minimizing the total flow-time on a single machine with an ......y. he, w. zhong, h. gu, improved...
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Minimizing the total flow-time on asingle machine with an unavailability
period
Julien Moncel (LAAS-CNRS, Toulouse – France)Jeremie Thiery (DIAGMA Supply Chain, Paris – France)
Ariel Waserhole (G-SCOP, Grenoble – France)
Project Management and Scheduling2–4 April 2012
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
Outline
1 Introduction
2 Literature review
3 Our contribution : theoretical results
4 Our contribution : experimental results
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
And now...
1 Introduction
2 Literature review
3 Our contribution : theoretical results
4 Our contribution : experimental results
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
The problem
Settings
One machine
One unavailability period [R,R + L]
No preemption
Total flow-time∑
i Ci
Denoted 1, h1||∑
i Ci
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
The problem
Why unavailable ?
Unavailability = planned maintenance, lunch break, commitmentfor other tasks, etc.
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
Similar problems (1)
1, h1||Cmax
Same settings with Cmax instead of∑
i Ci : NP-complete
Related to problem PARTITION
PARTITION
n numbers a1, . . . , an
is there a partition I ∪ J = {1, . . . , n} such that∑i∈I ai =
∑j∈J aj ?
(problem SP12 in the Garey-Johnson)
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
Similar problems (2)
1, h1|preemption|∑
iCi
Same settings with preemption : trivial (SPT)
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
And now...
1 Introduction
2 Literature review
3 Our contribution : theoretical results
4 Our contribution : experimental results
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
Complexity
Complexity
1, h1||∑
i Ci is NP-hard [Lee & Liman (1992)]
Proof using EVEN-ODD PARTITION
EVEN-ODD PARTITION
2n numbers a1, . . . , a2n such that ai < ai+1 for all i
is there a partition I ∪ J = {1, . . . , n} such that∑i∈I ai =
∑j∈J aj and |I ∩ {x2i−1, x2i}| = 1 for all i ?
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
Idea of proof
Settings
2n + 1 jobs
M � P two large constants
pi = M + ai for i = 1, . . . , 2n and p2n+1 = P
Z = 12
∑i ai
R = nM + Z and L = M
Settings that ensure
there always are n jobs before R (and n + 1 jobs after)
the problem reduces to minimizing the idle time before R
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
Approximation algorithms (1)
[Lee & Liman (1992)]
SPT : O(n log n) heuristic of relative error 27
[Sadfi et al. (2005)]
2-OPT with SPT : O(n2) heuristic of relative error 317
schedule jobs according to SPT
try all possible exchanges of 1 job before R with 1 job after R
output the best schedule
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
Approximation algorithms (2)
[He et al. (2006)]
2k-OPT with SPT : O(n2k) heuristic of relative error 25+2√2k+8
schedule jobs according to SPT
try all possible exchanges of ≤ k jobs before R with ≤ k jobsafter R
output the best schedule
This is a PTAS called MSPT-k
We improve the 25+2√2k+8
bound of [He et al. (2006)], and
provide a new bound that is asymptotically tight
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
Other approximation algorithms
[Breit (2007)]
An O(n log n) parameterized heuristic of best relative error 0.074
[Kacem & Mahjoub (2009)]
An FPTAS for 1, h1||∑
i wiCi
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
And now...
1 Introduction
2 Literature review
3 Our contribution : theoretical results
4 Our contribution : experimental results
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
Main results
Theorem (Improved bound)
An improved error bound of the PTAS MSPT-k is k+22k2+8k+7
. Thisimproves the computation of the bound made by[He et al. (2006)].
Theorem (Tightness of the new bound)
This error bound is asymptotically tight.
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
Notations (1)
pi processing time of job iCi completion time of job iC[i ] completion time of job scheduled at position i
R starting time of unavailability periodL duration of unavailability period
δ idle time of the machine before the unavailability period
S schedule obtained by SPTS ′ schedule obtained by MSPT-kS∗ optimal schedule
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
Notations (2)
S schedule obtained by SPTS ′ schedule obtained by MSPT-kS∗ optimal schedule
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
How to improve the bound (1)
Lemma
If S is a schedule better than the SPT schedule S, then δ ≤ δ.
Remark : the converse is not true
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
How to improve the bound (2)
Lemma
Let C[i ] and C ∗[i ] be completion times of job scheduled at position i
in the SPT and in the optimal solution (resp.). Then we have:∑i∈A
C[i ] ≤∑j∈Y
C ∗[j] + |Y |(δ − δ∗).
Lemma
Let t ≥ 1 be an integer. If (at least) t jobs of X are scheduled afterthe period of maintenance in the optimal solution, then we have:
n∑i=1
C ′i ≤n∑
i=1
C ∗i + (|Y | − (t + 1)) (δ − δ∗).
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
How to improve the bound (3)
Lemma
Let t ≥ 1 be an integer. If (at least) t jobs of B are scheduledafter the period of maintenance in the optimal schedule S∗, thenwe have:
n∑i=1
C ∗i ≥{|Y |(|Y |+ 1)
2+ t
}(δ − δ∗)
Lemma
Let p ≥ 1 and q ≥ 1 s.t. p ≥ q. If it is possible to exchange p jobsof B with q jobs of A, then it is possible to exchange p − q + 1jobs of B with 1 job of A.
