Prioritizing engineering characteristics in quality function deployment with incomplete information: A linear partial ordering approach

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Int. J. Production Economics 91 (2004) 235–249

Prioritizing engineering characteristics in quality function deployment with incomplete information: A linear partial ordering approach Chang Hee Hana, Jae Kyeong Kimb,*, Sang Hyun Choic a b

Department of Business Administration, Hanyang University, 1271 Sa-1dong, Ansan, Kyunggi-do, 425-791, South Korea School of Business Administration, Kyung Hee University, 1 Hoegi-Dong, Dongdaemoon-Gu, Seoul 130-701, South Korea c Department of Industrial and Systems Engineering, Gyeongsang National University, 900 gaqwa-dong, Ginju, Kyungsangnam-do 660-701, South Korea Received 7 March 2002; accepted 13 September 2003

Abstract Quality function deployment (QFD) has been used as the concurrent engineering tool to save the production cost and time. It is not easy to get information or knowledge for the prioritization of customer attributes and engineering characteristics during the QFD planning process. This research suggests a linear partial ordering approach for assessing the knowledge from participants and prioritizing engineering characteristics. The linear partial information will be used in extracting weights of customer attributes and relationship values of customer attributes between engineering characteristics. Using the linear partial ordering can reduce the cognitive burden of designers and engineers of QFD planning team. Four types of dominance relation that are frequently used in multi-attribute decision making with incomplete information are used to determine the priorities of engineering characteristics when the linear partial orderings of participants are given. The dominance relations between engineering characteristics can be established by solving a series of linear programming problem. r 2003 Elsevier B.V. All rights reserved. Keywords: Quality function deployment (QFD); Incomplete information; Linear partial ordering; Multi-attribute decision-making (MADM)

1. Introduction Many firms are facing rapid changes stimulated by technological innovations and changing customer demand. These firms realize that getting high-quality products to customer in a timely manner is crucial for their survival in the competitive marketplace. Product development process is a complex managerial process that involves multi-functional groups with different perspectives. Quality function deployment

*Corresponding author. Tel.: +82-2-961-9355; fax: +82-2-961-0622. E-mail address: [email protected] (J.K. Kim). 0925-5273/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2003.09.001

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(QFD) is a new product development process which stresses cross-functional integration. QFD ensures quality throughout each stage of the product development and production process. QFD was originally developed and used in Japan at the Kobe Shipyards of Mitsubishi Heavy Industries, Ltd. in 1972. QFD has been used successfully by Japanese manufacturers of customer electronics, home appliances, construction equipment, synthetic rubber, textile, software system, etc. Akao (1990) and Hauser and Clausing (1998) introduced basics of QFD process and a lot of successful cases of QFD. Also, Cohen (1995) explains the detailed process of QFD and provides several applications of QFD. Govers (2001) emphasized that special attention must be paid to product policy and cross functional project approach to make QFD a valuable technological and organizational aid for innovation projects. During the 1980s and 1990s many kinds of US companies began employing QFD after the initial success at Fuji-Xerox in 1983. Many successful cases across a broad range of industries have been reported until now. During the QFD planning process, product design team needs to know how to make a selection of design features. Due to the complexity of decision process, the design team will often rely upon ad hoc procedures to assist in this product development (Wasserman, 1993). Such procedures are often completely ‘‘arbitrary’’, however, and subject to the ‘‘whims’’ of the design team rather than to the need of customer (Franceschini and Rossetto, 1995). As many researchers have pointed out, more convenient methodology is needed to get information from design team and provide an unforced evaluation of the QFD tables. This paper proposes a methodology for prioritizing engineering characteristics in QFD with incomplete information. The methodology uses a linear partial ordering to assess the knowledge and information from QFD team and apply the MADM method to derive priorities or weights of engineering characteristics. The linear partial information will be used in extracting weights of customer attributes and relationship values between customer attributes and engineering characteristics. And then, dominance relations in multiattribute decision-making (MADM) procedure with linear partial information are used to derive the priorities of engineering characteristics. The dominance relations between engineering characteristics can be established by solving a series of linear programming problem. Due to the easiness in articulating preferential information, using linear partial information can reduce the cognitive burden of designers and engineers of QFD planning team and give a practical convenience in using QFD planning process. The remainder of this article is organized as follows. Section 2 includes the definition of QFD and short review of the prior researches in QFD, and explains the necessity of incomplete information. Five types of linear partial information will be introduced and discussed in Section 3. Also, a methodology based on MADM and incomplete information will be developed to prioritize engineering characteristics in Section 3. An illustrative example is shown in Section 4.

