Minimizing the weighted number of early and tardy jobs in a stochastic single machine scheduling problem

We study a static stochastic single machine scheduling problem in which jobs have random processing times with arbitrary distributions, due dates are known with certainty, and fixed individual penalties (or weights) are imposed on both early and tardy jobs. The objective is to find an optimal sequence that minimizes the expected total weighted number of early and tardy jobs. The general problem is NP-hard to solve; however, in this paper, we develop certain conditions under which the problem is solvable exactly. An efficient heuristic is also introduced to find a candidate for the optimal sequence of the general problem. Our illustrative examples and computational results demonstrate that the heuristic performs well in identifying either optimal sequences or good candidates with low errors. Furthermore, we show that special cases of the problem studied here reduce to some classical stochastic single machine scheduling problems including the problem of minimizing the expected weighted number of early jobs and the problem of minimizing the expected weighted number of tardy jobs which are both solvable by the proposed exact or heuristic methods.     

A single machine scheduling problem with availability constraints and sequence-dependent setup costs

We study a single machine scheduling problem with availability constraints and sequence-dependent setup costs, with the aim of minimizing the makespan. To the authors’ knowledge, this problem has not been treated as such in the operations research literature. We derive in this paper a mixed integer programming model to deal with such scheduling problem. Computational tests showed that commercial solvers are capable of solving only small instances of the problem. Therefore, we propose two ways for reducing the execution time, namely a valid inequality that strengthen the linear relaxation and an efficient heuristic procedure that provides a starting feasible solution to the solver. A substantial gain is achieved both in terms of the linear programming relaxation bound and in terms of the time to obtain an integer optimum when we use the enhanced model in conjunction with providing to the solver the solution obtained by the proposed heuristic.     

A note on the single machine scheduling to minimize the number of tardy jobs with deadlines

It is known that the single machine scheduling problem of minimizing the number of tardy jobs is polynomially solvable. However, it becomes NP-hard if each job has a deadline. Recently, Huo et al. solved some special cases by a backwards scheduling approach. In this note we present a dual approach—forwards greedy algorithms which may have better running time. For example, in the case that the due dates, deadlines, and processing times are agreeable, the running time of the backwards scheduling algorithm is O(n²)$O(n^2)$, while that of the forwards algorithm is O(nlogn)$O(n\log n)$.     

A production scheduling heuristic for an electronics manufacturer with sequence-dependent setup costs

In this paper, we develop a three-step heuristic to address a production scheduling problem at a high volume assemble-to-order electronics manufacturer. The heuristic provides a solution for scheduling multiple product families on parallel, identical production lines so as to minimize setup costs. The heuristic involves assignment, sequencing, and time scheduling steps, with an optimization approach developed for each step. For the most complex step, the sequencing step, we develop a greedy randomized adaptive search procedure (GRASP). We compare the setup costs resulting from the use of our scheduling heuristic against a heuristic previously developed and implemented at the electronics manufacturer that assumes approximately equal, sequence-independent, setup costs. By explicitly considering the sequence-dependent setup costs and applying GRASP, our empirical results show a reduction in setups costs for an entire factory of 14–21% with a range of single production line reductions from 0% to 49%.     

A neighbourhood scheme with a compressed solution space for the early/tardy scheduling problem

The single machine, distinct due date, early/tardy machine scheduling problem closely models the situation faced by Just-In-Time manufacturers. This paper develops a new method of finding good quality solutions to this scheduling problem by using the concept of a ‘compressed solution space’, based on a binary representation of the early/tardy scheduling problem, and tabu search. A heuristic which simultaneously sequences and schedules the jobs is developed to perform the conversion between the compressed and physical solution spaces. Results show that the compressed solution space performs well with small problems, and is superior to standard tabu search solution spaces for large-scale, realistically sized problems.     

