On-line machine scheduling with batch setups

On-line machine scheduling with batch setups

Department of Mechanical Engineering The University of Melbourne, VIC 3010, Australia

Abstract

We study a class of scheduling problems with batch setups for the online-list and online-time paradigms. Jobs are to be scheduled in batches for processing. All jobs in a batch start and complete to-gether, and a constant setup is prior to each batch. The objective is to minimize the total completion time of all jobs. We primarily consider the special cases of these problems with identical processing times, for which efficient on-line heuristics are proposed and their competitive performance is evaluated.

keyword: on-line scheduling; batch setups.

1

Introduction

This paper considers a class of on-line scheduling problems with batch se-tups, where n independent non-preemptive jobs are to be processed on a single machine or one ofm identical parallel machines. A machine can pro-cess at most one job at a time. Each jobimust be assigned to a batch, which consists of a set of jobs processed consecutively on a machine. A batch setup

sis incurred at the start of each batch. The completion time of a jobiis the time when the last job of the batch that includes jobicompletes its process-ing, that is, the completion time of the batch. The objective is to minimize the total completion time of all the jobs. Thus, our problem can be stated as on-line machine scheduling with sequence-independent batch setup times in the batch availability model, and also denoted as 1, P_m|s, F = 1|PC_i

[1, 2, 3], whereF represents the number of families.

This sort of problem is motivated by various real life applications in manu-facturing areas and storage systems. One application, mentioned in [4, 5], is the logging of tasks in a storage system. Log data can be consecutively

written on disks. Each write-on can be considered as a batch of jobs and a constant setup for disk-write occurs for each batch. Another example is from type production in flexible manufacturing systems [6]. All part-types must be mounted on a pallet for processing. The jobs on the same pallet form a batch and are completed at the same time. A standardized pallet setup time precedes each batch.

Previous work mainly considered the single machine scheduling problem. Coffman et al. [7] provided two main properties of optimal solutions for theirbatch-sizing problem to minimize the total flow time 1|s, F = 1|PFi:

(i) jobs are processed in shortest processing time order, and (ii) batch sizes, the numbers of jobs in batches, are in non-increasing order. Further, they proposed a dynamic programming algorithm which could optimally solve the problem inO(nlogn) time. Webster and Baker [8] summarized the results in [7, 9, 10], and for the problem with identical processing times, they showed that, ignoring the integer requirement, the optimal number of batchesη∗ =

q

1

4 +2nps −12 and the optimal batch size of thejth batch is ηn∗+s(η ∗₊₁₎

2p −jsp,

where p and s are the common processing and setup times. An on-line variant of the problem in the presence of release dates while minimizing the total flow time 1|ri, s, F = 1|

P

Fi was discussed by Gfeller et al. [4]. They

proposed a 2-competitiveGREEDY algorithm as well as two lower bounds for the special case with identical processing times. In the on-line general case, they showed that any on-line algorithm is at best (n₂ −²)-competitive for any ² > 0, and any on-line algorithm without unnecessary idle time cannot be better than n-competitive. Besides, they introduced an O(n5₎ dynamic programming algorithm for the off-line version of their problem with a fixed job sequence in [5]. For the parallel machine problem, Cheng et al. [6] studied the off-line problem of batching and scheduling simultaneously available jobs on identical parallel machines to minimize total completion time. They developed an O(mn(m+1)_{) dynamic programming algorithm to} solve the general problem. They also showed that the special case of identical processing times reduces to single machine scheduling problem, which can be solved optimally by previous algorithms [9, 11]. The comprehensive review papers [1, 2, 3, 8] provide general definitions and realistic applications for machine scheduling problems with setups for various machine configurations, performance measures and availabilities.

In this paper, we consider the problems of single machine and identical parallel machine scheduling with batch setups for the on-line list and online-time paradigms. More precisely, in the online-list paradigm, all jobs are available at time zero and are presented one by one in some sequence. Once a job is presented, an on-line algorithm must assign it to some machine and a batch immediately, without any the information about any subsequent jobs. In the online-time paradigm, a job becomes available at its release timeri

job upon its arrival or delay the decision-making until a later time. From the nature of scheduling with batch setups, we see that in the online-time setting jobs processed in a batch must arrive before the start of that batch’s processing. Furthermore, once that processing starts, no extra jobs can be added to that batch.

We evaluate on-line algorithms in terms of their competitive performance. Let I be a problem instance, A(I) be the objective function value by al-gorithm A for I and OP T(I) be the optimal off-line value. We say A is a c-competitive algorithm if _{OP T}A(I₍)_I) ≤ c for all I. Furthermore, we say A

has a competitive performance ratio of R_A if R_A = inf{c ≥ 1 : _{OP T}A(I₍)_I) ≤

c, for allI}.

The general notation is listed below.

Notation:

A an on-line algorithm;

I an instance of jobs;

m the number of parallel machines;

n the number of jobs;

s the constant setup requirement, which is incurred before the pro-cessing of each batch;

Ml thelth machine;

B_j(l) thejth batch on machine l;

S_j(l) the start time of the setup for batchB_j(l);

n(_jl) the number of jobs in batch B_j(l);

J_i theith presented or released job;

Ci the completion time of Ji, equal to the completion time of the

batchB(_jl) such that i∈B_j(l);

p_i the processing time ofJ_i;

ri the release time ofJi;

superscript∗ the corresponding optimal value.

Table 1 summarizes the the bounds proven in this paper.

Add Table 1 here.

2

Online-List Paradigm

Now we consider the problems in the online-list paradigm, where jobs are presented one by one and an on-line algorithm has to schedule a job to a machine and a batch immediately upon its presentation. It is clear that for identical processing times, it is unnecessary to have any idle times in this paradigm. So we shall assume that none of our algorithms for the problems with identical processing times allow idle time.

