Activity Scheduling for Cost-Time Investment Optimization in Project Management

Transcription

1 PROJECT MANAGEMENT 4 th Internatonal Conference on Industral Engneerng and Industral Management XIV Congreso de Ingenería de Organzacón Donosta- San Sebastán, September 8 th -10 th 010 Actvty Schedulng for Cost-Tme Investment Optmzaton n Project Management Dego Fernando Manotas Duque 1, Leonardo Rvera Cadavd 1 Escuela de Ingenería Industral y Estadístca. Facultad de Ingenería. Unversdad del Valle. Calle Cal, Colomba. manotas@pno.unvalle.edu.co Department of Industral Engneerng, Unversdad ICESI, Cal, Colomba. leonardo@ces.edu.co Keywords: Cost-Tme Profle, Project Management, Actvty Schedulng, Tme-Value of Money Abstract The Cost-Tme Profle s a tool that presents the combned mpact of dsbursements made durng the executon of a project, and ther tmng. The Cost-Tme Profle chart presents the accumulated cost at any gven tme, and the area under the curve s the Cost-Tme Investment. Ths Cost-Tme Investment has a quantfable mpact on the workng captal the company uses, and t s also useful for the determnaton of the Drect Cost of a product. Ths paper ams to develop a model to create the schedule of actvtes that mnmzes the Cost-Tme Investment for a project. 1. Introducton It s unversally recognzed that money has value, and t also has value related to ts poston n tme. If somebody owes money, they would prefer to pay ther loan later rather than earler. If some revenue s expected, t would be much better to receve t now nstead of next week. Ths prncple s more than applcable n ths era of Just-In-Tme, when actvtes are beng performed when needed and not before. Ths paper brefly presents the Cost-Tme Profle (CTP), whch s a smple, graphcal tool to consder both dmensons of cost (dollar amount and tmng), and the resultng Cost-Tme Investment (CTI - the area under the CTP curve, whch represents the combnaton of how much money and for how much tme t has been nvested on a project), to fnally propose a Mxed-Integer Program to schedule the actvtes of a project n such a way that the resultng CTI s mnmal. In the followng subsectons we wll present how to get there, startng from the concept and constructon of a Cost-Tme Profle and endng wth the MIP model to create an optmal schedule that mnmzes the CTI.. Methodology.1. What s a Cost-Tme Profle (CTP)? A Cost-Tme Profle s a graph that depcts the Accumulated Costs that have been expended durng the executon of a project at every tme unt durng the process. Ths way of presentng the nformaton follows the use of resources through tme, from the moment the executon begns untl the company recovers those nvested resources through the sale of the product. The area under the CTP s called the Cost-Tme Investment (CTI), because t presents how 1453

2 much money has been ted up n the manufacturng process and for how long before beng recovered through sales. The reader wll recognze the term Investment because the CTI shares the two common components of any fnancal nvestment: Money and Tme. Fgure 1 gves a smple llustraton of a Cost-Tme Profle. 00 A ccum ulated C ost (dollars) W at 3 W at Actvty C 100 Actvty A Wat 1 Actvty B Cost-Tme Investment T o tal C o st M atera ls Fgure 1. Example of a Cost-Tme Profle tm e (hours).. How do you buld a CTP? There are several mportant parts on a CTP. Actvtes: These are the parts that actvely add cost. They are represented by postveslope lnes. Materals: Some actvtes requre materals to be performed. The assumpton s that the materals arrve rght at the begnnng of the actvty and all at once, therefore they are represented by a vertcal lne (nstantaneous accumulaton of cost). Wats: These are moments when nothng that adds cost s actvely happenng. Wats are of nterest manly for manufacturng actvtes, because n projects there s always some cost happenng (overhead?), even f you are not actvely operatng on the project. Snce they do not add cost, they are represented by flat lnes n the CTP curve (the accumulated cost remans constant durng the wat). Total Cost: Ths s the heght of the curve at the end of the project. It represents the total accumulated cost of the project. Cost-Tme Investment (CTI): Ths s the area under the CTP curve, and t represents how much money and for how long has t been nvested n the project. Ths s a measure of the utlzaton of workng captal, and snce t s captal you are usng, ths captal wll undoubtedly have a cost that wll be whatever rate the company has to pay for ts workng captal. Workng Captal Cost (WCC): Ths s the cost the company has to pay for the use of captal. It can be expressed as n Equaton 1. WCC = CTI * Cost of Captal Rate (1) 1454

