Ministère de L Enseignement Supérieur et de la Recherche Scientifique. Université des Sciences et de la Technologie Houari Boumediene

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1 Ministèe de L Enseignement Supéieu et de la Recheche Scientifique Univesité des Sciences et de la Technologie Houai Boumediene Laboatoie de Recheche Opéationnelle, Combinatoie, Infomatique Théoique et Méthodes Stochastiques RECITS Recueil des Résumés Jounées Scientifiques du Laboatoie Alge, Algéie Avil 2014

2 Objectifs : Ces jounées scientifiques sont une oppotunité offete aux checheus du laboatoie, et pincipalement les doctoants, pou communique, échange et confonte leus ésultats avec leus confèes du même domaine de echeche. Elles se veulent une enconte qui pemetta de stimule et de cée un cade favoable pou le développement et la pomotion de leus echeches et de leus idées. Thèmes des jounées (Liste non limitative) : Recheche Opéationnelle Combinatoie Infomatique Théoique Méthodes Stochastiques Comité d oganisation : BELBACHIR Hacène, BOUDHAR Mouad, CHERGUI Mohamed El-Amine, MIHOUBI Miloud, SEDDIKI MERAD Djenat. 1

3 SOMMAIRE Pésentation du laboatoie RECITS (pa Mouad BOUDHAR) 4 Conféences Plénièes : Hacène BELBACHIR Dans quelles evues publie? et analyse scientométique de la poduction Algéienne en mathématique... 5 Adel BELOUCHRANI Time Fequency Aay Signal Pocessing: A synegistic elationship between time fequency methods and senso aay pocessing... 6 Abdelkade BOUYAKOUB Dénombabilité et non dénombabilité. 7 Mohamed el-amine CHERGUI Multiobjective Quadatic Intege pogamming and Applications... 8 Communications : Moussa AHMIA and Hacène BELBACHIR Peseving log-convexity fo s-pascal tiangle. 9 Abdelhakim AIT ZAI, Adel DABAH, M. AMARNI et Mouad BOUDHAR Etude compaative ente AG et PSO appliquées au Poblème d Odonnancement avec Blocage Kaim AMROUCHE and Mouad BOUDHAR Reentant flow shop with an exact time lag Amine BELKHIR and Hacène BELBACHIR Recuence elations and combinatoial identities fo -Lah numbes. 13 Mohamed BENDRAOUCHE and Mouad BOUDHAR Scheduling with ageements: new esults Athmane BENMEZAI and Hacène BELBACHIR A q-analogue fo bisnomials coefficients and genealized Fibonacci sequence. 16 Asma BOUMESBAH and Mohamed El-Amine CHERGUI Une méthode exacte pou la ésolution du poblème de l abe multi-objectif 17 2

4 Imad Eddine BOUSBAA and Hacène BELBACHIR Identités combinatoies des nombes de -Lah Hakim HARIK and Hacène BELBACHIR On Lucas numbe identities Wafaa LABBI, Mouad BOUDHAR and Amma OULAMARA Odonnancement sous containtes de pépaation.. 21 Fatma Zoha OUAIL, Mohamed El-Amine CHERGUI and Mustapha MOULAI Optimizing ove an Intege Efficient Set Yamina SAIDI and Miloud MIHOUBI Some combinatoial popeties of a class of -Stiling numbes of the fist kind 23 Meiem TIACHACHAT and Miloud MIHOUBI A class of the Stiling numbes and the values of the genealized Benoulli polynomials at non-negative integes

