Previous developments from the first author and other researchers were made on devising models for the total cost optimization of projects described by activity networks under stochastic conditions. Those models only covered the single resource case.
The present paper will discuss the case of multiple resources. More precisely, we introduce a strategy of allocation of those resources in order to minimize the waste arising from their latent idleness on their consumption within the same activity. On this strategy we elect one resource as “pivot” and write equations that describe all the quantities for the other resources, on the same activity.Fundação para a Ciência e a Tecnologia (FCT

Araújo, Maria Madalena Teixeira de

Elmaghraby, Salah

Moutinho, Rui

Tereso, Anabela Pereira

Universidade do Minho: RepositoriUM

Quantity oriented resource allocation strategy on multiple resources projects under stochastic conditions

International Conferenceon Industrial Engineering and Systems ManagementIESM 2009May 13-15MONTRE´AL - CANADAQuantity Oriented Resource Allocation Strategy onMultiple Resources Projects under StochasticConditions ?Anabela Tereso a, Madalena Arau´jo a, Rui Moutinho a,Salah Elmaghraby baUniversidade do Minho, 4800-058 Guimara˜es, PORTUGALbNorth Carolina State University, Raleigh, NC 27695-7906, USAAbstractPrevious developments from the first author and other researchers were made on devising models for the total costoptimization of projects described by activity networks under stochastic conditions. Those models only coveredthe single resource case.The present paper will discuss the case of multiple resources. More precisely, we introduce a strategy of allocationof those resources in order to minimize the waste arising from their latent idleness on their consumption withinthe same activity. On this strategy we elect one resource as “pivot” and write equations that describe all thequantities for the other resources, on the same activity.Key words: Project Management and Scheduling, Stochastic Activity Networks, Resource Allocation, MultipleResourcesList of AcronymsRCPSP: Resource Constraint Project Scheduling Prob-lemAoA: Activity-on-ArcDP: Dynamic ProgrammingEMA: Electromagnetic AlgorithmEVA: Evolutionary AlgorithmSRpco: Single Resource Project Cost OptimizationMRpco: Multiple Resources Project Cost OptimizationQOras: Quantity Oriented Resource Allocation StrategyWBras: Waste Balance Resource Allocation Strategy1 IntroductionThis paper follows the researches made by several contributors starting from the first research made bythe first author (see [1]). Those works address a RCPSP (Resource Constraint Project Scheduling Prob-lem) where we want to minimize the total project cost involving multimodal activities under stochasticconditions. More precisely, given an project on its planning phase, where all its activities and relatedrequirements are established, we want to determine which is the optimal allocation of resources. This? Corresponding author A. Tereso.Email addresses: anabelat@dps.uminho.pt (Anabela Tereso), mmaraujo@dps.uminho.pt (Madalena Arau´jo),rumout@gmail.com (Rui Moutinho), elmaghra@eos.ncsu.edu (Salah Elmaghraby).IESM 2009, MONTRE´AL - CANADA, May 13 - 15allocation yields the minimum total project cost, which is composed by the sum of the allocation costand the delay cost.All the researches so far, work with AoA (Activity-on-Arc) network representation of the project activitiesand their precedences. By treating multimodal activities, we deal with activities which their duration timeis measured as a function of the resource allocation. As an example, if a certain activity requires 6 dayswith 1 man allocated, it should require only 2 days when the allocation is 3 men. However, in the stochasticcontext, that relation between allocation and duration is not deterministic. This requires that we takethe Work Content of an activity as a probabilistic distribution. In particular, we have been using theexponential distribution for that matter.Since