HEURISTIC ALGORITHM FOR SINGLE RESOURCE CONSTRAINED PROJECT SCHEDULING PROBLEM BASED ON THE DYNAMIC PROGRAMMING

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1 Yugoslav Journal o Operaons Research Volume 19 (2009) Number 2, DOI: /YUJOR S HEURISTIC ALGORITHM FOR SINGLE RESOURCE CONSTRAINED PROJECT SCHEDULING PROBLEM BASED ON THE DYNAMIC PROGRAMMING Ivan STANIMIROVIĆ 1, Marko PETKOVIĆ 2 Predrag STANIMIROVIĆ 3 * and Mroslav ĆIRIĆ 4 1,2,3,4 Unversy o Nš, Deparmen o Mahemacs and normacs, Faculy o Scences and Mahemacs, 1 van.sanmrovc@gmal.com, 2 dexerons@neobee.ne, 3 pecko@pm.pm.n.ac.rs, 4 crcm@bankerner.ne Receved: May 2006 / Acceped: Ocober 2009 Absrac: We nroduce a heursc mehod or he sngle resource consraned projec schedulng problem, based on he dynamc programmng soluon o he knapsack problem. Ths mehod schedules projecs wh one ype o resources, n he non-preempve case: once sared an acvy s no nerruped and runs o compleon. We compare he mplemenaon o hs mehod wh well-known heursc schedulng mehod, called Mnmum Slack Frs (known also as Gray-Kdd algorhm), as well as wh Mcroso Projec. Keywords: Resource schedulng, dynamc programmng, knapsack problem, DELPHI. 1. INTRODUCTION In pracce mos organzaons work whn lmed resources, so projecs are subjec o he same consran. A new projec may seek an addonal use o resources, so s needed o ensure ha hey really would be avalable. * Correspondng auhor

2 282 I. Sanmrovć, M. Pekovć, P. Sanmrovć, M. Ćrć / Heursc algorhm On he oher hand, he me consrans are always presen, so projec manager mus work under ha boundary. Resource levelng s a way o resolve havng oo much work assgned o resources, known as resource over allocaon. The nework dagram can be used o nd opporunes or shorenng he projec schedule. Ths nvolves lookng a where we can cu he amoun o me akes o complee acves on he crcal pah, or example, by ncreasng he resources avalable o hese acves. Anoher soluon s o deny where any oher roues mgh have some slack. We may hen be able o reallocae resources o reduce he pressure on he eam members who are responsble or acves on he crcal pah. We consder he problem o resource dsrbuon a he pon o mnmzaon o he projec makespan, and nroduce a heursc mehod or solvng he sngle resource consraned projec schedulng problem, based on he knapsack problem and dynamc programmng. The oal number o avalable resource uns s consan and speced n advance. A un o resource canno be shared by wo or more acves. An acvy s ready o be processed only when all s predecessor acves are compleed and he number o resource uns requred by are ree and can be allocaed o. Once sared, an acvy s no nerruped and runs o compleon. When levelng resources, we do no change resource assgnmens, nor ask normaon. We only delay asks. In a dynamc programmng soluon o he knapsack problem, we calculae he bes combnaon or all knapsack szes up o M [12]: or j:=1 o N do or :=1 o M do -sze[j]>=o hen cos[]<(cos[-sze[j]]+val[j]) hen begn cos[]:=cos[-sze[j]]+val[j]; bes[]:=j end; In hs program, N s he number o ems, val[j] s he value o jh em, sze[j] s s volume, cos[] s he hghes value ha can be acheved wh a knapsack o capacy and bes[] s he las em ha was added o acheve ha maxmum (hs s used o recover he conens o he knapsack). Frs, we calculae he bes ha we can do or all knapsack szes when only ems o ype A (or j = 1) are aken, hen we calculae he bes ha we can do when only A s and B s (or j = 2) are aken, ec. The soluon reduces o a smple calculaon or cos[]. Suppose an em j s chosen or he knapsack: hen he bes value ha could be acheved or he oal would be val[j] + cos[ - sze[j]], where cos[ - sze[j]] s he opmal llng o he res o he knapsack. I hs value exceeds he bes value ha can be acheved whou an em j, hen we updae cos[] and bes[]; oherwse we leave hem alone. A smple nducon proo shows ha hs sraegy solves he problem [12].

