Mixed-integer Nonlinear Programming Formulation Research Articles

Simultaneous Stochastic Optimization of Mining Complexes and Mineral Value Chains

Recent developments in modelling and optimization approaches for the production of mineral and energy resources have resulted in new simultaneous stochastic optimization frameworks and related digital technologies. A mining complex is a type of value chain whereby raw materials (minerals) extracted from various mineral deposits are transformed into a set of sellable products, using the available processing streams. The supply of materials extracted from a group of mines represents a major source of uncertainty in mining operations and mineral value chains. The simultaneous stochastic optimization of mining complexes, presented herein, aims to address major limitations of past approaches by modelling and optimizing several interrelated aspects of the mineral value chain in a single model. This single optimization model integrates material extraction from a set of sources along with their uncertainty, the related risk management, blending, stockpiling, non-linear transformations that occur in the available processing streams, the utilization of processing streams, and, finally, the transportation of products to customers. Uncertainty in materials extracted from the related mineral deposits of a mining complex is represented by a group of stochastic simulations. This paper presents a two-stage stochastic mixed integer nonlinear programming formulation for modelling and optimizing a mining complex, along with a metaheuristic-based solver that facilitates the practical optimization of exceptionally large mathematical formulations. The distinct advantages of the approach presented herein are demonstrated through two case studies, where the stochastic framework is compared to past approaches that ignore uncertainty. Results demonstrate major improvements in both meeting forecasted production targets and net present value. Concepts and methods presented in this paper for the simultaneous stochastic optimization for mining complexes may be adopted and applied to the optimization of smart oil fields.

Optimization of a natural gas distribution network with potential future extensions

A model of a pipeline network for gas distribution is developed considering supply of gas, either from external gas networks or as injected biogas or gasified liquefied natural gas (LNG) at terminals. The model is based on mass and energy balance equations for the network nodes, equations of the pressure drop of a compressible gas in the pipes, as well as expressions of gas compression in compressor nodes. The model is applied within an optimization framework where the optimal supply of natural gas to the customers is studied under a multi-period mixed integer nonlinear programming (MINLP) formulation, considering possible extensions of the pipeline network to new sites as well as potential supply of the gas from LNG terminals. The natural gas network in Finland is used in a case study, which determines the network's size and operation conditions. The results illustrate that the model can tackle complex gas supply problems and that it finds interesting alternatives where the optimal gas flow is reversed between the periods. The findings reveal the conditions under which it is beneficial to upgrade existing connections by parallel pipelines, extend the pipeline to new sites, or to re-gasify LNG and inject it into the network.

Organic Rankine Cycle system performance targeting and design for multiple heat sources with simultaneous working fluid selection

This work presents a systematic approach toward the design of Organic Rankine Cycles (ORC) for the generation of power from multiple heat sources available at different temperature levels. The design problem is approached in a mixed-integer non-linear programming (MINLP) formulation where an inclusive and flexible ORC model is automatically evolved by a deterministic algorithm for global optimization. The basic building block of the model is the ORC cascade which consists of a heat extraction, a power generation, a condensation and a liquid pressurization section. The aim of the optimization is to determine the optimum number of ORC cascades, the structure of the heat exchanger network shared among different cascades, the operating conditions and the working fluid used in each cascade in order to identify an overall ORC structure that maximizes the power output. The approach is illustrated through a case study which indicates that a system of two waste heat sources is best exploited through two interconnected ORC utilizing different working fluids.

MINLP-based Analytic Hierarchy Process to simplify multi-objective problems: Application to the design of biofuels supply chains using on field surveys

Multi-objective optimization (MOO) is widely used in engineering systems design and planning. The solution of a MOO problem leads to a set of efficient points (Pareto set) from which decision-makers should identify the one that best fits their preferences. Generating this set requires large computational efforts, and the post-optimal analysis of the solutions becomes difficult as the number of objectives increases. This work introduces an approach based on the Analytic Hierarchy Process (AHP) to overcome these limitations. Through the definition of an aggregated objective function calculated using the AHP algorithm, a single-objective model is constructed that provides a unique Pareto solution of the original MOO model. The AHP is combined with a mixed-integer non-linear programming (MINLP) formulation that simplifies its application and is particularly suited to deal with many objectives (like those arising in sustainable engineering problems). The capabilities of the approach are demonstrated through a case study addressing the sustainable sugar/ethanol supply chain design problem.

