2.1. DC–capability mapping

Product design comprises three main stages, namely, conceptual design, embodiment design, and detail design (Ulrich & Eppinger, 2000). Among them, conceptual design is the most important stage, where 60% of the resources needed to manufacture a product are committed (Hundal, 1997). In reality, after a product has been designed, design changes can still be expected and are done iteratively. These design changes may lead to unnecessary delays in the launching of a new or improved product, wastage of resources, and increased costs. One way to minimize design changes is through a proper analysis of the capabilities of DCs with respect to product specifications (SP).

During the conceptual design stage, a design is often represented by a set of DCs that provides a rough description of the technology, procedures, and the shape of a target product (Ulrich & Eppinger, 2000). In this work, CP refers to the actual performance of the final product in relation to a design specification. For example, a vacuum cleaner may have a 150-in. water lift of suction head as its CP, and at least a 140-in. water lift of suction head as its design specification. The suction head of 150-in. water lift is the actual performance that meets the design specification of the vacuum cleaner. Accordingly, the relationships between DC and CP domains, and between CP and SP domains can be easily formalized (Fig. 1).

The relationships of design concepts, capabilities, and specifications.

As shown in Figure 1, the set of concept (DC_i) embodied in the DC domain decides the capabilities (CP_j) of a product in the CP domain through physical interactions among the various DCs. The relationships between DC and CP domains are highly convoluted, as many factors affecting the CP domain are not apparent. Using the proposed approach, these relationships can be represented by a so-called DC–capability mapping process. The CP domain embodies the actual performance parameters that may or may not be required by the specifications (SP_k) denoted in the SP domain. For the purpose of design analysis, a one–one relationship has to be established for each capability required by a specification so as to facilitate comparison. This implies that every design specification needs to be satisfied by a CP. Conversely, those CPs that are not associated with any of the design specifications are not needed for design analysis. More specifically, the transformation of DCs into CPs can be realized through the aforementioned DC–capability mapping process. Because of the vague description of conceptual design, experienced designers need to be engaged to evaluate the feasibility of the mapping. As a result, it is desirable to establish an approach that is able to extract design rules from past design information to support the mapping process, handle the vague descriptions of conceptual design, and deal with the incomplete information arising from UA. In this respect, the rough set theory can possibly be used to extract the necessary design decision rules for DC–capability mapping, and facilitates early design analysis of a creative design, in which the relationship between DCs are not apparent. Further analysis shows that even for a very simple product, it may have more than one design specifications. Consequently, because each design specification needs to be fulfilled by a CP, more than one CPs can be expected. Thus, it is imperative for the proposed approach to acquire the abilities to handle multiple CPs and deal with multidecision attributes in design decision rules extraction.

2.2. Selection of DCs

Essentially, the purpose of product design is to realize a design that is able to satisfy a set of design specifications (Ulrich & Eppinger, 2000) derived from customer needs. At the conceptual design phase, DCs are formulated and analyzed. Those DCs that can possibly meet design specifications are selected for further treatment. Basically, the analysis of DCs involves the establishment of CPs, which embody the expected performance of a product through the above-mentioned DC–capability mapping process. The suitability of a DC can be evaluated by comparing SPs with corresponding CPs. If all the SPs have been satisfied, designers will then proceed to the next phase of design. Otherwise, the DC is reviewed and modified.

In practice, without a prototype, it would be difficult to determine CPs during the conceptual design phase, as the relationship between DCs and CPs can be fairly complex. However, even with such a prototype, designers may still experience difficulties in determining CPs due to the vague descriptions of conceptual designs. As not all the design parameters can be explicitly determined during the conceptual design stage, only the main components of a design are often specified, making it highly ineffective for the realization of a prototype. In reality, an experienced designer may be able to identify suitable DCs. However, the decision made is often based on a combination of past design experience, intuition, and engineering knowledge. Using past design experience in deriving DCs implies that past designs may contain potentially useful knowledge for the design of a new or similar product. Thus, it would be advantageous if this knowledge could be extracted to provide the basis for the analysis of DCs.

3.1. Basic notions

The rough set theory provides the mathematical basis for the extraction of concepts from a data set called information system. Mathematically, an information system (I) can be expressed as follows:

where U is a nonempty finite set of objects and A is a nonempty finite set of attributes.

The set of attributes A can further be denoted as

where C is a nonempty set of condition attributes and D is a nonempty set of decision attributes.

