On the complexity of scheduling unit-time jobs with OR-precedence constraints

We present various complexity results for scheduling unit-time jobs subject to OR-precedence constraints. We prove that minimizing the total weighted completion time is strongly NP-hard, even on a single machine. In contrast, we give a polynomial-time algorithm for minimizing the makespan and the total completion time on identical parallel machines.     

Single machine scheduling with semi-resumable machine availability constraints

This paper considers the semi-resumable model of single machine scheduling with a non-availability period. The machine is not available for processing during a given time interval. A job cannot be completed before the non-availability period will have to partially restart after the machine has become available again. For the problem with objective of minimizing makespan, the tight worst-case ratio of algorithm LPT is given, and an FPTAS is also proposed. For the problem with objective of minimizing total weighted completion time, an approximation algorithm with worst-case ratio smaller than 2 is presented. Two special cases of the latter problem are also considered, and improved algorithms are given.     

带有拒绝的单机和同型机排序问题

研究了带有拒绝的单机和同型机排序问题. 对于单机情形, 工件的惩罚费用是对应加工时间的\alpha倍.如果工件有到达时间, 目标为最小化时间表长与惩罚费用之和, 证明了这个问题是可解的.如果所有工件在零时刻到达, 目标为最小化总完工时间与惩罚费用之和, 也证明了该问题是可解的.对于同型机排序问题, 研究了工件分两批在线实时到达的情形, 目标为最小化时间表长与惩罚费用之和.针对机器台数2和m, 分别给出了竞争比为2和4-2/m的在线算法.     

Scheduling on same-speed processors with at most one downtime on each machine

We consider the problem of scheduling a set of independent tasks on multiple same-speed processors with planned shutdown times with the aim of minimizing the makespan. We give an LPT-based algorithm, LPTX, which yields a maximum completion time that is less than or equal to 3/2 the optimal maximum completion time or 3/2 the time that passes from the start of the schedule until the latest end of a downtime. For problems where the optimal schedule ends after the last downtime, and when the downtimes represent fixed jobs, the LPTX maximum completion time is within 3/2 of the optimal maximum completion time. In addition, we show that this result is asymptotically tight for the class of polynomial algorithms assuming that P≠NP. We also show that the bound obtained previously for a similar problem, when no more than half of the machines are shut down at the same time, for the LPT algorithm is asymptotically tight in the class of polynomial algorithms if P≠NP.     

A composite algorithm for multiprocessor scheduling

A composite algorithm is developed for the classical problem of scheduling independent jobs on identical parallel machines with the objective of minimizing the makespan. The algorithm at first obtains a family of initial partial solutions and combines these partial solutions until a feasible solution is generated. Then local search procedures are used for improving the solution. The effectiveness of this approach is evaluated through extensive computational comparisons with recent improvement heuristics for different classes of benchmark instances.     

Considering manufacturing cost and scheduling performance on a CNC turning machine

A well known industry application that allows controllable processing times is the manufacturing operations on CNC machines. For each turning operation as an example, there is a nonlinear relationship between the manufacturing cost and its required processing time on a CNC turning machine. If we consider total manufacturing cost (F₁) and total weighted completion time (F₂) objectives simultaneously on a single CNC machine, making appropriate processing time decisions is as critical as making job sequencing decisions. We first give an effective model for the problem of minimizing F₁ subject to a given F₂ level. We deduce some optimality properties for this problem. Based on these properties, we propose a heuristic algorithm to generate an approximate set of efficient solutions. Our computational results indicate that the proposed algorithm performs better than the GAMS/MINOS commercial solver both in terms of solution quality and computational requirements such that the average CPU time is only 8% of the time required by the GAMS/MINOS.     

Total completion time with makespan constraint in no-wait flowshops with setup times

The m-machine no-wait flowshop scheduling problem with the objective of minimizing total completion time subject to the constraint that the makespan value is not greater than a certain value is addressed in this paper. Setup times are considered non-zero values, and thus, setup times are treated as separate from processing times. Several recent algorithms, an insertion algorithm, two genetic algorithms, three simulated annealing algorithms, two cloud theory-based simulated annealing algorithms, and a differential evolution algorithm are adapted and proposed for the problem. An extensive computational analysis has been conducted for the evaluation of the proposed algorithms. The computational analysis indicates that one of the nine proposed algorithms, one of the simulated annealing algorithms (ISA-2), performs much better than the others under the same computational time. Moreover, the analysis indicates that the algorithm ISA-2 performs significantly better than the earlier existing best algorithm. Specifically, the best performing algorithm, ISA-2, proposed in this paper reduces the error of the existing best algorithm in the literature by at least 90% under the same computational time. All the results have been statistically tested.     