2. Background 2.1. Quality function deployment QFD can be defined as an overall concept that provides a means of translating the needs of customers through the various stages of product planning, engineering and manufacturing into a final product. QFD is accomplished through a series of charts which are a conceptual map, providing the means for interfunctional communications. The chart is usually called a house of quality (HOQ). HOQ relates the variables of one design phase to the variables of the subsequent design phase. Four linked houses in Fig. 1 implicitly convey the voice of the customer through to manufacturing. In this paper, HOQ of the product planning phase (Phase I in Fig. 1) is described in detail. HOQ charts of other phases can be analyzed in a similar way. The ‘‘voice of the customer’’ is represented on the left side of HOQ. The customer attributes (CA) are usually very qualitative and vague. CA importance, the relative importance among the customer attribute, plays an important role in identifying critical customer attributes

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and prioritizing design efforts. The engineering characteristics (EC) are the design requirements that affect one or more of the customer attributes. EC importance, the importance ratings for the engineering characteristics, is calculated using CA importance ratings and weights assigned to the relationships between CAs and ECs. The matrix in the main body of the HOQ identifies the relationships between the CAs and ECs. The strength of the relationships are typically assessed by the design team in a subjective manner. The ‘‘roof’’ part of HOQ establishes the correlation among the ECs. The right side of HOQ lists a competitive benchmarking on each customer attribute for the company’s and competitor’s product. Target levels of engineering characteristics are determined using all the information in the HOQ (Fig. 2). Any approaches have been introduced for the determination of CAs’ and ECs’ importance rating in the QFD process. Analytic hierarchical process (AHP) has been used to prioritize CAs in an industrialized housing application (Armacost et al., 1994). Outranking method has been used to derive the ECs’ importance rating from quantified CAs’ importance rating and ordinal relationships between CAs and ECs (Franceschini and Rossetto, 1995). The concept of deployment normalization, the prioritization of ECs that is consistent with CAs’ importance rating, was advocated by Lyman (1990). Lyman’s deployment normalization is extended to properly account for dependencies that may exist between engineering characteristics by Wasserman (1993). Methods for determining ECs’ importance ratings are dependent upon the representation of the relationships between CAs and ECs. If the relationships are represented as 13-9 scale, we use the simple weighted sum. If the relationships are represented as ordinal ranking, we use the outranking method. However, when the relationships are represented by partial or incomplete information, it is not easy to find a well-known methodology.

Phase III

Process operations

Engineering characteristics

Phase II

Production requirement

Phase IV

Fig. 1. Linked houses convey the customer’s voice through to manufacturing.

. . .

CA Importance

CA1 CA2

EC1

EC2

. . .

Correlation among ECs

(Engineering Characteristics)

(Customer Attributes)

Customer attributes

Phase I

Process operations Parts characteristics

Parts characteristics

Engineering characteristics

ECm

Relationships between ECs and CAs

CAn

ECs Importance Target EC levels

Fig. 2. The house of quality.

Customer perception

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Recently, Karsak et al. (2002) combined the analytic network process and goal programming approach for determining the importance levels of engineering characteristics. Pullman et al. (2002) compared the QFD and conjoint analysis by applying each to the product design problem, and reached a conclusion, two methods can be used as complementary tools in product design process. Chan and Wu (2002) presented a literature review of QFD and establish reference bank of 650 QFD publications. This review may be useful for researchers and practitioners to find historical development of QFD. A number of cases are reported which uses QFD as an analysis tool in various area. In addition to the product development process, QFD was properly used in software development process. Haag et al. (1996) reported many cases using QFD in software development process. QFD is also suitable to the problems which have more qualitative information than product development problem, like information system planning and business reengineering. Erik et al. (1997) reported an application of QFD in business process reengineering process. Han et al. (1998) used QFD as a determination tool for the importance of information systems development. Considering the business environments and business strategies, they determine the importance of information systems by the sequential use of HOQ. Recently, Kim et al. (2000) use the QFD to measure the flexibility and efficiency of information technology. Lowe et al. (2000) used QFD as a tool to rapidly evaluate the feasibility of using the semi-solid metal forming process to manufacture products. Stehn and Bergstrom (2002) apply QFD to integrate the customer requirement and design problems in the R&D project aiming for an industrialized development of a multi-storey timber frame house system. 2.2. Incomplete information in the relationship matrix To establish relationships between CAs and ECs, each cell of the relationship matrix is filled with an ordinal scale as following levels: strong relationship, medium relationship, weak relationship. With the fundamental approach which makes use of the AHP’s 1-3-9 or 1-5-9 scale to denote weak, medium and strong relationships, ECs’ importance rating wj can be calculated from the weighted column sum of each CA by the quantified relationship values, wj ¼

m X

di  ri; j ;

ð2:1Þ

i¼1

where di is the degree of importance of the customer attribute i; i ¼ 1; 2; ym; ri;j is the quantified relationship between customer attribute i and engineering characteristic j; j ¼ 1; 2; y; n; and wj is the degree of importance rating of the engineering characteristic j: This quantification of relationships is the process of transforming an ordinal information into a cardinal scale that represents an ‘‘arbitrary passage’’, which may sometimes be dangerous (Franceschini and Rossetto, 1995). The operations of weighting and adding with arbitrary values are not meaningful, because it is easy to produce different results by choosing different scales from which to draw the ordinals (Satty, 1980, 1990). The use of the pre-determined arbitrary numerical scale leads to some drawbacks. First of all, there is low robustness in the variation of the values of the cardinal scale elements. Secondly the supporting tool using the scale cannot help the expert team’s behavior by bringing to them arguments which can strengthen or weaken their own convictions. Information in the QFD tables must be easy to provide and faithful to the real decision mechanisms that drive the development of a product. The main objective of such a supporting tool is to construct or to create some ability to help customers/expert team taking part in the decision process: either to argue and/or to transform their preferences information or to make decisions in conformity with their goals. Our suggested methodology allows us to face these issues.

