Fuzzy cash flow analysis using present worth criterion

Fuzzy cash flow analysis using present worth criterion

Chiu, Chui-Yu


In practice, most economic decision problems involve the uncertainty feature of cash flow modeling. If sufficient objective data is available, probability theory is commonly used in modeling cash flows and performing decision analysis. Unfortunately, decision makers rarely have enough information to perform the decision analysis, such that probabilities can never be known with certainty and the economic decision is attributable to many uncertain derivations. In this situation, most decision makers rely on experts’ knowledge in modeling cash flows when the probability information is not reliably justified [16].

Experts’ knowledge is a collection of information and techniques which may be learned from the experts’ past experience that are suitable for solving some specific problem domains. Along with the financial expert’s knowledge, subjective probability distributions are extensively employed in estimating future cash flows. However, in an uncertain economic decision environment, an expert’s knowledge about the cash flow information usually consists of a lot of vagueness instead of randomness. For example, to describe a sales profit which may be implicitly forecasted from past incomplete information, linguistic description like “around one million” is often used. To deal with the vagueness of human thought, Zadeh [18] first introduced the fuzzy set theory, which was oriented to the rationality of uncertainty due to imprecision or vagueness. In particular, Bellman and Zadeh [2] state that

To deal quantitatively with imprecision, we usually employ the concepts and techniques of probability theory and, more particularly, the tools provided by decision theory, control theory and information theory. In so doing, we are tacitly accepting the premise that imprecision — whatever its nature — can be equated with randomness. This, in our view, is a questionable assumption. Specifically, our contention is that there is a need for differentiation between randomness and fuzziness, with the latter being a major source of imprecision in many decision processes. By fuzziness, we mean a type of imprecision which is associated with fuzzy sets, that is, classes in which there is no sharp transition from membership to nonmembership.

A major contribution of fuzzy set theory is its capability of representing vague knowledge; the theory also allows mathematical operators and programming to apply to the fuzzy domain (Zadeh [19], Dubois and Prade [9], Zimmermann, Zadeh and Gaines [20]). Buckley [4] extends Dubois and Prade’s work by developing fuzzy analogues of the elementary compound interest problems in the mathematics of finance. Ward [15,16] develops fuzzy present worth analysis by introducing trapezoidal cash flow amounts. The result of the present worth formula is in complex nonlinear representations which require tedious computational effort. To make the fuzzy analysis more practical in engineering economic studies, we will develop an approximate form of the complex fuzzy present worth formula for a fundamental economic decision problem — the project analysis using present worth criterion in which cash flow information is modeled as a special form of fuzzy number — the triangular fuzzy number.


When dealing with uncertainty, decision makers may be provided with information characterized by vague language such as: high risk, low profit, a great amount of investment. By using such vague language, people are usually attempting to quantify uncertain events or objects. Developed by Zadeh[18], the fuzzy set theory is primarily concerned with quantifying the vagueness in human thoughts and perceptions. The transition from vagueness to quantification is performed by applying fuzzy set theory as depicted in Figure 1. (Figure 1 omitted)

To deal with the description about vagueness of an object, Zadeh proposed the membership function which associates with each object a grade of membership belonging to the interval [0,1].

Thus, a fuzzy set is designated as: (set omitted), mu sub A (x) epsilon [0,1] where mu sub A (X) is the degree of membership, ranging from 0 to 1, of a vague predicate, A, over the universe of objects, X. X is a space set which can be real numbers, natural numbers or integers. The term “predicate” is related to the vague description of event A.

More specifically, the more the object fits the vague predicate, the larger the degree of membership of the object. If an object has a degree of membership equal to 1, this reflects a complete fitness between the object and the vague predicate. Whereas, if the degree of membership of an object is 0, then the object does not belong to the predicate.

The membership function may be viewed as an opinion poll of human thought. For example, when a group of financial management people are asked of their opinion about the fuzzy predicate, “What is a high annual interest rate?,” everyone would agree that 20% is a high annual interest rate, while no one would say 3% is. As the interest rate gradually increases from 3% to 20%, more and more financial management people probably would agree that the rate is high. The membership can thus be explained as the percentage of financial management people who agree that a given interest rate is “high annual interest rate.”

