Axiomatic Based Decomposition for Conceptual Product Design

This paper describes a structured methodology for decomposing the conceptual design problem in order to facilitate the design process and result in improved conceptual designs that better satisfy the original customer requirements. The axiomatic decomposition for conceptual design method combines Alexander's network partitioning formulation of the design problem with Suh's Independence Axiom. The axiomatic decomposition method uses a cross‐domain approach in a House of Quality context to estimate the interactions among the functional requirements that are derived from a qualitative assessment of customer requirements. These interactions are used in several objective functions that serve as criteria for decomposing the design network. A new network partitioning algorithm is effective in creating partitions that maximize the within‐partition interactions and minimize the between‐partition interactions with appropriate weightings. The viability, usability, and value of the axiomatic decomposition method were examined through analytic comparisons and qualitative assessments of its application. The new method was examined using students in engineering design capstone courses and it was found to be useable and did produce better product designs that met the customer requirements. The student‐based assessment revealed that the process would be more effective with individuals having design experience. In a subsequent assessment with practicing industrial designers, it was found that the new method did facilitate the development of better designs. An important observation was the need for limits on partition size (maximum of four functional requirements.) Another issue identified for future research was the need for a means to identify the appropriate starting partition for initiating the design.


Introduction
The long-term health of a firm, even its very survival, is directly tied to its ability to innovate successfully-to provide customers with a continuing stream of attractive new products and services (Rosenblatt and Watson 1991).Whitney (1990) characterized the product realization process as the method by which a company identifies the need for a product, determines the customer's expectations, converts the expectations into specifications, and uses those specifications to design both the product and the methods for making, distributing, and supporting it.Product design, starting with conceptual design, is at the heart of this process.Suh (1990) described design as the creation of product solutions that satisfy perceived needs by mapping functional requirements in the functional domain into design parameters in the physical domain.Functional requirements (FRs) are the negotiable performance characteristics of a product.The degree to which FRs are realized in a design solution, driven by decisions made by the designer, ultimately determines the success of a product in the marketplace.FRs are related to associated customer requirements (CRs) that state the customers' needs in their own words, the "voice of the customer."In the design process, constraints are the non-negotiable bounds on an acceptable design solution.Design constraints may be functions of the laws of nature, the environment in which the product will function, governmental regulations, or corporate decisions or policies.Finally, design parameters (DPs) are those physical attributes of a design solution which, when varied by the designer, facilitate, or hinder satisfaction of one or more FRs.In a design problem, the objective is to select the DPs that determine the FRs that in turn maximize satisfaction of the CRs subject to relevant design constraints.To illustrate these concepts, consider the design of a pet food dispenser.One of the CRs is likely to be "refilling is easy".An associated FR would be "average time to refill dispenser without spillage".Assuming that the reservoir is in the shape of an upright cylinder, the associated DP might be the radius of the cylinder.Suh (1990) identified two fundamental principles or axioms that, when properly applied, consistently yield better design solutions: (1) maintain the independence of FRs (Independence Axiom), and (2) minimize the information content associated with the task of fulfilling an FR.The power of the Independence Axiom is that it encourages the designer to seek solutions that satisfy FRs independently, thereby removing the problem of managing interactions.Whitney (1990) observed that it is rarely possible to eliminate conflicts between FRs and, therefore, complexity is a major cause of difficulty during design.An example of this conflict can be illustrated using the pet food dispenser design problem introduced previously.In addition to the CR "refilling is easy", with the associated FR "average time to refill dispenser without spillage", assume that another appropriate CR is "maintain food freshness" with the associated FR "number of reservoir air changes per hour".Clearly, there is a potential conflict and interaction between FRs that result from the natural laws.Both the number of FRs and the interaction among FRs contribute to the complexity of a design problem.The consequences of design complexity include lower design quality, lengthy design cycle times and a process that appears confused and lacking direction (Whitney 1990).Whitney indicated that successful design requires that major conflicts be recognized early in the process, their existence and importance be agreed upon, and only then can they be resolved.Suh (1988) concluded that ". . . the conceptual nature of the axiom-atic approach may make it difficult for average designers to apply them to detailed design of components without a prescriptive algorithm specifically developed to solve a class of problems".
In order to adequately address design complexity, it is advantageous to have a structured methodology to identify and account for complexity early in the conceptual design process.This paper proposes a new structured approach that meets this need.It includes a method of assessing the complexity of the design problem as well as a decomposition approach that improves the manageability of the design complexity.The new approach is examined empirically through several design experiments.

