Uniform Parallel Machine Scheduling Problem Research Articles

Scheduling uniform parallel dedicated machines with job splitting, sequence-dependent setup times, and multiple servers

We address a uniform parallel machine scheduling problem with machine eligibility, job splitting, sequence-dependent setup times, and limited setup servers, which has been defined from a real application of piston manufacturing in Korea. The objective is to minimize the makespan. We first propose a mathematical programming model and derive four lower bounds. We then develop an efficient heuristic algorithm that assigns jobs to machines by considering processing times, setup times, and machine eligibility constraints, and then balances machine workloads by splitting jobs. We analyze a worst-case bound of the algorithm and perform extensive experiments with various scenarios. The gap from a lower bound is not significant for practical cases.

Modified Harris Hawks Optimizer for Solving Machine Scheduling Problems

Scheduling can be described as a decision-making process. It is applied in various applications, such as manufacturing, airports, and information processing systems. More so, the presence of symmetry is common in certain types of scheduling problems. There are three types of parallel machine scheduling problems (PMSP): uniform, identical, and unrelated parallel machine scheduling problems (UPMSPs). Recently, UPMSPs with setup time had attracted more attention due to its applications in different industries and services. In this study, we present an efficient method to address the UPMSPs while using a modified harris hawks optimizer (HHO). The new method, called MHHO, uses the salp swarm algorithm (SSA) as a local search for HHO in order to enhance its performance and to decrease its computation time. To test the performance of MHHO, several experiments are implemented using small and large problem instances. Moreover, the proposed method is compared to several state-of-art approaches used for UPMSPs. The MHHO shows better performance in both small and large problem cases.

Uniform Parallel Machine Scheduling with Dedicated Machines, Job Splitting and Setup Resources

We examine a uniform parallel machine scheduling problem with dedicated machines, job splitting, and limited setup resources for makespan minimization. In this problem, machines have different processing speeds, and each job can only be processed at several designated machines. A job can be split into multiple sections and those sections can be processed on multiple machines simultaneously. Sequence-independent setup times are assumed, and setup operations between jobs require setup operators that are limited. For the problem, we first develop a mathematical optimization model and for large-sized problems a constructive heuristic algorithm is proposed. Finally, we show that the algorithm developed is efficient and provides good solutions by experiments with various scenarios.

An uncertain parallel machine problem with deterioration and learning effect

An uncertain uniform parallel machine scheduling problem with job deterioration and a learning effect is considered. Job processing times, due dates, deterioration rates and learning rates are assumed to be uncertain variables. The objective functions are the total weight earliness, tardiness and makespan. Three mathematical programming models are presented, i.e., expected value model, pessimistic value model and measure chance model. These models can be converted into equivalent crisp models by the inverse uncertainty distribution method. A hybrid algorithm mixed with dispatching rules based on structural features is employed to solve the problem. Finally, computational experiments are presented to illustrate the effectiveness of proposed algorithms.

A new model and algorithm for uncertain random parallel machine scheduling problem

The paper presents a new model for uniform parallel machine scheduling problem with uncertainty and randomness simultaneously for processing times of jobs based on chance theory. The objective of the model is to minimize expected completion time. The constraint of the model is that uncertain random completion time of scheduling is less than or equal to expected completion time. The model is transformed to a crisp non-deterministic polynomial hard mathematical programming model based on chance theory. Firstly, simulation techniques of the objective function and the left chance constraint are proposed. Then, two heuristic methods to solve the crisp model are presented. Finally, they are integrated into two hybrid intelligent algorithms for searching the quasi-optimal schedule. Besides, the effectiveness of the model and its hybrid intelligent algorithms are verified by a numerical example generated randomly.

Scheduling on uniform parallel machines with periodic unavailability constraints

Scheduling problems under unavailability constraints has become a popular research topic in the last few years. Despite it’s important application in the real world, the uniform parallel machine scheduling problem was the least studied due to its complexity. In this paper, we investigated the uniform parallel machine scheduling problem under deterministic availability constraints. Each machine is subject to one unavailability period. Different versions of the problem regarding the type of jobs (identical and non-identical) and the performance measures (the total completion times and the makespan) were studied. For the case of identical jobs and for both performance measures, we developed linear programming models and optimal algorithms to provide a solution to the problem. For the case of non-identical jobs, we proved that the problem is NP-hard and propose a quadratic program. Because, this later cannot solve problems with very large number of jobs and machines, a heuristic was developed to find near optimal solutions to the problem especially with very large number of jobs and machines. The computational results showed that the heuristic’s performance is very high regardless the dimensions of problem instances.

