A WAY TO DEAL WITH THE PROJECT CRASHING PROBLEM

Transcription

1 Hamdjatou Kane Gilbert Nkubili Département des sciences administratives Université du Quebec en Outaouais in Canada Barthelemy Ateme-Nguema Département des sciences de la gestion Université du Quebec en Abitibi-Témiscamingue in Canada A WAY TO DEAL WITH THE PROJECT CRASHING PROBLEM Introduction C om pleting projects as quickly as possible rem ains a constant preoccupation o f all m anagers (L iberatore, Pollack-Johnson 2006). The literature is replete w ith affirm ations th at the businesses able to develop new products in the shortest tim e enjoy a substantial com petitive edge (Sw ink, 2003). P ractitioners, academ ics an d o th er pro fessio n als continue to search fo r w ays, m eans and tools fo r determ ining a priori h ow m uch a p ro ject can be accelerated, based on in h eren t constraints such as b u d g et and resources. C ontem porary m anagerial em phasis on g aining com petitive advan tages obliges p ro ject m anagers to scope the costs o f acceleratin g each activ ity o f pro jects underw ay o r upcom ing. In the co n text o f resource criticality, th e cost o f acceleratio n depen ds to a large degree on the type o f resources assigned, their availability, their quantity and so on. In this article, w e pro pose a n ew approach to solving the pro blem o f reaching an optim al com prom ise b etw een d u ration and resources fo r the acceleratio n o f pro jects in a contex t o f resource criticality, in w hich the substitution o f the resources is considered.

2 40 Hamdjatou Kane, Gilbert Nkubili, Barthelemy Ateme-Nguema 1. Review of the literature In conventional project management, the mandate is carried out within predeterm ined constraints such as specifications, deadlines and budget. E ven though the anticipated length o f a project in planning is usually longer than the critical path (Boctor 2005), projects seldom follow the classical project m anagem ent scheme or m eet deadlines (Gerk, Q assim 2008). In order to address this deficiency, project execution tim e can be accelerated to com pensate for potential delays. B y acceleration w e m ean finishing a project sooner than originally planned. In order to accelerate a project, w e m ust have all o f the inform ation relevant to its constituent activities, including the types and quantities o f the resources involved, unit costs, durations, priority relationships and so on. The usual m ethod o f accelerating a given activity w ithin a project is to assign to it m ore resources or resources w ith expertise and/or skills greater than those initially at the disposal o f the project m anager. This o f course involves increased costs. H owever, accelerating som e project activities can bring a reduction o f indirect costs (Dodin, Elim am 2008; Evensm o and Karlsen, 2008) in the form o f salary, am ortisem ent o f equipm ent and infrastructure, and so on. Some authors have proposed accelerating projects by superim posing activities (Roem er, A hm adi 2004). This approach is used essentially in the developm ent o f new products and services. The superim position o f activities consists o f carrying out in parallel (partially or wholly) activities that w ere organized sequentially in the project plan. A nother acceleration technique involves substituting certain activities w ith one and/or several other activities. There is thus the possibility o f accelerating a project by applying the conventional approach, superimposing activities and substituting certain activities (Gerk, Qassim 2008). Several approaches have been described in the literature for solving the tim e/cost trade-off problem. However, few of these take into consideration project particularities, that is, the peculiarities o f the project resources. W e m ay cite the cut search approach proposed by K ane, A zondekon (2008) and the linear program m ing m odel proposed by A lban (2008). H owever, m ost o f these approaches are deficient and difficult to apply to large com plex projects (i.e. 100 activities or m ore). Calculation tim e in particular increases considerably. There are new m ethods better suited to solving the acceleration problem in the case o f large projects for w hich m ore than two resources are available. These require m inimal tim e for calculation. W e recom m end a m ethod based on application o f the tabu algorithm and describe in the following section the principal steps of our approach.

