Mixed-integer Nonlinear Programming Formulation Research Articles

A trust region framework for heat exchanger network synthesis with detailed individual heat exchanger designs

We develop a trust region filter strategy for simultaneous optimal design of heat exchanger networks that includes detailed design of shell-and-tube heat exchangers. The strategy first solves a mixed-integer nonlinear programming (MINLP) formulation with shortcut models to generate candidate network topologies, which are then used in a non-isothermal mixing nonlinear programming (NLP) suboptimization with detailed optimal exchanger design models embedded using a modified trust region filter (TRF) algorithm. An integer cut based strategy is used to bound the solutions from MINLP and the NLP which aids in convergence to the solution of the overall simultaneous design problem. Under assumptions, the TRF based strategy can guarantee convergence to near optimal solutions of the overall design problem. The presented solution strategy is thus able to find optimal heat exchanger network designs based on the simultaneous optimization of the network topology and mass and energy balances, together with detailed shell-and-tube heat exchanger optimization, including the number of shell and tube passes, pressure drops, and tubes, tube lengths, etc. The proposed strategy is tested on three literature based case studies and their results are compared with previous studies to showcase its performance.

A two-stage stochastic supply chain scheduling problem with production in cellular manufacturing environment: A case study

An integrated decision in supply chain is a significant principle in order to compete in today’s market. This paper proposes a novel mathematical model in a two-stage supply chain scheduling to cooperate procurement and manufacturing activities. The supply chain scheduling along with the production approach of cellular manufacturing under demand, processing time, and transportation time uncertainties makes business environment sustainably responsive to the changing needs of customers. Uncertainties are formulated by queuing theory. In this paper, a new mixed-integer nonlinear programming formulation is used to determine types of vehicles to carry raw materials, suppliers to procure, priority of each part in order to process, and cell formation to configure work centers. The goal is to minimize total tardiness. A linearization method is used to ease tractability of the model. A genetic algorithm is developed due to the NP-hard nature of the problem. The parameters of the genetic algorithm are set and estimated by Taguchi’s experimental design. Numerous test problems are employed to validate the effectiveness of the modeling and the efficiency of solution approaches. Finally, a real case study and a sensitivity analysis are discussed to provide significant managerial insights and assess the applicability of the proposed model.

Energy-efficient timetabling and rolling stock circulation planning based on automatic train operation levels for metro lines

We investigate the energy-efficient train timetabling and rolling stock circulation planning problem for metro lines based on flexible selections of train operation levels, which involve running levels (various running times and speed profiles) and dwell levels (various dwell times at stations). The travel demand of passengers is involved in the energy-efficient train timetabling problem via service patterns, i.e., different operation headways are used in peak and off-peak hours to satisfy the passenger demand. A mixed integer nonlinear programming (MINLP) formulation is presented to optimize train timetables and circulation plans of rolling stocks simultaneously with the aim of minimizing the headway deviations relating to service patterns, the headway variations among neighboring train services, the needed depot movements for rolling stocks, and the overall energy consumption. To increase the regularity (i.e., the consistency of operation level selections of train services) of train timetables and to improve the computational efficiency, two extensions of the energy-efficient timetabling formulation are proposed, i.e. the energy-efficient train timetabling models with pre-fixed selections and optimal identical (or consistent) selections of the operation levels. The proposed MINLP formulations are transformed into mixed integer linear programming formulations, which could be effectively solved, e.g., by Cplex and Gurobi solvers. To investigate the effectiveness of the proposed models, we perform computational experiments based on the real-world data of the Beijing Yizhuang line. The numerical results illustrate that the energy consumption of the daily train timetable obtained by the proposed energy-efficient train timetabling model with flexible selection of operation levels is reduced by 9.90% when compared with that of the train timetables without operation level selections. Moreover, the energy savings of the energy-efficient train timetables with optimal identical selection and pre-fixed selection of operation levels are 7.67% and 5.75%, respectively. However, the computation time of the energy-efficient train timetabling models with operation level selections is significantly longer than that without operation level selections.

