EVOLUTIONARY OPTIMIZATION OF RESOURCE ALLOCATION IN REPETITIVE CONSTRUCTION SCHEDULES

Transcription

1 EVOLUTIONARY OPTIMIZATION OF RESOURCE ALLOCATION IN REPETITIVE CONSTRUCTION SCHEDULES SUBMITTED: October 2003 REVISED: September 2004 ACCEPTED: September 2005 at EDITOR: C. Anumba Khaled Nassar, Assoc Professor Unversty of Maryland Eastern Shore, Prncess Ann, USA emal: SUMMARY: Repettve constructon projects are common n a number of dfferent stuatons such as hghway constructon and housng projects. Several approaches and models have been presented for modelng repettve constructon projects as well as for optmzng resource allocaton for these types of projects. Ths paper presents a model that uses genetc algorthms to optmally assgn resources to repettve actvtes n order to mnmze the overall project duraton as well as the number of nterrupton days. The model presented n ths paper s a straghtforward formulaton that can be easly modfed and expanded for any problem through ts spreadsheet mplementaton, whch reduces the modelng tme for typcal problems extensvely. The model outperforms other models n terms of fndng a soluton wth less nterrupton days. In addton, users of the proposed model can nteractvely examne tradeoffs between the number of nterrupton days and the overall project duraton. KEYWORDS: optmzaton, repettve constructon schedules, resource allocaton problem 1. INTRODUCTION Repettve actvtes are common n a number of constructon projects. Repettve constructon s commonly found for example, n hgh-rse buldngs, housng projects, hghways, ppelne networks, brdges, etc. Several approaches and models have been presented for modelng repettve constructon projects, startng wth the tradtonal network technques such as Crtcal Path Method (CPM). A number of researchers however, recognzed early on the problems of usng conventonal network technques for schedule such projects. The man draw back of conventonal network technques s ther nablty to mantan crew work contnuty (Brrell 1980; Selnger 1980; Kavanagh 1985; Reda 1990; Russell and Wong 1993). In order to rectfy ths problem, a number of schedulng technques have been proposed for repettve constructon projects (Selnger 1980; Johnston 1981; Ardt and Albulak 1986; Charzanowsk and Johnston 1986; Russell and Caselton 1988; Al Sarraj 1990; El-Rayes and Moselh 1998). Dynamc programmng, for example, has also been extensvely used to optmze the schedulng of repettve projects, (Moselh and El- Rayes 1993; Eldn and Senouc 1994, Selnger 1980; Russell and Caselton 1988). These formulatons consder the overall project duraton (Selnger 1980; Moselh and El-Rayes 1993) or both the overall project duraton and the number of nterrupton days (Russell and Caselton 1988; Eldn and Senouc 1994; and El-Rayes and Moselh 2001). Mantanng crew work contnuty leads to maxmzng the learnng curve effect and mnmzng the dle tme of each crew (Ashley 1980; Brrell 1980). On the other hand, mantanng crew work contnuty mght result n longer project duraton. Selnger (1980) for nstance, suggested that ncreasng crew work nterruptons up to a specfc amount for some of the actvtes mght reduce the overall project duraton. Therefore, the tradeoff between the overall project duraton and the number of nterrupton days s a crucal factor n assgnng resources n repettve projects. In ths paper, an alternatve model that uses Genetc Algorthm (GA) to optmally assgn resources for repettve constructon projects. The model provdes an optmum duraton for repettve constructon projects whle applyng the mnmum crew work nterrupton days to the repettve constructon actvtes. The model proposed here s straghtforward n that t can be mplemented easly on any spreadsheet software and modfed and expanded for any problem. Furthermore, usng the proposed genetc algorthms optmzaton, the model acheves solutons that can be better than those acheved usng other models. ITcon Vol. 10 (2005), Nassar, pg. 265

