We show that every facet-defining inequality of the convex hull of a mixed-integer polyhedral set with two integer variables is a crooked cross cut (which we defined in 2010). We extend this result to show that crooke...
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We show that every facet-defining inequality of the convex hull of a mixed-integer polyhedral set with two integer variables is a crooked cross cut (which we defined in 2010). We extend this result to show that crooked cross cuts give the convex hull of mixed-integer sets with more integer variables if the coefficients of the integer variables form a matrix of rank 2. We also present an alternative characterization of the crooked cross cut closure of mixed-integer sets similar to the one on the equivalence of different definitions of split cuts presented in Cook et al. (1990) [4]. This characterization implies that crooked cross cuts dominate the 2-branch split cuts defined by Li and Richard (2008) [8]. Finally, we extend our results to mixed-integer sets that are defined as the set of points (with some components being integral) inside a closed, bounded and convex set. (C) 2011 Elsevier B.V. All rights reserved.
The problem is related to a fleet of military aircraft with a certain flying program in which the availability of the aircraft sufficient to meet the flying program is a challenging issue. During the pre- or after-fli...
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The problem is related to a fleet of military aircraft with a certain flying program in which the availability of the aircraft sufficient to meet the flying program is a challenging issue. During the pre- or after-flight inspections, some component failures of the aircraft may be found. In such cases, the aircraft are sent to the repair shop to be scheduled for maintenance jobs, consisting of failure repairs or preventive maintenance tasks. The objective is to schedule the jobs in such a way that sufficient number of aircrafts is available for the next flight programs. The main resource, as well as the main constraint, in the shop is skilled-workforce. The problem is formulated as a mixed-integer mathematical programming model in which the network flow structure is used to simulate the flow of aircraft between missions, hanger and repair shop. The proposed model is solved using the classical Branch-and-Bound method and its performance is verified and analyzed in terms of a number of test problems adopted from the real data. The results empirically supported practical utility of the proposed model.
Planning for a bus-based regional evacuation is essential for emergency preparedness, especially for regions threatened by hurricanes that have large numbers of transit-dependent people. While this difficult planning ...
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Planning for a bus-based regional evacuation is essential for emergency preparedness, especially for regions threatened by hurricanes that have large numbers of transit-dependent people. While this difficult planning problem is a variant of the vehicle routing problem, it differs in some key aspects, including the objective and the network structure (e.g., capacitated shelters). This problem is not well studied. In this paper we introduce a model specifically designed for bus-based evacuation planning, along with two mathematical programming formulations, which are used to develop a heuristic algorithm. Using these models, we analyze the differences in the structural properties of optimal solutions between this problem and traditional vehicle routing problems.
Gomory mixed-integer cuts (GMICs) are widely used in modern branch-and-cut codes for the solution of mixed-integer programs. Typically, GMICs are iteratively generated from the optimal basis of the current linear prog...
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Gomory mixed-integer cuts (GMICs) are widely used in modern branch-and-cut codes for the solution of mixed-integer programs. Typically, GMICs are iteratively generated from the optimal basis of the current linear programming ( LP) relaxation, and immediately added to the LP before the next round of cuts is generated. Unfortunately, this approach is prone to instability. In this paper we analyze a different scheme for the generation of rank-1 GMICs read from a basis of the original LP-the one before the addition of any cut. We adopt a relax-and-cut approach where the generated GMICs are not added to the current LP, but immediately relaxed in a Lagrangian fashion. Various elaborations of the basic idea are presented, that lead to very fast-yet accurate-variants of the basic scheme. Very encouraging computational results are presented, with a comparison with alternative techniques from the literature also aimed at improving the GMIC quality. We also show how our method can be integrated with other cut generators, and successfully used in a cut-and-branch enumerative framework.
In this paper a weighting method is developed to find the solution of a 0-1 Indefinite Quadratic Bilevel programming problem. The proposed approach converts the hierarchical system into a scalar optimization problem b...
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In this paper a weighting method is developed to find the solution of a 0-1 Indefinite Quadratic Bilevel programming problem. The proposed approach converts the hierarchical system into a scalar optimization problem by finding the proper weights using the Analytic Hierarchy Process (AHP). These weights are used to combine the objective functions of both levels into one objective. Here, the relative weights represent the relative importance of the objective functions. The reduced problem, that is, the scalar optimization problem is then linearized and it is solved with an appropriate optimization software. The algorithm is explained with the help of an example.
We consider a revenue management model for pricing a product line with several customer segments under the assumption that customers' product choices are determined entirely by their reservation prices. We highlig...
