Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT


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1 Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee several idepedet projects The problem is to miimise the time of all the projects implemetatio or the weighted sum of termiatio times The described approach is based o the represetatio of a project as the separate operatio The the problem of optimal resource allocatio betwee the operatios is solved Give resource allocatio for every project, oe ca solve the problem of allocatio iside the multiproject Allocatio ad aggregatio methods are explored INTRODUCTION The developmet of the society, ecoomy, eterprise ad the developmet of the certai ma may be represeted as the set of discrete processes with give termial goals uder the costraits o time ad resources It is coveiet for a ma to divide the process of his activity o local processes Projects are processes of chages, ie orepeated processes, which require for their implemetatio special methods of maagemet For example, regular morig exercisig is ot a project as it is repeatable ad does ot require special orgaisatioal efforts But learig ew morig exercises may be cosidered as a project Daily productio o the eterprise is ot the project too, but itroductio of ew techology is a project Evidetly, the differece is ot obvious  costructio of the buildig is a project, but there is o reaso to cosider the stadard block lodges productio with their istallatio o certai place as a project I the former Soviet Uio project maagemet was broadly used sice the ed of 60th ad was referred to as the etwork plaig ad maagemet The base of project maagemet is the presetatio of the project etwork
2 graphic, which reflects the depedece betwee differet operatios of the project I 70th the iterest to the methods of etwork plaig ad maagemet decreased, as the reasos of oefficiet maagemet were deeper  they laid i the basis of the publicpolitical ad ecoomic priciples of state Nowadays i Russia project maagemet outlives a secod birth The Russia Associatio of Project Maagemet (SOVNET) is the member of the Iteratioal Project Maagemet Associatio (INTERNET) Multiprojects are a importat class of projects Multiproject is a project, which cosists of several techologically idepedet projects, uited by the shared resources (fiacial or material, etc) This paper cosiders methods ad mechaisms of multuprojects maagemet The described approach is based o the presetatio of a project as the separate operatio The the problem of optimal resource allocatio betwee the operatios is solved Give resource allocatio o every project, oe ca solve the problem of allocatio for the multiproject (betwee the projects  operatios of the multiproject) Several allocatio ad aggregatio methods are explored below RESOURCE ALLOCATION BETWEEN INDEPENDENT OPERATIONS Cosider the multiproject, which cosists of idepedet projects Each project is aggregately described as the operatio with two characteristics: the volume of the project  W i ad the depedece betwee the rate of project implemetatio w i (t) = f i (u i (t)) ad the amout of resources u i (t) at time t The volume, the velocity ad the termiatio time T i are itercoected trough the followig equatio: Ti [ i ()] fi u t dt = Wi 0 Let the total amout of resources for the multiproject is give ad equals N(t) The problem is to allocate this resources betwee the operatios so, that 2 ()
3 the termiatio time T = max T be miimal If for ay t N(t)=N (eve arrival of i i resources) ad f i (u i ) are cocave fuctios, the resource allocatio problem has the solutio, defied i [, 2, 3] It is well kow that the optimal allocatio is characterised by the followig properties: à) each operatio is implemeted uder the fixed level of resources u i (t)=u i, i =, t [0, T], ie with a fixed rate; á) all the operatios termiate simultaeously Thus, w i = W i /T is a costat rate of ith operatio Deote ϕ i (w i )  the fuctio, iversed to fuctio f i (u i ), the u i = ϕ i (W i /T) is the amout of resources to termiate ith operatio at time T Miimal value of T is defied by the followig equatio ϕ i W T i = Let N(t) be piece wise costat fuctio: N(t) = N k, t [ ~ T k, ~ T k ), k N =, p, ~ T 0 = 0 Fix some k ad cosider the problem of multiproject s implemetatio at the time, less tha ~ T k Deote x iq  the volume of ith operatio, implemeted i the qth iterval Obviously (2) k xiq = Wi q= (3) As i the iterval [T q, T q ) the resources arrive evely, the values {x iq } i the optimal solutio satisfy the followig costrait: ϕ i x T iq q N = q ~ ~, q =, k, Tq = Tq Tq ad for the last iterval the followig coditio holds: ϕ i xik T = N k ~ ~, T = T T k, where T is the time of all the operatios termiatio 3
