Nonlinear Programming Formulation Research Articles

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.

Techno-Economic Optimization of a Low-Temperature Organic Rankine System Driven by Multiple Heat Sources

This study presents the optimization of organic Rankine cycle (ORC) which utilizes low temperature waste heat from the aluminum production process and two low temperature renewable energy sources, solar thermal energy and geothermal energy. As geothermal energy is present at lower temperature level compared to the other two heat sources, two separate ORC cycles are considered. Optimization of the proposed system is performed based on a non-linear programming (NLP) formulation by maximizing the thermodynamic and economic performance of the system. The main variables considered in the model include temperature, pressure, flowrate, mass enthalpy, and energy flows of all the streams in the system. To optimize the variables in the system, correlations were developed, which were formulated as NLP models and optimized by minimizing the sum of least squares. The results show that most of the generated power output can be provided by the waste heat, while the lowest by the solar energy due to the relatively low average solar irradiance at considered location. When monthly time periods are considered, the highest electricity production is generated by the working fluid R1245fa in June, and amounts to 830.4 kW from waste heat, 246.5 kW from geothermal energy and 149.4 kW from solar energy. The proposed system is economically feasible for all three studied working fluids and the discount rates of 2% and higher. The final conclusions indicate that the proposed ORC system utilizing waste heat, geothermal and solar thermal energy, can generate power in a more sustainable way.

A Comprehensive Approach for an Approximative Integration of Nonlinear-Bivariate Functions in Mixed-Integer Linear Programming Models

As decentralized energy supply units, microgrids can make a decisive contribution to achieving climate targets. In this context, it is particularly important to determine the optimal size of the energy components contained in the microgrids and their optimal operating schedule. Hence, mathematical optimization methods are often used in association with such tasks. In particular, mixed-integer linear programming (MILP) has proven to be a useful tool. Due to the versatility of the different energetic components (e.g., storages, solar modules) and their special technical characteristics, linear relationships can often only inadequately describe the real processes. In order to take advantage of linear solution techniques but at the same time better represent these real-world processes, accurate and efficient approximation techniques need to be applied in system modeling. In particular, nonlinear-bivariate functions represent a major challenge, which is why this paper derives and implements a method that addresses this issue. The advantage of this method is that any bivariate mixed-integer nonlinear programming (MINLP) formulation can be transformed into a MILP formulation using this comprehensive method. For a performance comparison, a mixed-integer quadratic constrained programming (MIQCP) model—as an MINLP special case—is applied and transformed into a MILP, and the solution of the transformed problem is compared with the one of the MIQCP. Since there are good off-the-shelf solvers for MIQCP problems available, the comparison is conservative. The results for an exemplary microgrid sizing task show that the method delivers a strong performance, both in terms of approximation error (0.08%) and computation time. The method and its implementation can serve as a general user-tool but also as a basis for further methodological developments and research.

Dynamical intervention planning against COVID-19-like epidemics.

COVID-19 has got us to face a new situation where, for the lack of ready-to-use vaccines, it is necessary to support vaccination with complex non-pharmaceutical strategies. In this paper, we provide a novel Mixed Integer Nonlinear Programming formulation for fine-grained optimal intervention planning (i.e., at the level of the single day) against newborn epidemics like COVID-19, where a modified SIR model accounting for heterogeneous population classes, social distancing and several types of vaccines (each with its efficacy and delayed effects), allows us to plan an optimal mixed strategy (both pharmaceutical and non-pharmaceutical) that takes into account both the vaccine availability in limited batches at selected time instants and the need for second doses while keeping hospitalizations and intensive care occupancy below a threshold and requiring that new infections die out at the end of the planning horizon. In order to show the effectiveness of the proposed formulation, we analyze a case study for Italy with realistic parameters.

Penyelesaian Masalah Pemrograman Kuadratik Menggunakan Metode Beale

Quadratic programming is a special form of nonlinear programming which has the general form of the objective function in the form of a quadratic functions and the constraints in the form of linear functions. One of the methods used to solve the quadratic programming model is Beale’s method. This method is a modification of the simplex method for linear programming problems. This study aims to determine the optmal results of paddy production data in Balikpapan city using Beale’s method. The quadratic programming model was formed using the least squares method with the objective functions by selecting two types of plants,namely lowland paddy and upland paddy. Furthermore, the quadratic programming model formed that is formed is solved using Beale’s method. The data used is data on the paddy production of balikpapan city in 2005-2019. Based on the calculatios, the objective function is with the constraint functions and . After calculating using Beale’s method, the optimal result for the maximum yield of lowland paddy and upland paddy is quintals with a harvested area of hectares of lowland paddy and a harvested paddy of hectares of upland paddy.

