AN ALGORITHM FOR ASSESSING DESIGN COMPLEXITY THROUGH A CONNECTIVITY VIEW

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1 Proceedings of the TMCE 28, April 21 25, 28, Izmir, Turkey, Edited by I. Horváth and Z. Rusák Organizing Committee of TMCE 28, ISBN ---- AN ALGORITHM FOR ASSESSING DESIGN COMPLEXITY THROUGH A CONNECTIVITY VIEW Joshua D. Summers Mechanical Engineering Clemson University joshua.summers@ces.clemson.edu Farhad Ameri Mechanical Engineering Clemson University fameri@clemson.edu ABSTRACT This paper describes an algorithm that is developed to quantify the connectivity of a graph. This connectivity is one of three aspects of complexity (connectivity, solvability, size) that has been identified in previous research. The complexity measures are applied against three components of design: the design problem, the design process, and the design artifact. This algorithm is illustrated step by step through an example. KEYWORDS design complexity, graph connectivity, coupling 1. INTRODUCTION Researchers in design have identified complexity as an important aspect of investigation. One of the common views of complexity that has been identified is the aspect of interconnectivity or the coupling between elements (1). Many of the approaches proposed in the literature have suggested that the coupling is a general measure of the depth of a tree or the number of relations between elements. Here, we propose a connectivity algorithm that may be used in both tree and graph formats. This connectivity algorithm is based upon minimal graph breaking. Coupling as a measure of complexity in design may be developed for the design problem, design artifact, and design process as an extension of the concept provided by (2). This coupling looks at the connections between variables or entities at multiple levels. This coupling measure requires that the representation of that which is being measures be in a graph-based format. Design processes may be represented in graph form where the tasks are nodes of a graph and they are connected through variable dependency (3). Bashir and Thompson s coupling measure is limited to trees (2). An extension for measuring the coupling of any type of graph is suggested here. It should be noted that this is not the only approach to measuring the coupling of graphs. The graph decomposition is a measure of how decomposable the graph may be. Removing relationships until the graph is separated into subgraphs demonstrates the coupling found in an entityrelation graph. This algorithm may be recursively applied to the divided graphs to continue the analysis. The algorithm attempts to break the graph into separate sub-graphs, which in turn are broken into sub-graphs. In order to illustrate the algorithm, two examples are provided. These examples are actually derived from two different graph based representations of design exemplars developed to retrieve boss protrusions from solid models (4). The specific details of the entities and relations are stripped from the models to focus exclusively on the connectivity algorithm. It should be noted that we argue in (1) that complexity should not be viewed singularly, but as a composite that includes size (amount of information represented), solvability (difficulty in evaluation/degree of freedom), and coupling (connectivity). Following the algorithm discussion and the provided examples, a brief analysis of the complexity of the algorithm is provided. It is argued that any measure of complexity is valuable only if it is readily calculated (5; 6). Finally, a brief outlook to future applications of this approach is provided in concluding remarks. 1

