Sequencing for Stochastic Scheduling
6.8 Non-probabilistic Approaches: Fuzzy and Robust Scheduling
precision for our purposes. If we repeat the simulation with a sample size of 10 000, the confidence interval drops to 0.24 for an estimated mean of 23.21 (or about 1.0% on either side). If this level of precision is adequate for our purposes, we would keep the sample size set at 10 000.
Having fine-tuned the simulation parameters, we can search for a sequence that minimizes the expected tardiness. Our search tool is the Evolutionary Solver, which we introduced in Chapter 4 and for which we provided a default set of search parameters. In this example, we specify the objective function cell as C11 (the estimated mean tardiness), which we want to minimize. The decision variables appear in row 14, and we impose the requirement that they satisfy the alldifferent constraint. Running the Evolutionary Solver produces the sequence 3-5-2-1-4, with (estimated) mean tardiness of 8.66.
Integrating simulation with the Evolutionary Solver creates a powerful search tool for solving stochastic scheduling problems. We refer to the ASP User’s Guide for additional detail.
6.8 Non-probabilistic Approaches: Fuzzy
widely applied; their main attraction lies in the mathematical challenges they pose.
Fuzzy logic is a practical and successful approach for controlling complex processes by the correct combination of possible adjustments that apply to multiple measurements. Suppose we need to control temperature and pressure, and we have several possible levers–or responses–with which to control them.
Furthermore, using any single lever to adjust temperature may affect pressure in an undesirable way and vice versa. We want to find the right combination of levers. Before the advent of fuzzy logic, the solution would attempt to build a controller that could respond to any combination of temperature and pressure correctly, often a very challenging task. The basic idea of fuzzy logic is that the usefulness of a particular response depends on the distance from target as per a continuous function with values between 0 and 1. But similar functions – known asmembership functions–apply to all possible responses. Using those membership values as weights (which usually requires normalization), one response is selected randomly and applied briefly; thus, a high weight response is more likely to be selected. A new measurement then induces new member- ship values, and a new brief response is selected accordingly. In effect, this creates a self-adjusting mixed response that has been shown by experience to work very well. It revolutionized the control of complex processes. That impres- sive practical success led to numerous attempts to apply fuzzy logic to other problems, including sequencing and scheduling decisions. However, selecting a sequence is not akin to the adjustment of a complex process by a mixture of responses: We cannot correct the choice very frequently based on new mea- surements. Nonetheless, when we wish to guide a random search with more than one objective, the fuzzy model may yet prove useful, as in each selection a different criterion may be selected based on membership weights.
Robust scheduling builds on the principles of decision theory. Decision theory models traditionally assume a finite set of possible actions and a finite set of out- come states, giving rise to a list of scenarios. Moreover, the models assume that the actual realization is included in these scenarios. We refer to this structure as ascenario model. This form is not appropriate when the number of possible actions or possible states is quite large or infinite, but in cases similar to Example 6.1, they are conceptually plausible.
A fundamental assumption in decision theory is that most decision makers are risk averse, which implies they are more concerned with avoiding excessive losses than with maximizing rewards, and in that respect they prefer conserv- ative choices. One way to capture this conservatism is to minimize the maxi- mum possible cost, often called the minimax cost criterion. The minimax cost criterion is especially attractive when we need protection against worst- case results and probabilities are not a major consideration. However, most of the literature on robust scheduling focuses on a relative measure of cost rather than an absolute measure – namely, regret. Regret is the difference
6.8 Non-probabilistic Approaches: Fuzzy and Robust Scheduling 155
between the result achieved by a particular decision in the random state that ultimately occurs and the best result that could have been achieved in that state.
Decision makers facing uncertainty know that it is impossible to predict the future or to always select the best possible schedule; however, they might well assume that their decisions will subsequently be judged based on hindsight, after uncertainties are resolved. Therefore, the argument goes, risk-averse decision makers may choose to minimize regret.
But what does minimizing regret actually mean in the context of stochastic scheduling problems? One interpretation could be that the objective is to minimize expected regret, with the expectation taken over the distribution of random outcomes. However, minimizing expected regret is not really a distinct research area for scheduling, due to the following well-known property.
Property 6.1 For any performance measure, the minimum expected regret is achieved by minimizing the expected value of the performance measure.
Therefore, this interpretation of minimizing regret does not lead to new concep- tual challenges because expected-value performance measures have been stud- ied extensively, as the foregoing sections demonstrate. In addition, the notion of conservative decision-making usually focuses on worst cases, not probability distributions. Instead of minimizing expected regret, we can apply the minimax criterion to regret–that is, seek to minimize the maximum regret. It turns out that even scheduling problems that are easy to solve when the objective is an expected value become difficult when the objective is minimax cost or minimax regret. As an illustration, consider Example 6.7.
