Sequencing for Stochastic Scheduling
6.2 Basic Stochastic Counterpart Models
We begin our coverage of stochastic scheduling with an examination of stochas- tic counterpart problems. The objective in such problems is the expected value of a performance measure such as total flowtime, maximum tardiness, total cost, and the like. To help clarify the nature of stochastic counterpart models, we explore a numerical example.
6 Sequencing for Stochastic Scheduling 130
∎ Example 6.1 Consider a problem containing n = 5 jobs with stochastic processing times. The due date and expected processing time for each job are shown in the following table.
Jobj 1 2 3 4 5
E[pj] 3 4 5 6 7
dj 8 5 15 20 12
Suppose that two factors influence these processing times, the weather and the quality of raw materials. Each factor has two equally likely conditions (good and bad), so together they define four states of nature: GG (when both conditions are good), GB, BG, and BB. Each job has a different processing time under each state of nature as follows.
State Jobj 1 2 3 4 5
GG pj 2.6 3.5 3.8 3.2 6.4
GB pj 2.8 3.9 4.4 5.5 6.6
BG pj 3.2 4.1 5.6 6.5 7.4
BB pj 3.4 4.5 6.2 8.8 7.6
Assume that the four states are equally likely, or in other words, each combination of five processing times, or eachscenario, occurs with probability 0.25, and we can interpret the table as a discrete probability distribution.
Suppose also that we are interested in total tardiness as a measure of performance. We begin by examining the EDD sequence, 2-1-5-3-4. As a first step, we reorder the columns of the given data set to produce Table 6.1.
Next, we calculate the job completion times for each state, as shown in Table 6.2.
Table 6.1
Sequence 2 1 5 3 4
State Processing times
GG 3.5 2.6 6.4 3.8 3.2
GB 3.9 2.8 6.6 4.4 5.5
BG 4.1 3.2 7.4 5.6 6.5
BB 4.5 3.4 7.6 6.2 8.8
6.2 Basic Stochastic Counterpart Models 131
From these results, we can compute the tardiness of each job for each state, as shown in Table 6.3.
As Table 6.3 shows, the total tardiness in the sequence depends on which state occurs, and the value of total tardiness ranges from a low of 1.8 to a high of 20.7.
Taking into account the fact that the four states are equally likely, we can calculate the mean tardiness as 11.1 by taking the average of the figures in the last column.
We could make similar calculations for several other expected-value performance measures, giving rise to the results shown in Table 6.4, all for the EDD sequence.
Table 6.2
State Completion times
GG 3.5 6.1 12.5 16.3 19.5
GB 3.9 6.7 13.3 17.7 23.2
BG 4.1 7.3 14.7 20.3 26.8
BB 4.5 7.9 15.5 21.7 30.5
Table 6.3
State Tardiness Total
GG 0.0 0.0 0.5 1.3 0.0 1.8
GB 0.0 0.0 1.3 2.7 3.2 7.2
BG 0.0 0.0 2.7 5.3 6.8 14.8
BB 0.0 0.0 3.5 6.7 10.5 20.7
Average 11.1
Table 6.4
Scenario F Cmax L Lmax T Tmax U
GG 57.9 19.5 −2.1 1.3 1.8 1.3 2.0
GB 64.8 23.2 4.8 3.2 7.2 3.2 3.0
BG 73.2 26.8 13.2 6.8 14.8 6.8 3.0
BB 80.1 30.5 20.1 10.5 20.7 10.5 3.0 Average 69.0 25.0 9.0 5.5 11.1 5.5 2.8 6 Sequencing for Stochastic Scheduling
132
In Table 6.4, we can recognize the expected tardiness value of 11.1, and we can see the expected value of the other 6 listed performance measures. Of course, if a different sequence is selected, then all these results can change. Thus, the example gives rise to seven stochastic counterpart problems, each aiming to minimize the relevant value in the last row of the table.
In stochastic counterpart models, it is convenient to assume that processing times are probabilistically independent. In words,independencemeans that the processing time realized for one of the jobs does not depend on which processing time is realized for any of the other jobs. Without this assumption, it is seldom possible to find analytic solutions that hold in general. In our coverage, however, we want to develop practical and flexible approaches to stochastic scheduling, so we do not necessarily limit ourselves by requiring independent processing times. For instance, in Example 6.1, we assumed independentfactorsthat influenced all processing times in the same direction, but the resulting processing times were not independent–they were correlated.
