MATHEMATICAL PROGRAMMING MODEL OF THE CRITICAL CHAIN METHOD

Transcription

1 MATHEMATICAL PROGRAMMING MODEL OF THE CRITICAL CHAIN METHOD TOMÁŠ ŠUBRT, PAVLÍNA LANGROVÁ CUA, SLOVAKIA Abstract Curretly there s creasgly dcated that most of classcal project maagemet methods s ot sutable for workg wth ll structured ad ot eough quatfed puts. Usg classcal methods (CPM, MPM, PERT, GERT) precse results from the accurate umbers are obtaed. Applyg such umbers o a real problem pots to the operformace of project durato through the crtcal path overru. New methods that are able to work wth ot eough quatfed puts start to ga groud. Besde soft approach based o fuzzy modelg oe rather ew method called Crtcal Cha wll be metoed ths paper. Crtcal Cha method has maly emprcal character, t has a relatvely suffcet software support, but t s stll watg for detal aalyss ad for precse mathematcal descrpto. The objectve of the followg paper s to preset oe possblty of how to model ad solve a Crtcal Cha problem usg dscrete programmg tools. 1. Itroducto Classcal approaches to project maagemet ad ther applcatos are commoly used, but there stll exst a lot of factors, whch ca egatvely fluece task durato, ad commesurate wth these duratos t flueces project durato too. Usg Theory of Costras ad ther drect applcato to project maagemet kow as Crtcal Cha, some egatve factors could be elmated ad so the project schedule ca be fshed tme wthout ay overru. The am of the Crtcal Cha method s to estmate task duratos ad to advert all factors, whch ca fluece (extet) these duratos. Amog them huma behavor ad recourse avalablty could be metoed. Both of these factors play a mportat role ad as we ca see they play a bg role durato estmatos. The Crtcal Cha method does t follow the durato of a task as a whole, but t dvdes ths durato to two dfferet parts. The frst part s a tme eeded for task completo ad the secod oe has a meag of a reserve, protectg the task agast delay arsg from tur up for the book. The tme estmate s gve lke a sum of these two parts. Whle the frst part s chageless, the secod part s very varable ad t s depedet o perso makg the estmate o hs/her kowledge, expereces ad propesty to rsk. These 256

2 estmatos are ot sgle umbers but rather statstcal ettes, reflectg the probablty of task completo a certa amout of tme. A aggressve estmate, reflectg oly the amout of work (the frst part) mght have a 50% level of cofdece, whle a loger realstc estmate, agast whch the resource s comfortable commttg to, mght be closer to a 85-95% rage of cofdece. The dfferece betwee 50% ad 95% estmates s a safety ad because of t the task estmate cludes a plety of safety. Eve thought ths safety s ofte the loger part of estmate, t wo t be used, because most cases t wll get wasted. The ma reaso of t s that work o task s usually started later tha plaed ad o the other had whe the work s fshed earler, saved tme s ot used, because the resources are ofte allocated o other places of projects. The Crtcal Cha methodology requres the schedule to be bult wth oly the tme for completo wthout ay safety. The ma dea of ths approach s to protect the project ad ot dvdual tasks, because the most mportat thk s fshg the project tme. The whole approach ca be descrbed followg steps: 1) Creatg the project etwork 2) Resource levelg, 3) Idetfyg the Crtcal Cha ( Crtcal Cha s a resource leveled crtcal path reflectg the task ad resource depedeces that determate project durato ) 4) Creatg ad sertg buffers. For purposes of our further approach let s precse a lttle bt more the fourth step dealg wth buffers. I the phase of buffers creatg there are used dvdual safetes for each task. The safety assocated wth the crtcal cha tasks ca be shfted to the ed of the cha, protectg the project deadle from varato the crtcal cha tasks. Ths cocetrated aggregato of safety s called project buffer ad t protects project fsh date from Crtcal Cha varato. The safety of dvdual tasks allows for everythg to be bad wth the same probablty as everythg to be well. It meas that oly 50% of cases the task s fshed earler ad other 50% later. So the project buffer ca be half durato smaller tha the sum of ts parts (tasks). Usg project buffer tasks o crtcal cha are protected, but we have to keep md, that there are ocrtcal tasks project schedule too. Accordg to the tradtoal approach they have to start as soo as possble ad usg ther slacks all other tasks are protected. The demert of the slack s that t ca get waste. I the Crtcal Cha method ths problem s elmated the same way usg aother type of buffer called feedg buffer. The feedg buffer s placed wheever ay feedg cha (ocrtcal cha) jos the crtcal cha ad ts fucto s to protect all tasks o the feedg cha. I the worst case t ca be expeded all feedg buffers, but the due date (project fsh date) s stll protected from project buffer. 257

