Single-machine scheduling with deteriorating jobs and setup times to minimize the maximum tardiness

In many realistic production situations, a job processed later consumes more time than the same job when it is processed earlier. Production scheduling in such an environment is known as scheduling with deteriorating jobs. However, research on scheduling problems with deteriorating jobs has rarely considered explicit (separable) setup time (cost). In this paper, we consider a single-machine scheduling problem with deteriorating jobs and setup times to minimize the maximum tardiness. We provide a branch-and-bound algorithm to solve this problem. Computational experiments show that the algorithm can solve instances up to 1000 jobs in reasonable time.     

Single-machine scheduling to stochastically minimize maximum lateness

We study the problem of scheduling a set of jobs on a single machine, to minimize the maximum lateness ML or the maximum weighted lateness MWL under stochastic order. The processing time P _i , the due date D _i , and the weight W _i of each job i may all be random variables. We obtain the optimal sequences in the following situations: (i) For ML, the {P _i } can be likelihood-ratio ordered, the {D _i } can be hazard-rate ordered, and the orders are agreeable; (ii) For MWL, {D _i } are exponentially distributed, {P _i } and {W _i } can be likelihood-ratio ordered and the orders are agreeable with the rates of {D _i }; and (iii) For ML, P _i and D _i are exponentially distributed with rates μ _i and ν _i , respectively, and the sequence {ν _i (ν _i +μ _i )} has the same order as {ν _i (ν _i +μ _i +A ₀)} for some sufficiently large A ₀. Some related results are also discussed. This work was partially supported by the Research Grants Council of Hong Kong under Earmarked Grants No. PolyU 5146/02E, CUHK 4170/03E, and NSFC Research Funds No. 70329001, 70518002.     

Single-machine scheduling with periodic maintenance to minimize makespan

We consider a single-machine scheduling problem with periodic maintenance activities. Although the scheduling problem with maintenance has attracted researchers’ attention, most of past studies considered only one maintenance period. In this research several maintenance periods are considered where each maintenance activity is scheduled after a periodic time interval. The objective is to find a schedule that minimizes the makespan, subject to periodic maintenance and nonresumable jobs. We first prove that the worst-case ratio of the classical LPT   algorithm is 2. Then we show that there is no polynomial time approximation algorithm with a worst-case ratio less than 2 unless P=NP$P=\text{<italic>NP</italic>}$, which implies that the LPT algorithm is the best possible.     

Single-machine scheduling to minimize total convex resource consumption with a constraint on total weighted flow time

In this paper, we consider single-machine scheduling problem in which processing time of a job is described by a convex decreasing resource consumption function. The objective is to minimize the total amount of resource consumed subject to a constraint on total weighted flow time. The optimal resource allocation is obtained for any arbitrary job sequence. The computational complexity of the general problem remains an open question, but we present and analyze some special cases that are solvable by using polynomial time algorithms. For the general problem, several dominance properties and some lower bounds are derived, which are used to speed up the elimination process of a branch-and-bound algorithm proposed to solve the problem. A heuristic algorithm is also proposed, which is shown by computational experiments to perform effectively and efficiently in obtaining near-optimal solutions. The results show that the average percentage error of the proposed heuristic algorithm from optimal solutions is less than 3%.     

Single-machine scheduling with periodic maintenance to minimize makespan revisited

In this paper, a single-machine scheduling problem with periodic maintenance and non-preemptive jobs is considered. The objective is to minimize the makespan. It shows that the classical list scheduling (LS) algorithm, the longest processing time first (LPT) algorithm, and the modified longest processing time first (MLPT) algorithm all have the same worst-case ratio and the same computational complexity for the considered problem. To compare the performances of three considered algorithms in detail, the worst-case ratios of algorithms are formed as single-variable functions of the total number of maintenance activities needed in the optimal schedule. Analysis results show that the bound associated with the LS algorithm is always dominated by the bound associated with the LPT algorithm, and the latter is always dominated by the bound associated with the MLPT algorithm.     

The two-stage assembly scheduling problem to minimize total completion time with setup times

We address the two-stage assembly scheduling problem where there are m machines at the first stage and an assembly machine at the second stage. The objective is to schedule the available n jobs so that total completion time of all n jobs is minimized. Setup times are treated as separate from processing times. This problem is NP-hard, and therefore we present a dominance relation and propose three heuristics. The heuristics are evaluated based on randomly generated data. One of the proposed heuristics is known to be the best heuristic for the case of zero setup times while another heuristic is known to perform well for such problems. A new version of the latter heuristic, which utilizes the dominance relation, is proposed and shown to perform much better than the other two heuristics.     

