Scheduling a single machine where setup times are sequence dependent using an ant colony heuristic Caroline Gagné 1 , Marc Gravel1 , Wilson L. Price2
1
Département d’informatique et mathématique, Université du Québec, Chicoutimi, Québec, G7H2B1
[email protected] [email protected] 2 Faculté des sciences de l’administration, Université Laval, Québec, Québec, G1K 7P4
[email protected] Abstract This paper presents an ant-colony heuristic for the solution of an industrial scheduling problem in an aluminum casting center. The authors first show how the ant-colony heuristic proposed in [1] can be adapted to the single machine scheduling problem with sequence dependent setup times. In this illustrations we minimize the total tardiness. It will be shown that this heuristic performs better, in most cases, than other recent heuristics on the published problem set of Rubin and Ragatz [2]. We then show how this heuristic can be adapted to the particular conditions found in aluminum production. We present an efficient representation of a continuous horizontal casting process which allows us to take account of multiple objectives. We have incorporated the methods proposed in software that has been implemented in the plant.
1
Problem Description
We treat a scheduling problem encountered in an Alcan aluminium foundry located in Québec. In this foundry, two holding furnaces are charged with molten metal from a transfer crucible coming from the refiners. These furnaces continuously feed liquid metal to the horizontal casting rig. A customer's alloy specification is produced by adding the required ingredients and grain refiners to the molten aluminium in the holding furnaces. Molten aluminum is poured into channels leading to a basin and a mold having the crossection of the desired ingots. The aluminum flows through the mold taking the proper crossectional shape and, at the same time, fuses. Since the casting is continuous, a large automated circular saw cuts the fused aluminum into the required ingot lengths as it is produced. If the crossection must be changed, then casting must be stopped and the mold changed. A change in the alloy being produced may also require a draining and cleaning to prevent contamination of the alloy to be cast next. Stoppages on the casting rig can be avoided if, while one holding furnace supplies metal for the pour, the second is prepared and loaded. The speed of a pour depends on the alloy type and on the number of pieces in the mold. The preparation time of a holding furnace is a function of the quantity of molten metal used, the quantity of solid metal used, and of any draining and cleaning required. A change of alloy will also affect the holding furnace preparation time. The metallurgical composition of the new alloy may require that the holding furnace be drained and cleaned before the pour. The basin is a holding chamber that retains a small amount of molten metal just before the mold and the availability of basins constitutes a further technological constraint. Molds may be attached only to specific basins and the one required may be undergoing cleaning from a previous usage.
A feasible sequence of orders is one that ensures that sufficient pure metal is available for all pours, that basins and molds are available when required for each order and that draining and cleaning of the rig is done when required. A desirable feasible sequence takes into account the objectives of customer service and efficiency. We model the objectives of the scheduler by treating the minimization of unweighted total tardiness for all orders, the minimization of unused production capacity over the planning horizon, the minimization of the total number of drainings for the furnaces and we include a penalty function encouraging efficient transportation of the product. This function favors sequences where orders for the same destination are consecutive and penalizes sequences where this is not the case. In its general form, we see this problem as one of scheduling of n orders with sequence dependent setup times on one machine while taking into account the technological and logical constraints on equipment and the management of the supply of liquid metal. In the literature, we find that this problem has been classed as NP-hard ([3], [4], [5]) and is more complex than many described in the survey paper of MacCarthy & Liu [6]. In our work, we have chosen the minimization of total tardiness as a basis for comparison with other methods.
2
Solution Algorithm: Ant Colony Optimization
The ant-colony metaheuristic was introduced in the doctoral thesis of Dorigo [7] and was inspired by studies of the behavior of ants ([8], [9], [10]). The works of Colorni, Dorigo & Maniezzo [11], Dorigo, Maniezzo & Colorni [12], Dorigo, Maniezzo & Colorni [13], Dorigo, Gambardella [1], Dorigo & Di Caro [14] offer detailed information on the workings of the algorithm and the choice of the various parameters. The present authors have constructed an ant-colony heuristic to address the scheduling problem described. The basic approach described in [1] was used as a basis for our heuristic. Our version of this heuristic includes the state-transition rule, the global updating rule, the local updating rule, and closely follows the parameter settings recommended by Dorigo and Gambardella. A modification, which we consider major was made to accommodate the nature of the problem at hand. In calculating the state transition probabilities, two distance matrices are used. The first is derived from the setup times in the usual manner for the TSP. The second is derived from computations of the tardiness that would result from the various state transitions. Both are used in a greedy manner and are combined in the state transition rule.
3
Numerical Results
Rubin et Ragatz [2] use a genetic algorithm to schedule n jobs on one machine so that the sum of tardiness is minimized where setups are sequence dependent. The problem set that is used to test this algorithm was graciously posted on Professor Rubin's website [http://www.bus.msu.edu/mgt/datafiles.htm] and was used as a basis for judging the performance of our ant-colony heuristic. The problem set consists of 32 cases ranging from 15 to 45 orders. In 16 cases, the problem was solved to optimality with a branch-and-bound algorithm [15]. For the remaining problems the best known solution obtained by the branch and bound algorithm was recorded. For the smaller problems (15 orders and 25 orders), the solution quality of the two approaches is similar with a slight advantage for the genetic heuristic. In one stubborn 25-order case, the genetic heuristic maintained a clear advantage. On average, for this particular problem, it was able to come about 5% closer to the best known solution. On the other hand, for a 35-order case the genetic heuristic averages more than 100% higher than the best known solution while the ant-colony heuristic improves this solution by an average of 2.7%.
