Total Size Of Jobs Research Articles

Integrated Scheduling Problem on a Single Bounded Batch Machine with an Unavailability Constraint

We consider a scheduling problem where a set of jobs are first processed on a machine with an unavailability interval and, then, delivered to the customer directly. We focus on an integrated schedule of production and distribution such that the sum of the maximum delivery time and total delivery cost is optimized. We study two classes of processing machines in the production part. In the first class, the serial-batch machine, the processing time of a batch is the sum of the processing times of its jobs. In the second class, the parallel-batch machine, the processing time of a batch is the maximum processing time of the jobs contained in the batch. The machine has a fixed capacity, and the jobs are processed in batches under the condition that the total size of the jobs in a batch cannot exceed the machine capacity. Two patterns of job’s processing, i.e., resumable and non-resumable, are considered if it is interrupted by the unavailability interval on the machine. In the distribution part, there are sufficient vehicles with a fixed capacity to deliver the completed jobs. The total size of the completed jobs in one delivery cannot exceed the vehicle capacity. We show that these four problems are NP-hard in the strong sense in which the jobs have the same processing times and arbitrary sizes, and we propose an approximation algorithm for solving these four problems. Moreover, we show that the performance ratio of the algorithm is 2 for the serial-batch machine setting, and the error bound is 71/99 for the parallel-batch machine setting. We also evaluate the performance of the approximation algorithm by the computational results.

A -approximation algorithm for scheduling on two parallel machines with job delivery coordination

Integration of production and distribution operations at the scheduling level is one of the most important issues concerned in recent years in supply chain management. We investigate the scheduling problem with job delivery coordination. Concretely, we are given a set of jobs with non-identical processing times and non-identical sizes to be first processed by two parallel-identical machines and then delivered to a common customer by a single vehicle, which has a limited capacity that can carry up to a fixed total size of jobs in one batch. The objective is to minimize the makespan, ie, the time when all the jobs have reached the customer and the delivery vehicle has returned to the machine. The problem does not admit a polynomial time approximation algorithm with worst-case ratio smaller than unless P = NP. By introducing some novel ideas and new methods, including algorithms solving multiple knapsack problem as subprocedures and the rediscovery of the profound nature of the bin-packing algorithm, we propose a polynomial time approximation algorithm with worst-case ratio arbitrary close to . The algorithm, acquired after long-term research containing many earlier related works, is the first algorithm for this kind of problems and its theoretical performance is nearly best.

Integrated scheduling of production and distribution for manufacturers with parallel batching facilities

We consider a class of integrated scheduling problems for manufacturers. The manufacturer processes job orders and delivers products to the customer. The objective is to minimize the service span, that is, the period lasting from the time when the order is received to the time when all the products have been delivered to the customer. In the production phase, parallel batch-processing facilities are used to process the jobs. Jobs have arbitrary sizes and processing times. Each facility has a fixed capacity and jobs are processed in batches with the restriction that the total size of jobs in a batch does not exceed the facility capacity. When all the jobs in a batch are completed, the batch is completed. In the distribution phase, the manufacturer uses a vehicle with a fixed capacity to deliver products. The transportation time from the manufacturer to the customer is a constant. Completed products can be delivered in one transfer if the total size does not exceed the vehicle capacity. We first consider the problem where jobs have the same size and arbitrary processing times. We propose approximation algorithms for the problem and we show that a worst-case ratio performance guarantee is respectively 2–1/m. Then we consider the problem where jobs have the same processing time and arbitrary sizes. An approximation algorithm is proposed with an absolute worst-case ratio of 13/7 and an asymptotic worst-case ratio of 11/9. Both the proposed algorithms can be executed in polynomial time.

Approximation algorithms for two-stage supply chain scheduling of production and distribution

In this paper, we investigate a class of two-stage optimisation problems for manufacturers in supply chains. The objective is to optimise production and outbound distribution. In the production part, the manufacturer has identical batch-processing machines with limited capacity in size. The processing times and sizes of jobs are determined by the customer. As long as the total size of jobs in a batch does not exceed the machine capacity, the batch can be processed on any machine. The processing of a batch cannot be interrupted until all the jobs in the batch have been completed. In the distribution part, the manufacturer assigns the completed jobs in groups and delivers them to the customer with its own vehicles. We propose the model of minimising service span, which lasts from the processing of the first job to the delivery of the last job. We investigate two types of machine configuration, including single machine and parallel machines. In the vehicle configuration, we consider two cases including infinite vehicles and a fixed number of vehicles. We show all the problems under investigation are NP-hard. For each problem, we design an effective polynomial time algorithm. We analyse the time complexity and show the performance guarantees of the algorithms.

