ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 610-519-4390, matthew.lberatore@vllanova.edu Bruce Pollack-Johnson, Department of Mathematcal Scences, Vllanova Unversty, Vllanova, PA 19085, 610-519-6926, bruce.polack-johnson@vllanova.edu ABSTRACT Ths paper presents a mathematcal programmng model that allows qualty to be explctly consdered n project plannng and schedulng, whle addressng the tradeoffs between qualty, tme, and cost. Usng a constructon example we show how ths model can be used to generate qualty level curves to llustrate the trade-offs among tme, cost, and qualty. These level curves can then be used by project managers to make project schedulng decsons that explctly model and consder qualty as well as tme and cost, so that better and more approprate decsons can be made for a partcular stuaton. Keywords: project management, qualty management, project plannng, project schedulng, mathematcal programmng INTRODUCTION Project management requres achevng cost, schedule, and performance targets, whle provdng an outcome that satsfes the clent. A measure of the value of the project to the clent s the level of qualty assocated wth the completed project. It follows that mportant manageral decsons relate to the level of qualty acheved for each of the project s tasks, snce n toto the qualty of the tasks defnes the qualty of the project. The emphass n project plannng and schedulng has been on the relatonshp between tme and cost, wth lttle attenton drected to qualty. In most stuatons there are alternate approaches for completng each task, each havng ts own tme, cost, and qualty. The manageral queston of nterest relates to these choces, whch have profound mpacts on the project outcome. The purpose of ths paper s to present a modelng framework that allows qualty to be explctly consdered n project plannng and schedulng, whle addressng the tradeoffs between qualty, tme, and cost. Ths paper also offers some manageral nsghts that are derved from the modelng framework through mproved understandng of these choces and tradeoffs. PROJECT QUALITY Qualty can be defned as a dynamc state assocated wth products, servces, people, processes, and envronments that meet or exceed specfcatons [2]. The ISO 9000 defnton of qualty [3] has been adopted by the Project Management Body of Knowledge [7]. Accordng to the PMBOK, qualty must address both the management of the project and the product of the
project. Our nterest n qualty relates to the qualty plannng process, and addresses qualty at the task and overall project levels. Paqun et al. [5] contend that a method for assessng qualty must enable project managers to elucdate and structure the clent s needs and expectatons. We are nterested n measurng planned qualty of work for dfferent desgns of specfc actvtes usng such a method. We assume that there are choces for completng tasks that vary n qualty, tme, and cost. The suggested approach requres the dentfcaton of the qualty attrbutes that are relevant for the project. An example usng the Analytc Herarchy Process (AHP) [8] [9] to evaluate the qualty of a task opton along wth further dscusson of ths approach can be found n [6]. MODELING FRAMEWORK We begn by formulatng a model of the qualty of each ndvdual task as a functon of the tme and cost allocated to t. We assume that there could be dfferent enttes who could do the task and that each entty could do the job wth dfferent allocatons of tme and budget. Each entty would have ts own qualty functon n terms of tme and cost. If those qualty functons are graphed on the same tme/cost/qualty axes, then the overall qualty functon for the task that we are nterested n s the maxmum, or the upper envelope, of the ndvdual entty qualty graphs. We assume that ths overall qualty functon for a task has two basc propertes: Holdng tme constant, qualty s nondecreasng n cost. Thus f tme s fxed, we assume that spendng more money on the task wll ncrease (or at least not decrease) the qualty. Holdng cost constant, qualty s nondecreasng n tme. Thus f cost s fxed, we assume that spendng more tme on the task wll ncrease (or at least not decrease) the qualty. If we normalze qualty to be on a 0-100 scale, and lmt tme and cost to reasonable values for the task at hand, based on the two nondecreasng assumptons above, we would expect the graph to show the qualty beng lowest at the corner of the doman wth the smallest values of tme and cost and hghest n the opposte corner (the hghest values of tme and cost). For a fxed qualty, we would expect a tradtonal tme/cost tradeoff curve, whch s normally a decreasng convex curve (to mantan the same level of qualty, to reduce the tme, one has to pay more money, such as n standard project actvty crashng [1]). Ths suggests a basc hll shape rsng out of a plan, although we would only be nterested n a one-quarter wedge of the hll. A famlar mathematcal functonal form that has ths shape s the bvarate normal dstrbuton * n probablty. We propose usng ths functonal form for the overall qualty functon for each task. Our qualty functon s normalzed so that the maxmum tme ( µ t ) and cost ( µ c ) values consdered reasonably correspond to a qualty of 100. The standard devaton parameters ( σ t and σ c ) gve a measure of how slowly the qualty drops from the top of the hll compared to the maxmum values for tme and cost, respectvely. Thus, our resultng qualty functon s gven by * Ths verson of the bvarate normal dstrbuton assumes ndependence of the two random varables. We have chosen ths verson for a smpler model, but the dependent verson could also be used, wth one more parameter, correspondng to the correlaton between the varables.
