A New Uniform-based Resource Constrained Total Project Float Measure (U-RCTPF) Roni Levi. Research & Engineering, Haifa, Israel

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1 Management Studes, August 2014, Vol. 2, No. 8, do: / / D DAVID PUBLISHING A New Unform-based Resource Constraned Total Project Float Measure (U-RCTPF) Ron Lev Research & Engneerng, Hafa, Israel Resource schedulng has become one of the most mportant actvtes n project management research and practce n the last decades. Recently, as part of ths development, formal tools and technques were developed to tackle the more complex problem of resource allocaton or schedulng; that s, makng sure both tme constrants and resource constrants are met. In project schedulng, the concept of float and crtcalty plays a central role, however, the recent lterature does not offer a general and useful measure for crtcalty (flexblty) n resource constraned projects. The resource constraned project total float measure (RCTPF) s defned as the sum of the total floats of actvtes, where the total float for an actvty s defned as the dfference of ts latest and earlest start tmes. The RCTPF calculates the total float of each actvty n order to maxmze the total float of the project. However, not only the exstence of float or ts amount s mportant, but n many cases the dstrbuton of the total amount of the float wthn the actvtes s even more sgnfcant. The followng paper presents an extended measure to the exstng RCTPF measure to solve ths problem. The new measure, unform-based resource constraned total project float measure (U-RCTPF), s geared toward the unform dstrbuton of the project total float among the varous project actvtes. The schedulng objectve s to mnmze the varaton of the float among the actvtes. In ths approach, a resource-constraned project s characterzed by ts best schedule, where best means a schedule n whch the RCTPF s maxmal and the unformly measure s mnmal. Keywords: project management, schedulng, resource allocaton, resource constrant, float, flexblty Introducton Crtcal path has long been central to the analyss of non-resource constraned projects. Ths ssue becomes more crucal when resource constrants are ntroduced. Even n smple resource constraned projects, alternatve resource allocatons are often possble, resultng n a choce of schedules wth dentcal project duratons, but dfferent crtcal sequences. An actvty may be crtcal n one schedule, but may have consderable float (flexblty) n another. In such stuatons, an analyss of floats plays an mportant and crucal role, makng the development of new float measures a central ssue n project schedulng. The desrablty of addtonal float measures has been noted n revews of project schedulng lterature conducted by Wlls (1985) and Ragsdale (1989). West (1967) proposed the crtcal sequence as an extenson Ron Lev, Ph.D., assocate professor, Research & Engneerng, Hafa, Israel. Correspondence concernng ths artcle should be addressed to Ron Lev, P.O. Box 34056, , Hafa, Israel. Tel: Fax: , E-mal: ronlev@netvson.net.l.

2 534 A NEW UNIFORM-BASED RESOURCE CONSTRAINED TOTAL PROJECT of the crtcal path. Ths concept was employed by Bowers (1995) n the development of a set of heurstcs for determnng resource constraned float. Raz and Marshall (1996) explored a defnton of resource constraned float nvolvng the generaton of two dfferent schedules. Bowers (2000) proposed a float defnton for multple alternatve resource constraned schedules. However, the recent lterature does not offer a general and useful measure for crtcalty (flexblty) n resource constraned projects. Lev (2005) presented a new and nnovatve approach RCTPF (as the sum of the total floats of actvtes, where the total float for an actvty s defned as the dfference of ts latest and earlest start tmes) to solve the problem). Ths algorthm calculates the total float of each actvty n order to maxmze the total float of the project. The RCTPF s n essence a flexblty measure whch s geared toward enhancng the schedule robustness by maxmzng the project total float. The greater the RCTPF s, the better the soluton (the robustness) s. Although t sgnfcantly mproves the flexblty of the soluton, ths measure s stll, n some cases, not necessarly the best soluton from the project robustness pont of vew. The followng paper presents an extended measure to the exstng RCTPF measure to solve ths problem. The new measure U-RCTPF s geared toward the unform dstrbuton of the project total float among the varous project actvtes. Ths paper argues that not only the exstence of float or ts amount s mportant, but n many cases the dstrbuton of the total amount of the float wthn the actvtes s even more sgnfcant. Ths dstrbuton of the total amount of the float wthn the actvtes can be analyzed and evaluated from several ponts of vew, all of them geared toward mprovng and enhancng the project robustness. The nterested reader s referred to as a state-of-the-art handbook of schedulng models by Demeulemeester and Herroelen (2002). The paper s organzed as follows: Secton two presents the new resource constraned total project float (RCTPF) measure for resource constraned projects; secton three shows that theoretcally the optmal schedule searchng process can be formulated as a mxed nteger lnear programmng (MILP) problem, whch may be solved drectly for small-scale projects n reasonable tme; secton four presents TU-RCTPF; and the last secton lsts some ssues that call for further nvestgaton. Resource Constrant Total Float Measure In the modelng process of RCTPF, the followng resource constraned project-schedulng problem (RCPSP) s consdered: A sngle project conssts of N real actvtes {1, 2,, N} wth a non-preemptable duraton of D perods. The actvtes are nterrelated by precedence and resource constrants: Precedence constrants as known from tradtonal CPM-analyss force an actvty not to be started before all ts predecessors are fnshed. These are gven by relatons j, where j means that actvty j cannot start before actvty s completed. Furthermore, actvty = 0 ( = N + 1) s defned to be the unque dummy source (snk). Resource constrants arse as follows: In order to be processed, actvty requres R r unts of resource type r {1,, R} durng every perod of ts duraton. Snce resource r {1,, R} s only avalable wth the constant perod avalablty of R r unts for each perod, actvtes mght not be scheduled at ther earlest (network-feasble) start tme but later. Let T denote the project s make-span and let T + 1 denote the start tme of the unque dummy snk. Let T denote an upper bound on the project s make-span (T T ). The tradtonal RCPSP approach mnmzes the startng tme of the unque snk and thus the make-span of

