Disjunctive Programming Research Articles

Integrated scheduling of rolling sector in steel production with consideration of energy consumption under time-of-use electricity prices

Due to increasing load and penetration of renewables, the electric grid is using time-of-use pricing for industrial customers. Involving energy-intensive processes, steel companies can reduce their production cost by accounting for changes in electricity pricing. In particular, steel companies can take advantage of processing flexibility to make better use of electric power, and thus reduce the energy cost. In this paper, we address a new integrated scheduling problem of multi-stage production derived from the rolling sector of steel production, with consideration of campaign decisions and demand-side management. The problem is formulated as a continuous time mixed-integer nonlinear programming (MINLP) model with generalized disjunctive programming (GDP) constraints, which is then reformulated as a mixed-integer linear programming (MILP) model. Numerical results are presented to demonstrate that the model is computationally efficient and compact.

Rigorous design of reaction-separation processes using disjunctive programming models

A systematic and efficient method for the rigorous design of complex chemical processes is significant in the chemical industry. In this paper, a superstructure-based optimization approach for the rigorous and simultaneous design of reaction and separation processes using generalized disjunctive programming (GDP) models is presented. In the reactor network, disjunctions for conditional reactors are introduced where the balance and reaction kinetic equations are applied only if the reactor is selected. Based on the proposed reactor disjunctions, two different reactor superstructures are developed and employed. In addition, the GDP representation of distillation columns is used to model the separation network. The reliability and efficiency of the proposed optimization method are demonstrated on two case studies, i.e., one cyclohexane oxidation process and one benzene chlorination process. The flowsheet structure and process-unit operating conditions are simultaneously optimized to minimize the total annual cost of the processes.

Synthesis of refrigeration system based on generalized disjunctive programming model

Abstract Refrigeration system holds an important role in process industries. The optimal synthesis cannot only reduce the energy consumption, but also save the production costs. In this study, a general methodology is developed for the optimal design of refrigeration cycle and heat exchanger network (HEN) simultaneously. Taking the heat integration between the external heat sources/sinks and the refrigeration cycle into consideration, a superstructure with sub-coolers is developed. Through defining logical variables that indicate the relative temperature positions of refrigerant streams after sub-coolers, the synthesis is formulated as a Generalized Disjunctive Programming (GDP) problem based on LP transshipment model, with the target of minimizing the total compressor shaft work in the refrigeration system. The GDP model is then reformulated as a Mixed Integer Nonlinear Programming (MINLP) problem with the aid of binary variables and Big-M Constraint Method. The efficacy of the process synthesis model is demonstrated by a case study of ethylene refrigeration system. The result shows that the optimization can significantly reduce the exergy loss as well as the total compression shaft work.

Stochastic group shop scheduling with fuzzy due dates

The group shop scheduling (GSS) problem is a general formulation that includes the job shop and the open shop scheduling problems. This paper is the first dealing with a GSS problem with random release and processing times and fuzzy due dates. While release and processing times are assumed to be ra ndom variables with known distributions, as in many real-world situations, job due dates are considered to be fuzzy tolerating a certain amount of delay. The objective is to maximize the expected total degree of satisfaction with respect to the fuzzy due dates. The problem is formulated as a stochastic disjunctive program and then two solution approaches are developed for it, both of which act within an ant colony optimization (ACO) framework. In the first approach, to estimate the performance of a constructed schedule, each random variable is replaced by its expectation, whereas in the other approach following a simulation optimization procedure, a discrete event simulation model is employed to estimate the performance of a schedule. The proposed approaches are then compared through computational experiments.

Pseudo basic steps: bound improvement guarantees from Lagrangian decomposition in convex disjunctive programming

An elementary, but fundamental, operation in disjunctive programming is a basic step, which is the intersection of two disjunctions to form a new disjunction. Basic steps bring a disjunctive set in regular form closer to its disjunctive normal form and, in turn, produce relaxations that are at least as tight. An open question is: What are guaranteed bounds on the improvement from a basic step? In this paper, using properties of a convex disjunctive program’s hull reformulation and multipliers from Lagrangian decomposition, we introduce an operation called a pseudo basic step and use it to provide provable bounds on this improvement along with techniques to exploit this information when solving a disjunctive program as a convex MINLP. Numerical examples illustrate the practical benefits of these bounds. In particular, on a set of K-means clustering instances, we make significant bound improvements relative to state-of-the-art commercial mixed-integer programming solvers.

