An optimal resource allocation approach to stochastic multimodal projects had been previously developed by applying a Dynamic Programming
model, which proved to be very demanding computationally.
Approximations to the initial model had been also developed, still within DP framework. Computing times were improved, but demonstrated the need for further developments. In this paper we report on the application of a recently developed technique for global optimization, the Electromagnetism Algorithm (EMA), to this problem and demonstrate its superior performance to previously attempted approximations

Araújo, Maria Madalena Teixeira de

Elmaghraby, Salah

Tereso, Anabela Pereira

Universidade do Minho: RepositoriUM

The Optimal Resource Allocation in StochasticActivity Networks via The ElectromagnetismApproachAnabela P. Tereso, M. Madalena T. AraújoUniversidade do Minho, 4800-058 GuimarãesPORTUGALanabelat@dps.uminho.pt; mmaraujo@dps.uminho.ptSalah E. ElmaghrabyNorth Carolina State University, Raleigh, NC 27695-7906USAelmaghra@eos.ncsu.eduOctober, 2004AbstractAn optimal resource allocation approach to stochastic multimodalprojects had been previously developed by applying a Dynamic Program-ming model [2] [3], which proved to be very demanding computationally.Approximations to the initial model had been also developed, still withinDP framework [4]. Computing times were improved, but demonstratedthe need for further developments. In this paper we report on the appli-cation of a recently developed technique for global optimization, the Elec-tromagnetism Algorithm (EMA) [1], to this problem and demonstrate itssuperior performance to previously attempted approximations.Key Words: Project Management and Scheduling, Stochastic ActivityNetworks, Resource Allocation, Electromagnetism Algorithm.1 Problem DefinitionThe problem treated in this paper may be stated as follows. We are given amultimodal activity network (that is, each activity can be performed at anynumber of levels of resource intensity applied to it, with resulting shorter orlonger duration), with stochastic work content {Wa} for each activity a ∈ A,where A is the set of activities, with |A| = n. We assume that the Wa’s are1stochastically independent. The total cost incurred in the performance of theproject is composed of two parts: the “resource cost” which is assumed to beproportional to the square of the intensity of resource usage for the durationof the activity, with constant of proportionality equal to ka representing themarginal cost per unit of the resource; and the “tardiness penalty” which isproportional to the amount of tardiness from a specified due date T , with con-stant of proportionality equal to cL representing the marginal cost per period.The duration of an activity, denoted by Ya, depends on its work content andon the amount of resource allocated to it; Ya =Wa/xa, a random variable. Forthe sake of simplicity of exposition we make the following assumptions: (i) Thework content of each activity is a random variable (r.v.) exponentially distrib-uted with parameter λa, which may be diﬀerent for diﬀerent activities; (ii) Theintensity of resource allocation is restricted to be within a lower and an upperbounds xa ∈ [la, ua] with 0 ≤ la ≤ ua < ∞; (iii) There is only one resource ofunlimited availability so that it does not impose any limitations on the numberof concurrent activities. The goal is to minimize the total cost by selecting theintensity xa of resource application to each activity a ∈ A.This problem was first treated by dynamic programming [2], and was testedon a set of examples [3] which alerted us to the impending diﬃculty of applyingthe approach to large scale problems. In that research, since our focus was on“proof of concept", we could not claim achieving the exact optimum, mainly dueto the need to discretize the decision space at a finite number of points (a max-imum of five was used). Still, successive implementation with increasingly finermesh should give the optimum to any desired degree of accuracy. Lowering oursights, our attention was then directed to devising approximation procedures,still within the framework of DP, that would not detract seriously from thequality of the solution while improving the computational performance. Theseapproaches are described in Tereso et al. [4]. Continued interest in seekingbetter approximations impelled us to apply a totally diﬀerent approach to thisproblem, in the hope of achieving our objective. We settled on the Electromag-netism Algorithm (EMA) of Birbil and Fang [1] as the most promising candidate2among the known meta-heuristic approaches, for three good reasons. First, itsexceptional performance in determining the optimum for all the problems onwhich it has been implemented, as reported by Birbil and Fang [1]. Second,the speed with which it converges to the optimal solution. Third, its novelty.This approach has been implemented successfully on numerous deterministicfunctions; but never in