Multi-level Linear Programming Problems Research Articles

An improved solution for fully neutrosophic multi-level linear programming problem based on Laplace transform

An improved solution for fully neutrosophic multi-level linear programming problem based on Laplace transform

Fully intuitionistic fuzzy multi-level linear fractional programming problem

In many real situations, it is frequently difficult to accurately determine the membership and non-membership degrees related to an element of the set with complete satisfaction because of the ambiguity in the input data. In instances like these, intuitionistic fuzzy (IF) numbers are crucial. We present a simple approach in this paper to solve the fully intuitionistic fuzzy multi-level linear fractional programming (FIFMLLFP) problem. By applying the new suggested approach to the problem under consideration, each level of the FIFMLLFP problem is converted into five crisp linear (MLMOLFP) problems, where each crisp problem has an additional bounded variables constraint and the upper problems' optimization variables are treated as parameters. This is done by using an iterative technique for linearizing fractional objectives. We converted the MLMOLFP problem into a multilevel multiobjective linear programming (MLMOLP) problem, in addition to the ε-constraint method that is used to reduce MLMOLP problem to a single objective linear programming problem. The method is demonstrated step-by-step with a numerical example.

Three different optimization techniques for solving the fully rough interval multi-level linear programming problem

Due to the importance of the multi-level fully rough interval linear programming (MLFRILP) problem to address a wide range of management and optimization challenges in practical applications, such as policymaking, supply chain management, energy management, and so on, few researchers have specifically discussed this point. This paper presents an easy and systematic roadmap of studies of the currently available literature on rough multi-level programming problems and improvements related to group procedures in seven basic categories for future researchers and also introduces the concept of multi-level fully rough interval optimization. We start remodeling the problem into its sixteen crisp linear programming LP problems using the interval method and slice sum method. All crisp LPs can be reduced to four crisp LPs. In addition, three different optimization techniques were used to solve the complex multi-level linear programming issues. A numerical example is also provided to further clarify each strategy. Finally, we have a comparison of the methods used for solving the MLFRILP problem.

On neutrosophic multi-level multi-objective linear programming problem with application in transportation problem

In this research, we use the harmonic mean technique to present an interactive strategy for addressing neutrosophic multi-level multi-objective linear programming (NMMLP) problems. The coefficients of the objective functions of level decision makers and constraints are represented by neutrosophic numbers. By using the interval programming technique, the NMMLP problem is transformed into two crisp MMLP problems, one of these problems is an MMLP problem with all of its coefficients being upper approximations of neutrosophic numbers, while the other is an MMLP problem with all of its coefficients being lower approximations of neutrosophic numbers. The harmonic mean method is then used to combine the many objectives of each crisp problem into a single objective. Then, a preferred solution for NMMLP problems is obtained by solving the single-objective linear programming problem. An application of our research problem is how to determine the optimality the cost of multi-objective transportation problem with neutrosophic environment. To demonstrate the proposed strategies, numerical examples are solved.

An adaptive method to solve multilevel multiobjective linear programming problems

This paper is a follow-up to a previous work where we defined and generated the set of all possible compromises of multilevel multiobjective linear programming problems (ML-MOLPP). We introduce a new algorithm to solve ML-MOLPP in which the adaptive method of linear programming is nested. First, we start by generating the set of all possible compromises (set of all non-dominated solutions). After that, an algorithm based on the adaptive method of linear programming is developed to select the best compromise among all the possible settlements achieved. This method will allow us to transform the initial multilevel problem into an ML-MOLPP with bonded variables. Then, apply the adaptive method which is the most efficient to solve all the multiobjective linear programming problems involved in the resolution process instead of the simplex method. Finally, all the construction stages are carefully checked and illustrated with a numerical example.

