Review Of Resource-Constrained Project Scheduling Problems (RCPSP .

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2014 International Transaction Journal of Engineering, Management, & Applied Sciences & Technologies.International Transaction Journal of Engineering,Management, & Applied Sciences & Technologieshttp://TuEngr.comA Review of Resource-Constrained Project SchedulingProblems (RCPSP) Approaches and SolutionsMohammad Abdolshah a*aEngineering Faculty, Islamic Azad University, Semnan Branch, Semnan, IranARTICLEINFOArticle history:Received 17 June 2014Received in revised form08 July 2014Accepted 10 July 2014Available online11 July 2014Keywords:Exact .A B S T RA C TResource-constrained project scheduling problems are oneof the most famous proposed problems in operational research andoptimization topic. Using of discrete models by consideringcomplexity of the problems requires designing efficient algorithmsfor solving them. On the other hand, this series of topics andgenerally project management are given attention in recent decades.Competition features of today’s world, lead in time implementationof project with required quality to be important. Those factors leadto be given attention to resource-constrained project schedulingproblems and their solutions theoretically and practically byacademic researches and practitioners. The purpose of the paper isdetermining different methods and approaches that are used forsolving the mentioned problems simultaneously or separately. Thevarious described models in literature that consist of more than 200published papers in most well-known journals, are collected andproposed in table format. In this research by studying these papers,in addition clarifying features of the developed models and thegaps, practitioners of projects implementation in variousorganizations can choose appropriate model for their projects byconsidering organizational conditions, types of resources and theirorganization’s activities’ technological specifications.2014 INT TRANS J ENG MANAG SCI TECH.1. IntroductionProject planning is determination of time sequencing or scheduling plan for conducting aseries of related activities that are constituents of project. In this case, Project disintegrate tosome activity by methods like work breakdown structure (WBS). These activities are connectedwith each other because there are various logical relations between them. Logical and Immediaterelations between each two activities are explained by controller like Finish to start (FS) relation,*Corresponding author (Mohammad Abdolshah). Tel: 98-231-4462198 E-mail address:abdolshah@gmail.com. 2014. International Transaction Journal of Engineering, Management, &Applied Sciences & Technologies. Volume 5 No.4ISSN 2228-9860 eISSN 1906-9642. OnlineAvailable at http://tuengr.com/V05/0253.pdf.253

start to start (SS) relation, finish to finish (FF) relation, and start to finish (SF) relation. Also, inmore complicated projects it is possible to define more controllers like parallel implementedbetween two activities(Hadju, 1997In fact dependence of activities is based on their priority ofimplementation; it means it is possible that implementation of an activity depends onimplementation of the others, this is called that project has priority constraints between activities.But in addition to these limitations, May bean other type of constraints, as resources constraintsexist in project. So in project planning in addition to considering priority constraints, planningshould be compatible with resources constraints. The objective of scheduling and sequencingactivities is optimal allocation of limited resources over time. In fact scheduling is determinationof activities which must be done in the specified time and sequencing, determine order ofactivities which must be done. Those project planning problems which do not have limitations ofresources or consider them, are known as project scheduling problems without resourceconstrained and those problems which have resource-constrained and these limitations areconsidered in planning project, called resource-constrained project scheduling problems(RCPSP). This problem is one of the most complicated problems of operation research which hasconsiderable progress in developing exact solution and innovative methods at recent decades andrecently new optimization methods are used to solve it” (Mohring et al, 2003).For implementingeach activity requires different resources such as time, capital, human power and etc. Theseresources are often divided into two categories: Renewable like human power and nonrenewable such as capital. Each activity can be implemented in several modes such as manually,semi-mechanized and mechanized. Implementation of each mood needs different type andamount of resources (Drexl et al, 1993). In resource-constrained project scheduling problems forimplementing each activity like i needs rik unit of resource k 1, ,m , at per unit of activity’sexecution time (di). Meanwhile k resource has bk constraints per unit of time. The parameters (di,ri , bk)are non-negative and determined. This problem’s objective often is determining start timeand mode of implementation of each activity for minimizing the project’s execution time. It isobvious that the problem solution must provide constraints that are related to activities’ logicalrelations, and consider resource constraints too. There are two optimal and heuristics approachfor solving the problem (Herroelen et al, 1998). The realistic solution instances of the problembecause of complexity, extension and difficulty with optimal approaches like mathematicalplanning, dynamic planning or branch and bound, is impractical (Brucker et al, 1998).2. Solving MethodsBefore suing of computer in project scheduling problems, researches scheduled projectsmanually so it was too time consuming and was not a good guaranty for achieving an optimalresult. In the last of 1950 decade, developing critical path techniques and evaluating andoverlooking the project led that projects had capability to be described by network diagrams as254Mohammad Abdolshah

