epsilon constraint method

The design and planning of cellular Manufacturing systems in most classical models is based on minimizing production costs or increasing producers' profits. With the expansion of production systems and the increase in demand for products, concerns about environmental issues and excessive consumption of non-renewable resources have increased. On the other hand, paying attention to workers and the safety of the work environment has been introduced as a vital issue in creating a sustainable Manufacturing system. In this article, a new multi-objective mathematical model for creating a sustainable cellular Manufacturing system is presented according to the mentioned items in order to minimize the production costs in the system in addition to optimizing the environmental effects of the Manufacturing system. In the following, the model was solved using the epsilon constraint method and various solutions were presented in the form of a Pareto front for decision making. Due to the NP-hardness of the proposed model and the inability of the GAMS software to find optimal solutions for large-scale problems, a non-dominant sorting genetic algorithm (NSGA-II) is presented to solve it. The results showed that the meta-heuristic method has reduced the solution time by at least three times compared to the epsilon constraint method; Also, reducing the level of environmental risks has led to an increase in production costs. Finally, the applicability of the proposed model in an agricultural equipment production workshop has been investigated as a case study.

Sustainable diets are important because of their less environmental impact, as well as having the micro-nutrients needed for healthy living in the present and future generations. Athletes always need special food programs to maintain their level of activity. Therefore, in this research, a mathematical model with three objective functions for the athlete's diet plan, consumed to taking into account the principles of sustainability, includes the objectives of 1- minimizing the cost of purchasing food, the cost of water consumption and the cost of producing carbon dioxide for food, 2- Maximizing the amount of protein intake and 3- Minimizing the amount of sugar and fat. For model validation, real information about nutrients and supplements required by the athletes' body has been used. The model was formulated in The GAMS computational software and solved by Cplex solver. The outputs of the model represent the performance of the model in the actual example.

The accumulation of a large number of production units with small capacities and transforming them into a larger entity will make them visible in electricity market and increase their economic efficiency. Virtual power plant is a Wide energy management system that incorporates interruptible loads, storage and distributed generation in order to create support services of system and restoring energy. The main objective of this paper is to provide a method for optimizing virtual power plant bid along with other competitors in a day-ahead market. To achieve this goal a bi-level mathematical optimization model is presented in which equilibrium constraints are also taken into account. The first level of this model maximizes profit of virtual power plant and the second level maximizes social welfare problem. Bi-level model with the use of the theory of duality and optimality conditions of Karush-Kuhn-Tucker converts to a linear programming model of mixed-integer. This two level problem is also solved in a two-objective method using Epsilon constraint in order to maximize profits and minimize emission of units in virtual power plants. Finally, using fuzzy decision making the best answer is chosen and profit in two-objective mode is 41429.7d and pollution is 1969.9 pound, which compare with single-objective mode is more reasonable number. Proposed model is tested on IEEE 24-bus system and the results show the effectiveness of proposed method.