A heuristic method for consumable resource allocation in multi-class dynamic PERT networks

Transcription

1 Yaghoubi e al. Journal of Indusrial Engineering Inernaional 0, 9:7 hp:// ORIGINAL RESEARCH Open Access A heurisic mehod for consumable resource allocaion in muli-class dynamic PERT neworks Saeed Yaghoubi *, Siamak Noori and Mohammad Mahdavi Mazdeh Absrac This invesigaion presens a heurisic mehod for consumable resource allocaion problem in muli-class dynamic Projec Evaluaion and Review Technique (PERT) neworks, where new projecs from differen classes (ypes) arrive o sysem according o independen Poisson processes wih differen arrival raes. Each aciviy of any projec is operaed a a devoed service saion locaed in a node of he nework wih exponenial disribuion according o is class. Indeed, each projec arrives o he firs service saion and coninues is rouing according o precedence nework of is class. Such sysem can be represened as a queuing nework, while he discipline of queues is firs come, firs served. On he basis of presened mehod, a muli-class sysem is decomposed ino several single-class dynamic PERT neworks, whereas each class is considered separaely as a minisysem. In modeling of single-class dynamic PERT nework, we use Markov process and a muli-objecive model invesigaed by Azaron and Tavakkoli-Moghaddam in 007. Then, afer obaining he resources allocaed o service saions in every minisysem, he final resources allocaed o aciviies are calculaed by he proposed mehod. Keywords: Projec managemen; Muli-class dynamic PERT nework; Queuing Inroducion Today, muli-projec scheduling which considers all projecs of organizaion as one sysem and herefore sharing limied resources among muliple projecs is mosly noiced, whereas up o 90% of organizaions handle he projecs in a muli-projec environmen (Payne 99). On he oher hand, for beer managemen, in some organizaions, he projec-zoriened approach is primarily adoped and he operaions of he organizaion are dependenly execued on projecs. In he lieraure of muli-projec scheduling, muli-projec resource consrained scheduling problem (MPRCSP) in saic and deerminisic condiions is a major opic of mos researches. Generally, wo approaches for analyzing MPRCSP exis. One approach by linking all projecs of organizaion synheically ogeher ino a large single projec, analyzes MPRCSP, while he oher approach by considering he projecs as independen componen and also resources consrains, proposes an objecive funcion ha conains all projecs (possibly appropriaely weighed) for analyzing MPRCSP. * Correspondence: [email protected] Deparmen of Indusrial Engineering, Iran Universiy of Science and Technology, Tehran, Iran Wies (967) and Prisker e al. (969) were he firs pioneering researchers in sudying MPRCSP who presened, respecively, a zero one programming approach and a heurisic model for analyzing his problem. Then Kurulus and Davis (98) and Kurulus and Narula (98), by employing he prioriy rules and describing measures, sudied he MPRCSP. Moreover, some invesigaions have been concenraed on MPRCSP by applying muli-objecive and muli-crieria approaches; for example, Chen (99) proposed he zero one goal programming model in MPRCSP for he mainenance of mineral processing, and Lova e al. (000) analyzed MPRCSP wih applying a lexicographically wo crieria. Also recenly, Kanagasabapahi e al. (009) by defining performance measures and considering maximum ardiness and mean ardiness of projecs sudied he MPRCSP in a saic environmen. On he oher hand, some researchers such as Tsubakiani and Deckro (990), Lova and Tormos (00), Kumanan e al. (006), Goncalves e al. (008), Ying e al. (009), and Chen and Shahandashi (009) proposed heurisic and meaheurisic algorihms for solving MPRCSP. Kruger and Scholl (008) exended he MPRCSP by considering ransfer imes and is cos. 0 Yaghoubi e al.; licensee Springer. This is an Open Access aricle disribued under he erms of he Creaive Commons Aribuion License (hp://creaivecommons.org/licenses/by/.0), which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied.