A location-routing problem for cross-docking networks: A biogeography-based optimization algorithm

This paper considers a location-routing problem in a distribution network with a set of part suppliers, cross-docking centers and assembly plants known as customers. We develop a mixed integer non-linear programming formulation for the problem in which the location for establishing the cross-docks is determined while simultaneously a fleet of vehicles are applied to transport goods from suppliers to the assembly plants via two transportation strategies: direct shipment and shipment through cross-dock (indirect shipment). In the second strategy, it is possible to have routes between suppliers. Not considering two problems of location and distribution planning simultaneously would result in increasing the costs of supplying parts since the transportation strategy has a huge effect on location of cross-docks. In the other words, if some loads can be directly shipped, then this kind of loads should not be taken into account in determining cross-docks location. Thus, a location- routing problem is presented for cross-docking system in this paper. The goal is to determine the location of cross-docks, allocating suppliers to them and routing decisions, so that the location cost and total shipping cost in the network are minimized, considering variable cost of servicing parts passed through cross-docks. The proposed model is NP-hard based on literature. Thus, a metaheuristic algorithm named Biogeography-based optimization (BBO) is utilized to solve the problem. In order to evaluate its efficiency, BBO results are compared with those of PSO, which is a well-known algorithm in the literature. Solving numerical examples for small size problem instances illustrates that the solving approach performs with a negligible gap relative to GAMS, while it performs much better than PSO in most cases in terms of total cost of the network and computational time.

A Novel Solution Approach to a Priority-Slot-Based Continuous-Time Mixed Integer Nonlinear Programming Formulation for a Crude-Oil Scheduling Problem

To satisfy component concentration constraints in crude oil operations, it is necessary to blend different oil types, resulting in a mixed integer nonlinear programming (MINLP) formulation for the scheduling problem of crude oil operations. Because of the intractability of such a nonlinear problem, approximate methods were proposed in the literature. However, by the existing methods, a composition concentration discrepancy may occur, leading to an infeasible solution; or a feasible solution cannot be found even if such a solution exists for some cases. Based on a priority-slot modeling method, this paper copes with the crude-oil scheduling problem suffering from composition concentration discrepancy. To find a solution without composition concentration discrepancy, a valid inequality is added to the MINLP model. Also, the model size is significantly reduced by properly determining the number of slots. Then, a novel solution method is proposed. By this method, the problem is iteratively solved and, at each...

Work-heat exchanger network synthesis (WHENS)

Research on heat integration has made significant advances in reducing utility consumption in chemical plants. However, the idea of work exchange between high and low-pressure process streams to reduce the consumption of the relatively expensive electricity has received limited attention. In this article, we present a more efficient mixed-integer nonlinear programming (MINLP) formulation to synthesize work-heat exchanger networks (WHENs). We propose a superstructure that explicitly considers constant-pressure streams for heat integration and enables an optimized selection of end-heaters and end-coolers to meet the desired temperature targets. Using a few examples, we demonstrate that simultaneous integration of work and heat in a chemical plant can offer significant savings in total annualized cost. In a case study from the literature, our approach yields a network with 3.1% lower total annualized cost, 10.6% more work exchange, and 81.0% more heat exchange than the best solution obtained from the existing literature approach. Furthermore, our approach successfully solves two case studies that previous literature approaches fail to solve.

A modelling framework for solving restricted planar location problems using phi-objects

This paper presents a general modelling framework for restricted facility location problems with arbitrarily shaped forbidden regions or barriers, where regions are modelled using phi-objects. Phi-objects are an efficient tool in mathematical modelling of 2D and 3D geometric optimization problems, and are widely used in cutting and packing problems and covering problems. The paper shows that the proposed modelling framework can be applied to both median and centre facility location problems, either with barriers or forbidden regions. The resulting models are either mixed-integer linear or non-linear programming formulations, depending on the shape of the restricted region and the considered distance measure. Using the new framework, all instances from the existing literature for this class of problems are solved to optimality. The paper also introduces and optimally solves a realistic multi-facility problem instance derived from an archipelago vulnerable to earthquakes. This problem instance is significantly more complex than any other instance described in the literature.

