Online Scheduling with Known Arrival Times

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Online Scheduling with Known Arrival Times

• Nicholas G Hall (Ohio State University)
• Marc E Posner (Ohio State University)
• Chris N Potts (University of Southampton)

14 May 2008 , CIRM, Marseille
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Motivation
Consider a typical make-to-order production
system that uses a periodic ordering and
scheduling process.
Orders are accepted, for example, weekly.
Between two orders, no new jobs become available.
Until an order arrives, its details (for example,
its processing time and value) are not known.
Meanwhile, scheduling decisions must be made for
the available jobs.
In order to keep resources available to process
new jobs, idle time may be inserted.
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A New Scheduling Environment
Data for currently available jobs are known.
New jobs may arrive only at known future times.
There is no restriction on the data for the new
jobs, which become known only on arrival.
The potential job arrival times effectively
define planning periods.
Therefore, we consider our problem to be an
online planning period scheduling problem.
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Online vs. Offline Scheduling
Classical offline all data are known at the
start of the planning horizon.
Classical online new jobs may arrive at any
time, and data become known only on arrival.
Our new environment interpolates between the
classical offline and classical online
environments.
If the number of potential job arrival times is
large and uniformly distributed, then the new
environment approaches the classical online
environment as a limiting case.
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Problem Definition
Consider a single machine scheduling problem
environment.
Let 0 t0 lt t1 lt lt tT denote known potential
job arrival times.
Each job j has a release date rj where rj ? t0,
t1 ,, tT, a processing time pj and a weight wj.
The goal is to minimize SwjCj , the total
weighted completion time of the jobs, a widely
used measure of customer service.
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Related Online Literature (1 of 2)
Consider the classical online version of our
problem. The competitive ratio compares the
solution value SwjCj given by the online
algorithm to the optimal offline solution value
C.
Several studies (e.g. Hoogeveen and Vestjens,
1996) have established that no online algorithm
can have a competitive ratio that is better than
2.
Anderson and Potts (2004) propose a best
possible online algorithm with a competitive
ratio of 2.
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Related Online Literature (2 of 2)
The following instance provides a lower bound on
the competitive ratio.
Job 1 arrives at time zero r1 0, p1 1, w1
e. Suppose that job 1 starts processing at
time t.
If t ? 1, then no other job arrives and therefore
SwjCj /C t1 ? 2.
If t lt1, then an adversary releases job 2
r2 t e, p2 e, w2 1. As
e ? 0, SwjCj /C ? (t1)/t gt 2.
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Research Question
The best possible algorithm for the classical
online problem has a competitive ratio of 2.
However, in our problem, we have more
information than in the classical online case
we know times when jobs cannot arrive.
Consequently, the best possible competitive
ratio may be smaller than 2.
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Main Result
We describe a simple online scheduling algorithm
which runs in O(n log n T2) time, and which
achieves a competitive ratio of
R max min Rs(v) where Rs(v) (ts1
ts ?(ts1 ts)2 4tvts1)/2ts1 This ratio
is best possible for the problem.
v1,,T
s0,,v-1
Note that R lies between (1?5)/2 and 2,
depending on the precise values of t1,,tT.
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Lower Bound for the Offline Problem
The job splitting lower bound of Belouadah,
Posner and Potts (1992) provides a lower bound zL
on the total weighted completion time of an
optimal offline schedule, which we use in our
analysis.
Choose an available job j with the largest wj/pj
and schedule it to start as early as possible.
If no job k with wk/pk gt wj/pj becomes
available during the processing of job j, then
process job j to completion.
Otherwise, choose the first job k with wk/pk gt
wj/pj to become available during the processing
of job j and split job j at time rk.
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Mathematical Program for Lower Bound
We introduce the following problem P(v)
Maximize R
subject to (ts p)/ts1 R, s
0,,v-1 (1) (tv p)/(t0
p) R (2)
p, R 0
Inequality (1) models a situation where job j
with rj t0 starts processing at time ts
and is still in process at time ts1when another
job with large weight and small processing time
arrives.
Inequality (2) models a situation where job j
with rj t0 starts at time tv, and no other
job is released.
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Lower Bound Theorem
Theorem. The competitive ratio of any online
algorithm is at least R max0ltvTR(v).
Proof. Let R R(v), and let p denote the
optimal value of p in problem P(v). Consider an
instance with 1 or 2 jobs, where r1 0, p1 p
and w1 1.
Let the algorithm start job 1 at time ts, where 0
s T.
If s lt v, then an adversary releases a second job
with large weight at time ts1, generating a
performance ratio of (ts p)/ts1 R.
If s v, then no other job is released,
generating a performance ratio of (ts p)/(t0
p) R. ?
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Optimality Conditions for P(v)
Lemma. In an optimal solution to problem P(v),
constraint (2) and at least one constraint of
(1) are satisfied at equality.
Proof. Let (p,R) denote an optimal solution to
P(v).