a stochastic setting such as ours. To the best of ourknowledge, this is the first application of the EMA to a stochastic optimizationproblem. This paper is a report on our findings with this approach.In section 2 we give a synopsis of the fundamental concepts of the EMA pro-cedure since we suspect that most readers will not be familiar with it. Section3 describes its application to our problem. Section 4 summarizes our computa-tional results. Finally, section 5 gives the conclusions and the directions of ourcontinued research on this paradigm.2 The Electromagnetism AlgorithmThe EMA [1] is based on the principles of electromagnetism: two particlesexperience forces of mutual attraction or repulsion depending on their charges.One may think of each point as being a “particle” in the n-dimensional space(of decision variables), with an associated “charge”. The particle is free to movein the space, with concomitant change in the decision variables. The particle’scharge determines the magnitude of the force of attraction or repulsion betweenit and any other point in the space. The strength of the attraction/repulsion isdirectly proportional to the product of their charges and inversely proportionalto the distance between them. The direction of the force between any twoparticles is along the line between them. The resultant (in vector sense) of all theforces acting on each particle from all other particles in the space determines themagnitude and direction of movement of the particle. All particles are displacedsimultaneously by an amount ∆ to be determined at each stage of calculations,which decreases with progress in the iterations and relative stability of the bestresponse. With the new layout of the particles in the space of decisions, the3same steps are repeated until iteration is halted according to any of several pre-specified conditions that usually relate to the incremental gain in the objectivefunction over the last few iterations or to the total number of iterations.3 Application of the ElectromagnetismAlgorithmIn our case, we have a set of activities with associated stochastic work con-tent, and the single resource to allocate to them. The correspondence betweenthe above narrative and our problem may be conceived as follows. We haveto contend with four structural parameters: (i) The size of the population ofparticles M. In our experimentation we fixed M = 15 for all networks. (ii) Thenumber of samples K of the vector of work contents. We selected K = 100.(iii) The number of iterations I of the application of the EMA to convergence(or abortion). In our experiments the stopping condition we selected was tohalt when the number of iterations performed is 25n, as suggested in Birbil andFang [1], since this number of iterations was found to be ample for convergence.(iv) The number of replications R of the experiment. We chose R = 4. Theimport of these parameters will become clear as our detailed description of theexperimental layout progresses.Observe that each particle x(m) is a vector of n elements, defined by thevector of resource allocationx(m) =³x(m)1 , · · · , x(m)n´; m = 1, · · · ,M. (1)Here,M is the size of the population of particles and n is the number of activitiesin the project. For instance, in a project of n = 30 activities, and M = 15, wetake 15 points in the hypercube defined by the n inequalities la ≤ xa ≤ ua. Weselect the M particles to span the feasible space of the resource allocation,as much as possible. Figure 1 illustrates a minuscule project of only n = 2activities, M = 5 particles, and the forces on particle #3. Each particle in thisspace is defined by its two dimensions, x(m) =³x(m)1 , x(m)2´, for m = 1, · · · , 5.The “charge” of each particle is a measure of the value of the objective function4x(5)x(1)x(2)x(3)x(4)F(3,5)F(3,2) F(3,1)F(3,5)Resultant ForceF(3)Each X(k) = (x1(k), x2(k))la = l for all aand ua = u for all aWe assume that v(3) < v(1), v(5),  therefore repulsed by themBut  v(3) > v(2), v(4), therefore attracted to themuulFeasible square of the allocationsFigure 1: Forces on particle #3at that point, which is denoted by v¡x(m)¢, m = 1, · · · ,M . This value isdetermined through Monte Carlo sampling of the vector of work content (wa)a∈Awhich, together with the allocation x(m), determine the “resource cost” as wellas the time of project completion, denoted by t(m)P through standard criticalpath calculations. Knowledge of t(m)P enables one to determine the penalty fortardiness beyond the specified project due date T.We define vmin as the minimal value among all M points,vmin = minmnv³x(m)´o.It is important, for stability reasons, to “normalize” and “scale” these values,which result in the charge q(m) at point x(m). This charge is evaluated as follows,q(m) = exp"−n× v(x(m))− vminPMk=1£v(x(k))− vmin¤# , m = 1, 2, ...,M.5Observe that a large v(x(m)) results in a small q(m), and conversely, a smallv(x(m)) results in a large q(m). Indeed, at vmin the charge is 1, the maximum.The charge