Uniform parallel machine scheduling problems with fixed machine cost

This paper considers two uniform parallel machine scheduling problems with fixed machine cost under the background of cloud manufacturing. The goal is to minimize the makespan with a given budget of total cost, \(\hat{U}\). All the jobs are homogeneous, i.e., the processing times of the jobs are identical. Non-preemptive and preemptive problems are studied. For the non-preemptive problem, we give a \(2[1+1{/}(h-1)]\)-approximation algorithm, where h is the number of the machine which can not be selected the first time. For the preemptive problem, we give an algorithm whose worst-case bound equals to \(1+1{/}(h-1)\). Preliminary experimental results indicate that the proposed algorithms are reasonably accurate compared with the lower bounds.

Uniform parallel machine scheduling problems with a truncation sum-of-logarithm-processing-times-based learning effect

Uniform parallel machine scheduling problems with a truncation sum-of-logarithm-processing-times-based learning effect

Uniform Parallel Machine Scheduling Problem with Controllable Delivery Times

AbstractThis paper considers the uniform parallel machine scheduling problem with controllable delivery times, which assumes that the delivery times of jobs are linear decreasing functions of the consumed resource. It aims to minimize the maximum completion time under the constraint that the total resource consumption does not exceed a given limit. For this NP-hard problem, we propose a resource allocation algorithm, named RAA, according to the feasible solution of the uniform parallel machine scheduling problem with fixed delivery times. It proves that RAA algorithm can obtain the optimal resource allocation scheme for any given scheduling scheme inO(nlogn)time. Some algorithms based on heuristic algorithm LDT, heuristic algorithm LPDT and simulated annealing are proposed to solve the uniform parallel machine scheduling problem with controllable delivery times. The accuracy and efficiency of the proposed algorithms are tested based on those data with problem sizes varying from 40 to 200 jobs and 2 to 8 machines. The computational results indicate that the SA approach is promising and capable of solving large-scale problems in a reasonable time.

Uniform parallel machine scheduling problems with a truncation sum-of-logarithm-processing-times-based learning effect

In internet manufacturing, a manufacturing cloud usually deals with multiple requests at the same time. Similar problems may appear if a manufacturing cloud is equipped with some kind of intelligence and can learn to process a later job more quickly. This research considers the problem of scheduling jobs on uniform parallel machines to minimise total weighted tardiness (TWT) under a truncation sum-of-logarithm-processing-times-based learning effect. In the proposed learning model, the actual job processing time is a function that depends not only on the processing times of the jobs already processed but also on a control parameter. An iterated local search (ILS) has been proposed to solve this problem. Computational results of applying the ILS to some cases show that the proposed ILS outperforms other metaheuristics (simulated annealing and ant colony optimisation) in terms of TWT. The present research also analyses the impacts of truncation learning effects on objective values.

Hedge against total flow time uncertainty of the uniform parallel machine scheduling problem with interval data

We consider a total flow time minimisation problem of uniform parallel machine scheduling when job processing times are only known to be bounded within certain given intervals. A minmax regret model is proposed to identify a robust schedule that minimises the maximum deviation from the optimal total flow time over all possible realisations of the job processing times. To solve this problem, we first prove that the maximal regret for any schedule can be obtained in polynomial time. Then, we derive a mixed-integer linear programming (MILP) formulation of our problem by exploiting its structural properties. To reduce the computational time, we further transform our problem into a robust single-machine scheduling problem and derive another MILP formulation with fewer variables and constraints. Moreover, we prove that the optimal schedule under the midpoint scenario is a 2-approximation for our minmax regret problem. Finally, computational experiments are conducted to evaluate the performance of the proposed methods.

Two-level decomposition algorithm for crew rostering problems with fair working condition

A typical railway crew scheduling problem consists of two phases: a crew pairing problem to determine a set of crew duties and a crew rostering problem. The crew rostering problem aims to find a set of rosters that forms workforce assignment of crew duties and rest periods satisfying several working regulations. In this paper, we present a two-level decomposition approach to solve railway crew rostering problem with the objective of fair working condition. To reduce computational efforts, the original problem is decomposed into the upper-level master problem and the lower-level subproblem. The subproblem can be further decomposed into several subproblems for each roster. These problems are iteratively solved by incorporating cuts into the master problem. We show that the relaxed problem of the master problem can be formulated as a uniform parallel machine scheduling problem to minimize makespan, which is NP-hard. An efficient branch-and-bound algorithm is applied to solve the master problem. Effective valid cuts are developed to reduce feasible search space to tighten the duality gap. Using data provided by the railway company, we demonstrate the effectiveness of the proposed method compared with that of constraint programming techniques for large-scale problems through computational experiments.

An FPTAS for Uniform Parallel-Machine Scheduling Problem with Deteriorating Jobs and Rejection

This paper studies uniform parallel-machine scheduling problem with deteriorating jobs and rejection. The processing time of each job is a linear nondecreasing function of its starting time. A job can be rejected by paying a penalty cost. The objective is to minimize the sum of the total load of the accepted jobs on all machines and the total rejection penalties of the rejected jobs. We propose a fully polynomial-time approximation scheme (FPTAS) for this problem.

