Single-machine and two-machine flowshop scheduling with general learning functions

We show that the O(n log n) (where n is the number of jobs) shortest processing time (SPT) sequence is optimal for the single-machine makespan and total completion time minimization problems when learning is expressed as a function of the sum of the processing times of the already processed jobs. We then show that the two-machine flowshop makespan and total completion time minimization problems are solvable by the SPT sequencing rule when the job processing times are ordered and job-position-based learning is in effect. Finally, we show that when the more specialized proportional job processing times are in place, then our flowshop results apply also in the more general sum-of-job-processing-times-based learning environment.     

Scheduling under a Non-renewable Resource Constraint

This paper examines a single-machine, non-renewable-resource-constrained scheduling problem where jobs have arbitrary processing times and resource requirements. Unit supply of a resource is assumed at each time period. Performance criterion is makespan. It is proved that this problem is identical to the two-machine flowshop problem, enabling the use of Johnson's algorithm. Immediate extensions of this result are presented.     

Flow shop scheduling jobs with position-dependent processing times

The paper is devoted to some flow shop scheduling problems, where job processing times are defined by functions dependent on their positions in the schedule. An example is constructed to show that the classical Johnson's rule is not the optimal solution for two different models of the two-machine flow shop scheduling to minimize makespan. In order to solve the makespan minimization problem in the two-machine flow shop scheduling, we suggest Johnson's rule as a heuristic algorithm, for which the worst-case bound is calculated. We find polynomial time solutions to some special cases of the considered problems for the following optimization criteria: the weighted sum of completion times and maximum lateness. Some furthermore extensions of the problems are also shown.     

Minimizing makespan with multiple-orders-per-job in a two-machine flowshop

We investigate a new scheduling problem, multiple-orders-per-job (MOJ), in the context of a two-machine flowshop. Lower bounds for the makespan performance measure are provided for combinations of lot-processing and item-processing machines. An optimization model is presented that addresses both job formation and job sequencing. We define a heuristic to minimize the makespan for the MOJ problem for two-machine item-processing flowshops. The heuristic obtains solutions within 2% of a tight lower bound and runs in O(HF) time, where H is the number of orders and F is the restricted number of jobs.     

A branch-and-bound algorithm for a two-machine flowshop scheduling problem with limited waiting time constraints

In this paper, we consider a two-machine flowshop scheduling problem in which the waiting time of each job between the two machines cannot be greater than a certain time period. For the problem with the objective of minimizing makespan, we identify several dominance properties of the problem and develop a branch-and-bound (B&B) algorithm using the dominance properties. Computational tests are performed on randomly generated test problems for evaluation of performance of the B&B algorithm, and results show that the algorithm can solve problems with up to 150 jobs in a reasonable amount of CPU time.     

A note on scheduling alternative operations in two-machine flowshops

In this note, we consider a class of problems for scheduling a set of jobs each of which consists of two consecutive operations. The jobs are to be processed in a two-machine flowshop in which either or both machines are versatile. A versatile machine can perform both operations of a job. The objective is to minimise the makespan. While the whole class of problems has been shown to be NP-complete, we provide a general pseudopolynomial dynamic programming scheme which solves the problems analytically. This also establishes that the problems are only NP-complete in the ordinary sense. The solution scheme can be modified to solve another class of similar two-machine flowshop scheduling problems.     

Minimizing makespan on a two-machine re-entrant flowshop

This paper focuses on a two-machine re-entrant flowshop scheduling problem with the objective of minimizing makespan. In the re-entrant flowshop considered here, all jobs must be processed twice on each machine, that is, each job should be processed on machine 1, machine 2 and then machine 1 and machine 2. We develop dominance properties, lower bounds and heuristic algorithms for the problem, and use these to develop a branch and bound algorithm. For evaluation of the performance of the algorithms, computational experiments are performed on randomly generated test problems. Results of the experiments show that the suggested branch and bound algorithm can solve problems with up to 200 jobs in a reasonable amount of CPU time.     