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3. Methodology This paper develops a methodology overcoming the quantification problem and using the information faithful to the real situation. A key feature of the methodology is that the decision maker is willing to or able to provide only incomplete information on parameters such as CAs’ weights and relationship values. The reasons may be that (1) a decision should be made under time pressure and lack of knowledge or data, (2) many of CAs and ECs are intangible or non-monetary because they reflect customers’ preferences and technical aspects, and (3) the decision maker has limited attention and information processing capabilities, especially on the judgment of numerical values under complex and uncertain environment (Kahneman et al., 1982). These may take the form of linear partial information (LPI) such as ranking, interval description, and so on. Before we explain the use of LPI in QFD, we describe the pairwise comparison concept which will be used as a technique for representing the degree of impacts between the ECs on the kth CA. With the fundamental approach, the degree of impacts is represented as 1-3-9 or 1-5-9 scale to denote weak, medium and strong relationships (Akao, 1990). The impact is represented as a function of variables of ECs which include first order effects from the relationship matrix of HOQ and second order effects from the roof of HOQ (Locascio and Thurston, 1994). In this paper, pairwise comparison method is used to evaluate the dominance relations of the degree of impacts between ECs on the kth CA. Firstly, we assign the zero value to rkj in order to represent that the effect of jth EC to the kth CA does not exist. Among ECs that have effects on the kth CA, all pairs between ECs are selected and each pair is compared to evaluate the degree of impact. These information of pairwise comparison is expressed as five types of LPI. We now examine LPI which is used as a quantified form to represent the relative impacts of the ECs’ on the kth CA, rkj : Also, LPI is used as a quantified form that captures the importance of CAs’ from pairwise comparison of customer’s perspective. Five types of LPI which are generally used in MADM and AHP are as follows: Form Form Form Form Form

1. 2. 3. 4. 5.

{di Xdj } {di  dj Xai } {di Xai dj } {ai pdi pai þ ei } {di  dj Xdw  dl } for jakal

where {ai } and {ei } are non-negative constants. Having no numerical values, Form 1 and 5 are the simplest forms of information (Kim et al., 1999). The ordinal ranking of CAs and relationships of relative impacts between ECs on each CA can be represented by the Form 1. Form 5 is a ranking of impact differences between adjacent parameters (CAs or relationships) obtained by ranking among the parameters, which can be subsequently constructed based on Form 1. A difficulty in taking the information of Forms 2 and 4 is to precisely justify their constants, since these forms contain non-scaled numerical values such as ai and ei (Kim and Han, 2000). But scaled numerical value is used in Form 3 as a constant ai which is widely used to assess the preference in AHP by asking pairwise comparison questions and represents a bound for the relative importance of the ith CA/relationship over the jth CA/relationship (Arbel, 1989; Salo and Hamamainen, 1992). Form 1 is a special case of Form 3 when ai ¼ 1: Therefore, in this paper we use the Forms 1, 3 and 5 as the input formats that will capture the importance of CA from customers’ perspective and represent the relationships between CAs and ECs. Now, since the five types of linear partial information are introduced as measures or tools for participants to articulate his/her preferences, we provide more discussion about the information from a view of preference measurement. Generally, measurement is concerned with developing a correspondence between the empirical system (e.g. preference or relationships between CAs and ECs) and abstract system (e.g. numbers) (Robert, 1979). Thus, measurement may be defined as assignment of numbers to