Optimal streaming of a single job in a two-stage flow shop

Lot streaming is moving some portion of a process batch ahead to begin a downstream operation. The problem to be considered in this paper is the following: a single job consisting of U units is to be processed on two machines in the given order. Given a fixed number of possible transfer batches between the two machines, the problem is to find the timing and the size of the transfer batches (or, sublots) so as to optimize a given criterion. The schedules can be evaluated based on job completion, sublot completion, or item completion times. In the single job lot streaming problem, minimizing job completion time corresponds to minimizing the makespan, for which formulas for optimal sublot sizes are available. In this paper, the results for the sublot and item completion time models are presented.     

No-wait flowshops with bicriteria of makespan and total completion time

We consider the m-machine no-wait flowshop scheduling problem with the objective of minimizing a weighted sum of makespan and total completion time. For the two-machine problem, we develop a dominance relation and embed it within a proposed branch-and-bound algorithm. For the m-machine problem, we propose a heuristic. Computational experiments show that the proposed heuristic outperforms the best existing multi-criteria heuristics and the best single criterion heuristics for makespan and total completion time. The efficiency of the dominance relation and branch-and-bound algorithm is also investigated and shown to be effective.     

A branch and bound algorithm for the one-machine scheduling problem with minimum and maximum time lags

We consider the one-machine scheduling problem with minimum and maximum time lags while minimizing the makespan. This problem typically arises in a manufacturing environment where the next job has to be carried out within a specific time range after the completion of the immediately preceding job. We describe a branch and bound algorithm, based on the input and output of a clique and the relevant propositions, for finding the optimal waiting times. The computational experiments give promising results, showing whether a given instance is feasible or infeasible. With the proposed branch and bound algorithm we can either find an optimal schedule or establish the infeasibility within an acceptable run time.     

Parallel machine scheduling with a convex resource consumption function

We consider some problems of scheduling jobs on identical parallel machines where job-processing times are controllable through the allocation of a nonrenewable common limited resource. The objective is to assign the jobs to the machines, to sequence the jobs on each machine and to allocate the resource so that the makespan or the sum of completion times is minimized. The optimization is done for both preemptive and nonpreemptive jobs. For the makespan problem with nonpreemptive jobs we apply the equivalent load method in order to allocate the resources, and thereby reduce the problem to a combinatorial one. The reduced problem is shown to be NP-hard. If preemptive jobs are allowed, the makespan problem is shown to be solvable in O(n²) time. Some special cases of this problem with precedence constraints are presented and the problem of minimizing the sum of completion times is shown to be solvable in O(n log n) time.     

具有周期维护最小化时间表长的两台平行机调度问题(英文)

本文讨论了具有周期维护的两台平行机调度问题,目标函数为最小化时间表长.设T为维护周期,t为每次对机器维护需要的时间,当t≤T/3时,本文证明了对于该问题由LPT算法得到的最坏误差界为2.     

Minimizing the sum of job completion times on capacitated two-parallel machines

This paper considers a scheduling problem with two identical parallel machines. One has unlimited capacity; the other can only run for a fixed time. A given set of jobs must be scheduled on the two machines with the goal of minimizing the sum of their completion times. The paper proposes an optimal branch and bound algorithm which employs three powerful elements, including an algorithm for computing the upper bound, a lower bound algorithm, and a fathoming condition. The branch and bound algorithm was tested on problems of various sizes and parameters. The results show that the algorithm is quite efficient to solve all the test problems. In particular, the total computation time for the hardest problem is less than 0.1 second for a set of 100 problem instances. An important finding of the tests is that the upper bound algorithm can actually find optimal solutions to a quite large number of problems.     

Two-machine flowshop scheduling problems involving a batching machine with transportation or deterioration consideration

This paper considers two scheduling problems for a two-machine flowshop where a single machine is followed by a batching machine. The first problem is that there is a transporter to carry the jobs between machines. The second problem is that there are deteriorating jobs to be processed on the single machine. For the first problem with minimizing the makespan, we formulate it as a mixed integer programming model and then prove that it is strongly NP-hard. A heuristic algorithm is proposed for solving this problem and its worst case performance is analyzed. The computational experiments are carried out and the numerical results show that the heuristic algorithm is effective. For the second problem, we derive the optimal algorithms with polynomial time for minimizing the makespan, the total completion time and the maximum lateness, respectively.     

