Safe Scheduling
7.4 The Stochastic E/T Problem
unscheduled job to it. The augmented partial sequence is then examined for dominance or fathoming, and the process continues until the best job sequence is discovered.
The B&B approach has implications for finding optimal solutions to the prob- lem of minimizingDsubject to feasibility constraints. As we observed earlier, the objective functionsH(d∗) in Eq. (7.9) andDin Eq. (7.4) are structurally sim- ilar and can be expressed in a common form by definingθappropriately. Thus, the B&B approach illustrated for the tightness/tardiness trade-off applies as well to minimizing the sum of stochastically feasible due dates.
Computational experiments with the B&B algorithm on random problem instances indicate that problems with well over 100 jobs can be solved in a matter of seconds, on average. Heuristic procedures such as the greedy algorithm can be used to solve very large instances quickly. In similar models with job-dependentθ values (for which Property 7.1 does not hold), another plausible heuristic rule would be to sequence the jobs in SEPT order followed by an API neighborhood search, which is asymptotically optimal for the case of commonθvalues.
G(d), or expected total E/T cost, becomes our objective function in the stochas- tic case, and we can express it as
H d =E G d =
n
j= 1
αjE Ej +βjE Tj 7 13
Setting due dates loosely, in such a way that idling occurs between jobs, can only reduce Eq. (7.13) because by ignoring flowtime the model does not account for the inefficiency that idling represents. Therefore, we assume that no idling is permitted. The optimal choice of due dates is again determined by a critical fractile rule, as stated in the following result.
∎Theorem 7.3 Assume all jobs are processed with no inserted idle time and the objective is to minimize the expected total E/T cost. Given the mean and the standard deviation of the normal distribution forCj, the optimal choice of the due datedjis given in standardized form by
Φ zj∗ = βj
αj+βj
A derivation of this property appears in Appendix B. For a given job sequence, we calculate optimal due dates by applying Theorem 7.3 separately to each job, thereby minimizing expected total E/T cost. Recall that in the problem of deter- mining the tightest feasible due dates, service levels are given. In the tightness/
tardiness trade-off, service levels are determined by the critical fractile (γ−1)/γ, whereγis given. In Theorem 7.3, service levels are also derived from a critical fractile property, but one in which job-dependent unit costs are given and jobs may therefore have different service levels.
From the result in Theorem 7.3, we can compute optimal due dates as we did in Eq. (7.8). In the stochastic E/T model, we obtain
dj=μBj+μj+zj∗ σ2Bj+σ2j
1 2 7 14
Then, as in Eq. (7.9) we can calculate the objective function:
H d∗ =
n
j= 1
αj+βj ϕ z∗j σ2B+σ2j
1 2 7 15
For convenience, we definecj= αj+βj ϕ z∗j andsj= σ2B+σ2j 1 2
so that the contribution from jobjto the objective function in Eq. (7.15) is simplycjsj.
7.4 The Stochastic E/T Problem 185
In the spirit of the deterministic counterpart, suppose that we also consider setting due dates equal to expected completion times:
dj=E Cj =μBj+μj
These due dates amount to using safety times of zero, or equivalently,z∗j = 0.
The expected E/T cost for job j is then obtained by replacing z∗j by 0 in Eq. (7.15). This cost must be at least as high as the optimal expected E/T cost associated with jobjin Eq. (7.15) becauseϕ(z) is maximized atz= 0.
As in the previous sections, we can get a feel for the calculations involved by examining a numerical example such as Example 7.4, which contains five jobs with nonidentical pairs of unit costs for earliness and tardiness. Sup- pose we fix the job sequence as 1-2-3-4-5. Then the critical fractiles and optimal due dates can be determined individually for each job. The relevant calculations are shown in Figure 7.7, and we elaborate on the details for job 3.
∎Example 7.4 Consider a problem containingn= 5 jobs with stochastic pro- cessing times as described in the following table.
Jobj 1 2 3 4 5
E(pj) 21 24 30 32 36
σj 4 3 2 3 5
αj 3 2 1 3 4
βj 6 8 5 9 4
cj 9 10 6 12 8
The processing times are independent, each drawn from a normal distribution with mean and standard deviation shown in the table.
Job 3 has a mean completion time equal to the sum of the first three mean processing times, or 75. To find the variance of its completion time, we sum the variances of the first three jobs, obtaining 29. The corresponding standard deviation is the square root of this figure, or about 5.39. Job 3 has a service-level target of 83.3%, corresponding to az-value of 0.967 in the standard normal dis- tribution. Thus, we can meet the service level by settingd3= 75 + 0.967(5.39) = 80.2. However, our objective function does not require the due date as such.
Instead, we use the formula in Eq. (7.15) or αj+βj ϕ z∗j sj= 6(0.25)(5.39)
= 8.073. Similar calculations apply for the other jobs. As Figure 7.7 shows, the sum of the five optimally calculated costs isH(d∗) = 84.00.
7 Safe Scheduling 186
Also shown in Figure 7.7 is the calculation of expected costs obtained by set- ting each due date equal to the expected completion time. The key difference in the cost calculation lies in substitutingϕ(0) forϕ z∗j in Eq. (7.15). With this substitution, the cost becomesH(d) = 102.04. This figure is over 20% greater than the optimal value, a substantial difference that suggests the cost penalty that might be incurred when we use a deterministic counterpart as a proxy for a stochastic problem.
As these calculations illustrate, we can determine optimal due dates provided that we already know the job sequence. Again, however, the larger problem is to find the optimal sequence. Just as a B&B approach worked for the tightness/tar- diness trade-off, we can attack the stochastic E/T problem in a similar fashion. The search tree is the same as the one in Figure 7.4. In addition, the components needed for a solution algorithm are (i) a dominance property to accelerate the search, (ii) a lower bound for partial sequences, and (iii) an effective heuristic pro- cedure to use at the start. These components exist and resemble those encoun- tered earlier when we addressed the problem of trading off tightness and tardiness.
Dominance. In the stochastic E/T problem, a pairwise dominance condition holds, similar to that in Property 7.2.
Figure 7.7 Detailed calculations for the jobs in Example 7.4.
7.4 The Stochastic E/T Problem 187