measures of uncertainty in water and other natural resource system design problems. generating techniques are the Weighting method [Geoffrion, 1967; Major, 1969; Hairnes, 1973], the e-constraint method [Hairnes and Nainis, 1973], adaptive search [Beeson and Meisel, 1971], multi-criteria simplex methods [Zeleny, 1974], and compromise programing [Starr and Zeleny, 1977; Duckstein and Opricovic, 1978]. Methods which rely on prior articulation of preferences are goal programing [Charnes and Cooper, 1961; Lee, 1972], utility function assessment [Raiffa, 1969; Keeney and Wood, 1977], Electre method [Roy, 1971; David and Duckstein, 1976], and surrogate worth tradeoff [Hairnes et al., 1975]. Techniques which rely on progressive articulation of preferences are the step method [Benayoun et al., 1971], sequential multiobjective problem solving [Monarchi et al., ], and the tradeoff development method [Goicoechea et al., 1976b, c]. Structuring a decision problem as a mathematical optimization problem often leads to a number of difficulties, such as described by Keeney and Wood [1977]: 1. The formulation of the objective function as a weighted linear function implies constant tradeoffs between objectives regardless of the level of each objective. This is rarely the situation.
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