Dynamic simulation, optimisation and economic analysis of fed-batch vs. perfusion bioreactors for advanced mAb manufacturing

The variation of antigen binding sites on monoclonal antibodies (mAbs) enables manufacturing of the latter for targeted treatment of a variety of ailments, including cancer types and autoimmune diseases. Fed-batch operation of biochemical reactors used to manufacture mAbs grants the ability to manipulate cell dynamics via intermittent feeding of nutrients/substrates. Regulatory authority agendas (e.g. the U.S. Food & Drug Administration) consider the development of continuous technologies (e.g. perfusion reactors) as essential towards improved technoeconomic potential and biopharmaceuticals quality control. This paper presents dynamic simulations for multiple operational scenaria of a fed-batch and a perfusion reactor, in which a hybridoma cell culture is used to secrete mAb glycoforms. A novel dynamic optimisation of the fed-batch as well as perfusion reactors has also been conducted towards maximising the total mAb mass produced. Nonlinear Programming (NLP) formulations with the use of APOPT and IPOPT solvers have been employed to derive the manipulation, state and controlled variable trajectories, to achieve the mAb production maximisation objective. The detailed technoeconomic analysis clearly highlights strong benefits for the fed-batch bioreactor (especially over a plant-life time horizon), but also showcases a definite cost-related promise of perfusion bioreactor technologies, given the higher mAb yield.

Robust dynamic optimization for nonlinear chemical processes under measurable and unmeasurable uncertainties

AbstractWe formulate an integrated framework for the robust dynamic optimization of nonlinear chemical processes under measurable and unmeasurable uncertainties. An affine decision rule is proposed to approximate the causal dependence of the wait‐and‐see decision variables on the gradually revealed measurable uncertainties. To overcome the computational intractability of the proposed model, a linearization technique based on the first‐order Taylor expansion is introduced around the nominal values of uncertainties to derive the robust dynamic counterpart, which can be discretized to a large‐scale nonlinear programming (NLP) formulation. Effects of first discretizing the dynamic models or introducing the affine decision rule are investigated. The proposed framework is also compared with the state‐of‐the‐art re‐optimization and traditional robust optimization approaches. An illustrative example and an industrial semi‐batch 2‐mercaptobenzothiazole production case are involved to demonstrate the advantages and applicability of the proposed framework.

Design and testing of a low-energy and -carbon prototype structure that adapts to loading through shape morphing

This paper presents an experimental validation of a new design method for adaptive structures that counteract the effect of loading through shape morphing. The structure is designed to morph into target shapes that are optimal to take external loads through combined optimization of structural layout and actuator placement. The prototype tested in this study is a simply supported spatial truss that spans 6.6 m with a span-to-depth ratio of 44/1. Shape adaptation is achieved through controlled length changes of 12 linear actuators that are strategically integrated into structural elements. A new mechanics-based framework has been formulated to enable real-time control. Linear-sequential shape optimization is employed to predict target shapes under loading. Actuator commands are computed through an iterative process that accounts for geometric nonlinearity. This formulation allows a reduction of computation time by four orders of magnitude compared with a nonlinear programming formulation. Experimental testing has shown that significant stress homogenization is achieved through shape adaptation, which results in up to 45% savings of material mass compared with a weight-optimized passive structure. Depending on the energy source employed for adaptation, the equivalent carbon is reduced by up to 29% and 43% in the case of non-renewable and 100% renewable energy mix, respectively.

Optimal design of multivariate acceptance sampling plans by variables

Quality-control via acceptance sampling is a technique well established for monitoring the quality of production by lots. The design of acceptance sampling plans for univariate characteristics that follow the Gaussian distribution is reported in various references. Formulations for finding statistically and economically optimal acceptance sampling plans have been consistently proposed. Contrarily, the extension of acceptance sampling to multivariate characteristics has limited applications and the methods available to design plans are still elusive as they involve complex numerical computation procedures. We propose optimization-based formulations to design acceptance sampling plans by variables for multivariate characteristics. First, we consider the independence of characteristics and address plans that satisfy all the controlled quality levels of each one. A Mixed Integer Nonlinear Programming formulation is introduced for such a purpose. Then, we extend the analysis to dependent characteristics and use Surrogate-based optimization to handle the problem. The formulations are demonstrated with simulated scenarios and an industrial case of practical interest.