Comparative evaluation of heuristic algorithms for the single machine scheduling problem with two operations per job and time-lags

This paper shows that the single machine scheduling problem with multiple operations per job separated by minimum specified time-lags is NP-hard in the strong sense. Seven simple and polynomially bounded heuristic algorithms are developed for its solution when each job requires only two operations. Empirical evaluation shows that the percentage deviation of the heuristic solutions from their lower bounds is quite low and the effectiveness of these heuristic algorithms in finding optimal schedules increases with an increase in the number of jobs.     

New solution methods for single machine bicriteria scheduling problem: Minimization of average flowtime and number of tardy jobs

We consider the bicriteria scheduling problem of minimizing the number of tardy jobs and average flowtime on a single machine. This problem, which is known to be NP-hard, is important in practice, as the former criterion conveys the customer’s position, and the latter reflects the manufacturer’s perspective in the supply chain. We propose four new heuristics to solve this multiobjective scheduling problem. Two of these heuristics are constructive algorithms based on beam search methodology. The other two are metaheuristic approaches using a genetic algorithm and tabu-search. Our computational experiments indicate that the proposed beam search heuristics find efficient schedules optimally in most cases and perform better than the existing heuristics in the literature.     

Fast approximation algorithms for bi-criteria scheduling with machine assignment costs

We consider parallel machine scheduling problems where the processing of the jobs on the machines involves two types of objectives. The first type is one of two classical objective functions in scheduling theory: either the total completion time or the makespan. The second type involves an actual cost associated with the processing of a specific job on a given machine; each job-machine combination may have a different cost. Two bi-criteria scheduling problems are considered: (1) minimize the maximum machine cost subject to the total completion time being at its minimum, and (2) minimize the total machine cost subject to the makespan being at its minimum. Since both problems are strongly NP-hard, we propose fast heuristics and establish their worst-case performance bounds.     

A hybrid heuristic and linear programming approach to multi-product machine scheduling

A new heuristic approach is presented for scheduling economic lots in a multi-product single-machine environment. Given a pre-defined master sequence of product setups, an integer linear programming formulation is developed which finds an optimal subsequence and optimal economic lots. The model takes explicit account of initial inventories, setup times and allows setups to be scheduled at arbitrary epochs in continuous time, rather than restricting setups to a discrete time grid. We approximate the objective function of the model and solve to obtain an optimal capacity feasible schedule for the approximate objective. The approach was tested on a set of randomly generated problems, generating solutions that are on average 2.5% above a lower bound on the optimal cost. We also extend the approach to allow shortages.     

Improved lower bounds for the early/tardy scheduling problem with no idle time

In this paper, we consider the single machine earliness/tardiness scheduling problem with no idle time. Two of the lower bounds previously developed for this problem are based on Lagrangean relaxation and the multiplier adjustment method, and require an initial sequence. We investigate the sensitivity of the lower bounds to the initial sequence, and experiment with different dispatch rules and some dominance conditions. The computational results show that it is possible to obtain improved lower bounds by using a better initial sequence. The lower bounds are also incorporated in a branch-and-bound algorithm, and the computational tests show that one of the new lower bounds has the best performance for larger instances.     

Multi-agent single machine scheduling

We consider the scheduling problems arising when several agents, each owning a set of nonpreemptive jobs, compete to perform their respective jobs on one shared processing resource. Each agent wants to minimize a certain cost function, which depends on the completion times of its jobs only. The cost functions we consider in this paper are maximum of regular functions (associated with each job), number of late jobs and total weighted completion time. The different combinations of the cost functions of each agent lead to various problems, whose computational complexity is analysed in this paper. In particular, we investigate the problem of finding schedules whose cost for each agent does not exceed a given bound for each agent.     

A note on exact and heuristic algorithms for the identical parallel machine scheduling problem

A recent paper (Davidovi? et al., J. Heuristics, 18:549?C569, 2012) presented a bee colony metaheuristic for scheduling independent tasks to identical processors, evaluating its performance on a benchmark set of instances from the literature. We examine two exact algorithms from the literature, the former published in 1995, the latter in 2008 (and not cited by the authors). We show that both such algorithms solve to proven optimality all the considered instances in a computing time that is several orders of magnitude smaller than the time taken by the new algorithm to produce an approximate solution.     