For the larger problems (35 and 45 orders) the ant colony heuristic consistently produces better solutions, more closely approaching the known optimum. For the solutions where the optimum is not known, we were able to improve the best known solution in 15 of 16 cases. It is difficult to compare solution times since different classes of computer were used in the two studies, but again, it seems clear that the ant-colony heuristic is quicker.
4
Multi-Objective Implementation
We will now describe our multi-objective implementation of the ant-colony metaheuristic for this industrial application. We take into account several objectives that are important to the scheduler: -
The first objective is the minimization of unused capacity due to setup times, drainings and time lost owing to technological constraints such as a lack of molten metal. The second objective is to minimise the total tardiness of the order book calculated in the usual manner [16]. The third objective is to minimise of the total number of drainings required when changing alloy. The fourth objective is to minimize a transportation penalty function that computes the total unused vehicle capacity.
For the decisions required at the local level, as an ant is selecting the next node to visit, we construct and use a matrix D that aggregates information on each of the four objectives that we have identified. Once a cycle is completed, a schedule (a tour in the TSP) found by the each ant is evaluated precisely for each objective. In the usual implementation of the ant-colony heuristic for the TSP, the trace would be updated using the tour length computed from the D-matrix defining the "distance" between pairs of nodes. In this implementation, guided by the scheduler, we first choose one of the four objectives as the primary function, which then becomes the sole reference in updating the trace. The heuristic that we have developed is presently incorporated in a system in use in an aluminium production plant and we obtain robust solutions in short computing times. A typical order book contains about 60 orders and may be solved in approximately 40 seconds. Schedulers we have encountered in this project use the results produced by the heuristic and feel that it well represents the constraints and objectives with which they are faced. In previous implementations [17], we also used a genetic algorithm to solve similar problems with good results. In comparison testing of the single objective models, we found the results of the antcycle heuristic to give much better quality results in much shorter computing times. Further tuning of the genetic algorithm could speed up its convergence and improve solution quality. We cite this comparison merely as evidence that the ant-cycle metaheuristic is quite competitive with other heuristics. We consider that the multiobjective procedure presented here is simple and robust and that it offers good quality solutions in short computation times. We have, as one might expect, produced solutions of a better overall quality than those found by single objective optimization. We have not, however, attempted to ensure that the solutions suggested are Pareto-efficient in order to reduce computing effort.
References [1]
Dorigo M., Gambardella L.M.; Ant colony system: a cooperative learning approach to the traveling salesman problem; IEEE Transactions on Evolutionary Computation, 1, 1, 53-66, 1997.
[2]
Rubin P.A., Ragatz G.L.; Scheduling in a sequence dependent setup environment with genetic search, Computers and Operations Research, 22, 2, 85-99, 1995.
[3]
Conway R.W., Maxwell W.L., Miller L.W.; Theory of scheduling, Addison-Wesley, 294 pages, 1967.
[4]
Du J., Leung J.Y.; Minimizing total tardiness on one machine is NP-hard, Mathematics of operations research, 15, 483-494, 1990.
[5]
Lenstra J.K., Rinnooy Kan A.H.G.; Computational complexity of discrete optimization problems, Annals of Discrete Mathematics, 4, 121-140, 1979.
[6]
Maccarthy B.L., Liu J.; Addressing the gap in scheduling research: a review of optimization and heuristic methods in production scheduling, International Journal of Production Research, 31, 1, 59-79, 1993.
[7]
Dorigo M.; Optimization, learning and natural algorithms, Ph.D. Thesis, Politecnico di Milano, Italy, 1992.
[8]
Deneubourg J.L., Pasteels J.M., Verhaeghe J.C.; Probabilistic behaviour in ants: A strategy of errors?, Journal of Therical Biology, 105, 259-271, 1983.
[9]
Deneubourg J.L., Goss S.; Collective patterns and decision-making, Ethology & Evolution, 1, 295-311, 1989.
[10] Goss S., Beckers R., Deneubourg J.L., Aron S., Pasteels J.M.; How trail laying and trail following can solve foraging problems for ant colonies, In: Behavioural Mechanisms of Food Selection, R.N. Hughes ed., NATO-ASI Series, vol. G20, Berlin: Springler-Verlag, 1990. [11] Colorni A., Dorigo M., Maniezzo V.; Distributed optimization by ant colonies, Proceedings of the European Conference on Artificial Life (ECAL'91), Elsevier Publishing, Paris, France, 134-142, 1992. [12] Dorigo M., Maniezzo V., Colorni A.; Positive feedback as a search strategy, Technical Report No 91-016, Politecnico di Milano, Italy, 20 pages, 1991. [13] Dorigo, M., Maniezzo, V., Colorni, A.; Ant system: Optimization by a colony of cooperating agents, IEEE Transactions on System, Man and Cybernetics - Part B, Vol. 26, No. 1, 29-41, 1996. [14] Dorigo M., Di Caro G.; The Ant Colony Optimization Meta-Heuristic, In: D. Corne, M. Dorigo and F. Glover Editors, New Ideas in Optimization, McGraw-Hill, 1999. [15] Ragatz, G.L.; A Branch-and-bound Method for Minimum Tardiness Sequencing On A Single Processor With Sequence Dependent Setup Times. Proceedings: Twenty-fourth Annual Meeting of The Decision Sciences Institute, 1375-1377, 1993. [16] Baker, Kenneth R.; Introduction to sequencing and scheduling, John Wiley & Sons, New York, 1974. [17] Gravel M., Price W., Gagné C.; Scheduling jobs in a Alcan aluminium factory using a genetic algorithm, International Journal of Production Research, 2000 (to appear).
