A rough set approach for estimating correlation measures in quality function deployment

quality (PPHOQ) – is of fundamental and strategic importance in the QFD system due to the fact that the CRs for a product are identified, incorporated into the production of a corporation’s competitive priorities, and converted into ECs. The QFD team then determines the target levels for ECs of improved products [23–26,28,31–35]. Many correlation influences among ECs 0020-0255/$ - see front matter � 2011 Elsevier Inc. All rights reserved. ⇑ Corresponding author at: School of Traffic, Transportation, and Logistics, Southwest Jiaotong University, Chengdu, Sichuan 610031, People’s Republic of China. Tel.: +86 28 87634876. E-mail address: [email protected] (Y.-L. Li). Information Sciences 189 (2012) 126–142 Contents lists available at SciVerse ScienceDirect Information Sciences journal homepage: www.elsevier .com/locate / ins doi:10.1016/j.ins.2011.12.002 1. Introduction Accomplishing new product development (NPD) work within a short timeframe and at minimum cost is a core factor in enhancing the competitiveness of the firm in the global market. The use of quality function deployment (QFD) as a customer- driven NPDmethodology in the NPD decision making context has now gained extensive international support [1–4]. QFD is a planning and problem-solving methodology known for effectively translating customer requirements (CRs) into engineering characteristics (ECs). Companies that direct their efforts toward meeting their CRs produce higher-quality products, engen- der a higher degree of customer satisfaction, and earn more revenue. All these benefits are achieved because internal con- flicts are minimized, development cycles are shortened, and market penetration is greater [5–9,11,12,17–22]. QFD generally utilizes four sets of matrices known as house of quality (HOQ) to translate CRs into ECs and subsequently into part characteristics, process plans, and production requirements. The first set of matrices – product planning house of a r t i c l e i n f o Article history: Received 2 March 2009 Received in revised form 27 February 2011 Accepted 2 December 2011 Available online 8 December 2011 Keywords: Quality function deployment House of quality Engineering characteristics Customer requirements Rough set Correlation measure a b s t r a c t Quality function deployment (QFD) is a planning and problem-solving methodology used to translate customer requirements (CRs) into engineering characteristics (ECs) in the course of new product development (NPD). Estimating the correlation measures among ECs is a crucial step in the product planning house of quality (PPHOQ) construction process because these measures seriously affect the planning of development efforts. This study presents a rough set-based approach used to estimate the correlation measures by reveal- ing the knowledge of a QFD team. The approach involves introducing the category factor of a correlation to express the influences of the correlation categories on the corresponding correlation measures. A case study of a two-cylinder washing machine is used to illustrate the proposed approach. The result shows that the novel approach is effective in revealing the related knowledge of the QFD team and facilitating NPD decision making. � 2011 Elsevier Inc. All rights reserved. A rough set approach for estimating correlation measures in quality function deployment Yan-Lai Li a,b,⇑, Jia-Fu Tang b, Kwai-Sang Chin c, Yi Han d, Xing-Gang Luo b a School of Traffic, Transportation, and Logistics, Southwest Jiaotong University, Chengdu, Sichuan 610031, People’s Republic of China bKey Lab of Integrated Automation of Process Industry of the Ministry of Education in Northeastern University, School of Information Science and Engineering, Northeastern University, P.O. Box 135, Shenyang, Liaoning 110004, People’s Republic of China cDepartment of Manufacturing Engineering and Engineering Management, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong, People’s Republic of China dCollege of Economics and Management, Zhejiang University of Technology, Hangzhou, Zhejiang 310023, People’s Republic of China Li and Gao [36] proposed a linear mathematical program to tackle design of experiment (DOE) data as a means of esti- into fuzzy linear regression. Consequently, they developed a pair of hybrid linear programming models with asymmetric tri- Y.-L. Li et al. / Information Sciences 189 (2012) 126–142 127 angular fuzzy coefficients and a pair of hybrid linear programming models with asymmetric trapezoidal fuzzy coefficients. This approach can be used to handle situations where the amount of known data is small and the correlations are vague; however, the results may be unreliable or unachievable due to data unavailability or because selecting the membership func- tion in a fuzzy set is difficult and affected by subjective experience. Some scholars (e.g., [26,14]) initially proposed estimating the correlation measure matrix based on the analytic network model (ANP) described by Saaty and Takizawa [61]. Moreover, recent ANP method developments have been applied in esti- mating correlation measures. Partovi [47,48] employed a better and more advanced ANP approach to estimate the correla- tion measure matrix. To effectively quantify and analyze the QFD team’s subjective experience, knowledge, and judgments in a systematic manner, Raharjo et al. [53] proposed a generic ANP in which the correlation measure matrix is obtained to en- able QFD practitioners to exploit the potential of QFD as a powerful NPD tool. The correlation roof was tackled symmetrically in the early stages of QFD. To solve the problem of a lack of symmetric correlations, Moskowitz and Kim [44] changed the structure of the roof to a square matrix. However, the square matrix is subject to another limitation in that the influence of one EC on another depends on the CRs in this structure. A single matrix is insufficient to model these influences; therefore, in recent research, the single matrix has been changed to a group of matrices in which one matrix corresponds to each CR [58,59]. Rough set theory is a relatively new and intelligent knowledge discovery tool. It is widely used to find data dependencies, evaluate the importance of attributes, reveal data patterns, reduce the number of redundant objects and attributes, and seek the minimum subset of attributes [10,13,15,27,29,30,41,43,45,60]. Moreover, its main advantage is that it does not require preliminary or additional information regarding the data, such as the membership grades or possibility values needed in fuz- zy set theory [49–52,68–74]. The correlation measure matrix corresponding to each CR can therefore be identified with the assistance of the intelligent characteristics of a rough set. The rest of the paper is organized as follows. Section 2 describes a hierarchical framework used to construct the first five parts of PPHOQ and estimate the correlation measures. Section 3 presents a rough set approach employed to estimate the correlation measure matrix corresponding to each CR. An equivalent matrix is then obtained by integrating the final impor- tance ratings of the CRs into a group of matrices. In Section 4, a case study of a two-cylinder washing machine is described to demonstrate the proposed method. Section 5 concludes the paper. 2. Constructing the first five parts of PPHOQ to estimate the correlation measures PPHOQ is a structured method used to define customer requirements and translate them into engineering characteristics. The motivation for using PPHOQ is to design a product that embeds the possible initial and potential abstract CRs. The con- struction of the first five parts of PPHOQ is depicted through one specific case to estimate a group of correlation measure matrices according to a unified 10-step PPHOQ model [5–7]. A QFD case study on the development of a type of two-cylinder washing machine in a corporation is then presented as an example. 