Minimizing the total flow-time on a single machine with an ......Y. He, W. Zhong, H. Gu, Improved algorithms for two single machine scheduling problems, Theoretical Computer Science

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


And now...

1 Introduction

2 Literature review




The problem

Settings

One machine

One unavailability period [R,R + L]

No preemption

Total flow-time∑

i Ci

Denoted 1, h1||∑

i Ci


The problem

Why unavailable ?

Unavailability = planned maintenance, lunch break, commitmentfor other tasks, etc.


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)


Similar problems (2)

1, h1|preemption|∑

iCi

Same settings with preemption : trivial (SPT)


And now...

1 Introduction

2 Literature review




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 ?


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


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


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


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


And now...

1 Introduction

2 Literature review




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.


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


Notations (2)

S schedule obtained by SPTS ′ schedule obtained by MSPT-kS∗ optimal schedule


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



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∗).



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.


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.


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


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


And now...

1 Introduction

2 Literature review




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)])


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.


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).


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


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.


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.

Documents

Minimizing the total flow-time on a single machine with an ......Y. He, W. Zhong, H. Gu, Improved algorithms for two single machine scheduling problems, Theoretical Computer Science