Worst-case and numerical analysis of heuristic algorithms for flowshop scheduling problems with a time-dependent learning effect

In this paper, we investigate a time-dependent learning effect in a flowshop scheduling problem. We assume that the time-dependent learning effect of a job was a function of the total normal processing time of jobs scheduled before the job. The following objective functions are explored: the makespan, the total flowtime, the sum of weighted completion times, the sum of the kth power of completion times, and the maximum lateness. Some heuristic algorithms with worst-case analysis for the objective functions are given. Moreover, a polynomial algorithm is proposed for the special case with identical processing time on each machine and that with an increasing series of dominating machines, respectively. Finally, the computational results to evaluate the performance of the heuristics are provided.     

Scheduling jobs with deterioration effect and controllable processing time

This article considered the single machine scheduling with controllable processing time (resource allocation) and deterioration effect concurrently. It discussed the minimization of three objectives, which involve the weighted sum of the makespan and the total resource consumption cost, the total resource consumption cost under the condition that the makespan (total flow time) is restricted to a fixed constant and the optimal resource allocation and the optimal job sequence is what we need to make decision. Considering the makespan constraint, it proved that these problems can be solved in polynomial time. A special case of the last problem can be solved in polynomial time with respect to the total flow time constraint.

     

Single-machine scheduling with sum-of-logarithm-processing-times-based learning considerations

Scheduling with learning effects has attracted growing attention of the scheduling research community. A recent survey classifies the learning models in scheduling into two types, namely position-based learning and sum-of-processing-times-based learning. However, the actual processing time of a given job drops to zero precipitously as the number of jobs increases in the first model and when the normal job processing times are large in the second model. Motivated by this observation, we propose a new learning model where the actual job processing time is a function of the sum of the logarithm of the processing times of the jobs already processed. The use of the logarithm function is to model the phenomenon that learning as a human activity is subject to the law of diminishing return. Under the proposed learning model, we show that the scheduling problems to minimize the makespan and total completion time can be solved in polynomial time. We further show that the problems to minimize the maximum lateness, maximum tardiness, weighted sum of completion times and total tardiness have polynomial-time solutions under some agreeable conditions on the problem parameters.     

On the complexity of bi-criteria scheduling on a single batch processing machine

This paper considers hierarchical bi-criteria scheduling on a single batch processing machine where the primary criterion is the makespan. We show that the problem where the secondary criterion is the total completion time can be solved in polynomial time for a given machine capacity and the problem where the secondary criterion is the (weighted) number of late jobs is (strongly) NP-hard.     

Single-machine group scheduling with resource allocation and learning effect

This paper addresses single-machine scheduling problems under the consideration of learning effect and resource allocation in a group technology environment. In the proposed model of this paper the actual processing times of jobs depend on the job position, the group position, and the amount of resource allocated to them concurrently. Learning effect and two resource allocation functions are examined for minimizing the weighted sum of makespan and total resource cost, and the weighted sum of total completion time and total resource cost. We show that the problems for minimizing the weighted sum of makespan and total resource cost remain polynomially solvable. We also prove that the problems for minimizing the weighted sum of total completion time and total resource cost have polynomial solutions under certain conditions.     

Flow shop scheduling to minimize makespan with decreasing time-dependent job processing times

In this study, we consider an n-job, m-machine flow shop scheduling problem with decreasing time-dependent job processing times. By the decreasing time-dependent job processing times, we mean that the processing time is a decreasing function of its execution starting time. When some dominant relationships between m − 1 machines can be satisfied, we show that the makespan minimization problem can be solved in polynomial time.     

Single-machine scheduling jobs with exponential learning functions

In a manufacturing system workers are involved in doing the same job or activity repeatedly. Hence, the workers start learning more about the job or activity. Because of the learning, the time to complete the job or activity starts decreasing, which is known as “learning effect”. In this paper, an exponential sum-of-actual-processing-time based learning effect is introduced into single-machine scheduling. By the exponential sum-of-actual-processing-time based learning effect, we mean that the processing time of a job is defined by an exponential function of the sum-of-the-actual-processing-time of the already processed jobs. Under the proposed learning model, we show that under a sufficient condition, the makespan minimization problem, the sum of the θth (θ > 0) power of completion times minimization problem, and some special cases of the total weighted completion time minimization problem and the maximum lateness minimization problem remain polynomially solvable.     