Rough set theory has been widely used to solve problems in different domains such as data-driven decision making (Kusiak et al., 2000; Nguyen et al., 2001; Lee & Vachtsevanos, 2002) where decision rules were derived based on historical information. These decision rules were then used to classify newly acquired information. Mollestad and Komorowski (1999) employed the notions of indiscernibility provided by the rough set theory to derive decision rules. Zhai et al. (2001) proposed a hybrid technique that combined the strength of genetic algorithms and rough set theory for the classification of concepts. In the domain of conceptual design, Arciszewski et al. (1993) applied the rough set theory to realize a learning system for the design of wind bracings in tall buildings. The rough set based learning system had demonstrated the capability of extracting the dependencies in the design of wind bracings that were poorly understood. Essentially, the aforementioned approaches were developed to enable the extraction of decision rules based on an information system with complete information. They are not able to handle the UA that is prevalent in the process of creative DC generation. Furthermore, as multiple decision attributes need to be dealt with simultaneously, changes in any of the attributes will require the entire rules set to be regenerated. The work of Mollestad and Komorowski (1999) is extended to extract decision rules from an information system with UA. For clarity, it is summarized as follows:

Step 1.

Based on the data registered in an information table, form a decision relative discernibility matrix comprising n columns and n rows, where n is the number of objects.

Step 2.

Complete the n × n decision relative discernibility matrix. A decision relative discernibility matrix is used to identify the attributes that discern a pair of objects if the corresponding decision(s) of the pair of objects are different. More formally, for each row and column (i, j) of the discernibility matrix, insert the condition attributes that discern objects i and j, where i ≤ n, j ≤ n, if the value of the decision attribute for objects i and j are different.

Step 3.

Form an antecedent clause of a rule by combining the condition attributes for each of the rows using Boolean operators. For each of the cells in the information system, combine all the entries using the OR Boolean operators to form a term. Subsequently, join all the terms in a row with the AND Boolean operators.

Step 4.

Simplify the antecedent clause obtained in Step 3 using Boolean algebra, if necessary.

Step 5.

For each of the antecedent clauses obtained in Step 4, form a rule by incorporating the decision attributes. Subsequently, insert the value registered by the condition and decision attributes corresponding to each row.

As an example, consider an information system (Table 1) comprising five objects such as five different designs of vacuum cleaners with four attributes each (three condition and one decision attribute). For simplicity, it is assumed that each attribute possesses two possible values, 0 and 1, which are identical to TRUE and FALSE, respectively, in a binary information system. Step 2 is then applied to form a decision relative discernibility matrix (Table 2).

An information system

A decision relative discernibility matrix

Note that the entries of the diagonal cells are empty because the same objects are being compared. Pairwise comparisons are then carried out. For example, on Row 1, Object X₁ is compared with other objects. Columns 2 and 3 are empty because Objects X₁, X₂, and X₃ have the same decision attribute. Columns 4 and 5 are expected to have entries because Object X₁, and Objects X₄ and X₅ have different decision attributes. Thus, Column 4 has a value of ABC because the values of the condition attributes A, B, and C of Object X₄ are different from those of Object X₁. Hence, Objects X₁ and X₄ can be discerned using the set of condition attributes A, B, and C. Similarly, Column 5 has a value of A, because the values of A of Objects X₁ and X₅ are different. From Tables 1 and 2, the antecedent clauses for the rules can be formed for each of the rows according to Step 3 as follows:

Row 1: (A + B + C)A; Row 2: B + C; Row 3: (A + C)(A + B); Row 4: (A + B + C)(A + C); and Row 5: A(B + C)(A + B).

Consider the antecedent clause of Row 1, (A + B + C)A. By applying the dual of absorption theorem of Boolean algebra, it can be readily shown that (A + B + C)A ≡ A. Thus, based on Step 4, the antecedent clauses can be simplified as follows:

Row 1: A; Row 2: B + C; Row 3: (A + C)(A + B); Row 4: A + C; and Row 5: A(B + C).

Based on Step 5, the complete rule set is given as follows:

More specifically, Rule 1 states that for an object N, if the value of Attribute A (condition) is 0, then the value of Attribute D (decision) is 0. Similarly, Rule 2 states that for N, if either the value of Attribute B (condition) is 1 or the value of Attribute C (condition) is 0, then the value of Attribute D (decision) is 0. These mappings are useful to relate the condition attributes to corresponding decision attributes, and can be used to classify new objects. As previously mentioned, the condition and decision attributes registered on an information system represent the DCs and CPs, respectively. In so doing, the mapping of DCs and CPs can be achieved.

