# Mixed-integer linear fractional programming (MILFP)

Author: Xiang Zhao (SysEn 6800 Fall 2020)

Steward: Allen Yang, Fengqi You

## Introduction

The mixed-integer linear fractional programming (MILFP) is a kind of mixed-integer nonlinear programming (MINLP) that is widely applied in chemical engineering, environmental engineering, and their hybrid field ranging from cyclic-scheduling problems to the life cycle optimization (LCO). Specifically, the objective function of the MINFP is shown as a ratio of two linear functions formed by various continuous variables and discrete variables. However, the pseudo-convexity and the combinatorial nature of the fractional objective function can cause computational challenges to the general-purpose global optimizers, such as BARON, to solve this MILFP problem. In this regard, we introduce the basic knowledge and solution steps of three algorithms, namely the Parametric Algorithm, Reformulation-Linearization method, and Branch-and-Bound with Charnes-Cooper Transformation Method, to efficiently and effectively tackle this computational challenge.

## Standard Form and Properties

Consider such standard form of the MILFP:

\displaystyle \begin{align}\max {c_0+\sum_{i}c_{1i}m_i+\sum_{j}c_{2j}y_j \over d_0+\sum_{i}d_{1i}m_i+\sum_{j}d_{2j}y_j} \\ s.t. \quad\a_{0k}+sum_{i}a_{1i}m_i+sum_{j}a_{2j}y_j\eq 0 \quad\\ m_i &\geq 0 \quad \end{align}