NONLINEAR OPTIMIZATION FOR PROJECT SCHEDULING AND RESOURCE ALLOCATION UNDER UNCERTAINTY

Transcription

1 NONLINEAR OPTIMIZATION FOR PROJECT SCHEDULING AND RESOURCE ALLOCATION UNDER UNCERTAINTY A Dssertaton Presented to the Faculty of the Graduate School of Cornell Unversty In Partal Fulfllment of the Requrements for the Degree of Doctor of Phlosophy by Al Mahmoudoff August 2006

2 2006 Al Mahmoudoff

3 NONLINEAR OPTIMIZATION FOR PROJECT SCHEDULING AND RESOURCE ALLOCATION UNDER UNCERTAINTY Al Mahmoudoff, Ph. D. Cornell Unversty 2006 Project plannng and schedulng when there are both resource constrants and uncertanty n task duratons s an mportant and complex problem. There s a long hstory of work on determnstc resource-constraned project schedulng problems, but efforts drected at stochastc versons of that problem are fewer and more recent. Incorporatng the ablty to reallocate resources among tasks to change the characterstcs of ther duraton probablty dstrbutons adds another mportant dmenson to the problem, and enables ntegraton of project plannng and schedulng. Among the small number of prevous works on ths subject, there are two very dfferent perspectves. Golenko-Gnzburg and Gonk (1997, 1998) have created a smulaton-based approach that operates the project through tme and attempts to optmze locally regardng decsons on startng specfc tasks at specfc tmes. Turnqust and Nozck (2004) have formulated a nonlnear optmzaton model to plan resource allocatons and schedule decsons a pror. Ths has the advantage of takng a global perspectve on the project n makng resource allocaton decsons, but t s not adaptve to the experence wth earler tasks when makng later decsons n the same way that the smulaton approach s. Although the soluton to ther model produces a baselne schedule (.e., tmes when tasks are planned to start), the formulaton puts much greater emphass on resource allocaton decsons. The paper by Turnqust and Nozck (2004) descrbes the problem formulaton

4 as a nonlnear optmzaton. For small problem nstances (up to about 30 tasks), good solutons can be found usng standard nonlnear programmng packages(e.g., NPSOL). However, for larger problems, the standard packages often fal to fnd any soluton n a reasonable amount of computatonal tme. One major contrbuton of ths dssertaton s the development of a soluton method that can solve larger problem nstances effcently and relably. In ths dssertaton, we recommend usng the partally augmented Lagrangan (PAL) method to solve the suggested nonlnear optmzaton. The test problems consdered here nclude projects wth up to 90 tasks, and solutons to the 90-task problems take about 2 mnutes on a desktop PC. A second contrbuton of ths dssertaton s exploraton of nsghts that can be ganed through systematc varaton of the basc parameters of the model formulaton on a gven problem. These nsghts have both computatonal and manageral mplcatons for practcal applcaton of the model.

5 BIOGRAPHICAL SKETCH Al Mahmoudoff graduated from Sharf Unversty of Technology, Tehran, Iran, n He receved hs Master of Engneerng degree from the Johns Hopkns Unversty s College of Engneerng to then pursue hs Ph.D. at Cornell Unversty at the feld of Transportaton Systems Engneerng.

6 To My Parents, v

7 ACKNOWLEDGMENTS The most grattude goes to my parents for rasng me and teachng me the values that I now stand for. Thanks to them for formng my vson. Wthout ther sacrfces I could never have the opportunty to undertake ths study. Thanks to my sster, Nma, who devoted her encouragement, trust and love to me durng all years of my studyng. I feel extremely grateful for day to day support from Kabeh throughout the past fve years. She consstently encouraged me to fnsh my thess partcularly snce I moved to start my job. Not only she encouraged me, but also she helped me wrte my thess and I wll never forget all her support. Specal thanks to Professor Turnqust for all hs support, patence and nsghtful suggestons. Wthout hs help, ths thess would be nfeasble. Hs unquenchable curosty, patence and overly enthusasm for research are the most valuable lessons I have learned from hm. Hs msson to set hgh standards and provdng opportunty to fulfll them has had a deep mpresson on me and I owe hm lots of grattude for showng me new corners of lfe n academa. I wll never forget hs response "Let's try..." whenever I sad "No Progress! Besdes of beng an excellent supervsor, Mark was as close as a relatve and a good frend to me. I acknowledge the support of professors n my commttee, Dr. Lnda Nozck, Dr. Peter Jackson and Dr. Charles Van Loan who montored my work and took effort n readng and provdng me wth valuable comments on earler versons of ths thess. Next there are all the frends I've made, ther frendshp, drnkng or... helped to make my stay n Ithaca a really happy one: Pantea, Carmen, Yao, Veronca, Yashoda, Wlkns, Adam and Nngxong. Specal thanks to Pantea for takng care of v

8 all admnstratve work of my thess whle I am not Ithaca and thanks to Yashoda for lettng me use her computer remotely for almost sx months. Ths materal s based upon work supported, n part, by Natonal Scence Foundaton under Grant No v

9 TABLE OF CONTENTS 1 Introducton Context and Objectves A Nonlnear Optmzaton for Resource Allocaton n Projects Outlne of the Dssertaton Revew of Pror Work n Project Schedulng and Resource Allocaton under Uncertanty Project Schedulng under Uncertanty Tme-Cost-Resource Tradeoffs Summary Solvng the Nonlnear Optmzaton Model Methodology Partally Augmented Lagrangans Equalty Constrants Inequalty Constrants Partal Elmnaton of Constrants Solvng the Sub-problem Applyng the Condtonal Gradent Method to Problem Summary of Computatonal Procedure Computatonal Testng of the Algorthm Overvew of the Computatonal Testng General Problem Set-up Actvty-on-Node Format Specfyng Shfted Webull Duraton Dstrbutons Lower and Upper Bounds on Varables Resource Consumpton Intal Values for Lagrange Multplers and Penalty Coeffcents Product Development Project Computatonal Results Soluton Interpretaton GGG Project Computatonal Results Interpretaton of the Soluton task Project from PSPLIB Computatonal Results Soluton Interpretaton Larger Test Problems from PSPLIB Conclusons from the Testng Parametrc Senstvty Analyss Introducton Senstvty Analyss for the Due Date, T Senstvty Analyss for the Due Date, T, wth Constant Total Resource Avalablty Senstvty of the Soluton to Resource Interval Defnton...93 v