2.1 Lower bounds for a single machine scheduling

The following proposition provides a lower bound on the competitive per-formance ratio for the a single machine scheduling problems with identical processing times and arbitrary processing times, respectively.

Proposition 1. No on-line algorithm is better than 1.049-competitive for the

1|online-list, pi =p, s, F = 1|

P

Ci problem.

Proof. Two problem instances are constructed to be scheduled by an on-line algorithm A. If A assigns the first two jobs in separate batches, then the third job shows up. Otherwise, no more job is presented. Without loss of generality, we assume that s < p. For the instanceI with two jobs, theA

schedule issJ1J2 and so

A(I) = 2(s+ 2p),

whereas an optimal off-line solution issJ1sJ2 and then

OP T(I) = 3(s+p).

On the other hand, for the instanceI0 with three jobs, theA schedule may be eithersJ₁sJ₂sJ₃ orsJ₁sJ₂J₃ and the objective function value satisfies

A(I0) ≥ min{6s+ 6p,5s+ 7p}

= 6s+ 6p by the assumption that s < p.

An optimal solution for I0 is sJ1J2sJ3 ass < p, and so

OP T(I0) = 4s+ 7p.

Now we minimize the following expression max ½ 2s+ 4p 3s+ 3p, 6s+ 6p 4s+ 7p ¾ ,

and find that

RA ≥ 2 √

109 + 34

3√109 + 21 ≈1.049,

where the optimum is achieved when s

p =

√ 109−3

2.2 Pm|online-list, pi =p, s, F = 1|

P

Ci

In this section, we consider scheduling jobs onmidentical parallel machines. First we define a positive integerλas follows:

λ = dηs

p e,

whereη is a constant and will be set to different values at various stages of the following discussion. Then, it immediately follows from the definition function that

p(λ−1)< sη ≤λp. (1) Now we introduce a simple heuristic for this case and then consider its com-petitive performance.

Heuristic Uniform-Batch-Size (U BS):

Setη = 2. Whenever a new job is presented, assign it to the machine with the least number of jobs. In case of a tie, choose the machine with a smaller index. Everyλjobs assigned to the same machine form a batch. (An exam-ple of aU BS schedule is given in Figure 1.

Add Figure 1 here.

Proposition 2. RU BS = 3₂ for the Pm|online-list, pi = p, s, F = 1|

P

Ci

problem.

Proof. Before we consider the competitive performance of heuristicU BS, we remind ourselves of some properties of optimal solutions. We recall Lemma 3 of [6] which states that, for the equal processing time problem, there exists an optimal solution for which the difference in the numbers of jobs between any two machines is not greater than 1. It is easy to see that, for any input instance, the number of jobs processed on some machine in theU BS sched-ule may be assumed to be the same as that in an optimal solution, since all the machines are identical. In addition, because all jobs have identical processing times, we also assume that the jobs assigned to some machine in theU BS schedule are processed on the same machine in the optimal solu-tion, and that their relative processing positions are also the same. In the remainder of the proof, we shall mainly consider the scenario of an arbitrary machine l for 1 ≤ l ≤ m, and similar arguments can be applied to other machines.

We leta+ 1 anddbe the number of batches on machinel and the number of jobs in the last batch, respectively. Obviously, the number of jobs in a

batch can never exceedλ. In addition, the firstabatches are the same size, and each of them contains exactly λ jobs. Therefore, the number of the jobs processed on machinel can be written as aλ+d, if a ≥0. Note that if a = −1, which implies that n < m, then the U BS schedule, which has each job processed on different machines, is exactly the same as the optimal solution, and then the problem becomes trivial. Thus, without loss of gen-erality, we may assume thata≥0 for the rest of the discussion.

Now we consider the jobs completed in the first a batches and in the last batch, respectively.

For batch B_j(l), 1≤j≤a if a≥1:

The completion time ofB_j(l), the jth batch processed on machine l for 1≤

j≤aand a≥1, isj(s+λp). Thus the total completion time of the jobs in batchesB₁(l), . . . , Ba(l) is given by

X

i∈B_j(l),1≤j≤a

Ci = aλ(a+ 1)(₂ s+λp).

We see that the number of the jobs in the first a batches is aλ, and the completion time of theith job in this set in an optimal solution is at least

s+ip. It follows that the optimal total completion time of these jobs satisfies

X

i∈B(_jl),1≤j≤a

C_i∗ ≥ aλs+ aλp(aλ+ 1)

2 = 1 2(a 2_λ2_{p+aλp+ 2aλs).} Let ∆1= 6 P i∈B_j(l),1≤j≤aCi∗−4 P i∈B(_jl),1≤j≤aCi. By (1) andη= 2, we have λp−p <2s≤λp. It follows that ∆1 ≥ a2λ(λp−2s) + 2aλ(2s+p−λp) +aλp >0. So we obtain P i∈B(_{j ,}l)1≤j≤aCi P i∈B_{j ,}(l)1≤j≤aC ∗ i < 3 2. For batch B_a(l₊₁) :

The completion time of batchB_a(l₊₁) iss(a+ 1) +p(aλ+d) and so the total completion time of the jobs in the batch is written as:

X

i∈B_a(l₊₁)

Ci = d[s(a+ 1) +p(aλ+d)] =aλdp+ads+ds+d2p.