Multi-level multi-objective linear fractional programming problem: a solution approach

Multi-level multi-objective linear fractional programming problem: a solution approach

Branch-and-cut solution approach for multilevel mixed integer linear programming problems

A multilevel programming problem is an optimization problem that involves multiple decision makers, whose decisions are made in a sequential (or hierarchical) order. If all objective functions and constraints are linear and some decision variables in any level are restricted to take on integral or discrete values, then the problem is called a multilevel mixed integer linear programming problem (ML-MILP). Such problems are known to have disconnected feasible regions (called inducible regions), making the task of constructing an optimal solution challenging. Therefore, existing solution approaches are limited to some strict assumptions in the model formulations and lack universality. This paper presents a branch-and-cut (B&C) algorithm for the global solution of such problems with any finite number of hierarchical levels, and containing both continuous and discrete variables at each level of the decision-making hierarchy. Finite convergence of the proposed algorithm to a global solution is established. Numerical examples are used to illustrate the detailed procedure and to demonstrate the performance of the algorithm. Additionally, the computational performance of the proposed method is studied by comparing it with existing method through some selected numerical examples.

Multi-level multi-objective linear fractional programming problem: A solution approach

Multi-level multi-objective linear fractional programming problem: A solution approach

The set of all the possible compromises of a multi-level multi-objective linear programming problem

In this paper, a new approach is developed to solve multi-level multi-objective linear programming problems (ML-MOLPP), by providing the set of all the compromises without taking into account the hierarchy of the problem. A simple criterion is developed for testing whether a given feasible solution is a possible compromise or not. We describe a method used to generate the set of all the compromises, based on the P.L. Yu and M. Zeleny’s method. Theoretical results, numerical example and flow diagram are reported.

On solving the multilevel rough interval linear programming problem

In real-world problems, the parameters of optimization problems are uncertain. A class of multilevel linear programming (MLLP) with uncertainty problem models cannot be determined exactly. Hence, in this paper, we are concerned with studying the uncertainty of MLLP problems. The main motivation of this paper is to obtain the solution to a multilevel rough interval linear programming (MLRILP) problem. To obtain that, we start turning the problem into its competent crisp equivalent using the interval method. Moreover, we rely on three methods to address the problem of multiple levels. First, by applying the constraint method in which upper levels give satisfactory solutions that are reasonable in rank order to the lower levels, second, by an interactive approach that uses the satisfaction test function, and third, by the fuzzy approach that is based on the concept of the tolerance membership function. A numerical example is given for illustration and to examine the validity of the approach. An application to deduce the optimality for the cost of the solid MLLP transportation problem in rough interval environment is presented.

Fuzzy harmonic mean technique for solving fully fuzzy multilevel multiobjective linear programming problems

This paper illustrates how the fuzzy harmonic mean technique can efficiently solve fully fuzzy multilevel multiobjective linear programming (FFMMLP) problems. First, at each level, the FFMMLP problem can be converted into three crisp multiobjective linear programming (CMLP) problems using the crisp linear technique. Then, the fuzzy harmonic mean technique is utilized to aggregate each crisp problem's multiobjective into a single objective. Second, the ensuing final, single-objective problem is constructed using the harmonic mean for each level. Finally, it is solved to obtain a fuzzy compromise solution for the FFMMLP problem in general. Two examples are given to obtain the application of the proposed method. One example is applying the proposed approach to a multilevel multiobjective production planning model for a supply chain under a fully fuzzy environment.

Solving the general fully neutrosophic multi-level multiobjective linear programming problems

Solving the general fully neutrosophic multi-level multiobjective linear programming problems

(2003-5760) Building fuzzy approach with linearization technique for fully rough multi-objective multi-level linear fractional programming problem

This paper presents a suitable solution procedure to solve the fully rough multi-objective multi-level linear fractional programming (FRMMFP) problem. First, an extension of interval method is presented to deal with roughness of the stated problem. Then, an iterative technique is proposed for linearization of fractional objectives. Finally, a modification of fuzzy approach is provided in the environment of the fully rough to solve the linear model. An example is provided for understanding the solution procedure of the proposed method.