Nevertheless, the small size of our example enabled us to enumerate the states of nature, treat the set of possible outcomes as a discrete probability distribution, and calculate the required expected values.
In relaxing condition C4, we assume that processing times are random variables with given distributions. The basic stochastic scheduling model containsnsuch random variables, and a small table such as the one in Example 6.1 may not be sufficient to fully capture the probability distributions involved.
However, the approach illustrated in our analysis of the example can still be applied if we rely on a table that is drawn from a much larger data set describing the probability distributions. Technically, we generate an r×n table of processing times resembling Table 6.1, where ris the number of scenarios, or the sample size, andn is the number of jobs. Typically, the scenarios are equally likely, but they could also be assigned probabilities. Row i contains nsampled processing times, one for each job, while columnjincludesrsamples for the processing times of jobj. If the table is exhaustive, as was the case in Table 6.1, then the data represent an exhaustive sample, and the table is essentially a discrete probability model. On the other hand, if the distribution is very large (or infinite, which would be the case for a continuous distribution), then anyr×ntable drawn from that distribution would be a sample. A sampling approach could accommodate both probabilistically independent and probabil- istically dependent cases.
In this chapter, we represent random processing times with the aid of anr×n table. As the foregoing discussion indicates, such a table may hold an exhaustive sample, which represents a discrete probability model, or it may hold a limited sample, which can represent a set of equally likely scenarios observed in histor- ical records or produced by simulation. This interpretation leads to a general technique ofsample-based analysis, in which we rely on discrete probability models or simulation outcomes in our analysis. In Chapter 7, we extend our
6.2 Basic Stochastic Counterpart Models 133
purview to continuous distributions as well. (See Appendix A for background on generating samples.)
In our second example, processing times are independent, and we can illustrate the simulation interpretation of the sample-based approach.
∎ Example 6.2 Consider a problem containing n = 5 jobs with stochastic processing times. The expected processing time for each job is shown in the following table. These match the values in Example 6.1.
Jobj 1 2 3 4 5
E[pj] 3 4 5 6 7
dj 8 5 15 20 12
Here, the processing times are randomly distributed with a range of 4. In other words, the processing time for job 1 occurs randomly between 1 and 5, the processing time for job 2 occurs randomly between 2 and 6, and so on.
For the purposes of illustration, we work with a sample of 10 scenarios corresponding to the realizations shown in Table 6.5.
At the top of the table, we calculate the average of the 10 processing time realizations for each of the jobs, mainly as a check on the accuracy of the sampling. For example, job 1 has an average processing time of 2.984 in the sample, very close to its expectation of 3. The other averages are also close to their expectations.
Table 6.5
Job 1 2 3 4 5
Average 2.984 3.891 5.122 6.195 7.280
Scenario
1 3.710 4.086 3.152 4.689 6.589
2 2.390 2.197 6.395 5.965 7.699
3 4.317 4.263 6.232 5.616 8.468
4 1.138 4.117 5.879 7.325 5.566
5 2.836 2.564 6.144 7.793 8.124
6 2.686 3.734 3.439 7.770 6.325
7 2.533 4.915 5.287 4.745 8.160
8 2.610 2.850 4.546 4.833 8.683
9 3.394 5.721 5.591 6.741 5.102
10 4.229 4.460 4.557 6.477 8.081
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The basic idea of sample-based analysis is to find scheduling decisions that are optimal for the sample. To the extent that the simulated sample mimics reality, the sample represents the range of possible realizations. By increasing the sample sizer, we can approximate the true optimal solution as precisely as we may wish. (Normally, a sample of size 10 is too small for the precision we seek, but a sample of 1000 is reliable enough for many applications.) We can even view the data in Example 6.1 as a special case of a sample in which the scenarios happen to be exhaustive. With this interpretation, a sample is operationally equivalent to a list of equally likely scenarios that represent possible random outcomes.
Starting with the sample in Table 6.5, we can explore the problem numerically. For example, suppose we adopt total flowtime as a measure of performance and begin with the sequence 1-2-3-4-5, which orders the jobs by SEPT. For each of the ten scenarios, we fix this sequence and compute the flowtime of each job under each scenario, as shown in Table 6.6. From these values, we compute the resulting value ofF, shown in the right-hand column of the table.
