Nonlinear Programming Formulation Research Articles

Spatiotemporal synchronous coupling algorithm for urban rail transit timetables design under dynamic passenger demand

This paper focuses on the problem of designing the schedule of urban rail transit systems under dynamic passenger demand considering limited train capacity and congestion. From the point of view of operators, the goal of timetable optimization consists of using as few as possible service trains to transport the arriving passengers to their destinations securely and quickly. On the other hand, passengers want to spend the minimum time travelling to their destinations, including the time they wait on platforms until they board the first and successive trains in case of transferring between lines. Due to the variability of passenger flows and the difficulty in exactly solving full-day timetabling problems in short computation times, it is difficult in practice to use the approaches proposed in the literature. Moreover, when the capacity of trains is considered and oversaturation conditions emerge, the problem becomes even more complex, converting into a challenge the design of schedules from an operational point of view. By considering fixed train dwell times at stations and train running times between stations, we first propose a non-linear programming formulation to model the problem of determining the most convenient interdeparture times of services at the first station of a line, aiming at minimizing the waiting time of passengers. Since the dynamic behaviour of passenger demand and the maximum sectional passenger flow are approximately synchronous, an innovative method, the spatiotemporal synchronization coupling algorithm, based on working with the time-varying maximum sectional passenger flow instead of the demand patterns is proposed to quickly solve in an approximate way the demand-driven timetabling problem. The proposed approach is applied to a real case of Chengdu Metro Line 1 in China. The results show the effectiveness of the algorithm, both in terms of computational effort and in terms of reducing the waiting time currently experienced by passengers.

A Recursive Conic Approximation for Solving the Optimal Power Flow Problem in Bipolar Direct Current Grids

This paper proposes a recursive conic approximation methodology to deal with the optimal power flow (OPF) problem in unbalanced bipolar DC networks. The OPF problem is formulated through a nonlinear programming (NLP) representation, where the objective function corresponds to the minimization of the expected grid power losses for a particular load scenario. The NLP formulation has a non-convex structure due to the hyperbolic equality constraints that define the current injection/absorption in the constant power terminals as a function of the powers and voltages. To obtain an approximate convex model that represents the OPF problem in bipolar asymmetric distribution networks, the conic relation associated with the product of two positive variables is applied to all nodes with constant power loads. In the case of nodes with dispersed generation, a direct replacement of the voltage variables for their expected operating point is used. An iterative solution procedure is implemented in order to minimize the error introduced by the voltage linearization in the dispersed generation sources. The 21-bus grid is employed for all numerical validations. To validate the effectiveness of the proposed conic model, the power flow problem is solved, considering that the neutral wire is floating and grounded, and obtaining the same numerical results as the traditional power flow methods (successive approximations, triangular-based, and Taylor-based approaches): expected power losses of 95.4237 and 91.2701 kW, respectively. To validate the effectiveness of the proposed convex model for solving the OPF problem, three combinatorial optimization methods are implemented: the sine-cosine algorithm (SCA), the black-hole optimizer (BHO), and the vortex search algorithm (VSA). Numerical results show that the proposed convex model finds the global optimal solution with a value of 22.985 kW, followed by the VSA with a value of 22.986 kW. At the same time, the BHO and SCA are stuck in locally optimal solutions (23.066 and 23.054 kW, respectively). All simulations were carried out in a MATLAB programming environment.

Dynamic coal blending under sustainability concerns: a case study from the Indian steel industry

PurposeIncreasing scarcity of natural resources and the adverse effects of unsustainable practices call for more and more efficient management strategies in the energy industry. The quality of the coke plays a significant role in the quality and durability of the output steel which is produced using the energy from the coal. This paper aims to investigate the dynamic coal blending problem under overall cost and coke quality constraints in the steel industry within a periodic cycle of operations.Design/methodology/approachConsidering the variability of the natural properties over a periodic cycle, this study proposes a multi-period mixed-integer non-linear programming formulation to optimize the total blending costs while taking various coke quality constraints into account. Besides, this study applies factorial design to investigate about the significant effect of coal proportions as well as improvement into the overall cost of blending.FindingsIn this case study, utilizing real data from a coal blending facility in India, through a factorial design, the authors obtain optimal desirable levels of coal proportions and their criticality levels towards the total cost of blending (TCB) or objective function. This analysis reflects the role of the coke quality constraints in the objective function value while characterizing the price of sustainability for the case study among other critical insights.Originality/valueObjective function (or TCB) includes basic coal cost, movement cost and environmental costs during the coal and coke processing at a coke-oven and blast furnace of steel industry. The price of sustainability provides managerial insights on that sacrifices the industry has to make in order to become more “sustainable”.

