Nonlinear Programming Formulation Research Articles

Optimizing drinking water distribution system operations

The operation of Water Distribution Systems (WDS) is often complex, especially when considering the changes in tariffs throughout the day. The cost of energy in these systems can reach 30% of total operating costs and its careful management can represent increased efficiency. The optimization of WDS scheduling operation appears as an effective method to reduce operating costs while ensuring a good service level to the population. In this paper we propose a new linear relaxation for a non-linear integer programming formulation for WDS in order to optimize its operation costs. This study makes five main contributions. First, our formulation includes new aspects related to the state of the system when the tanks are full, that were not considered before in mathematical programming models. Second, our linearization technique includes a variable number of breakpoints, resulting in significantly fewer binary variables for a given error level. Third, our relaxation reduces the search space of the solutions. Fourth, we have outperformed the best results for three benchmark instances from the literature. Lastly, we also provide a larger new real-life instance with specific conditions of energy tariffs, obtained from the WDS from the city of Florianópolis, southern Brazil, significantly outperforming the current solution employed by the utility provider.

Models for the two‐dimensional rectangular single large placement problem with guillotine cuts and constrained pattern

AbstractIn this paper, we address the constrained two‐dimensional rectangular guillotine single large placement problem (2D_R_CG_SLOPP). This problem involves cutting a rectangular object to produce smaller rectangular items from orthogonal guillotine cuts. In addition, there is an upper limit on the number of copies that can be produced of each item type. To model this problem, we propose a new pseudopolynomial integer nonlinear programming (INLP) formulation and obtain an equivalent integer linear programming (ILP) formulation from it. Additionally, we developed a procedure to reduce the numbers of variables and constraints of the integer linear programming (ILP) formulation, without loss of optimality. From the ILP formulation, we derive two new pseudopolynomial models for particular cases of the 2D_R_CG_SLOPP, which consider only two‐staged or one‐group patterns. Finally, as a specific solution method for the 2D_R_CG_SLOPP, we apply Benders decomposition to the proposed ILP formulation and develop a branch‐and‐Benders‐cut algorithm. All proposed approaches are evaluated through computational experiments using benchmark instances and compared with other formulations available in the literature. The results show that the new formulations are appropriate in scenarios characterized by few item types that are large with respect to the object's dimensions.

Multi-objective teaching-learning-based meta-heuristic algorithms to solve multi-skilled project scheduling problem

In this paper, a novel multi-objective multi-skilled resource-constrained project scheduling problem based on learning effect is presented in which efficiency of the workforce is proportional with the time they spend to execute that skill. A mixed integer nonlinear programming formulation is presented to optimize conflicting objectives. Two hybrid multi-objective teaching–learning based optimization (TLBO) algorithms are developed with a new solution structure based on primary concepts of multi-objective particle swarm optimization and multi-objective invasive weeds optimization algorithms to find near optimal solutions of the model’s complicated instances in a reasonable computational time. The proposed algorithms encompass a novel solution structure that satisfies all the constraints and generates feasible solutions. The classical multi-objective teaching-learning based optimization algorithm and NSGA II are developed with the same chromosome structure to validate performance of the developed algorithms. In addition, all the algorithms are employed to solve small- and large-scale benchmark instances that are available on literature of project scheduling problems as well as 30 randomly generated test problems. Taguchi grey relational analysis is used to tune parameters of the algorithms and enhance their performance for generating more qualified solutions. In addition, statistical test is employed to compare performance of the proposed algorithms in terms of some multi-objective metrics. Finally, hybrid multi-attribute decision making approach is employed to prioritize algorithms and determine the algorithm with higher performance on solving small, medium and large scale instances.

Optimal Design of Heat-Integrated Water Allocation Networks

Industrial operations consume energy and water in large quantities without accounting for potential economic and environmental burdens on future generations. Consumption of energy (mainly in the form of high pressure steam) and water (in the form of process water and cooling water) are essential to all processes and are strongly correlated, which requires development of systematic methodologies to address their interconnectivity. To this end, the subject of heat-integrated water allocation network design has received considerable attention within the research community in recent decades while further growth is expected due to imposed national and global regulations within the context of sustainable development. The overall mathematical model of these networks has a mixed-integer nonlinear programming formulation. As discussed in this work, proposed models in the literature have two main difficulties dealing with heat–water specificities, which result in complex formulations. These difficulties are addressed in this work through proposition of a novel nonlinear hyperstructure and a sequential solution strategy. The solution strategy is to solve three sub-problems sequentially and iteratively generate a set of potential solutions through the implementation of integer cut constraints. The novel mathematical approach also lends itself to an additional innovation for proposing multiple solutions balancing various performance indicators. This is exemplified with both a literature test case and an industrial-scale problem. The proposed solutions address a variety of performance indicators which guides decision-makers toward selecting the most appropriate configuration(s) among a large number of potential possibilities. Results exhibit that despite having a sequential solution strategy, better performance can be reached compared to previous approaches with the additional benefit of providing many potential solutions for further consideration by decision-makers to select the best case-specific solution.