Decomposition and Axiomatic Approaches to Design
Decomposition, dividing a problem into simpler subproblems, is the prototypical means of addressing complexity in design problems (Smith and Brown 1993).Alexander (1964) pioneered the use of networks to represent and decompose complex design problems in terms of key customer needs.In the network representation in Figure 1, vertices represent the FRs and edges represent interactions between the FRs.Edge length is inversely proportional to the strength of the interaction between FRs, with shorter edges indicating tighter coupling and longer edges indicating greater independence.Clusters of connected FRs can be thought of as sub-problems that are relatively independent of other FRs and other sub-problems.Alexander recommended that FRs be grouped so that there is a high level of interaction within groups and little interaction between groups.These smaller, relatively independent sub-problems can then be solved with minimal adverse effect on the rest of the design.Ulrich and Eppinger (1995) described several general approaches for design problem decomposition.Decomposing the design problem by key customer needs is an approach that can be used for innovative/creative products in which form, not working principles or technology, is the primary problem.User action decomposition is useful for products with simple technical functions and significant user interaction.Functional decomposition, decomposing the design problem based on the functions the product must perform, can be used for decomposing technical products.Finally, physical/architectural decomposition is useful when concepts already exist, either in incremental design situations or when developing product architecture.
A number of other decomposition approaches have been developed that could be applied to the product design problem.Courtois (1985) developed criteria for decomposition of complex systems.Pahl and Beitz (1991) developed function structure diagrams, modeling the flow of materials, energy, and signals between the functional units of a design.Pimmler and Eppinger (1994) proposed a decomposition approach for developing the architecture for a technical product in an incremental design situation.Yager (2000) developed intercluster measures that characterize the preferences for cluster size.Design process decomposition has received considerable attention in the literature (e.g., Steward 1981;Hippel 1990;Kusiak and Park 1990;Kusiak and Wang 1993a;Kusiak and Wang 1993b;Eppinger et al. 1994;Componation 1995;Krishnan et al. 1995;Krishnan 1996).Browning (2001) provided a comprehensive review of applying two types of Design Structure Matrices (DSM) to four types of decomposition and integration problems.Thebeau (2001) used DSM with cost and time objectives to develop a system architecture for a basic elevator system.Recently, Chen and Lin (2003) used a combination of DSM, the analytic hierarchy process, and cluster analysis to decompose a large interdependent task group into smaller and manageable task groups.These various research efforts have focused on understanding the basic relationships among design tasks, considering both precedence and interaction.The focus of the research area has been to better organize the design process through improved scheduling, enhanced project team assignment, and improved communication channels.Most of the methods require some sort of measure of the interaction among the items to be decomposed.
After determining the best set of design sub-problems, the design team must still develop overall system-level design solutions.Arif (1998) proposes two strategies for developing design solutions when starting from a set of design sub-problems.Using a parallel strategy, each sub-problem is solved independently and then integrated to develop final design solutions.Multiple designers or teams work concurrently on different sub-problems.The parallel strategy is ideal when sub-problems are independent or loosely coupled with clearly defined interfaces.When there are interactions between sub-problems, then design iteration will likely be required to resolve conflicts.Ulrich and Eppinger (1995) describe a parallel integration method that uses a combination table to systematically identify and assess promising combinations of conceptual design fragments that were developed by solving design sub-problems.
Using a progressive strategy, the design team proceeds from one sub-problem to the next.The design is developed progressively, with design features being added to the current design based on the incoming FRs in each subsequent sub-problem.The need for iteration is greatly reduced when using this strategy.Arif (1998) proposes a methodology for sub-problem sequencing when using a progressive design strategy.
It should be noted that design integration extends beyond the solution of the decomposed design problem.Integration analysis also addresses the development of product architecture-the arrangement of the product's functional elements into physical components, sub-systems and modules, and the coordination of their interactions.Noting that product architecture begins to emerge during the conceptual design phase, Pimmler and Eppinger (1994) proposed a three-stage methodology that integrates functional elements into chunks based on interactions between elements.Using the interaction information early in the design process, a design team can consider architectures that minimize coordination complexity.Hoekstra (1992) found that axiomatic design methods could enhance the creative process by allowing for a wide range of designs to be considered during the initial design conceptualization.However, he reaffirmed Suh's conclusion that with only a set of generalized axioms available, the designer lacks the structured approach required to focus his/her efforts in an organized way towards the solution of a problem.In follow-up research involving axiomatic methods, Salustri and Venter (1992) proposed a formal axiombased theory of design information called the hybrid model.Jung and Billatos (1993) applied the axiomatic design method to the development of an expert system for assembly.Sekimoto and Ukai (1994) explored aspects of creativity within the context of the axiomatic design method.Albano and Suh (1994) proposed the concept of axiomatic design as a framework for concurrent engineering.Hillstrom (1994) applied axiomatics along with conventional DFMA tools in the study of interface analysis for developing modular products.However, none of the methods used an explicit decomposition approach.
Although there has been a significant level of recent research in design decomposition focused on organizing design activities and developing architectures for large scale, function-dominated technical products where basic concepts are known, there has not been a comparable emphasis on developing structured methodologies that will facilitate and enhance the product conceptual design process.The axiomatic decomposition method described in this paper responds to this demand.

