A QFD-Based Mathematical Model for New Product Development Considering the Target Market Segment

Responding to customer needs is important for business success. Quality function deployment provides systematic procedures for converting customer needs into technical requirements to ensure maximum customer satisfaction. The existing literature mainly focuses on the achievement of maximum customer satisfaction under a budgetary limit via mathematical models. The market goal of the new product for the target market segment is usually ignored. In this study, the proposed approach thus considers the target customer satisfaction degree for the target market segment in the model by formulating the overall customer satisfaction as a function of the quality level. In addition, the proposed approach emphasizes the cost-effectiveness concept in the design stage via the achievement of the target customer satisfaction degree using theminimal total cost. A numerical example is used to demonstrate the applicability of the proposed approach and its characteristics are discussed.


Introduction
In a fast-changing business environment, business units must invent new products or improve the quality of products continually to meet customer needs. The introduction of new products into the market to respond to customer demands is important for business success. From a marketing perspective, market segmentation is a necessary means for finding target customers. The new products should be positioned at the target customers during the product positioning process. The target customer satisfaction level for product quality should then be determined to meet the target customer requirements. To do this, a systematic new product development (NPD) process is usually carried out by business units to design an appropriate product that achieves the determined target customer satisfaction level based on a limited budget or using the minimum cost expenditure.
Quality function deployment (QFD) has been widely adopted by practitioners since the 1970s to design new products or improve existing ones [1]. Besides product development/design, QFD has been applied to fields such as quality management/planning, decision-making, manufacturing, service, and education [2]. QFD provides a systematic procedure for converting customer needs into technical requirements to ensure customer satisfaction. A number of researchers have proposed various quantitative approaches for applying QFD to a variety of problems. Among quantitative approaches, mathematical programming is usually applied to construct models to find the optimum design requirements for maximizing customer satisfaction under the limits of budget, time, and/or other resources [3][4][5][6][7][8][9][10][11][12][13][14].
For developing new products, the quality level is usually determined by the R&D team to achieve the anticipated satisfaction degree of the target customers. However, the existing literature related to quantitative approaches for applying QFD in NPD processes focuses on finding the optimal solutions of design requirements for maximizing customer satisfaction under a budgetary limitation, not for achieving the desired customer satisfaction level in terms of product quality. Thus, the determined quality level of new products may not meet customer requirements so that the business unit may lose its competitive advantage in the market. In addition, the relationship between customer satisfaction and quality level for target customers is important for the developments of new products to determine the quality level in order to 2 Journal of Applied Mathematics achieve the target customer satisfaction. Nevertheless, such a relationship is not incorporated in existing QFD models. Based on the above considerations, how to use the minimal budget/resources to achieve the quality level in order to meet the desired customer satisfaction should be considered in QFD-related problems. In this regard, instead of considering the budget constraint, this paper takes into account the relationship between customer satisfaction and quality level for the target customers in the QFD model to satisfy the desired satisfaction level in which the minimum design cost is achieved.
The rest of this paper is organized as follows. Section 2 briefly introduces the concept of QFD and the related normalization formulations, which are used in the proposed model. From a marketing perspective, the relationship between customer satisfaction and quality level for target customers is described in Section 3. In Section 4, considering the desired customer satisfaction and the corresponding quality level for target customers as constraints, a nonlinear mathematical programming model is proposed to minimize the total design cost. Section 5 provides an illustrative example of new bike design to demonstrate the feasibility and applicability of the proposed approach. Numerical analyses of the design parameters in the proposed model are also made to explore the characteristics of the model and their managerial implications. Finally, conclusions are provided in the final section.