The membership function can also be viewed as an expert’s opinion. We use the term “expert” because an expert usually holds some required knowledge about relative problems while a lay person may not. For example, when a financial management people is asked “What is a high annual interest rate?,” the possibility of 20% being “high annual interest rate” would be higher than that of 3%, 5%, or 7%. Thus, the membership can be explained as the possibility of an interest fate being considered as a “high annual interest rate.” A rational mapping from interest rate to its degree of membership about the fuzzy set “high annual interest rate” is depicted in Figure 2. (Figure 2 omitted) This membership function looks like a typical cumulative probability function; however, in terms of the Y axis value, the values of membership function represent the possibility of a fuzzy event, while the values of cumulative probability function represent the accumulative probability of a statistical event.


A fuzzy number is a normal and convex fuzzy set with membership function mu(x) which both satisfies

normality: mu sub A (x) = 1, for at least one x epsilon R


convexity: mu sub A (x’) >/= mu sub A (x sub 1 ) mu sub A (x sub 2 )

where mu sub A (x) epsilon [0, 1] and (set omitted).

The normality implies that there is at least one x that strictly matches the predicate of fuzzy number A. The convexity indicates the convex shape of fuzzy set. To illustrate the fuzzy number, three fuzzy sets are presented in Figure 3 where only A is a fuzzy number which satisfies both normality and convexity. (Figure 3 omitted) B does not satisfy the normal condition because no object strictly matches the predicate of B, e.g., there is no object in B that has a membership function value equal to 1. C is not a convex fuzzy set, and thus is not considered as a fuzzy number.


Triangular fuzzy number (TFN) is a special type of fuzzy number. To perform a vague predicate, the triangular fuzzy number with three parameters, each representing the linguistic variable associated with a degree of membership of either 0 or 1, is shown to be very convenient and easily implemented. A triangular fuzzy number is designated as: P = (a, b, c). It is graphically depicted in Figure 4. (Figure 4 omitted) In terms of the predicate about the TFN itself, the parameters a, b, and c respectively denote the smallest possible value, the most promising value, and the largest possible value that describe a fuzzy event. Note that these parameters are analogical to the L, M, and H in triangular probability distribution, but the parameters in a triangular probability distribution represent the values associated with the probabilistic occurrence of an event, while the parameters in a TFN represent the values associated with the possibility, in human thinking, of an event.

Each TFN has linear representations on its left and right side such that its membership function can be defined as:

(Equation 1a)

mu sub p (x) = 0, x

(Equation 1b)

=(x – a)/(b – a) a

(Equation 1c)

=(c – x)/(c – b) b

(Equation 1d)

= o, x >c.

For each value of x increasing from a to b, its corresponding degree of membership linearly increases from 0 to 1. While x increases from b to c, its corresponding degree of membership linearly decreases from 1 to 0. The membership function is a mapping from any given x to its corresponding degree of membership.

Now consider an inverse mapping from any given degree of membership to its corresponding x values, one on the left side of the fuzzy number, another on the right side of the fuzzy number.

Thus a fuzzy number can always be defined by its corresponding left and right representation of each degree of membership. Now the TFN can be designated by another format such as:

(Equation 2 omitted)

where, instead of writing mu sub P (x), we write alpha as the given degree of membership which maps to its corresponding x values on each side of P. P sup 1(alpha) and P sup r(alpha) are the x values in a linear representation of, respectively, the left and right sides of P to which each alpha maps.

The TFN is mathematically easy to implement, and more importantly, it represents the rational basis for quantifying the vague knowledge about most decision problems, e.g., the estimate of cost-volume-profit, product revenue, interest rates, and so forth. In the remainder of this paper, we will use TFNs to model cash flows and perform economic decision analysis.