Optimal Control of Dynamical Systems using Calculus of Variations

This paper explores the application of calculus of variations techniques for optimizing the control of complex dynamical systems. Determining control strategies that minimize cost metrics and satisfy constraints is critical across engineering disciplines like robotics, aerospace, and process operations. Classical optimal control methods, such as Pontryagin's Maximum Principle, transform an optimal control problem into a calculus of variations framework amenable to analytical and numerical optimization. We present a unified framework for applying these techniques, enabling dynamical systems defined by ordinary and partial differential equations to be optimized by conversion into a nonlinear programming form. Analytical approaches provide theoretical guarantees on control performance while numerical methods, like direct collocation and transcription, enable large-scale optimal control problems to be efficiently solved. The connections between dynamical systems, calculus of variations, and modern numerical optimization methods establish a holistic methodology for control engineers and applied mathematicians to design optimal controllers for physical systems. Case studies on real-time trajectory optimization, adaptive path planning, and dynamic process operations demonstrate the efficacy of the proposed optimal control framework across robotics, aerospace, power systems, and other applications.

Optimal Control of Dynamical Systems using Calculus of Variations

This paper explores the application of calculus of variations techniques for optimizing the control of complex dynamical systems. Determining control strategies that minimize cost metrics and satisfy constraints is critical across engineering disciplines like robotics, aerospace, and process operations. Classical optimal control methods, such as Pontryagin's Maximum Principle, transform an optimal control problem into a calculus of variations framework amenable to analytical and numerical optimization. We present a unified framework for applying these techniques, enabling dynamical systems defined by ordinary and partial differential equations to be optimized by conversion into a nonlinear programming form. Analytical approaches provide theoretical guarantees on control performance while numerical methods, like direct collocation and transcription, enable large-scale optimal control problems to be efficiently solved. The connections between dynamical systems, calculus of variations, and modern numerical optimization methods establish a holistic methodology for control engineers and applied mathematicians to design optimal controllers for physical systems. Case studies on real-time trajectory optimization, adaptive path planning, and dynamic process operations demonstrate the efficacy of the proposed optimal control framework across robotics, aerospace, power systems, and other applications.

Multi-Objective Dispatch of PV Plants in Monopolar DC Grids Using a Weighted-Based Iterative Convex Solution Methodology

The design of an efficient energy management system (EMS) for monopolar DC networks with high penetration of photovoltaic generation plants is addressed in this research through a convex optimization point of view. The EMS is formulated as a multi-objective optimization problem that involves economic, technical, and environmental objective functions subject to typical constraints regarding power balance equilibrium, thermal conductor capabilities, generation source capacities, and voltage regulation constraints, among others, using a nonlinear programming (NLP) model. The main characteristic of this NLP formulation of the EMS for PV plants is that it is a nonconvex optimization problem owing to the product of variables in the power balance constraint. To ensure an effective solution to this NLP problem, a linear approximation of the power balance constraints using the McCormick equivalent for the product of two variables is proposed. In addition, to eliminate the error introduced by the linearization method, an iterative solution methodology (ISM) is proposed. To solve the multi-objective optimization problem, the weighted optimization method is implemented for each pair of objective functions in conflict, with the main advantage that in this extreme the Pareto front has the optimal global solution for the single-objective function optimization approach. Numerical results in the monopolar version of the IEEE 33-bus grid demonstrated that the proposed ISM reaches the optimal global solution for each one of the objective functions under analysis. It demonstrated that the convex optimization theory is more effective in the EMS design when compared with multiple combinatorial optimization methods.

