Complexity

Research Article

An Enhanced Two-Level Metaheuristic Algorithm with Adaptive Hybrid Neighborhood Structures for the Job-Shop Scheduling Problem

Algorithm 1

The procedure of decoding an operation-based permutation into a semiactive schedule.

	Step 1. Receive an operation-based permutation and a scheduling direction (forward or backward) from LOSAP (Algorithm 2).
	Step 2. If the scheduling direction received in Step 1 is forward, then the precedence relations of the operations of each job are unchanged. However, if it is backward, then the precedence relations of the operations must be reversed by using Steps 2.1 and 2.2.
	Step 2.1. For each job J_i (where i = 1, 2, …, n), let the operations O_i1, O_i2,…, O_im be renamed O_im, O_im_–1,…, O_i1, respectively.
	Step 2.2. Assign the precedence relations of all operations O_ij (where j = 1, 2, …, m) of each job J_i in ascending order of j values. Thus, O_i1 must be finished before O_i2 can start, O_i2 must be finished before O_i3 can start, and so on.
	Step 3. If the scheduling direction received in Step 1 is forward, then the operation-based permutation received in Step 1 is unchanged. However, if it is backward, then the order of all members in the operation-based permutation must be reversed. For example, the permutation (3, 2, 3, 1, 1, 2) with a backward scheduling direction must be changed into (2, 1, 1, 3, 2, 3).
	Step 4. Transform the permutation taken from Step 3 by changing the number i in its j^th occurrence (from left to right) into the operation O_ij (i = 1, 2, …, n and j = 1, 2, …, m). For example, the permutation (3, 2, 3, 1, 1, 2) must be transformed into (O₃₁, O₂₁, O₃₂, O₁₁, O₁₂, O₂₂).
	Step 5. Transform the permutation taken from Step 4 into a semiactive schedule by using Steps 5.1 to 5.3.
	Step 5.1. Let Φ_t be the partial schedule of the t scheduled operations. Thus, Φ₀ is empty. Now, let t ⟵ 1.
	Step 5.2. Let O_L ⟵ the leftmost as-yet-unscheduled operation in the permutation. Then, create Φ_t by scheduling O_L into Φ_t–1 at its earliest possible start time on its preassigned machine (the earliest possible start time of O_L is the maximum between the finished time of its immediate-preceding operation in its job and the finished time of the current last-scheduled operation on its machine).
	Step 5.3. If t < mn, then let t ⟵ t + 1 and repeat from Step 5.2. Otherwise, go to Step 6.
	Step 6. If the scheduling direction received in Step 1 is forward, then return Φ_mn (which is a completed forward semiactive schedule) as the final result. However, if it is backward, then modify Φ_mn to satisfy the original precedence relations by using Steps 6.1 and 6.2.
	Step 6.1. Let the operations O_im, O_im–1,…, O_i1 of each job J_i in the schedule Φ_mn be renamed O_i1, O_i2,…, O_im, respectively.
	Step 6.2. Turn the schedule modified from Step 6.1 back to front in order that the last-finished operation in the schedule becomes the first-started operation, and so on. After that, let the schedule be started at the time 0. Then, return the schedule modified in this step (which is a completed backward semiactive schedule) as the final result.