2 Yaghoubi e al. Journal of Indusrial Engineering Inernaional 0, 9:7 Page of hp:// In all of he menioned invesigaions, MPRCSP has been analyzed in saic and deerminisic environmen, while a few sudies have been concenraed on muliprojec scheduling under uncerainy condiions. Faemi- Ghomi and Ashjari (00) by considering a muli-channel queuing inroduced a simulaion model for muli-projec resource allocaion wih sochasic aciviy duraions. Nozick e al. (00) also presened a nonlinear mixedineger programming model o opimally allocae he resources, while he probabiliy disribuion of aciviy duraion and allocaed resources depend on ogeher. Moreover, even-driven approach and Criical Chain Projec Managemen (CCPM) approach were proposed for overcoming uncerainy in muli-projec sysem by Kao e al. (006) and Byali and Kannan (008), respecively. Normally, greaer resource allocaion will incur a greaer expense; herefore, we have a rade-off beween he ime o complee an aciviy and is cos (a funcion of he assigned resource). When he ask duraion is a linear funcion of he allocaed resource, we have he linear ime cos rade-off problem which was analyzed by Fulkerson (96). The nonlinear case of he ime cos rade-off problem was sudied by Elmaghraby and Salem (98, 98) and Elmaghraby (99, 996). Moreover, Elmaghraby and Morgan (007) presened a fas mehod for opimizing resource allocaion in a single-sage sochasic aciviy nework. Clearly, muli-projec scheduling is more elaborae han single-projec scheduling. On he oher hand, in some organizaions, no only he ask duraions are uncerain, bu also new projecs dynamically ener o he organizaion over he ime horizon. In such condiions, he organizaion is faced wih a muli-projec sysem in which he scheduling procedure would be more elaborae han before. Adler e al. (99) sudied such muliprojec sysem by considering he organizaion as a sochasic processing nework and using simulaion. They assumed ha he organizaion is comprised of a collecion of service saions (work saions) or resources, where several ypes (classes) of projecs exis in a sysem and one or more idenical parallel servers can be seled in each service saion. Therefore, such muliprojec sysem, denominaed as Dynamic PERT Nework, can be considered as a queuing nework and is aracive for organizaions having he same projecs, for example, mainenance projecs. Anavi-Isakow and Golany (00) by applying simulaion sudy used he concep of CONWIP (consan work-inprocess) in muli-class dynamic Projec Evaluaion and Review Technique (PERT) neworks. They described wo conrol mechanisms, consan number of projecs in process (CONPIP) and consan ime of projecs in process (CONTIP), namely, CONPIP mechanism resrics he number of projecs, while CONTIP mechanism limis he oal processing ime by all he projecs ha are acive in he sysem. In addiion, Cohen e al. (00, 007) by using cross enropy, based on simulaion, invesigaed he resource allocaion problem in muli-class dynamic PERT neworks, where he resources can work in parallel. Indeed, he number of resources allocaed and servers in each service saion are equal (e.g., mechanical work saion wih mechanics, elecrical work saion wih elecricians), and he oal number of resources in he sysem also is consan. On he oher hand, Azaron and Tavakkoli-Moghaddam (007) proposed an analyical muli-objecive model o opimally conrol consumable resources allocaed o he aciviies in a dynamic PERT nework, where only one ype of projec exiss in he sysem and he number of servers in every service saion is eiher or infiniy. They assumed ha he aciviy duraions are exponenially disribued random variables, resources allocaed affec he mean of service imes, and he new projecs are generaed according o a Poisson process. A risk elemen was considered in a dynamic PERT nework by Li and Wang (009) who, by using general projec risk elemen ransmission heory, proposed a muli-objecive risk-ime cos rade-off problem. Recenly, Azaron e al. (0) inroduced an algorihm in compuing opimal consan lead ime for each paricular projec in a repeiive (dynamic) PERT nework by minimizing he average aggregae cos per projec. In pracice, due o various projecs' requiremens, mos organizaions execue he projecs from several classes, where new projecs from differen classes arrive o he sysem dynamically over he ime horizon and service saions ha sochasically serve o projecs. In such condiions, organizaion is encounered wih muli-class dynamic PERT neworks, while projecs from differen classes differ in heir precedence neworks, he mean of heir service ime in every service saion, and also heir arrival raes. Reviewing he above-menioned invesigaions indicaed ha he muli-class dynamic PERT nework has been sudied only using simulaion. As he main conribuion of his paper, we presen a heurisic mehod for he consumable resource allocaion problem (or ime cos rade-off problem) in muli-class dynamic PERT neworks. Noe ha he resource allocaion problem in muli-class dynamic PERT neworks is a generalizaion of he resource allocaion problem in dynamic PERT nework, invesigaed by Azaron and Tavakkoli-Moghaddam (007) by considering only one ype (class) of projecs in muliprojec sysem. In his paper, i is assumed ha he new projecs from differen classes, including all heir aciviies, are enered

