Fuzzy capacity planning for an helicopter maintenance center

Transcription

1 Fuzzy capaciy planning for an helicoper mainenance cener Malek Masmoudi To cie his version: Malek Masmoudi. Fuzzy capaciy planning for an helicoper mainenance cener. i4e. Inernaional Conference on Indusrial Engineering and Sysems Managemen, May 0, Mez, France. (), pp <hal > HAL Id: hal hps://hal.archives-ouveres.fr/hal Submied on 6 Mar 0 HAL is a muli-disciplinary open access archive for he deposi and disseminaion of scienific research documens, wheher hey are published or no. The documens may come from eaching and research insiuions in France or abroad, or from public or privae research ceners. L archive ouvere pluridisciplinaire HAL, es desinée au dépô e à la diffusion de documens scienifiques de niveau recherche, publiés ou non, émanan des éablissemens d enseignemen e de recherche français ou érangers, des laboraoires publics ou privés.

2 Inernaional Conference on Indusrial Engineering and Sysems Managemen IESM 0 May 5 - May 7 METZ - FRANCE Fuzzy capaciy planning for an helicoper mainenance cener Malek MASMOUDI a, Alain HAIT a a Universié de Toulouse, Insiu Supérieur de l Aéronauique e de l Espace, 0 avenue Edouard Belin, F-3055 Toulouse Absrac Aircraf mainenance coss are becoming an imporan issue in he aeronauical indusry. In his paper we presen a acical planning model for an helicoper mainenance cener. The objecive is o guaranee a good service level, i.e. o limi aircraf visi duraion. Difficulies come from he numerous uncerainies on mainenance aciviies: addiional aciviies, procuremen delays... To cope wih hese uncerainies, a fuzzy muliprojec planning model is proposed. From he ask fuzzy daes and duraions, a periodic load char can be esablished. Then a parallel algorihm is adaped o his model in order o solve capaciy problems. Key words: Mainenance, Muliprojec, Fuzzy, Capaciy planning Inroducion Aircraf mainenance is a highly regulaed aciviy, due o he poenial criicaliy of he failures. Aircrafs mus follow a mainenance program in which several levels of inspecion appear, from ligh mainenance ha can be performed daily a he aircraf s basis, o heavy mainenance ha can las several monhs and requires specific equipmen. Our work focuses on he organizaion of a helicoper mainenance cener where heavy mainenance visis (HMV) are performed. An HMV conains planned mainenance asks and also correcive mainenance asks because problems are discovered during he inspecion of he helicoper a he beginning of he visi. Even planned asks may differ from one helicoper o anoher, according o equipmen, condiions of use, ec. Precedence consrains exis, due o echnical or accessibiliy consideraions. Hence a HMV may be seen as a projec involving various resources as operaors, equipmen and spare pars. Minimizing he overall visi duraion give a compeiive advanage o he company. Consequenly, he managemen of a mainenance cener is viewed as muliprojec managemen, where every projec duraion should be minimized while respecing capaciy consrains. A pariculariy of hese projec is he level of uncerainy, mainly due o unexpeced failures ha induce addiional work and procuremen delays. In case of imporan homogeneous flees, a global opimizaion of mainenance visis can be done, guaraneeing a general helicoper availabiliy level [8]. This is no he case in our projec dedicaed o civil cusomers whose mean number of helicoper is beween wo and hree, wih a grea heerogeneiy in he equipmens and condiions of use. This work is graned by he Helimainenance projec (FUI-Aerospace Valley). This paper was no presened a any oher revue. Corresponding auhor M. Masmoudi. addresses: [email protected] (Malek MASMOUDI), [email protected] (Alain HAIT).