Axiomatic Decomposition for
Conceptual Design Method

Overview
The prior research on using axiomatic approaches to support the design process identified important design principles, but does not provide sufficient structure to guide particular conceptual design activities.
As indicated, the decomposition methods have been used primarily to organize the design process.The use of decomposition in the context of axiomatic design provides the basis for a new structured methodology to facilitate conceptual design.This section describes an axiomatic approach for decomposing the conceptual design problem that integrates Alexander's network partitioning formulation of the design problem with Suh's Independence Axiom.An important contribution is the determination of interaction parameters using concepts related to the House of Quality.The axiomatic decomposition method is an iterative process that consists of four primary steps: • developing the conceptual design context, • decomposition of the conceptual design problem, • creating the conceptual design, and • conceptual design review: reflection and iteration.
The major focus in this research is on the first two elements of the process: the development of the design context and design problem statement, and the decomposition of this design problem into a number of manageable sub-problems.The decomposition step in itself is an iterative process that involves developing a set of alternative decompositions of the conceptual design problem.

Developing the Design Context
The axiomatic decomposition method begins with creating the design context leading to the development of the design problem statement.The design problem statement includes identification of the functional requirements for the product and measures of the interaction among the FRs.The FRs are developed from customer requirements.Specifically, the CRs in the customer domain are deployed to the FRs in the functional domain.
3.2.1.Identifying Customer and Functional Requirements.The axiomatic decomposition method begins by using proven House of Quality (HOQ) structured design techniques to map CRs into FRs (Hauser and Clausing 1988).HOQ is the first stage and the basic design tool used by Quality Function Deployment (QFD), a set of structured planning and communication routines that focus and coordinate organizational resources in designing, manufacturing, and marketing products that customers want to purchase.Griffin and Hauser (1993) and Armacost, Componation, Mullens, and Swart (1994) addressed methods for identifying and prioritizing the CRs.Hauser and Clausing (1988) and Akao (1990) described approaches for developing FRs using the HOQ.Key elements of the HOQ are illustrated in Figure 2. CR(s) from the customer domain (rows) are deployed to the FR(s) in the functional domain (columns).Relationships between the CRs and FRs are identified in the relationship matrix in the center of the house.In using the HOQ, it is important that at least one FR be strongly related to a given CR and vice versa.In real-world design applications, a CR is often related to multiple FRs, indicating conflicting objectives and the need for design tradeoffs.

Estimating Functional Requirement Interactions.
The axiomatic decomposition method requires a measure of the interaction between pairs of FRs.These interaction parameters required by the axiomatic decomposition method are closely related to the "roof" or technical interaction matrix in the HOQ, which characterizes the interactions between pairs of FRs.Using conventional HOQ approaches, negative interactions represent conflicting objectives that necessitate design tradeoffs, while positive interactions represent supporting objectives.In most HOQ models, the designer estimates the strength (and direction) of the interaction between pairs of FRs without any explicit consideration of the relationships among the CRs and FRs.
The axiomatic decomposition method uses the relationships between the CRs and FRs found in the relationship matrix to create a new way to measure interactions between pairs of FRs using a six-level, discrete, numeric interaction scale to represent the degree of FR interaction where a value of zero indicates no interac- tion between a given pair of FRs and a value of five indicates maximum interaction.The interaction value is negative if there is a negative interaction, or positive if the interaction is positive.If a particular FR only had an effect on a single CR, then the interaction between a pair of FRs would be equivalent to the interaction between the corresponding CRs.In practice, a given FR will affect one or more CRs in addition to the one to which it has a primary link.An interaction is said to exist when, by varying an FR over a wide range of feasible values (perhaps associated with a wide range of very loosely defined pre-conceptual design ideas), it is likely to affect the satisfaction of a CR other than the one to which it is directly linked.In other words, by varying some functional characteristic of the design (functional domain), the satisfaction of more than one CR is likely to be affected (customer domain).In conventional HOQ approaches, such a cross-effect between a FR and a CR is often represented symbolically on a four level scale indicating: substantial cross-effect, moderate cross-effect, minor cross-effect, and no cross-effect.
To estimate the level of each pairwise interaction between FRs, the axiomatic decomposition method uses a new cross-domain, pairwise assessment approach.The approach examines CR to FR cross-effects in the solution-neutral environment of the customerfunctional domains.The cross-domain structure using the pet food dispenser is illustrated in Figure 3.When considering design ideas intended to limit air changes in the food reservoir, it is likely that the time to refill without spillage will be increased.Likewise, when considering the attributes of design solutions that make refilling easier, it is likely that the number of air changes per hour may also be affected.
The cross-effects in the relationship matrix in the HOQ are used to estimate the interaction between FRs in the roof.To estimate the interaction parameter for any two FRs, the magnitude of the CR-FR cross-effects are mapped into the desired interaction value.Figure 4 shows the cross-effect-interaction mapping matrix showing all possible combinations of cross-effects and their associated interaction value magnitudes.In each cell, the two nodes on the left (numbers) represent two CRs and the two nodes on the right (letters) represent the corresponding primary FRs.The diagonal lines connecting the nodes represent the cross-effects and the numbers adjacent to the cross-effect lines indicate the relative magnitude of the cross-effect.The values (labels) corresponding to cross-effect magnitudes are shown on the right side of Figure 4.The small square in the cell includes the assigned relative magnitude of the value of the resulting pairwise interaction between the FRs.
Figure 4 maps all possible pairwise interactions into an interaction scale.The scale (0 to 5) is an arbitrary construct that includes sufficient numeric range to provide adequate discriminatory power.Suppose that the cross-effects for the pet food dispenser example shown in Figure 3 are captured in the partial HOQ given in Figure 5.Note that each cross effect is judged to be moderate and given the arbitrary value of two in the relationship matrix of the HOQ (Figure 5).We then look in the mapping matrix (Figure 4) for the cell with both cross-effects equal to two.The desired cell lies at the intersection of the third row and third column of the matrix and indicates a FR interaction magnitude of four.Then, the interaction of the two FRs shown in the roof of the HOQ (Figure 5) is assigned the value of four.Therefore, the components of Figure 4 provide a mapping from the cross-effect relationships (in the center of the HOQ) to the functional interactions in the roof of the HOQ.A designer may desire to use different scale values, but the general procedure illustrated in Figure 4 provides a reasonable method for assigning interaction values.For example, a design team may want to differentiate between (3,3), (3,2) and (3,1)   cross-effects, all assigned an interaction value of five in Figure 4.An alternative 10 point scale would provide the ability to assign each of the 10 possible combinations of interactions a unique score.The resulting FR interaction measures can now be used as a basis for decomposition of the FRs into appropriate groups to provide the design focus.

