This article presents a mathematical model for the problem of production and logistics in the forest industry. Specifically, a dynamic model of mixed-integer programming was formulated to solve three common problems i...
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This article presents a mathematical model for the problem of production and logistics in the forest industry. Specifically, a dynamic model of mixed-integer programming was formulated to solve three common problems in the forest sector: forest production, forest facilities location and forest freight distribution. The implemented mathematical model allows the strategic selection of the optimal location and size of a forest facility, in addition to the identification of the production levels and freight flows that will be generated in the considered planning horizon. A practical application of the model was carried out, validating its utility in the location of a sawmill. The model was optimally solved using LINGO, which also allowed to evaluate its response capacity in relation to changes in information considered in the initial planning, as well as the comparison of the decisions and the solution times for different scenarios such as demand, transportation costs, timber prices and yields of the sawn process. (c) 2004 Elsevier B.V. All rights reserved.
Many practical optimal control problems include discrete decisions. These may be either time-independent parameters or time-dependent control functions as gears or valves that can only take discrete values at any give...
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Many practical optimal control problems include discrete decisions. These may be either time-independent parameters or time-dependent control functions as gears or valves that can only take discrete values at any given time. While great progress has been achieved in the solution of optimization problems involving integer variables, in particular mixed-integer linear programs, as well as in continuous optimal control problems, the combination of the two is yet an open field of research. We consider the question of lower bounds that can be obtained by a relaxation of the integer requirements. For general nonlinear mixed-integer programs such lower bounds typically suffer from a huge integer gap. We convexify (with respect to binary controls) and relax the original problem and prove that the optimal solution of this continuous control problem yields the best lower bound for the nonlinear integer problem. Building on this theoretical result we present a novel algorithm to solve mixed-integer optimal control problems, with a focus on discrete-valued control functions. Our algorithm is based on the direct multiple shooting method, an adaptive refinement of the underlying control discretization grid and tailored heuristic integer methods. Its applicability is shown by a challenging application, the energy optimal control of a subway train with discrete gears and velocity limits.
We study the mixing inequalities that were introduced by Gunluk and Pochet [ Math. Program., 90 (2001), pp. 429-457]. We show that a mixing inequality which mixes n MIR inequalities has MIR rank at most n if it is a t...
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We study the mixing inequalities that were introduced by Gunluk and Pochet [ Math. Program., 90 (2001), pp. 429-457]. We show that a mixing inequality which mixes n MIR inequalities has MIR rank at most n if it is a type I mixing inequality and at most n - 1 if it is a type II mixing inequality. We also show that these bounds are tight for n = 2. Given a mixed-integer set P(I) = P boolean AND Z(I), where P is a polyhedron and Z(I) = {x is an element of R(n) : x(i) is an element of Z for all(i) is an element of I}, we define mixing inequalities for PI. We show that the elementary mixing closure of P with respect to I can be described using a bounded number of mixing inequalities, each of which has a bounded number of terms. This implies that the elementary mixing closure of P is a polyhedron. Finally, we show that any mixing inequality can be derived via a polynomial length MIR cutting-plane proof. Combined with results of Dash [On the complexity of cutting plane proofs using split cuts, IBM Research Report RC 24082, Oct. 2006] and Pudlak [J. Symbolic Logic, 62 (1997), pp. 981-998], this implies that there are valid inequalities for a certain mixed-integer set that cannot be obtained via a polynomial-size mixing cutting-plane proof.
The contribution of this paper focuses on the development of a security-based methodology for the solution of short-term SCUC when considering the impact of natural gas transmission system. The proposed methodology ex...
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The contribution of this paper focuses on the development of a security-based methodology for the solution of short-term SCUC when considering the impact of natural gas transmission system. The proposed methodology examines the interdependency of electricity and natural gas in a highly complex transmission system. The natural gas transmission system is modeled as a set of nonlinear equations. The proposed solution applies a decomposition method to separate the natural gas transmission feasibility check subproblem and the power transmission feasibility check subproblem from the hourly unit commitment (UC) in the master problem. Gas contracts are modeled and incorporated in the master UC problem. The natural gas transmission subproblem checks the feasibility of natural gas transmission as well as natural gas transmission security constraints for the commitment and dispatch of gas-fired generating units. If any natural gas transmission violations arise, corresponding energy constraints will be formed and added to the master problem for solving the next iteration of UC. The iterative process will continue until a converged feasible gas transmission solution is found. A six-bus power system with seven-node gas transmission system and the IEEE 118-bus power system with 14-node gas transmission system are analyzed to show the effectiveness of the proposed solution. The proposed model can be used by a vertically integrated utility or the ISO for the short-term commitment and dispatch of generating units with natural gas transmission constraints.
We derive polyhedral results for discrete-time mixed-integer programming (MIP) formulations for the production planning of multi-stage continuous chemical processes. We express the feasible region of the LP-relaxation...
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We derive polyhedral results for discrete-time mixed-integer programming (MIP) formulations for the production planning of multi-stage continuous chemical processes. We express the feasible region of the LP-relaxation as the intersection of two sets. The constraints describing the first set yield the convex hull of its integer points. For the second set, we show that for integral data the constraint matrix is kappa-regular, and that the corresponding polyhedron is integral if the length of the planning period is selected appropriately. We use this result to show that for rational data, integer variables can also assume integral values at the vertices of the polyhedron. We also discuss how these results provide insight and can be used to effectively address large-scale problems. Finally, we present computational results for a series of example problems. (C) 2009 Elsevier Ltd. All rights reserved.