3 IESM 0, METZ - FRANCE, May 5 - May 7 This paper deals wih acical planning for an helicoper mainenance cener. A a acical level, mainenance operaions are gahered ino macro-asks, and resources are considered by period in a acical horizon ha may cover several projecs from heir beginning o heir end. Given he level of uncerainy and he lack of hisorical informaion, a fuzzy model of he ask daes and duraions has been chosen []. This paper aims a solving capaciy problems a he acical level, based on a fuzzy represenaion of resource workload. Secion recalls fuzzy projec planning and describes he saring dae and ask duraion models chosen for our mainenance projecs. In Secion 3 we presen he way o build he resource workload chars from he ask fuzzy daes and duraions. Secion 4 presens a parallel algorihm adaped o our fuzzy model in order o solve capaciy issues. Finally, secion 5 presens an example of applicaion on hree helicoper mainenance projecs. Muliprojec planning under uncerainy. Fuzzy projec planning Zadeh [3] has defined a fuzzy se à as a subse of a referenial se X, whose boundaries are gradual raher han abrup. Thus, he membership funcion µã of a fuzzy se assigns o each elemen x X is degree of membership µã(x) aking values in [0; ]. To generalize some operaions from classical logic o fuzzy ses, Zadeh has given he possibiliy o represen a fuzzy profile by an infinie family of inervals called α-cus. Hence, he fuzzy profile à can be defined as a se of inervals A α = [A α min ; Aα max] = {x X/µÃ(x) α} wih α (0; ]. I became consequenly easy o uilize classical inerval arihmeic and adap i o fuzzy profiles. Dubois and Prade [5] and Chen [] have defined mahemaical operaions ha can be performed on rapezoidal fuzzy ses. Forin e al. [6] describe he algorihms o adap Criical Pah Mehod o fuzzy numbers. In he fuzzy case, forward propagaion is done using fuzzy arihmeics, leading o fuzzy earlies daes and a fuzzy end-of-projec even. Unforunaely, classical backward propagaion is no longer applicable because uncerainy would be aken ino accoun wice. Forin e al. propose algorihms and show ha some problems (e.g. minimal floa deerminaion) become NP-hard. A recen review of hese conceps can be found in Dubois e al. [4].. Projec release dae Figure presens an example of an equipmen inspecion dae deerminaion from helicoper exploiaion assumpions, fligh hours and calendar limis. From he updae, fligh hours evolve in a range going from no exploiaion o he physical limis of he aircraf, hrough pessimisic and opimisic exploiaion values. Inersecions of hese lines wih calendar and fligh hours limis define he four poins a H, b H, c H and d H of he rapezoidal fuzzy number H, inspecion dae according o he fligh hours. I is he same for fligh cycles. Fligh hours µ H() updae Physical limi Calendar limi Fligh hour limi No exploiaion ime a H b H c H d H Fig.. Equipmen fuzzy inspecion dae The fuzzy release dae of he projec is he fuzzy minimum of he inspecion daes of criical equipmens lised in he mainenance program, and of he helicoper iself. The uncerainy on his dae decreases along he ime, as informaion on acual exploiaion increases, so periodic updaes should be done.

4 IESM 0, METZ - FRANCE, May 5 - May 7.3 Macro ask duraions A he acical level, uncerainy on macro ask duraion is mainly due o unexpeced correcive mainenance. These addiional asks (work and delays) can represen an imporan par of he oal projec duraion. They generally appear during he srucural inspecion macro asks, bu he whole projec is impaced. Procuremen for correcive mainenance may inroduce delays in he planning. As he equipmens o be purchased are no known before inspecion, we consider scenarios: he equipmen is available on sie, a an European supplier, a a foreign supplier, or i may be found afer some research, or i is obsolee and mus be manufacured again. According o he informaion on he helicoper (age of he aircraf, condiions of use, ec.), some scenarios can be discarded from he beginning (e.g. new helicoper no obsolescence) and ask duraions can be refined. Hence ask duraions will be represened by rapezoidal fuzzy numbers ha may change a he planning updaes. 