Single-machine scheduling with learning effect and resource-dependent processing times in the serial-batching production

In this paper, we study a single machine scheduling problem by simultaneously considering the processing method of serial-batching, learning effect, resource-dependent processing times, and setup operations. We consider minimizing the makespan as the objective of the studied problem under the constraint that the total resource consumption does not exceed a given limit. For the special case where the resource allocation is given, we first propose the structural properties for job batching policies and batching sequencing, and an optimal batching policy is derived based on these properties. Then, we develop a novel hybrid GSA–TS algorithm which combines the Gravitational Search Algorithm (GSA) and the Tabu Search (TS) algorithm to solve the general case. Computational experiments with different scales show the effectiveness and efficiency of the proposed algorithm.     

Single-machine scheduling with a general sum-of-actual-processing-times-based and job-position-based learning effect

In this paper, we bring into the scheduling field a general learning effect model where the actual processing time of a job is not only a general function of the total actual processing times of the jobs already processed, but also a general function of the job’s scheduled position. We show that the makespan minimization problem and the sum of the kth power of completion times minimization problem can be solved in polynomial time, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.     

Single-machine scheduling problems with an aging effect

This paper considers single machine scheduling problems where the processing time of a job increases as a function of its position in the sequence. In this model, the later a given job is scheduled in the sequence, the longer its processing time. It is shown that the optimal schedule may be very different from that of the classical version of the problem. We introduce polynomial solutions for the makespan minimization problem, the sum of completion times minimization problem and the sum of earliness penalties minimization problem. For two resource constrained problems, based on the analysis of the problems, the optimal resource allocation methods are presented, respectively.     

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.     

Some single-machine scheduling with sum-of-processing-time-based and job-position-based processing times

The single-machine scheduling problems with position and sum-of-processing-time based processing times are considered. The actual processing time of a job is defined by function of its scheduled position and total normal processing time of jobs in front of it in the sequence. We provide optimal solutions in polynomial time for some special cases of the makespan minimization and the total completion time minimization. We also show that an optimal schedule to be a V-shaped schedule in terms of the normal processing times of jobs for the total completion time minimization problem and the makespan minimization problem.     

Single machine scheduling with general time-dependent deterioration, position-dependent learning and past-sequence-dependent setup times

The paper deals with single machine scheduling problems with setup time considerations where the actual processing time of a job is not only a non-decreasing function of the total normal processing times of the jobs already processed, but also a non-increasing function of the job’s position in the sequence. The setup times are proportional to the length of the already processed jobs, i.e., the setup times are past-sequence-dependent (p-s-d). We consider the following objective functions: the makespan, the total completion time, the sum of the δth (δ ≥ 0) power of job completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the δ th (δ ≥ 0) power of job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.     

Single machine scheduling with a general exponential learning effect

In this paper we consider the scheduling problem with a general exponential learning effect and past-sequence-dependent (p-s-d) setup times. By the general exponential learning effect, we mean that the processing time of a job is defined by an exponent function of the total weighted normal processing time of the already processed jobs and its position in a sequence, where the weight is a position-dependent weight. The setup times are proportional to the length of the already processed jobs. We consider the following objective functions: the makespan, the total completion time, the sum of the δ ? 0th power of completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the quadratic job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.     

Note on “A note on single-machine group scheduling problems with position-based learning effect”

In a recent paper, Lee and Wu [W.-C. Lee, C.-C. Wu, A note on single-machine group scheduling problems with position-based learning effect, Appl. Math. Model. 33 (2009) 2159–2163] proposed a new group scheduling learning model where the learning effect not only depends on the job position, but also depends on the group position. They investigate the makespan and the total completion time minimization problems on a single-machine. As for the total completion time minimization problem, they assumed that the numbers of jobs in each group are the same and the group normal setup and the job normal processing times are agreeable. Under the assumption conditions, they showed that the total completion time minimization problem can be optimally solved in polynomial time solution. However, the assumption conditions for the total completion time minimization problem do not reflect actual practice in many manufacturing processes. Hence, in this note, we propose other agreeable conditions and show that the total completion time minimization problem remains polynomially solvable under the agreeable conditions.     