Nonlinear programming reformulation of dynamic flux balance analysis models

Dynamic flux balance analysis (dFBA) models are very popular in systems biology because of the many possible applications. dFBA consists of dynamic mass balance equations that describe the concentration of external metabolites and an optimization model to compute the internal flux distribution of the cell. Here, we applied a nonlinear programming formulation for solving dFBA models. The ODE system is discretized using the orthogonal collocation technique and the optimization problem is replaced by its first-order optimality conditions. In addition, an adaptive mesh scheme is applied to improve performance by efficiently handling changes in the active set of constraints. This formulation ensures total differentiability of the model, which can be solved by large-scale solvers and take advantage of automatic differentiation packages. The new approach presents results equivalent to state-of-the-art methods and outperforms them in dynamic optimization problems by avoiding calling an external solver to handle the embedded optimization model.

Style: A new optimization model for Synthesis of uTility sYstems with steam LEvel placement

Moving industrial production to a more sustainable basis requires a step change in the efficiency of site steam and cogeneration systems, coupled with a switch away from fossil fuels to renewable energy sources. This paper presents the first step in the development of an optimization framework capable of identifying a roadmap to move utility systems to a more sustainable basis. Previous approaches to the design and optimization of steam and cogeneration systems have simplified the problem to the extent that many important practical issues have been neglected, restricting the scope of the options included. The use of grossly simplified models has been prompted by the mathematical difficulties of optimizing such complex energy systems. To overcome limitations in previous work and provide a sound basis for future developments, this paper proposes a new superstructure-based optimization model for the optimization of utility systems, accounting for optimum steam level placement. The latter is important for improving systems efficiency and reducing energy consumption. The optimization problem involves the selection of more realistic operating conditions of the steam mains (superheat conditions and pressure). The model accounts for water preheating, as well as superheating and de-superheating for process steam generation and use. Hot oil circuits are also included as hot utility option to overcome potential steam temperature and/or pressure limitations at high temperatures, and with it provide more flexibility in the framework. The general problem requires making several continuous and discrete decisions, where non-linearities and non-convexities from underlying physics and binary decisions exacerbate the complex nature of the problem, yielding a nonconvex Mixed Integer Non-Linear Programming (MINLP) formulation. However, MINLP formulations can become computationally intractable to solve. Thus, to guarantee tractability and fast conversion, the STYLE model develops a novel successive Mixed Integer Linear Programming formulation. The STYLE methodology is applied to two case studies to illustrate the advantages of the synthesis method and the benefits of optimizing steam levels for the reduction of overall energy consumption at industrial sites. The proposed approach addresses major shortcomings inherent in previous research and provides a foundation for future work to explore the next generation of sustainable utility systems.

Real-time optimization of train regulation and passenger flow control for urban rail transit network under frequent disturbances

Urban rail transit networks face frequent delays and oversaturation resulting from unexpected disturbances and high demands during rush hours, which inevitably leads to unstable train operations, poor service experiences, and potential accidents on crowded platforms. This paper explores the train regulation and passenger flow control strategies for large-scale urban rail transit networks. Specifically, we develop a mixed-integer nonlinear programming (MINLP) formulation to improve the punctuality and regularity in train operations, reduce the passenger waiting time, and alleviate the passenger flow burden of platforms, where the dynamic coupling interactions among network-wide passengers, trains and stations are systematically considered. To reduce the computational complexity and satisfy real-time requirements, we particularly devise a real-time optimization approach based on iterative nonlinear programming (INP) combined with Lagrangian relaxation (LR) under the rolling horizon (RH) framework, which effectively disposes of the intractable nonconvexity and constructs the desirable properties of decomposability and parallelism. The original problem is transformed into a sequence of line-level subproblems that can be solved quickly, so as to circumvent the computational burden. Moreover, to verify the effectiveness of our approach, we apply it to address numerical examples on a simplified network and the real-world Beijing Subway network. The realistic case studies illustrate that the proposed approach can reduce the train deviation from the original timetable, the passenger waiting time, and the passenger accumulation risk on the platform, by around 84.92%, 30.34% and 61.42% in comparison with the common rule-based strategy. Additionally, the computation time for the realistic large-scale experiment is acceptable for a real-time implementation.

