Mixed-integer linear optimization formulation

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. 

In this paper, we extend the work of [10] by simultaneously considering minimization of the total utility consumption, maximization of operational flexibility to source-stream temperatures, and even minimum number of matches as multiple design objectives. The flexible HEN synthesis problem is thus formulated as the one of multi-objective mixed-integer linear programming (MO-MILP). This formulation also assumes that the feasible region in the space of uncertain input parameters is convex, so that the optimal solution can thus be explored on the basis of the vertices... 

A process-synthesis problem can be formulated as a combination of tasks whose goal is the optimization of an economic objective function subject to constraints. Two types of mathematical techniques are the most used mixed-integer linear programming (MILP), and mixed-integer nonlinear programming (MINLP). 

The problem of portfolio selection is easily expressed numerically as a constrained optimization maximize economic criterion subject to constraint on available capital. This is a form of the knapsack problem, which can be formulated as a mixed-integer linear program (MILP), as long as the project sizes are fixed. (If not, then it becomes a mixed-integer nonlinear program.) In practice, numerical methods are very rarely used for portfolio selection, as many of the strategic factors considered are difficult to quantify and relate to the economic objective function. 

The complete formulation of the optimization model (decision variables, constraint equations and objective functions) will not be presented here. References (Portillo, 2009 Portillo et al., 2009) provide complete details of the Mixed Integer Linear Programming (MILP) model constraints and the objective functions. 

From the STN representation, we can go to a mathematical formulation of the batch scheduling problem. This class of optimization models is called mixed-integer linear programs (MILPs). 

Based on the given probabilities of success for each potential product, the problem is then to find the optimal product portfolio and investment decisions together with detailed production and sales plans so as to maximise the eNPV. The eNPV is simply the summation of all scenario NPVs, weighted by their associated probabilities. The derivation of the objective function is similar to the one in Papageorgiou et al. (2001). The overall problem is formulated as a two-stage, multi-scenario mixed integer linear programming (MILP) model. 

---