A Constraint Method in Nonlinear Multi-Objective Optimization

Author(s):  
Gabriele Eichfelder
2020 ◽  
Vol 12 (2) ◽  
pp. 687 ◽  
Author(s):  
Svetla Stoilova

The development of the transport plan must take into account various criteria impacting the transport process. The main objective of the study is to propose an integrated approach to determine the transport plan of passenger trains. The methodology consists of five steps. In the first step, the criteria for optimization of the transport plan were defined. In the second step, variants of the transport plan were formulated. In the third step, the weights of the criteria are determined by applying the step-wise weight assessment ratio analysis method (SWARA) multi-criteria method. The multi-objective optimization was conducted in the fourth step. The following multi-objective optimization approaches were used and compared: weighted sum method (WSM), compromise programming method (CP), and the epsilon–constraint method (EC). The study proposes a modified epsilon–constraint method (MEC) by applying normalization of each objective function according to the maximal value of the solution by individual optimization for each objective function, and hybrid methods: hybrid WSM and EC, hybrid WSM and MEC, hybrid CP and EC, and Hybrid CP and MEC. The impact of the variation of passenger flows on the choice of an optimal transport plan was studied in the fifth step. The Laplace’s criterion, Hurwitz’s criterion, and Savage’s criterion were applied to come to a decision. The approbation of the methodology was demonstrated through the case study of Bulgaria’s railway network. Suitable variant of transport plan is proposed.


2020 ◽  
Vol 12 (18) ◽  
pp. 7733
Author(s):  
Dong Yang ◽  
Qidong Liu ◽  
Jia Li ◽  
Yongji Jia

Cloud manufacturing is an emerging service-oriented paradigm that works by taking advantage of distributed manufacturing resources and capabilities to collaboratively perform a manufacturing task, with the consideration of QoS (Quality of Service) requirements such as cost, time and quality. Incorporating environmental concerns and sustainability into cloud manufacturing to produce a much greener product has become an urgent issue since there is fierce market competition and an increasing environment consciousness from customers. In this paper, we present a multi-objective optimization approach to selecting and scheduling cloud manufacturing services from the viewpoints of the economy and environment including carbon emissions and water resource. Subject to the carbon cap regulation, a multi-objective model for a cloud manufacturing task is built with the aim of minimizing total costs, carbon emissions, and water resource use. Transportation mode selections and carbon emissions from both cloud manufacturing services and transportation activities are taken into account in this model. The ε-constraint method is employed to obtain the exact Pareto front of optimal solutions. A case study from automobile cloud manufacturing is used to illustrate the effectiveness of the presented approach. Numerical experiments are conducted to compare the presented approach and the simple additive weighting method. The results show that the presented ε-constraint method can obtain a better and more diverse Pareto set of solutions and that it can solve the models in a reasonable time.


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