1. Introduction
In this article, we consider an online bi-criteria scheduling problem to minimize the maximum machine cost subject to the makespan achieving its minimum. Online means that jobs’ arrival is over time. It means, until when a job arrives, all information about it, including its arrival time, processing time and processing cost, is not known by us. For a minimization problem that is relevant to a single objective function, the competitive ratio of an online algorithm
A is defined to be
Here, is the objective value in algorithm A for any input instance I, OPT is the optimal objective value in the offline circumstance, respectively. We say algorithm A is the best possible if there doesn’t exist any algorithm such that .
Parallel-batch was first studied by Uzsoy et al. [
1,
2]. There are two classes of parallel-batch models that have been widely considered in the literature, the unbounded version
and the bounded version
, where
b is the batch capacity. That is, at most
b jobs can be processed simultaneously in one batch. The processing time of one batch is defined as the longest job in it. Since in this paper we consider that the jobs are with identical processing time, the processing time of the batches is 1.
In traditional scheduling theory, most problems are concerned with the minimization of one certain function. There have been many achievements such as, for minimizing maximum completion time when jobs have the same processing times, Zhang et al. [
3] provided two best possible online algorithmss with
and
-competitive ratio for the unbounded model
and bounded model
, respectively, and
satisfies
,
. When jobs have diverse processing times, Tian et al. [
4] and Liu et al. [
5] independently gave two best possible online algorithms with competitive ratio of
, and
is the positive solution of the equation
. Fang et al. [
6] presented a best possible online algorithm with a competitive ratio of
for a set of processing time scheduling problems, where
. For minimizing a maximum weighted completion time problem, Li et al. [
7] established a best possible online algorithm with a competitive ratio of
. For minimizing total weighted completion time problem, Cao et al. [
8] gave a best possible online algorithm with a competitive ratio of
, where
is the positive solution of
. Some reviews for parallel-batch scheduling research can be found in [
9,
10,
11,
12,
13,
14].
Today, with the rapid development of science and technology, minimization of one certain function doesn’t satisfy the needs of things. In addition, jobs’ objective functions may have certain kinds of aspects to minimize. In recent years, there have been some results about minimizing bi-criteria objective functions such as Ma et al. [
15], who considered an online trade-off scheduling problem that minimize makespan and total weighted completion time on a single machine, presenting a nondominated
competitive online algorithm for each
with
. Liu et al. [
16] considered the single machine online trade-off scheduling problem, which minimizes the makespan and maximum lateness. They established a nondominated
competitive online algorithm with
. Here, a
-competitive online algorithm is called nondominated if there is no other
-competitive online algorithm
such that
and either
or
. In addition, Lee et al. [
17] considered two bi-criteria scheduling problems: one is minimizing the maximum machine cost subject to the total completion time achieving its minimum, another is minimizing the total machine cost subject to the makespan achieves its minimum. As these two problems are strongly NP-hard, they proposed fast heuristics and found their worst-case performance bounds.
Another class of scheduling problems with bi-criteria is to minimize a secondary objective function
subject to a primary objective function
being at its minimum, and the objective is denoted by
. In practical production, the producer wants to reduce the cost of the machine as soon as it is finished. Given
m parallel batch machines
,
, and
n jobs
,
. Every machine has a fixed cost
, job
has cost
, and
. When job
is processed on machine
, this will result in different costs
,
,
. Suppose that
if job
j is processed on machine
i, otherwise
. Thus, the total machine cost, is named
, and
=
, and the maximum machine cost is named
, and
=
. In Lee et al. [
17], they studied the offline bi-criteria scheduling problems, for which the objective functions are minimizing
subject to the constraint that
is minimized and minimizing
subject to the constraint that
is minimized, where
is the completion time of job
,
is total completion time of jobs, and
is the maximum completion time of jobs. They considered three kinds of cost functions:
,
, and
. In our article, we consider online algorithms to minimize the maximum machine cost subject to the makespan achieving its minimum, and the objective function is denoted as
. Here, we assume that all machines have the same fixed cost
a, and we consider two kinds of costs:
and
,
,
. Since the jobs are processed in batches in our model, the cost of a batch processed on some machine is defined as the maximum cost of the jobs in it. Then, the cost of one machine is the total cost of the batches on it. This problem can be written in the three-field notation as
,
,
,
, when
or
,
,
.
