Showing papers in "Optimization and Engineering in 2009"
TL;DR: The method described, which is a variation on the K-means algorithm for clustering, seems to work well in practice, at least on data that can be fit well by a convex function.
Abstract: We consider the problem of fitting a convex piecewise-linear function, with some specified form, to given multi-dimensional data. Except for a few special cases, this problem is hard to solve exactly, so we focus on heuristic methods that find locally optimal fits. The method we describe, which is a variation on the K-means algorithm for clustering, seems to work well in practice, at least on data that can be fit well by a convex function. We focus on the simplest function form, a maximum of a fixed number of affine functions, and then show how the methods extend to a more general form.
220 citations
TL;DR: A nonlinear mixed integer model and a nonlinear programming model with favorable properties for gradient-based optimization methods, based on smooth component models for the network elements are developed.
Abstract: The topic of this paper is minimum cost operative planning of pressurized water supply networks over a finite horizon and under reliable demand forecast. Since this is a very hard problem, it is desirable to employ sophisticated mathematical algorithms, which in turn calls for carefully designed models with suitable properties. The paper develops a nonlinear mixed integer model and a nonlinear programming model with favorable properties for gradient-based optimization methods, based on smooth component models for the network elements. In combination with further nonlinear programming techniques (Burgschweiger et al. in ZIB Report ZR-05-31, Zuse Institute Berlin, 2005), practically satisfactory near-optimum solutions even for large networks can be generated in acceptable time using standard optimization software on a PC workstation. Such an optimization system is in operation at Berliner Wasserbetriebe.
117 citations
TL;DR: This paper shows how practical demands of the application dictate the various algorithmic choices that are made in the nonlinear optimization solver, with particular reference to the system in operation at the European Centre for Medium-Range Weather Forecasts.
Abstract: Variational data assimilation is used at major weather prediction centers to produce the initial conditions for 7- to 10-day weather forecasts. This technique requires the solution of a very large data-fitting problem in which the major element is a set of partial differential equations that models the evolution of the atmosphere over a time window for which observational data has been gathered. Real-time solution of this difficult computational problem requires sophisticated models of atmospheric physics and dynamics, effective use of supercomputers, and specialized algorithms for optimization and linear algebra. The optimization algorithm can be accelerated by using a spectral preconditioner based on the Lanczos method. This paper shows how practical demands of the application dictate the various algorithmic choices that are made in the nonlinear optimization solver, with particular reference to the system in operation at the European Centre for Medium-Range Weather Forecasts.
82 citations
TL;DR: A realistic ship design problem, namely the reduction of the amplitude of the heave motion of a ship advancing in head seas (a problem connected to both safety and comfort), is solved using the new codes and other global and local derivative-free optimization methods.
Abstract: The aim of this paper is to solve optimal design problems for industrial applications when the objective function value requires the evaluation of expensive simulation codes and its first derivatives are not available. In order to achieve this goal we propose two new algorithms that draw inspiration from two existing approaches: a filled function based algorithm and a Particle Swarm Optimization method. In order to test the efficiency of the two proposed algorithms, we perform a numerical comparison both with the methods we drew inspiration from, and with some standard Global Optimization algorithms that are currently adopted in industrial design optimization. Finally, a realistic ship design problem, namely the reduction of the amplitude of the heave motion of a ship advancing in head seas (a problem connected to both safety and comfort), is solved using the new codes and other global and local derivative-free optimization methods. All the numerical results show the effectiveness of the two new algorithms.
75 citations
TL;DR: This paper proposes a multi-objective optimization method based on meta-modeling predicting a form of each objective function by using support vector regression, and discusses a way how to select additional experimental data for sequentially revising a forms of objective function.
Abstract: Practical engineering design problems have a black-box objective function whose forms are not explicitly known in terms of design variables In those problems, it is very important to make the number of function evaluations as few as possible in finding an optimal solution So, in this paper, we propose a multi-objective optimization method based on meta-modeling predicting a form of each objective function by using support vector regression In addition, we discuss a way how to select additional experimental data for sequentially revising a form of objective function Finally, we illustrate the effectiveness of the proposed method through some numerical examples
66 citations
TL;DR: An asymmetric suboptimization method for performing multidisciplinary design optimization is introduced to improve the overall efficiency of aerostructural optimization, by simplifying the system-level problem, and thereby reducing the number of calls to a potentially costly aerodynamics solver.
