Single Machine Vector Scheduling with General Penalties

In this paper, we study the single machine vector scheduling problem (SMVS) with general penalties, in which each job is characterized by a d-dimensional vector and can be accepted and processed on the machine or rejected. The objective is to minimize the sum of the maximum load over all dimensions of the total vector of all accepted jobs and the rejection penalty of the rejected jobs, which is determined by a set function. We perform the following work in this paper. First, we prove that the lower bound for SMVS with general penalties is α(n), where α(n) is any positive polynomial function of n. Then, we consider a special case in which both the diminishing-return ratio of the set function and the minimum load over all dimensions of any job are larger than zero, and we design an approximation algorithm based on the projected subgradient method. Second, we consider another special case in which the penalty set function is submodular. We propose a noncombinatorial ee−1-approximation algorithm and a combinatorial min{r,d}-approximation algorithm, where r is the maximum ratio of the maximum load to the minimum load on the d-dimensional vector.

Approximation Algorithms for the Submodular Load Balancing with Submodular Penalties

In this paper, we study the submodular load balancing problem with submodular penalties. The objective of this problem is to balance the load among sets, while some elements can be rejected by paying some penalties. Officially, given an element set V, we want to find a subset R of rejected elements, and assign other elements to one of m sets A1,A2,⋯,Am. The objective is to minimize the sum of the maximum load among A1,A2,⋯,Am and the rejection penalty of R, where the load and rejection penalty are determined by different submodular functions. We study the submodular load balancing problem with submodular penalties under two settings: heterogenous setting (load functions are not identical) and homogenous setting (load functions are identical). Moreover, we design a Lovász rounding algorithm achieving a worst-case guarantee of m+1 under the heterogenous setting and a min{m,⌈nm⌉+1}=O(n)-approximation combinatorial algorithm under the homogenous setting.

Single-Machine Parallel-Batch Scheduling with Nonidentical Job Sizes and Rejection

We investigate the single-machine parallel-batch scheduling problem with nonidentical job sizes and rejection. In this problem, a set of jobs with different processing times and nonidentical sizes is given to be possibly processed on a parallel-batch processing machine. Each job is either accepted and then processed on the machine or rejected by paying its rejection penalty. Preemption is not allowed. Our task is to choose the accepted jobs and schedule them as batches on the machine to minimize the makespan of the accepted jobs plus the total rejection penalty of the rejected jobs. We provide an integer programming formulation to exactly solve our problem. Then, we propose three fast heuristic algorithms to solve the problem and evaluate their performances by using a small numerical example.

Approximation Algorithm for the Single Machine Scheduling Problem with Release Dates and Submodular Rejection Penalty

In this paper, we consider the single machine scheduling problem with release dates and nonmonotone submodular rejection penalty. We are given a single machine and multiple jobs with probably different release dates and processing times. For each job, it is either accepted and processed on the machine or rejected. The objective is to minimize the sum of the makespan of the accepted jobs and the rejection penalty of the rejected jobs which is determined by a nonmonotone submodular function. We design a combinatorial algorithm based on the primal-dual framework to deal with the problem, and study its property under two cases. For the general case where the release dates can be different, the proposed algorithm have an approximation ratio of 2. When all the jobs release at the same time, the proposed algorithm becomes a polynomial-time exact algorithm.

Single-machine batch scheduling problem with job rejection and resource dependent processing times

This paper addresses single-machine batch scheduling with job rejection and convex resource allocation. A job is either rejected, in which case a rejection penalty will be incurred, or accepted and processed on the machine. The accepted jobs are combined to form batches containing contiguously scheduled jobs. For each batch, a batch-dependent machine setup time, which is a function of the number of batches processed previously, is required before the first job in the batch is processed. Both the setup times and job processing times are controllable by allocating a continuously divisible nonrenewable resource to the jobs. The objective is to determine an accepted job sequence, a rejected job set, a partition of the accepted job sequence into batches, and resource allocation that jointly minimize a cost function based on the total delivery dates of the accepted jobs, and the job holding, resource consumption, and rejection penalties. An dynamic programming solution algorithm with running time O (n6) is developed for the problem. It is also shown that the case of the problem with a common setup time can be solved in O (n5) time.

An Optimal Algorithm for Scheduling with Variable Job Processing Times and Rejection

This paper studies a single machine scheduling problem with rejection. Each job has a variable processing time and a rejection penalty. The objective function is to minimize the sum of the makespan of the accepted jobs and the total rejection penalty of the rejected jobs. We show that the problem can be solved in polynomial time.

Improved Rejection Penalty Algorithm with Multiprocessor Rejection Technique

<p>This paper deals with multiprocessor scheduling with rejection technique where each job is provided with processing time and a given penalty cost. If the job satisfies the acceptance condition, it will schedule in the least loaded identical parallel machine else job is rejected. In this way its penalty cost is calculated. Our objective is to minimize the makespan of the scheduled job and to minimize the sum of the penalties of rejected jobs. We have merged ‘CHOOSE ‘and ‘REJECTION PENALTY’ algorithm to reduce the sum of penalties cost and makespan. Our proposed ‘Improved Reject penalty algorithm’ reduce competitive ratio, which in turn enhances the efficiency of the on-line algorithm. By applying our new on-line technique, we got the lower bound of our algorithm is is 1.286 which is far better from the existing algorithms whose competitive ratio is at 1.819. In our approach we have consider non-preemption scheduling technique.</p>

Single Machine Scheduling with an Availability Constraint and Rejection

This paper considers single machine scheduling with an availability constraint and rejection. It is assumed that the machine is not available for processing during a given time interval. A job is either rejected, in which case a rejection penalty has to be paid, or accepted and processed on the machine. The objective is to minimize the sum of the weighted total completion time of the accepted jobs and the total rejection penalty of the rejected jobs. For this NP-hard problem, we present a pseudo-polynomial dynamic programming algorithm and a fully polynomial-time approximation scheme (FPTAS).

Scheduling a Single Machine with Job Rejection and Multiple Common due Dates Assignment

This paper studies a single machine scheduling with job rejection and multiple common due dates assignment. A job is either rejected, in which a rejection penalty has to be paid, or accepted and processed on the machine. There is a cost if the job is completed prior or after the due date. The objective is to determine the optimal due dates, the set of jobs assigned to each due date and the optimal sequence of jobs to minimize a total costs based on earliness, tardiness, multiple common due dates and rejection cost. We provide dynamic programming algorithms and show that the problem is solvable in polynomial time.

Two-Machine Flowshop Problem with Release Dates, Rejection and Non-Availability Interval on the First Machine

This paper studies a two-machine flowshop problem with release dates, rejection and non-availability interval on the first machine. The non-availability interval often origins from equipments maintain or man-power. Usually, in order to pursue maximal profit, some jobs which can be rejected, and in this situation the rejection penalty should be paid. Our objective is to minimize the sum of the makespan of the accepted jobs and the total rejection penalty of the rejected jobs. For this demonstrated NP-hard in strong sense, we propose a heuristic method and further demonstrate that its worst case performance is 3.