Introduction
The previous chapter presented the relationship between portfolio components and organizational objectives. Figure 2.1 illustrated the manytomany relationships between organizational objectives and portfolio components while Table 2.1 showed that more than one portfolio component could contribute to a single objective. In Table 2.1, it could also be seen that a single component could contribute to multiple objectives—as in the case of Component 1 contributing to Objectives 1, 3, and (n). The model presented in the previous chapter showed how the qualitative assessment of multiple components, based on multiple criteria, could be taken as input, processed through the application of fuzzification, rules, aggregation, and defuzzification. This provided a quantitative output that represented the cumulative contribution of portfolio components to organizational objectives. This chapter extends the discussion on the use of the model presented earlier.
The goal of this chapter is to illustrate how the model could be reused to present an alternate perspective on the componenttoobjective relationship. Demonstrating how the total contribution of individual components to multiple objectives can be computed and discussing how this information can be used in the decisionmaking process achieves this.
The perspective presented in this chapter on the portfolio component contribution to strategic objectives is aimed at determining which components make the highest individual contribution to the objectives in the system. Once the cumulative contributions of the individual components are determined, the components will then be ranked in order of their individual contribution to multiple objectives. Finally, a weighting is applied to the organizational objectives based on which objectives the organization considers as more important than others. The weighting acts as a factor that influences the outcome of the rank order of the portfolio components. Components that contribute to more important objectives will receive a higher contribution score. Decision makers can use this information when deciding on which components to accelerate, suspend, or terminate.
Determining the Contribution of Single Portfolio Components to Multiple Objectives
In the previous chapter, it was determined that portfolio components could contribute to multiple objectives. Table 2.1 showed that portfolio component (PC) 1 contributes to multiple objectives (OBJ) 1, 3, and n. The data is repeated in Table 3.1 for ease of reference. The degree of contribution to each objective varies from one to the other. The degree of contribution of PC1 to OBJ1 could be 0.35 while its degree of contribution to OBJ3 could be 0.17, and its contribution to OBJ(n) could be 0.25. The total contribution that a component makes to multiple objectives does not need to be equal to 1. The fact that PC1 contributes to three objectives, rather than just one, intuitively suggests that it is an important component. However, it needs to be determined how important it is in relation to PC2, for example, which contributes to only one objective.
Vision 

Objective 
Objective 
Objective 
Objective 
Objective 

Portfolio 
Portfolio Component 1 
a 
d 
i 

Portfolio Component 2 
c 

Portfolio Component 3 
b 

Portfolio Component 4 
e 

Portfolio Component 5 
g 

Portfolio Component 6 
f 
h 

Portfolio Component (m) 
j 

Source: Enoch and Labuschagne.^{1} 
Let us assume that PC2 contributes to OBJ2 to a degree of 0.88. The contribution of PC2 is greater in terms of degree than PC1, which has a total contribution of 0.77 (0.35 + 0.17 + 0.25) but PC1 contributes to three objectives instead of just 1. The impact of decisions regarding PC1 in terms of the portfolio mix is likely to be greater. If the investment committee decides to cancel PC1, for example, it would imply that three objectives would be impacted. These three objectives will not be fully achieved as a result of PC1 being cancelled.
The model described previously can be reused to address the aspect of a single component contributing to multiple objectives. The following discussion describes how the model can be applied.
Figure 3.1 shows that input variables (PCVar1 and PCVar2) for portfolio component (PC) 1 are evaluated for each instance that PC1 makes a contribution to an objective. In this example, it is indicated that PC1 contributes to three objectives, and hence the figure shows three instances of the StageA process (fuzzification, inference engine, and output) for PC1.
The process of fuzzification, rule evaluation (inference engine), and determination of a qualitative output is described in the previous chapter. To avoid repetition, the process will not be reexplained here but will be used to illustrate the degrees of contribution of PC1 to each of the three objectives.
Degree of Contribution of a Single Component (PC1) to Multiple Objectives
The following section briefly describes the process of determining the individual contribution of a single component (PC1) to multiple objectives (OBJ1, 3, and [n]). For each contribution relationship, the process is followed:
 The fuzzified membership value following the evaluation of each input variable in terms of the components contribution to a specific objective
 The rule view from the MATLAB® tool following the evaluation of the input variables
 A table listing the satisfied rules associated with the evaluation of the input variables
 The output membership functions once the membership functions of the two input variables have been aggregated
 The output fuzzy region that equates the aggregation of the output membership functions
 The defuzzified value representing the degree of contribution to the specific objective
Degree of Contribution of PC1 to Objective 1
Let us assume that the input variables are evaluated as follows:
PCVar1 = Good (with a membership degree of 0.784)
PCVar2 = High (with a membership degree of 0.812)
Figure 3.2 illustrates the membership degrees for each of the variables through the shading of the membership functions.
Applying the rules in the inference engine will result in the rules given in Table 3.2 being satisfied.
Figure 3.3 shows the rule view of the output membership functions. The shaded triangles illustrate the degree of membership following the aggregation of the membership functions from the satisfied rules in Table 3.2. 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.
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 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. 
The output fuzzy region for the degree of contribution of PC1 to Objective 1 is illustrated in Figure 3.4.
The defuzzified value, using MoM, resulting from this output fuzzy region = 0.935.
The dark solid vertical line in the figure indicates this.
Degree of contribution of PC1 to Objective 3
Let us assume that the input variables are evaluated (see Figure 3.5) as:
PCVar1 = Average (with a membership degree of 1.0)
PCVar2 = Medium (with a membership degree of 1.0)
Applying the rules in the inference engine, the rules given in Table 3.3 will be satisfied.
The output membership function based on the satisfied rule is illustrated in Figure 3.6 while the output fuzzy region for the degree of contribution of PC1 to Objective 3 is illustrated in Figure 3.7.
Rule 5 
If PCVar1 is Average AND PCVar2 is Medium, THEN Contribution is Moderate. 
The defuzzified value resulting from this output fuzzy region = 0.5.
The dark solid vertical line in the figure indicates this.
Degree of contribution of PC1 to Objective (n)
Let us assume, as illustrated in Figure 3.8, that the input variables are evaluated as:
PCVar1 = Poor
PCVar2 = Medium
Applying the rules in the inference engine will result in the rules given in Table 3.4 being satisfied.
The output membership function based on the satisfied rules is illustrated in Figure 3.9 while the output fuzzy region for the degree of contribution of PC1 to Objective (n) is illustrated in Figure 3.10.
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 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. 
The defuzzified value resulting from this output fuzzy region = 0.295.
The dark solid vertical line in the figure indicates this.
Calculate the Cumulative Contribution of a Single Component to Multiple Objectives
The quantitative outputs of all PC1 contributions determined in the previous section must be aggregated to work out the total contribution of PC1 to the three objectives. Based on the preceding discussion, the defuzzified degrees of contribution for PC1 to the three objectives are:
 Degree of contribution to Objective 1 = 0.935
 Degree of contribution to Objective 3 = 0.5
 Degree of contribution to Objective (n) = 0.295
The total contribution of PC1 to the objectives in this system of portfolio components and objectives is equal to the sum of the individual contributions. Table 3.5 shows the quantitative contribution of PC1 to each of the three objectives, as well as the sum of the contributions, based on the preceding discussion.

