A new heuristic for three-machine flow shop scheduling

This paper considers the problem of sequencing n jobs in a three-machine flow shop with the objective of minimizing the makespan, which is the completion time of the last job. An O(n log n) time heuristic that is based on Johnson's algorithm is presented. It is shown to generate a schedule with length at most 5/3 times that of an optimal schedule, thereby reducing the previous best available worst-case performance ratio of 2. An application to the general flow shop is also discussed.

Shop scheduling problems under precedence constraints

The paper considers a scheduling model that generalizes the well-known open shop, flow shop, and job shop models. For that model, called the super shop, we study the complexity of finding a time-optimal schedule in both preemptive and non-preemptive cases assuming that precedence constraints are imposed over the set of jobs. Two types of precedence rela-tions are considered. Most of the arising problems are proved to be NP-hard, while for some of them polynomial-time algorithms are presented.

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 open shop scheduling problem with a given sequence of jobs on one machine

The paper considers the open shop scheduling problem to minimize the make-span, provided that one of the machines has to process the jobs according to a given sequence. We show that in the preemptive case the problem is polynomially solvable for an arbitrary number of machines. If preemption is not allowed, the problem is NP-hard in the strong sense if the number of machines is variable, and is NP-hard in the ordinary sense in the case of two machines. For the latter case we give a heuristic algorithm that runs in linear time and produces a schedule with the makespan that is at most 5/4 times the optimal value. We also show that the two-machine problem in the nonpreemptive case is solvable in pseudopolynomial time by a dynamic programming algorithm, and that the algorithm can be converted into a fully polynomial approximation scheme. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 705–731, 1998

Earliness penalties on a single machine subject to precedence constraints: SLK due date assignment

The paper considers the single machine due date assignment and scheduling problems with n jobs in which the due dates are to be obtained from the processing times by adding a positive slack q. A schedule is feasible if there are no tardy jobs and the job sequence respects given precedence constraints. The value of q is chosen so as to minimize a function ϕ(F,q) which is non-decreasing in each of its arguments, where F is a certain non-decreasing earliness penalty function. Once q is chosen or fixed, the corresponding scheduling problem is to find a feasible schedule with the minimum value of function F. In the case of arbitrary precedence constraints the problems under consideration are shown to be NP-hard in the strong sense even for F being total earliness. If the precedence constraints are defined by a series-parallel graph, both scheduling and due date assignment problems are proved solvable in time, provided that F is either the sum of linear functions or the sum of exponential functions. The running time of the algorithms can be reduced to if the jobs are independent. Scope and purpose We consider the single machine due date assignment and scheduling problems and design fast algorithms for their solution under a wide range of assumptions. The problems under consideration arise in production planning when the management is faced with a problem of setting the realistic due dates for a number of orders. The due dates of the orders are determined by increasing the time needed for their fulfillment by a common positive slack. If the slack is set to be large enough, the due dates can be easily maintained, thereby producing a good image of the firm. This, however, may result in the substantial holding cost of the finished products before they are brought to the customer. The objective is to explore the trade-off between the size of the slack and the arising holding costs for the early orders.

A greedy open shop heuristic with job priorities

The paper presents an improved version of the greedy open shop approximation algorithm with pre-ordering of jobs. It is shown that the algorithm compares favorably with the greedy algorithm with no pre-ordering by reducing either its absolute or relative error. In the case of three machines, the new algorithm creates a schedule with the makespan that is at most 3/2 times the optimal value.

Heuristics for short route job shop scheduling problems

We consider two “minimum”NP-hard job shop scheduling problems to minimize the makespan. In one of the problems every job has to be processed on at most two out of three available machines. In the other problem there are two machines, and a job may visit one of the machines twice. For each problem, we define a class of heuristic schedules in which certain subsets of operations are kept as blocks on the corresponding machines. We show that for each problem the value of the makespan of the best schedule in that class cannot be less than 3/2 times the optimal value, and present algorithms that guarantee a worst-case ratio of 3/2.