works and activities were defined by network structure. Nevertheless, within the techniques, onlytime was considered and limitation of using resources was not studied. Meanwhile project’sconstraint is one of the main problems of project planning in real world, during two recentdecades types of project scheduling planning techniques under resource constrained conditionswere proposed, implemented and controlled which generally are divided to exact andapproximate methods. In fact it can be told that resource-constrained project scheduling problemhas more than 40 years history. There are two approaches, optimal and heuristics, for solving theproblem (Herroelen et al, 1998). Each of the methods has disadvantages and advantages. Theexact methods have ability to obtain and guaranty optimal result. In these methods, all solvingproblem spaces are searched to find optimal answer from solving space. Although essentialcalculations for these methods are so many and as a results, they are so slow but guaranty thegeneral optimization of problem, in fact the realistic solution instances of the problem because ofcomplexity, extension and difficulty with optimal approaches like mathematical planning,dynamic planning or branch and bound, is impractical (Brucker et al, 1998). Of course theapplication of optimal approaches for solving smaller instances of the problem are reported inthe literature. For instance, the paper refers interested reader to (Deckro et al, 1991) aboutmathematical planning, to (Icmeli et al, 1996. Carruthers et al, 1996) for numerical methods suchas dynamic planning, to (Petrovic 1968, Demeulemeester 1998) about branch and boundmethods. And for overcoming the computational problems of the methods, approximate methodsare proposed. In these methods, Instead of the whole space of problem solution, a part of it issearched so they do not guaranty the optimal results and try to achieve a good approximateanswer but they are quick methods and at the right time they achieve a good answer for hugeproblems. Many of the heuristics solving approaches for resource-constrained project schedulingproblems are studied at 2006 (Kolisch et al, 2006). They categorized the approaches in 4 groupsas (1) Priority rule- based approaches like Random sampling (Coelho et al, 2003); (2)Approaches based on meta-heuristics methods such as genetic algorithm (Alcaraz et al, 2003.Tareghian et al, 2007), tabu search algorithm (Nonobe et al, 2002), simulated annealing (SA)algorithm (Valls et al. 2004) ant systems (Merkle et al, 2002); (3) Non – Standard metaheuristics approaches like scatter search algorithm (Fleszar et al, 2004); and at last (4)approaches based on other heuristics methods such as forward and backward Improvement (FBI)(Tormos et al. 2003), Network analysis (Sprecher, 2002). This paper categorizes solving modelsthat are discussed in past literature, as 3 diagrams3. Exact solving methodsRCPSP are as general format of sequence of operations of NP hard problems type. The*Corresponding author (Mohammad Abdolshah). Tel: 98-231-4462198 E-mail address:abdolshah@gmail.com. 2014. International Transaction Journal of Engineering, Management, &Applied Sciences & Technologies. Volume 5 No.4ISSN 2228-9860 eISSN 1906-9642. OnlineAvailable at http://tuengr.com/V05/0253.pdf.255

optimal solutions, which are mentioned in literature, are: Zero-one mathematical planning andnumerical implicit methods such as dynamic planning and branch and bound method. At recentdecades, solving the problems is improved widely which are tested in two series problem. Theseseries are: Series of 110 problems designed by Peterson and Series of 480 problems by Klisch.Algorithms are evaluated base on how many problems are solved by them at how much time.The series of Peterson problems include 110 problems instances that are designed by Peterson.Series of problems have 7 to 50 activities and 1 to 3 renewable resources. During last decades,this series was a criterion for evaluating validity and ability of optimal and close to optimalprocedure. In 1995, Klisch questioned validity of Peterson’s series that leads to develop ProGen.Network producer software that is able to produce RCPSP pattern with pre-determinate and 30types of activity and 4 types of renewable resource, see Figure 1.Exact solutionsMathematicalplanningNumerical methodsdeterminsticmethodesSynthetic kov chainBranch andBoundmethodCritical path PERTGoal onemethodSyntheticLinearplanningFigure 1: Exact solution categories3.1 Heuristics solutionsA brief definition of a heuristics method is a technique that search close solutions to optimalwith acceptable computational cost, but in fact unlike the exact solutions which guaranty findingthe optimal answer if there are, they do not guaranty for achieving to an optimal result.Heuristics methods sometimes find the optimal answer and most of the time they reach to goodanswer. And these methods usually require less time and memory than exact solutions. Theheuristics in scheduling often are defined as scheduling rules with dispatch rules. Often the rulesare complex to be defined and for a specific type of the problem with a special series ofrestrictions and assumptions, are appropriate. The heuristics are used for searchingcombinational space of permutations in sequences of tasks or determining the conceivability ofallocating resource, time and task during creation of scheduling or combining sequencing andscheduling. Heuristics scheduling are applied on series of tasks and determine at what timeMohammad Abdolshah256