Dubois and Prade [9,10] develop the extended operations which apply the basic arithmetic operations such as addition, substraction, multiplication, and division, to fuzzy numbers. The extended operations are very important because they allow us to manipulate functions or mathematical programming from engineering, sciences, and management in the fuzzy domain.

For the practical applications, we will study the operations on not only positive TFNs but also negative TFNs. In the real world engineering economic problems, any fuzzy receipts such as revenues and salvages can be represented in positive fuzzy numbers, while any disbursements such as investment and operating/maintenance costs can be represented in the form of negative fuzzy numbers.

The principle in Dubois and Prade’s extended operations indicates: the result of the operations on two fuzzy numbers yields a minimum value on be left representation, and a maximum value on the right representation. Following the principle, we will discuss some relative operations to our present worth derivations by using the left and right representations of fuzzy numbers.

Suppose P = [P sup 1(alpha) , P sup r(alpha) ], Q = [Q sup 1(alpha) , Q sup r(alpha) ].


For all possible values of two fuzzy numbers P and Q, e.g., P can be either positive where all the parameters are greater than 0, or negative where all the parameters are smaller than 0, we have

(Equation 3)

P (+)e =[P sup 1(alpha) + Q sup 1(alpha) , P sup r(alpha) + Q sup r(alpha) ]

Obviously it is the addition of individual left representations that yields the minimum value on left representation of the result. The addition of individual right representations yields the maximum value of right representation of the result.

To illustrate the addition on two TFNs, an example is given as:

P = (1, 2, 4) = [1 + alpha, 4 – 2 alpha]

Q = (2, 5 7) = [2 + 3 alpha, 7 – 2 alpha]


P (+) Q = [(1 + alpha + (2 + 3 alpha), (4 – 2 alpha) + (7 – 2 alpha)] = [3 + 4 alpha, 11 – 4 alpha].

The result of the addition operation is also a TFN with linear representations. We can calculate its three parameters by letting alpha be equal to 0 and 1 respectively. The smallest possible value of the result is the value of its left representation at alpha = 0. The most promising value is the value of both representations at alpha = 1. The largest possible value is the value of the right representation at alpha = 0. Hence we can represent the result in the form of TFN as P (+) Q = (3, 5, 11).

Using this result, we have the summation operation by adding all the individual left and right representations respectively as

(Equation 4)

Sigma P = [Sigma P sup 1(alpha) , Sigma P sup r(alpha) .


If P and Q are both positive, we have

(Equation 5)

P (*) Q = [P sup 1(alpha) * Q sup 1(alpha) , P sup r(alpha) * Q sup r(alpha)

To illustrate the multiplication operator, we use the values of P and Q from previous example, thus,

P (*) Q = [(1 + alpha)(2 + 3 alpha), (4 – 2 alpha)(7 – 2 alpha)] = [3 alpha sup 2 + 5 alpha + 2, 4 alpha sup 2 – 22 alpha + 28].

The results of addition and multiplication of the above examples are depicted in Figure 5. (Figure 5 omitted) Unlike the addition operation which results in a TFN with linear representations, the results of the product operation are in nonlinear representations.

Using the result, we can also have the product operation for positive fuzzy numbers as

(Equation 6)

Pi P = [Pi P sup 1(alpha) , Pi P sup r(alpha) ].


Two special cases are discussed in this operation:

1) If P and Q are both positive, we have

(Equation 7a)

P (/) Q = [P sup 1(alpha) / Q sup r(alpha) , P sup r(alpha) / Q sup 1(alpha) ].

2) If P is negative and Q is positive, we have

(Equation 7b)

P (/) Q = [P sup 1(alpha) / Q sup 1(alpha) , P sup r(alpha) / Q sup r(alpha) ].

Note that the result of 1) is a positive fuzzy number, whereas the result of 2) is a negative fuzzy number.


