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

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

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 C i Denoted 1 , h 1 || � i C i

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 , 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)

Introduction Literature review Our contribution : theoretical results Our contribution : experimental results Similar problems (2) 1 , h 1 | preemption | � i C i Same settings with preemption : trivial (SPT)

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

Introduction Literature review Our contribution : theoretical results Our contribution : experimental results Complexity Complexity 1 , h 1 || � i C i is NP-hard [Lee & Liman (1992)] Proof using EVEN-ODD PARTITION EVEN-ODD PARTITION 2 n numbers a 1 , . . . , a 2 n 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 2 i − 1 , x 2 i }| = 1 for all i ?

Introduction Literature review Our contribution : theoretical results Our contribution : experimental results Idea of proof Settings 2 n + 1 jobs M ≪ P two large constants p i = M + a i for i = 1 , . . . , 2 n and p 2 n +1 = P � Z = 1 i a i 2 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 ( 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

Introduction Literature review Our contribution : theoretical results Our contribution : experimental results Approximation algorithms (2) [He et al. (2006)] 2 k -OPT with SPT : O ( n 2 k ) heuristic of relative error 2 √ 5+2 2 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 2 k +8 bound of [He et al. (2006)], and √ 5+2 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 , h 1 || � i w i C i

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

Introduction Literature review Our contribution : theoretical results Our contribution : experimental results Main results Theorem (Improved bound) k +2 An improved error bound of the PTAS MSPT-k is 2 k 2 +8 k +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) p i processing time of job i C i 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 idle time of the machine before the unavailability period δ S schedule obtained by SPT S ′ schedule obtained by MSPT- k S ∗ optimal 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 ∗ 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: � � C ∗ [ j ] + | Y | ( δ − δ ∗ ) . C [ i ] ≤ i ∈ A j ∈ Y 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 � n C ′ C ∗ i + ( | Y | − ( t + 1)) ( δ − δ ∗ ) . ≤ i i =1 i =1

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: � | Y | ( | Y | + 1) � n � C ∗ ( δ − δ ∗ ) ≥ + t i 2 i =1 Lemma 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.

Introduction Literature review Our contribution : theoretical results Our contribution : experimental results The new bound The error bound ε k of MSPT- k satisfies � n i − � n i =1 C ′ i =1 C ∗ 2( | Y | − ( k + 1)) i ε k = � n ≤ | Y | ( | Y | + 1) + 2( k + 1) . i =1 C ∗ i 2( x − ( k +1)) x ( x +1)+2( k +1) , x ∈ N + For all k > 0, the function f k : x �→ f k ( x ) = reaches its maximum for x k = 2 k + 3. Then we have k + 2 | Y |∈ N + ε k ≤ f k ( x k ) = max 2 k 2 + 8 k + 7 . Hence k + 2 2 k 2 + 8 k + 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. k 2 = o ( M ) 3 k + 4 jobs with p i = 1 for i ∈ { 1 , 2 , .., k + 1 } p i = M for i ∈ { k + 2 , .., 3 k + 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 n � � C ′ i = M (2 k 2 +9 k +9)+ o ( M ) and C ∗ i = M (2 k 2 +8 k +7)+ o ( M ) i =1 i =1 � n i − � n i =1 C ′ i =1 C ∗ M ( k + 2) + o ( M ) k + 2 i ⇒ � n = M (2 k 2 + 8 k + 7) + o ( M ) → 2 k 2 + 8 k + 7 i =1 C ∗ i

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

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 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)])