Background
The basic concept of QFD is to transform customer voices into technical requirements to ensure customer satisfaction. The QFD processes are performed by applying the design information embodied in the relation matrix, called the house of quality (HOQ), as illustrated in Figure 1. In practice, QFD transforms customer needs into technical (or design/engineering) characteristics via the HOQ during the design or planning stage. The interior of the HOQ matches customer requirements (CRs) with the corresponding design requirements (DRs), identifying the relational intensity between each pair of CRs and DRs to ensure quality performance that can satisfy the target customers. If necessary, the roof of the HOQ, represented as a correlation matrix, is constructed to indicate the technical correlations among DRs. Figure 1 shows a HOQ example to signify the relational intensities between CRs and DRs as well as the correlation degrees among DRs. The importance degree of each CR is also displayed in the figure. In general, the HOQ contains the relational intensity between each pair of CRs and DRs and the technical correlation between each pair of DRs.
The legend in Figure 1 shows various designated symbols used to represent different degrees of relational intensity or technical correlation in the HOQ.
In general, the design information contained in the HOQ is integrated to further determine the importance of DRs and their achievement degrees in order to optimally satisfy customer satisfaction. To do this, a normalization process is usually applied by QFD researchers and practitioners. In the literature, the normalization model proposed by Wasserman [3] has been widely adopted (e.g., [6,7,9,10,13,[15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]). Based on Lyman's normalization concept [30], this model is developed by incorporating the correlations among DRs, formulated as from the vector space concept, where denotes the technical correlation between DR and DR . In the above equation, indicates the relational intensity between CR and DR , which is measured based on a 3-point scale, such as 1-3-9 or 1-5-9, for describing the weak-moderate-strong relationship. Although Wasserman's normalization model has been widely adopted, it has some weaknesses. For example, it assumes that customer requirements are mutually independent. However, this may not be true in practice. More importantly, this model may generate a relational intensity for a pair of CRs and DRs that does not exist in the original design information. The model may thus produce unreasonable outcomes.
Identifying such weaknesses in Wasserman's normalization model, L.-H. Chen and C.-N. Chen [31] proposed the following modified normalization model: where norm denotes the normalized relationship between CR and DR . Applying the above modified normalization model to integrate design information, the unreasonable outcomes from Wasserman's model can be avoided. Furthermore, considering the possibility that some CRs are correlated, as shown in Figure 2, L.-H. Chen and C.-N. Chen [31] also proposed the following normalization model to integrate CRs, similar to that in (2), in determining the normalized weights of CRs: where denotes the correlation between CR and CR and is the importance of CR . It is noted that the normalized weight norm in (3) is reduced to when CRs are mutually independent; that is, ∑ =1 = 1.

Satisfaction Function with Quality Level
The purpose of the development of new products or the improvement of existing products is mainly to satisfy customer needs or enhance customer satisfaction. Achieving customer satisfaction is always one of the main goals for the target market [32]. Customer satisfaction is considered one of the most important constructs for consumers [33]. Hence, customer satisfaction is important in marketing because of its great influence on customer purchase behavior. Based on an empirical study, Anderson and Sullivan [34] found that customer satisfaction can generally be described as a function of perceived quality from customers, considering an adjustment amount to reflect the customer perception of confirmation/disconfirmation. In their study, the customer satisfaction function is expressed as a concave down function of perceived quality. Following the concept of Anderson and Sullivan's study [34], the present study considers that customer satisfaction is positively related to the design quality of the product in terms of a concave down function, assuming that the quality designed for the product in the design stage will be perceived by the target customers in the future market; that is, the design quality level has a positive relationship with customer perceived quality. Figure 3 shows customer satisfaction (CS) as a concave down function of design quality (DQ).
For QFD applications, the design quality level of a product can be treated as the fulfillment level of design requirements (DRs), and therefore customer satisfaction can be described as a function of the DRs' fulfillment levels in the QFD processes. Let represent the cost required for DR with which the technical requirement for DR can be completely attained so that customer satisfaction can be fully achieved; (∈ [0, 1]) represents the decision variable taking the fulfillment level of the technical requirement for DR ,  denoting the quality level of DR required to meet the desired customer satisfaction. For simplification, suppose that the cost required to achieve the fulfillment level of DR , , is , that is, a proportional relation between the required cost and the fulfillment level. The value of = 0 means that the technical need for DR is only the basic requirement so that the cost is 0. In addition, the use of the whole budget for DR will achieve the full technical requirement, that is, = 1, and full customer satisfaction for this DR.
Let X = [ 1 , 2 , . . . , ] be the vector containing the fulfillment levels of the technical requirements for all DRs. For a new product, the overall DQ is defined as the average of the fulfillment levels of all DRs as where 0 ≤ DQ(X) ≤ 1. In other words, is interpreted as the design quality level of DR . In addition, based on previous studies (e.g., [3][4][5][6][7][8][9][10][11][12][13][14]), using the normalized information from the HOQ, norm and norm in (2) and (3), respectively, the customer satisfaction can be formulated as where 0 ≤ CS(X) ≤ 1. It is obvious that DQ(X) = 1 and CS(X) = 1 when = 1, ∀ = 1, 2, . . . , . As described above, based on the concept of Anderson and Sullivan's study [34], customer satisfaction is characterized as a concave down function of the design quality of a product for a specific market segment. Figure 3 shows such a function. The concave down curve illustrates the diminishing marginal effect; as design quality increases, the marginal effect of customer satisfaction decreases. This is characterized here using the following simple equation: 4 Journal of Applied Mathematics where (> 1) is a characteristic parameter used to describe the customer preference of a specific market segment for the design quality level of a specific product. The function in (6) characterizes features of the relationship between CS(X) and DQ(X), as described above. In practice, if the desired customer satisfaction degree is determined as the goal of satisfying the target market, (6) is employed to transform the desired value CS(X) into the corresponding value DQ(X) as the required design quality level.

