Disjunctive Programming Research Articles

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.

Optimization of installation and energy costs in water distribution systems with unknown flow directions

Abstract Water distribution networks (WDNs) are an important part of water distribution systems and are responsible for water transportation from the reservoirs to the demand nodes at adequate pressure and velocity. In the present paper, the synthesis of WDN is treated as an optimization problem with a mixed integer nonlinear programming formulation. The objective function to be minimized is the total network cost, considering installation and energy costs, with unknown flow directions, which is the novelty in the paper. Disjunctive programming and linearization techniques are used in the model formulation to avoid nonlinear and nonconvex problems. Two case studies are used to test the model's applicability. Results show that operational costs can represent a significant part of the total cost in sustainable networks. In the first case study, the total cost was better than the literature results (US$ 2,272,538.85 vs. US$ 2,272,387.49) and the operational costs represent ¼ of the total WDN costs. In the second case study, the operation cost corresponds to almost 2/3 of the total WDN cost. These results show the importance of considering operational costs in the WDN design. Also, the consideration of unknown flow directions can lead to better results for the network topology.

Incorporating Machine Learning in Computer-Aided Molecular Design for Fragrance Molecules

The demand for new novel flavour and fragrance (F&F) molecules has boosted the need for a systematic approach to designing fragrance molecules. However, the F&F-related industry still relies heavily on experimental approaches or on existing databases without considering the consequences resulting from changes in concentration, which could omit potential fragrances. Computer-aided molecular design (CAMD) has great potential to identify novel molecular structures to be used as fragrances. Using CAMD for this purpose requires models to predict the olfaction properties of molecules. A rough set-based machine learning (RSML) approach is used to develop an interpretable predictive model for odour characteristics in this work. New rule-based models are generated from RSML based on the dilution and a number of different topological indices which identify the structure-odour relationship of fragrance molecules. The most prominent rules are selected and formulated as constraints in a CAMD optimisation model. The combination of several rules was able to increase the coverage of different classes of molecules. To model the performance indicators that vary over a range of properties, a disjunctive programming model is also incorporated into the CAMD framework. A case study demonstrates the utilisation of this methodology to design fragrance additives in dishwashing liquid. The results illustrate the capability of the novel RSML and CAMD framework to identify potential fragrance molecules that can be used in consumer products.

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.

Active and Reactive Power Coordinated Two-Stage MG Scheduling for Resilient Distribution Systems Under Uncertainties

This paper proposes a two-stage microgrid (MG) scheduling approach that considers the dynamic MG formation and the coordinated optimization of active and reactive powers to enhance the distribution system resilience after blackouts. It aims to dispatch multiple devices operating in different timescales to sectionalize the distribution system into self-supplied MGs and supply the critical loads continuously with reliable power. In the hour-ahead operation stage, switches, capacitor banks and energy storage charging/discharging decisions are optimized based on hour-ahead interval predictions of renewable energy generations and load demands. In the intra-hour operation stage, the output of distributed energy resources and static Var compensators are redispatched to compensate the hour-ahead operation decisions after the uncertainty realization. Compared with the existing methods, the proposed MG scheduling method can restore more critical loads and sustain secure operation under random renewable energy generation and load demand. In addition, to address uncertainties, the MG scheduling problem is modeled as a robust optimization model, which is decomposed into a master problem and a subproblem by adopting the column-and-constraint generation algorithm. Furthermore, the subproblem is reformulated via the disjunctive programming method to reduce the computational complexity. The effectiveness of the proposed method is verified on the modified IEEE test systems.

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.

Synthesis and optimization of multilevel refrigeration systems using generalized disjunctive programming

The synthesis and optimization of multilevel refrigeration systems is challenging because it’s a highly non-linear, multi-variable, and multi-modal problem. This work presents a novel approach to develop a complete generalized disjunctive programming model for the synthesis and optimization of multilevel refrigeration systems. The model is based on the application of mass, energy, and thermodynamic model equations and developed to optimize the total shaft work requirement of the system. The presented model is very systematic, easily manageable, and can be extendable to include all design features for refrigeration systems. The model is solvable using advanced solution algorithms such as the Logic-based branch and bound because of it’s disjunctive nature. Thus the solution is sought in reduced space and avoids the full scale as is the case with Mixed integer linear/non-linear models. The model solution yields optimal temperature/pressure levels, mass flow rates, and the shaft work consumption. A case study of the precooling cycle of the propane pre-cooled mixed refrigerant (C3MR) LNG process is used to test the proposed model. The model results are validated by simulation using Aspen Hysys software. Shaft work savings of up to 12.3% are obtained from the model results against the base case. Preliminary estimations show that cost savings of up to 2.55 MM$/y are realizable against the base case.