Abstract: An asymmetric suboptimization method for performing multidisciplinary design optimization is introduced. The objective of the proposed method is to improve the overall efficiency of aerostructural optimization, by simplifying the system-level problem, and thereby reducing the number of calls to a potentially costly aerodynamics solver. To guide a gradient-based optimization algorithm, an extension of the coupled sensitivity equations is developed to include post-optimality information from the structural suboptimization. The optimization of an aircraft wing is performed using linear aerodynamic and structural analyses, and a thorough performance comparison is made between the new approach and the conventional multidisciplinary feasible method. The asymmetric suboptimization method is found to be the more efficient approach when it adequately simplifies the system-level problem, or when there is a large enough discrepancy between disciplinary solution times.
62 citations
TL;DR: In this paper, a moving horizon technique is employed, which splits the optimal control problem into a sequence of local optimal control problems that are combined by suitable continuity conditions, yielding a reference trajectory.
Abstract: The test-drive of an automobile along a given test-course can be modeled by formulating a suitable optimal control problem. However, if the length of the course is very long or if it has a very complicated structure, the numerical solution of the optimal control problem becomes very difficult. Therefore a moving horizon technique is employed, which splits the optimal control problem into a sequence of local optimal control problems that are combined by suitable continuity conditions. This approach yields a reference trajectory. A controller and differential GPS are integrated in a real-world car and allows a reference trajectory to be followed in real-time. A benefit of this approach is the very high accuracy obtained in reproducing the reference trajectory. Hence, it can be used for testing different setups of cars under the same conditions while excluding the comparatively large influence of a real-world driver. In this article, we will focus on a method for generating the reference trajectory and report our experiences with this algorithm. The method allows an locally optimal solution to be computed for various handling courses in a robust way.
57 citations
TL;DR: Ohsaki et al. as discussed by the authors considered semi-definite programming (SDP) formulations of certain truss topology optimization problems, where a lower bound is imposed on the fundamental frequency of vibration of the truss structure.
Abstract: We consider semidefinite programming (SDP) formulations of certain truss topology optimization problems, where a lower bound is imposed on the fundamental frequency of vibration of the truss structure. These SDP formulations were introduced in: [M. Ohsaki, K. Fujisawa, N. Katoh and Y. Kanno, Semi-definite programming for topology optimization of trusses under multiple eigenvalue constraints, Comp. Meth. Appl. Mech. Engng., 180: 203–217, 1999]. We show how one may automatically obtain symmetric designs, by eliminating the ‘redundant’ symmetry in the SDP problem formulation. This has the advantage that the original SDP problem is substantially reduced in size for trusses with large symmetry groups.
54 citations
TL;DR: The experiments show that SVM can be very efficient in processing unseen instances and may yield very high accuracy rate, in particular with the new proposed feature selection.
Abstract: Wireless capsule endoscopy (WCE) is a recently established imaging technology that requires no wired device intrusion and can be used to examine the entire small intestine non-invasively. Determining bleeding signs out of over 55,000 WCE images is a tedious and expensive job by human reviewing. Our goal is to develop an automatic obscure bleeding detection method by employing image color features and support vector machine (SVM) classifier. The bleeding lesion detection problem is a binary classification problem. We use SVMs for this problem and a new feature selection procedure is proposed. Our experiments show that SVM can be very efficient in processing unseen instances and may yield very high accuracy rate, in particular with our new proposed feature selection. More specifically, for this bleeding detection problem, training an SVM with 640 samples can be completed in as little as 0.01 second on a Dell workstation with dual Xeon CPUs, and classifying an image using the trained SVM can be done in as little as 0.03 milliseconds. The accuracy for both sensitivity and specificity can be over 99%.
43 citations
TL;DR: This paper presents subspace methods for solving large scale nonlinear systems of equations and nonlinear least square problems, which have the characteristic to force the next iteration in a low dimensional subspace.
Abstract: In this paper, we study large scale nonlinear systems of equations and nonlinear least square problems. We present subspace methods for solving these two special optimization problems. The subspace methods have the characteristic to force the next iteration in a low dimensional subspace. The main technique is to construct subproblems in low dimensions so that the computation cost in each iteration can be reduced comparing to standard approaches. The subspace approach offers a possible way to handle large scale optimization problems which are now attracting more and more attention. Actually, quite a few known techniques can be viewed as subspace methods, such as conjugate gradient method, limited memory quasi-Newton method, projected gradient method, and null space method.
