Linear Fractional Programming Research Articles

Problem fazi linearnog frakcionog programiranja pomoću leksikografskog metoda

Introduction/purpose: In solving real-life fractional programming problems, uncertainty and hesitation are often encountered due to various uncontrollable factors. To overcome these limitations, the fuzzy logic approach is applied to these problems. Methods: The discussion focused on solving the fuzzy linear fractional programming problem (FLFPP). First, the FLFP problem was converted into a lexicographic optimization problem, which was then solved to obtain the solution. Results: A numerical example was presented to simplify the explanation of the algorithm. While most researchers solve FLFPPs using the ranking function method, this approach reduces the efficiency of the fuzzy problem. Conclusion: This research contributes a comprehensive methodology for addressing fuzzy linear fractional programming problems using the lexicography method. The findings offer valuable insights for researchers, practitioners, and decision-makers grappling with optimization challenges in settings where imprecise information significantly influences the decision landscape.

Fractional programming for Stackelberg game under type-2 fuzzy environment

This study intends to present a Stackelberg game model for design of the fractional type-2 fuzzy programming. In achieving this aspiration, we develop a probabilistic fuzzy multi-objective fractional linear programming where the entire parameters are of type-2 fuzzy numbers apart from the right-hand side of the constraints are follow Weibull distribution. In the projected approach, the membership function allied with each objective function is generated by using the first-order Taylor series approximation and converted into a single objective function by assuming the weights of the objective functions are equal. Type conversion is made in two ways by existing methods, and using stochastic programming, the probabilistic constraints are transformed into a deterministic form. The accessible model incorporates the non-linear programming viewpoint of the decision-maker and is solved with the help of intuitionistic fuzzy programming (IFS). A comparison study on the optimum results by genetic algorithm (GA) and particle swarm optimization (PSO) with the LINGO 15.0 iterative scheme is offered to resolve the created bi-level programming problem (BLPP) in the course of the Stackelberg game. To make obvious the feasibility of the projected representation and solution methodology, realistic data are measured and results are presented through several discussions.

Solving Neutrosophic Multi-Objective Linear Fractional Programming Problem Using Central Measures – II

Solving Neutrosophic Multi-Objective Linear Fractional Programming Problem Using Central Measures – II

Extension of primal-dual interior point method based on a kernel function for linear fractional problem

Our aim in this work is to extend the primal-dual interior point method based on a kernel function for linear fractional problem. We apply the techniques of kernel function-based interior point methods to solve a standard linear fractional program. By relying on the method of Charnes and Cooper [3], we transform the standard linear fractional problem into a linear program. This transformation will allow us to define the associated linear program and solve it efficiently using an appropriate kernel function. To show the efficiency of our approach, we apply our algorithm on the standard linear fractional programming found in numerical tests in the paper of A. Bennani et al. [4], we introduce the linear programming associated with this problem. We give three interior point conditions on this example, which depend on the dimension of the problem. We give the optimal solution for each linear program and each linear fractional program. We also obtain, using the new algorithm, the optimal solutions for the previous two problems. Moreover, some numerical results are illustrated to show the effectiveness of the method.

Solving neutrosophic multi-objective linear fractional programming problem using central measures

In this article, we present different kind of mean technique to solve Multi-Objective Linear Fractional Programming Problem (MOLFPP) in Neutrosophic Environment. Here in these mean techniques the MOLFPP is converted into Single Objective Linear Programming problem(SOLPP) and then we obtained the optimal solution by simplex method in Neutrosophic Environment. The proposed method is illustrated with the help of a numerical example.

정수 선형 분수 계획법 문제에 대한 파라메트릭 접근법

In this paper, we focus on the integer linear fractional optimization problem, a special case of fractional programming where all functions in the objective function and constraints are linear and all variables are bounded and discrete. To derive the optimal solution, the parametric algorithms are considered, and the efficiency of the algorithms is investigated computationally. Unlike traditional parametric algorithms such as Newton's method, which create a unidirectional sequence approaching the optimal function value, our proposed algorithm generates both upper and lower bounds converging to this value. To demonstrate its effectiveness across various production and operations management problems, the suggested algorithms are used to solve the fractional knapsack problem by comparison to other algorithms (e.g., Newton’s method) under the randomized experimental conditions. The relative practical performance measured by the number of function calls demonstrates that the proposed algorithms are fast and robust for solving the linear fractional programs with discrete variables. Leveraging this algorithm holds the potential to overcome situations in traditional production and operations problems where non-fractional objective functions were previously unconsidered, thereby expecting to derive new outcomes and significance.

