A WAY TO DEAL WITH THE PROJECT CRASHING PROBLEM

Transcription

1 Hamdjatou ane Glbert Nkubl Département des scences admnstratves Unversté du Quebec en Outaouas n Canada Barthelemy Ateme-Nguema Département des scences de la geston Unversté du Quebec en Abtb-Témscamngue n Canada A WAY TO DEAL WITH THE PROJECT CRASHING PROBLEM Introducton Completng projects as quckly as possble remans a constant preoccupaton of all managers (Lberatore, Pollack-Johnson 2006). The lterature s replete wth affrmatons that the busnesses able to develop new products n the shortest tme enjoy a substantal compettve edge (Swnk, 2003). Practtoners, academcs and other professonals contnue to search for ways, means and tools for determnng a pror how much a project can be accelerated, based on nherent constrants such as budget and resources. Contemporary manageral emphass on ganng compettve advantages oblges project managers to scope the costs of acceleratng each actvty of projects underway or upcomng. In the context of resource crtcalty, the cost of acceleraton depends to a large degree on the type of resources assgned, ther avalablty, ther quantty and so on. In ths artcle, we propose a new approach to solvng the problem of reachng an optmal compromse between duraton and resources for the acceleraton of projects n a context of resource crtcalty, n whch the substtuton of the resources s consdered.

2 40 Hamdjatou ane, Glbert Nkubl, Barthelemy Ateme-Nguema 1. Revew of the lterature In conventonal project management, the mandate s carred out wthn predetermned constrants such as specfcatons, deadlnes and budget. Even though the antcpated length of a project n plannng s usually longer than the crtcal path (Boctor 2005), projects seldom follow the classcal project management scheme or meet deadlnes (Gerk, Qassm 2008). In order to address ths defcency, project executon tme can be accelerated to compensate for potental delays. By acceleraton we mean fnshng a project sooner than orgnally planned. In order to accelerate a project, we must have all of the nformaton relevant to ts consttuent actvtes, ncludng the types and quanttes of the resources nvolved, unt costs, duratons, prorty relatonshps and so on. The usual method of acceleratng a gven actvty wthn a project s to assgn to t more resources or resources wth expertse and/or slls greater than those ntally at the dsposal of the project manager. Ths of course nvolves ncreased costs. However, acceleratng some project actvtes can brng a reducton of ndrect costs (Dodn, Elmam 2008; Evensmo and arlsen, 2008) n the form of salary, amortsement of equpment and nfrastructure, and so on. Some authors have proposed acceleratng projects by supermposng actvtes (Roemer, Ahmad 2004). Ths approach s used essentally n the development of new products and servces. The supermposton of actvtes conssts of carryng out n parallel (partally or wholly) actvtes that were organzed sequentally n the project plan. Another acceleraton technque nvolves substtutng certan actvtes wth one and/or several other actvtes. There s thus the possblty of acceleratng a project by applyng the conventonal approach, supermposng actvtes and substtutng certan actvtes (Gerk, Qassm 2008). Several approaches have been descrbed n the lterature for solvng the tme/cost trade-off problem. However, few of these take nto consderaton project partculartes, that s, the peculartes of the project resources. We may cte the cut search approach proposed by ane, Azondekon (2008) and the lnear programmng model proposed by Alban (2008). However, most of these approaches are defcent and dffcult to apply to large complex projects (.e. 100 actvtes or more). Calculaton tme n partcular ncreases consderably. There are new methods better suted to solvng the acceleraton problem n the case of large projects for whch more than two resources are avalable. These requre mnmal tme for calculaton. We recommend a method based on applcaton of the tabu algorthm and descrbe n the followng secton the prncpal steps of our approach.

3 A WAY TO DEAL WITH THE PROJECT CRASHING PROBLEM Methodology The logcal procedure usually used to solve the duraton/cost trade-off problem s the smultaneous mathematcal equaton approach, one equaton representng tme and the other representng project cost. The tme taken to complete the actvtes as a whole depends on ndvdual actvty duraton and on the prorty relatonshps between the tasks. Cost depends on the resources assgned and of course on the allowed duraton of each task. Ths approach generally seeks to determne exact actvty completon tmes and total project tme. These methods dffer prmarly n terms of the optmzaton technques used to choose the actvtes to be accelerated and calculate by how much to accelerate them. Among these, we may cte the CPM/PERT method, the lnear exact and non-lnear programmng technques (Alban, 2008), the cut search approach (ane, Azondekon 2008), and algorthmc and heurstc technques (Bolduc, Laporte et al. 2010). However, methods based on exact mathematcal programmng run nto dffculty as project scale (number of actvtes or tasks) ncreases and the problem becomes of the so-called NP-hard type. The solvng of ths type of problem requres complex optmzaton technques and the tabu search s the technque that we feel provdes the best potental soluton for project acceleraton purposes. Ths algorthm technque also mnmzes problem-solvng tme Mathematcal model The am of ths mathematcal model s to calculated the total cost of the project. We descrbe below the parameters, the varables and the objectve functon of the model. Let t be the tme (from the begnnng of the project) at whch actvty s to begn accordng to the project plan and let x be the number of unts of tme by whch actvty s accelerated. The remanng parameters are as follows: d the normally expected duraton of actvty c the unt cost of acceleratng actvty u the maxmum number of tme unts by whch actvty can be accelerated n the number of actvtes (1 beng the frst actvty and n beng the last) T the normally expected duraton of the crtcal path of the un-accelerated project T m the calculated project duraton after the m th teraton, m = 1, 2, 3,..., M