3 Yaghoubi e al. Journal of Indusrial Engineering Inernaional 0, 9:7 Page of hp:// o he sysem according o independen Poisson processes wih differen arrival raes and each aciviy of any projec is execued a a devoed service saion seled in a node of he nework based on firs come, firs served (FCFS) discipline. Moreover, he number of servers in every service saion is eiher or infiniy, while he service imes (aciviy duraions) are independen random variables wih exponenial disribuions. Indeed, each projec arrives o he firs service saion and coninues is rouing according o precedence nework of is class. For deermining he resources allocaed o he service saions, he muli-class sysem is decomposed ino several single-class dynamic PERT neworks, whereas each class is considered separaely as a minisysem. For modeling of a single-class dynamic PERT nework, we use Markov process and a muli-objecive model, invesigaed by Azaron and Tavakkoli-Moghaddam (007), namely, we firs conver he nework of queues ino a sochasic nework. Then, by consrucing a proper finie-sae absorbing coninuous-ime Markov model, a sysem of differenial equaions is creaed. Moreover, we apply a muli-objecive model conaining four confliced objecives o opimally conrol he consumable resources allocaed o service saions in he single-class dynamic PERT nework and furher employ he goal aainmen mehod o solve a discree-ime approximaion of he primary muli-objecive problem. Finally, afer obaining he resources allocaed o service saions in every minisysem, he final resources allocaed o servers are calculaed by he proposed mehod. This paper is composed of five secions. The remainder is organized as follows. In Single class dynamic PERT neworks, we model he single-class dynamic PERT nework by employing a finie-sae coninuous-ime Markov process and apply a muli-objecive model o opimally conrol he consumable resources allocaed o service saions in a single-class dynamic PERT nework. In Consumable resource allocaion in muli-class dynamic PERT neworks, a heurisic model is proposed for consumable resources allocaed in he muli-class dynamic PERT neworks. We solve an example in An illusraive case secion and conclusion is given in Conclusions. Single-class dynamic PERT neworks In his secion, we represen a muli-objecive model o opimally conrol he consumable resources allocaed o he service saions invesigaed by Azaron and Tavakkoli- Moghaddam (007). For his purpose, we firs explain an analyical mehod o compue he disribuion funcion of projec compleion ime and hen we describe a muliobjecive model in single-class dynamic PERT nework. I is assumed ha a projec is represened as an aciviyon-node (AoN) graph, and also he new projecs, including all heir aciviies, are arrived o he sysem according o a Poisson process wih he rae of λ. Moreover, each aciviy of a projec is execued in a devoed service saion seled in a node of he nework based on FCFS discipline. Such sysem can be considered as a nework of queues, where he service imes represen he duraions of he corresponding aciviies. I is also assumed ha he number of servers in each service saion o be eiher one or infiniy, while he service imes (aciviy duraions) are independen random variables wih exponenial disribuions. Projec compleion ime disribuion For presening an analyical mehod o compue he disribuion funcion for single-class dynamic PERT nework, he mehod of Kulkarni and Adlakha (986) will be used because his mehod presens an analyical, simple, and easy approach o implemen on a compuer and a compuaionally sable algorihm o evaluae he disribuion funcion of he projec compleion ime. This mehod models PERT nework wih independen and exponenially disribued aciviy duraions by coninuous-ime Markov chains wih upper riangular generaor marices. The special srucure of he chain allows us o develop very simple algorihms for he exac analysis of he nework. I is assumed ha he service ime in service saion a is exponenially disribued wih he rae of μ a, where he arrival sream of projecs o each service saion is according o a Poisson process wih he rae of λ. Moreover, he number of server in node a is eiher one or infiniy and herefore he node a reas as an M/M/ or M/M/ model. The main seps of mehod o deermine a finie-sae absorbing coninuous-ime Markov process o compuing he disribuion funcion in single-class dynamic PERT nework are as follows: Sep. Compue he densiy funcion of he sojourn ime (waiing ime plus aciviy duraion) in each service saion. Sep.. If here is one server in service saion a, hen he queuing sysem would be M/M/, and he densiy funcion of ime spen in he service saion a would be exponenially expressed wih parameer μ a λ a (μ a λ) a ; herefore, w a () is calculaed as follows: w a ðþ¼ ðμ a λþ:e ðμ a λþ 0 ðþ Sep.. If here are infinie servers in service saion a, hen he queuing sysem would be M/ M/, and he densiy funcion of ime spen in he service saion a would be exponenially

