Online Batch Scheduling of Simple Linear Deteriorating Jobs with Incompatible Families

: We considered the online scheduling problem of simple linear deteriorating job families on m parallel batch machines to minimize the makespan, where the batch capacity is unbounded. In this paper, simple linear deteriorating jobs mean that the actual processing time p j of job J j is assumed to be a linear function of its starting time s j , i.e., p j = α j s j , where α j > 0 is the deterioration rate. Job families mean that one job must belong to some job family, and jobs of different families cannot be processed in the same batch. When m = 1, we provide the best possible online algorithm with the competitive ratio of ( 1 + α max ) f , where f is the number of job families and α max is the maximum deterioration rate of all jobs. When m ≥ 1 and m = f , we provide the best possible online algorithm with the competitive ratio of 1 + α max .


Background
In this paper, all jobs arrive over time, i.e., each job has an arrival time. Before the jobs arrive, we do not know any information, including arrival time, processing time, deterioration rate, etc. Due to the unknown information of the jobs, the online algorithm is not guaranteed to be optimal. Borodin and El-Yaniv [1] used the competitive ratio to measure the quality of an online algorithm. For a minimization scheduling problem, we define the competitive ratio of the online algorithm A as: ρ = sup{A(I )/OPT(I) : I is any instance such that OPT(I ) > 0 }.
where I is any job instance and A(I ) and OPT(I ) are the objective values obtained from the algorithm A and an optimal offline scheduling OPT, respectively. In this study, the objective was to minimize the makespan. An online algorithm A is called the best possible if no other online algorithms A * produce a smaller competitive ratio.
Parallel-batch means that one batch processing machine can process b jobs simultaneously as a batch. The processing time of a batch is the maximum processing time of all jobs in this batch. All jobs in a batch have the same starting time, processing time, and completion time. According to the number of jobs contained in a batch, Brucker et al. [2] divided the model into two cases: the unbounded model (b = ∞) and the bounded model (b < ∞).
Job families mean that one job must belong to some job family, and jobs of different families cannot be processed in the same batch. Online scheduling problems on parallel batch machines with incompatible job families have been studied extensively. Fu et al. [3] studied the online algorithm on a single machine to minimize the makespan. Li et al. [4] examined the online scheduling of incompatible unit-length job families with lookahead on a single machine. Tian et al. [5] analyzed the problem on m parallel machines. However, research is lacking on the parallel-batch online scheduling with incompatible deteriorating job families.
Traditional scheduling problems assume that the processing time of a job is fixed. However, in real life, one job will take longer when it has a later starting time. For example, in steel production and financial management [6,7], the processing time is longer when it starts later. In the steel-making process, strict requirements are placed on temperature. If the waiting time is too long, the temperature of molten steel will drop. So, it will take time to heat up again before further processing. Other examples are provided in cleaning and fire fighting. The scheduling problem of deteriorating jobs was first introduced by Browne and Yechiali [8] and Gupta and Gupta [9], independently. Both considered minimizing makespan on a single machine. Since then, this topic has attracted considerable attentions. Gawiejnowicz and Kononov [10] considered the general properties of scheduling with fixed job processing time and scheduling with job processing time as proportional linear functions of the job starting time. The relevant research includes [11][12][13][14][15][16][17][18], among many others. Recently, some works have been published about online algorithms for linear deteriorating jobs [19][20][21][22][23].
To minimize the makespan of the online scheduling problem on m parallel machines with linear deteriorating jobs, Cheng et al. [19] constructed an algorithm and proved that the bound of the competitive ratio of the algorithm is tight, where m = 2 and the largest deterioration rate of jobs is known in advance. Yu et al. [22] proved that no deterministic online algorithm is better than (1 + α max )-competitive when m = 2, where α max is the maximum deterioration rate of all jobs.

