Fuzzy Inference Systems for IT Project Portfolio Management

Overview of IT Project Portfolio Management

Despite the recognized need, the majority of organizations in the United States do not possess the information required to effectively manage their strategic initiatives because the requisite data is stored away in organizational silos (Project and portfolio management, 2014). The need for effective project portfolio management has become even more pronounced as the number and complexity of information technology projects has increased in recent years (Drake & Byrd, 2006). The corresponding growth in the range and scope of IT projects has made the effective management of project portfolios even more complex (Drake & Byrd, 2006). According to Drake and Byrd, "Translating strategic goals into successful projects would help ensure that IT investments resulted in increased business performance. Research into business-IT alignment answered some of the questions about how to translate IT investments in business to business performance" (2006, p. 1).

Today, practitioners are increasingly implementing organizational structures that are specifically designed to support the strategic alignment of IT projects with organizational goals, prioritizing them as the competitive environment changes (Drake & Byrd, 2006). In this regard, Drake and Byrd advise that, "This structure, IT project portfolio management, bridges the gap between project management and strategic management" (2006, p. 2). The function of IT project portfolio management is to evaluate strategic goals and organizational core competencies in order to formulate appropriate information systems for the enterprise — systems capable of communicating and storing information efficiently and effectively (Drake & Byrd, 2006). In the past, strategic information system planning was used for this purpose, but this approach failed to provide the robust attributes that are possible with efficient and effective project portfolio management techniques (Drake & Byrd, 2006).

There are two functions that comprise IT project portfolio management: (a) the planning of new projects and migration to new systems, and (b) the reassessment of ongoing projects (Drake & Byrd, 2006). The first function — the planning phase — can still be initiated using strategic information system planning, which is "the process of identifying which computer-based applications will assist an organization in executing its business plans and realizing its business goals" (Lederer & Sethi, 1988, p. 36). Following the planning phase, a portfolio of projects should be developed that aligns organizational information requirements with strategic objectives (Drake & Byrd, 2006).

The second function of IT portfolio management — the reassessment of ongoing projects and systems to determine whether they are achieving their objectives within established time or budgetary constraints — requires comprehensive examination at the portfolio level (Drake & Byrd, 2006). In this regard, Drake and Byrd emphasize that, "As the size and complexity of IT departments increase, so does the size and complexity of the projects they undertake. It takes a portfolio-level analysis to determine the progress and relevance of these projects" (2006, p. 2).

In addition, a risk assessment is required in order to better understand how project portfolio management should proceed (Drake & Byrd, 2006). According to Drake and Byrd, "Risk is the measure of probability and magnitude of an unwanted event happening. In risk management, identification of risks helps managers prevent and/or mitigate the effects of those risks" (2006, p. 2). Practitioners must identify which unwanted events can adversely affect the outcome of portfolio projects. This step is essential in order to prevent or mitigate the adverse impact of risks and to increase portfolio health, which is defined as the extent to which the projects in the portfolio are satisfying business requirements (Drake & Byrd, 2006).

Over the years, researchers have conducted a wide range of studies to identify and quantify project risk factors and portfolio risk (Drake & Byrd, 2006). For instance, McFarlan (1981) examined various risk factors associated with identifying a risk profile of corporations. Likewise, Shoval and Giladi (1996) identified a number of portfolio-level risks concerning the order of implementation for information service projects. Similarly, Jiang and Klein (1999) examined the selection criteria used for various information service projects that were deemed important to the executive leadership team when confronted with a new project portfolio.

The purpose of a study by Drake and Byrd (2006) was to explore the relevant literature, develop a list of important risk factors faced by IT project portfolios, and prioritize them along an emergent continuum. Based on this list, Drake and Byrd developed a framework for identifying, measuring, and mitigating risks at the portfolio level. This approach is consistent with previous research concerning financial portfolio management for a number of reasons. For instance, a number of researchers have determined that projects are investments organizations make for their future in the same way that stocks are an investment in the future (Benko & McFarlan, 2003). This financial element of portfolio management is based in part on the Modern Portfolio Theory first propounded by Markowitz (1959), which includes among its key principles the following:

An optimal portfolio generates the highest possible return for a given level of risk. Expected risk has two sources: (1) investment risk — the risk of the stock itself (unsystematic), and (2) relationship risk — the risk derived from how a stock relates to the other stocks in a portfolio (systematic).