Optimal Synthesis and Operation of Wastewater Treatment Process with Dynamic Influent

This work addresses the problem of optimal synthesis and operation of wastewater treatment processes considering multiscenario influent streams under different discharge standards and penalty rates of noncompliant emissions. We develop a general disjunctive programming model to address this problem, which is then reformulated as a multiperiod mixed-integer nonlinear programming model using hull reformulation for the disjunctions. To solve the resulting model to global optimality, we propose a Lagrangean-based decomposition algorithm. Numerical studies verify the effectiveness of the algorithm, and comparison studies provide useful management insights for policy makers.

Product-centric continuous-time formulation for pipeline scheduling

Continuous-time scheduling models for multiproduct pipelines typically view the contents of a pipeline as a set of batches. It facilitates tracking batch coordinates but prevents rigorously enforcing forbidden product sequences when dealing with intermediate dual-purpose nodes. Such models require the initial characterization of the pipeline and total number of batches as inputs, decisions that can be non-trivial. To overcome these limitations, this paper presents a new mixed-integer linear programming (MILP) formulation for straight pipelines with a single tuning parameter, the number of event points in the grid. It is derived from Generalized Disjunctive Programming (GDP) followed by a convex hull reformulation. Compared to a product-centric formulation based on the Resource-Task Network (RTN), it is much simpler, smaller in size and can be up to two orders of magnitude faster. Comparison to its batch-centric counterpart is not as favorable, the highlight being an 8% improvement in makespan for a benchmark problem.

Global optimization of disjunctive programs

We propose a new branch-and-bound framework for global optimization of disjunctive programs with general logical expressions. We do not assume the logical expressions to be in any normal form, and, under slightly stronger assumptions, we allow the use of negations and implications. In contrast to the widely used reformulation as a mixed-integer program, we compute the lower bounds and evaluate the logical expression in one step. Thus, we reduce the size of the problem and work exclusively with continuous variables, which is computationally advantageous. We present preliminary numerical results as proof of concept.

Optimal Pretreatment System of Flowback Water from Shale Gas Production

Shale gas has emerged as a potential resource to transform the global energy market. Nevertheless, gas extraction from tight shale formations is only possible after horizontal drilling and hydraulic fracturing, which generally demand large amounts of water. Part of the ejected fracturing fluid returns to the surface as flowback water, containing a variety of pollutants. For this reason, water reuse and water recycling technologies have received further interest for enhancing overall shale gas process efficiency and sustainability. Water pretreatment systems (WPSs) can play an important role for achieving this goal. This paper introduces a new optimization model for WPS simultaneous synthesis, especially developed for flowback water from shale gas production. A multistage superstructure is proposed for the optimal WPS design, including several water pretreatment alternatives. The mathematical model is formulated via generalized disjunctive programming (GDP) and solved by re-formulation as a mixed-integer n...

A generalized disjunctive programming framework for the optimal synthesis and analysis of processes for ethanol production from corn stover

The aim of this study is to analyze the techno-economic performance of process configurations for ethanol production involving solid-liquid separators and reactors in the saccharification and fermentation stage, a family of process configurations where few alternatives have been proposed. Since including these process alternatives creates a large number of possible process configurations, a framework for process synthesis and optimization is proposed. This approach is supported on kinetic models fed with experimental data and a plant-wide techno-economic model. Among 150 process configurations, 40 show an improved MESP compared to a well-documented base case (BC), almost all include solid separators and some show energy retrieved in products 32% higher compared to the BC. Moreover, 16 of them also show a lower capital investment per unit of ethanol produced per year. Several of the process configurations found in this work have not been reported in the literature.

Generalized disjunctive programming model for the multi-period production planning optimization: An application in a polyurethane foam manufacturing plant

A Generalized Disjunctive Programming (GDP) model for the optimal multi-period production planning and stock management is proposed in this work. The formulation is applied to a polyurethane foam manufacturing plant that comprises three stages: a first step that produces pieces with certain characteristics, a second process that involves the location of these pieces in a limited area and a third stage where pieces are stored in dedicated spaces. This article shows the GDP capabilities to provide a qualitative framework for representing the problem issues and their connections in a natural way, especially in a context where decisions integration is required. Due to the multi-period nature of the planning problem, a rolling horizon approach is suitable for solving it in reasonable computing time. It serves as a tool for analyzing the trade-offs among the different costs. Through the examples, the capabilities of the formulation and the proposed resolution method are highlighted.

