Generalized Disjunctive Programming Research Articles

Extensions to generalized disjunctive programming: hierarchical structures and first-order logic

Optimization problems with discrete–continuous decisions are traditionally modeled in algebraic form via (non)linear mixed-integer programming. A more systematic approach to modeling such systems is to use generalized disjunctive programming (GDP), which extends the disjunctive programming paradigm proposed by Egon Balas to allow modeling systems from a logic-based level of abstraction that captures the fundamental rules governing such systems via algebraic constraints and logic. Although GDP provides a more general way of modeling systems, it warrants further generalization to encompass systems presenting a hierarchical structure. This work extends the GDP literature to address two major alternatives for modeling and solving systems with nested (hierarchical) disjunctions: explicit nested disjunctions and equivalent single-level disjunctions. We also provide theoretical proofs on the relaxation tightness of such alternatives, showing that explicitly modeling nested disjunctions is superior to the traditional approach discussed in literature for dealing with nested disjunctions.

Linear model decision trees as surrogates in optimization of engineering applications

Machine learning models are promising as surrogates in optimization when replacing difficult to solve equations or black-box type models. This work demonstrates the viability of linear model decision trees as piecewise-linear surrogates in decision-making problems. Linear model decision trees can be represented exactly in mixed-integer linear programming (MILP) and mixed-integer quadratic constrained programming (MIQCP) formulations. Furthermore, they can represent discontinuous functions, bringing advantages over neural networks in some cases. We present several formulations using transformations from Generalized Disjunctive Programming (GDP) formulations and modifications of MILP formulations for gradient boosted decision trees (GBDT). We then compare the computational performance of these different MILP and MIQCP representations in an optimization problem and illustrate their use on engineering applications. We observe faster solution times for optimization problems with linear model decision tree surrogates when compared with GBDT surrogates using the Optimization and Machine Learning Toolkit (OMLT).

Disjunctive optimization model and algorithm for long-term capacity expansion planning of reliable power generation systems

This paper proposes a new optimization model and algorithm for long-term capacity expansion planning of reliable power generation systems. The model optimizes both investment decisions (e.g., size, location, and time to install, retire and decommission facilities) and hourly operation decisions (e.g., on/off status, operating capacity, and expected power output). It is also able to optimize both the main generation capacity and reserve capacity to improve power system reliability. The impact of operational strategies of generators (i.e., participating in electricity production vs. remaining as idle units during operation) on power system reliability is considered. Probability of equipment failures and capacity failure states are used to rigorously estimate the power system reliability depending on design and operation strategies. The optimization model is firstly formulated with Generalized Disjunctive Programming (GDP), which is then reformulated as a mixed-integer linear programming (MILP) model using the Hull relaxation. Two reliability-related penalties, such as downtime penalty and unmet demand (or load shedding) penalty, are included in the objective function to maximize reliability while minimizing the total net present cost. Furthermore, a bilevel decomposition with tailored cuts is developed to reduce computational times of the multi-scale optimization model. The effectiveness of the proposed model is shown by comparing it with the expansion planning model without explicitly considering reliability. We also show that the proposed bilevel decomposition is computationally efficient for solving large-scale problems with millions of variables and constraints through 5-years and 10-years planning case studies.

A generalized disjunctive programming model for the optimal design of reverse electrodialysis process for salinity gradient-based power generation

Reverse electrodialysis (RED) is an emerging electro-membrane technology that generates electricity out of salinity differences between two solutions, a renewable source known as salinity gradient energy. Realizing full-scale RED would require more techno-economic and environmental assessments that consider full process design and operational decision space from the RED stack to the entire system. This work presents an optimization model formulated as a Generalized Disjunctive Programming (GDP) problem that incorporates a finite difference RED stack model from our research group to define the cost-optimal process design. The solution to the GDP problem provides the plant topology and the RED units’ working conditions that maximize the net present value of the RED process for given RED stack parameters and site-specific conditions. Our results show that, compared with simulation-based approaches, mathematical programming techniques are efficient and systematic to assist early-stage research and to extract optimal design and operation guidelines for large-scale RED implementation.

A Benders decomposition framework for the optimization of disjunctive superstructures with ordered discrete decisions

AbstractThis study introduces the logic‐based discrete‐Benders decomposition (LD‐BD) for Generalized Disjunctive Programming (GDP) superstructure problems with ordered Boolean variables. The key idea is to obtain Benders cuts that use neighborhood information of a reformulated version of Boolean variables. These Benders cuts are iteratively refined, which guarantees convergence to a local optimum. A mathematical case study, the optimization of a network with Continuous Stirred‐Tank Reactors (CSTRs) in series, and a large‐scale problem involving the design of a distillation column are considered to demonstrate the features of LD‐BD. The results from these case studies have shown that the LD‐BD method exhibited good performance by finding attractive locally optimal solutions relative to existing logic‐based solvers for GDP problems. Based on these tests, the LD‐BD method is a promising strategy to solve optimal synthesis problems with ordered discrete decisions emerging in chemical engineering applications.

