Quantity Oriented Resource Allocation Strategy on Multiple Resources Projects under Stochastic Conditions


 Curtis Atkins
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1 Interntionl Conference on Industril Engineering nd Systems Mngement IESM 2009 My MONTRÉAL  CANADA Quntity Oriented Resource Alloction Strtegy on Multiple Resources Projects under Stochstic Conditions Anbel Tereso, Mdlen Arújo, Rui Moutinho, Slh Elmghrby b Universidde do Minho, Guimrães, PORTUGAL b North Crolin Stte University, Rleigh, NC , USA Abstrct Previous developments from the first uthor nd other reserchers were mde on devising models for the totl cost optimiztion of projects described by ctivity networks under stochstic conditions. Those models only covered the single resource cse. The present pper will discuss the cse of multiple resources. More precisely, we introduce strtegy of lloction of those resources in order to minimize the wste rising from their ltent idleness on their consumption within the sme ctivity. On this strtegy we elect one resource s pivot nd write equtions tht describe ll the quntities for the other resources, on the sme ctivity. Key words: Project Mngement nd Scheduling, Stochstic Activity Networks, Resource Alloction, Multiple Resources List of Acronyms RCPSP: Resource Constrint Project Scheduling Problem AoA: ActivityonArc DP: Dynmic Progrmming EMA: Electromgnetic Algorithm EVA: Evolutionry Algorithm SRpco: Single Resource Project Cost Optimiztion MRpco: Multiple Resources Project Cost Optimiztion QOrs: Quntity Oriented Resource Alloction Strtegy WBrs: Wste Blnce Resource Alloction Strtegy 1 Introduction This pper follows the reserches mde by severl contributors strting from the first reserch mde by the first uthor (see [1]). Those works ddress RCPSP (Resource Constrint Project Scheduling Problem) where we wnt to minimize the totl project cost involving multimodl ctivities under stochstic conditions. More precisely, given n project on its plnning phse, where ll its ctivities nd relted requirements re estblished, we wnt to determine which is the optiml lloction of resources. This Corresponding uthor A. Tereso. Emil ddresses: (Anbel Tereso), (Mdlen Arújo), (Rui Moutinho), (Slh Elmghrby).
2 lloction yields the minimum totl project cost, which is composed by the sum of the lloction cost nd the dely cost. All the reserches so fr, work with AoA (ActivityonArc) network representtion of the project ctivities nd their precedences. By treting multimodl ctivities, we del with ctivities which their durtion time is mesured s function of the resource lloction. As n exmple, if certin ctivity requires 6 dys with 1 mn llocted, it should require only 2 dys when the lloction is 3 men. However, in the stochstic context, tht reltion between lloction nd durtion is not deterministic. This requires tht we tke the Work Content of n ctivity s probbilistic distribution. In prticulr, we hve been using the exponentil distribution for tht mtter. Since the first pproch to this problem, on Mtlb using DP (Dynmic Progrmming) [2], the subsequent reserches improved the computtionl pproch by migrting the initil work to Jv [6]. Another Mtlb implementtion, but using globl optimiztion lgorithm [3], the EMA (Electromgnetic Algorithm), ws lso migrted to Jv [7]. Then, the use of the EVA (Evolutionry Algorithm) ws pplied to the sme problem [5]. All of these implementtions, only cover the SRpco (Single Resource Project Cost Optimiztion) models where the totl project cost C is obtined by [ ] C = E (c x W ) + c L mx 0, Υ n T A where the following nottion pplies: (1) A: Set of project ctivities; c: Quntity (of resource) cost per unit c L: Project dely cost per time unit x IR + : Allocted quntity of resource on ctivity such l x u W Exp (λ ): Work content of ctivity Υ n: Evluted reliztion time of lst node T : Schedule project reliztion time This pper ddresses the more generl cse the MRpco (Multiple Resources Project Cost Optimiztion) problem. In this cse, the project hs set of resources nd ech ctivity my require severl of them insted of just one. On the SRpco models, the behvior of n ctivity ws fully described by its single resource. However, on the new MRpco models we need to describe tht behvior for severl resources. Ech resource will hve its own work content nd lloction constrints ccording to ech ctivity s needs. 1.1 The Impct of Resource Multiplicity The extension of the project cost evlution from the SRpco model is quite strightforwrd. The multiplicity of resources simply induces sum of the lloction cost of ech resource of ech ctivity. Thus, the totl project cost C is [ ] C = E (c r x r Wr ) + c L mx 0, Υ n T A r R where the extended nottion pplies (refer to Eq. (1)): (2) R : project resources subset needed by ctivity c r: quntity cost per unit of resource r x r IR + : llocted quntity of resource r on ctivity such lr x r ur Wr Exp (λ r): Work content of resource r on ctivity To ech resource lloction to n ctivity is ssocited n individul durtion Yr the SRpco model. = W r The ctul ctivity durtion Y is, therefore, the mximum of those individul ones. Y r x r evluted similr to Y = mx r R Y r (4) (3)