Decomposing the Design Problem
When the functional requirements and the magnitude of their pairwise interactions have been identified, the design problem can be characterized as a design network using Alexander's (1964) structure.In that context, the FRs are the nodes of the design network and the interaction measures determine the length of the connecting arcs.Following both Alexander (1964) and Suh (1990) direction to reduce the complexity of the design problem, it is necessary to decompose the design problem into smaller, relatively independent subproblems by partitioning the network so that there is a high level of interaction within each partition and low interaction between partitions.The designer can then solve the sub-problems and synthesize their solutions with minimal conflicts.The scale of the resulting network partitioning problem is driven by the size of the solution space.The size of the solution space is a function of the number of FRs; specifically, the number of ways of grouping n FRs into k ϭ 1, 2, 3, . . ., n partitions.The total number of feasible solutions can be calculated by summing the Stirling numbers for k ϭ 1, 2, 3, . . ., n partitions as in equation ( 1) (Gawlik 1996.)Table 1 illustrates how the size of the solution space grows with increasing numbers of FRs.

Developing a Decomposition Objective Function.
A critical element of the axiomatic decomposition method is the mathematical expression of the objective function used to drive the partitioning.Experience with cluster analysis clearly indicates that the resulting groupings are dependent on the particular objective and the approach used and that no one method is dominant.Similarly, one would expect that several methods may be useful to decompose design elements (FRs) into appropriate partitions for product design.In the design context, Alexander sought to simultaneously: (1) place highly interactive FRs together within partitions, and (2) place more independent FRs in separate partitions.Related research has tended to follow Alexander's lead, with the objective of maximizing the number of significant interactions within partitions and minimizing the number of significant interactions between partitions.The new axiomatic design method seeks to consider not just the number of interactions, but also their relative strength.
In selecting candidate objectives for the new axiomatic decomposition method, several criteria should be considered.The first criterion that should be applied is applicability-that is the degree to which the partitioning problem can be meaningfully represented by the objective function.The second criterion for objective function selection is flexibility in selecting the level of partitioning symmetry-that is the extent to which the objective function responds in like manner to both a decrease in within partition interaction and a corresponding increase in between partition interaction.For example, a perfectly symmetrical objective function would respond the same to changing a within partition interaction with magnitude 5 to 0 as it would to a change in a between partition interaction with magnitude 0 to 5. Various researchers have pointed out that a higher between partition interaction is significantly less desirable than a low within partition interaction.Due to the varying nature of design problems, it is not reasonable to define a generic ratio for penalty on high between partition interaction to low within partition interaction.Therefore, the objective function selected should provide designers with the flexibility to vary this ratio based on the design problem.The third criterion that should be used is structural independence.An objective function is said to exhibit structural independence when the marginal impact of any interaction is independent of specific network configuration as long as its type (within or between partitions) remains the same.This implies that the objective function is independent of the number of within vs. between partition interactions.
The alternative partitioning objective functions considered in this paper exploit the natural analogy between Alexander's proposed objective and that of the group technology (GT) problem in manufacturing system design that seeks to group parts into families and machines into cells such that the machine utilization within cells is maximized while the number of part moves between cells is minimized.A broad range of objective functions have been considered for adaptation to the product design decomposition problem.The GT functions include "grouping efficiency" (Chandrasekharan and Rajago-  (Gawlik 1996) sums the partition scores associated with all pairs of functional requirements depending on whether FR pairs are located in the same or different partitions.The decomposition score to be maximized is computed as (2) where: I ϵ magnitude of the interaction (determined by the cross-effects analysis), and S(I) ϵ partition scores from transformation shown in Table 2 An interaction's impact on the objective function, S(I), depends on its location in the network structure of a solution.If two FRs with a large interaction are assigned to the same partition, then their interaction contributes to the objective.If the same two FRs with a large interaction are assigned to two different partitions, then their interaction detracts from the objective.It is important to emphasize that a negative S(I) is not related to a negative interaction between the FRs.A negative S(I) simply indicates that the relative positioning of the two FRs detracts from the partitioning objective, either placing highly interacting FRs in different partitions or placing minimally interacting FRs in the same partition.