Simultaneous synthesis of multi-plant heat exchanger networks using process streams across plants

Abstract Interplant heat integration using process streams is an alternative way to synthesize multi-plant heat exchanger networks (HENs). Compared with conventional way using steam, directly using process stream can recover more heat but with more pipelines. Due to the geographically separated locations of individual plants, distance related factors should be fully considered. In addition, the interaction between interplant heat integration and intra-plant heat integration is not fully addressed in traditional design methods. To counter these limitations, we present a new methodology for multi-plant HENs synthesis directly using process streams. The methodology makes use of a generalized disjunctive programming (GDP) formulation which gives rise to a mixed-integer nonlinear programming (MINLP) problem. For illustration purpose, two literature problems are conducted to verify the applicability of the proposed methodology. Our results can automatically offer the optimal design for multi-plant HENs, which are both more cost-optimal than the ones reported in the cited literatures.

Solving linear optimization over arithmetic constraint formula

Since Balas extended the classical linear programming problem to the disjunctive programming (DP) problem where the constraints are combinations of both logic AND and OR, many researchers explored this optimization problem under various theoretical or application scenarios such as generalized disjunctive programming (GDP), optimization modulo theories (OMT), robot path planning, real-time systems, etc. However, the possibility of combining these differently-described but form-equivalent problems into a single expression remains overlooked. The contribution of this paper is two folded. First, we convert the linear DP/GDP model, linear-arithmetic OMT problem and related application problems into an equivalent form, referred to as the linear optimization over arithmetic constraint formula (LOACF). Second, a tree-search-based algorithm named RS-LPT is proposed to solve LOACF. RS-LPT exploits the techniques of interval analysis and nonparametric estimation for reducing the search tree and lowering the number of visited nodes. Also, RS-LPT alleviates bad construction of search tree by backtracking and pruning dynamically. We evaluate RS-LPT against two most common DP/GDP methods, three state-of-the-art OMT solvers and the disjunctive transformation based method on optimization benchmarks with different types and scales. Our results favor RS-LPT as compared to existing competing methods, especially for large scale cases.

Continuous‐time scheduling formulation for straight pipelines

Pipelines represent the most cost‐effective way of transporting large quantities of refined petroleum products over large distances but can be challenging to operate. In this article, we propose a new mixed‐integer linear programming formulation for scheduling straight pipelines with multiple single and dual purpose nodes. The model allows for simultaneous injections and deliveries, and interacting pumping runs, in which a segment of the pipeline simultaneously receives material from its refinery and upstream segment. In contrast to previous batch centric models, it uses segment dependent coordinates. To make it efficient by design, we rely on generalized disjunctive programming and develop disjunctions for which the convex hull reformulation is simple (roughly the same number of variables and constraints as its big‐M counterpart). Through the solution of a set of test cases from the literature, we show a better utilization of the pipeline capacity that is translated into a lower makespan. © 2016 American Institute of Chemical Engineers AIChE J, 63: 1923–1936, 2017

A Hybrid Stochastic/Deterministic Unit Commitment Based on Projected Disjunctive MILP Reformulation

This letter presents a new expression of the chance constraint, representing ramping and operating reserve, in the context of the mixed integer linear programming (MILP) formulation for solving the hybrid stochastic/deterministic unit commitment (SDUC) problem. Based on projected disjunctive programming, the nonlinear chance constraint is converted into a set of linear constraints that has a more compact size. Numerical simulations highlight the primacy of the proposed MILP-SDUC reformulation as compared with other formulation approaches.

Stable-unstable semantics: Beyond NP with normal logic programs

AbstractStandard answer set programming (ASP) targets at solving search problems from the first level of the polynomial time hierarchy (PH). Tackling search problems beyond NP using ASP is less straightforward. The class of disjunctive logic programs offers the most prominent way of reaching the second level of the PH, but encoding respective hard problems as disjunctive programs typically requires sophisticated techniques such as saturation or meta-interpretation. The application of such techniques easily leads to encodings that are inaccessible to non-experts. Furthermore, while disjunctive ASP solvers often rely on calls to a (co-)NP oracle, it may be difficult to detect from the input program where the oracle is being accessed. In other formalisms, such as Quantified Boolean Formulas (QBFs), the interface to the underlying oracle is more transparent as it is explicitly recorded in the quantifier prefix of a formula. On the other hand, ASP has advantages over QBFs from the modeling perspective. The rich high-level languages such as ASP-Core-2 offer a wide variety of primitives that enable concise and natural encodings of search problems. In this paper, we present a novel logic programming–based modeling paradigm that combines the best features of ASP and QBFs. We develop so-calledcombined logic programsin which oracles are directly cast as (normal) logic programs themselves. Recursive incarnations of this construction enable logic programming on arbitrarily high levels of the PH. We develop a proof-of-concept implementation for our new paradigm.

