Safe Scheduling
7.3 Trading Off Tightness and Tardiness
7.3.3 A Branch-and-bound Solution
two functions are structurally similar. We can express their common form as follows:
Objective =
n
j= 1
μBj+μj+θ σ2Bj+σ2j 1 2
7 10
In this expression, which applies for the case of normal distributions, eitherθ= z, as in Eq. (7.4), for minimizing the sum of stochastically feasible due dates, orθ
=γϕ(z∗), as in Eq. (7.9), in the optimal tightness/tardiness trade-off. For a given θ, the same sequence will be optimal for both models, but the service levels, and therefore the due dates, will notbe the same for both models: The trade-off model requires lower service levels.
dominance properties, adjacent pairwise interchanges, lower bounds, and an initial solution.
Static Dominance. The basic dominance condition between two jobs, i and j, states that an optimal solution exists in which job i precedes job j.
In other words, when searching for an optimal sequence, we need not consider any sequences in which job j appears in sequence before job i.
A dominance property isstatic if it holds independently of the other jobs in the schedule. For example, a job dominates another if it has smaller mean and variance.
Property 7.2 Ifμi≤μjandσi≤σjthen an optimal schedule exists in which jobi precedes jobj.
Even when one job does not have a smaller mean and variance, it may still dom- inate another based on the relative differences in the two means and variances.
Property 7.3 Ifμi<μjandμj−μi≥θ(σi−σj) then an optimal schedule exists in which jobiprecedes jobj. (A sufficient condition isμj−μi≥θσi.)
P(0)
P(1) P(2)
P(23)
P(π) P(21)
P(n)
P(2n) . . .
. . .
. . . . . .
. . .
. . .
Figure 7.4 A portion of the branching scheme (B&B tree) for ann-job problem.
7.3 Trading Off Tightness and Tardiness 179
Properties 7.2 and 7.3 can be proven with pairwise interchange arguments.
Satisfying Property 7.2 implies satisfying Property 7.3, but not the other way around, so it is efficient to check Property 7.2 first. These static dominance con- ditions can be evaluated once at the outset and then applied as needed during the generation and evaluation of partial sequences. Furthermore, if jobjis domi- nated by jobiby virtue of the sufficient condition of Property 7.3, then every job beyond jobjin SEPT order is also dominated.
If we are augmenting a partial sequence and we find that jobidominates jobj while neither appears in the partial sequence, then we need not consider the augmented partial sequence constructed by appending jobjnext. That is, only jobs that are not dominated by other unscheduled jobs have to be considered.
Dominance conditions can reduce the search effort required to find an optimal schedule, but the extent to which dominance conditions apply may depend on the specific data in a given problem instance. For that reason, it is difficult to predict their effectiveness at curtailing the search effort.
Adjacent Pairwise Interchanges. We have already seen that requiring API sta- bility can be helpful for heuristic applications. The same often applies to B&B as well. Suppose that job j appears immediately after job i somewhere in the sequence, and consider the conditions under which it would be better to inter- change the two jobs. The completion times of jobs not involved in the inter- change are unaffected by the swap, so the overall objective is improved if and only if the total contribution from jobs i and jis improved. The mean time to process the jobs preceding i andjcan be denoted byμBand the variance of that time byσ2B.
For the sequencei-j, the contribution of the two jobs to the objective is h i,j = μB+μi +θ σ2B+σ2i
1 2+ μB+μi+μj +θ σ2B+σ2i +σ2j 1 2
The expressionh(j,i), for the reverse sequencej–i, is similar, and the change in the objective due to the interchange is
gij σ2B =h j,i –h i,j =μj−μi+θ σ2B+σ2j
1 2− σ2B+σ2i
1 2 7 11
Using Eq. (7.11), the interchange is undesirable (and thei–jorder is at least as good as the reverse) as long asgij(σ2B)≥0, which we call theAPI condition.
∎ Theorem 7.2 A necessary condition for a sequence to be optimal is that every pair of adjacent jobsiandj(withjfollowingi) satisfies the API condition, gij(σ2B)≥0.
Unfortunately, the API condition does not lead to a universal rule for determin- ing whetherjshould followibecause the condition depends onσ2Band therefore
7 Safe Scheduling 180
on the jobs making up the partial sequence that precedesiandj. Nevertheless, the API condition can be used to eliminate some partial sequences and therefore curtail an enumerative search. Specifically, suppose that we are about to aug- ment a partial sequence πby appending job i. Suppose further that the last job inπis jobj. If the API condition holds, then the augmented sequenceπi is dominated and can be eliminated.
For another perspective on the API condition, assume jobsiandjare in strict SEPT order (μi−μj< 0) butσ2i−σ2j> 0. In this case, Property 7.2 does not apply, and Property 7.3 may or may not hold, depending onθ. Under those conditions, Eq. (7.11) is monotone decreasing in σ2B, so there exists a threshold value σ2B i,j ≥0 for which the optimal sequence of jobsiandj, if adjacent, switches fromj–itoi–j. We can calculate this value in advance by,
σ2B i,j = max 0, θ 2
σ2i−σ2j
μj−μi
+μj−μi
2θ
2
−σ2i 7 12
As we add jobs to a partial sequence, the value ofσ2B increases. Once it reaches the threshold for (i,j), we never have to consider a partial sequence in which job iimmediately follows jobj.
