Mixed-integer Nonlinear Programming Formulation Research Articles

Computing Solutions in Infinite-Horizon Discounted Adversarial Patrolling Games

Stackelberg games form the core of a number of tools deployed for computing optimal patrolling strategies in adversarial domains, such as the US Federal Air Marshall Service and the US Coast Guard. In traditional Stackelberg security game models the attacker knows only the probability that each target is covered by the defender, but is oblivious to the detailed timing of the coverage schedule. In many real-world situations, however, the attacker can observe the current location of the defender and can exploit this knowledge to reason about the defender’s future moves. We show that this general modeling framework can be captured using adversarial patrolling games (APGs) in which the defender sequentially moves between targets, with moves constrained by a graph, while the attacker can observe the defender’s current location and his (stochastic) policy concerning future moves. We offer a very general model of infinite-horizon discounted adversarial patrolling games. Our first contribution is to show that defender policies that condition only on the previous defense move (i.e., Markov stationary policies) can be arbitrarily suboptimal for general APGs. We then offer a mixed-integer non-linear programming (MINLP) formulation for computing optimal randomized policies for the defender that can condition on history of bounded, but arbitrary, length, as well as a mixed-integer linear programming (MILP) formulation to approximate these, with provable quality guarantees. Additionally, we present a non-linear programming (NLP) formulation for solving zero-sum APGs. We show experimentally that MILP significantly outperforms the MINLP formulation, and is, in turn, significantly outperformed by the NLP specialized to zero-sum games.

Economic optimization of combined cycle district heating systems

This paper presents a methodology for economic optimization of combined cycle district heating systems. Heat and power requirements vary over 24h periods due to changing weather conditions and consumer requirements. System thermal performance is highly dependent on ambient temperature and operating load, because individual component performances are nonlinear functions of these parameters. Since electric grid charges are much higher for on-peak than off-peak periods, on-site fuel choices vary in prices, and cheaper fuel availabilities are limited by suppliers, opportunities arise to optimally schedule system operation, and minimize total daily running cost. For such problems a mixed-integer nonlinear programming formulation is proposed. Limited fuel availability constraints make problem solving difficult using classical techniques such as the branch-and-bound method. As an alternative, a genetic algorithm is proposed in which a genetic search is applied only on integer variables and a gradient search is applied on continuous variables. A comparative study using actual system operation data shows optimal scheduling can reduce total daily running cost by 11% and improve system operating efficiency by 6%.

Models and solution techniques for production planning problems with increasing byproducts

We consider a production planning problem where the production process creates a mixture of desirable products and undesirable byproducts. In this production process, at any point in time the fraction of the mixture that is an undesirable byproduct increases monotonically as a function of the cumulative mixture production up to that time. The mathematical formulation of this continuous-time problem is nonconvex. We present a discrete-time mixed-integer nonlinear programming (MINLP) formulation that exploits the increasing nature of the byproduct ratio function. We demonstrate that this new formulation is more accurate than a previously proposed MINLP formulation. We describe three different mixed-integer linear programming (MILP) approximation and relaxation models of this nonconvex MINLP, and we derive modifications that strengthen the linear programming relaxations of these models. We also introduce nonlinear programming formulations to choose piecewise-linear approximations and relaxations of multiple functions that share the same domain and use the same set of break points in the domain. We conclude with computational experiments that demonstrate that the proposed formulation is more accurate than the previous formulation, and that the strengthened MILP approximation and relaxation models can be used to obtain provably near-optimal solutions for large instances of this nonconvex MINLP. Experiments also illustrate the quality of the piecewise-linear approximations produced by our nonlinear programming formulations.

Simultaneous synthesis of work exchange networks with heat integration

The optimal integration of work and its interaction with heat can represent large energy savings in industrial plants. This paper introduces a new optimization model for the simultaneous synthesis of work exchange networks (WENs), with heat integration for the optimal pressure recovery of process gaseous streams. The proposed approach for the WEN synthesis is analogous to the well-known problem of synthesis of heat exchanger networks (HENs). Thus, there is work exchange between high-pressure (HP) and low-pressure (LP) streams, achieved by pressure manipulation equipment running on common axes. The model allows the use of several units of single-shaft-turbine-compressor (SSTC), as well as stand-alone compressors, turbines and valves. Helper motors and generators are used to respond to any demand and excess of energy. Moreover, between the WEN stages the streams are sent to the HEN to promote thermal recovery, aiming to enhance the work integration. A multi-stage superstructure is proposed to represent the process. The WEN superstructure is optimized in a mixed-integer nonlinear programming (MINLP) formulation and solved with the GAMS software, with the goal of minimizing the total annualized cost. Three examples are conducted to verify the accuracy of the proposed method. In all case studies, the heat integration between WEN stages is essential to improve the pressure recovery, and to reduce the total costs involved in the process.