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characteristics of objects in empirical system. Also, traditional preference measurement method assigns numbers to the decision maker’s preference. In order to assign the number, utility function is derived by the approximation of information articulated by decision maker (Keeney and Raiffa, 1976). In our linear partial ordering approach, ordinal rankings and inequalities are used as measures or tools for representing participant’s preference information. Although using linear partial ordering gives the practical convenience of participants or decision makers, several questions may be happened in a theoretical point of view. Firstly, have decision maker or participant’s preferences ordinal or interval, like five types of linear partial information? In traditional utility assessment method, the decision maker articulates his/her information as an ordinal or numerical form, and then the utility function is derived that is mostly approximated to the information of decision maker. This is, deriving the utility function is the process of aggregating ordinal or numerical information of decision maker. This means that decision maker can provide any type of information including linear partial information even in the traditional approach. Now, we can consider that the decision maker’s preference can be represented as not only utility function or numeric form but also ordinal or interval information. Second possible question is how can we operate or handle the linear partial information. This is, since ordinal scale does not represent the degree of the preference, and it represents just an ordering of objects, ordinal information can not be operated arithmetically. An answer to this question had been already established at the previous researches in the field of MADM with incomplete information. In that field, instead of doing any kind of operation, like arithmetic operation or curve fitting for approximation, etc. to the linear partial information, the set of linear partial ordering from decision maker is regarded as a constraint set of LP. The constraint set constructs a feasible region of LP that represents all possible preference values of decision maker (Park and Kim, 1997). 3.1. Mathematical model for prioritizing engineering characteristics The suggested methodology for prioritizing ECs is based on the pairwise dominance concept that uses LPI obtained from customers and expert team. We first calculate the absolute bounds of all ECs’ relative impact values on each CA under LPI constraints and then rank the ECs using dominance concept. If we cannot identify the relationship between all pairs of ECs using the absolute bounds, then pairwise comparison method is applied to the rest of ECs’ pairs. 3.1.1. The mathematical programming model Several notations are defined for representing a hierarchical structure of CAs. Firstly, we define the topmost customer attribute’s level in a hierarchy as L and the lowest level of a hierarchy as 0. Also, we define ml as the number of customer attribute at the level l ¼ f0; 1; 2; y; Lg: We define Yjl as the set of child nodes of jth attribute at the lth level. In convenience, we use Y 0 ; instead of Yj0 ; as the set of all lowest level attributes. The above definition is illustrated in the Fig. 3. Now, we will explain the linkage to CAs and ECs using Fig. 3. The customers attributes, CAs, are represented as Ci in HOQ and MADM framework. Similarly, the engineering characteristics, ECs, are represented as ei : The relationships between CAs and ECs are represented as the value of each alternative for an attribute in MADM framework. In order to represent the level of attribute tree, we use the notation L þ 1: For example, the level of CAs in HOQ of Fig. 3(a), and the level of the tree in MADM view of Fig. 3(b), is three. Thus, in the example of Fig. 3, the value of L is 2. According to the definition, we have l ¼ f0; 1; 2g; and the value of L is 2. And we can say that the attribute tree has 3 ðL þ 1Þ level of hierarchy. As shown in Fig. 3, let dil be the degree of importance rating of customer attribute i at the lth level. Then, the degree of importance rating of lowest level customer attribute can be represented as di0 ; in short, di : The relationship rij is redefined by the EC j’s degree of impact on the lowest level customer attribute i: The meaning of wej in the HOQ of Fig. 3(a) is the relative importance or weight of each ECs.

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ca

l=L+1=2

lowest level CA's

e1 e2 c1 c2 c3 c4 ci c m-1 cm

d1

.. e .. j

en

cb

... . . . ..

r ij

dn we1 we2

c1

c2

c3

l=1

j

ci

..

ej

..

cm

l=0

n

e1

House of Quality

.. rij

..we .. we

lowest level CA's importance rating

(a)

cc

d2

(b)

e2

e3

..

en

MADM view of HOQ

Fig. 3. Hierarchical structure of QFD: (a) House of quality and (b) MADM view of HOQ.

If decision makers or expert team of QFD planning process provide incomplete information about the parameters dil and rij ; we should define the sets of incomplete information of dil and rij : For representing the incomplete information of rij ; we define the set R as a constraints set which is compose of incomplete information from decision makers or expert team. The set R can be divided into sets Ri of each customers attribute information fromP expert team, P i ¼ 1; 2; y; m0 : The set Ri is composed of ERi ; a set of incomplete S 0 n and nj¼1 rij ¼ 1: Accordingly, the two sets can be defined as R ¼ m j¼1 rij ¼ 1g: i¼1 Ri and Ri ¼ ERi ,f The meaning of the set R is that all possible values of relationships of CAs and ECs, and the set R can be decomposed into the sets Ri which is mutually independent convex set. Now, we define three constraints set for representing the set of incomplete information of customer attributes. Firstly, let Cðdil Þ; l ¼ 1; 2; y; L; be a set of incomplete information from expert team. For example, if there is Y32 ; the set of child nodes of d32 ; and the attribute d32 means third attribute in level two in 2 2 hierarchy, then Cðd32 Þ means a set of incomplete information P aboutl1di in lY3 from expert team. Secondly, for all attribute j in level l; l ¼ 1; 2; yL; the equation, d ¼ d ; j is needed as a constraint for iAY Pjl i customer attributes. For convenience, we define the set, F ¼ f iAY l dil1 ¼ djl g for all attribute j in level l; j l ¼ 1; 2; yL: The meaning of the equation isP that the sum of the value of child nodes is equal to the value of m0 0 parent node. Thirdly, the one equation, i¼1 di ¼ 1; is involved in the constraints set of customer attributes. The meaning of the equation is that the sum of the value of the lowest level is equal to P attributes 0l 0 one. From the above three sets of constraints, we define the set, H ¼ Cðdil Þ,F,f m d ¼ 1g; and then i¼1 i use the set as a constraints set of customer attributes of hierarchy. When expert team or decision makers provide incomplete information about ECs and CAs, each ECs’ weight or priority can be represented as absolute bound, minimum and maximum possible values. Also, each pair of ECs’ weight or priority can be determined by comparing pairwise bounds, minimum and maximum possible pairwise values. In this paper, we use four types of dominance relationships, strict dominance, weak dominance, pairwise strict dominance, and pairwise weak dominance. Strict dominance and weak dominance is decided based on the minimum and maximum bounds of each ECs. Also, pairwise strict dominance and pairwise weak dominance is decided based on the minimum and maximum bounds of each pair of ECs. The pairwise dominances can be used as a supplementary tools of strict and weak dominance. This is, if the absolute bound of two ECs is almost same, the team or decision maker may hesitate to decide that one EC is more important than another EC. In this case, pairwise bounds or dominances can support or confirm expert team’s decision making.

