Minimizing the total machine workload for the wafer probing scheduling problem

The Wafer Probing Scheduling Problem (WPSP) is a practical generalization of the parallel-machine scheduling problem, which has many real-world applications, particularly, in the Integrated Circuit (IC) manufacturing industry. In the wafer probing factories, the jobs are clustered by their product types, which must be processed on groups of identical parallel machines and be completed before the due dates. The job processing time depends on the product type, and the machine setup time is sequentially dependent on the orders of jobs processed. Since the wafer probing scheduling problem involves constraints on job clusters, job-cluster dependent processing time, due dates, machine capacity, and sequentially dependent setup time, it is more difficult to solve than the classical parallel-machine scheduling problem. In this paper, we consider the WPSP and formulate the WPSP as an integer programming problem to minimize the total machine workload. We demonstrate the applicability of the integer programming model by solving a real-world example taken from a wafer probing shop floor in an IC manufacturing factory.     

Modelling and optimization of a bi-objective flow shop scheduling with diverse maintenance requirements

In real-world problems, machines cannot continuously operate and have to stop for maintenance before they fail. Lack of maintenance can also affect the performance of machines in processing jobs. In this paper, a permutation flow shop scheduling problem with multiple age-based maintenance requirements is modelled as a novel mixed-integer linear program in which the objectives are conflicting. In modelling the problem, we assume that infrequent maintenance can prolong job processing times. One of the objectives is to minimise the total maintenance cost by planning as few maintenance activities as possible to only meet the minimum requirements, and the other objective tries to minimise the total tardiness by sequencing the jobs and planning the maintenance activities in such a way that the processing times are not prolonged and unnecessary maintenance times are avoided. Because of this conflict, an interactive fuzzy, bi-objective model is introduced. Application of the model is illustrated through a case study for operations and maintenance scheduling of heavy construction machinery. An effective and efficient solution methodology is developed based on the structure of the problem and tested against commercial solvers and a standard GA. Computational results have verified the efficiency of the proposed solution methodology and show that unlike the proposed method, a generic metaheuristic that does not consider the unique structure of the problem can become ineffective for real-world problem sizes.     

Single-machine scheduling with sequence dependent setup to minimize total weighted squared tardiness

This paper addresses the NP-hard problem of scheduling N independent jobs on a single machine with release dates, due dates, sequence dependent setup times, and no preemption where the objective is to minimize the weighted sum of squared tardiness. A Lagrangian relaxation based approach is developed for single-machine scheduling with sequence dependent setup times that is based on a list scheduling concept in conjunction with Lagrangian relaxation. Sequence dependent setup times are formulated as capacity constraints, and then are relaxed using Lagrangian multipliers. The primal problem is decomposed into job-level subproblems which are solved optimally and an approximate dual problem is then solved using a sub-gradient technique. The result of the relaxation is a list of jobs sequenced by beginning times that is then improved via a three-way swap. Experimental results are compared with EDD (Earliest Due Date) and ATCS (Apparent Tardiness Cost with Setups) dispatching rules, a four-way swap local search, tabu search, and simulated annealing. The adopted approach results in superior solution quality with respect to EED, ATCS, four-way swap, and tabu search results. It has comparable solution quality to the simulated annealing results, but is substantially more computationally efficient. Overall, the approach is capable of dealing with realistically sized single machine scheduling problems with release dates, due dates, and sequence dependent setup times.     

Family-based scheduling of shops with functional layouts

Simulation studies of job shop scheduling have typically assumed that either setup times are zero (subsumed within the processing time), or that every part has such a unique setup that no setup advantages can be gained by better scheduling policies. These studies also assume that the shop has exactly one copy of every machine. Some researchers have proposed heuristics that explicitly consider setup times and parallel machines in the context of a one stage shop with static arrivals. In contrast, family-based scheduling centred around setup time reduction has been credited with achieving economic savings in batch production industries where GT is employed. We motivate this study by the case of an existing realworld semi-conductor testing facility that has family setups, parallel machines and dynamic job arrival. Using this setting, we investigate whether benefits can still be obtained by using a family-based scheduling philosophy in those environments which do not permit the physical creation of cellular layouts due to the presence of process related or other constraints. We propose and evaluate two new dispatching procedures in a functional job shop that is modelled after the semiconductor testing facility. Results show that a family-based scheduling philosophy centred around coordinating machine setups is advantageous at relatively high setup to processing time ratios, while classic job shop rules suffice otherwise. Based on these results, we present recommendations for managing such environments. We also suggest future research directions in this area.     

