Computational complexity of the single processor makespan minimization problem with release dates and job-dependent learning

In this paper, we analyse the single processor maximum completion time (makespan) minimization problem with distinct release dates of jobs and the sum-of-processing time-based learning effect. We prove that the considered problem is strongly NP-hard, if, in addition to jobs with the same learning ratio, there are jobs with constant job processing times. Such jobs are not affected by learning and model, for instance, required system upgrades or training courses.     

The strong NP-hardness of the maximum lateness minimization scheduling problem with the processing-time based aging effect

In this paper, we analyse the single machine maximum lateness minimization scheduling problem with the processing time based aging effect, where the processing time of each job is described by a non-decreasing function dependent on the sum of the normal processing times of preceded jobs. The computational complexity of this problem was not determined. However, we show it is strongly NP-hard by proving the strong NP-hardness of the single machine maximum completion time minimization problem with this aging model and job deadlines. Furthermore, we determine the boundary between polynomially solvable and NP-hard cases.     

具有学习效应的超前有奖延误受罚的排序问题(英)

本文考虑具有学习效应和共同交货期的单机排序问题.目标函数是加权超前有奖延误受罚总和.我们的目标是寻找一个最优序使得目标函数的值最小.由于该问题是NP-hard的,我们给出一些特殊情况下多项式时间可解的特例.同时在快速估计下界的基础上给出了分支定界算法来求一般情况下的最有排序.     

Minimizing the total completion time on a single machine with the learning effect and multiple availability constraints

This paper considers a single machine scheduling problem with the learning effect and multiple availability constraints that minimizes the total completion time. To solve this problem, a new binary integer programming model is presented, and a branch-and-bound algorithm is also developed for solving the given problem optimally. Since the problem is strongly NP-hard, to find the near-optimal solution for large-sized problems within a reasonable time, two meta-heuristics; namely, genetic algorithm and simulated annealing are developed. Finally, the computational results are provided to compare the result of the binary integer programming, branch-and-bound algorithm, genetic algorithm and simulated annealing. Then, the efficiency of the proposed algorithms is discussed.     

Scheduling problems with position dependent job processing times: computational complexity results

In this paper, we analyse single machine scheduling problems with learning and aging effects to minimize one of the following objectives: the makespan with release dates, the maximum lateness and the number of late jobs. The phenomena of learning and aging are modeled by job processing times described by non-increasing (learning) or non-decreasing (aging) functions dependent on the number of previously processed jobs, i.e., a job position in a sequence. We prove that the considered problems are strongly NP-hard even if job processing times are described by simple linear functions dependent on a number of processed jobs. Additionally, we show a property of equivalence between problems with learning and aging models. We also prove that if the function describing decrease/increase of a job processing time is the same for each job then the problems with the considered objectives are polynomially solvable even if the function is arbitrary. Therefore, we determine the boundary between polynomially solvable and strongly NP-hard cases.     

Heuristic constructive algorithms for open shop scheduling to minimize mean flow time

In this paper, we consider the problem of scheduling n jobs on m machines in an open shop environment so that the sum of completion times or mean flow time becomes minimal. For this strongly NP-hard problem, we develop and discuss different constructive heuristic algorithms. Extensive computational results are presented for problems with up to 50 jobs and 50 machines, respectively. The quality of the solutions is evaluated by a lower bound for the corresponding preemptive open shop problem and by an alternative estimate of mean flow time. We observe that the recommendation of an appropriate constructive algorithm strongly depends on the ratio n/m.     

Single-machine scheduling problems with actual time-dependent and job-dependent learning effect

In this study, we introduce an actual time-dependent and job-dependent learning effect into single-machine scheduling problems. We show that the complexity results of the makespan minimization problem and the sum of weighted completion time minimization problem are all NP-hard. The complexity result of the maximum lateness minimization problem is NP-hard in the strong sense. We also provide three special cases which can be solved by polynomial time algorithms.     

Minimizing makespan in a two-machine flowshop scheduling with batching and release time

This paper studies two-machine flowshop scheduling with batching and release time, whose objective is to minimize the makespan. We formulate the scheduling problem as a mixed integer programming model and show that it is a strongly NP-hard problem. We derive a lower bound and develop dynamic programming-based heuristic algorithms to solve the scheduling problem. Computational experiments are carried out to evaluate the performance of the heuristic algorithms. The numerical results show that some of the heuristic algorithms can indeed find effective solutions for the scheduling problem.     

Parallel machine scheduling with general sum of processing time based models

In this paper, we analyse the parallel machine makespan minimization problem with the general sum of processing time based learning or aging effects. First, we prove that an optimal solution to the single machine case can be found by priority rules. Next, for the considered parallel machine problem, we construct the exact dynamic programming algorithm that can operate on real-valued job processing times, which is the only exact algorithm for the analysed problem. The computational analysis confirms that it can solve optimally moderate problem instances.     

Maximizing the weighted number of on-time jobs in single machine scheduling with time windows

The problem of maximizing the weighted number of on-time jobs on a single machine with time windows (STW) is shown to be strongly NP-hard. An efficient heuristic is presented for STW. Computational experiments indicate that the performance of the heuristic is quite good.     

