Minimizing the total flow-time on a single machine with an - - PowerPoint PPT Presentation

Minimizing the total flow-time on a single machine with an unavailability period

Julien Moncel (LAAS-CNRS, Toulouse – France) J´ er´ emie Thiery (DIAGMA Supply Chain, Paris – France) Ariel Waserhole (G-SCOP, Grenoble – France) Project Management and Scheduling 2–4 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...

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Introduction

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Literature review

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Our contribution : theoretical results

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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, commitment for 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|

i Ci

Same settings with preemption : trivial (SPT)

Introduction Literature review Our contribution : theoretical results Our contribution : experimental results

And now...

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Introduction

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Literature review

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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 = 1

2

• 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 2

7

[Sadfi et al. (2005)] 2-OPT with SPT : O(n2) heuristic of relative error

3 17

schedule jobs according to SPT try all possible exchanges of 1 job before R with 1 job after R

• utput 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

2 5+2 √ 2k+8

schedule jobs according to SPT try all possible exchanges of ≤ k jobs before R with ≤ k jobs after R

• utput the best schedule

This is a PTAS called MSPT-k We improve the

2 5+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...

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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+2 2k2+8k+7. This

improves 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 i Ci completion time of job i C[i] completion time of job scheduled at position i R starting time of unavailability period L duration of unavailability period δ idle time of the machine before the unavailability period S schedule obtained by SPT S′ schedule obtained by MSPT-k S∗

• ptimal schedule

Introduction Literature review Our contribution : theoretical results Our contribution : experimental results

Notations (2)

S schedule obtained by SPT S′ schedule obtained by MSPT-k S∗

• ptimal 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 after the 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 scheduled after the period of maintenance in the optimal schedule S∗, then we 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 jobs

• f B with q jobs of A, then it is possible to exchange p − q + 1

jobs 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 = n

i=1 C ′ i − n i=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 → 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. Hence k + 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) and n

• i=1

C ∗

i = M(2k2+8k+7)+o(M)

⇒ n

i=1 C ′ i − n i=1 C ∗ i

n

i=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...

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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 chosen in [1, 100] duration L = mean of job processing times starting time D = proportion Rperc of the sum of the processing 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 0.00 25 0.96 3.92 0.08 0.63 0.00 0.07 50 0.68 2.10 0.07 0.49 0.01 0.09 75 0.43 1.52 0.05 0.32 0.01 0.10 100 0.38 1.07 0.07 0.46 0.02 0.13 200 0.21 0.65 0.05 0.20 0.02 0.09 300 0.12 0.47 0.03 0.13 0.01 0.08 500 0.09 0.27 0.02 0.10 0.01 0.04 750 0.07 0.17 0.01 0.07 0.01 0.03 1000 0.04 0.12 0.01 0.05 0.01 0.04 Av. 0.49 – 0.04 – 0.01 – Theor. 28.57 – 17.64 – 12.90 – 2000 0.02 0.07 0.01 0.03 0.00 0.02 5000 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) = worse percent deviation of A from the optimal. Av. = average value for the lines 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 M2 10 0.18 0.02 0.00 0.06 25 1.12 0.02 0.02 0.08 50 2.96 0.02 0.04 0.66 75 6.76 0.04 0.02 0.66 100 10.96 0.04 0.00 1.78 200 44.28 0.06 0.14 18.34 300 98.86 0.06 0.28 62.00 500 268.80 0.08 0.26 404.80 750 632.76 0.22 0.44 1 900.38 1000 1 160.28 0.20 0.80 7 904.16 Av. 222.70 0.08 0.20 1 029.29 2000 4 234.40 0.54 2.98 154 504.08 5000 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

• ther 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 single

machine 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 single

machine scheduling problems, Theoretical Computer Science 363 (2006) 257–265.

• I. Kacem, A. Ridha Mahjoub, Fully polynomial time

approximation scheme for the weighted flow-time minimization

• n 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 scheduling with scheduled maintenance, Acta Informatica 29 (1992), 375–382.

• C. Sadfi, B. Penz, C. Rapine, J. B

la˙ zewicz, P. Formanowicz, An improved approximation algorithm for the single machine total completion time scheduling problem with availability constraints, European Journal of Operational Research 161 (2005), 3–10.