3 I. Sanmrovć, M. Pekovć, P. Sanmrovć, M. Ćrć / Heursc algorhm 283 In hs paper we propose a new sraegy o solve he sngle resource consraned projec schedulng problem. In each sage o he schedulng we consder a schedule me and he correspondng elgble se o acves whch could be sared a he momen whou volaon o gven consrans whch dene he projec. As s proposed n [14] and [9], we acvae a subse o acves rom he elgble se solvng he knapsack problem maxmzng he resource ulzaon. Bu, nsead o Greedy randomzed adapve search procedure (GRASP), used n [9], we apply he dynamc programmng and Bellman s prncple (see [1]). The paper s organzed as ollows. In he second secon we sae mahemacal ormulaon o he problem and compare he projec duraon compued by our algorhm wh he early nsh o he projec. In he hrd secon he algorhm and several mplemenaon deals are descrbed. In he las secon we compare he mplemenaon o our algorhm wh known soware Mcroso Projec 2003 and Gray-Kdd algorhm. 2. MATHEMATICAL FORMULATION OF THE PROBLEM A projec consss o a se o acves J = {A } I, parally ordered by precedence consrans, where I = {1,,n} s a se o acves ndces and n s a number o acves requred by he projec. I s assumed ha he projec requres only one ype o resources. The enre projec s dened as he ordered par (J, R), where he naural number R denoes he resource maxmal uns avalable n he projec. Each o acves A s dened as he ordered rple A = (p,, r, P ), I, where p N represens he processng me (duraon) o acvy A, value r N s a number o resources needed or A, and P I s an array whch conans predecessors ndces or A. An acvy A s sad o be a predecessor o A j, when A j canno sar unl A has nshed. Ths ac s wren as P j or A < A j, where < denes he precedence relaonshp. Smlarly A j s sad o be a successor acvy o A. We assume ha he number r s xed or he lasng me o acvy A. Le F denoes he se ncludng all pars o acves wh predecessor and successor relaonshps. These pars dene a dgraph o he projec G = (J, F), where (A, A j ) F and only P j, A, A j J. A projec sars a me = 0. A schedule or he projec s an assgnmen o a sar s me o each acvy A. An acvy s sad o be scheduled when s assgned a sar me. The vecor denng sars o acves ncluded no he projec (J, R) s dened as he s s s ordered n-uple = ( 1,..., n ) o naural numbers, and s called sar vecor o he projec. Smlarly denoe he vecor o acves nshes by = ( 1,..., n ). The nsh o each acvy

4 284 I. Sanmrovć, M. Pekovć, P. Sanmrovć, M. Ćrć / Heursc algorhm s s compued rom s sar as = + p. Vecors s and are beng calculaed n our algorhm. A easble schedule s a schedule ha sases he gven precedence and resource consrans. An opmum schedule s a easble schedule ha opmzes he gven objecve uncon. Our goal s o nd sar me s or each acvy A due o mnmze he projec compleon me (makespan) o he projec, calculaed by Dur : = max{ I}. We dene he resource uns requred by A n he schedulng me as ollows: s r, r = = xr, 0, oherwse where x = 1 when he acvy A s sared and sll no nshed (n he me nerval s ), and x 0 = oherwse. Formally, he am o he general sngle resource consraned projec schedulng problem s o nd an opmal schedule, and can be ormulaed by he ollowng mahemacal model, n he me : mn Dur = max{ I} (2.1) s s..( j I)( P ) (2.2) j j ( N ) r R (2.3) I Our objecve s o mnmze he makespan o he projec. Each acvy needs o be sared aer all s predecessors acves nsh (condon (2.2)), and n every momen, he oal number o occuped resources s less han R (condon (2.3)). Proposon 2.1. Condon (2.2) can be wren n he ollowng equvalen orm: ( j I)( P) p (2.4) j j j Observe ha he condon (2.4) s equvalen wh he correspondng one rom [15]. Thereore, mahemacal model (2.1)-(2.3) s equvalen wh correspondng one, descrbed n [9], [15], n he case when one ype o resources s used. Schedule s o he projec (J, R) s easble condons (2.2) and (2.3) are sased. A easble schedule s opmal (2.1) s ullled. Nex lemma can be easly proven by he nducon. Ths lemma gves a soppng creron or ncreasng maxmal un s avalably o he resource,.e. he early nsh o he projec. Lemma 2.1 Le s be he easble schedule o he projec (J, R). Le us consder he early nsh τ or each acvy A j, j I, recursvely as ollows: 0 j