关于带运输的单机调度在线问题的研究

本文研究了带运输机的单机在线调度问题。问题假设工件实时在线到达,系统中有一台运输机,该运输机每次最多运输$k$个工件,每个工件需要先在单机上完成加工,然后再被运输机运往目的地,问题的优化目标为最小化完工时间,即所有工件被加工完并且运往目的地的时间最短。针对该问题,作者研究了工件满足一致性条件的模型,并且基于贪心思想给出了竞争比为$\frac{\sqrt{5}+1}{2}$的在线算法,并且证明该算法是最优在线算法。     

关于带运输的单机调度在线问题的研究

本文研究了带运输机的单机在线调度问题。问题假设工件实时在线到达,系统中有一台运输机,该运输机每次最多运输$k$个工件,每个工件需要先在单机上完成加工,然后再被运输机运往目的地,问题的优化目标为最小化完工时间,即所有工件被加工完并且运往目的地的时间最短。针对该问题,作者研究了工件满足一致性条件的模型,并且基于贪心思想给出了竞争比为$\frac{\sqrt{5}+1}{2}$的在线算法,并且证明该算法是最优在线算法。     

Two-agent single machine scheduling with forbidden intervals

We consider the resumable version of the two-agent single machine scheduling problems with forbidden intervals in which the jobs cannot be processed. The goal is to minimize the sum of the objective functions of the two agents. Polynomial and pseudo-polynomial time algorithms are presented for various combinations of regular scheduling objective functions.     

SINGLE MACHINE SCHEDULING WITH CONTROLLABLE PROCESSING TIMES AND COMPRESSION COSTS （Part Ⅱ Heuristics for the General Case）

A single machine scheduling problem with controllable processing times and compression costs is considered. The objective is to find an optimal sequence to minimize the cost ofcompletion times and the cost of compression. The complexity of this problem is still unknown.In Part Ⅱ of this paper,the authors have considered a special case where the compression timesand the compression costs are equal among all jobs. Such a problem appears polynomiafiy solvable by developing an O(n^2) algorithm. In this part(Part Ⅱ ),a general case where the controllable processing times and the compression costs are not equal is discussed. Authors proposehere two heuristics with the first based on some previous work and the second based on the algorithm developed in Part Ⅱ . Computational results are presented to show the efficiency and therobustness of these heuristics.     

A single machine batch scheduling problem with bounded batch size

We address a single-machine batch scheduling problem to minimize total flow time. Processing times are assumed to be identical for all jobs. Setup times are assumed to be identical for all batches. As in many practical situations, batch sizes may be bounded. In the first setting studied in this paper, all batch sizes cannot exceed a common upper bound. In the second setting, all batch sizes share a common lower bound. An optimal solution consists of the number of batches and their (integer) size. We introduce an efficient solution for both problems.     

单台机以总完工时间为目标的批排序问题

研究单台机,工件加工时间相等,大小不同的批排序问题,给出了一个最坏情况界为9+3~(1/2)/6≈1.7817的多项式时间近似算法,并证明了即使工件总大小不超过2,该问题也不存在FPTAS,除非P=NP.     

Single machine scheduling to minimize weighted earliness subject to no tardy jobs

This paper considers the single machine scheduling problem where the objective is to minimize the total weighted earliness subject to no tardy jobs. Known results for a well researched single machine scheduling problem where the objective is to minimize the weighted completion time subject to no tardy jobs have been used in analyzing this problem. Several important results are proved and both exact and approximate methods are developed to solve this problem.     

Improved algorithms for single machine scheduling with release dates and rejections

We consider bi-criteria scheduling problems on a single machine with release dates and rejections and both the makespan and the total rejection cost have to be minimized. We consider three scenarios: (1) minimize the sum of the two objectives: makespan and total rejection cost, (2) minimize the makespan subject to a bound on the total rejection cost and (3) minimize the total rejection cost subject to a bound on the makespan. We summarize the results obtained in the literature and provide for several cases improved approximation algorithms and FPTASs.