A heuristic algorithm for two-machine re-entrant shop scheduling

This paper considers the problem of sequencing n jobs in a two‐machine re‐entrant shopwith the objective of minimizing the maximum completion time. The shop consists of twomachines, M1 and M2 , and each job has the processing route (M1 , M2 , M1 ). An O(n log n)time heuristic is presented which generates a schedule with length at most 4/3 times that ofan optimal schedule, thereby improving the best previously available worst‐case performanceratio of 3/2.

A 3/2 algorithm for two-machine open shop with route-dependent processing times

This paper considers the problem of minimizing the schedule length of a two-machine shop in which not only can a job be assigned any of the two possible routes, but also the processing times depend on the chosen route. This problem is known to be NP-hard. We describe a simple approximation algorithm that guarantees a worst-case performance ratio of 2. We also present some modifications to this algorithm that improve its performance and guarantee a worst-case performance ratio of 3=2.

A polynomial algorithm for the three-machine open shop with a bottleneck machine

The paper considers the three‐machine open shop scheduling problem to minimize themakespan. It is assumed that each job consists of at most two operations, one of which is tobe processed on the bottleneck machine, the same for all jobs. A new lower bound on theoptimal makespan is derived, and a linear‐time algorithm for finding an optimalnon‐preemptive schedule is presented.

Heuristics for the two-stage job shop scheduling problem with a bottleneck machine

The paper considers the job shop scheduling problem to minimize the makespan. It is assumed that each job consists of at most two operations, one of which is to be processed on one of m⩾2 machines, while the other operation must be performed on a single bottleneck machine, the same for all jobs. For this strongly NP-hard problem we present two heuristics with improved worst-case performance. One of them guarantees a worst-case performance ratio of 3/2. The other algorithm creates a schedule with the makespan that exceeds the largest machine workload by at most the length of the largest operation.

Scheduling for parallel dedicated machines with a single server

This paper examines scheduling problems in which the setup phase of each operation needs to be attended by a single server, common for all jobs and different from the processing machines. The objective in each situation is to minimize the makespan. For the processing system consisting of two parallel dedicated machines we prove that the problem of finding an optimal schedule is NP-hard in the strong sense even if all setup times are equal or if all processing times are equal. For the case of m parallel dedicated machines, a simple greedy algorithm is shown to create a schedule with the makespan that is at most twice the optimum value. For the two machine case, an improved heuristic guarantees a tight worst-case ratio of 3/2. We also describe several polynomially solvable cases of the later problem. The two-machine flow shop and the open shop problems with a single server are also shown to be NP-hard in the strong sense. However, we reduce the two-machine flow shop no-wait problem with a single server to the Gilmore-Gomory traveling salesman problem and solve it in polynomial time. (c) 2000 John Wiley & Sons, Inc.

Group technology approach to the open shop scheduling problem with batch setup times

This paper studies the problem of scheduling jobs in a two-machine open shop to minimize the makespan. Jobs are grouped into batches and are processed without preemption. A 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. For this NP-hard problem, we propose a linear-time heuristic algorithm that creates a group technology schedule, in which no batch is split into sub-batches. We demonstrate that our heuristic is a -approximation algorithm. Moreover, we show that no group technology algorithm can guarantee a worst-case performance ratio less than 5/4.

An open shop scheduling problem with a non-bottleneck machine

This paper considers the problem of processing n jobs in a two-machine non-preemptive open shop to minimize the makespan, i.e., the maximum completion time. One of the machines is assumed to be non-bottleneck. It is shown that, unlike its flow shop counterpart, the problem is NP-hard in the ordinary sense. On the other hand, the problem is shown to be solvable by a dynamic programming algorithm that requires pseudopolynomial time. The latter algorithm can be converted into a fully polynomial approximation scheme that runs in time. An O(n log n) approximation algorithm is also designed whi finds a schedule with makespan at most 5/4 times the optimal value, and this bound is tight.

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.