10 5.5 Evaluatng Accuracy n the Objectve Functon Calculaton...95 v

11 LIST OF FIGURES Fgure 1-1 Probablty dstrbuton of task completon tme for dfferent resource 4 multplers. Fgure 1-2 Resource consumpton over tme for a task where start tme s s = 1, 8 and whose duraton s d = 1 Fgure 4-1 Project graph for Xu, et al. network 52 Fgure 4-2 Project graph for the Golenko-Gnzburg and Gonk network 65 Fgure 4-3 Project graph for the PSPLIB problem 74 Fgure 5-1 Probablty of success for the project wth changes n allowable tme, 92 under two dfferent assumptons regardng resources avalable per perod. Fgure 5-2 Probablty densty functon of makespan based on smulaton. 96 Fgure 5-3 Sample cumulatve dstrbuton functon of makespan based on 97 smulaton. Fgure C-1 Project graph for the 60.1 problem 113 Fgure C-2 Project graph for the 60.2 problem 120 x

12 LIST OF TABLES Table 4-1 Input data for product development example, part Table 4-2 Input data for product development example, part Table 4-3 Table 4-4 Table 4-5 Table 4-6 Objectve functon and computaton tmes for soluton methods on product development project. Soluton for task start tmes, duratons and resultng probabltes of successful completon from PAL algorthm for product development example. Resource multplers and resource constrants n the PAL algorthm soluton for product development example. Soluton for task start tmes, duratons and resultng probabltes of successful completon from NPSOL algorthm for product development example Table 4-7 Resource multplers and resource constrants n the NPSOL algorthm soluton for product development example. 60 Table 4-8 Input data for the GGG project, part Table 4-9 Input data for the GGG project, part Table 4-10 Objectve functon and computaton tmes for soluton methods 70 on product development project. Table 4-11 Soluton for task start tmes, duratons and resultng probabltes of successful completon from PAL algorthm for GGG example. 71 Table 4-12 Resource multplers and resource constrants n the PAL 72 algorthm soluton for GGG example. Table 4-13 Input Data for 30-node problem from PSPLIB, part Table 4-14 Input Data for 30-node problem from PSPLIB, part Table 4-15 Computatonal results for 30-node problem from PSPLIB 78 Table 4-16 Table 4-17 Table 4-18 Table 4-19 Table 5-1 PAL algorthm results for task start tmes, duratons and success probabltes n the 30-node test problem from PSPLIB. PAL algorthm results for resource multplers and Lagrange multplers n the 30-node problem from PSPLIB. NPSOL result for task start tmes, duratons and success probabltes for 30-node problem from PSPLIB NPSOL result for resource multplers and Lagrange multplers n the 30-node problem from PSPLIB. Probablty of success as T changes, wth resource avalablty constant n each nterval x

13 Table 5-2 Summary of experments wth constant total avalable resources. 91 Table 5-3 Effects on computed probablty of success for the project as 94 resource ntervals are adjusted. Table C-1 Input for test problem 60.1, part Table C-2 Input for test problem 60.1, part Table C-3 Results for test problem 60.1 from PSPLIB, part Table C-4 Results for test problem 60.1 from PSPLIB, part Table C-5 Input Data for test problem 60.2 from PSPLIB, part Table C-6 Input Data for test problem 60.2 from PSPLIB, part Table C-7 Results for test problem 60.2 from PSPLIB, part Table C-8 Results for test problem 60.2 from PSPLIB, part Table C-9 Precedence structure for 90-task problem from PSPLIB. 127 Table C-10 Input data for 90-task problem from PSPLIB, part Table C-11 Input data for 90-task problem from PSPLIB, part Table C-12 Input data for 90-task problem from PSPLIB, part Table C-13 Input data for 90-task problem from PSPLIB, part Table C-14 Computatonal results for 90-task test problem from PSPLIB, part 1. Table C-15 Results for 90-task problem from PSPLIB, part Table C-16 Results for 90-task problem from PSPLIB, part Table C-17 Results for 90-task problem from PSPLIB, part x

14 1. INTRODUCTION 1.1 Context and Objectves A project s a collecton of tasks whose completon s necessary to fulfll some well-defned objectve. There are typcally precedence constrants, requrng that some tasks are completed before others can begn, and the tasks generally requre specfed amounts of varous types of resources. When resources are lmted, the schedulng of the tasks requres careful coordnaton. A compact mathematcal formulaton of the resource-constraned project schedulng problem (RCPSP) can be wrtten qute easly, but exact soluton of that problem can be qute dffcult. The RCPSP has been studed extensvely snce the late 1960 s, leadng to a wde varety of algorthms for both exact and approxmate solutons. The vast majorty of the lterature on resource-constraned project schedulng s focused on a model formulaton that makes four mportant assumptons: (1) the objectve of nterest s the mnmzaton of the total tme requred to complete all tasks (often termed makespan); (2) tasks have known duratons that can be expressed as nteger multples of some basc tme perod; (3) tasks are not dvsble (.e., there s no preempton -- once started, a task cannot be stopped and then restarted at a later tme); and (4) the tasks have fxed and known resource requrements that are constant over the task s duraton. Varatons on the basc formulaton can accommodate changes n objectve (e.g., mnmzng the net present value of project costs), cases where tasks may be preempted, or cases where tasks may be performed n one of a few dfferent modes (.e., usng dfferent combnatons of resources, potentally wth dfferent resultng duraton for the task). An mportant mplcaton of assumpton (2) s that 1