5 I. Sanmrovć, M. Pekovć, P. Sanmrovć, M. Ćrć / Heursc algorhm 285 Then p, P = 0; 0 τ j = 0 j j 0 max{ τ + pj}, oherwse. Pj 0 0 τ, or each I. Also Dur max τ = τ. Pj Proo. Le us apply he opologcal sor o G,.e. nd he permuaon 1,, n o I such ha P { 1,..., j 1} or all j I. Then we sor values τ 0 n he non-ascendng order j τ τ.we wll connue proo by he nducon. For j = 1, ollows ha P 1 =, so n 0 s τ = p 1 p =. In he case j > 1 suppose ha he clam o lemma s sased or all 1 1 1,, j-1 and prove or j. In he case P j = ;, concluson ollows rom he same reasons as 0 0 n he case j = 1. Suppose now ha P j =, and consder τ = p + max τ. From he nducve j j k k P j 0 hypohess we have τ k or every k k P j. Also, n vew o (2.2) we have = + p + p τ + p (2.5) s 0 k k j j j j j 0 or all k P j. I we ake he maxmum n (2.5) over all k P j, we oban τ, so we j j nshed he proo by nducon. The second saemen o he lemma ollows mmedaely. The resul o hs lemma s used as he soppng creron n he ollowng sense: n each example, he maxmal uns o he resource s ncreased unl all o hree consdered algorhms reach he duraon equal o early nsh o he projec. We resae, n he recursve orm, he noon o he lae sar o acvy A, known n he leraure: τ, n 1 = θ = 0 n 1 mn{ θ j p}, = 1,..., n 1. The lae nsh τ 1 o A s equal o τ 1 1 = θ + p and s early sar θ 0 s equal o 0 0 θ = τ p. 3. ALGORITHM Le us rs menon some known heursc resource schedulng mehods rom [4]. To choose a subse o acves sasyng projec consrans no he schedule, several heurscs are known. Mercs or assgnng prores are: 1) Shores Task Frs 2) Mos Resources Frs 3) Mnmum Slack Frs (Gray-Kdd algorhm)

6 286 I. Sanmrovć, M. Pekovć, P. Sanmrovć, M. Ćrć / Heursc algorhm 4) Mos Crcal Followers 5) Mos Successors There are many papers comparng alernave heursc algorhms. Paerson and Davs n [6], [7] compared hese heurscs, n seral and parallel modes and acheved he resul he mos eecve algorhm s Mnmum Slack Frs. The smlar resuls are acheved n [10]. The Mnmum Slack Frs mehod s descrbed n [5] (p ). Kocheov and Solyar [9] devse an evoluonary algorhm whch com-bnes genec algorhm, pah relnkng, and abu search. In order o selec a subse o acves rom he elgble se no he schedule, hey solve he knapsack problem. The dea o usng he knapsack problem wh objecve uncon maxmzng he re-source ulzaon rao s nroduced n [13] and [14]. In order o solve he knapsack problem saed or he sake o resource ulzaon, n [9] and [14] use GRASP (greedy randomzed adapve search procedure) algorhm rom [8]. GRASP s an erave mul-sar algorhm. There are wo phases n every eraon: a greedy adapve randomzed consrucon phase and a local search phase. Sarng rom he easble soluon bul durng he greedy adapve randomzed consrucon phase, he local search explores s neghborhood unl a local opmum s ound. The bes soluon ound overall he deren eraons s kep as he resul [8]. A soluon x s sad o be n he basn o aracon o he global opmum local search sarng rom x leads o he global opmum. Once he neghborhood and objecve uncon are deermned, deren sarng soluons can be used o sar he local search n a mulsar procedure. I he sarng soluon s n he basn o aracon o he global opmum, local search nds he global opmum. Oherwse, a non-global local opmum s ound [11]. Usng greedy soluons as sarng pons or local search n a mul-sar procedure wll usually lead o good, hough, mos oen, subopmal soluons. Ths s because he amoun o varably n greedy soluons s small and s less lkely ha a greedy sarng soluon wll be n he basn o aracon o a global opmum. I here are no es n he greedy uncon values or, a deermnsc rule s used o break es, here s no varably and a mulsar procedure would produce he same soluon n each eraon [11]. In hs secon we wll nroduce an algorhm or resource schedulng, called DynamcRes, whch s based on he knapsack problem and he dynamc programmng. Ths heursc gves beer resuls wh respec o Gray-Kdd algorhm n mos cases. Algorhm DynamcRes s beng wren n he programmng language DELPHI. The algorhm requres a sequence o acves, and he ollowng parameers or each acvy A, I : - duraon o he acvy (neger p ), - array o ordnal numbers o s predecessors, denoed by P, - uns o resource requred (neger r ). Also, he npu parameer o he algorhm s oal number o resource uns avalable n he projec, denoed by R.