Next, we find the average value, 65.522, which is shown at the top of the right-hand column in Table 6.6. In this column, as in the previous table, we display an average at the top.
Having evaluated the objective function for the sequence 1-2-3-4-5, our task is now to examine other job sequences and find the one that minimizesF. That search may be tedious, but it is at least straightforward. It turns out that the value of 65.522 is the minimum possible value for this sample, indicating that the sequence 1-2-3-4-5 is optimal forF.
Table 6.6
Job 1 2 3 4 5 F
Scenario Flowtimes 65.522
1 3.710 7.796 10.948 15.637 22.225 60.317
2 2.390 4.587 10.982 16.947 24.646 59.553
3 4.317 8.581 14.813 20.428 28.897 77.036
4 1.138 5.255 11.134 18.459 24.026 60.012
5 2.836 5.400 11.544 19.336 27.460 66.575
6 2.686 6.420 9.859 17.629 23.954 60.548
7 2.533 7.448 12.735 17.480 25.641 65.838
8 2.610 5.460 10.006 14.839 23.522 56.437
9 3.394 9.115 14.706 21.447 26.549 75.211
10 4.229 8.689 13.246 19.722 27.804 73.690
6.2 Basic Stochastic Counterpart Models 135
Our main point is that sample-based analysis (conveniently implemented, for example, in a spreadsheet) is an appropriate general tool for solving stochastic counterpart problems, even though it relies on numerical calculations rather than analytic results. Later in this chapter, we describe a software alternative for implementing this type of analysis. In some cases, however, analytic results are available, sparing us the need to use sample-based analysis at all.
Consider the stochastic counterpart of theF-problem. In other words, proces- sing times are random, and the objective is to minimize the expected value of total flowtime. We can also consider the related problem of minimizing the expected value of total lateness, because of the algebraic relationship between flowtime and lateness.
∎Theorem 6.1 E[F] and E[L] are minimized by shortest expected processing time (SEPT) sequencing (E[p[1]]≤E[p[2]]≤ ≤E[p[n]]).
Proof. We first prove the theorem for E[F]. Repeating Eq. (2.1),
n
j= 1
Fj=
n
j= 1 j
i= 1
pi =
n
j= 1
n−j+ 1 pj
If we interpret p[j] as a random variable, this equation remains valid. Thus, the total flowtime is a weighted sum of random processing times (with deterministic weights). Therefore,
EF = E
n
j= 1
Fj =
n
j= 1
E Fj =
n
j= 1
n−j+ 1 E pj
By the same argument that we used as an alternative proof for Theorem 2.3, this sum is minimized by SEPT. To prove the result for E[L], note that Theorem 2.5 still holds–that is,L=F−Dso E[L] = E[F]–E[D] (whereD= dj). □
Theorem 6.1 shows how to solve two particular stochastic counterpart mod- els optimally, but it does not say that total flowtime and total lateness are mini- mized by SEPT in every scenario. Rather, we proved that SEPT minimizes them on average. In the 10 scenarios of Example 6.2, sequences other than SEPT are optimal. (In the first scenario, for example,Fis minimized by the sequence 3-1- 2-4-5.) But such an observation is made in hindsight, and we cannot rely on hindsight for sequencing decisions, so SEPT is the best we can doex ante, before the realizations are revealed. Thus, the theorem tells us that in Example 6.1, the sequence 1-2-3-4-5 is optimal for minimizing E[F], and it is not necessary to resort to sample-based optimization.
Using the same approach, we can also solve the weighted versions of these two problems, as stated in Theorem 6.2. We emphasize that Theorems 6.1 and 6.2 do not require the processing times to be stochastically independent.
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∎Theorem 6.2 E[Fw] and E[Lw] are minimized by shortest weighted expected processing time (SWEPT) sequencing (E[p[1]]/w[1]≤E[p[2]]/w[2]≤ ≤ E[p[n]]/w[n]).
Now consider the stochastic counterpart of minimizing maximum lateness, as approached by sample-based analysis. For every row in the sample, EDD is an optimal sequence. But this is true for any processing time outcomes–that is, we could sequence the jobs by EDD irrespective of their processing time realizations. In fact, this would be true for an exhaustive sample.
∎Theorem 6.3 E[Lmax] and E[Tmax] are minimized by earliest due date (EDD) sequencing (d[1]≤d[2]≤ ≤d[n]).
In other words, EDD sequencing remains optimal when our model contains stochastic processing times.