15 tme s treated dscretely, and the resultng mathematcal formulaton centers on bnary varables X j, defned to be 1 f task starts (or ends) n perod j, and 0 otherwse. Ths dssertaton s based on takng a dfferent vew of project schedulng, bult around a formulaton that makes several dfferent basc assumptons: (1) tasks have uncertan duratons and the tme requred for completon can be descrbed by a realvalued random varable; (2) allocatons of resources to tasks can be changed n a contnuous (as opposed to dscrete) way, and changes n those resource allocatons result n changes to the probablty dstrbuton of task duraton; (3) resource avalablty s measured over some set of predefned ntervals that correspond to natural work perods (e.g., days, weeks or months) and s thus represented as an area (e.g., person-days of avalablty wthn a gven week); and (4) the objectve s to allocate avalable resources among tasks n a way that maxmzes the probablty of successful project completon wthn an allowable tme wndow. Ths alternatve vew of project schedulng was frst proposed by Turnqust and Nozck (2004). Although the soluton to ther model produces a baselne schedule (.e., tmes when tasks are planned to start), the formulaton puts much greater emphass on resource allocaton decsons.e., how should avalable resources be allocated among tasks? Thus, ther model produces a fundamentally dfferent type of result than the tradtonal RCPSP. The paper by Turnqust and Nozck (2004) descrbes the problem formulaton as a nonlnear optmzaton. For small problem nstances (up to about 30 tasks), good solutons can be found usng standard nonlnear programmng packages(e.g., NPSOL). However, for larger problems, the standard packages often fal to fnd any soluton n a reasonable amount of computatonal tme. One major contrbuton of ths dssertaton s the development of a soluton method that can solve larger problem nstances effcently and relably. The test problems consdered here nclude projects 2

16 wth up to 90 tasks, and solutons to the 90-task problems take about 2 mnutes on a desktop PC. A second contrbuton of ths dssertaton s exploraton of nsghts that can be ganed through systematc varaton of the basc parameters of the model formulaton on a gven problem. These nsghts have both computatonal and manageral mplcatons for practcal applcaton of the model. Secton 1.2 descrbes the nonlnear optmzaton model for project resource allocaton under uncertanty n task duratons and resource requrements, defnng the doman for ths dssertaton. Secton 1.3 then summarzes the structure of the remander of the dssertaton. 1.2 A Nonlnear Optmzaton for Resource Allocaton n Projects Turnqust and Nozck (2004) defne a nonlnear optmzaton model of project plannng and schedulng that s bult on four core deas: 1. The key uncertanty s the tme requrement to successfully complete a task. Because tasks are assumed to requre resources at a constant rate as long as they are actve, ths uncertanty n duraton also drves uncertanty n resource requrements, and hence cost. If a fxed length of tme, t, s consdered, then F () t defnes a probablty that task s completed successfully wthn that amount of tme. 2. The probablty densty functon for the duraton of task, f ( t ), has parameters that can be affected by changes n resource allocaton to the task. As llustrated n Fgure 1-1, an ncrease n resources allocated to a task shfts the probablty dstrbuton to the left and reduces ts varance, whle a reducton n resources has the opposte effect. 3

17 0.12 Prob. Densty Base Reduce 25% Increase 25% Tme Perods Fgure 1-1: Probablty dstrbuton of task completon tme for dfferent resource multplers. 3. If an allowable duraton, T, s specfed for the project, a plan can be determned (.e., the antcpated start tmes for tasks and the allocaton of avalable resources to the tasks) so that the probablty of successful completon of the project wthn the allowable tme s maxmzed. By then varyng T, and resolvng the optmzaton problem, we can trace out the tradeoff curve of probablty of successful completon vs. T. Assocated wth each pont on ths curve s some allocaton of avalable resources that allows us to reach the maxmum probablty of successful completon at the gven allowable duraton. 4. Ths type of tradeoff s best done usng contnuous tme rather than dscrete tme ntervals. Thus, we need a representaton of resource use by each task that s defned over contnuous tme and that s senstve to decsons on adjustng allocaton of resources to tasks. 4

18 The plannng perspectve taken by ths model s qute dfferent from the operatng perspectve adopted n smulaton-based studes of stochastc project schedulng (to be dscussed more fully n Chapter 2). Ths model seeks to provde support to the project manager n creatng a plan n the face of uncertanty, and n understandng how the avalable resources are lkely to be allocated. It can also be used to assess the lkely effects of changes n the overall level of avalable resources for the project. It s not desgned to gude operatonal real tme decsons on how to assgn resources at specfc tmes durng the executon of the project. The model of uncertan task duraton s a shfted Webull dstrbuton (sometmes called the three-parameter Webull). Ths dstrbuton s characterzed wth 0 parameters: d (mnmum possble task allowable duraton); α, a scale parameter; and β, a shape parameter. The cumulatve dstrbuton functon for completon of task wthn an avalable duraton d s: 0 d d β ( ) α 0 F( d ) = 1 e d > d 0; α, β > 0 (1.1) The second key dea n ths model, that the dstrbuton of task duraton can be changed by resource reallocaton, s mplemented by makng the scale parameter α a functon of resource allocaton. The form of ths functon used by Turnqust and Nozck (2004) s: α = α c c > 0; ε 0 (1.2) 0 ε where c s a resource multpler for task. Values of c > 1 ndcate allocaton above normal level and values of c < 1 ndcate reducton below normal level. represents the nomnal scale factor for task ; that s, the value at a resource 0 α 5