3 A WAY TO DEAL WITH THE PROJECT CRASHING PROBLEM Methodology The logical procedure usually used to solve the duration/cost trade-off problem is the sim ultaneous m athem atical equation approach, one equation representing tim e and the other representing project cost. The tim e taken to com plete the activities as a w hole depends on individual activity duration and on the priority relationships betw een the tasks. C ost depends on the resources assigned and o f course on the allow ed duration o f each task. This approach generally seeks to determ ine exact activity com pletion tim es and total project tim e. These m ethods differ prim arily in term s o f the optim ization techniques used to choose the activities to be accelerated and calculate by how m uch to accelerate them. A m ong these, w e m ay cite the C PM /PERT m ethod, the linear exact and non-linear program m ing techniques (Alban, 2008), the cut search approach (Kane, A zondekon 2008), and algorithm ic and heuristic techniques (Bolduc, Laporte et al. 2010). H owever, m ethods based on exact m athem atical program m ing run into difficulty as project scale (num ber o f activities or tasks) increases and the problem becom es o f the so-called N P-hard type. The solving o f this type o f problem requires com plex optim ization techniques and the tabu search is the technique that w e feel provides the best potential solution for project acceleration purposes. This algorithm technique also minimizes problem -solving time Mathematical model T he aim o f this m athem atical m odel is to calculated the total cost o f the project. W e describe below the param eters, the variables and the objective function o f the m odel. L et ti be the tim e (from the beginning o f the project) at w hich activity i is to begin according to the pro ject plan and let x i be the num ber o f units o f tim e by w hich activity i is accelerated. T he rem aining param eters are as follow s: d i - the norm ally expected duration o f activity i ci - th e u n it cost o f accelerating activity i ui - the m axim um n um ber o f tim e units by w hich activity i can be accelerated n - the n um ber o f activities ( 1 being the first activity and n being the last) T - th e norm ally expected duration o f the critical path o f the un-accelerated pro ject T m - the calculated pro ject duration after the m tth iteration, m = 1, 2, 3,..., M

4 42 Hamdjatou Kane, Gilbert Nkubili, Barthélémy Ateme-Nguema T m = T a - the optim al duration attainable by accelerating the project P (j) - the set o f activities im m ediately preceding activity j Ckj - th e norm al cost p er u n it o f tim e o f resource k assigned to activity i aki - the cost p er tim e u n it o f obtaining v ia resource k a one-tim e-unit acceleration o f activity i K ki - th e total num b er o f k resources assigned to activity i N a - the to tal num ber o f activities th at can be accelerated Kki X c kid i - the norm ally expected cost o f com pleting activity i ki=1 Kki X a tdx i - the additional cost o f accelerating activity i by x i u nits o f tim e ki=1 B - the additional budget available for accelerating the project CNET - th e n et cost o f accelerating activity i by x i units o f tim e Cn - the norm ally expected total cost o f com pleting the project Cmax = C n + B - the total cost not to be exceeded due to pro ject acceleration CaToT - the net total additional cost o f accelerating the project CaTOTm - the net additional cost o f the project at the mth iteration Objective function: Na Kki T o m inim ize C atot = a i X Xi X (a k> - Cki ) ( 1) i=1 ki Subject to: a i = 1 if i is selected for acceleration; 0 i f i is not selected for acceleration (2 ) tj > - X v (i'. j )GP ( j ) (3) t n < T is th e instant o f the end o f the last activity o f the pro ject (4) T < Tm+1 < Tm avec TM = Ta (5) 0 < x i < u i ( 6 ) x i < u i (7) ( 8 )

5 A WAY TO DEAL WITH THE PROJECT CRASHING PROBLEM 43 W ith: t 1 = 0 is th e start o f the first activity o f the project t i > 0 V i K ki K ki K k K k K ki C NET = Z Ckr\d r - X )+Z = Z CUd i -Z CkiX +Z = ki=1 ki ki ki ki ( 9 ) K U K ki Z c kid i + x Z k - c n ) ki ki N Kti C =ZZ c A ( 1 0 ) i =1 ki = 1 T he problem am ounts to m inim izing the total pro ject duration w hile rem aining w ithin the lim its o f additional budget B. Tm (expected pro ject duration b ased on the critical path) is calculated using the C PM m ethod. The objective function for m inim izing Tm according to T < Tm+1 < Tm T consists o f choosing an activity on the critical path and accelerating it by one u n it o f tim e. H aving established the m ethod o f calculating pro ject duration and cost, our goal is to find a new approach to optim izing the solution to the problem o f finding the best tra d e -o ff betw een pro ject com pletion tim e and cost, w ith the aim o f obtaining the g reatest decrease in tim e at the low est cost. It is at this stage th at w e use the tabu m ethod to determ ine the project-accelerating option th at costs the least. In the follow ing section, w e present the approach based on the tabu algorithm. 2.2 The tabu algorithm The "tabu" algorithm is a local-search m eta-heuristic that explores the neighbourhood beyond the optim um solution obtained (X u et al. 2009). This search m ethod uses an iterative process to shift from the current solution tow ards a neighbouring solution that achieves a superior goal. In order to avoid futile cycles, that is, exploration o f solutions sim ilar to those previously exam ined, the search generates a "tabu" list o f shifts and solutions explored in previous iterations (Liu et al. 2010). In order to im prove the efficiency o f the iterative process, the tabu algorithm m aintains a follow-up o f the local inform ation as well as o f the search process itself (Bolduc et al. 2010). The other principles o f search w ith tabu, nam ely aspiration, intensification and diversification, are treated in detail in works published by Glover (1989; 1990) and by Glover and Laguna (1997).