5 Pésentation du Laboatoie RECITS Mouad BOUDHAR Laboatoie RECITS, Faculté de Mathématiques, Univesité des Sciences et de la Technologie Houai Boumediene, BP 32, El-Alia, Bab-ezzoua Alge RECITS est un laboatoie de echeche opéationnelle, de combinatoie, d infomatique théoique et de méthodes stochastiques agéé pa l'aêté ministéiel n 242 du 03 avil Il est localisé à la Faculté de Mathématiques de l Univesité des Sciences et de la Technologie Houai Boumediene Alge. Il est diigé pa le Pofesseu BOUDHAR Mouad Le Laboatoie compte 54 pesonnes dont : 2 Pofesseus, 5 Maîtes de Conféences A, 5 Maîtes de Conféences B, 8 Maîtes Assistants A, 9 Maîtes Assistants B et 23 Doctoants LMD. 18 des maîtes assistants et attachés de echeche pépaent une thèse de Doctoat. Le laboatoie est stuctué en cinq équipes de echeche : CATI : Combinatoie, Aithmétique et Infomatique Théoique RO-TOP : Recheche Opéationnelle pou le Tanspot et l Odonnancement en Poductique CEAPA : Combinatoie Enuméative, Analytique, Pobabiliste et Applications STEP : Séies Tempoelles, Econométie et Pobabilités O2M : Optimisation à Objectifs Multiples 4

6 Dans quelles evues publie? et analyse scientométique de la poduction Algéienne en mathématique Hacène Belbachi Laboatoie RECITS, Faculté de Mathématiques, Univesité des Sciences et de la Technologie Houai Boumediene, BP 32, El-Alia, Bab-ezzoua Alge Résumé : Nous donneons un panoama des evues et des gos éditeus dans le monde de la poduction scientifique et pésenteons quelques indicateus les concenant. Dans une seconde phase, nous focaliseons su la poduction en mathématiques avec quelques statistiques écentes. 5

7 Time Fequency Aay Signal Pocessing: A synegistic elationship between time fequency methods and senso aay pocessing Pofesso Adel Belouchani, Ecole Nationale Polytechnique adel.belouchani@enp.edu.dz, peso.enp.edu.dz/~belouchani Abstact: Conventional time-fequency analysis methods wee only ecently extended to data aays, and thee is the potential fo a geat synegistic development of new advanced tools by exploiting the joint popeties of time-fequency methods and aay signal pocessing methods. Conventional aay signal pocessing assumes stationay signals and mainly employs the covaiance matix of the data aay. This assumption is motivated by the cucial need in pactice fo estimating sample statistics by esoting to tempoal aveaging unde the additional hypothesis of egodic signals. When the fequency content of the measued signals is time vaying (i.e., nonstationay signals), this class of appoaches can still be applied. Howeve, the achievable pefomances in this case ae educed with espect to those that would be achieved in a stationay envionment. Instead of consideing the nonstationaity as a shotcoming, time fequency and aay pocessing (TFAP) takes advantage of the nonstationaity by consideing it as a souce of infomation in the design of efficient algoithms in such nonstationay envionments. This talk deals with this synegistic elationship between time-fequency methods and aay signal pocessing methods. The speake plans to addess a boad audience with geneal backgound in mathematics. Biogaphy speake: Adel Belouchani was bon in Algies, Algeia, on May 5, He eceived the State Engineeing degee in 1991 fom Ecole Nationale Polytechnique (ENP), Algies, Algeia, the M.S. degee in signal pocessing fom the Institut National Polytechnique de Genoble (INPG), Fance, in 1992, and the Ph.D. degee in signal and image pocessing fom Télécom Pais (ENST), Fance, in He was a Visiting Schola at the Electical Engineeing and Compute Sciences Depatment, Univesity of Califonia, Bekeley, fom 1995 to He was with the Depatment of Electical and Compute Engineeing, Villanova Univesity, Villanova, PA, as a Reseach Associate fom 1996 to Fom 1998 to 2005, he has been with the Electical Engineeing Depatment of ENP as Associate Pofesso. He is cuently and since 2006 Full Pofesso at ENP. His eseach inteests ae in statistical signal pocessing, (blind) aay signal pocessing, time-fequency analysis and time-fequency aay signal pocessing with applications in biomedical and communications. He is cuently Associated Edito of the IEEE Tansactions on Signal Pocessing. 6

8 Dénombabilité et non dénombabilité Abdelkade BOUYAKOUB Univesité d'oan Es-Sénia, Faculté des Sciences, BP 1524 El-Menawe, , Oan Résumé : Nous paleons des notions d infini, de dénombabilité et de non dénombabilité. On abode ensuite quelques notions de base su la théoie des ensembles pou temine avec l hypothèse du continu. 7

