# Project portfolio management

## using fuzzy logic to determine the contribution of portfolio components to organizational objectives

## Les Labuschagne, University of South Africa

**Abstract**

Organization success is dependent on the organization's ability to realise its objectives successfully. At a basic level, project portfolio management (PPM) focuses the organization on doing the right projects and efficiently allocating resources to those projects. Selecting the right projects is not enough. It is also necessary to understand the individual and collective contribution of these projects to the organizational objectives so that decision making regarding portfolio balancing and the determination of gaps in meeting objectives is better informed. The complexity associated with this problem increases with the number of projects and organizational objectives as there is a many-to-many relationship between projects and organizational objectives. Fuzzy logic was investigated as a possible technique for representing this complexity. This paper proposes a combined fuzzy model for determining the degree of contribution each project in the portfolio makes towards organizational objectives.

The research methodology followed a modelling approach, and an application illustrating how the model would work is described toward the end of this paper. The value of such a model lies in the ability to make informed decisions based on the relationship between projects and objectives. These decisions include cancelling, delaying, or even changing projects to better meet the strategic objectives.

Keywords: portfolio management; fuzzy logic; organizational performance; organizational objectives; modelling

**Introduction**

Earlier approaches to project portfolio management (PPM) were focused on categorizing the landscape of existing projects in organizations without paying much attention to driving strategy implementation (Berinato, 2001; Kersten & Verhoef, 2003; Ross, 2005; D'Amico, 2005; Jeffrey & Leliveld, 2004). Ward and Peppard (2004), however, illustrated that categories such as strategic, operational, high potential, and support could be used as a means for facilitating agreement between senior management on the portfolio of projects available and required. Projects could now be categorized according to their business contribution.

Subsequently, authors have given increasing focus to the role of single project management in achieving portfolio efficiency (Martinsuo & Lehtonen, 2007), alignment of the project portfolio to corporate strategy, vertical integration, and value creation through portfolio management (Thiry & Deguire, 2007), the translation of strategy into programmes and projects, organizational performance, and the role of the project/programme management office (Aubry, Hobbs, & Thuillier, 2008), project portfolio control and performance (Müller, Martinsuo, & Blomquist, 2008), and the influence of business strategy on PPM and its success (Meskendahl, 2010).

While many articles and books have been written on the topic of measuring organizational success, the contribution of projects and programmes to strategic objectives, and hence to organizational success, remains difficult to measure. Aubry et al. (2007) observed in literature that project success, and by implication, portfolio and organization success, “is measured by the business objectives” (p. 331). They noted “there is no consensus on the way to assess the value of performance in project management” and that “the financial approach alone cannot give a correct measure of the value of project management for the organisation” (p. 331). They go on to state that project success “is a vague approximation and, as such, a rather imperfect system for measuring results” (p. 331) and suggest that new approaches are needed. Indeed, project success cannot only be measured by delivering on time and within budget, but in terms of portfolio management, success should be measured on the contribution the projects (individually and collectively) make in achieving the organizational objectives.

In determining the contribution of projects to organizational objectives, the researcher had to consider the quantitative and qualitative measures of assessment of projects and a form of reasoning that would be suitable to model such a system. The research process toward developing a solution for determining the degree of the individual and collective contribution of projects to organizational objectives involved assessing various approaches employed in portfolio management, such as multicriteria utility theory (Stewart & Mohamed, 2002), but no approach could be found that addressed the problem directly. Fuzzy logic was identified as a possible approach to developing a model as it deals with approximate reasoning and degrees of truth values, and has the ability to handle numeric and non-numeric (linguistic) variables. This is important as the proposed model assesses the *degree* of contribution of projects. Fuzzy logic as a theoretical approach was examined. The application of fuzzy logic involved developing and testing various mathematical models until the model proposed in this paper was developed.

The remaining sections of this paper include a review of the literature, a description of the research approach, and a presentation of the proposed model as well as concluding remarks.

**The Relationship Between Strategy Definition and Strategy Execution**

Having a well-defined strategy and organizational objectives without the ability to execute them, or having efficient and effective operations without a strategy or organizational objectives limits the success organizations could have. Kaplan and Norton (2008), who stated, “A visionary strategy that is not linked to excellent operational and governance processes cannot be implemented support this. Conversely, operational excellence may lower costs, improve quality, and reduce process and lead times; but without a strategy's vision and guidance, a company is not likely to enjoy sustainable success.” This emphasizes the need not only to link strategy and execution but also to be able to assess the contribution of the components being executed to the strategy.

