## 3 The New Approach Framework
### Phase 3--Formulation and Optimization: Build Optimization Model and Formulation  
In Phase 3, a mathematical programming model is applied to formulate the overall customer satisfaction (OCS) with predefined information from the first two phases and additional input from the service demander. An optimal product configuration that maximizes OCS would be recommended. In Phase 1, the service provider needs to define ECs with prices (\(P_{jk}\)) and scores (\(S_{jk,m}\)), and then normalize the scores (\(S^{\prime}_{jk,m}\)). The normalized final score \(m\) of EC \(j\) (\(S^{\prime}_{j,m}\)) depends on the selection of parts (\(X_{jk}\)). In Phase 2, the service provider needs to construct a HOQ with the strength of the relationship between CRs and ECs (\(R_{ij}\)) and normalize them into the normalized strength of relationship (\(R^{\prime}_{ij}\)) to express the fulfillment level (\(F_{i}\)) by the final scores of all ECs for all CRs. Meanwhile, a Kano questionnaire must be completed to derive an expression of customer satisfaction, \(CS_{i}=f_{i}(F_{i},a_{i},b_{i})\). In Phase 3, the service demander needs to set a budget (\(B\)) and input the importance weight (\(w_{i}\)) for each CS. The total price of the selected parts must be no more than the budget. Overall customer satisfaction (\(S\)) can be expressed by the dot product of importance weights and customer satisfaction. The objective of the optimization model is to maximize the overall customer satisfaction under budgetary and other constraints. By integrating all the variables, the optimization model can be formulated as shown below. **Notation:**  
\(a_{i}\): _Constant a in \(CS_{i}-F_{i}\) Relationship Function_  
\(b_{i}\): _Constant b in \(CS_{i}-F_{i}\) Relationship Function_  
\(B\): _Budget_  
\(F_{i}\): _Fulfillment Level of CR\({}_{i}\)_  
\(CS_{i}\): _Customer Satisfaction of CR\({}_{i}\)_  
\(P_{jk}\): _Price of Part jk_  
\(R^{\prime\prime}_{ij,m}\): _Normalized Strength of Relationship between CR\({}_{i}\) and Score m of EC\({}_{j}\)_  
\(S\): _Overall Customer Satisfaction_  
\(S^{\prime}_{jk,m}\): _Normalized Score m of Part jk_  
\(S^{\prime}_{j,m}\): _Normalized Final Score m of EC\({}_{j}\)_  
\(w_{i}\): _Importance Weight of CR\({}_{i}\) in Overall Customer Satisfaction_  
\(X_{jk}\): _Selection of Part jk_  
**Decision Variables:**  
\(X_{jk}=\left\{ \begin{array}{ll}1&\quad\text{if }\text{Part jk is selected}\\ 0&\quad\text{otherwise}\end{array} \right.\)  
**Objective Function:**  
\(max\): \(S\): _Overall Customer Satisfaction_  
\(\sum\limits_{k}S^{\prime}_{jk,m}X_{jk}=S^{\prime}_{j,m}\): \(\forall j\): _Calculate Final Score m of EC_  
\(\sum\limits_{j}\sum\limits_{m}R^{\prime\prime}_{ij,m}S^{\prime}_{j,m}=F_{i}\): \(\forall i\): _Calculate Fullfillment Level of CR_  
\(CS_{i}=f_{i}(F_{i},a_{i},b_{i})\): \(\forall i\): _Calculate Customer Satisfaction of CR_  
\(\sum\limits_{i}w_{i}CS_{i}=S\): _Calculate Overall Customer Satisfaction_  
**Constraints:**  
\(\sum\limits_{j}\sum\limits_{k}P_{jk}X_{jk}\leq B\): _Total Price \(\leq\) Total Budget Available_  
\(\sum\limits_{k}X_{jk}=1\): \(\forall j\): _Only one component can be selected for an EC_The expression of one-dimensional CSs follows a linear function, \(y=ax+b\). The expression of positive or must-be CSs follows a smooth nonlinear function, \(y=ae^{x}+b\). Overall, this is a nonlinear optimization model. It cannot be solved by a linear programming algorithm. Instead, it can be solved by the generalized reduced gradient (GRG) method developed by [PERSON] et al. (2016). The GRG method can be considered as a nonlinear extension of the Simplex method, which solves a linear optimization problem by selecting a basis, determining a search direction, and performing a line search on each iteration. The GRG method computes or approximates derivatives of the problem functions many times during the course of the optimization. It is robust to process the nonlinear objective function and constraints.