2 2. COUPLING COMPLEXITY IN ENGINEERING DESIGN Coupling complexity in engineering design can be investigated from various perspectives including structural, functional and parametric connectivity. Structural complexity: From the structural perspective, coupling complexity is the measure of the level of interconnectedness of the individual components that collectively form an assembly. Products with modular architecture are less complex structurally as compared to the products with integral architecture. Design products with lower level of structural complexity are more desirable since a less coupled design facilitates mass customization as well as serviceability and reusability. A connectivity graph (7) is an appropriate representation that can be used for studying the structural complexity of products. A connectivity graph is a labeled graph which shows how different components in a product are physically attached to each other Functional Complexity: Coupling complexity from a functional point of view measures the connectivity of various functions, both primary and supporting, within the function structure of a product. In a purely decoupled design, there is a one-to-one mapping between the product s functions and the physical components that comprise the product. A function structure (8; 9) can serve as a useful representation in measuring the functional connectedness of products. Parametric complexity: Components within a product can be connected through their parameters such as geometric dimensions, material, or surface finish. Coupling complexity from a parametric perspective measures the level of connectivity of various components in terms of their attributes. Parametric associative graph is the most suitable representation for parametric complexity analysis. A parametric associativity graph is a non-directed graph which demonstrates how different components are connected through their parameters. Figure 1 shows three components that are connected through their geometric dimensions together with their corresponding connectivity graph. All of the above mentioned representations can be demonstrated as graphs. Therefore, a graph-based complexity analysis can be used for measuring various types of complexities in engineering design. p1 p5 Spreader plate s4 Spreader shaft s8 s7 s7 s8 s6 s6 s3 s2 Stir cap s5 c1 c2 s1 p2 Spreader Shaft s5 p5 p4 s4 c3 c4 p3 p4 Stir Cap p3 s3 s2 p2 Spreader Plate Figure 1: An example of parametric associativity graph 3. GRAPH COMPLEXITY IN ENGIEERING DESIGN In graph theory, complexity of graphs is studied using various methods. Some methods use the number of the spanning trees of a graph as the measure of the complexity of the graph (9). Graph complexity has also been defined to be the function of the number of vertices, edges, and c4 c3 s1 p1 c2 c1 2 Joshua D. Summers and Farhad Ameri

3 proper paths in a graph (1) or the number of Boolean operations necessary to construct the graph from a fixed generating set of graphs (11). In engineering design measuring graph complexity through graph decomposition is the basis of several research projects in different areas including design for assembly (DFA), concurrent engineering, and engineering knowledge representation. Saitou et al. (12) proposed a decomposition-based method for assembly synthesis of structural products. In their method, first a bitmap image of a structure obtained via structural topology optimization is transformed to a graph with the equivalent topology. The resulting graph is then decomposed into multiple sub-graphs each representing a structural member. This method enables designers to explore a large number of feasible decompositions in the system-level design phase. XXXWang et al. (Wang, 1997) developed a system for decomposition of unfolded sheet metal product based on the decomposition of a spanning tree of the face-adjacency graph of the product. Chen et al. (13) developed an expert system for concurrent design by organizing requirements information using a graph decomposition algorithm. In another research project, Chen et al. (14) applied graph decomposition techniques for decomposing a group of interrelated design knowledge to smaller sub-groups for easier construction of product design blackboard system. Our literature survey did not reveal any application of graph decomposition techniques in quantifying the complexities of design problem, process, and artifact. 4. ALGORITHM The algorithm manipulates the graph that is being examined in terms of connectivity. This manipulation begins with the elimination of all unary relations as these do not contribute directly to the connectivity complexity of the graph. Examples of unary relations that are used in the design exemplar include FIXED relations and ID relations. The FIXED relation sets the value of the variable (entity) that is associated with this relation so that the value may not be modified. The ID relation is one approach to tag identification information to individual entities in the design exemplar. Once the unary relations are removed from the graph, the incrementing variables are initialized. From this point forward, the graph connectivity algorithm is a recursive algorithm that is applied against all subgraphs that are generated in the process. A cumulative record, or score, is maintained to quantify the connectedness of the graph. The algorithm is found in Table 1. Table 1: Connectivity Algorithm 1. Eliminate Unary Relations (do not contribute to connectivity) 2. Initialize values: level = 1; total = ; 3. For each graph to be searched a. Initialize set size = 1 b. For all combinations of relations in a set size i. Remove set size relations from the graph AN ALGORITHM FOR ASSESSING DESIGN COMPLEXITY THROUGH A CONNECTIVITY VIEW 3 ii. iii. Check for separation If separated graphs, mark the relation set removed c. If no relation set removed, increment set size and return to 3.b d. For all relation sets marked, find the combination of sets that remove the most relations without duplicate removal (number of sets) e. Calculate score: level * set size * number of sets + total f. Submit each distinct graph to 3 This algorithm is predicated upon the assumption offered by (15; 4) that many problems (geometric and parametric) in engineering design may be captured in bi-partite graph representations. This assumption has been extended to the representations used for design artifacts (i.e. boundary representation solid models or bond graphs) and for design processes (i.e. task graphs). We offer this algorithm as one possible approach to measure one dimension of design complexity in a systematic and repeatable manner. In order to clearly illustrate this algorithm, two examples are offered in the following sections. These examples include all the intermediate steps together with some of the resulting sub-graphs Example 1 Consider an example of an entity-relationship graph (Figure 2). The entities are represented by circles and the relations are represented by squares. This graph will be decomposed and the connectivity measure proposed will be applied. This is applicable to any bi-partite entity-relationship graph as commonly found in design problem and design artifact representations. The entities and relations are labeled to assist in tracking the evolution of the algorithm.

