Polynomially Solvable Cases

This section discusses three polynomially solvable cases for the fixed-job-sequence prob-lem. Notice that the complexity result in this section is presented subject to the assump-tion of input size such that, for example, in the case of identical jobs, we have n copies of processing times and delay times for the n jobs (Orman & Potts, 1997).

4.4.1 1 |(p

, p

, p

), f js |C

In this subsection, the case where a_j= l_j= b_j= p_j for all j ∈ Nn is considered. Despite the strong NP-hardness of the 1|(pj, p_j, p_j)|Cmax problem (Orman & Potts, 1997), its fixed-job-sequence counterpart is polynomially solvable. An optimal schedule can be obtained by the following procedure.

Algorithm PSC1

Step 1. Set j = 1 and E = ∅.

Step 2. If p_j = p_j+1, then go to Step 4. Otherwise, append J_j+1 to the end of J_j, and set j = j + 1.

Step 3. If j = n, then output the schedule and stop. Otherwise, go to Step 2.

Step 4. Interleave Jj and Jj+1. Append Jj+2 to the end of Jj+1. Set E = E∪ {j, j + 1}

and j = j + 2. Go to Step 3.

Theorem 4.2. The 1|(pj, pj, pj), f js|Cmax problem can be solved in O(n) by Algorithm PSC1. The makespan of the optimal schedule is 2∑

j∈Ep_j + 3∑

j∈Nⁿ\Ep_j, where E is the set of jobs interleaving with each other.

Proof. It is obvious that no interleaving is possible for any two jobs other than two adjacent identical jobs, J_j and J_j+1 with p_j = p_j+1. By examining each pair of adjacent jobs, Algorithm PSC1 matches any un-interleaved J_j with J_j+1 if p_j = p_j+1. Since no more interleaving is possible, Algorithm PSC1 produces an optimal schedule. In the obtained optimal schedule, each interleaved pair of jobs, J_j and J_j+1 for {j, j + 1} ⊂ E, contributes 2(p_j + p_j+1) to the makespan. Any job J_h that cannot be interleaved contributes 3p_h to the makespan. Thus, C_max = 2∑

j∈Ep_j+ 3∑

j∈Nn\Ep_j. From Step 2 to Step 4, at most n iterations are required, each of which takes a constant time. The overall running time of Algorithm PSC1 is O(n).

4.4.2 1 |(p, p, b

), f js |C

Without the assumption of a fixed job sequence, this special case can be solved in O(n) time (Orman & Potts, 1997). Since a_j= l_j= p for j ∈ Nn, any job cannot be interleaved with more than one job. Subject to a given job sequence, we have the following property of this special case.

Property 4.2. If the interleaving of jobs exists in a feasible schedule for the problem 1|(p, p, bj), f js|Cmax, then the interleaved pair are some two consecutive jobs J_j and J_j+1, where b_j ≤ p, j ∈ Nn−1.

With property 4.2, a forward dynamic program can be designed. A job is called iso-lated if it is not interleaved with any other job. A subschedule of{J1, J₂, . . . , J_j} can be completely characterized by the 2-tuples (j, λ), where j and λ are the number of jobs in the subschedule and the interleaving status of job J_j, respectively. If λ = 0, then job J_j is isolated. If λ = 1, then job J_j is interleaved with job J_j−1. Denote the corresponding minimum makespan as f (j, λ) for 1≤ j ≤ n and λ ∈ {0, 1}.

Algorithm PSC2

Initialization: f (1, 0) = 2p + b₁ and f (1, 1) = ∞.

Recursive function: For 2≤ j ≤ n,

f (j, 0) = min{f(j − 1, 0), f(j − 1, 1)} + 2p + bj. (4.3)

f (j, 1) =

{ f (j− 1, 0) + p + bj − bj−1, b_j−1 ≤ p;

∞, otherwise. (4.4)

Goal: min

λ∈{0,1}f (n, λ).

Theorem 4.3. An optimal schedule for the 1|(p, p, bj), f js|Cmax problem can be produced in O(n) by Algorithm PSC2.

Proof. Eq. (4.3) indicates that any isolated job J_j adjoins J_j−1 which is either isolated or interleaved with J_j−2. In Eq. (4.4), job J_j can be interleaved with job J_j−1 if J_j−1 is

isolated and b_j−1 ≤ p. A subschedule in the state (j, λ) with value f(j, λ) dominates all other subschedules in the same state in the sense that it contributes the minimum value to the makespan among those of all subschedules in this state. The principle of optimality holds and Algorithm PSC2 can generate an optimal schedule. To obtain min_λf (n, λ), at most n−1 iterations are required, each of which takes a constant time. The overall running time of Algorithm PSC2 is O(n).

Corollary 4.1. The 1|(aj, p, p), f js|Cmax problem is solvable in O(n).

Proof. Orman & Potts (1997) proved that the coupled-task makespan problem and its reverse are equivalent. Given the fixed-job-sequence constraint, the equivalence still holds.

By virtue of lemma 4.3, this corollary follows.

4.4.3 1 |(p, l, p), fjs|C

Since all jobs are identical, any feasible schedule for problem 1|(p, l, p)|Cmax satisfies the fixed-job-sequence constraint. By the results of Orman & Potts (1997) for prob-lem 1|(p, l, p)|Cmax, the fixed-job-sequence problem 1|(p, l, p), fjs|Cmax can be solved in O(n).

By virtue of these three polynomially solvable cases, we can put the borderline between polynomially solvable problems and open problems in the complexity graph. In corre-spondence with the complexity graph of the coupled-task scheduling problems shown in Figure 4.11, that of the fixed-job-sequence problems is given in Figure 4.12. The strongly NP-hard problem 1|(pj, p_j, p_j)|Cmax becomes polynomially solvable when the fixed-job-sequence assumption is imposed. For each polynomially solvable case of the prototypical problem, the corresponding fixed-job-sequence problem is also solvable in O(n) time.

However, it cannot be concluded that a fixed-job-sequence problem is easier to deal with than its counterpart problem without the fixed-job-sequence assumption.

aj,lj,bj

Figure 4.11: Complexity graph of prototypical problems (Orman & Potts, 1997).

aj,lj,bj

Figure 4.12: Complexity graph of fixed-job-sequence problems.

4.5 Summary

This chapter has studied a single machine coupled-task makespan minimization prob-lem subject to a fixed job sequence. To schedule a given task sequence abiding by the fixed-job-sequence constraint, this study designed an O(n²) algorithm for determining its feasibility and a schedule with the minimum makespan, if such a feasible schedule exists.

Three polynomially solvable cases for the fixed-job-sequence problem were identified. A

complexity graph was also presented to depict the complexity statuses of the studied cases.

Although the complexity status of the 1|(aj, lj, bj), f js|Cmax problem remains open, the results presented in this study could inspire further research attention on this subject.

It is also interesting to investigate the complexity status of the open problems indicated in the complexity graph of the fixed-job-sequence problems. Further research could also be conducted in developing branch-and-bound procedures in which our proposed algorithm for plausible task sequences could be exploited. In addition, other different fixed-sequence constraints, e.g. given a fixed task-1 sequence or a fixed task-2 sequence, can be consid-ered.

Chapter 5