2.1. Part 1: Identifying CRs CRs drive PPHOQ; thus, the QFD teammust commit considerable effort to properly capturing these requirements [18–20]. CR candidates are collected via focus groups, individual interviews, listening and watching, and by using existing informa- tion. Once CR candidates have been collected, they are organized into several categories. An affinity diagram or cluster anal- ysis could be used to organize these candidates [21,22]. Suppose that, through appropriate methods, K customers denoted as Custf (f = 1,2, . . . ,K) have been selected, while N CRs have been identified based on the opinions of these K customers and the QFD team. These N CRs can be denoted as CR1,CR2, . . . ,CRi, . . . ,CRN. mating the correlation measures. Whether or not the DOE method can be used depends on the availability of a large volume of experimental data; the time taken and cost incurred in conducting the experiments are therefore major obstacles. Park and Kim [46] proposed a swing approach as an element of multiple-attribute decision making to investigate corre- lation measures. However, their approach is in most cases elaborate and sensitive, and produces similarly inexact results. Kim et al. [32], who first suggested the use of fuzzy regression to estimate correlation measures in QFD, proposed a fuzzy multi-criteria modeling approach for product planning. Their study employed a fuzzy linear regression model with symmet- ric triangular fuzzy numbers to investigate correlations. Meanwhile, Fung et al. [16] integrated the least-squares regression fall within the PPHOQ process and have degrees and directions that have serious effects on the development effort. The negative impacts of one EC on others represent design bottlenecks that call for special planning or breakthrough attempts. The correlation measures determine those technical areas that require close communication and collaboration and those areas that do not. Estimating correlation measures among ECs is thus a crucial step in the NPD process. CRs are transformed into ECs; thus, considerable effort must be made to acquire accurate correlation measures. This is a key factor in the ongoing success of the corporation [1–7,47,48,54–56]. Most QFD research [5–7] analyzes correlation measures by means of an elaborate and sensitive scale approach that occa- sionally results in inexact correlation measures. 128 Y.-L. Li et al. / Information Sciences 189 (2012) 126–142 2.2. Part 2: Determining the fundamental importance ratings of CRs CRs are usually accorded different degrees of importance according to the customer’s perspective [37–39]. The appropri- ate methods for obtaining customers’ perceptions are individual interviews and mail surveys [40,42]. Suppose that Custf (f = 1,2, . . . ,K) supplies a relative importance rating in terms of CRi, denoted as rirfi, according to a suitable scale. A carefully conducted market survey can be used to obtain the relative weights of these K customers, denoted as rwcf (f = 1,2, . . . ,K). Then, based on rirfi (f = 1,2, . . . ,K) and rwcf (f = 1,2, . . . ,K), the fundamental importance rating of CRi can be computed using the simple weighted sum denoted as irfuni . For convenience, the fundamental importance ratings of these N CRs can be de- scribed as IRfun ¼ irfun1 ; irfun2 ; . . . ; irfuni ; . . . ; irfunN � �T . 2.3. Part 3: Conduct competitive analysis of CRs CRs should be benchmarked from the perspective of the selected customers [6,7,44,46]. The performance estimations of a company in terms of these CRs could be obtained from the selected customers’ perceptions of the performance of the cor- poration and its competitors in terms of these N CRs. Based on these performance estimations, the normalized customer competitive priority ratings of these N CRs could be obtained, and are usually calculated using the scale method [57]. Spe- cifically, NCCPR = (nccpr1,nccpr2, . . . ,nccpri, . . . ,nccprN)T. The performance goals of these N CRs could be set according to the competitive and realistic goals of the corporation. In most cases, each goal performance level should be no lower than the current performance level. The normalized corpora- tion’s performance improvement ratios for these N CRs can then be obtained as NCPIR = (ncpir1,ncpir2, . . . ,ncpiri, . . . ,ncpirN)T. 2.4. Part 4: Determining the final importance ratings of CRs Because the fundamental importance rating, the normalized customer competitive priority rating, and the normalized corporation’s performance improvement ratio have been normalized, the situation described in this paper is deferent to most cases described in previous research. According to the specific competitive environment and technical ability [62– 67], the QFD team can determine the weights of these three aspects, and can subsequently derive the final importance rat- ings of the CRs. The results provided by the summation approach are therefore more accurate than those provided by the multiplication approach. The QFD team could use pair-wise qualitative comparisons to determine the weights of the funda- mental importance rating, the normalized customer competitive priority rating, and the normalized corporation’s perfor- mance improvement ratio as w1, w2, and w3, respectively. The team could then determine the final importance rating of CRi by using ir fin i ¼ w1 � irfuni þw2 � nccpri þw3 � ncpiri ði ¼ 1;2; . . . ;NÞ. For convenience, the final importance ratings of these N CRs could be described as IRfin ¼ irfin1 ; irfin2 ; . . . ; irfini ; . . . ; irfinN � �T . 2.5. Part 5: Generating ECs The CRs must be mapped into engineering terminology (i.e., ECs). The team could develop one or more ECs to satisfy a particular CR. It could then hold brainstorming sessions to generate creative ideas regarding EC candidates and evaluate these ideas to eliminate unfeasible candidates [39]. Some EC candidates may be redundant, and there could be conflicting relationships among EC candidates. Therefore, the process of selecting additional EC candidates is essential in generating ECs for PPHOQ. This step is largely based on the knowledge, experience, and intuition of the QFD team, and assumes that M ECs have been developed. According to Reich and Paz [59], there is a group of N matrices among which a correlation measure matrix CORi (i = 1,2, . . . ,N) corresponds to CRi (i = 1,2, . . . ,N). Fig. 1 shows the first six parts of the PPHOQ for the two-cylinder washing machine involved in these N CRs, the fundamental importance ratings of these CRs, along with their competitive analyses, their final importance ratings M ECs (ECj (j = 1,2, . . . ,M)), and a group of N correlation measure matrices. A case involving the practical development of a novel type of two-cylinder washing machine is cited to demonstrate the applicability of the proposed approach. A corporation denoted as Corp1 was in the process of developing a type of two-cyl- inder washing machine using QFD. The corporation constituted a multi-functional QFD team to develop the product econom- ically and effectively. In accordance with the marketplace survey undertaken and selected user feedback, 20 customers were chosen to help conduct the PPHOQ analysis. Twelve initial CR candidates were collected through a focus group and individual interviews, as well as on the basis of existing information. An affinity diagram was then successfully used to organize these candidates. The 12 CR candidates evolved into a two-level hierarchy of five CR items. The CRs and a brief description of each are shown in Table 1. Each of the 20 selected customers supplied relative importance ratings of the five CRs as RIRf = (rirf1,rirf2,rirf3,rirf4,rirf5) (f = 1,2, . . . ,20). Through a carefully conducted market survey, the relative weights of these 20 customers, denoted as rwcf (f = 1,2, . . . ,20), were subsequently obtained. Then, based on RIRf (f = 1,2, . . . ,20) and rwcf (f = 1,2, . . . ,20), the fundamental importance ratings of these five CRs were computed as follows using the simple weighted sum approach Fiv ration in the of Cor scale o six co was fo Bas could decide Accord impro mance By the no w1 = 0 the fin Th unfeas relatio 4 Oper Envi Y.