#
Modern Modeling Paradigms Using Generalized Disjunctive Programming^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

- Case 1: Generate model variants that focus on various portions of an end-to-end process;
- Case 2: Use higher-level (approximate) models to do preliminary analysis, and drill down into increasing model detail for promising options as part of the solution scheme.

## 3. Case Study

#### 3.1. Sets

$c\in C$ | Set of components (feeds, intermediates, inerts, and products) |

$i\in I$ | Set of end-to-end process sections |

$j\in J$ | Set of process section alternatives |

${J}_{i}$ | Set of alternatives available for process section i |

$k\in K$ | Set of streams |

$l\in L$ | Set of detail levels |

#### 3.2. Variables

${f}_{ck}$ | molar flow of component c on stream k |

${F}_{k}$ | total molar flow of stream k |

${T}_{k}$ | temperature of stream k |

${P}_{k}$ | pressure of stream k |

${z}_{i}$ | profit or cost contribution from section i |

${\zeta}_{ij}$ | profit or cost contribution from alternative j in section i |

x | other continuous state variables |

${Y}_{ij}$ | Boolean selection of process alternative j for section i |

${Y}_{i}^{l}$ | Boolean selection of detail level l for modeling process section i |

${Y}_{ij}^{l}$ | Boolean selection of detail level l for modeling process alternative j in section i |

#### 3.3. Functions

$g(f,F,T,P,x)$ | globally relevant constraints |

${r}_{ij}(f,F,T,P,x)$ | constraints relevant to selection of alternative $j\in {J}_{i}$ for process section $i\in I$ for any detail level |

${h}_{i}^{l}(f,F,T,P,x)$ | constraints describing process section $i\in I$ at detail level $l\in {L}_{i}$ |

${s}_{ij}^{l}(f,F,T,P,x)$ | constraints describing alternative $j\in {J}_{i}$ for process section $i\in I$ at detail level $l\in {L}_{ij}$ |

${\varphi}_{i}^{l}(f,F,T,P,x,\zeta )$ | calculation of profit or cost contribution for section $i\in I$ |

${\psi}_{ij}^{l}(f,F,T,P,x)$ | calculation of profit or cost contribution for selecting alternative $j\in {J}_{i}$ in section $i\in I$ |

$\Omega \left(Y\right)$ | Logical propositions between Boolean selections |

#### 3.4. Formulation

#### 3.5. Discussion

#### 3.6. Solution Strategies

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Big-M Reformulation (BM)

## Appendix B. Hull Reformulation (HR)

## Appendix C. Methanol Model

#### Appendix C.1. Feed Procurement

#### Appendix C.1.1. High Detail

#### Appendix C.1.2. Medium Detail

#### Appendix C.1.3. Low Detail

#### Appendix C.2. Product Sales

#### Appendix C.2.1. High Detail

#### Appendix C.2.2. Medium Detail

#### Appendix C.2.3. Low Detail

#### Appendix C.3. Compressors

#### Appendix C.3.1. High Detail

#### Appendix C.3.2. Medium Detail

#### Appendix C.3.3. Low Detail

#### Appendix C.4. Expansion Valve

#### Appendix C.4.1. High Detail

#### Appendix C.4.2. Medium Detail

#### Appendix C.4.3. Low Detail

#### Appendix C.5. Cooler

#### Appendix C.5.1. High Detail

#### Appendix C.5.2. Medium Detail

#### Appendix C.5.3. Low Detail

#### Appendix C.6. Heater

#### Appendix C.6.1. High Detail

#### Appendix C.6.2. Medium Detail

#### Appendix C.6.3. Low Detail

#### Appendix C.7. Reactors

#### Appendix C.7.1. High Detail

#### Appendix C.7.2. Medium Detail

#### Appendix C.7.3. Low Detail

#### Appendix C.8. Flash

#### Appendix C.8.1. High Detail

#### Appendix C.8.2. Medium Detail

#### Appendix C.8.3. Low Detail

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**Figure 1.**Example process illustrating the embedding of multiple detail levels within discrete options for each process section.

**Figure 2.**Simplified process diagram for the illustrative example. Three sections exist: procurement, production, and sales. In each section, decisions must be made in both discrete and continuous variable domains.

**Figure 3.**Methanol process flowsheet superstructure, adapted from [22], showing stream numbers in blue.

**Figure 4.**Solution flowsheet for Problem (P2), using two-stage feed compression, the cheap reactor, and single-stage recycle compression. At low procurement modeling detail, no feed selection decisions are made.

**Table 1.**Profit (1000 USD) at the fixed production section design of two-stage feed compression, the cheap reactor, and single-stage recycle compression compared to the profit achievable when the design is allowed to vary.

Procurement Detail | Sales Detail | Fixed Design | Best Design | Difference |
---|---|---|---|---|

low | low | 1793 | 1793 | |

low | med | 1564 | 1614 | single-stage feed compression |

low | high | 2617 | 2667 | single-stage feed compression |

med | low | 1793 | 1793 | |

med | med | 1564 | 1614 | single-stage feed compression |

med | high | 2617 | 2667 | single-stage feed compression |

high | low | 1709 | 1832 | single-stage feed compression |

high | med | 1746 | 1850 | single-stage feed compression |

high | high | 3133 | 3183 | single-stage feed compression |

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**MDPI and ACS Style**

Chen, Q.; Grossmann, I.
Modern Modeling Paradigms Using Generalized Disjunctive Programming. *Processes* **2019**, *7*, 839.
https://doi.org/10.3390/pr7110839

**AMA Style**

Chen Q, Grossmann I.
Modern Modeling Paradigms Using Generalized Disjunctive Programming. *Processes*. 2019; 7(11):839.
https://doi.org/10.3390/pr7110839

**Chicago/Turabian Style**

Chen, Qi, and Ignacio Grossmann.
2019. "Modern Modeling Paradigms Using Generalized Disjunctive Programming" *Processes* 7, no. 11: 839.
https://doi.org/10.3390/pr7110839