If each constraint (1) is satisfied as a strict
inequality, then the solution (p-e, R etv/p(p
-e)) is feasible for problem P(v).
If constraint (2) is satisfied as a strict
inequality, then the solution (p e, R e/tv) is
feasible for problem P(v).
In both cases, the optimality of (p,R) is
contradicted. ?
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Value of R
Theorem. R max min Rs(v), where Rs(v)
(ts1 ts ?(ts1 ts)2 4tvts1)/2ts1
v1,,T
s0,,v-1
Proof. The previous result shows that p(v) and
R(v) are obtained by solving (ts
p)/ts1 R and (tv p)/p R for some s,
which provides the value Rs(v).
We choose R(v) mins 0,,v-1Rs(v) since
other values of s give infeasible values of p and
R. ?
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Design of Online Algorithm
Need to protect against the arrival of a short
job with very large weight arriving at time t.
Thus, only process job j if
Cj/t R.
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Online Algorithm
Algorithm CSWPT
0. Compute the lower bound R on the competitive
ratio.
1. Let j an available job with largest wj/pj
ratio. Let tu the earliest future potential
job arrival time.
2. If job j cannot complete by time Rtu, then
insert idle time up to time tu and go to
Step 4.
3. Schedule job j. If it completes processing
before time tu, then go to Step 1.
4. If u lt T, or there are available jobs, then go
to Step 1.
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Analysis of Algorithm CSWPT
Restrict the instances to be considered to those
with wj/pj 0 or 1. Thus, each job is
classified as type 0 or type 1.
Establish that the completion time of any type 1
job under Algorithm CSWPT is no more than R
times its contribution to the job splitting lower
bound.
zC denotes the cost of the schedule sC delivered
by Algorithm CSWPT.
zL denotes the value of the lower bound provided
by schedule sL created by the job splitting
procedure.
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Characterizing a Worst-Case Instance
Lemma. Given any instance, there exists another
instance for which zC/zL is at least as large and
wj/pj 0 or 1.
Proof. Suppose that there is more than one
distinct nonzero wj/pj ratio. Choose a set of
jobs with nonzero wj/pj ratio such that this
ratio is not the largest.
If the value of zC/zL for jobs restricted to this
set is larger than the value of zC/zL for all of
the jobs, then increase the weight of all jobs in
this set so that they still have equal wj/pj
values.
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Characterizing a Worst-Case Instance
Otherwise, similarly decrease the weight of
jobs in this set.
Repeating this argument reduces the number of
distinct nonzero wj/pj ratios to one and does not
decrease zC/zL.
A rescaling of the weights allows us to
achieve wj/pj 0 or 1 without affecting the
value of zC/zL. ?
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Properties of Restricted Instances
In the job splitting procedure, only type 0 jobs
are split. Thus, each type 1 job j contributes
Cj(s L) to the lower bound.
Type 1 jobs are sequenced in the same order by
Algorithm CSWPT and by the job splitting
procedure.
For any type 1 job j, Cj(s L) Cj(s C).
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Type 1 Job j Starts after Type 0 Job i
Lemma. If a type 1 job j starts processing in sC
when a type 0 job i completes, then Cj(sC)/Cj(sL)
R.
Proof. Let job i start at time tl-1 or later but
before time tl. Then Cj(sC)
Ci(sC) pj Rtl
pj R(rj pj) since
job j cannot have been released when job i
started. Therefore,
Cj(sC)/Cj(sL) R. ?
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Type 1 Job j Starts after Type 1 Job i
Lemma. If a type 1 job j starts processing in sC
when another type 1 job i completes, then
Cj(sC)/Cj(sL) Ci(sC)/Ci(sL).
Proof. Cj(sC)/Cj(sL) (Ci(sC)
pj)/(Ci(sL) pj)
Ci(sC)/Ci(sL). ?
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Type 1 Job j Preceded by Idle Time
Lemma. If a type 1 job j starts processing in sC
immediately after a period of idle time, then
Cj(sC)/Cj(sL) R.
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Main Result
Theorem. Algorithm CSWPT has the best possible
competitive ratio of R.
Proof. The upper bound follows from the three
previous lemmas. From the lower bound theorem,
this result is best possible. ?
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Special Case Equally Spaced tu Values
Suppose that tu1 tu is constant, for u
0,,T-1. Then R (tT tT-1 ?(5tT2
tT-12 - 2tT-1tT))/2tT
Moreover, as T ? 8 with tT fixed, R ? 2.
Thus, we obtain the classical online result.
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Conclusions (1 of 2)
We consider an online planning period scheduling
problem, which falls within a wider class of
online planning period problems.
For the single machine total weighted completion
time problem, a lower bound on the competitive
ratio is given by the optimal value of a
mathematical program.
We provide a closed form expression for this
value, which is between (1 v5)/2 and 2, and
depends on the potential job arrival times.
We also describe a fast online algorithm with a
competitive ratio that matches this lower bound,
and which is therefore best possible.
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Conclusions (2 of 2)
For the case of equally spaced potential job
arrival times, as T ? 8 the competitive ratio
becomes arbitrarily close to 2.
There are many challenges for future research in
online planning period (scheduling) problems.
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Copies of the related paper are available by
request at
C.N.Potts_at_soton.ac.uk
This work is supported by The National Science
Foundation under grant DMI-0421823, and by EPSRC
under grant EP/D060931/1.
Thank you for your attention!

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