3 Clerly it mkes little sense to expend more of resource (nd incur higher cost) to hve the ctivity durtion under this resource less thn its durtion under ny other resource. Thus, it is desired to hve (in expecttion) E [Yi ] = E [ Yj ], i, j R (5) To ensure lloction vectors leding to the desired equlity we devise the following strtegy. 2 Quntity Oriented Strtegy The QOrs (Quntity Oriented Resource Alloction Strtegy) tries to secure equlity on individul durtions, yielded by the resources within n ctivity, by electing one of those resources s the pivot. Then, using the equlity on Eq. (5), the quntities for ll resources re, immeditely, known. For ese of explntion, we shll ssume the existence of only two resources (indexed by 1 nd 2) in the context of n ctivity A, i.e. #R = 2 Given the two resources, Eq. (4) becomes, Y = mx Y 1, Y2 (6) where, by Eq. (3) Y 1 = W 1 Y 2 = W 2 (7) From the equlity condition, it follows [ ] W E [Y1 ] = E 1 E [Y 2 ] = E [W 2 ] = E [W 1 ] (8) (9) (10) Thus, the equlity of the individul durtions (in expecttion) is E [W 1 ] = E [W 2 ] (11) nd putting ll the llocted quntities, sy, in terms of, we obtin = E [W 2 ] E [W 1 ]x 1 (12) From the lst eqution we cn put the lloction discussion. The process is not yet complete since we need to enforce tht ll the resulting llocted quntities lie within their fesible intervls. In order to guide us throughout the lloction method, we strt the following running exmple. Exmple Prt 1 of 2 Assume tht we hve only two resources (indexed s 1 nd 2) in n rbitrry fixed ctivity. Suppose, lso, tht [0.5, 2] E [W 1 ] = 1 [1.5, 3] E [W 2 ] = 2 (13) By using resource 1 s the pivot, we hve from Eq. (12) = 2 (14) In Tble 1 there re four lloctions resulting from the ppliction of Eq. (14) to set of four fesible lloctions of resource 1. As the tble shows, some evluted re out of bounds.
4 Tble 1 QOrs Problems while llocting resource by decision over nother = 2 in bounds? no (smller) yes yes no (greter) In order to correct the invlid lloctions we my simply truncte the vlues into the fesible region. So, whenever the invlid lloction is smller (greter) thn the lower (upper) bound, we enforce the vlue to be equl to the lower (upper) bound. Thus, let ˆ be the corrected vlue of the lloction of the resource 2: ˆ = min u 2, mx l 2, E [W 2 ] E [W1 ]x 1 (15) But, while the vlues for resource 2 re corrected, we must observe tht it is required to djust s well. Exmple Prt 2 of 2 Continuing our exmple, we must increse the from 1.0 to 1.5 = l 2 nd decrese from 4.0 to 3.0 = u 2, on the notble cses, respectively. Tble 2 synthesizes these corrections nd their implictions over the proportionlity reltion. When increses (decreses) the correct vlue for must decrese Tble 2 QOrs ˆ correction ˆ Correction Impliction on increse must decrese not needed no not needed no decrese must increse (increse) in proportion. By using simple eqution, derived from Eq. (14), we cn evlute the vlue of corresponding to the newly djusted ˆ. So, for ˆ = 1.5 it follows By similr resoning, when ˆ = 3.0, ˆ = = 2ˆ ˆ = 0.75 (16) Both vlues of ˆ re within the fesible region. Therefore, the proportion between the two resources is successfully restored. On Tble 3 re presented the finl corrected lloctions. Tble 3 QOrs Alloctions before nd fter correction ˆ ˆ As just seen on the exmple, the use of simple eqution ws sufficient to restore the proportionlity reltion. We will now formlize tht correction nd study its implictions on the other lloctions. We will do tht by observing the three possible scenrios:
5 = E[W 2 ] E[W 1 ] x 1 = E[W 2 ] E[W 1 ] x 1 u 2 u 2 l 2 l 2 l 1 u 1 l 1 u 1 () Alloction intervl of is subset of the rnge of the function of Eq. (12) (b) Rnge of function of Eq. (12) intersects the lloction intervl of but does not includes it u 2 = E[W 2 ] E[W 1 ] x 1 l 2 l 1 u 1 (c) Rnge of function of Eq. (12) does not intersect the lloction intervl of Fig. 1. QOrs Typicl scenrios on the corrective mechnism on the context of two resources Cse [ l2, u ] 2 All llocted quntities re within their fesible spce. Nothing is further required. Cse < l 2 Requires correction by putting x 2 = l2. The pivot lso needs correction in order to restore the reltion between the quntities. Thus, ˆ = E [W 1 ] E [W 2 ]l 2 (17) Cse > u 2 Anlogously to the x 2 < l 2 cse, it goes ˆ = u 2 nd ˆ = E [W 1 ] E [W 2 ]u 2 (18) After restoring the proportionlity reltion, there remins the possibility tht the resulting ˆ lying out of its fesible region in which cse we will need to truncte it, thus, where α = l 2 cse ˆ < l 2 or α = u 2 cse ˆ > u 2. ˆ = min u 1, mx l 1, E [W 1 ] E [W2 ]α (19) This trunction, however, my rise second need for restortion of the proportionlity reltion. But, this in turn will strt vicious cycle of corrections. Suffice it to sy tht this prticulr scenrio only occurs when the rnge of function on Eq. (12) does not intersect the fesible intervl of (see Fig. 1). Which mens tht it is impossible to secure the equlity between Y 1 nd Y 2. 3 Discussion nd Conclusion In the two resources only cse scenrio, it is still quite mngeble to ssess the impossibility of equl individul durtions. But, the generliztion to n rbitrry number cn be tricky s the corrective mechnism itself brings unexpected complexity.