Single machine batch scheduling problem with family setup times and release dates to minimize makespan

In this paper we consider the single machine batch scheduling problem with family setup times and release dates to minimize makespan. We show that this problem is strongly NP-hard, and give an time dynamic programming algorithm and an time dynamic programming algorithm for the problem, where n is the number of jobs, m is the number of families, k is the number of distinct release dates and P is the sum of the setup times of all the families and the processing times of all the jobs. We further give a heuristic with a performance ratio 2. We also give a polynomial-time approximation scheme (PTAS) for the problem.     

Single machine batch scheduling to minimize total completion time and resource consumption costs

In this paper we study the single-machine batch scheduling problem under batch availability, where both setup and job processing times are controllable by allocating a continuously divisible nonrenewable resource. Under batch availability a set of jobs is processed contiguously and completed together, when the processing of the last job in the batch is finished. We present polynomial time algorithms to find the job sequence, the partition of the job sequence into batches and the resource allocation, which minimize the total completion time or the total production cost (inventory plus resource costs).     

Iterated Local Search for single-machine scheduling with sequence-dependent setup times to minimize total weighted tardiness

We present an Iterated Local Search (ILS) algorithm for solving the single-machine scheduling problem with sequence-dependent setup times to minimize the total weighted tardiness. The proposed ILS algorithm exhibits several distinguishing features, including a new neighborhood structure called Block Move and a fast incremental evaluation technique, for evaluating neighborhood solutions. Applying the proposed algorithm to solve 120 public benchmark instances widely used in the literature, we achieve highly competitive results compared with a recently proposed exact algorithm and five sets of best solutions of state-of-the-art metaheuristic algorithms in the literature. Specifically, ILS obtains the optimal solutions for 113 instances within a reasonable time, and it outperforms the previous best-known results obtained by metaheuristic algorithms for 34 instances and matches the best results for 82 instances. In addition, ILS is able to obtain the optimal solutions for the remaining seven instances under a relaxed time limit, and its computational efficiency is comparable with the state-of-the-art exact algorithm by Tanaka and Araki (Comput Oper Res 40:344–352, 2013). Finally, on analyzing some important features that affect the performance of ILS, we ascertain the significance of the proposed Block Move neighborhood and the fast incremental evaluation technique.     

Single-machine scheduling with workload-dependent tool change durations and equal processing time jobs to minimize total completion time

We consider a single-machine tool change scheduling problem where tool change durations are workload-dependent. The processing times of all the jobs are the same. The objective is to determine the number of tool change activities, the start time and the completion time of each tool change activity jointly and schedule all the jobs to the machine such that the total completion time of the jobs is minimized. For the case where the tool change duration function is concave, we present a linear time optimal algorithm. For the case where the tool change duration function is convex, we convert it into a convex integer quadratic programming problem with fixed dimension and then propose two polynomial time algorithms for it. We also study some special cases for which optimal schedules can be obtained directly. For the case where the tool change duration function is linear, we present all the optimal schedules.     

On scheduling with non-increasing time slot cost to minimize total weighted completion time

This paper addresses a recent open scheduling problem which aims to minimize the summation of total weighted completion time and the total machine time slot cost. Focusing on the case of non-increasing time slot cost with non-preemptive jobs, we show that the problem can be solved in polynomial-time when the time slot cost decreases with certain patterns, including linearly decreasing, decreasing concave, and decreasing convex cases. Different methodologies are used for three cases. For the linearly decreasing case, we can classify all the jobs into three categories and schedule the job sets one by one. For the decreasing concave case, we calculate each job’s worst starting time and try to make them far away from their worst starting times. For the decreasing concave case, we calculate each job’s best starting time and let them start close to their best starting times. Finally, we show that the problem is NP-hard in the strong sense when the time slot cost decreases in an arbitrary way.     

Lot-sizing scheduling with batch setup times

This paper is concerned with scheduling independent jobs on m parallel machines in such a way that the makespan is minimized. Each job j is allowed to split arbitrarily into several parts, which can be individually processed on any machine at any time. However, a setup for uninterrupted s_j time units is required before any split part of job j can be processed on any machine. The problem is strongly NP-hard if the number m of machines is variable and weakly NP-hard otherwise. We give a polynomial-time -approximation algorithm for the former case and a fully polynomial-time approximation scheme for the latter. AMS Subject Classifications: 68M20 · 90B35 · 90C59     

Single-machine scheduling with periodic and flexible periodic maintenance to minimize maximum tardiness

This paper considers a single-machine scheduling problem with several maintenances periods. Specifically, two situations are investigated. In the first one, maintenance periods are periodically fixed: maintenance is required after a periodic time interval. In the second one, the maintenance is not fixed but the maximum continuous working time of the machine which is allowed is determined. The objective is to minimize the maximum tardiness. These problems are known to be strongly NP-hard. We propose some dominance properties and an efficient heuristic. Branch-and-bound algorithms, in which the heuristics, the lower bounds and the dominance properties are incorporated, are proposed and tested computationally.     

