Minimizing makespan in a two-stage hybrid flow shop scheduling problem with open shop in one stage

This paper considers a scheduling problem in two-stage hybrid flow shop, where the first stage consists of two machines formed an open shop and the other stage has only one machine. The objective is to minimize the makespan, i.e., the maximum completion time of all jobs. We first show the problem is NP-hard in the strong sense, then we present two heuristics to solve the problem. Computational experiments show that the combined algorithm of the two heuristics performs well on randomly generated problem instances.     

Optimal policies in stochastic shop scheduling

In this paper a survey is presented of some of the recent results in stochastic open shop, flow shop and job shop scheduling. The distributions of the processing times of the jobs are known in advance, but the actual processing times are not known in advance. The jobs may have due dates. Optimal preemptive and nonpreemptive policies are determined for the minimization of various objective functions, such as the expected makespan, the expected flow time and the expected number of late jobs. The effect of various degrees of dependence between the processing times of any given job on the various machines is investigated. Under given conditions bounds are obtained for the expected makespan in the different models.Partially supported by the National Science Foundation (NSF), under grant ECS-8115344 with the Georgia Institute of Technology.     

A note on the two-stage assembly flow shop scheduling problem with uniform parallel machines

We study the problem of minimizing the makespan in a two-stage assembly flow shop scheduling problem with uniform parallel machines. This problem is a generalization of the assembly flow shop problem with concurrent operations in the first stage and a single assembly operation in the second stage. We propose a heuristic with an absolute performance bound which becomes asymptotically optimal as the number of jobs becomes very large. We show that our results slightly improve earlier results for the simpler assembly flow shop problem (without uniform machines) and for the two-stage hybrid flow shop problem with uniform machines.     

Two-machine flow shop scheduling problem with an outsourcing option

We consider a two-machine flow shop problem in which each job is processed through an in-house system or outsourced to a subcontractor. A schedule is established for the in-house jobs, and performance is measured by the makespan. Jobs processed by subcontractors require paying an outsourcing cost. The objective is to minimize the sum of the makespan and total outsourcing costs. We show that the problem is NP-hard in the ordinary sense. We consider a special case in which each job has a processing requirement, and each machine a characteristic value. In this case, the time a job occupies a machine is equal to the job’s processing requirement plus a setup time equal to the characteristic value of that machine. We introduce some optimality conditions and present a polynomial-time algorithm to solve the special case.     

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.     

Bilevel programming approach applied to the flow shop scheduling problem under fuzziness

This paper presents a fuzzy bilevel programming approach to solve the flow shop scheduling problem. The problem considered here differs from the standard form in that operators are assigned to the machines and imposing a hierarchy of two decision makers with fuzzy processing times. The shop owner considered higher level and assigns the jobs to the machines in order to minimize the flow time while the customer is the lower level and decides on a job schedule in order to minimize the makespan. In this paper, we use the concepts of tolerance membership function at each level to define a fuzzy decision model for generating optimal (satisfactory) solution for bilevel flow shop scheduling problem. A solution algorithm for solving this problem is given. Mathematics Subject Classification: 90C70, 90B36, 90C99     

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.     

Heuristic constructive algorithms for open shop scheduling to minimize mean flow time

In this paper, we consider the problem of scheduling n jobs on m machines in an open shop environment so that the sum of completion times or mean flow time becomes minimal. For this strongly NP-hard problem, we develop and discuss different constructive heuristic algorithms. Extensive computational results are presented for problems with up to 50 jobs and 50 machines, respectively. The quality of the solutions is evaluated by a lower bound for the corresponding preemptive open shop problem and by an alternative estimate of mean flow time. We observe that the recommendation of an appropriate constructive algorithm strongly depends on the ratio n/m.     

Simulated annealing and genetic algorithms for minimizing mean flow time in an open shop

This paper considers the problem of scheduling n jobs on m machines in an open shop environment so that the sum of completion times or mean flow time becomes minimal. It continues recent work by Bräsel et al. [H. Bräsel, A. Herms, M. Mörig, T. Tautenhahn, T. Tusch, F. Werner, Heuristic constructive algorithms for open shop scheduling to minmize mean flow time, European J. Oper. Res., in press (doi.10.1016/j.ejor.2007.02.057)] on constructive algorithms. For this strongly NP-hard problem, we present two iterative algorithms, namely a simulated annealing and a genetic algorithm. For the simulated annealing algorithm, several neighborhoods are suggested and tested together with the control parameters of the algorithm. For the genetic algorithm, new genetic operators are suggested based on the representation of a solution by the rank matrix describing the job and machine orders. Extensive computational results are presented for problems with up to 50 jobs and 50 machines, respectively. The algorithms are compared relative to each other, and the quality of the results is also estimated partially by a lower bound for the corresponding preemptive open shop problem. For most of the problems, the genetic algorithm is superior when fixing the same number of 30 000 generated solutions for each algorithm. However, in contrast to makespan minimization problems, where the focus is on problems with an equal number of jobs and machines, it turns out that problems with a larger number of jobs than machines are the hardest problems.     