This paper presents a novel mathematical programming approach to the single-machine capacitated lot-sizing and scheduling problem with sequence-dependent setup times and setup costs. The approach is partly based on th...
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This paper presents a novel mathematical programming approach to the single-machine capacitated lot-sizing and scheduling problem with sequence-dependent setup times and setup costs. The approach is partly based on the earlier work of Haase and Kimms [2000. Lot sizing and scheduling with sequence-dependent setup costs and times and efficient rescheduling opportunities. International journal of Production Economics 66(2), 159-169] which determines during pre-processing all item sequences that can appear in given time periods in optimal solutions. We introduce a new mixed-integer programming model in which binary variables indicate whether individual items are produced in a period, and parameters for this program are generated by a heuristic procedure in order to establish a tight formulation. Our model allows us to solve in reasonable time instances where the product of the number of items and number of time periods is at most 60-70. Compared to known optimal solution methods, it solves significantly larger problems, often with orders of magnitude speedup. (C) 2008 Elsevier B.V. All rights reserved.
In real world maritime routing problems, many restrictions and regulations influence the daily operations. The effects of several of these restrictions have not yet been studied in depth from an operations research pe...
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In real world maritime routing problems, many restrictions and regulations influence the daily operations. The effects of several of these restrictions have not yet been studied in depth from an operations research perspective. This paper introduces the problem of allocating bulk cargoes to tanks in maritime shipping. A model and several variations are presented, and it is shown that the main problem consists of a number of complicating constraints. The problem studied is crucial when determining whether a given route is feasible for a given ship, and computational experiments are performed to assess the difficulty of solving realistically sized instances. The proposed formulation is hoped to provide a suitable starting point for research on stowage problems in maritime bulk shipping. (C) 2009 Elsevier Ltd. All rights reserved.
Latin hypercube designs (LHDs) play an important role when approximating computer simulation models. To obtain good space-filling properties, the maximin criterion is frequently used. Unfortunately, constructing maxim...
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Latin hypercube designs (LHDs) play an important role when approximating computer simulation models. To obtain good space-filling properties, the maximin criterion is frequently used. Unfortunately, constructing maximin LHDs can be quite time consuming when the number of dimensions and design points increase. In these cases, we can use heuristical maximin LHDs. In this paper, we construct bounds for the separation distance of certain classes of maximin LHDs. These bounds are useful for assessing the quality of heuristical maximin LHDs. Until now only upper bounds are known for the separation distance of certain classes of unrestricted maximin designs, i.e., for maximin designs without a Latin hypercube structure. The separation distance of maximin LHDs also satisfies these "unrestricted" bounds. By using some of the special properties of LHDs, we are able to find new and tighter bounds for maximin LHDs. Within the different methods used to determine the upper bounds, a variety of combinatorial optimization techniques are employed. mixed-integer programming, the traveling salesman problem, and the graph-covering problem are among the formulations used to obtain the bounds. Besides these bounds, also a construction method is described for generating LHDs that meet Baer's bound for the l(infinity) distance measure for certain values of n.
This paper deals with estimating the maximal production capacity of a hydrothermal system. We will show why this is of interest and how such a problem arises in a context of hedging against supply shortage, using fina...
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This paper deals with estimating the maximal production capacity of a hydrothermal system. We will show why this is of interest and how such a problem arises in a context of hedging against supply shortage, using financial and physical assets. Therefore, we will formulate the supply shortage hedging as a stochastic optimization problem with chance constraints and show on an example why introducing physical assets in the hedging problem is of economical interest. We then highlight the inherent mathematical difficulties, introduced by optimizing physical assets. Using maximal production capacities of the hydrothermal system improves tractability of the hedging problem. However, focusing on the hydraulic production maximal capacity, we illustrate the combinatorial and nontrivial nature of this subproblem. Finally, we show how our problem formulation may lead to embedded chance constraints.
mixed-integer linear programming (MILP) based techniques are among the most widely applied methods for unit commitment (UC) problems. The fuel cost functions are often replaced by their piecewise linear approximations...
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mixed-integer linear programming (MILP) based techniques are among the most widely applied methods for unit commitment (UC) problems. The fuel cost functions are often replaced by their piecewise linear approximations whereas it is more or less disturbing to use piecewise linear approximations without knowing the exact effect on solution deviation from the optima. Therefore, error analysis is important since the optimal solutions are different when different objective functions are adopted. Another important problem is balancing between solution quality and computation efficiency since better solution quality relies on finer discretization with exponentially increased computational efforts. A detailed error analysis is presented in this paper. It is found that the approximation error is inverse proportional to the square of the number of piecewise segments. Lower bounds on the minimum necessary number of discretization segments are also derived. A 2-Stage Procedure is then established to achieve a better balance between solution quality and computation efficiency. Numerical testing to 2 groups of UC problems is exciting. It is found that the operating cost increases no more than 0.6% in all cases while the CPU time is greatly reduced regarding other MILP approaches. The results are still valid in electric power market clearing computation. (C) 2009 Elsevier B.V. All rights reserved.
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