Robert N. Tomastik

The facet ascending algorithm for integer programming problems

Many practical large-scale optimization problems, such as scheduling a manufacturing system, can be modeled as integer programming problems. Because of their combinatorial nature, these problems are often very difficult to solve optimally, especially within a limited amount of time. Therefore, near-optimal solutions are often sought. Lagrangian relaxation is an effective method for decomposing a difficult problem into subproblems that are much easier to solve. A major part of this method is to optimize the dual function of the integer programming problem. Since the dual function is nondifferentiable, the subgradient method is frequently used as a method for maximizing (for a primal minimization problem) the dual function. However, this method can exhibit slow convergence due to iterations zigzagging across a set of nondifferentiable points. The new algorithm presented in this paper exploits the polyhedral concave nature of the dual function by ascending facets along nondifferentiable points, thus avoiding the zigzagging behaviour of the subgradient method. The algorithm is tested on a nonlinear integer programming problem for scheduling a simple manufacturing system. The computational results show this algorithm is a significant improvement over the subgradient method.<<ETX>>

Scheduling flexible manufacturing systems for apparel production

In mid to high volume apparel production, garments are typically grouped into production lots, and each lot is processed in its own manufacturing cell. A flexible manufacturing system used in this environment enables quick cell configuration, and the efficient operation of cells. The scheduling problem is to decide when to set up a cell and consequently begin garment production in the cell, and to decide the quantity of machines to allocate to each cell, under the constraints of limited machines. The time to process a production lot depends on the quantity of machines allocated to the cell in which the lot will be processed, and thus scheduling and resource allocation are highly coupled. In this paper, an accurate and low-order integer programming model is developed which integrates scheduling and resource allocation. Insight is provided into how the model relates to the operation of a real factory. The model is solved using the Lagrangian relaxation methodology, and a new bundle method is used for optimizing the Lagrangian dual function. The combination of an accurate low-order model, Lagrangian relaxation, and the bundle method is shown to be very practical.

A reduced-complexity bundle method for maximizing concave nonsmooth functions

Bundle methods have emerged as a promising concept for maximizing nonsmooth concave functions of many variables. A computationally-expensive step in conventional bundle methods is to find a trial direction, and current methods have exponential complexity, making them impractical for large problems. In this paper, a new version of the bundle method is developed, and this method has polynomial complexity in computing a trial direction.

Near-Optimal Scheduling of Manufacturing Systems with Presence of Batch Machines and Setup Requirements

Abstract Scheduling is a key factor for manufacturing productivity. Effective scheduling can improve on-time delivery, reduce inventory, cut lead time, and improve machine utilization. This study was motivated by the design and implementation of a scheduling system for a helicopter part production cell. The manufacturing is characterized by the presence of batch machines that can process multiple parts simultaneously, and the presence of machines requiring significant setup times. A novel mathematical optimization model with a separable structure is presented, and a solution methodology based on a combination of Lagrangian relaxation, dynamic programming, and heuristics is developed. Numerical results demonstrate that the method can generate near optimal schedules with quantifiable quality within a reasonable amount of computation time on a personal computer.

Optimization of Joint Replacement Policies for Multipart Systems by a Rollout Framework

Maintaining an asset with life-limited parts, e.g., a jet engine or an electric generator, may be costly. Certain costs, e.g., setup cost, can be shared if some parts of the asset are replaced jointly. Reducing the maintenance cost by good joint replacement policies is difficult in view of complicate asset dynamics, large problem sizes and the irregular optimal policy structures. This paper addresses these difficulties by using a rollout optimization framework. Based on a novel application of time-aggregated Markov decision processes, the ldquoOne-Stage Analysisrdquo method is first developed. The policies obtained from the method are investigated and their effectiveness is demonstrated by examples. This method and the existing threshold method are then improved by the ldquorollout algorithmrdquo for the total cost case and the average cost case. Based on ordinal optimization, it is shown that excessive simulations are not necessary for the rollout algorithm. Numerical testing demonstrates that the policies obtained by the rollout algorithms with either the ldquoOne-Stage Analysisrdquo or the threshold method significantly outperform traditional threshold policies.

An optimization method for joint replacement decisions in maintenance

Maintenance of safety-critical assets, such as jet engines, generators, etc. is very expensive. These assets usually consist of multiple life-limited parts which are grouped in different modules, and ask for shop or on-site visits subject to part expiration or random failures. The total cost includes shop (or on-site) visit costs, asset out-of-service costs, module removal costs, part replacement costs, and part waste costs. The problem in this paper is to decide when to perform maintenance and which parts to replace each time to optimize the total cost for a fixed-length contract in discrete time units. Because decisions for parts are correlated, optimization of joint replacement decisions encounters combinatorial explosion and it is usually difficult to find the optimal or good enough solution. This paper develops a new and separable model, where the solution is obtained iteratively from several linear or quadratic problems, providing linear formulations for all coupling relations (among parts, modules and the asset). Lower bounds are provided in the mean while. Numerical results show that this method achieves the optimal costs for small problems and achieves costs within 5% beyond the lower bounds for middle or large examples. Its computational time is linear to number of parts and modules and linear to the length of time horizon as well.

Joint replacement optimization for multi-part maintenance problems

A model of multi-part asset with dependent maintenance cost is presented. The problem is to minimize the long-run average cost per time unit. To share some costs, a good policy may jointly replace multiple parts when an asset is maintained. However, it is difficult to obtain an optimal joint replacement policy in view of combinatorial explosion of the states and stochastic system dynamics. To obtain optimal policies for small problems, a novel method is built by recent developed time aggregation Markov decision approach, which leads to analytical and computational simplifications as compared with traditional Markov decision approaches. One-stage and two-stage analysis methods are developed for large problems. The upper bound of one-stage analysis method for single part problems is obtained to show the insight that it can achieve near or true optimal policy. For multi-part problems, they are proved to satisfy certain necessary optimality conditions. These conditions can significantly simplify their implementation. Numerical results show that they are more efficient and effective than other near optimal methods.

Optimization-based scheduling of flexible manufacturing systems for apparel production

In the mid to high volume apparel production, garments are typically grouped into production lots, and each lot is processed in its own manufacturing cell. A flexible manufacturing system used in this environment enables quick cell configuration, and the efficient operation of cells. The scheduling problem is to decide when to set up a cell and consequently begin garment production in the cell, and to decide the quantity of machines to allocate to each cell. In this paper, an integer programming model is developed which integrates scheduling and resource allocation. The model is solved by using the Lagrangian relaxation technique, and a new bundle method is used to optimize the Lagrangian dual function. Testing is performed using data from a real factory, and numerical results show that high quality schedules are efficiently generated on personal computers.