A switched system approach for the direct solution of singular optimal control problems

Optimal control problems where the control variable appears linearly in the formulation lead to singular control arcs that introduce analytical and numerical complications during the solution process. In this work, the switching behavior (switching between singular and nonsingular control arcs) is exploited to solve singular optimal control problems using a direct method. The problem is reformulated as a switched system optimal control problem where each control arc defines a different subsystem, i.e., lower and upper limits and a singular arc. The switching between control arcs is handled using binary functions that depend on time. The switched system is embedded into a larger family by relaxing the binary functions to avoid integer variables. The variable switching times at which the control switches between arcs are mapped to fixed points in a new time domain using a time-scaling transformation. A direct transcription method along with control parametrization is applied to transform the embedded problem into a nonlinear programming formulation that contains few variables and constraints. A penalty term is introduced to obtain binary solutions for the optimal selection of control arcs. The proposed reformulation avoids the use of mesh refinement techniques and continuation methods, and allows to solve challenging problems in very short CPU times with high accuracy.

A two-stage Genetic Algorithm for joint coordination of spare parts inventory and planned maintenance under uncertain failures

The main problem of spare parts management is to maintain the minimal requirement of stock keeping units kept. This research proposes a joint optimization model of multi-item multi-period spare parts inventory management and planned maintenance under uncertain failures in order to balance inventory cost and spare parts availability. This paper presents a Mixed Integer Non-linear Programming formulation of the inventory optimization model under a minimum and maximum inventory policy with stock review intervals. Some studies in the literature have considered aggregating spare parts inventory management as they assume that it will reduce the ordering cost. We consider both independent and aggregate spare parts inventory policies and then combine the formulation with the predictive maintenance interval, which is a replacement action due to uncertain failures under predefined distribution. Furthermore, a novel two-stage Genetic Algorithm is proposed as a sim-heuristic approach to deal with the non-linearity, combinations, and stochasticity of the problem and solve large-scale instances. In the end, we perform a computational study on some instances and a real-world case study to demonstrate the proposed approach’s effectiveness and efficiency. The computational study shows that the independent policy results in lower cost than the aggregate policy, and the proposed Genetic Algorithm performs efficiently for large-scale problems.

An Analysis of the Methane Cracking Process for CO2-Free Hydrogen Production Using Thermodynamic Methodologies

The thermal cracking process of methane does not present the emissions of polluting gases, forming only hydrogen with a high degree of purity and solid carbon that can be commercialized for other industrial purposes globally. Thermodynamic methodologies based on Gibbs energy minimization and entropy maximization are used in the present study to simulate operating conditions of isothermal and adiabatic reactors, respectively. The chemical equilibrium and combined phases problem were written in a non-linear programming form and optimized with the GAMS software using the CONOPT 3 solver. The results obtained by the methodology described in this study present a good agreement with the data reported in the literature, with mean relative deviations lower than 1.08%. High temperatures and low pressures favor the decomposition of methane and the formation of products. When conditioned in an isothermal reactor, total methane conversions are obtained at temperatures above 1200 K at 1 bar. When conditioned to an adiabatic reactor, due to the lack of energy support provided by the isothermal reactor and taking into account that it is an endothermic process, high methane-conversion rates are obtained for temperatures above 1600 K at 1 bar. As an alternative, the combined effects of the addition of hydrogen to the feed combined with a system of extreme pressure variation indicate a possibility of conducting the thermal cracking process of methane in adiabatic systems. Setting the CH4/H2 ratio in the system feed at 1:10 at 1600 K and 50 bar, following severe depressurization through an isentropic valve, varying the pressure from 50 to 1 bar, the methane conversion varies from 0 to 94.712%, thus indicating a possible operational conformation for the process so that the amount of carbon generated is not so harmful to the process, taking into account that the formation of the same occurs only after the reaction and heating processes. Under the same operating conditions, it is possible to use about 40.57% of the generated hydrogen to provide energy for the process to occur.