Optimizing Gasoline Recipes and Blending Operations Using Nonlinear Blend Models

Gasoline is one of the largest-volume products of the oil industry that yields 60%–70% of the total refinery revenues. This work presents a novel continuous-time mixed-integer nonlinear programming (MINLP) formulation for the gasoline blend scheduling problem. It incorporates nonlinear blending correlations for an improved prediction of key blend properties, and nonlinear constraints for precisely tracking the inventory level in product tanks when multiple blenders are operated. The approach handles nonidentical blenders, multipurpose tanks, sequence-dependent changeovers, limited amounts of gasoline components, and multiperiod scenarios with component flow rates changing with the period. Operating rules for blenders and product/component tanks are also considered. A special model feature is the use of floating slots dynamically allocated to time periods while solving the problem. An approximate mixed-integer linear programming (MILP) formulation assuming ideal mixing provides a good initial point. By fix...

Source-based discrete and continuous-time formulations for the crude oil pooling problem

Abstract The optimization of crude oil operations in refineries is a challenging scheduling problem due to the need to model tanks of varying composition with nonconvex bilinear terms, and complicating logistic constraints. Following recent work for multiperiod pooling problems of refined petroleum products, a source-based mixed-integer nonlinear programming formulation is proposed for discrete and continuous representations of time. Logistic constraints are modeled through Generalized Disjunctive Programming while a specialized algorithm featuring relaxations from multiparametric disaggregation handles the bilinear terms. Results over a set of test problems from the literature show that the discrete-time approach finds better solutions when minimizing cost (avoids source of bilinear terms). In contrast, solution quality is slightly better for the continuous-time formulation when maximizing gross margin. The results also show that the specialized global optimization algorithm can lead to lower optimality gaps for fixed CPU, but overall, the performance of commercial solvers BARON and GloMIQO are better.

The Freight Train Routing Problem for Congested Railway Networks with Mixed Traffic

We consider the following freight train routing problem (FTRP). Given is a transportation network with fixed routes for passenger trains and a set of freight trains (requests), each defined by an origin and destination station pair. The objective is to calculate a feasible route for each freight train such that the sum of all expected delays and all running times is minimal. Previous research concentrated on microscopic train routings for junctions or inside major stations. Only recently approaches were developed to tackle larger corridors or even networks. We investigate the routing problem from a strategic perspective, calculating the routes in a macroscopic transportation network of Deutsche Bahn AG. In this context, macroscopic refers to an aggregation of complex and large real-world structures into fewer network elements. Moreover, the departure and arrival times of freight trains are approximated. The problem has a strategic character since it asks only for a coarse routing through the network without the precise timings. We provide a mixed-integer nonlinear programming (MINLP) formulation for the FTRP, which is a multicommodity flow model on a time-expanded graph with additional routing constraints. The model’s nonlinearities originate from an algebraic approximation of the delays of the trains on the arcs of the network by capacity restraint functions. The MINLP is reduced to a mixed-integer linear model (MILP) by piecewise linear approximation. The latter is solved by a state-of-the art MILP solver for various real-world test instances.

Mobile sensor networks for optimal leak and backflow detection and localization in municipal water networks

Leak and backflow detections are essential aspects of Water Distribution Systems (WDSs) monitoring and are commonly fulfilled using approaches that are based on static sensor networks and point measurements. Alternatively, we propose a mobile, wireless sensor network solution composed of mobile sensor nodes that travel freely inside the pipes with the water flow, collect and transmit measurements in near-realtime (called sensors) and static access points (called beacons). This study complements the tremendous progress in mobile sensor technology. We formulate the sensor and beacon optimal placement task as a Mixed Integer Nonlinear Programming (MINLP) problem to maximize localization accuracy with budget constraint. Given the high time complexity of MINLP formulation, we propose a disjoint scheme that follows the strategy of splitting the sensor and beacon placement problems and determining the respective number of sensors and beacons by exhaustive search in linear time.