7 I. Sanmrovć, M. Pekovć, P. Sanmrovć, M. Ćrć / Heursc algorhm 287 Oupu o algorhm are begnnngs o acves aer he resource schedulng s s s perormed,.e. he sar vecor ( 1,..., n ) o he projec acves as well as he correspondng makespan o he projec. We use he saus o each acvy, denoed wh Sa : 0, A s no sared, Sa = 1, A s nshed. In each sage o schedulng we have a schedule me, N and correspondng ses E() and A(), dened n he ollowng denons, resaed rom [9]: Denon 3.1. The se o acves whch could be sared a me whou volaon o any consrans s equal o E() = { j I Sa = 0 ( P ) Sa = 1}. (3.1) j j Denon 3.2. The se o acves whch are n progress a he me s s A() = { I }. (3.2) We now dene he noon called momen o he projec, useul n he algorhm descrpon. Ths value s represened by varable Momen n he algorhm. Denon 3.3. The momen o he projec s equal o 0 n s sar, and n each oher case he momen s equal o he mnmal me when a leas one o sared and uncompleed acves s nshed: 0, when projec sars ( = 0), Momen = mn{ j j A( )}, oherwse. We also dene momenary slack. Ths noon s acual or acves A j wh ndces belongng o E() a he me dened by momen. Denon 3.4. Momenary slack o he acvy A s equal o MSlack = Momen. Algorhm 3.1. DynamcRes Sep 1. Se he schedule me = 0, and Sa = 0 or all I. Sep 2. Compue he number o avalable resource uns by R () = R r (3.3) a A() For each j E() perorm he ollowng:

8 288 I. Sanmrovć, M. Pekovć, P. Sanmrovć, M. Ćrć / Heursc algorhm I 1 j a() j j, r R + p τ (3.4) where 1 τ j denoes he lae nsh or acvy A j, hen se s s =, = + p, R ( ) = R ( ) r ; (3.5) j j j j a a j oherwse, delay acvy A j. Ths means ha each acvy whch exceeds s own lae nsh, sarng rom he acual momen o he projec, needs o be sared here are enough resource uns avalable as s requred by. Sep 3. I E() = ; hen go o Sep 5, oherwse go o Sep 4. Sep 4. Solve he knapsack problem wh capacy R a (), where he values and volumes o arcles are equal o r, E(): mn R ( ) a E()\ A() s.. xr R (), E()\ A() a xr where x {0,1}, E(). For each sasyng x = 1 se s =, = s + p. Sep 5. Now se he new value or he varable Momen: (3.6) Momen = mn{ A( )} (3.7) and se Sa = 1 or all acves sasyng Momen. Sep 6. Se = Momen. I here are non sared acves (( I) Sa = 0), go o Sep 2. Oherwse, he makespan o he projec s equal o max { I}, and he schedule o he projec s he s = ( s 1,..., s n ). Remark 3.1 Gray-Kdd algorhm schedules acves whch have a mnmal value o he momenary slack MSlack. Oppose o hs algorhm, we propose he ollowng: se he begnnngs (lke n Sep 4.) or all acves whch concdes wh he opmal soluon o he knapsack problem (3.6). Remark 3.2. Problem (3.6) can be resaed n he ollowng equvalen orm: =