Generally speaking, the risk of an IT portfolio can be expected to resemble that of a financial portfolio because there is risk involved in individual projects and risk in how projects interrelate to each other; the term "relationship risk" is used to refer to risk that can impact the entire portfolio (Drake & Byrd, 2006, p. 2). According to Drake and Byrd, "These risks cannot be diversified away because the entire portfolio is affected by outside influences" (2006, p. 2). The analysis of relationship risk in project portfolios is typically more complicated than in financial portfolios, however, because projects can directly affect the outcomes of other projects in the portfolio (Drake & Byrd, 2006).

The difference in complexity between financial and project portfolios is especially salient in cases where projects rely on the completion of other projects before they can be implemented — for example, upgrading operating systems in order to support a new application (Drake & Byrd, 2006). In these cases, relationship risk operates in a highly unsystematic fashion that complicates ongoing evaluation because, unlike financial portfolios, each project may affect all of the others in a project portfolio (Drake & Byrd, 2006). Taken together, it is reasonable to suggest that there are three general areas of risk to consider when formulating the optimal project portfolio:

1. The risk of the projects themselves; 2. Risk from the relationships between projects; 3. Risk to the portfolio as a whole (Drake & Byrd, 2006, p. 2).

Beyond the foregoing, it is also imperative that all project objectives in the portfolio reflect an organization's high-level strategies (Project portfolio management, 2009). According to the editors of Research-Technology Management (2009), "Otherwise, the portfolio as a whole will suffer misalignment, thereby wasting time, energy, and resources" (Project portfolio management, p. 64). The next step in project portfolio management is implementation within the project portfolio, aligning R&D strategies with organizational strategies (Project portfolio management, 2009).

There are a number of tools and techniques available to facilitate this process. For instance, "Roadmapping is considered a powerful way to develop common language between different functions and for exploring possible future scenarios. Transferring the portfolio projects into the production and marketing environment is a further challenge" (Project portfolio management, 2009, p. 64). This is an important step because, as noted above, unlike financial portfolios, all projects in a portfolio stand to affect all other projects. In this regard, the editors of Research-Technology Management (2009) conclude that, "Roadmapping and knowledge management are considered to be effective tools in support of portfolio management, fostering organizational learning and the exploration and understanding of future scenarios. Roadmaps embed R&D projects in business projects instead of looking at 'independent' R&D projects" (Project portfolio management, 2009, p. 64).

The need for effective and efficient project portfolio management is also cited by Dinsmore and Cabanis-Brewin (2011), who emphasize that, "Executives without effective project portfolio management suffer from cross-functional resource conflicts with continual top management refereeing, poor or anemic organizational performance, and projects where the norm is to deliver late, over budget, or not within scope" (p. 315). Even when executives make the effort to plan their projects carefully, they may overlook the most important ones in their attempt to organize all of the enterprise's projects in a cohesive fashion. In this regard, Dinsmore and Cabanis-Brewin point out that, "Most executives are aware of the need for drastic change in multiproject management practices, but many place the emphasis in the wrong place. Unfortunately, a great deal of such investment is misdirected into multiyear efforts to implement software tools and time sheets before dealing with the highest leverage points" (2011, p. 315).

Foundations of Fuzzy Logic and Fuzzy Sets

There are three discrete roles that organizations must formally define in order to develop an effective project portfolio management system:

1. Governance: This executive role involves decision making, usually conducted by top management teams. In the most effective implementations, this role includes "C"-level executives (CFO, CEO, COO, CIO) who meet monthly to make decisions about which projects to approve or reject; when to activate projects; how many projects to activate and which to deactivate; due dates for projects; criteria for project proposals; priorities; resource allocation, including capital expenditure, personnel, and operating expense budgets; project reviews with approval to proceed to the next stage, to terminate a project, or to approve or reject project improvement plans; and investment in project management methodology and tools.

2. Management: Relative to project portfolio management, management's role is to ensure that the project management system is "in control." According to the late quality expert W. Edwards Deming, a system is in control when the goals of the system can be predictably met more than 95 percent of the time. Every project has three distinct goals — to be delivered on time, on budget, and within scope, according to original commitments. This role includes providing the project management processes for planning and execution to deliver projects according to their goals, usually through a Project Management Office or similar organization.