3 Resource load chars 3. Possibiliy heory To cope wih decision making on fuzzy area, Zadeh [4] developed he concep of he possibiliy approach based on fuzzy subses. The possibiliy heory inroduces boh a possibiliy measure (denoed Π) and a necessiy measure (denoed N), in order o express plausibiliy and cerainy of evens [5]. Le τ be a variable in he fuzzy inerval à and be a real value. To measure he ruh of he even τ, equivalen o τ ( ; ], we need he couple Π(τ ) and N(τ ) (Fig. ). Thus : Π(τ ) = sup µã(u) = µ [Ã;+ ) () = sup min(µã(u), µ ( ;] (u)) u u () N(τ ) = sup µã(u) = µ ]Ã;+ ) () = inf u> u à ( ;](u)) () Π(τ ) µã a A b A c A d A N(τ ) ( µã) µã a A b A c A d A Fig.. Possibiliy and Necessiy of τ wih τ Ã. Consequenly, le τ and σ wo variables in respecively fuzzy inervals à and B and a real value. To measure he ruh of he even beween τ and σ we need boh Π(τ σ) and N(τ σ). Thus: Π(τ σ) = µ () = µ () = min(µ (), µ [Ã; B] ()) [Ã;+ ) ( ; B] [Ã;+ ) ( ; B] (3) N(τ σ) = µ () = µ () = min(µ (), µ ]Ã; B[ ()) ]Ã;+ ) ( ; B[ ]Ã;+ ) ( ; B[ (4) µ [Ã; B] µã µ B µ ]Ã; B[ a A b A c A d A a B b B c B d B 3. Presence of a ask Fig. 3. Possibiliy and Necessiy of being beween à and B. The projec daes and duraions are represened by rapezoidal fuzzy numbers. Le S(a S, b S, c S, d S ) be he fuzzy sar dae of a ask T, F (a F, b F, c F, d F ) is finish dae and D(w, x, y, z) is duraion. Relaions beween hese values are: a F = a S + w, b F = b S + x, c F = c S + y, d F = d S + z

5 wih IESM 0, METZ - FRANCE, May 5 - May 7 a S b S c S d S and w x y z. We characerize he presence of a ask by he possibiliy (denoed Π()) and necessiy (N()) of even being beween he sar dae and he finish dae of he ask. Then we define he probabiliy of presence of a ask as a piecewise linear disribuion p() siuaed beween he possibiliy and he necessiy profile: N() p() Π(). The shapes of hese profiles vary according o he overlap configuraion of sar and finish dae (Fig. 4). Parameer H, ranging in [0, ], makes profile p() evolve from N() (H = 0) o Π() (H = ). The formal definiions of hese profiles have been presened in []. H H H Π() p() N() a S b S c S d S a F b F c F d F a S b S c S d S a F b F c F d F a S b S c S d S a F b F c F d F Fig. 4. Task presence profiles: wihou overlap (op), wih small (middle) and large overlap (boom). 3.3 Fuzzy resource usage profile Building a relevan resource usage profile for a ask wih fuzzy daes and duraions is no sraighforward. Mos of he ime, he problem parameers are fixed in order o obain a deerminisic configuraion. This leads o a scenario approach [0] where various significan scenarios may be compared in a decision process: lower and upper bounds, mos plausible configuraion, ec. We proposed in [] o build ask resource usage profiles in a way ha keeps rack of uncerainy on sar and finish daes. Hence he profile reflecs he whole possible ime inerval while giving a plausible repariion of he workload according o he duraion parameer value. To his aim, he resource usage profiles are defined as a projecion of he ask presence disribuions ono he workload space. The projecion of he possible profile should hen give he maximal resource profile L Π (), and he necessary profile he minimal resource profile L N (). As he surface of hese exreme resource profiles does generally no correspond o exreme loads r.w and r.z (where r is he resource requiremen of he ask, w he minimum and z he maximum duraion), we use probabiliy profiles o mach he exac loads. Figure 5 presens he exreme resource profiles L w () and L z () for he case wihou overlap (he oher cases can be found in []). To deermine hese profiles, parameers H w and H z are calculaed so ha + 0 r.p()d respecively equals o r.w and r.z. 3.4 Precedence consrains If he asks were independen, he sum of heir resource load profiles would give he overall projec load char. However, when considering a precedence consrain beween wo asks, heir load profiles may no overlap because he consrain expresses he fac ha he wo asks canno be performed simulaneously. Le us consider wo asks A and B so ha A precedes B. Their resource consumpions are denoed r A and r B. We assume ha he sar dae of B is equal o he finish dae of A (e.g. in case of forward earlies daes calculaion). This means ha beween he sar dae of A and he finish dae of B, an aciviy will occur successively induced by A hen B. So beween he necessiy peaks of A and B, we can affirm ha an aciviy will necessarily occur, induced by A or B. This necessary presence of A or B is projeced ono he resource load space using he minimal resource requiremen min(r A, r B ). Figure 6 presens an example of his case, and he load profiles L Π(A B) () and L N(A B) () for r A = and r B =.