4 Deote h q  the vector with compoets h iq = ϕ i (w iq ), i =, The vector h is «timeivariat» (its directio does ot alter with the chage of time iterval, while its legth chages): h = γ h q = 2 k q q,,, h { } = h i =, γ This characteristic of the optimal solutio allows to reduce the resource allocatio problem to the solutio of the oliear equatios system with ( + k) variables {h i }, i =,, {γ q }, ad T: [ ( h )] ϕ ξ γ = N i i q i q, q =, k (4) k q= ( ) ( ) ξ γ h T + ξ γ h T = W i q i q i k i i, i =, (5) Let f i (u i ) are cocave fuctios, the, give the total amout of resources (fiacig) for the iterval [0, T], maximum of the implemeted operatios volume is reached uder eve (homogeous) resources arrivemet The proved fact allows to optimise the schedule of resources arrivemet Such a tuig may be achieved by the shift of fiacig o later periods Hitherto we cosidered the problem of multiproject time miimisatio However, the other problem is ot of the less importat The matter is that after the termiatio of ay project oe should receive some icome The delay i the termiatio leads to the decrease of icome Assume that ith project returs after its termiatio the icome c i per time uit The the possible decrease of the icome (lost icome) is c i t i, while the total decrease equals C = ct i i (6) Thus the followig problem arises: to allocate resources betwee the projects to miimise (6) 4
5 2 THE GENERAL CASE Above we have cosidered the problem of resource allocatio uder the assumptio that the rate of project implemetatio is a cocave fuctio of the resources amout Now we tur to the geeral case Let f i (u i ) be some limited, cotiuos o the right fuctios, such that f i (0)=0 Defie the set Y of pairs (u, w) i the followig way: {(, ) : ( )} Y = u w > 0 w f u (7) i i i i i i Note, that if f i (u i ) is a cocave fuctio, the Y i is a covex set Geerally Y i is ot a covex set The covex shell of this set is the covex set Y ~ i, such that ay poit may be represeted as the covex liear combiatio of the poits from ~ the set Y i The border f i( u i ) of this set is, obviously, a cocave fuctio Cosider the resource allocatio problem for the multiproject with the followig ~ rates of projects implemetatio { f i( u i ) } Suppose, that we have some optimal allocatio {u 0 i } ad operatios termiatio times { t i } Theorem There exists feasible resource allocatio with the same operatios termiatio times t i 3 AGGREGATION METHODS FOR A COMPLEX OF OPERATIONS Cosider the methods of the complex of operatios descriptio i the form of some uique operatio Oe should determie the volume of the aggregated operatio Wý, which is referred to as the equivalet volume of the complex, ad the depedece betwee the rate of the aggregated operatio implemetatio ad the amout of resources, allocated for the complex: W ý (t) = F ý (N(t)) (8) Defiitio Aggregatio is idetified as ideal if for ay fuctio N(t) there exists optimal resource allocatio {u i (t)} betwee the operatios of the complex, such that 5
6 u () t Nt (), i meawhile miimal termiatio time of the complex satisfies the followig coditio: T mi ( ()) Fэ Nt dt = Wэ 0 The classical example of the ideal aggregatio is the followig : ( ) α i i i f u = u, α, i =, Really, for this case it was proved [3], that there exists equivalet volume Wý of the complex ad the fuctio F ý (N) = N α, such that for ay resources level N(t) the followig coditio holds T mi [ ()] FNt dt = Wэ Herewith, there exists optimal allocatio {u i 0 (t)}, such that 0 0 ui() t = Nt (), ad complex s termiatio time equals T mi Let for ay t N(t) = N, the the allocatio {u 0 i (t)} possesses the followig iterestig characteristics: Each operatio is implemeted without ay breaks with the costat amout of resources, ie H ( ) = [ i ] 0 0 i i i u t u, t t, t, where t H i, t 0 i are momets of the ith operatio begiig ad termiatio 2 The resources {u i } form the flow o the etwork graph Let us describe the algorithm of the equivalet volume of the complex determiatio First oe should defie the depedece betwee costs S = u τ ad operatio s time: 6 (9)
7 S α α α w w = α τ = τ τ Cosider the problem of optimisig the complex by cost: to determie operatios times to implemet the complex i time Ò with miimal costs = ( τ ) S S (,) It is well kow that ecessary ad sufficiet coditios for optimality is the followig: for ay evet i o the etwork (excludig iitial ad termial oes): j ds dτ ( τ) ds ki ( τ ki ) = dτ k ki (0) () But, as ds dτ ( τ) α = α u, the coditio () is equivalet to the flowece coditio for the resources at ay evet The miimum of the costs, uder the coditio that resources form the flow i the etwork, is equivalet to the miimum of the resources amout N To miimise the costs o the etwork the efficiet Kelly algorithm may be applied O each step of this algorithm coditio () is tested for the odes of the etwork, ad the time of the correspodig evet is corrected to satisfy this coditios, thus the value of (0) decreases Whe some solutio satisfies () for all the odes, the value of costs (0) is miimal ad, cosequetly, total amout of resources N mi is miimal too Equivalet volume of the complex is defied by W ý = TN α = S α mi T α Let us describe aother method of the equivalet complex s volume estimatio This method is based o the followig theorem: 7