Qtc (, ) = 100e t μ 2 2 t c μc ( σ ) ( σ ) + t c We have normalzed the ntal constant to 100 and elmnated the ½ n the exponent (whch means that each σ would be multpled by 2 to be nterpreted as the usual σ n the bvarate normal). If we hold ether varable constant, the margnal graph for the other wll be a bell curve (actually, a subset of a graph that s a constant multple of a normal dstrbuton curve). The upper envelope graph may not be smooth, but we are assumng that we can create a smooth functon that s a reasonable estmate of the upper envelope. In stuatons where n bds specfyng levels of qualty, tme, and cost ( qj, tj, c j) have been receved for a gven actvty, the four parameters of the bvarate normal functon can be determned usng nonlnear least squares estmaton. MODEL FORMULATION We start wth standard assumptons for modelng projects: that the project network has no cycles, that the start actvty (actvty 0, a dummy actvty) s the only actvty that s not an mmedate successor 1 of any actvty, and that the fnsh actvty (actvty N+1, also a dummy actvty) s the only actvty that has no successors..defne the followng parameters and varables: t = the duraton of actvty, for = 1,,N c = the cost of actvty, for = 1,,N q = the qualty of actvty, for = 1,,N S = the set of actvtes that are mmedate successors of actvty, for = 0,,N T UB = upper bound on the total project tme Q = lower bound on project qualty st = the scheduled start tme for actvty, for = 0,,N+1 t = lower bound on the duraton of actvty, for = 1,,N mn c mn = lower bound on the cost of actvty, for = 1,,N Relevant qualty measures could nvolve maxmzng average qualty, or maxmzng mnmum qualty, of the tasks. We select the latter, Q mn, as our qualty metrc, snce from a systems perspectve f the project s vewed as an ntegrated set of actvtes, the qualty of a project s only as hgh as ts weakest lnk. Q mn s defned as: mn q Q mn = (1) 1 N In our formulaton, we mnmze total project cost whle settng a lower bound on Q mn and an upper bound on total project tme. The nonlnear program s gven as equatons (2) (12): 1 It s common to use predecessors rather than successors for formulatons of ths type, but for ths example, the formulaton turns out to be much more concse and elegant usng successors.
Mnmze N c (2) = 1 Subject to: Qmn q, = 1,2,,... N (3) 2 2 { μt σt μt σt } Qt (, c) = 100 * exp [( t ) / )] [( c ) / )], = 1,2,,... N (4) Q mn 0 0 Q (5) st = (6) st st + t = 0,..., N, k S (7) st k N+ 1 T (8) UB st 0 = 1,..., N + 1 (9) tmn t, 1,2,..., μt = N (10) cmn c, 1,2,..., μc = N (11) q, t, c 0, = 1,2,..., N (12) Ths problem can be solved usng Lngo s global solver [4] and extends the standard cost tme tradeoff problem [1]. CONSTRUCTION EXAMPLE A general contractor plannng to start constructon of a new house has organzed the project nto actvtes as gven n Table 1. The correspondng project network dagram s shown n Fgure 1. She has receved bds for both duraton and cost from dfferent subcontractors. These bds were used to estmate the bvarate normal qualty functons for each actvty (Table 1). USING QUALITY LEVEL CURVE GRAPHS One way to evaluate the nteractve relaton among project tme, total cost, and qualty s to create level so-qualty graphs. Specfyng a value of Q, for dfferent total project tmes (upper lmts), usng our model we can then fnd the mnmum cost possble that fnshes the project wthn a gven tme and mantans a mnmum qualty of at least Q. A set of level so-qualty curves for the constructon example s shown n Fgure 2. The graph for a hgher qualty level les above and to the rght of that for a lower qualty level, although they can overlap n places for qualty levels that are very close together. There are several places where a level curve s horzontal. Ths could mean that to acheve a certan qualty level, a choce may need to be made at a longer project tme value that forces a soluton whch actually fnshes the project n strctly less than the upper lmt for the total tme, and therefore the same soluton s optmal at a shorter project tme lmt. Fgure 2 provdes a concse summary of the relatonshp among tme, cost, and qualty, and can be used to make wellnformed decsons about how to execute the project.