6 IESM 0, METZ - FRANCE, May 5 - May 7 H r Π() p() N() a S b S c S d S a F b F c F d F w z r.h z r.h w a S b S c S d S a F b F c F d F Fig. 5. Task presence (op) and resource load (boom) profiles. A B A B Load Fig. 6. Task presence (op) and resource load (boom) profiles. The projeced necessiy and possibiliy load profiles of he sequence A B can be defined as follow: L N(A B) () = max(r A.N A (), r B.N B (), min(r A, r B ).N A B ()) L Π(A B) () = max(r A.Π A (), r B.Π B ()) Again, we should check if he resource load profiles mach wih he effecive load for various duraions and adap probabiliy profiles o his aim. Moreover, his should be checked globally on he sequence because of he area where we do no know which ask is effecively execued. Consequenly, he load profiles should be checked for each pah in he projec graph. Furher work will sudy his aspec. 3.5 Global resource load char A he acical level, planning decisions are aken according o he capaciy of some criical resources: i is called rough cu capaciy planning. The planning horizon is decomposed in periods on which he load is evaluaed and compared o he available capaciy. The fuzzy load char is esablished by periods, using he ask load profiles. For each period, four values are given, corresponding o scenarios wih he four duraion values w, x, y, z of he asks. Load char can be represened like he workload plan suggesed by Grabo e al. [7] for fuzzy MRPII (Fig. 7). 4 Capaciy planning Schedule Generaion Schemes (SGS) are he core of many heurisics for he RCPSP. The so-called serial SGS performs aciviy incremenaion and he parallel SGS performs ime incremenaion []. In boh procedures,

7 IESM 0, METZ - FRANCE, May 5 - May 7 R 3 4 Fig. 7. Fuzzy load char asks are ranked in some order and scheduled according o resources availabiliies. Hapke and Slowinski [0] have proposed a parallel scheduling procedure for fuzzy projecs. I is based on fuzzy prioriy rules and fuzzy ime incremenaion. The parallel procedure ha we propose mainly differs from he laer on he resource availabiliy es. Acually, as a acical capaciy planning ool, our es relies on a periodic resource load char where availabiliy is aken as a whole on each period. 4. Parallel algorihm The fuzzy parallel procedure is adaped from he parallel SGS by considering fuzzy daes and fuzzy number comparison. I can be described as follows: Begin Choose a prioriy rule; Iniialize (resource, period) capaciy value; Iniial ime := projec saring dae; Repea Sep : compose he se Q( ) of asks ready for scheduling a ; Sep : Schedule a, according o he prioriy rule, each ask from Q( ) ha respecs resource availabiliy; When a ask is scheduled, calculae is finishing dae, updae he earlies saring dae of is direc successors and he resource availabiliies; Sep 3: increase ime ; Unil all asks are scheduled End 4.. Fuzzy prioriy rules Prioriy heurisics using crisp or fuzzy ime parameers were found efficien by many researchers eiher for one projec or muliprojec scheduling [,0,]. I is generally useful o perform parallel scheduling wih a se of rules insead of one as he compuaional complexiy is low [9,0]. Some rules ha appears o be good in minimizing makespan are presened in Table 3. The aim of his paper is no o find he bes rule, oherwise many oher ineresing rules could be used, like he Minimum Wors Case Slack (MINWCS), he Minimum Toal Work Conen(MINTWC) and some dynamic and combined rules presened in []. 4.. Time incremenaion and resource availabiliy A ask is ready o schedule a ime when all is predecessors have been compleed a ime. In Deerminisic parallel SGS, a dynamic ime progression is used. When a ime no ask can be scheduled, curren ime is