4 Yaghoubi e al. Journal of Indusrial Engineering Inernaional 0, 9:7 Page of hp:// expressed wih parameer μ a ; herefore, w a () is calculaed as follows: w a ðþ¼μ a :e μ a 0 ðþ Sep. Transform he single-class dynamic PERT nework, represened as an AoN graph, o a classic PERT nework represened as an aciviy-on-arc (AoA) graph. When considering AoN graph, subsiue each node wih a sochasic arc (aciviy) whose lengh is equal o he sojourn ime in he corresponding queuing sysem. For his purpose, node a in he AoN graph is replaced wih a sochasic aciviy. Assume ha b,b,,b n are he incoming arcs o node a and d, d,,d m are he ougoing arcs from i in he queuing nework. Then, node a is subsiued by aciviy (v,w), whose lengh is equal o he aciviy duraion a. Furhermore, all arcs b,b,,b n erminae o node v while all arcs d,d,,d m originae from node w [for more deails, see Azaron and Modarres (00 )]. Sep. Calculae he disribuion funcion of he projec compleion ime (longes pah ime) in he AoA obained in sep, while aciviy duraions are disribued exponenially. For he compuaion of longes pah ime disribuion in AoA, he mehod of Kulkarni and Adlakha (986) is applied. Le G =(V, A) be he PERT nework, obained in sep, wih a single source and a single sink, in which V represens he se of nodes and A represens he se of arcs of he sochasic nework (AoA). Le s and behesourceandsink nodes, respecively, and lengh of arc aϵ A be a random variable ha exponenially disribued wih parameer γ a.fora ϵ A, he saring node and he ending node of arc a are denoed as α(a) and β(a), respecively. Definiion. Le I(v) be he se of arcs ending a node v and O(v) be he se of arcs saring a node v, which are defined as follows: X; X ¼ n a A : αðaþ X; βðaþ X o : ðþ An (s,) cuðx; X Þ is denominaed a uniformly di reced cu (UDC), if ðx; X Þ ¼ Ø, i.e., here are no wo arcs in he cu belonging o he same pah in he projec nework. Each UDC is clearly a se of arcs, in which he saring node of each arc belongs o X and he ending node of ha arc belongs o X. Example. Consider he sochasic nework shown in Figure aken from Azaron and Tavakkoli-Moghaddam (007). According o he definiion, he UDCs of his nework are (,); (,); (,,6); (,,6); and (,6). Definiion. (E,F), subses of A, is defined as admissible -pariion of a UDC D if D = E F and E F = φ, and also I(β(a)) F for any a F. Again consider Example ha (,,6) is one of he UDCs. For example, his cu can be decomposed ino E = {, 6} and F = {}. In his case, he cu is an admissible -pariion because I(β()) = {, } F. Furhermore, if E = {6} and F = {, }, hen he cu is no an admissible -pariion because I(β()) = {} F = {, } Definiion. Along he projecs execuion a ime, each aciviy can be in one and only one of he acive, dorman or idle saes, which are defined as follows: (i) Acive. An aciviy a is acive a ime, ifiis being performed a ime. (ii)dorman. An aciviy a is called dorman a ime, if i has compleed bu here is a leas one unfinished aciviy in I(β(a)) a ime. (iii) Idle. An aciviy a is denominaed idle a ime, if i is neiher acive nor dorman a ime. In addiion, Y() and Z() are defined as follows: YðÞ¼ fa A; a is acive a ime g; ð6þ Iv ðþ¼fa A : βðaþ ¼ vg ðv V Þ; ðþ Ov ðþ¼fa A : αðaþ ¼ vg ðv V Þ: ðþ s Definiion. For X V such ha s X and X = V X, an(s,) cu is defined as follows: Figure Example. 6

5 Yaghoubi e al. Journal of Indusrial Engineering Inernaional 0, 9:7 Page of hp:// Table All admissible -pariion cus of he example nework Cu Value (,) (,) (, a ) (,,6) (, a,6) 6 (,,6 a ) 7 (, a,6 a ) 8 (,,6) 9 ( a,,6) 0 (, a,6) (,,6 a ) ( a,,6 a ) (,,6) (,6) ( a,6) 6 (,6 a ) 7 (ø,ø) A superscriped roman leer a is applied o denoe a dorman aciviy and all ohers are acive. Z ðþ¼fa A; a is dorman a ime g; ð7þ and X() =(Y(), Z()). A UDC is divided ino E and F ha conain acive and dorman aciviies, respecively. All admissible -pariion cus of nework of Figure are presened in Table. The se of all admissible -pariion cus of he nework is defined as S and also S ¼ S fðϕ; ϕþg. Noe ha X() = (ø,ø) presens ha he all aciviies are idle a ime and herefore he projec is finished by ime. Iisdemon- sraed ha {X(), 0} is a finie-sae absorbing coninuous-ime Markov process (for more deails, see Kulkarni and Adlakha (986)). If aciviy a erminaes wih he rae of γ a,andi(β(a)) F {a}, namely, here is a leas one unfinished aciviy in I (β(a)), hen E = E {a}, F = F {a}. Furhermore, if by erminaing aciviy a, all aciviies in I(β(a)) become idle (I (β(a)) F {a}), hen E =(E {a}) O(β(a)), F = F I (β(a)). Namely, all aciviies in I(β(a)) will become idle and also he successor aciviies of his aciviy, O(β(a)), will become acive. Therefore, he componens of he infiniesimal generaor marix denoed by