3. Project Portfolio Management: The person or persons undertaking this role provide information and recommendations to the governance group for improved return on investment and monitor the execution of projects. There is usually a close relationship between the individual responsible for strategic planning and the portfolio manager. While strategic planners identify the ideas necessary to meet organizational goals, the portfolio manager ensures that corresponding programs and projects are sufficient to accomplish those ideas, maps and tracks project execution against strategies, raises a flag when there is danger of missing a goal, and informs strategic planning when a strategy is not practical relative to available project resources (Dinsmore & Cabanis-Brewin, 2011, p. 315).

As noted in the introductory chapter, this study proposed a solution to accurately estimate completion time and cost for project tasks. The solution is based on a fuzzy inference system grounded in fuzzy logic (Guan & Zurada, 2008). According to Guan and Zurada, "Fuzzy logic has been widely studied in the relatively 'hard sciences' such as different fields of engineering with varying degrees of success" (p. 395). As an extension of Boolean logic, fuzzy logic deals with the vagaries involved in partial truths, indicating the degree to which a proposition is true or not (Sabeghi et al., 2006). According to Sabeghi and his associates, "Whereas classical logic holds that everything can be expressed in binary terms (0 or 1, black or white, yes or no), fuzzy logic replaces Boolean truth values with the degree of truth" (2006, p. 2). The degree-of-truth measure is frequently used to describe the imprecision involved in various modes of reasoning that serve as a basis for human decision-making in uncertain and imprecise environments (Sabeghi et al., 2006).

The general objective of fuzzy logic is to provide a map of an input space to an output space, with the main mechanism for accomplishing this objective being a series of IF-THEN statements, which are termed "rules" (Foundations of fuzzy logic, 2014, para. 2). According to Wang (1996), "The membership function of a fuzzy set corresponds to the indicator function of the classical sets. It can be expressed in the form of a curve that defines how each point in the input space is mapped to a membership value or a degree of truth between 0 and 1" (p. 37). Although the most common shape of a membership function is triangular, trapezoidal and bell curves have also been used (Wang, 1996). The input space is referred to in most cases as "the universe of discourse" (Wang, 1996, p. 37).

According to one fuzzy logic vendor, "All rules are evaluated in parallel, and the order of the rules is unimportant. The rules themselves are useful because they refer to variables and the adjectives that describe those variables" (Foundations of fuzzy logic, 2014, para. 2). In other words, fuzzy logic rules are described in linguistic terms so that humans can understand the input space and how the output vector is generated. In this regard, MathWorks advises that, "Before a system can be built that interprets rules, practitioners must define all the terms they plan on using and the adjectives that describe them. In other words, 'To say that the water is hot, you need to define the range that the water's temperature can be expected to vary as well as what we mean by the word hot'" (Foundations of fuzzy logic, 2014, para. 2).

The fuzzy inference method interprets the values in the input vector and, following the established set of rules, assigns quantifiable values to the output vector (Foundations of fuzzy logic, 2014). According to Hamzeh, Fakhaire, and Lucas (2007), "Fuzzy inference is the process of formulating the mapping from a given input set to an output using fuzzy logic. The basic elements of fuzzy logic are linguistic variables, fuzzy sets, and fuzzy rules" (p. 211). As noted above, fuzzy sets and rules employ linguistic terms to facilitate the interpretation of generated results. In this regard, Hamzeh and his associates note that, "The linguistic variables' values are words, specifically adjectives like 'small,' 'little,' 'medium,' 'high,' and so on" (2007, p. 212).

The fuzzy logic process begins with the fuzzy set concept. According to MathWorks, "A fuzzy set is a set without a crisp, clearly defined boundary. It can contain elements with only a partial degree of membership. Any statement can be fuzzy" (Foundations of fuzzy logic, 2014, para. 3). Practitioners who have been confronted with vague and nebulous answers to straightforward questions will welcome the ability of fuzzy inference systems to provide answers that span a continuum. In this regard, MathWorks adds that, "The major advantage that fuzzy reasoning offers is the ability to reply to a yes-no question with a not-quite-yes-or-no answer. Humans do this kind of thing all the time (think how rarely you get a straight answer to a seemingly simple question), but it is a rather new trick for computers" (Foundations of fuzzy logic, 2014).