3.3.3.Non-Linear Objective Function.The linear objective function (LOF) using Table 2 provides the same penalty for two FRs with a high interaction being in-cluded in the different partitions as that for two FRs with no interaction being placed in the same partition.As indicated previously, Alexander, Suh, and others indicated that the between partition penalty should be higher (asymmetry).The increased penalty weights on high interaction FRs being in different partitions is achieved by applying equation (3) for I ϭ 3, 4, or 5 and p Ͼ 1.
The resulting scores in Table 3 are used with equation (2) to compute the objective function value for a given assignment of FRs to partitions.3.3.4.Grouping Index Objective Function.Arif (1998) adapted Nair and Narendran's (1996) Grouping Index measure for use in conceptual product design.According to Alexander (1964), FRs should be partitioned so that the FRs within a partition are heavily related and the FRs in different partitions are not related.When interactions are measured on a 0 to 5 scale (5 being the highest) all within partition interactions should be 5 and all between partition interactions should be 0. (Note that this ideal value depends on the construction of the scale in Tables 2 and 3.) This ideal is incorporated in the modified GI function so that it can be interpreted as a measure of "distance" from the ideal as follows. where: a ϵ ⌺ weights of all within partition interactions b ϵ number of within partition interactions c ϵ ⌺ weights of all between partition interaction q ϵ weight parameter for within partition and between partition interactions The first term in the numerator of X is the sum of the distances of all within partition interactions from their ideal value of 5. Similarly the second term in the numerator is the sum of the distances of all between partition interactions from their ideal value of 0. The denominator in X is simply the ideal score.
The coefficient q is the weight given to within partition interactions and (1-q) is the weight assigned to between partition interactions.The value of q gives flexibility to the designer in selecting the weights for between and within partition interaction.The ability to change the value of q based on the importance given to within and between partition interactions makes it more applicable to the design problem.A common value for q found in the literature is q ϭ 0.5.Sensitivity analyses (Arif 1998) varying the value of q on three different networks suggest that partition solutions are insensitive to the value of q except at higher levels (i.e., q Ն 0.7).In general, increasing q increased the penalty for between partition interactions, resulting in an increase in the number of partitions in the solutions.Therefore, a designer wishing to limit the number FRs in sub-problems and increase the number of partitions might choose to increase q beyond a value of q ϭ 0.5.

The Network Partitioning Algorithm.
The network partitioning problem considers the network of FRs connected by interaction magnitudes and seeks to decompose the network into partitions that represent the "best" grouping of FRs using the given criterion or objective.The conceptual design network partitioning problem is included in the computationally difficult class of problems known as NP-complete (Garey and Johnson 1979).Therefore, a heuristic algorithm was developed to construct appropriate partitions of the design network.The algorithm is summarized in Figure 6 and described in detail in Gawlik (1996).
The network partitioning algorithm yields a partitioned network of interacting FRs where there is a high level of interaction within each partition and low interaction between partitions.Each partition corresponds to a conceptual design sub-problem.Because each subproblem is relatively independent, the designer is assured that the design solutions being considered will have a minimal impact on other sub-problems.Any significant between-partition interactions that do exist are known and can be considered as solutions are developed and synthesized.

Creating the Conceptual Design
The process of developing, synthesizing, and selecting a single, coherent, functional design concept from the decomposed design problem is the essence of new product development.There are a number of welldefined development approaches that are described by Pflueger (1991), Rickards (1980), VanGundy (1981), Pahl and Beitz (1991), Pugh (1990), and Ulrich and Eppinger (1995).Designers may select whatever approaches are appropriate in the design context and apply them to the decomposed design sub-problems.The result of this step is a selected conceptual design, representing the "best" solution attainable using the current design problem statement.