3 Resource buffers usually the form of a advace warg, are placed wheever a resource has a job o the crtcal cha, ad the prevous crtcal cha actvty s doe by a dfferet resource. A very moot pot deals wth the sze of safety for dvdual tasks. The statemet that the project buffer s a half of sum of safety of Crtcal Cha ad the feedg buffer s approxmately oe thrd of feedg cha seams to be very smplfed. Ther sze depeds o may factors. For example t s very mportat f the task s uque or f t s commo ad was used may tme before. The uque task wll probably have loger safety tme that a commo oe. We have to keep md, that there are also huge dffereces safety betwee short ad log tasks because. These factors ca fluece each other ad so t s very dffcult to estmate ths safety for each task. The Crtcal Cha schedule s prmary based o sertg buffers ad o Crtcal Cha detfcato. Compared wth tradtoal crtcal path methods the Crtcal Cha schedule s ofte loger, but there exsts a hgh probablty of project completo o tme. Mathematcal Programmg Model of a Actvty o Node Crtcal Path Problem as a Fudamet for CC Model Defto Eve thought the majorty of project maagemet mathematcal methods are based o actvty o arc (AOA) approach we shall apply a actvty o ode (AON) etwork model for the formal project terpretato because all software realzato of CC method are derved from t. I AON graphs all project tasks are represeted by odes ad ther relatoshps are represeted by arcs. Whe fdg crtcal path, each project task ca ether be crtcal or ocrtcal. The arc jog two tasks (odes) s a part of a crtcal path f ad oly f both tasks at ts eds are crtcal too. Let s defe two kds of varables x ad x j, x = 1 whe task s crtcal, x = 0 otherwse ad x j = 1 whe the arc j les o a crtcal path, x j = 0 otherwse. The frst ad the last task must le o a crtcal path ad so x 1 = 1, x = 1. Each task has a parameter t the meag of ts durato ad each arc has oly oe parameter t j the meag of a lag (or lead tme whe egatve),.e. the tme terval betwee the ed of prevous task ad the begg of the ext oe. The set of feasble solutos X, ( x X, x X s defed usg followg costrats: j ) 258

4 x + xj = 0; = 1,2,...,( 1) j R dex of predecessor or of a start ode xj xj = 0; j = 2,3,..., P j dex of successor or of a ed ode (1) j x1 = 1; x = 1 x {0;1} xj {0;1} x, x j varables represetg odes (tasks) x j varables represetg arcs (task depedeces) P j set of j-th task predecessors The objectve fucto dffers could be descrbed by the followg way: ( tx + t x ) max j j = 1 j R t -th task durato (2) t j tme terval betwee the ed of -th task ad Wth respect to a specal type of costrat coeffcet matrx A = (a j ) m. a { 1;0;1} ad to a type of RHS vector b = (b) m b {0;1} (matrx s completely j umodal) the bvalet soluto s guarateed whe solvg ths mathematcal programmg problem usg stadard smplex algorthm. Solvg ths problem we obta a set of crtcal tasks ad cosequetly the whole crtcal path. The objectve fucto value determes a legth of crtcal path. For aalyzg ocrtcal tasks parameters we ca use sestvty aalyss of cost coeffcets. Applyg stadard cost coeffcet sestvty aalyss algorthms for lear problems we obta tervals of stablty for both types of costs coeffcets,.e. t t ; t ad t t ; t j j j. Usg tervals of stablty for task durato t we ca defe a total slacks for ocrtcal tasks. A ocrtcal task s represeted ether by a o-basc varable or by a basc varable wth zero value. A task becomes crtcal whe ts durato (ts cost coeffcet value) exceeds a maxmum lmt (upper boud of stablty terval). The total slack ca be expressed as s 1 = t t (3) s 1 s a total slack of -th task ad t s a upper boud of a terval of stablty for t. CC Mathematcal Programmg Model For the Crtcal Cha mathematcal model let s assume, that the project s represeted by a classcal AON etwork graph (most of all t meas t s cotuous ad t has oe startg ode ad oe edg ode) ad s really resource leveled. Furthermore let s assume each project eds by a dummy ode coected wth last project task by a dummy arc (ths last arc wll be used for project buffer sertg later ths text) ad so the umber of odes CC model () s oe ut hgher tha the umber of project task s. We have to keep md 259

