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System Dynamics (SD) has been suggested as the ideal quantitative complement to qualitative modeling approaches such as the Sensitivity Model (SM) and Methodology of Network Thinking (MNT). The purpose of this paper is to present a set of tools that contribute to the critical issue of linking both stages. It outlines an interactive analytical procedure based on matrix operations and algorithms from graph theory. Expected contributions to the modeling process are a better and more efficient understanding of qualitative feedback diagrams and, while client groups are actively participating, building confidence in the process is encouraged ñ before expert-based and resource-intensive quantitative modeling begins.
System dynamicists as well as related soft systems modelers have been paying increasing attention to combining the respective strengths of their approaches to meet their clientsí needs (eg Lane 1994). Over the last decade, the SD community in particular has made considerable progress in developing tools and techniques to support processes of group modeling. Team processes have become an effective way to increase ownership, consensus and commitment (eg Morecroft and Sterman 1994, Vennix 1996). However, many client groups do not have the time, the resources or simply don't wish to engage in quantitative modeling. Bridging gaps between qualitative and quantitative modeling approaches has become an increasingly important topic.
The Sensitivity Model (SM, Vester and von Hesler 1987) and the Methodology of Network Thinking (MNT, Gomez and Probst 1987) are qualitative approaches for covering the initial steps of a group modeling process. Both are established methodologies to facilitate communication, elicit knowledge, surface issues, find consensus and identify levers. For the quantitative stage they may be complemented by SD. The respective limitations and strengths of qualitative and quantitative modeling call for a synthesis of the complementary elements as suggested in the combination named Integrative Systems Modeling (ISM, Rios and Schwaninger 1996).
While enhancing understanding is a desirable goal in itself, most clients do wish to gather more quantitative information ñ like figures concerning possible future developments in various scenarios (predictions under specific conditions) or optimum decision (paths). Before issues of quantitative modeling shift the responsibility from clients to technically more advanced modelers, however, it is increasingly necessary to give clients the chance to interpret the qualitative model, allocate weights to various aspects, consolidate their ideas, and most of all to build confidence by gaining a deeper understanding.
Some of these requirements may be accomplished on the basis of a qualitative model. We propose an interactive analytical procedure applying tools based on matrix operations as well as algorithms from graph theory, the promising potential of which C. Kampmann (1996) has demonstrated by earlier research on SD models.
The many purposes for building a model can be derived from the use intended as well as process-related issues. During the last decades a range of tools and methodologies have been developed to support the process which usually works in four learning stages that build on each other:
| Modeling to | Tools and Methods | Results |
| 1. Communicate, to define and structure a problem | Description of context, system-in-focus and problem | Set of variables; Cross-Impact Matrix; Feedback Diagram |
| 2. Explain, identify influences and control levers | Qualitative structural analysis | Portfolio of Variables; Strategy matrix; Subsystems; Dominant Loops |
| 3. Explore and predict | Quantitative Models | Scenarios; Simulations; Conditional forecasts |
| 4. Optimise | Operations Research tools and models | (Optimum) Solutions |
Table 1: A modeling process for problem solving: use, tools and results
In the first stage modeling needs to support communication, to define and structure a problem from different perspectives (Checkland 1989). Describing the problem context and situation and the different goals usually represents the recommended start. This stage centers on describing the system-in-focus thus delivering all relevant variables and parameters whilst creating a common language among the participants. During the second stage the main interest is on defining relationships between variables. The visualisation of the system-in-focus as a network of nodes, representing variables, and edges, representing influences, are first results. Further detailing of the systemic structure can then take place in a cross-impact matrix (CIM) which indicates the strength and direction of the links with values different than zero (usually four intensities, Schlange 1995). SM and MNT chose similar avenues towards the development of a qualitative network with weighted influences that will allow early interpretation by the participants. In the third stage, more in-depth analysis is undertaken based on quantitative SD models. They can be used to develop scenarios, perform simulation runs and establish conditional forecasts. Finally, in a fourth stage the modeling process is completed by Operations Research techniques to find optimum solutions.
Qualitative Analysis of system structure serves three purposes. First, participants usually want to gather results from their efforts. Therefore, a qualitative feedback model needs to be commented and interpreted. Second, acceptance of and confidence in a model needs understanding and faith in the basic qualitative work. Third, an early review helps to eliminate misunderstandings and errors before intensive quantitative modeling begins.
Qualitative structural analysis is based on a feedback diagram in its representation as CIM. Software packages like Gamma for MNT rely on a direct classification of the variables. The SM-Tools package offers a more sophisticated interpretation aid by indicating the relative (in-)dependence of a particular variable in relation to the system as a whole as well as its interpendency in a 'portfolio of variables' (Schlange 1995).
An interpretation that solely relies on a direct classification of relationships between variables, however, does not take account of inherent feedback structures. The Mic-Mac-procedure (Godet 1991) is an algorithm which provides an indirect classification that allows to discern over- and underrated variables. A combination of the two approaches allows to encompass indirect influences and serves to identify key variables of a given system model (Gausemeier et al. 1995).
CI-analysis is taken still one step further by an interpretive model that presents the structural information from various points of view. By positioning variables in a Strategy Matrix according to their leverage and steering potentials actors' 'strategic levers' may be identified which have a high impact on a system's dynamic evolution (Schlange 1995a, Jüttner and Schlange 1996).