43 citations
TL;DR: A novel time series classification algorithm is presented, which adopts triangle distance function as similarity measure, extracts some meaningful patterns from original data and uses traditional machine learning algorithm to create classifier based on the extracted patterns.
Abstract: Multivariate time series classification is of significance in machine learning area. In this paper, we present a novel time series classification algorithm, which adopts triangle distance function as similarity measure, extracts some meaningful patterns from original data and uses traditional machine learning algorithm to create classifier based on the extracted patterns. During the stage of pattern extraction, Gini function is used to determine the starting position in the original data and the length of each pattern. In order to improve computing efficiency, we also apply sampling method to reduce the searching space of patterns. The common datasets are used to check our algorithm and compare with the naive algorithms. Experimental results are shown to reveal that much improvement can be gained in terms of interpretability, simplicity and accuracy.
TL;DR: In this paper, it was shown that the Slater condition is indeed necessary for the S-lemma and then established a regularized form of the Slemma in the absence of a Slater condition.
Abstract: The celebrated S-lemma establishes a powerful equivalent condition for the nonnegativity of a quadratic function over a single quadratic inequality. However, this lemma fails without the technical condition, known as the Slater condition. In this paper, we first show that the Slater condition is indeed necessary for the S-lemma and then establishes a regularized form of the S-lemma in the absence of the Slater condition. Consequently, we present characterizations of global optimality and the Lagrangian duality for quadratic optimization problems with a single quadratic constraint. Our method of proof makes use of Brickman’s theorem and conjugate analysis, exploiting the hidden link between the convexity and the S-lemma.
TL;DR: In this article, a restricted version of the robust counterpart approach is introduced, referred to as Affinely Adjustable Robust Counterpart (AARC), where decision variables are allowed to depend on past values of uncertain parameters.
Abstract: The problem of production management can often be cast in the form of a linear program with uncertain parameters and risk constraints. Typically, such problems are treated in the framework of multi-stage Stochastic Programming. Recently, a Robust Counterpart (RC) approach has been proposed, in which the decisions are optimized for the worst realizations of problem parameters. However, an application of the RC technique often results in very conservative approximations of uncertain problems. To tackle this drawback, an Adjustable Robust Counterpart (ARC) approach has been proposed by Ben-Tal et al. In ARC, some decision variables are allowed to depend on past values of uncertain parameters. A restricted version of ARC, introduced by Ben-Tal et al. and which can be efficiently solved, is referred to as Affinely Adjustable Robust Counterpart (AARC).
TL;DR: This study proposes a new spanning tree-based Genetic Algorithm using determinant encoding for solving this problem and demonstrates that the proposed GA outperforms the other previously published GA in the solution quality and convergence rate.
Abstract: The design of configuration and the transportation planning are crucial issues to the effectiveness of multi-stage supply chain networks. The decision makers are interested in the determination the optimal locations of the hubs and the optimal transportation routes to minimize the total costs incurred in the whole system. One may formulate this problem as a 0-1 mixed integer non-linear program though commercial packages are not able to efficiently solve this problem due to its complexity. This study proposes a new spanning tree-based Genetic Algorithm (GA) using determinant encoding for solving this problem. Also, we employ an efficient heuristic that fixes illegal spanning trees existing in the chromosomes obtained from the evolutionary process of the proposed GA. Our numerical experiments demonstrate that the proposed GA outperforms the other previously published GA in the solution quality and convergence rate.
TL;DR: A unified formulation for multi-response optimal design problem using Semi-Definite Programming (SDP) that can generate D-, A- and E-optimal designs and solves a one-shot optimization model whose solution selects the optimal design points among all possible points in the design space.
