Sequencing for Stochastic Scheduling
6.9 Summary
be listed among the scenarios. That requirement rules out sampling and simu- lation. Second, solving for either minimax cost or minimax regret under the sce- nario-based approach is also a challenging combinatorial problem. Third, when using ranges, there is no“robust”means of identifying minimum and maximum values for the ranges guaranteed to work, let alone work for the worst case.
Finally, and most importantly, the minimax regret criterion is not an effective risk management tool. Instead of ameliorating potentially crippling risks, as proper risk management should, it provides expensive insurance against rela- tively affordable risks and often does so while increasing the risk of the worst-case scenario.
We also showed that in some cases the deterministic approach may be less applicable. For example, dynamic programming cannot handle the maximum cost problem because E[max{gj}] is not an additive performance measure. Given such difficulties in the stochastic environment, it is important to identify effi- cient solutions, or at least partial solutions, where they exist, and that has been the main thrust of the chapter. For problems that are beyond the reach of the methods we have introduced thus far, it remains important to develop practical heuristic approaches.
To handle general processing time distributions that are not necessarily sta- tistically independent, we introduced sample-based analysis and showed how to compare sequences numerically. This approach is inherently more time con- suming than the use of deterministic counterparts, but it remains practical and can be implemented in a spreadsheet. Nevertheless, sample-based analysis is intrinsically a heuristic approach, because a simulated sample cannot repre- sent a model perfectly; it can identify solutions that are likely to be optimal, but it cannot guarantee optimality. Furthermore, the use of large samples is an addi- tional computational burden. For instance, suppose we have a stochastic prob- lem that we wish to solve by sample-based analysis with a sample of 1000 (roughly 210) using dynamic programming. Then, every function evaluation takes 1000 times longer than would be the case in the deterministic counterpart.
If we assume the computational requirement in dynamic programming is roughly proportional to 2n, then a given computational effort will solve for 10 fewer jobs in the stochastic case than in the deterministic counterpart.
For example, if 25 deterministic jobs can be sequenced in half a minute of com- putation time, only 15 stochastic jobs can be sequenced in the same time.
A similar reduction in tractable problem size occurs with branch-and-bound approaches.
The availability of user-friendly simulation software, such as ASP, expands the set of models that we can analyze with a sample-based approach. Thus, if we can determine an optimal sequence easily but encounter computational difficulty in evaluating the optimal value of the objective, we can enlist the help of ASP to make the evaluation easier. More importantly, we can integrate the Evolution- ary Solver to produce a flexible and effective heuristic procedure for solving sto- chastic sequencing problems.
We also discussed two other approaches to scheduling under uncertainty, specifically fuzzy scheduling and minimizing maximum regret. These methods do not rely on probability distributions for processing times. Dubois et al. (2003) provide a widely cited survey on fuzzy scheduling, and the seminal paper on minimizing maximum regret in a scheduling context is due to Daniels and Kou- velis (1995). However, with respect to these two topics, the gap between research and practice remains substantial. The literature on minimizing the maximum cost – which we also discussed – is much more sparse. In addition, arguably, it emphasizes the importance of the worst case too much even when it is extremely unlikely. (Similar criticism applies to regret as well.)
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In the next chapter we discuss a more practical approach designed to control the likelihood of a very bad outcome.
Exercises
6.1 Consider a problem containing n = 5 jobs with stochastic processing times. The randomness in the processing times can adequately be repre- sented by three states of nature: good, normal, and bad, with probabilities of 0.2, 0.5, and 0.3, respectively.
State Jobj 1 2 3 4 5
Good pj 5 3 7 6 8
Normal pj 7 6 8 10 12
Bad pj 9 12 10 15 14
dj 10 22 40 31 25
a) Find the minimum value of the expected total flowtime, along with the sequence that achieves it. Compare the optimal value with that of the deterministic counterpart.
b) Find the minimum value of the expected maximum tardiness, along with the sequence that achieves it. Compare the optimal value with that of the deterministic counterpart.
c) Find the minimum value of the expected number of jobs tardy.
Compare the optimal value with that of the deterministic counterpart.
6.2 Consider a problem containing n= 5 jobs with stochastic processing times, each of which follows a normal distribution with known mean and standard deviation.