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We consider a revenue management model for pricing a product line with several customer segments under the assumption that customers' product choices are determined entirely by their reservation prices. We highlight key mathematical properties of the maximum utility model and formulate it as a mixed-integer programming problem, design heuristics and valid cuts. We further present extensions of the models to deal with various practical issues arising in applications. Our computational experiments with real data from the tourism sector as well as with the randomly generated data show the effectiveness of our approach.
Persistent calls come from within the graduate medical education community and from external sources for regulating the resident duty hours in order to meet the obligations about the quality of resident education, the...
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Persistent calls come from within the graduate medical education community and from external sources for regulating the resident duty hours in order to meet the obligations about the quality of resident education, the well-being of residents themselves, and the quality of patient care services. The report of the Accreditation Council for Graduate Medical Education (ACGME) proposes common program requirements for resident hours. In this paper, we first develop a mixed-integer programming model for scheduling residents' duty hours considering the on-call night, day-off, rest period, and total work-hour ACGME regulations as well as the demand coverage requirements of the residency program. Subsequently, we propose a column generation model that consists of a master problem and an auxiliary problem. The master problem finds a configuration of individual schedules that minimizes the sum of deviations from the desired service levels for the day and night periods. The formulation of this problem is possible by representing the feasible schedules using column variables, whereas the auxiliary problem finds the whole set of feasible schedules using constraint programming. The proposed approach has been tested on a series of problems using real data obtained from a hospital. The results indicate that high-quality schedules can be obtained within a few seconds. (C) 2010 Elsevier Ltd. All rights reserved.
This paper addresses the problem of storage assignment in a warehouse characterized by multi-command picking and served by milkrun logistics. In such a logistic system, vehicles circulate between the warehouse and the...
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This paper addresses the problem of storage assignment in a warehouse characterized by multi-command picking and served by milkrun logistics. In such a logistic system, vehicles circulate between the warehouse and the production facilities of the plant according to a pre-defined schedule, often with multiple cycles (routes) serving different departments. We assume that a request probability can be assigned to each item and each cycle, which leads to a special case of the correlated storage assignment problem. A MIP model is proposed for finding a class-based storage policy that minimizes the order cycle time, the average picking effort, or a linear combination of these two criteria. Computational experiments show that our approach can achieve an up to 36-38% improvement in either criterion compared to the classical COI-based strategy. (C) 2010 Elsevier B.V. All rights reserved.
Persistent calls come from within the graduate medical education community and from external sources for regulating the resident duty hours in order to meet the obligations about the quality of resident education, the...
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Persistent calls come from within the graduate medical education community and from external sources for regulating the resident duty hours in order to meet the obligations about the quality of resident education, the well-being of residents themselves, and the quality of patient care services. The report of the Accreditation Council for Graduate Medical Education (ACGME) proposes common program requirements for resident hours. In this paper, we first develop a mixed-integer programming model for scheduling residents' duty hours considering the on-call night, day-off, rest period, and total work-hour ACGME regulations as well as the demand coverage requirements of the residency program. Subsequently, we propose a column generation model that consists of a master problem and an auxiliary problem. The master problem finds a configuration of individual schedules that minimizes the sum of deviations from the desired service levels for the day and night periods. The formulation of this problem is possible by representing the feasible schedules using column variables, whereas the auxiliary problem finds the whole set of feasible schedules using constraint programming. The proposed approach has been tested on a series of problems using real data obtained from a hospital. The results indicate that high-quality schedules can be obtained within a few seconds. (C) 2010 Elsevier Ltd. All rights reserved.
The fixed-charge transportation problem is a fixed-charge network flow problem on a bipartite graph. This problem appears as a subproblem in many hard transportation problems, and has also strong links with the challe...
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ISBN:
(纸本)9783642208072
The fixed-charge transportation problem is a fixed-charge network flow problem on a bipartite graph. This problem appears as a subproblem in many hard transportation problems, and has also strong links with the challenging big-bucket multi-item lot-sizing problem. We provide a polyhedral analysis of the polynomially solvable special case in which the associated bipartite graph is a path. We describe a new class of inequalities that we call "path-modular" inequalities. We give two distinct proofs of their validity. The first one is direct and crucially relies on sub-and super-modularity of an associated set function, thereby providing an interesting link with flow-cover type inequalities. The second proof is by projecting a tight extended formulation, therefore also showing that these inequalities suffice to describe the convex hull of the feasible solutions to this problem. We finally show how to solve the separation problem associated to the path-modular inequalities in O(n(3)) time.
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