the first approach to this problem, on Matlab using DP (Dynamic Programming) [2], the subse-quent researches improved the computational approach by migrating the initial work to Java [6]. AnotherMatlab implementation, but using a global optimization algorithm [3], the EMA (Electromagnetic Algo-rithm), was also migrated to Java [7]. Then, the use of the EVA (Evolutionary Algorithm) was applied tothe same problem [5]. All of these implementations, only cover the SRpco (Single Resource Project CostOptimization) models where the total project cost C is obtained byC = E[∑a∈A(c × xa ×Wa) + cL ×max0,Υn − T] (1)where the following notation applies:A: Set of project activities;c: Quantity (of resource) cost per unitcL: Project delay cost per time unitxa ∈ IR+: Allocated quantity of resource on activitya such la 6 xa 6 uaWa ∼ Exp (λa): Work content of activity aΥn: Evaluated realization time of last nodeT : Schedule project realization timeThis paper addresses the more general case – the MRpco (Multiple Resources Project Cost Optimization)problem. In this case, the project has a set of resources and each activity may require several of theminstead of just one. On the SRpco models, the behavior of an activity was fully described by its singleresource. However, on the new MRpco models we need to describe that behavior for several resources. Eachresource will have its own work content and allocation constraints according to each activity’s needs.1.1 The Impact of Resource MultiplicityThe extension of the project cost evaluation from the SRpco model is quite straightforward. The multi-plicity of resources simply induces a sum of the allocation cost of each resource of each activity. Thus,the total project cost C isC = E[∑a∈A∑r∈Ra(cr × xar ×W ar ) + cL ×max0,Υn − T] (2)where the extended notation applies (refer to Eq. (1)):Ra: project resources subset needed by activity acr: quantity cost per unit of resource rxar ∈ IR+: allocated quantity of resource r on activ-ity a such lar 6 xar 6 uarW ar ∼ Exp (λar): Work content of resource r on ac-tivity aTo each resource allocation to an activity is associated an individual duration Y ar evaluated similar tothe SRpco model.Y ar =W arxar(3)The actual activity duration – Ya – is, therefore, the maximum of those individual ones.Ya = maxr∈RaY ar  (4)IESM 2009, MONTRE´AL - CANADA, May 13 - 15Clearly it makes little sense to expend more of a resource (and incur a higher cost) to have the activityduration under this resource less than its duration under any other resource. Thus, it is desired to have(in expectation)E [Y ai ] = E[Y aj], ∀i, j ∈ Ra (5)To ensure allocation vectors leading to the desired equality we devise the following strategy.2 Quantity Oriented StrategyThe QOras (Quantity Oriented Resource Allocation Strategy) tries to secure equality on individual du-rations, yielded by the resources within an activity, by electing one of those resources as the pivot. Then,using the equality on Eq. (5), the quantities for all resources are, immediately, known.For ease of explanation, we shall assume the existence of only two resources (indexed by 1 and 2) in thecontext of an activity a ∈ A, i.e. #Ra = 2Given the two resources, Eq. (4) becomes,Ya = maxY a1 ,Y a2 (6)where, by Eq. (3)Y a1 =W a1xa1Y a2 =W a2xa2(7)From the equality condition, it followsE [Y a1 ] = E[W a1xa1]=E [W a1 ]xa1(8)E [Y a2 ] =E [W a2 ]xa2(9)(10)Thus, the equality of the individual durations (in expectation) isE [W a1 ]xa1=E [W a2 ]xa2(11)and putting all the allocated quantities, say, in terms of xa1 , we obtainxa2 =E [W a2 ]E [W a1 ]xa1 (12)From the last equation we can put the allocation discussion. The process is not yet complete since weneed to enforce that all the resulting allocated quantities lie within their feasible intervals.In order to guide us throughout the allocation method, we start the following running example.Example – Part 1 of 2Assume that we have only two resources (indexed as 1 and 2) in an arbitrary fixed activity a. Suppose,also, thatxa1 ∈ [0.5, 2] E [W