Q =[q{(e, F), (E, F )}], where (E,F) and E ; F S are obained as follows: A coninuous-ime Markov process, {X(), 0}, is wih finie-sae space S and since q{(ø,ø),(ø,ø)} = 0, he projec is compleed. In his Markov process, all of saes excep X() = (ø,ø) ha is he absorbing sae are ransien. Furhermore, he saes in S should be numbered such ha his Q marix be an upper riangular one. I is assumed ha he saes are numbered as ; ; ; N ¼ j S j such ha X() =(O(s), ø) and X() = (ø,ø) are sae (iniial sae) and sae N (absorbing sae), respecively. Le T be he lengh of he longes pah or he projec compleion ime in he PERT nework, obained in sep. Obviously, T = min{ 0:X()=N X(0) = }. Chapman-Kolmogorov backward equaions can be used o calculae F()=P(T ). If i is defined P i ðþ¼px ð ðþ¼nx0 j ð Þ ¼ iþ i ¼ ; ; ; N; ð9þ hen F()=P (). The sysem of linear differenial equaions for he vecor P ðþ¼½p ðþ P ðþ :::: P N ðþ Š T is presened by P ðþ¼ dpðþ ¼ Q:PðÞ d Pð0Þ ¼ ½0 0 :::: Š T ; ð0þ where P () and Q represen he derivaion of he sae vecor P() and he infiniesimal generaor marix of he sochasic process {X(), 0}, respecively. Muli-objecive consumable resource allocaion In his secion, we apply a muli-objecive model o opimally conrol he consumable resources allocaed o he service saions in a single-class dynamic PERT nework, represening as a nework of queues, where we allocae more resources o service saion and where he mean service ime will be decreased and he direc cos of service saion per period will be increased. This means he mean service ime in each service saion is a nonincreasing funcion and he direc cos of each service saion per period is a non-decreasing funcion of he amoun of resource allocaed o i, i.e., he oal direc coss of service saion per period and he mean projec compleion ime are dependen ogeher and an appropriae rade-off beween hem is required. Also, he variance of he projec compleion ime should be accouned in he model because he mean and he variance are wo complemenary conceps. The las objecive ha should be considered is he probabiliy ha he 8 γ a if : a E; IðβðaÞÞ F fag; E ¼ E fag; F ¼ F fag q ðe; FÞ; E ; F >< γ ¼ a if : a E; IðβðaÞÞ F fag; E ¼ ðe fagþ OðβðaÞÞ; F ¼ F IðβðaÞÞ γ a if : E ¼ E; F ¼ F >: a E 0 oherwise: ð8þ

6 Yaghoubi e al. Journal of Indusrial Engineering Inernaional 0, 9:7 Page 6 of hp:// projec compleion ime does no exceed a cerain hreshold for on-ime delivery performance. Le x a be he resource allocaed in service saion a (a A); also, d a (x a ) represens he direc cos of service saion a (a A) per period in he PERT nework, obained in Projec compleion ime disribuion secion sep, while i is assumed o be a non-decreasing funcion of amoun of resources x a allocaed o i. Thus, he projec direc cos (PDC) would be PDC = P a Ad a (x a ). Also, he mean service ime in he service saion a, g a (x a ), is assumed o be a non-increasing funcion of he amoun of resource x a allocaed o i ha would be equal o P i ð0þ ¼ 0 P N ðþ¼ i ¼ ; ; ; N ð6þ Le B and C be he se of arcs in he PERT nework, obained in Projec compleion ime disribuion secion sep ha here are, respecively, one server and infinie servers seled on he corresponding service saions (A = B C). The nex consrain should be saisfied o show he response in he seady-sae: μ a λ μ a 0 a B a C ð7þ g a ðx a Þ ¼ ð a AÞ ðþ μ a Le U a be he maximum amoun of resource available o be allocaed o he aciviy a (a A), L a be he minimum amoun of resource needed o execue he aciviy a, x =[x a : a A] T,andJ be he amoun ha represens he resource available o be allocaed o all of aciviies. Moreover, we define u as a hreshold ime ha projec compleion ime does no exceed i. In pracice, d a (x a ) and g a (x a ) can be obained by employing linear regression based on he previous similar aciviies or applying hejudgmensofexpersinhisarea. Therefore, he objecive funcions are given by he following:. Minimizing he projec direc cos per period Min f ðþ¼ x X d a A aðx a Þ ðþ. Minimizing he mean of projec compleion ime (P () is he densiy funcion of projec compleion ime) Min f ðþ¼et x ð Þ ¼ 0 ð P ðþ Þd ¼ 0 P ðþþd ðþ. Minimizing he variance of projec compleion ime Min f ðþ¼var x ðtþ ¼ 0 P ðþþd 0 P ðþþd ðþ. Maximizing he probabiliy ha he projec compleion ime does no exceed a cerain hreshold Max f ðþ¼p x ðuþ ¼ PT u ð Þ ðþ The infiniesimal generaor marix Q would be a funcion of he conrol vecor μ =[μ a :a A] T (x =[x a :a A] T ). Therefore, he nonlinear dynamic model is P ðþ¼q ðμþ:pðþ In he mahemaical programming, we do no use such consrains. Hence, ε, following he esablishmen of consrain μ a λ þ ε a B μ a ε a C ð8þ Consequenly, he appropriae muli-objecive opimal conrol problem is Min f ðþ¼ x X d a ðx a Þ a A Min f ðþ¼ x 0 P ðþþd Min f ðþ¼ x 0 P ðþþd 0 P ðþþd Max f ðþ¼p x ðuþ s: : P ðþ¼q ðμþ:pðþ P i ð0þ ¼ 0 i ¼ ; ; ; N P N ðþ¼ g a ðx a Þ ¼ a A μ a μ a λ þ ε a B μ a ε a C x a L a a A x a U a x a J: a A a A ð9þ This sochasic programming is impossible o solve, herefore, based on he definiion of inegral hinking, we divide ime inerval ino R equal porions wih he lengh of Δ. Indeed, we ransform he differenial equaions ino difference equaions. Thus, he corresponding discree sae model can be given as follows (for more deails see Azaron and Tavakkoli-Moghaddam, 007):