19 multpler of 1. The parameter ε s the elastcty of the scale parameter wth respect to changes n the resource multpler; n other words, the percentage change n α that results from a one percent change n c. Combnng equatons (1.1) and (1.2) results n an expresson for F( d, c) that 0 0 depends on four basc nput parameters for each task : α, ε, d, β 0 d d β ( ) 0 ε α c 0 0 F( d, c ) = 1 e d > d 0; c, α, β > 0 ε 0 (1.3) The overall probablty of success for the project wll be some functon of the collecton of F( d, c ) values for all tasks n the project. Turnqust and Nozck (2004) denote ths functon as Z(F) where F represents the set of all F( d, c ) values. In the project schedulng problem the objectve s to determne startng tmes of all tasks ( s ), resource multplers ( c ) and allowable duratons ( d ) that maxmzes Z(F), subject to constrants on resource avalablty, overall duraton of the project (T) and precedence requrements among the tasks. All varables are assumed to be contnuous. The precedence constrants are of the form: sn T (1.4) sj s + d (1.5) Here s N s the startng tme of the last task (dummy) representng project completon and T s a parameter defnng the allowable duraton for the project. For each task, resource consumpton at each pont of tme s defned as rk ( t ) : r () t k ca / d f s t s + d = 0 otherwse k (1.6) 6

20 where A k s the nomnal total requrement for resource k by task (.e., total person-days or dollars, etc.) Turnqust and Nozck (2004) suggest that equaton (6) be replaced by a nonlnear smooth functon. Ths removes the task start tme, s, and duraton d, from condtonng statements on the form of the functon. The functon that Turnqust and Nozck (2004) suggest s: t s t s t s d t s d wd wd wd wd ca k e e e e rk () t = t s 2 t s t s d t s d d wd wd wd wd e + e e + e (1.7) where w s a constant. As we see n Fgure 1-2, ths functon s approxmately ca k equal to n the range s t s + d and zero elsewhere. The begnnng and end d of the task have ramp up and ramp down perods whose length s determned by the constant w. Renewable resources are avalable n lmted amounts over defned perods (and are then renewed n the followng tme perod). If we ndex these perods by m, th and defne τ as the begnnng pont of the m nterval for resource k, the resource mk constrant can be wrtten as follows: N τ m+ 1, k rk ( t) dt Rmk k = 1,..., K; m= 1,..., Mk (1.8) = 1 t= τ mk 7

21 1.2 Fracton of Nomnal Rate Tme Fgure 1-2: Resource consumpton over tme for a task where start tme s s = 1, and whose duraton s d = 1. For the r ( t ) functon suggested by Turnqust and Nozck (2004), the ntegral k can be wrtten n closed form as follows: τ m 1, k v1 v2 ( e + 1)( e + 1) 3 4 ( e + )( e + ) + ca kw rk () t dt = ln (1.9) v v mk τ where 2( τ m + 1, k s) 1 = v2 wd v v 3 2( τ mk s ) = wd v 4 2( τ mk s d ) = wd 2( τ m + 1, k s d) = wd Ths allows the resource constrants n the problem to be wrtten n a straghtforward, although nonlnear, way. Dependng on the mnmum allowable duraton, lower and upper bounds for startng tmes can be constructed (the same dea as earlest start tme and latest fnsh tme n 8

22 determnstc schedulng problems). Also based on computatonal experence, there should be lower and upper bounds for resource multplers denoted by c up, and c low,. In summary, the formulaton of the problem s to choose s, d and c for all tasks,, so as to: Maxmze Z(F) Subject to: (1.3)-(1.5), (1.7), (1.9) and s s s low, up, c c c low, up, d d 0 problem (P). Throughout ths dssertaton, ths optmzaton problem wll be referred to as One convenent form for Z(F) s: N 1 = Z ( F) F ( d ) (1.10) = 1 Ths expresses the probablty of success for the entre project wthn tme T as the product of the success probabltes for all the ndvdual tasks wthn ther allowable duratons (determned wthn the soluton). If all the tasks do complete successfully wthn ther determned allowable duratons, then the project wll certanly be completed successfully wthn tme T. However, t s also possble for some task to requre a lttle more than ts allowed tme wthout jeopardzng the project f some other task that s n sequence wth t fnshes n less than ts allowed tme, and ths case s not consdered n (1.10). Thus equaton (1.10) represents a lower bound on the probablty of successful completon of the project. The real probablty 9

23 of fnshng the project wthn tme T requres evaluaton of a convoluton whose structure depends on both the precedence structure n the project and the avalablty of resources over tme. Evaluaton of ths probablty s very complex. For cases where resources are unlmted and ndvdual task duraton dstrbutons are known, there are several algorthmc methods for ether approxmatng the dstrbuton of completon tme for a project, or computng bounds on t. Ludwg, et al. (2001) provde a thorough computatonal evaluaton of several of these methods. Guo, et al. (2001) provde a very effectve method based only on the means and varances of the task duratons, usng Clark s approxmaton (Clark, 1961) for maxma of sets of Normally dstrbuted random varables. However, when resource lmtatons requre that some tasks be postponed beyond the pont where they become precedence-feasble, these technques do not work well. When resource constrants are present, there does not appear to be any avalable method of drectly evaluatng Z(F) n problem (P) short of smulaton. In the soluton of problem (P), the functon Z(F) s lkely to requre evaluaton many tmes, and performng many replcates of a smulaton each tme s computatonally prohbtve. To avod ths complcaton, we wll use (1.10) as a conservatve approxmaton of the desred probablty. The verson of problem (P) wth (1.10) as the expresson for Z(F) wll be referred to as problem (P1). 1.3 Outlne of the Dssertaton Chapter 2 dscusses relevant pror lterature on resource-constraned schedulng, efforts to nclude uncertanty n project schedulng problems, and approaches to makng task duratons senstve to resource allocaton. Chapter 3 then descrbes an algorthm for solvng problem (P1). Experments to test the algorthm are descrbed n Chapter 4. Chapter 5 focuses on parametrc analyss n a sngle example 10

24 problem to llustrate how changng the underlyng problem parameters affects the nature of the soluton. These results show how the solutons to problem (P1) can help project managers make effectve plans for allocatng resources under uncertanty. Chapter 6 then provdes conclusons and drectons for contnung work. 11