Project Management Institute (2008) describes the process of translating the organization's strategy into components that will be executed to achieve the strategy. In so doing, the need for project portfolio management is illustrated. According to the institute (Project Management Institute, 2008), organizations build strategy to define how their vision will be achieved. The vision is enabled by the mission, which directs the execution of the strategy. The organizational strategy is a result of the strategic planning cycle, where the vision and mission are translated into a strategic plan. The strategic plan is subdivided into a set of initiatives that are influenced by market dynamics, customer and partner requests, shareholders, government regulations, and competitor plans and actions. These initiatives establish projects and programmes, which through their execution ultimately achieve the organizational objectives. Projects, programmes, and other work make up the portfolio and are therefore referred to as portfolio components. Linking the organization's objectives directly to the portfolio components reveals that there is a many-to-many relationship between objectives and components.

This relationship can be illustrated in the following way:

**Figure 1: Many-to-many relationship between organizational objectives and portfolio components.**

In Figure 1, each portfolio component (PC) contributes to one or more objectives. For example, PC1 could contribute to partly achieving objectives 1, 3, and (n), while the remainder of objective 1 is achieved through the execution of PC3. PC2 could contribute to fully achieving objective 2, and objective (n) could be achieved by components 2, 3, and (n). The degree of contribution of each component varies from one to the other.

An alternate depiction of this relationship is given in Figure 2.

**Figure 2: Relationship between organizational objectives and portfolio components.**

It is also important to understand the relationships between portfolio components. While PC1 and PC3 contribute to the achievement of objective 1, they do not necessarily have to be related to each other in any other way. They could be singular, independent projects managed by different teams and not dependent on each other through deliverables or resources. On the other hand, for objective 3, PC1, PC4, and PC6 could be run as a programme where all components are related to each other and have interdependency through, for instance, deliverables and/or resources. Each component contributes to objectives to varying degrees. For example, the degree of contribution of PC1 to objective 1 is represented by (a), and the degree of contribution of PC3 to objective 1 is represented by (b). The degree of contribution of these two components to objective 1 is not equal. Additionally, PC1 also contributes to objectives 3 and (n) and the degree of contribution to each of these objectives (including objective 1) is represented by (a), (d), and (i). The degree of contribution of a single component (PC1) to each of the three objectives is not equal. The degrees of contribution, represented by the letters (a) to (j) in Figure 2, therefore vary for each component-to-objective relationship. The challenge is in understanding the degree of contribution of each component to each objective, as well as the collective contribution of components to a single objective.

Understanding the degree of contribution of portfolio components to the achievement of organizational objectives aids the organization in also understanding the impact of decisions made in relation to those components. When certain constraints are applied to the portfolio, such as a reduction in budget or a change in strategy, the organization needs a mechanism to aid in management decision making regarding rebalancing the portfolio. For example, if there is a reduction in the available funds for portfolio components, the organization can choose to stop or slow down components that make a *low* contribution to organizational objectives. Alternatively, a change in strategy may reprioritize certain objectives, resulting in the fast tracking of associated components that make a *medium* or *high* contribution. *Low, medium*, and *high* refer to the qualitative assessment of the degree of contribution of components.

In addition, assessing the degree of contribution of portfolio components to objectives will also achieve the benefit of determining gaps in the portfolio. If the combined contribution of components 5 and 6 to objective 4 is determined to be less than 100%, it may be necessary for the organization to consider doing additional portfolio components in order to close the gap and achieve the objective fully.

The challenge is in being able to quantitatively assess the individual and collective contribution of portfolio components to organizational objectives. In this paper, the researchers explore fuzzy logic as a means to do this.

**Research Design**

For this paper, modeling was chosen as the research approach. According to Egger and Carpi (n.d.), “modeling involves developing physical, conceptual, or computer-based representations of systems. Scientists build models to replicate systems in the real world through simplification, to perform an experiment that cannot be done in the real world, or to assemble several known ideas into a coherent whole to build and test hypotheses” (p. 1). They suggest that as a research method, it is necessary to define the system that is being modelled. This involves determining the boundaries for the model as well as the variables and their relationships. Once a model is built, it can be tested using a given set of conditions (Egger & Carpi, n.d.).

Cooper and Schindler (2011) define a model as “a representation of a system that is constructed to study some aspect of that system or the system as a whole” (p. 67). They point out that “models differ from theories in that a theory's role is explanation whereas a model's role is representation” (p. 67). They further state that “a model's purpose is to increase our understanding, prediction and control of the complexities of the environment” (p. 67).

They also suggest that in business research, three types of models are typically found. These are descriptive, predictive, and normative. Descriptive models are used more frequently to describe complex systems. Predictive models are used to forecast future events. Normative models are used for control—informing decision makers about the actions to be taken. The model described in this paper can be described as a predictive model as it can be used to predict the degree of contribution of portfolio components based on initial qualitative assessments.