3 To buld a CTP, we need to know several thngs. Frst, we need to know when s each element of the CTP happenng (actvtes, wats and materals releases). We also need to know how much does each of these elements cost. Puttng the two prevous ponts together we can determne how much money s beng spent as cost at every tme unt n the process. Fnally, we tally the costs and buld the CTP wth the accumulated cost at every tme unt, and present ths nformaton n graphcal form. The area under the curve (obtaned addng the accumulated cost at each tme unt for all the tme unts) represents the Cost-Tme Investment. In the followng subsectons, a bref dscusson on how to complete these steps for the constructon of the CTP wll be presented..3. Applcablty of CTP to Project Management CTP does not consder ndrect costs because t s truly hard to assgn these costs drectly to actvtes or ndvdual unts of a product. However, n project management t s reasonable to assume that all resources, people and equpment that take part n the project are drectly and completely assgnable to the project. Therefore, CTP s especally applcable to projects and the full spectrum of costs s ncluded n them..4. What s the contrbuton of each actvty to the Cost-Tme Investment? [3] In order to create an optmal schedule of actvtes, t s necessary to characterze the contrbuton to the area under the CTP curve that each actvty makes. If we consder that materals are always released at the begnnng of an actvty, we can see that Fgure presents the typcal areas that we would see as a result of a gven actvty. For Fgure we have that: Y = End date of the project X = Tme when actvty fnshes T = Duraton of actvty MAT =Materals requred to perform actvty CR = Cost rate of actvty (cost per tme unt) Accumulated Cost Actvty Area Area 3 (T * CR ) Materals Area 1 MAT T tme X - T X Fgure : Areas that an actvty contrbutes to the CTI. Y Area 1 occurs when we ncorporate the expendture n materals, because they reman spent untl the end of the project (Y). Area 1 can be calculated wth Equaton. 1455

4 Area 1 = [ Y (X T) ] * MAT () Area occurs when the actvty s actvely happenng. See Equaton 3. Area CR * T * T CR * T (3) Area 3 occurs from the end of the actvty untl the end of the project. Ths s the cost added by the actvty projected untl the end date of the project. Area 3 can be found usng Equaton 4. Area 3 = () * (Y X) (4) Addng Equatons, 3 and 4 and organzng some terms we obtan the total contrbuton that each actvty makes to the Cost-Tme Investment. See Equaton 5. Area Contrbuton Y X MAT T * MAT CR * T (5).5. Tradtonal LP model for Project Management The tradtonal LP model for project management (found n Operatons Research or Management Scence textbooks) consders precedence relatonshps between actvtes, and ts objectve functon s to mnmze the total completon tme of a project. Fgure 3 presents the text of such a model. These models have two problems for ts use wth Cost-Tme profles: Ther objectve functon mnmzes completon tme of the project, not total Cost-Tme Investment They consder precedence constrants, but they do not recognze resource dependences that happen when we have lmted resources that are shared by dfferent actvtes. These shared resources sometmes force that two actvtes that could be performed smultaneously wll have to be scheduled one after the other. In subsectons.6 and.7 the solutons to these two problems wll be presented. 1456

5 MODEL 1 Set: Actvtes {A, B, C, } (n actvtes). Indces:, j on Actvtes. Parameters: T Executon tme for Actvty PD,j 1 f Actvty j s an mmedate predecessor of Actvty 0 otherwse Varables: X Completon tme for Actvty. Y Completon tme for the whole project Objectve Functon: Mnmze the completon tme for the project Mn Z = Y Constrants: Completon tme for the project must be greater than or equal to the completon tme for any actvty Y X ; for all. Any actvty must only begn after ts mmedate predecessors have been completed. X X j + PD,j * T j ; for all j (Ths constrant s only enforced when the PD,j s 1, that s, when there exsts a relatonshp of precedence between and j). Fgure 3: Tradtonal Project Management LP model..6. Changng the Objectve Functon If the objectve of the model s the mnmzaton of the Cost-Tme Investment, then the objectve functon must be to mnmze the sum of the CTI contrbutons of all the actvtes. That s, the objectve functon should be based n Equaton 5. Fgure 4 presents ths updated model. 1457

