Dependency matrices

We have concluded that the independence of the elements that form system is a key factor in modularity. Well-known tools have been developed for the evaluation of the independence. The roots of the best-known method lie in project management. In 1981, Donald V. Steward criticized the defective ability of the critical path method and PERT (Program Evaluation and Review Technique) to show the order in which the design tasks ought to be performed to avoid unnecessary iterations. He introduced a theory which he called the Design Structure System [Steward 1981].

This analysis tool is based on a square matrix in which the subtasks of design are entered on the

horizontal rows in their order of execution. The same tasks are entered on the vertical columns in the same order. After this, the dependencies are marked in the matrix by proceeding from the first row and the first task. Each task in the columns is examined, and it is discovered whether “the task on the horizontal row requires the result of the task in the vertical column as its initial value.” If the answer is yes, there is a dependency between the tasks. On the diagonal line, the task is compared to itself, and no significant markings are entered there.

FIGURE 40. Steward's sample matrix. For illustrative purposes, the matrix also shows the iteration loop (which is not part of the presentation). [Steward 1981]

A matrix filled as illustrated above is a directional matrix: each relation marked in it is an arrow that shows the direction of the design data flow. All relations in the triangle below the diagonal line point forward in the order of the execution of the tasks and it is thus possible to execute them one after another following the cascade model. The relations above the diagonal line refer to iterations back to the previously executed task, and they are problematic dependencies for the management of the project. Steward's method seeks to eliminate these dependencies by reorganizing the tasks so that iterations are minimized and the ones that might remain are as short as possible. This is a task of mechanical optimization in which we aim to form a bottom triangle matrix, and to bring the relations as close to the diagonal line as possible. The task of reorganization can be carried out as an imperative procedure which can be written in the form of a genetic algorithm (as for example in [Rogers 1997]). A more detailed method is created by Marimont [Marimont 1959] and Sargent &

Westerberg [Sargent & Westerberg 1964]. Steward proves the competency of the method in his two articles, [Steward 1965] and [Steward 1981B], but we will pass these proofs as they are not relevant for the scope of the present study.

When the matrix arrangement nears the optimum, the relations causing iterations form limited task groups that Steward calls “blocks”, but whose later much-used name is cluster. The tasks in a cluster are the smallest possible number of tasks that must be simultaneously examined when design cannot be performed iteratively on the basis of estimations. Thus, the project tasks are divided into independent clusters. This is called partitioning.

FIGURE 41. Steward's sample matrix shown organized, the procedure of which is called partitioning. Each squared area is an independent design entity, and they can be executed one after another without iterations, following the cascade model. In this sample figure, there is only one big subtask. [Steward 1981]

Steward further introduces a method for tearing apart the blocks/clusters. For tearing, we need to examine the internal dependencies in the cluster. For this, Steward suggests drawing a diagram that is usually called a shunt diagram, which shows how the iteration proceeds within the cluster of tasks. In the figure above, the iteration was drawn in the figure. The resulting shunt diagram is shown in the figure below. We must note that Task 4 also exists in another, simpler iteration with Task 12. This is illustrated in the figure on the right. According to Steward, these diagrams show that, for example, replacing the dependency between Task 2 and Task 7 by using an estimate would not shorten the iteration and remove hardly any of the dependency. Instead, is would be beneficial to tear apart the dependency between Task 5 and Task 9 if technically possible so that the risks can be managed.

FIGURE 42. Steward's shunt diagrams on iteration [Steward 1981]

A better-known example of dependency matrices is the research on the composition of engine design teams, conducted by McCord and Eppinger for General Motors [McCord & Eppinger 1993]. This case is so well known that it is only shown here as figures. The data on the dependencies between the various element entities in the engine was collected via interviews. The importance of the dependency was divided into three strength classes. An engine is an integrated product in the sense that it contains a number of strong dependencies, which means that partitioning was not entirely successful. As it was a question of the composition of design teams, two clusters could share some elements. Thus, teams were formed and the participants reported that the design task was carried out successfully and in a company-specific record time.

FIGURE 43. The GM engine case: the original dependency matrix and the first organized matrix

FIGURE 44. The final divided matrix and a matrix in which partial cluster overlap is accepted

FIGURE 45. The proposed division for system teams which is said to have contributed to the success of the development project

McCord and Eppinger call their matrix a design structure matrix. As this example is better known than the original, the matrix is usually referred to by this name.