3.2. Unavailable information (UA)

In product design, analogy is frequently used to generate DCs. Thus, DCs can be adapted from existing designs that are known to be able to perform the required functions. However, existing designs may not be able to meet all the requirements of the design specifications. Additionally, the attributes of some of the DCs may not be relevant to the new design. These discrepancies, as previously mentioned, create a different type of incomplete information, known as UAs, within the information system that is constructed based on the information obtained from existing designs.

In the rough set theory, Grzymala–Busse and Hu (2000) proposed a simple method based on the probability of occurrences of attribute values to repair an incomplete information system. Such a method requires prior knowledge of data distribution, which may not be readily available for most data sets. In addition, data with a small population may yield erroneous results that are caused by the inaccuracy in estimating the probability. Kryszkiewicz (1998, 1999) assigned a “null” value to replace all the incomplete attribute values. The null value represents all the possible values attainable by the attribute with incomplete value. Subsequently, the original indiscernibility relation is expanded into a tolerance relation known as similarity relation. Rules can then be derived from the incomplete information system by replacing the indiscernibility relation with a similarity relation. Felix and Ushio (1999) solved the incomplete information problem by introducing rough discernibility relations, that is, surely discernible and possibly indiscernible. These relations were used for rules derivation in an incomplete information system to replace the original indiscernibility relation. Wang (2002) proposed a similar approach to the handling of information by extending the relations of rough set theory. Interestingly, Pawlak (1998) cautioned researchers not to complicate the operations of the rough set theory unnecessarily, as it might lead to mathematical complications and philosophical problems resulted from classification without sharp boundaries. Liang and Xu (2000) used the information entropy theory to facilitate the handling of incomplete information. The incomplete variable was able to take any possible values. Khoo and Zhai (2001) employed pseudo-objects to replace objects with incomplete information. Basically, a pseudo-object is created for every combination of possible case. After the information system has been repaired, standard rough set operations can be performed. The approaches reviewed above appear to be able to handle incomplete information. However, they are based on the same basic assumption that the incomplete attribute values exist but are not known. Thus, they are not able to tackle situations where the incomplete information is due to nonrelevant attributes, that is, UA, where the incomplete attribute value does not exist. In such situations, no actual value could be assigned to the nonrelevant attributes. This phenomenon is relatively common in DCs generation, where existing DCs are used to generate new or modified designs. This work introduces a so-called dissimilar objects algorithm (DOA) to tackle the UA.

Further analysis shows that the work of Mollestad and Komorowski (1999) can be used to extract design decision rules from an information system with multidecision attributes, that is, multiple CPs. However, for an information system with many decision attributes, it becomes unnecessarily cumbersome. Accordingly, this work also proposes a so-called attribute decomposition algorithm (ADA) to simplify the rules extraction process.

4.1. DOA

The proposed DOA extends the work of Mollestad and Komorowski (1999) to deal with UA. The algorithm is summarized as follows:

1. Replace any unavailable information in the information system with “UA.”
2. Derive the rules using the approach proposed by Mollestad and Komorowski by treating “UA” as a new value for the attributes.

More specifically, consider an information system that consists of n + p objects. The first n objects are complete, that is, all the attributes of each object have corresponding values. The remaining p objects contain one or more attributes with UA.

4.1.1. A complete information system

Consider a simplified information system comprising the first n objects. Mathematically, the information system can be expressed as

where C and D denote the condition and decision attribute sets, respectively.

It is important to note that there is no UA here. Based on the work of Mollestad and Komorowski, an n × n discernibility matrix (M_[C,D]) with respect to Decision Attribute D can be formed. An entry in the matrix, m_[C,D], in row i and column j is given by m_[C], if the decisions of Objects X_i and X_j (i ≠ j) are not the same. Otherwise, a 1 is assigned. Thus, m_[C] is the condition attribute that discerns Objects X_i and X_j, while the value 1 represents the Boolean value TRUE. More formally, the relative discernibility matrix with respect to Decision Attribute D can be expressed as

where m_[C,D](X_i, X_j) is defined by Eq. (2).

where m_[C](X_i, X_j) is defined by Eq. (3).

for 1 ≤ i ≤ n, 1 ≤ j ≤ n.

Based on the established matrix, a set of rules, R_[C,D], can be derived. Essentially, the antecedent clause of a rule, r_[C,D]i, in the rules set is formed by first combining the entries in each cell of Row i into a term using an OR operator as mentioned in Section 3.1. Subsequently, these terms are joined using the AND operators to form a rule.