Research Problem
Our contribution is to extend the online scheduling problem on m parallel batch machines with simple linear deteriorating job families to minimize the makespan. Here, batch capacity is unbounded, i.e., b = ∞. We use f-family to denote there are f job families. We constructed the best possible online algorithm with the competitive ratio of (1 + α max ) f when m = 1, where f is the number of job families and α max is the maximum deterioration rate of all jobs. When m ≥ 1 and m = f , we created the best possible online algorithm with the competitive ratio of 1 + α max .
We examined the online batch scheduling of simple linear deteriorating job families. The actual processing time p j of job J j is assumed to be a linear function of its starting time s j , i.e., p j = α j s j , where α j > 0 is the deterioration rate, which is unknown until it arrives. The objective was to minimize makespan. Assume that the arrival time of all jobs is greater than or equal to t 0 > 0; otherwise, jobs arriving at time 0 can be completed at time 0. We used the three-field notation α|β|γ [24] to represent one scheduling problem. This paper is organized as follows. In Section 2, we consider the problem 1|online,r j ,p-batch, b = ∞,f-family, p j = α j t|C max , where f is the number of job families. We prove the lower bound and provide the best possible online algorithm with the competitive ratio of (1 + α max ) f . In Section 3, we consider the problem Pm|online,r j ,p-batch, b = ∞,m-family, p j = α j t|C max , where m is the number of machines. We prove the lower bound and provide the best possible online algorithm with the competitive ratio of 1 + α max , where α max is the maximum deterioration rate of all jobs.
Throughout this paper, we use σ and π to denote the schedules obtained from an online algorithm and an optimal offline schedule, respectively. Let C max (σ) and C max (π) be the objective values of σ and π, respectively, and α max be the maximum deterioration rate of all jobs. Let be an arbitrary small positive number.

Single Batch Machine (m = 1)
In this section, we consider the online scheduling on an unbounded batch machine and the jobs belong to f incompatible deteriorating job families. The number of job families, f , is known in advance. We prove the lower bound and provide the best possible online algorithm with the competitive ratio of (1 + α max ) f . Theorem 1. For problem 1|online,r j ,p-batch, b = ∞,f-family, p j = α j t|C max , the competitive ratio of any online algorithm is not less than (1 + α max ) f .

Proof.
Let H be any online algorithm and I be a job instance provided by the adversary. In instance I, all the jobs have the same deterioration rate of α.
At time t 0 , f jobs from the f different job families arrive by the adversary. If a job is scheduled by H to process at time t and t is in the time interval [t 0 , (1 + α) f t 0 ), then at time t + , the adversary releases a copy of this job, which belongs to the same job family. Let s be the starting time of the first job whose completion time is at least ( (1) In this case, there are still f jobs from f distinct job families that are not processed at time We assume that the jobs processed in the time interval [t 0 , (1 + α)s) belong to k distinct job families, Let s i be the last starting time of the jobs in F i that start before or at time s for 1 ≤ i ≤ k, and satisfy s 1 < s 2 < · · · < s k . Clearly, s k = s. From the construction of instance I, we know that the last arrival time of the jobs in Construct a schedule π below: the jobs in F i (1 ≤ i ≤ f ) form a batch starting at time s i , where: We can see that π is feasible, and the maximum completion time of the jobs in π is the completion time of the jobs in F k . So, Since s k = s, we have C max (π ) = max{(1 + α)(s + ), (1 + α) f t 0 }. and C max (π) ≤ C max (π ). If C max (π ) = (1 + α)(s + ), then by Equation (2) we know that: (1) and (2), we know that: According to the constructing of I, s k is the last starting time of the jobs in time interval [t 0 , (1 + α) f t 0 ), and s k + is the last arrival time of all jobs.
Since s is the starting time of the first job whose completion time is at least (1 + α) f t 0 , we obtain By the definition of s, f jobs from f distinct job families have not been processed at time s. So, Thus, The result follows.
Before introducing the online algorithm, given a batch B, we define some notations in the following: The online algorithm, called A 1 (Algorithm 1), can be stated as follows. Without causing confusion, assume that B i (t) = B i in the following.
Step 2: of the next job. Go to Step 2.
Step 5: If new jobs arrive after t, then reset t as the arrival time of the first new job. Go to Step 1. Output: Job schedule σ.

Example 1.
To make the algorithm more intuitive, we present an instance I 1 in Table 1, where F 1 and F 2 are two different families. As shown in Figure 1, σ is the schedule generated by A 1 and π is an optimal offline schedule for I 1 , where We have C max (σ) = 81 and C max (π) = 9.
The following two lemmas are the competition ratio analyses of Algorithm 1.

