Three-dimensional Gantt Chart Based Resource-constrained Multiple Projects Scheduling and Critical Chain Identification

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1 Three-dmensonal Gantt Chart Based Resource-constraned Multple Proects Schedulng and Crtcal Chan Identfcaton J. Q. Wang,, S. F. Zhang,, J. Chen,, S. Wang,, Y. F. Zhang, Insttute of System Integrated & ngneerng Management, Northwestern Polytechncal Unversty, X an, Chna Key Laboratory of Contemporary Desgn and Integrated Manufacturng Technology, Mnstry of ducaton, Northwestern Polytechncal Unversty, X an, Chna (wangq@nwpu.edu.cn) Abstract - Amng at resources-constraned multple proects schedulng problem (RCMPSP), a threedmensonal representaton s presented through extendng the two-dmensonal representaton of resources-constraned schedulng problem by addng resource dmenson to the two-dmensonal Gantt chart. It can clearly dsplay the resources consumpton of every actvty and provde valuable reference for concernng bottleneck resources and solvng the resources conflct. In addton, a hybrd algorthm whch combned the merts of bdrectonal schedulng and parallel schedule generaton scheme (PSGS) s proposed to optmze the RCMPSP. Furthermore how to use the presented three-dmensonal Gantt chart to represent the schedule and dentfy crtcal chan s also descrbed n detals. The results of sngle proect wth sngle resource schedulng, sngle proect wth multple resources schedulng, multple proects wth sngle resource schedulng and multple proects wth multple resources schedulng show that the proposed approach s effectve and explct for resource constraned proect schedulng and crtcal chan dentfcaton. Keywords - Three-dmensonal Gantt chart, crtcal chan, resource-constraned multple proects schedulng problem (RCMPSP) I. INTRODUCTION Proect schedulng, as specal case of the ob shop schedulng, s a typcal combnatonal optmzaton problem and belongs to the NP-hard problem []. Proect schedulng optmzaton and crtcal chan dentfcaton are two basc and hot ssues n proect management. Amng at resource-constraned proect schedulng methods, Raner Kolsch [] revsted seral and parallel resource-constraned proect schedulng method and stated that samplng method can mprove the performance of sngle-pass schedulng sgnfcantly, and parallel schedule generaton scheme (PSGS) can work well n the condton of sngle-pass, small-scale schedulng and low resource strength, whle seral schedule generaton scheme (SSGS) can do better when facng wth largescale schedulng and hgh resource strength condton. Robert Klen [] clamed that bdrectonal plannng strategy can overcome the maor dsadvantage of PSGS. Vcente Valls [] proposed that ncorporatng ustfcaton technque nto other algorthms can mprove the solvng qualty sgnfcantly wthout consumng more computng tme //$. I Amng at the proect schedulng representaton, network dagram s a popular approach wth the development of PRT technque and crtcal path. In network dagram system arc represents actvty, node represents event, and network structure represents nnerrelatons between actvtes. Apparently network dagram only deals wth tme aspect wthout consderng resource constrants []. However resource constrants play an mportant and key rule n proect schedulng. Later some research on crtcal chan n theory of constrants (TOC) put resource constrants on an mportant poston. But for the multple proects, the exstng approaches whch only consder proect dmenson and tme dmenson cannot explctly and effectvely llustrate the thrd dmenson, resource dmenson. In addton, the current crtcal chan dentfcaton was mostly based on crtcal path method (CPM) of proect management []. Actually CPM does not consder the resource constrants. Therefore the method that frst get crtcal path and then dentfy crtcal chan by excludng resource conflct does not really work when t s dffcult to