Table 1: Task, Immedate Successor, and Qualty Functon Informaton for Constructon Project IMMEDIATE TASK DESCRIPTION SUCCESSORS ( t, t, c, c) QUALITY PARAMETERS μ σ μ σ 0 START 1 dummy actvty 1 Excavate and Pour Footers 2 Not estmated one bd* 2 Pour Concrete Foundaton 3 Not estmated one bd* 3 Erect Rough Wall & Roof 4,5,6 (4, 1.79, 48.6, 42) 4 Install Sdng 11 (13, 19, 79.2, 99.4) 5 Install Plumbng 7 (3, 1.62, 26.6, 20.4) 6 Install Electrcal 7 (10.9, 12.8, 29.7, 77.9) 7 Install Wallboard 8,9 (5, 2.73, 16.8, 8.05) 8 Lay Floorng 10 (8.09, 7.18, 64, 67.5) 9 Do Interor Pantng 10 (4.57, 4.3, 16.8, 12.7) 10 Install Interor Fxtures 13 Not estmated one bd* 11 Install Gutters & Downspouts 12 (2, 12, 17.7, 18.9) 12 Do Gradng & Landscapng 13 (3.36, 2.4, 21.5, 12.3) 13 FINISH -- dummy actvty *for those actvtes havng one bd, the qualty, tme, and cost (q, t, c) estmates were used drectly n the analyss: actvty 1: (70, 3, 26.6); actvty 2: (70, 1, 7.2); actvty 10: (70, 3, 7.2) Fgure 1: Project Network Dagram for Constructon Example 4 11 12 0 1 2 3 5 8 13 7 10 6 9 CONCLUSIONS In standard project plannng and schedulng, qualty s acknowledged to be mportant at dfferent levels, but prevously has not been explctly modeled. In many stuatons there are alternate optons for accomplshng project actvtes, and these nvolve dfferng levels of tme, cost, and qualty. In such stuatons t makes sense to model the relatonshp between cost, tme, and qualty, and determne ther levels for each actvty that best acheves the project s objectves. We have presented a nonlnear programmng model for the qualty/tme/cost problem, and have shown how qualty level curves can be a very useful management tool n makng fnal project schedulng decsons that explctly model and ncorporate qualty.
Fgure 2: Iso-Qualty Curves for Constructon Example total project cost 250 240 230 220 210 200 190 65 66 67 68 69 70 180 20 21 22 23 24 25 26 27 project completon tme REFERENCES [1] Brucker, P., Drexl, A., Mohrng, R., Neumann, K., Pesch, E. Resource-constraned project schedulng: Notaton, classfcaton, models, and methods, European Journal of Operatonal Research, 1999, 112(1), 3-41. [2] Goetsch, D. L., Davs, S. B. Qualty management (5 th ed.). Upper Saddle Rver, NJ: Pearson Prentce-Hall, 2006. [3] Internatonal Organzaton for Standards, ISO 9000:2000, 2000. [4] Lndo Systems. Lngo Verson 9.0. Chcago, 2004. [5] Paqun, J. P., Coullard, J., Ferrand, D. J. Assessng and controllng the qualty of a project end product: The earned qualty method, IEEE Transactons on Engneerng Management, 2000, 47(10), 88-97. [6] Pollack-Johnson, B., Lberatore, M. Incorporatng Qualty Consderatons nto Project Tme/Cost Trade-off Analyss and Decson Makng, IEEE Transactons on Engneerng Management, 2006, 53(4), 534 542. [7] Project Management Insttute. A Gude to the project management body of knowledge 3 rd ed. Newtown Square, PA, 2004. [8] Saaty, T. L. A Scalng method for prortes n herarchcal structures, Journal of Mathematcal Psychology, 1977, 15, 234-81. [9] Saaty, T. L. The analytc herarchy process. Pttsburgh: RWS Publcatons, 1996.