2 2. RESOURCE ASSIGNMENT IN REPETITIVE CONSTRUCTION SCHEDULES In order to explan the problem of resource assgnment n repettve constructon schedules, consder the concrete brdge example whch was frst presented n (Selnger 1980). Russell and Caselton (1988), and Moselh and EL-Rayes (1993 & 2001) also analyzed ths same example. In the secton 4 of ths paper, the results from the model presented n ths paper are compared to the solutons from these studes. The project conssts of four smlar sectons or unts, and each ncludes the followng repettve actvtes n sequence: Excavatons, Foundatons, Columns, Beams, and Slabs. Each repettve actvty s performed by a crew that progresses from the frst to the fourth secton sequentally. The job logc (.e., relatonshps) among succeedng actvtes, for the project selected here, s fnsh to start wth no lag tme. The same technque however can also work wth other knds of relatonshps (start-to-fnsh, start-to-start, etc ). TABLE 1: Sample Problem (from Selnger 1980) Actvty Unt j Excavaton Foundaton Columns Beams Slabs Opton opton opton opton opton opton opton opton opton opton opton opton opton # 1 # 1 # 2 # 3 # 1 # 2 # 3 # 1 # 2 # 3 # 4 # 1 # Dfferent crew formatons optons are avalable for each actvty as lsted n Table 1 (excavaton for example has only 1 crew opton, whle the beams actvty has 4 crew optons). Each crew formaton has ts own producton rate. In addton to determnng the optmum crew assgnment that mnmzes the overall project duraton, one also needs to determne the best nterrupton vector assocated wth ths assgnment. Prevous researchers have defned the nterrupton vector s the number of days that the assgned crews wll be nterrupted between the dfferent work unts (Caselton 1988, Moselh and EL-Rayes 1993, 2001). Allowng crew work nterruptons for some of the actvtes mght reduce the overall project duraton. The resource assgnment problem n repettve constructon schedules therefore s to determne the optmum crew formaton and nterrupton vector that wll reduce the overall project duraton. In prevous formulatons, some of them utlze an arbtrary set of nterrupton vectors prepared pror to schedulng (Russell and Caselton 1988; Eldn and Senouc 1994). El-rayes and Moselh 2001 appled an nterrupton to ther model that automatcally generates a set of calculated nterrupton vectors durng schedulng to make the nterrupton more feasble and bounded. In the model presented n ths paper however, ths s not requred, as wll be presented n the next secton. 3. THE DEVELOPED MODEL 3.1 Model descrpton The developed model s a two-state varable model because t consders mnmzng the total overall duraton of the project as well as mnmzng the number of nterrupton days (n the nterrupton vector). The model s presented n a way that s smlar to tradtonal CPM calculatons and therefore can be easly developed for any project. In addton, the model s developed n a table format whch facltates ts mplementaton n any spreadsheet software as well as utlzng the already developed optmzaton routnes found n such software, such as the genetc algorthm solver n excel, whch s utlzed n ths case. Table 2 presents the calculatons for the sample problem descrbed above. The calculatons are dvded nto three man parts: Frstly, part one calculates the earlest start date for each of the actvtes. Secondly part 2 calculates the latest start date. Fnally, the thrd part s used to determne when to actually schedule the start of each actvty to mnmze the overall duraton and also mnmze the nterruptons. The Interrupton Vector ITcon Vol. 10 (2005), Nassar, pg. 266

3 column (column 13) shows the number of days that each actvty s delayed, and the fnsh day of the last actvty s the overall total duraton of the project (106.8 days). TABLE 2.: Table used n the Proposed Model The nputs to the problem presented here are shown n TABLE 1, whch shows the dfferent crew duratons for the dfferent actvtes. The crew duraton s the amount of tme that each crew n would take to fnsh each unt j of actvty. 3.2 Model Explanaton Part One of the Model Now, consder part 1 of the model n TABLE 2. Column 1 contans the earlest possble start date accordng to crew avalablty, SCA. For the frst actvty n the frst unt (no succeedng actvty) ths s equal to zero. For the next unt, SCA s equal to the early fnsh date of the precedng unt of the same actvty. It s determned usng the followng formula: SCA = Early Fnsh Date n ( j 1) (column7) (1) The early fnsh date for a unt j of an actvty s based on the crew n selected. The crew used for each actvty s shown n column (column 5). For example, the earlest possble tme that the thrd unt of the foundaton actvty can start (40.1) s determned from the earlest fnsh date of the second unt. Note that snce crew number 1 s used (as shown n the no column, number 5) for the foundaton actvty, the duraton, D, for any of the unts of the actvty consdered s actually the crew duraton (shown n Table 1). If a dfferent crew were selected then the duraton (column 4) for the unts would reflect the new crew selected n column 5. ITcon Vol. 10 (2005), Nassar, pg. 267