Design Review
An important element in the axiomatic decomposition method is the design review.The designer must either freeze the concept and proceed with detailed design or continue concept development.Critical factors which should guide this decision include: (1) how well does the latest concept satisfy the FRs and constraints in the design problem statement, (2) how independent is the latest concept, and (3) are time and resources available for further concept development.If the decision is made to continue concept development, the designer should re-assess the underlying interaction structure of the design problem before proceeding and then repeat the partitioning.

Summary
The new axiomatic decomposition method extends proven structured design methodologies, creating a new method that formalizes and facilitates design problem decomposition.Specific innovations include: the development of an explicit approach for estimating interactions between functional requirements, the identification of relevant objective functions for the network partitioning problem, the development of a heuristic algorithm for solving the partitioning problem, and the integration of these elements into a design methodology.While the axiomatic decomposition method is a logical approach to product design, experimentation was required to demonstrate both the technical performance of methodological components as well as the overall usability and effectiveness of the method.These experiments and their results are summarized in the following section.

Overview
Several designed experiments were conducted to assess the performance of the axiomatic decomposition method.The goals of the experiments were (1) to assess the viability of the partitioning algorithm and the proposed objective functions, (2) to examine the usability of the method, and (3) to determine the value of the new method to facilitate the development of better conceptual designs.To assess viability, two analytical experiments were conducted using several sets of test design networks.Usability of the method was examined by evaluating the performance of undergraduate students using the method in engineering design projects.Finally, the value of the method was assessed by an evaluation using experienced industrial designers.The results of these assessments are summarized below.

Viability Assessment
The performance of the network partitioning algorithm was tested using a replicated full factorial (2 3 ) designed experiment (Gawlik 1996).Sixteen test networks were randomly generated, varying the number of FRs between six and fifteen and interaction values between zero and five.The LOF was chosen as the objective function for the test problem formulation, since this experiment was performed prior to consideration of alternative objective functions.The measure selected for performance measurement was the percent of an estimated optimal solution that the algorithm was able to attain.An estimated optimal solution was used as the baseline for performance measurement, since the true optimal solution could not be found for larger networks (i.e., a means for efficiently enumerating all feasible partitioning solutions without repeating solutions could not be found).Instead, a very large scale sampling technique based on the Binomial distribution was used to randomly generate feasible solutions until there was a 98% probability that the true optimal was included in the sample and, thus, had been identified.Three factors were hypothesized to affect performance: the size of the network (number of FRs), the mean value of the interaction magnitudes, and variance of the interaction magnitudes.These values and the experimental results for each of the sixteen test networks are shown in Table 4.
The results indicated that no main effects or interactions were significant.The results also indicated that the partitioning algorithm found solutions that closely approached the estimated optimum solution in almost every case.Results for the 16 test problems averaged over 98% with a high of 100% (the algorithm found the estimated optimal solution) and a low of 87%.The results suggest that the new network partitioning algorithm is a viable method for decomposing a design network.

Objective Function Viability Assessment.
Both the Grouping Index function and the nonlinear objective function were developed to provide the asymmetry recommended by both Suh and Alexander.An experimental evaluation was conducted to assess the viability of using the three functions in the decomposition algorithm.The three objective functions considered are the LOF (Equation ( 2)), the power function with an exponent (p) equal to 1.75 in Equation ( 3), and the GI function with q ϭ 0.5 in Equation ( 4).The parameter p in the power function was given a value of 1.75 because it produced a scoring pattern nearest to that suggested by Gawlik (1996).Three different design problems were selected for the analysis.Each problem had an increasing number of non-zero interactions.The first two problems included six nodes and the third problem included seven nodes.The third design problem is illustrated in Figure 7 where the number next to each arc is the magnitude of the interaction between FRs on a scale of 0 to 5 (5 being highest).
The results of partitioning the three networks using the three objective functions are summarized in Table 5.The table is structured in three major horizontal sections, one for each network.Within each network section are three lines of results, one for each of the objective functions used to partition the network.For each network/objective function combination the optimal solution (obtained by complete enumeration) is given, including the number of partitions and the value of the objective function (shaded).The values of the other objective functions for the same solution are also provided for comparison.The solutions (partitions) using the LOF function and the power function (p ϭ 1.75) are identical for all three networks.The solution using the GI (q ϭ 0.5) function is the same for the second network, but different for the first and third networks.In addition, the solutions using the GI function for the first two networks had no between-partition interaction greater than 3.This demonstrates the ability of the asymmetric GI function to discriminate against higher betweenpartition interactions.
The partitioned network solutions described in this section were all optimal, since the relatively small number of FRs allowed complete enumeration.Arif (1998) also compared the optimal results from the three networks (Table 5) with solutions obtained using the heuristic algorithm.The two non-linear objective functions were tested including a sensitivity analysis of the objective function parameters p and q.Results are summarized in Table 6.The algorithm worked very well for two of the three test problems, returning optimal solutions for all problem configurations except one.The third problem was problematic, resulting in only two optimal solutions and negative values of the objective function for several power function configurations.These results suggest that the effec-tiveness of the heuristic is dependent on the objective function and the network.Note however that in most of the test problems, the algorithm obtained the overall optimal solution.Together, the objective function assessment and the partitioning algorithm evaluation support the viability of the method from a constructive perspective.The remaining assessments examine the practical aspects of partitioning for conceptual design decomposition.