带准备时间的单机指数时间学习效应排序问题

研究带有准备时间的单机学习效应模型,其中工件加工时间具有指数时间学习效应,即工件的实际加工时间是已经排好的工件加工时间的指数函数。学习效应模型考虑工件的实际加工时间同时依赖于工件本身的加工时间和已加工工件的累计加工时间,目标函数为最小化总完工时间。这个问题是NP-难的,提出了一个数学规划模型来求解该问题的最优解。通过分析几个优势性质和下界,提出分支定界算法来求解此问题,并设计启发式算法改进分支定界算法的上界值。通过仿真实验验证了分支定界算法在求解质量和时间方面的有效性。     

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.     

Hyper-heuristic approaches for the response time variability problem

We propose two classes for the implementation of hyper-heuristic algorithms. The first is based on constructive heuristics, whereas the second uses improvement methods. Within the latter class, a general framework is designed for the use of local search procedures and metaheuristics as low-level heuristics. A dynamic scheme to guide the use of these approaches is also devised. These ideas are tested on an NP-hard scheduling problem known as the response time variability problem (RTVP). An intensive computational experiment shows, especially in the second class where the new best results are found, the effectiveness of the proposed hyper-heuristics.     

Polynomial time algorithms for two special classes of the proportionate multiprocessor open shop

This paper describes a complex scheduling problem taken from a hospital diagnostic testing center that schedules hundreds of patients in an open shop environment consisting of multiple facilities and multiple processors. This scheduling problem, known as the multiprocessor open shop (MPOS) problem, is strongly NP-hard with few published results. Realizing that in many MPOS environments processing times are stage-dependent, not both job and stage-dependent, this paper examines a new class of problems for the MPOS—proportionate ones. This paper exploits the structural nature of the proportionate MPOS and defines new terms. Despite the enormous complexity of the MPOS problem, this work demonstrates that polynomial time algorithms exist for two special cases. Since other applications of this problem exist in service and manufacturing environments, solving the proportionate MPOS problem is not only significant in the theory of optimization, but also in many real-world applications.     

Two-machine flow-shop scheduling with equal processing time on the second machine for minimizing total weighted completion time

We consider the classical two-machine flow-shop scheduling for minimizing the total weighted completion time. For this problem, the computational complexity of a version in which the jobs have a common processing time on the second machine, has not been addressed. We show that the problem is unary NP-hard, answering an open problem posed in Zhu et al. (2016). Then we present an approximation algorithm for the problem with worst-case performance ratio at most 2.     

储存时间有上限的两阶段供应链排序问题

研究一类储存时间有上限的两阶段供应链排序问题.两阶段是指工件先加工,后运输:加工阶段是一台加工机器逐个加工工件;运输阶段是无限台车辆分批运输完工的工件.工件的运输完成时刻与完工时刻之差定义为工件的储存时间,且有相应的储存费用,且任意工件的储存时间都不超过某一常数.若工件的运输完成时刻早于(晚于)交货期窗口的开始(结束)时刻,则有相应的提前(延误)惩罚费用.目标是极小化总提前惩罚费用、总延误惩罚费用、总储存费用、总运输费用以及与交货期窗口有关的费用之和.先证明该问题是NP-难的,后对单位时间的储存费用不超过单位时间的延误惩罚费用的情形给出了伪多项式时间算法.     

单可变资源最小化加权完工时间和排序问题的强NP-困难性

Baker和Nuttle提出了下述单可变资源排序问题：扎个工件利用某个单资源进行加工使得工件的完工时间的某个函数达到最小,而资源的可利用率是随着时间而变化的．当最小化的目标函数是工件的加权完工时间和时,Baker和Nuttle猜测该问题是NP-困难的．最近,Yuan、Cheng和Ng证明该问题在一般意义下是NP-困难的,但是问题的精确复杂性仍然是悬而未决的．本文我们证明了该问题是强NP-困难的．     

Pseudopolynomial time solvability of a quadratic Euclidean problem of finding a family of disjoint subsets

In this paper, a strongly NP-hard problem of finding a family of disjoint subsets with given cardinalities in a finite set of points from a Euclidean space is considered. Minimization of the sum over all required subsets of the sum of the squared distances from the elements of these subsets to their geometric centers is used as the search criterion. It is proved that if the coordinates of the input points are integer and the space dimension and the number of required subsets are fixed (i.e., bounded by some constants), the problem is a pseudopolynomial time solvable one.     

Single machine scheduling with time deteriorating job values

This paper considers the problem of scheduling n jobs on a single machine where the job value deteriorates with its starting time. The objective of the problem is to maximize the cumulative value of all jobs. The problem is motivated from the real-life applications, such as movie scheduling, remanufacturing of high-technology products, web object transmission, and banner advertising. Unrestricted, truncated, and capacity constrained job value functions are considered. Some special cases of the problem, such as the unrestricted linear job value function, are polynomially solvable. The general problem is shown to be unary NP-hard and is modelled as a time index integer program. For the NP-hard cases, several heuristics are proposed. Results of the empirical evaluation of the relative performance of the proposed heuristics on critical parameters are reported.     

Single machine bicriteria scheduling with equal-length jobs to minimize total weighted completion time and maximum cost

Bicriteria scheduling problems are of significance in both theoretical and applied aspects. It is known that the single machine bicriteria scheduling problem of minimizing total weighted completion time and maximum cost simultaneously is strongly NP-hard. In this paper we consider a special case where the jobs have equal length and present an $O(n^{3}\log n)$ algorithm for finding all Pareto optimal solutions of this bicriteria scheduling problem.