Project Networks With Mixed-Time Constraints

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1 Project Networs Wth Mxed-Tme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa Abstract : A Project can be vewed as a well-defned collecton of tass jobs or actvtes that must be performed n some order and when all tass are completed the desred project objectve s acheved. For example n the constructon of a buldng there are many tass such as: ste preparaton; pourng of footng and slab; the constructon of walls; and roofng; whch must be completed n some order. A fundamental problem n project management s that of determnng the mnmum project duraton tme and the schedule whch acheves ths. The Crtcal Path Method (CPM) and Project Evaluaton and Revew Technque (PERT) are fundamental tools for determnng ths schedule. In ths paper we concentrate on tme constrants that specfy when each actvty can be n progress. More precsely the Tme Constrant Crtcal Path Problem can be stated as: Gven a project networ G n whch each actvty has a specfed duraton tme and a tme constrant specfyng when the actvty can be carred out fnd the mnmum total project duraton tme and the schedule whch acheves ths. The tme constrants can be specfed as: tme-wndows; tme-schedule; or normal tme. Chen et al. (1997) consdered these and presented a two-phase algorthm for determnng the crtcal path. We develop a new mxed nteger lnear program (MILP) formulaton for ths problem. We also consder the tme-cost trade-off problem for networs wth mxed-tme constrants. The method for tradtonal networs extends to ths problem. In addton we develop a MILP model for determnng the mnmum cost schedule. Keywords : Project networ; schedulng; crtcal paths. 1. INTRODUCTION Project Management s concerned wth projects such as the constructon of a buldng the plannng and launchng of a new product the nstallaton of a new manufacturer s faclty the mplementaton of perodc mantenance n a plant faclty etc. The basc characterstc of a project conssts of a welldefned collecton of tass (or actvtes) that must be performed n some technologcal sequence. Wthn the specfed sequence the tass may be started and stopped ndependently of each other. When all the tass are completed the project s completed. A project can be represented as a networ n whch the arcs represent actvtes and the nodes represent events (ponts where a group of actvtes are accomplshed and where a new set of actvtes can be ntated). Ths yelds the actvty- on-arc model. An alternatve networ model s the actvty-onnode model n whch the arcs represent the predecessor restrctons and the nodes represent the actvty. In ths paper we consder only the actvtyon-arc model. In the early days the schedulng of a project was done only wth lmted plannng. The tool used for solvng ths problem was the Gantt chart whch specfes the start and fnsh tme for each actvty on a horzontal scale. In 1950 s the crtcal path method (CPM) and the project evaluaton and revew technque (PERT) were developed. CPM was frst developed by E.I. du Pont de Neumours & Company as an applcaton to constructon projects and was later extended to a more advanced status by Mauchy Assocates; PERT was developed for the U.S. Navy for schedulng the research and development actvtes for the Polars mssle program. PERT and CPM are two-phase labellng methods that are computatonally very effcent. In project management there s usually a due date for the project completon. Therefore n some stuatons a project could be completed n a shorter tme than the normal program. The method of reducng the project duraton by shortenng the actvty tme at a cost s called crashng. Another cost assocated wth projects s the ndrect cost. When both the cost components are consdered we have the mportant tme-cost trade-off problem. Ths can be modelled as a mathematcal program. By assumng that the drect cost of an actvty vares lnearly wth tme the problem can be expressed as a lnear program. The soluton of ths lnear program smultaneously determnes the

