A Multi-Objective Optimization Approach to Product Configuration Design with the Consideration of Uncertain Information

Abstract:

The current product configuration methods can only be applied to the situation when the configuration information is specific or fuzzy. In order to address this problem, a new multi-objective optimization approach to configuration design with the consideration of several types of uncertain information was proposed. The uncertain configuration information was uniformly described with interval numbers. Targeting on optimizing the performance, cost and term of configured products, three mathematical models was established, and some adaptations were made to these models according to the interval number. A multi-objective optimization model was generated by integrating the three models. The non-dominated sorting genetic algorithm II was used to solve the model and a Pareto optimal set of product configuration schemes was obtained. A general optimum selection method was put forward based on the fuzzy set theory, and the optimization sequence of the Pareto solutions can be founded using the method. The proposed approach can effectively deal with the problem of product configuration optimization under uncertain information.

• Periodical: Applied Mechanics and Materials (Volumes 236-237)
• Pages: 1078-1084
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.236-237.1078
• Online since: November 2012
• Authors: Meng Zhang, Guo Xi Li, Yue Hui Yan, Bao Zhong Wu
• Keywords: Configuration Optimization, Multi-Objective, NSGA-II, Pareto Optimal Set, Uncertain Information

References

[4] and [6], taking product performance, cost and term as the optimization objectives, built a multi-objective configuration optimization model to retrieve the reasonable configuration options to meet customer requirements for performance, cost and time. In this article, on the basis of these studies, we extend the fruits for further development in order to enable them to be suitable to the situation when configuration information is uncertain.

[1] Performance-oriented configuration optimization model Let denotes the vector of the product performance, where is the performance r and R is the total number of the performance items. The corresponding weight vector is . The correlation between the module instance and product performance is shown in Fig. 1. Fig. 1 The correlativity matrix of the module instance and the product performace In the above figure, represents the correlativity value of and , that can be described by inguistic assessment information. The vague language need to be converted to the triangular fuzzy number, and then normalized. Consequently, the correlation value in the expression of interval numbers is listed in Table 1. Table 1 The quantitative correlation values Inguistic infor triangular fuzzy number normalized fuzzy nubler interval number Very Strong (10, 12, 12) (0. 833, 1, 1).

DOI: 10.7554/elife.21592.008

. 9165, 1] Strong (8, 10, 12) (0. 667, 0. 833, 1).

. 75, 0. 9165] Less strong (6, 8, 10) (0. 5, 0. 667, 0. 833).

. 5835, 0. 75] Common (4, 6, 8) (0. 333, 0. 5, 0. 667).

. 4165, 0. 5835] Less weak (2, 4, 6) (0. 167, 0. 333, 0. 5).

. 25, 0. 4165] Weak (0, 2, 4) (0, 0. 167, 0. 333).

. 0835, 0. 25] Very weak (0, 0, 2) (0, 0, 0. 167).

, 0. 0835] The objective of the product configuration is to make the product performace realize the optimality. Accordingly, the performance-oriented configuration optimization model is shown as follows [6].

[6] is a decision variable corresponding to the module instance. represents that the instance is in the configuration while means that the instance does not participate in the configuration. Due to the interval number , the above model need to be adapted.

[7] A dual-objective optimization model can be further gotten by transforming the Equation (7).

[8] Use the linear weighting method to convert the model to a single-objective optimization model.

[9] and , respectively, denote the importance of the objective function and , where and . It is known that the solution of the model (9) belongs to the non-inferior solution set of the model (8).

[2] Cost-oriented configuration optimization model Let denote the cost matrix of the module instance, where is the cost of , including the manufacturing cost and the assembly cost. The goal of the product configuration is to make the product cost the lowest. The cost-oriented configuration optimization model is built [6].

DOI: 10.32657/10356/54837

[10] can be a specific value, an interval nubmer or a fuzzy number. It need to be unified expressed by the interval number, thus the Equation (10) is adapted.

[11] .

[12] .

[13] and denote the importance of the objective function and , respectively, where and.

[3] Time-oriented configuration optimization model Let represent the duration matrix of the module instance, where denotes the duration of the module instance , including the production duration and the assembly duration. The objective of the product configuration is to make the product term the shortest. The time-oriented configuration optimization model is established [6].

