# ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING

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3 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.

4 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.

5 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 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.

6 Fgure 2: Iso-Qualty Curves for Constructon Example total project cost 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), [2] Goetsch, D. L., Davs, S. B. Qualty management (5 th ed.). Upper Saddle Rver, NJ: Pearson Prentce-Hall, [3] Internatonal Organzaton for Standards, ISO 9000:2000, [4] Lndo Systems. Lngo Verson 9.0. Chcago, [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), [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), [7] Project Management Insttute. A Gude to the project management body of knowledge 3 rd ed. Newtown Square, PA, [8] Saaty, T. L. A Scalng method for prortes n herarchcal structures, Journal of Mathematcal Psychology, 1977, 15, [9] Saaty, T. L. The analytc herarchy process. Pttsburgh: RWS Publcatons, 1996.

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