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.

Scheduling Vehicles with Spatial Conflicts

When scheduling the movement of individual vehicles on a traffic network, one must ensure that they never get too close to one another. This is normally modelled by segmenting the network and forbidding two vehicles to occupy the same segment at the same time. This approximation is often insufficient or too restraining. This study develops and systematises the use of conflict regions to model spatial proximity constraints. By extending the classical disjunctive programming approach to job-shop scheduling problems, we demonstrate how conflict regions can be exploited to efficiently schedule the collective movements of a set of vehicles, in this case aircraft moving on an airport ground network. We also show how conflict regions can be used in the short-term control of vehicle speeds to avoid collisions and deadlocks. The overall approach was implemented in a software system for air traffic management at airports and successfully evaluated for scheduling and guiding airplanes during an extensive human in the loop simulation exercise for the Budapest airport. Through simulations, we also provide numerical results to assess the computational efficiency of our scheduling algorithm.

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.

Multiobjective optimization of the supply chain for the production of biomass-based fuels and high-value added products in Mexico

Due to the growing concern about the increase in the concentration of greenhouse gases in the atmosphere, as well as the depletion of oil reserves, biofuels have become increasingly important. Human activity, mainly in the industrial and transportation sectors, is one of the main causes of the above scenario. Bioethanol and biobutanol have emerged as possible solutions to this problem, however, their production is not economically competitive. Therefore, in addition to biofuel production, the generation of high value-added products in a biorefinery scheme is mandatory to improve the economic potential of the biomass-based industry. Mexico is a country with a high agricultural production, which implies a high production of agricultural waste. Therefore, this work proposes the development of a mathematical model for the supply chain of biofuels and bioproducts production. This model is a multi-objective optimization problem through generalized disjunctive programming. The GDP model is relaxed and solved in its equivalent mixed integer linear programming form using GAMS software. The model analyzes the production of bioethanol and/or biobutanol to satisfy at least 10% of the gasoline demand in the country, as well as the production of high value-added products; considering economic, environmental, economic and social objectives. Pareto-optimal solutions were obtained using the ε constraint method and selecting the best option by the Pythagorean theorem of the Eucladian distance. According to the results, biobutanol production is the process that maintains the greatest compromise between the four objective functions, achieving a balance. On the other hand, bioethanol production is only feasible if the water footprint is disregarded, as high values are obtained, which leads to a problem of water distribution to carry out the process.

Assembly planning by disjunctive programming and geometrical reasoning

The challenge in modeling and solving assembly planning problems lies in integrating combinatorial optimization techniques for finding efficient task sequences and resource assignments with geometrical reasoning to ensure the geometrical and technological feasibility of the assembly plans. This paper proposes a Benders decomposition approach that separates the macro-level planning problem, responsible for task sequencing and resource assignment, from micro-level validation on detailed geometrical and technological models. Feedback from the micro to the macro level is provided in the form of disjunctive constraints generated during search, which precludes the repeated occurrence of the collisions encountered in earlier iterations. A disjunctive programming approach is proposed to solve the macro-level planning problem with the added constraints. The efficiency of the approach is demonstrated both in industrial case studies and in computational experiments on generated problem instances.

Hybrid model generation for superstructure optimization with Generalized Disjunctive Programming

We propose a novel iterative procedure to generate hybrid models (HMs) within an optimization framework to solve design problems. HMs are based on first principle and surrogate models (SMs) and they may represent potential plant units embedded within a superstructure. We generate initial SMs with simple algebraic regression models and refine them by adding Gaussian Radial Basis Functions in three steps: initial SM refinement, domain exploration, and, after solving the optimal design problem, further domain exploitation, until the convergence criterion is fulfilled. The superstructure optimization problem is formulated with Generalized Disjunctive Programming and solved with the Logic-based Outer Approximation algorithm. We addressed methanol synthesis and propylene plant design problems. Compared to rigorous model-based optimal design, the proposed HMs gave the same configuration, objective function and decision variables with maximum relative differences of 1 and 7 %, respectively. A sensitivity analysis shows that the proposed strategy reduced CPU time by 33 %.