Models are developed through the use of inductive and deductive reasoning. “Inductive reasoning allows the modeler to draw conclusions from the facts or evidence in planning the dynamics of the model. The modeler may also use existing theory, managerial experience, judgment, or facts deduced from known laws of nature…deductive reasoning serves to create particular conclusions derived from general premises” (Cooper & Schindler, 2008, p. 67). In this instance, inductive reasoning was used in the development of the proposed model.

With regard to research model classification, the University of Southampton (n.d.) suggests that there is “no common agreement on the classification of research models” (p. 1) but proposes the following five categories:

Physical model: A physical model is a physical object shaped to look like the represented phenomenon, usually built to scale, such as small-scale versions of vehicles or buildings. Theoretical model: This generally consists of a set of assumptions about some concept or system; is often formulated, developed, and named on the basis of an analogy between the object or system that it describes and some other object or different system; and it is considered an approximation that is useful for certain purposes.

Mathematical model: A mathematical model refers to the use of mathematical equations to depict relationships between variables. It is an abstract model that uses mathematical language to describe the behaviour of a system.

Mechanical model: A mechanical (or computer) model tends to use concepts from the natural sciences, particularly physics, to provide analogues for social behaviour. It is often an extension of mathematical models.

Symbolic interactionist model: This is generally a simulation model. That is, it is based on artificial (contrived) situations, or structured concepts that correspond to real situations. It is characterised by symbols, change, interaction, and empiricism, and is often used to examine human interaction in social settings.

The model proposed in this paper uses fuzzy logic and is therefore aligned with the mathematical model category. Cox (2005) states that fuzzy logic “from a modeling perspective, can represent such elastic and imprecise concepts as high risk, a long duration, a tall person, and a large transaction volume,” and that it “provides a way of finding the degree to which an object is representative of a concept or the degree to which a state is representative of a process” (p. 67). He adds that “these degrees play a subtle but critical role in the evaluation of fuzzy models and fuzzy systems” in that “they represent not only the degree of membership in a concept…but such important modeling concepts as supporting evidence, numeric elasticity, and semantic ambiguity” (p. 67). This paper proposes the use of a fuzzy logic model to describe the degree of contribution one or more portfolio components make towards achieving the organizational objectives. The model is described in more detail in the next section.

**Proposed Fuzzy Model**

Fuzzy logic is a technique that can deal with qualitative and quantitative information. It is a technique that can take subjective information and make it more objective and has proven to be very successful in a wide range of applications (Sowell, 2005).

The various disciplines in which fuzzy logic has been used successfully include, but are not limited to, decision support, control theory, artificial intelligence, genetic algorithms, and mechanical engineering (Sowell, 2005).

The use of fuzzy logic in research related to PPM is also gaining popularity. At the time of writing this paper, a number of articles on the use of fuzzy logic had been written on the area of project selection (Laarhoven & Pedrycz, 1983; Chen & Gorla, 1998; Machacha & Bhattacharya, 2000; Wang & Hwang, 2005; Huang, Chu, & Chiang, 2006; Chen & Cheng, 2009). This paper focuses on quantitatively assessing the contribution of those selected projects to organizational objectives. In order to comply with the submission requirements for this paper, the researcher chose to place a more detailed description of *the fuzzy logic process* in the appendix. In addition, authors such as Earl Cox (1995) have written numerous books on the application of fuzzy logic.

The relationships between organizational objectives and portfolio components make up a complex system. “A complex system is a system (whole) comprising of numerous interacting entities (parts) each of which is behaving in its local context according to some rule(s) or force(s)” (Caldart & Ricart, 2004, p. 97). Earlier, Baccarini (1996) proposed that “project complexity be defined as consisting of many varied interrelated parts and can be operationalized in terms of differentiation and interdependency.” He later described types of complexity as being organizational (vertical and horizontal differentiation as well as the degree of operational interdependencies) and technological (the transformation processes which convert inputs into outputs).

The relationships among portfolio components, the integration, and interdependency between portfolio components, and the varying degrees of contribution add to the complexity of the total system of portfolio components and organizational objectives.

Complex business systems are built around multiple fuzzy models representing the combined intelligence of several experts (Cox, 1995). A combination of multiple fuzzy models is required to address the problem of representing portfolio component contribution to strategic objectives. The reason for using multiple models is to allow for the variability in the number of portfolio components contributing to the organizational objectives. For each portfolio component, values for the input variables are entered into the model, the rules are applied, and a qualitative output value is derived. The fuzzification and application of fuzzy rules is done for each portfolio component and the contribution is determined by aggregating the qualitative outputs and then applying defuzzification to produce a crisp value that represents the quantitative contribution of portfolio components to objectives.

Combining fuzzy models tends to increase the overall information entropy (disorder and loss of information) associated with the entire system. To maintain the information in the complete system, the combination of solution fuzzy regions using the additive aggregation method before defuzzification is used (Cox, 1995).