q(j) of particle j determines the force of attraction or repulsionbetween particle j and the other particles. For each pair of particles x(j) andx(k) suppose that v¡x(j)¢< v¡x(k)¢, which implies that q(j) > q(k). Thenparticle x(k) is “attracted” to particle x(j) by a force given byF (j, k) ="³x(j) − x(k)´× q(j)q(k)°°x(j) − x(k)°°2#, ∀ j, k, (2)and particle x(j) is “repulsed” by particle x(k) by a force of the same magnitudein the opposite direction. The direction of the attraction/repulsion force isalong the line between the two particles with the arrow pointing from x(k) tox(j) for particle k and the reverse for particle j (i.e., along the same line awayform k). The (vector) resultant force F (m) on each particle m is calculated byconventional methods to yield the magnitude of the force and its direction ofmovement; see Fig.1. The force F (m) is then normalized to yield,F (m) = vector sum (F (j,m)) , j 6= m, m = 1, · · · ,M.This procedure is repeated for each particle x(m), m = 1, · · · ,M to yield thevalues of the forces at all M particles in the population. Each particle x(m) isthen moved in the specified direction by a random step given by,x(m0) ← x(m) + β · (RNG)(m) · F (m)normβ ∈ (0, 1) ,in which β is selected randomly and (RNG)(m) is the “range" of movement ofthe particle; a measure of the allowed feasible movement toward the upper orlower bounds, depending on the movement of the point.The movement of the particles continues X(1) → X(2) → · · · → X(I) untilthe stopping condition is satisfied (with I = 25n). The allocation that yieldedvmin when iteration is halted is selected as the “optimal” allocation.For each replication (R) of the experiment, a set of K = 100 W ’s is gen-erated and stays fixed thereafter. For each of the M particles consecutively6generated during the EMA iterative process, the corresponding value of the ob-jective function is evaluated, for each of the W ’s generated. In other words, weevaluate the value of the objective function for each of the particles, for eachof the 100 W ’s. The value of the particle is based on the average of the 100values, v(m) =PKk=1 v(k)/K. It is these values that are used in the EMA todecide on the forces acting on the particle. Finally, when we stop the movementof the particles (i.e., after 25n movements) we have one particle vmin whichaverage value (over the 100 W ’s) is minimal. And to gain some idea aboutthe variability of vmin the whole experiment (with M particles and diﬀerent Ksamples of the work content) was replicated R times. In our experimentation wetook R = 4. The minimum over these four replications is the value we quote inTable 1.4 Experimental Layoutand Computational ResultsThe program outlined above was implemented in Matlab on a set of fourteenprojects that ranged in size from 3 to 76 activities. The networks under studycan be seen in the internet1, or requested by e-mail from the lead author2. Theresults obtained (on a Pentium IV, 3 GHz, 1 GB RAM) are in the same webpages cited, and are summarized in Table 1. In all networks, the allocation toany activity, xa, is constrained to be in the interval [0.50, 1.50]. The resource costproportionality constant ka is normalized at 1, the same for all activities. Thetardiness penalty cL is as given in Table 1. The “PERT Expected Value” is thelength of the so-called “critical path” following the PERT model. This, togetherwith the due time T give an idea about the “tightness” of the imposed due time(recall that the PERT expected duration is an underestimate of the exact ex-pected duration), and was maintained in the range (T/E (PERT )) ∈ [1.03, 1.09]throughout. The value of K was fixed at 100 for all experiments; that is, 100work content samples were generated for each iteration in each experiment. The1www.dps.uminho.pt/pessoais/anabelat (Topic: research).2 anabelat@dps.uminho.pt.7experiment was repeated R = 4 times. The values quoted in Table 1 are the bestin the four replications, as explained above. The restriction to 2 and 3 discretepoints in the DP approach was necessary to keep the computing time withinreasonable limits. Still, DP failed to reach a solution for networks of n ≥ 24within 48 hrs. This is where the superiority of the EMA is manifested. In fact,the EMA gives a better “best value" than DP in several instances of even smallnetworks. The cause lies in the discretization required by the DP approach,which clouds the assertion of optimality of its results.PERT DP DP DP EMA EMANet n Dur’n T cL Points Best Cost DP Time Best Cost Time1 3 15 16 2 3 43.32 0.1 sec 36.57 14.0 sec2 5 115 120 8 3 304.62 1.0 sec 277.53 32.4 sec3 7 62.9 66 5 3 209.94 3.0 sec 207.33 1.1 min4 9 100 105 4 3 387.20 1.1 min 379.67 1.8 min5 11 26.67 28 8 3 106.76 7.4 min 115.19 2.3 min6 11 62.08 65 5 3 280.85 45.5 min 286.30 2.7 min7 12 44.72 47 4 3 182.91 5.8 hr 183.19 3.5 min8 14 35.5 37 3 2 116.49 31.9 min 122.67 4.2 min9 14 178.57 188 6 2 645.39 6.5 hr 710.27 5.0 min10 17 44.98 49 7 2 160.12 4.2 hr 137.53 7.5 min11 