Analysis of Linear Programming

1.       Objective of the Paper

The main objective of this paper is to develop methods for selecting certain subsets from the set N of non-dominated points for multiple-objective linear programming problems. Moreover, it also focusses on the selection of subsets from the set of the non-dominated vectors in multiple objective linear programming. In this paper one subset is considered to be the set of points x in N and for this the objective function’s maximum deviation from an ideal vector is the minimum. The methods for creating the set of all non-dominated maximum points have been developed by numerous writers in the past. Furthermore, there are also some methods that have been developed for the description of the complete non-dominated set N as the union of maximal non-dominated faces. These methods have their importance because in the multiple objective linear programming problem, these are useful and can be used for the multiple purposes. When the set N is created then the usage of this set remains a problem in the decision process. This is why this paper has filled this gap and provide a solid grasp to resolve this problem. In this paper, different methods can be developed with the selection of the subsets of non-dominated set. In this paper, the most important part is the analysis of the relation of the compromise solutions along with the non-dominated solutions. For this purpose, a general framework is presented for the better understanding of the relation of non-dominated solutions and compromise solutions. Furthermore in this paper the compromise solutions are also discussed that are defined by w-norm and this main focuses can be on the linear multiple objective problem.

2.       Advanced Techniques and Methods Used in the Computations

In recent time, the techniques and methods have been changed in the computations and now more advanced and latest methods are used in the computations. This paper focuses on the selection of subsets from the set of the non-dominated vectors in multiple objective linear programming. It also provides a unique method that can be used to find out all the non-dominated compromise solutions when the Q(x) has alternate optima. Furthermore the solution of linear program O in order to obtain v can also be find out and then the non-dominated set that can be for the multiple objective program P is also created using one of the methods that are mentioned in this paper. It can be helpful to most of the people that can easily find solutions to the related problems. In this paper the optimal tableau is also discussed for the linear program Q and that can be used to find out the initial tableau that is required by the methods and to create the non-dominate set for P. Furthermore the details on the computational considerations in the solution of P are also discussed. A special case has also been comprehensively described in which the original multiple objective problem has three objectives. Then this multiple objective program can be reduced to the ordinary linear program. This happens because there are at least two of the three objectives can be constant. Furthermore, this paper also examines the extension of this method and for this a subset can be found out of NO, which can deliver the information relating to the objective function trade among the set of non-dominated points. The trade-off point is also discussed in this paper that is the important topic. The trade-off points provide the information about all the possible trade-offs and these points are also important because they are non-dominated compromise points. A method has been provided in the paper for the determination of the trade-off set in which an algorithm has been discussed that can generate the trade-off compromise set. This algorithm contains a finite sequence that can be helpful for the solutions. In the last a multiple objective program is used and an example is provided for the better understanding of the concept and this problem has five objectives where the main motive is to produce a procedure that can be interactive and also based on the Algorithm. This paper can contribute in the area of the selection of subsets from the set of the non-dominated vectors in multiple objective linear programming, with the useful and unique information that can help the readers to more focus on this topic and to develop a thought further to expand this area of study.

3.       Results and Conclusion Described in the Paper

The main objective of this paper was to study and analyze the develop methods for selecting certain subsets from the set N of non-dominated points for multiple-objective linear programming problems. There is a gap in this area of focus and this is why this study provides a comprehensive analysis on this topic that can not only help the readers to understand the topic easily, but it can also open their mind to further research and work on the similar areas and to expand it more. Furthermore, it also focusses on the selection of subsets from the set of the non-dominated vectors in multiple objective linear programming. There are different methods that have been discussed along with the examples for the clear understanding. These methods have their importance when it comes to the multiple objective linear programming. For this reason, this paper provide a solid grasp to resolve these problems so that one can solve easily. In this paper, different methods can be developed with the selection of the subsets of non-dominated set.