Usability Assessment
To assess usability, two separate experiments were conducted using student subjects enrolled in engineering design classes (Gawlik 1996).The hypothesis to be tested is that students using the axiomatic decomposition method would produce superior designs.The first experiment involved students in their final semester of a two-year Electrical Engineering Technology program offered by a community college.Their individual assignment was to design a prototype lawn sprinkler system controller.The second experiment involved Mechanical Engineering students in the first semester of their two-semester senior capstone engineering design class at a university.Their individual assignment was to design a semi-automated system for deploying and retracting a car cover.
4.3.1.Method.The (quasi-) experiments were conducted in conjunction with the normal conduct of the classes by the regularly assigned instructors.The students in each class were divided into three groups of approximately equal size according to the normal practices of the individual instructors.Student assignment to groups was random for the Engineering Technology class, while students in the Mechanical Engineering class were distributed in groups based on the student's current overall grade point average (GPA) to provide "balance" among the groups.The intent in both experiments was to achieve groups with relatively equal design capability.Design methodology information was provided to the student subjects in three lecture-style classroom presentations.
All students received the usual design methodsrelated information normally provided by the instructors.The Group 1 students did not receive any additional design guidance.The Group 2 students also received additional information on Suh's axiomatic method and how to apply it to concept design.The identification of interacting functional requirements was stressed through the use of selected examples, including the use of the design matrix to identify acceptable designs.Finally, the Group 3 students received the same information as Group 2 plus information on the new axiomatic decomposition method and its application using the LOF.Emphasis was placed on the assessment of interaction magnitude using crosseffects and completion of the interaction matrix. 1 100 GI with q ϭ 0.3 2 100 GI with q ϭ 0.5 2 100 GI with q ϭ 0.7 1 93.98 The completed design concepts were rated by practicing engineers using conceptual design rating sheets to structure the rating process.All of the engineer raters had a Masters degree in an engineering discipline, but not necessarily in the specific technical area related to the design problems.Each conceptual design received two independent ratings.The same two engineers rated all conceptual designs for each class.The two scores received for each design were averaged and used in a one-way ANOVA to determine if the mean scores among the three subgroups were significantly different.
4.3.2.Results.The experimental results are summarized in Table 7.For the experiment performed in the Engineering Technology class, the difference among the means is not statistically significant at the significance level ␣ ϭ 0.05 (F ϭ 0.137 Ͻ F 2, 33, 0.95 ϭ 3.29).The difference between the means is more pronounced for the Mechanical Engineering class.The ANOVA results (F ϭ 3.55 Ͼ F 2, 34, 0.95 ϭ 3.28) indicate that the mean scores for the Mechanical Engineering groups are not equal at a significance level of ␣ ϭ 0.05.Using Scheffe's paired comparison method, the axiomatic decomposition method (Group 3) is statistically different from the no design method group (Group 1).
Observations of the students throughout the design projects provide valuable insight concerning the usability of the axiomatic decomposition method.Initially, the concept of identifying and assessing functional requirement interactions seemed to be most difficult for the students.As students proceeded with their designs, they had difficulty with the interaction matrices.Students from the two year technology program had noticeably more difficulties than those from the four year engineering program.Students in Groups 2 and 3 frequently sought clarification on the design methods.Nearly all students who sought individual instruction showed some improvement in their ability to understand the concept of interactions and apply it to their design problem.Only about thirty percent of the students took advantage of opportunities for individual assistance.
It is believed that the difficulty experienced by the student subjects had several causes.First, in order to identify functional requirement interactions, designers must have a fundamental understanding of the general sciences as well as the natural laws and physical relationships that govern them (i.e., weight of a body and the friction force on it).This understanding is essential to design in general and axiomatics in particular.It is believed that this lack of understanding of natural laws and principles was a primary reason that the students from the two year program had more difficulties than those from the four year program.A second cause of student difficulty was their lack of experience and training in the design methodologies themselves.Suh's axiomatic method provides some fundamental principles of design.However, these principles require instruction, guidance, and practice for even a highly qualified designer.The presentation of Suh's axiomatic method and the new axiomatic decomposition method in a lecture-style classroom setting was generally of limited effectiveness.If a deeper understanding of axiomatic methods is necessary, other learning techniques should be considered.
Considering the limitations associated with using students, the results of the two experiments suggest that the axiomatic decomposition method is usable, although significant training and education may be required, and leads to generation of better conceptual designs.