25 2. REVIEW OF PRIOR WORK IN PROJECT SCHEDULING AND RESOURCE ALLOCATION UNDER UNCERTAINTY Ths dssertaton develops a soluton procedure for the nonlnear optmzaton dentfed as problem (P1) n Chapter 1, and explores the mplcatons of that model for effectve management of projects under uncertanty, ncludng both schedulng and resource allocaton ssues. Thus, the revew of pror work touches on three dfferent areas work on project schedulng under uncertanty, work on tme-cost-resource tradeoffs n adjustng project schedules, and soluton approaches for nonlnear optmzaton problems. Ths chapter focuses on pror work n the frst two areas, related to project schedulng. Relevant references for the algorthmc development are contaned n the dscusson of the soluton procedure n Chapter Project Schedulng under Uncertanty The resource-constraned project schedulng problem (RCPSP) s an NP-hard optmzaton problem (Blazewcz, et al., 1983) that has been studed very extensvely. For example, the revew by Brucker, et al. (1999) lsts more than 200 papers. A recent revew by Kolsch and Hartmann (2005) focusng only on the latest developments n heurstc soluton methods ncludes 33 papers publshed snce Other mportant revews and comparsons of algorthms for the RCPSP nclude Herreolen et al. (1998), Hartmann and Kolsch (2000), Kolsch and Padman (2001), and Neumann et al. (2002). The vast majorty of ths lterature focuses on determnstc schedulng, and wll not be revewed here. Recognton of uncertanty n task duratons entered schedulng efforts qute a long tme ago (.e., PERT n the late 1950 s), but untl 12

26 relatvely recently there was lttle lterature on resource-constraned project schedulng under uncertanty. A recent revew of work n ths area (Herroelen and Leus, 2005), however, gves ndcaton of growng nterest. Herroelen and Leus (2005) dstngush four dfferent approaches to project schedulng under uncertanty that have appeared to date: reactve schedulng, stochastc project schedulng, fuzzy project schedulng, and robust (proactve) schedulng. They also note recent work on senstvty analyss n the machne schedulng lterature, and ndcate that extenson of that work nto project schedulng offers sgnfcant future research potental. Among the four approaches dentfed n the exstng lterature, reactve schedulng does not try to cope wth uncertanty a pror, but rather reacts to the occurrence of schedule dsruptons by tryng to repar an orgnal schedule after the dsrupton occurs. Schedules (ncludng the ntal baselne schedule and any modfcatons constructed later) are bult usng determnstc assumptons. A varety of reschedulng methods and objectves can be consdered. For example, El Sakkout and Wallace (2000) develop a procedure wth a mnmum perturbaton objectve that mnmzes the sum of the (weghted) absolute dfferences between the start tmes of tasks n the repared schedule and ther orgnally scheduled start tmes. Calhoun et al. (2002) focus on mnmzng the number of tasks whose start tmes are changed. Stochastc project schedulng ams at creatng a schedulng polcy that can be mplemented to make decsons as the project proceeds. For example, Iglemund and Radermacher (1983) and Möhrng, et al. (1984, 1985) defne a polcy Π as a mechansm for makng decsons at the decson ponts, t = 0 and at completon of tasks. A decson at tme t s to start some subset of precedence-feasble and resourcefeasble tasks, explotng only nformaton that has become avalable up to tme t. In general, the objectve n creatng a schedulng polcy s to mnmze the expected 13

27 duraton (makespan) of the project. Because the focus of stochastc project schedulng s to create a polcy for makng decsons as the project unfolds, no baselne schedule s created. Stochastc project schedulng represents an operatng perspectve, rather than a plannng perspectve. To estmate expected makespan correctly, t s mportant that a polcy be nonantcpatve (.e., uses only nformaton avalable at the tme at whch a decson s made). For example, Tsa and Gemml (1998) suggest a tabu search procedure for estmatng the expected makespan n stochastc resource-constraned project schedulng. Duratons of the tasks are assumed to follow a beta dstrbuton. To compute the expected makespan, they sample task duratons from the beta dstrbuton and then apply the suggested tabu search to the drawn task duratons. At the end, the expected makespan s estmated as the mean of all the computed makespans. Ths method volates the requrement of beng non-antcpatve because n each schedulng sample, t s assumed that the duratons of all tasks are known at the begnnng of the schedulng process. As a result, the estmated makespan n each sample of the project s a lower bound on the actual makespan for that nstance of task duratons, and the overall expected makespan wll be underestmated (Fernandez, et al., 1996). Möhrng and Stork (2000) focus on classes of polces that they term prorty polces, pre-selectve polces, and lnear pre-selectve polces. Prorty polces are polces that order all tasks accordng to a prorty lst and at every decson pont, start as many tasks as possble n the order of that lst. Pre-selectve polces defne, for each possble resource conflct, a pre-selected task that s postponed f the correspondng resource conflct happens wthn the executon of the project. Lnear pre-selectve polces are a subset of pre-selectve strateges that uses a prorty scheme to determne the task that wll be postponed. Pre-selectve polces are conceptually appealng, but the computatonal requrements for fndng optmal polces are qute 14