Proposed QFD Budget Planning Model
In contrast to previous QFD-related studies that maximized customer satisfaction under a limited budget, this paper aims to find the minimum design budget to achieve the market goal in terms of the target customer satisfaction. To this end, some constraints are considered in the model. Based on (6), the target design quality level (0 < ≤ 1) corresponding to the target customer satisfaction (0 < ≤ 1) is determined by the management; that is, = , > 1. The achieved customer satisfaction, formulated as (5), based on the fulfillment level of the technical requirement, that is, the fulfilled design quality level, for each DR, should not be less than the target customer satisfaction ; similarly, the fulfilled overall design quality of (4) should not be less than the target design quality level . More importantly, the fulfilled overall design quality should not be less than the quality level reflected by the achieved customer satisfaction for the target market in (6); that is, ) . The right-hand side in the above inequality relation can be interpreted as the product quality perceived by the target customers, since it is derived based on the relations between CRs and DRs in the HOQ. Thus, this constraint ensures that the realized overall quality achieved from the product design is better than or equal to that perceived by the target customers.
In addition to the restrictions on the target design quality level and customer satisfaction, from the designers' viewpoint, in general there is a minimum satisfaction level for each customer requirement CR . The constraint of ∑ =1 norm ⋅ ≥ , 0 < ≤ 1, = 1, . . . , , reflects this condition, where can be determined based on the normalized weights of CR using the equation = 1 norm , 1 > 0, to reveal each CR's importance. The fulfilled design quality level for each DR , , is usually required to not be less than a minimum level, say , 0 < ≤ 1, where can be set as = 2 (∑ =1 norm ⋅ norm ) , 2 > 0, considering the normalized importance of DR and the target design quality level . Note that the parameters 1 and 2 can be determined subjectively by the management in weighing the corresponding CRs and DRs, respectively, under the constraints of 0 < ≤ 1 and 0 < ≤ 1. A nonlinear programming model is formulated in (7), in which constraints (7a) to (7g) are used for the specifications described above. The objective function (7) in the model reflects the proportional relation between the required cost and the fulfillment level of each DR.