Value Assessment
To assess the practical value of the axiomatic decomposition method, it is important to obtain input from experienced industrial designers.A final "experiment" was conducted to assess the value of using the axiomatic decomposition method to design a real product.The product that was the subject of the conceptual design was an automatic pet feeder.
4.4.1.Partitioning.Previous assessments of customer requirements for an automatic pet feeder led to a set of 10 functional requirements using the House of Quality approach.Cross-effects comparisons as described above were used to obtain estimates of the FR interaction magnitudes.The network decomposition algorithm was used to solve nine different network decomposition problems generated by using each of the three objective functions over a range of selected parameter values (LOF; power function with p of 1.75, 2.0, 2.25, 2.5, 2.75, 3.0; and GI with q of 0.5 and 0.7).The resulting solutions were not all unique, yielding the five different partition structures shown in Table 8.The conceptual design network with one of the five partition structures (Partition Set # 1) is illustrated in  4.4.2.Method.Two highly experienced Industrial Designers were asked to develop conceptual designs for an automatic pet feeder using the decomposed design networks.Although it was expected that using the partitions would be beneficial, the "experiment" was really an exploratory analysis.Both of the designers were given the networks in the same order.The designers were asked to consider the interactions while designing and to design the product using one partition at a time.No specific instructions were given on how to use the networks.The designers developed the designs independently of each another.No time limit was imposed.A joint evaluation by the designers was used to conduct the qualitative assessment.

Results.
The designers strongly agreed that partitioning the conceptual design problem gave them more insight into the design.One design resulting from the use of Partition Set 1 is shown in Figure 9.The two participating designers indicated that they actually do partitioning informally and that it would be helpful if it is done using a structured process.To reduce the time commitment for the volunteer Industrial Designers, they were not involved in the assessment of the interactions that led to the five different partitions.They felt that they would have more insight into the product had they had that experience.
The designers noted that the choice of the first partition was an important factor in the design effort.The first partition acted as a driver for the rest of the design process and the rest of the design is built on top of this driving partition.The designers proceeded in a serial manner, progressively adding details to the design using the FRs in each partition.They typically selected the first partition as the smallest or the topmost.On realizing that their choice of the driving partition was wrong, they immediately moved to an- other partition.A missing element in the structured axiomatic decomposition method is some measure of the importance of each partition.The designers showed a natural tendency to avoid large partitions in the beginning, but it was felt that an initial choice of a larger partition might be helpful in building a stronger foundation for the design.The designers found themselves most comfortable with partitions containing between two and four FRs.The designers felt that a partition size of three to four FRs was the most that they could handle.When the number of FRs exceeded four, the designers reported that they subdivided the partition.For 10 to 15 functional requirements, the designers felt that they can handle up to four FRs in a partition.This observation suggests that the size of partition when decomposing a product design problem should be constrained between two and four FRs.This is consistent with the conclusions of Michelena and Papalambros (1995) who suggest that the size of the largest design partitions should be minimized.
In summary, the industrial design "experiment" verified the value of partitioning, and in particular, of using a structured approach.The partitions generated by the axiomatic decomposition method provided valuable information for the industrial designers to develop a conceptual design for a new product.

Conclusions
The axiomatic decomposition method described in this paper provides an effective method to decompose a design problem early in the conceptual design stage.
The method creates a number of relatively independent design subproblems, each consisting of a set of highly related functional requirements.Experiments using practicing industrial designers and capstone engineering design students found the method to be useable and to produce better product designs.The experiments also suggested that the method is more effective for more experienced designers and that design sub-problem size should be limited to a maximum of four functional requirements.The research also yielded several techniques that may be used independently of the overall design method.These included a cross-domain technique for identifying and quantifying interactions in the roof of the House of Quality and a partitioning technique to rearrange columns in the roof of the House of Quality to emphasize related design challenges and potential tradeoffs.
Several research questions remain.The first involves identifying the best "starting" partition, providing a better set of functional requirements for designers to use in the initial design.The second research question involves developing a method for identifying the preferred objective function for use in the partitioning algorithm.In the application with industrial designers, three objective functions were explored, yielding five unique sets of partitioned networks.

References
Figure 1 Alexander's (1964) Network Representation of the Design Problem.
Figure 2House of Quality.

Figure 3
Figure 3Assessing Interactions Between FRs for Pet Food Dispenser.

Figure 5
Figure 5Partial HOQ for Pet Food Dispenser Design Problem.
Figure 7Network Representation of Design Problem #3.

Figure 8 .
Figure 8.These five sets of partitions were used by the industrial designers in the empirical assessment.

Figure 9
Figure 9Concept Sketches for Pet Food Dispenser Using Partition Set 1.