3 A NEW UNIFORM-BASED RESOURCE CONSTRAINED TOTAL PROJECT 535 the project. In ths paper, wthout loss of generalty, the assumpton s that make-span T s the resource whch constraned mnmal make-span and fxed the poston of the unque dummy snk n perod T + 1. Let X, where X X X, denote the start tme of actvty, for {1,, N}. It can be noted that X X denotes the earlest (latest) startng tme of actvty. Because preempton s not allowed, the ordered set X = {X 1,, X N } defnes a schedule of the project. Let X s, where X Accordng to the appled notaton: s X, denote a zero-one decson varable: 1 f actvty s started n perod s Xs, {1,, N} (1) 0 otherwse X s Xs, X s 1, {1,, N} (2) s s Let PS = { j j, {1,, N}, j {1,, N}} denote the set of predecessor-successor relatons. A schedule s network feasble, f t satsfes the predecessor-successor relatons: X D X j, f j PS (3) For a network feasble schedule X, let A X t X D, t 1,..., T (workng) actvtes n perod t and let Utr rr A be the amount of resource r used n perod t. t denote the set of actve t, t 1,..., T, r 1,..., R (4) A network feasble schedule X s resource feasble, f t satsfes the resource constrants: U tr R, t 1,..., T r, r 1,..., R (5) The objectve of the presented approach s to fnd a resource feasble schedule, n whch the total float (the schedulng flexblty) s maxmal: N Maxmze RCTPF X X (6) 1 It s easy to see that the presented performance measure RCTPF s rregular, therefore, the usual modelng practces cannot be appled to formulate the model and a new approach s needed. The result of ths revson wll be a new MILP formulaton of RCPSP. A Mxed Integer Lnear Programmng Formulaton The presented MILP formulaton s based on the forbdden (resource constrant volatng) set concept. A forbdden set F of actvtes s dentfed such that: (1) All actvtes n the set may be executed concurrently, (2) the usage of some resource by these actvtes exceeds the resource avalablty, and (3) the set does not contan another forbdden set as a proper subset, for example, Bell and Park (1990). A resource conflct can be repared explctly by nsertng a network feasble precedence relaton between two forbdden set members, whch wll guarantee that not all members of forbdden set can be executed