When the degree of uncertainty about a decision problem increases, the capability of the precise description of that model decreases. In order to provide an adequate approach to describe such uncertainty, Zadeh [19] proposed the “linguistic approach.” The linguistic variable is introduced to represent the value of natural or artificial languages such as: “Very,” “Little,” “Around.” Coupled with fuzzy set theory, the value of a linguistic variable can be quantified and extended to mathematical operations. This makes it possible for the decision model under uncertainty to be considered rational.

In quantifying the vagueness of the uncertainty about cash flow information, the expert’s reasoning usually results in linguistic values such as: the required investment amount, P, is ‘more or less’ equal to $1 million. Such a predicate can be conveniently represented by triangular fuzzy numbers with membership function mu sub p (x) depicted in Figure 6. (Figure 6 omitted) In Figure 6, two triangular fuzzy numbers, P and P’, both represent the predicate with the most promising amount of $1 million. P, with tighter range, is more confident, or certain, than P’.

Similarly, the periodic cash flow at time t can be represented by a triangular fuzzy number such as P sub t = (p sub t0 , p sub t1 , p sub t2 ). We can also characterize the TFN by its left and right representations at each degree of membership, alpha, as

(Equation 8 omitted)

P can be either disbursement (p sub t 0).

In the same manner, the discount rate during time t is also modeled as a TFN.

(Equation omitted)

Also, at each or, the TFN can be denoted by its left and right representations such as

(Equation 9 omitted)

Thus, a general formulation of present worth of a fuzzy cash flow is

(Equation 10 omitted)

By definition, the present worth is the summation of the equivalent amount at present time of each periodic cash flow. Under several assumptions:

1) all Ps and Rs are end-of-year (end-period) fuzzy predicates.

2) n is the project evaluation period in multiples of years (months, weeks, and so forth.)

3) P can be either positive or negative TFNs.

4) all Rs are positive TFNs except R sub 0 = (0, 0, 0).

Some fuzzy discounted cash flow analyses were examined by Kaufmann and Gupta [14], and Ward [16,17]. In their studies, either the periodic cash flow or the discount rates are specified as fuzzy numbers. In real-world applications, it is more realistic to deal with problems of both periodic cash flow and discount rate as being fuzzy numbers. Buckley [5] considered both categories — periodic cash flow and discount rates — as fuzzy numbers simultaneously, and developed various economic equivalence formulas in elementary financial calculations. Here we develop the present worth formula of this fuzzy problem based on extended operations (Dubois and Prade [10])and Buckley [5]. Our derivation is actually constructed by two procedures: the derivation of equivalent amount at present time of each periodic cash flow, and the summation of these equivalent amounts which result in the exact present worth formula.

1. The equivalent amount at present time of each periodic cash flow is written as

(Equation 11 omitted)

First consider the product of (1 + R sub t’ ) using the property of multiplication of fuzzy numbers in Eq. (6). Since R >/= 0, which implies (1 + R sub t’ ) is positive, we have

(Equation 12 omitted)

Note that the result is also a positive fuzzy number. Next, using the division property of fuzzy numbers, we consider two possible amounts of P sub t

a) when P sub t is positive, using Eq. (7a), we have

(Equation 13 omitted)

b) when P sub t is negative, using Eq. (7b), we have

(Equation 14 omitted)

Using the maximum and minimum symbols, we can denote the result in single format as

(Equation 15 omitted)

When P sub t is positive, max{P sup 1(alpha) , 0} is equal to P sup 1(alpha) , and min {P sup 1(alpha) , 0} is equal to 0 which becomes redundant. The same logic happens on P sup r(alpha) .

2. Using the property of summation of TFNs in Eq. (4), the exact present worth formula can be derived by the summation of all equivalent amounts at present time. We have

(Equation 16 omitted)

Using the same logic of present worth derivation, the future worth formula of the fuzzy project can be derived. (See Appendix A)


The result of the present worth formula is in complex nonlinear representations which require tedious computational effort. For the reason of simplicity, Kaufmann and Gupta [14] studied the approximation of operations on fuzzy numbers. Similarly, we can use a TFN as an approximate form of the complex present worth formula in Eq. (16). By letting alpha equal to 0 and 1 on the left and right representations of the present worth formula, we can calculate the three parameters of the TFN respectively. We discuss these in the following.