9 Multiobjective Quadatic Intege Pogamming and Applications 1 M.E.A. Chegui 1 and F.Z. Ouail 2 Laboatoie RECITS, USTHB, Faculty of Mathematics, mohamedelaminec@yahoo.com 2Laboatoie LaROMaD, USTHB, Faculty of Mathematics, ouailfatmazoha@yahoo.f In optimization, it is widely known that linea pogamming is a poweful tool to solve many eal wold applications. Nevetheless, models of the eal wold ae not always of a linea fom; de facto, in pat they can be expessed as nonlinea. In addition to that, most eveyday decisions ae made on the basis of conflicting objectives that we want to be as good as possible. Thus, and contay to the case of single objective, finding the optimal solution is no longe the goal but a set of best compomise solutions instead. In this study we deal with nonlinea multiobjective optimization and popose though it an exact method to solve the Multiobjective Quadatic Intege Linea Pogamming Poblem (MQILP). We fist conducted an extensive liteatue eview, looking fo othe studies elated to ou topic but to no avail. Thee is only the Multiobjective Quadatic Assignment Poblem that has been studied using metaheuistics [1] and no exact method exists till now fo the MQILP. Howeve, the most epoted applications with only one objective wee the Quadatic Assignment Poblem [2] and the Quadatic Knapsack Poblem [3]. Also, we have noticed in finance in the pesence of continuous vaiables, the famous Mean-Vaiance Makowitz s Model fo the potfolio selection poblem ([4], [5]). In this pesentation, a genealization of ou method in the linea case [6] is descibed to geneate the entie set of efficient solutions fo MQILP. Based on Banch & Cut technique and Taylo Seies Appoximation of the objective functions, ou method equies only a single optimization pe iteation and exploes the feasible domain smatly. This is made possible by using the inceasing diections of the quadatic functions given by the coesponding gadients, what allows us to delete consideably non efficient solutions without have to compute them. Applications to the Potfolio Selection Poblems ae done. In the 1950s, Makowitz [4] established the moden potfolio theoy and poposed a model selection pocedue of seveal assets, based on statistical citeia, in ode to obtain optimal potfolios. Makowitz showed that the investo seeks to optimize its choice based on isk/etun elationships. Mostly, the "efficient potfolio" is detemined fo a given level of etun. Wheeas, consideing simultaneously the conflicting objective functions is moe judicious. We popose to adapt ou geneal method to solve a modified vesion of the classical Mean-Vaiance (MV) potfolio selection model; that is the Limited Asset Makowitz (LAM) model, noted (P). Indeed, the MV model was modified in the liteatue by seveal authos by imposing new constaints on the numbe of assets to be included and thei weights in the potfolio. Also, due to intege aspect of the decision vaiables, this poblem falls into the class of NP-had poblems and its esolution can be made by using ou method. Refeences 1. Richad O. Day, Mak P. Kleeman and Gay B (2003). Lamont. Solving the Multi-objective Quadatic Assignment Poblem Using a fast messy, Genetic Algoithm, in Poceedings of the 2003 Congess on Evolutionay Computation, Volume 3, pp , IEEE Pess. 2. Kut M. Ansteiche & Nathan W. Bixius (2008). Solving Quadatic Assignment Poblems using Convex Quadatic Pogamming Relaxations, Optimization method and softwae, Volume 16, pp D. Pisinge (2007). The Quadatic Knapsack Poblem- A suvey, Volume 155, Issue 5, pp Makowitz H.M. (1959). Potfolio Selection: Efficient Divesification of Investments. J. Wiley, pp Steue R. E., M. Wimme and M. Hischbege (2013). Oveviewing the Tansition of Makowitz Bi-citeion Potfolio Selection to Ti-citeion Potfolio Selection, J. Business Economics, fothcoming. 6. Mohamed El-Amine Chegui, Mustapha Moulai and Fatma Zoha Ouail (2008), Solving the Multiple Objective Intege Linea pogamming Poblem, Modelling, Computation and Optimization in Infomation Systems and Managment Sciences, Volume 14, pp

