1. Introduction
In many real-world producing and scheduling problems, the due-window assignment is a growing concern, especially for the earliness and tardiness penalty problems (Tian [
1]). Under the common/slack due-window (CONDW/SLKDW), if a job is earlier (later) than the due-window, it will incur an early (delayed) penalty (Xu et al. [
2], Shabtay et al. [
3] and Zhang et al. [
4]). For any enterprise in production, it is necessary to specify the delivery time period of the product, which is the due window. Qian and Zhan [
5] and Mao et al. [
6] thought over the single-machine scheduling problems with a due-window and delivery time, where the delivery time is the past-sequence-dependent delivery time (denoted by
). Mao et al. [
7] considered the single-machine scheduling with deteriorating jobs and
. They proved that the makespan minimization can be solved in polynomial time. Ren et al. [
8] explored the single-machine scheduling with learning effects and exponential
. Moreover, minimizing the total cost is an important optimization objective in the problem of due-window assignment in single-machine scheduling. The total cost is a weighted sum function of the earliness and tardiness, the start of due window and the size of due window, where the weights depend only on their positions in the sequence (i.e., position-dependent weights, Lin [
9]). Wang et al. [
10] considered single-machine position-dependent weights scheduling problem with
and truncated sum-of-processing-times-based learning effect. Recently, under common and slack due-window assignments, Wang et al. [
11] examined single-machine problems with due-window assignment and positional-dependent weights. They demonstrated that some versions of common and slack due-window assignments are polynomially solvable. Nevertheless, in real-world production processes, job processing times need to be controlled by allocating additional resources, which is the scheduling with resource allocation (see Lu et al. [
12] and Wang et al. [
13]). Zhao et al. [
14] and Sun et al. [
15] considered the resource allocation scheduling with slack due-window assignment.
In addition to the due-window assignments, there are more factors to consider in actual scheduling problems, namely the cost of job rejection and/or learning effect. In the classical scheduling models, all jobs must be processed; nevertheless, in actual production, due to the requirements of processing capacity and profitability, vendors need to reject some jobs that take longer to process or are less profitable, thus incurring rejection costs. The learning effect is due to the accumulation of experience, which shortens the processing times of jobs (Lv and Wang [
16] and Zhang et al. [
17]). Li and Chen [
18] conducted single-machine scheduling problems with job rejection and a deteriorating maintenance activity. Mor [
19] considered single-machine scheduling problems with job rejection and deteriorating effects. Geng et al. [
20] studied proportionate flow shop scheduling with job rejection. Under common due-date assignment, they proved that some problems are polynomially solvable. Toksari and Atalay [
21] and Liu et al. [
22] considered single-machine scheduling problems with job rejection and learning effects. Atsmony and Mosheiov [
23] studied single-machine scheduling with job rejection. For the maximum earliness/tardiness cost minimization under an upper bound on the total permitted rejection cost, they proposed a pseudo-polynomial dynamic programming algorithm and a heuristic algorithm. The learning effect is specifically seen in the fact that workers repeat their work many times and their skills become more proficient (Sun et al. [
24], Zhao [
25] and Chen et al. [
26]). Li et al. [
27] considered the flow shop scheduling with truncated learning effects. For the makespan objective, they proposed the branch-and-bound and heuristic algorithms.
Currently, in the practical application of due-window assignment scheduling problems, there are multiple optimization objectives and, therefore, some limitations. Recently, Wang et al. [
28] considered the single-machine resource allocation scheduling with the CONDW assignment. Under the linear and convex resource allocations, they showed that four versions of general earliness–tardiness cost and resource consumption cost are polynomially solvable. The goal of this paper is to further extend the results of Wang et al. [
28] by combining delivery times and optional outsourcing (job rejection), i.e., under CONDW/SLKDW assignment; the method of outsourcing enables the operations manager to improve the overall performance; obviously, the outsourced jobs have outsourcing costs (see Fang et al. [
29] and Freud and Mosheiov [
30]). The main contributions of this paper are as follows: (i) single-machine scheduling with learning effects, delivery times and outsourcing costs is studied and simulated with data; (ii) for a given schedule, the optimal solution properties are given; (iii) for the weighed sum of scheduling cost, resource consumption cost and outsourcing measure, we prove that the problem can be solved in polynomial time. The paper is organized as follows:
Section 2 provide a description of the problem. In
Section 3, we prove that four problems are polynomially solvable.
Section 4 conducts the experiment. In
Section 5, we summarize the conclusions.
2. Problem Statement
In this paper, the mathematical notations used are listed in
Table 1. There are
n independent and non-preemptive jobs
are processed on a single machine, all jobs are available at time zero and the machine processes at most one job at the same time. First, we determine whether the job is processed or not, and classify the jobs into an acceptable set
and unacceptable (rejected) set
.
Under linear resource allocation, the actual processing time of job
in position
r can be expressed as
where
is the position-dependent learning index,
and
is the upper bound of
. Under convex resource allocation,
can be expressed as
where
is a constant. Let
be the due-window of job
, where
(resp.
) is the starting (resp. finishing) time, and
is the due-window size of job
. Under the common due-window (CONDW) assignment, it is assumed that
and
is the size of the common due-window. Under slack due-window (SLKDW) assignment, it is assumed that
and
is the size of the slack due-window. Let
be some job placed in the
jth position, as in Qian and Zhan [
5], the past-sequence-dependent delivery time (denoted by
) of job
is
where the delivery rate is
; then, the completion time of job
is
Let the number of early job
and tardy job
be
and
Let the earliness and tardiness of job
be
and
, the goal of this paper is to determine the optimal schedule
,
(
),
(
) (such
D can be obtained) and resource allocation
for the jobs in the acceptable set. This paper is to minimize the weighed sum of scheduling cost (
), resource consumption cost (
) and outsourcing measure (
), i.e.,
, where
,
,
,
and
are positional-dependent weights (Wang et al. [
31] and Qian et al. [
32]);
and
are the unit time weight for the due-window starting time
and size
D;
is the weight of resource consumption cost;
is the weight of unacceptable set of jobs (where
is the rejected cost of job
). If the above parameters are given (i.e.,
,
,
,
,
,
,
,
,
,
,
,
,
,
X are the assumptions made), the problems can be denoted by
and
Table 2 lists the relevant models studied.
5. Conclusions
This paper studied resource allocation single-machine scheduling problems with delivery times and due-window assignments under the premise that jobs have learning effects and a rejection cost. Under common and slack due-window assignments, we proved that the problems presented in this paper can be solved in
time, i.e., these problems are polynomially solvable. The managerial implications of our approach are as follows: the single-machine scheduling with delivery times, variable processing times, outsourcing cost and due-window allocation combine to affect the order in which jobs are processed and, thus, production decisions. We conducted computational experiments on randomly generated data. As can be seen in
Table 8, our approach is effective and helps to improve resource utilization, reduce resource consumption costs, reduce inventory backlogs and, thus, reduce overheads such as warehousing costs in practical applications.
However, the current algorithms have some limitations because there are many dynamic factors in scheduling in real production, and the existing algorithms may perform well under specific conditions, but may need to be adjusted differently under different production environments and scheduling goals. In future research, the scheduling problem presented in this paper can be further generalized to existing algorithms by taking the deterioration effects (i.e., deteriorating jobs) into account (see Sun et al. [
33], Lv et al. [
34], Miao et al. [
35] and Lu et al. [
36]), or applied to flow shops (see Rossit et al. [
37] and Panwalkar and Koulamas [
38]).