Proposed Nonlinear Programming Model. Consider
It is noted that constraint (7a) can be represented as If the normalized importance of DR to the contribution of all CRs, norm , is defined as where ∑ =1 norm = 1, then (7a) and (7g) can be, respectively, expressed as In (6), parameter is important as it specifies the relation between the target design quality level and the target customer satisfaction and the relation between the fulfilled overall design quality and the quality level reflected by the achieved customer satisfaction, that is, the perceived quality level. As mentioned, parameter is related to the customer behavior of a specific market segment for a specific product. The value of can be either determined subjectively by the management or obtained using quantitative approaches. To do this, the simple regression approach is suggested in this paper. Under the consideration that customer satisfaction is Journal of Applied Mathematics 5 a concave down function of the perceived quality level of the product, a linearly transformed function can be expressed as ln = ln . Using a group of paired estimations ( , ) based on the experience and knowledge from the QFD team members and/or managers of the marketing department, the simple regression function ln = (1/ ) ln can be constructed to determine the value.
The procedure for applying the proposed model is as follows.
Step 1. Determine the desired product to be developed or improved for the target market segment.
Step 2. Construct the corresponding HOQ to reflect the relations between CRs and DRs.
Step 3. Integrate the basic design information in the HOQ to obtain the normalized values norm , norm , and norm using the normalization model in (2) (and (3), if needed).
Step 4. Considering that is a concave down function of , subjectively determine the value of for meeting the relationship = , > 1. Alternatively, collect a group of paired estimations ( , ), ∀ , ∈ [0, 1], for the desired product in the target market segment. Decide the parameter (> 1) in ln = (1/ ) ln by applying a simple regression analysis.
Step 5. Determine the target customer satisfaction for NPD based on the target market segment and then obtain the target design quality level based on = using the value determined in Step 4.
Step 6. Let management subjectively decide the values of parameters 1 and 2 .
Step 7. Construct the proposed nonlinear programming model using the integrated information from the HOQ obtained in Step 2.
Step 8. Solve the model to find the optimal design quality level of each DR so that the target customer satisfaction is achieved for the target market with the minimum budget.

Numerical Illustration
This section demonstrates the applicability of the proposed approach using a constructed example of a manufacturer of bicycles. Based on this example, parameter analyses are provided to further illustrate the proposed approach.

Example.
With rising awareness of environmental protection, the municipal government has encouraged citizens to take mass transportation to work. The marketing department of the bicycle firm has observed that an increasing number of citizens are willing to bike the short distance between home and mass transportation stations. Therefore, the marketing department plans to launch a new bike to cater to this trend. For this plan, an important issue for the NPD project team is to determine the minimum budget needed to achieve the market goal of the NPD during the design stage.
Firstly, the newly developed bike is positioned as simple and easy for biking the short distance between home and mass transportation station and aims to satisfy commuters. The NPD team selects QFD as the design platform. In the second step, the HOQ is constructed for specifying the relations between CRs and DRs. The following six basic requirements are identified: (1) ride comfort, (2) ride safety, (3) easy handling, (4) easy movement, (5) low maintenance, and (6) durability. Based on consensus, the team members propose nine design requirements that correspond to the six basic customer requirements. The nine design requirements are (1) frame, (2) suspension, (3) derailleur, (4) brake, (5) wheels, (6) handlebars, (7) saddle, (8) pedals, and (9) total weight. The degree of relational intensity for the relationship between CRs and DRs is defined by weak-moderate-strong; the correlation among CRs or among DRs is also defined by weak-moderatestrong, but allowing for negative correlations. Furthermore, the NPD team identifies the CRs' importance degrees for the product to emphasize its easy handling and ride safety. The basic QFD design information for the product is shown in Figure 4.
The main work in the third step is to integrate the design information into the HOQ to obtain the normalized relation strengths or weights, namely norm , norm , and norm , for establishing the proposed nonlinear programming model to achieve the market/quality goal at the minimum design cost. As described before, L.-H. Chen and C.-N. Chen's normalization models, (2) and (3), are employed. To do this, the degree of relational or correlation intensity described as weak-moderate-strong must be numerically scaled. As commonly used in the literature, the rating scale 1-3-9 is employed for relational intensity, that is, weak-moderatestrong, between CRs and DRs, and ±(0.1-0.3-0.9) is used for correlation, that is, weak-moderate-strong, among CRs or among DRs with negative correlations allowed. Figure 5 shows the normalized results for the product using the basic design information in Figure 4.
After the integrated information from the HOQ has been obtained, the characteristic parameter for the market segment of commuters should be determined to characterize the functional curve between customer satisfaction and design quality level. Based on the experience of the marketing experts, is set to 2, that is, CS(X) = [DQ(X)] 1/2 , as shown in Figure 6. Based on the determined function, the fifth step is to set up the target design quality level . From a market competitive analysis, the target customer satisfaction is set as = 0.8, so the target design quality level is obtained as 0.64 based on the function = , = 2. To formulate the proposed nonlinear programming model for the minimum cost in the sixth step, the parameters 1 and 2 should be determined to specify the minimum satisfaction level for each customer requirement CR and the minimum design quality level for each DR . The values 1 = 3 and 2 = 3 are set by the NPD team in this example.