The Robust Redundancy Allocation Problem in Series-Parallel Systems With Budgeted Uncertainty

We propose a robust optimization framework to deal with uncertain component reliabilities in redundancy allocation problems in series-parallel systems. The proposed models are based on linearized versions of standard mixed integer nonlinear programming (MINLP) formulations of these problems. We extend the linearized models to address uncertainty by assuming that the component reliabilities belong to a budgeted uncertainty set, and develop robust counterpart models. A key challenge is that, because the models involve nonlinear functions of the uncertain data, classical robust optimization approaches cannot apply directly to construct their robust optimization counterparts. We exploit problem structure to develop robust counterparts and exact solution methods, and present computational results demonstrating their performance.

On Efficient Resource Allocation for Cognitive and Cooperative Communications

Cooperative communication (CC) can offer high channel capacity and reliability in an efficient and low-cost way by forming a virtual antenna array among single-antenna nodes that cooperatively share their antennas. It has been well recognized that the selection of relay nodes plays a critical role in the performance of multiple source-destination pairs. Unfortunately, all prior work has made an unrealistic assumption that spectrum resources are unlimited and each source-destination pair can communicate over a dedicated channel with no mutual interference. In this paper, we study the problem of maximizing the minimum transmission rate among multiple source-destination pairs using CC in a cognitive radio network (CRN). We jointly consider the relay assignment and channel allocation under a finite set of available channels, where the interference must be considered. In order to improve the spectrum efficiency, we exploit the network coding opportunities existing in CC that can further increase the capacity. Such max-min rate problems for cognitive and cooperative communications are proved to be NP-hard and the corresponding MINLP (Mixed-Integer Nonlinear Programming) formulations are developed. Moreover, we apply the reformulation and linearization techniques to the original optimization problems with nonlinear and nonconvex objective functions such that our proposed algorithms can produce high competitive solutions in a timely manner. Extensive simulations are conducted to show that the proposed algorithms can achieve high spectrum efficiency in terms of providing a much improved max-min transmission rate under various network settings.

Heat integrated resource conservation networks without mixing prior to heat exchanger networks

Abstract This paper presents a generic approach for the synthesis of heat integrated resource conservation networks (HIRCNs) of the fixed flow rate problem, where process sources linked directly to process sinks without any prior mixing. The mixed integer non-linear program (MINLP) formulation complemented by floating pinch concept was developed to determine the optimum fresh material resources as well as hot and cold energy utilities. The proposed approach is applicable for both concentration- and property-based direct reuse/recycle system with variable operating parameters (i.e. flow rates, temperatures and properties). Three literature case studies are solved to illustrate the proposed approach.

Formulation and solution of distribution system voltage and VAR control with distributed generation as a mixed integer non-linear programming problem

Coordinated voltage and VAR control (VVC) can provide major economic benefits for distribution utilities. Incorporating distributed generations (DG) for VVC can improve the efficiency and reliability of distribution systems. This paper presents an approach to formulate and solve distribution system VVC with DG units as a mixed integer non-linear programming (MINLP) problem. The method can be utilized to create an effective control scheme for both the traditional VVC devices and DG units. The MINLP formulation is based on three-phase power flow formulation, and is solved with an open-source BONMIN optimization solver with outer approximation (OA) algorithm. BONMIN is interfaced with Matlab via a third-party optimization toolbox. The proposed approach is applied to several distribution feeder models with promising results.

Threshold Boolean form for joint probabilistic constraints with random technology matrix

We develop a new modeling and solution method for stochastic programming problems that include a joint probabilistic constraint in which the multirow random technology matrix is discretely distributed. We binarize the probability distribution of the random variables in such a way that we can extract a threshold partially defined Boolean function (pdBf) representing the probabilistic constraint. We then construct a tight threshold Boolean minorant for the pdBf. Any separating structure of the tight threshold Boolean minorant defines sufficient conditions for the satisfaction of the probabilistic constraint and takes the form of a system of linear constraints. We use the separating structure to derive three new deterministic formulations for the studied stochastic problem, and we derive a set of strengthening valid inequalities. A crucial feature of the new integer formulations is that the number of integer variables does not depend on the number of scenarios used to represent uncertainty. The computational study, based on instances of the stochastic capital rationing problem, shows that the mixed-integer linear programming formulations are orders of magnitude faster to solve than the mixed-integer nonlinear programming formulation. The method integrating the valid inequalities in a branch-and-bound algorithm has the best performance.