Single-machine scheduling with an availability constraint to minimize the weighted sum of the completion times

In this article, we consider a single-machine scheduling problem with one unavailability period, with the aim of minimizing the weighted sum of the completion times. We propose three exact methods for solving such a problem: a branch-and-bound method based on new properties and lower bounds, a mixed integer programming model, and a dynamic programming method. These methods were coded and tested on randomly generated instances, and their performances were analyzed. Our numerical experiments show that the branch-and-bound method and the dynamic programming method are complementary. Using these approaches, we are able to solve problems with up to 3000 jobs within a reasonable computation time.     

Family scheduling with batch availability in flow shops to minimize makespan

This paper addresses a batch scheduling problem in flow shop production systems, where job families are formed based on setup similarities. In order to improve setup efficiency, we consider batching decisions in our solution procedure. Due to its high practical relevance, the batch availability assumption is also adopted in this study. In the presence of sequence-dependent setup times, it is proved that a permutation flow shop is generally not optimal. Therefore, our objective is to determine solutions with inconsistent batches, which essentially lead to non-permutation schedules, to minimize makespan. After examining structural properties, we develop a tabu search algorithm with multiple neighbourhood functions. Computational results confirm the remarkable benefits of batching decisions. Our algorithm also outperforms some well-known and well-performing approaches.     

Deteriorating job scheduling to minimize the number of late jobs with setup times

Deteriorating jobs scheduling problems have been widely studied recently. However, research on scheduling problems with deteriorating jobs has rarely considered explicit setup times. With the current emphasis on customer service and meeting the promised delivery dates, we consider a single-machine scheduling problem to minimize the number of late jobs with deteriorating jobs and setup times in this paper. We derive some dominance properties, a lower bound, and an initial upper bound by using a heuristic algorithm to speed up the search process of the branch-and-bound algorithm. Computational experiments show that the algorithm can solve instances up to 1000 jobs in a reasonable amount of time.     

Job shop scheduling with setup times, deadlines and precedence constraints

In the last 15 years several procedures have been developed that can find solutions of acceptable quality in reasonable computing time to Job Shop Scheduling problems in environments that do not involve sequence-dependent setup times of the machines. The presence of the latter, however, changes the picture dramatically. In this paper we adapt one of the best known heuristics, the Shifting Bottleneck Procedure, to the case when sequence dependent setup times play an important role. This is done by treating the single machine scheduling problems that arise in the process as Traveling Salesman Problems with time windows, and solving the latter by an efficient dynamic programming algorithm. The model treated here also incorporates precedence constraints, release times and deadlines. Computational experience on a vast array of instances, mainly from the semiconductor industry, shows our procedure to advance substantially the state of the art. Paper presented in New York at MISTA 2005. E. Balas supported by the National Science Foundation through grant DMI-9802773 and by the Office of Naval Research through contract #N00014-97-1-0133.     

Parallel-machine scheduling to minimize tardiness penalty and power cost

Traditional research on machine scheduling focuses on job allocation and sequencing to optimize certain objective functions that are defined in terms of job completion times. With regard to environmental concerns, energy consumption becomes another critical issue in high-performance systems. This paper addresses a scheduling problem in a multiple-machine system where the computing speeds of the machines are allowed to be adjusted during the course of execution. The CPU adjustment capability enables the flexibility for minimizing electricity cost from the energy saving aspect by sacrificing job completion times. The decision of the studied problem is to dispatch the jobs to the machines as well as to determine the job sequence and processing speed of each machine with the objective function comprising of the total weighted job tardiness and the power cost. We give a formal formulation, propose two heuristic algorithms, and develop a particle swarm optimization (PSO) algorithm to effectively tackle the problem. Since the existing solution representations do not befittingly encode the decisions involved in the studied problem into the PSO algorithm, we design a tailored encoding scheme which can embed all decisional information in a particle. A computational study is conducted to investigate the performances of the proposed heuristics and the PSO algorithm.