Approximative procedures for no-wait job shop scheduling

In this article we consider the no-wait job shop problem with makespan objective. Based on a decomposition of the problem into a sequencing and a timetabling problem, we propose two local search algorithms. Extensive computational tests in which the algorithms compare favorably to the best existing strategies are reported. Although not specifically designed for that purpose, our algorithms also outperform one of the best no-wait flow shop algorithms in literature.     

Three is easy, two is hard: open shop sum-batch scheduling problem refined

For the two-machine open shop sum-batch problem to minimize the makespan an optimal schedule is known to contain one, two or three batches on each machine, and finding a two-batch optimal schedule is NP-hard. We adapt the open shop algorithm by de Werra for finding a three-batch optimal schedule in linear time.     

A note on the complexity of flow shop scheduling with transportation constraints

This note investigates two-machine flow shop scheduling with transportation constraints to minimize makespan. Recently, Soukhal et al. [A. Soukhal, A. Oulamara, P. Martineau, Complexity of flow shop scheduling problems with transportation constraints, European Journal of Operational Research 161 (2005) 32–41] proved that this problem is strongly NP-hard when the capacity of the truck is limited to two or three parts. The considered problem with blocking constraints is also proved to be strongly NP-hard by Soukhal et al. Unfortunately, their proofs contain mistakes. We point out their proofs’ invalidity and then show that, when the capacity of the truck is limited to two parts, the problem is binary NP-hard, and when the capacity of the truck is limited to three parts the problem is strongly NP-hard even if the jobs have a common processing time on machine one and all jobs have the same transportation time. We show also that the last result can be generalized to any fixed c (c ? 3) parts.     

An empirical analysis of the optimality rate of flow shop heuristics

If a certain optimization problem is NP-hard or even harder, one could expect that the chances of solving it optimally should rather decrease with an increase of the problem size. We reveal, however, that the opposite occurs for a strongly NP-hard problem, which requires sequencing n jobs through an m machine flow shop so as to minimize the makespan. In particular, we empirically examine optimality rates (the probability of being optimal) of the famous NEH heuristic of Nawaz et al. [Nawaz, M., Enscore, Jr., E., Ham, I., 1983. A heuristic algorithm for the m-machine, n-job flow-shop sequencing problem. Omega, The International Journal of Management Science 11, 91–95] and two improved versions of NEH. By using millions of simulation trials and a new effective lower bound on the shortest makespan, we observe relatively high optimality rates of the three heuristics for small values of m. Rather surprisingly, for larger values of n, the heuristics become more frequently optimal as n increases. Neither theoretical nor empirical studies of optimality rates of flow shop heuristics have been conducted so far, and – to the best of our knowledge – no similar studies are known in the field of operations research.     

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.     

A projective algorithm for preemptive open shop scheduling with two multiprocessor groups

We study a multiprocessor extension of the preemptive open shop scheduling problem, where the set of processors is partitioned into processor groups. We show that the makespan minimization problem is polynomially solvable for two multiprocessor groups even if preemptions are restricted to integral times.     

Two-machine flow shop scheduling problems with no-wait jobs

In this paper we provide a fairly complete complexity classification of various versions of the two-machine permutation flow shop scheduling problem to minimize the makespan in which some of the jobs have to be processed with no-wait in process. For some version, we offer a fully polynomial-time approximation scheme and a -approximation algorithm.     

Complexity of flow shop scheduling problems with transportation constraints

In most manufacturing and distribution systems, semi-finished jobs are transferred from one processing facility to another by transporters such as Automated Guided Vehicles, robots and conveyors, and finished jobs are delivered to warehouses or customers by vehicles such as trucks.This paper investigates two-machine flow shop scheduling problems taking transportation into account. The finished jobs are transferred from the processing facility and delivered to customers by truck. Both transportation capacity and transportation times are explicitly taken into account in these models. We study the class of flow shop problems by analysing their complexity. For the makespan objective function, we prove that this problem is strongly NP-hard when the capacity of a truck is limited to two or three parts with an unlimited buffer at the output of the each machine. This problem with additional constraints, such as blocking, is also proven to be strongly NP-hard.     

Online scheduling of ordered flow shops

We consider online as well as offline scheduling of ordered flow shops with the makespan as objective. In an online flow shop scheduling problem, jobs are revealed to a decisionmaker one by one going down a list. When a job is revealed to the decision maker, its operations have to be scheduled irrevocably without having any information regarding jobs that will be revealed afterwards. We consider for the online setting the so-called Greedy Algorithm which generates permutation schedules in which the jobs on the machines are at all times processed without any unnecessary delays. We focus on ordered flow shops, in particular proportionate flow shops with different speeds and proportionate flow shops with different setup times. We analyze the competitive ratio of the Greedy Algorithm for such flow shops in the online setting. For several cases, we derive lower bounds on the competitive ratios.     

Two-stage open shop scheduling with a bottleneck machine

It is known that for the open shop scheduling problem to minimize the makespan there exists no polynomial-time heuristic algorithm that guarantees a worst-case performance ratio better than 5/4, unless P≠NP. However, this result holds only if the instance of the problem contains jobs consisting of at least three operations. This paper considers the open shop scheduling problem, provided that each job consists of at most two operations, one of which is to be processed on one of the m⩾2 machines, while the other operation must be performed on the bottleneck machine, the same for all jobs. For this NP-hard problem we present a heuristic algorithm and show that its worst-case performance ratio is 5/4.