QUANTITATIVE ASSESSMENT OF CONSTRUCTION RISK

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1 QUANTITATIVE ASSESSMENT OF CONSTRUCTION RISK T. KASPROWICZ 1 Abstract: Costructo rs assessmet s the fal ad decsve stage of rs aalyss. Whe hghly chageable codtos of wors executo are redcted, rs should be evaluated the favorable, moderate, ad dffcult radom codtos of costructo. Gve the radom codtos, the schedule ad cost estmate of the costructo are develoed. Based o these values, the rs of fal deadle delay ad the rs of total cost crease of costructo comleto are calculated. Next, the charts of the rss are elaborated. Rs chages are show the charts ad are aalyzed the rage [1, 0]. Keywords: rs codtos, schedulg, cost estmatg, rs assessmet, rs charts 1. INTRODUCTION I ractce, a costructo 2 rs ofte has to be scrutzed because of the fluece of radom evets over the course ad results of wors 3 executo. Costructo s the ma stage of costructo roect 4. There are a umber of ublcatos o dfferet methods of costructo rs aalyss. The evoluto of such cocets, a overvew of research ad alcatos ertag to relablty of costructo roducto, the use of resources, robust terares, ad cotgecy of tme ad cost has bee reseted ublcato [9]. Authors have descrbed areas of maagemet advsory systems relato to the cycle of rs aalyss. They also ot out emergg treds whch stadardze the vestmet rocess maagemet, cludg rs aalyss. The vestgato of rs maagemet 1 Prof. DSc., PhD., Eg., Mltary Uversty of Techology, Faculty of Cvl Egeerg ad Geodesy, Kals Str. 2, Warsaw, Polad, e-mal: tadeusz.asrowcz@wat.edu.l 2 Costructo the buldg of structures ad facltes 3 Costructo wor a actvty volvg metal or hyscal effort doe order to buld structures ad facltes 4 Costructo roect dvdual or collaboratve actvtes laed ad desged to buld structures ad facltes; Lfe cycle of costructo roect: lag hase, desg hase, costructo hase, oerato hase; Uauthetcated Dowload Date 10/29/17 6:58 AM

2 56 T. KASPROWICZ small roects Sgaore terms of barrers ad mact of rs maagemet o roect erformace has bee reseted the artcle [2]. Based o the results of a questoare survey, the authors oted to a ostve correlato betwee rs maagemet ad the mrovemet qualty, cost, ad schedule erformace of small costructo roects. Publcatos [1, 3, 6, 8] reset methods assocated wth costructo-related rs maagemet. I artcle [1] a systematc aroach to extedg the Techque for Order Preferece by Smlarty to Ideal Soluto has bee roosed for a rs assessmet model selecto, the cotext of a fuzzy decso-mag evromet. I artcle [3] a Rs Maagemet Maturty System has bee roosed. The system cossts of three fuctoal comoets: caabltes, assessmet, ad evoluto. It s based o a tegrated alcato of theores ad tools drawg from exerece systems egeerg, emergece ad adatato techques, ad mult-crtera decso mag. The resultg arasal of roect rs s a multdmesoal assessmet corresodg to secfc rs characterstcs. I artcle [6] the author has descrbed the rearato of Proect Rs Assessmet Methodology ad ts mtgato comlex costructo roects. The author has develoed a fucto whch maes t ossble to categorze ay rss to oe of the fve categores. It s a combato of robablty ad the mact of oe of these tems: eole ad ther safety or budget, cost, schedule ad lag or qualty ad erformace. The relatosh betwee rss macts ad ther level of lelhood has bee also vestgated. I aer [8] the author has roosed the method of detfcato ad relmary quatfcato of rs. Rs reducto techques, the method of rs allocato costructo schedules, the cotgecy la develomet, ad the assessmet of the effectveess of rs aalyss has also bee reseted. Ths