Recall that we assume that the processing order in the optimal solution is the same as that in theU BSschedule, because all jobs are identical. That is

to say, in the optimal off-line schedule, the jobs in the batchesB₁(l), . . . , Ba(l)

precede the jobs inB_a(l₊₁) . Hence, the optimal completion time of theith job in B_a(l₊₁) cannot be smaller than s+p(aλ+i), and then the optimal total completion time of the jobs inB_a(l₊₁) satisfies

X i∈Ba(l+1) C_i∗ ≥ ds+aλdp+dp(d+ 1) 2 = 1 2(2aλdp+ 2ds+d 2_p+dp). Again, we let ∆2 = 6 P i∈Ba(l+1) C ∗ i −4 P i∈B(al+1) Ci. Since d ≤ λ, similar to

the preceding case we have

∆2 ≥ 2ad(λp−2s) +d(2s+p−dp) + 2dp >0.

Hence we prove that

P i∈B_a(l₊₁) Ci P i∈B(_al₊₁) C ∗ i < 3 2. It is derived from the above cases that

P i∈B_j(l)Ci P i∈B(_jl)C ∗ i < 3 2. This inequality is applicable to any machine, and so we draw the conclusion that

P_n i=1Ci P_n i=1Ci∗ < 3 2.

Furthermore, we see thatR_{U BS}≤ 3 2.

Now we consider a lower bound of the competitive performance ratio for

U BS. Again, we consider the situation of an arbitrary machine l. First we recall some properties of optimal solutions for the corresponding single machine scheduling problem from [8]: the optimal number of batchesη∗ ₌

q

1

4 +2nps − 12 and the size of the jth batch is ηn∗ + s(η ∗₊₁₎

2p − jsp. We can

rewrite the number of jobs in the jth batch as: n∗

j = 2ηn∗ − js_p. Next, we

consider an instance for whichs=p andn= m(G₂2+G) for a large integerG

such thatG2_{+Gis divisible by 4. Then, byU BS, the parameterλequals 2} and (G2₄+G) batches are formed and processed on machinel. It follows that the total completion time of the jobs on machinelis given by

X i∈Ml Ci = 3₁₆s(G2+G)(G2+G+ 4) = 3sG4 16 +O(G 3_).

As for an optimal solution, by the optimal properties,Gbatches are formed and the number of jobs in the jth batch is G+ 1−j. So the optimal total

completion time is obtained by X Ml C_i∗ = η∗ X j=1 n∗_j(js+s j X k=1 n∗_k) = η∗ X j=1 (G+ 1−j) " js+s j X k=1 (G+ 1−k) # asη∗=G, = s 2  _(2G2_{+ 5G+ 3)} η∗ X j=1 j−(3G+ 4) η∗ X j=1 j2+ η∗ X j=1 j3   = sG 4 8 +O(G 3_). So we have P i∈MlCi P i∈MlC ∗ i → 3 2, asG→ ∞. The proof is now complete.

We note thatU BS can be applied to the 1|online-list,pi=p, s, F = 1|

P

Ci

problem as well and, besides, the results of Proposition 2 is also valid for that problem.

3

Online-Time Paradigm

In this section, we adopt the problem in the online-time scheduling environ-ment. We recall the statement of the problems in the online-time paradigm that any job in a batch must arrive before that batch’s processing starts, and no extra jobs can be added to that batch, once its processing starts. In this section, we shall discuss parallel machine scheduling first. We propose a new heuristic for thePm|online-time,ri, pi=p, s, F = 1|

P

Ci and a lower

bound of the competitive ratio for the case of m = 2. Furthermore, we consider the problem of a single machine scheduling, for which we introduce an on-line heuristic as well as a lower bound.

3.1 Pm|online-time, ri, pi =p, s, F = 1|

P

Ci

3.1.1 A New Heuristic

We remind ourselves of the definition expression of the parameter λ, that is,λ=dηs_p e. Now we set η= 1 for the following heuristic.

Heuristic Sync (Sy):

Do not schedule until all machines are idle and some jobs are available. Let

nt be the number of unscheduled jobs at some scheduling time t. Start a

new batch on each machine, and then assign min{λ,dnt

me} jobs to each of

machines 1, . . . , nt−mbn_mtc and assign min{λ,bn_mtc} jobs to each of the

re-maining machines. In case there are more thanmλjobs, select the earliest released jobs.

Remarks:

• Without loss of generality, we may assume that the first released job arrives at time 0. Suppose that there is an instanceI for which the first job is available after time 0, that is,r1 >0. Then we can obtain another instance I0 which is derived by decreasing the release time of each job in I by r1. It is easy to see that the on-line and the opti-mum total completion times both decrease byr1nfromI toI

0

. Thus,

Sy(I)

OP T(I) <

Sy(I0)

OP T(I0). As we are interested in the worst performance of

Sy, we may assume that r₁ = 0.

• We notice that there may exist a scenario in which, at timet, a batch

B1

j starts on machine 1 whilst there is no batch that starts on machines

l0, . . . , mfor some l0 with 2≤l0 ≤m. For convenience, we may add a dummy batch on each of those machines such that n(_jl) = 0 for

l0≤l≤m.

Now we give some preliminaries for the following discussion ofSy’s perfor-mance.

Classifications:

• Let ¯I ={I1, I2, . . .} be a set of time intervals defined as follows: Ma-chine 1 is never idle during any interval I_q and all machines are idle throughout the time period betweenIq and Iq+1.

• Let ¯BI ={BI1, BI2, . . .} be a set of time intervals such that a batch-interval BIj contains the processing period ofB(1)_j , the jth batch on

machine 1, that is,BIj = [Sj(1), Sj(1)s+n(1)j p], where Sj(1) and n(1)j are

the start time and the number of jobs of batchB1

j.