2 optmal duraton of a project and the approprate tme of each actvty n the networ so that project cost s mnmzed. The early wor on the tme constrant project management problem was carred out by Fulerson (1961) and Kelley (1961). The tme-cost tradeoff problem has more recently been studed by many authors ncludng : Phllps and Dessouy (1977) De et al. (1995) Sunde and Lchtenberg (1995) Demeulemeester et al. (1996) Baer (1997) and Demeulemeester et al. (1988). Soluton methods for ths problem nclude : a mnmal cut approach; Dynamc Programmng; a heurstc cost-tme tradeoff Lnear Programmng (LP); and the Branch and Bound procedure. EF : earlest fnsh tme on actvty (j). Phase I : Forward-pass Procedure (Calculates the early start and fnsh tmes) Step1 Set E = 0. Step2 For each arc (j) drected nto node j: () If (j) s a normal-tme arc then ES = E and EF = ES + t. () If (j) s a tme-schedule arc wth tmes ts 1 ts 2... ts... ts v then Wth regard to the Crtcal path problem recently Chen and Tang (1997) presented a mxed-tme constrant model. These tme constrants consst of two types namely tme-wndow (nterval) constrant and tme-schedule (lst of start tmes). They presented an effcent lnear tme algorthm for determnng the crtcal path that s smlar to the tradtonal CPM. In ths paper we wll develop mxed nteger lnear programmng models for determnng the crtcal path and the mnmum cost schedule n mxed-tme constraned networs. 2 THE TWO-PHASE METHOD In ths secton we present the essental ngredents of the two-phase method developed by Chen et al. (1979) wrtten n the style of the standard algorthm. We adopt the followng basc notaton. Step3 ES = ts (j) ; where ts -1 < E ts. and EF = ES + t. () If (j) s tme-wndow arc wth tme [L U ] then ES L = E f f f and EF = ES + t. E L E L E > U U For node j f EF has been calculated for all then E = j max{εf}. s : source node. Step4 (Stoppng rule) If = d stop EF = E d. Otherwse go d : destnaton node. to step 2. The longest path s found. A 1 A 2 A 3 : set of actvtes or arcs havng normal tmes. : set of actvtes or arcs havng a prescrbed schedule of start tmes (tme-schedule). : set of actvtes or arcs havng a prescrbed nterval of start tmes (tme-wndow). Phase II : Bacward-Pass Procedure (Calculates the late start and fnsh tmes) Step 1 Set LF j = ES j. Step 2 For each arc(j) drected nto node j : () If ( j ) s normal tme arc wth tmes ts 1 ts 2... ts... ts v then LS j = LF j t j. ts : schedule departure tmes for actvty (j). () If ( j ) s tme-schedule arc then tw : tme wndow departure tme for actvty t (j) ; usually wrtten n [L U ]. : tme duraton of actvty (j). E : earlest occurrence tme event. ES : earlest start tme on actvty (j). LS ts = LF ts + t otherwse. ()If ( j ) s tme- wndow arc [L U ] then.

3 U y = 1 for ( j) A 2. (6) LF U + t LS = LF L LF U y j s bnary j. (7) otherwse. [The restrctons (4) (7) ensure that the tmeschedule Step 3 For node f LS has been calculated for all j then For (j) A 3 L = mn {LS }. j E j - x t for (j) A 3. (8) Step 4 Stoppng rule: If j = s stop LF = L S otherwse go to Step 2. 3 MILP FORMULATIONS In ths secton the crtcal path problem wth mxed-tme constrants wll be formulated as a MILP (Mxed Integer Lnear Program). We start wth a new MILP for determnng the earlest project completon tme and follow ths wth an MILP to determne the latest allowable occurrence tme. L x U for (j) A 3. (9) [The two above restrctons ensure that the tmewndow Determnng the latest allowable occurrence tme We have the followng addtonal notaton T : Project Completon date (due date). LS : Latest start tme of actvty (j). LSS : Latest start tme of actvty (j) A 2 A 3 ; Determnng the earlest project completon tme We begn wth some further notaton. Let: D = { d 1 d 2... d } for (j) A 2. [Start tme of actvty (j) ] LSS = LF t ; and LF ts t. N + = Neghbor set of (wth arcs drected out of ). Formulaton: 1 f actvty ( j) starts at tme d y = Maxmze L s (10) 0 else. subject to L d = T (11) : occurrence tmes on tme-schedule ; = 12 K (last number of tmeschedule). [The above restrctons ensures a completon tme of T] x : earlest start tme of actvty (j). Formulaton: Mnmze E d (1) subject to E S = 0 (2) [The above restrcton ensures a start tme s zero] E j - E t for (j) A 1. (3) [The above restrcton ensures that normal tme E j x t for (j) A 2. (4) x d y = 0 for ( j).(5) A 2 L j L t for (j) A 1. (12) [The above restrctons ensure that the normal tme For each and j N + ; (j) A L LS (13) and LS L + m z m (14) j [m s large postve nteger] z =1 (15) z s bnary. (16)

4 [The restrctons (13)-(16) ensure that the event j gves the smallest latest start(j)] For (j) A 1 LS = L j t. (17) [The above restrctons ensure that the normal tme For (j) A 2 LSS = L j t (18) LS LSS (19) d y = LS 0 (20) y = 1 (21) Fgure 1 : Start and Fnsh Tmes. A 2 = {(13)(35)(46)(47)} and A 3 = {(34)}. Applyng the two-phase method or usng the MILP s descrbed n Secton 3 yelds : Actvty t(j) E ES j EF j LF LS j (1-2) (1-3) (2-4) (2-5) (3-4) (3-5) (4-5) (4-6) (4-7) (5-6) (5-7) (6-8) (7-8) Table 1 : Start and Fnsh Tmes y s bnary for all j. (22) [The restrctons (18)-(22) ensure that tme schedule For (j) A 3 LSS = L j t (23) LS LSS (24) L LS U. (25) [The restrctons (23)-(25) ensure that tme wndow 4 EXAMPLE Consder the 8-node networ dsplayed n Fgure 1 wth A 1 = {(12)(24)(25)(45)(56)(63)(68)(78)} The crtcal path of length 21 s : {(12) (24) (46) (68)}. 5 TIME COST TRADE-OFF The modfed two-phase method of Secton 2 can be used n the tradtonal way to resolve the tmecost trade-off problem for networs wth mxedtme constrants. That s Step 1: Generate a prelmnary schedule usng normal resources (a modfed two-phase method wth mx-tme constrants). Step 2: Fnd the job along the crtcal path wth the least cost slope. Ths s the job that can be crashed wth least expense. If the cost of shortenng the schedule by one perod s less than the fxed ndrect cost for one perod then the job s expedted up to the pont where no further shortenng s possble (ether because the job duraton