Four-Objective Optimal Scheduling of Energy Hub Using a Novel Energy Storage, Considering Reliability and Risk Indices

Energy hub as a novel model which has provided more efficiency, flexibility, and reliability. Also, due to using different multi carriers as renewable and non-renewable sources and storage devices in energy hubs, optimal operation scheduling becomes more challenging. In this paper, the day-ahead scheduling of energy hub is studied in the presence of wind turbine, electrical, thermal, and a novel water-based storage called Pico hydel energy storage. A useful method using the Monte-Carlo and k-mines clustering method is developed to estimate the uncertainty associated with wind speed, load, and market price. On the other hand, Conditional Value at Risk and Second-Order Stochastic Dominance (SSD) method are applied as a risk-averse method. To find the best operation performance of energy hub, multiple objectives consist of economic, emission pollution, electrical, and load profile deviation are defined. As well, the demand response program for gas, electricity, water, and thermal are implemented to improve the energy hub performance. Mixed-integer non-linear program formulation for the day-ahead scheduling of energy hub has been solved by executing CPLEX solver of GAMS optimization software in five different cases. Simulation results show that the hub's operation cost and emission reduce up to 4.8% and 5.6% by implementing four objective functions in the presence of the Pico hydel energy storage and demand response programing.

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.

Relax-tighten-round algorithm for optimal placement and control of valves and chlorine boosters in water networks

In this paper, a new mixed integer nonlinear programming formulation is proposed for optimally placing and operating pressure reducing valves and chlorine booster stations in water distribution networks. The objective is the minimization of average zone pressure, while penalizing deviations from a target chlorine concentration. We propose a relax-tighten-round algorithm based on tightened polyhedral relaxations and a rounding scheme to compute feasible solutions, with bounds on their optimality gaps. This is because off-the-shelf global optimization solvers failed to compute feasible solutions for the considered non-convex mixed integer nonlinear program. The implemented algorithm is evaluated using three benchmarking water networks, and they are shown to outperform off-the-shelf solvers, for these case studies. The proposed heuristic has enabled the computation of good quality feasible solutions in most instances, with bounds on the optimality gaps that are comparable to the order of uncertainty observed in operational water network models.

MINLP formulations for continuous piecewise linear function fitting

We consider a nonconvex mixed-integer nonlinear programming (MINLP) model proposed by Goldberg et al. (Comput Optim Appl 58:523–541, 2014. https://doi.org/10.1007/s10589-014-9647-y) for piecewise linear function fitting. We show that this MINLP model is incomplete and can result in a piecewise linear curve that is not the graph of a function, because it misses a set of necessary constraints. We provide two counterexamples to illustrate this effect, and propose three alternative models that correct this behavior. We investigate the theoretical relationship between these models and evaluate their computational performance.

A Lagrangean relaxation approach to lifetime maximization of directional sensor networks

AbstractWe consider the directional sensor network lifetime maximization problem (DSLMP). Given a set of directional sensor and target locations, the problem consists in assigning, at each time unit of a given time horizon, the action radius, the aperture angle, and the orientation direction to all sensors. The objective is to maximize the number of time units when all targets are covered, under certain constraints on sensor available energy. We present a mixed integer nonlinear programming formulation and tackle it by Lagrangean decomposition and subgradient optimization. The algorithm is equipped with a repairing heuristics aimed at finding good‐quality feasible solutions to DSLMP. The results of the application of the proposed approach to a number of problem instances are also reported.

3-D Dynamic UAV Base Station Location Problem

We address a dynamic covering location problem of an unmanned aerial vehicle base station (UAV-BS), in which the location sequence of a single UAV-BS in a wireless communication network is determined to satisfy data demand arising from ground users. This problem is especially relevant in the context of smart grid and disaster relief. The vertical movement ability of the UAV-BS and nonconvex covering functions in wireless communication restrict utilizing classical planar covering location approaches. Therefore, we develop new formulations to this emerging problem for a finite time horizon to maximize the total coverage. In particular, we develop a mixed-integer nonlinear programming formulation that is nonconvex in nature and propose a Lagrangean decomposition algorithm (LDA) to solve this formulation. Because of the high complexity of the problem, the LDA is still unable to find good local solutions to large-scale problems. Therefore, we develop a continuum approximation (CA) model and show that CA would be a promising approach in terms of both computational time and solution accuracy. Our numerical study also shows that the CA model can be a remedy to build efficient initial solutions for exact solution algorithms. Summary of Contribution: This paper addresses a facet of mixed integer nonlinear programming formulations. Dynamic facility location problems (DFLPs) arise in a wide range of applications. However, classical DFLPs typically focus on the two-dimensional spaces. Emerging technologies in wireless communication and some other promising application areas, such as smart grids, have brought new location problems that cannot be solved with classical approaches. For practical reasons, many research attempts to solve this new problem, especially by researchers whose primary research area is not OR, have seemed far from analyzing the characteristics of the formulations. Rather, solution-oriented greedy heuristics have been proposed. This paper has two main objectives: (i) to close the gap between practical and theoretical sides of this new problem with the help of current knowledge that OR possesses to solve facility location problems and (ii) to support the findings with an exhaustive computational study to show how these findings can be applied to practice.

