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

Transcription

1 Minimizing the total flow-time on a single machine with an unavailability period Julien Moncel (LAAS-CNRS, Toulouse France) Jérémie Thiery (DIAGMA Supply Chain, Paris France) Ariel Waserhole (G-SCOP, Grenoble France) Project Management and Scheduling 2 4 April 2012

2 Outline 1 Introduction 2 Literature review 3 Our contribution : theoretical results 4 Our contribution : experimental results

3 And now... 1 Introduction 2 Literature review 3 Our contribution : theoretical results 4 Our contribution : experimental results

4 The problem Settings One machine One unavailability period [R, R + L] No preemption Total flow-time i C i Denoted 1, h 1 i C i

5 The problem Why unavailable? Unavailability = planned maintenance, lunch break, commitment for other tasks, etc.

6 Similar problems (1) 1, h 1 C max Same settings with C max instead of i C i : NP-complete Related to problem PARTITION PARTITION n numbers a 1,..., a n is there a partition I J = {1,..., n} such that i I a i = j J a j? (problem SP12 in the Garey-Johnson)

7 Similar problems (2) 1, h 1 preemption i C i Same settings with preemption : trivial (SPT)

8 And now... 1 Introduction 2 Literature review 3 Our contribution : theoretical results 4 Our contribution : experimental results

9 Complexity Complexity 1, h 1 i C i is NP-hard [Lee & Liman (1992)] Proof using EVEN-ODD PARTITION EVEN-ODD PARTITION 2n numbers a 1,..., a 2n such that a i < a i+1 for all i is there a partition I J = {1,..., n} such that i I a i = j J a j and I {x 2i 1, x 2i } = 1 for all i?

10 Idea of proof Settings 2n + 1 jobs M P two large constants p i = M + a i for i = 1,..., 2n and p 2n+1 = P Z = 1 2 i a i 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

11 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(n 2 ) 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 output the best schedule

12 Approximation algorithms (2) [He et al. (2006)] 2k-OPT with SPT : O(n 2k ) heuristic of relative error k+8 schedule jobs according to SPT try all possible exchanges of k jobs before R with k jobs after R output the best schedule This is a PTAS called MSPT-k 2 We improve the 5+2 bound of [He et al. (2006)], and 2k+8 provide a new bound that is asymptotically tight

13 Other approximation algorithms [Breit (2007)] An O(n log n) parameterized heuristic of best relative error [Kacem & Mahjoub (2009)] An FPTAS for 1, h 1 i w ic i

14 And now... 1 Introduction 2 Literature review 3 Our contribution : theoretical results 4 Our contribution : experimental results

15 Main results Theorem (Improved bound) An improved error bound of the PTAS MSPT-k is k+2 2k 2 +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.

16 Notations (1) p i C i C [i] R L δ S S S processing time of job i completion time of job i completion time of job scheduled at position i starting time of unavailability period duration of unavailability period idle time of the machine before the unavailability period schedule obtained by SPT schedule obtained by MSPT-k optimal schedule

17 Notations (2) S S S schedule obtained by SPT schedule obtained by MSPT-k optimal schedule

18 How to improve the bound (1) Lemma If Ŝ is a schedule better than the SPT schedule S, then δ δ. Remark : the converse is not true

19 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 ).

20 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: Lemma n i=1 C i { Y ( Y + 1) 2 } + t (δ δ ) Let p 1 and q 1 s.t. p q. If it is possible to exchange p jobs of B with q jobs of A, then it is possible to exchange p q + 1 jobs of B with 1 job of A.

21 The new bound The error bound ε k of MSPT-k satisfies ε k = n i=1 C i n n i=1 C i i=1 C i 2( Y (k + 1)) Y ( Y + 1) + 2(k + 1). For all k > 0, the function f k : x f k (x) = 2(x (k+1)) x(x+1)+2(k+1), x N+ reaches its maximum for x k = 2k + 3. Then we have Hence max ε Y N + k f k (x k ) = k + 2 2k 2 + 8k + 7 k + 2 2k 2 + 8k + 7. is an (improved) relative error bound for MSPT-k.

22 Why is the new bound tight? (1) Family of extremal instances k N and M N s.t. k 2 = o(m) 3k + 4 jobs with p i = 1 for i {1, 2,.., k + 1} p i = M for i {k + 2,.., 3k + 4} R = M and L = 1 Such that the SPT schedule is

23 Why is the new bound tight? (2) n i=1 C i = M(2k 2 +9k+9)+o(M) and n i=1 C i = M(2k 2 +8k+7)+o(M) n i=1 C i n n i=1 C i i=1 C i = M(k + 2) + o(m) M(2k 2 + 8k + 7) + o(m) k + 2 2k 2 + 8k + 7

24 And now... 1 Introduction 2 Literature review 3 Our contribution : theoretical results 4 Our contribution : experimental results

25 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 R perc of the sum of the processing times, R perc {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)])

26 n M0(m) M0(w) M1(m) M1(w) M2(m) M2(w) Av Theor 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.

27 n OPT M0 M1 M Av Table: Mean running time (in ms).

28 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

29 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), Y. He, W. Zhong, H. Gu, Improved algorithms for two single machine scheduling problems, Theoretical Computer Science 363 (2006) I. Kacem, A. Ridha Mahjoub, Fully polynomial time approximation scheme for the weighted flow-time minimization on a single machine with a fixed non-availability interval, Computers and Industrial Engineering 56 (2009),

30 References (2) C.-Y. Lee, S. D. Liman, Single machine flow-time scheduling with scheduled maintenance, Acta Informatica 29 (1992), C. Sadfi, B. Penz, C. Rapine, J. B lażewicz, 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.