aroach eables verfcato of the solutos ad develomet of rs-related relevace coeffcets for assessg roect cost ad tme. Ths artcular method of rs detfcato wors as well as cost ad tme of the roect s comarable to the PERT method whch has bee reseted artcles [4, 5]. The most mortat ractcal cosequece of the author s research s that stead of the tme-cosumg ad costly rocess of selectg the roer actvty durato dstrbutos, laers should devote more effort to adequately determe the actvty durato tmes. O the bass of these ublcatos oe ca coclude that there s stll a owledge ga betwee ractcal eeds ad avalable theoretcal solutos. There s o uversal method of roect rs assessmet whch s commoly used. I artcular, ths ales to rs assessmet of wors o the costructo stes. The method of costructo rs assessmet that s roosed ths aer s a attemt to address ths ga. Based o the costructo rs detfcato the schedules ad cost estmates are develoed. The rs of fal deadle delay ad the rs of total cost crease of costructo comleto are calculated. Rss are aalyzed the terval [1, 0]. Uauthetcated Dowload Date 10/29/17 6:58 AM

3 QUANTITATIVE ASSESSMENT OF CONSTRUCTION RISK RULES OF CONSTRUCTION RISK ASSESSMENT Costructo rs should be evaluated accordg to the codtos of wors executo ad resource requremets. I the case of hgh chageable crcumstaces, favorable, moderate, ad dffcult codtos of costructo should be cosdered. Requremets of obect structure for such codtos are descrbed by usg the desg characterstcs model of structure S as follows: (1) S = G, L G = Y, U, P coheret ad a-cyclc ugrah wth a sgle tal ode ad a sgle fal ode that descrbes terdeedece ad ermssble sequeces of wors executo, { y,, y,, y,, y } Y = set of the odes of the grah reresetg tal ad fal evets for each wor 1 m u U ; the odes mea: y tal ad y fal ode of the wor u executo; { u,, u,, u,, u } U = set of the arcs (arrows) of the grah reresetg relatvely 1 l deedet wors (actvtes) that are costraed by tal y Y ad fal y Y odes; P Y U Y, y, u, y P three-term relato that assgs to each arc u U the tal ode y Y ad fal ode y Y ; + L : U R fucto defed o the set U of arcs of the grah G that determes the certa l or exected E [ L ] values of the varables L ad descrbes the sze of wors u U. Requremets of wors executo are descrbed by usg the costructo techology model : (2) = { H, K, T, S } 1 2 { H, H,, H } r, H s H =, set of ratoal or otmal teams T ) + : ( H U R fucto defed o the set H of teams r H for wors u r U executo, r H whch determes the exected values E [ T,r ] of radom varables T, r ad descrbes the exected durato of wors u U r U. + K : ( H U ) R fucto defed o the set H r of teams H whch determes the exected values E [ K,r ] of radom varables K, r ad descrbes the exected cost of wors u U r U. Uauthetcated Dowload Date 10/29/17 6:58 AM

4 58 T. KASPROWICZ 3. SCHEDULING CONSTRUCTION Based o the desg characterstcs model of structure model = { H, K, T, S } S = G, L ad the costructo techology, the schedules ad cost estmates ca be develoed. I the rs cod- tos, the schedules determe the exected fal deadles E [T ], E [ T ], E [ T ] ad the cost est- mates determe the exected total costs E [K], E [ K ], E [ K ] for the moderate, favorable, ad dffcult costructo codtos, resectvely. These values ca be calculated usg the sutable software acage for wors-schedulg ad software acage for cost estmato. I order to exla the roblem, the schedules of wors executo are develoed as follows: [ m calculate the exected earlest fal deadle E V ] of costructo S comleto by usg resources, ad the exected early start E V ], that s calculate the values of the