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Now, we will provide two mathematical programming models that are used for deriving absolute and pairwise bound. The previously defined sets, R and H; can be used as a constraints set of the following formulations. The following model (3.1) is formulated for deriving minimum absolute bound for engineering characteristic j; ecj ; under the constraints sets. Also, we can derive maximum absolute bound by substituting ‘min’ by ‘max’ in model (3.1). The following notations ecj ; ymin ðecj Þ; and jmin ðecj Þ are used in the formulation of (3.1) and (3.2). The notation, ecj ; j ¼ 1; 2; y; n; means engineering characteristics. The notation ymin ðecj Þ means minimum absolute bound for engineering characteristic j; and ymax ðecj Þ means maximum absolute bound for engineering characteristic j: jmin ðecj Þ means minimum pairwise bound for for engineering characteristic j; and jmax ðecj Þ is maximum pairwise bound for for engineering characteristic j: Pm 0 ymin ðecj Þ ¼ min i¼1 di  rij s:t:

rij AR; dil AH; rij X0; i ¼ 1; 2; ym0 ;

j ¼ 1; 2; y; n;

dil X0;

l ¼ 0; 1; 2; yL:

i ¼ 1; 2; yml ;

ð3:1Þ

Similar to model (3.1), the following model (3.2) can be used for deriving minimum pairwise bound between engineering characteristics j and k: By substituting ‘min’ by ‘max’ in model (3.2), maximum pairwise bound can be derived from Pm 0 jmin ðeck ; ecl Þ ¼ min i¼1 di  ðrik  ril Þ s:t: rik ; ril AR; dil AH;

ð3:2Þ

rik ; ril X0 dil X0;

i ¼ 1; 2; y; m0 ;

i ¼ 1; 2; y; ml ;

k; l ¼ 1; 2; y; n;

kal;

l ¼ 0; 1; 2; y; L:

If the value of either dil or rij is known precisely, then formulations (3.1) and (3.2) become linear programming (LP) problems. Thus it can be easily solved. Assume that both values of dil and rij are imprecisely or incompletely identified, because the objective functions are sum-product type of the unknown variables, then (3.1) and (3.2) generally become non-LP problems. Consequently the mathematical models (3.1) and (3.2) are not easily solved, so the appropriate solution method is needed for deriving minimum or maximum bounds. Now, we provide a solution method for the above non-LP problems. Assume that the LPI of values for each CA are functionally independent, then (3.1) and (3.2) are separable for each CA (Kim and Han, 2000; Ahn et al., 2000), thus yielding a set of the following LPs: Pm 0  ymin ðecj Þ ¼ min i¼1 di  xi ðecj Þ s:t:

dil AH; dil X0;

i ¼ 1; 2; y; ml ;

l ¼ 0; 1; 2; y; L

with x i ðecj Þ ¼ min

rij

s:t:

rij ARi ; rij X0:

ð3:3Þ

0 This is, firstly we should derive minimum value, x i ðecj Þ; for all customer attribute i; iAY ; and then  minimize ymin ðecj Þ by substituting the xi ðecj Þ in the objective function by the derived value of x i ðecj Þ:

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Similar to the (3.3), pairwise bound can be derived from the following model: Pm 0  ymin ðeck ; ecj Þ ¼ min i¼1 di  xi ðeck ; ecj Þ s:t:

dil AH; dil X0;

i ¼ 1; 2; y; ml ;

l ¼ 0; 1; 2; y; L

with x i ðeck ; ecj Þ ¼ min s:t:

rij rik ; ril ARi ;