4 42 Hamdjatou ane, Glbert Nkubl, Barthelemy Ateme-Nguema T m = T a the optmal duraton attanable by acceleratng the project P(j) the set of actvtes mmedately precedng actvty j C the normal cost per unt of tme of resource k assgned to actvty a the cost per tme unt of obtanng va resource k a one-tme-unt acceleraton of actvty the total number of k resources assgned to actvty N a the total number of actvtes that can be accelerated = 1 = 1 c d a x the normally expected cost of completng actvty the addtonal cost of acceleratng actvty by x unts of tme B the addtonal budget avalable for acceleratng the project C NET the net cost of acceleratng actvty by x unts of tme C n the normally expected total cost of completng the project C max = C n + B the total cost not to be exceeded due to project acceleraton C atot the net total addtonal cost of acceleratng the project C atotm the net addtonal cost of the project at the m th teraton Objectve functon: N a To mnmze C = α x ( a c ) Subject to: atot = 1 α = 1 f s selected for acceleraton; 0 f s not selected for acceleraton (2) t j ( j) P( j) t x, (3) t n T s the nstant of the end of the last actvty of the project (4) (1) T T m +1 T m avec T M = Ta (5) 0 x (6) u x u (7) C atot B (8)

5 A WAY TO DEAL WITH THE PROJECT CRASHING PROBLEM 43 Wth: t = 0 1 s the start of the frst actvty of the project t 0 C C NET n = = 1 c d + x = N c ( d x ) + ( a c ) c = 1 = 1 d a x = c d c x + a x = (9) (10) The problem amounts to mnmzng the total project duraton whle remanng wthn the lmts of addtonal budget B. T m (expected project duraton based on the crtcal path) s calculated usng the CPM method. The objectve functon for mnmzng T m accordng to T T m +1 Tm T conssts of choosng an actvty on the crtcal path and acceleratng t by one unt of tme. Havng establshed the method of calculatng project duraton and cost, our goal s to fnd a new approach to optmzng the soluton to the problem of fndng the best trade-off between project completon tme and cost, wth the am of obtanng the greatest decrease n tme at the lowest cost. It s at ths stage that we use the tabu method to determne the project-acceleratng opton that costs the least. In the followng secton, we present the approach based on the tabu algorthm. 2.2 The tabu algorthm The "tabu" algorthm s a local-search meta-heurstc that explores the neghbourhood beyond the optmum soluton obtaned (Xu et al. 2009). Ths search method uses an teratve process to shft from the current soluton towards a neghbourng soluton that acheves a superor goal. In order to avod futle cycles, that s, exploraton of solutons smlar to those prevously examned, the search generates a "tabu" lst of shfts and solutons explored n prevous teratons (Lu et al. 2010). In order to mprove the effcency of the teratve process, the tabu algorthm mantans a follow-up of the local nformaton as well as of the search process tself (Bolduc et al. 2010). The other prncples of search wth tabu, namely aspraton, ntensfcaton and dversfcaton, are treated n detal n works publshed by Glover (1989; 1990) and by Glover and Laguna (1997).