7 Yaghoubi e al. Journal of Indusrial Engineering Inernaional 0, 9:7 Page 7 of hp:// Min f x Min f x ðþ¼ XR r¼0 ðþ¼ XR r¼0 rδðp ðr þ Þ P ðþ r Þ ðrδþ ðp ðr þ Þ P ðþ r Þ " # XR r¼0 rδðp ðr þ Þ P ðþ r Þ h u i Max f ðþ¼p x Δ s: : Prþ ð Þ ¼ Pr ðþþqðμþpr ðþδ r ¼ 0; ; ; ; R P i ð0þ ¼ 0 i ¼ ; ; ; N P N ðþ¼ r r ¼ 0; ; ; R P i ðþ r i ¼ 0; ; ; N ; r ¼ ; ; ; R g a ðx a Þ ¼ a A μ a μ a λ þ ε μ a ε x a L a a C a A a B x a U a a A X x a J: a A ð0þ Goal aainmen mehod We now need o use a muli-objecive mehod o solve (0), and we acually use goal aainmen echnique for his purpose. The goal aainmen mehod needs o deermine a goal, b j, and a weigh, c j, for every objecive, namely, he oal projec direc cos per period, he mean projec compleion ime, he variance of projec compleion ime, and he probabiliy ha he projec compleion ime does no exceed a cerain hreshold. c j srepresenheimporance of objecives, whereas, if an objecive has he smalles c j, hen i will be he mos imporan objecive. c j s(j =,,) are commonly normalized such ha X c j ¼. Goal aainmen mehod is acually a j¼ variaion of goal programming mehod inending o minimize he maximum weighed deviaion from he goals. Since goal aainmen mehod has a fewer variables o work wih, compared o oher simple and ineracive muli-objecive mehods, i will be compuaionally faser and more suiable o solve our complex opimizaion problem. Therefore, he appropriae goal aainmen formulaion of he resource allocaion problem is given by Min z s: : X d a A aðx a Þ c z b X R r¼0 X R r¼0 rδðp ðr þ Þ P ðþ r Þ c z b ðrδþ ðp ðr þ Þ P ðþ r Þ " # XR rδðp ðr þ Þ P ðþ r Þ c z b h r¼0 u i P c z b Δ Prþ ð Þ ¼ Pr ðþþqðμþpr ðþδ r ¼ 0; ; ; ; R P i ðþ¼0 0 i ¼ ; ; ; N P N ðþ¼ r r ¼ 0; ; ; R P i ðþ r i ¼ 0; ; ; N ; r ¼ ; ; ; R g a ðx a Þ ¼ μ a a A μ a λ þ ε a B μ a ε a C x a L a a A x a U a a A X x a J a A ðþ Consumable resource allocaion in muli-class dynamic PERT neworks In pracice, due o various projecs' requiremens, mos organizaions execue he projecs from several classes, where new projecs from differen classes arrive o he sysem dynamically over he ime horizon and service saions ha sochasically serve o projecs. In such condiions, organizaion is encounered wih muli-class dynamic PERT neworks, while projecs from differen classes differ in heir precedence neworks, he mean of heir service ime in every service saion, and also heir arrival raes. In his secion, we presen a heurisic algorihm for consumable resource allocaion problem in muli-class dynamic PERT neworks, while his problem is a generalizaion of he resource allocaion problem in dynamic PERT nework, as invesigaed by Azaron and Tavakkoli-Moghaddam (007) by considering only one ype (class) of projecs in muli-projec sysem. For his purpose, i is assumed ha we have K differen classes from projecs, where he new projecs of class i (i =,, K) arrive o sysem according o Poisson process wih he rae of λ i, and he service imes in service saion a for he projecs of class i are exponenially disribued wih he rae of μ i a. On he basis of his algorihm, muli-class dynamic PERT neworks problem is decomposed ino K single-class dynamic PERT nework

8 Yaghoubi e al. Journal of Indusrial Engineering Inernaional 0, 9:7 Page 8 of hp:// problem, while each class is considered separaely as a minisysem. In every minisysem i (i =,, K), all projecs from differen classes are considered as class i! wih he arrival rae of λ ¼ XK λ i. Then every minisysem is solved by using he seps found in he secions of Single class dynamic PERT neworks as a single-class dynamic PERT nework problem. Le G i =(V i,a i ) be a direced sochasic nework of class i (i =,, K), in which V i represens he se of nodes and A i represens he se of arcs of he nework in class i. Les i and i be he source and sink nodes in he AoA graph of class i, respecively. Le x i a represen he amoun of resource o be allocaed o he service saion a (a A i ) in minisysem i, while all projecs from differen classes are considered! as class i wih he arrival rae of λ ¼ XK λ i and i¼ x i ¼ x i a : a A T, where A ¼ [ K i¼ A i. Then he resources allocaed o service saion a in minisysem i should be compued by he model represened in Muli-objecive consumable resource allocaion secion. Therefore, he resources allocaed o service saion a would be X λ a A i :x i x a ¼ i a a A ðþ λ i Xa Ai Le x =[x a : a A] T ; herefore, he resources allocaed o service saions are calculaed as follows: x ¼ XK i¼ λ i λ xi i¼ ðþ Moreover, Le z i represen he objecive funcion of minisysem i using goal aainmen formulaion. Therefore, he objecive funcion of muli-class sysem would be z ¼ XK i¼ λ i λ zi ðþ Consequenly, he proposed algorihm is as follows. Proposed