БИЗНЕС-СТРАТЕГИЧЕСКОЕ ПРИНЯТИЕ РЕШЕНИЙ С УЧЕТОМ МОДЕЛЕЙ МАРКЕТИНГА И ЦЕПОЧКИ ПОСТАВОК

A supply chain model that includes the components of business strategic decisions is an es-sential tool for gaining a competitive advantage in today's global marketplace. Enterprise models of this type need to cover not only the supply chain but also the demand chain, since understanding the market is critical to formulating good business policy. To work effectively, marketing activities must be coordinated with other functional areas of the company. Specifically, managers must evaluate the trade-off between marketing and supply chain decisions to improve the performance of an overall metric: shareholder value. The article formu-lates a mathematical model that takes into account the main business functions. A mixed integer nonlinear programming formulation is presented for the integrated optimization of strategic supply chain decisions and marketing decisions. This allows you to more accurately form a financial model and estimate the value of the enterprise. Keywords

A Hybrid Approach Based on SOCP and the Discrete Version of the SCA for Optimal Placement and Sizing DGs in AC Distribution Networks

This paper deals with the problem of the optimal placement and sizing of distributed generators (DGs) in alternating current (AC) distribution networks by proposing a hybrid master–slave optimization procedure. In the master stage, the discrete version of the sine–cosine algorithm (SCA) determines the optimal location of the DGs, i.e., the nodes where these must be located, by using an integer codification. In the slave stage, the problem of the optimal sizing of the DGs is solved through the implementation of the second-order cone programming (SOCP) equivalent model to obtain solutions for the resulting optimal power flow problem. As the main advantage, the proposed approach allows converting the original mixed-integer nonlinear programming formulation into a mixed-integer SOCP equivalent. That is, each combination of nodes provided by the master level SCA algorithm to locate distributed generators brings an optimal solution in terms of its sizing; since SOCP is a convex optimization model that ensures the global optimum finding. Numerical validations of the proposed hybrid SCA-SOCP to optimal placement and sizing of DGs in AC distribution networks show its capacity to find global optimal solutions. Some classical distribution networks (33 and 69 nodes) were tested, and some comparisons were made using reported results from literature. In addition, simulation cases with unity and variable power factor are made, including the possibility of locating photovoltaic sources considering daily load and generation curves. All the simulations were carried out in the MATLAB software using the CVX optimization tool.

Approach for improving the life-cycle profit exploding remanufacturing conditions

This work presents a mixed-integer nonlinear programming (MINLP) formulation for the multi-period planning of a Closed-Loop Supply Chain (CLSC) with the emphasis placed on remanufacturing conditions. The approach considers the life-cycle of products related to processes industry. It is taken into account that two types of products flow through the CLSC, products with traditional and modular design (TD and MD). While products with traditional design (PTD) uses only new raw materials, products with modular design (PMD) utilizes partially or totally materials that are recovered from discarded products. The manufacturing process connected with the PMD is characterized by improving the manufacturing and material separation. As part of the problem, it must be determined the selling prices, the amount of manufactured products, the return rates of discarded products and the level of materials recovery. Demand functions and customer utility measures for products with traditional and modular design are considered. The objective function is to maximize the total life-cycle profit taking into account the sum of the profits from selling products with traditional and modular designs. The MINLP approach is applied to the planning problem of a firm dedicated to manufactures carpets that wants to produce according to a new modular design where recovered materials must be used. The performance of the model is evaluated solving several case scenarios. The results obtained with the model for the carpet design allow to achieve important practical conclusions, such as the consideration of both designs, traditional and modular, has a positive impact on the total profit and allows the company to have a greater presence in the market.

