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Mathematical models and algorithms for a high school timetabling problem

COMPUTERS & OPERATIONS RESEARCH 2015年 61卷 56-68页

作者： Al-Yakoob, Salem M. Sherali, Hanif D. Kuwait Univ Dept Math Coll Sci Safat 13060 Kuwait Virginia Tech Grado Dept Ind & Syst Engn 0118 Blacksburg VA 24061 USA

This paper investigates a high school timetabling problem in a case study related to Kuwait's public educational system, which is concerned with assigning teachers to classes and time-slots. Because a direct solution to an initially formulated comprehensive mixed-integer programming model for generating weekly teacher schedules was found to be untenable for practical-sized realistic test instances, we propose in this paper two decomposition approaches to the underlying problem. A two-stage modeling solution approach is presented first, where the initial stage determines weekly time-slots for the classes, based on which, the second stage then assigns teachers to classes. Instead of generating weekly schedules within the model itself, we propose another mixed-integer programming formulation that selects valid combinations of weekly schedules from the set of all feasible schedules, and we design a column generation solution framework to exploit its inherent special structure. Computational results are presented for the proposed solution approaches using several real as well as simulated realistic test problems pertaining to high schools in Kuwait (C) 2015 Elsevier Ltd. All rights reserved.

关键词： Timetabling Scheduling mixed-integer programming Two-stage approach Column generation Linear programming-based heuristic

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A polyhedral characterization of the inverse-feasible region of a mixed-integer program

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OPERATIONS RESEARCH LETTERS 2015年第6期43卷 575-578页

作者： Lamperski, Jourdain B. Schaefer, Andrew J. MIT Ctr Operat Res Cambridge MA 02139 USA Rice Univ Dept Computat & Appl Math Houston TX 77251 USA

Given a feasible solution x(0) to a mixed-integer program (MIP), the inverse MIP problem is to find an objective d such that x(0) is optimal for the MIP with objective function d, and among all such objectives, the distance from a given target objective is minimized. By using a novel expression for the MIP value function, we formulate the inverse MIP problem as a linear program (LP), albeit of exponentially large size. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Inverse optimization mixed-integer programming Superadditive duality Value functions

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Convergence of the Surrogate Lagrangian Relaxation Method

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2015年第1期164卷 173-201页

作者： Bragin, Mikhail A. Luh, Peter B. Yan, Joseph H. Yu, Nanpeng Stern, Gary A. Univ Connecticut Dept Elect & Comp Engn Storrs CT 06269 USA Southern Calif Edison Rosemead CA 91770 USA

Studies have shown that the surrogate subgradient method, to optimize non-smooth dual functions within the Lagrangian relaxation framework, can lead to significant computational improvements as compared to the subgradient method. The key idea is to obtain surrogate subgradient directions that form acute angles toward the optimal multipliers without fully minimizing the relaxed problem. The major difficulty of the method is its convergence, since the convergence proof and the practical implementation require the knowledge of the optimal dual value. Adaptive estimations of the optimal dual value may lead to divergence and the loss of the lower bound property for surrogate dual values. The main contribution of this paper is on the development of the surrogate Lagrangian relaxation method and its convergence proof to the optimal multipliers, without the knowledge of the optimal dual value and without fully optimizing the relaxed problem. Moreover, for practical implementations, a stepsizing formula that guarantees convergence without requiring the optimal dual value has been constructively developed. The key idea is to select stepsizes in a way that distances between Lagrange multipliers at consecutive iterations decrease, and as a result, Lagrange multipliers converge to a unique limit. At the same time, stepsizes are kept sufficiently large so that the algorithm does not terminate prematurely. At convergence, the lower-bound property of the surrogate dual is guaranteed. Testing results demonstrate that non-smooth dual functions can be efficiently optimized, and the new method leads to faster convergence as compared to other methods available for optimizing non-smooth dual functions, namely, the simple subgradient method, the subgradient-level method, and the incremental subgradient method.