A bicriteria flowshop scheduling with a learning effect

In many situations, a worker’s ability improves as a result of repeating the same or similar tasks; this phenomenon is known as the learning effect. In this paper the learning effect is considered in a two-machine flowshop. The objective is to find a sequence that minimizes a weighted sum of total completion time and makespan. Total completion time and makespan are widely used performance measures in scheduling literature. To solve this scheduling problem, an integer programming model with n² + 6n variables and 7n constraints where n is the number of jobs is formulated. Because of the lengthy computing time and high computing complexity of the integer programming model, the problem with up to 30 jobs can be solved. A heuristic algorithm and a tabu search based heuristic algorithm are presented to solve large size problems. Experimental results show that the proposed heuristic methods can solve this problem with up to 300 jobs rapidly. According to the best of our knowledge, no work exists on the bicriteria flowshop with a learning effect.     

Bicriteria hierarchical optimization of two-machine flow shop scheduling problem with time-dependent deteriorating jobs

We study a two-machine flowshop scheduling problem with time-dependent deteriorating jobs, i.e. the processing times of jobs are an increasing function of their starting time. The objective is to minimize the total completion time subject to minimum makespan. We propose a mixed integer programming model, and develop two pairwise interchange algorithms and a branch-and-bound procedure to solve the problem while using several dominance conditions to limit the size of the search tree. Several polynomial-time solvable special cases are discussed. Finally, numerical studies are performed to examine the effectiveness and the efficiency of the proposed algorithms.     

Generalization of Johnson's and Talwar's scheduling rules in two-machine stochastic flow shops

The paper deals with the classical problem of minimizing the makespan in a two-machine flow shop. When the job processing times are deterministic, the optimal job sequence can be determined by applying Johnson's rule. When they are independent and exponential random variables, Talwar's rule yields a job sequence that minimizes the makespan stochastically. Assuming that the random job processing times are independent and Gompertz distributed, we propose a new scheduling rule that is a generalization of both Johnson's and Talwar's rules. We prove that our rule yields a job sequence that minimizes the makespan stochastically. Extensions to m-machine proportionate stochastic flow shops, two-machine stochastic job shops, and stochastic assembly systems are indicated.     

NP-hard cases in scheduling deteriorating jobs on dedicated machines

In this paper problems of time-dependent scheduling on dedicated machines are considered. The processing time of each job is described by a function which is dependent on the starting time of the job. The objective is to minimise the maximum completion time (makespan). We prove that under linear deterioration the two-machine flow shop problem is strongly NP-hard and the two-machine open shop problem is ordinarily NP-hard. We show that for the three-machine flow shop and simple linear deterioration there does not exist a polynomial-time approximation algorithm with the worst case ratio bounded by a constant, unless P=NP. We also prove that the three-machine open shop problem with simple linear deterioration is ordinarily NP-hard, even if the jobs have got equal deterioration rates on the third machine.     

Scheduling with a position-weighted learning effect

In general, human learning takes time to build up, which results from a worker gaining experience from repeating similar operations over time. In the early stage of processing a given set of similar jobs, a worker is not familiar with the operations, so his learning effect on the jobs scheduled early is not apparent. On the other hand, when the worker has gained experience in processing the jobs his learning improves. So a worker’s learning effect on a job depends not only on the total processing time of the jobs that he has processed but also on the job position. In this paper we introduce a position-weighted learning effect model for scheduling problems. We provide optimal solutions for the single-machine problems to minimize the makespan and the total completion time, and an optimal solution for the single-machine problem to minimize the total tardiness under an agreeable situation. We also consider two special cases of the flowshop problem.     

Single-machine scheduling problems with a learning effect

In many situations, the skills of workers continuously improve when repeating the same or similar tasks. This phenomenon is known as the “learning effect” in the literature. In most studies, the learning phenomenon is implemented by assuming the actual job processing time is a function of its scheduled position [D. Biskup, Single-machine scheduling with learning considerations, Eur. J. Oper. Res. 115 (1999) 173–178]. Recently, a new model is proposed where the actual job processing time depends on the sum of the processing times of jobs already processed [C. Koulamas, G.J. Kyparisis, Single-machine and two-machine flowshop scheduling with general learning functions, Eur. J. Oper. Res. 178 (2007) 402–407]. In this paper, we extend their models in which the actual job processing time not only depends on its scheduled position, but also depends on the sum of the processing times of jobs already processed. We then show that the single-machine makespan and the total completion time problems remain polynomially solvable under the proposed model. In addition, we show that the total weighted completion time has a polynomial optimal solution under certain agreeable solutions.     

