AC : THE DIAGRAMMATIC AND MATHEMATICAL APPROACH OF PROJECT TIME-COST TRADEOFFS

Transcription

1 AC : THE DIAGRAMMATIC AND MATHEMATICAL APPROACH OF PROJECT TIME-COST TRADEOFFS Kun-jung Hsu, Leader Unversty Amercan Socety for Engneerng Educaton, 2008 Page

2 Ttle of the Paper: The Dagrammatc and Mathematcal Approach of Project Tme-cost Tradeoffs Abstract A potental project management nvolvng tme used of a project can always be tradeoff by addtonal resources nput. Such a tradeoff may come from dfferent optons of the actvty of the project whch can be choce. The stuaton of Pay more - Save Tme s common for project management related decson problems. The avalable technology of shortenng the duraton of each actvty s often the sources of the tme-cost tradeoffs problem. And the problem solvng processes always rely upon the technques of crtcal path method (CPM) calculaton and mathematc programmng, for example lnear programmng, or nteger programmng etc. The paper ncludes an ntroducton to the concepts of CPM method, tme-cost tradeoff, and the uses of mathematcal programmng n spreadsheet. The dagrammatc expresson of crtcal path method and mathematcal method wll be combned n ths paper, by whch a more clear and effcent exposton of solvng the tme-cost tradeoffs problem wll be exhbted. As a more effcent tool, the paper dscusses such new educaton pedagogy. Introducton Prompted by the present emphass on tme-based competton n ndustry, there are more and more ssues focus on the problem of tme-cost tradeoffs. A potental project management nvolvng tme used of a project can always be tradeoff by addtonal resources nput. Such a tradeoff may come from dfferent optons of the actvty on the crtcal path of the project. The stuaton of Pay more - Save Tme s common for project management related decson problems. The avalable technology of shortenng the duraton of each actvty s often the sources of the tme-cost tradeoffs problem. And the problem solvng processes always rely upon the technques of crtcal path method (CPM) calculaton and mathematc programmng, for example lnear programmng, or nteger programmng etc. Methods of crtcal path method that are frequently used nclude Early-Start, Early-Fnsh, Late-Start, Late-Fnsh calculaton of each actvty. Further, one can use forward method and backward method to fnd the zero float tme actvty, and defne the crtcal path of the project. The project total duraton and ts respectve total project cost needed thus can be obtaned. After advanced evaluaton, f an actvty on the crtcal path can be shortened by more resources nput, one can obtan the other project tme and ts respectve cost. We can use the same way to calculate each possble combnaton of project tme/cost one by one, and fnally obtan a project s tme-cost tradeoffs curve. But now ncludng the mathematcal programmng wll be more effcent for the problem. Usng mathematcal method to solve the tme-cost trade-off problem has been studed extensvely n the project management lterature. Mathematcal approaches convert CPM Page

3 network and tme-cost relatonshps of the project nto constrants and objectve functons. Lnear programmng and nteger programmng are the two major mathematcal approaches used to solve the tme-cost trade-off problems n project schedulnng. By assumng lnear relatonshp between tme and cost for project actvtes, the lnear programmng had been developed three decades ago [3, 4], and well-developed later. The general phlosophy of lnear programmng convert the project tme-cost trade-off problems to mnmzng the objectve cost functon, subject to nequalty tme constrants, and then solve the problem. Computerzed CPM procedure and the applcaton of project management system had been developed by many researchers, for example, [1], [2], [8], and [7]. Computerzed CPM procedure usng spreadsheets to solve the tme-cost tradeoffs problem also already was ntegrated as parts of the standard OR textbook, for example, [5] and [6]. The advantages of lnear programmng algorthms used to obtan the optmal solutons nclude effcency and accuracy. To smplfyng the mathematcal formulaton and ts applcaton n tme-cost tradeoff problem, t wll be helpful f ncludng the vsualzaton spreadsheet expresson. Bascally, crtcal path of a project schedule can be easly calculated n spreadsheet form, the dagrammatc CPM network thus can be extended. But usng mnmum cost prncple to solve tme-cost solutons of all possble combnatons s tme consumng. On the other hand, usng mathematcal programmng to solve the tme-cost trade-off problems, or expressed the problem framework n spreadsheet s relatve easer. How the dagrammatc expresson of crtcal path method and mathematcal method of solvng tme-cost tradeoff wll be a good way for us to combne. Such a new educaton pedagogy, an effcent tool combnes dfferent methods wth nteracton, thus worth us to address n detal. The paper begns wth a typcal ntroducton of the CPM method. The tme-cost tradeoff problem s explored to help facltatng the decson-makng process n tme-based competton framework. Then the paper ntegrates the CPM method and mathematcal programmng n spreadsheet. Fnally, how the solutons of tme-cost tradeoffs can nteract wth the respectve CPM dagrams were presented. The Crtcal Path Method Let ES (Early-Start) represents as the earlest an actvty can start; EF (Early-Fnsh) represents as the earlest an actvty can fnsh; LS (Late-Start) represents as the latest an actvty can start wthout delayng project completon, LF(Late-Fnsh) represents as the latest an actvty can fnsh wthout delayng project completon. One can use uses these nformaton and CPM network calculatons to determne when each actvty must take place n order to fnsh the project n the least amount of tme [9, 10, 11]. Methods of crtcal path method that are frequently used nclude Early-Start, Early-Fnsh, Late-Start, Late-Fnsh calculaton of each actvty. These nformaton and technque allow us to dentfy crtcal actvtes whch must start and fnsh on exact dates and non-crtcal actvtes whose start and fnsh tmes can vary. Because a crtcal path s the longest paths from project start to fnsh, and the total float s the maxmum tme an actvty can be delayed wthout delayng completon of the project. And total float s the maxmum amount of tme n whch an Page