Design and optimization of low thrust transfer trajectory for engineering constraints

Multimode electric propulsion system is widely used in deep space exploration. Its thrust is output in stages with the change of input power, which also means that continuous power changes lead to discontinuous thrust changes. In the design and optimization of low-thrust trajectory, the discontinuity of thrust is often neglected and regarded as a continuous parameter. The optimal design of low-thrust transfer trajectory constrained by multimode electric propulsion is studied, and the influence of thrust grading characteristics on the trajectory is analyzed in this paper. A specific optimization strategy is proposed, including three main steps: model discretization based on direct method, fast generation of initial value based on particle swarm optimization (PSO), and optimization based on sequential quadratic programming (SQP). The dynamic model and optimization model of low-thrust transfer trajectory is established. Then, the optimization model is discretized based on the direct method and the original optimal control problem is transformed into a parameter optimization problem in the form of nonlinear programming (NLP). A hybrid optimization algorithm based on PSO and SQP is proposed to solve the new NLP problem. PSO is used to search the initial value efficiently and guarantee the global performance of the initial value and SQP is used for local optimization based on initial value, which can give full play to the excellent global optimization performance of PSO and the outstanding local optimization performance of SQP. The effectiveness of the proposed optimization strategy is verified by a simulation example of the main belt asteroid Metis (1974 QU2) rendezvous mission with once Mars gravity assist. The results show that the optimal control curve obtained is in accordance with the optimal control form of bang-bang control derived from the theory, and the thruster has just two situations at any time: shutdown or startup with maximum thrust. In the working power range, increasing the number of thrust stages can improve the rate of energy utilization, thereby improving the efficiency of thruster, shortening the startup time and reducing fuel consumption. Meanwhile, ineffective grading should be avoided in the non-working power range.

Mixed-integer bilinear and piecewise-linear models for designing switching strategies of multilevel power converters

With the adoption of multilevel converters in the industry, the importance of their optimal operation has increased. Able to operate with more than three voltage levels, multilevel converters are often modulated by vectors defined in time, which can be optimized to reduce the number of commutations or total harmonic distortion. This papers presents two formulations that do not use trigonometric functions for selecting optimal switching angles, focusing on the minimization of the current weighted total harmonic distortion. The first relies on bilinear terms to yield a mixed-integer nonlinear programming (MINLP) formulation that can be tackled with MINLP algorithms. The second uses piecewise-linear approximation to enable the application of mixed-integer quadratic programming. Computational results from the application of these formulations are reported, in conjunction with a simulation of the optimal angles that were computed for a 5-level Neutral-Point Clamped converter.

Multi-level departments-to-offices assignment with different room types

When a company moves to a new building, each department has to receive the room capacities it requires. Construction of large office buildings can take multiple years while organizational structures are subject to frequent change. At the time of building design, often only a rough capacity planning is carried out. An explicit assignment of departments to rooms is postponed until the building is almost finished. In addition to hard constraints with regard to the required number and type of rooms each department needs, various preferences have to be considered for an efficient level and room assignment. To support intra- and interdepartmental interactions, we consider the distances between the assigned rooms of each department and the proximity of related departments. Additionally, preferences with respect to the spatial location in the building are taken into account. We propose a new non-linear mathematical programming formulation for the multi-level departments-to-offices assignment problem with different room types. Standard solvers cannot produce relevant solutions from a practical perspective in an acceptable time. To solve a realistic problem, an innovative heuristic approach is proposed. Computational results across a large set of test instances demonstrate the general applicability of our approach. Finally, results and trade-offs between the different objectives are discussed extensively in a case study.

Automated learning of chemical reaction networks

The identification of kinetic models from process measurements relies on obtaining expensive data from experimental designs in order to estimate kinetic rate parameters and discern which mechanisms fit the data most appropriately. This paper proposes an approach that utilizes a mixed-integer nonlinear programming formulation to estimate and identify kinetic rate parameters from a postulated superset of reactions. The proposed methodology can be applied to steady state systems, or dynamic systems through explicit derivative estimation. An adaptive design of experiments methodology is proposed to facilitate intelligent sampling across the experimental domain. The proposed methodology is applied to three problem settings: isothermal CSTRs where initial concentrations are varied to learn the reaction network, dynamic chemical looping combustion reactors where semi-empirical mechanistic models are used to model catalyst conversion, and an expanded example that considers alternative reaction mechanisms and stoichiometric relationships and applies the proposed adaptive sampling technique to a higher dimensional domain.