10 Peseving log-convexity fo s-pascal tiangle Moussa AHMIA et Hacène BELBACHIR UFAS, Faculty of Sciences, Setif, Algeia USTHB, Faculty of Mathematics, Algies, Algeia ahmiamoussa@gmail.com, hacenebelbachi@gmail.com Abstact Peseving log-convexity popety is established fo the genealized Pascal tiangles. We genealize a esult of H. Davenpot and G. Pólya: "On the poduct of two powe seies", whee they poof that the odinay Pascal tiangle satisfy the peseving log-convexity popety. Key wods : log-convexity, linea tansfomations, odinay multinomials. Let s 1 and n 0 be two integes, and k 0, 1,..., sn, the element n k s tiangle is defined as the k th coefficient in the development n (0.1) 1 x x 2 x s n x k. k k0 s of genealized Pascal A sequence of nonnegative numbes x k k is log-convex (LX fo shot) if x i1 x i1 xi 2 i 0. This is equivalent to x i1 x j1 x i x j fo all j i 1. Let us conside the following two linea tansfomations of sequences (0.2) z n ns k0 n x k k, n 0, s fo all and (0.3) t n ns k0 n x k k y nsk, n 0. s 1. We say that the linea tansfomation (0.2) has the peseving log-convexity popety (by shot PLX) if it peseves the log-convexity of sequences, i.e. the log-convexity of x n implies that of z n 2. We say that the linea tansfomation (0.3) has the double PLX popety if it peseves the logconvexity of sequences, i.e. the log-convexity of x n and y n implies that of t n. We establish ou main esult. Theoem 0.1. We have the following 1. If the sequence of nonnegative numbes x k k is log-convex, then so is the sequence z k k. 2. If the sequences of nonnegative numbes x k k and y k k ae log-convex, then so is the sequence t k k. 9 1

11 Refeences [1] M. Ahmia, H. Belbachi, "Peseving log-concavity and genealized tiangles". K. Takao(ed.), Diophantine analysis and elated fields NY: Ameican Institute of Physics (AIP). AIP Confeence Poceedings 1264, (2010). [2] H. Belbachi, S. Bououbi, A. Khelladi, Connection between odinay multinomials, Fibonacci numbes, Bell polynomials and discete unifom distibution, Annales Mathematicae et Infomaticae, (2008). [3] B. A. Bondaenko, Genealized Pascal tiangles and pyamids, thei factals, gaphs and applications, The Fibonacci Association, Santa Claa, Tanslated fom Russian by R. C. Bollinge (1993). [4] H. Davenpot, G. Pólya, On the poduct of two powe seies, Canadian J. Math. 1, 1 5 (1949). [5] Li. Liu, Yi. Wang, On the log-convexity of combinatoial sequences, Advances in Applied Mathematics vol.39, Issue.4, (2007). [6] N. J. A. Sloane, The online Encyclopedia of Intege sequences, Published electonically at

12 Dépatement d Infomatique, Faculté d Electonique et Infomatique, Univesité des sciences et de la Technologie Houai Boumediene, USTHB. 2 Laboatoie RECITS, Faculté de Mathématiques, Univesité des sciences et de la Technologie Houai Boumediene USTHB,

13 Reentant flow shop with an exact time lag Amouche Kaim 1,2, Mouad BOUDHAR 2 1 Faculté des Sciences Economique et de Gestion, Univesité d Alge 3, Alge, Algéie amouche-kaim@hotmail.com 2 Laboatoie RECITS, Faculté de Mathématiques, Univesité USTHB, B.P. 32, Bab-Ezzoua, El-Alia 16111, Alge, Algéie mboudha@yahoo.f Abstact : This pape focuses on a chain-eentant flow shop with two machines and exact time lag L, in which each task may be pocessed in this ode M 1,M 2,M 1 and thee is an identical time lag between the completion time of the fist opeation and the stat time of the second opeation on the fist machine. The objective is to minimize the total completion time. As the poblem is poved to be NP-Had, we pesent an intege pogam to solve small instances, we popose some heuistics to solve the geneal poblem and we give a subpoblem that can be solved in polynomial time based on the maximum weight matching poblem. Keywods : flow shop, time lags, eentance, makespan, complexity, heuistics. Réféences [1] FD. Vagas-Villamil, DE. Rivea. A model pedictive contol appoach fo eal-time optimization of eentant manufactuing lines. Comput Ind, 45 :45-57, [2] MY. Wang, SP. Sethi, SL. Van De Velde. Minimizing makespan in a class of eentant shops. Ope. Res, 45 : , [3] AJ. Oman, CN. Potts. On the Complexity of Coupled-task Scheduling. Discete Applied Mathematics, 72 :141 54,