6 44 Hamdjatou ane, Glbert Nkubl, Barthelemy Ateme-Nguema In each teraton, our tabu algorthm frst explores the entre solutons space (the project plan as a whole) and thus defnes the zone n whch t wll subsequently ntensfy the search for the actvty to be accelerated. The actvty confguraton scheme, duraton and the total cost of the project are then updated, based on acceleraton of the actvty thus dentfed. The process s stopped when the condtons regardng project duraton relatve to predefned budgetary constrants are met or when t has been determned that no actvty wthn the predefned zone of search can be accelerated to obtan a desrable result (Fgure l). Fgure 1. The project acceleraton algorthm wth Tabu

7 A WAY TO DEAL WITH THE PROJECT CRASHING PROBLEM Applcatons In order to valdate the proposed project acceleraton algorthm, we performed tests on real projects nvolvng numerous actvtes (over 100). The project ncludes 172 actvtes requrng four dfferent types of resource. Fgures 2a and 2b show the project network, whle Table 1 n appendx provdes the tme and resource-assocated costs for each actvty. Fgure 2a. The project network Source: Doerner et al

8 46 Hamdjatou ane, Glbert Nkubl, Barthelemy Ateme-Nguema Fgure 2b. The project network Source: Doerner et al Results We used MATLAB 6.0 to mplement our algorthm. The results obtaned show that the project can be accelerated wth mnmal ncreases n cost. Furthermore, the calculaton tme was relatvely short. The expected project duraton before acceleraton was 83 weeks for a total cost of $1,483,600. We defned the normal unt cost of each resource arbtrarly and set the unt cost of acceleraton 50% hgher (see Table 1 n appendx).

9 A WAY TO DEAL WITH THE PROJECT CRASHING PROBLEM 47 If we also suppose that any actvty can be accelerated wthout lmts and that we have at our dsposal a budgetary ncrease of up to 20% wth whch to accelerate the project, we can obtan a reducton n project duraton from 83 to 47 weeks at an addtonal cost of $293,450. Ths s a 43.3% reducton of project tme for a cost ncrease of 19.77%. The algorthm acheved ths result by carryng out 113 teratons, whch requred about 3 seconds of calculaton tme. These results ndcate that the algorthm s effectve and could be used on a daly bass by professonals to accelerate large-scale projects nvolvng relatvely large numbers of tasks. Fgure 3. Cost varaton VERSUS project duraton Concluson In ths study, we have presented the results obtaned usng a new tool developed to solve the problem of fndng the optmal trade-off between project duraton and cost n the context of resource crtcalty. The tool we have developed apples prncples of tabu search to optmze the process of dentfyng project actvtes to be accelerated. Usng our project acceleraton algorthm on projects prevously treated usng other approaches, we demonstrated that the tool dentfes the same soluton or n some cases a better soluton and wth a shorter calculaton tme. Ths work thus proposes a new avenue to explore wth more n-depth studes for the mprovement of project management.

10 48 Hamdjatou ane, Glbert Nkubl, Barthelemy Ateme-Nguema Appendx Table 1 Duraton, acceleraton and assocated resource costs of actvtes (correspondng to arrows n the CPM/PERT dagram) for the project Act. Path Pqte d u C k1 C k2 C k3 C k4 a k1 a k2 a k3 a k4 1 (1,2) (1,3) (1,4) (2,5) 1 2 l (2,7) (3,5) (3,6) (3,8) (4,6) (4,8) 3 2 l (5,12) 4; (6,9) 7; (6,12) 7; (7,9) (8,9) 8; (8,10) 8; (8,11) 8; (9,13) 11;14; (10,16) (11,15) (12,14) 11; (13,18) (14,17) (14,20) (15,18) 19 2 l (15,19) (16,19) (17,24) (18,21) (18,22) 22; (19,23) 26; (20,22) 24 l (21,28) (22,27) 30; (23,26) (24,25) (25, (25,30) (26,30) (26,31) (26,32) (27,31) (28,29) (29,35) (30,33) 37; (31,36) 38; (32,34) 40; (32,36) (33,38) (34,37)

11 A WAY TO DEAL WITH THE PROJECT CRASHING PROBLEM 49 Table 1(contnued) Act. Path Pqte d u C k1 C k2 C k3 C k4 a k1 a k2 a k3 a k4 51 (35,38) (35,39) (35,40) (36,37) (37,42) 46; (38,41) 50; (39,41) 49; (40,42) (41,43) (42,69) (43,44) 55; (43,45) (43,46) (43,53) (43,88) (44,47) (44, (44,52) (45,49) (46,51) (46,54) (47,48) (47,49) (48,55) (49,55) (50,56) 69; (51,56) (51,57) (52,58) (52,59) (53,59) (54,60) (55,61) (56,61) 74; (57,61) 76; (57,62) (58,63) (59,66) (59,81) 80; (60,96) 80; (61,64) (62,64) 83;84; (62,65) (63,66) (64,67) (64,87) 91; (65,67) 91; (65,84) (66,68) (67,112) (67,113) 95; (68,112) 95; (68,113) (69,70) (69,71) (69,73) (69,74) (69,88) (70,72)