Algorihm: Sep. Conver he muli-class dynamic PERT neworks problem ino K single-class dynamic PERT nework problem (minisysem). For his purpose, in every minisysem i, consider he all projecs from differen classes as class i wih he arrival rae of λ ¼ XK i¼ λ i!. Sep. For every minisysem i, compue he resources allocaed o service saions, x i =[x a i :a A] T, and objecive funcion, z i, by he model represened in Muliobjecive consumable resource allocaion secion. Sep. Calculae he resources allocaed o service saions, x =[x a :a A] T,byx ¼ XK i¼ λ i λ xi and also obain he objecive funcion of muli-class sysem by z ¼ XK i¼ λ i λ zi. An illusraive case In his secion, o illusrae he proposed heurisic algorihm, we consider he hree classes of projecs depiced in Figure. The assumpions are as follows: The sysem has hree differen classes of projecs. The new projecs from differen classes, including all heir aciviies, are generaed according o Poisson processes wih he raes of λ =., λ =., λ = per year. The aciviy duraions in service saion a for he projecs of class i (i =,, ) are exponenially disribued. Table shows he characerisics of he aciviies in differen classes, where he ime uni and he cos uni are, respecively, in year and in housand dollars. There are infinie servers in service saion and oher saions have only one server seled in he nodes. The capaciy of he sysem is infinie. The hreshold ime ha he projec compleion ime does no exceed i for every minisysem is. The maximum amoun of consumable resource available o be allocaed o all service saions is J =. In all experimens, he value of ε is equal 0.0. The goals and he weighs of objecives for goal aainmen mehod in every class, are b =,b =, b =0.,b =0.9andc =0.,c =0.,c =0., c = 0., respecively. Noe ha he goals and he weighs of objecives for differen classes can be differen. In his example, hree-class dynamic PERT neworks problem is decomposed ino hree single-class dynamic PERT nework problem, while each class is considered separaely as a minisysem. In every minisysem i(i =,,), all projecs from differen classes are considered as class i wih he arrival rae of λ = (= ). Then, every

9 Yaghoubi e al. Journal of Indusrial Engineering Inernaional 0, 9:7 Page 9 of hp:// Projec : λ Sar 6 Finish Projec : λ Sar 6 Finish Projec : λ Sar Finish Figure The precedence relaions for he hree projec ypes. 6 minisysem is solved as single-class dynamic PERT neworkproblembyusinghesepsfoundinhesecionsof Single class dynamic PERT neworks. The objecive is o obain he resources allocaed o he differen aciviies by using proposed algorihm in Consumable resource allocaion in muli-class dynamic PERT neworks. For his purpose, he sysem saes and ransiion raes for every minisysem are deermined (depiced in Figure ), where he saes are showed in nodes and he rae diagrams are represened on arcs. We obain he infiniesimal generaor marix Q(μ) for forming he corresponding discree sae model Table Characerisics of he aciviies Aciviy (a) d a (x a ) Projec ype g a (x a ) L a U a.x + \0. 0.0x x x.x x x x.8x x x x x x x x.x x x x 6.x x x x 6 in every class. We consider he menioned goals and weighs and also he various combinaions of R and Δ. To do so, we employ LINGO 8 on a PC Penium, CPU GHz. The opimal allocaed resources in every minisysem, he compuaional imes, CT (mm/ss), and also he values of all objecives for he differen combinaions of R and Δ, are shown in Table for every minisysem. So based on proposed algorihm in Consumable resource allocaion in muli-class dynamic PERT neworks secion he allocaed consumable resources are x ¼ :77:þ:87:þ ¼ :9, x =.9797, x =.666, x =.976, x =.9, and x 6 =.07 (z =.799). To illusrae he performance of proposed algorihm, we evaluae is resuls wih a simulaion mehod ha randomly allocae he resources o service saions and simulae he muli-class dynamic PERT nework hrough Mon Carlo simulaion. Noe ha Faemi Ghomi and Hashemin (999) proposed condiional Mone Carlo simulaion and crude Mone Carlo simulaion o compue he nework compleion ime disribuion funcion. However, he simulaion mehod will be as follows: Simulaion mehod: Sep. n= Sep. Generae one iniial soluion x =[x a : a A] T, randomly, where x a [L a, U a ] and x a J a A Sep. Simulae he muli-class dynamic PERT nework (Figure ) according o he iniial soluion Sep. Calculae he objecive funcion, z, for he iniial soluion Sep. Repea Generae a new soluion, x new ¼ x new a randomly, where x new a ½L a ; U a T, : a A Š and x a J a A