14 Recuence elations and combinatoial identities fo -Lah numbes Amine Belkhi Hacène Belbachi USTHB, Faculty of Mathematics RECITS Laboatoy, DG-RSDT Po. Box 32, El Alia, 16111, Algies, Algeia. o hbelbachi@usthb.dz Abstact: We give a new combinatoial intepetation fo -Lah numbes. We also expess -Lah numbes in tems of Lah numbes. Finally, we give an application elated to ising and falling factoial powes. Keywods: Lah numbes, -Lah numbes. 1 Intoduction Fo positive integes n, k, the -Lah numbes, see fo instance [2], count the numbe of patitions of the set 1, 2,...,ninto k odeed lists with the estiction that the elements 1, 2,...,belong to distinct lists. Note that as fo Lah numbes, the -Lah numbes satisfy fo n>, the ecuence elation with n k n k = =1ifn = k = and n n 1 n 1 +(n + k 1) k 1 k =0ifn<. k. (1) Ou motivation is to investigate combinatoial appoach to give two coss ecuence elations fo -Lah and some popeties. 2 Main esults The elation (1) gives a ecuence elation with fixed. Bellow, we establish a coss ecuence elation with espect to. Theoem 1 [1] The -Lah numbes satisfy the ecuence elation n k (n + 2) n 1 =(k +1) (k + 1) k 1 + (n + 2) n 1. (2) (k + 2) k 1 1. The main diffeence with The following theoem gives a coss ecuence elation fo n k elation (2) is that the coefficients appeaing in the ecuence elation ae integes. 13

15 2 A. Belkhi and H. Belbachi Theoem 2 [1] Fo any nonnegative integes 0 k n, we have n n k ( ) n n i 1 = (i + 1)!. (3) k i k 1 1 i=0 As an application of the -Lah numbes, we give a genealized identity consideing ising and falling factoial powes. Theoem 3 [1] Fo any nonnegative integes, 0 k n, we have n n + (x +2) n = x k, (4) k + whee x k = x (x 1) (x k +1) and x k = x (x +1) (x + k 1). k=0 Réféences [1] H. Belbachi and A. Belkhi, Coss ecuence elations fo -Lah numbes, As Combinatoia., 110, , (2013). [2] A. Z. Bode, The -Stiling numbes,discete Math., 49, , [3] G.-S. Cheon, J.-H. Jung, -Whitney numbes of Dowling lattices, Discete Math., 312 (15), ,

16 Schedulingwithageements:newesults MohamedBendaouche 1,2 andmouadboudha 2 1 FacultyofSciences,SaadDahlebUnivesity, RoutedeSoumaaBlida,BP270,Blida,Algeia 2 RECITSLaboatoy,FacultyofMathematics,USTHBUnivesity, BP32ElAlia,BabEzzoua,Algies,Algeia Abstact:Inthiswok,weconsidethepoblemofschedulingwithageements(SWA).This consistsinschedulingasetofnjobsnonpeemptivelyonmidenticalmachinessubjectto constaints that only some specific jobs can be scheduled concuently. These constaints aegivenbyanageementgaphandtheaimistominimizethemakespan.weextendtwo NPhadnessesultsfoSWAinthecaseoftwomachines,pocessingtimesatmost3fo bipatiteageementgaphstoanyclassofageementgaphswithaspecificstuctue.we establishcomplexityesultsfoswainthecaseofsplitandcomplementofbipatitegaphs. Finallysomeinappoximabilityandappoximabilityesultsaeestablished. Themainesultsobtained: Theoem1: TheSWApoblemwithtwomachinesandpocessingtimesin{1,2,3}isNP had fo any class of ageement gaphs G=(A U B,E) whee A is an independent set with A =2a and B is a set of ode 2ab inducing a subgaph with a specific stuctue. The paametesaandbaetwoabitayintegessuchthatbislessoequaltoa. Theoem 2: The SWA poblem fo abitay split ageement gaphs G=(K;S,E), can polynomiallybesolvedino(n 3 )whenthepocessingtimesofthejobsofkaeequaltoone andthoseofsaeabitay. Theoem 3: The SWA poblem with abitay complement of bipatite ageement gaphs, unitpocessingtimesandeleasedatesin{0,1,2}isnphadevenifm=n. Theoem 4: The SWA poblem is polynomial in the case of complement of bipatite ageementgaphs,twodistincteleasetimes0and,identicalpocessingtimespandm=n. 15