12 50 Hamdjatou ane, Glbert Nkubl, Barthelemy Ateme-Nguema Table 1(contnued) Act. Path Pqte d u C k1 C k2 C k3 C k4 a k1 a k2 a k3 a k4 110 (70,88) (71,74) (72,77) (73,78) (74,75) (74,76) 107; (75, ; (76,79) (77,99) (78,80) (78,81) (79,99) (80,82) 115; (80,83) (81,83) (82,84) 89; ,85) (83,86) (84,87) 123; I (85,87) (86,112) (86,113) (87,111) (88,89) 128; (88,90) 65;108; (89,91) 65;108; (89,92) (90,108) (90,109) (91,93) (92,96) (93,94) (93,95) (93,97) (94,98) (94,99) (95,100) (96,100) (97,101) 90; (98,102) (98,103) (99,107) (100,108) 118;121; (101,108) (101,109) (102,104) h00 h (102,105) (102,108) I (102,109) (103,106) (103,107) (104,111) (105,111) (106,112) (106,113) (107,110) 151; (108,111) 137;152;153; (109,111) 138;154; (110,112) (110,113)

13 A WAY TO DEAL WITH THE PROJECT CRASHING PROBLEM 51 Table 1(contnued) Act. Path Pqte d u C k1 C k2 C k3 C k4 a k1 a k2 a k3 a k4 170 (111,114) 132;161;162;166; (112,1140) 100;102;130;163; (113,114) 101;103;131;164; References Alban T. (2008), Accélératon des projets et allocaton des ressources: des enjeux de la geston mult-projet. Mémore de Maîtrse, Unversté du Québec en Outaouas. Boctor Fayez F. (2005), Le problème d'allocaton et de nvellement des ressources. ASAC 2005 Conference, Toronto (Ontaro), Canada. Bolduc M.C., Laporte G., Renaud J., Boctor F.F. (2010), A tabu search heurstc for the splt delvery vehcle routng problem wth producton and demand calendars. European Journal of Operatonal Research, 202 (1): Dodn B., Elmam A.A. (2008), Integraton of equpment plannng and project schedulng. European Journal of Operatonal Research, 184(3): Evensmo J., arlsen J. T. (2008), Loong for the Source Where Do Crash Costs Come From?. Cost Engneerng, 50(7). Gerk J.E.V., Qassm R.Y. (2008), Project Acceleraton va Actvty Crashng, Overlappng, and Substtuton. IEEE Transactons On Engneerng Management, 55(4): Glover F.W. (1989), Tabu Search Part I. ORSA Journal on Computng, INFORMS: Insttute for Operatons Research 1: 190. Glover F.W. (1990), Tabu Search Part II. ORSA Journal on Computng, INFORMS: Insttute for Operatons Research 2: 4. Glover F.W. Laguna M. (1997), Tabu Search. luwer Academc Publshers, Boston. ane H., Azondekon S.H. (2008), Un algorthme pour la résoluton d un problèrne d accélératon de projet avec des ressources multples. 7e Conférence Intematonale de Modélsaton et Smulaton MOSIM'08 du 31 mars au 2 avrl 2008-Pars-France, Modélsaton, Optmsaton et Smulaton des systemes: Communcatons, Coopératon, et Coordnaton. Lberatore M.J., Pollack-Johnson B. (2006), Extendng project tme-cost analyss by removng precedence relatonshps and actvtystreamng. Internatonal Journal of Project Management, 24(6): Lu J., Cheng H., Sh X., Xu J. (2010), A Tabu Search Algorthm for Fast Restoraton of Large Area Breakdown n Dstrbuton Systems. Energy and Power Engneerng, 1-5.

14 52 Hamdjatou ane, Glbert Nkubl, Barthelemy Ateme-Nguema Lu S. (2003), Fuzzy actvty Tmes In crtcal Path and Project Crashng problems. Cybernetcs and Systems: An Internatonal Journal, 34: Rakotomalala H.L. (2002), Le comproms durée/ressource en geston de projet dans un contexte de multplcté des ressources : Analyse et mpact sur la performance de projet. Mémore de Maîtrse, Unversté du Québec en Outaouas. Roemer T.A., Ahmad R. (2004), Concurrent Crashng and Overlappng n Product Development. Operatons Research, INFORMS: Insttute for Operatons Research 52: Swnk M. (2003), Completng projects on-tme: how project acceleraton affects new product development. Journal of Engneerng and Technology Management, 20(4): Xu., Feng Z., Jun. (2009), A Tabu Search algorthm for schedulng jobs wth controllable processng tmes on a sngle machne to meet duedates. Computers & Operatons Research, 37: Wakas S., June L.W. (2009), Cost Reducton for the Project Completon n Shortest Possble Duraton by Stretchng Noncrtcal Actvtes. Australan Journal of Basc and Appled Scences, 3(4):