Non-Iterative Rigid 2D/3D Point-Set Registration Using Semidefinite Programming.

We describe a convex programming framework for pose estimation in 2D/3D point-set registration with unknown point correspondences. We give two mixed-integer nonlinear program (MINLP) formulations of the 2D/3D registration problem when there are multiple 2D images, and propose convex relaxations for both the MINLPs to semidefinite programs that can be solved efficiently by interior point methods. Our approach to the 2D/3D registration problem is non-iterative in nature as we jointly solve for pose and correspondence. Furthermore, these convex programs can readily incorporate feature descriptors of points to enhance registration results. We prove that the convex programs exactly recover the solution to the MINLPs under certain noiseless condition. We apply these formulations to the registration of 3D models of coronary vessels to their 2D projections obtained from multiple intra-operative fluoroscopic images. For this application, we experimentally corroborate the exact recovery property in the absence of noise and further demonstrate robustness of the convex programs in the presence of noise.

A specialized branch-and-bound algorithm for the Euclidean Steiner tree problem in n-space

We present a specialized branch-and-bound (b&b) algorithm for the Euclidean Steiner tree problem (ESTP) in $$\mathbb {R}^n$$Rn and apply it to a convex mixed-integer nonlinear programming (MINLP) formulation of the problem, presented by Fampa and Maculan. The algorithm contains procedures to avoid difficulties observed when applying a b&b algorithm for general MINLP problems to solve the ESTP. Our main emphasis is on isomorphism pruning, in order to prevent solving several equivalent subproblems corresponding to isomorphic Steiner trees. We introduce the concept of representative Steiner trees, which allows the pruning of these subproblems, as well as the implementation of procedures to fix variables and add valid inequalities. We also propose more general procedures to improve the efficiency of the b&b algorithm, which may be extended to the solution of other MINLP problems. Computational results demonstrate substantial gains compared to the standard b&b for convex MINLP.

A game-based meta-heuristic for a fuzzy bi-objective reliable hub location problem

Nowadays, offering fast and reliable delivery service has become a vital issue associated with all shipment delivery systems. Due to unpredictable variability in travel times, configuration of transportation systems plays a key role in ensuring of meeting the delivery service requirement. This paper tries to investigate the effect of delivery service requirement on the configuration of the transportation system through a hub-and-spoke network. The primary goal of this paper is to study a bi-objective single allocation p-hub center-median problem (BSpHCMP) by taking into account the uncertainty in flows, costs, times and hub operations. The proposed problem is modeled through a bi-objective mixed-integer non-linear programming (BMINLP) formulation that simultaneously locates p hubs, allocates spokes to the located hubs, and assigns different transportation mode to the hub-to-hub links. Then, a fuzzy-queuing approach is used to model the uncertainties in the network. Additionally, an efficient and powerful evolutionary algorithm based on game theory and invasive weed optimization algorithm was developed to solve the proposed BSpHCMP model and obtain near optimal Pareto solutions. Several experiments besides a real transportation case show the applicability of the proposed model as well as the superiority of the proposed solution approaches compared to NSGA-II and PAES algorithms.

Energy Efficiency Optimization for Mobile Ad Hoc Networks

Tremendous traffic demands for ubiquitous access and emerging multimedia applications significantly increase the energy consumption of battery-powered mobile devices. This trend leads to that energy efficiency (EE) becomes an essential aspect of mobile ad hoc networks (MANETs). In this paper, we explore EE optimization as measured in bits per Joule for MANETs based on the cross-layer design paradigm. We model this problem as a nonconvex mixed integer nonlinear programming (MINLP) formulation by jointly considering routing, traffic scheduling, and power control. Because the nonconvex MINLP problem is NP-hard in general, it is exceedingly difficult to globally optimize this problem. We, therefore, devise a customized branch and bound (BB) algorithm to efficiently solve this globally optimal problem. The novelties of our proposed BB algorithm include upper and lower bounding schemes and branching rule that are designed using the characteristics of the nonconvex MINLP problem. We demonstrate the efficiency of our proposed BB algorithm by offering numerical comparisons with a reference algorithm that uses the relaxation manners proposed in [1] – [3] . Numerical results show that our proposed BB algorithm scheme, respectively, decreases the optimality gap 81.98% and increases the best feasible solution 32.79% compared with the reference algorithm. Furthermore, our results not only provide insights into the design of EE maximization algorithms for MANETs by employing cooperations between different layers but also serve as performance benchmarks for distributed protocols developed for real-world applications.