• We divide an interval Iq into batch-groups in the following way. A

batch-groupGq_k inI_q, k= 1,2, . . ., ends with a batch-interval BI_j in which the number of jobs is smaller than mλ, or possibly ends with the last batch-interval ofIq.

Add Figure 2 here.

Proposition 3. 2−_m1 ≤RSy ≤2 for the Pm|online-time, ri, pi =p, s, F =

1|PCi problem.

Proof. Consider an arbitrary intervalI_q. For a batch-groupGq_k we leta+ 1 be the number of batches on machine 1 and let h be the index of its first batch. Also, we letN_k(l) be the total number of jobs in the batch-group on machine l. If a ≥ 1, by the definition of batch-groups we see that each of batches 1, . . . , aon any machine must haveλjobs. Hence,N_k(l)=aλ+n(_hl₊)_a, wheren(_hl₊)_ais the number of jobs in the last batch ofGq_kon machinel. Then, the total completion time of the jobs inGq_k on machinelof theSy schedule is

X

i∈Gq_k,Ml

Ci = S(1)_h N_k(l)+λ(s+λp)×a(a₂+ 1)+n(_hl₊)_a[a(s+λp) +s+n(_hl₊)_ap].

On the other hand, a lower bound for the optimal total completion time of the jobs inGq_k satisfies

X i∈Gq_k C_i∗ ≥ X i∈Gq_k ri+s m X l=1 N_k(l) +p 2 × d P_m l=1N (l) k m e(d P_m l=1N (l) k m e+ 1)( m X l=1 N_k(l)−mb P_m l=1N (l) k m c) +p 2 × b P_m l=1N (l) k m c(b P_m l=1N (l) k m c+ 1)(m− m X l=1 N_k(l)+mb P_m l=1N (l) k m c).

We notice that the difference betweenNl1

k andNkl2 forl1, l2 with 1≤l1, l2 ≤

m, is at most 1. Also, because all the jobs have the same processing time and the machines are all identical, we can rewrite the above lower bound as

X i∈Gq_k C_i∗ = m X l=1 X i∈Gq_k,Ml C_i∗ ≥ m X l=1   X i∈Gq_k,Ml ri+sN_k(l)+ pN_k(l)(N_k(l)+ 1) 2   ≥ m X l=1 " r_minN_k(l)+sN_k(l)+ pN (l) k (N (l) k + 1) 2 # ,

where rmin = minri for i ∈ Gq_k. We let ∆l be the difference between

2P_i∈Gq k,MlC

∗

i and

P

haveλp−p < s≤λp. In addition toN_k(l)=aλ+n(_hl₊)_a, we have ∆l ≥ 2rminN_k(l)+ 2sN_k(l)+pN_k(l)(N_k(l)+ 1)−S_h(1)N_k(l)−aλ₂ (s+λp)(a+ 1) −n(_hl₊)_a(as+aλp+s+n(_hl₊)_ap) = 2rminN_k(l)+sN_k(l)−S_h(1)N_k(l)+ (λp−s)(a 2_λ 2 +an (l) h+a) +aλ(s+p−λp) 2 + aλp 2 +pn (l) h+a > 2r_minN_k(l)+sN_k(l)−S_h(1)N_k(l).

Ifk= 1, thenrmin=S_h1 and thus

∆l > rminN_k(l)+sN_k(l) >0.

Otherwise, from the definition ofGq_kas well as heuristicSy, we see that the earliest released job inGq_k must arrive after the start time of B1

h−1. That is to say,rmin > S1_h−1 =S_h1 −s−n1_h−1p. In this scenario, if h≥ 3 orq ≥2,

then we haver_min > S1

h−1 > s+p. Furthermore, we can obtain ∆_l > 2rminN_k(l)+sN_k(l)−S_h(1)N_k(l)

> rminN_k(l)+sN_k(l)+ (S_h(1)−s−n1h−1p)Nk−S_h(1)Nk

> (s+p)N_k(l)−n(1)_h−1pN_k(l)

≥ (s+p)N_k(l)−λpN_k(l)

> 0, sincen(1)_h−1 ≤λand λp < s+p.

Therefore, for the cases of k = 1 ∀q, h ≥ 3 ∀k and q ≥ 2, we obtain the following result: 2 X i∈Gqk C_i∗− X i∈Gqk C_i = m X l=1 ∆_l>0.

Thus, for the rest of the proof we need only consider the case ofq = 1 and

h = 2 for G1

2, that is, the situation in which the first batch processed on machine 1 inG1₂ isB₂1. For this case, we consider the first two batch-groups together. Again, we let a+ 1 be the number of batches on machine 1 in

G1

2. Let u and v be the smallest superscripts of batches B1(l1) and B (l2) a+2 such that n(l1) 1 < n (1) 1 and n (l2) a+2 < n (1) a+2 respectively. If n (l) 1 = n (1) 1 and/or

n(_al₊₂) =n(1)_a+2 for alllwith 1≤l≤m, then letu=m+ 1 and/or v=m+ 1. We note that there are m2 possible cases with regard to the values of u

n(1)_a+2 =· · ·= n(_am₊₂−1) =n(_am₊₂) + 1, that is, u = 2 and v= m. Then, for the other cases, analogous arguments can lead to the same result as below. By the assumption that the first job is released at time 0, the total comple-tion time of the jobs in the two batch-groups is given by

X i∈G1 1∪G12 C_i = m X l=1 X i∈G1 1∪G12,Ml C_i = m X l=1 [n(₁l)(s+n(₁l)p) +aλ(s+n(1)₁ p) +aλ(a+ 1)(s+λp) 2 +n(_al₊₂) (as+aλp+ 2s+n(1)₁ p+n(_al₊₂) p)].

