Vector scheduling with rejection on a single machine

In this paper, we study a vector scheduling problem with rejection on a single machine, in which each job is characterized by a d-dimension vector and a penalty, in the sense that, jobs can be either rejected by paying a certain penalty or assigned to the machine. The objective is to minimize the sum of the maximum load over all dimensions of the total vector of all accepted jobs, and the total penalty of rejected jobs. We prove that the problem is NP-hard and design two approximation algorithms running in polynomial time. When d is a fixed constant, we present a fully polynomial time approximation scheme.     

Due-date assignment and parallel-machine scheduling with deteriorating jobs

In this paper we study the problem of scheduling n deteriorating jobs on m identical parallel machines. Each job's processing time is a nondecreasing function of its start time. The problem is to determine an optimal combination of the due-date and schedule so as to minimize the sum of the due-date, earliness and tardiness penalties. We show that this problem is NP-hard, and we present a heuristic algorithm to find near-optimal solutions for the problem. When the due-date penalty is 0, we present a polynomial time algorithm to solve it.     

A note on scheduling jobs with equal processing times and inclusive processing set restrictions

We consider the problem of scheduling n jobs on m parallel machines with inclusive processing set restrictions. Each job has a given release date, and all jobs have equal processing times. The objective is to minimize the makespan of the schedule. Li and Li (2015) have developed an O(n²+mn log?n) time algorithm for this problem. In this note, we present a modified algorithm with an improved time complexity of O(min{m, log?n} ? n log?n).     

Scheduling an unbounded batching machine with family jobs and setup times

We consider the problem of scheduling n jobs on an unbounded batching machine that can process any number of jobs belonging to the same family simultaneously in the same batch. All jobs in the same batch complete at the same time. Jobs belonging to different families cannot be processed in the same batch, and setup times are required to switch between batches that process jobs from different families. For a fixed number of families m, we present a generic forward dynamic programming algorithm that solves the problem of minimizing an arbitrary regular cost function in pseudopolynomial time. We also present a polynomial-time backward dynamic programming algorithm with time complexity O (mn(n/m+1) ^m ) for minimizing any additive regular minsum objective function and any incremental regular minmax objective function. The effectiveness of our dynamic programming algorithm is shown by computational experiments based on the scheduling of the heat-treating process in a steel manufacturing plant.     

Two-stage flexible flow shop scheduling subject to fixed job sequences

This paper investigates the scheduling problem in a two-stage flexible flow shop, which consists of m stage-1 parallel dedicated machines and a stage-2 bottleneck machine, subject to the condition that n _l jobs per type l∈{1, …, m} are processed in a fixed sequence. Four regular performance metrics, including the total completion time, the maximum lateness, the total tardiness, and the number of tardy jobs, are considered. For each considered objective function, we aim to determine an optimal interleaving processing sequence of all jobs coupled with their starting times on the stage-2 bottleneck machine. The problem under study is proved to be strongly NP-hard. An O(m²Π_l=1 ^m n _l ²) dynamic programming algorithm coupled with numerical experiments is presented.     

A note on parallel-machine due-window assignment

We consider a due-window assignment problem on identical parallel machines, where the jobs have equal processing times and job-dependent earliness-tardiness costs. We would like to determine a ‘due window’ during which the jobs can be completed at no cost and to obtain a job schedule in which the jobs are penalized if they finish before or after the due window. The objective is to minimize the total earliness and tardiness job penalty, plus the cost associated with the size of the due window. We present an algorithm that can solve this problem in O(n³) time, which is an improvement of the O(n⁴) solution procedure developed by Mosheiov and Sarig.     

工件有到达时间及可拒绝下的同类平行机排序问题的近似算法

本文研究工件有到达时间且可拒绝下的同类平行机排序问题。在该问题中, 给定一个待加工工件集, 每个工件在到达之后, 可以被选择安排到$m$台同类平行机器中的某一台机器上进行加工, 也可以被选择拒绝加工, 但需支付一定的拒绝惩罚费用。目标函数是最小化接受工件集的最大完工时间与拒绝工件集的总拒绝费用之和。当$m$为固定常数时, 设计了一个伪多项式时间动态规划精确算法; 当$m$为任意输入时, 设计了一个近似算法, 当接受工件个数大于$(m-1)$时, 该算法近似比为3, 当接受工件个数小于$(m-1)$时, 该算法近似比为$(2+\rho)$, 其中$\rho$为机器加工速度最大值和最小值的比值。最后通过算例演示了算法的运行。     