关键词： Non-smooth optimization Subgradient methods Surrogate subgradient method Lagrangian relaxation mixed-integer programming

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Optimal allocation of water to competing stakeholders in a shared watershed

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ANNALS OF OPERATIONS RESEARCH 2015年第1期229卷 657-676页

作者： Roozbahani, R. Abbasi, B. Schreider, S. RMIT Univ Sch Math & Geospatial Sci Melbourne Vic 3001 Australia

This paper proposes a mathematical model for allocating water to stakeholders of a shared watershed. Each stakeholder in the basin has a water demand and a water profit;however, the available water cannot meet the demands of all stakeholders. This shortage raises a conflict between stakeholders as they use a common resource. To reach an agreement between the stakeholders in water allocation, first a model was developed to obtain the highest possible profit that a stakeholder can achieve if the stakeholder is allowed to utilize as much as possible water after satisfying the basin environmental demands (flows). Then, another model was introduced which allocates water to each stakeholder such that the minimum ratio of stakeholders' profits to their highest possible profits is maximized. It is shown that the obtained solution is non-dominated in terms of considering each stakeholder profit as an objective, which means that none of the objective functions can be improved in value without degrading some of the other objective values. The proposed method is applied to the Sefidrud River basin, which is one of the biggest rivers in Iran. The stakeholders of this basin are eight administrative provinces that compete for utilizing more water while the Basin's water resources could not satisfy all stakeholders' water requirements. The model's results show that it can successfully be used for sustainable conflict resolution in a shared basin because the model satisfies the environmental water requirement in the entire basin and provides equitably the same ratio of the stakeholders' highest possible profits for them. For the case of this study, the proposed approach allocates water to the stakeholders in such a way that they could obtain at least 65 % of their highest possible profits in average.

关键词： Water allocation mixed-integer programming Multi-objective optimization Shared watersheds

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Bilevel programming for price-based electricity auctions: a revenue-constrained case

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EURO JOURNAL ON COMPUTATIONAL OPTIMIZATION 2015年第3期3卷 163-195页

作者： Fernandez-Blanco, Ricardo Arroyo, Jose M. Alguacil, Natalia Univ Castilla La Mancha ETSI Ind Dept Ingn Elect Elect Automat & Comunicac Ciudad Real 13071 Spain

This paper describes the application of bilevel programming to a class of real-life problems in the field of electric power systems. Within the context of electricity markets, market-clearing procedures, i.e., auction models, are used by an independent entity to schedule generation offers and consumption bids as well as to determine market-clearing prices. This paper addresses a mathematically challenging type of auction, denoted as price-based market clearing, wherein, as a distinctive feature, market-clearing prices are explicitly incorporated in the formulation of the optimization process. This paper shows that bilevel programming provides a suitable modeling framework for price-based market clearing. Furthermore, based on practical modeling aspects, an equivalent single-level primal-dual transformation into a mixedinteger program can be implemented. Such transformation relies on the application of duality theory of linear programming. The bilevel programming framework for price-based market clearing is applied to a revenue-constrained auction model similar to those used in several European electricity markets. As a major contribution, bilinear terms associated with both generation revenue constraints and the duality-based transformation are equivalently converted into linear forms with no additional binary variables. Simulation results show the effective performance of the proposed approach and its superiority over current industry practice.

关键词： Bilevel programming mixed-integer programming Price-based market clearing Revenue-constrained auction model

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On solving a hard quadratic 3-dimensional assignment problem

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MATHEMATICAL programming COMPUTATION 2015年第2期7卷 219-234页

作者： Mittelmann, Hans D. Salvagnin, Domenico Arizona State Univ Sch Math & Stat Sci Tempe AZ USA Univ Padua DEI Padua Italy

We address the solution of a very challenging ( and previously unsolved) instance of the quadratic 3-dimensional assignment problem, arising in digital wireless communications. The paper describes the techniques developed to solve this instance to optimality, from the choice of an appropriate mixed-integer programming formulation, to cutting planes and symmetry handling. Using these techniques we were able to solve the target instance with moderate computational effort (2.5 million nodes and 1 week of computations on a standard PC).

关键词： mixed-integer programming Quadratic assignment Symmetry

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Synchronisation of customers’ orders using discounts

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International Journal of Systems Science: Operations and Logistics 2016年第3期3卷 148-162页

作者： Buhayenko, Viktoryia van Eikenhorst, Erik CORAL Department of Economics and Business School of Business and Social Sciences Aarhus University Aarhus V Denmark Faculty of Economics Informatics and Social Sciences Molde University College Molde Norway

Conventionally, price discounts are offered to enhance demand, for example, to get rid of excess inventory. This paper, however, studies the possibility of using discounts in situations when total demand is non-sensitive to changes in prices. By introducing a discount in some periods, the supplier is able to give his customers the incentive to change their order patterns in a way that minimises his total cost. With the new order schemes, all parties are at least as well off as before. A mono-objective reformulation for this conflicting bi-objective problem is proposed and a mixed-integer-based heuristic is developed for the reformulated problem. The heuristic algorithm is based on a separation between determining the order/set-up pattern and the required discounts. Extensive numerical studies show that the application of the heuristic results in a benefit for the whole supply chain. To increase the supplier's profit, a modification of the heuristic which differs in the discount approximation used is developed. © 2015, © 2015 Taylor & Francis.