ð3:4Þ

rik ; ril X0: Then, we can obtain two types of bound from each formulation. From the result of (3.3), we can obtain the absolute bound for each wj ; ymin ðecj Þpecj pymax ðecj Þ: From the solution of (3.4), we can obtain the pairwise bound for each pair eck and ecl ; jmin ðeck ; ecl Þpeck  ecl pjmax ðeck ; ecl Þ: Based on the derived absolute bound and pairwise bound, we suggest the dominance relationships between ECs. 3.1.2. Dominance relationships between ECs After obtaining the absolute bounds, ymin ðecj Þ and ymax ðecj Þ; for all ECs under consideration, we can consider the following three cases between the weight bounds of EC i and j: Case 1: ymin ðeci ÞXymax ðecj Þ: Case 2: ymax ðeci ÞXymax ðecj Þ and ymin ðeci ÞXymin ðecj Þ: Case 3: ymax ðeci ÞXymax ðecj Þ and ymin ðeci Þpymin ðecj Þ: An illustration of each case is shown in Fig. 4. Case 1 represents that the lowest weight value for ith EC is greater or equal than the best value for jth EC. If so, we can say that ith EC is strictly or absolutely preferred to jth EC, denoted as strict dominance notated by ECi >S ECj : In Case 2, ith EC is weakly preferred to jth EC, denoted as weak dominance notated by ECi >W ECj : In Case 3, however, it is difficult to determine the dominance relationship between EC i and j with the absolute bound. In this case the dominance relationships may be determined by applying pairwise dominance relationships which can be divided into the following two cases. Case P1: jmin ðeck ; ecl Þ > 0: Case P2: jmin ðeck ; ecl Þp0 and jmin ðeck ; ecl ÞXjmin ðecl ; eck Þ: We define Case P1 as pairwise strict dominance which has been used by Fishburn (1965). We can find the fact that if the strict dominance (Case P1) between EC k and l holds, then pairwise strict dominance (Case P1) for the pair of EC k and l holds, and not vice versa, since jmin ðeck ; ecl ÞXymin ðeck Þ  ymax ðecl Þ: Case P2 is defined as pairwise weak dominance which has been used by Kmietowicz and Pearman (1984) and Weber (1985). Simply stated, the lowest possible value for eck is greater than the lowest possible value for ecl. We present two useful characteristics of pairwise weak dominance, of which formal proof are omitted. Firstly,

absolute weight bound

Case 1 ai

Case 2 aj

ai

aj

Case 3 ai

aj

1.0 0.5 0 Fig. 4. Three cases of absolute weight bounds between pairwise ECs’.

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when eci dominates pairwise weakly ecj ; then selecting ai cause less regret than selecting ecj (Kmietowicz and Pearman, 1984). Next, pairwise weak dominance can be always identified for eci and ecj ; thus pairwise weak dominance technique can be used as a certain decision rule for a final choicemaking (Park and Kim, 1997). After we get pairwise dominance relationships using formulation (3.3) and (3.4), we use the dominance graph approach (Park and Kim, 1997) to find the relation of all ECs’. 3.2. Characteristics of the method In this paper, only the prioritization method of ECs’ in HOQ of the product planning phase is described in detail. But HOQ charts of other phases can be analyzed in a similar way. The result of weight bounds of ECs’ in phase I is used as the input data or constraints for column in phase II, and the LPI of part characteristics is newly provided by the expert team. To derive the importance rating of part characteristics, we run the model using these new information. The sequential processes are performed until we get the priority of the production requirements in phase IV using the same model. 3.2.1. Normalization Lyman (1990) suggests the problem that the ECs’ weight is not consistent with CAs’ importance rating and recommends a normalization transformation method on the relationship values contained in traditional relationship matrix to remedy the problem. Normalization transformation is performed by dividing each of the relationship values in a given row by the row sum of the relationship values. This is represented by the following formula: n X rij r0ij ¼ Pn and r0ij ¼ 1 ð3:5Þ r j¼1 ij j¼1 P 0 0 and the normalized ECs’ importance rating is calculated using w0j ¼ m i¼1 di  rij : Using the Lyman’s normalization method means that the relationship values, rP ij ; are transformed into the relative impact, r0ij ; and all ECs’ total impact on each CA becomes sum-to-one, nj¼1 r0ij ¼ 1: In our model, we use three types of LPI, Forms P1, 3 and 5 in Section 3.1, as a mean to represent the relative impact of EC j on CA i: And the constraint, f nj¼1 rij ¼ 1gi¼1;2;y;m0 ; is included in the constraints set, R; then the above normalization concept is in itself satisfied by our formulations. 3.2.2. Information types of QFD When we use the subjective judgments in the QFD process, the determination problem of CAs’ and ECs’ important rating can be classified into four categories according to the information types of CA importance rating and relationship values. The information types provided by customer and design team can be complete or incomplete. Table 1 illustrates the applicable methods in the view of the information types provided by customers and design team. Simple weighted sum and AHP requires the complete data of CA weights and relationship values to prioritize the ECs. However, obtaining such cardinal information is very difficult and requires the considerable cognitive burden of information providers. In these days, there has been a direction to use incomplete information of relationship between ECs such as outranking method (Franceschini and Rossetto, 1995). Our proposed method is more user-friendly, because it can be used when both information types are incomplete. 3.2.3. Workload or cognitive load of articulating LPI At a glance, our suggested LPI form is more difficult, so providing LPI by decision maker needs more cognitive load. However, many researches, like AHP, use the pairwise comparison approach. Using the LPI