Minimizing the makespan in a two-machine cross-docking flow shop problem

This paper studies a two-machine cross-docking flow shop scheduling problem in which a job at the second machine can be processed only after the processing of some jobs at the first machine has been completed. The objective is to minimize the makespan. We first show that the problem is strongly NP-hard. Some polynomially solvable special cases are provided. We then develop a polynomial approximation algorithm with an error-bound analysis. A branch-and-bound algorithm is also constructed. Computational results show that the branch-and-bound algorithm can optimally solve problems with up to 60 jobs within a reasonable amount of time.     

Single-machine and two-machine flowshop scheduling problems with truncated position-based learning functions

Scheduling with learning effects has received growing attention nowadays. A well-known learning model is called ‘position-based learning’ in which the actual processing time of a job is a non-increasing function of its position to be processed. However, the actual processing time of a given job drops to zero precipitously as the number of jobs increases. Motivated by this observation, we propose two truncated learning models in single-machine scheduling problems and two-machine flowshop scheduling problems with ordered job processing times, respectively, where the actual processing time of a job is a function of its position and a control parameter. Under the proposed learning models, we show that some scheduling problems can be solved in polynomial time. In addition, we further analyse the worst-case error bounds for the problems to minimize the total weighted completion time, discounted total weighted completion time and maximum lateness.     

Two-Stage Production Scheduling with Separated Set-up Times and Stochastic Breakdowns

The problem of scheduling a two-machine flowshop, where set-up times are considered as separate from processing times and machines suffer random breakdowns, is addressed with respect to the makespan objective. A dominance relation for minimizing makespan with probability 1 is established. Furthermore, it is shown that Yoshida and Hitomi's algorithm for the deterministic problem stochastically minimizes makespan when random breakdowns are present.     

A branch and bound algorithm for the two-stage assembly scheduling problem

This paper develops a branch and bound algorithm for the two-stage assembly scheduling problem. In this problem, there are m machines at the first stage, each of which produces a component of a job. When all m components are available, a single assembly machine at the second stage completes the job. The objective is to schedule the jobs on the machines so that the maximum completion time, or makespan, is minimized. A lower bound based on solving an artificial two-machine flow shop problem is derived. Also, several dominance theorems are established and incorporated into the branch and bound algorithm. Computational experience with the algorithm is reported for problems with up to 8000 jobs and 10 first-stage machines.     

Nonpreemptive flowshop scheduling with machine dominance

Flowshop scheduling deals with processing a set of jobs through a set of machines, where all jobs have to pass among machines in the same order. With the exception of minimizing a makespan on two machines, almost all other flowshop problems in a general setup are known to be computationally intractable. In this paper we study special cases of flowshop defined by additional machine dominance constraints. These constraints impose certain relations among the job processing times on different machines and make the studied problems tractable.     

Some single-machine scheduling problems with actual time and position dependent learning effects

In this paper we study some single-machine scheduling problems with learning effects where the actual processing time of a job serves as a function of the total actual processing times of the jobs already processed and of its scheduled position. We show by examples that the optimal schedules for the classical version of problems are not optimal under this actual time and position dependent learning effect model for the following objectives: makespan, sum of kth power of the completion times, total weighted completion times, maximum lateness and number of tardy jobs. But under certain conditions, we show that the shortest processing time (SPT) rule, the weighted shortest processing time (WSPT) rule, the earliest due date (EDD) rule and the modified Moore’s Algorithm can also construct an optimal schedule for the problem of minimizing these objective functions, respectively.     

A bicriteria m-machine flowshop scheduling with sequence-dependent setup times

In this study, a bicriteria m-machine flowshop scheduling with sequence-dependent setup times is considered. The objective function of the problem is minimization of the weighted sum of total completion time and makespan. Only small size problems with up to 6 machines and 18 jobs can be solved by the proposed integer programming model. Also the model is tested on an example. We also proposed three heuristic approaches for solving large jobs problems. To solve the large sizes problems up to 100 jobs and 10 machines, special heuristics methods is used. Results of computational tests show that the proposed model is effective in solving problems.