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Lexicographic and ɛ-Constraint Methods
Published in Maria Isabel Gomes, Nelson Chibeles Martins, Mathematical Models for Decision Making with Multiple Perspectives, 2022
Maria Isabel Gomes, Nelson Chibeles Martins
The Lexicographic and the ɛ-constraint methods are two of the most well-known methods in Multi-Objective Linear Programming (MOLP). Both methods fit in to the category of Reduced Feasible Region Methods, which then falls within the category of “a priori” preference aggregation methods within a broader classification system for MOLP methods (as presented in Chapter 7). In short, both methods transform the MOLP problem into a single objective problem by optimizing one objective function and taking the remaining objectives (all at once, or step by step) as constraints. The Lexicographic method asks for the Decision Maker (DM) to sort all objective functions from the most important to least important one. Then, it takes one objective function at a time and finds its optimal solution within a reduced feasible region. This reduction is made considering the previous solved objectives and setting them greater or equal to some aspirations level proposed by the DM. The H-constraint method asks the DM for the most important objective and solves the single objective problem considering only this function, while all the remaining objective functions are set as constraints. The right-hand side of these new constraints are achievement levels suggested by the DM for each objective.
A New Method to Solve Multi-Objective Linear Fractional Problems
Published in Fuzzy Information and Engineering, 2021
Mojtaba Borza, Azmin Sham Rambely
Because of the importance of the LFPP and also MOLFPP, many studies have been accomplished to come out with efficient methods and techniques for these optimisation problems. Chakraborty and Gupta [7] developed a method to address MOLFPP. In their method, the multi-objective problem is transformed into a multi-objective linear programming problem (MOLPP). Subsequently, the membership functions are specified after identifying the fuzzy aspiration levels of the linear objectives. Finally, the MOLPP is changed into a linear programming problem (LPP) using a max–min technique. Motivated by Chakraborty and Gupta’s methodology, Veeramani and Sumathi [8] and De and Deb [9] introduced approaches to deal with LFPP with fuzzy coefficients and MOLFPP, respectively. Following the methodology of Dinkelbach [10], Güzel [11] and Nayak and Ojha [12] developed approaches to MOLFPP. In fact, Nayak and Ojha attempted to improve the results of Guzel’s approach by employing the ε-constrain technique. However, applying the ε-constrain technique encompasses some difficulties in practice when the decision maker is trying to specify the value of ε for each constraint. Pal et al. [13] transformed the MOLFPP into a LPP using a fuzzy goal programming approach in addition to suitable variable transformations. Toksari [14] introduced an approach to tackle the MOLFPP where the membership functions of the objectives are defined and then linearised using the first-order Taylor series about the individual optimal solutions. For some examples, Borza et al. [15] reported that the results of using the first-order Taylor series proposed by Toksari are to some extent more accurate than the results of the fuzzy goal programming used by Pal et al. Nayak and Ojha [16] introduced a method dealing with the MOLFPP with fuzzy coefficients where the fuzzy problem is altered into interval valued LFPP using the concept of α-cuts. In their method, the fuzzy problem is reduced into the MOLFPP. Afterwards, they reach a MOLPP employing the first-order Taylor series. Finally, weighted sum technique is utilised to transform the MOLPP into a LPP. Borza and Rambely [17] designed a non-iterative method to obtain the global optimal solution of the sum of the linear fractional programming problem (S-LFPP) by the use of variable transformation. Liu et al. [18] constructed an iterative algorithm for the large-scale S-LFPP using a branch and bound technique.