Abstract: Optimal design of multi-response experiments for estimating the parameters of multi-response linear models is a challenging problem. The main drawback of the existing algorithms is that they require the solution of many optimization problems in the process of generating an optimal design that involve cumbersome manual operations. Furthermore, all the existing methods generate approximate design and no method for multi-response n-exact design has been cited in the literature. This paper presents a unified formulation for multi-response optimal design problem using Semi-Definite Programming (SDP) that can generate D-, A- and E-optimal designs. The proposed method alleviates the difficulties associated with the existing methods. It solves a one-shot optimization model whose solution selects the optimal design points among all possible points in the design space. We generate both approximate and n-exact designs for multi-response models by solving SDP models with integer variables. Another advantage of the proposed method lies in the amount of computation time taken to generate an optimal design for multi-response models. Several test problems have been solved using an existing interior-point based SDP solver. Numerical results show the potentials and efficiency of the proposed formulation as compared with those of other existing methods. The robustness of the generated designs with respect to the variance-covariance matrix is also investigated.
TL;DR: This paper develops an integrated production-recycling system over a finite time horizon where the dynamic demand is satisfied by production and recycling.
Abstract: This paper develops an integrated production-recycling system over a finite time horizon. Here, the dynamic demand is satisfied by production and recycling. The used units are bought back and then either recycled or disposed of which are not repairable. The used units are collected continuously from the customers. Recycling products can be used as new products which are sold again. The rate of production and disposal are assumed to be function of time. The setup cost is reduced over time due to “Learning curve” effect. The optimum results are presented both in tabular form and graphically.
TL;DR: In this paper, a new theoretical formulation is presented for the shape optimization problem associated with maximizing or minimizing the diffusive scalar transport from a two-dimensional body, where the objective function is the length of the object in the transformed domain and the variables of the optimization are the parameters of Schwarz-Christoffel transformation.
Abstract: A new theoretical formulation is presented for the shape optimization problem associated with maximizing or minimizing the diffusive scalar transport from a two-dimensional body. In particular, we consider the diffusive transport of heat from an isothermal body into a medium with constant temperature at the far-field. The formulation also applies to mass and momentum transport. The diffusion problem, which is governed by the Laplace equation, is addressed using conformal mapping techniques where the two-dimensional domain is mapped onto a simpler domain where an analytical solution can be readily obtained. The objective function of the optimization problem is the length of the object in the transformed domain and the variables of the optimization are the parameters of the Schwarz-Christoffel transformation. The length of the object in the transformed domain is related to the scalar displacement, which corresponds to a far-field temperature drop or rise (slip velocity in case of momentum transport), that depends on the shape of the body and it quantifies the enhancement or reduction in transport rate. The mathematical formulation is validated by addressing two fundamental shape optimization problems associated with maximizing or minimizing the transport rate (drag in case of momentum transport) from a two-dimensional body of unit span: i) for a given surface area to obtain the shape that maximizes the transport rate from a body, ii) for a given volume to obtain the shape that minimizes the transport rate from a body. For both cases we compute numerically that the cylinder is the optimal shape. The versatility of the formulation is further demonstrated by including constraints with respect to the length of the body.
TL;DR: In this paper, a method for finding the optimal control of a linear time varying delay system with quadratic performance index is discussed, which is based upon expanding various time functions in the system as their truncated hybrid functions.
Abstract: A method for finding the optimal control of a linear time varying delay system with quadratic performance index is discussed. The properties of the hybrid functions which consists of block-pulse functions plus Taylor series are presented. The method is based upon expanding various time functions in the system as their truncated hybrid functions. Operational matrices of integration and delay are presented and are utilized to reduce the solution of an optimazation problem to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.
TL;DR: This work proposes a method, called the LP-Newton method, for linear programming that is based on the zonotope formulation and the minimum-norm-point algorithm of Philip Wolfe and is a finite algorithm even for real-number input data with exact arithmetic computations.
Abstract: Although linear programming problems can be solved in polynomial time by the ellipsoid method and interior-point algorithms, there still remains a long- standing open problem of devising a strongly polynomial algorithm for linear pro- gramming (or of disproving the existence of such an algorithm). The present work is motivated by an attempt toward solving this problem. Linear programming problems can be formulated in terms of a zonotope, a kind of greedy polyhedron, on which linear optimization is made easily. We propose a method, called the LP-Newton method, for linear programming that is based on the zonotope formulation and the minimum-norm-point algorithm of Philip Wolfe. The LP-Newton method is a finite algorithm even for real-number input data with exact arithmetic computations. We show some preliminary computational results to exam- ine the behavior of the LP-Newton method.
TL;DR: By using the proposed method, an optimum solution is attained with high accuracy and a small number of function evaluations, and the effectiveness and validity of ARPSO are examined.