Optimal design and operation of an islanded water-energy network including a combined electrodialysis-reverse osmosis desalination unit

This paper proposes an optimization framework to simultaneously optimize the saline and freshwater water sources as well as decentralized renewable and conventional energy sources, which consequently results in an economic and efficient water-energy system operation. The inextricable links between water and energy systems are taken into considerations, and a mixed integer nonlinear programming formulation is used to solve the proposed optimization model. A hybrid electrodialysis-reverse osmosis is integrated in the water distribution system and its electricity consumption is minimized. Also, the day-ahead economic dispatch problem of the energy system incorporates the random behavior of solar and wind generation units, along with charging and discharging modes of battery storage systems. The outcomes of the proposed models highlight: 1) integrating the hybrid desalination module in the water-energy system will reduce peak demands by more than 30% if the water and energy systems are solely supplied by saline water and renewable energy sources, respectively; 2) presence of an extra water storage tank also decreases peak demands by 13%. The impact of key elements of the combined system such as pumps’ speeds, water tank constraints, and presence of renewable sources on the output cost, generated energy, and hydraulic parameters is investigated through case studies.

One-step optimization strategy in the simulated moving bed process with asynchronous movement of ports: A VariCol case study

The VariCol process is a variant of the conventional simulated moving bed (SMB) process, distinguished by the asynchronous shifting of the inlet and outlet ports of the chromatographic column train. This feature allows for a more flexible operation in column utilization and can also achieve higher separation performances. However, to take full benefit out of it, the operating parameters, such as the strategy for port switching, must be optimal. in this paper, a novel methodology for optimizing those parameters, based on a single NLP (non-linear programming), is proposed. The main advantage of this approach is that it significantly reduces the complexity of the original MINLP (mixed-integer non-linear programming) formulation currently discussed in the literature. The proposed optimization problem is built, considering that the average column configuration of three zones provides the necessary and sufficient information to describe the VariCol process. Several optimization scenarios for the enantioseparation of 1,1´-bi-2-naphthol and aminoglutethimide were considered to evaluate the proposed methodology and to compare the performance of VariCol and SMB processes. The results have shown that with the single NLP approach, it is possible to explore the optimal solution in all the VariCol process domains with less computational effort than other optimization strategies reported in the literature. That is a great advantage, especially in the context of real-time applications.

Three-dimensional guillotine cutting problems with constrained patterns: MILP formulations and a bottom-up algorithm

In this paper, we address the Constrained Three-dimensional Guillotine Cutting Problem (C3GCP), which consists of cutting a larger cuboid block (object) to produce a limited number of smaller cuboid pieces (items) using orthogonal guillotine cuts only. This way, all cuts must be parallel to the object’s walls and generate two cuboid sub-blocks, and there is a maximum number of copies that can be manufactured for each item type. The C3GCP arises in industrial manufacturing settings, such as the cutting of steel and foam for mattresses. To model this problem, we propose a new compact mixed-integer non-linear programming (MINLP) formulation by extending its two-dimensional version, and develop a mixed-integer linear programming (MILP) version. We also propose a new model for a particular case of the problem which considers 3-staged patterns. As a solution method, we extend the algorithm of Wang (1983) to the three-dimensional case. We emphasise that the C3GCP is different from 3D packing problems, namely from the Container Loading Problem, because of the guillotine cut constraints. All proposed approaches are evaluated through computational experiments using benchmark instances. The results show that the approaches are effective on different types of instances, mainly when the maximum number of copies per item type is small, a situation typically encountered in practical settings with low demand for each item type. These approaches can be easily embedded into existing expert systems for supporting the decision-making process.

Location-routing problem with time-dependent demands

This study, by considering the time-dependent demand (TDD) characteristic, investigates the location-routing problem with time-dependent demands (LRPTDD) as an extension of the location-routing problem (LRP). The demand in each customer site is represented by a constant demand rate over a known production period. In the LRPTDD, the locations are selected, and the routes are constructed to pick up all the demands and minimize the total distance. The picked load depends on the vehicle arrival time at the site; thus, the “time-dependency” characteristic of the LRPTDD is based on the vehicle arrival time. A mixed-integer nonlinear programming (MINLP) formulation is presented. A simulated annealing (SA) algorithm for the LRPTDD is developed. The computational study demonstrates the competitiveness of the proposed SA heuristic against other well-known algorithms for LRPs, and most importantly, its effectiveness for the LRPTDD.