which task must be done. If a task can be done in more than one implementation condition or onseries of resources, heuristics determines which resource or implementation is used. Theheuristics solutions are be used for major problems al schedulingSerialdouble schedulingConstructiveSearch- basedParallelSingle passpriority ruleMulti orhoodsearchforward backwardimprovmentdecompositionBased on exactmethodsrandom samplingsampling methoditeratationHeuristicsmethodsbaised randomsamplingregret bsedrandomsamplingcolumngenerationexact methodrelaxationLagrangeHybridCombinationwithith severalheuristicsFigure 2: Heuristics methods categories3.2 Meta-heuristics SolutionsDuring last 20 years, a new type of estimated algorithm has been created which essentiallytries to combine basis heuristics methods with an objective of efficient and effective search insearch space in frameworks of upper level. The meta-heuristics methods are the last generationsof heuristics algorithms and widely used for solving RCPSP too. In fact, the meta-heuristics arestrategies in order to guiding search process. Participant techniques in meta-heuristics algorithmsare in range of simple procedure, local search to complex learning processes.3.2.1 Trajectory MethodsIt works on single solutions and includes meta-heuristics based on local search. It means thatalgorithm start form primary condition (primary solution) and describes a trajectory in searchspace. Each movement is take place if the result solution is better actual one. Upon finding localminimal, the algorithms end such as Tabu search, iterated local search and variableneighborhood search. Their common features are describing a trajectory in search space duringsearch process.3.2.2 Population methodsThey do search process which combine meta-heuristics evolution with exact methods or*Corresponding author (Mohammad Abdolshah). Tel: 98-231-4462198 E-mail address:abdolshah@gmail.com. 2014. International Transaction Journal of Engineering, Management, &Applied Sciences & Technologies. Volume 5 No.4ISSN 2228-9860 eISSN 1906-9642. OnlineAvailable at http://tuengr.com/V05/0253.pdf.257

other meta-heuristics, and combination of types of heuristics and meta heuristics in order toachieving optimal answers, can be observed in meta-heuristics methods, Figure 3.Figure 3: Meta Heuristics methods categories4. ApproachesMost of the studies in planning and project scheduling assume that there are completeinformation for solving scheduling problem which must be solve and the obtained basisscheduling plan will be implemented in a static environment too. Although there are manyuncertainly in a relation with activities implementation that take place with implementation ofproject gradually which includes the following categories in diagram? In this section, there isreview of basis approaches in project planning and scheduling at exact and unreliabilityconditions. It will be discussed about application potential of each of the methods in projectuncertainly planning with definitive network structure.Figure 4 show types of RCPSPapproaches.Types of approacheson RCPSPDeterministicReactiveNon- DeterministicProactive(Robust)StochasticNumber of igure 4: Types of RCPSP approaches4.1 Deterministic approachIn this approach, all problems’ parameters are assumed definitive and determined and it has258Mohammad Abdolshah