Lemma 1.
Suppose the machine has an idle time immediately before s(B) in σ. Then,

Proof.
Since an idle time occurs immediately before s(B) in σ, from Algorithm 1, we have: s(B) = max{r l , (1 + α(B 1 )) k t 0 }. If s(B) = r l , then for each B i ∈ B with 1 ≤ i ≤ k, J(B i ) arrives at time r l . From Equation (4),

Proof.
Since the machine has no idle time immediately before s(B) in σ, s(B) is the completion time of some batch, say B * , in σ. We have s(B) = (1 + α(B * ))s(B * ). From the definition of s, we know s(B * ) < r l .
We suppose that B is divided into two sets, B 1 and B 2 , such that: , then from Equation (4) we have: According to the definition of B 2 and Algorithm 1, each batch in set B 2 and B * belongs to the different family. Then, at most f − 1 batches exist in B 2 . Hence, By Lemmas 1 and 2, and Theorem 1, we can reach the final conclusion.

Theorem 2.
For problem 1|online,r j ,p-batch, b = ∞,f-family, p j = α j t|C max , Algorithm 1 has a competitive ratio of (1 + α max ) f and is the best possible.

Parallel Batch Machines (m ≥ 1)
In this section, we consider the online scheduling on m parallel batch machines and the jobs belong to m incompatible deteriorating job families. We prove the lower bound and construct the best possible online algorithm with a competitive ratio of 1 + α max . Theorem 3. For problem Pm|online,r j ,p-batch, b = ∞,m-family, p j = α j t|C max , no online algorithm exists with a competitive ratio less than 1 + α max .
Proof. Let H be any online algorithm and I be a job instance provided by the adversary. In the instance I, all the jobs have a deterioration rate of α.
At time t 0 , m jobs J 1 , J 2 , · · · , J m from different families arrive. Suppose that job J j starts processing at time s j in σ, j = 1, 2, · · · , m.
Before providing the online algorithm, we define some notations used in the following: U(t): the set of the unprocessed jobs at time t.
Step 3: If m(t) = m, and t < (1 + α max (t))t 0 , then reset t = t * , such that t * is either the arrival time of the next job or (1 + α max (t))t 0 . Go to Step 1.
Step 6: If new jobs arrive after t, then reset t as the arrival time of the first new job. Go to Step 1. Output: Job schedule σ.