Scheduling jobs with equal processing times and time windows on identical parallel machines

We present a linear programming approach to the problem of scheduling equal processing time jobs with release dates and deadlines on identical parallel machines. The known algorithm with complexity O(n ³log log n) of B. Simons schedules all the jobs while minimizing both the maximum completion time and the mean flow time. Our approach permits also to minimize the weighted sum of completion times and total tardiness in polynomial time for the problems without deadlines. The complexity status of these problems was open. Contract/grant sponsor: Alexander von Humboldt Foundation.     

Permutation flowshop problems with bi-criterion makespan and total completion time objective and position-weighted learning effects

In this paper, we study the problem of minimizing the weighted sum of makespan and total completion time in a permutation flowshop where the processing times are supposed to vary according to learning effects. The processing time of a job is a function of the sum of the logarithms of the processing times of the jobs already processed and its position in the sequence. We present heuristic algorithms, which are modified from the optimal schedules for the corresponding single machine scheduling problem and analyze their worst-case error bound. We also adopt an existing algorithm as well as a branch-and-bound algorithm for the general m-machine permutation flowshop problem. For evaluation of the performance of the algorithms, computational experiments are performed on randomly generated test problems.     

Single-machine scheduling with deteriorating jobs under a series–parallel graph constraint

This paper considers single-machine scheduling problems with deteriorating jobs, i.e., jobs whose processing times are an increasing function of their starting times. In addition, the jobs are related by a series–parallel graph. It is shown that for the general linear problem to minimize the makespan, polynomial algorithms exist. It is also shown that for the proportional linear problem of minimization of the total weighted completion time, polynomial algorithms exist, too.     

Single-machine scheduling with truncated sum-of-processing-times-based learning effect including proportional delivery times

The single-machine scheduling problem with truncated sum-of-processing-times-based learning effect and past-sequence-dependent job delivery times is considered. Each job’s delivery time depends on its waiting time of processing. For some regular objective functions, it is proved that the problems can be solved by the smallest processing time first rule. For some special cases of the total weighted completion time and the maximum lateness objective functions, the thesis shows that the problems can be solved in polynomial time.     

Approximation algorithms for minimizing the total weighted number of late jobs with late deliveries in two-level supply chains

We study a supply chain scheduling problem in which n jobs have to be scheduled on a single machine and delivered to m customers in batches. Each job has a due date, a processing time and a lateness penalty (weight). To save batch-delivery costs, several jobs for the same customer can be delivered together in a batch, including late jobs. The completion time of each job in the same batch coincides with the batch completion time. A batch setup time has to be added before processing the first job in each batch. The objective is to find a schedule which minimizes the sum of the weighted number of late jobs and the delivery costs. We present a pseudo-polynomial algorithm for a restricted case, where late jobs are delivered separately, and show that it becomes polynomial for the special cases when jobs have equal weights and equal delivery costs or equal processing times and equal setup times. We convert the algorithm into an FPTAS and prove that the solution produced by it is near-optimal for the original general problem by performing a parametric analysis of its performance ratio.     

Online <Emphasis Type="Italic">k</Emphasis>-Server Routing Problems

In an online k-server routing problem, a crew of k servers has to visit points in a metric space as they arrive in real time. Possible objective functions include minimizing the makespan (k-Traveling Salesman Problem) and minimizing the sum of completion times (k-Traveling Repairman Problem). We give competitive algorithms, resource augmentation results and lower bounds for k-server routing problems in a wide class of metric spaces. In some cases the competitive ratio is dramatically better than that of the corresponding single server problem. Namely, we give a 1+O((log k)/k)-competitive algorithm for the k-Traveling Salesman Problem and the k-Traveling Repairman Problem when the underlying metric space is the real line. We also prove that a similar result cannot hold for the Euclidean plane. An extended abstract of this work has appeared in the proceedings of the 4th Workshop on Approximation and Online Algorithms, September 2006. Research of V. Bonifaci partly supported by the Dutch Ministry of Education, Culture and Science through a Huygens scholarship. Research of L. Stougie partly supported by MRT Network ADONET of the European Community (MRTN-CT-2003-504438) and the Dutch BSIK/BRICKS project.     