Abstract: This paper proposes a new technique for particle swarm optimization called adaptive range particle swarm optimization (ARPSO). In this technique an active search domain range is determined by utilizing the mean and standard deviation of each design variable. In the initial search stage, the search domain is explored widely. Then the search domain is shrunk so that it is restricted to a small domain while the search continues. To achieve these search processes, new parameters to determine the active search domain range are introduced. These parameters gradually increase as the search continues. Through these processes, it is possible to shrink the active search domain range. Moreover, by using the proposed method, an optimum solution is attained with high accuracy and a small number of function evaluations. Through numerical examples, the effectiveness and validity of ARPSO are examined.
TL;DR: The authors present a robust methodology to update the homotopy parameter based on the theory of probability-one homotopies for nonlinear programming and show that the method is robust and effective in its implementation for achieving sequential approximate optimization feasibility.
Abstract: In a previous work the authors developed an interior point approach for trust region managed sequential approximate optimization. The interior point approach insures that approximate feasibility is maintained throughout the optimization process. In the case of an infeasible starting point, a relaxation of the constraints allows the algorithm to operate without modification. The relaxation is controlled by a homotopy parameter. A primary advantage resides in the fact that all the constraints (contrasted with just the active or most violated constraints) influence the optimization, since the relaxation fades at the same time for all violated constraints. Adjustment of the parameter was performed in an heuristic fashion. In this paper the authors present a robust methodology to update the homotopy parameter based on the theory of probability-one homotopies for nonlinear programming. Results show that the method is robust and effective in its implementation for achieving sequential approximate optimization feasibility.
TL;DR: The available reduction techniques for PNS problems are reviewed as well as a further reduction algorithm is presented, and the performance of the new algorithm is examined by empirical analysis.
Abstract: Process Network Synthesis (PNS) has an enormous practical impact. The problem is very difficult to solve, determining the cost optimal network of operating units with fixed charge belongs to the complexity class of NP-hard problems. Therefore, it is important to develop reduction algorithms to minimize the size of the problem. In the present work the available reduction techniques for PNS problems are reviewed as well as a further reduction algorithm is presented. The performance of the new algorithm is examined by empirical analysis.
TL;DR: An iterative global approximation technique based on the Kriging method that seems a promising alternative to the classic a-priori building of response surface allowing better accuracy and saving of sample points is proposed.
Abstract: An iterative global approximation technique based on the Kriging method is proposed. The technique is validated through analytical test cases and then applied to solve two practical optimization problems: the optimization of aluminium-foam filled absorbers against crashworthiness requirements and the optimization of composite stiffened panels against buckling and strength constraints. The absorbers of the first application consist of two co-axial aluminium alloy tubes filled with lightweight aluminium foam. They were optimized to collapse at a controlled force level and to be the lightest possible. Explicit Finite element analyses were performed to evaluate the structural behavior of the absorbers in the sample points used to build the approximation. In the second application stiffened panels were optimized against buckling and strength constraints. The Tsai-Wu criterion was used to estimate first-ply failures as strength limit of the structure. Non-linear Riks analyses were performed with ABAQUS/Standard to evaluate the shell behavior in the sample points used to build the response surfaces. Basing on the obtained results the proposed iterative procedure seems a promising alternative to the classic a-priori building of response surface allowing better accuracy and saving of sample points.
TL;DR: This paper presents several formulations and comparative test results for problems involving the general paradigm of coupled sets of components (CSoC), which is general enough to include systems of systems (SoS) under any of the various definitions, as well as multidisciplinary design optimization (MDO).
Abstract: This paper presents several formulations and comparative test results for problems involving the general paradigm of coupled sets of components (CSoC). This paradigm is general enough to include systems of systems (SoS) under any of the various definitions, as well as multidisciplinary design optimization (MDO). It is assumed that a CSoC involves a (potentially inactive) coordinating component, or “Central Authority”, and one or more, potentially interacting, subordinate components. The formulations differ in the amount of control given to the CA versus the autonomy granted to the subordinate components. In this generality, satisfaction of equilibrium conditions replaces the optimality condition in defining a solution. A solution still requires feasibility of all constraints. The desirability of a particular equilibrium point depends on the specific problem being studied and the component algorithms used to obtain that equilibrium. These concepts are illustrated using comparative test results for several of the formulations on a NASA-generated, public domain, aircraft conceptual design problem.
