epsilon constraint

Disaster management in the post-disaster phase is crucial for minimizing damages. Relief logistics enable the rapid deployment of relief personnel and necessary materials to affected areas and the rescue of victims. During natural disasters like earthquakes, physical infrastructures such as roads and bridges are often destroyed, making access to affected areas extremely difficult or even impossible. For this reason, helicopters are the most suitable means of transport to assist the injured. In this context, another critical issue is the difference in service times between helicopters. Naturally, shorter service times result in higher costs. Therefore, it is essential to strike a balance between the dual objectives of time and cost.

This paper proposes a mathematical model for post-disaster relief logistics following a catastrophic earthquake in a mountainous region. The model aims to plan the deployment of helicopters to affected areas and manage the rescue and transportation of injured individuals to temporary facilities. The issue of uncertainty regarding the affected population and the demand for rescue personnel is also addressed. Initially, a deterministic mathematical model is proposed. Subsequently, the model is adapted using the chance-constraint programming method to incorporate the stochastic nature of the aforementioned parameters, converting them into deterministic constraints. Additionally, two approaches, the LP-metric method and the epsilon constraint method, are employed to solve the bi-objective model concerning time and cost.

A key finding of this research is the formulation of decision variables in the mathematical model. One critical decision variable is the capacity allocated for preparing and dispatching relief personnel at each temporary facility. This designed capacity must not exceed a maximum allowable value due to technical constraints and, operationally, must also accommodate the total number of personnel deployed by all helicopters from the facility across multiple trips. Similar constraints apply to the capacity for treating injured individuals at each temporary facility. Specifically, this capacity must not exceed a predefined maximum and must also meet or exceed the population of injured individuals transported to the facility by various helicopters over multiple trips. Two additional important constraints addressed in this research include adherence to the maximum flight hours of each helicopter and ensuring a minimum level of service to cover the affected population. These constraints are made feasible through the defined decision variables. Another significant finding pertains to the modeling of the trade-off between helicopter service costs and times, represented as a bi-objective model. Moreover, given the uncertainty of the two exogenous parameters—relief force demand and the population of affected areas—these parameters are assumed to follow a normal distribution with specific means and standard deviations, and their associated constraints are ultimately converted into deterministic forms.

The proposed model is solved using the epsilon constraint method and the LP-metric method for a case study involving the Tabriz fault. In a pilot scenario, 13 affected areas were considered, with 3 potential locations for establishing temporary facilities and 5 types of helicopters. The results indicate that increased demand for relief personnel and affected areas leads to higher costs and longer service times, demonstrating the logical functionality of the developed model.

After large-scale natural disasters, supply shortages and inequitable distribution because of various damages that hinder the functioning of the humanitarian supply chain. For this purpose, in this paper, this problem is presented as a two-level planning model to minimize the rate of unmet demand, potential environmental risks, emergency costs at the upper level of the decision chain and, to maximize the perceived satisfaction of the survivors in the lower level of the decision chain is modeled. Formulates a decision hierarchy. To solve the proposed model, the modified epsilon method has been used. According to the obtained results, the risk coefficient of waste γ imposes significant effects on the performance of environmental sustainability. And it has no effect on the weighted total unmet demand rate, total emergency costs. Also, by applying changes in the epsilon value, the acceptable distance for determining the non-dominated solutions of the multi-objective problem is presented. Therefore, this study provides valuable results to managers and decision makers to make valuable decisions after a crisis to control the situation.

Portfolio optimization and selection is one of the most important issues in the financial world, so investors are trying to make decisions that are most in line with the real world. But the uncertainty in data and parameters, and the contradiction in the investor's goals, adds to the complexity of the stock portfolio optimization problem, and the other hand because of the efficient market, it is necessary to use multi-period models that, unlike single-period models, allow the investor to review their wealth at the beginning of each period. This paper introduces a new approach to optimizing a multi-period portfolio optimization based on fuzzy general theory and using scenario tree to deal with uncertainties. In addition to considering all of the above constraints, It has made it possible for the investor to be able to apply his manner by changing the parameter to optimistic-pessimistic, and there is no need to model in credibility, necessity or possibility mode. Then the proposed model is solved by the Epsilon constraint method. Finally, using the data of 17 companies from different industries operating in the Tehran Stock Exchange Market in 1398, we examine the validity of the model and its efficiency.

Strategic development goals is achieved by investment potential utilization of each region. Some of these goals may be inconsistent. Therefore, Governments want to align a combination of different investment objectives and sometimes contradictory as far as possible. This study aimed to explain the mathematical model for investment projects by providing an integer linear programming model with three objectives reducing costs, increasing employment and reducing greenhouse gas emissions - carbon dioxide. Afterward, the model was tested with extracted data from 51 proposed projects. Among common methods, the Augmented Epsilon Constraint is used. The cost reduction goal was chosen as the main objective by surveying experts and applying the Intuitionistic fuzzy approach. By applying the intuitionistic fuzzy approach and regarding experts viewpoint, Cost reduction is chosen as the main objective. the proposed model was evaluated in pareto front with twenty-one cut Augmented Epsilon constraints and optimal solution obtained, so, 45 projects is introduced as the basic variables of the problem. Finally, The Augmented epsilon constraint approach was compared with the classical epsilon constraint approach. The results showed in the classic approach, only 35 projects are accepted from 51 projects. Further examination revealed the Augmented approach outperforms the classical approach.

In this paper, in order to increase the performance of the proposed supply chain network, the multi-period, multi-product model with multiple simultaneous objectives has been considered. To design the network, use has been made of the multi-objective mixed integer mathematical programming model, where the objectives of the problem include minimization of production costs, reduction of products delivery time, and increase in the reliability of the supply chain. To solve the model and present non-dominated solutions, the Epsilon Constraint Method has been used; to this end, the model’s code has been written in the LINGO software. Considering the multiplicity of Pareto solutions, to prevent the decision maker from confusion, use has been made of Data Envelopment Analysis for evaluation of non-dominated solutions. To solve the Data Envelopment Analysis model, GAMS software has been used. The outputs of this problem include the optimal number of facilities at each level of the supply chain and the optimal amount of goods delivery from each level to another.

In this paper an uncertain multi objective closed-loop supply chain is developed. The first objective function is maximizing the total profit. The second objective function is minimizing the use of row materials. In the other word، the second objective function is maximizing the amount of remanufacturing and recycling. Genetic algorithm is used for optimization; and for finding the pareto optimal line، Epsilon-constraint method is used. Finally a numerical example is solved with proposed approach and performance of the model is evaluated in different sizes. The results show that this approach is effective and useful for managerial decisions.