The rule set derived from the matrix can thus be expressed as

4.1.2. UA in an information system

Mathematically, Eqs. (1)–(5) summarize the basic notions of the work by Mollestad and Komorowski, which are targeted at a complete information system. For an n + p objects information system with p nonrelevant attributes, Eqs. (1)–(5) can be extended and rewritten as follows [Eqs. (6)–(10)].

where m_[C,D](X_i, X_j) is defined as

where m_[C](X_i, X_j) is defined as

for 1 ≤ i ≤ n + p, 1 ≤ j ≤ n + p;

Rules are then generated by treating UA as valid value of attributes according to the DOA. In other words, Eqs. (6)–(10) have taken into account the additional objects with UA. Now, let those rules that contain UA be eliminated from the rules set R. This results in n + q number of rules remaining in the rules set where 0 ≤ q ≤ p such that

• q = 0 when all the rules corresponding to the objects with UA are eliminated and
• q = p when all the rules corresponding to the objects with UA are retained.

Hence, there are at least n rules remaining, as there are n objects without any UA entry. If those rules with a UA entry are eliminated, Eq. (9) can be rewritten as follows:

where 0 ≤ q ≤ p.

Hence, the set of rules derived from the information system containing p additional objects with UA entry has at least the same number of rules as those derived previously [Eq. (4)]. Each of the rules of Eq. (10) is then examined. By grouping the first n terms, Eq. (10) can be rewritten as follows:

The antecedent clause of Eq. (12) is then given by

It is important to note that the first term in Eq. (12a),

, is identical to the antecedent clause of Eq. (5). As the first and second terms of Eq. (12a) are joined using an AND operator, it can be deduced that the antecedent clause of Eq. (12) is indeed a subset of that of Eq. (5). Hence, the rules derived from such an information system with objects having UA based on the proposed DOA is more stringent compared to those obtained by Mollestad and Komorowski, that is, without UA. Accordingly, it is likely to be more robust, and potentially more useful for the evaluation of CP. It appears that by including objects with some nonrelevant attributes would result in a larger search space for the analysis of novel conceptual designs. A larger search space, in turn, is likely to provide a means to explore DCs beyond the domain application, which is important for the generation of innovative design solutions or, perhaps, more robust DCs.

4.2. ADA

The ADA postulates that it is possible to decompose an information system with n number of decision attributes into n information systems with single decision attribute each, and subsequently combine the rules obtained using Boolean operators into decision rules for DC–capability mapping.

The ADA is summarized as follows:

Step 1.

Form m decision relative discernibility matrices, where m is the number of decision attributes. Each matrix consists of n columns and n rows, where n is the number of objects.

Step 2.

Complete the n × n discernibility matrices. For each row and column (i, j) of the discernibility matrix, insert the condition attributes that discern objects i and j where i ≤ n and j ≤ n, if the value of decision attributes for objects i and j are different, that is, discernable. Otherwise, insert Boolean expression 1 that represents TRUE. Note that a Boolean expression 1 is introduced for consistency.

Step 3.

Form an antecedent clause of a rule by combining the condition attributes for each of the rows using Boolean operators. For each of the cells in the information system, combine all the entries using the OR Boolean operators to form a term. Subsequently, join all the terms in a row with the AND Boolean operators.

Step 4.

Simplify the antecedent clauses obtained in Step 3 using Boolean algebra, if necessary.

Step 5.

For each of the antecedent clauses obtained in Step 4, form a rule by incorporating the decision attributes. Subsequently, insert the value registered by the condition and decision attributes corresponding to each row.

Step 6.

Combine the rules using the AND Boolean operators.

Intuitively, the ADA can be employed to simplify the procedure used to treat the multidecision attributes discernibility matrix. For clarity, consider an information system (Table 3) with six objects such as six different vacuum cleaner designs and five attributes (three condition and two decision attributes). Compared with the information system depicted in Table 1, it has nonbinary values 1, 2, and 3, representing the three possible values that Attributes A and D may take. As for Attributes B, C, and E, they may have two possible values, 1 and 2. Based on the proposed algorithm, two decision relative discernibility matrices can be constructed (Tables 4, 5).