Let N12 denote the total number of jobs in the two batch-groups. By hy-pothesis,N12=amλ+mn11+mn1a+2−m. Then the optimal total completion time of these jobs satisfies

X i∈G1 1∪G12 C_i∗ ≥ sN₁₂+p 2d N₁₂ m e(d N₁₂ m e+ 1)(N12−mb N₁₂ m c) +p 2b N12 m c(b N12 m c+ 1)(m−N12+mb N12 m c) = m(aλ+n(1)₁ −1 +n(1)_a+2)[s+p 2(aλ+n (1) 1 +n(1)a+2)]. We let ∆ = 2P_i∈G1 1∪G12C ∗ i − P i∈G1 1∪G12Ci. Since λp≥sand n (1) a+2≥1, ∆ ≥ amλ(s+λp−as−λp) 2 +amn(1)_a+2(λp−s) + (n₁(1)−1)(sm+amλp+mpn(1)_a+2) +mpn(1)₁ −mp+ 2pn(1)_a+2−pn(1)₁ +aλp+as+s > amλ(a−1)(λp−s) 2 +mpn (1) 1 −mp+p−pn(1)1 ≥ p(m−1)(n(1)₁ −1) ≥ 0.

Consequently, we have shown that P_i∈IqCi < 2

P

i∈IqC ∗

i for all l. Thus

RSy ≤2.

Next we provide a lower bound forRSy. Consider an instanceI consisting

of m jobs with r1 = 0, ri =²for 2≤i≤m and s < p. The Sy schedule is

sJ1sJ2 (on machine 1) and [idle]sJi (on machinei−1) for 3≤i≤m, where

the idle period equalss+p. On the other hand, an optimal solution is sJ1 (on machine 1) and [²]sJ_i (on machinei) for 2≤i≤m. Thus,

Sy(I) = (s+p) + 2(m−1)(s+p),

It follows that

RSy ≥ _{OP T}Sy(I₍)_I) →2−_m1.

Now the proof of this proposition is complete.

We notice that an obvious variant of heuristic Sy can be applied to the 1|ri, pi =p, s, F = 1|

P

Ci problem with a competitive ratio of 2.

3.1.2 A Lower Bound for P2|ri, pi =p, s, F = 1|

P

Ci

Proposition 4. No on-line algorithm is better than √22+2₆ -competitive for the P2|ri, pi =p, s, F = 1|

P

Ci problem.

Proof. We construct instances based on the scheduling mechanism of an on-line algorithmA. The constructed instances all start with 2 jobs available at time 0 and all haves= 2p. We letX1andX2be the start times ofJ1and

J₂ by A, respectively. Without loss of generality, we assume thatX₁≤X₂. Now we consider the following possible cases.

1. J₁ and J₂ are assigned to the same machine.

In this case, there are no further jobs. Let the instance of this case be

I. Then the total completion time satisfiesA(I)≥2(X₁+s+ 2p) and the optimum isOP T(I) = 2(s+p). Hence

A(I) OP T(I) = X1+s+ 2p s+p ≥ s+ 2p s+p = 4 3, as s= 2p.

2. J1 and J2 are scheduled to different machines.

Depending on the value of X2, either no further job arrives or two jobs arrive at timeX2+², where² is a small positive number. LetI

0

denote the instance consisting of two jobs, and then we haveA(I0) =

X1+X2+ 2(s+p) whereasOP T(I

0

) = 2(s+p). Besides, letI00be the instance of four jobs. For I00 A can either schedule J3 and J4 on the same machine after J1 orJ2, or put them on different machines. We remind ourselves that jobs processed in a batch must arrive before the start of the batch processing. Thus, J3, J4 must be assigned to new batches for r₃ =r₄ =X₂+² > X₂ ≥X₁. Also, we note that if jobs 3 and 4 are placed on the same machine, the total completion time is smaller if they are processed within a batch, sinces= 2p. Thus,

A(I00) ≥ min{3(X₁+s+p) + 2(s+ 2p) +X₂+s+p,

3(X₂+s+p) + 2(s+ 2p) +X₁+s+p,

2(X1+s+p) + 2(X2+s+p) + 2(s+p)}

by the assumption thatX1≤X2. An optimal solution of instance I

00

issJ₁J₂ on M₁ and [X₂+²]sJ₃J₄, where [X₂+²] is an idle period for

M2. Thus, we haveOP T(I 00 ) = 2(s+ 2p) + 2(X2+s+ 2p). The maximum of A(I 0 ) OP T(I0) and A(I00)

OP T(I00) is greater than or equal to max ½ X1+X2+ 2(s+p) 2(s+p) , min ½ 3X1+X2+ 6s+ 8p 2X₂+ 4s+ 8p , X1+X2+ 3s+ 3p X₂+ 2s+ 4p ¾¾ .

SinceX1 ≥0 ands= 2p, we rewrite the above expression as:

max ( A(I0) OP T(I0 ), A(I00) OP T(I00 ) ) ≥ max ½ X₂+ 6p 6p ,min ½ X₂+ 20p 2X2+ 16p, X₂+ 9p X2+ 8p ¾¾ .

We minimize the maximum, and by some algebra we find that max ( A(I0) OP T(I0 ), A(I00) OP T(I00 ) ) ≥ √ 22 + 2 6 ,

where the minimum is achieved whenX1 = 0 andX2 = (

√

22−4)p. From the discussion above, we see that RA ≥ min{4₃,

√ 22+2 6 } = √ 22+2 6 (≈ 1.115) for any on-line algorithm A. Hence the result follows.