Figure 3 describes the total system or model being proposed. In the figure, stages A and B represent separate fuzzy models, which are combined to form a single conceptual fuzzy logic model.

**Figure 3: Combined fuzzy logic model.**

The model is described in more detail in the following sections.

**Stage A**

For each portfolio component that contributes to an organizational objective (in this case, portfolio components 1 and 2), the model considers input values for the input linguistic variables PCVar1 and PCVar2. The input values are passed through a fuzzification process, after which the rules in the inference engine are applied to determine a qualitative value of contribution for each portfolio component. Linguistic variables are variables of the system whose values are words from a natural language, instead of numerical values. Each input variable is qualified by values, such as *poor, average*, and *good* for PCVAR1 and *low, medium*, and *high* for PCVAR2. The output variable (contribution) is qualified by the values *very low, low, moderate, high*, and *very high*. Membership functions are used in the fuzzification process to quantify a linguistic variable value.

An important characteristic of fuzzy logic is that a numerical value does not have to be fuzzified using only one membership function. In other words, a value can belong to multiple sets at the same time.

The process for stage A of the fuzzy model is illustrated in Figure 4, followed by an explanation of the steps involved.

**Figure 4: Illustration of stage A of the combined fuzzy model.**

*Phase 1—Input Variables*

*Phase 1—Input Variables*

For illustrating the model, only two input variables are used. In a typical organization, a group of portfolio management experts could decide on a number of input variables to be used for evaluating the contribution of portfolio components to organizational objectives. The model is designed to cater for more than two input variables but for illustrative purposes, only two are used. The two input variables are described in the following.

1. Portfolio Component Variable 1 (PCVar1)

To give some meaning to the following example, PCVar1 represents “value.” The value that a portfolio component is expected to deliver is an important criterion when determining the portfolio component's contribution. Value considers the strategic alignment of the portfolio component—in particular, the decision maker's perception of how the component serves the organization's objectives in the long term—as well as the financial attractiveness of the component. That is, the economic feasibility, which is measured by the component cost, contribution to profitability, and the component's growth rate (Deng & Wibowo, 2009; Ghasemzadeh & Archer, 2000; Santhanam & Kyparisis, 1995).

2. Portfolio Component Variable 2 (PCVar2)

In this example, PCVar2 represents durability of competitive advantage. If the portfolio component is delivering a product for which a competitor already exists, then the portfolio component will be rated “low.” If the product can be copied within two years, then the portfolio component will be rated as “medium.” If the likelihood of copying the product extends beyond two years, then the portfolio component is rated as “high,” as the contribution of the portfolio component to an objective related to market share is high.

*Phase 2—Fuzzification*

*Phase 2—Fuzzification*

Fuzzy logic starts with the concept of a fuzzy set. A *fuzzy set* is a set without a clearly defined boundary. It can contain elements with only a partial degree of membership. For each input variable in this example, three membership functions are defined. The qualitative categories for the membership functions for PCVar1 are *poor, average*, and *good*, while the qualitative categories for the membership functions for PCVar2 are *low, medium*, and *high*. The membership functions for PCVar1 and PCVar2 are illustrated in Figures 5 and 6, respectively.

**Figure 5: PCVar1 – Value.**

**Figure 6: PCVar2 - Durability of competitive advantage.**

In Figures 5 and 6, the x-axis represents the domain and the y-axis represents the membership values.

The membership function is a curve (triangular in this case) that defines how each point in the input space (domain) is mapped to a membership value (or degree of membership) between 0 and 1 (y-axis).

The definition of the membership functions would be done by the portfolio management experts in the organization in accordance with their knowledge and experience in portfolio management and the organization. This will be done before the model is used for the first time. The membership functions will vary from one organization to the next.

The domain is not numeric since the input values are qualitative. Subjective information can now be modelled mathematically as the qualitative inputs can be converted into quantitative values.

The next step in the fuzzification process is to take the qualitative inputs, PCVar1 (represented by ‘a’ in Figure 5) and PCVar2 (represented by ‘b’ in Figure 6), and determine the degree to which these inputs belong to each of the respective membership functions. In an organization, the portfolio management experts would evaluate the PCVar1 of a portfolio component and determine to what degree it is *poor, average*, or *good*.

As an example, in Figure 5, this is represented by the dark bold vertical line, which intersects *poor* at a membership value of 0.4 and *average* at a membership value of 0.2. In other words, PCVar1 is assessed as being poor to a degree of 0.4 as well as average to a degree of 0.2 simultaneously.

Similarly, the portfolio management experts would evaluate PCVar2 of the same portfolio component and determine to what degree it is *low, medium*, or *high*. In Figure 6, the dark bold vertical line intersects *low* at a membership value of 0.8 and *medium* at a membership value of 0.2. In this example, the input variable PCVar2 is assessed as being low (to a degree of 0.8) as well as medium (to a degree of 0.2) simultaneously.