28 severe. Lmtng the search to lnear pre-selectve polces can reduce the computatonal burden somewhat, but ths lne of work remans more theoretcal than practce-orented. Golenko-Gnzburg and Gonk (1997) ntroduce an optmzaton-smulaton technque for stochastc project schedulng. Ther technque smulates the operaton of the project, wth task duratons beng sampled from gven probablty dstrbutons. At decson ponts n the smulaton, they solve an optmzaton problem to determne what task(s) to start next, wth the objectve of mnmzng the total expected project duraton, but assumng that all tasks that are stll n the future relatve to the current smulated tme have stll unknown duratons. They assume that all resources are renewable resources and that resources have constant avalablty throughout the project tme horzon. At each decson pont, f all precedence-feasble tasks can be suppled by avalable resources, then they all begn. But n case of a resource conflct, each task s assgned a prorty level that s based on ts contrbuton to the expected duraton of the project. For each task, contrbuton to the expected duraton of the project s the product of ts average duraton and the probablty of the task beng on the crtcal path n the course of project s realzaton. Ths probablty s computed va an nner smulaton, operated separately from the outer smulaton of the entre project s progress. Usng the prorty levels and resource requrements of all tasks ready to be started, a multdmensonal knapsack problem s solved to dentfy the tasks to be scheduled. In the knapsack problem, the goal s to dentfy a combnaton of tasks that most contrbute to reducng expected project duraton when schedulng them at the same tme does not volate the resource constrants. Ths knapsack problem can be 15

29 solved wth ether exact or heurstc methods, and the soluton to the knapsack problem determnes the task(s) to be scheduled at that pont n tme. Golenko-Gnzburg and Gonk (1998) follow ther prevous work wth a new assumpton that the allocated resources to tasks are also varable. They assume that assgnng more or fewer resources to any task results n changng the densty functon of the task s duraton. To mplement ths assumpton, they defne a speed of operatng for each task. Speed of operatng for each task s a random varable whose mean s a lnear functon of resources assgned to the task (between a mnmum and maxmum value). As n ther earler work (Golenko-Gnzburg and Gonk, 1997), the goal s to mnmze the expected project duraton but both startng tmes and resource allocatons are varable. The same basc structure of ther 1997 paper a smulaton of project operaton wth an optmzaton embedded n t to choose among tasks competng for resources s also used n the 1998 paper. To handle the extenson to varable resource allocatons, they offer several methods. The most reasonable method of these s to use the average of mnmum and maxmum values of resource requrements for the nner smulaton to compute task prortes. Havng computed prorty levels for competng tasks, a nonlnear knapsack allocaton problem s solved (at each decson pont) to dentfy the tasks to be scheduled among competng tasks. In the knapsack problem, the goal s to dentfy a combnaton of tasks that most contrbute to reducng expected project duraton subject to resource constrants and bounds on resource allocatons. They prove that ths problem s NP-complete. They develop two algorthms to tackle ths complex problem. Frst, a lookover algorthm sngles out feasble solutons and offers a precse soluton to the problem. Second, a heurstc algorthm offers an approxmate soluton wth less computatonal tme relatve to the lookover algorthm. In 16

30 summary, at each decson pont the soluton to the knapsack problem, solved by one of these methods, determnes the task(s) to be scheduled n that pont and the resources allocated to them. In both papers (Golenko-Gnzburg and Gonk, 1997; 1998), there s no baselne schedule. Each smulaton operates the project through ts duraton and makes schedulng decsons as they arse. The outcome of the smulaton s a sample realzaton of the project, and produces samples from the start tme dstrbutons for each task, as well as a sample of project duraton. By repeatng the smulaton many tmes, hstograms of the task start tmes and project duraton can be constructed. In a further extenson, Golenko-Gnzburg et al. (2003) offer a heurstc based on a dfferent representaton of a project as an alternatve actvty network. Ths representaton s an extenson of GERT (Prtsker and Happ, 1966) and VERT (Moeller and Dgman, 1981) models of decsons n network-based actvtes. It s smulaton based, and uses the core concepts developed by Golenko-Gnzburg and Gonk (1997, 1998), but adds a dmenson of uncertanty n the outcome of tasks, whch they refer to as an alternatve network representaton. Another avenue for approachng stochastc project schedulng s artfcal ntellgence. Knotts et al. (2003) present a soluton approach based on agent technology. They create one agent for each task n the project. It s the responsblty of the agent to acqure the resources requred for the task to whch t has been assgned. The defned agents have certan characterstcs and react to changes n the schedule n certan ways. In other words, agents are asked to schedule tasks n the course of smulaton. Ther result shows nterestng promse for applcaton of agent technology n large project schedulng networks. The thrd type of approach defned by Herroelen and Leus (2005) to project schedulng under uncertanty s fuzzy project schedulng. Advocates of the fuzzy 17

31 approach argue that unknown actvty duratons have to be estmated by human experts, often n a non-repettve settng or based on lmted hstorcal data, so that the core problem s mprecson n the estmates, rather than uncertanty n a probablstc sense. The fuzzy schedulng lterature then recommends usng fuzzy numbers to model task duratons, rather than probablty dstrbutons. The output of a fuzzy schedulng procedure s a fuzzy schedule, whch ndcates fuzzy startng and endng tmes for ndvdual tasks. Study of fuzzy models of the RCPSP was ntated by Hapke, et al. (1994) and Hapke and Slownsk (1996), who extended prorty rule-based schedulng methods orgnally developed for determnstc problems to deal wth fuzzy parameters. The book edted by Hapke and Slownsk (2000) collects several mportant contrbutons usng the fuzzy approach. Wang (2002, 2004) llustrates recent thought along ths lne, as appled to product development projects, and a comparson of the fuzzy approach to a probablty-based approach s offered by Zhang, et al. (2005). The fourth approach to schedulng under uncertanty s robust, or proactve, schedulng. The focus of ths approach s on creatng a robust baselne schedule, rather than on creatng a polcy that can be operated as the project unfolds and uncertantes are resolved. Thus, we can thnk of ths as a plannng approach, rather than an operatng approach. Much of the robust schedulng work s n the machne schedulng lterature, and nvolves creatng some type of tolerance to machne breakdowns n accomplshng the schedule. Drect extenson of much of the machne schedulng work to project schedulng s somewhat dffcult because the uncertantes n task duraton are generally not amenable to treatment usng the standard knd of analyss done for producton machnes (mean tme between falures, mean tme to repar, etc.). 18