4 536 A NEW UNIFORM-BASED RESOURCE CONSTRAINED TOTAL PROJECT concurrently. An nserted explct conflct reparng relaton (as ts sde effect) mght be able to repar one or more other conflcts mplctly, at the same tme. Let j denote that actvty j s a drect or ndrect successor of actvty. An j explct reparng relaton mght be replaced wth a p q relaton, where p and q j, p q j, and there s a forbdden set n whch p q s an explct reparng relaton. Let Pred() denote the set of mmedate (drect) predecessors of actvty and let author denote wth ExpRel(F) (ImpRel(F)) the set of mplct (explct) reparng relatons of forbdden set F. Consderng a smple RCPSP example wth sx actvtes, the actvtes are numbered one through sx (plus the dummy actvtes zero and seven). There are two resource types. Only one unt s avalable from every resource type. The mnmal resource feasble make-span s T = 9. The total number of network feasble schedules s R = 2,520. The total float of the project s 26, whch s an upper bound of the resource feasble total project float RCTPF. The presented earlest CPM schedule s not n feasble resource. There s over-utlzaton n perod 1(2). Fgure 1 and Table 1 llustrate the essence of the example problem. In Fgure 1, the actvtes are represented by bars, the network relatons by lnes. The unque dummy source (snk) s represented by the >(<) symbol. Let E = {0 1, 0 2, 2 3, 0 4, 1 5, 3 6, 4 7, 5 7, 6 7} denote the set of network relatons. R 1 Fgure 1. A smple project wth sx actvtes and two resources. Table 2 shows the forbdden sets and ther explct (mplct) reparng sets n the presented earlest CPM schedule. In ths schedule, every conflct s feasble. A feasble conflct may be vsble or hdden. It can be noted that a hdden conflct s nvsble n the earlest CPM schedule, but mght be vsble n a shfted schedule. In the earlest schedule, only the last conflct s vsble.

5 A NEW UNIFORM-BASED RESOURCE CONSTRAINED TOTAL PROJECT 537 Table 1 A Smple Project: Duratons, Earlest (Latest) Start Tmes, Immedate Predecessors, Resource Avalabltes D X X R 1 Pred() {0} {0} {2} {0} {1} {3} {4, 5, 6} Table Forbdden Sets and Ther Explct and Implct Reparng Relatons n the Earlest Schedule F Status ExpRel(F ) ImpRel(F ) 1 {4, 5} Feasble Hdden {4 5, 5 4} 2 {5, 6} Feasble Hdden {5 6, 6 5} 3 {4, 6} Feasble Hdden {4 6, 6 4} 4 {1, 3} Feasble Hdden {1 3,3 1} {1 2} 5 {1, 6} Feasble Hdden {1 6} {1 2,1 3, 5 6} 6 {1, 2} Feasble Vsble {1 2, 2 1} {3 1} The proposed RCTPF measures an rregular measure of performance, therefore the tradtonal vsble conflct orented approach must be replaced by a feasble conflct orented one. In other words, every feasble resource usage conflct has to be repared regardless of whether t s vsble or hdden. Let author denote wth Y the set of dfferent conflct reparng relatons and characterze a resource feasble soluton by the set nserted explct repars, whch s a subset of Y. In the smple problem: Y = {1 2, 1 3, 1 6, 2 1, 3 1, 4 5, 4 6, 5 4, 5 6, 6 4, 6 5} (7) The total number of the reparng relatons s Y = 11. Ths smple project has four feasble conflct free solutons (soluton sets), whch are shown n Fgure 2. In the soluton sets, every movable actvty can be shfted wthout affectng the resource feasblty. It s easy to realze that from manageral pont of vew, soluton 1 s the best schedule, because n ths soluton, the resource constraned total float (the schedulng flexblty) s maxmal. In soluton 4, every actvty s crtcal (unmovable), so an actvty delay wll delay the completon of the entre project. The relaton set, whch descrbes a soluton set, s a non-redundant subset of the unon of the orgnal network relatons and the addtonal conflct reparng relatons. RCTPF = RCTPF(Y) (8) Accordng to the results of the smple example, t s easy to realze that the proposed RCTPF s a functon of the conflct reparng relatons: Therefore, the tradtonal MILP descrpton of RCPSP wll be reformulated accordng to ths fact. To author s knowledge, ths s the frst tme that a MILP model has been proposed for RCPSP n functon of