工件有到达时间及可拒绝下的同类平行机排序问题的近似算法

本文研究工件有到达时间且可拒绝下的同类平行机排序问题。在该问题中, 给定一个待加工工件集, 每个工件在到达之后, 可以被选择安排到$m$台同类平行机器中的某一台机器上进行加工, 也可以被选择拒绝加工, 但需支付一定的拒绝惩罚费用。目标函数是最小化接受工件集的最大完工时间与拒绝工件集的总拒绝费用之和。当$m$为固定常数时, 设计了一个伪多项式时间动态规划精确算法; 当$m$为任意输入时, 设计了一个近似算法, 当接受工件个数大于$(m-1)$时, 该算法近似比为3, 当接受工件个数小于$(m-1)$时, 该算法近似比为$(2+\rho)$, 其中$\rho$为机器加工速度最大值和最小值的比值。最后通过算例演示了算法的运行。     

Single Stage Minimum Absolute Lateness Problem with a Common Due Date on Non-Identical Machines

In this paper we consider scheduling n single operation jobs with a common due date on m non-identical machines (in parallel) so as to minimize the sum of the absolute lateness. We reduce the problem to a transportation problem that can be solved by a polynomial time algorithm. Furthermore, we consider the problem in the case of identical machines and we give a heuristic algorithm to minimize makespan among all schedules that minimize the absolute lateness problem.     

A Heuristic Decomposition Algorithm for Scheduling Problems on Mixed Graphs

We consider a scheduling problem where a set of n jobs has to be processed on a set of m machines and arbitrary precedence constraints between operations are given. Moreover, for any two operations i and j values a _ij >0 and a _ji >0 may be given where a _ij is the minimal difference between the starting times of operations i and j when operation i is processed first. Often, the objective is to minimize the makespan but we consider also arbitrary regular criteria. Even the special cases of the classical job shop problem J//C_max belong to the set of NP-hard problems. Therefore, approximation or heuristic algorithms are necessary to handle large-dimension problems. Based on the mixed graph model we give a heuristic decomposition algorithm for such a problem, i.e. the initial problem is partitioned into subproblems that can be solved exactly or approximately with a small error bound. These subproblems are obtained by a relaxation of a subset of the set of undirected edges of the mixed graph. The subproblems are successively solved and a proportion of the results obtained for one subproblem is kept for further subproblem definitions. Numerical results of the algorithm presented here are given.     

Two-stage assembly-type flowshop batch scheduling problem subject to a fixed job sequence

This paper discusses a two-stage assembly-type flowshop scheduling problem with batching considerations subject to a fixed job sequence. The two-stage assembly flowshop consists of m stage-1 parallel dedicated machines and a stage-2 assembly machine which processes the jobs in batches. Four regular performance metrics, namely, the total completion time, maximum lateness, total tardiness, and number of tardy jobs, are considered. The goal is to obtain an optimal batching decision for the predetermined job sequence at stage 2. This study presents a two-phase algorithm, which is developed by coupling a problem-transformation procedure with a dynamic program. The running time of the proposed algorithm is O(mn+n⁵), where n is the number of jobs.     

Balancing Workloads and Minimizing Set-up Costs in the Parallel Processing Shop

This paper presents an optimal scheduling algorithm for minimizing set-up costs in the parallel processing shop while meeting workload balancing restrictions.There are M independent batch type jobs which have sequence dependent set-up costs and N parallel processing machines. Each of the M jobs must be processed on exactly one of the N available machines. It is desirable to minimize total changeover costs with the restriction that each machine workload assignment T _n be within P units of the average machine assignment. The paper describes a static problem in which all jobs are available at time zero. The sequence dependent change over costs are identical for each machine. An extension of the algorithm handles nonidentical processor problems.A combinatorial programming approach to the problem is used. For the special case of identical processors, the problem can be treated as a multi-salesman travelling salesman problem. A general branch and bound algorithm and numerical results are given.     

On the asymptotic optimality of a solution of the euclidean problem of covering a graph by <Emphasis Type="Italic">m</Emphasis> nonadjacent cycles of maximum total weight

In the problem of covering an n-vertex graph by m cycles of maximum total weight, it is required to find a family of m vertex-nonadjacent cycles such that it covers all vertices of the graph and the total weight of edges in the cover is maximum. The paper presents an algorithm for approximately solving the problem of covering a graph in Euclidean d-space R^d by m nonadjacent cycles of maximum total weight. The algorithm has time complexity O(n³). An estimate of the accuracy of the algorithm depending on the parameters d, m, and n is substantiated; it is shown that if the dimension d of the space is fixed and the number of covering cycles is m = o(n), then the algorithm is asymptotically exact.     