Computer-aided naphtha liquid–liquid extraction: Molecular reconstruction, sustainable solvent design and multiscale process optimization

A new integrated sustainable solvent and process design framework is proposed based on GDP (generalized disjunctive programming) to separate the naphthenic and aromatic components from naphtha. First, the detailed molecular composition information of naphtha is obtained using a molecular reconstruction model based on feasible bulk properties. Two group-based virtual molecules are then presented to express naphtha components for the liquid–liquid extraction process design. Second, four separation performance indicators and eight SHE (safety, health and environment) metrics are screened and described for the sustainable solvent design on the basis of the UNIFAC model and functional groups. The solvent structures are modelled through the improvement of the octet rule. Last, a conceptual liquid–liquid extraction process is proposed and expressed as a CAMPD (computer aided molecular and process design) optimization problem combining with the rigorous process mechanism model, sustainable solvent design model and molecular information model of naphtha. Especially, GDP is tailored to express the discrete choices in the simultaneous optimization design of solvents and processes and address the challenge of the large-scale framework solution. Compared with the process using the traditional solvent sulfolane, the content of the separated naphthenic and aromatic components is improved by 10.37%. At the same time, the sustainability is enhanced according to SHE assessment.

Disjunctive programming model for the synthesis of property-based water supply network with multiple resources

Industrial water supply networks can use multiple water resources (i.e., municipal water, surface water, municipal treated wastewater, groundwater, and seawater) with their respective pre-treatment technologies to meet the industrial water demand. The water quality of each water resource varies, and the associated pre-treatment technologies provide different technical and economic efficiencies that directly impact the overall cost of the water supply network. This paper proposed a superstructure-based mathematical modeling optimization to design the multi-resources industrial water supply network with optimal pre-treatment technologies. The established model includes the relative equations among different pre-treatment units and water sinks, flow rate constraints, property constraints, logic constraints, and cost constraints. These constraints are used to describe water treatment technologies selection through general disjunctive programming (GDP) and transform them into a mixed-integer nonlinear programming (MINLP) model for the problem solution. The minimum annualized cost of the water supply network is taken as an objective function. The optimization and comparison analysis of the coastal oil refinery water supply system is illustrated. The results show that an integrated water supply network of surface water and municipal treated wastewater has the lowest annualized cost of 12.3 million CNY/y. The cost is reduced by 7.44 million CNY/y compared to municipal water as the single water resource. This integrated water supply network saves freshwater resources, provides substantial economic gains, and encourages municipal treated wastewater reuse.

Optimal plate design problem in steel production

In this paper, we address a new optimal plate design problem in steel production, in which slab selection is jointly considered. On the basis of the underlying features, the problem is formulated as a mixed integer nonlinear programming (MINLP) model with generalized disjunctive programming (GDP) constraints. A logic-based outer approximation (L-OA) algorithm is proposed to solve the problem. Specifically, a two-stage heuristic method is designed to initialise the L-OA algorithm. Numerical results are presented to demonstrate that the proposed L-OA algorithm and the heuristic method are effective and computationally efficient.

GDP-based approach for optimal design of forest biorefinery supply chain considering circularity and conversion facilities co-location

The Forest Biorefinery Supply Chain (FBSC) design problem is addressed. A general mathematical framework based on a Generalized Disjunctive Programming (GDP) formulation is proposed as an efficient decision-making tool for the optimal FBSC. The approach simultaneously tackles the dynamic capacity allocation and the facilities’ co-location, features that are explicitly modelled. The FBSC superstructure promotes the circularity of the network by including flexible processing recipes and the use of multiple biomass residues as biorefineries raw materials as well as paper recycling. The proposed case study copes with the production of paper and biofuels in Argentina, showing the advantage of the integration of biorefineries with the existing forest industries as well as the addition of value to biomass industrial waste and by-products. Additionally, a set of scenarios are considered and tested. By these examples, key features of the proposed approach are highlighted as essential to reach a profitable FBSC.

Generalized Disjunctive Programming Model for Optimization of Reverse Electrodialysis Process

Reverse electrodialysis (RED), an emerging electrochemical technology that uses ion-selective membranes to directly draw electricity out from salinity differences between two solutions, i.e., salinity gradient energy (SGE), has the potential to be a clean and steady renewable source to reach a sustainable water and energy supply portfolio. Although RED has made notable advances, full-scale RED progress demands more techno-economic and environmental assessments that consider full process design and operational decision space from module- to system-level. This work presents an optimization model formulated as a Generalized Disjunctive Programming (GDP) problem to define the cost-optimal RED process design for different deployment scenarios. We use a predictive model of the RED stack developed and validated in our research group to fully capture the behavior of the system. The problem addressed is to determine the RED plant's topology and the working conditions for a given design of each RED stack which renders the cost-optimal design for the defined problem and scenario. Our results show that, compared with simulation-based approaches, mathematical programming techniques are an efficient and systematic approach to provide decision-making support in early-stage applied research and to obtain design and operation guidelines for full-scale RED implementation in real scenarios.

