Heuristic Methods for the Single-machine Problem
4.3 Random Sampling
solution to the optimal solution. Three performance measures were tallied: a count of the number of times the optimum was found, the average ratio of the heuristic solution to the optimal solution, and the maximum solution ratio.
The results are summarized in Table 4.1. As the table clearly shows, most basic dispatching procedures were not especially effective in solving theTw-problem.
WMDD was the clear winner, followed by the greedy and insertion techniques.
A decent heuristic usually gets within 10% of the optimum, and a really good one reliably gets within 1–2%. So WMDD is a decent heuristic, but there is still room for testing other kinds of heuristic approaches. In addition, WMDD applies only to theTw-problem, so we must study additional heuristics if we want to tackle other objective functions.
The combination of WMDD and the correction of Eq. (4.2) is not a pure dis- patching or construction procedure because it involves revisiting earlier decisions after later ones reveal they could be improved. Indeed, the correction step is a rudimentary example of asearch technique. Search techniques, such as those we describe in later sections, are fundamentally different than construction pro- cedures. Whereas construction methods start from scratch and build one sched- ule, search procedures assume that a solution has already been built, and they examine a series of alternative solutions in an effort to find improvements.
The essence of a sampling procedure is easy enough to describe. Using some random device, construct and evaluate N sequences, and identify the best sequence in the sample. We can view random sampling as a solution method that lies on a continuum between a specialized heuristic procedure and an optimizing procedure. Many heuristic procedures, such as the greedy algorithm described earlier, generate one sequence, while an optimizing procedure, such as branch and bound, enumerates alln! sequences, at least implicitly. A random sampling procedure constructs some intermediate number of sequences and selects the best one. The design of a sampling scheme must resolve two tactical questions:
1) How to specify a particular device for carrying out sampling?
2) How to draw conclusions about the best sequence in the sample?
Much of the literature on sampling techniques has attempted to provide some insight into the answers to these questions, which we next explore in more detail.
It is not easy to draw substantive conclusions about the best sequence found in the sample. The ideal information is the likelihood that a sample contains an opti- mum or the distance from optimality. Unfortunately, these relationships are gen- erally known only qualitatively: A larger sample is more likely than a smaller sample to contain an optimum, and the best sequence in a larger sample also tends to be closer to the optimal value. But without quantitative information about these relationships, there is virtually no logical way to select a sample size. In principle, there is a certain probabilitypthat on a particular trial, a specified sampling pro- cedure will construct an optimum for a given problem. Therefore, because sam- pling is essentially done with replacement, the probability that an optimum will be found in a sample of sizeNis [1−(1−p)N]. The difficulty is to estimatep.
In the basic single-machine problem, there is perhaps one situation in which we can draw a quantitative conclusion. Suppose that a sequence is constructed by assigning the first position in sequence, then the second, and so on. In order to assign the first sequence position, suppose that a random device is used and that each job is assigned to this position with probability 1/n. After this assign- ment, suppose that each remaining job is assigned to second position with probability 1/(n−1). If we continue in this manner, then we will assign each position by an equally likely selection device. In this structure, all of the n!
sequences are equally likely to be included in the sample. If the optimum is unique, then p = 1/n!, so in this procedure we can conclude that the best sequence in a sample of size N is an optimum with probability [1−(1−1/
n!)N]. On the subject of how close to optimal the best sequence in the sample may be, it is still not possible to provide quantitative conclusions. In order to suggest the kind of behavior that might occur, a set of random sampling experiments was conducted with the 20-job test problems. Three different sample sizes were tested, and the results are shown in Table 4.2 and compared with the random dispatching and the greedy algorithm from Table 4.1. (The
4 Heuristic Methods for the Single-machine Problem 78
random dispatching procedure is equivalent to random sampling with a sam- ple size of N= 1.)
Table 4.2 shows that solution efficiency improves with sample size, which we should have expected. We also observe that the sampling procedure is not nearly as effective as the greedy heuristic even for a sample size of 500, which involves a computational effort much greater than that of the greedy heuristic.