4 Column number 2, on the other hand, contans the earlest possble start date accordng to job logc, SJL. Ths represents the earlest date the actvty could start f there were no constrants due to the crew duraton. Ths s calculated based on the scheduled fnsh date or the same unt of the precedng actvty, and s gven by: 1 1 SJL = Scheduled Fnsh date (Column12) (2) Snce an actvty can not start untl ts predecessor has fnshed or untl the crew workng on the actvty s avalable, the earlest start date (column 6) for each actvty would have to be the latest of the (SJL or SCA). Ths s calculated accordng to the followng formula: Early Start Date = MAX( SJL, SCA ) For any gven actvty, f the earlest start date accordng to job logc, SJL, were later than the earlest start due to crew avalablty, SCA, then ths would mean that the actvty could start mmedately snce the crew would also be avalable to work mmedately. In ths case, there s no dle tme for the actvty. On the other hand, f the start earlest start date accordng to job logc, SJL, were earler than the earlest start due to crew avalablty, SCA, then the actvty would have to st dle untl the crew has fnshed workng on the precedng unt. The tme the actvty sts dle n that case s the dfference between the SJL and the SCA. Idle tme s shown n column 3 and s calculated accordng to for the followng formula: Idle 0,f ( SJL SCA ) 0 = SJL SCA,f( SJL SCA ) > 0 Fnally, the early fnsh date (column 7) for any unt n an actvty s smply the sum of the early start date and the duraton. Ths s gven by the followng formula: Early Fnsh Date (column7) = Early Start Date (column6) + D (column 4) Ths concludes the frst part of the model, whch determnes the early starts and early fnsh dates, as well as the amount of dle tme for any unts of an actvty due to lack of resources (because the crews would be stll workng on the prevous unt of the actvty) Part Two of the Model In the second part of the model the late start and late fnsh dates are calculated. The dea here s that snce some of the unts n an actvty would have to st dle anyways (due to the crew avalablty), one could shft the start of precedng unts accordngly. The amount that each unt could be delayed wthout delayng the overall project fnsh date s called the shft. The shft (column 8) s equal to the sum of all the dle tmes of the precedng unts n an actvty and s gven by the followng formula: shft = Idle j j+ 1 Snce the shft determnes the amount that each unt could be delayed wthout delayng the overall project fnsh date, the late start date (column 9) would smply equal the early start plus the shft, and the late fnsh s equal to the late start plus the duraton: Late Start Date (column 9) = Early Start Date (column 6) + shft (column 8) (3) (4) (5) (6) (7) Late Fnsh Date Part Three of the Model (column 10) = Late Start Date (column 9) + D (column Now that the late and early start dates for each unt have been calculated, the thrd part of the model s used to determne the overall fnsh date of the project gven specfc nterruptons (or shfts) for each unt of the actvtes. The work on specfc unts can be nterrupted for a number of days wthout ncreasng the overall project duraton. The number of days that each actvty can be nterrupted s shown n the Interrupton column 4) (8) ITcon Vol. 10 (2005), Nassar, pg. 268

5 (column 13). In fact by varyng the number of nterrupton days (.e. the nterrupton vector) and the crews, a reducton n the overall project duraton can be acheved. The scheduled start date of any unt n the project would equal the late start date mnus those nterrupton days. Although the scheduled start date can also be determned as the earlest start plus the nterrupton days, the way the spreadsheet s setup makes t more readable to use the late start mnus the nterrupton days. The scheduled start and fnsh dates are therefore defned as: Scheduled Start Date Scheduled Fnsh Date (column 11) = Late Start Date (column 12) = Scheduled Start Date (column 9) j j+ 1 Inter (column 11) + D (column 13) (9) (column 4) (10) TABLE 3: Comparson between the results of the proposed model and other smlar models for the same problem Actvty Unt J Excavaton Foundaton Columns Beams Slabs Inter S Inter S Inter S Inter S Inter S One-state varable formulaton (Selnger 1980) Fnsh Optmum Formaton Two-state varable formulaton (Russel and Cselton 1988) Fnsh Optmum Formaton Two-state varable formulaton (El-Rayes and Moselh 2001) Fnsh Optmum Formaton Proposed Two-state varable formulaton Fnsh Optmum Formaton ITcon Vol. 10 (2005), Nassar, pg. 269