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Table 1 Category of the solution methods in prioritization of EC Relationship CA Complete Incomplete

Complete

Incomplete

Simple weighted sum AHP Proposed method

Outranking method Proposed method Proposed method

of this paper is based on the well-known pairwise comparison that have been widely used for comparison of alternatives in a selection problem. The lesser cognitive load of using LPI is already justified by a lot of prior researches, so we can say that using LPI is more convenient than using traditional approach. However, the computational aspect of using LPI is more difficult than that of traditional approach. Hence, using LPI needs a computer program, like decision support system. We develop just a prototype system for reducing the computational load, but it has to be further extended to be widely used in real applications. 3.2.4. Practical usage of dominance relation Results from the proposed model can be used to determine the priorities of engineering characteristics. Especially, when there is a conflict between teams or participants in determining design priority of engineering characteristics during the QFD process, we can determine which engineering characteristics are firstly considered according to the dominance relation. For example, when we determine raw material of a product in product design process, if EC2 ; the length of product, strictly dominates EC3 ; the width of a product, we can determine the raw material that is suitable for the length of product. As an another example (Han et al., 1998), when we determine information system development priority, if there are conflicts between departments of a company, we can determine the priority in accordance with the results of dominance relation from QFD process. To compare the strict and weak dominance, we can assume two situations. First one is occurring strict dominance between two engineering characteristics, for example ‘EC1 strictly dominates EC2’, and another situation is occurring weak dominance, for example ‘EC3 weakly dominates EC4’. The above two cases mean that EC1 is more important than EC2 and EC3 is more important than EC4. Also, the difference of importance in the first case, strict dominance, is relatively higher than the second case of weak dominance. Practically, if the EC1 strictly dominates EC2, many customer attributes are related with EC1. Also, to satisfy the customer requirements, EC1 is more seriously considered than the EC2. In case of weak dominance, the difference of importance of EC1 and EC2 is relatively lower than strict dominance. Relatively small numbers of customer attributes are related with EC1. Accordingly, when we determine the values or specifications of ECs, we have to put more concentration on the strict dominance case than weak dominance case. Also, we propose pairwise dominance as a supplementary tool which can define dominance relations between engineering characteristics. Although the pairwise strict/weak dominance can make priority between engineering characteristics, discriminating power of pairwise dominance is weaker than strict/weak dominance. Hence, we use the pairwise dominance as a supplementary tool of strict/weak dominance. If we use the pairwise dominance, we can differentiate the pairwise strict dominance and pairwise weak dominance similar to the strict and weak dominance which are mentioned previously.

4. An illustrative example Now we explain with the pencil example of Wasserman (1993) how the proposed method can be applied to prioritize customer attributes and engineering characteristics in QFD. There are five ECs and four CAs

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for the product planning matrix for a pencil shown in Fig. 5. From a market survey, the important customer attributes—easy to hold, does not smear, point lasts, and does not roll—are listed on the left-hand side of the matrix. For each customer attribute, the design team must respond to this need by identifying the important engineering characteristics which if the characteristics are fulfilled, would fulfill this need. In the example of Wasserman, he uses a complete information such as 100 point scale for CAs’ weights and a 1-3-9 scale to denote weak, medium, and strong relationships between CA and EC as shown numerical values in Fig. 5. We use LPI suggested in Section 3 to describe the relationships of CAs’ weights and ECs. Suppose that the expert team’s LPI is d1 ¼ d4 ; 0:5d2 pd1 p0:7d2 ; 0:5d3 pd2 p0:6d3 ; d2  d1 pd3  d2 : This information is modified into the relative ordinal scale from the specific values in Wasserman’s example. Naturally, it is easier to provide the above information than the specific values for CAs’ weights. Also, the LPI, fRg; of the relationships between ECs on four customer attributes is given by Table 2. As the next step, we formulate the QFD problem into the mathematical model and identify the dominance relations between ECs under the LPI fH0 g and fRg: Firstly, with the LPI Ri ; the minimum and maximum value of each rij are derived by using #ıi ðaj Þ in formula (3.3). The example formulation for deriving minimum value of r11 ; x 1 ðec1 Þ; is shown as following: r11 1:5r11  r14 p0; 3r11 þ r14 p0; r11 þ r13 þ r15 ¼ 0; r11 þ r12 þ r13 þ r14 þ r15 ¼ 1; r11 ; r12 ; r13 ; r14 ; r15 X0:

Relationship Symbols : very strong rel. : strong rel.