Single machine scheduling problem with controllable processing times and resource dependent release times

We consider two single machine scheduling problems with resource dependent release times and processing times, in which the release times and processing times are linearly decreasing functions of the amount of resources consumed. The objective is to minimize the total cost of makespan and resource consumption function that is composed of release time reduction and processing time reduction. In the first problem, the cost of reducing a unit release time for each job is common. We show that the problem can be solved in polynomial time. The second problem assumes different reduction costs of job release times. We show that the problem can be reduced polynomially from the partition problem and thus, is NP-complete.     

Single-machine scheduling problems with past-sequence-dependent delivery times and position-dependent processing times

This paper considers single-machine scheduling problems with job delivery times where the actual job processing time of a job is defined by a function dependent on its position in a schedule. We assume that the job delivery time is proportional to the job waiting time. We investigate the minimization problems of the sum of earliness, tardiness, and due-window-related cost, the total absolute differences in completion times, and the total absolute differences in waiting times on a single-machine setting. The polynomial time algorithms are proposed to optimally solve the above objective functions. We also investigate some special cases of the problem under study and show that they can be optimally solved by lower order algorithms.     

On a borderline between the NP-hard and polynomial-time solvable cases of the flow shop with job-dependent storage requirements

The paper is concerned with the two-machine flow shop, where each job requires an additional resource (referred to as storage space) from the start of its first operation till the end of its second operation. The storage requirement of a job is determined by the processing time of its first operation. At any point in time, the total consumption of this additional resource cannot exceed a given limit (referred to as the storage capacity). The goal is to minimise the makespan, i.e. to minimise the time needed for the completion of all jobs. This problem is NP-hard in the strong sense. The paper analyses how the parameter - a lower bound on the storage capacity specified in terms of the processing times, affects the computational complexity.

     

Single-machine scheduling problems with an actual time-dependent deterioration

In this paper, we consider the single machine scheduling problems with an actual time-dependent deterioration effect. By the actual time-dependent deterioration effect, we mean that the processing time of a job is defined by increasing function of total actual processing time of jobs in front of it in the sequence. We show that even with the introduction of an actual time-dependent deterioration to job processing times, makespan minimization problem, total completion time minimization problem, the total lateness, and the sum of the quadratic job completion times minimization problem remain polynomially solvable, respectively. We also show that the total weighted completion time minimization problem, the discounted total weighted completion time minimization problem, the maximum lateness minimization problem, and the total tardiness minimization problem can be solved in polynomial time under certain conditions.     

A note on single-machine scheduling with decreasing time-dependent job processing times

In this note we consider some single-machine scheduling problems with decreasing time-dependent job processing times. Decreasing time-dependent job processing times means that its processing time is a non-increasing function of its execution start time. We present polynomial solutions for the sum of squared completion times minimization problem, and the sum of earliness penalties minimization problem subject to no tardy jobs, respectively. We also study two resource constrained scheduling problems under the same decreasing time-dependent job processing times model and present algorithms to find their optimal solutions.     

同时具有学习效应和退化效应的单机排序问题

本文给出了一种同时具有一般化学习效应和退化效应的单机排序模型。在此模型中,工件的实际加工时间既与工件所在位置又与其开工时间有关,且工件在加工之后具有一个配送时间。其中学习效应是工件所在位置的函数,退化效应是工件开工时间的函数。证明了极小化最大完工时间和极小化总完工时间问题是多项式可解的,在满足一定的条件下,极小化加权总完工时间和极小化最大延误问题也是多项式可解的。推广了一些已有文献中的结论。     

两类加工时间是一般函数的单机排序问题

考虑了两类有一般加工时间函数的排序问题. 工件的加工时间分别为基本加工时间与开工时间函数、位置函数的和. 对加工时间依赖开工时间的模型,证明了一定条件下极小化最大完工时间和极小化总完工时间是多项式可解的. 对加工时间依赖开工位置的模型,给出极小化最大完工时间和极小化总完工时间的最优序,同时证明了极小化加权总完工时间的一个最优排序性质并给出一个贪婪算法.     