4 removing one relation at a time and then verifying whether two or more distinct graphs are formed. The first occurrence of a divided graph comes at the first level with one removed relation repeated for three relations. Figure 2: Entity-Relationship Graph (Initial Graph) The first step in the algorithm is to remove all the unary relations from the graph (Figure 3). In this example, there is only one unary relation (a relation that is associated with only one entity). This unary () relation is marked in gray in Figure 3. The unary relations are removed as they do not contribute at all to the coupling between entities. (passes) (fails) Figure 4: Search for Single Relation Three relations may be individually removed (,, ). Thus, four graphs are created and the algorithm steps to the next level. From the algorithm, a subtotal for the connectedness complexity calculation may be determined (Figure 5). The level is one. The set size is one (number of relations to remove as a set). The number of sets removed is three. The product of these three values yields a running total of six (1 * 1 * 3 = 3). This subtotal is added to the running total (in this case ) to calculate a new running total (3). Figure 3 - Entity-Relationship Graph (Removed Unary Relations) An exhaustive search is used to determine whether there are any single relations that may be removed to create separate graphs (Figure 4). This is done by Figure 5: (level 1, size 1, number 3) Total = 1*1*3 = 3 The resulting sub-graphs are illustrated in Figure 6. Graph 1.1 is the only sub-graph that has additional relations that need to be removed. The algorithm is recursively applied against this sub-graph. 4 Joshua D. Summers and Farhad Ameri

5 Graph 1.1 Graph 1.2 Graph 1.3 Figure 6: Resulting Sub-Graphs Graph 1.3 The algorithm requires that all single sets of relation be examined first for decoupling (Figure 7). (fails) (fails) Figure 8 continued (fails) (fails) It is found that no single relation may be removed to separate the graph. Therefore, pairs of relations are exhaustively explored to determine if there are any separating sets. Figure 9 shows four pairs (out of 28 pairs) explored in this level. (fails) (fails) {,} (passes) {,} (passes) (fails) (fails) Figure 7: Attempt at Single Relation Suppression {,} (fails) {,} (fails) Figure 9: Pairs of Relations Suppressed AN ALGORITHM FOR ASSESSING DESIGN COMPLEXITY THROUGH A CONNECTIVITY VIEW 5

6 There are many pairs that could be removed to separate the graph ({,}, {,}, {,}, {,}, {,}, {,}, {,}, {,}, {,}, {,}, {,}, {,}, {,}, {,}, {,}, {,}, {,}, {,}, {,}, {,}, {,}, {,}). All of these pairs are illustrated in Figure 1. Thus, the set of these pairs that covers as many of the relations as possible without overlapping is used (Figure 11). Four pairs may be selected that include all of the relations ({,}, {,}, {,}, {,}). All relations are at this point removed, yielding only sub-graphs of single entities, thus terminating the algorithm. Based upon this, the complexity value is determined for this second level, the size of the relation removal (pairs), and the number of pairs removed (4). The running total is now 3+16=19. Figure 1: Possible Pairs ignored. The solvability and size aspects of complexity are discussed in detail in (1; 4). Figure 12: Entity-Relationship Graph (Initial Graph) The next step is to suppress single relations in the graph to determine if there are any that connect two distinct sub-graphs. Two single relations that are explored in this level are illustrated in Figure 13. Figure 11: (level 2, size 2, number 4) Total = 2*2*4 = Example 2 This algorithm is repeated for a second example (Figure 12). This example has the same number of entities and relations (same size from the complexity perspective). The primary difference between the first and second examples is the connectivity for the graph. The first step in the algorithm is to remove all unary relations of which there are none in this example. This example is used to illustrate how the connectivity can be determined through this complexity-calculating algorithm. At this point, the solvability of the relation-entity problem is (fails) (passes) Figure 13: Single Relations Suppressed Three single relation sets may be suppressed and create distinct sub-graphs (,, ). This yields a complexity calculation of 3 (Figure 14). 6 Joshua D. Summers and Farhad Ameri

