Parametric Solution Algorithms for Large-Scale Mixed-Integer Fractional Programming Problems and Applications in Process Systems Engineering

Globally convergent exact and inexact parametric algorithms for solving large-scale mixed-integer fractional programs and applications in process systems engineering

An algorithm is developed which ranks the feasible solutions of an integer linear and linear fractional programming problems with bounded variables in decreasing order of objective values. Numerical illustration is also included.

We describe an improvement of Chergui and Moulaï’s method (2008) that generates the whole efficient set of a multiobjective integer linear fractional program based on the branch and cut concept. The general step of this method consists in optimizing (maximizing without loss of generality) one of the fractional objective functions over a subset of the original continuous feasible set; then if necessary, a branching process is carried out until obtaining an integer feasible solution. At this stage, an efficient cut is built from the criteria’s growth directions in order to discard a part of the feasible domain containing only nonefficient solutions. Our contribution concerns firstly the optimization process where a linear program that we define later will be solved at each step rather than a fractional linear program. Secondly, local ideal and nadir points will be used as bounds to prune some branches leading to nonefficient solutions. The computational experiments show that the new method outperforms the old one in all the treated instances.

Most of the current methods for solving linear fractional programming (LFP) problems depend on the simplex type method. In this paper, we present a new approach for solving linear fractional programming problem in which the objective function is a linear fractional function, while constraint functions are in the form of linear inequalities. This approach does not depend on the simplex type method. Here first we transform this LFP problem into linear programming (LP) problem and hence solve this problem algebraically using the concept of duality. Two simple examples to illustrate our algorithm are given. And also we compare this approach with other available methods for solving LFP problems.

For a linear fractional programming problem, Sharma and Swarup have constructed a dual problem, also a linear fractional program, in which the objective functions of both primal and dual problems are the same. Craven and Mond have extended this result to a nonlinear fractional programming problem with linear constraints, and a dual problem for which the objective function is the same as that of the primal. This theorem is now further extended from linear to differentiable convex constraints.

In this chapter, we consider several special problems which appear in linear fractional programming. Thus, in Section 8.1, we consider a problem of linear fractional programming in which the variables of the objective function appear in absolute-value, and the constraints lack the condition for non-negativity. We show that, under certain conditions, it is possible to apply the Simplex algorithm developed for the standard linear fractional problem. In Section 8.2, we consider the separable linear fractional programming problem, and we reduce it to the solution of a sequence of quadratic programming problems. In Section 8.3, we study the linear fractional programming problem with disjunctive constraints for which we present a solution method and we construct the Dual problem.

An urban transportation system formulated in terms of a multiobjective mixed integer linear fractional programming (MOMILFP) problem under uncertainty is considered. The system is based on two means of public transportation i.e., trams and buses. One takes into account the factors of both passengers' and operator's concern, whose objectives are, generally, in conflict. The real uncertainty and imprecision of data is modeled by L‐R type fuzzy numbers. To solve the fuzzy MOMILFP problem an interactive method is utilized. As an illustration of that approach an application to Poznan's urban transportation system is presented.

ABSTRACTIn this paper, we consider a Fuzzy Stochastic Linear Fractional Programming problem (FSLFP). In this problem, the coefficients and scalars in the objective function are the triangular fuzzy number and technological coefficients and the quantities on the right side of the constraints are fuzzy random variables with the specific distribution. Here we change an FSLFP problem to an equivalent deterministic Multi-objective Linear Fractional Programming (MOLFP) problem. Then by using Fuzzy Mathematical programming approach transformed MOLFP problem is reduced single objective Linear programming (LP) problem. A numerical example is presented to demonstrate the effectiveness of the proposed method.