6 44 Hamdjatou Kane, Gilbert Nkubili, Barthelemy Ateme-Nguema In each iteration, o u r tabu algorithm first explores the entire solutions space (the project plan as a w hole) and thus defines the zone in w hich it w ill subsequently intensify the search for the activity to be accelerated. The activity configuration schem e, duration and the total cost of the pro ject are th en updated, based on acceleration o f the activity thus identified. The process is stopped w hen the conditions regarding project duration relative to predefined budgetary constraints are m et o r w hen it has been determ ined th at no activity w ithin the predefined zone o f search can be accelerated to obtain a desirable result (Figure l). Figure 1. The project acceleration algorithm with Tabu

7 A WAY TO DEAL WITH THE PROJECT CRASHING PROBLEM Applications In ord er to validate the proposed pro ject acceleration algorithm, w e perform ed tests on real projects involving num erous activities (over ). T he pro ject includes 172 activities requiring four different types o f resource. Figures 2 a and 2b show the pro ject netw ork, w hile T able 1 in appendix provides the tim e and resource-associated costs for each activity. Figure 2a. The project network Source: Doerner et al

8 46 Hamdjatou Kane, Gilbert Nkubili, Barthelemy Ateme-Nguema Figure 2b. The project network Source: Doerner et al Results W e used M A T L A B 6.0 to im plem ent o u r algorithm. The results obtained show th at the pro ject can be accelerated w ith m inim al increases in cost. F urtherm ore, the calculation tim e w as relatively short. The expected project duration before acceleration w as 83 w eeks for a total cost o f $1,483,600. W e defined the norm al u n it cost o f each resource arbitrarily and set the unit cost o f acceleration 50% higher (see Table 1 in appendix).

9 A WAY TO DEAL WITH THE PROJECT CRASHING PROBLEM 47 I f w e also suppose th at any activity can be accelerated w ithout lim its and th at w e have at o u r disposal a budgetary increase o f up to 2 0 % w ith w hich to accelerate the project, w e can obtain a reduction in pro ject duration from 83 to 47 weeks at an additional cost o f $293,450. This is a 43.3% reduction o f project time for a cost increase o f 19.77%. The algorithm achieved this result by carrying out 113 iterations, which required about 3 seconds o f calculation time. T hese results indicate th at the algorithm is effective and could be used on a daily basis by professionals to accelerate large-scale projects involving relatively large num bers o f tasks. Variation de la durée vs coût g Û. = o La durée totale du projet Figure 3. Cost variation VERSUS project duration Conclusion In th is study, w e have presented the results obtained using a new tool developed to solve the problem o f finding the optim al trad e-o ff betw een project duration and cost in the context o f resource criticality. The tool we have developed applies principles o f tab u search to optim ize the process o f identifying project activities to be accelerated. U sing o u r pro ject acceleration algorithm on projects previously treated using other approaches, we dem onstrated that the tool identifies the same solutio n o r in som e cases a b etter solution and w ith a shorter calculation tim e. T his w ork thus proposes a new avenue to explore w ith m ore in-depth studies for the im provem ent o f pro ject m anagem ent.

