Mixed-Integer Linear Optimization

This chapter provides an introduction to the basic notions in Mixed-Integer Linear Optimization. Sections 5.1 and 5.2 present the motivation, formulation, and outline of methods. Section 5.3 discusses the key ideas in a branch and bound framework for mixed-integer linear programming problems. A large number of optimization models have continuous and integer variables which appear linearly, and hence separably, in the objective function and constraints. These mathematical models are denoted as Mixed-Integer Linear Programming MILP problems. In many applications of MILP models the integer variables are 0 — 1 variables (i.e., binary variables), and in this chapter we will focus on this sub-class of MILP problems. A wide range of applications can be modeled as mixed-integer linear programming MILP problems. These applications have attracted a lot of attention in the field of Operations Research and include facility location and allocation problems, scheduling problems, and fixed-charge network problems. The excellent books of Nemhauser and Wolsey (1988), and Parker and Rardin (1988) provide not only an exposition to such applications but also very thorough presentation of the theory of discrete optimization. Applications of MILP models in Chemical Engineering have also received significant attention particularly in the areas of Process Synthesis, Design, and Control. These applications include (i) the minimum number of matches in heat exchanger synthesis (Papoulias and Grossmann, 1983; see also chapter 8) (ii) heat integration of sharp distillation sequences (Andrecovich and Westerberg, 1985); (iii) multicomponent multiproduct distillation column synthesis (Floudas and Anastasiadis, 1988); (iv) multiperiod heat exchanger network, and distillation system synthesis (Floudas and Grossmann, 1986; Paules and Floudas, 1988); flexibility analysis of chemical processes (Grossmann and Floudas, 1987); (v) structural properties of control systems (Georgiou and Floudas, 1989, 1990); (vi) scheduling of batch processes (e.g., Rich and Prokapakis, 1986, 1986; Kondili et al., 1993; Shah et al, 1993; Voudouris and Grossmann, 1992, 1993); and (vii) planning and scheduling of batch processes (Shah and Pantelides, 1991, Sahinidis et al, 1989, Sahinidis and Grossmann, 1991). In addition to the above applications, MILP models are employed as subproblems in the mixed-integer nonlinear optimization approaches which we will discuss in the next chapter. In this section, we will present the formulation of Mixed-Integer Linear Programming MILP problems, discuss the complexity issues, and provide a brief overview of the solution methodologies proposed for MILP models.