Designing optimal mixtures using generalized disjunctive programming: Hull relaxations

A general modeling framework for mixture design problems, which integrates Generalized Disjunctive Programming (GDP) into the Computer-Aided Mixture/blend Design (CAMbD) framework, was recently proposed (S. Jonuzaj, P.T. Akula, P.-M. Kleniati, C.S. Adjiman, 2016. The formulation of optimal mixtures with Generalized Disjunctive Programming: A solvent design case study. AIChE Journal 62, 1616–1633). In this paper we derive Hull Relaxations (HRs) of GDP mixture design problems as an alternative to the big-M (BM) approach presented in this earlier work. We show that in restricted mixture design problems, where the number of components is fixed and their identities and compositions are optimised, BM and HR formulations are identical. For general mixture design problems, where the optimal number of mixture components is also determined, a generic approach is employed to enable the derivation and solution of the HR formulation for problems involving functions that are not defined at zero (e.g., logarithms). The design methodology is applied successfully to two solvent design case studies: the maximization of the solubility of a drug and the separation of acetic acid from water in a liquid–liquid extraction process. Promising solvent mixtures are identified in both case studies. The HR and BM approaches are found to be effective for the formulation and solution of mixture design problems, especially via the general design problem.

Mathematical optimization of a supply chain for the production of fuel pellets from residual biomass

One of the main concerns of humankind in the last years is the availability of energy sources. Research has been focused on finding clean and renewable ways to satisfy the energy demand worldwide. In the particular case of the state of Guanajuato, Mexico, clay industry burns each year about 15,000 m3 of fuel oil and residual oils, and 96,000 t of wood derivatives. As a way to reduce the environmental impact of clay industry, the use of solid fuel pellets, obtained from vegetable residual material, is proposed. The raw material for the pellets is obtained from agribusiness and from the cities of the state. The solid biofuel has high density, low content of humidity, a homogeneous shape and high energy density. Nevertheless, special care must be taken about the location of the production facility and hubs, in order to make the production of the biofuel economically feasible. Furthermore, to have an environmentally friendly fuel, the supply chain and the production process must minimize the global environmental impact. In this work, a mathematical programming model is proposed to determinate the optimal location of the production facilities, the hubs, and the best distribution logistics. The problem is modelled using a general disjunctive programming approach, and then relaxed into a mixed-integer non-linear programming (MINLP) problem. It has been determined that the main plant should be located in the city of Irapuato, while secondary plants must be established in the cities of Leon, Irapuato, Abasolo and Salamanca. Moreover, it has been estimated that, when the residual biomass is converted into pellets, about 72,548 t/year of equivalent CO2 are avoided in the main plant, together with 24,182 of equivalent CO2 avoided per secondary facility.

Fermentable sugars from Eucalyptus globulus: Process optimization

Abstract In the last years, bioethanol production from lignocellulosic materials is receiving more attention from researchers since it is an abundant raw material in certain regions, it is cheaper, does not compete with foods like sugarcane or corn, and can reduce up to 86% of carbon dioxide emissions when compared with gasoline. However, the process economy is linked to the pretreatment needed to make the cellulose accessible for further steps of enzymatic hydrolysis and fermentation. Then, the motivation of this work is to propose a MILP model to perform an optimal synthesis of the pretreatment for obtaining fermentable sugars from the Eucalyptus globulus specie. For this purpose, a General Disjunctive Program (GDP) is formulated. The obtained results suggest that there is a small difference between a two-step pretreatment process (pretreatment with posthydrolysis) and one-step pretreatment of diluted acid process.

Source-based discrete and continuous-time formulations for the crude oil pooling problem

Abstract The optimization of crude oil operations in refineries is a challenging scheduling problem due to the need to model tanks of varying composition with nonconvex bilinear terms, and complicating logistic constraints. Following recent work for multiperiod pooling problems of refined petroleum products, a source-based mixed-integer nonlinear programming formulation is proposed for discrete and continuous representations of time. Logistic constraints are modeled through Generalized Disjunctive Programming while a specialized algorithm featuring relaxations from multiparametric disaggregation handles the bilinear terms. Results over a set of test problems from the literature show that the discrete-time approach finds better solutions when minimizing cost (avoids source of bilinear terms). In contrast, solution quality is slightly better for the continuous-time formulation when maximizing gross margin. The results also show that the specialized global optimization algorithm can lead to lower optimality gaps for fixed CPU, but overall, the performance of commercial solvers BARON and GloMIQO are better.