Batch scheduling problem for a machinery factory with fixed-position layout

We consider a batch scheduling problem for a two-stage flow shop with fixed-position layout. In the first stage, a fixed number of jobs are assembled on a batch machine with a family batch setup time and a common processing time. In the second stage, the assembled jobs are individually performed for system integration on a discrete machine. The finished job is immediately packed and shipped if the payment has been made; otherwise, it is moved to a temporary storage area, incurring additional removal time. This study develops a mixed integer programming (MIP) to solve the problem of minimising the total completion time and proposes two heuristics for large-size problems. Computational results show that the proposed methods can be applied to resolve real-world problems similar to those in this study.     

Scheduling algorithms to minimise the total family flow time for job shops with job families

This paper considers the job scheduling problem in which jobs are grouped into job families, but they are processed individually. The decision variable is the sequence of the jobs assigned to each machine. This type of job shop scheduling can be found in various production systems, especially in remanufacturing systems with disassembly, reprocessing and reassembly shops. In other words, the reprocessing shop can be regarded as the job shop with job families since it performs the operations required to bring parts or sub-assemblies disassembled back to like-new condition before reassembling them. To minimise the deviations of the job completion times within each job family, we consider the objective of minimising the total family flow time. Here, the family flow time implies the maximum among the completion times of the jobs within a job family. To describe the problem clearly, a mixed integer programming model is suggested and then, due to the complexity of the problem, two types of heuristics are suggested. They are: (a) priority rule based heuristics; and (b) meta-heuristics. Computational experiments were performed on a number of test instances and the results show that some priority rule based heuristics are better than the existing ones. Also, the meta-heuristics improve the priority rule based heuristics significantly.     

Heuristics for permutation flow shop scheduling with batch setup times

In this paper we consider permutation flow shop scheduling problems with batch setup times. Each job has to be processed on each machine once and the technological routes are identical for all jobs. The set of jobs is divided into groups. There are given processing timest _ij of jobi on machinej and setup timess _rj on machinej when a job of ther-th group is processed after a job of another group. It is assumed that the same job order has to be chosen on each machine. We consider both the problems of minimizing the makespan and of minimizing the sum of completion times, where batch or item availability of the jobs is assumed. For these problems we give various constructive and iterative algorithms. The constructive algorithms are based on insertion techniques combined with beam search. We introduce suitable neighbourhood structures for such problems with batch setup times and describe iterative algorithms that are based on local search and reinsertion techniques. The developed algorithms have been tested on a large collection of problems with up to 80 jobs.Supported by Deutsche Forschungsgemeinschaft (Project ScheMA) and by the International Association for the Promotion of Cooperation with Scientists from the Independent States of the Former Soviet Union (Project INTAS-93-257)     

A general framework for scheduling equipment and manpower at container terminals

In this paper, we propose a general model for various scheduling problems that occur in container terminal logistics. The scheduling model consists of the assignment of jobs to resources and the temporal arrangement of the jobs subject to precedence constraints and sequence-dependent setup times. We demonstrate how the model can be applied to solve several different real-world problems from container terminals in the port of Hamburg (Germany). We consider scheduling problems for straddle carriers, automated guided vehicles (AGVs), stacking cranes, and workers who handle reefer containers. Subsequently, we discuss priority rule based heuristics as well as a genetic algorithm for the general model. Based on a tailored generator for experimental data, we examine the performance of the heuristics in a computational study. We obtain promising results that suggest that the genetic algorithm is well suited for application in practice.     

An efficient heuristic approach for parallel machine scheduling with job splitting and sequence-dependent setup times

In this paper, we consider a simplified real-life identical parallel machine scheduling problem with sequence-dependent setup times and job splitting to minimize makespan. We propose a heuristic to solve this problem. Our method is composed of two parts. The problem is first reduced into a single machine scheduling problem with sequence-dependent setup times. This reduced problem can be transformed into a Traveling Salesman Problem (TSP), which can be efficiently solved using Little's method. In the second part, a feasible initial solution to the original problem is obtained by exploiting the results of the first part. This initial solution is then improved in a step by step manner, taking into account the setup times and job splitting. We develop a lower bound and evaluate the performances of our heuristic on a large number of randomly generated instances. The solution given by our heuristic is less than 4.88% from the lower bound.     

Iterative algorithms for batching and scheduling to minimise the total job tardiness in two-stage hybrid flow shops

This study considers the batching and scheduling problem in two-stage hybrid flow shops in which each job with a distinct due-date is processed through two serial production stages, each of which has identical machines in parallel. Under the fundamental trade-off that large batch sizes with less frequent changeovers may reduce setup costs and hence increase machine utilisation, while small batch sizes may reduce job flow times and hence improve scheduling performance, the problem is to determine the number of batches, the batch compositions, the allocation of batches to the parallel machines at each stage, and the sequence of the batches allocated to each machine for the objective of minimising the total job tardiness. A mixed integer programming model is developed for the reduced problem in which the number of batches is given, and then, three iterative algorithms are proposed in which batching and scheduling are done repeatedly until a good solution is obtained. To show the performance of the algorithms, computational experiments were done on a number of test instances, and the results are reported. In particular, we show that the number of batches decreases as the ratio of the batch setup time to the job processing time increases.     