Integrating stochastic programming and reliability in the optimal synthesis of chemical processes

Plant availability and operating uncertainties are critical considerations for the design and operation of chemical processes as they directly impact service level and economic performance. This paper proposes a two-stage stochastic programming GDP (Generalized Disjunctive Programming) model with reliability constraints to deal with both the exogenous and endogenous uncertainties in process synthesis, where the reliability model is incorporated into the flowsheet superstructure optimization. The proposed stochastic programming model anticipates the market uncertainties through scenarios for selecting the optimal flowsheet topology, equipment sizes and operating conditions, while considering the impact of selecting parallel units for improving plant availability. An improved logic-based outer approximation algorithm is applied to solve the resulting hybrid GDP model, which effectively avoids numerical difficulties with zero flows and provides high quality design solutions. The applicability of the proposed modeling framework and the efficiency of solution strategy are illustrated with two well-known conceptual design case studies: methanol synthesis process and toluene hydrodealkylation process. The model, which integrates reliability (endogenous uncertainty) and exogenous uncertainty, shows the best economic performance with the increasing operational flexibility and plant availability.

Pyomo.GDP: an ecosystem for logic based modeling and optimization development

We present three core principles for engineering-oriented integrated modeling and optimization tool sets—intuitive modeling contexts, systematic computer-aided reformulations, and flexible solution strategies—and describe how new developments in Pyomo.GDP for Generalized Disjunctive Programming (GDP) advance this vision. We describe a new logical expression system implementation for Pyomo.GDP allowing for a more intuitive description of logical propositions. The logical expression system supports automated reformulation of these logical constraints to linear constraints. We also describe two new logic-based global optimization solver implementations built on Pyomo.GDP that exploit logical structure to avoid “zero-flow” numerical difficulties that arise in nonlinear network design problems when nodes or streams disappear. These new solvers also demonstrate the capability to link to external libraries for expanded functionality within an integrated implementation. We present these new solvers in the context of a flexible array of solution paths available to GDP models. Finally, we present results on a new library of GDP models demonstrating the value of multiple solution approaches.

Model Predictive Control of Discrete-Continuous Energy Systems via Generalized Disjunctive Programming

Generalized Disjunctive Programming (GDP) provides an alternative framework to model optimization problems with both discrete and continuous variables. The key idea behind GDP involves the use of logical disjunctions to represent discrete decisions in the continuous space, and logical propositions to denote algebraic constraints in the discrete space. Compared to traditional mixed-integer programming (MIP), the inherent logic structure in GDP yields tighter relaxations that are exploited by global branch and bound algorithms to improve solution quality. In this paper, we present a general GDP model for optimal control of hybrid systems that exhibit both discrete and continuous dynamics. Specifically, we use GDP to formulate a model predictive control (MPC) model for piecewise-affine systems with implicit switching logic. As an example, the GDP-based MPC approach is used as a supervisory control to improve energy efficiency in residential buildings with binary on/off, relay-based thermostats. A simulation study is used to demonstrate the validity of the proposed approach, and the improved solution quality compared to existing MIP-based control approaches.

Efficient formulation for transportation scheduling of single refinery multiproduct pipelines

Multiproduct pipeline transportation scheduling is a complex operations research problem that is characterized by the movement of the cargo rather than the carrier. Hence, it cannot be solved using vehicle routing methods. While most formulations for short-term scheduling adopt a continuous-time representation, they often lead to suboptimal solutions because of the dependence on the number of time slots in the grid, which is difficult to predict. Furthermore, some of these formulations have poor linear relaxations due to the presence of inefficient big-M constraints. In this paper, we develop a discrete-time mixed integer linear programming (MILP) model for the detailed scheduling of a straight pipeline with a single refinery and multiple depots. The proposed formulation rigorously detects interface material generated between adjacent products and considers planned shutdowns in pipeline segments due to maintenance operations, as well as local market demands occurring at multiple intermediate due dates. The main novelty is that continuous tasks can span multiple time slots to enforce minimum batch sizes on injection and delivery nodes, which allows for the model to generate better schedules than those obtained with previously proposed formulations. To ensure an efficient model by design, we rely on generalized disjunctive programming (GDP) and on the convex hull reformulation of disjunctions, which results in stronger and often more computationally efficient formulations. We present numerical results for a set of benchmark instances and show that the proposed model applies to large-scale industrial cases.