varables E V ],, V ],, V ] that mmze the fucto E [T ] : [ 1 m = m = 1 [ T ] = m V ], (3) uder the costrats: E [ V ] V ] T ] dla u U, = 1,2,...,, y, u, y P V ], V ] 0 dla = 1,2, m, [ m calculate the exected earlest fal deadle E V ] of costructo S comleto by usg resources, ad the exected late start E V ], that s calculate the values of the varables E V ],, V ],, V ] that mze the fucto E [T ] : [ 1 m = m = 1 [ T ] = V ], (4) uder the costrats: E [ V ] V ] T ] dla u U, = 1,2,...,, y, u, y P Vm ] V m] V ], V ] 0 dla, = 1,2, m., The above roblems oe ca solve by usg ay stadard comuter rogram. Uauthetcated Dowload Date 10/29/17 6:58 AM

5 QUANTITATIVE ASSESSMENT OF CONSTRUCTION RISK CONSTRUCTION RISK ASSESSMENT Costructo rs assessmet here s strctly coected wth the schedules ad wth the cost estmates of costructo comleto. Rs assessmet always refers to the secfc codtos of costructo, ad s calculated accordace wth the ormal dstrbuto of the radom varables T, T, T, K, K ad K. Tag ths to cosderato, the rs of fal deadle delay ad the rs of total cost crease are calculated. Based o the solutos of schedulg roblems, the crtcal aths ( U CP ) are determed. For wors u CP U, the varaces of fal deadle 2 ( T ) D, D 2 ( T ), D 2 ( T ) are calculated. For wors u U, the varaces of total cost D 2 ( K ), D 2 ( K ), D 2 ( K ) are calculated. Based o the values E [T ], E [ T ], E [ T ], E ], E ], E ], D 2 ( T ), D 2 ( T ), 2 ( D T ) ad E [K ], E [ K ], E [ K ], D 2 ( K ), D 2 ( K ), D 2 ( K ), E ], E ], [ T,r [ T, r [ T, r [ K,r [ K, r E ] oe ca calculate the rs of fal deadle delay of costructo comleto ad the rs of [ K, r total cost crease of costructo comleto. The rss of fal deadle delay ( t), ( t) ad ( t) for the secfed values of fal deadle t ad the ormal dstrbuto of the radom varable T oe ca calculate as follows: (5) ( t) = P[ T ] t] = 1 P[ T t] t T ] 1 Z D 2 [ T ] t secfed value of deadle of costructo S comleto whch s robable ad ca be cosdered uder the cotract egotatos; Z stadardzed ormal radom varable. The rss of the total cost crease ( ) K ] ), ( ) K ] ) ad ( t) for secfed values of total cost ad ormal dstrbuto of the radom varable K oe ca calculate as follows: (6) ( ) = P[ K] ] = 1 P[ K ] K] 1 Z D 2 [ K] secfed value of the cost of costructo S comleto, whch s robable ad ca be cosdered uder the cotract egotatos. Uauthetcated Dowload Date 10/29/17 6:58 AM

6 60 T. KASPROWICZ Usg the formulas (5) ad (6) the rss (t), (t), (t), (), () ad () for successve values t ad ca be reseted o the charts of rs of fal deadle delay, ad the charts of rs of total cost crease of costructo comleto. These charts are referred to as cotgecy charts of costructo delays ad cotgecy charts of costructo cost overrus. These charts characterze chages of the rs of fal deadle delay (t) ad chages of the rs of the total cost crease () the terval [1, 0] for cosecutve values of the total costs [ m, ] ad the fal deadles t [ t m, t ]. It meas that: ( m ) = 1, ( ) = 0, ( t m ) = 1, ( t ) = 0. The dfferet values t ad ca be aalyzed durg the cotract egotatos as the fal deadle ad total cost of costructo comleto. Tag to accout the addtoal codtos of decsomag, the values that are accetable gve stuato should be settled the cotract. 