关键词： Discounts dynamic lot-sizing heuristics mixed-integer programming supplier's profit maximisation

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Energy Demand Response of Process Systems through Production Scheduling and Control 9

Energy Demand Response of Process Systems through Production...

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9th IFAC Symposium on Advanced Control of Chemical Processes ADCHEM 2015

作者： Tong, Chudong El-Farra, Nael H. Palazoglu, Ahmet Univ Calif Davis Dept Chem Engn & Mat Sci Davis CA 95616 USA

Demand response has become a topic of significant research, development, and deployment over the last few years. The energy demand management is a critical task in industrial process systems for the potential benefits to be realized by promoting the interaction and responsiveness of process operation. However, the dynamic behavior, especially transition trajectories, of the underlying process is seldom taken into account during this task. Furthermore, the incorporation of energy constraints related to electricity pricing and availability is one of the key challenges in this process. The purpose of this study is thus to present a novel optimization formulation for energy demand management in dynamic process systems that takes transition behavior and cost explicitly into account, while simultaneously handling time-sensitive electricity prices. This is accomplished by bringing together production scheduling and transition control through a real-time optimization framework. The dynamic formulation is cast as a mixed-integer nonlinear programming problem and demonstrated using a continuous stirred tank reactor example where the energy required is assumed to be roughly proportional to the material flow. (C) 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

关键词： Demand Response Energy systems Production Scheduling Control mixed-integer programming Chemical Processes

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Damper placement optimization in a shear building model with discrete design variables: a mixed-integer second-order cone programming approach

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EARTHQUAKE ENGINEERING & STRUCTURAL DYNAMICS 2013年第11期42卷 1657-1676页

作者： Kanno, Yoshihiro Univ Tokyo Dept Math Informat Tokyo 1138656 Japan

Supplemental damping is known as an efficient and practical means to improve seismic response of building structures. Presented in this paper is a mixed-integer programming approach to find the optimal placement of supplemental dampers in a given shear building model. The damping coefficients of dampers are treated as discrete design variables. It is shown that a minimization problem of the sum of the transfer function amplitudes of the interstory drifts can be formulated as a mixed-integer second-order cone programming problem. The global optimal solution of the optimization problem is then found by using a solver based on a branch-and-cut algorithm. Two numerical examples in literature are solved with discrete design variables. In one of these examples, the proposed method finds a better solution than an existing method in literature developed for the continuous optimal damper placement problem. Copyright (c) 2013 John Wiley & Sons, Ltd.

关键词： optimal damper placement aseismic design transfer function structural control mixed-integer programming global optimization

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Transmission Network Restoration Considering AC Power Flow Constraints

Transmission Network Restoration Considering AC Power Flow C...

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IEEE International Conference on Smart Grid Communications (SmartGridComm)

作者： Arab, Ali Khodaei, Amin Khator, Suresh K. Han, Zhu Univ Houston Dept Ind Engn Houston TX 77204 USA Univ Denver Dept Elect & Comp Engn Denver CO 80208 USA Univ Houston Dept Elect & Comp Engn Houston TX 77004 USA

ISBN: (纸本)9781467382892

This paper develops an efficient model for power system restoration after natural disasters based on AC power flow constraints. The objective is to derive the optimal restoration schedule in order to minimize the real power load interruptions in the post-disaster phase. The load criticality is represented by using the value of lost load which prioritizes the loads to be restored. A linear AC formulation is further proposed to allow the calculation of voltage angle and reactive power in the network, hence ensuring a more practical solution compared to DC power flow based models. mixed-integer programming is used to formulate the proposed restoration model. Numerical analysis on the IEEE 118-bus test system demonstrates the effectiveness and the applicability of the proposed restoration model.

关键词： AC power flow natural disaster mixed-integer programming power system restoration

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