∎Example 6.7 Consider a problem containingn= 5 jobs with stochastic pro- cessing times. The randomness in processing times is described by four equally likely states of nature: GG, 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 2.7 3.7 3.4 7.8 6.3
GB pj 2.5 4.9 5.3 4.7 8.2
BG pj 2.6 2.9 4.5 4.8 8.7
BB pj 3.4 5.7 5.6 6.7 5.1
What is the best sequence using a minimax criterion for total flowtime?
For instance, the GG state-specific solution is determined by shortest-first sequencing of the five processing times that occur in state GG, or 1-3-2-5-4.
The corresponding total flowtime is 58.6 (listed under Best in Figure 6.9). When
6 Sequencing for Stochastic Scheduling 156
GB occurs, the best sequence is 1-4-2-3-5, with total flowtime of 64.8, and sim- ilarly for BG (1-2-3-4-5, with total flowtime 56.4) and BB (1-5-3-2-4, with total flowtime 72.3). These results give rise to the state-specific Regret Table shown at the bottom right of Figure 6.9.
In the state-specific Regret Table, each row corresponds to one of the four states, and each column corresponds to one of the sequences that produces minimum total flowtime in one of the states. The table entry is zero if the total flowtime for that sequence in that state is actually the minimum for that state.
(For instance, in state BG, the sequence 1-2-3-4-5 produces the total flowtime of 56.4, and this is the minimum for BG.) The table entry is positive if the sequence does not produce the minimum; in that case, the entry is the difference between the total flowtime produced and the minimum value for the state. That differ- ence is the quantitative representation of regret. (For instance, in state BG, the sequence 1-4-2-3-5 produces total flowtime of 58.6, whereas the minimum for that state is 56.4, so the regret is measured as the difference, 2.2.) Thus, the Regret Table shows that, of the four sequences identified so far, the minimax regret is 2.8, associated with 1-2-3-4-5, as determined by comparing the four values below the Regret Table. However, in this example, the minimax regret is actually produced by a sequence that is not one of the four identified so far. The optimal sequence is 1-3-2-4-5. Its maximum regret is computed by comparing its Total Flowtime in each state with the Best Flowtime in each state and recording the differences. The largest difference, 2.7, represents its
Figure 6.9 Calculations for Example 6.7, showing results for the sequence (1-3-2-4-5).
6.8 Non-probabilistic Approaches: Fuzzy and Robust Scheduling 157
maximum regret. As the example shows, it is possible that none of the four state-specific optimal sequences minimizes the maximum regret, and for that reason it may be necessary to expand the Regret Table and evaluate all possible sequences to find the optimum, perhaps using one of the optimization methods described in Chapter 3. Finding the minimax regret is known to be an NP-hard problem for theF-problem.
When minimax cost is the objective in Example 6.7, a similar analysis applies.
Once again, in this example, the minimax cost is not attained by one of the state- specific minimax cost sequences. (Finding the minimax cost is also known to be NP-hard for theF-problem with scenarios.) Instead, an enumerative search is required to find the optimal sequence, which is 1-3-5-2-4, obtaining a maximum flowtime of 72.8. In brief, the sequence 1-3-5-2-4 achieves maximum flowtime of 72.8 and maximum regret of 11.3, whereas the sequence 1-3-2-4-5 achieves maximum flowtime of 75.0 and maximum regret of 2.7. Thus, we can see from this example that minimax cost and minimax regret are, in general, optimized by different sequences. To achieve optimal results on one measure may well require a sacrifice in the other measure. For this reason, it is not obvious which criterion to use, even when agreement exists that a risk-averse approach is desirable.
Our analysis so far essentially relies on the scenario model in which the set of scenarios must be finite and exhaustive. In practice, however, the assump- tion that we can listallpossible scenarios is rarely applicable, and simulated scenarios are not usable for this purpose because they are not guaranteed to include the actual outcome. Instead, therange modelis an alternative formu- lation that is suited to instances in which a finite set of exhaustive scenarios does not exist. In the range model, we specify a range for each realization.
(These ranges represent intervals that are deemed likely to occur, but because probability distributions are not specified, no clear guidelines exist for deter- mining them.) In this setting, given any regular performance measure repre- senting a proxy for cost, solving for the minimax value simply requires substituting the longest possible processing times into a deterministic model.