1. when alpha = 0,

P sup 1(alpha) sub t = p sub t0 , P sup r(alpha) sub t = p sub t2

R sup 1(alpha) = r sub t0 , P sup r(alpha) = r sub t2

2. when alpha = 1,

P sup 1(alpha) sub t = P sup r(alpha) sub t = p sub t1 ,

R sup 1(alpha) sub t = R sup r(alpha) sub t = r sub t1 ,

Substituting these to the present worth formula, the approximate form of the present worth formula, PWA, can be derived as

(Equation 17 omitted)

Note again, PWA is represented using its three parameters, and it is easier to implement because they are in linear representations. In Example 1, we will calculate the exact present worth and the approximate present worth of a project modeled as TFNs.


Consider a three-year project with the estimated cash flow and interest rates specified by TFNs as

P sub 0 = (-110, -100, -90) and R sub 1 = (6%, 7%, 8%)

P sub 1 = (-80, -60, -40) — R sub 2 = (6%, 7%, 8%)

P sub 2 = (110, 130, 140) — R sub 3 = (6%, 8%, 10%)

P sub 3 = (100, 110, 130)

Then we can denote this information by the left and right representations as

P sub 0 = [-110 + 10 alpha, -90 – 10 alpha] and R sub 1 = [1.06 + .01 alpha, 1.08 – .01 alpha]

P sub 1 = [-80 + 20 alpha -40-20 alpha] –R sub 2 = [1/06 + .01 alpha, 1.09 – 0.2 alpha]

P sub 2 = [110 + 20 alpha 140 – 10 alpha] –R sub 3 = [1.06 + .02 alpha, 1.10 – .02 alpha]

P sub 3 = [100 + 10 alpha, 130-20 alpha]

Using the exact present worth formula in Eq. (16), we can calculate the present worth as

PW = [PW sup t(alpha) , PW sup r(alpha) ]


PW sup t(alpha) = -110 + 10 alpha + (-80 + 20 alpha) / (1.06 + .01 alpha) + (110 + 20 alpha) / (1.08 -.01 alpha)(1.09 -.02 alpha) + (100 + 10 alpha) / (1.08 – .01 alpha)(1.1 – .02 alpha)

PW sup t(alpha) = -90 -10 alpha +(40 – 20 alpha) / (1.08 -.01 alpha) + (140 – 10 alpha) / (1.06+.01 alpha)(1.06+.01 alpha) + (130 – 20 alpha) / (1.06 +.01 alpha)(1.06 +.01 alpha)(1.06 + .02 alpha)

Using the PWA formula in Eq. (17), we can also calculate the approximate form of PW as

PWA = (PWA sub 0 , PWA sub 1 , PWA sub 2 )


PWA sub 0 = -110 -80 / (1.08) + 110 / (1.08)(1.09) + 100 / (1.08)(1.09)(1.1) = -14.8048

PWA sub 1 = -100 – 60 / (1.07) +130 / (1.07)(1.07) + 110 / (1.07)(1.07)(1.08) = 46.4336

PWA sub 2 = -90 -40 / (1.08) + 140 / (1.06)(1.06) + 130 / (1.06)(1.06)(1.06) = 106.713

Since the result is a TFN, we can also designate PWA by its left and right linear representations as

PWA =[PWA sup 1(alpha) ) PWA sup r(alpha) ]

=[ PWA sub 0 + (PWA sub 1 – PWA sub 0 ) alpha, PWA sub 2 + (PWA sub 1 – PWA sub 2 ) alpha]

= [-14.8048 + 61.2384, 106.713 -60.2794 alpha]

PW and PWA are graphically depicted in Figure 7. (Figure 7 omitted) The most promising present worth is 46.4396. The largest possible present worth amount is 106.7. The smallest possible present worth amount is -14.8. The area with negative present worth indicates that there is a slight possibility that the project is exposed to not being justified as an acceptable project. The possibility can be computed as 24% by interpolation method for mu(PW = 0).


