Mixed-integer Linear Fractional Programming Research Articles

English

English

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

This paper is concerned with the parametric algorithms for solving large-scale mixed-integer linear and nonlinear fractional programming problems, as well as their application in process systems engineering. By developing an equivalent parametric formulation of the general mixed-integer fractional program (MIFP), we propose four exact parametric algorithms based on the root-finding methods, including bisection method, Newton's method, secant method and false position method, respectively, for the global optimization of MIFPs. We also propose an inexact parametric algorithm that can potentially outperform the exact parametric algorithms for some types of MIFPs. Extensive computational studies are performed to demonstrate the efficiency of these parametric algorithms and to compare them with some general-purpose mixed-integer nonlinear programming methods. The applications of the proposed algorithms are illustrated through two case studies on process scheduling. Computational results show that the proposed exact and inexact parametric algorithms are more computationally efficient than several general-purpose solvers for solving MIFPs.

Global optimization of large‐scale mixed‐integer linear fractional programming problems: A reformulation‐linearization method and process scheduling applications

Mixed‐integer linear fractional program (MILFP) is a class of mixed‐integer nonlinear programs (MINLP) where the objective function is the ratio of two linear functions and all constraints are linear. Global optimization of large‐scale MILFPs can be computationally intractable due to the presence of discrete variables and the pseudoconvex/pseudoconcave objective function. We propose a novel and efficient reformulation–linearization method, which integrates Charnes–Cooper transformation and Glover's linearization scheme, to transform general MILFPs into their equivalent mixed‐integer linear programs (MILP), allowing MILFPs to be globally optimized effectively with MILP methods. Extensive computational studies are performed to demonstrate the efficiency of this method. To illustrate its applications, we consider two batch scheduling problems, which are modeled as MILFPs based on the continuous‐time formulations. Computational results show that the proposed approach requires significantly shorter CPU times than various general‐purpose MINLP methods and shows similar performance than the tailored parametric algorithm for solving large‐scale MILFP problems. Specifically, it performs with respect to the CPU time roughly a half of the parametric algorithm for the scheduling applications. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4255–4272, 2013

Design of Sustainable Product Systems and Supply Chains with Life Cycle Optimization Based on Functional Unit: General Modeling Framework, Mixed-Integer Nonlinear Programming Algorithms and Case Study on Hydrocarbon Biofuels

We propose a life cycle optimization framework for the design of sustainable product systems and supply chains considering the concept of “functional unit” under economic and environmental criteria. This general modeling framework integrates the life cycle analysis methodology with multiobjective optimization and measures both the economic and environmental performances based on a standard quantity of functional unit associated with final products. The Pareto-optimal frontier returned by the multiobjective optimization problem reveals the trade-off between the economic and environmental objectives. We also present tailored optimization algorithms for efficiently solving the mixed-integer linear fractional programming problems, which result from the life cycle optimization framework. We apply the proposed life cycle optimization framework to a case study on the hydrocarbon biofuels through a spatially explicit model for the county-level supply chain in Illinois. The Pareto-optimal results show that the env...

Sustainable scheduling of batch processes under economic and environmental criteria with MINLP models and algorithms

We address the bi-criterion optimization of batch scheduling problems with economic and environmental concerns. The economic objective is expressed in terms of productivity, which is the profit rate with respect to the makespan. The environmental objective is evaluated by means of environmental impact per functional unit based on the life cycle assessment methodology. The bi-criterion optimization model is solved with the ε-constraint method. Each instance is formulated as a mixed-integer linear fractional program (MILFP), which is a special class of non-convex mixed-integer nonlinear programs. In order to globally optimize the resulting MILFPs effectively, we employ the tailored reformulation-linearization method and Dinkelbach's algorithm. The optimal solutions lead to a Pareto frontier that reveals the tradeoff between productivity and environmental impact per functional unit. To illustrate the application, we present two case studies on the short-term scheduling of multiproduct and multipurpose batch plants.

Dinkelbach's algorithm as an efficient method to solve a class of MINLP models for large-scale cyclic scheduling problems

In this paper we consider the solution methods for mixed-integer linear fractional programming (MILFP) models, which arise in cyclic process scheduling problems. We first discuss convexity properties of MILFP problems, and then investigate the capability of solving MILFP problems with MINLP methods. Dinkelbach's algorithm is introduced as an efficient method for solving large-scale MILFP problems for which its optimality and convergence properties are established. Extensive computational examples are presented to compare Dinkelbach's algorithm with various MINLP methods. To illustrate the applications of this algorithm, we consider industrial cyclic scheduling problems for a reaction–separation network and a tissue paper mill with byproduct recycling. These problems are formulated as MILFP models based on the continuous time Resource-Task Network (RTN). The results show that orders of magnitude reduction in CPU times can be achieved when using this algorithm compared to solving the problems with commercial MINLP solvers.

Fractional Programming: Theory, Methods and Applications

Introduction. 1. Fractional Programming Applications. 2. Convex, Quasiconvex, Pseudoconvex, Logarithmic Convex, alpham-Convex, and Invex Functions. 3. Methods for Solving Linear Fractional Programming Problems. 4. Nonlinear Fractional Programming. 5. Duality in Fractional Programming. 6. Fractional Programming with Multiple Objective Functions. 7. Fractional Programming in the Complex Space. 8. Special Linear Fractional Programming Problems. 9. Integer and Mixed Integer Linear Fractional Programming. 10. Fractional Transportation Problem. Bibliography. Subject Index. Author Index.

A model of an urban transportation system formulated as a multiobjective mathematical programming problem under uncertainty

AbstractAn 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.