10 48 Hamdjatou Kane, Gilbert Nkubili, Barthelemy Ateme-Nguema Appendix Duration, acceleration and associated resource costs of activities (corresponding to arrows in the CPM/PERT diagram) for the project Act. Path Pqte di ui Ck1 Ck2 Ck3 Ck4 ak1 ak2 ak3 ak4 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; (<M3) 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) Table 1

11 A WAY TO DEAL WITH THE PROJECT CRASHING PROBLEM 49 Table l(continued) Act. Path Pqte di Ui Ck1 Ck2 Ck3 Ck4 ak1 ak2 ak3 ak4 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 Hamdjatou Kane, Gilbert Nkubili, Barthelemy Ateme-Nguema Table (continued) Path Pqte di u Ck1 Ck2 Ck3 Ck4 (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) (80,83) (81,83) (82,84) 89; ,85) (83,86) (84,87) 123; (85,87) (86,112) (86,113) (87,111) (88,89) 128; >,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 l(continued) Act. Path Pqte di Ui Ck1 Ck2 Ck3 Ck4 ak1 ak2 ak3 ak4 170 (111,114) 132;161;162;166; (112,1140) 100;102;130;163; (113,114) 101;103;131;164; References A lban T. (2008), A ccélération des p ro jets et allocation des ressources: des enjeux de la gestion m ulti-projet. M ém oire de M aîtrise, U niversité du Q uébec en O utaouais. B octor F ayez F. (2005), L e pro b lèm e d'allocation et de nivellem ent des ressources. A SA C 2005 Conference, Toronto (O ntario), Canada. Bolduc M.C., Laporte G., Renaud J., Boctor F.F. (2010), A tabu search heuristic fo r the split delivery vehicle routing problem with production and dem and calendars. European Journal o f Operational Research, 202 (1): D odin B., E lim am A.A. (2008), Integration o f equipm ent plannin g a n d pro ject scheduling. European Journal o f Operational Research, 184(3): E vensm o J., K arlsen J. T. (2008), L o o kin g fo r the Source - W here D o C rash Costs Come From?. Cost E ngineering, 50(7). G erk J.E.V., Q assim R.Y. (2008), P ro ject A cceleration via A ctivity Crashing, O verlapping, a n d Substitution. IE E E T ransactions O n E ngineering M anagem ent, 55(4): G lover F.W. (1989), Tabu Search - P a rt I. O R SA Journal on C om puting, IN FO R M S: Institute for O perations R esearch 1: 190. G lover F.W. (1990), Tabu Search - P a rt II. O R SA Journal on C om puting, IN FO R M S: Institute for O perations R esearch 2: 4. Glover F.W. Laguna M. (1997), Tabu Search. Kluwer Academic Publishers, Boston. Kane H., A zondekon S.H. (2008), Un algorithm e p o u r la résolution d un pro b lèm e d accélération de p ro jet avec des ressources multiples. 7e Conférence Internationale de M odélisation et Sim ulation - M O SIM '08 - du 31 m ars au 2 avril 2008-Paris-France, M odélisation, Optim isation et Sim ulation des systemes: Communications, Coopération, et Coordination. Liberatore M.J., Pollack-Johnson B. (2006), E xtending pro ject tim e-cost analysis by rem oving precedence relationships and activitystream ing. In ternational Journal o f Project M anagem ent, 24(6): L iu J., C heng H., Shi X., X u J. (2010), A Tabu Search A lgorithm f o r F a st R estoration o f L arge A rea B reakdow n in D istribution System s. E nergy and P ow er E ngineering, 1-5.

14 52 Hamdjatou Kane, Gilbert Nkubili, Barthelemy Ateme-Nguema Liu S. (2003), F uzzy activity Times In critical P ath a n d P roject Crashing problem s. Cybernetics and Systems: A n International Journal, 34: R akotom alala H.L. (2002), L e com prom is durée/ressource en gestion de p ro je t dans un contexte de m ultiplicité des ressources : A nalyse et im pact su r la perfo rm a n ce de p ro jet. M ém oire de M aîtrise, U niversité du Q uébec en O utaouais. R oem er T.A., A hm adi R. (2004), C oncurrent C rashing a n d O verlapping in P ro d u ct D evelopm ent. O perations R esearch, IN FO R M S: Institute for O perations R esearch 52: Sw ink M. (2003), C om pleting p ro jects on-tim e: how p ro je c t acceleration affects new p ro d u c t developm ent. Journal o f E ngineering and T echnology M anagem ent, 20(4): X u K., Feng Z., Jun K. (2009), A Tabu Search algorithm fo r sch eduling jo b s w ith controllable p ro cessin g tim es on a single m achine to m eet duedates. Com puters & O perations R esearch, 37: W akas S., June L.W. (2009), C ost R eduction f o r the P ro ject C om pletion in Shortest Possible D uration by Stretching N oncritical Activities. A ustralian Journal o f Basic and A pplied Sciences, 3(4):