Minimizing resource consumption on uniform parallel machines with a bound on makespan

With the crucial issue of environmental protection, managing natural resources efficiently and/or reducing the amount of carbon emissions have become more important than ever. In this paper, we introduce a uniform parallel machine scheduling problem where the objective is to minimize resource consumption given that the maximum completion time does not exceed a certain level. We show that the problem is strongly NP-hard. A tight lower bound and a particle swarm optimization algorithm are then developed. Finally, some computational results are provided.

Uniform Parallel-Machine Scheduling with Time Dependent Processing Times

We consider several uniform parallel-machine scheduling problems in which the processing time of a job is a linear increasing function of its starting time. The objectives are to minimize the total completion time of all jobs and the total load on all machines. We show that the problems are polynomially solvable when the increasing rates are identical for all jobs; we propose a fully polynomial-time approximation scheme for the standard linear deteriorating function, where the objective function is to minimize the total load on all machines. We also consider the problem in which the processing time of a job is a simple linear increasing function of its starting time and each job has a delivery time. The objective is to find a schedule which minimizes the time by which all jobs are delivered, and we propose a fully polynomial-time approximation scheme to solve this problem.

Heuristic algorithms for scheduling on uniform parallel machines with heads and tails

This paper considers the uniform parallel machine scheduling problem with unequal release dates and delivery times to minimize the maximum completion time. For this NP-hard problem, the largest sum of release date, processing time and delivery time first rule is designed to determine a certain machine for each job, and the largest difference between delivery time and release date first rule is designed to sequence the jobs scheduled on the same machine, and then a novel algorithm for the scheduling problem is built. To evaluate the performance of the proposed algorithm, a lower bound for the problem is proposed. The accuracy of the proposed algorithm is tested based on the data with problem size varying from 200 jobs to 600 jobs. The computational results indicate that the average relative error between the proposed algorithm and the lower bound is only 0.667%, therefore the solutions obtained by the proposed algorithm are very accurate.

A simulated annealing approach to minimize the maximum lateness on uniform parallel machines

This paper considers the uniform parallel machine scheduling problem which is to minimize the maximum lateness. This problem is equivalent to the uniform parallel machine scheduling problem, which is to minimize the maximal completion time of n jobs whose release times are zero, processing times depend on the speed of the machine to which they are assigned, and their delivery times are different. This problem is NP-hard, even if the machines’ speeds are identical and all the delivery times equal to zero. We propose a simulated annealing algorithm, named LPDT-SA, to obtain solutions with high quality for large-sized problems. A heuristic algorithm LPDT is built to generate initial solutions. An effective method for solution representation is designed, which is efficient to realize the swap and insertion neighborhood, and simultaneously avoid some obvious inferior solutions, therefore the efficiency of the proposed simulated annealing algorithm is improved. A large set of instances are generated randomly to test the solution quality of LPDT-SA and assess its runtime. The results and analysis of experiments are reported and discussed.

A new dynamic programming formulation for scheduling independent tasks with common due date on parallel machines

This paper deals with a scheduling problem of independent tasks with common due date where the objective is to minimize the total weighted tardiness. The problem is known to be ordinary NP-hard in the case of a single machine and a dynamic programming algorithm was presented in the seminal work of Lawler and Moore [E.L. Lawler, J.M. Moore, A functional equation and its application to resource allocation and sequencing problems, Management Science 16 (1969) 77–84]. In this paper, this algorithm is described and discussed. Then, a new dynamic programming algorithm is proposed for solving the single machine case. These methods are extended for solving the identical and uniform parallel-machine scheduling problems.

Scheduling unit-length jobs with machine eligibility restrictions

We consider uniform parallel machine scheduling problems with unit-length jobs where every job is only allowed to be processed on a specified subset of machines. We develop efficient methods to solve problems with various objectives, including minimizing a total tardiness function, a maximum tardiness function, total completion time, the number of tardy jobs, the makespan, etc.

JOB ASSIGNMENT ALGORITHMS ON UNIFORM PARALLEL MACHINES

Uniform parallel machine scheduling problems with a makespan measure cannot generally be solved within polynomial time complexity. This paper considers special problems with a single type of job on the uniform parallel machines, where each machine is available at a given ready time. Also the machine can be restricted on the number of jobs to be processed. The objective is to develop job assignment or batching algorithms which minimize makespan. When all the machines are available at time zero and have no restriction on the number of assignable jobs, a lower bound and optimal solution properties are derived. Based upon these properties, a polynomial algorithm is suggested to find the optimal job assignment on each machine. Three generalized problems are considered under the following situations: (1) some machines have capacity restrictions on the production batch, (2) each machine has its ready time, and (3) the jobs require series-parallel operations. The generalized problems arc also characterized and polynomial algorithms are developed for the same aim of optimal job assignment, except for the case of series-parallel operations. A heuristic algorithm is suggested with numerical tests for the series-parallel operations problem