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18 Jounées Scientifiques du Laboatoie RECITS, avil 2014 Une méthode exacte pou la ésolution du poblème de l abe multi-objectif Asma BOUMESBAH et Mohamed El-Amine CHERGUI Laboatoie RECITS aboumesbah@usthb.dz, mchegui@usthb.dz Dans note étude, nous nous intéessons à la généation de l ensemble de tous les abes efficaces d un gaphe connexe dont chacune des aêtes est munie d un vecteu de poids de dimension pou toute aête de et. Ce poblème est connu pou ête de type NP-difficile même pou [1]. À note connaissance, peu d appoches de ésolution valides ont été implémentées, paticulièement pou des poblèmes compotant deux objectifs, [2],[4]. Soit un gaphe connexe, d'ode n et de taille m, le modèle mathématique associé au poblème de l'abe multi-objectif s écit : où = ê à ; Une méthode exacte est pésentée pou le poblème de l abe multi-objectif. Elle est basée su un pincipe de sépaation pa appot à des aêtes paticulièes du gaphe, induisant une étape de constuction des containtes du poblème, de poche en poche, dans une stuctue aboescente. Ceci a pou effet de patitionne le gaphe initial en sous-gaphes de petites dimensions, pemettant ainsi d'établi, en chaque feuille f de l'aboescence constuite, le pogamme multi-objectif linéaie discet de echeche des abes efficaces elativement à chaque sous-gaphe (MOST f ). La ésolution du pogamme MOST f fait appel à une méthode exacte, pa exemple celle décite dans [3], généant l'ensemble des solutions efficaces elativement à la feuille f. Les ésultats de l'expéimentation infomatique, en cous de éalisation, su des instances (n,m,) pou des valeus de sont concluants. Réféences [1] P.M. Cameini, G. Galbiati and F. Maffioli, The complexity of weighted multi-constained spanning tee poblems, Colloquium on the Theoy of Algoithms, Pecs, [2] Calos Gomes da Silva and João C. N. Clímaco, A note on the computation of odeed suppoted nondominated solutions in the biciteia minimum spanning tee poblems, Jounal of telecommunications and infomation technology pp.11-15, [3] M.E-A. Chegui, M. Moulaï and F.Z. Ouaïl, Solving the Multiple Objective Intege Linea Pogamming Poblem Modeling, Computation and Optimization in Infomation Systems and Management Sciences Communications in Compute and Infomation Science, Volume 14, Pat 1, 69-76, [4] Host W. Hamache and Günte Ruhe, On spanning tee poblems with multiple objectives, Annals of Opeations Reseach, vol. 52, pp ,

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21 G n = V (G) m = E(G) N(v) vgdeg(v) N(v) Link(v) vp n v 1 v n n 2 v 1 v n v i v i+1 i = 1,, n - 1v 1,,v n vgg - vgvg v G G/e euve v 0 uvv 0 {L n } Ln = L n-1 + L n-2 n 2L n n al 20

