No-wait flowshops with bicriteria of makespan and total completion time

We consider the m-machine no-wait flowshop scheduling problem with the objective of minimizing a weighted sum of makespan and total completion time. For the two-machine problem, we develop a dominance relation and embed it within a proposed branch-and-bound algorithm. For the m-machine problem, we propose a heuristic. Computational experiments show that the proposed heuristic outperforms the best existing multi-criteria heuristics and the best single criterion heuristics for makespan and total completion time. The efficiency of the dominance relation and branch-and-bound algorithm is also investigated and shown to be effective.     

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.     

Two-machine open shop scheduling with special transportation times

The paper considers a problem of scheduling n jobs in a two-machine open shop to minimise the makespan, provided that preemption is not allowed and the interstage transportation times are involved. In general, this problem is known to be NP-hard. We present a linear time algorithm that finds an optimal schedule if no transportation time exceeds the smallest of the processing times. We also describe an algorithm that creates a heuristic solution to the problem with job-independent transportation times. Our algorithm provides a worst-case performance ratio of 8/5 if the transportation time of a job depends on the assigned processing route. The ratio reduces to 3/2 if all transportation times are equal.     

Approximation algorithms for two-machine flow shop scheduling with batch setup times

In many practical situations, batching of similar jobs to avoid setups is performed while constructing a schedule. This paper addresses the problem of non-preemptively scheduling independent jobs in a two-machine flow shop with the objective of minimizing the makespan. Jobs are grouped into batches. A sequence independent batch setup time on each machine is required before the first job is processed, and when a machine switches from processing a job in some batch to a job of another batch. Besides its practical interest, this problem is a direct generalization of the classical two-machine flow shop problem with no grouping of jobs, which can be solved optimally by Johnson's well-known algorithm. The problem under investigation is known to be NP-hard. We propose two O(n logn) time heuristic algorithms. The first heuristic, which creates a schedule with minimum total setup time by forcing all jobs in the same batch to be sequenced in adjacent positions, has a worst-case performance ratio of 3/2. By allowing each batch to be split into at most two sub-batches, a second heuristic is developed which has an improved worst-case performance ratio of 4/3. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

The two-machine no-wait general and proportionate open shop makespan problem

We consider the two-machine no-wait open shop minimum makespan problem in which the determination of an optimal solution requires an optimal pairing of the jobs followed by the optimal sequencing of the job pairs. We show that the required enumeration can be curtailed by reducing the pair sequencing problem for a given pair set to a traveling salesman problem which is equivalent to a two-machine no-wait flow shop problem solvable in O(n log n) time. We then propose an optimal O(n log n) algorithm for the proportionate problem with equal machine speeds in which each job has the same processing time on both machines. We show that our O(n log n) algorithm also applies to the more general proportionate problem with equal machine speeds and machine-specific setup times. We also analyze the proportionate problem with unequal machine speeds and conclude that the required enumeration can be further curtailed (compared to the problem with arbitrary job processing times) by eliminating certain job pairs from consideration.     

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.     

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.     

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.     

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.     

Customer order scheduling to minimize the number of late jobs

In the order scheduling problem, every job (order) consists of several tasks (product items), each of which will be processed on a dedicated machine. The completion time of a job is defined as the time at which all its tasks are finished. Minimizing the number of late jobs was known to be strongly NP-hard. In this note, we show that no FPTAS exists for the two-machine, common due date case, unless P = NP. We design a heuristic algorithm and analyze its performance ratio for the unweighted case. An LP-based approximation algorithm is presented for the general multicover problem. The algorithm can be applied to the weighted version of the order scheduling problem with a common due date.     

Flow-shop scheduling with a learning effect

The paper is devoted to some flow-shop scheduling problems with a learning effect. The objective is to minimize one of the two regular performance criteria, namely, makespan and total flowtime. A heuristic algorithm with worst-case bound m for each criteria is given, where m is the number of machines. Furthermore, a polynomial algorithm is proposed for both of the special cases: identical processing time on each machine and an increasing series of dominating machines. An example is also constructed to show that the classical Johnson's rule is not the optimal solution for the two-machine flow-shop scheduling to minimize makespan with a learning effect. Some extensions of the problem are also shown.     

A dynamic programming algorithm for scheduling jobs in a two-machine open shop with an availability constraint

This paper studies a two-machine open shop scheduling problem with an availability constraint, ie we assume that a machine is not always available and that the processing of the interrupted job can be resumed when the machine becomes available again. We consider the makespan minimization as criterion. This problem is NP-hard. We develop a pseudo-polynomial time dynamic programming algorithm to solve the problem optimally when the machine is not available at time s>0. Then, we propose a mixed integer linear programming formulation, that allows to solve instances with up to 500 jobs optimally in less than 5?min with CPLEX solver. Finally, we show that any heuristic algorithm has a worst-case error bound of 1.     