4 actvty can be delayed wthout nterferng wth future events. (TF) equal to zero and fnd the crtcal path. So we need to fnd the total float When we do the CPM network calculaton, one can use forward pass technque and backward technque to fnd the zero float tme actvty, and defne the crtcal path of the project. Forward pass technque s a process of fndng earlest start (ES) tmes and earlest fnsh (EF) tmes for all actvtes; by whch, the forward pass wll gve us an early-start schedule - the earlest the project can fnsh wth the gven logc and actvty duratons. And backward pass technque s a process of fndng latest start (LS) tmes and latest fnsh (LF) tmes for all actvtes. Let represents as begnnng node of actvty, and j represents as the endng node of actvty. One can calculate the total float of an actvty (LS -ES ), we can determne the crtcal path(s). As an llustratve example, Fgure 1 showed the network of an example faclty project wth ten actvtes. Table 1 showed the normal tme vs. crash tme scenaros of all actvtes of the project network, and ther tme and costs to complete the actvtes. Fgure 1: Illustratve example of a buldng constructon project network Followng the crtcal path method descrbes above, one can apply Excel to calculate the total float of each actvty, thus draw the crtcal paths of the normal and crash scenaros. Fnd ES, EF, LS, LF, FF, and TF for the arrow dagram n Fgure 2 and Fgure 3. Fgure 2 showed the crtcal paths dagram of the normal tme; the normal project duraton s 130 weeks. And Fgure 4 showed the crtcal paths dagram of the crash tme; the mnmum project duraton s 90 weeks. The double arrows n Fgure 2 and Fgure 3 ndcate the crtcal paths of the network. Followng the crtcal path method descrbes above, the project total duraton and ts respectve total project cost needed can be obtaned. After advanced evaluaton, f an actvty on the crtcal path can be shortened by more resources nput, one can thus obtan the other project tme and ts respectve cost. Bascally, we can use the same way to calculate each possble combnaton of project tme/cost one by one, and fnally obtan a project s tme-cost tradeoffs curve. But now ncludng the mathematcal programmng wll be more effcent for the problem. To smplfyng the mathematcal formulaton and ts applcaton n tme-cost tradeoff problem, t wll be helpful f ncludng the vsualzaton spreadsheet expresson. Page

5 Table 1: Actvty optons of the project scenaros: normal tme vs. crash tme Actvty # Actvty Descrpton Optons Duraton Cost A Ste Preparaton and Foundaton CREW1+EQUIP1+METHOD ,600 CREW2+EQUIP2+METHOD ,000 B Column and Beam Formwork METHOD ,800 METHOD ,000 C Renforcement Erecton EQUIPMENT ,000 EQUIPMENT ,000 D Exteror Enclosure and Roofng: METHOD1+RAILROAD 45 80,000 shop work and delvery METHOD2+TRUCK 50 60,000 E F Concrete work and Curng METHOD ,000 METHOD ,000 Exteror Enclosure and Roofng CRANE1+CREW ,000 Installaton CRANE2+CREW ,000 Fgure 2: The CPM dagram of the project normal tme Fgure 3: The CPM dagram of the mnmum project tme Page