18 106.11 110 10 2 339.07 30.2 hr 375.43 9.7 min12 24 212.05 223 12 2 - > 48hr 1212.00 18.5 min13 38 143.99 151 5 2 - > 48 hr 834.77 1.0 hr14 76 115.01 121 4 2 - > 48 hr 559.19 6.2 hrTable 1. Experimental Results5 Conclusions and Future ResearchThe results demonstrate:1. The superiority of the EMA in value over the DP approach for many8networks. As stated before, this is due to the large “mesh” used in DP(only 3 points in the first seven networks and 2 points in the remaining(larger) seven networks). Finer mesh should yield better results, at thecost of (severely) increased computational burden.2. The computational superiority of the EMA over DP for the larger networksby several orders of magnitude.3. The advantage of the use of EMA with repeated experimentation. This ad-vantage is twofold. First, it does provide an estimate of the expected cost ofthe project. Second, one can use the replication of the experiment R timesfor any given network to estimate the range of variation of the expectedvalue. As an example, this was done with Net 5 in the above table. Theresults of the best values were as follows: 127.71, 115.19, 122.62, 132.06,for an average of 124.40 and variance ≈ (7.25)2 . The best average cost is115.19 which corresponds to the allocationx∗ = (1.50, 1.50, 0.64, 1.38, 1.50, 1.45, 0.69, 1.47, 1.50, 1.50, 0.99) . (3)(Observe that 5 of the 11 activities were at the upper bound of resourceallocation, which may hint at securing a smaller expected value if the re-source capacity is increased — an important indication to management.)More importantly, assuming normality, one can ascertain that with prob-ability of approximately 97.7% the optimal expected value of this projectis larger than 109.90. To be sure, all four values lie above this value. Fur-thermore, by adopting the x∗ of (3) we are only 115.19 − 109.90 = 5.29units away from the lower bound on cost, or less than 4.8%.4. The assumption of exponential distribution of the work content was madefor ease of discretization of the work content in the DP approach, and iscentral in the analysis of Markov activity networks. It is irrelevant to theEMA approach which would accept any distribution, since the randomvariables enter the approach only via the sampled W ’s.9Matlab is not the ideal language for industrial applications, and we areconfident that if the logic of EMA is coded in another language such as C orC++ by an expert programmer the time required to reach a solution will improveby at least one order of magnitude. It is important to achieve eﬃcient codesbecause, while it is true that the result given by EMA is the optimal vector x∗,this is a static policy which may be, and in all probability shall be, modifieddynamically later. To change the picture from a static optimum to a dynamicone, we propose that only the allocations to the activities along the cutsetat the start of the project are implemented. The resource allocation to otheractivities should await the progress of the project. As the project progressessome of the activities in the initial cutset will be completed and the status ofthe project will change. At which time the new information is incorporated intothe model and the EMA is re-run. This process continues until the project iscompleted. Therefore, in the span of time in which the project is “alive" theEMA procedure may be run several times, each on a diﬀerent network, possiblywith updated values on the distributions. The networks get smaller and smalleras activities are completed, and consequently the time required to run EMAalso gets smaller.Apart from improved programming of the logic of the EMA, our researchis currently directed towards the improvement of the performance of the EMAitself, possibly through improved local search, improved dynamic update of thestep size, and improved layout of the particles in the feasible space.10References[1] S.I. Birbil, and S-C. Fang, An Electromagnetism-Like Mechanism for globaloptimization, Journal of Global Optimization 25, 2003, pp. 263—282.[2] A.P. Tereso, M.M. Araújo, and S.E. Elmaghraby, Adaptive Resource Alloca-tion in Multimodal Activity Networks, International Journal of ProductionEconomics 92, 2004, pp. 1-10 .[3] A.P. Tereso, M.M. Araújo, and S.E. Elmaghraby, Experimental Results of anAdaptive Resource Allocation Technique to Stochastic Multimodal Projects,Research Report, Universidade do Minho, Guimarães, Portugal, 2003 (Sub-mitted for publication).[4] A.P. Tereso, M.M. Araújo, and S.E. Elmaghraby, Basic Approximationsto an Adaptive Resource Allocation Technique to Stochastic MultimodalProjects. Research Report, Universidade do Minho, Guimarães, Portugal,2003 (Submitted for publication).11

The optimal resource allocation in stochastic activity networks via the electromagnetism approach

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The optimal resource allocation in stochastic activity networks via the electromagnetism approach

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