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Weekly UpdatesINTRODUCTION
In decision situations where long-run historical data are available, a traditional economic analysis often uses the relative frequency interpretation of probability to model uncertainty (Montgomery and Runger, 1994). However, for economic investment evaluations of new technologies, new product offerings, and highly integrated systems, historical frequency data are often nonexistent. In fact, the information used to develop cash flows and interest rates for most modern projects is either vague or incomplete. In such situations, decision-makers often rely on subjective estimates such as "the annual production cost will be approximately seven hundred thousand dollars" to capture investment information. Zadeh's (1965) fuzzy set theory was developed precisely to represent variables expressed in this vague manner. However, due to the "probability paradigm" held by some decision-makers (thinking only in terms of frequency and probability), vagueness in human thought is still typically captured with subjective probability distributions or represented by expected values. This research seeks to provide some insight on the cost, risk, or overall downside of this view of capturing uncertainty by directly comparing probability and fuzzy approaches to modeling vague economic parameters.
LITERATURE AND BACKGROUND
Most investment situations involving new products or emerging technologies have future cash flows that are uncertain to the decision-maker. To deal with the imprecision, uncertainty, or vagueness in these cash flows, the concepts and techniques of probability theory are usually employed (Lai and Hwang, 1994). This is not surprising since it was not until the 1960s that the fuzzy set theory was introduced for modeling vagueness in human thought by Zadeh (1965). Since the introduction of fuzzy sets, various works have been published on the economic analysis of uncertain investments using the fuzzy concept (Buckley, 1987; Chiu and Park, 1994, 1998; Kaufmann and Gupta, 1988; Wang and Liang, 1995; Ward, 1989).
Probabilistic Economic Analysis
Probabilistic methods have commonly been proposed in the literature to model and evaluate both probabilistic (Luce and Raiffa, 1957) and uncertain (Choobineh, 1989; Giacotto, 1984; Sullivan and Orr, 1982) investment alternatives. In the latter case, due to the lack of objective probabilistic information, subjective measures are used to construct probability distributions for the uncertain investment parameters. Two of the most popular probabilistic techniques used to model and evaluate economic investment problems are Monte Carlo simulation (Meimban et al., 1992; Sullivan and Orr, 1992) and closed-form analysis (Buck, 1982; Tufecki and Young, 1987). Other probabilistic techniques include the accuracy range system (Weiss 1987), propagation of errors technique (Yoon 1990), analysis of semi-variance (Ouerderni and Sullivan, 1991), the stochastic sensitivity analysis (Eschenbach and Gimpel, 1990), and an analysis by partial means (Buck and Askin, 1986).
Fuzzy Set Economic Analysis
The fuzzy set technique has grown considerably over the past four decades as a tool for modeling and evaluating uncertain/vague economic investments. Fuzzy set theory was originally called multi-valued logic in the 1920s to 1930s to deal with Heisenberg's uncertainty principle in quantum mechanics (Kosko, 1993). However, this theory did not receive much attention until Zadeh published "Fuzzy Sets" in 1965. Since that time there have been many applications of the use of fuzzy theory and fuzzy logic in engineering economy contexts. Buckley (1987) illustrated how to find the fuzzy present and future values of fuzzy cash flows using fuzzy interest rates over a crisp or fuzzy project life. Ward (1989) and Kaufmann and Gupta (1988) presented examples of fuzzy discounted cash flow analysis. Chui and Park (1994, 1998) used the fuzzy concept to find a fuzzy present worth from fuzzy cash flows and fuzzy interest rates and developed a capital budgeting model to select fuzzy projects under a limited capital constraint. Wang and Liang (1995) used the fuzzy concept to find a fuzzy benefit/cost ratio. Hartman and Hercek (1999) and Esogbue and Hearns (1998) utilized fuzzy logic in studying the asset replacement problem, and Nachtman and Needy (2001, 2003) considered fuzzy's use in handling uncertainty in activity based costing (ABC) systems. In addition, fuzzy logic or theory has been used to develop fuzzy annual worth (Liou and Chert, 2006) and present worth criterion (Omitaomu and Badiru, 2007), in selecting R&D projects (Kuchta 2001), in product and process cost estimation in multi-attribute utility theory (Ting et al., 1999), and in a unique multi-criteria decision-making model (Gogus and Boucher, 1998).