TL;DR: The major conclusion of the paper is that the optimal operation of express lines does not improve the average waiting time significantly, but the effect of non-optimal operation can be very unfavorable.
Abstract: Improving the waiting process at checkouts in stores is an important goal of operations management in the era of time-based competition. The paper presents a method for evaluating the effect of express lines on the waiting process. An optimization model is developed which minimizes the average waiting time in line with respect to the maximum number of items allowed in the express lines. The research is based on a real case of a do-it-yourself superstore, but the methodology applied at the store can be used generally. The optimization model includes sensitivity analyses. Sensitivity analyses show how the optimal value of the limit parameter changes if major parameters of the model change. The results of these analyses help managers make decisions about short and medium-term operations of express lines. The major conclusion of the paper is that the optimal operation of express lines does not improve the average waiting time significantly, but the effect of non-optimal operation can be very unfavorable. Therefore, a successful implementation of express lines requires a thorough analysis of operational issues and a careful consideration of perceptional issues as well.
TL;DR: In this article, a constrained min-max optimization model is proposed to enlarge the small signal stability margin via adjusting some system parameters, and the well developed and efficient SQP method is applied to solve it.
Abstract: In this article, we consider the small signal stability margin problem arising from power electric systems. To enlarge the small signal stability margin via adjusting some system parameters, we develop a constrained min-max optimization model. For this model, we first reformulate it as a classical nonlinear programming problem, and then apply the well developed and efficient SQP method to solve it. The numerical experiments for examples taken from power systems show that the new model and the reformulation are really effective.
TL;DR: In this article, a canonical dual approach for solving nonconvex quadratic programming problems with both linear inequality constraints and box constrains is presented, where the primal problem is reformulated as a concave maximization dual problem with zero duality gap, which can be solved under certain conditions.
Abstract: This paper presents a canonical dual approach for solving nonconvex quadratic programming problems subjected to both linear inequality constraints and box constrains. It is proved that the constrained nonconvex primal problem can be reformulated as a concave maximization dual problem with zero duality gap, which can be solved under certain conditions. Both global and local extremimality conditions are presented by the triality theory. Several applications are illustrated.
TL;DR: In this paper, a gate service Geom/G/1 queue system with single vacation is analyzed and the expected number of customers in the system at the beginning instant of a service period is obtained.
Abstract: In this paper, a gate service Geom/G/1 queue system with single vacation is analyzed. Firstly, a gate service policy and single vacation are introduced in detail, the expected number of customers in the system at the beginning instant of a service period is obtained. Next, the Probability Generating Function (P.G.F.) of the stationary queue length is obtained by using regeneration cycle approach. Then the P.G.F. of the waiting time is derived under the condition of the independence between the arrival process and the waiting time. Moreover, the P.G.Fs. of the service period, the vacation period and the idle period are obtained, and the mean length of three periods are also obtained. Cost model is formulated to determine the optimal expected vacation length. Finally, based on numerical results, the relations of the performance measures and traffic intensity are discussed.
TL;DR: In this article, the authors present an application of the canonical duality theory for solving a class of nonconvex and nonsmooth optimization problems, which can be converted into a one-dimensional canonical dual problem.
Abstract: This paper presents an application of the canonical duality theory for solving a class of nonconvex and nonsmooth optimization problems. It is shown that by use of the canonical dual transformation, these difficult optimization problems in Rn can be converted into a one-dimensional canonical dual problems, which can be solved to obtain all extremal points. Both global and local extremality conditions can be identified by the triality theory. Applications are illustrated.
TL;DR: In this paper, the problem of finding a suitable placement of solar cells on the roof of a building was considered and an enhanced version of the classical bottom-left heuristic was proposed.
Abstract: We consider the problem of finding a suitable placement of solar cells on the roof of a building. Since roofs may be of nonrectangular shape and may have obstacles like chimneys, dormers or windows on them, finding an optimal layout is not a trivial task. Mathematically, this setting belongs to the class of cutting and packing problems, with an additional difficulty in the shape of the disposable roof area: existing approaches do not cover our situation where we deal with nonconvex roofs. We develop a mixed integer linear model for this problem and present an enhanced version of the classical bottom-left heuristic. Examples demonstrate that this new approach considerably improves traditional strategies.