5. RISK ASSESSMENT OF SMALL BRIDGE CONSTRUCTION I order to exla ths method, a small brdge erecto s cosdered. The costructo of the brdge s descrbed the desg documetato ad the rmary cost estmato. The desg characterstcs model of brdge structure S has bee develoed accordace wth formula (1). The brdge costructo techology model has bee develoed accordace wth the formula (2). Detaled data for the costructo of the brdge descrbed by usg these models has bee talled Table 1. O the bass of ths data, the schedules ad cost estmates of the brdge costructo the moderate, favorable, ad dffcult codtos of costructo have bee develoed. Here, all schedules have bee made as solutos of the schedulg roblems of brdge costructo accordg to the equatos (3) ad (4) for the moderate, favorable, ad hard costructo codtos. For solvg these roblems MS Excel was used. Results of the calculatos are talled Table 2. The solutos of the roblems determe ES the exected early start E V ], ad LS the exected [ late start E V ] for each ode y Y for the wors U U, y, u, y P. The solutos deter- me also the exected fal deadles E [T ], E [ T ] ad E [ T ] of brdge comleto. For each so- luto exected total costs E [K], E [ K ] ad E [ K ] of brdge comleto have bee calculated. The, usg formula (5) ad tag to accout the wor that belogs to the crtcal ath, for subsequet values of t the rs of fal deadle delay of costructo comleto has bee calculated. Next, usg formula (6) ad tag to accout all wors of brdge costructo, for subsequet values of the rss of the total cost crease of costructo comleto have bee calculated. [ Uauthetcated Dowload Date 10/29/17 6:58 AM

7 Table 1. Data for the desg characterstcs model of brdge structure S ad brdge costructo techology model No Ttle u y y E [V ] E [V ] Durato of the wors mlemetato (days ) Cost of the wors mlemetato (PLN) Dffcult Modertate codtos Dffcult H r Favorable codtos codtos codtos E [T ] T T T E [T E [K ] K E ] E [T ] K K E [K ] [K ] y y Moderate codtos Favorable codtos Dummy actvtes Lad develomet ad E [V recostructo wors 1 ] E [V 2 ] Temorary suort for E [V structural assembly ad erecto 2 ] E [V 7 ] Abutmet left E [V 3 ] E [V 5 ] Abutmet rght E [V 4 ] E [V 6 ] Embamet left E [V 5 ] E [V 8 ] Embamet rght E [V 6 ] E [V 8 ] Brdge sa E [V 7 ] E [V 8 ] Btumous avemet E [V 8 ] E [V 9 ] Fshg wors E [V 9 ] E [V 10 ] Exected total cost of the costructo comleto Table 2. Results of the calculatos No Ttle u y y E [V ] E [V ] H ES E [V ] Exected early start (ES ) ad exected late start (LS ) Exected values E [V ] (days ) r Moderate codtos LS E [V ] Favorable codtos ES E [V ] LS E [V ] Dffcult codtos ES E [V ] LS E [V ] Exected cost E [K ] of the wors executo (PLN ) Moderate codtos Favorable codtos Dffcult codtos Lad develomet ad recostructo wors Temorary suort for structural assembly ad erecto E [V 1 ] E [V 2 ] E [V 2 ] E [V 7 ] Abutmet left E [V 3 ] E [V 6 ] Abutmet rght E [V 4 ] E [V 7 ] Embamet left E [V 5 ] E [V 8 ] Embamet rght E [V 6 ] E [V 8 ] Brdge sa E [V 7 ] E [V 8 ] Btumous avemet E [V 8 ] E [V 9 ] Fshg wors E [V 9 ] E [V 10 ] Exected fal deadle of the costructo roect comleto E [V 10 ] Exected total cost of the costructo roect comleto 1 2 QUANTITATIVE ASSESSMENT OF CONSTRUCTION RISK 61 Uauthetcated Dowload Date 10/29/17 6:58 AM

8 62 T. KASPROWICZ The rss have bee show o the rs charts of costructo comleto (Fgs. 1 ad 2). 1,10 1,00 0,90 0,80 0,70 0,60 0,50 0,40 0,30 0,20 0,10 0, Rs of the fal deadle delay Fal deadle of the brdge costructo comleto Favorable codtos Moderate codtos Dffcult codtos Determstc fal deadle of the costructo comleto Fg. 1. Rs charts of the fal deadle delay of brdge costructo comleto 1,10 1,00 Rs of the total cost crease 0,90 0,80 0,70 0,60 0,50 0,40 0,30 0,20 0,10 0, Total cost of the costructo comleto Favorable codtos Moderate codtos Dffcult codtos Determstc total cost of the costructo comleto Fg. 2. Rs charts of the total cost crease of brdge costructo comleto 