Jobj 1 2 3 4 5
μj 17 20 24 25 30
σj 3 4 2 5 3
dj 60 80 70 50 90
a) Find the minimum value of the expected total flowtime, along with the sequence that achieves it. Compare the optimal value with that of the deterministic counterpart.
b) Find the minimum value of the expected maximum tardiness, along with the sequence that achieves it. Compare the optimal value with that of the deterministic counterpart.
Exercises 163
6.3 Shown below is a sample of 10 observations for the processing times ofn= 5 jobs in a sequencing problem.
Jobj 1 2 3 4 5
dj 60 80 70 50 90
Job 1 Job 2 Job 3 Job 4 Job 5
Sample 1 17.79 23.80 19.74 26.90 32.63
Sample 2 15.65 18.34 25.40 14.98 26.80
Sample 3 22.59 18.62 21.75 25.53 30.56
Sample 4 15.29 20.98 22.85 31.80 33.86
Sample 5 15.56 20.39 24.09 22.45 28.16
Sample 6 19.00 18.05 20.28 25.71 28.99
Sample 7 18.00 19.16 20.75 25.02 30.86
Sample 8 18.37 19.06 25.86 24.14 23.24
Sample 9 14.35 14.68 22.69 26.55 24.73
Sample 10 16.61 22.99 20.99 26.12 28.43
a) Find the minimum value of the expected maximum tardiness, along with the sequence that achieves it.
b) Find the minimum value of the maximum expected tardiness, along with the sequence that achieves it.
c) Find the minimum value of the expected number of jobs tardy.
6.4 Show that SWEPT is optimal for minimizing E[Fw].
6.5 Consider a problem containing n = 5 jobs with stochastic processing times. The randomness in the processing times can adequately be repre- sented by three states of nature: S1, S2, and S3, with probabilities of 0.3, 0.4, and 0.3, respectively.
State Jobj 1 2 3 4 5
S1 pj 5 3 7 6 8
S2 pj 7 6 8 10 12
S3 pj 9 12 10 15 14
dj 10 22 40 31 25
6 Sequencing for Stochastic Scheduling 164
In addition, the parameters of the cost functiongj(Tj) =δ(Tj)(aj+bjTj) are given in the following table.
Jobj 1 2 3 4 5
aj 2.0 3.0 4.0 5.0 1.0
bj 0.6 0.3 0.1 0.4 0.3
a) Find the minimum value of the expected maximum cost.
b) Compare the value in (a) with that of the deterministic counterpart.
The proof of the new version of Theorem 6.8 already shows that.
Furthermore, careful reading of the new version of the Theorem shows that there is no assumption the two jobs must be adjacent (which is why the proof includes the case of intermediate jobs).
6.6 Consider a problem containing n= 5 jobs with stochastic processing times, each of which follows a normal distribution with known mean and standard deviation.
Jobj 1 2 3 4 5
μj 17 20 24 25 30
σj 3 4 2 5 3
dj 60 80 70 50 90
a) Find the minimum value of the expected maximum tardiness, along with the sequence that achieves it. Use ASP and Evolutionary Solver to produce a solution.
b) Find the minimum value of the maximum expected tardiness, along with the sequence that achieves it. Does the sequence match the sequence in (a)?
6.7 Consider the range model with total flowtime criterion and minimax regret objective. We say that two ranges are orderedif the one with the smaller or equal lower limit also has a smaller or equal upper limit, with at least one inequality strict. It can be shown that jobs with ordered ranges should appear in that order in the optimal minimax regret sequence. (Note that two unequal ranges arenotordered only if one is strictly nested within the other, with no equal limit, as is the case for any two ranges in Example 6.8.)
Exercises 165
a) Suppose all ranges have a lower limit of 0 (or any other equal value).
Show that the minimax cost solution also attains the minimax regret.
b) Suppose processing times are stochastically ordered where the cdf of job j isFj(x). Let aand bbe small positive numbers such that a + b< 1, and let the range limits for jobjbeFj−1 a andFj−1 b –where ifFj−1 x =ythenFj(y) =x–so each range has a confidence level of 1− a−b; for instance, for a 95% confidence interval, we may seta=b= 2.5%. Show that SEPT minimizes expected flowtime, minimax cost, and minimax regret.
c) Suppose all processing times are lognormal with the samecv. Does the result of (b) apply?
6.8 It can be shown that any two lognormal random variables with the samecv are stochastically ordered by SEPT. Revisit Example 6.6 but now assume all jobs havecv= 0.6. Show that job 3 must be first. Are there any addi- tional dominance conditions in the example that may apply by virtue of stochastic dominance?
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