-L. Li et al. / Information Sciences 189 (2012) 126–142 129 CR5: minor damage to clothes Minimize damage to clothes in the washing and rinsing process ational CRs CR2: shorter washing time Perform washing and rinsing work in minimum time Fig. 1. The first six parts in PPHOQ involved in the N correlation measure matrixes. Table 1 Five CRs in the PPHOQ of the two-cylinder washing machine. Level 1 Level 2 Description Quality CRs CR1: thorough washing Thoroughly wash and clean clothes in the washing process CR : thorough rinsing Eliminate washing and dirty matter in the rinsing process gs osCRN s Fundam ental im portance ratin Perform ance im provem ent rati Com petitive priority ratings CR1 CRN-1 CR2CR S EC1 EC2 EC4 EC5 Final im portance rating EC1 EC1 EC2 ECM-1 ECM EC2 EC4 EC1 EC2 ECM-1 EC1 EC2 ECM-1 ECM EC1 EC2 EC1 2 EC1 EC2 ECM-1 ECM EC5 ECM EC4 COR1 COR2 EC CORN EC5CORN-1 ECM-1 ECM IRfun ¼ irfun1 ; irfun2 ; irfun3 ; irfun4 ; irfun5 � �T ¼ ð0:262;0:159; 0:183;0:226;0:170ÞT: e corporations denoted as Corps (s = 2,3, . . . ,6) were identified as the main competitors of Corp1. Each of these corpo- s produced a similar two-cylinder-type washing machine. To understand the product market and its relative position market, the QFD team asked the 20 selected customers to rate the performance estimations for the washing machine p1, as well as for the similar products of its five competitors. The products were rated in terms of the five CRs using a f one to five. Based on the customer assessments of the performance estimations of the similar products made by the mpanies in terms of the five CRs, a performance comparison matrix X = [xis]5�6 was obtained. This performance matrix llowed to calculate the normalized customer competitive priority ratings of these five CRs using the scale method: NCCPR ¼ ðnccpr1;nccpr2;nccpr3;nccpr4; nccpr5ÞT ¼ ð0:234;0:210;0:113;0:184;0:259ÞT: ed on the available resources and the relative performance of the six companies based on these five CRs, the QFD team set an improvement goal for each CR to ensure they were better satisfied. After considering various factors, the team d on the appropriate performance estimation goals for these CRs as PG = (pg1,pg2,pg3,pg4,pg5)T(pgiP xi1, i = 1,2, . . . ,5). ing to the current and goal levels of Corp1 for these five CRs, the team could obtain the corporation’s performance vement ratio for the five CRs using the equation ncpiri = pgi/xi1 (i = 1,2, . . . ,5). The normalized corporation’s perfor- improvement ratios with respect to these five CRs were normalized as follows: NCPIR ¼ ðncpir1;ncpir2; ncpir3; ncpir4;ncpir5ÞT ¼ ð0:194;0:208;0:143;0:352;0:103ÞT: using pair-wise qualitative comparisons, the QFD team determined the weights of the fundamental importance rating, rmalized customer competitive priority rating, and the normalized corporation’s performance improvement ratio as .50, w2 = 0.25, and w3 = 0.25, respectively. Using ir fin i ¼ w1 � irfuni þw2 � nccpri þw3 � ncpiri, the team determined al importance ratings of the five CRs as follows: IRfin ¼ irfin1 ; irfin2 ; irfin3 ; irfin4 ; irfin5 � �T ¼ ð0:242;0:184;0:154;0:237;0:183ÞT: e QFD team brainstormed to generate EC candidates, and evaluated these candidates to eliminate those that were ible. Ten original EC candidates were determined. Some EC candidates were redundant, and there were conflicting nships between the EC candidates selected. Finally, based on the knowledge, experience, and intuition of the team, ronmental CRs CR3: quiet washing Minimize noise in the work environment in the washing and rinsing process EC1 EC2 E C3 E C4 E C5 130 Y.-L. Li et al. / Information Sciences 189 (2012) 126–142 C o m p etitiv e prio rity rating s CR1 CR4 CR3 CR2 CR S 0.262 0.226 0.183 0.159 0.234 0.184 0.113 0.210 0.194 0.352 0.143 0.208 0.242 0.237 0.154 0.184 P erform ance im provem ent ratio F undam ental im portance rating Final im portance rating Table 2 Five ECs in the PPHOQ on the case of the two-cylinder washing machine. Level 1 Level 2 Description Unit Quality ECs EC1: washing quality Percentage of clothes cleanliness in the washing process Percentage (%) EC2: clothes damage rate Percentage of clothes damage in the washing and rinsing process Percentage (%) EC4: rinsing quality Percentage of successful rinsing Percentage (%) Operational ECs EC3: washing time Work time required for the washing and rinsing process Minutes Environmental ECs EC5: noise quantity Measure of maximum noise in the washing and rinsing process Decibels EC1 EC2 EC3 EC4 EC5 EC1 EC2 EC3 EC4 EC5 EC1 EC2 EC3 EC4 EC5 EC1 EC2 EC3 EC4 EC5 EC1 EC2 EC3 EC4 EC5 EC1 EC2 EC3 EC4 EC5 EC1 EC2 EC3 EC4 EC5 COR1 COR2 COR3 COR4 COR5 EC1 EC2 EC3 EC4 EC5 EC1 EC2 EC3 EC4 EC5 the EC candidates were subject to an additional selection process to generate a two-level hierarchy of five ECs items for PPHOQ. The ECs and a brief description of each are shown in Table 2. Fig. 2 shows the first six parts of the PPHOQ for the two-cylinder washing machine involved in these five CRs, the fun- damental importance ratings of the five CRs, their competitive analyses, their final importance ratings, the five ECs, and the five correlation measure matrices corresponding to the five CRs. 3. Estimating the correlation measures using a rough set 3.1. Building decision systems to estimate the correlation measures According to rough set theory, the set of CRs could be defined as SCR = {CR1,CR2, . . . ,CRi, . . . ,CRN}, and the set of ECs could be defined as SEC = {EC1,EC2, . . . ,ECj, . . . ,ECM}. Corresponding to CRi (i = 1,2, . . . ,N), the set of decision-making attributes is composed of ECj, denoted as Dij = {ECj} (j = 1,2, . . . ,M). The set of conditional attributes corresponding to Dij is composed of ECs in SEC, but not in Dij, and is denoted as Cij = {ECkjECk 2 SEC,ECk R Dij}. For CRi, the QFD team graded the relative performance of each EC in Cij and the total correlation effect on ECj arising from all the ECs in Cij based on QFD team discussions. The QFD team then determined the different combinations of relative per- formance for all ECs in Cij. The team determined the total correlation effect of each combination on ECj corresponding to CRi, then obtained the grade of each effect. Data were then collected, from which the sample set Uij = {uij1,uij2, . . . ,uijQij} could be constructed, where Qij is the number of elements in Uij. Corresponding to CRi, the decision system DSij = (Uij,Cij [ Dij)(Cij \ Dij =£) used to acquire the correlation measures of ECk(ECk 2 Cij) on ECj was built. Corresponding to CRi, the team constructed a total of M decision systems denoted as DSi1,DSi2, . . . ,DSij, . . . ,DSiM. In the case of the two-cylinder washing machine, according to rough set theory, the set of CRs was defined as SCR = {CR1,CR2,CR3,CR4,CR5}. Meanwhile, the set of ECs was defined as SEC = {EC1,EC2,EC3,EC4,EC5}. Corresponding to CRi (i = 1,2, . . . ,5), the set of decision-making attributes was denoted as Dij = {ECj} (j = 1,2, . . . ,5) and the corresponding set of con- ditional attributes was denoted as Cij = {ECkjECk 2 SEC, ECk R Dij} (i = 1,2, . . . ,5, j = 1,2, . . . ,5). The estimation process for COR1 was presented and discussed as an illustration. For CR1 and ECj (j = 1,2, . . . ,5), the QFD team graded the relative performance of each EC in C1j and the total correlation effect on ECj arising from all ECs in C1j in light of QFD team discussions. The QFD team then determined the different combinations of the relative performance of all ECs in C1j. The team determined the total CR5 0.170 0.259 0.103 0.183s s s Fig. 2. The first six parts in PPHOQ involved in the five correlation measure matrixes. Y.