We present aggregate contribution as a solution for two longstanding issues with goal modeling languages such as i* and GRL which support decomposition and contribution links for intentional elements. An aggregate contribution summarizes the contributions of constituent parts (child elements) at the parent level, giving the modeler an overview of the actual impact of the parent while still enabling the correct evaluation of the goal model.
Goal models are graphs of intentional elements such as goals and tasks connected by
various relationships and optionally assigned to actors. Common relationships
supported by many goal modeling notations (e.g., i* [
In Section 2, we briefly discuss the two aforementioned issues and further motivate
our proposed solution with the help of an example. Section 3 then explains how we
address these two issues in detail. We use GRL to visualize the examples, but the
solution applies to any other goal modeling notation with decompositions,
contributions, and a propagation-based evaluation mechanism. The paper concludes and
briefly discusses future work in Section 4.
“Sloth” (Lack of Support for Relationship Abstractions) and “Wrath” (Analysis
Anomalies) are two of the eight “deadly sins” [
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ambiguities as the parent task does not seem to be contributing to the goal. To
overcome this slothful [
When an intentional element (task A), that is decomposed into A1 and A2, contributes to another intentional element (goal G), two cases arise. First, task A does not have any individual contribution to goal G and contributes only through its constituent parts, i.e., A1 and A2. In this case, only contributions of A1 and A2 should be considered while evaluating goal G (i.e., G should evaluate to 30 in Figure 1(c)). Second, task A, along with contributing through its parts, also has its own individual contribution to goal G. Here, contributions of all elements (A, A1, A2) should be considered for the evaluation of goal G (i.e., G should evaluate to 60 in Figure 1(c)). The second case often occurs when the parent captures contributions common to all its parts.
The new concept of aggregate contribution clearly differentiates these two cases by showing the range of contributions for a parent element based on its parts and the decomposition type. For an AND decomposition, all parts contribute for the parent. For an OR decomposition, at least one part but possibly all parts contribute for the parent. For an XOR decomposition, exactly one part contributes for the parent. An aggregate contribution is a range of values, because it shows the result of all possible combinations of the contributions of its constituent parts (1 for AND, 2N-1 for OR, and N for XOR with N being the number of parts). A new icon ( ) and its textual equivalent “A:” are used to denote a contribution link with aggregate contribution, followed by the actual value of the aggregate contribution.
All common GRL evaluation algorithms (quantitative, qualitative, mixed) first
calculate for a given element its evaluation value coming from decomposition links, then
contribution links, and finally dependency links [
Quantitative Evaluation Algorithm. This algorithm uses quantitative contribution values (e.g., task A1 contributes 10 to goal G in Figure 1(c)) and quantitative evaluations of contributing elements (e.g., 100 for task A1) to determine an element’s evaluation that is coming from contribution links (e.g., for task A). An aggregate contribution is only calculated for a decomposed parent element that has an outgoing contribution link and is decomposed into one or more constituent parts. A modeler, therefore, has control over whether to show an aggregate contribution by explicitly adding the outgoing contribution link for the decomposed parent element. In addition, an aggregate contribution may also be indicated with a Boolean flag added to the metaclasses representing the goal model or intentional elements in the GRL metamodel. The aggregate contribution calculated for this algorithm is referred to as quantitative aggregate contribution (QNAC). position types, showing the cases for only positive contributions, only negative contributions, and mixed contributions of parts. It considers that the contributions by an element’s parts may consist of both individual and range contributions, both positive or negative. A range contribution may exist because a part of an element may itself be a decomposed element with an aggregate contribution, an example of which is shown in Figure 3(b). An aggregate contribution is a range represented by “A:[Min,Max]”, except where the minimum and maximum values are the same which is simply represented by “A:Max”.