A Decomposition Algorithm for Solving Multi-Level Large-Scale Linear Programming Problems With Neutrosophic Parameters in the Constrains

A Decomposition Algorithm for Solving Multi-Level Large-Scale Linear Programming Problems With Neutrosophic Parameters in the Constrains

Neutrosophic goal programming strategy for multi-level multi-objective linear programming problem

Neutrosophic set theory plays an important role in dealing with the impreciseness and inconsistency in data encountered in solving real life problems. This article aims to present a novel goal programming based strategy which will be helpful to solve Multi-Level Multi-Objective Linear Programming Problem (MLMOLPP) with parameters as neutrosophic numbers (NNs). Difficulty in decision making arises due to the presence of multiple decision makers (DMs) and impreciseness in information. Here each level DM has multiple linear objective functions with parameters considered as NNs which are represented in the form $$c + dI$$ , where c and d are considered real numbers and the symbol I denotes indeterminacy. The constraints are also linear with the parameters as NNs. Firstly the NNs are changed into intervals and the problem turns into a multi-level multi-objective linear programming problem considering interval parameters. Then interval programming technique is employed to obtain the target interval of each objective function. In order to avoid decision deadlock which may arise in hierarchical (multi-level) problem, a possible relaxation is imposed by each level DM on the decision variables under his/her control. Finally a goal programming strategy is presented to solve the MLMOLPP with interval parameters. The method presented in this paper facilitates to solve MLMOLPP with multiple conflicting objectives in an uncertain environment represented through NNs of the form $$c + dI$$ , where indeterminacy I plays a pivotal role. Lastly, a mathematical example is solved to show the novelty and applicability of the developed strategy.

Adjustable robust optimization through multi-parametric programming

Adjustable robust optimization (ARO) involves recourse decisions (i.e. reactive actions after the realization of the uncertainty, ‘wait-and-see’) as functions of the uncertainty, typically posed in a two-stage stochastic setting. Solving the general ARO problems is challenging, therefore ways to reduce the computational effort have been proposed, with the most popular being the affine decision rules, where ‘wait-and-see’ decisions are approximated as affine adjustments of the uncertainty. In this work we propose a novel method for the derivation of generalized affine decision rules for linear mixed-integer ARO problems through multi-parametric programming, that lead to the exact and global solution of the ARO problem. The problem is treated as a multi-level programming problem and it is then solved using a novel algorithm for the exact and global solution of multi-level mixed-integer linear programming problems. The main idea behind the proposed approach is to solve the lower optimization level of the ARO problem parametrically, by considering ‘here-and-now’ variables and uncertainties as parameters. This will result in a set of affine decision rules for the ‘wait-and-see’ variables as a function of ‘here-and-now’ variables and uncertainties for their entire feasible space. A set of illustrative numerical examples are provided to demonstrate the potential of the proposed novel approach.

Interactive programming approach for solving multi-level multi-objective linear programming problem

Interactive programming approach for solving multi-level multi-objective linear programming problem

Solving Fully Rough Interval Multi-level Multi-objective linear Fractional Programming Problems via FGP

Solving Fully Rough Interval Multi-level Multi-objective linear Fractional Programming Problems via FGP

Fully Fuzzy Multi-Level Linear Programming Problem

Fully Fuzzy Multi-Level Linear Programming Problem

Multi-level linear programming subject to addition-min fuzzy relation inequalities with application in Peer-to-Peer file sharing system

Multi-level linear programming problem subject to addition-min fuzzy relation inequalities is introduced to characterize a kind of optimization models in BitTorrent-like Peer-to-Peer file sharing systems. Based on some theorems, which contribute to the resolution of the proposed problem, we develop a novel algorithm to find the unique optimal solution. A practical application example is presented to illustrate the feasibility and efficiency of the algorithm.