rich position in RCPSP literature and is used for relaxation of the assumption in most of thepapers. These kinds of papers because of simplifying real conditions have defects andpractically, restrict efficiency of model in real projects.Figure 5: FRCPSP categories.4.2 Proactive (Robust) approachObjective of the proactive scheduling is producing basis-scheduling stable so in order to beprotected against interruptions during implementation of project. Temporary protections (Gao1995)increase duration of activities based on unreliability of amount of resources, which areused for activities. Resources that have possibility of failure or violation are called probable toviolation resources. Protected duration of the activity includes main duration added to waitingduration of violation. Then basic scheduling is provided by problem solution with protecteddurations.4.3 Reactive approachIn Reactive scheduling, uncertainly are not given attention at creating basis scheduling butwhen uncertainly occur, the approach tries to answer, correcting and re-optimize the basisscheduling. Generally, the approach’s main correction is on correcting and optimizing the basisscheduling if unanticipated events are occurred. The basis scheduling can be designed based onvarious strategies. On the other hand, answering to occurred changes can be based on very*Corresponding author (Mohammad Abdolshah). Tel: 98-231-4462198 E-mail address:abdolshah@gmail.com. 2014. International Transaction Journal of Engineering, Management, &Applied Sciences & Technologies. Volume 5 No.4ISSN 2228-9860 eISSN 1906-9642. OnlineAvailable at http://tuengr.com/V05/0253.pdf.259

simple techniques such as Right shift rule (Sadeh et al. 1993) that they are influenced because ofthe defect in resources or precedence relations, transferred to the right which means theirimplementation time are postponed, it’s obvious that the method is not a such good idea becauseit does not reschedule. The similar strategies are called schedule repair actions.4.4 Fuzzy approachFans of activity ambiguous express the probability distribution function of activities leads toambiguity and imprecise of estimation. The probability distribution function of an activity isambiguous as long as information of its past, was not gained. A human expert should estimatethe probability distribution function of an activity that often is non-recurring and exclusive.4.5 Stochastic project approachObjective of stochastic project scheduling with resource constrained, is project schedulingwhich is such that despite of activity duration uncertainly, precedence relations (Finish to startwith zero lag) and renewable resource-constrained, minimizes make span. The studies onstochastic project scheduling are partly sporadic. Most of the studies are known as “stochasticproject scheduling with resource-constrained” which are studied in next section.5. Review of Solutions and Approaches of Resource-constrained ProjectSchedulingIn order to review researches procedure and researches’ opportunities, all of the researchesare studied as two perspectives “ solution methods and approaches” in more than 200 papers ofvalid journals and after removing the similar articles, the chosen articles was studied andextracted their points and the results are shown by following tables. Tables 1 and 2 show theresults of research about types of solution methods and approaches in RCPCP literature. Thereare brief explanations about important results of research in considerations column.Table 1: RCPSP researches based on solution methodsAuthorsYearExactmethodSolutionsNot exact methodHeuristics1. D.C.2012Paraskevopoulos et al.AILS,SAILS2. Chen Fang,Ling Wang2012SSGS3. MohamedHaouari et al2012 Dynamicprogramming, lower260Mohammad os solution methodology, namely SAILS, operates on theevent list and relies on a scatter search framework. The latterincorporates an Adaptive Iterated Local Search (AILS), as animprovement method, and integrates an event-list basedsolution combination method.Encode the virtual frog as the extended activity list (EAL) anddecode it by the SFLA-specific serial schedule generationscheme (SSSGS) and To enhance the exploitation ability, acombined local search including permutation-based localsearch (PBLS) and forward–backward improvement (FBI) isperformed.Propose three classes of lower bounds that are based on theconcept of Enhanced energetic reasoning