On scheduling parallel machines with two setup classes

We examine I he problem of scheduling jobs at a bank of parallel identical machines where each job requires either a major or a minor setup depending on the job that was processed immediately before it. Several heuristic procedures are proposed and tested with respect to makespan, average flowtime and time spent on setups.     

Minimizing production flow time in a process and assembly job shop

Scheduling is one of the most important issues in the planning and operation of production systems, but in medium to large shops, the generation of consistently good schedules has proven to be extremely difficult. The problem is that optimal scheduling solutions involve costly and impractical enumeration procedures. In the literature, most scheduling problems only address jobs with serial or sequential operations. Rarely do they consider jobs in which machining and assembly operations are simultaneously involved. This lack of attention to scheduling problems that involve both machining and assembly goes against what one would normally find in most job shops. In this paper, the problem of scheduling a set of N final products on M machines in a job shop environment that involve both machining and assembly operations is addressed. The objective pursued is the minimization of production flow time (makespan). A mathematical model is developed in an effort to obtain optimal solutions. Because this type of model grows exponentially as the size of the problems increases, an heuristic solution approach is developed to solve the problems more efficiently. The models are tested and compared on several test problems.     

An integrated scheduling and material-handling approach for complex job shops: a computational study

We suggest an extension of the shifting bottleneck heuristic for complex job shops that takes the operations of automated material-handling systems (AMHS) into account. The heuristic is used within a rolling horizon approach. The job-shop environment contains parallel batching machines, machines with sequence-dependent setup times, and re-entrant process flows. Jobs are transported by an AMHS. Semiconductor wafer fabrication facilities (wafer fabs) are typical examples for manufacturing systems with these characteristics. Our primary performance measure is total weighted tardiness (TWT). The shifting bottleneck heuristic (SBH) uses a disjunctive graph to decompose the overall scheduling problem into scheduling problems for single machine groups and for transport operations. The scheduling algorithms for these scheduling problems are called subproblem solution procedures (SSPs). We consider SSPs based on dispatching rules. In this paper, we are also interested in how much we can gain in terms of TWT if we apply more sophisticated SSPs for scheduling the transport operations. We suggest a Variable Neighbourhood Search (VNS) based SSP for this situation. We conduct simulation experiments in a dynamic job-shop environment in order to assess the performance of the suggested algorithms. The integrated SBH outperforms common dispatching rules in many situations. Using near to optimal SSPs leads to improved results compared with dispatching based SSPs for the transport operations.     

Integrated production scheduling and vehicle routing problem with job splitting and delivery time windows

In this paper, we study a production scheduling and vehicle routing problem with job splitting and delivery time windows in a company working in the metal packaging industry. In this problem, a set of jobs has to be processed on unrelated parallel machines with job splitting and sequence-dependent setup time (cost). Then the finished products are delivered in batches to several customers with heterogeneous vehicles, subject to delivery time windows. The objective of production is to minimize the total setup cost and the objective of distribution is to minimize the transportation cost. We propose mathematical models for decentralized scheduling problems, where a production schedule and a distribution plan are built consecutively. We develop a two-phase iterative heuristic to solve the integrated scheduling problem. We evaluate the benefits of coordination through numerical experiments.     

An extension of modified-operational-due-date priority rule incorporating job waiting times and application to assembly job shop