Parameter
(2) The 1 and 2 values in (7e) and (7g) affect the minimum satisfaction level for each customer requirement CR and the fulfilled minimum design quality level for each DR , respectively, and therefore affect the total design cost. Larger values of 1 and/or 2 increase the total design cost. With = 2 and 2 = 3, Figure 10 shows the impact of the 1 value on the total design cost. The total design cost remains unchanged in a range of 1 values that depends on because the fulfilled minimum design quality level at 2 = 3 confines the influence of the minimum satisfaction level on the total design cost at some levels of 1 regardless of the target customer satisfaction level . It is also noted that if the 1 value is sufficiently large ( 1 = 3.5) and 2 = 3, the total design cost is the same (1469) for all three levels of the target customer satisfaction, since the aggregated results from the required minimum satisfaction level for each customer requirement CR surpass each target customer satisfaction. The high required minimum satisfaction level due to 1 = 3.5 resulted in the largest total design cost. With = 2 and 1 = 3, the total design cost changes with 2 , as shown in Figure 11.
(3) From a design viewpoint, if the minimum satisfaction level for each customer requirement CR and the fulfilled minimum design quality level for each DR are not required, that is, 1 and 2 are fixed at 0, the minimal total design cost can be obtained from the proposed model (7) at each setting of the targeted customer satisfaction level . For example, with 1 = 2 = 0 and = 0.6, 0.7, and 0.8, respectively, the total design costs are 700, 878, and 1063, respectively, regardless of the value. This indicates that to achieve the target degree of customer satisfaction for a certain market segment, the investment amount of the total design cost has the lowest bound.

Conclusions
A mathematical model was proposed for determining the design quality level required to achieve the target customer satisfaction for the target market segment with the minimum total design cost. In the proposed model, the customer satisfaction is expressed as a function of the quality level, considering the consumer behavior of the target market segment. Moreover, the proposed model allows the NPD team to set the minimum satisfaction level for each CR and/or the fulfilled minimum design quality level for each DR, making the design processes flexible. A numerical example was provided to demonstrate the feasibility and applicability of the proposed approach. In future research, the relation between customer satisfaction and quality level will be further investigated to make the proposed model more applicable in the real world.

CR :
The th customer requirement DR : The th design requirement : Relational intensity between CR and DR : Technical correlation between DR and DR : Normalized relational intensity between CR and DR in Wasserman's normalization model norm : Normalized relational intensity between CR and DR in L.-H. Chen and C.-N.
Chen's normalization model : OriginalimportanceweightofCR : Correlation between CR and CR norm : Normalized importance weight of CR in L.-H. Chen and C.-N. Chen's integration model when CRs' correlations exist norm : Normalized importance weight of DR : RequiredcostforDR to fully achieve customer satisfaction : Technical fulfillment level for DR X: Design vector containing the fulfillment levels of all DRs DQ(X): Overall design quality level by design vector X CS(X): Fulfilled customer satisfaction by design vector X : Target design quality level : Target customer satisfaction : The characteristic parameter to describe the customer preference of a specific market segment for the design quality level of a specific product : Minimum required satisfaction level for CR 1 : Design parameter to determine : Minimum fulfilled quality level for DR 2 : Design parameter to determine .