3.2 1|online-time, ri, pi =p, s, F = 1|

P

Ci

Now we turn our attention to the shop type of a single machine. We notice that the on-line scheduling environment of the case considered in this section is the same as the corresponding problem 1|online-time,ri, s, F = 1|

P

Fi

in [4], except for the objective function. The two objectives of the total completion time and the total flow time are equivalent when all jobs are available at the same time or for off-line scheduling problem. We cannot say which one is more appropriate, since one may be more realistic in some cases, and may be less in others. As an essential part of our discussion, we consider this 1|online-time, ri, pi =p, s, F = 1|

P

Ci problem in this section.

In [4], the authors proposed a 2-competitive on-line heuristic for the problem, and also showed that no on-line algorithm could achieve better results. We shall prove that for 1|online-time,ri, s, F = 1|

P

Cithis lower bound on the

competitive ratio for all on-line algorithms is √5+1₂ , and further develop a simply implemented heuristic, which guarantees a competitive ratio bounded between 5

3 and 1 +

q

3 5.

3.2.1 A Lower Bound

Proposition 5. No on-line algorithm is better than √5+1₂ -competitive for the 1|online-time, ri, pi=p, s, F = 1|

P

Ci problem.

Proof. We consider the following scenario. The first job is released at time 0 withp=²s, where²is an arbitrary small number which we shall let tend to zero. Suppose that an on-line algorithm A allocates the machine to J1 at time X1. Depending on the value of X1, either no further job arrives or

n−1 jobs arrive at timeX1+². In the latter case, as ²→0,Acan at best assign the lastn−1 jobs in one batch immediately succeeding the first batch, whereas an off-line optimal solution may have all n jobs processed in one batch starting at timeX1+². Hence the on-line total completion time of the

njobs is greater than or equal ton(X1+s) + (n−1)[s+ (n−1)p], whilst the off-line optimum is not greater thann(X1+²+s+np). A may choose the best value of X1 to minimize max{X1_s+₊s_p+p,n(X1+_ns₍)+(_X1+n−_²+1)[_s+s+(_np)n−1)p]}. Now we let²→0, thenp→0 and finally letn→ ∞. It follows that

max ½ X1+s+p s+p , n(X1+s) + (n−1)[s+ (n−1)p] n(X1+²+s+np) ¾ → max ½ 1 +X1 s ,1 + s X1+s ¾ .

Some algebra shows that the minimum is obtained when X₁ = s( √

5−1) 2 and equals √5+1₂ . Therefore, we conclude that RA ≥

√ 5+1

2 for all on-line algorithms.

3.2.2 A New Heuristic

Now we introduce a new heuristic for the 1|ri, pi =p, s, F = 1|

P

Ci

prob-lem, which is a variant of heuristic Sy, and then evaluate its performance. Again we recall the definition function of λ, λ = dηs_pe, and further we set

η=

q

5

3 for the following heuristic.

Heuristic Wait-Half-Setup (W HS):

Keep the machine idle until time s

2. If both the machine and some un-scheduled job(s) are available, then assign theλearliest released jobs to the machine to form a new batch. If the number of available jobs is smaller than

λ, then schedule all of them to the machine in a new batch.

Proposition 6. 5₃ ≤RW HS ≤ 1 + q 3 5 for the 1|ri, pi = p, s, F = 1| P Ci problem.

Proof. Let c= 1 + 1_η. The proof of this proposition is analogous to that of Proposition 3, and thus we adopt its notation. We assume, without loss of generality, that the first job is released at time 0. We say a batch isfull if it consists ofλjobs; otherwise, we say it isnon-full. Then we recall that, in an arbitrary processing intervalIq without idle time, batch-group Gq_k contains

a group of batches ending with a non-full batch or possibly with the last batch ofIq. We let a+ 1 be the number of batches in Gq_k and let h be the

index of the first batch. As we now consider the a single machine scheduling case, we omit the superscript of the symbols for batches. Thus, the number of jobs inGq_k is given byNk =aλ+na+h, wherena+h is the number of jobs

in the last batch ofGq_k. Furthermore, the total completion time of the jobs inGq_k can be written as:

X

i∈Gq_k

Ci = ShNk+

aλ(a+ 1)(s+λp)

2 +na+h(as+s+pNk). (2)

A lower bound for the optimum satisfies

X i∈Gq_k C_i∗ ≥ X i∈Gq_k ri+sNk+ pN_k(N_k+ 1) 2 ≥ rminNk+sNk+ pN_k(N_k+ 1) 2 , (3)

wherermin = minri fori∈Gq_k.

With regard to the values ofh,kand q, we consider the following cases. (A summary of the analysis is given in Table 1??.) We let ∆l be the difference

betweencP_i∈Gq kC

∗

i and

P

i∈Gq_kCi in Casel, as set out below.

Add Table 1 here.

1. h= 1 (k= 1, q= 1)

By the assumption that the first job arrives at time 0, we havermin= 0

andS1 = s₂ in this case. By (2) and (3), the difference ∆1 satisfies

∆1 ≥ aλ       s η − λp 2 + p(1 +η) 2η | {z } δ1(1) +a 2( λp η −s | {z } δ2(1) )       +na+h      a( λp η −s) + s η − s 2 − pn_a+h(η−1) 2η + p(1 +η) 2η | {z } δ₃(1)      .

Now we consider the values ofδ₁(1),δ₂(1) andδ(1)₃ respectively. If follows from (1) thatδ₂(1) ≥0 and also

δ₁(1) > p(λ−1) η2 − λp 2 + p(1 +η) 2η = λp( 1 η2 − 1 2) +p( η2_+η−2 2η2 ) > 0, sinceη = q 5 3.