*Phase 3—Inference Engine*

*Phase 3—Inference Engine*

A number of rules are determined by a knowledgeable group of individuals in the organisation who can determine the outputs based on specific conditions within the inference engine. This would also be done before using the model for the first time. An example of a rule would be:

**IF** PCVar1 is *poor* AND PCVar2 is *low*, **THEN** Contribution is *very low*.

The number of rules for a system with two input variables, each having three values, is nine. A system with four variables, each having three values, would have 81 or 3^{4} rules. The Mamdani style of inference is used here (MathWorks, 2011). The Mamdani method is the most commonly used fuzzy inference technique and was among the first control systems built using fuzzy set theory.

The following rules were applied to the input variables in the inference engine:

Rule 1 | If PCVar1 is Poor AND PCVar2 is High, THEN Contribution is Moderate. |

Rule 2 | If PCVar1 is Poor AND PCVar2 is Medium, THEN Contribution is Low. |

Rule 3 | If PCVar1 is Poor AND PCVar2 is Low, THEN Contribution is Very Low. |

Rule 4 | If PCVar1 is Average AND PCVar2 is High, THEN Contribution is High. |

Rule 5 | If PCVar1 is Average AND PCVar2 is Medium, THEN Contribution is Moderate. |

Rule 6 | If PCVar1 is Average AND PCVar2 is Low, THEN Contribution is Low. |

Rule 7 | If PCVar1 is Good AND PCVar2 is High, THEN Contribution is Very High. |

Rule 8 | If PCVar1 is Good AND PCVar2 is Medium, THEN Contribution is High. |

Rule 9 | If PCVar1 is Good AND PCVar2 is Low, THEN Contribution is Moderate. |

**Table 1: Fuzzy rules.**

*Rule Evaluation*

*Rule Evaluation*

The next step in the fuzzy logic process is to take the fuzzified inputs (for the above example these would be: μ(PCVar1 = poor) = 0.4, μ(PCVar1 = average) = 0.2, μ(PCVar2 = low) = 0.8 and μ(PCVar2 = medium) = 0.2), and apply them to the antecedents of the fuzzy rules. If a given fuzzy rule has multiple antecedents, the fuzzy operator (AND or OR) is used to obtain a single value that represents the result of the antecedent evaluation. The rules used here have been developed for illustration purposes. In an organization, a group of portfolio management experts would need to design the rules and agree on the consequent values for the respective input value combinations before using the model for the first time.

The rules transform the input variables into an output that will indicate the degree of contribution of the portfolio component. This output variable is also defined with membership functions (*very low, low, medium, high, very high*). Once the rules have been defined according to expert knowledge, they become the knowledge base of the model. The following matrix represents the knowledge base associated with the rules described in Table 1.

PCVar2 | ||||

Low | Medium | High | ||

PCVar1 | Poor | Very Low | Low | Moderate |

Average | Low | Moderate | High | |

Good | Moderate | High | Very High |

**Table 2: Knowledge base associated with fuzzy rules.**

*How the Rule Base Works*

*How the Rule Base Works*

The next step is to compute the degree of membership to the membership functions (*very low, low, moderate, high, or very high*) of the output variable (Contribution). Once a variable is fuzzified (refer to the section on fuzzification described earlier), it takes a value between 0 and 1 indicating the degree of membership to a given membership function of that specific variable. The degrees of membership of the input variables have to be combined to get the degree of membership of the output variable. In this instance where there is more than one input variable, the degree of membership for the output value will be the *minimum* value of the degree of membership for the different inputs. Referring back to Figures 5 and 6 as well as Tables 1 and 2, input (a) for PCVar1 has a membership degree of 0.4 to the membership function *poor*, which applies to rules 1, 2 and 3 (Table 1), and a membership degree of 0.2 to the membership function *average*, which applies to rules 4, 5, and 6. Similarly, input (b) for PCVar2 has a membership degree of 0.8 to the membership function *low*, which applies to rules 3, 6, and 9, and a membership degree of 0.2 to the membership function *medium*, which applies to rules 2, 5, and 8. When a rule is totally satisfied (indicated by ✓ in Figure 7), it will have an output with a membership degree to an output membership function equal to the lower degree among the inputs. The rules satisfied in this example are:

Rule 2 | IF PCVar1 is Poor (degree of 0.4) AND PCVar2 is Medium (degree of 0.2), THEN Contribution is Low (degree of 0.2)…the lowest degree among the inputs. |

Rule 3 | IF PCVar1 is Poor (degree of 0.4) AND PCVar2 is Low (degree of 0.8), THEN Contribution is Very Low (degree of 0.4). |