a1 ] = 1 xa2 ∈ [1.5, 3] E [W a2 ] = 2 (13)By using resource 1 as the pivot, we have from Eq. (12)xa2 = 2xa1 (14)In Table 1 there are four allocations resulting from the application of Eq. (14) to a set of four feasibleallocations of resource 1. As the table shows, some evaluated xa2 are out of bounds.IESM 2009, MONTRE´AL - CANADA, May 13 - 15Table 1QOras – Problems while allocating a resource by decision over anotherxa1 xa2 = 2xa1 xa2 in bounds?0.5 1.0 no (smaller)1.0 2.0 yes1.5 3.0 yes2.0 4.0 no (greater)In order to correct the invalid allocations we may simply truncate the values into the feasible region. So,whenever the invalid allocation is smaller (greater) than the lower (upper) bound, we enforce the value tobe equal to the lower (upper) bound. Thus, let xˆa2 be the corrected value of the allocation of the resource2:xˆa2 = minua2 ,maxla2 , E [W a2 ]E [W a1 ]xa1 (15)But, while the values for resource 2 are corrected, we must observe that it is required to adjust xa1 aswell.Example – Part 2 of 2Continuing our example, we must increase the xa2 from 1.0 to 1.5 = la2 and decrease from 4.0 to3.0 = ua2 , on the notable cases, respectively. Table 2 synthesizes these corrections and their implicationsover the proportionality relation. When xa2 increases (decreases) the correct value for xa1 must decreaseTable 2QOras – xˆa2 correctionxa2 xˆa2 Correction Implication on xa11.0 1.5 increase must decrease2.0 2.0 not needed no3.0 3.0 not needed no4.0 3.0 decrease must increase(increase) in proportion. By using a simple equation, derived from Eq. (14), we can evaluate the valueof xa1 corresponding to the newly adjusted xˆa2 . So, for xˆa2 = 1.5 it follows1.5 = 2xˆa1 ⇔ xˆa1 = 0.75 (16)By similar reasoning, when xˆa2 = 3.0, xˆa1 = 1.5.Both values of xˆa1 are within the feasible region. Therefore, the proportion between the two resourcesis successfully restored. On Table 3 are presented the final corrected allocations.Table 3QOras – Allocations before and after correctionxa1 xa2 xˆa1 xˆa20.50 1.00 0.75 1.501.00 2.00 1.00 2.001.50 3.00 1.50 3.002.00 4.00 1.50 3.00As just seen on the example, the use of a simple equation was sufficient to restore the proportionalityrelation. We will now formalize that correction and study its implications on the other allocations. Wewill do that by observing the three possible scenarios:IESM 2009, MONTRE´AL - CANADA, May 13 - 15la1 ua1la2ua2xa2xa1xa2 =E[Wa2 ]E[Wa1 ]xa1(a) Allocation interval of xa2 is a subset of therange of the function of Eq. (12)la1 ua1la2ua2xa2xa1xa2 =E[Wa2 ]E[Wa1 ]xa1(b) Range of function of Eq. (12) intersectsthe allocation interval of xa2 but does not in-cludes itla1 ua1la2ua2xa2xa1xa2 =E[Wa2 ]E[Wa1 ]xa1(c) Range of function of Eq. (12) does notintersect the allocation interval of xa2Fig. 1. QOras – Typical scenarios on the corrective mechanism on the context of two resourcesCase xa2 ∈[la2 , ua2]All allocated quantities are within their feasible space. Nothing is further required.Case xa2 < la2 Requires correction by putting xa2 = la2 . The pivot also needs correction in order to restorethe relation between the quantities. Thus,xˆa1 =E [W a1 ]E [W a2 ]la2 (17)Case xa2 > ua2 Analogously to the xa2 < la2 case, it goes xˆa2 = ua2 andxˆa1 =E [W a1 ]E [W a2 ]ua2 (18)After restoring the proportionality relation, there remains the possibility that the resulting xˆa1 laying outof its feasible region in which case we will need to truncate it, thus,xˆa1 = minua1 ,maxla1 , E [W a1 ]E [W a2 ]α (19)where α = la2 case xˆa2 < la2 or α = ua2 case xˆa2 > ua2 .This truncation, however, may raise a second need for restoration of the proportionality relation. But,this in turn will start a vicious cycle of corrections. Suffice it to say that this particular scenario onlyoccurs when the range of function on Eq. (12) does not intersect the feasible interval of xa2 (see Fig. 1).Which means that it is impossible to secure the equality between Y a1 and Ya2 .3 Discussion and ConclusionIn the two resources only case scenario, it is still quite manageable to assess the impossibility of equalindividual durations. But, the generalization to an arbitrary number can be tricky as the correctivemechanism itself brings unexpected complexity.IESM 2009, MONTRE´AL - CANADA, May 13 - 15xa1 xa2 xˆa2 xˆa1decision truncation restorationFig. 