Finally, we considerδ₃(1). A batch may contain at mostλjobs, and so

n_a+h ≤λ. Thus,δ3 must satisfy

δ₃(1) > s η − ηs 2 + p η >0.

Consequently, we see that ∆1 >0.

2. h= 3,k= 2 andq = 1 In this case, batch-groupG1

1 contains two batches B1 andB2. By the definitions ofW HS andGq_k, we see thatB1 is full whereasB2 is non-full, that is, n₁ =λ, n₂ ≤λ−1, and also the earliest released job in

G1₂must arrive after the start time ofB2, that is,rmin> S2 = 3₂s+λp. Thus, we rewrite (2) and (3) for this case as:

X i∈Gq_k Ci = (5₂s+λp+n2p)Nk+ aλ(a+ 1)(s+λp) 2 +na+h(as+s+pNk) ≤ (5s 2 + 2λp−p)Nk+ aλ(a+ 1)(s+λp) 2 +na+h(as+s+pNk), X i∈Gqk C_i∗ ≥ (3s 2 +λp)Nk+sNk+ pN_k(N_k+ 1) 2 . Then we obtain ∆2 ≥ Nk · (1 +1 η)( 3s 2 +λp)−2s−2λp+p ¸ + ∆1 > Nk · 3s 2η + λp η − s 2 −p(λ−1) ¸ > sNk 2η (3 +η−2η 2_{) by (1),} > 0. 3. h≥2,k= 1

In the scenario ofk= 1 andh≥2,Bh is the first batch ofIq, and the

jobs processed in or after B_h cannot arrive before time S_h and thus the smallest release date of Gq_k satisfies rmin ≥ Sh > 3₂s. Then the

following inequality bounds the value of ∆3. ∆₃ ≥ S_hN_k(1 +1

η)−ShNk+ sN_k

2 + ∆1 >0. 4. h≥4,k≥2

When h ≥4 and k 6= 1, we observe that the batch immediately pre-ceding ofGq_k,Bh−1, is a non-full batch and thus we havenh−1 ≤λ−1 and rmin > Sh −s−pnh−1 > 5₂s. Also by ((1-2)), the difference ∆4 for this case is bounded below by

∆4 ≥ Nk(rmin+rmin_η −Sh+₂s) + ∆1 > Nk(Sh−s−pnh−1+5_2ηs−Sh+ s₂) > sNk 2η (5−η−2η 2₎ > 0. 5. h= 2,k= 2 andq = 1

From this case onwards, we shall consider the batch-groupsGq_k−1 and

Gq_ktogether instead ofGq_k only. That is to say, in this case we consider

G1

1 and G12 consisting of the first a+h batches. The on-line total completion time is X i∈G1 1∪G12 C_i = (N_k−1+N_k)(3s 2 +pNk−1) + aλ(a+ 1)(s+λp) 2 +na+h(as+s+pNk). Now we provide two lower bounds for the optimal total completion time of the jobs in G1₁ and G1₂. If the optimal solution starts all the jobs inG1

1 at or after time 32s, then a lower bound for the optimum is given by

LB₁(5) = 3s(Nk−1+Nk)

2 +

p(Nk−1+Nk)(Nk−1+Nk+ 1)

2 .

Otherwise, if the optimal solution starts processing some of the jobs in

G1

1 before time 32s, then it must start another batch for the processing of the jobs inG1

2. Thus, another lower bound is

IfLB₁(5)≤LB(5)₂ , then the difference ∆5 between c P i∈G1 1∪G12C ∗ i and P i∈G1 1∪G12Ci satisfies ∆5 ≥ 3s(Nk−₂1_η+Nk) +p(Nk−1+Nk)(₂N_ηk−1+Nk+ 1)+p(Nk−1₂+Nk) −pN 2 k−1 2 − aλ(as+s+λp) 2 −na+h(as+s+ n_a+hp 2 ) = Nk−1 2η    p(2Nk+η+ 1) + (_|pNk−1−pηN_{z k−1+ 3s_} δ₁(5) )     +aλ 2       3s η −s−λp+p | {z } δ₂(5) +p η +a( λp η −s)       +na+h       3s 2η −s− pna+h(η−1) 2η + p(1 +η) 2η | {z } δ3(5) +a(λp η −s)      .

We consider the signs of δ₁(5), δ₂(5) and δ(5)₃ . Since n_a+h ≤ λ and

Nk−1=nh−1≤λ−1, by (1) we see that δ₁(5) > s(3 +η−η2)>0, δ₂(5) > s η(3−η−η 2_)>0, δ₃(5) > s 2η(3−η−η 2_)>0. Thus, we obtain ∆5 >0.

Otherwise, ifLB₁(5) > LB₂(5), then we have

∆0₅ ≥ s(1 +η)(Nk−Nk−1) 2η + ∆5 > Nk−1(δ (5) 1 −sη−s) 2η > sNk−1(2−η2) 2η > 0. 6. h= 3,k= 2 andq = 2

As for Case 5, we consider the jobs in G2

1 and G22. For q = 2, there must be machine idle time preceding G2

1. Also, by W HS we see that the earliest released job in G2₁ arrives at S2 the start time of B2 and the earliest released job inG2

2 must arrive after time S2. We letr

0

min

be the smallest release time of the jobs inG2

1 andG22, and then we have

r0_min ≥S2 > 3₂s+pn1. Thus we can write the on-line total completion time and a lower bound for the optimum as:

X i∈G2 1∪G22 Ci = (S2+s+pNk−1)(Nk−1+Nk) + aλ(s+λp)(a+ 1) 2 +na+h(as+s+pNk), X i∈G2 1∪G22 C_i∗ ≥ (r_min0 +s)(Nk−1+Nk) +p(Nk−1+Nk)(₂Nk−1+Nk+ 1).