6. CONCLUSIONS The method of costructo rs assessmet that has bee reseted the aer allows cotractors to exame chages of costructo rs deedg o codtos of costructo ad deedg o dfferet values of the fal deadle t ad the total cost of brdge comleto. All accetable chages of these quattes the terval (1, 0] are show the rs charts. The smallest values t m ad m of the fal deadles ad the total costs are calculated for the robabltes of exceedg ( t) = 1 ad ( ) = 1. For these values, the fal deadle ad the total cost of costructo Uauthetcated Dowload Date 10/29/17 6:58 AM

9 QUANTITATIVE ASSESSMENT OF CONSTRUCTION RISK 63 comleto wll most lely be exceeded. The greatest values t ad of the fal deadles ad total costs are calculated for the robabltes of exceedg ( t) = 0 ad ( ) = 0. For these values, the fal deadle ad the total cost of costructo comleto wll most lely ot be exceeded. Such values ad the rages of ther chages have bee calculated for the favorable, moderate, ad dffcult codtos of brdge comleto. O the bass of such aalyss oe ca evaluate the degree of threat that the fxed fal deadle of the costructo comleto ca exceed, ad estmate the threat that the cotracted total cost of costructo comleto ca surass. I Table 3 are show the rss of fal deadle delay (t), (t) ad (t), ad the rss of total cost crease (), () ad () of brdge comleto for the fal deadle t ad the total cost, equal to the determstc values that are values whch are calculated wthout rs aalyss. Ufortuately these rss are uow the determstc aroach. Table 3. The rs of fal deadle delay ad the rs of total cost crease of brdge costructo comleto for the determstc values of the fal deadle ad total cost of costructo Codtos of brdge costructo Rs of fal deadle delay of brdge comleto for 488 days Rs of total cost crease of brdge comleto for PLN Favorable 0,53 0,53 Moderate 0,64 0,64 Dffcult 0,77 0,77 Tag to accout the results of calculatos reseted the Table 3, t s mortat to clarfy that the rss of the fal deadle delay ad the rss of the total cost crease of costructo comleto are equal because of the relatvely short durato of the costructo ad the stable rces of labor ad costructo roducts durg wors executo. I such stuatos, ths cocluso s also true for other ermssble values of the fal deadle ad total cost of costructo comleto. Fally, oe ca cofrm that the method of costructo rs assessmet s theoretcally ready to use. But, ractce, oe should say that there are two basc codtos order to use t. Frst, the method of detfcato of radom evets ad ther mact o executo ad results of the wors should be mroved, for examle by usg a exert system. I artcular ths ales to the quattatve estmato of the mact of the dsrutos o wors executo. Secodly, t s ecessary to develo software that would be coveet to use ths stuato. The software should be fully comatble wth the cost estmato software. At reset, these codtos are the rocess of beg aalyzed. Uauthetcated Dowload Date 10/29/17 6:58 AM

10 64 T. KASPROWICZ 7. REFERENCES [1] AmrReza, Karm Azar, Neda Mousav S. Fard Mousav, SeyedBagher Hosse, Rs assessmet model selecto costructo dustry. Exert Systems wth Alcatos 38 (2011) [2] Bo-Gag Hwag, Xabo Zhao, L Pg Toh, Rs maagemet small costructo roects Sgaore: Status, barrers ad mact. Iteratoal Joural of Proect Maagemet 32 (2014) [3] Guagshe Ja, Xaochua N, Zhe Che, Baoa Hog, Yutg Che, Fagu Yag, Che L,Measurg the maturty of rs maagemet large-scale costructo roects. Automato Costructo 34 (2013) [4] Hadu M., Boor O., Sestvty aalyss PERT etwors: Does actvty durato dstrbuto matter? Automato Costructo 65 (2016) [5] Hadu M., Effects of the alcato