8 IESM 0, METZ - FRANCE, May 5 - May 7 Table Prioriy rules giving good resuls in makespan minimisaion Rule Name Formula Rule Name Formula EST Early Sar Time min(ẽs j ) LIS Leas Immediae Succesors min( S j ) EFT Early Finish Time min(ẽf j ) MIS Mos Immediae Succesors max( S j ) LST Lae Sar Time 3 min( L s j ) MTS Mos Toal Successors3 max( S j ) LFT Lae Finish Time 3 min( L f j ) GRD Greaes Resource Demand p j Kk= r jk MINSLK Minimum slack 3 min( f j ) SASP Shores Aciviy from Shores Projec 3 min( p jl ) MAXSLK Maximum slack 3 max( f j ) LALP Longes Aciviy from Longes Projec 3 max( p jl ) SPT Shores Processing Time 3 min( p j ) GRPW Greaes Rank Posiional Weigh 3 max( p j + i S j p i ) LPT Longues Processing Time 3 min( p j ) LRPW Leas Rank Posiional Weigh min( p j + i S j p i ) : used by Slowinski in [0] for a Fuzzy RCPSP, : used by Kolish in [] for Deerminisic RCPSP, 3 : used by Browning in [] for Muli-projecs RCPSP (RCMPSP), p j :duraion, L f j :Las finishing, Ẽf j :Earlies finishing, L s j :Las saring, Ẽs j :Earlies saring, f j :Margin r jk : is he requiremen for resource R k, S j : direc successors, S j : oal successors increased o he leas value from he finishing imes of scheduled asks ha are no ye finished and he saring imes of asks ha are candidaes for scheduling. To cope wih fuzzy scheduling, Hapke and Slowinski have generalized he deerminisic ime progression o fuzzy area and compare fuzzy numbers using weak and srong inequaliies. A firs assumpion gives ha a ask has been compleed a ime if is srongly greaer han or equal o he fuzzy finish ime of he ask. When a ime no ask can be scheduled, he ime is increased o he earlies dae, in he sense of weak inequaliy, from he se of hose ones in which any resource is released or any ask is ready o be scheduled [0]. Moreover, due o heir resource represenaion, Hapke and Slowinski may implicily add precedence consrains beween asks o solve capaciy problems. However in a acical poin of view, i can someimes be acceped o mainain a se of asks scheduled simulaneously when uncerainy is large, wihou deciding oo early which sequence will be respeced. So in our algorihm, resource needs are considered globally on he ime periods, using ask resource load profiles. 4. Fuzzy ask preempion Preempion can be a way o solve resource capaciy problems a an aggregaed level of planning. In case of deerminisic projecs, preempion is provided by cuing macro-asks ino elemenary work pars [3]. Obviously, he elemenary duraion value is unique in he deerminisic case and is equal o. Thus, any deerminisic duraion is a muliplicaion of. In he same way, any rapezoidal fuzzy number à = [a, b, c, d] is equal o a unique linear combinaion of he elemenary numbers Ĩ0=[,,, ], Ĩ=[0,,, ], Ĩ=[0, 0,, ] and Ĩ3=[0, 0, 0, ], lised from he mos necessary o he less possible equal o (Fig. 8): à = aĩ0 + (b a)ĩ + (c b)ĩ3 + (d c)ĩ4 (5) µĩ0 (x) µĩ (x) µĩ (x) µĩ3 (x) 0 x 0 x 0 x 0 x Fig. 8. Elemenary rapezoidal fuzzy numbers The decomposiion formula () is applied o asks fuzzy duraions in AOA graph. The elemenary arcs are assigned in he order of hem possibiliy o be equal o. Thus, he Ĩ0 are assigned firs, hen he Ĩ, afer ha he Ĩ and finally he Ĩ3(Fig. 9). For example, he duraion of ask (34) on he lef graph (before preempion) is equal o [,, 3, 3]. According o he formula (5), we have 34 = Ĩ0 + Ĩ + Ĩ. Thus, he ask (34) can be replaced in he righ graph (afer preempion) by (45), (56) and (56) wih 45 = Ĩ0, 56 = Ĩ and 67 = Ĩ.