5 that each task s protected by a reserve (safety). The amout of such reserve tme s mplcated objectve fucto cost coeffcets. Accordg to the assumptos metoed above, a task durato t ca be dvded to two parts: q task reserve, t meas task protecto from delay ad (t - q ) the pure (or clarfed) durato. Furthermore let s assume that all depedecy lags (t j ) are equal to 0. (I the ext phase of the model these lags (objectve fucto cost coeffcets of arcs) wll be used for CC buffers. Acceptg ths a mathematcal programmg model for the frst phase (crtcal cha defto ad project buffer establshmet) ca be defed as follows (( t q ) x + t x ) max j j = 1 j R ( x X, x X) j q task durato reserve Solvg ths model we obta a crtcal path ts structure ad ts legth. I ths phase of model solvg the crtcal path determes a crtcal cha as well. Project Buffer Accordg to the coclusos of CC method, the amout of tme eeded for project fsh date protecto the form of project buffer s determed by the formula PB = q x (5) = 1 Because x = 1 oly o a crtcal path, ths formula guaratees summarzato of CC reserves oly. I the ext step of the model ths buffer wll be terpreted as a cost coeffcet of last arc x -1, t meas t = q x. Feedg Buffers 1, = 1 The am of feedg buffer s to protect crtcal cha from potetal delays o ocrtcal cha actvtes (supportg paths) ad t has to be placed wheever a o-crtcal actvty jos the crtcal cha. Accordg to the classcal CC approach a feedg buffer protects oly the logest o-crtcal sequece (cha). Let s modfy ths assumpto ad suppose, that feedg buffer protects all o-crtcal actvtes lyg betwee two crtcal oes, t meas all zero varables placed the etwork betwee two ozero oes (two odes). Such set of task wll be called feedg subgraph. We suppose that realstc value for each feedg buffer wll be a value summarzg all reserves o a feedg subgraph plus the largest total slack o ths subgraph. The reaso for such preposto s that respectg ay smaller slack could precocously cosume buffer. (4) 260

6 For descrpto of feedg buffer let s defe a feedg subgraph P r, ts startg odes (represeted by varables x r( s) k ), ad ts uque ed ode (represeted by varable Each varable ca be a startg (ed) pot of a feedg subgraph satsfyg: r( s) ( k k ) (, j):( x = x x x B j R) r( f ) ( k k ) ( j, j):( x = x x x j B j R) B set of basc ozero varables dexes R set of zero varables x ). d f d Usg these varables ad terms a r-th feedg subgraph ca be defed as a set of all paths, startg ay ode V r(s) ad edg a uque ode V r(f). Example: Let s have a followg graph ad let s suppose, that crtcal cha s formed by tasks A, E, I r ( f ) k (5) B D C F H G A E I Dummy O ths graph two feedg subgraph are defed: P 1 as {B-D-F-H; C-H}, ad P 2 as {C-G}. Cost coeffcets of arcs G-E ad H-I wll be reserved for placemet of feedg buffers ad cost coeffcet of arc I-Dummy for placemet of project buffer. I our approach let s model feedg buffers usg cost coeffcet of last arc each subgraph P r r, t meas as a cost coeffcet of arc ( f x ), j R ad accordg to prevous defto t could be descrbed as r( f ) FB r = tk, j = f ( q) P max f ( q ) = ( q ) + ( s ) Rr r R k, j R r set of dexes of r-th feedg cha varables f(q ) fucto for FB calculato For f(q ) calculato we assume summarzato of all feedg cha tasks safetes (reserves) but as a acceptable calculato we ca take a sum of all reserves o the logest feedg path (classcal Goldratt s approach) or applyg the maxmum sze of task reserve o the whole feedg subgraph etc. Defg all these buffers we obta a mathematcal programmg model wth the same defto of feasble soluto set, but wth aother objectve fucto cost coeffcets. Solvg ths model we obta ether the same structure of basc varables vector or we obta aother basc soluto. I such case (majorty of cases) a classcal crtcal path cludes serted buffers too. If the legth of CP s far away from former CC legth, we have to thk (6) 261

7 about the sze of task reserves. After soluto aalyss we should beg wth workg o a project, track ts progress, cosume buffers ad potetally chage the CC structure too. 2. Cocluso The objectve of our paper was to show oe possblty of how to solve Crtcal Cha method models usg mathematcal programmg tools. The ma effect of ths approach s AON graph terpretato of ths model. Because of real actvtes are o odes ad buffers are modeled usg arcs, o chages graph structure (ad o chages mathematcal programmg model defto) eed to be doe. All buffers sertg are made oly usg chages cost coeffcets of arcs. Usg stadard Excel lear programmg tools a optmal soluto of CC model ca be obtaed. Refereces 1. Goldratt,E.: Crtcal Cha. Great Bergto, MA, Northrver Press, Kerzer,H.: Project Maagemet: A system Approach to Plag, Schedulg ad Cotrollg. Joh Wley & Sos, New York, Herroele,W., Leus,R.: O the Merts ad Ptfalls of Crtcal Cha Schedulg. I.: Joural of Operatos Maagemet 19 (2001) , Elsever Scece, Šubrt,T.: Mult Crtera Bvalet Programmg Models for Actvty o Node Networks. I.: Proceedgs of the 19th Iteratoal Coferece Mathematcal Methods Ecoomcs 2001 Coferece, UHK, Hradec Králové, 2001 Dr. Ig. Tomáš Šubrt, Ig. Pavlía Lagrová Dept. of Operatos Research ad System Aalyss, Faculty of Ecoomcs ad Maagemet, Czech Uversty of Agrculture Prague, subrt@pef.czu.cz, lagrova@pef.czu.cz 262