Two-machine shop scheduling: Compromise between flexibility and makespan value

In this paper, we consider the problem of providing flexibility to solutions of two-machine shop scheduling problems. We use the concept of group-scheduling to characterize a whole set of schedules so as to provide more choice to the decision-maker at any decision point. A group-schedule is a sequence of groups of permutable operations defined on each machine where each group is such that any permutation of the operations inside the group leads to a feasible schedule. Flexibility of a solution and its makespan are often conflicting, thus we search for a compromise between a low number of groups and a small value of makespan. We resolve the complexity status of the relevant problems for the two-machine flow shop, job shop and open shop. A number of approximation algorithms are developed and their worst-case performance is analyzed. For the flow shop, an effective heuristic algorithm is proposed and the results of computational experiments are reported.     

A heuristic for minimizing the expected makespan in two-machine flow shops with consistent coefficients of variation

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 job processing times are independently and Weibull distributed random variables, we present a new job sequencing rule that includes both Johnson’s and Talwar’s rules as special cases. The proposed rule is applicable as a heuristic whenever the job processing times are characterized by their means and the same coefficient of variation. Simulation results show that it leads to very encouraging results when the expected makespan is minimized.     

A combined branch-and-bound and genetic algorithm based approach for a flowshop scheduling problem

In this paper, we study the application of a meta-heuristic to a two-machine flowshop scheduling problem. The meta-heuristic uses a branch-and-bound procedure to generate some information, which in turn is used to guide a genetic algorithm's search for optimal and near-optimal solutions. The criteria considered are makespan and average job flowtime. The problem has applications in flowshop environments where management is interested in reducing turn-around and job idle times simultaneously. We develop the combined branch-and-bound and genetic algorithm based procedure and two modified versions of it. Their performance is compared with that of three algorithms: pure branch-and-bound, pure genetic algorithm, and a heuristic. The results indicate that the combined approach and its modified versions are better than either of the pure strategies as well as the heuristic algorithm.     

Total completion time minimization in two-machine job shops with unit-time operations

We study the problem of minimizing total completion time in two-machine job shop with unit-time operations. We propose an efficient algorithm for the problem. The algorithm is polynomial with respect to a succinct encoding of the problem instances, where the number of bits necessary to encode a job with k operations is O(log(k + 1)). This result answers a long standing open question about the complexity of the problem.     

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.     

Johnson’s rule,composite jobs and the relocation problem

Two-machine flowshop scheduling to minimize makespan is one of the most well-known classical scheduling problems. Johnson’s rule for solving this problem has been widely cited in the literature. We introduce in this paper the concept of composite job, which is an artificially constructed job with processing times such that it will incur the same amount of idle time on the second machine as that incurred by a chain of jobs in a given processing sequence. This concept due to Kurisu first appeared in 1976 to deal with the two-machine flowshop scheduling problem involving precedence constraints among the jobs. We show that this concept can be applied to reduce the computational time to solve some related scheduling problems. We also establish a link between solving the two-machine flowshop makespan minimization problem using Johnson’s rule and the relocation problem introduced by Kaplan. We present an intuitive interpretation of Johnson’s rule in the context of the relocation problem.     

Total flowtime and makespan for a no-wait <Emphasis Type="Italic">m</Emphasis>-machine flowshop with set-up times separated

This paper addresses the m-machine no-wait flowshop problem where the set-up time of a job is separated from its processing time. The performance measures considered are the total flowtime and makespan. The scheduling problem for makespan reduces to the travelling salesman problem (TSP), and the scheduling problem for total flowtime reduces to the time-dependent travelling salesman problem (TDTSP). Non-polynomial time solution methods are presented, along with a polynomial heuristic.     

Two-machine flowshop scheduling with availability constraints

The majority of the scheduling literature carries a common assumption that machines are available all the time. However, this availability assumption may not be true in real industry settings, since a machine may become unavailable during certain periods of time when, for instance, a machine breakdown or a preventive maintenance activity is scheduled. Although the problem is realistic and important, it is relatively new and unstudied. In this paper, we study the two-machine flowshop problem under the assumption that the unavailable time is known in advance. We assume that if a job cannot be finished before the next down period of a machine then the job will have to partially restart when the machine has become available again. We call our model semiresumable. Our model contains two important special cases: resumable where the job can be continued without any penalty and nonresumable where the job needs to totally restart. We study the problem where an availability constraint is imposed only on one machine as well as on both machines. We provide complexity analysis, develop a pseudo-polynomial dynamic programming algorithm to solve the problem optimally and also propose heuristic algorithms with an error bound analysis.