Scenario tree reduction for optimisation under uncertainty using sensitivity analysis

This work addresses the optimal management of a system through a two-stage stochastic Non-Linear Programming (NLP) formulation. This approach uses a scenario-based mathematical formulation to tackle uncertain information. Accurate representation of uncertainty usually involves increased number of scenarios, which may result in large-scale optimisation models. Thus, the proposed formulation aims to reduce the number of scenarios through a sensitivity analysis approach. The proposed model investigates the use of scenario reduction techniques to reduce computational requirements while maintaining good quality of the final optimal solution.

Optimal site selection for modular nuclear power plants

Small Modular Reactor (SMR) is a small, compact version of a conventional Nuclear Power Plant (NPP) and holds much promise for the future. Installing SMRs in a region requires a series of carefully planned steps out of which site selection is a critical one. This paper proposes a novel mathematical model for evaluating potential land sites for their suitability of hosting a modular NPP. Most existing decision making tools for NPP site selection rely on qualitative information from the experts. These tools require significant resources and therefore can only be applied to a limited number of selected sites. The proposed model is a Mixed Integer Non-linear Programming (MINLP) formulation which considers a variety of factors like cost, cooling water availability, earthquake risk, etc. to identify best locations for the SMRs in a distributed power system. A case study based on a virtual EIP simulator is conducted to demonstrate the capabilities of the model by finding the optimal locations of modular NPPs. The model offers a preliminary platform for carrying out further extensive studies.

Production scheduling and linear MPC: Complete integration via complementarity conditions

Competitive global market conditions and the availability of real-time pricing data call for agile and flexible operations in the chemical process industries. As the operating paradigm shifts, process scheduling decisions may consider a shorter time scale, and interactions with the process control layer become significant. The integration of these two decision-making layers is an active area of research. In this work, we propose a novel framework for the integration of production scheduling and linear model predictive control. Our approach explicitly represents the closed-loop response of the system by embedding the KKT conditions of the controller in a nonlinear programming formulation of the optimal scheduling problem. We present two case studies which demonstrate that control-informed scheduling leads to superior realized performance compared with a hierarchical, sequential decision-making structure.

Integrated resource optimization with WDM-based fronthaul for multicast-service beam-forming in massive MIMO-enabled 5G networks

Massive MIMO (mMIMO) is a technology with high potential in future 5G radio access networks (RAN). As a crucial feature in mMIMO, beam-forming provides a remarkable enhancement for the wireless transmission. However, beam-forming introduces extra wireless resource and fronthaul bandwidth consumption for utilizing multiple antennas to transmit the same data. Moreover, this problem will be highlighted in the multicast-service beam-forming because more identical data will be transmitted in networks. In this paper, we focus on the entire resource consumption of a single cell for the multicast-service beam-forming in next-generation RAN and propose a flexible wavelength-division-multiplexing-based fronthaul to support the high-capacity and resilient transport. A mixed-integer nonlinear programming formulation and two heuristic algorithms are developed to minimize the utilized optical bandwidth, radio resource blocks and antennas. Simulation comparisons are performed among different resource allocation strategies. Numerical results demonstrate that our strategies can efficiently reduce the total resource consumption.

Optimal exact designs of experiments via Mixed Integer Nonlinear Programming

Optimal exact designs are problematic to find and study because there is no unified theory for determining them and studying their properties. Each has its own challenges and when a method exists to confirm the design optimality, it is invariably applicable to the particular problem only. We propose a systematic approach to construct optimal exact designs by incorporating the Cholesky decomposition of the Fisher Information Matrix in a Mixed Integer Nonlinear Programming formulation. As examples, we apply the methodology to find D- and A-optimal exact designs for linear and nonlinear models using global or local optimizers. Our examples include design problems with constraints on the locations or the number of replicates at the optimal design points.

Optimization of natural resource management: Application to French copper cycle

An innovative resource flow optimization model is proposed that aims at helping decision makers to choose the best resource management policies at national and international scales. This model has been developed and validated on French copper cycle, but can easily be extended to any country or metal. A mathematical formulation of all the in- and outflows of the resource, as well as stocks in the technosphere and waste management, including recycling, has been developed. The complexity of resource cycle led to the introduction of 47 decision variables and resulted in a Mixed-Integer Non-Linear Programming formulation. The resource cycle efficiency has been assessed through four indicators regarding costs, greenhouse gas emissions, energy consumption and resource losses. The NSGA II genetic algorithm methodology has been selected as an optimization strategy. Monobjective optimizations have firstly been conducted for each of the four criteria: the results showed that very different management strategies are needed depending on the targeted criteria. Multiobjective optimizations have thus been conducted coupled with the decision support tool TOPSIS to find an optimal compromise solution. As expected, the selected solution highlights the importance of developing a recycling channel for Waste from Electrical and Electronic Equipment in France.