Simultaneous Synthesis and Design of Reaction–Separation–Recycle Processes Using Rigorous Models

Simultaneous synthesis and design of reaction–separation–recycle processes using rigorous models is highly desirable to improve process efficiency. However, it often leads to a large-scale highly challenging optimization problem. In this work, we propose a computationally efficient optimization framework for the challenging problem. The reactor and separator networks are modeled using the generalized disjunctive programming, which are reformulated into a highly nonconvex mixed-integer nonlinear programming (MINLP) formulation using the convex-hull method. To solve the complex MINLP model, a systematic solution approach is proposed in which an initialization strategy is first proposed to generate a feasible solution for a partially relaxed synthesis problem using the hybrid steady-state and time-relaxation optimization algorithm. A successive relaxed MINLP solution strategy is then adopted to solve the original model to local optimality. The computational results demonstrate that the proposed framework obtains better solutions with less computational effort, by ∼1 order of magnitude, than the existing algorithms.

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.

Optimal design of ethylene and propylene coproduction plants with generalized disjunctive programming and state equipment network models

In this work, we propose a superstructure optimization approach for the optimal design of an ethylene and propylene coproduction plant. We formulate a superstructure that embeds ethane and propane steam cracking technologies, propane dehydrogenation and olefin metathesis processes. We represent the superstructure with a Generalized Disjunctive Programming model, and solve the problem through a custom implementation of the Logic-based Outer Approximation algorithm in GAMS. We propose a state-equipment-network representation to model potential distillation trains, as well as alternative acetylene reactor configurations. Rigorous models are formulated for distillation columns, compressors, turboexpanders, vessels and several process equipment units. The objective function is to maximize the net present value. We analyze four international price scenarios for raw material and utility costs, while considering global ethylene and propylene prices. We obtain the optimal scheme for each case. Numerical results show that the combination of ethane steam cracking, olefin metathesis and ethylene dimerization is the most profitable configuration under low ethane price scenarios, whereas the combination of ethane and propane steam cracking together with propane dehydrogenation is the optimal solution when the propane price is on the order of ethane price.

A nanosatellite task scheduling framework to improve mission value using fuzzy constraints

Task scheduling is an effective approach to increase the value of a satellite mission, which leads to improved resource management and quality of service. This work improves the energy prediction model and a task scheduling formulation, expressed in integer programming to maximize the number of tasks performed in nanosatellite missions. A realistic battery model is introduced in the formulation to extend battery lifetime. This is achieved by a disjunctive program that makes battery charge and discharge more efficient. Furthermore, fuzzy constraints are designed to limit the current rates (for charge and discharge) and the depth of discharge for battery lifetime preservation. Each battery access is penalized in the objective function, thereby stimulating energy consumption to match energy input. For simulation purposes, the varying power input was based on two-line element data of the CubeSat FloripaSat-I, operating in an orbit with J2 perturbation and an attitude that keeps one face of the nanosatellite towards the Earth for the entire orbit, similar to a remote sensing mission. The effectiveness of the task-scheduling methodology was shown by means of simulated experiments of representative scenarios. With the improvements proposed here, a robust and realistic framework for optimal offline scheduling of nanosatellite missions is achieved.

Sparsity constrained optimization problems via disjunctive programming

In this paper, we consider the problem of minimizing a continuously differentiable function subject to sparsity constraints. We formulate this problem as an equivalent disjunctive constrained optimization program. Then, we extend some of the well-known constraint qualifications by using the contingent and normal cones of the sparsity set and show that these constraint qualifications can be applied to obtain the first-order optimality conditions. In addition, we give the first-order sufficient optimality conditions by defining a new generalized convexity notion. Furthermore, we present the second-order necessary and sufficient optimality conditions for sparsity constrained optimization problems. Finally, we provide some examples and special cases to illustrate the obtained results.

Modeling and Analysis of Organic Waste Management Systems in Centralized and Decentralized Supply Chains Using Generalized Disjunctive Programming

Although the decentralized and centralized strategies are widely known in today's supply chain realm, few studies have investigated their impacts on the bio-based waste management context. This study aims to formulate a multi-echelon structure for evaluating the influence of these supply chain strategies on optimal utilization of the biomass feedstocks. The proposed supply chain models, connecting the economic, social, and environmental aspects, tend to lower the total system costs, reduce the inadvertent effects of transportation and production processes, and respond to dynamic demands. The optimization problem is formulated as a mixed-integer linear programming model using generalized disjunctive programming to select the most robust integrated supply chain strategy. The results indicate that, although the centralized model leads to higher investment and operating costs than the decentralized model, centralization is still a more profitable alternative as well as being capable of the production of a broader portfolio of bio-based products.