9 I. Sanmrovć, M. Pekovć, P. Sanmrovć, M. Ćrć / Heursc algorhm 289 max xr E()\ A() s.. xr R (), E()\ A() a (3.8) Problem (3.8) s a one-dmensonal varan o he knapsack problem saed n [9] and [14], and a varan o he classcal 0-1 knapsack problem (see, or example [2]). In order o solve (3.8), nsead o he GRASP algorhm used n [9] and [14], we use he dynamc programmng. Thereore, we speak o ems represenng requremens o he resource. The se o weghs (volumes) as well as he se o values s equal o {r I}. To avod rval cases we assume r Rand r R, or each I. Comparng problem (3.8) I wh he dynamc programmng soluon o he knapsack problem, resaed rom [12], he ollowng analoges are evden: - he objecve uncon cos s equvalen wh x r ; E()\ A() - he knapsack sze M s equvalen wh he number o avalable resource uns, denoed by R a (), whch s dened n (3.3) and (3.5); - number N o deren ypes o ems s replaced by he cardnal number o he se I, denoed by n; - val[j] = sze[j] = r j ; - Assgnmen bes[] = j mples x j = 1 n (3.6) and (3.8). I s known ha he knapsack problem exhbs opmal subsrucure, and s opmal soluon conans whn opmal soluon o sub problems [3]. Typcally, he oal number o dsnc sub problems s a polynomal n he npu sze. When a recursve algorhm revss he same problem over and over agan, we say ha he opmzaon problem has overlappng sub problems. Dynamc programmng algorhms ypcally ake advanage o overlappng sub problems by solvng each sub problem once and hen sorng he soluon n a able [3]. Also, s known ha greedy algorhms do no always yeld opmal soluons [3]. Remark 3.3 Usng he man dea o he Gray-Kdd algorhm, n he case when we have more soluons or he knapsack problem, we use he soluon conanng acvy wh he mnmal value or MSlack. Here we consder an example o dscuss abou he derence beween he Gray- Kdd algorhm and DynamcRes algorhm. Example 3.1 Duraons, uns o he resource requred and predecessors o all acves n he projec are arranged n he ollowng able.

10 290 I. Sanmrovć, M. Pekovć, P. Sanmrovć, M. Ćrć / Heursc algorhm Acves A B C D E F G H I Duraon Uns Predecessors A B,C B B E F,G Assume ha R = 12 resource uns are avalable n he projec. Observe he rs acual nerval: [0, 2]. We should numerae acves A, B and C, n accordance o Gray-Kdd algorhm. Acvy B s crcal and we mark by he number 1. Acvy C has a slack o 2 days and s marked by he number 2. Fnally, acvy A has he bgges slack o 4 days and s assgned by he number 3. Acves A, B and C requre all ogeher 13 uns o he resource, so we pu he sar o acvy A a he momen 2, and schedule acves B, C. On he oher hand, usng algorhm DynamcRes we ll he knapsack o capacy 12 by objecs whose values and capaces are r 1 = 5, r 2 = 5, r 3 = 3. Thereore, he maxmal volume s lled wh volumes r 1 = 5, r 2 = 5. In accordance wh algorhm DynamcRes, s necessary o move he sar o acvy C a he momen 2. In boh algorhms, he remander o he projec sars a he momen 2. Gray-Kdd and DynamcRes gve deren soluons wh he same duraon o he projec (17 days). Here are gven wo deren soluons o he projec schedulng, rs usng DynamcRes algorhm (Fgure 3.1) and he oher usng Gray-Kdd algorhm (Fgure 3.2). Fgure 3.1. Soluon gven by DynamcRes Fgure 3.2. Soluon gven by Gray-Kdd Acves are denoed by leers A, B,..., I, and s shown how many resources each o hem requres. Here, cells n SrngGrd assgned wh he sgn > represen he oal slack or each o acves. Acves ha are sared aer her lae nsh are colored whe, oherwse are blue. Theorem 3.1 Schedule s produced by DynamcRes algorhm s easble (sases condons (2.2) and (2.3)).

