This study investigates different resource management policies in resource constrained multi-project problem environments. The problem environment under investigation has alternative modes for activities, a set of renewable and nonrenewable resources used by activities and further considerations such as general resource budget. The characterization of the way resources are used by individual projects in the multiproject environment is called resource management policy in this study. The solution approaches in the literature for multi-project problems generally defines the resources as a pool that can be shared by all the projects which in fact creates a general assumption for the resource usage characteristics. This resource management policy is referred as resource sharing policy in this study. Resource sharing policy can be invalid in

some certain cases where sharing assumption is not feasible because of some characteristics of resources and/or projects which require different resource management policies for the multi-project environment. According to the characteristics of resources and projects, resource management policies such as

resource dedication, relaxed resource dedication and generalized resource management policies can be defined. In this paper, these resource management policies will be defined and their mathematical formulations will be presented and discussed

Besikci , Umut

Beşikçi , Umut

Bilge, Umit

Bilge, Ümit

Ulusoy, Gunduz

Ulusoy, Gündüz

English

Beşikçi, Umut

Sabanci University Research Database

Different Resource Management Policies inMulti-Mode Resource Constrained Multi-ProjectSchedulingUmut Bes¸ikciDepertment of Industrial EngineeringBog˘azic¸i UniversityIstanbul, TurkeyEmail: umut.besikci@boun.edu.tr¨Umit BilgeDepertment of Industrial EngineeringBog˘azic¸i UniversityIstanbul, TurkeyEmail: bilge@boun.edu.trGu¨ndu¨z UlusoyFaculty of Engineering and Natural SciencesSabancı UniversityIstanbul, TurkeyEmail: gunduz@sabanciuniv.eduAbstract—This study investigates different resource manage-ment policies in resource constrained multi-project problemenvironments. The problem environment under investigation hasalternative modes for activities, a set of renewable and nonre-newable resources used by activities and further considerationssuch as general resource budget. The characterization of theway resources are used by individual projects in the multi-project environment is called resource management policy inthis study. The solution approaches in the literature for multi-project problems generally defines the resources as a pool thatcan be shared by all the projects which in fact creates ageneral assumption for the resource usage characteristics. Thisresource management policy is referred as resource sharingpolicy in this study. Resource sharing policy can be invalid insome certain cases where sharing assumption is not feasiblebecause of some characteristics of resources and/or projectswhich require different resource management policies for themulti-project environment. According to the characteristics ofresources and projects, resource management policies such asresource dedication, relaxed resource dedication and generalizedresource management policies can be defined. In this paper,these resource management policies will be defined and theirmathematical formulations will be presented and discussed.Index Terms—Multi-project scheduling, resource managementpolicies, resource portfolio problem.I. INTRODUCTIONMulti-project management constitutes an important part inof the business in both manufacturing and services and withits complex nature it is an important research topic in theliterature. Multi-project scheduling problems consist of severalprojects that involve finish to start zero time lag and non-preemptive activities with multiple modes. There are bothrenewable and nonrenewable resources in the environmentconstraint with a general resource budget. Uncertainty in themulti-project environment is not considered in this paper.The general approach for multi-project scheduling problemsis considering the available resources as a shared pool whichis open to all projects. With this approach, the individualprojects in the multi-project problem can be combined byadding dummy start and finish activities and the overall multi-project problem can be solved over this big combined network.Solution approaches based on this methodology in fact