7 The complexity calculation algorithm is recursively applied against Graph 1.1. First, single relations are suppressed (Figure 16). (passes) (fails) Figure 14: (level 1, size 1, number 3) Total = 1*1*3 = 3 The resulting sub-graphs are illustrated in Figure 15. Two sub-graphs are of interest (Graph 1.1 and Graph 1.2) as they have remaining relations. (fails) Figure 16: Single Relations Suppressed for Graph 1.1 This results in being suppressed (Figure 17) and creating two sub-graphs (Figure 18). Graph 1.1 Figure 17: (level 2, size 1, number 1) Total = 2*1*1 = 2 Graph 1.3 Graph 1.2 Graph 1.4 Figure 15: Resulting Sub-Graphs for Level 1 Graph Graph Figure 18: Resulting Sub-graphs AN ALGORITHM FOR ASSESSING DESIGN COMPLEXITY THROUGH A CONNECTIVITY VIEW 7

8 Of these two sub-graphs, only Graph requires further decoupling (Figure 19). (passes) (passes) Figure 19: Single Relations Suppressed for Graph Applying the algorithm at this third level, three distinct sub-graphs (,, and ) are created by removing single relations (Figure 2). This results in three distinct graphs that each contain only one entity (Figure 21). (fails) (fails) {,} (passes) (fails) (fails) {,} (fails) Figure 22: Pairs of Relations Removed Figure 2: (level 3, size 1, number 2) Total = 3*1*2 = 6 Graph Graph Graph Figure 21: Sub-graphs Generated Returning to Graph 1.2, the algorithm is applied once more. Single relations are suppressed (Figure 23). As no single relations may be removed to create disjoint sub-graphs, the algorithm continues with relation pairs (Figure 22). This results in four possible pairs ({,}, {,}, {,}, {,}) that may be suppressed to create distinct disjoint sub-graphs where five relations are involved. At this point, only two pairs can be suppressed without overlapping: {,} and one of the other three. Thus, the calculation for Graph 1.2 continues (Figure 24). The resulting sub-graphs are shown in Figure 25. (fails) (fails) Figure 23: Single Relations Suppressed for Graph 1.2 Figure 24: (level 2, size 2, number 2) Total = 2*2*2 = 8 Graph Graph Graph Figure 25: Resulting Sub-Graphs 8 Joshua D. Summers and Farhad Ameri