Example 2.
To make the algorithm more intuitive, we present an instance I 2 in Table 2. Figure 2 depicts the schedule generated by Algorithm 2 and Figure 3 is an optimal offline schedule for I 2 .
Lemma 3. Suppose that only one job J i exists in batch B i of σ, i = 1, 2, · · · , n, then the value of C max (σ)/C max (π) does not decrease.
Proof. From Algorithm 2, the start time of batch B i is only related to the maximum deterioration rate of the jobs in this batch and the maximum deterioration rate of all jobs that have arrived. So, the value of C max (σ) does not change when we assume each batch B i has only one job J i . The reduction in the number of jobs may decrease the value of C max (π), so the value of C max (σ)/C max (π) does not decrease.
In the following, we assume that only one job J i exists in batch B i of σ, i = 1, 2, · · · , n. Per Lemma 3, this does not influence the competition ratio analysis of Algorithm 2. Lemma 4. α l = α 1 (s l ).
In this case, at time s l , some batches must have a start time less than s l being processed. Let B a be the last such batch to start, then s a < s l ≤ (1 + α a )s a . Hence, Suppose that r l ≤ s a . Since s l > s a , the batch with a larger deterioration rate has higher priority in σ, then α l ≤ α a ≤ α 1 (s a ) per Algorithm 2. At time s a , J l does not start processing. This indicates that there is no machine that can process J l at time s a , i.e., m(s a ) < f (s a ) ≤ m. From Algorithm 2, we know that s a ≥ (1 + α max (s a ))(1 + α 1 (s a ))t 0 . By Lemma 4, we have α l = α 1 (s l ). Hence, By the definition of B a , we have α max (s a ) = α max (s l ), so This contradicts s a < s l . Hence r l > s a . Thus, C max (π) ≥ (1 + α l )r l > (1 + α l )s a . From Equation (5), we have In the following, we discuss the case where B l is not a regular batch. This implies that no machine is idle immediately before time s l , where s l > max{(1 + α max (s l ))(1 + α 1 (s l )t 0 , r l }. Renumber the m last batches starting on the m machines before time s l to B l,1 , B l,2 , · · · , B l,m , such that s l,1 ≤ s l,2 ≤ · · · ≤ s l,m . By Lemma 4, we have α l = α 1 (s l ). So, If s l,1 = s l,2 = · · · = s l,k = · · · = s l,m < s l , then J l,1 , J l,2 , · · · , J l,m belong to m different job families and one of them belongs to the same family with J l . Then r l > s l,1 and C max (π) ≥ (1 + α l )r l > (1 + α l )s l,1 . From Equation (6), we have: In the following, we suppose that s l,i < s l,i+1 for some i ∈ {1, 2, · · · , m − 1}. Let k be the index that satisfies s l,1 = s l,2 = · · · = s l,k < s l,k+1 ≤ · · · ≤ s l,m < s l , then α l,1 ≥ α l,2 ≥ · · · ≥ α l,k and J l,1 = J 1 (s l,1 ). If k ≥ 2, then we observe that any two jobs from {J l,1 , J l,2 , · · · , J l,k } belong to different job families. Define I 1 = {J l,1 , J l,2 , · · · , J l,k } and I 2 = {J l,k+1 , · · · , J l,m }.
Proof. Since s l,1 = s l,2 = · · · = s l,k < s l,k+1 ≤ · · · ≤ s l,m < s l , there is no idle machine immediately before time s l , and B l,1 , B l,2 , · · · , B l,m is the m last batches starting on the m machines before time s l , then m(t) < m for any time t ∈ [s l,k+1 , s l,m ]. From Algorithm 2, we have: s l,j ≥ (1 + α max (s l,j ))(1 + α l,j )t 0 for any job J l,j ∈ I 2 .
Suppose that, for any job J l,h ∈ I 2 \{J l,k+1 }, r l,h ≤ s l,k+1 . Since r l,k+1 ≤ s l,k+1 and r l ≤ s l,k+1 , then the arrival time of all jobs from I 2 {J l } is less than s l,k+1 . Thus, any two jobs from I 2 {J l } belong to distinct job families.
Claim At least one job in I 2 {J l } has an arrival time greater than s l,k .
Otherwise, if the arrival time of all jobs is less than or equal to s l,k , then all jobs in I 1 I 2 {J l } are available at time s l,1 , and each job independently forms a batch in σ. We obtain that every two jobs from I 1 I 2 {J l } belong to distinct job families. Since I 1 I 2 {J l } = {J l,1 , J l,2 , · · · , J l,m } {J l }, then f (s l,1 ) = m + 1 > m. This contradicts f (s l,1 ) ≤ m. The claim follows.
From Lemmas 5, 7 and 8, and Theorem 3, we obtain the following theorem.
Theorem 4. For problem Pm|online,r j ,p-batch, b = ∞,m-family, p j = α j t|C max , Algorithm 2 has a competitive ratio of 1 + α max and is the best possible.

Conclusions and Future Research
In this paper, we outlined two best possible online algorithms. The first algorithm for problem 1|online,r j ,p-batch, b = ∞,f-family, p j = α j t|C max is a simple delay algorithm. We obtained the delay time by analyzing the properties of the unprocessed jobs, providing the best possible online algorithm with the competitive ratio of (1 + α max ) f . The second algorithm for problem Pm|online,r j ,p-batch, b = ∞,m-family, p j = α j t|C max is a more complex delay algorithm. We obtained the different delay times depending on the number of idle machines and provide the best possible online algorithm with the competitive ratio of 1 + α max . The results are shown in Table 3.
In future research, the general linear deterioration effect, such as p j = α j s j + β j , is worthy of research. In additional, for the online scheduling problem on m parallel machines with linear deteriorating jobs to minimize the makespan, Yu et al. [22] only proved that no deterministic online algorithm is better than (1 + α max )-competitive when m = 2, where α max is the maximum deterioration rate of all jobs. However, no best possible online algorithm has been reported. This is also a topic for further study.

Conflicts of Interest:
The authors declare no conflict of interest.