This "new trick for computers" involved in fuzzy inference systems relies on the categorization of data. According to Hahn and Ramscar (2001), this is a natural human tendency that makes fuzzy inference especially useful. For instance, Hahn and Ramscar advise that, "One of the central concerns of cognitive psychology is the process of categorization (and subsequent classification) by which humans organize and represent their knowledge of the world" (2001, p. 37). In the business world, it is axiomatic that in order to improve something it must first be measured, and in order to measure something it must first be categorized. This natural process is highly congruent with the tenets of fuzzy inference systems. For example, Hahn and Ramscar add that, "Classical category theory views the world as being comprised of natural partitions, and the purpose of categorization is to allocate objects into the appropriate partitions" (2001, p. 37). Pursuant to this perspective, all instances of a category share some type of commonality that sets them apart from other instances in ways adequate for defining the category (Qi & Zhu, 2008). According to Qi and Zhu, "Thus, an instance that possesses all of the defining features belongs to the category with full membership, while an instance that lacks any of the defining features must be excluded from the category completely" (2008, p. 223).

This same process applies to fuzzy sets, which are collections of couples of elements (Hamzeh et al., 2007). Fuzzy sets are used to generalize the concept of classical sets, thereby allowing the sets' elements to enjoy partial memberships (Hamzeh et al., 2007). In this regard, Hamzeh et al. advise that, "The degree to which the generic element 'x' belongs to the fuzzy set A (expressed by the linguistic statement x is A) is characterized by a membership function (MF), fA(x)" (2007, p. 3). According to MathWorks, "A membership function (MF) is a curve that defines how each point in the input space is mapped to a membership value (or degree of membership) between 0 and 1" (Foundations of fuzzy logic, 2014, para. 3).

The ability to assign a degree of membership to the elements of a set is one of the main strengths of fuzzy inference systems. The importance of this degree of membership can be discerned from an exchange between a researcher and a congressman concerning the ability of pilots to continue flying beyond their 60th birthday. During his tenure with the National Institute on Aging, Butler (2005) reports working with Congressman Claude Pepper on age-related issues, including a mandatory retirement age for pilots. During one working session, Butler reports that the congressman became animated when discussing the problems with the Experienced Pilots Act requirement, asking: "Why can't pilots over 60 fly commercial planes?" When Butler reported that older pilots had been prohibited from flying in the past, leaving a lack of relevant data to support extending the age-60 mandatory retirement, the congressman responded: "If a man can fly a plane safely at the age of 59 years and 364 days, why can't he do it one or two days later?" (cited in Butler, 2005, p. 85).

Although age is the issue in this anecdotal account, other human attributes are also frequently used in fuzzy inference systems. For example, MathWorks reports that, "One of the most commonly used examples of a fuzzy set is the set of tall people. In this case, the universe of discourse is all potential heights, say from 3 feet to 9 feet, and the word tall would correspond to a curve that defines the degree to which any person is tall" (2014, para. 4). Just as pilots deemed capable of flying on the day before their 60th birthday were considered incapable the day after, defining tall people according to a finite parameter fails to account for the full continuum of height. In this regard, MathWorks advises that, "If the set of tall people is given the well-defined (crisp) boundary of a classical set, you might say all people taller than 6 feet are officially considered tall; however, such a distinction is clearly absurd" (Foundations of fuzzy logic, 2014, para. 4). Therefore, some gradation in height is required to accurately describe humans along the full continuum. According to MathWorks:

"It may make sense to consider the set of all real numbers greater than 6 because numbers belong on an abstract plane, but when we want to talk about real people, it is unreasonable to call one person short and another one tall when they differ in height by the width of a hair." (Foundations of fuzzy logic, 2014, para. 4)