The divergence between exact and approximate form of fuzzy operations is studied by Kaufmann and Gupta [14]. Due to the fact that PWA is easier to manipulate than PW, we want to examine the deviation between PW and PWA. If the deviation is not significant, we can use the PWA as a surrogate of PW in the project analysis. To study the deviation between the exact PW and its approximate form, PWA, at each degree of membership, alpha, we can calculate the deviations on their left and right presentations as d sub 1 and d sub r respectively.

(Equation 18a)

d sub 1 = PWA sub 0 + (PWA sub 1 – PWA sub 0 )alpha -PW sup t(alpha)

(Equation 18b)

d sub r = PWA sub 2 + (PWA sub 1 – PWA sub 2 )alpha -PW sup r(alpha)

The deviations of PW and PWA are depicted in Figure 8. (Equation 8 omitted) We can compute the maximum deviation as a measurement of the fitness between PW and PWA. Since PW sup t(alpha) and PW sup r(alpha) are in complex nonlinear representations, it is difficult to find out the maximum deviation using the derivative method; instead, we can find out the maximum deviation by very small increments of alpha, e.g., 0.001.

Using the result from the numerical example, the maximum deviation on the left PW representation, d sub 1 *, can be found at alpha = 0.508 with a value of 0.2111, and the maximum deviation on the right PW representation, d sub r * =, is located at alpha = 0.499 with a value of 0.1833.

We further define the left and right maximum percentage of deviations, d sub 1(%) * and d sub r(%) *, as

(Equation 19a)

d sub 1(%) * = 100% * d sub 1 * / (PWA sub 1 – PWA sub 0 )

(Equation 19b)

d sub r(%) * = 100% * d sub r * (PWA sub 2 – PWA sub 1 )

which represent the measurement in percentage of its deviation over the width of left and right representation of PWA. This measurement gives a clearer idea of how PW and PWA deviate. If the percentage of deviation is not significant, e.g., within 1%, then PWA can serve as a surrogate of PW.

From the result of the example, we can calculate d sub 1(%) * = 0.345% and d sub r(%) * = 0.304%. In Figure 7, we also observe that there is no significant deviation between PW and PWA because of the negligibly small amounts of d sub 1(%) * and d sub r(%) *. This implies that PWA is a “reliable” approximate form of PW. In this situation, using PWA’s obviously has the advantage of saving a lot of mathematical effort in the fuzzy present worth analysis.

Since the present worth formula consists of two factors, the periodic cash flows and the discount rates, the deviation between PW and PWA may be relative to the range of these two factors. To examine which factor affects the deviation, we use a computer program to implement the deviations of various simulated problems. The simulation is performed under different ranges of periodic cash flows and discount rates. From the results of the simulation, we can conclude the deviation between PW and PWA that:

1) The range of periodic cash flows does not significantly affect the deviation of PW and PWA.

2) The range of discount rates significantly affects the deviation of PW and PWA; the deviations increase significantly as the range of discount rates becomes larger.

3) When the confident width of discount rate is within an absolute range of +/- 4%, the maximum deviation percentage is relatively small, which indicates PWA can be a surrogate of PW.

4) In the real world applications, when the discount rates are usually estimated within the width of +/- 4%, we can use PWA in project analysis, thus expending less computing effort.

This conclusion leads to our use of PWA in the remainder of this paper for the fuzzy present worth analysis and capital budgeting analysis.


One fundamental economic decision problem is the project selection among a set of mutually exclusive projects. We focus on the problem of evaluating projects under uncertainty in which the cash flow information is modeled as TFNs. Taking the results of the PWA formula, the problem is equivalent to ranking a set of TFNs and finding the most dominant one.

There are a number of methods that are devised to rank TFNs as partially summarized in Appendix B. Most methods are tedious in graphic manipulation requiring complex mathematical calculation. Hence we propose a method, the weighted method, that is easily implemented and effective in comparing TFNs.


