scholarly journals Parallel-Batch Scheduling and Transportation Coordination with Waiting Time Constraint

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Hua Gong ◽  
Daheng Chen ◽  
Ke Xu
Keyword(s):  
Waiting Time ◽  
Processing Time ◽  
Optimal Algorithm ◽  
Dynamic Programming Algorithm ◽  
Time Constraint ◽  
Batch Scheduling ◽  
Programming Algorithm ◽  
Np Hard ◽  
Processing Times ◽  
Equal Processing Times

This paper addresses a parallel-batch scheduling problem that incorporates transportation of raw materials or semifinished products before processing with waiting time constraint. The orders located at the different suppliers are transported by some vehicles to a manufacturing facility for further processing. One vehicle can load only one order in one shipment. Each order arriving at the facility must be processed in the limited waiting time. The orders are processed in batches on a parallel-batch machine, where a batch contains several orders and the processing time of the batch is the largest processing time of the orders in it. The goal is to find a schedule to minimize the sum of the total flow time and the production cost. We prove that the general problem is NP-hard in the strong sense. We also demonstrate that the problem with equal processing times on the machine is NP-hard. Furthermore, a dynamic programming algorithm in pseudopolynomial time is provided to prove its ordinarily NP-hardness. An optimal algorithm in polynomial time is presented to solve a special case with equal processing times and equal transportation times for each order.

scholarly journals Two-Machine Job-Shop Scheduling with Equal Processing Times on Each Machine

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 301 ◽  
Author(s):  
Evgeny Gafarov ◽  
Frank Werner
Keyword(s):  
Dynamic Programming ◽  
Job Shop ◽  
Job Shop Scheduling ◽  
Dynamic Programming Algorithm ◽  
Programming Algorithm ◽  
Scheduling Problem ◽  
Single Track ◽  
Shop Scheduling ◽  
Processing Times ◽  
Equal Processing Times

In this paper, we consider a two-machine job-shop scheduling problem of minimizing total completion time subject to n jobs with two operations and equal processing times on each machine. This problem occurs e.g., as a single-track railway scheduling problem with three stations and constant travel times between any two adjacent stations. We present a polynomial dynamic programming algorithm of the complexity O ( n 5 ) and a heuristic procedure of the complexity O ( n 3 ) . This settles the complexity status of the problem under consideration which was open before and extends earlier work for the two-station single-track railway scheduling problem. We also present computational results of the comparison of both algorithms. For the 30,000 instances with up to 30 jobs considered, the average relative error of the heuristic is less than 1 % . In our tests, the practical running time of the dynamic programming algorithm was even bounded by O ( n 4 ) .

A STOCHASTIC BATCHING AND SCHEDULING PROBLEM

2001 ◽  
Vol 15 (4) ◽  
pp. 465-479 ◽  
Author(s):  
Ger Koole ◽  
Rhonda Righter
Keyword(s):  
Optimal Policy ◽  
Semiconductor Manufacturing ◽  
Processing Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Flow Time ◽  
Batch Scheduling ◽  
Scheduling Problem ◽  
Processing Times ◽  
Deterministic Problem

We consider a batch scheduling problem in which the processing time of a batch of jobs equals the maximum of the processing times of all jobs in the batch. This is the case, for example, for burn-in operations in semiconductor manufacturing and other testing operations. Processing times are assumed to be random, and we consider minimizing the makespan and the flow time. The problem is much more difficult than the corresponding deterministic problem, and the optimal policy may have many counterintuitive properties. We prove various structural properties of the optimal policy and use these to develop a polynomial-time algorithm to compute the optimal policy.

scholarly journals Batch Scheduling with Proportional-Linear Deterioration and Outsourcing

2017 ◽  
Vol 2017 ◽  
pp. 1-5
Author(s):  
Cuixia Miao ◽  
Fanxiao Meng ◽  
Juan Zou ◽  
Binglin Jia
Keyword(s):  
Dynamic Programming Algorithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Third Party ◽  
Batch Scheduling ◽  
Programming Algorithm ◽  
Release Dates ◽  
Linear Deterioration ◽  
Total Penalty ◽  
Maximum Completion Time