6 MODEL Set: Actvtes {A, B, C, } (n actvtes). Indces:, j on Actvtes. Parameters: T Executon tme for Actvty PD,j CR MAT T 1 f Actvty j s an mmedate predecessor of Actvty 0 otherwse Cost Rate of Actvty Materals released at the begnnng of Actvty Duraton of Actvty Varables: X Completon tme for Actvty. Y Completon tme for the whole project Objectve Functon: Mnmze the sum of contrbutons to the total area by the actvtes Mn Z Y X MAT T * MAT Constrants: Y X ; for all. X X j + PD,j * T j ; for all j Fgure 4: Project Management Model wth CTI mnmzaton as the objectve functon..7. Shared resources constrants [3] Projects have precedence constrants between actvtes. You should have the structure of a buldng before you can put up ts walls. In ths case, t s clear whch actvty needs to be completed before the dependent one starts. However, there are cases where actvtes do not have precedence relatonshps (you can landscape the front yard and the back yard n any sequence), but they cannot be performed smultaneously because they use the same resources (we only have one lawn mower, for example). When these resources are shared, t s not mmedately evdent whch actvty should be completed before the other. Therefore, a couple of new constrants and bnary varables wll be ntroduced to consder ths type of stuaton. Constructon of a logcal statement usng bnary varables: In the case of shared resources we need to tell the MIP that the actvtes that share a resource (let s call them a and b) wll happen one before the other, but NEVER at the same tme. So, t must be true that: (XOR s the exclusve OR, meanng that one proposton or the other should happen, but NEVER both). Ths XOR can be acheved usng a bnary varable called y1 j, and two other new elements appear: SR j : Ths s a parameter. Its value s 1 f actvtes and j share a resource and 0 otherwse. M: M s a BIG number that wll be used to enforce or relax a constrant accordng to the values of the y1 j (6) 1458

7 So, we wll present the two new constrants and then offer some comments about the way they would work. X T + (X j * SR,j ) (M * y1,j * SR,j ) (If j needs to happen before ) (7.1) Xj Tj + (X * SRj,) (M * y1j, * SRj,) (If needs to happen before j) (7.) y1,j + y1j, = 1 ; for all j (y1j and y1j are bnary varables) (8) Now, for the sake of argument, let s examne what happens n dfferent stuatons wth Equatons 7.1 and 7.. Actvtes and j do not share resources: When ths s the case, the parameters SR j and SR j must both be zero, therefore Equatons 7.1 and 7. transform n the followng way X T and Xj Tj These two constrants do not add any trouble to the MIP, because t s naturally obvous that an actvty s fnshng tme should be greater than or equal to ts executon tme. (7.1) (7.) Actvtes and j share a resource: When ths s the case, the parameters SR j and SR j must both be one. In ths case we also need to consder the values of y1 j and y1 j, and we know from equaton 8 that one of them must be a one and the other one zero. Assumng that y1 j = 1, and y1 j = 0, then Equatons 7.1 and 7. transform n the followng way X T + Xj (M * y1,j), that for all practcal purposes wll be X M (7.1) Xj Tj + X (7.) The real effect of usng y1 j and y1 j wll be that the constrant n Eq. 7.1 wll be relaxed (not enforced), and equaton 7. wll be held, therefore the net result wll be that actvty j wll happen after actvty. Had the y1 j and y1 j been the other way around (y1 j = 0, and y1 j = 1), actvty would happen after actvty j. Integratng all these nto the model, we obtan Model 3, presented n Fgure

8 MODEL 3 Set: Actvtes {A, B, C, } (n actvtes). Indces:, j on Actvtes. Parameters: T Executon tme for Actvty PD,j 1 f Actvty j s an mmedate predecessor of Actvty 0 otherwse SR,j 1 f Actvtes and j share a Resource or an Operator. SR,j = SR j, for all and j. Also, SR, = 0. CR Cost Rate of Actvty MAT Materals released at the begnnng of Actvty T Duraton of Actvty Varables: X Completon tme for Actvty. Y Completon tme for the whole project y1,j Bnary Varables to assst n the logcal ether-or constructon for Resource and Operator sharng Objectve Functon: Mnmze the sum of contrbutons to the total area by the actvtes Mn Z Y X MAT T * MAT Constrants: Y X ; for all. X X j + PD,j * T j ; for all j y1,j + y1 j, = 1 ; for all j X T + (X j * SR,j ) (M * y1,j * SR,j ) ; for all j Fgure 5: Model wth CTI mnmzaton as the objectve functon and shared resources 3. Conclusons and Future Research The model presented n the prevous secton could be of great use for project orented organzatons, especally n those cases when the company works wth ts own resources before percevng revenue (a real estate development company that bulds governmentsubsdzed housng, for example). In these cases the mnmzaton of the completon s not as mportant an objectve as t would be the mnmzaton of the Cost-Tme Investment (and the Cost-Tme Profle would penalze lengthy projects anyway). It s nterestng to explore dfferent objectves and constrants that mght be ncluded n Project Management models, snce the specfc characterstcs of companes and ther projects are so vared that specfc models are always applcable to some companes. Also, models wth varable actvty tmes (a la PERT) wll be studed applyng Monte Carlo smulaton to explore the mpact of specfc actvtes on the resultng CTI and the dfferences n crtcal paths that could result. 4. References Fooks, J. H. (1993). Profles for Performance: Total Qualty Methods for Reducng Cycle Tme. Addson-Wesley, Readng, MA. 1460

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