A Greedy Randomized Adaptive Search Procedure for Technicians and Interventions Scheduling for Telecommunications

MIC 2007: The Seven th Metaheuristics International Conference -1 A Greedy Randomized A daptiv e Searc h Pro cedu re for T ec hnicians and In terv en tions Sc heduling for T elecomm unications Sylv ain Boussier † Hideki Hashimoto ∗ Mic hel V asquez † ∗ Graduate Sc ho ol of Informatics, Kyoto Univ ersit y Ky oto 606-85 01, Japan hasimoto @amp.i.ky oto-u.ac.jp † LGI2P , Ecole des Mines d’Al ` es P arc S cientifique Georges Besse, F 30035 N ˆ ımes, F rance { Sylvain.Bo ussier,M ichel.Vasquez } @ema.fr 1 In tro duction The su b ject of the 5th c hallenge p rop osed b y the F r enc h S o ciet y of Op erations Research and De- cision Analysis (R OAD EF) consists in scheduling tec hnicians and interv en tions for telecomm un i- cations (see http ://www.g -scop.inp g.fr/ChallengeROADEF2007/ or http://w ww.roadef .org/ ). The aim of this problem is to assign inte rve n tions to teams of tec hnicians. Eac h int erv en tion has a giv en priorit y and the ob jectiv e i s t o minimize z = 28 t 1 + 14 t 2 + 4 t 3 + t 4 where t λ is th e end ing time of th e last scheduled in terv ention of priorit y λ . A schedule is sub jected to the follo wing constrain ts: (1) a team sh ould not c han ge during one d a y , (2) tw o interv en tions assigned on the same day to the same team are done at different times, (3) all predecessors h a ve to b e completed b efore starting an in terv entio n, (4) the working da ys hav e a limited dur ation H max and (5) to w ork on an in terve n tion, a team has to meet the r equiremen ts. Let us note that it is allo w ed to hire an interv ent ion to an external company but the total amount of cost for hired interv ent ions cannot exceed a giv en b u dget A . It is easy to sh o w that this p roblem b elongs to the family of NP-hard p roblems. 2 Description of the algorithm and notations With exp erim entatio n, w e noti ced that some natural criteria lik e the co efficien t ( linke d to the priority of intervention ) in the ob jectiv e f u nction or ev en the ratio coefficient/ duration are not alw ays the more efficien t to insert in terv entions. The main idea of our appr oac h is to find the b est order to insert int erv en tions. The p rop osed algorithm is divid ed in th ree phases: (1) find int erv en tions to b e h ired and delete them from the problem, (2) fi nd the tw o b est orders to insert in terv entio ns, (3) generate solutions with a GRASP s tarting with those t w o insertion orders. In w hat follo w s , Ω t and Ω I are respectiv ely the set of tec hnicians and the set of in terv entions; R ( I , i, n ) is the num b er of tec h nicians of lev el n in domain i requir ed for interv ent ion I ; C ( t, i ) is Mon treal, Canada, June 25–29, 2007 -2 MIC 2007: The Seven th Metaheuristics International Conference the skill lev el of tec hnician t in domain i ; S ( x i , x j ) = 1 if interv en tion i is a p redecessor of j , 0 otherwise; T ( I ) is the dur ation of interv enti on I and cos t ( I ) is the cost for hir in g I . 2.1 Prepro cessing heuristics for hired in terv en t ions The aim of the prepro cessing heuristics is to s elect a su bset of interv en tions to b e hired. The fi rst phase consists in computing a lo w er b ound of the min im u m n um b er of tec hnicians needed for eac h in terv entio n I ( mintec ( I )) b y solving the linear program ( P 1 ( I )). ( P 1 ( I ))      Minimize P t ∈ Ω t x t sub ject to , P t/C ( t,i ) ≥ n,t ∈ Ω t x t ≥ R ( I , i, n ) ∀ i, n , x t ∈ { 0 , 1 } t ∈ Ω t Let us consider w I = mintec ( I ) × T ( I ), then we solv e the p recedence constrained kn ap s ac k p roblem ( K P ) for eac h in terv ention I by considering that the wei gh t of eac h in terv en tion is w I . The subset of interv en tions to b e hired is giv en by the solution of th e p roblem ( K P ): I is hir ed if x I = 1 and I is not hir ed