[14] is the function between the product term and the cumulative period of modules, which can be gotten by the least squares fitting method on the basis of a large number of statistical data. Similarly, can be a specific value, an interval nubmer or a fuzzy number. Some adaptations are performed to Equation (14).

[15] .

[16] .

[17] and denote the importance of the objective function and , respectively, where and . It should be noted that product performance, cost and term are generic product configuration optimization evaluation criteria, and in the implementation process of the actual configuration, other evaluation criteria can be considered. There are diverse constraints in the optimization models, including the configuration constraint, the selection constraint, the correlation constraint, the cost constraint, the time constraint and the weight constraint.

DOI: 10.32657/10356/54837

[1] The configuration constraint. During the configuration process, only one or zero module instance is allowed to be selected for each functional module, that can be expressed as.

[18] .

[2] The selection constraint. This constraint is used by customers to pre-determine some module instances to be selected or rejected. The corresponding module instance decision variable will be set to 1 or 0. indicates that the customer choose while means that the customer is not allowed to choose.

[3] The correlation constraints. In the actual configuration process, there are correlation constraints among the module instances belonging to different functional modules, including the selective constraints and the exclusive constraints. The former refers to the matching relations between different module instances, i. e., the constraint - If then - means that if is selected, then must be selected too. The latter refers to non-coexistence of some instances, i. e., the exclusive constraint - If then - represents that if is selected, then can not be selected.

DOI: 10.1787/9789264232907-table65-en

[4] The cost constraint. The price of the configured product shall not exceed the maximum price the customer can afford, , and the constraint can be expressed as follows.

[19] denotes the profit margin of the enterprise. The above equation is adapted as follows.

[20] .

[21] .

[5] The time constraint. The term of the configured products shall not exceed the time period allowed by the customer, , that can be expressed as follows.

[22] Some adaptations are made to the above equation.

[23] .

[24] .

[6] The weight constraint.

[25] Pareto optimality solving and solutions sorting Product configuration optimization is a multi-objective optimization problem. The different objectives are in conflict with each other, then a solution does not exist which can make all goal to reach optimality. Actually, a group of non-dominated solutions ought to be obtained, namely Pareto optimal solution set [] TAN C K, GOH K C, LIU D S, et al. A competitive and cooperative co-evolutionary approach to multi-objective particle swarm optimization algorithm design [J]. European Journal of Operation Research, 2010, 202: 42-43. ]. In recent years, multi-objective optimization algorithm is developing rapidly. The genetic algorithm for the multi-objective problem has been widely applied in the field of scientific research and engineering practice, especially Niched of Pareto Genetic Algorithm (NPGA), Neighborhood Cultivation Genetic Algorithm (NCGA), Fast and Elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) and so on. NSGA-II is the most successful and most widely used. The gamultiobj, function of Matlab is a multi-objective optimization algorithm based on NSGA-II. In this paper, the NSGA-II algorithm is adopted to resolve the problem of product multi-objective configuration optimization, resulting in a Pareto optimal solution set of the configuration schemes. When the Pareto set is gotten, the elements in the set should be sorted. Using the method of optimal selection in the Pareto set based on the fuzzy set theory [] ABIDO M A. Multiobjective evolutionary algorithms for electric power dispatch problem [J]. IEEE Transactions on Evolutionary Computation, 2006, 10(3): 315-329. ], the integrated Pareto selection mechanism can be established. Let the member function denote the proportion of the objective i of the non-dominated solution s.

[26] represents the objective i and can be , or here. is the minimum value of while is the maximum value. For the non-dominated solution s in the Pareto set, we define a dominated function.

[27] M denotes the size of the Pareto set. The greater the value of is, the better the performance of the solution s is. The preferred sequence of the solutions in the set could be obtained by sorting the solution in accordance with in descending order. Summary For the uncertainty of the product configuration information, a standard method of uniform expression for various forms of information was established. On the basis of that, three product configuration optimization goals with the consideration of uncertain information were quantitatively analyzed, and consequently the multi-objective optimization model of product configuration was developed. Using the NSGA-II algorithm to solve it, the Pareto optimal set can be obtained in which the Pareto solutions arrange in descending order. The proposed approach in this research can effectively be applied to the configuration optimization problem that the configuration information is uncertain. The precise configuration and the fuzzy configuration are two special cases of this method. This paper provides a supportive methodology to solve the practical configuration under the mass customization mode. References.