9 IESM 0, METZ - FRANCE, May 5 - May 7 [0,,, ] [0,,, ] [0, 0, 0, ] [,,, ] [,, 3, 3] 4 [0, 0,, ] [,,, ] 5 [,,, ] [0, 0,, ] 3 [0,,, ] 6 5 [,,, ] [0, 0,, ] 7 [,,, ] 8 [0, 0,, ] [0, 0, 0, ] Fig. 9. Task preempion 5 Experimenal resuls The parallel algorihm have been applied o he helicoper mainenance planning problem. In helicoper mainenance, hree caegories of human resources are considered: avionics, srucure and mechanics expers. They work generally on various helicopers a he same ime. Table conains he daa of an example wih hree projecs. Table Example of daa for helicoper HMV. Tasks Name Tasks Id Projec Projec Projec 3 Predecessors Resources Duraion Duraion Duraion R R R3 Waiing for he release dae A [7, 8, 9, 0] [0,,, 3] [7, 8, 9, 0] Firs check when receiving he helicoper B [] [] [] A Removal srucural and mechanical pars C [3] [3] [3] B Removal avionics D [3] [3] [3] B Supplying procedure for finishing E [3,3,5,6] [,,,] [,,,3] C Firs par of mechanical inspecion F [7] [5] [3,3,3,3] C Supplying procedure for assembling ask G [5,5,5,6] [,,,3] [,3,4,4] C Supplying procedure during srucural inspecion H [5,5,5,6] [,,,3] [,3,4,4] C Subconraced srucure-cleaning I [] [] [] C Subconraced avionic ess and repairs J [,3,4,5] [,3,4,5] [,3,4,5] D Firs par of srucural inspecion K [] [7] [5] I Second par of srucural inspecion L [,,3,4] [,, 3, 4] [,, 3, 4] H-K Subconraced paining M [] [,,, ] [] L Second par of mechanical inspecion N [] [,,,] [] F Assemble helicoper pars O [] [] [] G-J-M-N Finishing before fly es P [] [] [] E-O Tes before delivering helicoper Q [] [] [] P Possible addiional work on helicoper R [,,, 3] [,,, 3] [,,, 3] Q The objecive is o minimize he immobilizaion of helicopers i.e. he makespan of each projec. The capaciy is consan a each period and equal o 3, and 3 for, respecively, R (Mechanics exper), R(Avionics exper) and R3(Srucure exper). The use of he 6 fuzzy prioriy rules presened in Table 3 gives differen planning wih differen projec makespans. Planning is performed wih preempion and wihou preempion for opimisic (smalles duraions) and pessimisic (larges duraions) cases. The Fig. 0 shows he load char of he earlies planning wih infinie capaciy and he bes resul obained for each case using our parallel algorihm. 6 Conclusion This paper have presened a fuzzy capaciy planning approach for helicoper mainenance. A parallel algorihm has been adaped o consider resource consrains hrough a periodic workload represenaion ha accouns for he ask fuzzy daes and duraions. We also proposed o include fuzzy ask preempion o he algorihm. Fuure work will be dedicaed o model improvemen and validaion. A firs, resource load profiles should consider precedence consrains in order o ge more realisic load chars. Then he capaciy planning approach will be used wih real daa from Helimainenance projec in order o compare he resuls, following he updaes of he planning along he ime. Finally, his approach should be included in a broader decision suppor sysem for an helicoper mainenance cener.

10 IESM 0, METZ - FRANCE, May 5 - May 7 Table 3 Example of ask sar and finish imes for MIS rule in pessimisic case wihou preempion. Task Id Fuzzy sar ime fuzzy finish ime projec projec projec3 projec projec projec3 A [0, 0, 0, 0] [0, 0, 0, 0] [0, 0, 0, 0] [7, 8, 9, 0] [0,,, 3] [7, 8, 9, 0] B [7, 8, 9, 0] [0,,, 3] [7, 8, 9, 0] [8, 9, 0, ] [,, 3, 4] [8, 9, 0, ] C [8, 9, 0, ] [3, 5, 7, 9] [3, 4, 5, 6] [,, 3, 4] [6, 8, 0, ] [6, 7, 8, 9] D [8, 9, 0, ] [,, 3, 4] [8, 9, 0, ] [,, 3, 4] [4, 5, 6, 7] [,, 3, 4] E [,, 3, 4] [6, 8, 0, ] [6, 7, 8, 9] [4, 5, 8, 0] [7, 9,, 4] [7, 9, 30, 3] F [, 3, 4, 5] [0,,, 3] [6, 7, 8, 9] [9, 0,, ] [5, 6, 7, 8] [9, 30, 3, 3] G [,, 3, 4] [6, 8, 0, ] [6, 7, 8, 9] [6, 7, 8, 0] [7, 0,, 5] [8, 30, 3, 33] H [,, 3, 4] [6, 8, 0, ] [6, 7, 8, 9] [6, 7, 8, 