of actvty caledars o the dstrbuto of roect durato PERT etwors. Automato Costructo (2013). htt://dx.do.org/ /.autco , Elsever. [6] Koor J., Eterrse s rs assessmet of comlex costructo roects. Archves of Cvl Egeerg. Vol. LXI, Issue 3, DOI: /ace [7] Ru Zou, Yog Lu, Le Lu, Huacheg Guo, REILP Aroach for Ucertaty-Based Decso Mag Cvl Egeerg [8] Sorua D., Method of Costructo Proects Rs Assessmet. LAMBERT Academc Publshg GmbH & Lcesors, Saarbrüce [9] Turss Z., Gazler M., Dzadosz A., Relablty, Rs Maagemet, Ad Cotgecy of Costructo Processes ad Proects. Joural of Cvl Egeerg ad Maagemet, ISSN rt/issn ole, 2012 Volume 18(2): LIST OF FIGURES AND TABLES Fg. 1. Rs charts of fal deadle delay of brdge costructo comleto Rys. 1. Wyresy ryzya oó ea ostateczego termu zao czea budowy mostu Fg. 2. Rs charts of total cost crease of brdge costructo comleto Rys 2. Wyresy ryzya wzrostu całowtego osztu zao czea budowy mostutable 1. Data for the desg characterstcs model of brdge structure S ad brdge costructo techology model Tabela 1. Dae modelu techolog ostruc mostu S modelu techolog budowy mostu Table 2. Results of the calculatos Tabela 2. Wy oblcze Table 3. Rs of fal deadle delay ad rs of total cost crease of brdge costructo comleto for the determstc values of fal deadle ad total cost of costructo Tabela 3. Ryzyo oó ea termu o cowego ryzyo wzrostu całowtego osztu zao czea budowy mostu dla determstyczych warto c ostateczego termu całowtego osztu budowy Receved Revsed Uauthetcated Dowload Date 10/29/17 6:58 AM

11 QUANTITATIVE ASSESSMENT OF CONSTRUCTION RISK 65 KWANTYTATYWNA OCENA RYZYKA BUDOWY Słowa luczowe: waru ryzya, harmoogramowae, osztorysowae, ocea ryzya, wyresy ryzya STRESZCZENIE Realzaca robót budowlaych a lacu budowy est odstawowym etaem rzeds wz ca budowlaego. W ratyce, z utu wdzea realzac robót a lacu budowy lub rzebudowy, mota u, remotu lub rozbór obetu budowlaego, ryzyo czasu osztów cz sto owo by dołade badae ze wzgl du a wływ zdarze losowych a rzebeg wy wyoaa robót. Jest wele ublac a temat metod detyfac ocey ryzya. Jeda oublowae metody e rozw zu roblemów ocey ryzya, tóre geerowae est rzez roces realzac robót a lacu budowy ego otoczee. Prooowaa metoda est ocec lo cowe ocey ryzya robót budowlaych a lacu budowy. Jest to o cowy eta aalzy ryzya. Porzedza go detyfaca uwaruowa ryzya, tóre zale od losowe techologcze charaterysty ostruc, losowe charaterysty dost ych zasobów oraz losowych waruów realzac robót. W erwszym rou aalzy ryzya s detyfowae charaterysty losowe oraz oblczae s oczeway czas oczewae oszty oszczególych robót budowlaych. W drugm rou, wyzaczae s harmoogramy osztorysy robót dla rzewdywaych waruów losowych budowy. W trzecm rou, badae oceae est ryzyo oó ea ostateczego termu zao czea budowy (t m ) ryzyo rzeroczea całowtych osztów budowy ( m ). Ryzya te badae s w rzedzale [1, 0]. Ryzya (t) = 1 () = 1 wyzacza warto c o cowe czasu t m osztów m uo czea budowy, tóre arawdoodobe b d rzeroczoe. Zatem warto c rówe mesze od t m m s edouszczale e owy by ustalae, ao warto c umowe. Ryzya (t) = 0 () = 0 wyzacza warto c o cowe czasu t m osztów m uo czea budowy, tóre arawdoodobe e b d rzeroczoe formale mog by ustalae, ao warto c umowe. Oczyw ce, wszyste w sze warto c o cowe rówe e b d rzeroczoe ta e mog by wyzaczoe, ao umowe warto c czasu osztów zao czea budowy. W rzyadu sle zmeych waruów realzac detyfac oce ryzya rooue s rzerowadz dla omy lych, rzec tych trudych waruów budowy. W a dym rozatrywaym rzyadu ryzyo aalzowae owo by w zale o c od losowych wymaga techologczych ostruc obetu oraz techologczo-orgazacyych losowych uwaruowa zasobów, tóre