Systematic modeling of discrete‐continuous optimization models through generalized disjunctive programming

Abstract

Discrete-continuous optimization problems in process systems engineering are commonly modeled in algebraic form as mixed-integer linear or nonlinear programming models. Since these models can often be formulated in different ways, there is a need for a systematic modeling framework that provides a fundamental understanding on the nature of these models, particularly their continuous relaxations. This paper describes a modeling framework, Generalized Disjunctive Programming (GDP), which represents problems in terms of Boolean and continuous variables, allowing the representation of constraints as algebraic equations, disjunctions and logic propositions. We provide an overview of major research results that have emerged in this area. Basic concepts are emphasized as well as major classes of formulations that can be derived. These are illustrated with a number of examples in the area of process systems engineering. As will be shown, GDP provides a structured way for systematically deriving mixed-integer optimization models that exhibit strong continuous relaxations. IntroductionMixed-integer optimization provides a powerful framework for the mathematical modeling of many optimization problems that involve discrete and continuous variables. Over the last few years there has been a pronounced increase in the development of Mixed-integer Linear/ Nonlinear Programming (MILP/MINLP) models in Process Systems Engineering [10][12] [21][23] [29] . Synthesis models can be formulated as MILP for high level targeting, and as MINLP for more detailed superstructure optimization [11] . For planning and scheduling, the majority tends to be MILP models [6][9] [24][28] although gradually there is also an increasing trend to MINLP, especially as process models are incorporated [17] . MILP/MINLP models, however, are based on algebraic formulations which are not unique. Although there has been significant progress in software for solving mixed-integer problems, especially MILP, how one formulates a model can have a major impact in the performance and capability to find a solution. Therefore, deriving "good" formulations or finding potential improvements in existing models is commonly regarded as an art and strongly depends on the modeler skills.

Perspectives in multilevel decision-making in the process industry

“…All contract options can be modeled using generalized disjunctive programming (Grossmann and Trespalacios, 2013) and be formulated as mixed-integer linear programming (MILP) models via convex hull reformulation (Balas, 1998). The same four types and conditions are used for sales contracts.…”

The formulation of optimal mixtures with generalized disjunctive programming: A solvent design case study

“…In GDP, conversely, the logic is captured inside the disjunctions by relating Boolean variables (Y j;k ) to equations in the continuous form (h j;k ðxÞ), whereas the logic that connects the disjunctive sets is expressed through the relations XðYÞ. 64 To formulate a general mixture design problem (Eq. 1) as a GDP, several characteristics of the constraints must be taken into account.…”

“…Once an appropriate GDP formulation has been obtained, it can be converted into an MINLP problem using different approaches, such as big-M or hull relaxation, that result in relaxations of varying strength. 54,64,65 The big-M (BM) formulation 66 is the simplest representation of a GDP problem in a mixed-integer form.…”

A Convex Hull Formulation for the Design of Optimal Mixtures

“…Within this approach, Generalized Disjunctive Programming (GDP) (Raman and Grossmann, 1994) was employed to formulate the CAM b D problem, in order to address the difficulties arising from the complexity of the model and facilitate the problem formulation. Although GDP has been applied to process network systems, scheduling and distillation column design (Grossmann and Trespalacios, 2013), it had not previously been used in formulating mixture problems. In our initial work, the GDP problems were reformulated as MINLP problems using the Big-M (BM) approach.…”