solve resource conflct under the unclear precedence relatons among proect actvtes. Furthermore when there s more than one crtcal path among all proect actvtes, t may dentfy dfferent crtcal chan usng dfferent crtcal path, whch wll affect the real crtcal chan dentfcaton. The paper focus on the resource-constraned multple proects schedulng problem (RCMPSP) and dentfcaton of crtcal chan, the remanng parts of ths paper are organzed as follows. Secton presents a new presentaton of proect schedulng. Secton gves a solvng approach for RCMPSP and dentfcaton of crtcal chan. Secton shows some case studes and dscussons and Secton draws some conclusons. II. THR-DIMNSIONAL GANTT CHART RPRSNTATION In the feld of ob shop schedulng, Chrstopher V. Jones [] extended the representaton of the well known Gantt chart to three-dmensonal coordnate system, n whch machnes, obs and tme are three orthogonal axes. Introducng Gantt chart and three-dmensonal presentaton nto proect schedulng, a new threedmensonal Gantt chart representaton s presented for solvng proect schedulng and crtcal chan dentfcaton, whch s llustrated n Fg.. Here the coordnate axes are

2 composed of tme axs T, proects axs P and resources axs R. Rn R R R Pn P P Proects P P R T T T T Tme Fg.. Three-dmensonal coordnate system of proect schedulng. Resource-constraned proect schedulng s dvded nto four knds of scheme, namely sngle proect wth sngle resource, sngle proect wth multple resources, multple proects wth sngle resource and multple proects wth multple resources []. Apparently the former two schemes are the specal case of the latter two schemes. Therefore only facng wth multple proects, sngle resource or multple resources, the proect axs s necessarly added n coordnate system. Tn ft() : forward schedulng tme of the th actvty bt() : backward schedulng tme of the th actvty. How to fully utlze resource and effectvely obtan the optmum proect schedulng s the key ssue before dentfyng crtcal chan. Forward schedulng can fnd the earlest begnnng tme of every actvty and the longest path from the actvtes begnnng tme to the end tme, and backward schedulng can fnd the latest begnnng tme of each actvty and analyze float tme of each actvty [9]. What s more, bdrectonal plannng mergng the merts of forward schedulng and backward schedulng can easly get the optmal schedulng plan. At the same tme, parallel schedule scheme has the merts of runnng fast and can obtan better results than the seral one when appled to large proects [] snce t searches n a smaller soluton space than the seral schedulng scheme. Here, for the resource-constraned proect schedulng problem, a schedulng algorthm s presented, whch combned the merts of bdrectonal plannng and parallel schedule generaton scheme n order to accelerate convergence speed and use resources more effectvely. The flow dagram of hybrd schedulng algorthm for RCMPSP s descrbed n Fg.. III. SCHDULING ALGORITHM AND CRITICAL CHAIN IDNTIFICATION A. Hybrd Schedulng Algorthm The notatons used are defned as follows., : actvty ndex,,,... J,,,... J, where J and J s the total number of actvtes s : start tme of the th actvty c : fnsh tme of the th actvty T : plannng horzon FS : forward schedulng set durng n perod, n,..., T n BS : backward schedulng set durng n perod n S : earlest start tme of the th actvty LF : latest fnsh tme of the th actvty FD : set of the forward schedulng set BD : set of the backward schedulng set mn:, postve real number : predetermned th crtcal actvty bef, : the th crtcal actvty bef, : predetermned th crtcal actvty J : predetermned last crtcal actvty U : predetermned crtcal actvty set S : feasble schedulng scheme LB : delvery tme for forward schedulng LB c J T LB T c J Fg.. Flow dagram of hybrd schedulng algorthm for RCMPSP. The solvng steps of hybrd schedulng algorthm for RCMPSP are descrbed as follows.