Because this type of differentiation with respect to height is unviable in real-world applications, a more appropriate way to define the set of tall people is along a continuum that can be achieved with a fuzzy inference system. The output axis of the corresponding membership function is a number termed the "membership value," which falls between 0 and 1 (Foundations of fuzzy logic, 2014). The curve that is described is termed a "membership function" and is frequently designated as μ (Foundations of fuzzy logic, 2014). According to MathWorks, "This curve defines the transition from not tall to tall. Both people are tall to some degree, but one is significantly less tall than the other" (Foundations of fuzzy logic, 2014, para. 5). The membership function of fuzzy sets corresponds to the classical sets' indicator function (Hamzeh et al., 2007). As noted above, the membership function can be expressed in the form of a curve that describes how every point in the input space corresponds to a membership value or degree of truth — a number between 0 and 1 (Hamzeh et al., 2007). According to Hamzeh and his colleagues, "The most common shape of a membership function is triangular, although trapezoidal and bell curves are also used. This operation normalizes all inputs to the same range and has a direct effect on system performance and accuracy" (2007, p. 214). MathWorks emphasizes that, "The only condition a membership function must really satisfy is that it must vary between 0 and 1. The function itself can be an arbitrary curve whose shape we can define as a function that suits us from the point of view of simplicity, convenience, speed, and efficiency" (Foundations of fuzzy logic, 2014).

Some indication of the difference between classical sets and fuzzy sets can be discerned from the following examples. A classical set can be expressed as: A = {x | x > 6}. A fuzzy set is an extension of a classical set. If X is the universe of discourse and its elements are denoted by x, then a fuzzy set A in X is defined as a set of ordered pairs: A = {x, μA(x) | x ∈ X}, where μA(x) is called the membership function (or MF) of x in A. The membership function maps each element of X to a membership value between 0 and 1 (Foundations of fuzzy logic, 2014). These membership functions are typically depicted as different types of graphs. According to MathWorks:

Building the Fuzzy Inference System

"The simplest membership functions are formed using straight lines. Of these, the simplest is the triangular membership function, which is nothing more than a collection of three points forming a triangle. The trapezoidal membership function has a flat top and is just a truncated triangle curve. These straight-line membership functions have the advantage of simplicity." (Foundations of fuzzy logic, 2014, para. 5)

The membership function is used to describe the various relationships that exist within a fuzzy set, employing linguistic expressions of the fuzzy inference system's input and output. In this regard, Hamzeh et al. note that, "Generally, these rules are natural language representations of human or expert knowledge and provide an easily understood knowledge representation scheme" (2007, p. 4). It should be noted, however, that acquiring expert knowledge in certain domains is a challenging enterprise. As Qi and Zhu (2008) emphasize, "Knowledge acquisition directly from domain experts has always been considered as a bottleneck for the development of knowledge-based approaches and systems" (p. 224).

Fuzzy inference systems are typically comprised of a series of IF-THEN rules that are invoked in parallel in those instances in which the IF conditions are satisfied, with the outcome tied to the extent to which the antecedents of the rules are satisfied (Guan & Zurada, 2008). In this regard, Dewitte and Willey (1995) report that, "Given a set of conditions, many of such rules are activated; the level of activation is determined by the minimum of the truth values of each of the antecedents" (p. 12). In other words, depending on the number between 0 and 1 assigned to the elements of the fuzzy sets, the responses generated will fall along a continuum based on the prescribed fuzzy rules. For instance, Dewitte and Willey report that, "The response under a given condition is determined by the weighted average of the consequences of the different rules which fired" (1995, p. 12). Consequently, a major constraint in the application of fuzzy inference systems is developing appropriate fuzzy sets and fuzzy rules (Guan & Zurada, 2008).

In cases where two-valued or binary logic is involved, the IF-THEN rules used in fuzzy inference systems are fairly straightforward: "If the premise is true, then the conclusion is true, whereas with fuzzy approach, if the antecedent is true to some degree of membership, then the consequent is also true to that same degree" (Hamzeh et al., 2007, p. 214). Formulating appropriate IF-THEN rules is a fundamental requirement for efficient and effective fuzzy inference system operations. As MathWorks points out, "Fuzzy sets and fuzzy operators are the subjects and verbs of fuzzy logic. These IF-THEN rule statements are used to formulate the conditional statements that comprise fuzzy logic" (Foundations of fuzzy logic, 2014, para. 5). A single fuzzy IF-THEN rule assumes the following form:

