# Configuration Equilibrium Model of Product Variant Design Driven by Customer Requirements

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## Abstract

**:**

## 1. Introduction

## 2. Research Background

#### 2.1. Solutions to Customer Requirements (CRs)

#### 2.2. Decision-Making of the Scheme

## 3. Product Variant Design Driven by CRs

#### 3.1. Customer Dynamic Requirements Acquisition and Transformation

_{1},value

_{1}>∨<R

_{2},value

_{2}>∨…∨<R

_{f},value

_{f}>…∨<R

_{n},value

_{n}>,

_{f}is the fth characteristic parameter of the product, and value

_{f}is the corresponding value of the fth characteristic parameter (expressed by a parameter or option value), f = 1, 2, …, n.

#### 3.2. Modularization and Parameterized Variant Design Method

_{1}basic modules and M

_{2}optional modules. The instance library of the module can be expressed as:

_{ij}represents the jth module of the ith type module, where i = 1 represents a basic module, and i = 2 represents an optional module. P

_{ijk}represents the kth instance of the module, where k = 1,2,..., N

_{ij}, and N

_{ij}is the number of instances of the module. The instance library of module M

_{ij}contains (N

_{ij}− 1) existing common instances and a virtual parametric variant instance ${\overline{P}}_{ij{N}_{ij}}$. The expression of the product instance library is shown in Figure 2.

## 4. Configuration Model Based on the Bayesian Nash Equilibrium

#### 4.1. The Model of the Bayesian Nash Equilibrium Theory

_{g},X

_{m},U

_{m}(S

_{R},C

_{p})}.

_{g}(g = 1,2) is gth player, X

_{m}(m = 1,2,...) is set of strategies, and U

_{m}(S

_{R},C

_{p}) is the payoff set corresponding to the strategy set X

_{m}. S

_{R}is the degree of customer satisfaction, and C

_{p}is the product cost.

#### 4.2. Determination of Product Strategy Set

_{1},R

_{2},…,R

_{n}). The corresponding weight vectors are W = (ω

_{1},ω

_{2},…,ω

_{n}). The customer satisfaction matrix CS between module instances and product characteristic parameters is constructed as follows:

_{ijkf}is the customer satisfaction of product characteristic parameter R

_{f}for the kth instance of module M

_{ij}. Here, i = 1,2; j = 1,2,…,M

_{i}; k = 1,2,…N

_{ij}; f = 1,2,…n. If a module is not related to the product characteristic parameter R

_{f}, the customer satisfaction of the characteristic parameter R

_{f}for all instances of the module set to 0.

_{ij}in the module layer, that is, <M

_{ij}, add >. The customer satisfaction of the characteristic parameter for module instance is expressed as S

_{ijkf}= 1. The deletion of optional modules matched with the CRs in the module layer can reduce the cost or delivery time of the product.

_{d}and the value of value

_{d}, the function of the customer satisfaction and the product characteristic parameter value can be established as shown in Figure 4, where Q is the overflow value of the customer satisfaction. The value can be determined by the designer according to the actual situation. If the product characteristic parameter value is discrete, the product designer can determine the correlation between the product characteristic parameter value and the customer satisfaction according to the actual situation.

_{M}= {s

_{ij}

_{1f},s

_{ij}

_{2f},s

_{ij}

_{3f}, ...,1}, can be obtained according to the value of its own characteristic parameters of each instance in the instance library of the module M

_{ij}. The virtual variable instances can completely satisfy the CRs, so the satisfaction is set to 1. The description of this process is shown as follows:

_{ijk}is a binary decision variable, a

_{ijk}= 1 represents the kth instance of module M

_{ij}was selected, otherwise, a

_{ijk}= 0, k = 1,2, …, N

_{ij}. d

_{ijk}is the delivery time of the kth instance of module M

_{ij}, $\overline{D}$ is the expected product delivery time, and ${\overline{D}}^{\prime}$ is the last tolerated delivery time. Among them, the delivery time of the virtual parameterized variant instance includes the design cycle and manufacturing cycle of the new instance, and its value can be obtained based on the historical data of the firm.

_{f}is the weight of the characteristic parameter R

_{f},${{\displaystyle \sum}}_{f=1}^{n}{\omega}_{f}=1$.

_{△}, is set according to the empirical value. The product model satisfying the condition, S

_{i}≥ S

_{△}, is searched in the product configuration model. The searched product model constitutes a product strategy set X

_{m}= {x

_{1},x

_{2},…,x

_{m}}.

#### 4.3. Construction of Payoff Function

_{1}and U

_{2}of customers and enterprises respectively. Among them, customer satisfaction and product cost are required to be close to maximization and minimization.