3 * * T LB: c J FS BS F B FS BS S T ft bt, S, c xample of sngle proect sngle resource s gven, where only one knd of renewable resource s used n a sngle proect, and there are four unts of resource to be used n these actvtes, as llustrated n Fg.. Usng the proposed proect schedulng approach, the optmum schedule s obtaned n Fg.. The crtcal chan s hghlghted n red lne on Fg.. LB c J * * * F( FS( ), ft( )) S ( FS ) T LF ( BS ) FS * BS BS S FS BS T c J T LB * BSend th F( FS( ), ft( )) B( BS( ), bt( )) * * B( BS( ), bt( )) (,) (,) (,) (,) (,) (,) 9 (,) (,) (,) (,) (,) (d,u ) B. Identfcaton of Crtcal Chan Accordng to the obtaned schedulng soluton, all actvtes whch both satsfed precedence and resource constrans are found gradully startng from the farthest actvty. Untl the earlest actvty s found, the whole crtcal chan s ganed. The dentfcaton steps of crtcal chan are gven as followng. Here end s defned as the farthest actvty from orgn of coordnates n the three-dmensonal representaton, or the farthest actvty from the resources axs or proects axs n the two-dmensonal representaton. Intalzaton: CC, U {, bef,, bef,... end, bef }, m, n L length ( U) Whle ml If end s the farthest actvty then CC() J end nd If m satsfed both precedence and resource dependences then CC() n m nd mm nn nd Output crtcal chan CC IV. CAS STUDIS AND DISCUSSIONS Accordng to the four knds of resource-constraned proect schedulng problem, four correspondng examples are gven to verfy the presented approach, where d represents duraton of actvty, u represents the quantty of resources r requred per perod and r represents resource category. () Sngle proect wth sngle resource Resource Fg.. Network dagram of example. 9 Tme Fg.. Schedulng scheme of example. () Sngle proect wth multple resources xample of sngle proect multple resources s gven, there needed three categores of resource R, R and R. The constant avalablty of the resource R s lmted to three unts per tme unt, R four unts and R fve unts. The detaled descrpton s shown n Fg.. Usng the proposed proect schedulng approach, the optmum schedule s obtaned n Fg.. The crtcal chan s hghlghted n red lne on Fg.. (,,R) (,,R) (,,R) 9 (,,) (,,R) (,,R) (,,) (,,R) (,,R) (,,R) (d,u, r ) Fg.. Network dagram of example.

4 9 9 Resource ( unts) Resource ( unts) Resource ( unts) p p p (,,R) (,,R) (,,R) (,,R).... (,,R) (,,R) (,,R). (,/,R/R).. (,,R).. (,,R) (,,R) (,,R) (d, u, r )... Fg.. Network dagram of example. Tme Fg.. Schedulng scheme of example. In the above two stuatons, the present lterature always use the form of resource profle, when multple proects appear, t seems there no effectve way to express the schedulng results. Now, we wll use threedmensonal Gantt chart to represent. () Mult ple proects wth sngle resource xample of multple proects sngle resource s gven, there are P, P and P three proects, proect P s composed of actvty.,.,. and., proect P s actvty.,.,.,. and., and proect P s actvty.,. and., resource R has totally eght unts per tme unt. The detaled descrpton s as Fg.. Usng the proposed proect schedulng approach, the optmum schedule s obtaned n Fg. 9 and Fg. s lateral vew. The crtcal chan s hghlghted n red lne on Fg.. () Multple proects wth multple resources xample of multple proects multple resources s gven, there are R, R and R three resource types, resource R has constant avalablty of three unts per tme unt, resource R has constant avalablty of four unts per tme unt, and resource R has totally two unts per tme unt. The detaled stuaton s as Fg.. Usng the proposed proect schedulng approach, the optmum schedule s obtaned n Fg. and Fg. s lateral vew. The crtcal chan s hghlghted n red lne on Fg.. (,) (,) (,) (,) p... (,). (,) (,). (,) p.. (,).. (,) (,) (,) (d, u ) p... Fg.. Network dagram of example. Resource ( Unts) Resource ( Unts) Resource ( Unts) Proect Proect Proect 9 9 Tme Fg. 9. Schedulng scheme of example. Proect Proect Proect Fg.. Lateral vew of schedulng scheme of example. Proect Proect Proect 9 9 Tme Fg.. Schedulng scheme of example.