6 The values used n the Interrupton Vector column (column 13), as well as those n the column (column 5), whch specfy whch of the dfferent crews wll be used for each actvty, wll determne the overall project duraton. These values were determned through an optmzaton routne as descrbed n the next secton. The spreadsheet cells n the Interrupton Vector column (column 13) were used as the varables n the optmzaton problem, whch was used to mnmze the overall project duraton Computerzed Solutons usng Evolutonary Add-n One of the advantages of the formulaton descrbed above s that t can be easly developed for any sze problem on any commercal spreadsheet applcaton. Spreadsheets provde a transparent way of modelng the problem, gvng the user a complete understandng and control of the dfferent varables and constrants n the problem. For the majorty of constructon problems, whose sze usually ranges from small to medum, spreadsheet solutons would be the most effcent and economcal method to solve the problem. Most contractors already use spreadsheets extensvely. Furthermore, the model can be easly modfed and expanded for any problem by smple formula draggng whch reduces the modelng tme for typcal problems extensvely. The developed spreadsheet uses standard spreadsheet formulae to model the problem. For example, an HLOOKUP functon s used to look up duraton of each unt based on the crew used. Spreadsheet, such as Excel, provdes a solver add-n, whch can be used for optmzng problems lke the one presented here. However, tradtonal solvers can not be used to optmze ths problem, because the spreadsheet nvolves dscontnuous and non-smooth functons such as the MAX() and LOOKUP() functons. Fortunately, a number of commercal genetc solver add-ns are avalable for excel that can maxmze the speed and robustness for solvng optmzaton problems such as the one presented above, e.g. Evolver and the Premum Solver whch s used here. The Premum Solver Platform works wth exstng Excel Solver models to solve much larger problems up to hundreds of tmes faster. Ths evolutonary solver uses the powerful optmzaton capabltes of generc algorthm (GA) technques. GA technques are nspred by bologcal systems that mprove ftness through evoluton. Ths s done through mmckng Darwnan prncples of natural selecton by creatng an envronment where hundreds of possble solutons to a problem can compete wth one another and only the "fttest" survve. Just as n bologcal evoluton, each soluton can pass along ts good "genes" through "offsprng" solutons so that the entre populaton of solutons wll contnue to evolve better solutons. In terms of our applcaton, good genes would represent the schedule wth the least duraton and wth the least nterrupton days. The objectve of the model presented here s to determne the optmum crew work formaton and nterrupton vector that wll mnmze the overall project duraton, whle also keepng the number of nterrupton days to a mnmum. Therefore the objectve functon s set as the overall project duraton plus the sum of the nterrupton days. The crew work formaton type and work nterrupton for each unt were set as the changng varables. It s a requrement when usng evolutonary solver that ndependent varable must have an upper and lower lmt. Therefore upper lmt and lower lmt constrants were specfed for all varables. For example, the upper lmt on the crew varable was set as 3 for the foundatons, 3 for the columns, 4 for the beams and 2 for the slabs. Another constrant to satsfy s that the actvty n each repettve unt cannot start at an earler date than that establshed n part 1 of the model because of the job logc and crew avalablty constrants. After runnng the optmzaton, the optmum overall project duraton was determned as wth assocated total nterrupton of 14 days. 4. COMPARISON TO EXISTING MODELS The same problem was analyzed by usng dynamc programmng formulatons. The ntal problem formulaton by Slnger (1980) Russell and Caselton (1988) dd not consder nterrupton as a second state varable and resulted n a mnmum duraton of days. Russel and Caselton (1988), whch expanded the formulaton allowng 16 nterrupton days, provded an mprovement by mnmzng the overall duraton days to days. The last formulaton (El-Rayes and Moselh (2001) provded an addtonal mprovement by reducng the overall project duraton to days, and nterruptng the crew work contnuty by a total of 15 days, as shown n Table 4.2. ITcon Vol. 10 (2005), Nassar, pg. 270

7 Optmum Project Duraton Total Project Durato, days Selnger (1980) Russell & El-Rayes & Caselton (1988) Moselh (2001) Proposed EV Model FIG. 1: Comparson of optmum project duraton Total Interrupton Total Project Iterruptons, days Russell & Caselton (1988) 15 El-Rayes & Moselh (2001) 14 Proposed EV Model FIG. 2: Comparson of total nterruptons (days) TABLE 4: Optmum Project Duraton & Interrupton Formulaton Model Total Duraton Interrupton (days) (days) Selnger (1980) Russell & Caselton (1988) El-Rayes & Moselh (2001) Proposed GA Model The results obtaned usng dynamc formulatons (Selnger 1980; Russell and Caselton 1988; El-Rayes and Moselh 2001) are compared to those provded by the spreadsheet model as shown n Table 3, Fg 1, Fg 2 and Table 4, confrm that the GA model provdes a more effcent soluton than those obtaned by others, the mnmum overall duraton s the same as that of (El-Ryes and Moselh 2001), but wth less nterruptng days for crew work contnuty. In addton, one of the advantages of the model descrbed here s the ablty to nteractvely modfy the varable n the model, such as the nterrupton vector, and observe the effects on the overall duraton. For example, by ncreasng the nterrupton days for the fourth unt n the foundaton actvty from 5 to 7, the overall duraton of the project can stll be further reduced to 105.5, whle ncreasng the nterrupton days by those two days. Ths allows users to examne tradeoffs between the number of nterrupton days and the overall project duraton. 5. DISCUSSIONS AND CONCLUSIONS In ths paper a model for resource allocaton n repettve constructon schedules was descrbed. Usng the proposed spreadsheet GA mplementaton, users can mnmze the overall project duraton whle keepng the nterruptons as low as possble. Whle most optmzers keep only the best soluton found durng ts search, the ITcon Vol. 10 (2005), Nassar, pg. 271