mog by wyorzystae do wyoaa robót. W racy, wymagaa techologcze ostruc obetu s osywae za omoc modelu techolog ostruc obetu S = G, L. W modelu tym graf G osue zbór wsółzale o techologcz rzedmarowaych robót. Natomast fuca L osue rozmar rzedmarowaych robót. Welo c te ore lae s a odstawe aalzy roetu budowlaego. Wymagaa techologcze, orgazacye zasobowe robót s osywae za omoc modelu techolog budowy = { H, K, T, S }. Zbór H zawera racoale lub otymale brygady robocze, tóre zorgazowae (zaroetowae) zostały do wyoaa robót S. Warto- c fuc K T osu odowedo oszty czas realzac robót S rzez brygady robocze zboru H. Warto c tych fuc s ore lae a odstawe aalzy modelu S, dysozycyych zasobów waruów wyoaa robót. Wyorzystu c dae osae za omoc ta zdefowaych model s oracowywae harmoogramy osztorysy robót w waruach robablstyczych rzec tych, omy lych trudych. Mog oe by oracowywae za omoc tyowych rogramów do harmoogramowaa osztorysowaa robót. W racy, w celu rzygotowaa daych do bezo rede ocey ryzya, formułowae rozw zywae s roblemy harmoogramowaa robót dla model S, ao zagad- Uauthetcated Dowload Date 10/29/17 6:58 AM

12 66 T. KASPROWICZ ea rogramowaa lowego. W te sosób wyzaczae s ostatecze oczewae o cowe termy E [T ], E [ T ], E [ T ] oraz oczewae całowte oszty E [K], E [ K ], E [ K ] wyoaa robót w waruach rzec tych, o- my lych trudych. Załada c w tych waruach rozład ormaly losowych zmeych czasu trwaa T, T, T oraz całowtych osztów K, K K budowy, dla oszczególych harmoogramów s wyzaczae warace osta- 2 T 2 2 teczych termów D ( ), D ( T ), D ( T ) warace całowtych oszów D ( ), D ( K ), D ( K ) zao czea budowy. Na odstawe tych warto c oblczae est ryzyo czasu osztów budowy. Dla c gu douszczalych warto c t termu o cowego wyoaa robót est oblczae ryzyo oó ea ostateczego termu zao czea budowy za omoc zale o c: ( t) w waruach rzec tych, ( t) w waruach omy lych ( t) w waruach trudych. Dla c gu warto c całowtych osztów wyoaa robót est oblczae ryzyo rzeroczea całowtych osztów budowy za omoc zale o c: ( ) K] ) w waruach rzec tych, ( ) K ] ) w waruach omy lych ( ) K ] ) w waruach trudych. Nast e dla c gu warto c o cowych t, warto c E [T ], E [ T ], E [ T ], E ], E ], E ] 2 K 2 2 [ K,r [ K, r [ K, r oraz D ( ), D ( K ), D ( K ), E ], E ] E ] s oracowywae wyresy ryzya oó ea ostateczego termu (t), (t) (t) oraz wyresy ryzya wzrostu całowtych osztów (), () () zao czea budowy. Ja wya z rzedstawoych zale o c, mar ryzya est rawdoodobe stwo oó ea ostateczego termu zao czea budowy (t) rawdoodobe stwo wzrostu całowtych osztów () uo czea budowy dla ore loych warto c o cowego termu t całowtych osztów. W celu ore lea graczych warto c czasu osztów uo czea budowy wyzaczae s rzedzały douszczale (realzowale) zmeo c o cowego termu t [ t m, t ] całowtych osztów [ m, ] uo czea budowy. Warto c t m m, dla tórych ( t m ) = 1 ( m ) = 1, wyzacza arótszy czas amesze oszty budowy, tóre b d arawdoodobe rzeroczoe. Warto c t 2 K 2 [ T,r 2 [ T, r [ T, r, dla tórych ( t ) = 0 ( ) = 0, wyzacza lmty czasu osztów budowy, tóre arawdoodobe b d erzeroczoe. Wszyste warto c t zawarte w rzedzałach ( t, t ] oraz (, ] s douszczale (realzowale). Jeda ryzyo zw zae z ch dotrzymaem mo e s m m zacze ró. Oczyw ce wszyste warto c t, zao czea budowy w sze od t mog by rówe rzy te, ao umowe warto c dyretywe zao czea budowy, gdy ryzyo ch rzeroczea wyos zero, czyl: ( t > t ) = 0 ( > ) = 0. Zaomo lmtów dolych t m m, górych t oraz wyresów ryzya umo lwa wyoawcy westorow ustalee warto c otratowych o cowego termu całowtych osztów zao czea budowy dla ore loego ozomu ryzya. Uauthetcated Dowload Date 10/29/17 6:58 AM