Facility Location-Network Design Problem: Reliability and Investment Budget Constraint

The simultaneous consideration of combined facility location and network design is an attractive empirical topic for locating public and private facilities and for reducing transportation costs. Moreover, applying a reliable facility location problem is one of the most effective ways to protect against occasional failures of the system, because of considering unexpected failures and possible customer reassignment when locating facilities. Accordingly, this study concerns a reliable combined facility location–network design problem regarding the investment budget constraint and proposes a mixed integer nonlinear programming formulation to model this constraint. The proposed model is linearized by suitable techniques. Also, a practical case study subject to sustainable design in urban health care development is presented in detail to illustrate the application of the proposed mathematical model. Finally, a sensitivity analysis is conducted to provide insights into the behavior of the proposed model in response to the changes of key parameters of the problem.

Simultaneous model discrimination and parameter estimation in dynamic models of cellular systems

BackgroundModel development is a key task in systems biology, which typically starts from an initial model candidate and, involving an iterative cycle of hypotheses-driven model modifications, leads to new experimentation and subsequent model identification steps. The final product of this cycle is a satisfactory refined model of the biological phenomena under study. During such iterative model development, researchers frequently propose a set of model candidates from which the best alternative must be selected. Here we consider this problem of model selection and formulate it as a simultaneous model selection and parameter identification problem. More precisely, we consider a general mixed-integer nonlinear programming (MINLP) formulation for model selection and identification, with emphasis on dynamic models consisting of sets of either ODEs (ordinary differential equations) or DAEs (differential algebraic equations).ResultsWe solved the MINLP formulation for model selection and identification using an algorithm based on Scatter Search (SS). We illustrate the capabilities and efficiency of the proposed strategy with a case study considering the KdpD/KdpE system regulating potassium homeostasis in Escherichia coli. The proposed approach resulted in a final model that presents a better fit to the in silico generated experimental data.ConclusionsThe presented MINLP-based optimization approach for nested-model selection and identification is a powerful methodology for model development in systems biology. This strategy can be used to perform model selection and parameter estimation in one single step, thus greatly reducing the number of experiments and computations of traditional modeling approaches.

Identification of reservation capacity in critical peak pricing electricity demand response program for sustainable manufacturing systems

SUMMARY Electricity demand response is considered an effective approach to balance the electricity demand and supply with existing infrastructure of generation, transmission, and distribution. A majority of existing literature on the electricity demand response has mainly centered on the commercial and residential building sectors while the application for the industrial sector is largely neglected. This paper presents a methodology for the application of a typical demand response program, Critical Peak Pricing (CPP) program, for the manufacturing enterprises. The configuration of the reservation capacity (in kilowatt, kW) in the CPP program, which plays a critical role in the cost of the final bill charge, will be identified by optimal production scheduling for the typical manufacturing systems with multiple machines and buffers. Mixed Integer Nonlinear Programming formulation is used to establish the mathematical model with the objective to minimize the electricity bill cost as well as the potential penalty cost due to the non-fulfillment of the target production. An approximate technique is introduced to find a near optimal solution, and a numerical case study is used to illustrate the effectiveness of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.

Integrated Inventory Control and Facility Location Decisions in a Multi-Echelon Supply Chain Network with Hubs

This paper develops mathematical models to coordinate facility location and inventory control for a four-echelon supply chain network consisting of multiple suppliers, warehouses, hubs and retailers. The hubs help in reducing transportation costs by consolidating products from multiple warehouses and directing the larger shipments to the retailer. The integrated models studied in this paper simultaneously determines three types of decisions: (i) facility location—the number and location of warehouses and hubs, (ii) allocation—assignment of suppliers to located warehouses and retailers to located warehouses via the location hubs, and (iii) inventory control decisions at each located warehouse. The goal is to minimize the facility location, transportation and the inventory costs. A mixed integer nonlinear programming formulation is first presented. The nonlinear integer programming formulation is then transformed into a conic mixed integer program and a novel and compact conic mixed integer programming formulation. Computational runs are conducted using commercial solvers to compare the performance of the different formulations. The compact conic mixed integer programming formulation was found to significantly outperform the other formulations by achieving significant computational savings. The results demonstrate that large scale instances of certain multi-echelon supply chain network design problems can be solved using commercial solvers through intelligent reformulation of the model.

Retrofit of steam power plants in a petroleum refinery

A comprehensive mathematical model is developed for the operational optimization and retrofit of industrial steam systems. The problem of maximizing the cost reduction given by the difference in annual operating costs minus the annualized investment cost of the retrofit is formulated as a mixed integer nonlinear program (MINLP) based on a collection of unit models. The steam power plant of a traditional petroleum refinery is analyzed and retrofitted based on the proposed MINLP formulation. The operating conditions are firstly optimized to investigate whether there is room for improvement without modifying existent units or layout. Thereafter, retrofit possibilities on internal steam turbine or layout are taken into account to enhance steam utilization efficiency. Feasible method for inspiring the total site steam integration among adjacent companies is finally deliberated, where the steam ejector is adopted to upgrade the lower pressure import steam for being directed into the existent steam headers. The results of various degrees of retrofit on the steam system of a practical refinery verify the applicability of the proposed approach.