Probabilistic Ranking Methods
There are several approaches demonstrated and suggested in the literature for ranking projects using probabilistic economic measures. As an example, Buck (1982) suggests two methods for ranking probabilistic present worth. One suggestion is to rank projects by their riskiness. Using this method, prospective projects are ranked from worst to best based upon the probability that the project will not yield a positive net present worth. The second ranking method suggested by Buck involves a trade-off between the expected present worth and the risk. The ranking value for this method is P' = E(P) - [R.sup.*]p(P < 0) where p(P < 0) is the probability that the resulting present worth is not positive and R is the trade-off rate of dollars per p (P < 0). Lohmann and Baksh (1993) identify a ranking decision procedure for dealing with a project's probabilistic net present worths. This ranking procedure ranks on the expected net present worth as the primary criterion. A secondary criterion is a profitability risk restriction that rejects all projects having a probability greater than [beta] of a negative net present worth. Farrar (1962) proposes a method for ranking based upon a linear combination of the mean value and the variance.
Fuzzy Set Ranking Methods
In the field of fuzzy sets, more than a dozen fuzzy ranking methods have been suggested in the literature. Bortolan and Degani (1985) provide a good review of these methods. Since that review, some newer methods have also been suggested (Campos and Gonzalez, 1989; Choobineh and Huishen, 1993; Dubois and Prade, 1987; Fortemps and Roubens, 1996; Gonzalez, 1990; Kim and Park, 1990; Lee et al., 1994; Liou and Wang, 1992).
Lee et al. (1994) and Kim and Park (1990) group the approaches taken by fuzzy ranking methods into two types. One approach defines a ranking function that maps fuzzy values into a crisp, real line where a natural ordering exists. The second approach involves obtaining a fuzzy set of optimal alternatives where a membership function indicates the degree to which the corresponding alternative may be considered best. In the second approach, a subjective aspect for ranking fuzzy sets is maintained. On the contrary, the first approach defuzzifies fuzzy sets to provide a crisp rating for ranking. Bortolan and Degani (1985) criticize the fuzzy ranking methods that take the first approach by noting that they defuzzify an intrinsically fuzzy rating into a crisp rating. However, Fortemps and Roubens (1996) argue that most of the fuzzy ranking methods taking the second approach can produce different rankings and that counterintuitive results cannot be prevented. Table 1 lists the fuzzy ranking methods suggested in the literature and categorizes them by those that use the first approach and those that use the second approach.
For the purpose of this research we focus on those methods from the first category. Our goal is to produce a comparison of ranking between probabilistic and fuzzy methods.
METHODOLOGY
In this study, the results from a multivariable analysis of a newly developed dimensionless comparison ratio are displayed using trellis graphics (Becker et al., 1997) to reveal the present worth ranking differences between the probabilistic and fuzzy set paradigms. These differences are gauged by sensitivity of the comparison ratio to changes in the investment parameters. Results are summarized by the percentage of project dominance required to guarantee consistent ranking among the paradigms for a set of competing projects (scenarios).
Axum[R]6 by Mathsoft is used to produce the trellis displays, which consist of sensitivity charts laid out into a three-way rectangular array of columns, rows, and pages. Each row, column, and page of a trellis display represents a value or set of values for different cash flow types, degrees of uncertainty, interest rates, and cash flow timings.
Modeling Vague Investment Parameters
The uncertainty range, bias. and timing of cash flows can vary considerably from fairly certain investments (such as those in high-efficiency HVAC systems) to uncertain investments (such as research on breakthrough drugs). To establish sensitivity trends in the comparison ratio to various investment parameters, three levels of low, moderate, and high are subjectively chosen for each parameter to model a range of typical investments.
The uncertainty range in cash flows and interest rate are modeled with pessimistic, most promising, and optimistic values. This three-point estimate method of capturing uncertainty is recommended for several applications such as project scheduling, capital budgeting, and cost estimation (Grant et al., 1990; Meredith and Mantel, 1995: Newnan et al., 2004).
Modeling Vague Cash Flows
In some circumstances with projects, investment costs occur over a moderate period of time before positive cash flows are realized. Such cases include the construction of nuclear power plants, highly integrated computer networks, and theme parks. Thus, in this study, a model that incorporates both present and future costs and benefits with various levels and types of uncertainty is adopted.
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