5 9 9 Resource ( Unts) Resource ( Unts) Resource ( Unts) Proect Proect Proect Tme Fg.. Lateral vew of schedulng scheme of example. To the multple proects, for convenence of expressng the gotten crtcal chan, the lateral vew of Fg. and Fg. s used. It can be seen that the proposed approach can clearly show the resource conflct and utlzaton for proect manager. Meanwhle t s easy to know the resource consumpton amount, the sequence of actvtes and actvtes work tme, so provdes gudance for gettng proect schedule and crtcal chan. V. CONCLUSION Resource constrants play an mportant and key rule n proect schedulng. How to ntutvely represent the relatonshp among proect, resource and tme, effectvely optmze the resource-constraned multple proects schedulng problem and dentfy the crtcal chan are the core ssues. Ths paper descrbes a new representaton of resource-constraned proect schedulng. The proposed three-dmensonal representaton can dsplay resource consumpton amount of each actvty not only for sngle proect or multple proects but also for sngle resoure or multple resources, whch provdes a helpful gudence for solvng resource representaton and resources conflct. In addton an easy way to dentfy crtcal chan s presented, whch frstly gets optmum schedule by usng the hybrd algorthm desgned n ths paper, and then search the longest actvty through begnng from the last actvty to get a former actvty whch both satsfed technologcal relatons and resource constrans gradually untl the earlest actvty was found. Accordng to the four knds of resource-constraned proect schedulng problem, four correspondng examples are gven to verfy the presented approach. The results of sngle proect wth sngle resource schedulng, sngle proect wth multple resources schedulng, multple proects wth sngle resource schedulng and multple proects wth multple resources schedulng show that the proposed approach s effectve and explct for resource constraned proect schedulng and crtcal chan dentfcaton. It s a hghly effectve tool for representng the resources utlzaton and resource conflct, whch wll help proect manager manage proect schedule and crtcal chan effectvely. ACKNOWLDGMNT Ths work s supported by the Natonal Natural Scence Foundaton of Chna (Grants No. ), Natural Scence Basc Research Plan n Shaanx Provnce of Chna (Grants No. 9JQ9) and graduate startng seed fund of Northwestern Polytechncal Unversty (Grants No. Z). RFRNCS [] J. Blazewcz, J. K. Lenstra, and A. H. G. Rnnooy Kan, Schedulng ect to resource constrants: classfcaton and complexty, Dscrete Appled Mathematcs, vol., no., pp. -, 9. [] R. Kolsch, Seral and parallel resource-constraned proect schedulng methods revsted: Theory and computaton, uropean Journal of Operatonal Research, vol. 9, no., pp. -, 99. [] R. Klen, Bdrectonal plannng: mprovng prorty rulebased heurstcs for schedulng resource-constraned proects, uropean Journal of Operatonal Research, vol., no., pp. 9-,. [] V. Valls, F. Ballestín and S. Quntanlla, Justfcaton and RCPSP: A technque that pays, uropean Journal of Operatonal Research, vol., no., pp. -,. [] R. P. Mohanty, Proect Schedulng Methods: An valuaton, Management Research News, vol., no., pp. 9-, 9. [] N. F. Cu, and J. Lu, A Heurstc Algorthm of Crtcal Chan Based on Proect, Industral ngneerng and Management, vol., no., pp. -, 9. [] Chrstopher V. Jones, The Three-Dmensonal Gantt Chart, Operatons Research, vol., no., pp. 9-9, 9. [] O. Icmel, S. Selcuk renguc, and Chrstopher J. Zappe, Proect Schedulng Problems: A Survey, Internatonal Journal of Operatons & Producton Management, vol., no., pp. -9, 99. [9] Y. L. Chen, D. Rnks, and K. Tang. Crtcal path n an actvty network wth tme constrants, uropean Journal of Operatonal Research, vol., no., pp. -, 99. [] S. Hartmann, and R. Kolsch, xpermental evaluaton of state-of-the-art heurstcs for the resource-constraned proect schedulng problem, uropean Journal Operatonal Research, vol., no., pp. 9-,.