1. Introduction
The scheduling problem with single criterion has been the subject of considerable research.
However, it has been gradually recognized that practitioners usually consider multiple criteria in scheduling jobs since a single criterion seldom represents the total cost. Therefore, extensive research on scheduling has been done on the topic of multiple criteria in the past two decades. This inspires us to apply ACO algorithm to solve the single machine problem with bi-criteria of makespan and total weighted tardiness. The makespan is as important as the total weighed tardiness.
This is because the makespan represents the degree of the resource utilized by the system. Both of the makespan and total weighted tardiness are what the decision maker is concerned about.
Therefore, in this chapter we try to choose makespan and total weighted tardiness as the criteria to be minimized.
2. Literature Review
Many researchers have been working on multiple criteria scheduling, with the majority of work being on bi-criteria scheduling. Using two criteria usually makes the problem more realistic than using a single criterion. One criterion can be chosen to represent the manufacturer’s concern while the other could represent consumer’s concern.
There are several papers that review the multiple criteria scheduling literature. Nagar et al. [1], and T’kindt and Billaut [2] review the problem in its general form whereas Lee and Vairaktarakis [3]
review a special version of the problem, where one criterion is set to its best possible value and the other criterion is tried to be optimized under this restriction. Also, Hoogeveen [4] studies a number of bi-criteria scheduling problems.
In solving a multiple-criteria scheduling problem, there is a difficulty in dealing with several different criteria due to their inconsistency in dimension. Fortunately, three useful methods have been proposed to solve this difficulty. They are the weighting method, priority method, and efficient solution method. The difficulties of applying the first two methods are how actually to find credible weights and satisfactory priorities [5]. The efficient solution method resolves the difficulties by generating the complete set of possibly optimal solutions for any objective function involving the chosen criteria. Briefly speaking, a schedule is efficient if it cannot be dominated by any other schedules. It is particularly useful in scheduling because the generated set is relatively small, which makes it easier for the decision maker to select the most appropriate solution based on the actual situation. To provide the decision maker with more flexibility, the efficient solution method is used here to deal with the multiple criteria.
3. Apply ACO to 1/sij/
∑
w T Cj j, maxIn order to increase the efficiency of the ACO algorithm to solve the problem 1/sij/
∑
w T Cj j, max, we change some procedures in our ACO algorithm.1. Update of pheromone trial: Now we may have different efficient solutions (non-dominated), so how we use the local and global update of pheromone trial is a difficulty. In our algorithm, the choice of which efficient solution used to update is determined by a random manner.
2. The timing of applying local search: Because of so many efficient solutions, if we apply the same timing of applying local search as previous ACO algorithm, it will take too much time. In order to decrease the time of local search, we try to use only two times local search.
These two times local search are aimed at all efficient solutions we have, and one is in a half of maximum iterations and the other is in the end.
3. Δτt( , )i j in global update of pheromone trial: In single criteria, the amount ( , ) 1/ *
t i j T
τ
Δ = , where T is the objective value of the global best solution. But now we * have multiple criteria, we need a different rule to calculate our objective value. We let
*
1 max 2 j j
T =w C +w
∑
W T , where w is the weight for the associated criterion. In the i weights of all the criteria are constant, the search will always be in the same direction. In order to search for various directions, we use the variable weights, proposed by Murata, Ishibuchi, ＆ Tanaka [6], by assigning a random number X to each weight i w as iIn the remaining of this section, we compare the ACO algorithm with constructive-type heuristic of Apparent Tardiness Cost with Setups (ATCS), because ATCS is a dispatching rule which consider makespan and total weighted tardiness these two criteria in single machine with sequence-dependent setup times.
In the experiments, ACO and ATCS were tested on the problem instance 91~110 provided by Cirirello. As for the performance measure, the mean relative percentage error (MRPE) is used. Let M and WT represent the values of makespan and total weighted tardiness associated with the ACO algorithm, and M , and ′ WT ′ the values associated with the ATCS. The MRPE of each criterion for the ACO algorithm can be computed as:
min( , )
Because the objective function is to be minimized, the smaller the MRPE value the better the algorithm. The comparisons of the ACO algorithm with ATCS are summarized in Table 7. Since the ACO algorithm generates a set of efficient schedules, each efficient schedule will be compared with the single schedule yielded by ATCS. The comparison is done on the average value of the set of efficient schedules. From Table 1 it can be seen that the ACO algorithm performs better than ATCS in all two criteria.
5. Conclusion
In this part we try to apply ACO algorithm to solve the single machine problem with bi-criteria of makespan and total weighted tardiness. In order to fit multiple criteria, our ACO algorithm also has three distinctive features, including the choice of which efficient solution used to update is determined by a random manner, the change of timing for applying local search , and a different rule to calculate our objective value for Δτt( , )i j . We compare the ACO algorithm with constructive-type heuristic of Apparent Tardiness Cost with Setups (ATCS) and the ACO algorithm outperforms ATCS in all two criteria of our test problems.
In the above two parts we have developed the structural model of applying ACO to different
Table 1
Comparison of the ACO algorithm with ATCS
ACO ATCS
scheduling problems. Since the versatile and robust nature of ACO, it shows that the potential and dependable algorithm proposed here is well worth exploring in the context of solving different scheduling problems. The proposed ACO heuristic can be extended to deal with other difficult scheduling problems, such as multiple identical machines with sequence-dependent setup times or with re-entrant jobs. In the chapter on the multiple objective scheduling problems, a further improvement is possible. Future studies could also extend the ACO algorithm to more complex scheduling environments such as flow-shop, job-shop (in ext part), or open shop.
References
[1] Nagar A, Haddock J, Heragu S. Multiple and bicriteria scheduling: A literature survey. European Journal of Operations Research 1995;81:88-104.
[2] T’kindt V, Billaut JC. Some guidelines to solve multicriteria scheduling problems. IEEE International Conference on Systems, Man and Cybernetics Proceedings 1999;6:463-468.
[3] Lee CY, Vairaktarakis GL. Complexity of single machine hierarchical scheduling: A survey.
Pardalos PM. Complexity in Numerical Optimization. Singapore, World Scientific Publishing Co, 1993.
[4] Hoogeveen JA. Single machine bicriteria scheduling. Ph.D. Thesis, CWI, Amsterdam 1992.
[5] Ignizo JP. Linear Programming in Single and Multiple Objective Systems. NJ: Prentice-Hall 1982.
[6] Murata T, Ishibuchi H, Tanaka H. Multi-objective genetic algorithm and its applications to flowshop scheduling. Computers and Industrial Engineering 1996;30:957-968.
Part III. Ant Colony Optimization combined with Taboo Search