6 Mathematcal Approach of Tme-cost Tradeoff problem Let s denote the normal and crash tme-cost ponts as the coordnates (D, C D ) and (d, C d ) respectvely. Supposng the optons of the actvty can be effectve combnaton, so that all ntermedate tme-cost trade-offs also are possble and that le on the lne segment between these two ponts. For the present, t wll be assumed that the resources are nfntely dvsble, so that all tme between d and D are contnuous feasble, and the tme-cost relatonshp of the actvty s gven by the lnear lne. The CPM method of tme-cost trade-off approach s to determne just whch tme-cost combnaton should be used for each actvty to meet the scheduled project competton at a mnmum cost. Based on all normal actvty tme-cost opton, the mnmzng total costs prncple of crash tme acton can be expressed as: Mnmzng total project costs = C S d C ; (1) D where d = the reducton tme of actvty ; S represents as the slope of actvty. The aggregaton of all normal actvty costs of the total project, C D s constant, so the basc nformaton we need to address n ths queston s how the mnmum total reducton cost. Whenever we crash each possble actvty, we choose d to mnmze the total addtonal crash cost, where the total tme of the crtcal path s T. To take the project completon tme nto account, we add an auxlary varable y whch expresses the earlest start tme of actvty. For any actvtes wth predecessor () /successor (j) relatonshp, we denote j. So all t presents as nequalty constrant, y j y D d, for all actvty tme-cost trade-off relatonshp. The nequalty constrant showed that an actvty cannot start untl each of ts mmedate predecessors s fnshed. The objectve functon and constrants of all actvtes for lnear programmng to approach the tme-cost trade-off problem then can be wrtten as follows: Mn. C S.t. d ; for all actvty. (2b) * d (2a) y j y D d ; for all actvty, each precedence j. (2c) y FINISH T ; (2d) and 0 ; for all actvty. d * where D = duraton of actvty ; * d = the maxmum reducton tme of actvty. Page

7 Illustratve Example of the Dagrammatc and Mathematcal Approach Followng the llustratve example showed n Fgure 1 and Table 1 n secton of Crtcal Path Method. Usng crtcal path method (CPM), one can attans the maxmum normal project duraton s 130 weeks, and the mnmum project crash duraton s 90 weeks. But f we hope to attan a tme-cost tradeoff curve, we need to calculate all of the possble combnatons for the normal and crash scenaros. Applyng the lnear programmng method descrbes n above secton, we can use Excel to fnd the soluton of a gven project tme. Table 2 showed the basc nput data of actvtes for tme-cost tradeoffs model. Table 3 showed all the solutons of tme-cost and ts respect actvty tme reducton from T= days, where T=1 day. The smulaton results of all tme-cost combnaton were plotted n Fgure 4. Table 2: Basc nput data of actvtes for tme-cost tradeoffs model Tme Cost Actvty # Normal Crash Normal Crash Maxmum Tme Reducton Crash Cost per day Added A ,000 24, ,200 B ,000 39, C ,000 60, ,500 D ,000 80, ,000 E ,000 36, ,600 F ,000 45, Y F E D C B A Fgure 4: Tme-cost tradeoff curve of the llustratve example Page

8 Table 3: Smulaton results of all tme-cost tradeoffs for the llustratve example Project Fnsh Tme Reducton of Actvty # Tme A B C D E F Total Cost , , , , , , , , , , , ,400 If we hope to reveal the story behnd each lnear segment of the tme-cost tradeoff curve n Fgure 4, we need to combne the nformaton of Table 2, Table 3, and CPM dagram of the specfc project duraton now. For example, among actvtes on crtcal path of the normal tme network showed n Fgure 2, actvty-f has the least crash cost per day added ( S =-500, see Table 2). So Table 3 and Fgure 5 showed that n order to crash the project tme from 130 days to 120 days, the project manager need to spend more resources n actvty-f. Ths s the case of lnear segment AB n Fgure 4. Whenever the maxmum tme reducton of actvty-f s exhausted, the strategy of crashng the project tme shfts to the second lower unt crash cost, actvty-b ( S =-700, see Table 2, Table 3, and Fgure 6). Fgure 5: The CPM dagram of the project tme 120 days Page