_{Ri}is the customer satisfaction with strategy x

_{i}, C

_{Pi}is the product cost with strategy x

_{i}. In order to improve the accuracy of the payoff value and facilitate the comparison of payoff values, when the satisfaction is 1, the customer’s payoff is set to 10, then the payoff function of the customer to strategy x

_{i}is:

_{Ai}is the additional cost of product x

_{i}and C

_{r}is the cost of customer requirements, then the payoff function of firm to strategy x

_{i}is:

_{ijk}is the cost of the kth module M

_{ij}. The cost of virtual parameterized variant instances includes the design cost and manufacturing cost, and its value can be obtained based on firm historical data. r

_{i}is the acceptance coefficient of the firm to the cost.

_{i}is equal to or lower than the rated value, the firm accepts the strategy; otherwise the firm does not accept the strategy. Therefore, if ${C}_{pi}\le {C}_{\Delta}$, r

_{i}= 1; otherwise, r

_{i}= 0.

_{1}must be selected, and Z modules (Z ≤ M

_{2}) are selected in the optional modules of M

_{2}. Then each module selects an instance accordingly.

_{i}= 1; otherwise, the value is 0.

#### 4.4. Game Tree and Payoff Matrix

#### 4.5. Calculation of Nash Equilibrium Based on an Improved Simulated Annealing (SA)

_{m}.

_{T}= g.

_{1}and U

_{2}of the customer and firm, as well as the comprehensive payoff U(x

_{i}). Then the strategy x

_{i}of the minimum value of U(x

_{i}) is a Nash equilibrium solution. If the customer’s satisfaction with the product is higher and the cost is relaxed, the firm can increase the value of the threshold S

_{△}in the selection process of the strategy set, and further search to obtain a new Nash equilibrium solution.

## 5. Case Study

#### 5.1. Transformation of CRs

_{1}<rated load, 500 kg>R

_{2}<continuous running time, 10 h>R

_{3}<stop precision, ±5 mm>and R

_{4}<automatic charging function, add>. At the same time, it inherits the unchanged characteristic parameters of the product such as R

_{5}<body shape, L1800 × W650 × H1020>.

#### 5.2. Variant Design Modules

#### 5.3. Determination of Strategy Sets

_{1}<rated load, 500 kg> as an example, the characteristic parameter R

_{1}is the larger-the-better type of parameter, and the designer can construct the customer satisfaction function as shown in Figure 3a. According to the characteristic parameter R

_{1}locking to the car body module, the satisfaction of the module instance to the characteristic parameter R

_{1}is S

_{M}= {0.6, 0.9, 1.1, 1}. The value of customer satisfaction with the characteristic parameter R

_{1}is set to 0 in the other unrelated modules. Similarly, the customer satisfaction matrix CS between module instances and characteristic parameters of the product can be constructed as shown below.

_{1}= 0.241, ω

_{2}= 0.176, ω

_{3}= 0.172, ω

_{4}= 0.196, and ω

_{5}= 0.215. From the data in Table 1, it can be seen that the parameterized variant instances of the module meet the requirements well, but at the same time the cost and delivery time of other modules increase. By considering the cost and delivery time, a Nash equilibrium game model between customer and firm payoff is established. The customer satisfaction threshold of the AGV is set by the firm designer to S

_{△}= 0.84 according to past experience. After verification, there are nine configuration schemes that meet the requirements. The corresponding strategy sets are X

_{4}= {x

_{1}, x

_{2}, x

_{3}, x

_{4}, x

_{5}, x

_{6}, x

_{7}, x

_{8}, x

_{9}} as shown in Table 3.

#### 5.4. Calculation of Nash Equilibrium Based on SA

_{T}= 50. Thereafter, by setting the corresponding satisfaction threshold, the simulated annealing algorithm is used to calculate the optimal strategy scheme, which is repeated many times, and the solution is set as a constraint to find the next one. The results are listed in Table 4.

_{1}is the minimum (0.252). Therefore, it is the Nash equilibrium solution, and the configuration scheme x

_{7}(P

_{111}, ${\overline{P}}_{124}$, P

_{132}, P

_{211}) is the best one.

_{9}is larger than those of the other strategies. From the composition of the module, the strategy does not include other variation modules, and the inherent modules themselves are expensive, resulting in high costs. Due to the limitation of module parameters, customer satisfaction has not been significantly improved. Relatively speaking, there are few competitive advantages and it is unlikely for firms to choose this strategy.

_{1}is the highest, and the virtual variant instance module is adopted. Compared with x

_{7}the cost is high, but it is more suitable for pursuing high-quality customers.

## 6. Conclusions

- Based on the variant requirements of products, a virtual parameterized variable example is presented to realize the product variant design which combines modularization and parameterization.
- A Bayesian Nash equilibrium game model with customer satisfaction and reduced cost as the objectives is established based on the configuration decision problem of the product variant module, and the equilibrium solution of the scheme decision is realized.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Table 1.**Comparison between the material annealing process and the optimization problem solving process.