-L. Li et al. / Information Sciences 189 (2012) 126–142 131 correlation effect of each combination on ECj corresponding to CR1, then obtained the grade of each effect. The team subse- quently collected each combination and the grade of its integrated effect before constructing the sample set U1j. Correspond- ing to CR1, the decision systems DS1j = (U1j,C1j [ D1j)(C1j \ D1j = ;) used to acquire the correlation measures of ECk(ECk 2 C1j) on ECj were built. Therefore, corresponding to CR1, the team constructed a total of five decision systems: DS11, DS12, DS13, DS14, and DS15. The DS11 building and analysis process is cited as an example. On the strength of its experience and expert knowledge, corresponding to CR1 and EC1, the team determined the grades of the relative performance of each EC in C11 = {EC2,EC3,EC4,EC5} based on the relative market survey and in-depth team discussions. The team classified the relative performance of EC2 in terms of the ‘‘clothes damage rate’’ into the highest level, the next highest level and the high level, denoting these three levels as 1, 2, and 3, respectively. The team graded the relative performance of EC3 into long, medium, Table 3 The sample set U11. Numb. EC2 EC3 EC4 EC5 D11 Numb. EC2 EC3 EC4 EC5 D11 1 1 3 1 2 1 27 1 2 1 2 1 2 1 3 2 2 2 28 1 2 2 2 1 3 1 3 3 2 2 29 1 2 3 2 2 4 2 3 1 2 3 30 2 2 1 2 2 5 2 3 2 2 3 31 2 2 2 2 2 6 2 3 3 2 3 32 2 2 3 2 3 7 3 3 1 2 3 33 3 2 1 2 3 8 1 2 1 1 1 34 3 2 2 2 3 9 1 2 2 1 1 35 3 2 3 2 3 10 1 2 3 1 2 36 1 1 1 2 1 11 2 2 1 1 2 37 1 1 2 2 1 12 2 2 2 1 2 38 1 1 3 2 1 13 2 2 3 1 3 39 2 1 1 2 2 14 3 2 1 1 3 40 2 1 2 2 2 15 3 2 2 1 3 41 2 1 3 2 2 16 3 2 3 1 3 42 3 1 1 2 2 17 1 1 1 1 1 43 3 1 2 2 3 18 1 1 2 1 1 44 3 1 3 2 3 19 1 1 3 1 1 45 1 3 1 1 1 20 2 1 1 1 2 46 1 3 2 1 2 21 2 1 2 1 2 47 1 3 3 1 2 22 2 1 3 1 2 48 2 3 1 1 3 23 3 1 1 1 2 49 2 3 2 1 3 24 3 1 2 1 3 50 2 3 3 1 3 25 3 1 3 1 3 51 3 3 1 1 3 26 3 3 3 1 3 52 3 3 2 1 3 and short ‘‘washing time,’’ denoting these three levels as 1, 2, and 3, respectively. The relative performance of EC4 was then classified into the highest, next highest, and high levels of ‘‘rinsing quality,’’ denoting these three levels as 1, 2, and 3, respec- tively. The team also classified the relative performance of EC5 into high and low ‘‘noise quantity’’ grades and denoted these two grades as 1 and 2, respectively. The team identified the total correlation effect on EC1 that arose from all the ECs in C11 as belonging to one of three grades (i.e., high, medium, and low), denoted as 1, 2, and 3, respectively. The QFD team then deter- mined the different combinations of relative performance among all the ECs in C11. It determined the total correlation effect of each combination on EC1 corresponding to CR1, then obtained the grade of each effect. The team then collected each com- bination and the grade of the integrated effect of the corresponding combination, leading to construction of the sample set U11 as shown in Table 3. Taking the first sample in Table 3 as an illustration, corresponding to CR1 and EC1, the team deter- mined the relative performance of EC2, EC3, EC4, and EC5 as 1, 3, 1, and 2, respectively. The team denoted the combination of these relative performance levels of EC2, EC3, EC4, and EC5 as the first combination. It then assessed the grade of the total correlation effect on EC1 arising from all four ECs in C11 corresponding to CR1 as belonging to the high level, denoted as 1. The team collected the first combination and the grade of the total correlation effect of the first combination as the first sample in Table 1. Taking the forty-third item in Table 3 as another example, corresponding to CR1 and EC1, the team deter- mined the relative performance of EC2, EC3, EC4, and EC5 as 3, 1, 2, and 2, respectively, and denoted the combination of these relative performance levels of EC2, EC3, EC4, and EC5 as the forty-third combination. The team then acquired the grade of the total correlation effect on EC1 arising from all four ECs in C11 corresponding to CR1 as belonging to the low level, denoted as 3. The team collected the forty-third combination and the grade of the total correlation effect of the first combination as the forty-third sample in Table 3. Corresponding to CR1, the decision system DS11 = (U11,C11 [ D11)(C11 \ D11 = ;) was built to ac- quire the correlation measures of ECk(ECk 2 C11) on EC1. 3.2. Relative cores and relative reductions of the decision systems The concepts of relative core and relative reduction are very important in rough set theory, and an essential distinction is made between objects that may definitely be classified into a certain category. For DSij, its relative reduction involves the classification task of mapping a set of variables Cij to a labeling set Dij. A reduction denoted as RedDij ðC ijÞ is defined as any in eve exten Cle If c X If cX Here, requir being In CRi, w 132 Y.-L. Li et al. / Information Sciences 189 (2012) 126–142 and the set of decision-making attributes Dij, Uij is simplified as the new sample set NUij. The simplified decision system NDSij = (NUij,NCij [ Dij)(NCij \ Dij = ;) corresponding to DSij, a set used to determine the correlation measures, is then built. In the case of the two-cylinder washing machine, there was no EC5 in RedD11 ðC11Þ; therefore, EC5 in C11 was redundant. The new set of conditional attributes was then defined as NC11 = {EC2,EC3,EC4}. Simultaneously, another set corresponding to hereas no correlation of ECv ðECv 2 NC ijÞ on ECj corresponds to CRi. Based on the new set of conditional attributes NCij FDS11 ¼ ðEC2 ^ EC3 ^ EC4Þ. According to Equation (6), this can be derived using the rough set toolbox of related computational software. The reduction of DS11 was obtained as RedD11 ðC11Þ ¼ fEC2;EC3;EC4g. Based on Eq. (1), the relative core of DS11 was determined as CoreD11 ðC11Þ ¼ \RedD11 ðC11Þ ¼ fEC2;EC3; EC4g. The relative core CoreD11 ðC11Þ constituted the most important subset of conditional attributes because none of its elements could be removed without affecting the classification power of conditional attributes. The new set NCij, the collection of all conditional attributes comprising these reductions, can be defined as NC ij ¼ ECujECu 2 ½RedDijðC ijÞ�1 [ ½RedDijðC ijÞ�2 [ � � � [ ½RedDij ðC ijÞ�Wij ;u ¼ 1;2 . . . ;M;u – j n o : ð7Þ Simultaneously, another set corresponding to NCij is defined as NC ij ¼ fECv jECv 2 C ij; ECv R NC ij; v ¼ 1;2 . . . ;M;v – jg: ð8Þ According to the definitions of relative core and relative reduction, correlations of ECu(ECu 2 NCij) on ECj can correspond to denoted as ½RedDij ðC ijÞ�1; ½RedDij ðC ijÞ�2; . . . ; ½RedDij ðC ijÞ�Wij . the case of the two-cylinder washing machine, the set of all prime implications of DS11 was computed as any pair of discernible objects. The set of all prime implications of FDSij determinesWij reductions of DSij, theseWij reductions ðrij ;sijÞ2U ij�U ij FDSij describes constraints that must hold to preserve discernibility among all pairs of discernible objects from Uij. It es that at least one attribute be kept from each non-empty element of the discernibility matrix corresponding to aijðrij; sijÞ ¼ ;: ð5Þ The discernibility function for DSij is a propositional formula of some Boolean variables defined by FDSij ¼ Y X aijðrij; sijÞ: ð6Þ rijsij ¼ ;, then this function is depicted as ECh1 _ ECh2 _ � � � _ EChij ¼ aijðrij; sijÞ: ð4Þ rijsij ¼ fECh1 ;ECh2 ; . . . ; EChijg– ;, then the Boolean function is defined as follows: matrix of DSij completely depicts the ability of DSij to identify the objects in Uij. Therefore, all reductions of the system are hidden in some discernibility function induced by the discernibility matrix. ; ECjðuijrij Þ ¼ ECjðuijsij Þ arly, crij sij is the subset of conditional attributes that can be used to discern object rij and object sij in DSij. The discernible 3.3. Discernibility matrices and discernibility functions of the decision systems The indiscernibility matrix expresses the fact that one is unable to discern some objects by employing the available infor- mation (or knowledge) due to a lack of information (or knowledge). The indiscernibility matrix used to discern between the perceived objects in DSi is important in constructing many entities such as reductions, decision rules, or decision algorithms. The discernibility matrix of DSij is expressed by MDij ¼ ðcrijsijÞQij�Qij ; for rij ¼ 1;2; . . . ;Qij; sij ¼ 1;2; . . . ;Qij; ð2Þ where crijsij ¼ fECljECl 2 C ij ^ EClðuijrijÞ – EClðuijsij Þ ^ ECjðuijrij Þ– ECjðuijsij Þ; l ¼ 1;2; . . . ;Mg ECjðuijrijÞ – ECjðuijsij Þ ( ) ð3Þ t. According the definition of relative core, its calculation is fundamental in determining all the relative reductions. ative core is the most important subset of conditional attributes because none of its elements can be removed without affect- ing the classification power of conditional attributes. Certainly, the geometry of reductions can be compounded to a greater ry relative reduction. For instance, each element of the core belongs to a specific relative reduction. In a sense, the rel- RedDij ðC ijÞ#C ij such that cðC ij;DijÞ ¼ cðRedDij ðC ijÞ;DijÞ where c(Cij,Dij) is the degree to which Dij depends on Cij and cðRedDij ðC ijÞ;DijÞ is the degree to which Dij depends on RedDij ðC ijÞ. An attribute ECj 2 Cij, which belongs to the intersection of all reductions, is a core attribute in Cij with respect to Dij. The relative core of DSij is defined as CoreDij ðC ijÞ ¼ \ RedDijðC ijÞ: ð1Þ Here, DSij may have several relative reductions. Because the relative core is the intersection of all reductions, it is included 16 3 1 2 3 Y.-L. Li et al. / Information Sciences 189 (2012) 126–142 133 NC11 was obtained as NC11 ¼ fEC5g. Based on the definition of relative reduction, there were correlations of EC2, EC3, EC4 in NC11 on EC1. In contrast, there was no correlation of EC5 in NC11 on EC1. According to NC11 and D11, U11 was simplified as the new sample set NU11 shown in Table 4. Corresponding to DS11, the simplified decision system NDS11 = (NU11,NC11 [ D11) (NC11 \ D11 = ;) used to determine the correlation measures of EC2,EC3, and EC4 on EC1 was built. 17 3 1 3 3 18 1 2 1 1 19 1 2 2 1 20 1 2 3 2 21 2 2 1 2 22 2 2 2 2 23 2 2 3 3 24 3 2 1 3 25 3 2 2 3 26 3 2 3 3 3.4. Es Na ECj. Th attribu NCij an reflect To identi entire by Dij. where uniqu the po The co Dij wi Th bute i to CRi to this of ECu NCij is is pro fined 13 2 1 2 2 14 2 1 3 2 15 3 1 1 2 Table 4 New sample set NU11. Numb. EC2 EC3 EC4 D11 1 1 3 1 1 2 1 3 2 2 3 1 3 3 2 4 2 3 1 3 5 2 3 2 3 6 2 3 3 3 7 3 3 1 3 8 3 3 2 3 9 1 1 1 1 10 1 1 2 1 11 1 1 3 1 12 2 1 1 2 timating the correlation measures turally, the correlation measure of ECu(ECu 2 NCij) on ECj is measured. in that ECu can provide the degree of the effect on erefore, the magnitude of the correlation measure is defined as the importance rating of the corresponding conditional te ECu in the multi-attribute decision-making process. This is constructed using the new set of conditional attributes d the set of decision-making attributes Dij. Therefore, corresponding to CRi, the importance rating of ECu in NDSij can the relative magnitude of the correlation measure of ECu on ECj. determine the importance ratings of these conditional attributes, an important data analysis issue that arises is how to fy dependencies among these attributes. Dependency can be defined as follows: with respect to NDSij, NCij depends ly on Dij, if and only if IND(NCij) # IND(Dij). This means that the partition generated by NCij is finer than that generated It can be said that Dij depends on NCij to the degree k = c(NCij,Dij) (0 6 k 6 1), denoted by k ¼ cðNC ij;DijÞ ¼ PosNCij ðDijÞ ��� ��� jNU ijj ; ð9Þ PosNCij ðDijÞ, a positive region of the partition NUij/Dij with respect to NCij, is the set of all elements of NUij that can be ely classified into the blocks of the partition NUij/Dij through NCij. If k = 1, Dij is totally dependent on NCij. If k = 0, then sitive region of the partition NUij/Dij with respect to NCij is empty. If k < 1, Dij depends partially (to degree k) on NCij. efficient k expresses the ratio of all elements of NUij, which can properly be classified into blocks of the partition NUij/ th respect to NCij and referred to as the degree of dependency. e importance rating of ECu(ECu 2 NCij) could be determined by calculating the change in dependency when this attri- s removed from NCij. In NDSij, every conditional attribute in NCij may have different importance ratings corresponding . This measure enables ECu to be evaluated using a multi-valued scale that assigns a real number from the interval [0,1] attribute. If the dependency change resulting from removing ECu(ECu 2 NCij) from NCij is large, the importance rating is correspondingly and proportionally high. If the dependency change resulting from removing ECu(ECu 2 NCij) from small, then the importance rating of ECu is proportionally low. Therefore, the importance rating of ECu(ECu 2 NCij) portional to the dependency change. The importance rating of ECu in the simplified decision system NDSij can be de- as the re neous ECkðEC For measu In were Sim trix of to COR Ac alent equiv Table 7 Correla cor11 134 Y.-L. Li et al. / Information Sciences 189 (2012) 126–142 iju iju lative magnitude of the correlation measure of ECu(ECu 2 NCij) on ECj corresponding to CRi can be considered simulta- biju ¼ bðECu;DijÞECu2NC ij ¼ cðNC ij;DijÞ � cððNC ij � fECugÞ;DijÞ: ð10Þ In the case of the two-cylinder washing machine, the importance ratings of conditional attributes EC2, EC3, and EC4 with respect to EC1 were determined according to Equation (10) using the rough set toolbox of related computational software as shown in Table 5. Nevertheless, biju reflects only the relative magnitude of the correlation measure of ECu on ECj corresponding to CRi. To determine the correlation measure, the categories of the correlations among ECs must be considered. Generally, the two cat- egories of the correlations of ECs in NCij on ECj are the positive and negative categories [66]. The category factor of a corre- lation of ECu(ECu 2 NCij) is therefore introduced on ECj, corresponding to CRi as follows: Definition 1. Corresponding to CRi, cfiju reflects the effect of the correlation category of ECu(ECu 2 NCij) on ECj, and its value may only be selected from 1 and �1. These two numbers (i.e., 1 and �1) represent positive correlation and negative correlation, respectively. If the correlation category of ECu(ECu 2 NCij) on ECj corresponding to CRi is a positive correlation, the value of cfiju is 1. If the correlation category of ECu(ECu 2 NCij) on ECj corresponding to CRi is a negative correlation, the value of cfiju is �1. In the case of the two-cylinder washing machine, in light of the team’s experience and expert knowledge of this product, the cat- egory factors of the correlations of ECu(ECu 2 NC11) on EC1 were identified, as shown in Table 6. The values of the three cat- egory factors reflect the effects of the corresponding correlation categories with respect to the correlation measures. By introducing the concept of the category factor of a correlation based on the integration of b and cf , the category and tion measures of ECk(ECk 2 C11) on EC1. EC2 EC3 EC4 