Design and Optimization of Flexible Hydrogen Systems in Refineries

With the increasing demand for hydrogen resulting from fierce market competition and stringent environmental legislation, the hydrogen system has become an important component of a refinery. It is vital for the hydrogen system to be operated economically and safely under varying operating conditions. This calls for a systematic approach to the design and optimization of flexible hydrogen systems, which is the aim of this article. The hydrogen distribution network is designed at the minimum total annual cost subject to constraints on the flow rates and pressures of both existing and new equipment during the payback period. Varying hydrogen demands, different pipeline levels, and the possibility of hydrogen units being shut down are considered as operating conditions in the design optimization task, leading to the formulation and solution of a mixed-integer nonlinear programming (MINLP) problem. Using a linearization method, the MINLP formulation is approximated by a mixed-integer linear programming (MILP) ...

Embedding structural information in simulation-based optimization

This paper proposes and explores an algorithm designed to find optimal settings for a process network. Emphasis is put on the system being divisible into components, as this underlying assumption motivates the algorithm in its entirety in that rather simple relations between the system components are modeled as explicit structural constraints, while the significantly more complex relations within each component are approximated based on the underlying simulator data. Although the approach taken in this paper is rather broadly applicable we are, in particular, interested in its application to production optimization problems in the oil and gas industry. We give limited numerical results for one such example that clearly indicates the advantages of our approach. We show the advantages of both decomposing the problem of interest and accounting for the structure from the point of view of exploiting, where ever possible, the explicitly analytic aspects of the problem. The advantage of doing the former is that the considered subproblems are significantly smaller than the overall problem. The advantage of the latter is that one can use derivatives for the analytic parts whereas they are unavailable for the simulators. The underlying approach is a trust-region one with a mixed integer nonlinear program formulation. There are some significant differences in the details of the algorithm from those generally available for such problems.

A solution procedure for mixed-integer nonlinear programming formulation of supply chain planning with quantity discounts under demand uncertainty

Quantity discount policy is decision-making for trade-off prices between suppliers and manufacturers while production is changeable due to demand fluctuations in a real market. In this paper, quantity discount models which consider selection of contract suppliers, production quantity and inventory simultaneously are addressed. The supply chain planning problem with quantity discounts under demand uncertainty is formulated as a mixed-integer nonlinear programming problem (MINLP) with integral terms. We apply an outer-approximation method to solve MINLP problems. In order to improve the efficiency of the proposed method, the problem is reformulated as a stochastic model replacing the integral terms by using a normalisation technique. We present numerical examples to demonstrate the efficiency of the proposed method.

Optimal operation scheduling of wind power integrated with compressed air energy storage (CAES)

This study optimises and compares the operation of a conventional gas-fired power generation company with its operation in combination with wind power and compressed air energy storage (CAES). A mixed integer non-linear programming (MINLP) formulation is developed for the optimisation problem. Limits in ramp rate, capacity, and minimum on/off time, as well as start-up cost constraints, are considered for the modelling of conventional units. Injected and produced power constraints, storage, air balance and CAES-operation limits are considered in the CAES modelling. Two objective functions (profit maximisation and cost minimisation) are modelled. Without considering capital costs, it is found that the use of CAES results in 43% higher operational profits and 6.7% lower costs in a market environment.

Integrated Approach for Simultaneous Mass and Property Integration for Resource Conservation

The objective of this work is to develop an integrated, systematic, and cost-effective design technique to synthesize a resource conservation network. On the basis of different characteristics for the process constraint and environmental discharge limit, mass and property integration techniques are adapted in this proposed approach. A conceptual framework is first developed to serve as the basis of the mathematical optimization model. A mixed-integer nonlinear programming (MINLP) formulation is developed to synthesize a resource conservation network with minimum total annualized cost. A case study on the production of phenol from cumene hydroperoxide is presented to illustrate the proposed approach.

MINLP formulations for solving strip packing problems in LCD mother glass production

Mother glass production for liquid crystal displays involves the cutting of small rectangles from a large rectangle (strip) to minimize the length of the strip used. This is called the strip packing problem. The main concept for formulating this problem is to generate cutting patterns that can be used to produce items from the strip. This problem is a non-convex mixed integer nonlinear programming (MINLP) problem due to the bilinear terms in the demand and objective function. To obtain global optimal solutions, this problem should be transformed into several linear forms. Numerical examples based on the strip packing problem in LCD mother glass production are provided. Different objective functions for two problems are presented and compared.