as-sumes a specific resource usage characterization or a specificrelation between projects and resources in the multi-projectenvironment. This resource related characterization of themulti-project environment is named as resource managementpolicy in this study. The resource management policy for theaforementioned multi-project environment is called resourcesharing policy and this type of renewable resources is namedas shared resources.Different multi-project environments can require differentresource management policies according to the certain char-acteristics of the projects and/or resources. For example, theprojects that are geographically distributed such that sharingof resources is too costly or infeasible, require another re-source management policy to model the multi-project problemenvironment realistically. This resource management policyis named as resource dedication policy. The renewable re-source that cannot be shared among different projects iscalled dedicated resources. Under this resource managementpolicy, resources cannot be shared among projects and mustbe dedicated among different projects such that the dedicatedresources cannot be shared. Resource dedication policy isdiscussed in detail in [1].Another resource management policy can be relaxed re-source dedication policy where the resource dedication conceptis relaxed in the following way: the renewable resources thatare dedicated to a project can be transferred to other projectswhich have a starting time later than the finish time of theaforementioned project. This relaxed resource dedication pol-icy can be feasible when resource sharing is not allowed duringthe course of projects but transferring of renewable resourcesto another project is possible when one of the projects startsafter the other one finishes. This type of renewable resourcesis called relaxed dedicated resources.The final resource management policy discussed in thisstudy is characterizing resources in the multi-project environ-ment by the most general approach. Three types of renewableresources are defined in this resource management policy,namely shared resources, dedicated resources and relaxeddedicated resources with the corresponding characteristics asdescribed above.Another assumption of the general approaches for multi-project scheduling is a predetermined or given general resourcecapacities set. These general resource capacities values can beincluded as another decision for the multi-project schedulingproblem. With this approach, the multi-project environmentwould have a conceptually higher decision level which can bedefined as deciding the general resource capacities accordingto the project requirements in the multi-project environment.The resource management policies discussed above can beintegrated with this general resource capacities decision. Thisproblem is defined as Resource Portfolio Problem (RPP) whichcan take different forms with different resource managementpolicies.The remainder of this paper continues with the discussionsof shared resources, resource dedication, relaxed resourcededication and generalized resource management policies inRPP.II. RESOURCE MANAGEMENT POLICIESA. Shared Resources PolicyResource sharing policy considers the general resourcecapacities as a shared resource pool where all the individualprojects have unlimited access. This point of view for the re-source management brings specific advantages for the solutionof the multi-project problem. The projects in the multi-projectenvironment can be integrated by adding a general start nodethat all nodes of the projects that do not have a predecessorare assigned as successors of this start node, and a generalend node that all nodes that do not have a successor areassigned as predecessors of this end node. Then this combinedproject network can be solved using different approaches forproject scheduling (see i.e [5], [2], [6], [3]). A mathematicalformulation for shared resource policy can be obtained fromthe mathematical model proposed by [7].B. Resource Dedication PolicyResource dedication policy in a multi-project environmentbecomes a requirement when resource cannot be shared amongprojects because of the characteristics of projects and/or re-sources. This resource management approach did not takemuch attention in the project scheduling literature. [4] identi-fies different multi-project scheduling environments and spec-ifies resource dedication problem