So far, we have only touched upon the internal structure of products. The next innovation was to discover that the clusters could also represent a module. This is a natural conclusion, the cluster-internal dependency is great and independent of the environment. At this point, however, the method essentially changes: it is no longer reasonable to present the matrix as directional. The product structure of modular products is usually considered static, which means it is not important which of the elements needs the other. While the matrix becomes symmetrical in proportion to the diagonal line, we attain new mathematical methods for the optimization. The most used of these methods is the so-called bandwidth algorithm which optimizes the points indicating the relations as close to the diagonal line as possible (into what resembles a band). An example of using this method could be Erkki Ahola's dissertation in which bus components of a similar life cycle were combined as replaceable units [Ahola 2000]. Another example of this will be found from among the industrial sample cases in Chapter 10.1.

A symmetrical dependency matrix is not by far the only approach when using matrices in product design and modelling. Johan Malmqvist has presented a summary of the presentation and analysis methods of the matrices [Malmqvist 2002]. The figure below illustrates the presentation types of the matrix methods: what is being compared and to what. The matrices presented above were of the intra-domain type, that is, they have examined the relations of the internal elements in the product.

Such a matrix is a square matrix by nature. The relations may be non-directional, in which case the matrix is symmetrical, or directional, in which case causality is related to the dependency.

FIGURE 46. The types of matrix methodologies, according to Malmqvist [Malmqvist 2002]

In another presentation, the focus remains on product-internal relations, but the objects proper are the elements in different domains, for example, on different levels of abstraction or domains. An example of this is Nam Pyo Suh's presentation of the linking of elements in the functional and the physical domain with a matrix presentation in axiomatic design [Suh 2001]. According to Suh's axiomatic design, the design world consists of four domains in which the items within the same product exist as having a different form depending on the context of examination. Suh's theory is discussed in greater detail in Chapter 12.3.

In product-level matrices, the elements in and the properties of the product range are compared.

Malmqvist presents the brand modularity matrix by Sudjanto & Otto [Sudjanto & Otto 2001] as an example of this. Malmqvist classifies methods in which several matrix presentations are linked together as belonging to the matrix methodology class, as for example the Quality Function Deployment method (see e.g. [Clausing 1994]). In some methods, various different forms may exist, such as in the Konfigurations- & Verträglichkeits matrix method by Luca Bongulielmi [Bongulielmi 2003]. The K/V matrices consist of two matrices of the intra-domain type and one matrix of the inter-domain type, but as these are linked together, the method can be classified as belonging to the matrix methodologies class. The matrices in this method are shown in the figure below.

FIGURE 47. The Konfigurations- & Verträglichkeits matrix method consists of three matrices that are linked together. The matrix is used in the management of configuration data. [Bongulielmi 2003]

Malmqvist recognizes seven methods of analysis. Of these methods, we are already familiar with partitioning and clustering, but the slight difference between the two methods was not yet explained.

- Clustering, in which the elements are grouped as clusters with strong internal relations and weak cluster-external relations

- Partitioning, in which the iterations in the process are minimized (design).

- Coverage, in which the completeness and the coverage of the entity is examined - Index computation, in which indices are computed to produce deductions

- Interaction focuses on the contents of the relations and guides the redesign

- Change propagation, in which the effect of the changes can be estimated by examining the relations

- Alignment, in which the relations of the product and the organization structure are compared

Of these methods, clustering, partitioning, index computation, and alignment can be considered in producing a modular structure (synthesis). In analyzing a modular structure, all of the methods are applicable. Erixon's method, to be presented in the following, begins with an analysis of the index computation type. In the industrial examples, we will introduce an alignment tool, the Late Point Differentiation analysis, in section 10.1.

Problems of the matrix methods

The simple logic of matrix methods makes them easy to use. The most problematic issue in developing modularity lies in problem setting. What relations are to be examined and how to interpret the results? The methods per se remain neutral in this issue, which means that the matrix methods are only auxiliary tools for designing a modular structure, not methods proper. A very

detailed level of examination must also often be selected in matrix presentations to be able to unambiguously define the relations. This, in turn, often leads to an excessive amount of work compared to the value of the end-result.