EC5 k (k = 2,3,4,5) �0.458 0.341 0.201 0.000 Category factors of the correlations of ECu(ECu 2 NC11) on EC1. EC2 EC3 EC4 cf11u (u = 2,3,4) �1 +1 +1 Table 5 Importance ratings of ECs in NC11. EC2 EC3 EC4 b11u (u = 2,3,4) 1.000 0.745 0.438 Table 6 ly. This aids in obtaining a more accurate correlation measure. Therefore, when j– k, the correlation measure of k 2 C ij;NC ij [ NC ij ¼ C ijÞ on ECj corresponding to CRi can be defined as corijkj–k ¼ cfijk � bijkP ECk2NCij bijk ! ECk 2 NC ij; 0 ECk 2 NC ij: 8>>< >>: ð11Þ convenience, the correlation measure of ECj on itself can be regarded as zero. Corresponding to CRi, the correlation re matrix of ECk(ECk 2 SEC) on ECj(ECj 2 SEC) can be defined as CORi ¼ corijk corijkj–k j– k corijj ¼ 0 j ¼ k ����� ! M�M ; i ¼ 1;2; . . . ;N: ð12Þ the case of the two-cylinder washing machine, the correlation measures of ECk(ECk 2 C11) on EC1 corresponding to CR1 determined according to Eq. (11), as shown in Table 7. ilar to DS11, the other four decision systems DS1j (j = 2,3,4,5) were built and analyzed. Corresponding to CR1, the ma- the correlation measures among ECs, denoted as COR1, was estimated according to Eq. (12) as shown in Fig. 2. Similar 1, the other four matrices of the correlation measures CORi (i = 2,3,4,5) were estimated according to Eq. (11). cording to Reich and Levy [58] and Reich and Paz [59], the multiple roofs of PPHOQ can be simplified into one equiv- roof by multiplying each matrix by the final importance rating of the associated CR, then adding them together as one alent matrix. The matrix of equivalent correlation measures can therefore be obtained as follows: Bas PPHO betwe In Fig. 4, Y.-L. Li et al. / Information Sciences 189 (2012) 126–142 135 2. Determining the fundamental importance ratings of CRs 3. Conduct competitive analysis of CRs 4. Determining the final importance ratings of CRs Constructing the first five parts of PPHOQ 1. Identifying CRs START ECOR ¼ XN i¼1 irfini � CORi � � : ð13Þ ed on the above analysis, the summary of the rough set method developed to estimate the correlation measures in Q is shown in Fig. 3. Based on the first six parts of PPHOQ, the QFD team is able to estimate the relationship measures en CRs and ECs before obtaining the initial priority ratings of the ECs. the case of the two-cylinder washing machine, the QFD team simplified the five PPHOQ roofs, which are shown in into one equivalent roof by multiplying each matrix by the final importance rating of the associated CR and adding 5. Generating ECs Building decision system for estimating the correlation i=1 The relative core and relative reduction The discernibility matrix and discernibility function ijDS ijNCijNC ijuβijucfijucf ijkcor i=i+1 i>NNO YES ( )fin 1 N i i i ir = = ∗∑ECOR COR ECOR END Fig. 3. The methodology developed to estimate the correlation measures. 136 Y.-L. Li et al. / Information Sciences 189 (2012) 126–142 them together as one equivalent matrix. The equivalent correlation measure matrix was obtained by ECOR ¼P5i¼1 irfini � CORi� �. Fig. 5 shows the first six parts of the PPHOQ for the two-cylinder washing machine involved in the five CRs; the fundamental importance ratings, competitive analyses, and final importance ratings for these five CRs; the five ECs; and the equivalent correlation measures among ECs. From Fig. 5, it is easy to verify the equivalent correlation measures among ECs. To illustrate, the equivalent correlation measure ecor13 of EC3 on EC1 was 0.291. Here, ecor13 indicates that lengthening the ‘‘washing time’’ (EC3) by one minute led to a 0.291% enhancement in ‘‘washing quality’’ (EC1). This could be exploited to make efficient improvements in customer satisfaction. As another example, the equivalent correlation measure ecor32 of EC3 on EC2 was �0.242. This translates into ecor32, indicating that shortening the ‘‘washing time’’ (EC3) by one minute inherently resulted in a 0.242% increase in the ‘‘clothes damage rate’’ (EC2). This could bias the solution to the initial priority rating of EC2. The negative equivalent corre- lation measures of one EC on the others quantitatively represent the bottlenecks of the design process. This calls for special planning or breakthrough attempts, in contrast with positive equivalent correlation measures. Therefore, the equivalent cor- relation measures could help the QFD team to locate and analyze the correlations among ECs in a quantitative, effective, and efficient manner. Based on the first six parts of PPHOQ, the QFD team then estimates the relationship measures between CRs and ECs before obtaining the initial priority ratings of five ECs. Fig. 4. Five matrices of the correlation measures. 4. Com Th Y.-L. Li et al. / Information Sciences 189 (2012) 126–142 137 approach, the fuzzy liner regression approach with trapezoidal fuzzy number coefficients, and the ANP approach. This sec- tion presents the results of comparisons among these four approaches and discusses the proposed rough set approach. By considering competition requirements and technological feasibility, the maximum target value tvmaxj and the mini- mum target value tvminj of ECj (j = 1,2, . . . ,5) were determined in line with the DOE principle. Taking the estimation of the correlation measures between EC1 and ECj (2,3, . . . ,5) as an example, the QFD team determined the value of EC1 as the target variable and the value of ECj (j = 2,3, . . . ,5) as the decision variable. The team simulated an experiment plan corresponding to EC1. The simulated experiment matrix corresponding to EC1 was then obtained, after which the correlation measures be- tween EC1 and ECj (j = 2,3,4,5) were determined using relative computational software. The same procedure was used to assess the other correlation measures between ECj1 ðj1 ¼ 2;3;4;5Þ and ECj2 ðj2 ¼ 1;2; . . . ;5; j2 – j1Þ. The correlation measure matrix was obtained using DOE as follows: doe doe � � 0:000 �0:365 0:315 0:167 0:044 �0:345 0:000 �0:302 �0:296 0:000 0 BBBBB 1 CCCCC Ap team Bas Follow trapez guitie ized v regres provid opera parison e QFD team also estimated the four matrices of correlation measures among ECs using the DOE approach, the swing Fig. 5. First six parts of PPHOQ involved in the equivalent correlation measure matrix. COR ¼ corj1 j2 5�5 ¼ 0:294 �0:175 0:000 0:365 0:000 0:126 �0:285 0:443 0:000 �0:043 0:087 0:042 �0:056 0:088 0:000 BBB@ CCCA : plying the step-by-step procedure of the swing approach in assessing the correlation measures among ECs, the QFD estimated the correlation measure matrix using the following: CORswa ¼ corswaj1j2 � � 5�5 ¼ 0:000 0:345 0:287 0:214 0:085 0:286 0:000 0:231 0:245 0:000 0:246 0:214 0:000 0:308 0:056 0:214 0:298 0:323 0:000 0:078 0:245 0:085 0:076 0:043 0:000 0 BBBBBBBB@ 1 CCCCCCCCA : ed on thorough tests, the QFD team obtained the values of ECs with respect to the products of Corps (s = 1,2, . . . ,6). ing Fung et al. [16], the values of five ECs and the RP estimation of five CRs were normalized. Fung et al. proposed that oidal fuzzy number coefficients offer more flexibility in handling a wider variety of systematic uncertainties and ambi- s that cannot be modeled efficiently using triangular fuzzy number coefficients. This proposal is based on the normal- alues of five ECs and the normalized relative performance estimations of five CRs. Therefore, a pair of hybrid fuzzy liner sion models with trapezoidal fuzzy number coefficients was used to estimate the correlation measures among ECs. To e further comparison, the expected values of these correlation measures were obtained using the fuzzy expected value tor as follows: 1 2 A g ploit t chine He exact cision ments Th doe In swa Th j2 ¼ 1 1;2; . . In adganpj1 j2 was d 138 Y.