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
The new bound
The error bound εk of MSPT-k satisfies
εk =
∑ni=1 C ′i −
∑ni=1 C ∗i∑n
i=1 C ∗i≤ 2(|Y | − (k + 1))
|Y |(|Y |+ 1) + 2(k + 1).
For all k > 0, the function fk : x 7→ fk(x) = 2(x−(k+1))x(x+1)+2(k+1) , x ∈ N+
reaches its maximum for xk = 2k + 3. Then we have
max|Y |∈N+
εk ≤ fk(xk) =k + 2
2k2 + 8k + 7.
Hencek + 2
2k2 + 8k + 7
is an (improved) relative error bound for MSPT-k.
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
Why is the new bound tight ? (1)
Family of extremal instances
k ∈ N and M ∈ N s.t. k2 = o(M)
3k + 4 jobs with
pi = 1 for i ∈ {1, 2, .., k + 1}pi = M for i ∈ {k + 2, .., 3k + 4}
R = M and L = 1
Such that the SPT schedule is
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
Why is the new bound tight ? (2)
n∑i=1
C ′i = M(2k2+9k+9)+o(M) andn∑
i=1
C ∗i = M(2k2+8k+7)+o(M)
⇒∑n
i=1 C ′i −∑n
i=1 C ∗i∑ni=1 C ∗i
=M(k + 2) + o(M)
M(2k2 + 8k + 7) + o(M)→ k + 2
2k2 + 8k + 7
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
And now...
1 Introduction
2 Literature review
3 Our contribution : theoretical results
4 Our contribution : experimental results
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
Settings
Tested algorithms
MSPT-k for k = 0, 1, 2
Random instances
job processing times : integers randomly and uniformly chosenin [1, 100]
duration L = mean of job processing times
starting time D = proportion Rperc of the sum of theprocessing times, Rperc ∈ {0.1, 0.3, 0.5, 0.7, 0.9}number n of jobs ranged from 10 to 5000
(Classical settings for this problem, see e.g. [Breit (2007)] or[Sadfi et al. (2005)])
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
n M0(m) M0(w) M1(m) M1(w) M2(m) M2(w)10 1.86 8.36 0.03 0.40 0.00 0.0025 0.96 3.92 0.08 0.63 0.00 0.0750 0.68 2.10 0.07 0.49 0.01 0.0975 0.43 1.52 0.05 0.32 0.01 0.10
100 0.38 1.07 0.07 0.46 0.02 0.13200 0.21 0.65 0.05 0.20 0.02 0.09300 0.12 0.47 0.03 0.13 0.01 0.08500 0.09 0.27 0.02 0.10 0.01 0.04750 0.07 0.17 0.01 0.07 0.01 0.03
1000 0.04 0.12 0.01 0.05 0.01 0.04Av. 0.49 – 0.04 – 0.01 –
Theor. 28.57 – 17.64 – 12.90 –2000 0.02 0.07 0.01 0.03 0.00 0.025000 0.01 0.03 0.00 0.01 0.00 0.01
Table: Percent deviations. M0, M1, M2 = SPT, MSPT-1, MSPT-2.A(m) = mean percent deviation of A from the optimal, A(w) = worsepercent deviation of A from the optimal. Av. = average value for thelines n = 10 to n = 1000, Theor. = theoretical value of the error bound.
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
n OPT M0 M1 M210 0.18 0.02 0.00 0.0625 1.12 0.02 0.02 0.0850 2.96 0.02 0.04 0.6675 6.76 0.04 0.02 0.66
100 10.96 0.04 0.00 1.78200 44.28 0.06 0.14 18.34300 98.86 0.06 0.28 62.00500 268.80 0.08 0.26 404.80750 632.76 0.22 0.44 1 900.38
1000 1 160.28 0.20 0.80 7 904.16Av. 222.70 0.08 0.20 1 029.29
2000 4 234.40 0.54 2.98 154 504.085000 30 511.22 1.60 15.92 5 614 721.00
Table: Mean running time (in ms).
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
Conclusion
DP, SPT, and MSPT-1 already very efficient
MSPT-2 dominated by DP, SPT, MSPT-1
other tests : FPTAS of [Kacem & Mahjoub (2009)],dominated by DP, SPT, MSPT-1
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
References (1)
J. Breit, Improved approximation for non-preemptive singlemachine flow-time scheduling with an availability constraint,European Journal of Operational Research 183 (2007),516–524.
Y. He, W. Zhong, H. Gu, Improved algorithms for two singlemachine scheduling problems, Theoretical Computer Science363 (2006) 257–265.
I. Kacem, A. Ridha Mahjoub, Fully polynomial timeapproximation scheme for the weighted flow-time minimizationon a single machine with a fixed non-availability interval,Computers and Industrial Engineering 56 (2009), 1708–1712.
Introduction Literature review Our contribution : theoretical results Our contribution : experimental results
References (2)
C.-Y. Lee, S. D. Liman, Single machine flow-time schedulingwith scheduled maintenance, Acta Informatica 29 (1992),375–382.
C. Sadfi, B. Penz, C. Rapine, J. B lazewicz, P. Formanowicz,An improved approximation algorithm for the single machinetotal completion time scheduling problem with availabilityconstraints, European Journal of Operational Research 161(2005), 3–10.