6 ˆ ˆ decision trunction restortion Fig. 2. QOrs Corrective mechnism scheme (two resources) with resource 1 s pivot ˆ ˆ ˆ? ˆ x 3 ˆx 3 ˆ x 3 ˆx 3 Moment A Moment B decision trunction restortion blncing? blncing moment Fig. 3. QOrs Corrective mechnism scheme (three resources) with resource 1 s pivot Fig. 2 represents the corrective mechnism with only two resources, which my be stisfctory. However, with three or more resources new problem hs to be ddressed, nmely tht of incomptibility between the results secured from the two nonpivotl resources. The expnsion of the decision eqution is strightforwrd: E [W 1 ] = E [W 2 ] = E [W 3 ] x 3 (20) from which we secure: = E [W 2 ] E [W 1 ]x 1 nd x 3 = E [W 3 ] E [W 1 ]x 1 (21) When the corrective procedure is pplied new problem rises. Fig. 3 gives schemtic of wht cn hppen nd the corrective ction tht my be tken. At first the two ˆ s tht result from pplying the correction to nd x 3 independently re different, leding to inconsistency in the decision, which needs to be treted. A new mechnism must evlute suitble lloction of the pivot resource in order to chieve the desired result. Nturlly, the issue becomes incresingly more complex s more resources re involved. Although the QOrs is intuitive, it is incpble of enforcing equl durtions nd is difficult to generlize to more thn two resource. Insted of pursuing wys to improve QOrs, we decided to diverge our pproch nd look for other strtegies of multiple resources lloction, nmely the WBrs (Wste Blnce Resource Alloction Strtegy) presented lst yer in EngOpt2008 [4]. References [1] Anbel P. Tereso. Gestão de Projectos Alocção dpttiv de recursos em redes de ctividdes multimodis. PhD thesis, Universidde do Minho, [2] Anbel P. Tereso, M. Mdlen T. Arújo, nd Slh E. Elmghrby. Adptive Resource Alloction in Multimodl Activity Networks. Interntionl Journl of Production Economics, 92(1):1 10, November [3] Anbel P. Tereso, M. Mdlen T. Arújo, nd Slh E. Elmghrby. The Optiml Resource Alloction in Stochstic Activity Networks vi the Electromgnetism Approch. Ninth Interntionl Workshop on Project Mngement nd Scheduling (PMS 04), April 2004.
7 [4] Anbel P. Tereso, M. Mdlen T. Arújo, Rui Moutinho, nd Slh E. Elmghrby. Project mngement: multiple resources lloction. Rio de Jneiro  Brzil, June Interntionl Conference on Engineering Optimiztion (EngOpt2008). Full text published in the proceedings (ISBN ). [5] Anbel P. Tereso, Lino A. Cost, Rui A. Novis, nd Mdlen T. Arújo. The Optiml Resource Alloction in Stochstic Activity Networks vi the Evolutionry Approch: A Pltform Implementtion in Jv. Beijing, Chin, My 30 June Interntionl Conference on Industril Engineering nd Systems Mngement (IESM 2007). Full pper published in the proceedings (ISBN ). [6] Anbel P. Tereso, João Ricrdo M. Mot, nd Rui Jorge T. Lmeiro. Adptive Resource Alloction to Stochstic Multimodl Projects: A Distributed Pltform Implementtion in Jv. Control nd Cybernetics Journl, 35(3): , [7] Anbel P. Tereso, Rui A. Novis, nd M. Mdlen T. Arújo. The Optiml Resource Alloction in Stochstic Activity Networks vi the Electromgnetism Approch: A Pltform Implementtion in Jv. Reykjvc, Icelnd, July st Europen Conference on Opertionl Reserch (EURO XXI). Submitted to the Control nd Cybernetics Journl (under revision).
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