-L. Li et al. / Information Sciences 189 (2012) 126–142 0:239 0:121 0:216 0:000 0:094 0:145 0:034 0:101 0:020 0:000 BB@ CCA comparison with cordoej1 j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ 1;2; . . . ;5Þ, the absolute deviation of cor ganp j1 j2 and cordoej1j2 could be defined as . Using the equation adganpj1 j2 ¼ cor ganp j1 j2 � cordoej1 j2 ��� ���, the matrix of absolute deviation adganpj1 j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ 1;2; . . . ;5Þ etermined as ADflr ¼ adflrj1 j2 � � 5�5 ¼ 0:122 0:000 0:078 0:161 0:064 0:053 0:027 0:000 0:139 0:000 BBBBBBB CCCCCCC: 0:158 0:043 0:132 0:045 0:000 e absolute deviation of corflrj1 j2 and cor doe j1j2 could be denoted as adflrj1 j2 , in comparison with cor doe j1 j2 ðj1 ¼ 1;2; . . . ;5; ;2; . . . ;5Þ. Using the equation adflrj1 j2 ¼ corflrj1 j2 � cordoej1 j2 ��� ���, the matrix of absolute deviation adflrj1j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ . ;5Þ was derived as 0:000 0:107 0:130 0:042 0:044 0 1 adj1 j2 . Using the equation adj1 j2 ¼ corswaj1 j2 � cordoej1 j2� �, the matrix of absolute deviation adj1 j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ 1;2; . . . ;5Þwas obtained as ADswa ¼ adswaj1j2 � � 5�5 ¼ 0:000 0:710 0:028 0:047 0:041 0:631 0:000 0:533 0:541 0:000 0:048 0:389 0:000 0:057 0:056 0:088 0:583 0:120 0:000 0:121 0 BBBBBBBBB@ 1 CCCCCCCCCA : ADrs ¼ adrsj1 j2 � � 5�5 ¼ 0:023 0:000 0:121 0:030 0:000 0:026 0:052 0:000 0:062 0:000 0:101 0:043 0:095 0:000 0:043 0:020 0:080 0:035 0:031 0:000 BBBBBBB@ CCCCCCCA : comparison with cordoej1 j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ 1;2; . . . ;5Þ, the absolute deviation of corswaj1 j2 and cordoej1j2 could be denoted as swa �� �� swa corj1 j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ 1;2; . . . ;5Þ. Using the equation adj1 j2 ¼ ecorj1 j2 � corj1 j2� �, the matrix of absolute deviation adrsj1 j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ 1;2; . . . ;5Þ was calculated as 0:000 0:049 0:024 0:025 0:044 0 B 1 C 0:106 0:057 0:087 0:045 0:000 re, CORdoe was obtained by conducting simulated experiments and analyzing the resulting data. CORdoe was the most among ECOR, CORdoe, CORswm, CORflr, and CORganp. Therefore, CORdoe could serve as benchmark for comparing the pre- of ECOR, CORswm, CORflr, and CORganp. Nevertheless, because the DOE approach depends on many simulated experi- , time and cost are major obstacles. e absolute deviation of ecorj1 j2 and cor doe j1 j2 could be defined as adrsj1 j2 , in comparison with rs doe �� �� CORganp ¼ corganpj1 j2 � � 5�5 ¼ 0:395 0:000 0:164 0:198 0:045 0:246 0:267 0:000 0:413 0:000 0:114 0:298 0:314 0:000 0:112 BBBBBBB@ CCCCCCCA : 5�5 0:365 �0:164 0:0227 0:000 0:051 �0:058 0:076 0:045 0:068 0:000 BB@ CCA eneric ANP model could be proposed to estimate the correlation measures, thereby enabling QFD practitioners to ex- he potential of QFD. The QFD team constructed the generic ANP model of the PPHOQ of the two-cylinder washing ma- before estimating the correlation measures using pair-wise comparisons as follows: 0:000 0:387 0:294 0:116 0:0000 1 CORflr ¼ corflrj j � � ¼ 0:000 �0:258 0:185 0:125 0:088 �0:223 0:000 �0:224 �0:457 0:064 0:347 �0:148 0:000 0:226 0:000 0 BBBBB 1 CCCCC: ADganp ¼ adganpj1 j2 � � 5�5 ¼ 0:000 0:752 0:021 0:051 0:044 0:740 0:000 0:566 0:494 0:045 0:048 0:442 0:000 0:048 0:000 0:012 0:583 0:129 0:000 0:155 0:019 0:015 0:143 0:043 0:000 0 BBBBBB@ 1 CCCCCCA : From the four matrices of ADrs, ADswa, ADflr, and ADganp, the maximum value mvrs of adrsj1 j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ 1;2; . . . ;5; j2 – j1Þwas identified as 0.121; the maximum valuemvswa of adswaj1 j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ 1;2; . . . ;5; j2 – j1Þwas calculated as 0.710; the maximum value mvflr of adflr ðj ¼ 1;2; . . . ;5; j ¼ 1;2; . . . ;5; j – j Þ was determined as 0.239; and the maxi- Y.-L. Li et al. / Information Sciences 189 (2012) 126–142 139 ADrs ADswa ADflr ADganp [0.00,0.05] 14 7 6 10 (0.05,0.10] 4 3 4 1 (0.10,0.20] 2 4 8 3 (0.20,0.40] 0 1 2 0 (0.40,1.00] 0 5 0 6 j1 j2 1 2 2 1 mum value mvganp of adganpj1j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ 1;2; . . . ;5; j2 – j1Þ was computed as 0.752. Therefore, the magnitudes of mvrs, mvswa, mvflr, and mvganp could be ranked according to the following order: mvganp �mvswa �mvflr �mvrs, where ‘‘�’’ means ‘‘larger than.’’ From the maximum absolute deviation perspective, the swing approach and the generic ANP approach provided similarly exact results. These approaches are less precise than both the proposed rough set approach and the fuzzy liner regression approach with trapezoidal fuzzy number coefficients, considering that the two previous approaches were used to estimate the correlation measures without considering the effluences of the correlation categories. Furthermore, the proposed rough set approach producedmore exact results than did the other three approaches because it revealed the QFD team’s knowledge of the correlation measures among ECs. This was accomplished by introducing the category factor of a correlation to express the effects of the correlation categories. The average absolute deviation of adrsj1 j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ 1;2; . . . ;5; j2 – j1Þ was denoted as aad rs, and aadrs was cal- culated as 0.046 using the equation aadrs ¼ 120� P5 j1¼1 P5 j2¼1 j2 – j1 adrsj1 j2 ! . The average absolute deviation of adswaj1 j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ 1;2; . . . ;5; j2 – j1Þ was denoted as aad swa, and aadswa was calculated as 0.219 according to the equation aadswa ¼ 120� P5 j1¼1 P5 j2¼1 j2–j1 adswaj1 j2 ! . The average absolute deviation of adflrj1 j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ 1;2; . . . ;5; j2 – j1Þ was denoted as aadflr, and aadflr was computed as 0.097 using the equation aadflr ¼ 120� P5 j1¼1 P5 j2¼1 j2–j1 adflrj1 j2 ! . The average absolute deviation of 20 elements adganpj1 j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ 1;2; . . . ;5; j2 – j1Þwas denoted as aad ganp, and aadganp was cal- culated as 0.211 based on the equation aadganp ¼ 120� P5 j1¼1 P5 j2¼1 j2–j1 adganpj1 j2 ! . Therefore, the magnitudes of aadrs, aadswa, aadflr, and aadganp could be ranked according to the following order: aadrs � aadflr � aadganp � aadswa, where ‘‘�’’ means ‘‘smaller than.’’ From the perspective of average absolute deviation compared with the result of the DOE approach, the results of the swing and generic ANP approaches were similarly exact. These approaches were also less precise than the proposed rough set approach and the fuzzy liner regression approach with trapezoidal fuzzy number coefficients. This is because the two former approaches do not consider the effects of correlation categories on the correlation measures. In addition, the result of the proposed rough set approach was more exact than were those of the other three approaches, because it revealed the QFD team’s knowledge with respect to the correlation measures among ECs. This was aided by the introduction of the cat- egory factor of a correlation to express the effects of the correlation categories on the correlation measures. To derive the distributions of adrsj1 j2 ; ad swa j1j2 ; adflrj1 j2 , and ad ganp j1 j2 , where j1 = 1,2, . . . ,5, j2 = 1,2, . . . ,5, and j2– j1, the set [0,1] was divided into five minor subsets. The resulting five minor subsets were [0.00,0.05], (0.05,0.10], (0.10,0.20], (0.20,0.40], and (0.40,1.00], respectively. According to the form of division, the distributions of adrsj1 j2 ; ad swa j1 j2 ; adflrj1 j2 , and adganpj1 j2 are shown in Table 8. Table 8 indicates that all elements of adrsj1 j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ 1;2; . . . ;5; j2 – j1Þ belonged to the first three subsets and that 90% belonged to the [0.00,0.05] and (0.05,0.10] subsets. All elements of adflrj1 j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ 1;2; . . . ;5; j2 – j1Þ belonged to the first four subsets, whereas 30%, 20%, 40%, and 10% of the elements belonged to the [0.00,0.05], Table 8 Element distributions of four matrices. 