10 Yaghoubi e al. Journal of Indusrial Engineering Inernaional 0, 9:7 Page 0 of hp:// (,*) (,*) (,) (,) (*,) (*,) a) Projec ype : (,) (,) (,) (,) () () 6 (6) (6) (ø,ø) (ø,ø) (,) (,) (,*) (,*) (,*) (,*) (,) (,) (,*) (,*) (,) (,) (,*) (,*) b) Projec ype : (,) (,) (,) (,) (,) (,) (6) (6) 6 (ø,ø) (ø,ø) (,) (,) (*,) (*,) (,*) (,*) (,*) (,*) (*,) (*,) (,*) (,*) (,) (*,6) (*,6) 6 c) Projec ype : (,) (,) (,) (,) (,6) (,6) () () (ø,ø) (ø,ø) (,) (,) 6 (,6*) (,6*) (,6) (,6) 6 (,6*) (,6*) Figure The admissible -pariion cus and rae diagram for every single class in example. Simulae he muli-class dynamic PERT nework according o he new soluion Calculae he objecive funcion, z new,forhenew soluion Calculae Δz = z new z If Δz 0 hen x = x new, z = z new n ¼ n þ Unil n N, where N is number of simulaion run Sep 6. Presen x and z We se he number of simulaion run o be N = 8,000. According o simulaion mehod, he simulaion resuls are x sim ¼ :78, x sim ¼ :769, x sim ¼ :0, x sim ¼ :, x sim ¼ :06, x sim 6 ¼ :7 (z sim =.0). In Figure, a comparaive analysis of our proposed model in Consumable resource allocaion in muli-class dynamic PERT neworks, and simulaion resuls for his example is presened. Based Table The compuaional resuls Projec ype R value Δ x x x x x x 6 z f f f f CT : : : : : : : : :

11 Yaghoubi e al. Journal of Indusrial Engineering Inernaional 0, 9:7 Page of hp:// Figure Comparaive analysis of our proposed model and simulaion resuls. on his example, he objecive funcion values obained hrough boh algorihms are close. Conclusions In his aricle, we proposed a heurisic mehod for he consumable resource allocaion problem (or ime cos rade off problem) in muli-class dynamic PERT neworks. I was assumed ha he new projecs of differen classes, including all heir aciviies, are enered o he sysem according o an independen Poisson processes, and each aciviy of any projec is execued a a devoed service saion seled in a node of he nework according o is class. The number of servers in every service saion is eiher or infiniy, while he service imes (aciviy duraions) are independen random variables wih exponenial disribuions. Indeed, each projec arrives o he firs service saion and coninues is rouing according o is precedence nework of corresponding class. Such sysem was considered as a queuing nework, while he discipline of queues is firs come, firs served. For deermining he resources allocaed o he service saions, he muli-class sysem was decomposed ino several single-class dynamic PERT neworks, whereas each class was considered separaely as a minisysem. For modeling of single-class dynamic PERT nework, we used Markov process and a muli-objecive model, invesigaed by Azaron and Tavakkoli-Moghaddam (007), namely, we firs convered he nework of queues ino a sochasic nework. Then, by consrucing a proper finie-sae absorbing coninuous-ime Markov model, a sysem of differenial equaions was creaed. Nex, we applied a muli-objecive model conaining four confliced objecives o opimally conrol he resources allocaed o service saions in single-class dynamic PERT nework and furher used he goal aainmen mehod o solve a discree-ime approximaion of he primary muli-objecive problem. Finally, afer obaining he resources allocaed o service saions in every minisysem, he final resources allocaed o servers were calculaed by proposed algorihm. In muli-objecive model, he oal projec direc cos was considered as an objecive o be minimized and he mean projec compleion ime, as anoher effecive objecive, should also be accouned ha o be minimized. The variance of he projec compleion ime was anoher effecive objecive in he model because he mean and he variance are wo complemenary conceps. The probabiliy ha he projec compleion ime does no exceed a cerain hreshold was considered as he las objecive. For obaining he bes opimal allocaed consumable resource in single-class dynamic PERT nework, we considered he various combinaions of porions for specific ime inerval. Based on he presened example, If he lengh of every porion is decreased, he accuracy of soluion is increased, i.e., he value of objecive is decreased and he compuaional ime, CT, is also increased. Compeing ineress The auhors declare ha hey have no compeing ineress. Auhors' conribuions SY sudied he associaed lieraure review and presened he mahemaical model and simulaion algorihm. He was also responsible for revising he manuscrip. SN managed he sudy and paricipaed in he problem solving approaches. MMM approved he model and analyzed he required daa and conduced he experimenal analysis. All auhors read and approved he final manuscrip. Auhors' informaion SY is an Assisan Professor of Indusrial Engineering a Iran Universiy of Science and Technology (IUST), Tehran, Iran. His curren research ineress include sochasic processes and heir applicaions, projec scheduling and supply chains. He has conribued aricles o differen inernaional journals, such as European Journal of Operaional Research, Inernaional Journal of Sysem Science, Iranian Journal of Science & Technology, and Inernaional Journal of Advanced Manufacuring Technology. SN is he Associae Professor of Indusrial Engineering Deparmen in IUST since Ocober 990. The eaching experiences are mainly in projec managemen, human resource managemen, and engineering economics. MMM also is he Assisan Professor of Indusrial Engineering Deparmen in IUST. His research ineress

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