In sum, interpretation of systemic characteristics of variables is well supported by the tools mentioned. A major drawback remains, however, since participants will not gain any insight into a system's relationship structure representing their problem and its context. They essentially have to accept results of these steps in a black box manner. If they happen to be counterintuitive or even plain unwanted the consolidation of a team's ideas and their confidence in the effort tend to diminish. As a consequence, further quantitative modeling steps may even be cancelled. Building confidence calls for more relevant insights into systemic structure.
Some team members have called the task of analyzing a large, strongly interconnected feedback diagram one of 'sorting ends in a pot of cooked spaghettis'. Questions regarding feedback structure raised in a group modeling process were for instance: (1) Which variable has the highest direct influence on all other variables? Why is this so? (2) Which is the least directly influenced variable? (3) Which variable in our system directly influences (is influenced by) most other variables via the smallest number of relay variables (on the shortest path)? (4) Which are the dominant loops within our system? (5) Are there any independent subsystems that influence (are influenced by) the rest without being impacted by (able to impact) the rest? Answers may be found by applying selected algorithms from graph theory (Sedgewick 1995).
Floyd's algorithm provides answers to questions one to three.
It serves to find shortest paths in a weighted digraph network.
The CIM is rebased, replacing the strength of an influence with
an indicator of distance. A strong influence, denoted by the strength
signal three, will be replaced with a one and a weak one with
a three. Taking the Aachen RITTS case as illustration (Rios and
Schwaninger 1996), the results of the algorithm and some tentative
interpretation are shown in Table 2. Reading the rows shows that
from every variable there exists an influence path to any other
variable except to the variables W and X, which cannot be influenced
from within the system in focus. The smaller a value in the matrix,
the stronger the direct influence from one variable in a row on
a variable in a column. The number of variables influenced by
a variable is the row-sum of the number of all columns greater
than zero (column AV for Active influence on other Variables).
The sum of the numbers in a row indicates the total strength of
the influence on the system (column A-SoL for Active Sum
of Linkages). The smaller the number the stronger
the influence. Dividing the A-SoL by AV and multiplying by ((n+1)-AV)
provides an indicator for the influence of one variable on the
whole system (IoS for Influence on System
first column from right). The same principles apply for identifying
the influence from the system (IfS) on a variable in the bottom
row. The smaller the number IoS, the more direct the impact on
the system and viceversa. Of course, in a dense CIM one may just
as well take strong and medium linkages into account in order
to reduce the number of relationships. To find out the number
of relay variables (question 3) every linkage greater than zero
has to be replaced with a one.
Table 2: Results of Floyd's algorithm
Question four aims at discerning dominant feedback loops within a system. An algorithm is used that delivers all directed cycles in a CIM. It has been implemented in a mathematica package for the analysis of feedback loop gains and the system's behaviour of a fully completed SD model (Kampmann 1996). In the case illustration a total of 1940 directed cycles can be found. Of the strength 2 and 3 there are 366 directed cycles, but only 3 cycles of the strength 3: O-Z, O-Z-P and H-Q-L-K-J which now can be discerned as dominant loops. More detailed examination and interpretation may follow to build confidence. For quantitative model-building this information would provide firsts hints where to start and concentrate.
The fifth question asks for independent subsystems that influence (are influenced by) the rest without being influenced by (able to influence) the rest? This calls for an algorithm that finds strongly connected components (SCC) in a feedback structure. Within a strongly connected component feedback loops exist between the variables in the SCC but not to other variables outside the SCC. Tarjan's algorithm provides the means to identify subsystems in a digraph. Using all 90 links in the CIM with strengths greater than zero results in one large subsystem consisting of 24 variables except variable W and X, which influence the system from the outside; a result to be expected in a dense CIM. Taking into account only the 82 links of strength greater than one, the outcome is one SCC including variables A-E-F-G-H-I-J-K-L-M-N-O-P-Q-R-U-Z, all other being singular SCCs. Taking into account only the strongest 24 links (value 3), the case illustration yields just the system's dominant feedback loop as mentioned above.
The much smaller example in Figure 1 (Sedgewick 1995) displays the potential to answer the fifth question. SCC 1 as a subsystem influences SCC 2 which in turn affects SCC 3 via two links.
Even a hierarchy of subsystems may be identified that can help to derive at more or less independent subsystems not only by applying general understanding and logical reasoning, but from analyzing the underlying structure in its relationships as well.
Figure 1: Subsystems as Strongly Connected Components
An exploratory study of client's opinions on group modeling came up with three findings: 1. Client participation creates ownership and commitment. 2. Communication generates learning. 3. Quantification and simulation improve decision quality (Akkermans and Vennix 1996). We believe, our tools have the potential to enhance a team's understanding of the model creation and to build confidence in their problem structuring efforts. Gaining early insights into path lengths, dominant loops and system structure may be seen as preparation for the more technical modeling stages. Speeding up communication and learning with tools that support the analysis of decisive issues may help to bridge the gap between qualitative modeling in a group effort and the subsequent expert modeling like a relay race. In order to ensure efficient proceedings towards SD modeling, however, there still remains the facilitator's task to make sure that the variables in a qualitative system are levels and do not mix wildly with rates. But like in engineering most aspects of a successful design are decided in the very early stages of the process, when good questions and communication are essential.
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