Relocation scheduling subject to fixed processing sequences

This study addresses a relocation scheduling problem that corresponds to resource-constrained scheduling on two parallel dedicated machines where the processing sequences of jobs assigned to the machines are given and fixed. Subject to the resource constraints, the problem is to determine the starting times of all jobs for each of the six considered regular performance measures, namely, the makespan, total weighted completion time, maximum lateness, total weighted tardiness, weighted number of tardy jobs, and number of tardy jobs. By virtue of the proposed dynamic programming framework, the studied problem for the minimization of makespan, total weighted completion time, or maximum lateness can be solved in \(O(n_1n_2(n_1+n_2))\) time, where \(n_1\) and \(n_2\) are the numbers of jobs on the two machines. The simplified case with a common job processing time can be solved in \(O(n_1n_2)\) time. For the objective function of total weighted tardiness or weighted number of tardy jobs, this problem is proved to be NP-hard in the ordinary sense, and the case with a common job processing length is solvable in \(O(n_1n_2\min \{n_1,n_2\})\) time. The studied problem for the minimization of number of tardy jobs is solvable in \(O(n^2_1n^2_2(n_1+n_2)^2)\) time. The solvability of the common-processing-time problems can be generalized to the m-machine cases, where \(m\ge 3\).     

Scheduling a Single Server in a Two-machine Flow Shop

We study the problem of scheduling a single server that processes n jobs in a two-machine flow shop environment. A machine dependent setup time is needed whenever the server switches from one machine to the other. The problem with a given job sequence is shown to be reducible to a single machine batching problem. This result enables several cases of the server scheduling problem to be solved in O(n log n) by known algorithms, namely, finding a schedule feasible with respect to a given set of deadlines, minimizing the maximum lateness and, if the job processing times are agreeable, minimizing the total completion time. Minimizing the total weighted completion time is shown to be NP-hard in the strong sense. Two pseudopolynomial dynamic programming algorithms are presented for minimizing the weighted number of late jobs. Minimizing the number of late jobs is proved to be NP-hard even if setup times are equal and there are two distinct due dates. This problem is solved in O(n ³) time when all job processing times on the first machine are equal, and it is solved in O(n ⁴) time when all processing times on the second machine are equal. Received November 20, 2001; revised October 18, 2002 Published online: January 16, 2003     

Scheduling a batching machine with convex resource consumption functions

In various industries jobs undergo a batching, or burn in, process where different tasks are grouped into batches and processed simultaneously. The processing time of each batch is equal to the longest processing time among all jobs contained in the batch. All to date studies dealing with batching machines have considered fixed job processing times. However, in many real life applications job processing times are controllable through the allocation of a limited resource. The most common and realistic model assumes that there exists a non-linear and convex relationship between the amount of resource allocated to a job and its processing time. The scheduler?s task when dealing with controllable processing times is twofold. In addition to solving the sequencing problem, one must establish an optimal resource allocation policy. We combine these two widespread models on a single machine setting, showing that both the makespan and total completion time criteria can be solved in polynomial time. We then show that our proposed approach can be applied to general bi-criteria objective comprising of the makespan and the total completion time.     

A note on machine scheduling with sum-of-logarithm-processing-time-based and position-based learning effects

Recently, Biskup [2] classifies the learning effect models in scheduling environments into two types: position-based and sum-of-processing-time-based. In this paper, we study scheduling problem with sum-of-logarithm-processing-time-based and position-based learning effects. We show that the single machine scheduling problems to minimize the makespan and the total completion time can both be solved by the smallest (normal) processing time first (SPT) rule. We also show that the problems to minimize the maximum lateness, the total weighted completion times and the total tardiness have polynomial-time solutions under agreeable WSPT rule and agreeable EDD rule. In addition, we show that m-machine permutation flowshop problems are still polynomially solvable under the proposed learning model.