22 Odonnancement sous containtes de pépaation Wafaa LABBI 1, Mouad BOUDHAR 2 et Amma OULAMARA 3 1 Faculté des Sciences Economique et de Gestion, Univesité d Alge 3, Alge, Algéie fawalab@yahoo.f 2 Faculté de Mathématiques, Univesité USTHB, B.P. 32, Bab-Ezzoua, El-Alia 16111, Alge, Algéie mboudha@yahoo.f 3 Univesity of Loaine - Ile de Saulcy, 57000, Metz, Fance amma.oulamaa@loia.f Abstact : In this pape, we conside the poblem of scheduling a set of n jobs on two identical machines with pepaation constaints. Each job equies befoe its execution a set of esouces and a non-negligible pepaation time. The objective is to minimize the makespan. This poblem is NP-had. We pove the NP-hadness of two specific cases whee in the fist case pepaation times take only thee values wheeas in the second case pepaation times and the elease dates take only two values, espectively. Then we pesent some special cases and heuistic algoithms along with an expeimental study. Keywods : Scheduling, identical machines, esouces, pepaation times, makespan, complexity, heuistics. Réféences [1] Abdelkhodaee A.H., With A. Scheduling paallel machines with a single seve : some solvable cases and heuistics, Computes & Opeations Reseach, 29(3), , [2] Blazewicz J., Bacelo J., Kubiak W. and Rock H. Scheduling tasks on two pocessos with deadlines and additional esouces. Euopean Jounal of Opeational Reseach, 26, , [3] Blazewicz J., Ecke K.H., Pesch E.,Schmidt G. and Weglaz J. Scheduling Compute and Manufactuing Pocesses. Spinge-Velag, Belin, [4] Hasani K., Kavchenko S.A, Wene F.Block models fo scheduling jobs on two paallel machines with a single seve. Compute and Opeations Reseach, 41, 94-97,

23 Optimizing ove an Intege Efficient Set Fatma Zoha OUAIL 1*, Mohamed El-Amine CHERGUI 1 and Mustapha MOULAI 1 1 Faculty of Mathematics, USTHB BP 32 EL ALIA BAB EZZOUAR, Algies-ALGERIA ouail.fatmazoha@yahoo.f mchegui@usthb.dz mmoulai@usthb.dz ABSTRACT The poblem of optimizing a linea function ove the efficient set of a Multiple Objective Intege Linea Pogamming (MOILP) poblem is known to be difficult, that is, not only fo the discete aspect of the decision vaiables but also due the unknown stuctue of the feasible egion (the efficient set). In geneal, the poblem of finding a most pefeed efficient point can be witten as a mathematical pogamming poblem: whee is a continuous linea function and denote the set of efficient points of MOILP poblem mathematically descibed as: whee, Despite its impotance in eal life applications, a bibliogaphical eseach allowed us to conclude that solving (O) have not yet eceived much attention as it should be. To ou knowledge, the only two methods poposed dealing with this topic wee given in [1] and [2]. In this wok, we popose a new methodology fo solving (O) based on a banch and bound technique. Using efficient cuts and tests, ou appoach succeeds to find the optimal solution fo (O) without having to compute all efficient solutions. Futhemoe and fom expeiencing ou method on seveal instances, we have noticed that ove the hole set of efficient solutions, just a few ones ae found and the algoithm succeeds to find the ight optimal solution of pogam (O). Howeve, as the method exploes the domain though efficient cuts, the moe the numbe of constaint is bigge, the moe the domain becomes smalle, which allows to the method to go faste. Also, the numbe of citeia seems to influence ou method in a good way. That is, if we aise the citeia s numbe, ou algoithm uns moe apidly. Refeences [1] M. Abbas and D. Chaabane, Optimizing a linea function ove an intege efficient set, Euopean Jounal of Opeational Reseach 174, , 2006.O.C. Zienkiewicz, R.C. Taylo, The finite element method, Vol. I, 4th Edition. McGaw Hill, [2] J. Joge, An algoithm fo optimizing a linea function ove an intege efficient set, Euopean Jounal of Opeational Reseach 195, Jounal of Global Optimization 31, 1 16,