A new heuristic approach for the P-median problem

The Euclidean p-median problem is concerned with the decision of the locations for public service centres. Existing methods for the planar Euclidean p-median problems are capable of efficiently solving problems of relatively small scale. This paper proposes two new heuristic algorithms aiming at problems of large scale. Firstly, to reflect the different degrees of proximity to optimality, a new kind of local optimum called level-m optimum is defined. For a level-m optimum of a p-median problem, where m<p, each of its subsets containing m of the p partitions is a global optimum of the corresponding m-median subproblem. Starting from a conventional local optimum, the first new algorithm efficiently improves it to a level-2 optimum by applying an existing exact algorithm for solving the 2-median problem. The second new algorithm further improves it to a level-3 optimum by applying a new exact algorithm for solving the 3-median problem. Comparison based on experimental results confirms that the proposed algorithms are superior to the existing heuristics, especially in terms of solution quality.     

Variational Analysis on Local Sharp Minima via Exact Penalization

In this paper we study local sharp minima of the nonlinear programming problem via exact penalization. Utilizing generalized differentiation tools in variational analysis such as subderivatives and regular subdifferentials, we obtain some primal and dual characterizations for a penalty function associated with the nonlinear programming problem to have a local sharp minimum. These general results are then applied to the ? _p penalty function with 0 ≤ p ≤ 1. In particular, we present primal and dual equivalent conditions in terms of the original data of the nonlinear programming problem, which guarantee that the ? _p penalty function has a local sharp minimum with a finite penalty parameter in the case of \(p\in (\frac {1}{2}, 1]\) and \(p=\frac {1}{2}\) respectively. By assuming the Guignard constraint qualification (resp. the generalized Guignard constraint qualification), we also show that a local sharp minimum of the nonlinear programming problem can be an exact local sharp minimum of the ? _p penalty function with p ∈ [0, 1] (resp. \(p\in [0, \frac {1}{2}]\)). Finally, we give some formulas for calculating the smallest penalty parameter for a penalty function to have a local sharp minimum.     

Scheduling jobs with release dates,equal processing times,and inclusive processing set restrictions

We consider the problem of scheduling a given set of n jobs with equal processing times on m parallel machines so as to minimize the makespan. Each job has a given release date and is compatible to only a subset of the machines. The machines are ordered and indexed in such a way that a higher-indexed machine can process all the jobs that a lower-indexed machine can process. We present a solution procedure to solve this problem in O(n²+mnlogn) time. We also extend our results to the tree-hierarchical processing sets case and the uniform machine case.     

Minimizing total tardiness in an unrelated parallel-machine scheduling problem

We consider a problem of scheduling n independent jobs on m unrelated parallel machines with the objective of minimizing total tardiness. Processing times of a job on different machines may be different on unrelated parallel-machine scheduling problems. We develop several dominance properties and lower bounds for the problem, and suggest a branch and bound algorithm using them. Results of computational experiments show that the suggested algorithm gives optimal solutions for problems with up to five machines and 20 jobs in a reasonable amount of CPU time.     

Asymptotically optimal approach to the approximate solution of several problems of covering a graph by nonadjacent cycles

We consider the m-Cycle Cover Problem of covering a complete undirected graph by m vertex-nonadjacent cycles of extremal total edge weight. The so-called TSP approach to the construction of an approximation algorithm for this problem with the use of a solution of the traveling salesman problem (TSP) is presented. Modifications of the algorithm for the Euclidean Max m-Cycle Cover Problem with deterministic instances (edge weights) in a multidimensional Euclidean space and the Random Min m-Cycle Cover Problem with random instances UNI(0,1) are analyzed. It is shown that both algorithms have time complexity O(n ³) and are asymptotically optimal for the number of covering cycles m = o(n) and \(m \leqslant \frac{{n^{1/3} }}{{\ln n}}\), respectively.     

An exact algorithm for the bi-objective timing problem

The timing problem in the bi-objective just-in-time single-machine job-shop scheduling problem (JiT-JSP) is the task to schedule N jobs whose order is fixed, with each job incurring a linear earliness penalty for finishing ahead of its due date and a linear tardiness penalty for finishing after its due date. The goal is to minimize the earliness and tardiness simultaneously. We propose an exact greedy algorithm that finds the entire Pareto front in \(O(N^2)\) time. This algorithm is asymptotically optimal.     

Minimizing the number of early jobs on a proportionate flowshop

A proportionate flowshop is a special case of the classical flowshop, where the job processing times are machine-independent. We study the problem of minimizing the number of early jobs in this machine setting. This objective function has hardly been investigated on a single machine, and never on a flowshop. We introduce an efficient iterative solution algorithm. In each iteration, a single job is moved to the first position (and is added to the set of early jobs), and the remaining jobs are rescheduled such that the maximum earliness is minimized. The algorithm guarantees an optimal solution in O(n³) time, where n is the number of jobs.