Sincer0_min≥S2> 3₂s+pn1, the difference ∆6 satisfies

∆6 ≥ (Nk−1+Nk)(r 0 min−S2+2r 0 min−s 2η ) + ∆5 >0. 7. h= 3,k= 3 andq = 1

In this case, G1₂ and G1₃ consist of B2 and B3, . . . , Ba+3, respectively. From the values of h,k and q, we see thatG1

1 andG12 both contain a non-full batch, namely, N1 =n1 ≤λ−1 and N2 =n2 ≤λ−1. This implies that the jobs in the batch-groupsG1

2 andG13 must arrive after time s₂ and 3₂s+pNk−2, respectively. Following observations similar to those in Case 5, we can write the on-line total completion time of the jobs inG1

2 and G13 and two lower bounds for the optimum as:

X i∈G1 2∪G13 Ci = (Nk−1+Nk)(5₂s+pNk−2+pNk−1) +aλ(a+ 1)(₂ s+λp) +n_a+h(as+s+pN_k), X i∈G1 2∪G13 C_i∗≥min          (Nk−1+Nk)(5₂s+pNk−2) +p(Nk−1+Nk)(N₂k−1+Nk+ 1) | {z } LB1(7) , (Nk−1+Nk)(3₂s+pNk−2) +sNk+p(Nk−1+Nk)(₂Nk−1+Nk+ 1) | {z } LB₂(7)          ,

where LB₁(7) and LB₂(7) are defined similar to those in Case 5. We note that the termpNk−2(Nk−1+Nk) ofLB2(7) comes from the pro-cessing requirements of the jobs inG1

1. Since we may assume that jobs are processed in order of non-decreasing release dates in an optimal solution, the jobs inG1

2 and G13 must be processed after those in G11. An argument analogous to that in Case 5 leads to the same result that the difference ∆7 betweenc

P i∈G1 2∪G13C ∗ i and P i∈G1 2∪G13Ci is greater than 0.

We conclude from the above arguments that the inequalitycP_i∈Gq k∪G q k+1C ∗ i− P

i∈Gq_k∪Gq_k+1Ci>0 is valid for allh,q andk. Hence we obtain c

P_n

i=1Ci∗−

P_n

i=1Ci>0.

Now we consider the following instance I to determine a lower bound for the competitive ratio RW HS. The first job of I is available at time 0 and

n−1 jobs arrives at time s₂+². We let the positive number²→0 and then the identical processing time p → 0. The on-line and the optimal off-line objective function values are given below:

W HS(I) = 3s 2 +p+ (n−1)(5s+ 2np) 2 , OP T(I) = n(3s+ 2²+ 2np) 2 .

Let²→0, then p→0 and finallyn→ ∞, and it follows that

W HS(I)

OP T(I) → 5 3. Hence the result follows.

4

Conclusions and Future Work

In this paper we discuss four problems of scheduling machines with a com-mon batch setup in the batch availability model. For both the online-list and the online-time scenarios, we introduce lower bounds and new on-line heuristics to each case with equal processing requirements. Furthermore, we establish the lower and upper bounds on the competitive performance ratio for each heuristic. Table 2summarizes the the bounds proven in this paper. Future work can extend the above discussion in various ways. With tighter lower bounds on the optimal objective function values or better defined properties of the optimal (off-line) solutions, can the competitive ratios of the proposed heuristics be improved? Also, given the general case with ar-bitrary processing times, is there any on-line algorithm which can guarantee

r-competitive for any r? Moreover, issues with other objectives like the minimization of total weighted flow time or the total weighted completion time remain to be approached in on-line scheduling environments.

References

[1] A. Allahverdi, J. N. Gupta, T. Aldowaisan, A review of scheduling re-search involving setup considerations, Omega, The International Jour-nal of Management Science 27 (1999) 219–239.

[2] A. Allahverdi, C. T. Ng, T. C. E. Cheng, M. Y. Kovalyov, A survey of scheduling problems with setup times or costs, European Journal of Operational Research 187 (2008) 985–1032.

[3] C. N. Potts, M. Y. Kovalyov, Scheduling with batching: a review, Eu-ropean Journal of Operational Research 120 (2000) 228–249.

[4] B. Gfeller, L. Peeters, B. Weber, P. Widmayer, Online single machine batch scheduling, LNCS 4162 (2006) 424–435.

[5] B. Gfeller, L. Peeters, B. Weber, P. Widmayer, Single machine batch scheduling with release times, Institute of Theoretical Computer Sci-ence, ETH Zurich Technical Report 514 (2006) 1–23.

[6] T. C. E. Cheng, Z. L. Chen, M. Y. Kovalyov, B. M. T. Lin, Parallel-machine batching and scheduling to minimize total completion time, IIE Transactions 28 (1996) 953–956.

[7] E. G. Coffman, M. Yannakakis, M. J. Magazine, C. Santos, Batch sizing and job sequencing on a single machine, Annals of Operations Research 26 (1990) 135–147.

[8] S. Webster, K. R. Baker, Scheduling groups for jobs on a single machine, Operations Research 43 (4) (1995) 692–703.

[9] D. F. Shallcross, A polynomial algorithm for a one machine batching problem, Operations Research Letters 11 (1992) 213–218.

[10] C. Santos, M. J. Magazine, Batching in single operation manufacturing systems, Operations Research Letters 4 (1985) 99–103.

[11] E. G. Coffman, A. Nozari, M. Yannakakis, Optimal scheduling of prod-ucts with two subassembles on a single machine, Operations Research Letters 37 (3) (1989) 426–436.