Peter B. Luh

A practical approach to job-shop scheduling problems

The use of Lagrangian relaxation to schedule job shops, which include multiple machine types, generic precedence constraints, and simple routing considerations, is explored. Using an augmented Lagrangian formulation, the scheduling problem is decomposed into operation-level subproblems for the selection of operation beginning times and machine types, with given multipliers and penalty coefficients. The multipliers and penalty coefficients are then updated at the higher level. The solution forms the basis of a list-scheduling algorithm that generates a feasible schedule. A procedure is also developed to evaluate the quality of this feasible schedule by generating a lower bound on the optimal cost. Numerical examples are taken from a representative industrial job shop. High-quality schedules are efficiently generated every other day over a three-week period, with costs generally within 4% of their respective lower bounds. The methodology compares favorably with knowledge-based scheduling. >

Schedule generation and reconfiguration for parallel machines

A methodology for scheduling independent jobs with due dates on identical, parallel machines is presented. The jobs have different levels of importance and various processing times on the machines, and the objective is to minimize the total weighted job tardiness of the schedule. Since the problem is NP hard, the goal is not to obtain the optimal schedule. Rather, an efficient near-optimal algorithm based on Lagrangian relaxation is presented. This approach provides a lower bound on the cost, which can be used as a measure of suboptimality. According to an implementation for a work center at Pratt and Whitney, most schedules generated are within 1% of the optima with reasonable CPU times. Furthermore, the method provides valuable job interaction information, which shop floor management uses to answer what if questions, to reconfigure the schedule to accommodate dynamic changes, and to schedule new jobs. >

Scheduling products with bills of materials using an improved Lagrangian relaxation technique

A bill of materials specifies the sequence in which parts are to be processed and assembled in order to manufacture a deliverable product. In practice, a bill of materials may be quite complex, involving hundreds of parts to be processed on a number of limited resources, making scheduling difficult. This has forced many practitioners to turn to Material Requirements Planning (MRP) and heuristic rules to perform scheduling. These methods are seldom integrated, resulting in unreliable completion times for products and, hence, low customer satisfaction. This paper addresses the issue of integrally scheduling parts that are related through a bill of materials for the purpose of improving the on-time performance of products as well as reducing work-in-process (WIP) inventory. The technique presented here is based on an existing Lagrangian relaxation (LR) approach for the scheduling of independent parts in a job shop. An auxiliary problem formulation with a modified subgradient method is adopted to improve the computation time of the existing LR approach. This improved LR approach allows the bill of material constraints to be considered directly in the problem formulation. >

Job shop scheduling with group-dependent setups, finite buffers, and long time horizon

Scheduling is a key factor for manufacturing productivity. Effective scheduling can improve on-time delivery of products, reduce inventory, cut lead times, and improve the utilization of bottleneck resources. This study was motivated by the design and implementation of a scheduling system for the manufacturing of Toshibas gas insulated switchgears. The manufacturing is characterized by significant machine setup times, strict local buffer capacities, the option of choosing a few alternative processing routes, and long horizon as compared to the time resolution required. This problem has been recognized to be extremely difficult because of the combinatorial nature of integer optimization and the large size of the real problem. Our goal is thus to obtain near-optimal schedules with quantifiable quality in a computationally efficient manner. To achieve this goal, a novel integer optimization formulation with a separable structure is developed, and a solution methodology based on a combined Lagrangian relaxation, dynamic programming, and heuristics is developed. The method has been implemented using the object-oriented programming language C++, and numerical testing shows that the method generates high-quality schedules in a timely fashion to achieve on-time delivery of products and low inventory. Through explicit consideration of setups, tanks with the same processing requirements tend to be processed together to avoid excessive setups. The integrated treatment of machines and buffers facilitates the smooth flow of parts through the system. The embedded routing selection mechanism also balances the load among candidate routes. Finally, the newly developed time step reduction technique implicitly establishes two time scales to reduce computational requirements without much loss of modeling accuracy and scheduling performance, thereby enabling the resolution of long horizon problems with controllable computational requirements.

An effective method to reduce inventory in job shops

Inventory plays a major role in deciding the overall manufacturing costs, and a good scheduling system should balance the on-time delivery of products versus low work-in-progress (WIP) inventory. In this paper, the constant work-in-process (CONWIP) concept is applied to job shop scheduling to effectively control WIP inventory. A new mathematical formulation of CONWIP-based job shop scheduling with a separable structure is presented. By using a synergistic combination of Lagrangian relaxation, dynamic programming, and heuristic methods, good schedules are obtained in a reasonable amount of computation time. Results show that the new method can directly control the maximum WIP levels while maintaining good on-time delivery performance.

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.

A macro-level scheduling method using Lagrangian relaxation

A macro-level scheduling method is developed to provide high-level planning support for factories with multiple coordinating cells. The key challenges are the large problem size, complicated product process plans, stringent cell coordination requirements and possible resource overload in view of the overtime and reserve resources. To model the scheduling problem with manageable complexity, detailed operations of a product within a cell are aggregated as a single operation whose processing time is related to the amount of resource allocated. Overload variables are introduced in the resource capacity constraints, and penalties are imposed for having overload. The goal is to properly allocate resources, efficiently handle complicated process plans, and well coordinate cells to ensure on-time delivery, low working-in-process inventory, and small resource overload. The formulation obtained is separable and can be efficiently solved by Lagrangian relaxation.

Scheduling Products with Bills of Materials

Abstract A product that is comprised of a complex arrangement of parts and assemblies is often described by a bill of materials. The bill of materials specifies the hierarchical order in which parts and raw materials are to be assembled in order to manufacture a deliverable product. The processing of a part can be further comprised of a sequence of operations, each requiring processing by a particular resource type for a certain processing time. Since parts have complex interdependent relationships, local scheduling decisions for one part have countless repercussions on the decisions for other parts, making the scheduling process extremely difficult. In an effort to simplify this task, many practitioners attempt to decompose the scheduling problem into independent subproblems for each part which are easier to solve. The problem with this approach, however, is that bill of material constraints are often ignored, yielding schedules in which products are unnecessarily tardy. To improve the ability of a manufacturer to meet promised delivery dates for products, the bill of materials must be integrated into the part scheduling problem. This will ultimately enhance the manufacturers credibility and competitiveness in the marketplace. In this paper, an integer programming formulation of the product scheduling problem is presented, and a combined Lagrangian relaxation and heuristic approach is used to solve the problem. The issue of integrally scheduling parts that are related through bills of materials is addressed by including part precedence constraints in the formulation, while an objective function containing penalties that reflect the hierarchical nature of parts in the bill of materials is used to avoid oscillation in the solution process. Test results using data from a real manufacturing shop show that this method can obtain near-optimal schedules efficiently with quantifiable performance. Furthermore, preliminary results indicate that the penalties in the objective function reduce solution oscillation and can provide a mechanism for reducing work-in-process inventory. Lastly, by comparing this method to another algorithm in which parts are scheduled independently, it is shown that integration of the bills of materials into the part scheduling problem can improve scheduling performance by reducing product tardiness.

Scheduling of design projects with uncertain number of iterations

Design projects are typically broken down into inter-related tasks that are worked on by designers equipped with various resources. In the creative but uncertain design process, certain tasks may have to be iterated a few times to meet the design criteria, and this very often has major impact on designer/resource planning and on project completion. This paper presents a novel integer programming formulation for the scheduling of design projects with uncertain number of iterations. A combined Lagrangian relaxation and backward dynamic programming algorithm is developed to solve the problem with manageable complexity. Testing results demonstrate that good and robust schedules can be generated in computationally efficient manner.