The weighted method compares the projects’ present worth by assigning relative weights to criteria that determine the preference of projects.

Since the present worth of a project is described by three parameters — the lowest possible present worth, the most promising present worth, and the highest possible present worth — the comparison of projects is fundamentally dependent on the basis of these parameters. Hence, as analogous to the dominance of triangular probability distribution, the average of these three parameters serves as a criterion of comparison. Moreover, with the inherit nature of the largest possible estimate of present worth, the most promising present worth is also considered as a criterion. Thus, the evaluation of the present worth in the form of a TFN (a, b, c) is determined by assigning relative weights to each criterion.

(Equation 20 omitted)

w sub 1 (a + b + c / 3) + w sub 2 b

where w sub 1 and w sub 2 represent the relative weights of each criterion. By assigning w sub 1 equal to 1, Eq. (20) can be reduced to

(Equation 21)

(a + b + c / 3) + wb

The value of w should be determined by the nature and the magnitude of the “most promising” present worth. If the magnitude of the most promising present worth is important, a larger number of w, such as 0.3, is recommended. Otherwise, a smaller weight, such as 0.1, is recommended.

In the Example 2, we will employ project dominance method to determine the preference of a set of projects which provides the basis for further economic decision analysis.


Suppose a set of four mutually exclusive alternatives are modeled as triangular fuzzy numbers, and their present worth are calculated by the PWA formula as in Eq. (17) which result in the following TFNs:

A sub 1 = (2350, 2725, 2850) A sub 2 = (2250, 2650, 2800) A sub 3 = (2325, 2600, 2900) A sub 4 = (2200, 2425, 2725)

The present worth are depicted in Figure 9. (Figure 9 omitted) Graphically it is difficult to interpret the preference of these four projects, especially of A sub 2 and A sub 3 . Along with the weighted method, we had selected several conventional dominance methods (see Appendix B) in calculating each project’s relative index value, and obtain the preference of the projects as in Table 1.


In general, A sub 1 is the superior project except, in Chang’s Method where A sub 3 has the largest mathematical expected value due to large width. A sub 4 is not preferred by any of the methods. The preference between A sub 2 and A sub 3 differs because of the various aspects emphasized in the methods. Most methods suggest A sub 3 >/= A sub 2 , whereas only Dubois and Prade’s ‘Possibility of Dominance’ method indicates A sub 2

Summarizing it all, we shall propose that A sub 1 is the preferred project, and the dominance sequence among projects is A sub 1 > A sub 3 > A sub 2 > A sub 4 .

As indicated by Bortolan and Degani [3], there exists no dominance rule that has been approved to compare any and all cases of TFNs. Deciding which rule to use depends on the nature of the problem set and the prospect of individual dominance methods. However, in Example 2, the weighted method has shown to be easily implemented and effective in comparing fuzzy projects.


Economic decision problems always involve the uncertainty of cash flow modeling. For decision making under uncertainty, management relies on experts’ knowledge in modeling cash flows. The cash flow modeling using triangular fuzzy numbers allows experts’ linguistic predicate about engineering proposals to be considered as rational.

In this paper, we propose an exact present worth formulation of cash flows that are modeled as triangular fuzzy numbers. The derivation of the present worth formula involves complex arithmetic operations — addition, summation, multiplication, production, division — on positive and negative triangular fuzzy numbers. With much less computational effort, an approximate form of present worth is also derived In real world applications where the future estimated discount rates do not have large ranges, usually within an absolute range of +/- 4%, the approximate present worth is shown to be reliable as a surrogate of exact present worth.

The present worth formula provides a general frame of fuzzy project analysis. Taking the result of the present worth formula, the fuzzy project evaluation can be achieved by applying some dominance rules on TFNs. Several dominance methods are selected and discussed. Most methods are tedious in graphic manipulation requiring complex mathematical calculation. Hence, we suggest an easily implemented project dominance method. In Example 2, a set of mutually exclusive fuzzy projects is evaluated and the results are implemented by applying selected dominance methods.


