Lagrangean relaxation for minimizing the weighted number of late jobs on parallel machines p.1/18. PMS 2002 Valencia, Spain

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1 Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines PMS 00 Vaencia Spain Stéphane DauzèrePérès and Marc Sevaux Ecoe des Mines de Nantes CNRS IRCCyN Logistic and Production Systes University of Vaenciennes France CNRS LAMIH Production Systes Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.8

2 Parae achine scheduing Set of obs to be sequenced on Preeption is not aowed. Characteristics : reease date : processing tie : due date : weight Variabes : start tie ( ) : copetion tie ( : binary variabe (ob ) is ate or not) parae achines. Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.8

3 Conditions If If is on tie or eary ( is ate or tardy ( ). ). Obective Miniize the weighted nuber of ate obs Standard cassification: Hard in a strong sense [Lenstra et a. 77] ey reark Late obs schedued arbitrariy after obs on tie Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.8

4 # # "! # ( % Copexity resuts is Hard [Garey Johnson 979] #*) Hard [Garey Johnson 979] Heuristics and MIP fouation [Ho Chang 995] Hybrid genetic agortih [Liu et a. 998] ' % $&% Hard [Lenstra et a. 997] Constraint propagation technique [Baptiste et a. 998] Lagrangean reaxation [DauzèrePérès Sevaux 999] Genetic Loca search [Sevaux DauzèrePérès 000] Lagrangean reaxation and B&B [Péridy et a. 00] ' % $&% Two soving ethods Constraint propagation technique [Baptiste et a. 000] Heurisitics and etaheuristics [Sevaux Thoin 00] Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.48

5 + + + The aster sequence Consider the one achine probe. Let be the set of sequences in which every ob is sequenced ust after a ob such that one of the two foowing conditions hods: () or. () and 0 sequenced before. Theore:There is aways an optia sequence (of eary obs) in. A aster sequence is a sequence containing every sequence in. There are at ost 4positions in the aster sequence. Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.58

6 < < < < < < < < JD < < An Exape Jobs For instance the subset of sequences containing 5 obs is: D <@? <@C <BA D ;< <BA <@C <@? D ;< <@C <BA <@? > :;< D <@A <EC <E?F D ;< <EC <@A <E? D ;< <@A <E? <EC ;< D <HA <GC <G?I D ;< <GC <HA <G? ;< <BA <@? <@C ;< D <@? <@C <BA> ;< Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.68

7 M L M L M L + Buiding a Master Sequence Jobs are preordered in nondecreasing order of reease dates FOR each ob DO FOR each ob N such that DO The aster sequence has the foowing for: Every sequence of obs in can be constructed fro. Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.78

8 P Q4 O S R O V XW V T Advantages of using Reduction of the nuber of possibe sequences Subsequence of Subsequence of Vaid for the weighted case with at ost 5 obs with at ost 5 obs Can be reduced by preprocessing UT Guidance for a sart branch and bound foruation can be derived using Lagrangean reaxation Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.88

9 Z ] ] n q r x s w v z N z N s u u Z\[^] Y MIP foruation for 4 4 ih P n p n T o T k kih G T { T { k k gf ih G ced rs t @ _ `B`B`H`B`H`B`B`H`B`H`B`B`H` `B`B`H`B`H`B`B`H`B`H`B`B`H`Ba Q y n S y k k T }~ b k k T }~ b o and ih ˆ ˆŠŠ f r ƒ t t wr v wv Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.98

10 Z ] ] T q 4 r x s w v P z N p z N s u u Z [^] MIP foruation for R Œ 4 n ih kih G T ih n k n T o T k k rs t u G gf c d b T T f _ `B`B`H`B`H`B`B`H`B`H`B`B`H` `B`B`H`B`H`B`B`H`B`H`B`B`H`Ba y n { y { k k T }~ b k k T }~ b o and ih ˆ ˆŠŠ f r ƒ t t wr v wv Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.08

11 ˆ ˆ q T S T q R r x s w v z N 4 z N Lagrangean reaxation By reaxing Constraint (9) the ode becoes: gf x r r Q kih G T ih T ih 5 f f n T o { c d b { k k G rs t u }~ Ž b r T f _ `B`B`H`B`H`B`B`H`B`H`B`B`H`B`H` `H`B`B`H`B`H`B`B`H`B`H`B`B`H`B`Ha n T k y Œ n y Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.8

12 q r x w us v ž ƒ t q 4 r x w us v ž ƒ t ª y ««n. ª y Soving the reaxed probe T š rs t œ gf f c bd r w r rÿ vž o T T T rs tœ w r rÿ vž To iniize () one has to deterine the position and achine with the saest coefficient (depends on the sign of ). Done in tie. n T n n V and n «V coefficient coefficient n Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.8

13 q X. Coputing a ower bound The ower bound is coputed at each iteration by WW f k ˆ f f gf T if T T WW T if o o T T WW T Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.8

14 The Lagrangean reaxation agorith Step Initiaization of the Lagrangean utipiers intiaize the iteration counter. Step Initiaize various paraeters and set y n Step Sove the reaxed probe. Step 4 Copute the ower bound. Step 5 Copute an upper bound by a sequencing the obs in the order of the vaues (greedy agorith). Step 6 Update the Lagrangean utipiers using a subgradient search. Step 7 Check for the stopping conditions if they are not et goto Step.. Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.48

15 Nae First resuts Cpu LB UB Tabu Cpu PB50_* PB50_ PB50_* PB50_ PB50_ PB50_ PB50_ PB50_ PB50_ PB50_ *PB50_: opt 5; PB50_: opt 9. Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.58

16 Nae Cpu LB UB Tabu Cpu PB50_ PB50_ PB50_ PB50_ PB50_5* PB50_6* PB50_ PB50_ PB50_ PB50_ *PB50_5: opt 0; PB50_6: opt. Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.68

17 Nae Cpu LB UB Tabu Cpu PB50_ PB50_ PB50_ PB50_ PB50_ PB50_ PB50_ PB50_ PB50_ PB50_ Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.78

18 Concusion & perspectives Sipe extension of the one achine probe A new ower bound for the Give poor resuts (Lower and upper bound) A sipe insertion heuristic does better than the upper bound Add specific rues to shorten the aster sequence (through a preprocessing step) Find vaid inequaities Copare with the resoution of the sae MILP foruation Extend the set of test probes (if it works!) Lagrangean reaxation for iniizing the weighted nuber of ate obs on parae achines p.88