if x is A then y is B

where A and B are linguistic values defined by fuzzy sets on the ranges (universes of discourse) X and Y, respectively. The if-part of the rule, "x is A," is called the antecedent or premise, while the then-part, "y is B," is called the consequent or conclusion (Foundations of fuzzy logic, 2014, para. 5). A useful example of such a rule is: "If service is good then tip is average." The concept good is represented as a number between 0 and 1, so the antecedent is an interpretation that returns a single number between 0 and 1. By contrast, the concept average is represented as a fuzzy set; therefore, "the consequent is an assignment that assigns the entire fuzzy set B to the output variable y" (Foundations of fuzzy logic, 2014, para. 5).

A more straightforward way of expressing the above rule is: "If service == good then tip = average" (Foundations of fuzzy logic, 2014, para. 6). In this regard, MathWorks advises that, "In general, the input to an if-then rule is the current value for the input variable (in this case, service) and the output is an entire fuzzy set (in this case, average). This set will later be defuzzified, assigning one value to the output" (Foundations of fuzzy logic, 2014).

The interpretation of IF-THEN rules requires several steps. The first step requires an evaluation of the antecedent that involves fuzzifying the input and applying any requisite fuzzy operators. The second step involves applying the generated results to the consequent — a process termed "implication" (Foundations of fuzzy logic, 2014). According to MathWorks, "In the case of two-valued or binary logic, if-then rules do not present much difficulty. If the premise is true, then the conclusion is true. If the antecedent is true to some degree of membership, then the consequent is also true to that same degree" (Foundations of fuzzy logic, 2014, para. 6). Therefore: in binary logic, p → q (p and q are either both true or both false); in fuzzy logic, 0.5p → 0.5q (partial antecedents provide partial implication).

The antecedent of a rule can have multiple parts — for example: "if sky is gray and wind is strong and barometer is falling, then …" — in which case all parts of the antecedent are calculated simultaneously and resolved to a single number using logical operators (Foundations of fuzzy logic, 2014). Likewise, the consequent of a rule can also consist of multiple elements: "if temperature is cold then hot water valve is open and cold water valve is shut" — in which case all consequents are affected equally by the result of the antecedent (Foundations of fuzzy logic, 2014, para. 6).

Although different fuzzy logic applications use different fuzzy operators, the approach used by MATLAB is illustrative of what takes place in the analytical process. In this regard, MathWorks reports that: "The consequent specifies a fuzzy set be assigned to the output. The implication function then modifies that fuzzy set to the degree specified by the antecedent. The most common ways to modify the output fuzzy set are truncation using the min function (where the fuzzy set is truncated) or scaling using the prod function (where the output fuzzy set is squashed)" (Foundations of fuzzy logic, 2014, para. 6).

In sum, fuzzy logic can provide practitioners with a more realistic view of a modeled phenomenon, but the accuracy of this view depends on the accuracy of the fuzzy rules that are used. This point is made by Yazdani-Chamzini and Yakhchali (2012) who report, "Fuzzy logic has the ability to express the ambiguity of human thinking and translate expert knowledge into computable numerical data. But despite the fact that fuzzy inference systems are widely applied, extracting the rules of a fuzzy inference system is not easily realized" (p. 994). The difficulties generally associated with extracting the rules of a fuzzy inference system typically relate to the complexity of the phenomenon being modeled. In this regard, Sun and Alexandre (1997) note that: "Frequently, the imprecision on definitions is related to the complexity of the categories, and at this level fuzziness is a tool for coping with the problem of complexity. The kind of knowledge applied in this case is not symbolic, but imprecise knowledge defined by a set of relaxed conditions and expressed through fuzzy sets" (p. 70).

Despite these constraints, fuzzy inference systems have been used to good effect in a wide range of industrial and research settings (Guan & Zurada, 2008). In this regard, Malhotra and Malhotra (1999) report that, "Fuzzy systems are attracting growing interest among both researchers and practitioners. These systems offer advantages over traditional computational methods by offering greater flexibility, greater tolerance of imprecise data, and an ability to model nonlinear information of arbitrary complexity" (p. 25). By using off-the-shelf fuzzy toolboxes provided by software vendors such as Brainmaker and MATLAB, practitioners can develop fuzzy inference systems that accurately model the phenomenon of interest (Malhotra & Malhotra, 1999). According to Malhotra and Malhotra, "The software is user-driven and offers an interactive interface that does not require any programming. The user can fill in the mathematical functions and their parameters, and write the rules in English. The software develops the fuzzy inference system" (p. 25).

