# A dynamic resource constrained task scheduling problem

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4 ET4: Arc Removing (AR). It is a reduction rule and doesn t noise the solution quality, but reduces the processing time, by removing some redundant arcs of the input graph. An arc (r,s) can be removed if, after this, we can find a path K from r to s in the resulting graph. So the arc (r,s) is said an explicit precedence of an implicit precedence (the path K), and can be removed. Some preliminary tests prove that in the most cases it reduces the processing time, mainly in instances with a very large number of arcs. 3.2 Local Search Local search (LS) is an improvement procedure that tries to modify an initial solution by changing small segments of it, called neighborhood. Two local search algorithms are proposed: the first one (LS1) analyzes on each task t i by time. It does the same calculation made by CT. If a task t i has c i > S i it is removed from the current solution. The analysis begins with the tasks activated lately, coming to the task activated early. The second LS (LS2) is a generalization of the LS1 and works on a set of all successors of a task t i (F i ) by time. If the total cost of the set F i is more than the total profit generated by F i, the whole set F i is removed from the current solution. 3.3 Evolutionary Algorithms Evolutionary Algorithms (EA) and the Genetic Algorithms (GA), its most popular representative, are heuristics that works simultaneously with a population of solutions, trying to combine them to generate better solutions. Basically we need to generate a population of feasible solutions called initial population. Then, a population subset is chosen to combine and generate a new population. From these two populations, a subset of solutions is chosen to form the new generation of EA. Then, new solutions are chosen to combine and generate a new population and so on. The EA stops when a number of generations are achieved. Readings on EA can be made at [2][5][7][8][10][11]. The EAs have been used in several areas to solve problems considered intractable (NP-Complete and NP- Hard) although its basic versions do not demonstrate too much efficiency in the high complexity problems [10][11]. In order to improve the GA and EA performance researchers have proposed hybrid versions [5][8][10][11]. Two EA (EA1 and EA2) are proposed to solve the DRCTSP. The EA1 is a basic version and the EA2 is a hybrid version including all the enhancement techniques (ET) and the local searches proposed. The Table 1 shows the differences between them. The column Size indicates the size of the populations. Gens is the maximum number of generations. Init. LS shows the Local Searches used after the initial population generation. Pop. LS is the local searches used after the generation of a new population. The last column indicates the enhancement techniques used. EA Size Gens Init. LS Pop. LS ETs EA AR EA LS1, LS2 LS1 AR, CT, RM, PP Table 1: EAs structures Both EAs generates the initial population using the ADDR heuristic. The α parameter is very important to obtain good solutions. We defined a α range of values ([0,05 ; 1,0], with 0,05 increments) within one value is chosen. For each value of α, the ADDR runs 20 times. The chosen α is the value that generates the best solution from them all. Similarly, the CT value ([0,2 ; 1,0] with 0,1 increments), the RM value ([1,0 ; 1,4] with 0,1 increments) and the PW value ( use or don t use ) are chosen. To choose the solutions that will combine, the population is partitioned in three parts. The PA class is composed by the 20% best solutions. The PC class by the 20% worst solutions. The PB class by the others solutions. Then to select two parents, a solution from PA is chosen randomly and combined with a randomly chosen solution from PB. Thirty new solutions (offsprings) are generated for each iteration. From these 60 solutions (30+30) only the 30 best

6 Large 145,8% 112,2% Large 149,1% 120,3% Table 4: Average outperform from EA2 to EA1 Table 5: Average outperform from EA2 to EA1 without time limit with time limit EA1: A Instances EA2: A Instances EA1: A Instances EA2: A Instances Small 0,0% 100,0% 100,0% 100,0% Medium 0,0% 89,0% 55,6% 100,0% Large 0,0% 100,0% 73,3% 100,0% Table 6: Percentage of reached target values In the last test (Table 6), a target value was defined for each instance used as stop criteria for the EA1 and EA2. This target value is the EA1 result (average of the three runs) obtained in the first test. The cell values indicate the percentage of times that the target was reached. The very bad results obtained in table 6 by EA1 with A-instances (compared to the B-instances) is due the fact that A- instances have less constraints than B, once A-instances have less precedence among the tasks than B). Thus, the space solution of A-instances is bigger than B, what does A-instances more difficult to get an optimal solution. 5. Concluding Remarks This paper presented a new model of Resource-Constrained Task Scheduling Problem (RCTSP) called DRCTSP. To solve this problem are presented: a mathematical formulation; enhancement techniques-et and evolutionary heuristics. Initially the tables 2 and 3 show that EA2 using all enhancement techniques (ET) and the two local searches proposed significantly improve the average performance of the standard version of EA (EA1). Additionally, we did a set of computational tests with large scale instances, where the objective was to evaluate the behavior of the hybrid version (EA2) including the ETs and LSs procedures. The average results of the Tables 4, 5 and 6 shows that EA2 also clearly outperforms the EA1 in the large instances. 6. References [1] Remy, J. Resource constrained scheduling on multiple machines. Information Proc. Letters, v. 91, p , [2] Boeres, C.; Rios, E.; Ochi, L. S. Hybrid Evolutionary Static Scheduling for Heterogeneous Systems. Proc. of the IEEE Conf. on Evolutionary Computation (CEC2005), Book 3, pp , Edinburgh, [3] Kalinowski, T.; Kort, I.; Trystam, D. List scheduling of general task graphs under LogP. Parallel Computing, 26, p , [4] Bouleimen, K.; Lecoco, H. A new efficient simulated annealing algorithm for the resource-constrained project scheduling problem and its multiple mode version. European Journal of Operational Research, v. 149, p , [5] Gonçalves, J. F.; Mendes, J. J. M.; Resende, M. G. C. A genetic algorithm for the resource constrained multi-project scheduling problem. European Journal of Operational Research (submitted), [6] Brucker, P. et al. Resource-constrained project scheduling: Notation, classification, models, and methods. EJOR, v. 12, p. 3 41, [7] Holland, J. H. Adaptation in Natural and Artificial Systems. Ann Arbor, MI: [s.n.], [8] Gonçalves, J.; Mendes, J.; Resende, M.G.C. A hybrid genetic algorithm for the job shop scheduling problem. European Journal of Operational Research, v. 167, p , [9] Resende, M.G.C.; Ribeiro, C.C. Greedy randomized adaptive search procedures (GRASP). In: Handbook of Metaheuristics [edited by F. Glover and G. Kochenberger], Kluwer Academic Publishers, p , [10] Santos, H. G.; Ochi, L. S.; Marinho, E. H.; Drummond, L. Combining an Evolutionary Algorithm with Data Mining to solve a Vehicle Routing Problem. To appear in Neurocomputing Journal - ELSEVIER, [11] Trindade, A. R.; Ochi, L. S. Um algoritmo evolutivo híbrido para a formação de células de manufatura em sistemas de produção. To appear in Pesquisa Operacional, SOBRAPO, Partly supported by FAPERJ: E-26/ /2004

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