Eng. Desg. Req'ts

15

Does Not Swear

25

Point Lasts

45

Does Not Roll

15

Technical Importance

Hexagonality

Lead Dust Generated

Length of Pencil Easy to hold

Time Between Sharpening

a1 a2 a3 a4

a5 Minimal Erasure Residue

: weak rel

Customer Wants

min s:t:

: 9 point : 3 point : 1 point

Absolute

105

Relative

5.7 11.4 34.1 14.6 34.1

210 630 270

630

Fig. 5. QFD matrix for a pencil example (Wasserman, 1993).

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Table 2 LPI on the Relationships between ECs CA1 Set of constraints

CA2

CA3

CA4

r12 ¼ r13 ¼ r15 ¼ 0

r21 ¼ r14 ¼ 0

r34 ¼ 0

r42 ¼ r43 ¼ r45 ¼ 0

1:5r11 pr14 p3r11 X r1j ¼ 1

r23 ¼ r25

r33 ¼ r35

1:5r22 pr23 p3r22 X r2j ¼ 1

1:5r32 pr33 p3r32

3r41 pr44 p6r41 X r4j ¼ 1

j

1:5r31 pr32 p3r31 X r3j ¼ 1

j

j

j

Table 3 The value interval of each relationship Values of rij

EC1

EC2

EC3

EC4

EC5

CA1

min max

0.25 0.4

0 0

0 0

0.6 0.75

0 0

CA2

min max

0 0

0.1429 0.25

0.3750 0.4286

0 0

0.3750 0.4386

CA3

min max

0.0455 0.1429

0.1304 0.2308

0.3214 0.4091

0 0

0.3214 0.4091

CA4

min max

0.1429 0.25

0 0

0 0

0.75 0.8571

0 0

The maximum value of r11 can be derived by substituting ‘max’ for ‘min’ of the above formulation, and minimum and maximum value of rij for all pair of CAs and ECs can be derived by using their constraints set in Table 2. The results are shown in Table 3. Then, we can obtain the absolute bound for each EC in order to find the strict and weak dominance relation. An example of the formulation for the lower bound of EC1 is min s:t:

   d1  x 1 ða1 Þ þ d2  x2 ða1 Þ þ d3  x3 ða1 Þ þ d4  x4 ða1 Þ d1 ¼ d 4 ;

0:5d2  d1 p0; d1  0:7d2 p0; 0:5d3  d2 p0; d2  0:6d3 p0; d1 þ d2 þ d3 þ d4 ¼ 1; di X0

for i ¼ 1; 2; 3; 4:

By substitution of the value x i ða1 Þ into the objective function and solving the LP, ymin ða1 Þ is 0.071835.  ymax ða1 Þ is obtained by using max and xþ i ða1 Þ in the place of min and xi ða1 Þ: Similarly, the absolute bound

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for the other EC can be derived. Thus the following yields: EC1’s absolute bound ¼ ½0:0718; 0:1704; EC2’s absolute bound ¼ ½0:0886; 0:1779; EC3’s absolute bound ¼ ½0:2239; 0:3117; EC4’s absolute bound ¼ ½0:1688; 0:2766; EC5’s absolute bound ¼ ½0:2239; 0:3117: With the absolute bounds, the dominance relation between alternatives can be identified by dominance condition in Case 1 and Case 2; EC3 ¼ EC5 >W EC4 >W EC2 >W EC1: In this example, we get the dominance relations by using only absolute bounds and have no the alternative pairs corresponding to Case 3. If the pairs exist, then the dominance relation of the pairs can be obtained by applying the pairwise dominance condition. We can get the same results as Wasserman’s example although using LPI instead of pre-determined arbitrary numerical scale. Our proposed method can help expert team to minimize the burden of providing the information, and use the information faithful to the expert team’s feeling. If we proceed to next phase of the linked houses in Fig. 1, then the absolute bounds are used as constraints and the similar procedure can be applied with the new information about parts characteristics at phase II. Finally, our proposed method can be applied to the series of HOQ. The normalization concept is included in the our formulation and the consistence of CAs’ importance rating is satisfied. Also, our method is applicable to the decision situation when given all possible types of the decision makers’ LPI on both values of CAs weights and of ECs in any complicated hierarchical tree.

5. Conclusions Although QFD planning has been widely used for product design problem and other areas, assessing information from participants is still difficult process in QFD planning. Proposed methodology uses linear partial ordering to assess information in QFD. Especially, using the linear partial information makes it possible that the participants easily provide various types of information. In extracting information or knowledge of participants in QFD planning process, using linear partial information can reduce cognitive burden of designers and engineers. Based on the dominance concepts of MADM with incomplete information, we develop a methodology for prioritizing engineering characteristics and explain the methodology with an example. Although solving a number of LPs and computational complexity are major drawbacks of the methodology, development of computer program, like decision support system, and development of computer technology will overcome the drawbacks. Applying the methodology to real world problem, and then comparing the result with the result of traditional methodology will be an interest topic for further research. Also, applying to various application fields of QFD will be expected.

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