The fuzzy inference process consists of the following five components: (1) fuzzification of the input variables; (2) application of the fuzzy operator (AND or OR) in the antecedent; (3) implication from the antecedent to the consequent; (4) aggregation of the consequents across the rules; and (5) defuzzification (Fuzzy inference process, 2014, para. 2). These steps are described in Table 1 below.

Table 1: Description of the Fuzzy Inference Process

Step 1 — Fuzzify inputs: The first step in the fuzzy inference process is to take the inputs and determine the degree to which they belong to each of the appropriate fuzzy sets via membership functions. In Fuzzy Logic Toolbox software offered by MathWorks, the input is always a crisp numerical value limited to the universe of discourse of the input variable (in this case, the interval between 0 and 10) and the output is a fuzzy degree of membership in the qualifying linguistic set (always the interval between 0 and 1). Fuzzification of the input amounts to either a table lookup or a function evaluation.

Step 2 — Apply fuzzy operator: After the inputs are fuzzified, the degree to which each part of the antecedent is satisfied for each rule is known. If the antecedent of a given rule has more than one part, the fuzzy operator is applied to obtain one number that represents the result of the antecedent for that rule, which is then applied to the output function. The input to the fuzzy operator is two or more membership values from fuzzified input variables; the output is a single truth value. Two built-in AND methods are supported: min (minimum) and prod (product). Two built-in OR methods are also supported: max (maximum) and the probabilistic OR method probor. The probabilistic OR method (also known as the algebraic sum) is calculated according to the equation: probor(a,b) = a + b − ab.

Step 3 — Apply implication method: Before applying the implication method, the rule's weight must be determined. Every rule has a weight (a number between 0 and 1) applied to the number given by the antecedent. Generally, this weight is 1 and thus has no effect on the implication process, though it can be adjusted to weight one rule relative to others. After weighting, the implication method is implemented. A consequent is a fuzzy set represented by a membership function that weights appropriately the linguistic characteristics attributed to it. The consequent is reshaped using a function associated with the antecedent. The input for the implication process is a single number given by the antecedent, and the output is a fuzzy set. Implication is implemented for each rule. Two built-in methods are supported — min (minimum), which truncates the output fuzzy set, and prod (product), which scales the output fuzzy set.

Step 4 — Aggregate all outputs: Because decisions are based on the testing of all rules in a fuzzy inference system (FIS), the rules must be combined in some manner in order to make a decision. Aggregation is the process by which the fuzzy sets representing the output of each rule are combined into a single fuzzy set. Aggregation occurs once for each output variable, just prior to defuzzification. The input of the aggregation process is the list of truncated output functions returned by the implication process for each rule; the output is one fuzzy set for each output variable. Three built-in methods are supported: max (maximum), probor (probabilistic OR), and sum (the sum of each rule's output set).

Step 5 — Defuzzify: The input for the defuzzification process is a fuzzy set (the aggregate output fuzzy set) and the output is a single number. As much as fuzziness helps rule evaluation during intermediate steps, the final desired output for each variable is generally a single number. However, the aggregate of a fuzzy set encompasses a range of output values and so must be defuzzified to resolve a single output value. Perhaps the most popular defuzzification method is the centroid calculation, which returns the center of area under the curve. Five built-in methods are supported: centroid, bisector, middle of maximum (the average of the maximum value of the output set), largest of maximum, and smallest of maximum. (Fuzzy inference process, 2014)

Conclusion

Fuzzy logic can provide practitioners with a more realistic view of a modeled phenomenon, but the accuracy of this view depends on the accuracy of the fuzzy rules that are used. Despite the constraints associated with extracting rules for complex systems, fuzzy inference systems have been applied to good effect across a wide range of industrial and research settings. As Malhotra and Malhotra (1999) note, these systems offer advantages over traditional computational methods through greater flexibility, greater tolerance of imprecise data, and the ability to model nonlinear information of arbitrary complexity (p. 25).