Optimization of triple-pressure combined-cycle power plants by generalized disjunctive programming and extrinsic functions

A new mathematical framework for optimal synthesis, design, and operation of triple-pressure steam-reheat combined-cycle power plants (CCPP) is presented. A superstructure-based representation of the process, which embeds a large number of candidate configurations, is first proposed. Then, a generalized disjunctive programming (GDP) mathematical model is derived from it. Series, parallel, and combined series-parallel arrangements of heat exchangers are simultaneously embedded. Extrinsic functions executed outside GAMS from dynamic-link libraries (DLL) are used to estimate the thermodynamic properties of the working fluids. As a main result, improved process configurations with respect to two reported reference cases were found. The total heat transfer areas calculated in this work are by around 15% and 26% lower than those corresponding to the reference cases.This paper contributes to the literature in two ways: (i) with a disjunctive optimization model of natural gas CCPP and the corresponding solution strategy, and (ii) with improved HRSG configurations.

Multi-periodic Refinery Scheduling Based on Generalized Disjunctive Programming

Refinery complex production process usually involves some distinct or implied production rules and expert experiences. Representation and utilizing these heuristic rules is conducive to efficient scheduling optimization. In this paper, the heuristic rules were formulated using disjunctive form and logical proposition, and a discrete-time based multi-periodic generalized disjunctive programming (GDP) scheduling model was built. Due to the combining of logic expressions and algebra expressions, heuristic rules were utilized by the proposed model, while scheduling optimization was ensured. The formulated model was used to deal with a refinery scheduling problem. Case study shows the model’s feasibility and efficiency.

Design optimization of shell and tube heat exchangers sizing with heat transfer enhancement

Heat transfer enhancement (HTE) is an efficient technology to improve the performance of shell and tube heat exchangers (STHEs). While there exist many commercial software packages, heat exchanger design still highly depends on users’ experiences, resulting in long computing time and inferior solutions. Therefore, a systematic optimization methodology is required in order to achieve an efficient and accurate design solution. This paper presents a generalized disjunctive programming (GDP) model for the optimization of STHE design. It is formulated as a mixed integer non-linear programming (MINLP) problem, involving selection for 12 technology combinations (four tube-side techniques, three shell-side techniques), and all discrete decisions for each selection are modeled by disjunctions. The model is then applied to minimize the total capital cost. Both global optimization solver BARON, and general MINLP solver DICOPT are tested. The results show that the developed method provides a better design solution compared with conventional STHE design procedures.

Optimal production and maintenance scheduling for a multiproduct batch plant considering degradation

Performance decay due to asset degradation is an important constraint in industrial production and therefore needs to be actively considered. This paper focuses on short-term scheduling for multiproduct batch processes with sequence-dependent degradation and is motivated by a case study in which the sequence of multiple-grade batch runs impacts evolution of fouling. A continuous-time scheduling formulation is proposed to incorporate realistic features of the case study processes. The precedence scheduling concept for the sequential process is employed to model sequences of multiproduct orders and maintenance and is implemented using the general disjunctive programming method. The scheduling formulation is applied to the case study and further analyzed through comprehensive computational tests, which illustrates the efficacy of the proposed formulation.

Modern Modeling Paradigms Using Generalized Disjunctive Programming

Models involving decision variables in both discrete and continuous domain spaces are prevalent in process design. Generalized Disjunctive Programming (GDP) has emerged as a modeling framework to explicitly represent the relationship between algebraic descriptions and the logical structure of a design problem. However, fewer formulation examples exist for GDP compared to the traditional Mixed-Integer Nonlinear Programming (MINLP) modeling approach. In this paper, we propose the use of GDP as a modeling tool to organize model variants that arise due to characterization of different sections of an end-to-end process at different detail levels. We present an illustrative case study to demonstrate GDP usage for the generation of model variants catered to process synthesis integrated with purchasing and sales decisions in a techno-economic analysis. We also show how this GDP model can be used as part of a hierarchical decomposition scheme. These examples demonstrate how GDP can serve as a useful model abstraction layer for simplifying model development and upkeep, in addition to its traditional usage as a platform for advanced solution strategies.

Microscale bioprocess design and optimisation

Lignocellulosic biomass is a raw material to produce biofuels and bioproducts. Thus, the proper use of such materials may help to development of a bio-based industry, additionally giving a second use to such waste material. Mexico is a country with a high agricultural production, implying a high production of agricultural residues. Since there are many possibilities for making use of the residues, the question arising is: which are the best products to be obtained from the lignocellulosic biomass available in the country? To answer this question, in this work a mathematical model for the supply chain of the production of biofuels/bioproducts is proposed. The mathematical model is formulated using generalized disjunctive programming (GDP) and relaxed through the convex hull approach. The resulting MILP is solved using the software GAMS, aiming to maximize the profit.