as a tactical level problemin the case where projects are characterized as independentfrom each other. [1] investigates resource dedication policywith given general resource capacities and propose solutionapproaches for the problem.The mathematical model for RPP with resource dedicationpolicy is given below.Sets:V set of projects, v ∈ VJv set of activities of project v, j ∈ JvPv set of all precedence relationships of project vMvj set of modes for activity j of project v, m ∈MvjK set of renewable resources, k ∈ KI set of nonrenewable resources, i ∈ IT set of time periods, t ∈ TParameters:Evj Earliest finish time of activity j of project vLvj Latest finish time of activity j of project vdvjm Duration of activity j, operating on mode mrvjkm Renewable resource k usage of activity j ofproject v, operating on mode mwvjim Nonrenewable resource i usage of activity j ofproject v, operating on mode mddv Assigned due date for project vcv Relative weight of project vcrk Unit cost of renewable resource kcwi Unit cost of nonrenewable resource itb Total resource budgetDecision Variables:xvjmt =1 if activity j, operating on mode m,in project v is finished at period t0 otherwiseBRvk = Amount of renewable resource k dedicatedto project vBWvi = Amount of nonrenewable resource idedicated to project vTCv = Weighted tardiness cost of project vRk = Total amount of required renewableresource kWi = Total amount of required nonrenewableresource iMathematical Model RPP-RDmin. z =∑v∈VTCv (1)Subject to∑m∈MvjLjv∑t=Evjxvjmt = 1 ∀ j ∈ Nv and ∀ v ∈ V (2)∑m∈MvjLvb∑t=Evb(t− dvbm)xvbmt ≥∑m∈MvjLva∑t=Evatxvamt∀ (a, b) ∈ P and ∀ v ∈ V (3)∑j∈Nv∑m∈Mvjt+dvjm−1∑q=trvjkmxvjmq ≤ BRvk∀ k ∈ K ∀ t ∈ T ∀ v ∈ V (4)∑j∈Nv∑m∈MvjLvj∑t=Evjwvjimxvjmt ≤ BWvi∀ i ∈ I and ∀ v ∈ V (5)∑v∈VBRvk ≤ Rk ∀ k ∈ K (6)∑v∈VBWvi ≤ Wi ∀ i ∈ I (7)∑i∈IcwiWi +∑v∈VcrkRk ≤ TB (8)TCv ≥ cv(LvN∑t=EvN∑m∈MvNxvNmt − ddv)∀ v ∈ V (9)xvjmt ∈ {0, 1} ∀ j ∈ J , ∀ t ∈ T , ∀ m ∈Mand ∀ v ∈ V (10)BRvk ∈ Z+ ∀ v ∈ V and ∀ k ∈ K (11)BWvi ∈ Z+ ∀ v ∈ V and ∀ i ∈ I (12)Rk ∈ Z+ ∀ k ∈ K (13)Wi ∈ Z+ ∀ i ∈ I (14)TCv ∈ Z+ ∀ v ∈ V (15)The objective function (1) is determined as minimization ofthe total weighted tardiness cost for all projects. Constraint(2) forces all the activities of all the projects to finish onceand only once. Constraint set (3) satisfies the precedencerelations between activities. Constraint set (4) and (5) set therenewable and nonrenewable resource dedication values foreach project respectively. Constraint (6) and (7) calculatesthe total renewable and nonrenewable resource requirementsrespectively. Constraint set (8) limits the total renewable andnonrenewable resource costs with the general resource budget.And finally constraint (9) calculated the weighted tardinessvalues for each project.The resource dedication concept is achieved by constraintsets (4) and (6). Constraint set (4) sets BRvk for the cor-responding resource and project as the maximum resourceusage over all time periods which is the required amount ofresources that must be dedicated to the project. Constraint set(6) ensures that the resource dedication values for all projectscannot exceed the general renewable resource capacity. Withthis approach renewable resources cannot be shared amongdifferent projects.C. Relaxed Resource Dedication PolicyThe complete resource dedication policy is not a verygeneral case even though it has its own merits of usage.In some cases, the renewable resources that are dedicatedto an already finished project can be used by the projectsthat are subject to start. This approach would have certainbenefits when renewable resources can be transferred in theaforementioned way and the resource budget is limited.Mathematical model for RPP under relaxed resource dedi-cation policy is given below.Additional Parameters:Ω A big numberAdditional Decision Variables:fv = Release time of project vSvv′k = Amount of renewable resource k given toproject v‘ from project vyvv′ =1 if project v′ is released afterproject v is finished0 otherwiseModel RPP-RRDmin. z =∑v∈VTCv (1)Subject to∑j∈Nv∑m∈Mvjt+dvjm−1∑q=trvjkmxvjmq ≤ BRvk +∑v′∈VSRv′vk∀ k ∈ K ∀ t ∈ T ∀ v ∈ V (14)BRvk +∑v′∈VSRv′vk ≥∑v′∈VSRvv′k∀ k ∈ K and ∀ v ∈ V (15)fv′ − fv −LvN∑t=EvN∑m∈MvNtxvNmt ≤ Ω(yvv′ )∀ v, v′∈ V (16)fv +LvN∑t=EvN∑m∈MvNtxvNmt − fv′ ≤ Ω(1− yvv′ )∀ v, v′∈ V (17)SRvv′k ≤ Ω(yvv′ )∀ v, v′∈ V and ∀ k ∈ K (18)TCv ≥ Cv(fv +LvN∑t=EvN∑m∈MvNxvNmt − ddv)∀ v ∈ V (19)yvv′ ∈ {0, 1} ∀ v ∈ V (20)fv ∈ Z+ ∀ v, v′∈ V (21)SRvv′k ∈ Z+ ∀ v, v′∈ V (22)(2), (3), (5), (6), (7), (8), (10), (11), (12), (13)The resource sharing policy that allows resource trans-fer between projects added the following changes to themathematical model. First of all, renewable resource usageconstraints for each project (14) have a resource usage capacityas the sum of dedicated resource value and the total transferredresource to the project from the other projects. In constraint(15) the total resource that can be transferred by a projectis limited with the total resources dedicated to this projectand the total resource it gained from transfers. Constraint(16) and (17) sets decision variable yvv‘ to 1 if project v isfinished before project v‘ is released and 0 otherwise. Thusthe SRvv‘k values will only have positive values if project vis finished before project v‘ is released with constraint (18),it will be set to 0 otherwise. The weighted tardiness for eachproject is calculated according to the release times of projectin constraint (19).D. Generalized Resource Management PolicyThe generalized resource management policy incorporatesall the characteristics of the aforementioned resource man-agement policies by identifying three different types of re-newable resources: shared, dedicated and relaxed dedicated.The mathematical formulation for this resource managementpolicy will have three different types of renewable resourceusage constraints for different renewable resource types. Forthe corresponding renewable resource types different renew-able resource usage and general renewable resource capacityconstraints can be used for the mathematical formulation ofthe generalized resource management policy.III. CONCLUSIONTo realistically model different multi-project environmentsdifferent resource management policies should be taken intoaccount. In this study, different resource management policiesare tried to be identified, namely shared resources policy,resource dedication policy, relaxed resource dedication policyand the generalization of these three resource managementpolicies. With these different resource management policies,multi-project scheduling problem environments can cover veryimportant project and resource characteristics that must betaken into account. In addition to this, mathematical formula-tions for resource dedication and relaxed resource dedicationpolicies are given and discussed in a multi-project schedulingenvironment with a general resource budget. Solution method-ologies for these different resource management policies willhave different approaches because of the characteristics thatresource management policies incorporated to the problemenvironment.ACKNOWLEDGMENTWe gratefully acknowledge the support given by theScientific and Technological Research Council of Turkey(TUBITAK) through Project Number MAG 109M571 andBogazici University Scientific Research Projects (BAP)through Project Number O9HA302D.REFERENCES[1] U. Besikci, U. Bilge and G. Ulusoy, ”Resource Dedication Problemin a Multi-Project Environment”, submitted to Flexible Services andManufacturing Journal[2] J. F. Goncalves, J. M. M Mendes, M. G. C Resende, ”A genetic algorithmfor resource constrained multi-project scheduling problem”, EuropeanJournal of Operational Research, Vol. 189, pp. 1171-1190, 2008.[3] S. Hartmann and D. Briskorn, ”A Survey of Variants and Extensionsof the Resource-Constrained Project Scheduling Problem” to appear inEuropean Journal of Operational Research[4] W. Herroelen, ”Project Scheduling - Theory and Practice”, Productionand Operations Management, Vol. 14, No. 4, pp. 413-432, 2005.[5] A. Lova, C. Maroto, P. Tormos, ”A multicriteria heuristic method toimprove resource allocation in multiproject environment”, EuropeanJournal of Operational Research, Vol. 127, pp. 408-424, 2000.[6] M. L. Mittal and A. Kanda, ”Two phase heuristics for scheduling ofmultiple projects”, International Journal of Operational Research, Vol.4, Iss. 2, pp. 159-177, 2009.[7] F. B. Talbot, ”Resource-constrained project scheduling with time-resourcetrade-offs: The nonpreemptive case”, Management Science, Vol. 28 Iss.10, pp. 1199-1210, 1982.

Different resource management policies in multi-mode resource constrained multi-project scheduling

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Different resource management policies in multi-mode resource constrained multi-project scheduling

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