We consider the bounded parallel-batch scheduling with proportional-linear deterioration and outsourcing, in which the actual processing time is pj=αj(A+Dt) or pj=αjt. A job is either accepted and processed in batches on a single machine by manufactures themselves or outsourced to the third party with a certain penalty having to be paid. The objective is to minimize the maximum completion time of the accepted jobs and the total penalty of the outsourced jobs. For the pj=αj(A+Dt) model, when all the jobs are released at time zero, we show that the problem is NP-hard and present a pseudo-polynomial time algorithm, respectively. For the pj=αjt model, when the jobs have distinct m (<n) release dates, we provide a dynamic programming algorithm, where n is the number of jobs.

scholarly journals Stochastic Single Machine JIT Scheduling with Geometric Processing Times and Due Dates

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Yuncheng Luo
Keyword(s):  
Single Machine ◽  
Optimal Schedule ◽  
Dynamic Programming Algorithm ◽  
Programming Algorithm ◽  
Scheduling Problem ◽  
Due Dates ◽  
Common Mean ◽  
Processing Times ◽  
Tardy Jobs ◽  
Special Case

In this paper, we investigate a static stochastic single machine JIT scheduling problem in which the jobs’ processing times are stochastically independent and follow geometric distributions whose mean is provided, due dates are geometrically distributed with a common mean, and both the unit penalty of earliness/tardiness and the fixed penalty of earliness/tardiness are deterministic and different. The objective is to minimize the expected total penalties for quadratic earliness, quadratic tardiness, and early and tardy jobs. We prove that the optimal schedule to minimize this problem is V-shaped with respect to the ratio of mean processing time to unit tardiness penalty under the specific condition. Also, we show a special case and two theorems related to this JIT scheduling problem under specific situations where the optimal solutions exist. Finally, based on the V-shaped characteristic, a dynamic programming algorithm is designed to achieve an optimal V-shaped schedule in pseudopolynomial time.

TWO-MACHINE FLOW SHOP SCHEDULING WITH INDIVIDUAL OPERATION'S REJECTION

2014 ◽  
Vol 31 (01) ◽  
pp. 1450002 ◽  
Author(s):  
QIANG GAO ◽  
XIWEN LU
Keyword(s):  
Approximation Scheme ◽  
Flow Shop ◽  
Dynamic Programming Algorithm ◽  
Flow Shop Scheduling ◽  
Programming Algorithm ◽  
Polynomial Time Approximation Scheme ◽  
Time Approximation ◽  
Shop Scheduling ◽  
Processing Times ◽  
Total Penalty

A two-machine flow shop scheduling problem with rejection is considered in this paper. The objective is to minimize the sum of makespan of accepted operations and total penalty of rejected operations. Each job has two operations that could be rejected, respectively. Operations on the first machine have penalties α1 times of their processing times and operations on the second machine have penalties α2 times of their processing times. A [Formula: see text]-approximation algorithm is presented for the case where min{α1, α2} < 1 and max{α1, α2} ≥ 1. A dynamic programming algorithm is provided for general α1 and α2. A fully polynomial-time approximation scheme (FPTAS) is designed for all NP-hard cases.

scholarly journals Supply Chain Batching Problem with Identical Orders and Lifespan

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Shanlin Li ◽  
Maoqin Li ◽  
Hong Yan
Keyword(s):  
Dynamic Programming ◽  
Processing Time ◽  
Recursion Formula ◽  
Dynamic Programming Algorithm ◽  
Setup Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Programming Algorithm ◽  
Short Lifespan ◽  
The Difference

In the real world, there are a large number of supply chains that involve the short lifespan products. In this paper, we consider an integrated production and distribution batch scheduling problem on a single machine for the orders with a short lifespan, because it may be cheaper or faster to process and distribute orders in a batch than to process and distribute them individually. Assume that the orders have the identical processing time and come from the same location, and the batch setup time is a constant. The problem is to choose the number of batches and batch sizes to minimize the total delivery time without violating the order lifespan. We first give a backward dynamic programming algorithm, but it is not an actually polynomial-time algorithm. Then we propose a constant time partial dynamic programming algorithm by doing further research into the recursion formula in the algorithm. Further, using the difference characteristics of the optimal value function, a specific calculating formula to solve the problem with the setup time being integer times of the processing time is obtained.