For the online scheduling problem to minimize a primary objective function and a secondary objective function , we say that an online algorithm A is -competitive if it is -competitive when minimizing and -competitive when minimizing . In the case that is the competitive ratio of A for minimizing and is the competitive ratio of A for minimizing , we also say that the online algorithm A has a competitive ratio of . Suppose that the best possible competitive ratio is when minimizing . We say that the online algorithm A is the best possible, if and there is no other online algorithm such that and .
This paper is organized as four sections as follows. In
Section 2, the parameters and notations are introduced. In
Section 3, the lower bounds of the competitive ratio are presented. In
Section 4, two best possible online algorithms with a competitive ratio of
are showed, where
is the positive root of the equation
.
The objective considered in this paper is to minimize the maximum machine cost subject to the makespan being at its minimum. In addition, the algorithms studied in this paper are extensions of the results about makespan in the literature.
3. The Lower Bound
Theorem 1. For problem , when or , , , there exists no online algorithm with a competitive ratio less than .
Proof. Supposing that the fixed cost of each machine is
a, the cost of each job is 1, which means
. Let
be the set of the best possible solutions of the objective function
. Then, for
, we prove that there exists no online algorithm that satisfies
, subjected to the constraint
We use adversary strategy to prove this conclusion. Let
be an arbitrary online algorithm, and
is an arbitrarily small positive number. Suppose the first job
arrives at 0 and starts at
. From
, we can know that
. Otherwise, we have
a contradiction. Job
arrives at
and starts at
,
. We claim that
. Otherwise,
Then, , a contradiction.
Hence, n jobs are processed as n batches on m machines.
When , after n jobs are processed, there must be not less than jobs on one machine. Because the cost of each job is 1, the maximum machine cost is . In , all jobs can form one batch starting at the last time when the job arrives, so . Then, we get .
When , similarly after n jobs are finished, there must be one machine that does not have less than jobs. Since each job’s cost is 1, the maximum machine cost is . In , all jobs can form one batch starting at the time which the last job arrives, so . Then, we get .
Therefore, for problem , when or , , , and there exists no online algorithm in which the competitive ratio is less than .
4. Best Possible Online Algorithms
Here, there are two online algorithms for this problem.
Algorithm
At current time t, if some machine is idle, , when ; then, start the jobs in as a batch on the idle machine that has the minimum machine cost at the moment. Otherwise, do nothing but wait.
Algorithm
At current time t, if some machine is idle, , when ; then, start the jobs in as a batch on the idle machine that has the minimum number of batches at the moment. Otherwise, do nothing but wait.
Following the notation in Zhang et al. [
3], we also call batches that start at
regular batches. From Lemma 1 of Zhang et al. [
3], we have
Lemma 2. All batches generated by algorithm and are regular batches.
Lemma 3. When , Then, ; When , then .
Proof. The offline optimal objective case of the maximum machine cost is: all jobs can form one batch starting at the last arrival time on an arbitrary machine. Thus, when , the maximum machine cost is ; when , the maximum machine cost is . □
Theorem 2. For problem , when or , , , algorithm is a best possible online algorithm with a competitive ratio of .
Proof. When
, suppose that the schedule generated by algorithm
is
. From Lemmas 1 and 2, we can know that
In the following, we prove
. Suppose, in
, that the machine
has the maximum machine cost, Then,
We distinguish the following cases:
Case 1 The number of batches on machine
is no more than
. Thus,
In addition, by (2), we have
Case 2 The number of batches on machine is more than . Thus, there must be one machine that has less than batches. Suppose machine is the machine that has less than batches; let be the last batch to process on machine .
Firstly, if machine
is idle directly before
, let the total cost of batches that start before
on
be
. From algorithm
, because the number of batches on machine
is no more than
, so
Moreover, from algorithm
, the total cost of batches that start before
on machine
is not greater than the total cost of batches start before
on machine
, that is
then from
, we have
In addition, by
, we have
Secondly, if machine is busy directly before , let be the last batch that is on machine such that machine is idle directly before . Supposing that there are k batches between and , we denote them as .
Claim must exist and is not the first batch on .
Otherwise, does not exist or it is the first batch on . This means that machine is busy when , ,⋯ start on machine . Since the number of batches on machine is more than , the number of batches on machine must be not less than , contradicting the assumption that the number of batches on machine is less than . Thus, the claim holds.