Single-machine scheduling with time-and-resource-dependent processing times

We consider single-machine scheduling problems in which the processing time of a job is a function of its starting time and its resource allocation. The objective is to find the optimal sequence of jobs and the optimal resource allocation separately. We concentrate on two goals separately, namely, minimizing a cost function containing makespan, total completion time, total absolute differences in completion times and total resource cost; minimizing a cost function containing makespan, total waiting time, total absolute differences in waiting times and total resource cost. We show that the problems remain polynomially solvable under the proposed model.     

Scheduling interfering job sets on parallel machines

We consider bicriteria scheduling on identical parallel machines in a nontraditional context: jobs belong to two disjoint sets, and each set has a different criterion to be minimized. The jobs are all available at time zero and have to be scheduled (non-preemptively) on m parallel machines. The goal is to generate the set of all non-dominated solutions, so the decision maker can evaluate the tradeoffs and choose the schedule to be implemented. We consider the case where, for one of the two sets, the criterion to be minimized is makespan while for the other the total completion time needs to be minimized. Given that the problem is NP-hard, we propose an iterative SPT–LPT–SPT heuristic and a bicriteria genetic algorithm for the problem. Both approaches are designed to exploit the problem structure and generate a set of non-dominated solutions. In the genetic algorithm we use a special encoding scheme and also a unique strategy – based on the properties of a non-dominated solution – to ensure that all parts of the non-dominated front are explored. The heuristic and the genetic algorithm are compared with a time-indexed integer programming formulation for small and large instances. Results indicate that the both the heuristic and the genetic algorithm provide high solution quality and are computationally efficient. The heuristics proposed also have the potential to be generalized for the problem of interfering job sets involving other bicriteria pairs.     

Bi-criteria single machine scheduling with a time-dependent learning effect and release times

This paper deals with a bi-criteria single machine scheduling problem with a time-dependent learning effect and release times. The objective is to minimize the weighted sum of the makespan and the total completion time. The problem is NP-hard, thus a mixed integer non-linear programming formulation is presented, and a set of dominance properties are developed. To solve the problem efficiently, a procedure is then proposed by incorporating the dominance properties with an ant colony optimization algorithm. In the proposed algorithm, artificial ants construct solutions as orders of jobs based on the heuristic information as well as pheromone trails. Then, the dominance properties are added to obtain better solutions. To evaluate the algorithm performance, computational experiments are conducted.     

Unrelated parallel machines scheduling with deteriorating jobs and resource dependent processing times

We consider unrelated parallel machines scheduling problems involving resource dependent (controllable) processing times and deteriorating jobs simultaneously, i.e., the actual processing time of a job is a function of its starting time and its resource allocation. Two generally resource consumption functions, the linear and convex resource, were investigated. The objective is to find the optimal sequence of jobs and the optimal resource allocation separately. This paper focus on the objectives of minimizing a cost function containing makespan, total completion time, total absolute differences in completion times and total resource cost, and a cost function containing makespan, total waiting time, total absolute differences in waiting times and total resource cost. If the number of unrelated parallel machines is a given constant, we show that the problems remain polynomially solvable under the proposed model.     

A bicriteria m-machine flowshop scheduling with sequence-dependent setup times

In this study, a bicriteria m-machine flowshop scheduling with sequence-dependent setup times is considered. The objective function of the problem is minimization of the weighted sum of total completion time and makespan. Only small size problems with up to 6 machines and 18 jobs can be solved by the proposed integer programming model. Also the model is tested on an example. We also proposed three heuristic approaches for solving large jobs problems. To solve the large sizes problems up to 100 jobs and 10 machines, special heuristics methods is used. Results of computational tests show that the proposed model is effective in solving problems.     

Flow shop scheduling algorithms for minimizing the completion time variance and the sum of squares of completion time deviations from a common due date

We consider two problems of m-machine flow shop scheduling in this paper: one, with the objective of minimizing the variance of completion times of jobs, and the other with the objective of minimizing the sum of squares of deviations of job completion times from a common due date. Lower bounds on the sum of squares of deviations of job completion times from the mean completion time of jobs for a given partial sequence are first presented. Using these lower bounds, a branch and bound algorithm based on breadth-first search procedure for scheduling n jobs on m-machines with the objective of minimizing completion time variance (CTV) is developed to obtain the best permutation sequence. We also present two lower bounds and thereafter, a branch and bound algorithm with the objective of minimizing the sum of squares of deviations of job completion times from a given common due date (called the MSD problem). The computational experience with the working of the two proposed branch and bound algorithms is also reported. Two heuristics, one for each of the two problems, are developed. The computational experience on the evaluation of the heuristics is discussed.