An exact algorithm for the single liner service design problem with speed optimisation

ABSTRACT This paper models a single liner service design and operations problem. The model selects the ports to be included, their sequence, the sailing speed of vessels, the number of vessels and the amounts of cargo to transport by the service. The objective is to maximise profit. First, a relaxation with a mixed-integer nonlinear programming (MINLP) formulation is proposed. We show how to obtain the optimal speed value. Once this value is obtained, the mathematical programming formulation becomes a mixed-integer linear program (MILP). Then, a two-step exact algorithm is presented to solve the problem. Using real data, the optimal solution was found in less than 1 min for small-size problems and in few hours for relatively large-size problems. More tests were carried out on randomly generated data sets with up to 25 ports. The results of these tests are rather promising, and they enabled us to identify the performance limits of the algorithm.

Practical approximations and heuristic approaches for managing shiftable loads in the multi-period optimal power flow framework

The presence of smart buildings (SBs) is expected to grow manifold in the next decade, leading to both challenges and opportunities in managing increasingly “active” distribution systems. SBs are host to a number of devices and technologies capable of providing residential flexibility in order to maintain acceptable operating conditions within the distribution system. A prevalent class of devices with significant potential for flexibility provision are shiftable loads (SLs), which represent a large amount of potential SB-provided flexibility. However, the accurate modelling of SLs within the multi-period optimal power flow (MP-OPF) framework results in complex non-convex mixed-integer nonlinear programming (MINLP) problems. For realistic feeders, MINLP problems are intractable and hence require some heuristic form of circumvention. This work evaluates the limitations of the MINLP formulation and proposes new practical alternatives, namely a flexible NLP approximation and a multi-faceted heuristic algorithm. The proposed approaches are generic and applicable for any problem size or type of SL, while they are shown to work efficiently, both speed-wise and solution-quality-wise, outperforming state-of-the-art MINLP solvers. A simple scoring and ranking scheme is also proposed in order to make a direct comparison between the different approaches.

A multi-objective bi-level location problem for heterogeneous sensor networks with hub-spoke topology

An heterogeneous sensor network (hSN) consists of different types of sensor nodes with varying coverage capability, range, and sensing quality. hSNs are used by military and other defense organizations for a number of purposes such as monitoring critical facilities, surveilling an area of interest, tracking enemies, or protecting borders. Therefore, the deployment problem of such networks to form an effective coverage and surveillance is of high importance. In this study, we consider the problem of locating an hSN along a two-dimensional belt-shaped border region which includes a number of critical facilities that need protection against intruders. Assuming a hub-spoke location topology, we allow the use of cooperative and gradual covering, multiple types of sensors, critical facilities and intruders. We developed two competing multi-objective mixed integer non-linear program formulations as well as their equivalent mixed integer linear program (MILP) reformulations which adopt a goal programming approach. The MILP formulations are then solved by a commercial optimizer for a set of problem instances using the branch-and-cut (B&C) procedure with various branching and node selection strategies as well as different user defined optimality gap and goal settings. Our results indicate that the performance of formulations are significantly affected by the B&C variable selection strategy as well as the goal levels determined by the decision-makers. We also observe that, using proper settings, the majority of the problem instances can be solved within a maximum of 10% optimality gap in reasonable computing times.

An MINLP-Based Approach for the Design-for-Control of Resilient Water Supply Systems

The resilience of a water distribution system is defined as its ability to maintain continuous customer supply, and can be improved by increasing its redundancy in energy and/or connectivity. This can, however, negatively impact other aspects of network performance, such as leakage management. In this article, we consider the design-for-control (DfC) problem of adding new connections (from a predefined set of candidate pipes) to water supply systems to improve their resilience to failure events while minimizing the impact on leakage management under normal operating conditions. We present a mixed-integer nonlinear programming (MINLP) formulation of the problem of optimal link addition for the minimization of average zone pressure (AZP), a surrogate measure of pressure dependent leakage. We implement a method based on spatial branch-and-bound to solve the problem on a case study network from the literature and an operational network part of an urban water system in the U.K. Finally, we validate the improvement in network resilience resulting from the addition of new connections by performing an a posteriori critical link analysis, using the hydraulic resilience measure of reserve capacity.