9 Both Graph and Graph contain only one relation, each of which can be suppressed. This results in the complexity calculations of 3 and 3 (Figure 26 and Figure 27). Figure 26: (level 3, size 1, number 1) Total = 3*1*1 = 3 Figure 27: (level 3, size 1, number 1) Total = 3*1*1 = 3 The total complexity calculation for the second example is: = 25. Based upon the coupling calculation, the second example is more complex as it is more coupled than the first. A1.1.1 A1.1.2 A1.1 A2.1 A2.1.1 A2.1.2 A2.7 A1.3 A1.2 A2.6 A2.8 A2.2.1 A2.2.2 A2.2 A2.4 A2.5 A2.2.3 A2.4.3 A2 A2.4.4 A2.3.3 A2.3.4 A2.3.2 Figure 28: The connectivity graph for water sprinkle Example 3 The first two examples provided in this section were based on bi-partite graphs derived from design exemplars built to query solid models (4). To demonstrate practical applications of the proposed algorithm, a real-world example is presented below. Figure 28 shows the connectivity graph created for a water sprinkler. As can be seen in this figure, the left-hand side nodes of the graph represent the components of the water sprinkler while the righthand side nodes denote the physical connections (such as twist lock, snap fit and clip fit) between the components. Figure 29(a) shows the bi-partite graph corresponding to the connectivity graph of the sprinkler. The decomposition steps for one of the resulting intermediate sub-graphs are shown in Figure 29(b). The overall complexity score for the sprinkler (based on the connectivity graph) is 62. Interference Fit Twist Lock Snap Fit Slide Fit Clip Fit Hinged Contact/Friction Temporary Plated Metal Steel A2.3.1 Black Plastic Yellow Plastic Rubber/Nylon A2.3 A1 A1.1 A1.1.1 A1.1.2 A1.2 A1.3 A2 A2.1 A2.1.1 A2.1.2 A2.2 A2.2.1 A2.2.2 A2.2.3 A2.3 A2.3.1 A2.3.2 A2.3.3 A2.3.4 A2.4 A2.4.3 A2.4.4 A2.5 A2.6 A2.7 A2.8 Spike Assembly Plastic Cap Sub-Assembly Retainer Cap Internal Cap O-Ring Spike Internal Housing O-Ring Sprinkler Head Assembly Distance Deflection Arm Sub-Assembly Distance Deflection Arm Distance Deflection Arm Spring Rotational Deflection Arm Sub-Assembly Central Pivot Rod Rotational Deflection Arm Rotational Torsional Spring Rotational Control Sub-Assembly Rotation Control Arm Linkage Spring Rotation Direction Arm Rotation Impact Tab Sprinkler Head Bearing Sub-Assembly Sprinkler Head Bearing Compression Spring Rubber Washer Nylon Washer Sprinkler Head Housing Rotation Adjustment Tabs Distance Adjustment Knob Water Stream Diffusion Pin AN ALGORITHM FOR ASSESSING DESIGN COMPLEXITY THROUGH A CONNECTIVITY VIEW 9

10 twist-3 twist-2 hinged-1 hinged-2 hinged-2 twist-1 hinged-1 hinged-2 A2.2.1 A2.2.2 cont-1 cont-2 A2.2.2 cont-1 Inter-1 cont-4 cont-4 A1.1.1 A1.1.2 A1.2 A1.3 A2.1.1 A2.1.2 A2.2.1 A2.2.2 A2.2.3 A2.3.1 A2.3.2 A2.3.3 A2.3.4 A2.4.3 A2.4.4 A2.5 A2.6 A2.7 A2.8 (a) Inter-2 Inter-3 Inter-4 Inter-5 Inter-6 Inter-7 Inter-8 cont-1 cont-2 cont-3 cont-4 clip-1 clip-2 slide-1 slide-2 slide-3 slide-4 snap-1 snap-2 snap-3 snap-4 temp-1 temp-2 A2.4.3 A2.5 A2.5 Level 3, size 2, number 1 Total = 25+ 3*2*1 =31 Level 5, size 2, number 1 Total = 35+ 5*2*1 =45 slide-1 slide-2 slide-4 temp-2 hinged-2 cont-1 cont-4 slide-1 A2.5 Level 4, size 1, number 1 Total = 31+ 4*1*1 =35 Level 6, size 2, number 1 Total = 45+ 6*2*1 = 57 Figure 29: (a) The bi-partite graph corresponding to the connectivity graph of the water sprinkler (b) Decomposition of an intermediate sub- graph 5. ANALYSIS The algorithm detailed in this paper offers an approach for calculating a quantifiable value of complexity based upon graph connectedness. This is important when comparing design problems, design artifacts, and design processes, each of which may be represented in a graph based representation. The algorithm is invariant to the types of nodes and arcs in the graph. The algorithm is recursive. In the worst case, the complexity of the algorithm is dependent upon the number of relations in the graph. Table 2 illustrates the total number of combinations that are required for each graph, as determined by hand. (b) cont-4 slide-1 slide-1 temp-2 Table 2: Possible Combinations for Graphs Based upon Number of Relations Number of Relations Total Combinations The worst case situation is where the graph is fully connected and cannot be broken into sub-graphs, except by removing all the relations. The numbers derived here suggest that the complexity of the algorithm ranges from O(ln(N)N 2 ) and O(N 3 ), as evidenced by Figure 3. 1 Joshua D. Summers and Farhad Ameri