Use of future worth as the criterion to evaluate fuzzy projects is justifiable in economic perception. Following the same logic of present worth derivation, the future worth formula can be derived by applying the summation, multiplication and product operations to fuzzy numbers. The result of the exact future worth, FW, is

(Equation 22 omitted)

The approximate future worth formula, FWA, can be derived as

(Equation 22 omitted)

Since they are based on same index, e.g., discount rates and periodic cash flows, deviation between fuzzy projects’ FW and FWA have similar behavior as those between PW and PWA.


There are a number of methods that are devised to rank mutually exclusive projects including both Buckley [5] and Ward [16]. However, to focus on ranking TFNs, we have selected the following methods to discuss and implement in the case study.


Chang [6] defines the mathematical expectation of a TFN (a, b, c) as

(Equation 24)

(c – a)(a + b + c) / 6

With this method, the preference of a set of projects is determined by comparing their corresponding PWA’s mathematical expectations. The project with the largest mathematical expectation is considered as the most dominant project.


Kaufmann and Gupta [14] suggest three criteria for ranking TFNs with parameters ( a, b, c). The dominance sequence is determined according to priority of:

1) comparing the ordinary number

(Equation 25)

(a +2b + c) / 4,

2) comparing the mode, (the corresponding most promising value), b, of each TFN,

3) comparing the range, c – a, of each TFN.

The preference of projects is determined by the amount of their ordinary numbers. The project with the larger ordinary number is preferred. If the ordinary numbers are equal, the project with the larger corresponding most promising value is preferred. If projects have the same ordinary number and most promising value, the project with the larger range is preferred.


Jain [12] proposes a dominance rule using maximizing set, M, of n TFNs. The maximizing set, M, is a linear representation which connects the point from the largest possible value, rmax, in the set with degree of membership equal 1, to the point of smallest possible value in the set with degree of membership equal 0.

The preference of projects is determined by the height of the intersections of M and the right representation of each project. This method is graphically depicted in Figure 10 where A sub 1 > A sub 2 > A sub 3 .


Dubois and Prade [9] suggested the possibility of the dominance of project A sub i over all other projects as

(Equation 26 omitted)

The preference of the projects is determined by the intersection (height of degree of membership) of each project’s right representation and the maximum left representation in the set. The maximum left representation is the left representation with largest x values corresponding to each degree of membership. Figure 11 represents the dominance sequence of A sub 1 > A sub 2 > A sub 3 using Dubois and Prade’s PD method. The maximum left representation is the left representation of A sub 1 . The preference is thus determined by the height of intersections where PD(A1) > PD(A2) > PD(A3).

In an ordinary decision problem domain, the above methods come out with consistent dominance sequence. However, in some extreme cases, the results may not be consistent. More detailed comparisons of dominance rules for fuzzy numbers can be found in Bortolan and Degani’s work [3] as well as Buckley’s [4].


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CHUI-YU-CHIU is an Associate Professor of Industrial Engineering and Management at National Taipei Institute of Technology, Taipei, Taiwan. His area of teaching and research interests include fuzzy set theory, artificial intelligence, and engineering economics. He received a BS degree from Tunghai University, both M.S. and Ph.D. degrees in industrial engineering from Auburn University.

CHAN S. PARK is an Alumni Professor of Industrial Engineering at Auburn University. He is a graduate of Hanyang University (BS), Purdue University (MSIE), and Georgia Institute of Technology (PH.D.). He is currently Manuscripts Editor for The Engineering Economist. Dr. Park has written numerous technical papers in the areas of economic decision analysis and computer applications. He authored two books in the area of engineering economics Advanced Engineering Economics from John Wiley (1990) and Contemporary Engineering Economics from Addision Wesley (1993). He also received many research awards on his manufacturing economics work (Alfred V. Bodine/SME Award from SME, E.L. Grant Award from the ASEE, and Outstanding Publications Award from IIE).

Copyright Institute of Industrial Engineers, Inc. Winter 1994

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