An optimization approach for real time evacuation reroute planning

This paper addresses evacuation route management in the case of incidents arising during an evacuation and aims to minimize further delays they may cause. An evacuation reroute planning approach is developed for decision makers to utilize alternative routes in real time for the evacuees whose evacuation paths are affected by an incident during the evacuation process. Assuming that real time traffic information is available, a preprocessing algorithm is performed to update the evacuation networks. A multi-commodity network flow optimization model is then utilized to develop alternative paths and corresponding flow rates. Due to the underlying optimization model being a mixed integer nonlinear programming formulation, a linear reformulation of the model is developed to improve the computational performance. In the numerical results, the performance of the proposed decision making tool is tested using a case study.

Modeling and optimization methods of integrated production planning for steel plate mill with flexible customization

With diversified requirements and varying manufacturing environments, the optimal production planning for a steel mill becomes more flexible and complicated. The flexibility provides operators with auxiliary requirements through an implementable integrated production planning. In this paper, a mixed-integer nonlinear programming (MINLP) model is proposed for the optimal planning that incorporates various manufacturing constraints and flexibility in a steel plate mill. Furthermore, two solution strategies are developed to overcome the weakness in solving the MINLP problem directly. The first one is to transform the original MINLP formulation to an approximate mixed integer linear programming using a classic linearization method. The second one is to decompose the original model using a branch-and-bound based iterative method. Computational experiments on various instances are presented in terms of the effectiveness and applicability. The result shows that the second method performs better in computational efforts and solution accuracy.

Mixed Integer Nonlinear Programming Framework for Fixed Path Coordination of Multiple Underwater Vehicles Under Acoustic Communication Constraints

Mixed integer nonlinear programming (MINLP) techniques are increasingly used to address challenging problems in robotics, especially multivehicle motion planning (MVMP). The main contribution of this paper is a discrete time, distributed receding horizon mixed integer nonlinear programming (RH–MINLP) formulation of the underwater multivehicle path coordination problem with constraints on vehicle kinematics, dynamics, collision avoidance, and acoustic communication connectivity, and the application of state-of-the-art MINLP solution techniques. Each vehicle robot starts from a fixed start point and moves toward a goal point along a fixed path, so as to avoid collisions and remain in communication connectivity with other robots. Acoustic communication connectivity constraints account for the attenuation due to signal propagation and delays arising from multipath propagation in noisy communication environments, and specify intervehicle connectivity in terms of a signal-to-noise ratio (SNR) threshold. Scenarios including up to four robots are simulated to demonstrate: 1) the effect of communication connectivity requirements on robot velocity profiles; and 2) the dependence of the solution computation time on the communication connectivity requirement. Typically the optimization improved connectivity at no appreciable cost in journey time (as measured by the arrival time of the last arriving robot). Results also demonstrate the responsive nature of robot trajectories to safety requirements with collision avoidance being achieved at all times despite overlapping and intersecting paths.

Global optimization of multiphase flow networks using spline surrogate models

A general modelling framework for optimization of multiphase flow networks with discrete decision variables is presented. The framework is expressed with the graph and special attention is given to the convexity properties of the mathematical programming formulation that follows. Nonlinear pressure and temperature relations are modelled using multivariate splines, resulting in a mixed-integer nonlinear programming (MINLP) formulation with spline constraints. A global solution method is devised by combining the framework with a spline-compatible MINLP solver, recently presented in the literature. The solver is able to globally solve the nonconvex optimization problems. The new solution method is benchmarked with several local optimization methods on a set of three realistic subsea production optimization cases provided by the oil company BP.