11 I. Sanmrovć, M. Pekovć, P. Sanmrovć, M. Ćrć / Heursc algorhm 291 Proo. Sar me s j or each acvy A j can be deermned n wo deren ways, applyng Sep 2 and Applyng Sep 4. Frsly we very Condon (2.2). Usng Sep 2 n a xed schedule me we schedule all acves whose ndces sasy j E() and (3.4) and se s j = or hese acves. On he oher hand, rom Sep 5, s sased Momen max{ Sa = 1}. Aer seng = Momen n Sep 6 s sased max{ Sa = 1}. From (3.5) we conclude ha s j = max{ Sa = 1}, or each j E() sasyng (3.4). Thereore, we conclude s ( Pj) j. In he second case, accordng o Sep 4 o DynamcRes algorhm, we conclude ha n he knapsack problem are ncluded only hese acves A j sasyng j E()\A(). Thereore, n accordance wh (3.1), all predecessors ncluded n he knapsack are nshed (( Pj) Sa = 1). Moreover, n Sep 4 we schedule acves correspondng o he opmal soluon o he knapsack problem. Analogous o he prevous case, accordng o Sep 5, condon (2.2) s sased or all acves ncluded. We now very condon (2.3). I s clear ha condon (2.3) can be wren as ( N) r R (3.9) A() In vew o (3.3), hs condon s laer equvalen wh ( N) R ( ) 0. (3.10) a Now, Sep 2 sases condon (3.10) because o (3.5) and condon (3.4). In he sequel we prove ha Sep 4 also sases (3.10). Denoe by A () he se o ndces o jus sared acves: s A'( ) = { I = }. Snce s = mples A (), we conclude A() = A() A'() aer Sep 4. A hs momen, he number o avalable resources s R () = R () r (3.11) a a A'( ) Accordng o condon n he knapsack problem (3.6), we conclude R a () 0. Takng no accoun (3.3), we ge (3.12) R () = R r r 0 a A() A'() Denoe by A () he se o jus nshed acves n Sep 6:

12 292 I. Sanmrovć, M. Pekovć, P. Sanmrovć, M. Ćrć / Heursc algorhm A''( ) = { I = Momen}. Now, n Sep 2 s sased A() = A() A'()\ A''() and applyng (3.12) we oban R () = R r r + r R r r 0. a A() A'() A''() A() A'() The proo s complee. 4. NUMERICAL EXPERIMENTS In hs secon we compare hree algorhms or resources schedulng: algorhm ncluded n MS Projec, Gray-Kdd algorhm and DynamcRes algorhm. Accordng o our assumpons o he algorhm, n MS Projec s assumed ha resources levelng canno spl ask and check box Level only whn avalable slack n Resource Levelng opons s cleared. Example 4.1. Consder he projec dened as ollows: Acves A B C D E F G H Duraon Uns Predecessors A B C A G Increasng he number o avalable resource uns we ge he ollowng able. The early nsh or hs example s 14 days, so we sop searchng or new soluons when all hree algorhms reach hs lm, usng he resul o lemma 2.1. Duraon o he projec Max. uns Ms Projec Gray-Kdd DynamcRes Daa presened n he able are llusraed by he ollowng char:

13 I. Sanmrovć, M. Pekovć, P. Sanmrovć, M. Ćrć / Heursc algorhm 293 Fgure 4.1. Comparson o hree mehods Average values, geomercal means and sandard devaons or projec duraons are presened n he able. Sascal uncon Ms Projec Gray-Kdd DynamcRes AVERAGE Geomercal mean Sandard devaon The average values are mnmal or DynamcRes algorhm. Ths means ha DynamcRes produces, generally, mnmal values or makespan o he projec. Also, mehod DynamcRes produces maxmal value or he sandard devaon, because he projec makespan correspondng o hs algorhm decreases mos rapdly. Example 4.2. Acves, her duraons, maxmal uns o he resource and predecessors o he projec are dened n he able: Acves A B C D E F G H Duraon Uns Predecessors A B C D,F G Increasng he number o avalable resource uns we ge he ollowng able. I can easly be calculaed ha he early nsh here s 22 days.

14 294 I. Sanmrovć, M. Pekovć, P. Sanmrovć, M. Ćrć / Heursc algorhm Duraon o he projec Max. uns Ms Projec Gray-Kdd DynamcRes We llusrae he daa n he able by he ollowng char: Fgure 4.2. Comparson o hree mehods Average values, geomercal means and sandard devaons or projec duraons are presened n he able. The conclusons are smlar as n he prevous example. Sascal uncon Ms Projec Gray-Kdd DynamcRes AVERAGE Geomercal mean Sandard devaon Example 4.3. Projec s dened by he ollowng daa: Acves A B C D E F G H I Duraon Uns Predecessors A A,C B D B E F,G

15 I. Sanmrovć, M. Pekovć, P. Sanmrovć, M. Ćrć / Heursc algorhm 295 Increasng he number o avalable uns o he resource, we ge he ollowng able. The early nsh or hs example s 15 days. Duraon o he projec Max. uns Ms Projec Gray-Kdd DynamcRes The daa n he able are llusraed by he ollowng char. Fgure 4.3. Comparson o hree mehods Average values, geomercal means and sandard devaons or projec duraons are presened n he able.