11 Figure 3: Complexity of Algorithm In order to further evaluate the performance of the proposed algorithm, its results were compared against the results obtained based on the spanning tree method, a common method for calculating graph complexity in graph theory. To this end, both methods were applied for calculating the complexity of the bipartite graph G as shown in Figure 31. G G Figure 31: A sample graph (G) together with its ' complete bi-partite graph ( G ) and complete regular '' graph ( G ). Based on the proposed algorithm, the complexity of the graph G equals to 4 and the complexity of the complete bi-partite graph corresponding to G (i.e., ' G ) equals to 9. Therefore, after normalization, the relative complexity of graph G is: C rel = C G 4 = = C ' G To calculate the complexity of the graph G based on the spanning tree method, the matrix-tree theorem is applied. Using the matrix-tree theorem (16), the number of spanning trees of graph G can be computed. A spanning tree of graph G is a tree composed of all the vertices and some of the edges of G. As the complexity of a graph increases, so does the number of distinct spanning trees within the graph. Based on the matrix-tree theorem, first it is necessary to construct the matrix Q for graph G such G Series1 (Ln N)N^2 that for i j, if vertex i is adjacent to vertex j in G, qij equals -1 otherwise q equals and for i=j, ij qij equals the degree of vertex i in graph G. Accordingly, for graph G: 2 1 Q * Matrix Q is then constructed by arbitrarily deleting one row and one column form Q. For instance, by removing first row and first column in the graph Q: * Q Finally, the number of spanning trees that exist in graph G can be obtained by calculating the * determinant ofq : * C = t(g) = Q = 4 G '' In the complete graphg, the maximum number of n2 spanning trees equals n, where n is the number of vertexes in the graph (17). Therefore, the relative complexity of graph G (with n=6) based on the spanning tree method is calculated as following: C rel = C G 4 = 6 2 C ' 6 G 4 = Its can be seen that the relative complexity of the graph G based on the proposed method in this paper is significantly higher than the relative complexity based on the spanning tree method. One reason is that the complete bi-partite graph, by definition, is less complex than a complete regular graph with equal number of vertices. It can be concluded that the proposed algorithm works better for bi-partite graphs since it yields better differentiation in terms of AN ALGORITHM FOR ASSESSING DESIGN COMPLEXITY THROUGH A CONNECTIVITY VIEW 11