Online hierarchical scheduling on two machines with known total size of low-hierarchy jobs

In this paper, we consider three versions of semi-online hierarchical scheduling problems on two identical machines, with the purpose of minimizing the makespan. In the first version, we assume that the total size of jobs with lower hierarchy is given and we get the tight bound . In the second one, assume that the total size of jobs in each hierarchy is given and we get the tight bound . In the third one, we assume that the total size of jobs with lower hierarchy is known in advance and a buffer of size K is given to store at most K jobs temporarily. We propose an optimal algorithm with competitive ratio using K=1 and show that a bigger buffer size is not helpful.

A branch and bound based heuristic for makespan minimization of washing operations in hospital sterilization services

In this paper, we address the problem of parallel batching of jobs on identical machines to minimize makespan. The problem is motivated from the washing step of hospital sterilization services where jobs have different sizes, different release dates and equal processing times. Machines can process more than one job at the same time as long as the total size of jobs in a batch does not exceed the machine capacity. We present a branch and bound based heuristic method and compare it to a linear model and two other heuristics from the literature. Computational experiments show that our method can find high quality solutions within short computation time.

Semi-online scheduling problems on two uniform machines under a grade of service provision

This paper studies the semi-online scheduling on two uniform machines under a grade of service (GoS) provision, where one machine is available for all jobs and the other one is only available for partial jobs. The objective is to minimize the makespan. We consider three variants, where the optimal makespan, the total size of jobs, and the largest job size are known in advance respectively, and design optimal algorithms for all of them.

An improved ant colony optimization for scheduling identical parallel batching machines with arbitrary job sizes

In this paper we consider the problem of scheduling parallel batching machines with jobs of arbitrary sizes. The machines have identical capacity of size and processing velocity. The jobs are processed in batches given that the total size of jobs in a batch cannot exceed the machine capacity. Once a batch starts processing, no interruption is allowed until all the jobs are completed. First we present a mixed integer programming model of the problem. We show the computational complexity of the problem and optimality properties. Then we propose a novel ant colony optimization method where the Metropolis Criterion is used to select the paths of ants to overcome the immature convergence. Finally, we generate different scales of instances to test the performance. The computational results show the effectiveness of the algorithm, especially for large-scale instances.

Scheduling algorithm for flow shop with two batch-processing machines and arbitrary job sizes

This article considers the problem of scheduling two batch-processing machines in flow shop where the jobs have arbitrary sizes and the machines have limited capacity. The jobs are processed in batches and the total size of jobs in each batch cannot exceed the machine capacity. Once a batch is being processed, no interruption is allowed until all the jobs in it are completed. The problem of minimising makespan is NP-hard in the strong sense. First, we present a mathematical model of the problem using integer programme. We show the scale of feasible solutions of the problem and provide optimality properties. Then, we propose a polynomial time algorithm with running time in O(nlogn). The jobs are first assigned in feasible batches and then scheduled on machines. For the general case, we prove that the proposed algorithm has a performance guarantee of 4. For the special case where the processing times of each job on the two machines satisfy p 1 j = ap 2 j , the performance guarantee is for a > 0.

Solving single batch-processing machine problems using an iterated heuristic

This article considers the problem of minimising total weighted tardiness on a batch-processing machine with compatible product families, job release dates and non-identical job sizes. A batch-processing machine can process several jobs as long as the total size of jobs in the batch does not exceed the machine's capacity. All jobs in the batch are started and completed at the same time. Batch processing time is equal to the longest job processing time among jobs in the batch. This article proposes a two-phase heuristic, in which a population-based reasoning approach is developed to determine the sequence of jobs based on their intensity values at positions in the first phase, and a dynamic programming algorithm to divide the ordered jobs into batches sequentially. An iterated heuristic is also proposed to improve solution quality further. Computational results show that high-quality solutions can be obtained by the proposed heuristic algorithms in a very short time.

An effective hybrid multi-objective genetic algorithm for bi-criteria scheduling on a single batch processing machine with non-identical job sizes

This paper addresses the problem of scheduling jobs with non-identical sizes on a single batch processing machine. A batch processing machine is one which can process multiple jobs simultaneously as a batch as long as the total size of jobs being processed does not exceed the machine capacity. The batch processing time is equal to the longest processing time among all jobs in the batch. For the simultaneous minimization of the bi-criteria of makespan and maximum tardiness, we propose two different multi-objective genetic algorithms based on different representation schemes. While the first algorithm do search via generating sequences of jobs using genetic operators and then batching jobs keeping their order in the sequence, the second algorithm uses the idea of generating batches of jobs directly using genetic operators and ensures feasibility through using heuristic procedures. The type of representation used in the second algorithm allows introducing heuristics with the ability of biasing the search towards each objective and also allows hybridization with a local search heuristic that gives the ability of finding Pareto-optimal or locally efficient Pareto-solutions. Computational results show that the non-dominated solutions obtained by the latter algorithm are very superior in closeness to the true Pareto-optimal solutions and to keep diversity in the obtained Pareto-set, as the problem size increases.