Optimum scheduling of machines, automated guided vehicles and tools without tool delay in a multi‐machine flexible manufacturing system using symbiotic organisms search algorithm

AbstractThis article deals with machines, automated guided vehicles (AGVs) and tools simultaneous scheduling in multimachine flexible manufacturing system (FMS) with the lowest possible number of copies of every tool type without tool delay considering jobs' transport times among machines to minimize makespan (MSN). The tools are kept in a central tool magazine (CTM), which is shared by and serves many machines. Tool transporter (TT) and AGVs shift tools and jobs among machines respectively. This concurrent scheduling problem is extremely complex in nature as it involves determining the lowest possible tool copies of every type of tool, allocation of AGVs and tools copies to job‐operations, job‐operations sequencing on machines, and associated trip operations including the dead heading trip and loaded trip times of AGVs. This paper proposes nonlinear mixed integer programming (MIP) formulation to model this problem and symbiotic organisms search algorithm (SOSA) for solving this problem. Verification is carried out using an industrial problem in a manufacturing firm. The results show that employing two copies for one tool type and one copy each for the remaining tool types results in no tool delay and reduction in MSN which causes a reduction in cost, and SOSA is providing better solution than Jaya algorithm.

Isotope-assisted metabolic flux analysis as an equality-constrained nonlinear program for improved scalability and robustness.

Stable isotope-assisted metabolic flux analysis (MFA) is a powerful method to estimate carbon flow and partitioning in metabolic networks. At its core, MFA is a parameter estimation problem wherein the fluxes and metabolite pool sizes are model parameters that are estimated, via optimization, to account for measurements of steady-state or isotopically-nonstationary isotope labeling patterns. As MFA problems advance in scale, they require efficient computational methods for fast and robust convergence. The structure of the MFA problem enables it to be cast as an equality-constrained nonlinear program (NLP), where the equality constraints are constructed from the MFA model equations, and the objective function is defined as the sum of squared residuals (SSR) between the model predictions and a set of labeling measurements. This NLP can be solved by using an algebraic modeling language (AML) that offers state-of-the-art optimization solvers for robust parameter estimation and superior scalability to large networks. When implemented in this manner, the optimization is performed with no distinction between state variables and model parameters. During each iteration of such an optimization, the system state is updated instead of being calculated explicitly from scratch, and this occurs concurrently with improvement in the model parameter estimates. This optimization approach starkly contrasts with traditional “shooting” methods where the state variables and model parameters are kept distinct and the system state is computed afresh during each iteration of a stepwise optimization. Our NLP formulation uses the MFA modeling framework of Wiechert et al. [1], which is amenable to incorporation of the model equations into an NLP. The NLP constraints consist of balances on either elementary metabolite units (EMUs) or cumomers. In this formulation, both the steady-state and isotopically-nonstationary MFA (inst-MFA) problems may be solved as an NLP. For the inst-MFA case, the ordinary differential equation (ODE) system describing the labeling dynamics is transcribed into a system of algebraic constraints for the NLP using collocation. This large-scale NLP may be solved efficiently using an NLP solver implemented on an AML. In our implementation, we used the reduced gradient solver CONOPT, implemented in the General Algebraic Modeling System (GAMS). The NLP framework is particularly advantageous for inst-MFA, scaling well to large networks with many free parameters, and having more robust convergence properties compared to the shooting methods that compute the system state and sensitivities at each iteration. Additionally, this NLP approach supports the use of tandem-MS data for both steady-state and inst-MFA when the cumomer framework is used. We assembled a software, eiFlux, written in Python and GAMS that uses the NLP approach and supports both steady-state and inst-MFA. We demonstrate the effectiveness of the NLP formulation on several examples, including a genome-scale inst-MFA model, to highlight the scalability and robustness of this approach. In addition to typical inst-MFA applications, we expect that this framework and our associated software, eiFlux, will be particularly useful for applying inst-MFA to complex MFA models, such as those developed for eukaryotes (e.g. algae) and co-cultures with multiple cell types.