12 complexity as compared to the spanning tree method which results in complexity values less than.1 for bi-partite graphs. 6. SUMMARY This paper has presented an innovative algorithm that may be used to assess the connectivity dimension of complexity. The algorithm has been applied to two graphs that are originally developed to represent design exemplars to capture boss-protrusion characteristics in solid modeling. These examples are equivalent from the perspective of size (same number of relations and the same number of entities). Recognizing that many different aspects of engineering design may be represented in a graphical format (design problems, design artifacts, and design processes), it is possible to use this algorithm in a representation independent manner. A brief analysis of the complexity of the algorithm suggests that this algorithm and complexity calculation is on the order of O(ln(N)N 2 ), a rather expensive proposition. For small graphs, this is an acceptable expense. For larger graphs, this algorithm may prove too cumbersome for determining the graph s connectivity. For this reason, additional strategies ought to be explored. However, as an initial attempt (one of the first systematic approaches proposed in the design literature), we feel that this algorithm is of value. This algorithm has been employed to examine the complexity of comparable design exemplars in the field of computer aided design (4; 18). Further, the algorithm has been employed in the investigation of large scale systems and their respective complexities as a means of comparing and evaluating (19). These examples of application are not sufficient to fully validate the utility of this approach in the design community; however it is a good first step. REFERENCES 1. Developing Measures of Complexity for Engineering Design. Summers, Joshua D. and Shah, Jami J. Chicago, IL : ASME, 23. Design Engineering Technical Conferences. pp. DTM Models for Estimating Design Effort and Time. Bashir, H. and Thomson, V. 21, Design Studies, Vol. 22, pp Complexity Analysis of Computational Engineering Design Processes. Ahn, J. and Crawford, R. s.l. : ASME, Design Theory and Methodology Conference. pp Summers, Joshua D. Development of a Domain and Solver Independent Method for Mechanical Engineering Embodiment Design. Mechanical and Aerospace Engineering, Arizona State University. Tempe, AZ : s.n., 24. PHD Dissertation. 5. Balazs, M. Design Simplification by Analogical Reasoning. Computer Science, Worcester Polytechnic Institute. Worcester, MA : s.n., PHD Dissertation. 6. Balazs, M. and Brown, David. Design Simplification by Analogical Reasoning. [book auth.] Cugini and Wozny. From Knowledge Intensive CAD to Knowledge Intensive Engineering. Norwell, MA : Kluwer Academic Press, 22, pp A Driver for Selection of Functionally Inequivalent Concepts at Varying Levels of Abstraction. Teegavarapu, Sudhakar, et al. 27, Journal of Design Research, p. in press. 8. Ullman, David. The Mechanical Design Process. New York, NY : McGraw-Hill, Inc., Pahl, G. and Beitz, W. Engineering Design: A Systematic Approach. Springer-Verlag : New York, NY, Graph Complexity and the Laplacian Matrix in Blocked Experiments. Constantine, G. 1-2, 199, Linear and Multilinear Algebra, Vol. 28, pp Combinatorial Graph Complexity. Minoli, D. 6, 1976, Nature, Vol. 59, pp Graph Complexity. Pudlak, P., Rodl, V. and Savicky, P. 5, 1988, Acta Inform, Vol. 25, pp Decomposition-Based Assembly Synthesis Based on Structural Considerations. Saitou, K. and Yetis, F. 4, 22, Journal of Mechanical Design, Vol. 124, pp CONDENSE: A Concurrent Design Evaluation System for Product Design. Chen, C., Occena, L. and Fok, S. 3, 21, International Journal of Production Research, Vol. 39, pp Knowledge Organization of Product Design Blackboard Systems Via Graph Decomposition. Chen, C., Wu, T. and Occena, L. 7, 22, Knowledge- Based Systems, Vol. 15, pp Bettig, B. A Graph-Based Geometric Problem Solving System for Mechanical Design and Manufacturing. Mechanical and Aerospace Engineering, Arizona State University. Tempe, AZ : s.n., PhD Thesis. 17. Tutte, W. Graph Theory. Cambridge, MA : Addison- Wesley, Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. 12 Joshua D. Summers and Farhad Ameri

13 Cambridge, MA : Addison-Wesley, 199. ISBN: Empirical Studies for Evaluation and Investigation of a New Knowledge Representation Structure in Design Automation. Summers, J. and Shah, J. Montreal, Cananda : ASME, 22. IDETC/CIE Conferences. Vol. CIE Martin, P. A Framework for Quantifying Complexity and Understanding its Sources: Applications to Two Large-Scale Systems. Engineering Systems Division, MIT. Cambridge, MA : s.n., 24. MS Thesis. AN ALGORITHM FOR ASSESSING DESIGN COMPLEXITY THROUGH A CONNECTIVITY VIEW 13