Rule 5 | IF PCVar1 is Average (degree of 0.2) AND PCVar2 is Medium (degree of 0.2), THEN Contribution is Moderate (degree of 0.2). |

Rule 6 | IF PCVar1 is Average (degree of 0.2) AND PCVar2 is Low (degree of 0.8), THEN Contribution is Low (degree of 0.2). |

**Table 3: The satisfied rules.**

Figure 7 shows the graphical representation (rule view) of the rules in the system. The MATLAB^{®} tool from MathWorks^{®} was used to build the simple fuzzy system and generate the rule view using the Fuzzy Logic Toolbox^{®}. In Figure 7, each row, numbered 1 to 9, represents a rule in the system. The two input variables are shown alongside each other and the output variable is to the right of the figure. The red (vertical) lines indicate the points of intersection on the relevant membership functions associated with the membership values for each input variable.

**Figure 7: Rule view.**

How the output values are derived is described in the next section.

*Phase 4—Outputs*

*Phase 4—Outputs*

The output is the aggregation or sum of the membership functions from the satisfied rules. Aggregation is the process of unification of the outputs of all rules. We take the membership functions of all rule consequents and combine them into a single fuzzy set. The input of the aggregation process is the list of consequent membership functions, and the output is one fuzzy set for each output variable. Among the satisfied rules, the membership degree of each output membership function will be the *higher* among the rules that have as a result that membership function.

Referring to Figure 8, the shading in the triangles indicates if there is a degree of membership.

For the membership function *very low*, the degree of membership is 0.4 (based on the result of rule 3 in Table 3).

For the membership function *low*, the degree of membership is 0.2 (based on the higher result of rules 2 and 6 in Table 3).

For the membership function *moderate*, the degree of membership is 0.2 (based on the result of rule 5).

For the membership function *high*, the degree of membership is 0.

For the membership function *very high*, the degree of membership is 0.

**Figure 8: Output of rules.**

To calculate the quantitative contribution of a single portfolio component with two input variables, the aggregated output must be defuzzified in order to get a single output value. The most popular defuzzification method is the centroid method (Cox, 1995), which returns the centre of the area under the curve labelled “output” in Figure 8.

Mathematically, this centre of gravity (COG) can be expressed as:

where *COG* is the defuzzified output, *μ _{i}*(

*x*) is the aggregated membership function and

*x*is the output variable. In this example, the output value 0.278 represents the contribution of the portfolio component to an objective. An output value of 1 would imply that the objective is fully achieved (100%); hence, the output value in this example (0.278) indicates that the portfolio component contributes to the objective to a degree of 27.8%.

This implies that if this were the only portfolio component selected to achieve an organizational objective, then only 27.8% of the objective would be achieved. The organization would need to select other portfolio components or amend the scope of the component such that more or the entire objective is achieved.

However, we want to determine the cumulative contribution of two or more components and so, before we defuzzify the qualitative output of a single component, we move to stage B where the contribution of multiple components is considered.

**Stage B**

**Figure 9: Stage B of the combined fuzzy model.**

*Phase 5—Additive Aggregation*

*Phase 5—Additive Aggregation*

The aggregation in stage A is the unification of the outputs of all rules per portfolio component. The aggregation in stage B is the aggregation (sum) of the outputs of all portfolio components before defuzzification.

In order to maintain the information in the complete system, the fuzzy regions (outputs of portfolio components in stage A) are combined using the additive aggregation method before defuzzification. Using the bounded sum method (Cox, 1995), the process adds the truth membership values of the consequent fuzzy set and the solution fuzzy set at each point along their mutual membership functions. The bounded sum method is applied so that the composite membership value can never exceed 1.0 (Cox, 1995). Figures 10a–10d illustrate the aggregation of the portfolio component outputs into a single aggregated output before defuzzification.

The additive technique adds the consequent fuzzy sets (stage A outputs) to the solution variable's output fuzzy region. The process adds the truth membership value of the consequent fuzzy sets and the solution fuzzy set at each point along their mutual membership functions. (For a detailed explanation of aggregation and implication techniques, refer to Cox [1995], Chapter 2).

Using the output of the example used earlier for one portfolio component, Figure 10a shows the first step in the aggregation process.

**Figure 10a: Additive aggregation—first portfolio component.**

For the second portfolio component (Figure 3), let us assume the stage A process is followed as was done for the first portfolio component, and an output for the second portfolio component is derived, such that the output membership value is equal to 1 for the membership function *high*. Figure 10b shows how the second output is added to the final output (solution fuzzy region).

**Figure 10b: Additive aggregation—second portfolio component.**

The combined output of both portfolio components is illustrated in the following Figure 10c:

**Figure 10c: Additive aggregation—combining both portfolio components.**

To summarise, Figure 10a showed the addition of the consequent fuzzy set for portfolio component 1 being added to the final output region (cumulative contribution). Figure 10b showed the addition of the consequent fuzzy set for portfolio component 2 being added to the final output region. Figure 10c showed the combined view of Figures 10a and 10b.