The well-known priority dispatching rule MOD (Modified Operational Due Date) in job shop scheduling considers job urgency through ODD (Operational Due Date) and also incorporates SPT (Shortest Processing Time)-effect in prioritising operationally late jobs; leading to robust behaviour in Mean Tardiness (MT) with respect to tightness/looseness of due dates. In the present paper, we study an extension of the MOD rule using job-waiting-time based discrimination among operationally late jobs to protect long jobs from excessive delays by incorporating an ‘acceleration property’ into the scheduling rule. Formally, we employ a weighted-SPT dispatching priority index of the form: (Processing time)/(Waiting time)^α for operationally late jobs, while the priority index is ODD for operationally non-late jobs; and the latter class of jobs has a lower priority than the former class. In the context of Assembly Job Shop scheduling, some existing literature includes considerable focus around the concept of ‘Staging Delay’, i.e., waiting of components or sub-assemblies for their counterparts for assembly. Some existing approaches attempt dynamic anticipation of staging delay problems and re-prioritisation of operations along converging branches. In the present paper, rather than depending on such a centralised and largely backward scheduling approach, we consider a partially decentralised approach, endowing jobs with a priority index yielding an ‘acceleration property’ based on a ‘look-back’ in terms of waiting time, rather than ‘look-ahead’. For the particular case, in our proposed rule, whenα is set at zero and when all jobs at a machine are operationally late, our rule agrees with MOD as both exhibit the SPT effect. In simulation tests of our priority scheme for assembly job shops, in comparison with leading heuristics in literature, we found our rule to be particularly effective in: (1) minimising conditional mean tardiness, (2) minimising 99-percentile-point of the tardiness distribution, through proper choice ofα. We also exploit an interesting duality between the scheduling and queueing control versions of the problem. Based on this, some exact and heuristic analysis is given to guide the choice ofα, which is also supported by numerical evidence.     

Probabilistic asymptotic analysis of stochastic online scheduling problems

In the stochastic online scheduling environment, jobs with unknown release times and weights arrive over time. Upon arrival, the information on the weight of the job is revealed but the processing requirement remains unknown until the job is finished. In this paper we consider the objective of minimizing the total weighted completion time. With the assumptions that job weights are bounded, machine capacity is adequate, and processing requirements are bounded and identical and independently distributed across the machines and jobs, we show that any nondelay algorithm is asymptotically optimal for the stochastic online single machine problem, flow shop problem, and uniform parallel machine problem. Our simulation studies of these stochastic online scheduling problems show that two generic nondelay algorithms perform very well as long as the number of jobs is larger than 100.     

Lagrangian relaxation and cut generation for sequence-dependent setup time flowshop scheduling problems to minimise the total weighted tardiness

The Lagrangian relaxation and cut generation technique is applied to solve sequence-dependent setup time flowshop scheduling problems to minimise the total weighted tardiness. The original problem is decomposed into individual job-level subproblems that can be effectively solved by dynamic programming. Two types of additional constraints for the violation of sequence-dependent setup time constraints are imposed on the decomposed subproblems in order to improve the lower bound. The decomposed subproblem with the additional setup time constraints on any subset of jobs is also effectively solved by a novel dynamic programming. Computational results show that the lower bound derived by the proposed method is much better than those of CPLEX and branch and bound for problem instances with 50 jobs and five stages with less computational effort.     

An adaptive scheduling algorithm for a parallel machine problem with rework processes

The development of a scheduling methodology for a parallel machine problem with rework processes is presented in this paper. The problem is to make a schedule for parallel machines with rework probabilities, due-dates, and sequence dependent setup times. Two heuristics are developed based on a dispatching algorithm and problem-space-based search method. In order to evaluate the efficacy of the proposed algorithms, six performance indicators are considered: total tardiness, maximum lateness, mean flow-time, mean lateness, the number of tardy jobs, and the number of reworks. This paper shows how these algorithms can adaptively capture the characteristics of manufacturing facilities for enhancing the performance under changing production environments. Extensive experimental results show that the proposed algorithms give very efficient performance in terms of computational time and each objective value.     

Coordinated scheduling of a single machine with sequence-dependent setup times and time-window constraints

In this paper we consider the selection and scheduling of several jobs on a single machine with sequence-dependent setup times and strictly enforced time-window constraints on the start times of each job. We demonstrate how to develop network-based algorithms to sustain the desired work in process (WIP) profile in a manufacturing environment. Short-term production targets are used to coordinate decentralised local schedulers and to make the objectives of specific areas in line with the chain objectives. A wide range of test problems with two different network structures are simulated. The effectiveness, efficiency, and robustness of the proposed algorithms are analysed and compared with an exhaustive search approach.     

Makespan minimisation for a parallel machine scheduling problem with preemption and job incompatibility

In this paper, an extension of the graph colouring problem is introduced to model a parallel machine scheduling problem with job incompatibility. To get closer to real-world applications, where the number of machines is limited and jobs have different processing times, each vertex of the graph requires multiple colours and the number of vertices with the same colour is bounded. In addition, several objectives related to scheduling are considered: makespan, number of pre-emptions and summation over the jobs’ throughput times. Different solution methods are proposed, namely, two greedy heuristics, two tabu search methods and an adaptive memory algorithm. The latter uses multiple recombination operators, each one being designed for optimising a subset of objectives. The most appropriate operator is selected dynamically at each iteration, depending on its past performance. Experiments show that the proposed algorithm is effective and robust, while providing high-quality solutions on benchmark instances for the graph multi-colouring problem, a simplification of the considered problem.