Furthermore, if those longest times are not finite, the minimax cost is unbounded, in which case minimax cost scheduling is neither challenging nor interesting. But the optimization of minimax regret remains challenging (and perhaps that is why the termrobust schedulingis typically interpreted as driven by minimax regret rather than minimax cost). Nonetheless, the minimax regret solution can be found by limiting attention to extreme realizations in which each processing time lies at either the minimum or the maximum of its range. Thus, even if we allow for an infinite number of possible realizations, the range model allows us to consider only 2nrealiza- tions to identify a minimax regret solution. That is, conceptually, we can treat extreme solutions as exhaustive scenarios. The following example illustrates the range model.
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∎Example 6.8 Consider a problem containingn= 4 jobs, with random pro- cessing times symmetrically distributed between the following minimum and maximum values.
Jobj 1 2 3 4
Minimum 46 42 40 10
Maximum 48 54 58 90
Expected 47 48 49 50
In keeping with the philosophy of risk-averse decision-making, we don’t need to know the probability distribution for each processing time, but knowledge of expected processing times allows us to compare the minimax regret solution to the SEPT sequence. Because it is sufficient in the range model to consider only extreme realizations, the example problem can be formulated as a sce- nario-based model with 2n= 16 scenarios. In the data set, the range maxima and the expected values appear in increasing order when we sequence the jobs 1-2-3-4. This sequence is therefore both the minimax cost sequence and the SEPT sequence, with E[Fmax] = 560 and E[F] = 480. Because we use ranges, sol- ving for E[Fmax] here is not more difficult than solving for SEPT, but the two optimal sequences need not agree in general.
Next consider minimax regret. First, we list the 16 extreme scenarios and for each one find the best possible total flowtime by taking the processing times in shortest-first order. Then, for each job sequence, we compare its total flowtime to the best possible value and record the largest difference for all 16 realizations– that is, the maximum regret. Finally, we search for the minimum value among the 4! possible job sequences to identify the minimax regret. In the example, the minimax regret is 104, achieved by the sequence 1-2-4-3. A comparison of the 16 extreme realizations for this sequence reveals a maximum flowtime of 592.
The summary of these results in Table 6.9 reveals that the two minimax objec- tives are in conflict, optimized by different sequences. The minimax cost sequence (1-2-3-4) guarantees total flowtime of at most 560, but with maximum regret of 134. The minimax regret sequence (1-2-4-3) can reduce this value to 104, but only with exposure to a total flowtime of 592. In other words, the min- imax regret sequence achieves an advantage (30) over the minimax cost sequence, but only by risking a larger disadvantage (32) in the worst case. In addition, the minimax regret sequence sacrifices a small amount (1) in the expected flowtime. However, because E[F] is an average over all extreme states, it makes sense to give a reduction in expected flowtime more weight than a reduction in maximum regret. Furthermore, extensive numerical experience suggests that the minimax regret sequence can carry a much higher risk in terms of expected flowtime than in this example, as we discuss next.
6.8 Non-probabilistic Approaches: Fuzzy and Robust Scheduling 159
We ran tests on problem instances containingn= 7 jobs, with ranges gener- ated randomly. Table 6.10 summarizes the results. We observed, for example, that the same sequence is optimal for all three objectives in about 30% of the instances. In the remaining 70%, at least two of the objectives conflicted. In addition, expected flowtime and minimax regret were in conflict only 35% of the time, whereas minimax cost was in conflict with one of the other two objec- tives almost two-thirds of the time. When expected flowtime and minimax regret conflict, the sacrifice in one objective to optimize the other favors expected flowtime by almost an order of magnitude, if we value the measures as equivalent. The strong suggestion is that the SEPT sequence is the best choice for risk-neutral decision makers and can also serve as a decent heuristic for either minimax criterion, but the minimax cost sequence is preferred in risk- averse situations.
In summary, several major problems arise in the pursuit of minimax regret that make it impractical. First, in the scenario-based approach, practicability is significantly reduced by the requirement that all possible final outcomes must
Table 6.9
Sequence Maximum cost Maximum regret Expected flowtime
1-2-3-4 560 134 480
1-2-4-3 592 104 481
Difference 32 30 1
Table 6.10
Conflicts Percent (%)
0 30 One sequence optimizes all three objectives
2 35 Expected flowtime and minimax regret are optimized by the same sequence
2 5 Expected flowtime and minimax cost are optimized by the same sequence
2 7 Minimax cost and minimax regret are optimized by the same sequence
3 24 Expected flowtime, minimax cost, and minimax regret are each optimized differently
100
35 Expected flowtime and minimax regret are in conflict 65 Expected flowtime and minimax cost are in conflict 63 Minimax regret and minimax cost are in conflict 6 Sequencing for Stochastic Scheduling
160
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.