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.

Concurrent Scheduling of Machines and AGVS in Multi-Machine FMS with Alternative Routing Using Symbiotic Organisms Search Algorithm

This paper addresses machines and automated guided vehicles (AGVs) simultaneous scheduling with alternative machines in a multi-machine flexible manufacturing system (FMS) to produce the best optimal sequences for the minimization of makespan (MKSN). This problem is highly complex to solve because it involves the selection of machines for job operations (jb-ons), the sequencing of jb-ons on the machines, the assignment of AGVs, and associated trips such as AGVs’ deadheaded trip and loaded trip times to jb-ons. This paper offers a nonlinear mixed integer programming (MIP) formulation for modeling the problem and the symbiotic organisms search algorithm (SOSA) for solving the problem. For verification, a manufacturing company’s industrial problem is employed. The results show that SOSA outperforms the existing methods and the Jaya algorithm, and using alternate machines for the operations can reduce the MKSN.

Maintenance scheduling optimisation of Reverse Osmosis Networks (RONs) via a multistage Optimal Control reformulation

State-of-the-art approaches for membrane cleaning scheduling have focused on the Mixed-Integer Nonlinear Programming (MINLP) formulation so far, a strategy leading to a combinatorial problem that does not capture accurately the dynamic behaviour of the system. In this work, the Reverse Osmosis (RO) cleaning scheduling problem is solved using a novel approach based on the Multistage Integer Nonlinear Optimal Control Problem (MSINOCP) formulation. The approach produces an automated solution for the membrane cleaning scheduling, which also obviates the need for any form of combinatorial optimisation.Two different simulations, for 26 and 52 periods of operation (each period with a duration of one week), are carried out to illustrate the application of the proposed framework and the total cost is 1.17 and 2.48107€, respectively. The RO network configuration considers 2 stages, each with 3 individual RO modules. The results show evidently that the new proposed solution framework can solve successfully this type of problems, even for large scale configurations, long time horizons and arbitrary realistic complexity of the underlying dynamic model of the RO process considered.

Uncertainty Propagation Based MINLP Approach for Artificial Neural Network Structure Reduction

The performance of artificial neural networks (ANNs) is highly influenced by the selection of input variables and the architecture defined by hyper parameters such as the number of neurons in the hidden layer and connections between network variables. Although there are some black-box and trial and error based studies in the literature to deal with these issues, it is fair to state that a rigorous and systematic method providing global and unique solution is still missing. Accordingly, in this study, a mixed integer nonlinear programming (MINLP) formulation is proposed to detect the best features and connections among the neural network elements while propagating parameter and output uncertainties for regression problems. The objective of the formulation is to minimize the covariance of the estimated parameters while by (i) detecting the ideal number of neurons, (ii) synthesizing the connection configuration between those neurons, inputs and outputs, and (iii) selecting optimum input variables in a multi variable data set to design and ensure identifiable ANN architectures. As a result, suggested approach provides a robust and optimal ANN architecture with tighter prediction bounds obtained from propagation of parameter uncertainty, and higher prediction accuracy compared to the traditional fully connected approach and other benchmarks. Furthermore, such a performance is obtained after elimination of approximately 85% and 90% of the connections, for two case studies respectively, compared to traditional ANN in addition to significant reduction in the input subset.

Minimum cost b-matching problems with neighborhoods

In this paper, we deal with minimum cost b-matching problems on graphs where the nodes are assumed to belong to non-necessarily convex regions called neighborhoods, and the costs are given by the distances between points of the neighborhoods. The goal in the proposed problems is twofold: (i) finding a b-matching in the graph and (ii) determining a point in each neighborhood to be the connection point among the edges defining the b-matching. Different variants of the minimum cost b-matching problem are considered depending on the criteria to match neighborhoods: perfect, maximum cardinality, maximal and the a–b-matching problems. The theoretical complexity of solving each one of these problems is analyzed. Different mixed integer non-linear programming formulations are proposed for each one of the considered problems and then reformulated as Second Order Cone formulations. An extensive computational experience shows the efficiency of the proposed formulations to solve the problems under study.