More generally, we should think in terms of selection devices that are not equally likely, and we should recognize that such mechanisms might yield a value of p much larger than 1/n!. The following is an example of a simple method for performingbiased random sampling. We begin by ordering the jobs according to some ranking rule. To assign the first position in sequence, we select the job in jth position on the ordered list with probability p1j(j= 1, 2,
…, n). These probabilities are “biased” in the sense that they favor the first job on the list to the second, the second to the third, and so on. Next, we remove the assigned job from the list, and we assign the second position by selecting the job injth position on the updated ordered list with probabilityp2j(j= 1, 2,…,n
−1). In this approach, we use a discrete distributionpkjat thekth stage. A typical approach would use a set ofpkjvalues that follow a truncated geometric distri- bution. In this case the selection device corresponds to
pkj=πjQk, j= 1, 2,…,n+ 1–k
whereQk is a normalizing constant. With this structure, the first job on the ordered list has the highest probability of being selected, the second job has the second highest probability, and so on. In addition, the probabilities decrease in a geometric manner, but the nature of the decrease can be controlled by selecting the parameterπ. For example, if there are eight jobs and we setπ= 0.6, then the probabilities are as follows:
j 1 2 3 4 5
pkj 0.297 0.238 0.190 0.152 0.122 Table 4.2
Algorithm Optimizing frequency Average ratio Maximum ratio
Random 0 of 12 1.86 2.51
Sampling (N= 20) 0 of 12 1.59 2.08
Sampling (N= 100) 0 of 12 1.51 1.90
Sampling (N= 500) 0 of 12 1.41 1.72
Greedy 0 of 12 1.22 1.39
4.3 Random Sampling 79
A larger value ofπ would make the jobs early on the list more likely to be selected, while a smaller πwould distribute the selection probabilities more equally. Thus, we can bias the random selection process of the basic sampling method toward a given job ordering and thereby improve the efficiency of the sampling. For the 20-job test problems, Table 4.3 compares some biased sam- pling plans with the equally likely plan described in Table 4.2.
The results suggest that biased random sampling improves on pure random sampling. In other words, the intelligent choice of a job ordering and a bias in the randomization are worth more than a large amount of sampling. In Table 4.3, the performance of random sampling with a sample size of 500 was virtually matched by the performance of biased sampling with a sample size of only 20. In fact, with SWPT as the initial ordering, the biased sampling pro- cedure was even better on average.
In short, random sampling is a procedure for obtaining good solutions to combinatorial problems with simple, straightforward logic and limited compu- tational effort. In more complicated problems, both in and out of the scheduling field, sampling techniques have provided effective heuristic procedures. How- ever, as the results in our next computational experiments suggest, sampling is not always competitive with other general-purpose heuristic procedures.
Its virtues are ease of implementation and flexibility. The flexibility derives from many tactical options. These options include the initial ordering of the jobs for biased sampling, the selection of a probability distribution for assigning prob- abilities to positions, and the determination of sample size. The art of applying random sampling lies in specifying these tactics in order to arrive at an effective sampling procedure. Different tactics may perform well in different types of pro- blems, so it may take some experimentation to determine the tactics that are best suited to any particular application. Finally, random sampling is potentially useful in combination with other heuristics. For example, each random sample
Table 4.3
Algorithm
Optimizing frequency
Average ratio
Maximum ratio
Sampling (N= 500) 0 of 12 1.41 1.72
Sampling (π= 0.8,N= 20; MST) 0 of 12 1.46 1.76 Sampling (π= 0.8,N= 20; EDD) 0 of 12 1.42 1.62 Sampling (π= 0.8,N= 20; SWPT) 0 of 12 1.30 1.82 Sampling (π= 0.8,N= 100; SWPT) 0 of 12 1.25 1.60 Sampling (π= 0.8,N= 500; SWPT) 0 of 12 1.21 1.53
Greedy 0 of 12 1.22 1.39
4 Heuristic Methods for the Single-machine Problem 80
could be subjected to the insertion heuristic; this combination is very likely to improve on the basic sampling procedure.