6 538 A NEW UNIFORM-BASED RESOURCE CONSTRAINED TOTAL PROJECT conflct reparng relatons. In the proposed model, the total number of zero-one varables and the formulaton s based on well-known bg-m constrants (Schmdt & Grossmann,1996). Soluton 1 Soluton 2 R 1 R R 1 R > < , 4 5, 5 6, RCTPF , 4 6, 5 4, RCTPF 8 Y1 1 Y2 2 Soluton 3 Soluton 4 R 1 R 1 1 3, 2 1, 4 5, 5 6, RCTPF 7 1 3, 2 1, 4 6, 5 4, RCTPF 0 Y3 3 Fgure 2. Resource feasble solutons. Y4 4 New Unform-based Resource Constraned Total Project Float Measure The U-RCTPF measures both the avalable total float of the project actvtes and the dfferences among these total floats. In ths new measure (U-RCTPF), a resource-constraned project s characterzed by two complementary measures: the orgnal RCTPF and the unformalty measure of the feasble schedule. Accordng to ths measure, a resource-constraned project s characterzed by ts best schedule, where best means a schedule n whch the RCTPF s maxmal and the unformalty measure measure s mnmal. In cases n whch the RCTPF and the unformalty measure wll not reach ther maxmum and mnmum values smultaneously (n the same feasble schedule), a second best schedule wll be the one preferred by the project manger accordng to the nature of the actvtes. The RCPSP s used as a bass for modelng the U-RCTPF. The current objectve s to fnd the U-RCTPF, whch s the RCTPF wth the project total float unformly dstrbuted among the actvtes that have a total float. Let M denote the number of resource feasble schedules that have total float (RCTPF 0) and let K denote the total number of the actvtes n the resource feasble schedule j, j {1,, M} that have float. Wthout

7 A NEW UNIFORM-BASED RESOURCE CONSTRAINED TOTAL PROJECT 539 loss of generalty, the assumpton s that all the resource feasble schedules M have K actvtes. Let TF, {1,, K} denote the total float of actvty and let TFav denote the average float of the actvtes: (σ). TFav = k TF \ K k {1,, K} (9) 1 Let TFvar denote the total dfferences of the actvtes float from the average schedule float TFav: TFvar = k TF TFav k {1,, K} (10) k 1 Naturally, the average schedule float TFav can be calculated as the varance (σ2) or the standard devaton Let V sj defne the unformalty measure of the feasble schedule j: V sj = TFvar,j/RCTPF j j {1,, M} (11) It can be noted that V sj = 0 means that the total schedule float s dstrbuted equally among the actvtes havng float. Havng calculated the unformalty measure Vs, each feasble schedule wll be characterze by two characterstcs: the RCTPF and the V sj. Now, the defnton of the new U-RCTPF measure s as follows: U-RCTPF = RCTPF (V sj ) (12) The best schedule s a schedule n whch the RCTPF s maxmal and the Vs s mnmal. However, n most cases, these two measures do not reach ther optmum at the same feasble schedule. For that reason, the second best schedule wll be the one preferred by the project manger accordng to the nature of the actvtes Conclusons Ths paper proposed an extenson of the new model of the RCTPF. The proposed new model, U- RCTPF, s geared towards provdng a unform-based float dstrbuton among the project actvtes. The proposed model can be formulated as a MILP problem, whch can be solved for small-scale problems n reasonable tme. The presented new resource conflcts orented MILP model, s based on the reformulaton of the tradtonal tme orented resource constraned project schedulng MILP model, and can be used as a new resource constraned project schedulng model on ts own rght. In the presented new model, the objectve functon can be replaced by any other objectve, whch can be descrbed as a functon of the earlest (latest) startng tme varables. An nterestng problem, for example, would be to develop a decson support system for the resource constraned hammock actvtes. An mplct enumeraton algorthm for medum sze problems s also recommended. References Bell, C. E., & Park, K. (1990). Solvng resource-constraned project schedulng problems by a search. Naval Research Logstcs, 37, Bowers, J. A. (1995). Crtcalty n resource constraned networks. Journal of Operatonal Research Socety, 46, Bowers, J. A. (2000). Interpretng float n resource constraned projects. Internatonal Journal of Project Management, 18,

8 540 A NEW UNIFORM-BASED RESOURCE CONSTRAINED TOTAL PROJECT Demeulemeester, E. L., & Herroelen, W. S. (2002). Project schedulng: A research handbook. Boston: Kluwer Academc Publshers. Lev, R. (2005). Crtcalty n resource constraned projects A new flexblty orented approach (Unversty of Pecs, Hungary). Ragsdale, C. (1989). The current state of network smulaton n project theory and practce. Omega, 17, Raz, T., & Marshall, B. (1996). Effect of resource constrants on float calculaton n project networks. Internatonal Journal of Project Management, 14, Schmdt, C. W., & Grossmann, I. E. (1996). A mxed nteger programmng model for stochastc schedulng n new product development. Computers and Chemcal Engneerng, 20, West, J. D. (1967). Heurstc model for schedulng large projects wth lmted resources. Management Scence, 13, Wlls, R. J. (1985). Crtcal path analyss and resource constraned schedulng Theory and practce. European Journal of Operatonal Research, 23,