otherwise. ( K P )            Maximize P I ∈ Ω I w I x I sub ject to , P I ∈ Ω I cost ( I ) · x I ≤ A, x I ≤ x j ∀ I , j ∈ Ω I /S ( x I , x j ) = 1 x I ∈ { 0 , 1 } I ∈ Ω I 2.2 Best insertion orders searc h phase A p riorit y order p giv e a wei gh t ω I ( p ) to eac h interv ent ion I . Initially , ω I ( p ) = 28 for in terv entions of p riorit y 1, 14 for priorit y 2, 4 for p riorit y 3 and 1 for p riorit y 4, in that case p = (1 , 2 , 3 , 4). Then w e try sev eral runs of a simple greedy algorithm (which insert the in terven tions with the higher w eigh t first) with th e 24 p ossible p ermutatio ns of th e 4 p riorities of the p roblem ( p = (2 , 3 , 4 , 1), p = (3 , 4 , 1 , 2), p = (3 , 1 , 4 , 2), etc.) and we ke ep th e t wo p ermutations p 1 and p 2 that giv e the b est gr e e dy solution . 2.3 Greedy Randomized Adaptiv e Searc h P ro cedure F or eac h p ermutat ion p = p 1 , p 2 , r u n successiv ely the follo win g algorithm: (1) assign a criteria C I = ω I ( p ) to eac h interv ent ion I , (2) rep eat the follo wing phases until the endin g time is reac h ed: • Greedy phase : select the in terv ention I with the maximum criteria and insert it the earliest da y p ossible, in the team wh ic h requires the less additional tec h nicians to p erform it an d at the minimum starting time p ossible. • Lo cal searc h phase : if the gr e e dy algorithm find a b etter solution, we try to improv e it with lo c al se ar ch . The lo c al se ar c h is divid ed in t wo phases: (1) the critic al p ath phase wh ic h search to decrease endin g times of p riorities ( t λ , λ = 1 , 2 , 3 , 4) and (2) the p acking phase which seeks to sc hedu le interv en tions more efficien tly without increasing the end in g times of p riorities ( t λ ). Mon treal, Canada, June 25–29, 2007 MIC 2007: The Seven th Metaheuristics International Conference -3 instance in t. tec. dom. lev. b est ob j ob j. gap data1-setA 5 5 3 2 2340 2340 0 data2-setA 5 5 3 2 4755 4755 0 data3-setA 20 7 3 2 11880 11880 0 data4-setA 20 7 4 3 13452 13452 0 data5-setA 50 10 3 2 2 8845 28845 0 data6-setA 50 10 5 4 1 8795 18870 0,003 data7-setA 100 20 5 4 3 0540 30840 0,009 data8-setA 100 20 5 4 1 6920 17355 0,025 data9-setA 100 20 5 4 2 7692 27692 0 data10-set A 100 15 5 4 38 296 40020 0,043 data1-setB 200 20 4 4 34395 43860 0,215 data2-setB 300 30 5 3 15870 20655 0,231 data3-setB 400 40 4 4 16020 20565 0,221 data4-setB 400 30 40 3 25305 26025 0,027 data5-setB 500 50 7 4 89700 120840 0,257 data6-setB 500 30 8 3 27615 34215 0,192 data7-setB 500 100 10 5 3330 0 35640 0,065 data8-setB 800 150 10 4 3303 0 33030 0 data9-setB 120 60 5 5 28200 29550 0,045 data10-set B 120 40 5 5 34680 34920 0,006 T able 1: Results obtained on b enchmarks provided by F rance T elecom • Up date phase : Increase the criteria of the last in terv entio ns of eac h priorit y I λ ( λ = 1 , 2 , 3 , 4) and their p redecessors J ∈ P r ed ( I λ ) so that C I λ := C I λ + ω I λ ( p ) and C J := C J + ω I λ ( p ). 3 Computational results Computational results led us to the 1 st p osition in the Junior category an d to the 4 th p osition in All catego ry of the Ch allenge R O ADEF 2007. The results on t w o data sets pr o vided b y F rance T elecom are exp osed in table 1. T he description of the data p er column is the follo win g: instanc e : The n ame of the instance. int. : The num b er of int erv en tions. te c. : The num b er of tec h nicians. do m. : The n um b er of domains. lev. : The n um b er of lev els. b e st obj. : The b est ob jectiv e v alue foun d by all the c hallengers. obj. : Th e ob j ective v alue found by our algorithm. gap. : The gap v alue to our ob jectiv e v alue and the b est one. References [1] Resende MGC, Rib eiro CC. (2003) Greedy rand omized adaptiv e search pro cedures. In : Glo ve r F, Ko c h en b erger G, editors. H andb o ok of M etaheuristics . Dordr ech t: Kluw er Academic Publishers, 219-4 9. Mon treal, Canada, June 25–29, 2007