Type-2 fuzzy chance-constrained linear fractional programming model for a water resource management system: A case study of Taiyuan city, China

Abstract Uncertainties arising from extreme climate events and human activities pose a challenge to the efficient allocation of water resources. In this study, a type-2 fuzzy chance-constrained linear fractional programming (T2F-CCLFP) is developed to support the water resource management system under uncertainty by incorporating type-2 fuzzy sets, chance-constrained programming, and fractional programming into a comprehensive multi-objective optimization framework. The model enables the trade-off between economic, social, and environmental sustainability and provides water supply solutions associated with different levels of fuzzy uncertainty and risk of violating constraints. The T2F-CCLFP model is applied to Taiyuan, Shanxi Province, China, to support its water resource management. Results reveal that: (i) the industrial structure is transitioning toward diverse industries from energy and heavy industry dominance; (ii) external water transfer will be the major water-supply sources for the city in the future, accounting for 55 and 50% of the total water supply in 2025 and 2030, respectively; (iii) the water-supply security of the city is enhanced by provoking the utilization of reclaimed water (the annual growth rate is 13.9%). The results are helpful for managers in adjusting the current industry structure, enhancing water supply security, and contributing to the sustainable development of socio-economic and water systems.

A Linearization to the Multi-objective Linear Plus Linear Fractional Program

A Linearization to the Multi-objective Linear Plus Linear Fractional Program

Fully fuzzy DEA: a novel additive slacks-based measure model

AbstractThe subject of assessing the relative efficiency of Decision-Making Units (DMUs) has been explored extensively in the literature on Data Envelopment Analysis (DEA). While typical DEA models require exact and conclusive data, the observed values of the inputs and outputs in real-world problems may be inaccurate or ambiguous. This paper presents a novel fully fuzzy DEA (FFDEA) model based on the additive slacks-based measure (ASBM). FFDEA models imply that all inputs, outputs, and variables, are fuzzy. It is demonstrated how fuzzy ranking and fuzzy arithmetic approaches can be used to handle the trapezoidal fuzzy number assumptions. To solve the fuzzy ASBM model under variable returns-to-scale assumptions, we proposed two different methods, first method use of lexicographic method for solving multi-objective linear programming (MOLP) approach, which provides an efficiency measure and a fuzzy goal operating point for each DMU under assessment. In alternative method, we convert the fuzzy linear fractional programming problems to an equivalent MOLP problem. A case study is provided to illustrate the applicability of the proposed approach for selecting sustainable suppliers, we compare the results with other fuzzy DEA approaches, while shedding light on future research venues.

A Transportation Problem Considering Fixed Charge and Fuzzy Shipping Costs

This paper considers a fixed-charge transportation problem with fuzzy shipping costs. Whereas the shipping costs of routes are fuzzy intervals, with increasing linear membership functions, fixed costs, supplies, and demands are deterministic numbers. By defining a membership function associated with the objective values and utilizing Bellman-Zadeh’s max-min criterion instead of the main problem, a new crisp nonlinear mixed-integer programming problem is formulated. To solve this non-linear programming problem, at first, we find optimality conditions for a feasible solution, then, using these optimality conditions, reformulate the non-linear programming problem as a linear mixed-integer fractional programming problem, and finally transform the fractional programming problem as a mixed-integer linear programming problem, which can be solved using the existence methods. An illustrative example is solved to explain the presented details.

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.