An information system with multiple decision attributes

Decision relative discernibility matrix D

Decision relative discernibility matrix E

These matrices can be obtained by considering Decision Attributes D and E separately using the work of Mollestad and Komorowski with the exception of filling up the blank space with 1, which denotes the situation where two objects have the same decision value. Decision rules are then derived based on each of the matrices. The rules derived from Decision Attributes D and E relative discernibility matrices using ADA after Step 5 are tabulated in Table 6. Subsequently, they are combined using the AND operator and treated using Boolean algebra to form a composite rule (Table 7). For clarity, consider Rule 2 of Table 6: IF [(A = 1) AND (C = 2)] THEN (D = 2) and IF (A = 1) OR (C = 2) THEN (E = 1) are combined using an AND operator to form the composite rule, IF {[(A = 1) AND (C = 2)] AND [(A = 1) OR (C = 2)]} THEN {(D = 2) AND (E = 1)}. Subsequently, the composite rule is simplified using the dual of absorption theorem of Boolean algebra to form the rule, IF [(A = 1) AND (C = 2)] THEN {(D = 2) AND (E = 1)}.

Sets of rules based on D only or E only

Sets of rules based on combined D and E

4.2.1. Theoretical basis of the ADA

Consider an information system I = {U, (C, D)} comprising n number of objects where C and D denote the sets of condition and decision attributes, respectively. Based on the work of Mollestad and Komorowski, an n × n discernibility matrix, M_[C,D], with respect to Decision Attribute D can be formed. Let the entry, m_[C,D], in Row i and Column j be given by m_[C], if the values of decision attributes of Objects X_i and X_j are different. Otherwise, let the entry of m_[C,D] be 1. Here, m_[C] is the attribute that discerns Objects X_i and X_j. More formally, the relative discernibility matrix with respect to Decision Attribute D can be expressed as

for 1 ≤ i ≤ n, 1 ≤ j ≤ n;

for 1 ≤ i ≤ n, 1 ≤ j ≤ n.

From the established matrix, a rule set, R_[C,D], can be derived. Basically, a rule, r_[C,D]i, in a rule set can be derived by first combining the entries in each of the cells of Row i into a Boolean expression using the OR operators. These Boolean expressions are then combined using the AND operators to form a composite rule in accordance to the work of Mollestad and Komorowski. More formally, the rule set derived from the matrix can be expressed as

For an information system with q number of decision attributes, Eq. (14) can be rewritten as

for 1 ≤ i ≤ n, 1 ≤ j ≤ n, 1 ≤ k ≤ q.

Similarly, Eq. (17) can formally be expressed as

for 1 ≤ i ≤ n, 1 ≤ j ≤ n, 1 ≤ k ≤ q.

For simplicity, let D = {d₁, d₂}. This implies that D comprises two decision attributes d₁ and d₂. Hence, the rules derived from an information system with two decision attributes simultaneously can be expressed as follows:

Accordingly, based on the ADA, the rules extracted from the information system with one decision attribute, d₁, are given by

To show that the rules derived from the information system with two decision attributes simultaneously based on the work of Mollestad and Komorowski and those derived using the ADA are indeed the same, it is necessary to prove that their antecedent and consequent clauses of the rules are identical, namely,

The consequent clauses are the same because those rules derived using the ADA yield d₁(X_i) _ d₂(X_j) [Eqs. (23) and (24)], and the work of Mollestad and Komorowski produces d₁(X_i) AND d₂(X_j) [Eq. (21)]. Thus, only the antecedent clause needs to be examined further.

Rearranging the left-hand side of Eq. (25) by pairing each of

,

The possible pairing combinations are depicted in Table 8.

Possible combinations of d1 and d2 using the ADA

For the rules derived simultaneously from an information system with two decision attributes, there are also four possible cases (Table 9). It is apparent that the rightmost column of Tables 8 and 9 are identical for all the possible cases.

Possible combinations of d1 and d2 for simultaneous derivation

Because both the antecedent and consequent clauses of the rules are identical, it can be deduced that Eq. (24) is true. In other words, the rules derived by both methods are identical for an information system with two decision attributes.

The above can be extended to an information system with more than two decision attributes. Consider an information system with q decision attributes, where q > 2. The information system can be split into two information systems with q − 1 and one decision attribute, respectively. For an information system with q decision attributes, m_[C,D] is given by Eq. (18), while r_[C,D] is given by Eq. (19). For an information system with q − 1 decisions, m_[C,D] and r_[C,D] are respectively defined as

for 1 ≤ i ≤ n, 1 ≤ j ≤ n, 1 ≤ k ≤ q − 1.

for 1 ≤ i ≤ n, 1 ≤ j ≤ n, 1 ≤ k ≤ q − 1.

The above process is performed repeatedly until it is left with an information system with two decision attributes, which was earlier shown to be valid.