32 Tavares, et al. (1998) do apply the core dea of robust machne schedulng n a project framework, however, by examnng the dea of expandng the baselne schedule to accommodate uncertan task duratons, and the tradeoffs between schedule rsk and cost. They suggest that the start tme of each task,, be set to s ( α ) = es + α( ls es ), where es and ls denote, respectvely, the earlest and latest start tmes for task, gven a project deadlne T. They refer to α as a float factor. The float factor model assumes unlmted resources n the constructon of a robust baselne schedule. A somewhat more sophstcated approach to allocatng tme n a baselne schedule s offered by Herroelen and Leus (2004), where a lnear programmng model s used to allocate avalable tme n a schedule to mnmze the expected cost of schedule delays from potental dsruptons. Lke the model proposed by Tavares, et al. (1998), ths procedure assumes unlmted resources. An extenson of the dea to the case where resources are lmted appears n Leus and Herroelen (2004), but the resultng problem s NP-complete and thus computaton for realstc problem nstances s problematc. The dea of allocatng avalable tme wthn an allowable project performance perod T among varous tasks s related to the noton of defnng an allowable duraton for each task n the model created by Turnqust and Nozck (2004). The papers by Tavares, et al. (1998), Herroelen and Leus (2004) and Leus and Herroelen (2004) all focus on a tradeoff between the probablty of dsrupton to specfc tasks and the cost of delayed start tmes on subsequent tasks. Ths s dfferent from a concern wth the probablty of completng the project successfully wthn the perod T, as expressed by Turnqust and Nozck (2004). 19

33 An alternatve way of thnkng about robust schedules s represented by the recent work of Vazr, et al. (2005). Lke Golenko-Gnsburg and Gonk (1998) and Turnqust and Nozck (2004), they are concerned wth allocatng resources to tasks to change the probablty dstrbutons of task duraton. The model uses a combnaton of smulaton, parallel prorty-rule schedulng and smulated annealng to create a soluton that specfes a planned allocaton of resources and resultng dstrbutons of start tmes for each task and a dstrbuton of makespan for the project (or set of concurrent projects) as a whole. Ths end result s smlar to what could be produced by smulatng the Golenko-Gnsburg and Gonk (1998) approach many tmes, but the focus s not on creatng a polcy for operatng the project as t unfolds, but on creatng a baselne plan for resource allocaton and a resultng dstrbuton of makespan. Thus, ths procedure can be consdered as a type of robust schedulng, rather than as a varaton on stochastc project schedulng. Problem (P1) defned n Chapter 1 s also a robust schedulng model. It shares several formulaton elements wth the model studed by Vazr, et al. (2005), but dffers n ts objectve functon and n ts focus on dentfyng a baselne schedule, rather than a probablty dstrbuton of makespan. Lke the models of Golenko- Gnsburg and Gonk (1998) and Vazr, et al. (2005), an mportant element of (P1) s the ablty to change allocatons of resources to tasks and affect ther duraton probablty dstrbutons. Ths dstngushes these three models from the other efforts n the robust schedulng category. The focus on resource allocaton creates a connecton to a second body of pror work that dealng wth tme-cost-resource tradeoffs n project schedulng, and ths work s revewed n the followng secton. 20

34 2.2 Tme-Cost-Resource Tradeoffs Tme-cost tradeoffs are an mportant part of the determnstc project schedulng lterature. When resources are assumed to be avalable n unlmted quanttes, the cost-crashng model of reducng task duraton has qute a long hstory, and s usually formulated as a lnear programmng problem (Ahuja, et al., ). If addtonal resources allocated to a specfc task have dmnshng margnal returns, the problem generally becomes nonlnear (Deckro and Hebert, 2003). When the resources are constraned, dscrete tradeoff possbltes between tme and resources can be addressed through a multmode formulaton of the RCPSP (Ahuja, et al., ). Tasks can be performed n one of several modes, and each mode s assocated wth a specfc duraton and resource requrements. When task duratons are uncertan, changes n resource allocaton to a task affect one or more characterstcs of the probablty dstrbuton for duraton, and the treatment of tme-cost-resource tradeoffs becomes more complex. Burt (1977) was one of the frst to consder how allocaton of resources to tasks mght affect the probablty dstrbutons for task duraton. He developed a model that would consder ether unform or symmetrc trangular dstrbutons for task duraton, n whch the allocaton of addtonal resources would shft the rght-hand end pont of the dstrbuton to the left. Hs model provded a smple mechansm for lookng at resource allocatons to tasks, where the effects of addtonal resources have a specfc effect on both the mean and varance of task duraton. Hs procedure was lmted to lookng at parallel sequences of seral tasks, and to allocatng a sngle non-renewable resource (e.g., overall budget), but t was an mportant begnnng n examnng the general problem. More recently, Gerchak (2000) studed a related problem, where allocatng more of a sngle lmted resource to an actvty can reduce the varablty of ts duraton wthout affectng ts mean duraton. Hs objectve was to construct analytc 21

35 results for allocatng a sngle resource (e.g., budget) to two actvtes n sequence so as to mnmze the varance n the sum of ther duratons. Özdamar and Alanya (2001) studed software development projects, and used a fuzzy duraton model to represent uncertanty n task duratons. They consder a specal resource (whch they term a consultant ) whose tme can be allocated to tasks n a way that shfts ther duraton functons toward smaller values. They convert the possble allocatons of consultant tme nto a dscrete set of modes for undertakng the tasks (wth a dfferent duraton functon for each mode), and then solve (heurstcally) a multple-mode schedulng problem wth a resource constrant reflectng the consultant s avalable tme. Another recent effort focused on dscrete modes of task performance s the work of Tereso, et al. (2004). They formulate a dynamc programmng model for allocatng resources to tasks to mnmze total expected costs. Drect soluton of ths problem s very demandng computatonally, so they also suggest some approxmatons that can smplfy the computatons. One of the nterestng aspects of ther work s that they treat the basc uncertanty n the tasks as beng n the requred work content. The allocaton of resources aganst the random work content then results n a random duraton that s affected by the resource allocaton. Ther focus on costs n the objectve brngs the three elements of tme, costs and resources together, but the underlyng assumpton s that resources are not lmted. Addtonal work n the area of tme-cost tradeoffs ncludes the efforts of Gutjahr, et al. (2000), Leu et al. (2001) and Laslo (2003). Gutjahr et al. (2000) study stochastc dscrete tme-cost tradeoffs usng an optmzaton model as an extenson of PERT. They develop an nteger program to optmze the selecton of measures for crashng tasks to acheve the optmal tradeoff between tme and cost. Leu, et al. (2001) use fuzzy numbers to represent uncertan task duratons, and pose a relatonshp 22