*Phase 6—Aggregated Output*

*Phase 6—Aggregated Output*

The aggregated output, also known as the solution fuzzy region, is illustrated in Figure 10d.

**Figure 10d: Aggregated output.**

The solution fuzzy region (cumulative contribution) is described as satisfying the membership functions *very low* to *high* such that:

- The membership function
*very low*has a membership value of 0.4. - The membership function
*low*has a membership value of 0.2. - The membership function
*moderate*has a membership value of 0.2. - The membership function
*high*has a membership value of 1.0. - The membership function
*very high*has a membership value of 0.0.

Now that the aggregated output (solution fuzzy region) has been determined, the quantitative value representing cumulative contribution must be determined through the process of defuzzification.

*Phase 7—Defuzzification*

*Phase 7—Defuzzification*

The last step in the fuzzy inference process is defuzzification. Fuzziness helps us to evaluate the rules, but the final output of a fuzzy system has to be a crisp number. The input for the defuzzification process is the aggregate output fuzzy set and the final output is a single number. There are several defuzzification methods, but the most popular one is the centroid technique (Cox, 1995). It finds the point where a vertical line would slice the aggregate set into two equal masses. This is represented by the vertical red line in Figure 11.

**Figure 11: Aggregated fuzzy output.**

*Phase 8—Final Output*

*Phase 8—Final Output*

As described earlier, the application of the centroid technique (Cox, 1995) results in a quantitative value. In this instance, the centroid technique is applied to the aggregated fuzzy output to produce a quantitative value. The quantitative value (result) represents the combined contribution of the portfolio components. In this example, the combined contribution is 0.448, implying that the cumulative contribution of the two portfolio components is 44.8% of the objective. This would mean that if these were the only portfolio components considered for achieving this objective, the organization would fail in doing so.

**Interpretation and Utility of the Model**

From the previous discussion, while the portfolio components make a contribution to the organizational objective, it can be seen that there is still a gap in fulfilling the objective completely, i.e., 100%. This is indicated by the fact that the degree of contribution is not equal to 1. There is still potential for additional portfolio components to be added in order to achieve the objective fully. Alternatively, the scope of the selected portfolio components could be amended such that their contribution can be improved towards meeting the objective. The results obtained from the model can assist in decision making regarding the composition of the portfolio.

**Value of the Model**

The ability to quantitatively determine the cumulative contribution of portfolio components in achieving objectives after making qualitative assessments of those components using multiple criteria improves the decision-making capability of decision makers when considering the portfolio mix and the potential to achieve organizational objectives. Decisions regarding the portfolio composition still lie with people but the model acts as a tool for making better-informed decisions. For example, if the organization, due to budget constraints, wants to determine which portfolio component can be stopped, it would use the model to test the effect on the whole system by removing individual components and, based on the results, make the decision as to which components can be stopped.

**Conclusion**

Organizational success is dependent on the organization's ability to realise its objectives successfully. At a basic level, portfolio management focuses the organization on doing the right projects and efficiently allocating resources to those projects. Selecting the right projects is not enough. It is also necessary to understand the individual and collective contribution of these projects to the organizational objectives so that decision making regarding portfolio balancing and the determination of gaps in meeting objectives is better informed.

Previous research has revealed that there is a lack of understanding of the link between portfolio management and organizational objectives. Decisions regarding the portfolio and its components are made subjectively and the implications of the decisions for the organizational objectives are not fully understood.

The degree of contribution of portfolio components to organizational objectives is an important aspect of project portfolio management as it brings us closer to ensuring organization success through the successful execution of the right components. The model proposed in this paper can assist organizations in determining gaps in terms of components required to achieve organizational objectives as well as aid in the decision making regarding the portfolio composition when confronted with imposed constraints such as a reduction in budget.

The fuzzy logic model assists with the subjective evaluation of portfolio components in terms of criteria relevant to individual organizations. The model proposed here addresses the complexity of the problem by combining fuzzy models and allowing the assessment of a variable number of components. In addition, the model can be expanded to incorporate additional input variables should an organization choose to do so.

While the MATLAB tool was used to illustrate the model using two portfolio components, the need for a new tool or enhancements to existing tools has been identified to harness the true potential of the model, enabling easier input of portfolio component evaluations, allowing for additional criteria and enabling the simulation of outcomes of decisions regarding the portfolio components. For example, if it is decided to stop a portfolio component due to resource constraints, such a tool would need to illustrate in a dashboard format the implications of the decision on the achievement of the objectives to which the component contributes.