8 Theorem 2 Equivalet volume of the complex is a covex homogeous fuctio of operatios volumes W The theorem allows to obtai overestimates of the equivalet volume of the complex, due to the presetatio of the complex i the form of the covex liear combiatio of other complexes with kow equivalet volumes Example Cosider the complex of operatios, preseted of fig The precise value of complex s equivalet volume equals W ý = 3 4 9, 2 Cosider two complexes of operatios, preseted o fig 2 à, b It easy to see that the average of this complexes volumes gives the first complex For the secod complex we obtai ad for the complex from fig 2b: Thus for the origial complex: 2 2 W э = , 7, W э = ,6 2 ( ) Wэ Wэ + Wэ = 26, 2 The deviatio from the precise estimate is 2,4 or approximately 2,5% Accuracy of the estimate may be improved by the selectio of differet values W, W 2 ad α, such that W = αw + (  α)w 2 8
9 0 3 2 Fig (5) (6) 0 3 (8) 2 (0) Fig 2à (5) (6) 0 3 (8) 2 (5) Fig 2b 9
10 For example, if α, the: W ý 8, W ý 2 3/(α), W ý = 2, ie the deviatio is approximately 9% 4 THE LINEAR CASE Cosider the liear depedece betwee the rates of operatios ad the amout of resources: f i u, u a ( ui) = a, u a i i i i i i Deote τ i = w i /u i  the miimal time of ith operatio Let us costruct the itegral graphic of resource utilisatio o the complex of operatios uder the assumptio that all the operatios are iitialised at the latest time momets Graphic of resources utilisatio is a aggregated descriptio of the project Really, give the aggregated descriptios of all the projects (rightshifted graphics of the projects), we are able to solve the problem of the optimal resource allocatio for the multiproject both for the criteria of time miimisatio ad for the lost icome miimisatio Time miimisatio problem for the multiproject is solved rather simple Let S i (t,t) be the itegral graphic of ith project resources utilisatio uder the coditio of its termiatio at time momet T, S(, tt) = S (, tt)  the itegral graphic of multiproject resources utilisatio, M(t)  the itegral graphic of total amout of resources (fig 3) i 0
11 M(t) M(t) S(t, T) S i (T i ) S(t, T) T Fig 3 T 0 t To determie the miimal time T 0 of the multiproject implemetatio oe should shift the graphic S(t, T) to the left (see fig 3) util it will touch the graphic M(t) Havig got itegral feasible graphic of resource allocatio for the multiproject, oe ca split it ito the graphics of resource allocatio for certai projects (the process of decompositio) The lost icome miimisatio problem is ot charactarized by such a «easy» solutio method If the umber of projects is ot very high, it may be solved by the aalysis of all the possible combiatios It s worth otig that if the priority of the projects is fixed, the the allocatio is obtaied by the shift (i give sequece) of the itegral graphics S i (t, T) to the left util touchig the graphic M(t) If the umber of the projects is high eough, heuristic rule, give i sectio 2, may be applied
12 CONCLUSION The paper cotais the developmet of the optimal resource allocatio methods for multiprojects The idea of the decompositio, based o aggregatio of the projects ad the cosequet solutio of resource allocatio problems for the set of idepedet operatios, tured to be extremely fruitful It is importat that for most of the practically used i the project maagemet depedecies of operatios rates from the amout of resources (liear, expoetial, etc), the ideal aggregatio is possible ad adequate Moreover, if the parameters of the projects are ot commo kowledge, the the gametheoretical problems of truthtellig ad coordiated plaig arise Some approaches to the costructio of project maagemet mechaisms, which are efficiet, coordiated ad omaipulable, are described i [4,5] The aggregatio problem for the projects with much more complex depedecies of operatios rates from the amout of resources requires further studies The solutio of the applied problems of projects fiacig i the framework of the developed approach, also seems rather perspective REFERERNCES BURKOV VN Resource allocatio problem as the problem of optimal time maagemet Automatio ad Remote Cotrol, 966, N 7 2 BURKOV VN Applicatio of the optimal cotrol theory to the resource allocatio problems Proceedigs of III Iteratioal Coferece o Automatio Cotrol, Odessa, «Productio maagemet», BURKOV VN Optimal cotrol of operatios complexes Proceedigs of IV Iteratioal IFAC Cogress, Warsaw, 969 2
13 4 Burkov VN, Ealeev AK, Stimulatio ad decisiomakig i the active system theory: Review of problem ad ew results, Mathematical Social Scieces, Vol 27 (994), pp Burkov VN, Novikov DA How to maage the projects Moscow, SINTEG, 997 3
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