STOCHASTIC BATCH SCHEDULING AND THE “SMALLEST VARIANCE FIRST” RULE

2007 ◽  
Vol 21 (4) ◽  
pp. 579-595 ◽  
Author(s):  
Michael Pinedo
Keyword(s):  
Completion Time ◽  
Processing Time ◽  
Random Variable ◽  
Combinatorial Problems ◽  
Set Partitioning ◽  
Batch Scheduling ◽  
Matching Problem ◽  
Batch Mode ◽  
Processing Times ◽  
Expected Makespan

Consider a single machine that can process multiple jobs in batch mode. We havenjobs and the processing time of jobjis a random variableXjwith distributionFj. Up tobjobs can be processed simultaneously by the machine. The jobs in a batch all have to start at the same time and the batch is completed when all jobs have finished their processing (i.e., at the maximum of the processing times of the jobs in that batch). We are interested in two objective functions, namely the minimization of the expected makespan and the minimization of the total expected completion time. We first show that under certain fairly general conditions, the minimization of the expected makespan is equivalent to specific deterministic combinatorial problems, namely the Weighted Matching problem and the Set Partitioning problem. We then consider the case when all jobs have the same mean processing time but different variances. We show that for certain special classes of processing time distributions theSmallest Variance Firstrule minimizes the expected makespan as well as the total expected completion time. In our conclusions we present various general rules that are suitable for the minimization of the expected makespan and the total expected completion time in batch scheduling.

Single Machine Scheduling and Due Date Assignment with Past-Sequence-Dependent Delivery Times and Position-Dependent Processing Times

2014 ◽  
Vol 1006-1007 ◽  
pp. 498-503 ◽  
Author(s):  
Yu Fang Zhao
Keyword(s):  
Single Machine ◽  
Dynamic Programming Algorithm ◽  
Machine Scheduling ◽  
Single Machine Scheduling ◽  
Programming Algorithm ◽  
Due Date ◽  
Due Date Assignment ◽  
Processing Times ◽  
Delivery Times ◽  
Tardy Jobs

This paper considers single machine scheduling and due date assignment problems in which the jobs need to be delivered to customers after processing. It is assumed that the delivery times are proportional to the length of the already processed jobs, and a job's processing time depends on its position in a sequence. The objective functions include total earliness, the weighted number of tardy jobs and the cost of due date assignment. We analyze the problems with two different due date assignment methods and conclude that the problems are polynomial time solvable. We provide a dynamic programming algorithm with O(n3) times for the problems.

OPTIMAL SEMI-ONLINE ALGORITHMS FOR m-BATCH-MACHINE FLOW SHOP SCHEDULING

2012 ◽  
Vol 04 (04) ◽  
pp. 1250051 ◽  
Author(s):  
MING LIU ◽  
CHENGBIN CHU
Keyword(s):  
Online Algorithms ◽  
Processing Time ◽  
Flow Shop ◽  
Optimal Algorithm ◽  
Flow Shop Scheduling ◽  
Shop Scheduling ◽  
Batch Machine ◽  
Processing Times ◽  
Batch Processing Machine ◽  
Over Time

This paper deals with semi-online scheduling on m-batch-machine flow shop. The objective is to minimize the makespan. A parallel batch processing machine can handle up to B jobs simultaneously. We study an unbounded model where B = ∞. The jobs that are processed together construct a batch, and all jobs in a batch start and complete at the same time. The processing time of a batch is given by the longest processing time of any job in the batch. The problem is online in the sense that jobs arrive over time. Let pi, j(i = 1,…,m) denote the processing time of job Jj on machines Mi, respectively. Let Jj+1 be the following job of Jj in a job instance. We study semi-online problem with jobs' nondecreasing processing times. We focus on the case where p1, j = ⋯ = pm, j for i = 1, …, m and pi, j+1 ≥ βpi, j (β ≥ 1). For this problem, we propose an optimal algorithm [Formula: see text] with a competitive ratio [Formula: see text].

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