Lower Bounds. Suppose that we have a partial sequenceπand we wish to com- pute a lower bound on the value of the objective function that can be obtained by completing the sequence. The component of the objective function corre- sponding to the jobs jinπhas already been determined from Eq. (7.10). Let π denote the set of unscheduled jobs. In the setπ, we take the set of means μj in smallest-first order and take the set of standard deviations σj in smal- lest-first order and treat these values as if they were paired. Then we calculate the values ofHj(dj) generated by these fictitious jobs and add them to the com- ponent for the partial sequence. This total is a lower bound on the value that could be achieved by completing the partial sequence in the best possible way. (A formal proof follows a pairwise interchange argument.)
Thus, if we ever encounter a partial sequenceπfor which the lower bound on the value of the objective function is greater than or equal to the value for a known sequence, we conclude that completing π can never lead to a full sequence with a better value than the known sequence. Such a condition, which amounts to fathoming P(π), tells us that we do not need to solve P(π), and we can thus eliminate it and curtail the tree search.
Initial Solution. Given that we are using a B&B algorithm, it makes sense to begin by finding a good initial solution that can be effective at fathoming partial sequences with relatively few jobs. That is, we can implement a heuristic pro- cedure to construct a feasible solution before the tree search begins, on the chance that a good feasible solution may eliminate some partial sequences and reduce the search effort. For this purpose, we can implement a sorting rule, such as SEPT, or, with additional computational effort, the greedy heuristic.
7.3 Trading Off Tightness and Tardiness 181
As an example, we apply the B&B approach to Example 7.3 using these tools, starting with a heuristic initial solution. Suppose we implement SEPT, which corresponds to the sequence 1-2-3-4-5 and yields an objective function of 482.6.
Next, we investigate static dominance properties. In this example, Property 7.2 reveals that job 4 dominates job 5 and job 3 dominates both job 4 and job 5. Property 7.3 reveals that jobs 1 and 2 dominate job 5.
We begin the tree search starting with P(0), the empty sequence. The candi- dates for first position in sequence are only jobs 1, 2, and 3 because the other jobs are dominated. Thus, the first level of the tree contains P(1), P(2), and P(3).
The detailed calculation of the lower bound for P(2) is shown in Figure 7.5. For this calculation, the initial partial sequence contains job 2 followed by four fic- titious jobs (F1–F4) characterized by increasing means and variances. Similar calculations provide lower bounds for P(1) and P(3), as shown in the full search tree of Figure 7.6. These three lower bounds are well below the objective for the initial heuristic, so no fathoming is possible yet, and we proceed to Level 2.
Branching from P(1), we generate nodes for P(12) and P(13). (The static dom- inance properties eliminate the other two partial sequences that begin with job 1.) To evaluate P(12), we first apply the API condition and discover that P(21) dominates P(12). Similarly P(31) dominates P(13). Thus, P(12) and P(13) are eliminated. Branching from P(2) yields one undominated augmented sequence, P(21), and branching from P(3) yields three undominated augmented
Figure 7.5 Lower bound calculation for partial sequence P(2).
7 Safe Scheduling 182
sequences. The lower bounds for these four remaining partial sequences are shown in Figure 7.6.
At Level 3, the only undominated candidate to follow P(21) is job 3, but P(213) is dominated by P(231), so the P(213) branch can be fathomed. The partial sequences that survive the API condition are P(321), P(324), and P(341). Their lower bounds are shown in Figure 7.6.
At Level 4, we branch from P(321), yielding the full sequence 3-2-1-4-5 with objective 475.1, the best value thus far. This value allows us to fathom P(341) because its lower bound is larger than 475.1. The remaining node is P(324), which leads to the full sequence 3-2-4-1-5. Its value of 475.8 is larger than the best solution yet found, so the node is eliminated. Alternatively, we could check to see that 3-2-4-1 is not API stable, allowing its elimination one level earlier.
The full tree search is shown in Figure 7.6, demonstrating that only three complete sequences need to be evaluated during the B&B algorithm (as com- pared with 120 feasible permutations that would be evaluated by complete enumeration).
In summary, the B&B algorithm builds job sequences starting from time 0, augmenting partial sequences in all possible ways. For each partial sequence encountered, we first check dominance properties, allowing us to eliminate dominated partial sequences and to pursue only those that remain. If the dom- inance condition fails, we then compute the corresponding lower bound to see whether the partial sequence can be compared with the value of an existing solution and fathomed. If the lower bound does not permit fathoming, then the partial sequence remains active and is ultimately augmented by adding some
P(21)
P(213) P(312) P(314) P(321) P(324) P(341) P(342) P(31)
P(3)
P(32) P(34)
472.7 469.7 471.5
475.3 473.7
475.1 P(32145) P(32415)
475.8 475.1
464.5 P(2)
P(23) P(1)
476.8
P(13) P(12)
P(0)
472.0
481.8 482.6
SEPT
Figure 7.6 The B&B tree for Example 7.3.
7.3 Trading Off Tightness and Tardiness 183
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.