36 between the characterstcs of the fuzzy task duratons and the cost of the task. They then develop a genetc algorthm to construct the tradeoff between overall project cost and project duraton. Laslo (2003) uses a chance-constrant approach to study tmecost tradeoffs of a sngle actvty. However, none of these efforts drectly reflects resource constrants or the varyng effects of dfferent resources on actvty duratons. By far, the two most closely related efforts to what s done n ths dssertaton are those of Golenko-Gnsburg and Gonk (1998) and Vazr, et al. (2005). Recall that n problem (P1) changes n resources allocated to task affect the scale parameter of a shfted Webull dstrbuton: α = α c c > 0; ε 0 (2.1) 0 ε where c s a resource multpler for task and ε s an elastcty parameter. The resultng cumulatve dstrbuton functon for task completon (.e., the probablty of successful completon wthn a gven duraton d ) s then: 0 d d β ( ) 0 ε α c 0 0 F( d, c ) = 1 e d > d 0; c, α, β > 0 ε 0 (2.2) If we let μ and σ denote the mean and standard devaton, respectvely, of the duraton of task, ths dstrbuton mples that: μ = d ε 1 ( ) α Γ c + 1 β (2.3) σ = 0 ε ( α c ) 2 1 Γ + 1 Γ + 1 β β 2 (2.4) 23

37 24 where Γ( ) s the gamma functon. Because the shape parameter β remans constant as resources allocated to the task change, both the mean and standard devaton change lnearly wth α. Thus, we can also wrte (2-3) and (2-4) as follows: ( ) c d d ε μ μ + = (2.5) c ε σ σ = 0 (2.6) where 0 μ and 0 σ are nomnal mean and standard devaton values (.e., when c = 1), gven by: + Γ + = d β α μ (2.7) Γ + Γ = β β α σ (2.8) Vazr, et al. (2005) also use the concept of resource multplers wth elastctes, but n a slghtly dfferent way. They assume that task duratons are Normally dstrbuted, wth the mean and varance of duraton gven by: = K k k c λ k μ μ 0 (2.9) ( ) ( ) = K k k c γ k σ σ (2.10)

38 The subscrpt k denotes ndvdual resources n a resource set K. Thus, Vazr, et al. (2005) defne separate multplers for each resource on each task, and have separate elastctes for each resource. They also have separate elastctes for mean and varance of duraton wth respect to each resource. Apart from the dsaggregaton by resource type, the other dfference between (2-9) and (2-5) s that the model of Vazr, et al. (2005) has no mnmum duraton for the task ( d ). Although (2-10) s defned n terms of varance and (2-6) n terms of standard devaton, the elastctes can be converted by usng a factor of 2. Thus, n addton to dsaggregaton by resource type, the mportant defntonal dfference between problem (P1) and the Vazr model of changes n varablty of task duraton s whether or not dfferent elastctes are used for the mean and standard devaton. Obvously, the model used by Vazr, et al. (2005) allows somewhat greater specfcty n allocaton of resources among tasks, but t also has many more parameters that need to be estmated. Golenko-Gnsburg and Gonk (1998) use a dfferent type of model for the effects of resource allocaton on task duraton. They argue that what s uncertan about task progress s the speed at whch a known (and requred) amount of work wll be accomplshed. Speed (v ) s lnearly related to the resources of varous types (r k ) appled to the task, but the relatonshp has random coeffcents (a k ), so the resultng duraton (D ) to accomplsh a fxed amount of work (Q ) s a random varable dependent on r k : 0 Q Q D ( r 1,..., rk ) = = (2.11) v a r k k k They requre the allocated resources r k to be nteger multples of some base amount. Ths s clearly a dfferent approach to representng the effects of resource 25

39 allocaton on uncertan task duraton. It has the property of dmnshng margnal returns for addtonal resources on a task, whch (2-5) and (2-6) also have when ε < 1. It s relatvely easy to manpulate n a smulaton envronment (where Golenko- Gnsburg and Gonk (1998) used t), but less attractve for representaton of the probablty of completng a task n a gven tme (whch s what equaton (2-2) does ncely). Ths representaton shares wth the Vazr, et al. (2005) model the characterstc that dfferent resources may have dfferent margnal effects on task duraton, and that ndvdual resources can be adjusted ndependently for each task. The mplcaton of the model n problem (P1), by contrast, s that resources come n packages and that all resources on a task can be adjusted up or down, but the rato of unts of one to unts of another remans fxed. Ether assumpton may be most approprate n dfferent stuatons. 2.3 Summary Although the formulaton of problem (P1) s not a drect contrbuton of ths dssertaton, one of the man objectves here s to explore the mplcatons of that model for project management. Thus, t s mportant to place that formulaton n context, and the man purpose of ths Chapter s to establsh the context. Most of the work on project schedulng under uncertanty s relatvely recent. The revew by Herroelen and Leus (2005) defnes a useful classfcaton of the work nto four man approaches: reactve schedulng, stochastc project schedulng, fuzzy schedulng and robust schedulng. Problem (P1) s n the doman of robust schedulng, beng concerned prmarly wth the creaton of a baselne schedule to serve as an effectve plan for the project, rather than on creaton of a strategy for operatng durng the progress of the project. 26