tions ↓
AND Min
Max Σ + Σ Σ + Σ
OR Min
Max min( , )
Σ + Σ Σ + Σ Σ + Σ max( , ) Σ + Σ
XOR Min
Max ) , ( x Legend: : Individual quantitative contribution value of part i tively, of a range contributed by part i , : Left (minimum) and right (maximum) quantitative contribution values, respec
, , respectively, of a range contributed by part i, when it contributes negatively : Individual quantitative contribution value of part i, when i contributes negatively : Individual quantitative contribution value of part i, when i contributes positively
: Left (minimum) and right (maximum) quantitative contribution values, respectively, of a range contributed by part i, when it contributes positively
: Left (minimum) and right (maximum) quantitative contribution values, Note:
All calculated minimums and maximums are capped to -100 and 100, respectively, to respect valid GRL contribution and evaluation values. Note that aggregate contribution as proposed here abstracts from the contributions of constituent parts only, but does not abstract from all combinations of contributions and evaluations of constituent parts. In that sense, the abstraction is based on the static goal model specification and does not consider the evaluation of the goal model. This can clearly be seen in Figure 2(a). Propagating the evaluation of B (50) to A with only the aggregate contribution (20) results in 10 and not the correct 25. It is possible to consider an aggregate contribution as an abstraction of the evaluation instead of the static goal model specification. In that case, however, AND as well as XOR decompositions have to be treated like an OR decomposition in Table 1 (i.e., the aggregate contribution in Figure 2(a) would be “A:[-10,30]” and in Figure 2(c) “A:[-5,35]”; this would ensure that the evaluation of B always falls into the range defined by the aggregate contribution of A on B and the evaluation of A).
As discussed earlier, a decomposed element may also contribute individually in addition to its parts. In Figure 3(a), B contributes to A individually as well, unlike the examples in Figure 2. In this case, the QNAC value for B is calculated the same way as for the examples in Figure 2 and the contribution of B is added on top. Thus, the evaluation of goal A is 40 (10 as shown in Figure 2(c) plus (30 * 100)/100 from B). To calculate the QLAC value abstracting contributions only for an AND decomposition (e.g., the QLAC value of B contributing to A in Figure 4(b)), the count of each type of qualitative contribution of the constituent parts of B to A is determined first. Then, the dominant for each strength is identified, i.e., a positive contribution and a negative contribution of the same strength (bottom-left to top-right diagonal in bold in Figure 4(a); e.g., “Make” and “Break”) cancel each other out (= “None”) and any remaining ones are dominant. Afterwards, the strongest contribution (most positive or most negative depending on the dominant) is combined with the strongest opposite contribution (most negative or most positive, respectively) based on Figure 4(a), until only either positive or negative contributions are left. Finally, the highest (in case only positive contributions are left) or the lowest (in case only negative contributions are left) contribution value is selected as the QLAC value of B to A. If all contribution values cancel each other out, the QLAC value is considered to be “Unknown”.
For example, Figure 4(b) shows an AND decomposition, where the parts of B contribute as follows to A: “Make”, “Help”, “Some Negative”, and “Hurt”. One “Make” and zero “Break” result in one “Make” being dominant. Zero “Some Positive” and one “Some Negative” result in one “Some Negative” being dominant. One “Help” and one “Hurt” cancel each other out. Since “Make” is the strongest remaining contribution, it is combined with the strongest opposite contribution, i.e., “Some Negative”, resulting in one “Help” according to Figure 4(a) (row 2, column 3).
Similar to the quantitative evaluation algorithm, decomposition hierarchies of arbitrary depths and various combinations of decomposition types are handled with a bottom-up approach.
Mixed Evaluation Algorithm. This algorithm uses qualitative contribution values
and quantitative evaluations of contributing elements to determine an element’s
evaluation that is coming from contribution links [
This paper discusses the issues “Sloth” (Lack of Support for Relationship Abstractions) and “Wrath” (Analysis Anomalies) in GRL and proposes the concept of aggregate contribution to address them. An aggregate contribution indicates the contributions of constituent parts for a decomposed parent element, given a sense of the overall combined contribution of the parent when all its parts are taken into account. The calculation of aggregate contribution for different decomposition types and evaluation algorithms is defined and detailed with the help of illustrative examples. In future work, we plan to find ways to abstract other goal modeling concepts such as dependencies and indicators for parent elements.