bounds4. Ling Wangn,ChenFang20125. Thomas S.2012Kyriakidis etal.6. KoorushZiarat 2011i etal.7. Shu-ShunLiu& ChangJung Wang2011EDAIn the EDA the individuals are encoded based on the activitymode list (AML) and decoded by the multi-mode serialschedule generation scheme (MSSGS), and a novelprobability model and an updating mechanism are proposedfor well sampling the promising searching region.Present new mixed-integer linear programming modelsSSGSBeealgorithmsProposed algorithms iteratively solve the RCPSP by utilizingintelligent behaviors of honeybees. Each algorithm has threemain phases: initialization, update, and termination.A generic model is proposed to maximize the total profit ofselected projects for construction and R&D departmentsgiven scheduling problems with various resource constraintsduring specified time intervalsThe procedure is basically a combination of four descentprocedures that use simulation to evaluate the objectivefunctionMILPCP8. FilipDeblaere 2011et al.Simulation-basedDescent(SBD),SSGS9. SiamakBarada 2011ran et al.10. MohammadRanjbar et al.SSGS,MFBI,MPBLS2011 Branchandboundalgorithm11. R. Čapek et al 2011Linearprogramming model12. MariemTrojet 2011et al.13. Ling Wang,2011Chen FangCPIRSASSGS14. José Coelho,MarioVanhoucke201115. Ruey-MawChen2011SSGS16. Shanshan Wu 2011et al.17. Mahdi Mobini 2011et al.SSGSSSGS18. OumarKone et 2011al.MILP19. LucioBianco& 2011MassimilianoCaramiaLowerbound20. AgustínBarrios et al.201121. AnuragAgarw 2011DGAHMAPresents a hybrid met heuristic algorithm based on scattersearch and path linking algorithms to solve the stochasticMRCPSPPresent a branch-and-bound algorithm in which the branchingscheme starts from a graph representing a set of conjunctionsIn the search tree; each node is branched to two child nodesbased on the two opposite directions of each undirected arc ofdisjunctions.A heuristic algorithm based on priority schedule constructionwith an un-scheduling step is proposed for the nested versionof the problem and it is used to solve the case study of thewire harnesses production.Provide a decision support framework under the constraints asa margin of cooperation/ negotiation with subcontractorsHybridIndividuals are encoded based on the extended active listEDA(EAL) and decoded by serial schedule generation scheme(HEDA)(SGS), a Forward–Backward iteration (FBI) and apermutation based local search method (PBLS) areincorporated into the EDA based search to enhance theexploitation abilityA novelThe algorithm splits the problem into a mode assignment stepmetaand a single mode project-scheduling step. The modeheuristicassignment step is solved using a fast and efficient SATsolver.JPSOThe justification technique is combined with PSO as theproposed justification particle swarm optimization (JPSO),which includes other designed mechanisms.CBIIAThe proposed CBIIA is based on the traits of an artificialimmune system, chaotic generator and parallel mutationAIAThe proposed algorithm benefits from local searchmechanisms as well as mechanism that enhances the diversityof the search directionsMake a comparative study of several-mixed integer linearprogramming (MILP) formulations for resource-constrainedproject scheduling problems (RCPSPs).The lower bound is based on a relaxation of the resourceconstraints among independent activities and on a solution ofthe relaxed problem suitably represented by means of anAON acyclic network.The heuristic is a two-phased genetic algorithm with differentrepresentation, fitness, crossover operator, etc., in each ofthem.Neurogene A new hybrid of a neural network approach and the genetic*Corresponding author (Mohammad Abdolshah). Tel: 98-231-4462198 E-mail address:abdolshah@gmail.com. 2014. International Transaction Journal of Engineering, Management, &Applied Sciences & Technologies. Volume 5 No.4ISSN 2228-9860 eISSN 1906-9642. OnlineAvailable at http://tuengr.com/V05/0253.pdf.261