Kaibel column: Modeling, optimization, and conceptual design of multi-product dividing wall columns

In this work, we present the modeling, optimization, and conceptual design of a dividing wall column for the separation of four products, commonly referred to in the literature as a Kaibel column, by implementing three different formulations: an NLP, an MINLP, and a GDP formulation. For its solution, we propose a rigorous tray-by-tray model and compared it to results from a commercial software, followed by its reformulation to include a mixed-integer nonlinear programming and a general disjunctive programming formulation to respond to the conceptual design problem attached to these complex configurations. Considering the proposed rigorous model and the two formulations, the Kaibel column is solved, obtaining four high-purity products and new optimal tray locations for the feed and two side product streams, when the mixed-integer nonlinear programming formulation is applied. The use of these optimally located side streams showed reductions in the energy consumption when compared to cases were non-optimal fixed tray locations are used. When the general disjunctive programming problem was solved, the minimum number of trays needed in the main column and dividing wall are obtained, showing a great reduction of the remixing effects in the Kaibel column, and with that, a more energy efficient configuration. The models were coded in Pyomo using the solver IPOPT for the solution of the nonlinear programming problem, the solver Bonmin for the solution of the mixed-integer nonlinear programming problem, and GDPopt for the solution of the general disjunctive programming optimization problem.

Scheduling Upstream Operations at Inland Petroleum Refineries Using a Precedence-Based Formulation

Several types of crude oils arrive at inland oil refineries to be transformed into different intermediate and finished products. Incoming crude oils should be cleverly blended to meet some quality specifications before processing them in the crude distillation units (CDUs). This requires a careful allocation of the available crude oils to the refinery tanks and a proper sequence of the tanks feeding the same CDU. This work introduces a mixed-integer nonlinear programming (MINLP) formulation that uses general-precedence sequencing variables to choose the best ordering of operations in every tank. By using a rigorous objective function, the MINLP model provides the exact operating cost of the best solution found. The replacement of nonlinear component balances by tailor-made linear constraints in the MINLP leads to a tight mixed-integer linear (MILP) model that usually yields a good MINLP feasible point. A nonlinear programming (NLP) formulation that results by fixing the 0–1 variables to their MILP-values ...

An exact algorithm for the minimum squared load assignment problem

In this study, we consider an assignment problem with the objective to minimize the sum of squared loads over all agents. We provide mixed integer nonlinear and linear programming formulations of the problem and present a branch and bound algorithm for their solution. The results of our computational experiment have shown the satisfactory behavior of our branch and bound algorithm.

Ordered p-median problems with neighbourhoods

In this paper, we introduce a new variant of the p-median facility location problem in which it is assumed that the exact location of the potential facilities is unknown. Instead, each of the facilities must be located in a region around their initially assigned location (the neighborhood). In this problem, two main decisions have to be made simultaneously: the determination of the potential facilities that must be open to serve the customers’ demand and the location of the open facilities in their neighborhoods, at global minimum cost. We present several mixed integer non-linear programming formulations for a wide family of objective functions which are common in Location Analysis: ordered median functions. We also develop two math-heuristic approaches for solving the problem. We report the results of extensive computational experiments.

New algorithmic framework for conditional value at risk: Application to stochastic fixed-charge transportation

This paper introduces a new algorithmic scheme for two-stage stochastic mixed-integer programming assuming a risk averse decision maker. The focus is the minimization of the conditional value at risk for a hard combinatorial optimization problem. Some properties of a mixed-integer non-linear programming formulation for conditional value at risk are studied as well as their algorithmic implications. This yields to a procedure for obtaining lower and upper bounds on the optimal value of the problem that may lead to an optimal solution. The new developments are applied to a fixed-charge transportation problem with stochastic demand, and they are computationally tested. The corresponding results are thoroughly presented and discussed.

Synthesis of Constraints for Mathematical Programming With One-Class Genetic Programming

Mathematical programming (MP) models are common in optimization of real-world processes. Models are usually built by optimization experts in an iterative manner: an imperfect model is continuously improved until it approximates the reality well-enough and meets all technical requirements (e.g., linearity). To facilitate this task, we propose a genetic one-class constraint synthesis method (GOCCS). Given a set of exemplary states of normal operation of a business process, GOCCS synthesizes constraints in linear programming or nonlinear programming form. The synthesized constraints can be then paired with an arbitrary objective function and supplied to an off-the-shelf solver to find optimal parameters of the process. We assess GOCCS on three families of MP benchmarks and conclude promising results. We also apply it to a real-world process of wine production and optimize that process.