Nonisothermal reactor networks optimization using metaheuristics in a bi-level approach

In the present article, an optimization approach for the synthesis of reactor networks is proposed. A stage-wise superstructure containing a continuous stirred tank reactor (CSTR) and a plug flow reactor (PFR) is used to formulate the mathematical model without the use of differential equations and with a mixed-integer nonlinear programming (MINLP) formulation. To solve the optimization problem a bi-level meta-heuristic approach combining simulated annealing (SA) and particle swarm optimization (PSO) is used. Two case studies involving complex nonisothermal homogeneous reaction systems were used to test the applicability of this technique, and a methodology based on temperature profiles was developed to guide the search for operating temperatures during the optimization. The results achieved improvements in the objective function when compared to the majority of results reported in the literature, which demonstrates that the developed methodology is suitable to perform the reactor network synthesis in homogeneous systems.

Simultaneous Scheduling of Machines and Tools in a Multi-Machine FMS with Alternate Machines Using Crow Search Algorithm

Scheduling jobs and tools is a significant problem for manufacturing systems. Inefficient job scheduling and tool loading planning may result in under utilization of capital intensive machines and a high level of machine idle time. Therefore, efficient scheduling of jobs and tools enables a manufacturing system to increase machines’ utilization and decrease their idle times. This paper addresses machines’ and tools’ joint scheduling with alternate machines in a multimachine flexible manufacturing system (FMS) to minimize makespan (MSN). Only one copy of each type of tool is made available in FMS where tools are expensive. The tools are stored in the central tool magazine (CTM), which shares and serves them to several machines in order to reduce the cost of duplicating the tools in every machine. The problem is to select machines from alternate machines for job-operations, allocation of tools to the job-operations and job-operations’ sequencing on machines for MSN minimization. This paper presents nonlinear mixed integer programming (MIP) formulation to model this simultaneous scheduling problem and crow search algorithm (CSA) built on the crows’ intelligent behavior for solving this problem. The results show that CSA is providing better solutions than Jaya algorithm and the usage of alternate machines for the operations can reduce MSN.

Joint optimization of lot-sizing and pricing with backlogging

Lot-sizing and pricing are two important manufacturing decisions that impact together the profit of a company. Existing works address the joint lot-sizing and pricing problem without backlogging, although it is a usual strategy that permits to satisfy customer demand with delay. In this work, we study a new multi-product joint lot-sizing and pricing problem with backlogging and limited production capacity. The objective is to maximize the total company profit over a finite planning horizon. For the problem, a mixed integer nonlinear programming (MINLP) formulation is given. Then, several optimality properties are provided and a tighter MINLP model is established based on these properties. According to the NP-hard nature and non-linearity of the model, a model based heuristic that focuses on efficiently solving small-sized instances is proposed and a genetic algorithm (GA) with new progressive repair strategy is developed to address large-sized instances. Managerial insights are drawn based an illustrative example. Numerical experiments are conducted on 64 benchmark based instances and 105 randomly generated instances with up to 10 products and 12 periods, which validates the MINLP formulation and shows the efficiency of the proposed solution methods.

Distribution Grid Optimal Power Flow (D-OPF): Modeling, Analysis, and Benchmarking

In the power distribution systems, optimal power flow (D-OPF) is formulated as a non-convex and non-linear programming (NLP) problem. Convex relaxation and linear approximation models have been increasingly adopted to achieve computational efficiency for D-OPF. Despite the benefits of scalability and global optimality, each method is based on certain assumptions, performs differently, and may lead to solutions that are physically not meaningful. In this context, this work numerically evaluates the relative performance of second-order cone programming (SOCP), semi-definite programming (SDP), and linear programming (LP) formulations of D-OPF in terms of their feasibility, optimality, and scalability with respect to NLP-based formulations. We also compare the bus injection (in bus voltage and current variables) and branch flow (in active and reactive power flow variables) based on NLP formulations. The performance is evaluated using small (123-node), medium (730-node), and large (2522-node) sized distribution feeders. Case studies, which are backed up by visualization of the analytical models for the solution space to the extent possible, show that (1) the feasibility and exactness of relaxed D-OPF formulations depend upon the problem type, (2) some NLP formulations are computationally more tractable than others, (3) different NLP formulations can converge to different local solutions, and (4) the approximate linear model may underestimate or overestimate the cost function (depending upon the problem type) and may lead to AC-infeasible solutions.