al et al.22. Francisco2011Ballestıín,RosaBlanco23. FilipDeblaere 2011 Branchet al.andbound24. TarunBhaskar 2011et al.SSGSIDASPEA2,NSGA2and PSATSDCSGSHeuristicHUDD-PS26. José Fernando 2011Gonçalves etal.FBI,SSGSGeneticalgorithm27. Vincent VanPeteghem,MarioVanhoucke28. Reza Zamani2011SSGSScattersearchalgorithm2011SSGSA hybriddecompositionprocedure30. BehzadAshtia 2011ni et al.31. Francisco2011Ballestín et al.32. Jie Zhu et al.201133. MohammadJaberi201134. HongZhang,FengXing201035. E. Klerides, E. 2010Hadjiconstantinou36. Qi Hao et al.201037. Svio B.Rodrigues,Denise S.Yamashita38. SondaElloumi,Philippe2010262Propose and evaluate a number of dedicated exact reactivescheduling procedures as well as a TABU search heuristic forrepairing a disrupted schedulePropose a non-recursive heuristic method based on priorityrule for a new scheduling scheme and call it priority rule asSchedule Performance IndexDifferent approaches to solving the continuous part of theproblem were presented an exact approach requiring solving aconvex mathematical programming problem, a heuristicapproach to the continuous resource allocation problem(heuristic HUDD-PS), and the approach based on thecontinuous resource discretization.Active schedules are constructed using a priority-ruleheuristic in which the priorities of the activities are defined bythe genetic algorithm. A forward-backward improvementprocedure is applied to all solutions.Combination of improvement methods and the introduction oftwo local searches into one overall solution procedure leads topromising computational resultsSPI25. GrzegorzWali 2011góra29. Olivier2011Lambrechts etal.ticalgorithms approachapproachExtensive computational results help decide which algorithmsor techniques are the most promising for the problem.Timebufferingusing Mohammad AbdolshahThe procedure finds an initial schedule for the project, andrefines it through a decomposition process, To achieve furtherreduction, the refined schedule is over-refined by a geneticalgorithmSuggest to either implement time buffering based on the firstsurrogate objective function or using the STC heuristicA hybridrank-basedevolutionaFLCA two-phase local-search procedure is developed to producehigh-quality pre-processor policies for SRCPSP instance, firstphase is devoted to finding good priority listsWorks on a population consisting of several distance-orderpreserving activity lists representing feasible or infeasibleschedules. The algorithm uses the conglomerate-basedcrossover operatorDuring the genetic process of the proposed GA, an offspringgenerator was introduced to generate a feasible activity listfrom parent chromosomesA Potts mean field feedback artificial neural network isdesigned and integrated into the scheduling scheme so as toautomatically select the suitable activity for each stage ofproject schedulingPresent a fuzzy-multi-objective particle swarm optimizationto solve the fuzzy TCQT problem. The time, cost and qualityare described by fuzzy numbers and a fuzzy multi-attributeutility methodology incorporated with constrained fuzzyarithmetic operations is adopted to evaluate the selectedconstruction methodsPropose a path-based two-stage stochastic integerprogramming approach in which the execution modes aredetermined in the first stage while the second stage performsactivity scheduling according to the realizations of activitydurationsA dynamic algorithm based on partial task networks ,practicalheuristics for conflict detection, project prioritization andconflict resolutionThe new algorithm consists of a hybrid method where aninitial feasible solution is found heuristicallyIntroduce clustering algorithms to compute densities. In thisway enforce that neighbor solutions belong to the samecluster and are assigned the same density.

Fortemps39. AnisKooli etal.201040. Jairo R.MontoyaTorres et icalgorithm41. SiamakBarada 2010ran et al.SSGS,PSGS42. Moslem2010Shahsavar etal.43. C.U.2010Fündeling, N.TrautmannA hybridscattersearchGeneticalgorithmA novelmethod ofSGS44. Ruey-MawChen et al.2010A novelPSO45. Wang Chen et 2010al.SSGSACOSS46. Vincent Van 2010Peteghem,MarioVanhoucke47. E. Klerides, E. 2010 IntegerHadjiconstantiprogrammnouingSSGSGA48. AndreiHorbach49. Angela H. L.Chen, ChiuhCheng Chyu2010Lowerbounds2010 Branchandbound50. WANG Hong 2010et al.51. Reza edure52. JiupingXu&Z 2010he Zhang53. Isabel Correia 2010et al.54. Wang2010Xianggang1 &Huang WeiThe icalgorithmNew feasibility tests for the energetic reasoning areintroduced based on new integer programming (IP)formulations.Propose an alternative representation of the chromosomesusing a multi-array object-oriented model in order to takeadvantage of programming features in most commonlanguages for the design of decision support systemsThe path re-linking algorithm and two operators likecrossover and prominent permutation-based are applied tosolve the problemGenetic algorithm (GA) is designed using a new three-stageprocess that utilizes design of experiments and responsesurface methodology.Present a priority-rule method based on a novel schedulegeneration scheme and a consistency test for efficientscheduling of individual activities that iteratively determinesa feasible resource-usage profile for each activityThe delay local search enables some activities delayed andaltering the decided start processing time. The bidirectionalscheduling rule which combines forward and backwardscheduling to expand the searching area in the solution spacefor obtaining potential optimal solution.Algorithm combines a local search strategy, ant colonyoptimization (ACO), and a scatter search (SS) in an iterativeprocessApply a bi-population genetic algorithm, which makes use oftwo separate populations and extend the serial schedulegeneration scheme by introducing a mode improvementprocedure.Propose a path-based two-stage stochastic integerprogramming approach in which the execution modes aredetermined in the first stage while the second stage performsactivity scheduling according to the realizations of acti

decades types of project scheduling planning techniques under resource constrained conditions were proposed, implemented and controlled which generally are divided to exact and approximate methods. In fact it can be told that resource-constrained project scheduling problem has more than 40 years history.

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