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**APPENDIX**

**Fuzzy Logic Overview**

Fuzzy logic is a tool capable of modelling complex, uncertain and vague data and is considered appropriate to deal with uncertainty in a portfolio component environment.

Fuzzy logic was introduced by Lotfi Zadeh in 1965 in a paper entitled “Fuzzy sets” in the journal *Information and Control*. The paper is considered to have been completed more than two years before this date but was not accepted for publication by journals due to its radical idea. Zadeh later laid the foundation for fuzzy logic and reasoning by proposing the idea of the fuzzy algorithm. The theory has advanced in concepts and application over the decades. In the early 1990s, fuzzy logic was applied to home electronics products and the general public became aware of it (Tanaka, 1997).

Fuzzy logic is a broad theory including fuzzy set theory, fuzzy logic, fuzzy measures, and others. It is designed to deal with vagueness and imprecision. It is very effective in areas where human reasoning is needed, which is usually imprecise (Hosmer, 1993). Fuzzy logic can help in making subjective opinions more objective.

Cebeci and Beskese (2002) stated “the theory of fuzzy logic builds on the idea of non-statistical uncertainty” and argued that “conventional probabilistic models provide inappropriate descriptions of certain kinds of uncertainties” (p. 93). In particular, “linguistic imprecision is believed to be a major cause of…lexical uncertainty. While conventional stochastic uncertainty deals with the uncertainty of whether a certain event will occur or not, lexical uncertainty…deals with the uncertainty of the definition of the event itself. Humans often evaluate various concepts differently.” (p. 93)

When evaluating the performance and value of portfolio components within a portfolio, stakeholders and members of investment committees make subjective assessments. The RAG (red, amber, green) status is one such example of a subjective evaluation. In some instances, parameters (threshold values) are assigned to indicate when a status changes from green to amber to red, but in most cases, evaluation of some factors is done subjectively. Fuzzy evaluation and reasoning techniques or approaches offer a way of dealing with subjective evaluation.

**Fuzzy Logic Systems**

A fuzzy logic system receives a crisp input and delivers either a fuzzy set or a crisp value. It contains four components: a rule set, a fuzzifier, an inference engine, and a defuzzifier. Rules may be provided by experts or can be extracted from numerical data. The rules are expressed as a collection of IF-THEN statements. These statements are related to fuzzy sets associated with linguistic variables (Mendel, 1995).

The fuzzifier maps the input crisp numbers into the fuzzy sets to obtain degrees of membership. It is needed in order to activate rules, which are in terms of the linguistic variables. The inference engine of the fuzzy logic system maps the antecedent fuzzy (IF part) sets into consequent fuzzy sets (THEN part). This engine handles the way in which the rules are combined. In practice, only a very small number of rules are actually used in engineering applications of fuzzy logic (Guo & Peter, 1994). In most applications, crisp numbers must be obtained at the output of a fuzzy logic system. The defuzzifier maps output fuzzy sets into a crisp number, which becomes the output of the fuzzy logic system.

**Membership Functions**

A membership function is a curve that defines how each point in the input space is mapped to a membership value between 0 and 1. The input space is also referred to as the universe of discourse (MATLAB User Guide).

**Fuzzification**

The first step in fuzzy logic processing involves a domain transformation called fuzzification. Crisp inputs are transformed into fuzzy inputs. To transform crisp input into fuzzy input, membership functions must first be defined for each input. A member of a fuzzy set belongs to that set to a certain degree. The degree of membership is determined by the membership function. Once membership functions are defined, fuzzification takes a real time input value, such as temperature, and compares it with the stored membership function information to produce fuzzy input values.

**Figure 1: Fuzzy sets for temperature.**

This figure illustrates five fuzzy sets: cold, cool, normal, warm, and hot. The bold (black) line shows that a temperature of 32°C belongs to the HOT fuzzy set with a membership degree of 0.7. It also belongs to the WARM fuzzy set with a membership degree of 0.2.

**Fuzzy Rules**

Fuzzy logic systems use rules to indicate the relationship between variables (observations) and outcomes (actions). The rules have an IF (precondition)…THEN (consequence) structure. The precondition can consist of multiple conditions linked with AND or OR conjunctions and negated with a NOT. The computation of fuzzy rules is called fuzzy inference (Shull, 2006). The set of rules must be determined in terms of the problem to be solved. These rules are typically defined by experts in the specific domain.

**Defuzzification**

Defuzzification converts the fuzzy output set into a crisp output. Techniques such as the Centroid method, where the crisp value of the output variable is computed by finding the value of the center of gravity of the membership function, or the Maximum method, where the crisp value of the output variable is the maximum truth-value (membership weight) of the fuzzy subset, are used. Defuzzification is the last step in the fuzzy logic process and the output is interpreted for the context in which it is used (Cox, 1995)

©2012 Project Management Institute

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