Delay-Sensitive and Availability-Aware Virtual Network Function Scheduling for NFV

Network Function Virtualization (NFV) has been emerging as an appealing solution that transforms from dedicated hardware implementations to software instances running in a virtualized environment. In NFV, the requested service is implemented by a sequence of Virtual Network Functions (VNF) that can run on generic servers by leveraging the virtualization technology. These VNFs are pitched with a predefined order, and it is also known as the Service Function Chaining (SFC). Considering that the delay and resiliency are two important Service Level Agreements (SLA) in a NFV service, in this paper, we first investigate how to quantitatively model the traversing delay of a flow in both totally ordered and partially ordered SFCs. Subsequently, we study how to calculate the VNF placement availability mathematically for both unprotected and protected SFCs. After that, we study the delay-sensitive Virtual Network Function placement and routing problem with and without resiliency concerns. We prove that this problem is NP-hard under two cases. We subsequently propose an exact Integer Nonlinear Programming (INLP) formulation and an efficient heuristic for this problem in each case. Finally, we evaluate the proposed algorithms in terms of acceptance ratio, average number of used nodes and total running time via extensive simulations.

Minimizing risk in the scheduling of crudes in an oil refinery

This paper focuses on formulating and solving the optimization of crude oil operations scheduling carried out in a system composed of a refinery and a marine terminal. The main challenge lies in coordinating the decisions made at both facilities and, at the same time, dealing with the uncertainties inherent in this activity. To tackle this problem, we present a continuous-time mixed-integer non-linear programming (MINLP) formulation. Furthermore, uncertainty in the ship arrival times is considered through a two-stage stochastic programming approach. Finally, the Conditional Value-at-Risk (CVaR) risk measure is employed to weigh the risk of having high costs relative to the worst scenarios. In this way, the proposed model is capable of supporting the decision-making process under uncertainty in an integral way.

Optimization of Power Dispatch With Load Scheduling for Domestic Fuel Cell-Based Combined Heat and Power System

This study proposes an optimization scheme of thermal power and electric power dispatch integrated with load scheduling for the domestic fuel cell-based combined heat and power (DFCCHP) system. The scheme is implemented in home energy management systems installed in smart homes. To provide accurate energy cost evaluation for the optimization scheme, the nonlinear electric efficiency characteristics of the fuel cell are approximated with a polynomial expression. Along with the inclusion of thermal power dispatch in the scheme, the nonlinear relationship between the thermal power and the electric power of the fuel cell is incorporated to design temperature constraints in the DFCCHP system. On top of the nonlinearity and complexity of the fuel cell, residential electric load scheduling and other energy sources including the power grid, PV panels, and battery energy storage are considered to ensure that the scheme takes a comprehensive energy management approach. Because of the nonlinearity that exists in the modeling of the DFCCHP system, a mixed-integer nonlinear programming formulation is utilized to solve the optimization problem. The scheme is tested in a day-ahead environment with time-varying electricity prices and natural gas prices. The optimization aims to minimize the electricity cost and natural gas cost. It is shown in the simulation that optimization scheme dispatches the electric power and thermal power in an optimal way so that the energy cost due to time-varying electricity prices and natural gas prices is minimized. The electricity cost optimization puts both power purchase and power selling into consideration. It is also shown in the simulation that the household loads are scheduled in an optimal way to the time slots with lower electricity prices in accordance with the optimal thermal and electric power dispatch.

The Unit Re-Balancing Problem

We describe the problem of re-balancing a number of units distributed over a geographic area. Each unit consists of a number of components. A value between 0 and 1 describes the current rating of each component. By a piecewise linear function, this value is converted into a nominal status assessment. The lowest of the statuses determines the efficiency of a unit, and the highest status its cost. An unbalanced unit has a gap between these two. To re-balance the units, components can be transferred. The goal is to maximize the efficiency of all units. On a secondary level, the cost for the re-balancing should be minimal. We present a mixed-integer nonlinear programming formulation for this problem, which describes the potential movement of components as a multi-commodity flow. The piecewise linear functions needed to obtain the status values are reformulated using inequalities and binary variables. This results in a mixed-integer linear program, and numerical standard solvers are able to compute proven optimal solutions for instances with up to 100 units. We present numerical solutions for a set of open test instances and a bi-criteria objective function, and discuss the trade-off between cost and efficiency.