9 Fgure 6: the CPM dagram of the project tme 119 days But whenever one hopes to crash the project tme from 116 days to 115 days, actvty-b wll be not a good choce. The CPM dagram Fgure 7 showed that f we crash actvty one day more, actvty-b should not an actvty on the crtcal path and more. In ths case, even actvty-b has least unt crash cost, t doesn t work for crashng the project tme. The soluton of crashng project tme as T = 115 now shft to the second lower unt crash cost of the crtcal path actvty-a (Fgure 8, Table 4). Table 3 showed that the solutons of crashng project tme from 116 days to 108 days are crashng the actvty-a, whch n the lnear segment CD n Fgure 4. After the maxmum tme reducton of actvty-a n exhausted, the crashng shfts to actvty-e (the case of DE n Fgure 4). One can uses the same concept descrbed above and extends t to explore the rest segments of Fgure 4, whch need combne the nformaton of LP solutons of tme-cost tradeoffs and CPM dagram of specfc project duraton. Table 4: Comparson dfferent cases of specfc actvty tme reducton: T= T=116 T=115 T=116 Actvty Tme Cost Tme Cost Tme Cost A 28 15, , B 40 32, , C 40 40, , D 50 60, , E 34 20, , F 14 45, , Total cost 212, , Page

10 Fgure 7: T= ; the case of crashng actvty-b Fgure 8: T= ; the case of crashng actvty-a Concludng Remarks Prompted by the present emphass on tme-based competton n ndustry, there are more and more ssues focus on the problem of tme-cost tradeoffs. A potental project management nvolvng tme used of a project can always be tradeoff by addtonal resources nput. Such a tradeoff may come from dfferent optons of the actvty on the crtcal path of the project. The avalable technology of shortenng the duraton of each actvty s often the sources of the tme-cost tradeoffs problem. And the problem solvng processes always rely upon the technques of CPM calculaton. After advanced evaluaton, one can use the CPM to calculate each possble combnaton of project tme/cost one by one, and fnally obtan a project s tme-cost tradeoffs curve. But t wll be more effcent for the problem solvng f one ncludes mathematcal programmng now. Readng the sgnfcant meanng of lnear segments of tme-cost tradeoff curve, t wll be helpful f ncludng the dagrammatc CPM and the solutons of LP together. Ths paper presents a dagrammatc approach n spreadsheet form, whch can provde an easy-to-use tool and calculate the crtcal path n a more easy way. How a tme-cost Page

11 trade-off problem can be represented as a spreadsheet form, then use mathematcal programmng to obtan the tme-cost tradeoff curve. The dagrammatc expresson of crtcal path method and mathematcal method wll be combned wth nteracton way, by whch a more clear and effcent exposton of solvng the tme-cost tradeoffs problem. Bblography 1. Burns, S-A, Lu, L., and Feng, C-W., 1996, LP/IP hybrd Method for Constructon Tme-Cost Trade-off Analyss, Constructon Management and Economcs, 14: Elmaghraby, S.E., Pulat, P.S., 1979, Optmal Project Compresson. wth Due Dated Events, Nay. Research Logstcs Q., 26 (2), Fulkerson, D. R., A Network Flow Computaton for Project Cost Curves, Management Scence, Vol. 7, No. 2. (Jan., 1961), pp Kelley Jr., James E., 1961 Crtcal-Path Plannng and Schedulng: Mathematcal Bass, Operatons Research, Vol. 9, No. 3. (May - Jun., 1961), pp Hller, F. S. and Leberman, G. J., 2007, Introducton to Operaton Research, (8th ed.), McGraw-Hll Company. 6. Hller, F. S. and Hller, M., 2008, Introducton to Management Scence: a modelng and case studes approach wth spreadsheets, (2 th ed.), McGraw-Hll Company. 7. Lu, L., Burns, S-A and Feng, C-W., 1995, Constructon Tme-Cost Trade-off Analyss Usng LP/IP hybrd Method. J. Constr. Engrg. and Mgmt., 121(4) Moder, J. J., Phlps, C. R., and Davs, Edward W., 1983, Project Management wth CPM, PERT and Precedence Dagrammng, 3d ed., Van Nostrand Renhold, New York. 9. Oberlender, G. D., 2002, Project Management for Engneerng and Constructon, 2e, The Mc.Graw-Hll Conyanes, Inc., 10. Pnedo, C., 2002, Schedulng Theory, Algorthms, and Systems, Prentce-Hall, Inc. 11. Stevens, J. D., 1990, Technques for Constructon Network Schedulng, Stanford Unversty. Page