Downlink Precoding for DP-UPA FDD Massive MIMO via Multi-Dimensional Active Channel Sparsification

In this paper, we consider user selection and downlink precoding for an over-loaded single-cell massive multiple-input multiple-output (MIMO) system in frequency division duplexing (FDD) mode, where the base station is equipped with a dual-polarized uniform planar array (DP-UPA) and serves a large number of single-antenna users. Due to the absence of uplink-downlink channel reciprocity and the high-dimensionality of channel matrices, it is extremely challenging to design downlink precoders using closed-loop channel probing and feedback with limited spectrum resource. To address these issues, a novel methodology – active channel sparsification (ACS) – has been proposed recently in the literature for uniform linear array (ULA) to design sparsifying precoders, which substantially reduces channel feedback overhead. Pushing forward this line of research, we aim to facilitate the potential deployment of ACS in practical FDD massive MIMO systems, by extending it from ULA to DP-UPA with explicit user selection and making the current ACS implementation simplified. To this end, by leveraging Toeplitz matrix theory, we start with the spectral properties of channel covariance matrices from the lens of their matrix-valued spectral density function. Inspired by these properties, we extend the original ACS using scalar-weight bipartite graph representation to the matrix-weight counterpart. Building upon such matrix-weight bipartite graph representation, we propose a multi-dimensional ACS (MD-ACS) method, which is a generalization of original ACS formulation and is more suitable for DP-UPA antenna configurations. The nonlinear integer program formulation of MD-ACS can be classified as a generalized multi-assignment problem (GMAP), for which we propose a simple yet efficient greedy algorithm to solve it. Simulation results demonstrate the performance improvement of the proposed MD-ACS with greedy algorithm over the state-of-the-art methods based on the QuaDRiGa channel models.

Flow-Capture Location Model with Link Capacity Constraint Over a Mixed Traffic Network

Abstract This paper constructs and settles a charging facility location problem with the link capacity constraint over a mixed traffic network. The reason for studying this problem is that link capacity constraint is mostly insufficient or missing in the studies of traditional user equilibrium models, thereby resulting in the ambiguous of the definition of road traffic network status. Adding capacity constraints to the road network is a compromise to enhance the reality of the traditional equilibrium model. In this paper, we provide a two-layer model for evaluating the efficiency of the charging facilities under the condition of considering the link capacity constraint. The upper level model in the proposed bi-level model is a nonlinear integer programming formulation, which aims to maximize the captured link flows of the battery electric vehicles. Moreover, the lower level model is a typical traffic equilibrium assignment model except that it contains the link capacity constraint and driving distance constraint of the electric vehicles over the mixed road network. Based on the Frank-Wolfe algorithm, a modified algorithm framework is adopted for solving the constructed problem, and finally, a numerical example is presented to verify the proposed model and solution algorithm.

Robust supplier-selection and order-allocation in two-echelon supply networks: A parametric tolerance design approach

In this paper, we study the integrated supplier-selection and order-allocation problem in a two-echelon make-to-order supply chain motivated by the belt conveyor industry. In our setting, the Original Equipment Manufacturer (OEM) can source its single product from multiple suppliers in the first echelon who themselves source the sub-components of the product from multiple other suppliers in the second echelon. Within the second echelon, the suppliers fall into clusters of product supply networks based on their capability to produce a part. OEM in our setting is interested in determining the optimal tolerances for its supply network members to operate so as to minimize the variability introduced into the total supply chain costs. To obtain the optimal supplier-selection and order-allocations under a periodic review demand policy, we model the problem as a mixed-integer non-linear programming (MINLP) formulation. We further incorporate the Taguchi Method of Tolerance Design (TMTD) into our model, leading to robustness of our solutions while maintaining optimality in supplier-selection and order-allocation decisions. We test our model using a real-world case study and derive critical insights. Our numerical results demonstrate a 51% improvement in coefficient of variation of the objective function when our method is applied to the real life case study. Additionally, we show that as the level of confidence of the decision-maker on the range for a critical parameter increases, the gains from our methodology significantly increase.