8 evolutonary optmzer keeps a large set of results, called a populaton of canddate soluton. The populaton s used to help create new startng soluton ponts not necessarly n the neghborhood of the current best soluton, and thus helpng to avod Evolutonary Solver gettng trapped at a local optmum. Therefore, the model users can experment wth the dfferent varables n the model to perform a senstvty analyss on the model. Whle the model presented here can easly be mplemented for any smlar problem as has been found to outperform smlar models n terms of mnmzng nterrupton days, a number of ponts need to be ponted out. The model s non-determnstc as t reles n part on randomly determned startng pont, ths may yeld dfferent solutons for dfferent runs. The model also has the same restrctons as smlar models n the lterature: Labor requrement and feasble crew sze are determned n advance. Each crew formaton has a unque daly output and assocated duraton for completng the actvty as well as the assumpton that work nterruptons may reduce total constructon tme, and so general expenses. 6. REFERENCES Al Sarraj, Z. M. (1990). Formal development of-lne Balance Technque. J. Constr. Engrg. and Mgmt., ASCE, 116(4), Ardt, D., and Albulak, M. Z. (1986). Lne of-balance Schedulng n Pavement Constructon. J. Constr. Engrg. and Mgmt., ASCE, 112(3), Ashley, D. B. (1980). Smulaton of Repettve-Unt Constructon. J. Constr. Engrg. and Mgmt., ASCE, 106(2), Beans, J. C. (1994). Genetc Algorthms and Random Keys for Sequencng and Optmzaton. ORSA J. on Comp., 6(2), Brrell, G. S. (1980). Constructon Plannng - Beyond the Crtcal Path. J. Constr. Engrg. and Mgmt., ASCE, 106(3), Chrzanowsk, E. N. Jr., and Johnston, D. W (1986). Applcaton of Lnear Schedulng. J. Constr. Engrg. and Mgmt., ASCE, 112(4), Eldn, N. N., and Senouc, A. B. (1994). Schedulng and Control of Lnear Projects. Can. J. Cv. Engrg., Ottawa, 21, El-Rayes, K., and Moselh, O. (1998). Resource-Drven Schedulng of Repettve Actvtes on Constructon Projects. J. Constr. Engrg. and Mgmt., ASCE, 107(2), El-Rayes, K., and Moselh, O. (2001). Optmzng Resource Utlzaton for Repettve Projects. J. Constr. Engrg. and Mgmt., ASCE, 16(4), Frontlne System Inc., developers of Spreadsheet Solver/Optmzer n Mcrosoft Excel, Constructon Goldberg, D. E. (1989). Genetc Algorthm n Search, Optmzaton, and Machne Learnng, Addson- Wesley, Readng, Mass. Johnston, D. W. (1981). Lnear Schedulng Method for Hghway Constructon. J. Constr.Dv., ASCE, 107(2), Kavanagh, D. P. (1985). SIREN: A Repettve Constructon Smulaton Model. J. Constr.Dv., ASCE, 111(3), O Bren, J. J., Kretzberg, F. C., and Mkes, W. F. (1985). Network Schedulng Varatons for Repettve work. J. Constr. Engrg. and Mgmt., ASCE, 111(2), Reda, R. M. (1990). RPM: Repettve Project Modelng. J. Constr. Engrg. and Mgmt., ASCE, 116(2), Russell, A. D., and Caselton, W. F. (1988). Extensons to Lnear Schedulng Optmzaton. J. Constr. Dv., ASCE, 114(1), ITcon Vol. 10 (2005), Nassar, pg. 272

9 Russell, A. D., and Wong, W. C. M. (1993). New Generaton of Plannng Structures. J. Constr. Dv., ASCE, 119(2), Senouc, A., and Eldn, N. (1996). A Tme-Cost Trade-Off Algorthm for Nonseral Lnear Projecrs. Can. J. Cv. Engrg., Ottawa, 23, ITcon Vol. 10 (2005), Nassar, pg. 273