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25 JOURNEES SCIENTIFIQUES DU LABORATIORE RECITS, ALGER,21-22 AVRIL A class of the Stiling numbes and the values of the genealized Benoulli polynomials at non-negative integes Meiem Tiachachat 1 Miloud Mihoubi 2 1 Lab: RECITS, USTHB, Faculty of math. 2 Lab: RECITS, USTHB, Faculty of math. tiachachatmeiem@yahoo.f 1 mmihoubi@usthb.dz 2 Abstact : The main object of this pape is to intoduce and study a class of the Stiling numbes on giving some of thei applications on the genealized Benoulli numbes and polynomials. Keywods : The associated -Stiling numbes; the genealized Benoulli polynomials. Classification MSC2010 : 11B73; 11B68; 11B83. 1 Intoduction Recall that, the the -Stiling numbe of the second kind, { } n k, counts the numbe of patitions of the set [n] intok non-empty subsets such that the elements of the set [] aeindiffeent subsets [2], with { } n k 1 = { } n k 0 := { n k} ae the Stiling numbes of the second kind. In a pevious pape [5], the authos showed that B n (α) () can be expessed in tems of the - Stiling numbes of the second kind, whee B n (α) (x) then-th high ode Benoulli polynomials (see fo example [4, 7]). In this pape, we show that the s-aasiociated -Stiling numbes intoduced below ae linked to the values at non-negative integes of the genealized Benoulli polynomials B n [s 1,α] (x) defined ecently by Kut [3] by α B n [s 1,α] (x) tn n! = t s n 0 exp (t) s 1 exp (xt), s 1. t j j! j=0 The paticula case α = 1 was intoduced by Natalini and Benadini [6] (see also [1]). Fo s 1, we define the s-associated -Stiling numbes of the second kind by } s Definition 1 The numbe { n k counts the numbe of patitions of a n-set into k blocks such that the fist elements ae in diffeent blocks and each block fom the othe k blocks is of cadinality s. Fom this definition, we assume that { } n s =0, n < sk o k<, k { } 0 s = δ k,0, n 0 and k { } n s = n, n s. 24

26 Jounées Scientifiques du Laboatoie RECITS, avil 2 M. Tiachachat and M. Mihoubi 2 2 Main esults On using combinatoial aguments, the s-associated -Stiling numbes of the second kind admit an expession given by the following theoem: Theoem 1 Fo n sk s 1, the s-associated -Stiling numbes of the second kind have exponential geneating function { } n + s t n k + n! = 1 t i k exp (t). k! i! i s n k Upon using combinatoial aguments, we can deive seveal thee ecuence elations genealized those of the -Stiling numbes. An application to the genealized Benoulli polynomials is given by the following theoems. Fo, m be non-negative integes and α, y be eal numbes, we give below two expessions in tems of the s-associated -Stiling numbes of the second kind fo B n [s 1,α] (). The following theoem gives a simplified expession fo B n [s 1,α] () fo all non-negative integes. Theoem 2 We have B [s 1,α] n () = n j=0 { (s!) α+j n! n + + sj (n + sj)! j + } s+1 ( α) j and fo α = k be a non-positive intege, we have B [s 1, k] n () = n!k! (n+sk)! { n++sk k+ Othe expession of the genealized Benoulli polynomials can be obtained by using the Melzak s fomula is given by the following theoem. Theoem 3 Let, n, s be non-negative integes such that s 1. Then, we have ( ) n ( ) α + n B n [s 1,α] () =α (s!) α+j n ( 1) j { n!j! n + + sj n j α + j (n + sj)! j + Refeences j=0 [1] G. Betti, P. Natalini, P.E. Ricci, Genealizations of the Benoulli and Appell polynomials. Abst.Appl.Anal.7 (2004) [2] A. Z. Bode, The -Stiling numbes. Discete Math., 49 (1984) [3] B. Kut, A futhe genealization of the Benoulli polynomials and on the 2D-Benoulli polynomials B 2 n (x, y). Appl. Math. Sci. 4 (2010) [4] Y.L. Luke, The Special Functions and Thei Appoximations, vol. I. Academic Pess, New Yok, London, [5] M. Mihoubi and M. Tiachachat, The values of the high ode Benoulli polynomials at integes and the -Stiling numbes. axiv: v1 [math.nt] 23 Jan 2014 [6] P. Natalini, A. Benadini, A genealization of the Benoulli polynomials. J. Appl. Math. 3 (2003) [7] H. M. Sivastava, An explicit fomula fo the genealized Benoulli polynomials. J. Math. Anal. Appl. 130 (1988) } s } s.. 25