Interval division and linearization algorithm for minimax linear fractional program

Interval division and linearization algorithm for minimax linear fractional program

Investing in Wind Energy Using Bi-Level Linear Fractional Programming

Investing in wind energy is a tool to reduce greenhouse gas emissions without negatively impacting the environment to accelerate progress towards global net zero. The objective of this study is to present a methodology for efficiently solving the wind energy investment problem, which aims to identify an optimal wind farm placement and capacity based on fractional programming (FP). This study adopts a bi-level approach whereby a private price-taker investor seeks to maximize its profit at the upper level. Given the optimal placement and capacity of the wind farm, the lower level aims to optimize a fractional objective function defined as the ratio of total generation cost to total wind power output. To solve this problem, the Charnes-Cooper transformation is applied to reformulate the initial bi-level problem with a fractional objective function in the lower-level problem as a bi-level problem with a fractional objective function in the upper-level problem. Afterward, using the primal-dual formulation, a single-level linear FP model is created, which can be solved via a sequence of mixed-integer linear programming (MILP). The presented technique is implemented on the IEEE 118-bus power system, where the results show the model can achieve the best performance in terms of wind power output.

Efficient algorithm for globally computing the min–max linear fractional programming problem

In this paper, we consider the min–max linear fractional programming problem (MLFP) which is NP-hard. We first introduce some auxiliary variables to derive an equivalent problem of the problem (MLFP). An outer space branch-and-bound algorithm is then designed by integrating some basic operations such as the linear relaxation method and branching rule. The global convergence of the proposed algorithm is proved by means of the subsequent solutions of a series of linear relaxation programming problems, and the computational complexity of the proposed algorithm is estimated based on the branching rule. Finally, numerical experimental results demonstrate the proposed algorithm can be used to efficiently compute the globally optimal solutions of test examples.

Parametric approach for multi-objective enhanced interval linear fractional programming problem

The design (decision) variables in the presented article of a multi-objective interval fractional optimization problem based on a linear function are assumed to take the form of a closed interval using the concept of the parametric form of an interval. The original problem is initially changed into equivalent multi-objective interval linear programming with the design variables as closed intervals. Further, it is made free from interval uncertainty by changing into a classical single-objective problem using the weighted-sum method. The solutions of the model are theoretically justified by its existence. Finally, a numerical example and a case study on the agricultural planting structure optimization problem with hypothetical data are presented to support the recommended technique for the model.

Advantages, sensitivity and application efficiency of the new iterative method to solve multi-objective linear fractional programming problem

Advantages, sensitivity and application efficiency of the new iterative method to solve multi-objective linear fractional programming problem

Taylor Series Approach to Grey Multiobjective Integer linear Fractional Programming Problem and A case study for environmental economic energy dispatch

Taylor Series Approach to Grey Multiobjective Integer linear Fractional Programming Problem and A case study for environmental economic energy dispatch

Solving Triangular Fuzzy Linear Fractional Programming Problem

Solving Triangular Fuzzy Linear Fractional Programming Problem

Optimization of fuzzy K-objective fractional assignment problem using fuzzy programming model

The assignment problem is one of the core combinatorial optimization problems in the optimization branch, and the theory and applications of fractional programming have made great strides in recent years. Usually, the possible coefficient values of linear fractional programming and real-world problems are frequently only known to the decision in vague or uncertain terms. Hence, it would be more acceptable to interpret the coefficients for as fuzzy numerical information. In this paper, a fuzzy bi-objective fractional assignment (FBOFAP) has been formulated. A problem has been defined. Here, triangular shapes are used to indicate the parameters. The fuzzy problem is turned into a typical crisp problem through α-cut using a fuzzy number and then the compromise solution is generated by fuzzy programming.

A Case Study on Solutions of Linear Fractional Programming Problems

In some decision making problems, objective function can be defined as the ratio of two linear functional subjects to given constraints. These types of problems are known as linear fractional programming problems. The importance of linear fractional programming problems comes from the fact that many real life problems can be expressed as the ratio of physical or economical values represented by linear functions, for example traffic planning, game theory and production planning etc. In this article, correspond to a production planning problem the mathematical model developed, is a linear fractional programming and in order to solve it, various fractional programming techniques has been used. Finally result is compared with the solution obtained by graphical method. To illustrate the efficiency of stated method a numerical example has given.