In fuzzy sets, the elements of a universal set may belong to that set completely, may not belong to that set, or may belong partially. Therefore, fuzzy sets are called generalizations of crisp sets. Due to the diverse nature of problems in real-world situations, fuzzy set theory is found to be unable to address some of the other features of ambiguity. Therefore, researchers extended fuzzy sets to picture fuzzy sets, orthopair, intuitionistic, Pythagorean, Fermatean, neutrosophic, spherical fuzzy sets, etc. Real-world problems can be solved using the method of fractional programming. Decision-makers are trying to get good results using generalized fuzzy numbers in fractional programming problems; hence, this review paper provides a specific framework for generalized fuzzy sets and generalized fuzzy fractional programming problems. Furthermore, it is a platform for researchers who want to work on generalized fuzzy parameter-based fractional programming problems. This research paper summarizes all the basic concepts and prior research on fractional programming and generalized fuzzy linear fractional programming problems.

Several methods currently exist for solving fuzzy linear fractional programming problems under non negative fuzzy variables. However, due to the limitation of these methods, they cannot be applied for solving fully fuzzy linear fractional programming (FFLFP) problems where all the variables and parameters are fuzzy numbers. So, this paper is planning to fill in this gap and in order to obtain the fuzzy optimal solution we propose a new efficient method for FFLFP problems utilized in daily life circumstances. This proposed method is based on crisp linear fractional programming and has a simple structure. To show the efficiency of our proposed method some numerical and real life problems have been illustrated.

This paper proposes a new algorithm for computing the stationary distribution vector in continuous-time upper block-Hessenberg Markov chains. To this end, we consider the last-block-column-linearly-augmented (LBCL-augmented) truncation of the (infinitesimal) generator of the upper block-Hessenberg Markov chain. The LBCL-augmented truncation is a linearly-augmented truncation such that the augmentation distribution has its probability mass only on the last block column. We first derive an upper bound for the total variation distance between the respective stationary distribution vectors of the original generator and its LBCL-augmented truncation. Based on the upper bound, we then establish a series of linear fractional programming (LFP) problems to obtain augmentation distribution vectors such that the bound converges to zero. Using the optimal solutions of the LFP problems, we construct a matrix-infinite-product (MIP) form of the original (i.e., not approximate) stationary distribution vector and develop a sequential update algorithm for computing the MIP form. Finally, we demonstrate the applicability of our algorithm to BMAP/M/$\infty$ queues and M/M/$s$ retrial queues.

This paper presents the use of a Taylor series for multiobjective integer linear fractional programming problem having grey parameters in the right-side of the constraints (GMOILFP).To deal with the grey parameters in the right-side of the constraints the positioned programming should be used.An equivalent grey multiobjective linear fractional programming problem (GMOLFP) is formulated using Gomory's cutting plane method.The Taylor series, which is a series expansion that a representation of a function, is applied to convert the fractional functions into polynomials.In the proposed approach a white value of each grey parameter is determined, Taylor series is applied and the functions are unified by using the nonnegative weighted sum method.Thus, the problem is reduced to a single linear objective with grey parameter in the right side of the constrains.An algorithm for solving GMOILGP problem with grey interval coefficients using positioned programming and Taylor series polynomials is proposed.A numerical example is provided to demonstrate the efficiency and feasibility of the proposed approach.A special case study for handling the environmental economic energy dispatch problem also is included in this paper, the main goal of this problem is how to schedule committed generators to meet the load required to minimize the pollution emissions and fuel cost.The model formulation for a special case study problem is presented, the mathematical model will be considered as (GMOLFP) and the problem is solved according to the proposed solution algorithm.

On Ratio Invexity in Mathematical Programming

In this paper, the linear fractional programming problem subject to a system of fuzzy relation inequalities (FRI) with the max-Hamacher composition operator is studied. First, the structure of its feasible domain is investigated, and its feasible solution set determined. Then, some sufficient conditions are given that under them, some of the optimal components of the problem are directly determined. The optimal solution of the linear fractional programming problem might not be any of the minimal solutions. However, in the process of obtaining the optimal solution, we need to compute the minimal solutions. Therefore, some reductions are presented that under them, we can compute the minimal solutions fast. The original problem can be transformed into some traditional linear fractional programming subproblems and eventually optimized in a small search space. Finally, an algorithm is designed to solve the problem based on the above reductions. An application and some numerical examples are provided to illustrate the procedure.

On the polynomial mixed 0–1 fractional programming problems