0] [7, 0,, 5] [8, 30, 3, 33] I [,, 3, 4] [6, 8, 0, ] [6, 7, 8, 9] [, 3, 4, 5] [7, 9,, 3] [7, 8, 9, 30] J [,, 3, 4] [4, 5, 6, 7] [,, 3, 4] [3, 5, 7, 9] [6, 8, 0, ] [3, 5, 7, 9] K [, 3, 4, 5] [7, 0,, 5] [3, 3, 36, 38] [3, 4, 5, 6] [4, 7, 9, 3] [36, 37, 4, 43] L [5, 6, 7, 8] [9, 30, 3, 3] [35, 37, 4, 44] [6, 7, 30, 3] [30, 3, 34, 36] [36, 38, 44, 48] M [6, 7, 30, 3] [30, 3, 34, 36] [36, 38, 44, 48] [7, 8, 3, 33] [3, 3, 36, 38] [37, 39, 45, 49] N [9, 0,, ] [7, 8, 9, 30] [9, 30, 3, 3] [0,,, 3] [8, 9, 30, 3] [30, 3, 3, 33] O [7, 8, 3, 33] [3, 3, 36, 38] [37, 39, 45, 49] [8, 9, 3, 34] [3, 33, 37, 39] [38, 40, 46, 50] P [8, 9, 3, 34] [3, 33, 37, 39] [38, 40, 46, 50] [9, 30, 33, 35] [33, 34, 38, 40] [39, 4, 47, 5] Q [9, 30, 33, 35] [33, 34, 38, 40] [39, 4, 47, 5] [30, 3, 34, 36] [34, 35, 39, 4] [40, 4, 48, 5] R [30, 3, 34, 36] [34, 35, 39, 4] [40, 4, 48, 5] [3, 33, 36, 39] [35, 37, 4, 44] [4, 44, 50, 55] Prioriy lis (rule MIS): C C C 3 B B B 3 A D E F G H I J K L M N 0 P Q A D E F G H I J K L M N O P Q A 3 D 3 E 3 F 3 G 3 H 3 I 3 J 3 K 3 L 3 M 3 N 3 O 3 P 3 Q 3 R R R 3 Sequence in scheduling: A A A 3 B C D B D J E F H I G K J C B 3 E F H I D 3 N K G J 3 C 3 L E 3 F 3 G 3 H 3 I 3 N M O L N 3 P Q R M O K 3 P Q R L 3 M 3 O 3 P 3 Q 3 R 3 References Fig. 0. Bes Resuls wih and wihou preempion for opimisic and pessimisic cases [] Tyson R. Browning and Ali A. Yassine. Resource-consrained muli-projec scheduling: Prioriy rule performance revisied. Inernaional Journal of Producion Economics, 6(): 8, 00. [] Shu-Jen Chen and Ching-Lai Hwang. Fuzzy ses. Fuzzy muliple aribue decision making: Mehods and applicaions, 375, 99. [3] Ronald de Boer. Resource-consrained muli-projec managemen -a hierarchical decision suppor sysem. Thesis, BETA Insiue for Business Engineering and Technology applicaion, 998. [4] Didier Dubois, Jérôme Forin, and Pavel Zieliński. Inerval PERT and is fuzzy exension. In Sudies in Fuzziness and sof compuing, volume 5/0, pages Springer, 00. [5] Didier Dubois and Henri Prade. Possibiliy heory: an approach o compuerized processing of uncerainly. Inernaional Journal of General Sysems, 988. [6] Jérôme Forin, Pavel Zieliński, Didier Dubois, and Hélene Fargier. Inerval analysis in scheduling. In Proc. h Inernaional

11 IESM 0, METZ - FRANCE, May 5 - May 7 Conference on Principles and pracice of consrain programming, Lecure Noes in Compuer Science, volume 3709, pages 6 40, 005. [7] Bernard Grabo, Lauren Genese, Gabriel Reynoso Casillo, and Sophie Véro. Inegraion of uncerain and imprecise orders in he MRPII mehod. Inernaional Journal of Inelligen Manufacuring, 005. [8] R.A. Hahn and A. M. Newman. Scheduling unied saes coas guard helicoper deploymen and mainenance a clearwaer air saion. Compuers and Operaions Research, 35(6):89 843, 008. [9] Maciej Hapke and Roman Slowinski. A dss for resource consrained projec scheduling under uncerainy. Journal of Decision Sysems, (): 7, 993. [0] Maciej Hapke and Roman Slowinski. Fuzzy prioriy heurisics for projec scheduling. Fuzzy Ses and Sysems, Vol83, pp. 9-99, 996. [] Rainer Kolish and Sonke Harmann. Heurisic algorihms for solving he resource-consrained projec scheduling problem: Classificaion and compuaional analysis. In Jan Weglarz, edior, Projec scheduling: recen models, algorihms and applicaions. Kluwer academic publishers, 999. [] Malek Masmoudi and Alain Haï. A acical model under uncerainy for helicoper mainenance planning. In 8 h Inernaional Conference of Modeling and Simulaion, MOSIM 0, 00. [3] Lofi Zadeh. Fuzzy ses. Informaion and Conrol, 8: , 965. [4] Lofi Zadeh. Fuzzy ses as basis for a heory of possibiliy. Fuzzy ses and sysems, 978.