Let the total cost of batches start before
on machine
be
. Then, from the definition of
and
, the number of batches that start before
on machine
is no more than
. Thus,
Furthermore, by algorithm
, the total cost of batches starting before
on machine
is not greater than the total cost of batches starting before
on machine
, then
Thus, from
, we know that
In addition, by
, we have
We know that . When , similar to the above discussion, the conclusion also holds.
When
, suppose the schedule produced by algorithm
is
. From Lemmas 1 and 2, we obtain that
. In the following, we want to prove that
. Supposing that machine
has maximum machine cost in
, Then,
We distinguish the following cases:
Case 3 The number of batches is no more than
on machine
. Then,
From (5) and (6), we have
Case 4 The number of batches is more than on machine . Thus, there must be one machine that has fewer than batches. Supposing that machine is the machine for which the number of batches on it is less than , let be the last batch to process on machine .
If machine
is idle directly before
, we denote the total cost of batches start before
on machine
as
, by algorithm
because the number of batches on machine
is no more than
, hence
Furthermore, by algorithm
, the total cost of batches starting before
on machine
is not greater than the total cost of batches starting before
on machine
; then,
In addition, by
, we have
If machine
is busy directly before
, let
be the last batch on machine
such that machine
is idle directly before
. Similarly, suppose there are
k batches between
and
, we denote them as
. From the discussion of case 2 in
situation, such
must exist and is not the first batch on
. Denoting the total cost of batches starting before
on machine
is
; thus, by the definition of
and
, the number of batches starting before
on machine
is no more than
. Then,
Moreover, by algorithm
, the total cost of batches starting before
on machine
is not greater than the total cost of batches starting before
on machine
, so
Therefore, from
, we get
In addition, by
, we have
We know that . When , similar to the above discussion, the conclusion also holds.
Overall, for problem , when or , , , algorithm is a -competitive online algorithm.
Combining Theorem 1, we obtain that algorithm is a best possible online algorithm. □
Lemma 4. In algorithm , there are at most batches on each machine.
Proof. When , there is at most one batch on each machine, so the conclusion holds naturally. When , suppose, after the kth batch has been processed, that there are at most batches on each machine, and . In the following, we have an induction on k, to prove that, after the th batch has been processed, there are at most batches on each machine. The batches are denoted by , such that .
Case 1, and
,
, where
are integers—as after the
kth batch has been processed, there are
machines in which their batch numbers are
, and other
machines in which their batch numbers are
. Let
be the release time of
kth batch, and
be the latest release time of jobs in
th batch. Then, we get
; otherwise, jobs in the
th batch will process with jobs in the
kth batch, a contradiction. Furthermore, by algorithm
, we can get that the starting time of the
th batch is
. In addition, because
We use to represent the starting time of the th batch. From (9), it shows that, when the th batch starts, the th batch has been completed. Then, l batches that start before the th batch also have been completed. We define these l batches as . Then, when the th batch starts, batches have been completed. In addition, because of , when the th batch starts, there is at least one machine that is idle. In addition, it can be known, by algorithm , that the number of batches on this idle machine is . Hence, after the th batch is completed, the number of batches on this idle machine is . Moreover, because when , there are machines whose batch numbers are , and other machines that have batches. This means that there are at most batches on each machine. The result follows.
Case 2 and , where q is an integer. After the th batch is processed, one machine has batches, and the number of batches on other machines is still . Furthermore, when . Thus, every machine has at most batches. The results follow. □
Theorem 3. For problem , when or , , , algorithm is the best possible online algorithm with a competitive ratio of .
Proof. When
, suppose the schedule generated by algorithm
is
. From Lemma 1 and Lemma 2, we have
In the following, we want to prove .
Suppose machine
has the maximum machine cost, Then,
From Lemma 4, we know that the number of batches on machine
is no more than
; then, from
, we get
In addition, Lemma 3 shows that
When , suppose the schedule produced by algorithm is . From Lemmas 1 and 2, we can get —the following to prove .
Assume that machine
is the machine with the maximum cost, Then,
From Lemma 4, we know that the number of batches on machine
is no more than
; then, from
, we get
To sum up, for problem , when or , , , algorithm is an online algorithm with a competitive ratio of .
Combining Theorem 1, it implies that algorithm is a best possible online algorithm. □