140 Y.-L. Li et al. / Information Sciences 189 (2012) 126–142 (0.05,0.10], (0.10,0.20], and (0.20,0.40] subsets. All elements of adswaj1 j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ 1;2; . . . ;5; j2 – j1Þ belonged to all five subsets, and 35%, 15%, 20%, 5%, and 25% of the elements belonged to the [0.00,0.05], (0.05,0.10], (0.10,0.20], (0.20,0.40], and (0.40,1.00] subsets, respectively. All elements of adganpj1 j2 ðj1 ¼ 1;2; . . . ;5; j2 ¼ 1;2; . . . ;5; j2 – j1Þ belonged to all five sub- sets, and 50%, 5%, 15%, 0%, and 30% of the elements belonged to the [0.00,0.05], (0.05,0.10], (0.10,0.20], (0.20,0.40], and (0.40,1.00] subsets, respectively. Moreover, the percentages of adswaj1j2 and ad ganp j1 j2 elements that belonged to the (0.20,0.40] and (0.40,1.00] subsets were the same (i.e., 30%). Therefore, from the element distribution perspective, the results of the swing and generic ANP approaches displayed a similar degree of exactness. These approaches were less precise than the pro- posed rough set approach and the fuzzy liner regression approach with trapezoidal fuzzy number coefficients because the two former approaches do not consider the effects of correlation categories on the correlation measures. The proposed rough set approach provided more exact results than the other three approaches. In other words, the proposed approach revealed the QFD team’s knowledge of the correlation measures among ECs by introducing the category factor of a correlation to ex- press the effects of the correlation categories. Based on the comparative analyses of the aforementioned three perspectives, the swing and generic ANP approaches pro- vided similarly exact results in comparison with the DOE approach. Their results were much less precise than were those of the proposed rough set approach and the fuzzy liner regression approach with trapezoidal fuzzy number coefficients, be- cause the two latter approaches determined the correlation measures without considering the effluences of the correlation categories. This illustrates the importance of the proposed category factor. The proposed rough set approach is more exact than both the fuzzy liner regression approach with trapezoidal fuzzy number coefficients and the DOE approach. The pro- posed rough set approach could therefore provide the most exact results from among the approaches considered, and may be more precise than the fuzzy liner regression with trapezoidal fuzzy number coefficients, swing, and generic ANP ap- proaches. This finding can be attributed to the ability of the proposed rough set approach to reveal the QFD team’s knowl- edge with respect to the correlation measures among ECs. This knowledge was identified by introducing the category factor of a correlation to express the effects of the correlation categories. Upon completing the correlation measure estimation process in the two-cylinder washing machine case, a meeting was held with the QFD team to review the results and solicit feedback on the utility of the proposed approach. The QFD teamwas intrigued by the additional information the approach provided, as well as its potential as an accurate means of obtaining correlation measures. The team commented that the proposed approach was easy to implement and that its results were surprisingly accurate. Thus, the team was willing to make the effort required to provide all the necessary numbers. They col- lectively expressed their enthusiasm for the proposed approach and indicated that they would be willing to use it to develop future products and ensure designs were more cost-effective. One drawback of the rough set method is the complexity of the calculations required. Nevertheless, this drawback is overcome when the rough set algorithm is implemented recursively using appropriate computational software. 5. Conclusion Estimating the correlation measures among ECs is a crucial aspect of QFD analysis due to their serious effect on planning development efforts. Nevertheless, in estimating such correlation measures, previous studies barely address the QFD team’s knowledge. This paper describes a rough set-based approach developed to estimate correlation measures in the product development context. A novel approach to constructing the decision systems required to estimate correlation measures is proposed on the basis of the basic principles underlying rough set theory. By using the decision systems constructed, the relative magnitudes of the correlation measures can be directly computed from raw data without considering subjective adjustments, assumptions, or membership functions, with the objective of revealing the QFD team’s knowledge with respect to correlation influences among ECs. The novel concept of the category factor of a correlation is introduced to express the effects of the correlation categories on correlation measures. A formula for estimating correlation measures is presented based on the integration of the relative magnitude and category factor of a correlation. The proposed rough set-based ap- proach provides the QFD team with flexibility in making judgments and freely expressing its opinions within a rigorous and systematic framework. The case study of the development of the two-cylinder washing machine shows that the proposed rough set approach effectively and logically identifies the process of estimating the correlation measures. Moreover, the proposed approach pro- vides a rigorous, reliable, and systematic means of revealing the QFD team’s related knowledge. Extending the work pre- sented in this paper by considering the benchmarking information discussed and comparing it with data on competitors, using related methods of incomplete decision systems in rough set theory, or providing a means of allowing for objectives and constraints in terms of the possibility of their attainment may be research topics worthy of attention in future. Acknowledgements The authors express their sincere appreciation to the editor-in-chief and the anonymous reviewers for their invaluable comments and suggestions on earlier versions of this manuscript. This research was financially supported by the National Science Foundation of China (Nos. 70971017, 71021061, and 71171039), a grant from the City University of Hong Kong (No.7002571) and the Science Foundation of Zhejiang Province (No. Y1100854). Y.-L. Li et al. / Information Sciences 189 (2012) 126–142 141 References [1] Y. Akao, G.H. Mazur, The leading edge in QFD: past, present, and future, International Journal of Quality & Reliability Management 20 (2003) 20–35. [2] R.L. Armacost, P.J. Componation, M.A. Mullens, W.W. Swart, An AHP framework for prioritizing customer requirements in QFD: an industrialized housing application, IIE Transactions 26 (1994) 72–79. [3] J.A. Carnevalli, P.A.C. Miguel, F.A. 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A rough set approach for estimating correlation measures in quality function deployment 1 Introduction 2 Constructing the first five parts of PPHOQ to estimate the correlation measures 2.1 Part 1: Identifying CRs 2.2 Part 2: Determining the fundamental importance ratings of CRs 2.3 Part 3: Conduct competitive analysis of CRs 2.4 Part 4: Determining the final importance ratings of CRs 2.5 Part 5: Generating ECs 3 Estimating the correlation measures using a rough set 3.1 Building decision systems to estimate the correlation measures 3.2 Relative cores and relative reductions of the decision systems 3.3 Discernibility matrices and discernibility functions of the decision systems 3.4 Estimating the correlation measures 4 Comparison 5 Conclusion Acknowledgements References