16 296 I. Sanmrovć, M. Pekovć, P. Sanmrovć, M. Ćrć / Heursc algorhm Sascal uncon Ms Projec Gray-Kdd DynamcRes AVERAGE Geomercal mean Sandard devaon In hs case, he maxmal sandard devaon produces he Ms Projec. Bu, hs ac s caused by he maxmal makespan (equal o 31) or he mnmal number o Max. uns (equal o 7). Example 4.4. Consder he ollowng projec: Acves A B C D E F G H I J Duraon Uns Predecessors A B A,B C,E D,E D,F G,H G,H Increasng he number o avalable resource uns, we ge he ollowng able. Duraon o he projec Max. uns Ms Projec Gray-Kdd DynamcRes The ollowng char llusraes he daa n he able.

17 I. Sanmrovć, M. Pekovć, P. Sanmrovć, M. Ćrć / Heursc algorhm 297 Fgure 4.4. Comparson o hree mehods able. Here are gven some o sascal uncons or projec duraons represened n he Sascal uncon Ms Projec Gray-Kdd DynamcRes AVERAGE Geomercal mean Sandard devaon CONCLUSION A new algorhm or solvng he sngle resource consraned projec schedulng problem s nroduced. A each schedule me we consder he elgble se o acves whch could be sared whou volaon o gven consrans (dened by predecessor relaonshps). Usng acves rom hs se we consder he adequae knapsack problem and use he Bellman s prncple. The possbly or he arher research can be he generalzaon o DynamcRes algorhm or solvng he mulple resource consraned projec schedulng problems, and he comparson wh GRASP algorhm rom [11]. REFERENCES [1] Bellman, R., Dynamc Programmng, Prnceon Unversy Press, [2] Borgward, K.H., and Brzank, J., Average savng eecs n enumerave mehods or solvng кnapsack problems, Journal o Complexy, 10 (1994)

18 298 I. Sanmrovć, M. Pekovć, P. Sanmrovć, M. Ćrć / Heursc algorhm [3] Cormen, T. H., Leserson, C. E., and Rves, R. L., Inroducon o Algorhms, MIT Press & McGraw-Hll, [4] Osgood, N., Smulaon and Resource Smulaon and Resource- Based Schedulng, Presenaon, [5] Perć, J., Operaons Research, Savremena Admnsracja, Beograd, (In Serban) [6] Paerson, J.H., A comparson o exac approaches or solvng he mulple consraned resource projec schedulng problem, Managemen Scence, 30 (7) (1984) [7] Daves, E.M., An expermenal nvesgaon o resource allocaon n mulacvy projecs, Operaonal Research Quarerly, 24 (11) (1976) [8] Feo, T., and Resende, M.G.C., Greedy randomzed adapve search procedures, Journal o Global Opmzaon, 6 (1995) [9] Kocheov, Y., and Solyar, A., Evoluonary local search wh varable neghborhood or he resource consraned projec schedulng problem, n: Proceedngs o he 3rd Inernaonal Workshop o Compuer Scence and Inormaon Technologes, Russa, [10] Lawrence, S.R., A compuaonal comparson o heursc schedulng echnques, Techncal Repor, Graduae School o Indusral Admnsraon, Carnege-Mellon Unversy, [11] Psouls, L.S., and Resende, M.G.C., Greedy randomzed adapve search procedures, AT&T Labs Research Techncal Repor, January 18, [12] Sedgewck, R., Algorhms, Addson-Wesley publshng company, Massachuses, Menlo Park, Calorna, London, Amserdam, Don Mlls, Onaro, Sydney, [13] Valls, V., Ballen, F., and Qunanlla, S., A populaon-based approach o he resourceconsraned projec schedulng problem, Techncal repor , Unversy o Valenca, [14] Valls, V., Ballen, F., and Qunanlla, S., A populaon-based approach o he resourceconsraned projec schedulng problem, Annals o Operaons Research, 131 (2004) [15] Verma, S., An opmal breadh-rs algorhm or he preempve resource-consraned projec schedulng problem, Second World Conerence on POM and 15h Annual POM Conerence, Cancun, Mexco, Aprl 30 - May 3, 2004.