Single machine scheduling with exponential sum-of-logarithm-processing-times based learning effect

In this paper we consider the single machine scheduling problems with exponential sum-of-logarithm-processing-times based learning effect. By the exponential sum-of-logarithm-processing-times based learning effect, we mean that the processing time of a job is defined by an exponent function of the sum of the logarithm of the processing times of the jobs already processed. We consider the following objective functions: the makespan, the total completion time, the sum of the quadratic job completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the quadratic job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.     

Single machine past-sequence-dependent setup times scheduling with general position-dependent and time-dependent learning effects

In this paper we consider the single machine past-sequence-dependent (p-s-d) setup times scheduling problems with general position-dependent and time-dependent learning effects. By the general position-dependent and time-dependent learning effects, we mean that the actual processing time of a job is not only a function of the total normal processing times of the jobs already processed, but also a function of the job’s scheduled position. The setup times are proportional to the length of the already processed jobs. We consider the following objective functions: the makespan, the total completion time, the sum of the θth (θ ? 0) power of job completion times, the total lateness, the total weighted completion time, the maximum lateness, the maximum tardiness and the number of tardy jobs. We show that the problems of makespan, the total completion time, the sum of the θth (θ ? 0) power of job completion times and the total lateness can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem, the maximum lateness minimization problem, maximum tardiness minimization problem and the number of tardy jobs minimization problem can be solved in polynomial time under certain conditions.     

Multi-criteria approximation schemes for the resource constrained shortest path problem

In the resource constrained shortest path problem we are given a directed graph along with a source node and a destination node, and each arc has a cost and a vector of weights specifying its requirements from a set of resources with finite budget limits. A minimum cost source-destination path is sought such that the total consumption of the arcs from each resource does not exceed its budget limit. In the case of constant number of weight functions we give a fully polynomial time multi-criteria approximation scheme for the problem which returns a source-destination path of cost at most the optimum, however, the path may slightly violate the budget limits. On the negative side, we show that there does not exist a polynomial time multi-criteria approximation scheme for the problem if the number of weight functions is not a constant. The latter result applies to a broad class of problems as well, including the multi-dimensional knapsack, the multi-budgeted spanning tree, the multi-budgeted matroid basis and the multi-budgeted bipartite perfect matching problems.     

Single-machine scheduling simultaneous with position-based and sum-of-processing-times-based learning considerations under group technology assumption

This paper considers the problems of scheduling with the effect of learning on a single-machine under group technology assumption. We propose a new learning model where the job actual processing time is linear combinations of the scheduled position of the job and the sum of the normal processing time of jobs already processed. We show that the makespan minimization problem is polynomially solvable. We also prove that the total completion time minimization problem with the group availability assumption remains polynomially solvable under agreeable conditions.     

Group Scheduling with Controllable Setup and Processing Times: Minimizing Total Weighted Completion Time

The following single machine scheduling problem is studied. A partition of a set of n jobs into g groups on the basis of group technology is given. The machine processes jobs of the same group contiguously, with a sequence independent setup time preceding the processing of each group. The setup times and the job processing times are controllable through the allocation of a continuously divisible or discrete resource to them. Each job uses the same amount of the resource. Each setup also uses the same amount of resource, which may be different from that for the jobs. Polynomial-time algorithms are constructed for variants of the problem of finding an optimal job sequence and resource values so as to minimize the total weighted job completion time, subject to given restrictions on resource consumption. The algorithms are based on a polynomial enumeration of the candidates for an optimal job sequence and solving the problem with a fixed job sequence by linear programming. This research was supported in part by The Hong Kong Polytechnic University under grant number G-T246 and the Research Grants Council of Hong Kong under grant number PolyU 5191/01E. In addition, the research of M.Y. Kovalyov was supported by INTAS under grant number 00-217.