5 cannot be reduced further or because some other job has become crtcal along a parallel path). [ The above two restrctons ensure that normal tme. For (j) A 2 (tme schedule arcs) : Step 3: Repeat Step 2 untl no further shortenng of crtcal jobs s uneconomcal. E j = ES +x. (32) (.e. reduce the savngs that would result). We can also use the followng MILP formulaton. l x u. (33) Consder a project networ wth n nodes labelled E j - ES 0. (34) n where node 1 represents the start of the project and node n the end of t. In addton to the earler notaton we let x : duraton of actvty (j). E : realzaton of event. a : cost slope of actvty(j). ES d y = 0. (35) y = 1. (36) y s bnary. (37) [the restrctons (32) (37) ensure that the tmeschedule l : Lower bound on the duraton of actvty For (j) A 3 (tme wndows arcs). (j). E j - ES x. (38) u : Upper bound on the duraton of actvty(j). l x u. (39) 1 2 D = { d d... d } for (j) A2. L ES U. (40) TW = [L U ] for (j) A 3. [the two above restrctons ensure that the tme wndow. y 1 = 0 f actvty( j) start to leave node to node else j The above MILP s easly solved by a pacage such as CPLEX. f : fxed cost (per unt tme). Example : Consder the project networ of Fgure 1 wth normal and crash cost data : We assume a lnear cost-duraton for each actvty. So we can wrte the cost of actvty(j) as Actvty Normal Crash c (j) Tme Tme = b + a x. The MILP formulaton s: Mnmze [ a x +f (En) + b ] (26) subject to For (j) A 1 (normal tme arcs). E 1 = 0. (27) E n E 1 T. (28) E j = ES + x. (29) l x u. (30) E j - ES 0. (31) t c t c Slope (12) (13) (24) (25) (34) (35) (45) (46) (47) (56) (57) (68) (78) Total Table 2 : Normal and Crash Data Suppose the ndrect cost assocated wth the project s $100 per day. Then the applcaton of the above method yelds :

6 Total tme Drecton Indrect Total Crash actvtes (4-6) (4-6) * (4-6)(2-4) (4-5)(6-8) Table 3 The optmal soluton s for a 19-day schedule. 6 CONCLUSION Ths paper addresses project management problems n networs wth mxed-tme constrants. We allow actvtes to be restrcted to tme-wndow and tmeschedule constrants. We present mxed nteger lnear programmng models that can be effcently solved by avalable commercal software such as CPLEX. 7 REFERENCES Baer B.M /tme trade-off analyss for the crtcal path method: a devaton of the networ flow approach Journal of the Operatonal Research Socety vol. 48 pp Chen Y-L. Rns D. and Tang K Crtcal path n an actvty networ wth tme constrants European Journal of Operatonal Research 100 pp De M. Dunne E.J.Ghosh J.B. and WellsC.E The dscrete tme-cost tradeoff problem revsted European Journal of Operatonal Research 81 pp Demeulemeester E.L. Herroelen W.S. and Elmaghraby S.E Optmal procedures for the dscrete tme/cost tradeoff problem n project networ European Journal of Operatonal Research 88 pp Demeulemeester E. Reyc B.D. FoubertB. HerroelenW. and Vanhouce M.1998 New computatonal results on the dscrete tme/cost trade-off problem n project networs Journal of the Operatonal Research Socety vol. 49 pp Fulerson D A networ flow computaton for project cost curves Management Scence 7 pp ILOG Inc. CPLEX Dvson 1997 Usng the CPLEX Callable LbraryVerson5.5. Kelley J Crtcal-path plannng and schedulng: Mathematcal bass Operatons Research vol. 9 no. 3 pp Phllps J.P. and Dessouy M.I Solvng the project tme/cost tradeoff problem usng the mnmal cut concept Management Scence vol. 13 no. 6 ppb359-b377. Sunde L. and Lchtenberg S Net-present value cost/tme tradeoff Internatonal Journal of Project Management vol. 13 no. 1 pp45-49.

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