2. QOras – Corrective mechanism scheme (two resources) with resource 1 as pivotxa2 xˆa2 xˆa1xa1xa3 xˆa3 xˆa1 Moment A?xa2 xˆa2xa1 xˆa1xa3 xˆa3 Moment Bdecision truncation restoration balancing? balancing momentFig. 3. QOras – Corrective mechanism scheme (three resources) with resource 1 as pivotFig. 2 represents the corrective mechanism with only two resources, which may be satisfactory. However,with three or more resources a new problem has to be addressed, namely that of incompatibility betweenthe results secured from the two non-pivotal resources.The expansion of the decision equation is straightforward:E [W a1 ]xa1=E [W a2 ]xa2=E [W a3 ]xa3(20)from which we secure:xa2 =E [W a2 ]E [W a1 ]xa1 and xa3 =E [W a3 ]E [W a1 ]xa1 (21)When the corrective procedure is applied a new problem arises. Fig. 3 gives a schematic of what canhappen and the corrective action that may be taken. At first the two xˆa1 ’s that result from applying thecorrection to xa2 and xa3 independently are different, leading to inconsistency in the decision, which needsto be treated. A new mechanism must evaluate a suitable allocation of the pivot resource in order toachieve the desired result. Naturally, the issue becomes increasingly more complex as more resources areinvolved.Although the QOras is intuitive, it is incapable of enforcing equal durations and is difficult to generalizeto more than two resource.Instead of pursuing ways to improve QOras, we decided to diverge our approach and look for other strate-gies of multiple resources allocation, namely the WBras (Waste Balance Resource Allocation Strategy)presented last year in EngOpt2008 [4].References[1] Anabela P. Tereso. Gesta˜o de Projectos – Alocac¸a˜o adaptativa de recursos em redes de actividades multimodais. PhDthesis, Universidade do Minho, 2002.[2] Anabela P. Tereso, M. Madalena T. Arau´jo, and Salah E. Elmaghraby. Adaptive Resource Allocation in MultimodalActivity Networks. International Journal of Production Economics, 92(1):1–10, November 2004.[3] Anabela P. Tereso, M. Madalena T. Arau´jo, and Salah E. Elmaghraby. The Optimal Resource Allocation in StochasticActivity Networks via the Electromagnetism Approach. Ninth International Workshop on Project Management andScheduling (PMS’04), April 2004.IESM 2009, MONTRE´AL - CANADA, May 13 - 15[4] Anabela P. Tereso, M. Madalena T. Arau´jo, Rui Moutinho, and Salah E. Elmaghraby. Project management: multipleresources allocation. Rio de Janeiro - Brazil, June 1-5 2008. International Conference on Engineering Optimization(EngOpt2008). Full text published in the proceedings (ISBN 978857650156-5).[5] Anabela P. Tereso, Lino A. Costa, Rui A. Novais, and Madalena T. Arau´jo. The Optimal Resource Allocation inStochastic Activity Networks via the Evolutionary Approach: A Platform Implementation in Java. Beijing, China, May30 – June 2 2007. International Conference on Industrial Engineering and Systems Management (IESM’ 2007). Fullpaper published in the proceedings (ISBN 978-7-89486-439-0).[6] Anabela P. Tereso, Joa˜o Ricardo M. Mota, and Rui Jorge T. Lameiro. Adaptive Resource Allocation to StochasticMultimodal Projects: A Distributed Platform Implementation in Java. Control and Cybernetics Journal, 35(3):661–686,2006.[7] Anabela P. Tereso, Rui A. Novais, and M. Madalena T. Arau´jo. The Optimal Resource Allocation in Stochastic ActivityNetworks via the Electromagnetism Approach: A Platform Implementation in Java. Reykjavc, Iceland, July 2006. 21stEuropean Conference on Operational Research (EURO XXI). Submitted to the “Control and Cybernetics Journal”(under revision).

Quantity oriented resource allocation strategy on multiple resources projects under stochastic conditions

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