Objective 1 
Objective 3 
Objective (n) 
Total 
PC1 
0.935 
0.5 
0.295 
1.73 
Similarly, to determine a rank order of component contributions to the organizational objectives, the total contribution of the remaining portfolio components to multiple objectives can be calculated and the total contributions compared.
Determine the Relative Contribution of Single Portfolio Components to Multiple Objectives
The previous section assumed that each objective is equally weighted. In reality, objectives can be prioritized and a weighting applied to each objective to distinguish their importance in the system. This is essential to consider when looking at the individual component contributions to multiple objectives as it influences the importance of the individual components to each other in the system.
Let us assume that the objective in Table 3.5, cumulative contribution of PC1, is weighted as follows:
Objective 1 = 1.0
Objective 3 = 0.7
Objective (n) = 0.5
The higher the weighting, the more important a particular objective is compared to other objectives. In the example, Objective 1 has the highest weighting (1.0) while Objective (n) has the lowest weighting (0.5) implying that Objective 1 is considered by the organization to be most important while Objective (n) is considered to be least important.
The product of the objective weighting and the portfolio component contribution results in a new portfolio contribution value per objective and, by implication, a new total contribution value for PC1. This is illustrated in Table 3.6.
By applying the weighting assigned to each objective to the portfolio component contribution, the contributions are normalized and components can be more realistically compared. The same process is applied to the remaining components in the system after which the components can be ranked from highest to lowest.
Table 3.7 shows the rank order of portfolio components based on their total individual contribution to objectives.

Objective 1 
Objective 3 
Objective (n) 
Total 
PC1 
0.935 
0.35 
0.148 
1.433 
Vision 

OBJ 1 
OBJ 2 
OBJ 3 
OBJ 4 
OBJ (n) 
Total 
Rank 

Portfolio 
PC 1 
0.935 
0.350 
0.148 
1.433 
1 

PC 2 
0.655 
0.655 
4 

PC 3 
0.700 
0.700 
3 

PC 4 
0.455 
0.455 
7 

PC 5 
0.550 
0.550 
6 

PC 6 
0.375 
0.675 
1.050 
2 

PC (m) 
0.650 
0.650 
5 
The ranked order of components indicates to decision makers the importance of components in terms of the impact of decisions made. If the decision makers decide to cancel PC1, for example, and PC1 is the highest ranked component, it would mean that a significant portion of the objectives would not be achieved. Knowledge of the ranked order of components enables decision makers to understand where to allocate resources. The ranked order also helps to focus attention appropriately on the relevant components.
Conclusion
This chapter provided an alternate perspective of the contribution of portfolio components to organizational objectives. Here, the contribution of individual components to multiple objectives was considered.
The conceptual model from the previous chapter was reused to determine the individual component contribution to multiple objectives. The individual component contributions were then aggregated. This allowed for the ranking of portfolio components, with those components contributing to more objectives being ranked highly. In addition, by applying a higher weighting to organizational objectives that had a higher priority, their respective components contribution value was adjusted to a higher contribution value. This influenced their position in the rank order of components. The rank order of components provides additional information to decision makers and ensures better understanding of individual components and so enables betterinformed decisions regarding those components.