Physical Annealing | Optimization Problem |
---|---|

State of matter | Solution |

The lowest energy state of matter | Optimal solution |

Annealing process | Solution procedure |

Temperature | Controls parameter |

Energy | Objective function |

Constant temperature process | Metropolis Sampling process |

Module Name | Module Instance | Cost (1000 Yuan) | Delivery Time (Days) | Module Attributes |
---|---|---|---|---|

M_{11} car body module | P_{111} | 8 | 4 | basic module |

P_{112} | 12 | 5 | ||

P_{113} | 14 | 6 | ||

${\overline{P}}_{114}$ | 13 | 7 | ||

M_{12} battery module | P_{121} | 4.5 | 2.5 | basic module |

P_{122} | 3 | 2 | ||

P_{123} | 5 | 3.5 | ||

${\overline{P}}_{124}$ | 6 | 4 | ||

M_{13} navigation module | P_{131} | 12 | 7 | basic module |

P_{132} | 8 | 6.5 | ||

${\overline{P}}_{133}$ | 15 | 8 | ||

M_{21} automatic charging module | P_{211} | 10 | 5 | optional module |

P_{212} | 9 | 7 |

Scheme | Component Module | Delivery Time (Days) | Customer Satisfaction S |
---|---|---|---|

x_{1} | ${P}_{111},\text{}{\overline{P}}_{124}$, P_{131}, P_{211} | 20 | 0.8692 |

x_{2} | P_{111}, P_{121}, ${\overline{P}}_{133}$, P_{211} | 19.5 | 0.8684 |

x_{3} | P_{112}, P_{123}, P_{132}, P_{211}, | 20 | 0.8637 |

x_{4} | P_{112}, P_{122}, ${\overline{P}}_{133}$, P_{211}, | 20 | 0.8625 |

x_{5} | P_{113}, P_{122}, P_{131}, P_{211} | 20 | 0.8548 |

x_{6} | P_{112}, P_{122}, ${\overline{P}}_{133}$, P_{212} | 20.5 | 0.8528 |

x_{7} | P_{111}, ${\overline{P}}_{124}$, P_{132}, P_{211} | 19.5 | 0.8520 |

x_{8} | P_{111}, P_{123}, P_{131}, P_{211}, | 20 | 0.8516 |

x_{9} | P_{113}, P_{121}, P_{132}, P_{211} | 20 | 0.8431 |

Strategy | Component Module | Payoff Value |
---|---|---|

x_{7} | ${P}_{111},\text{}{\overline{P}}_{124}$, P_{132}, P_{211} | 0.252 |

x_{3} | P_{112}, P_{123}, P_{132}, P_{211} | 0.419 |

x_{8} | P_{111}, P_{121}, P_{131}, P_{211} | 0.760 |

x_{1} | P_{111}, ${\overline{P}}_{124}$, P_{131}, P_{211} | 1.375 |

x_{9} | P_{113}, P_{121}, P_{132}, P_{211} | 1.458 |

Firm | ||||
---|---|---|---|---|

Strategy | Respective Strategies | Implement | Non-Implement | |

Customer | x_{7} | accept | (8.520,8.889) | (0,0) |

refuse | (0,0) | (0,0) | ||

x_{3} | accept | (8.637,9.556) | (0,0) | |

refuse | (0,0) | (0,0) | ||

x_{8} | accept | (8.516,9.556) | (0,0) | |

refuse | (0,0) | (0,0) | ||

x_{1} | accept | (8.692,9.778) | (0,0) | |

refuse | (0,0) | (0,0) | ||

x_{9} | accept | (8.431,9.889) | (0,0) | |

refuse | (0,0) | (0,0) |

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**MDPI and ACS Style**

Yang, Q.; Bian, X.; Stark, R.; Fresemann, C.; Song, F.
Configuration Equilibrium Model of Product Variant Design Driven by Customer Requirements. *Symmetry* **2019**, *11*, 508.
https://doi.org/10.3390/sym11040508

**AMA Style**

Yang Q, Bian X, Stark R, Fresemann C, Song F.
Configuration Equilibrium Model of Product Variant Design Driven by Customer Requirements. *Symmetry*. 2019; 11(4):508.
https://doi.org/10.3390/sym11040508

**Chicago/Turabian Style**

Yang, Qin, Xianjun Bian, Rainer Stark, Carina Fresemann, and Fei Song.
2019. "Configuration Equilibrium Model of Product Variant Design Driven by Customer Requirements" *Symmetry* 11, no. 4: 508.
https://doi.org/10.3390/sym11040508