The routing open-shop problem on a network: Complexity and approximation

Transporting jobs through a two‐machine open shop

“…i.e., schedule S (4) [2] is optimal due to (7). Further, if the batches H 1 and H 2 are found in Step 2 of the algorithm, then p(H 1 ) ≤ 4 3 T .…”

“…In particular, in the two-machine case, the algorithms presented in [2] deliver a worst-case ratio of 3 2 . A similar model in the open shop setting is considered in [3] and [4]. In particular, the two-machine problem is shown to be NP-hard even if the network consists of two nodes, i.e., the jobs are placed at two locations, and a 6 5 -approximation algorithm is developed for this problem.…”

“…τ , and we use (17) and (9) to deduce K(S (4) Figs. 6c and 6d), then the schedule is optimal due to p(N 3 ) ≤ Fig.…”

“…It is clear that finding schedule S (4) [2] requires O(n) time. The statement below analyzes the worst-case performance of this algorithm.…”

“…6b), then we have two options. If batches H 2 and H 3 are defined as in Step 1, then K(S (4) [3] ) = a(N a ) + (p k + 2τ ), and we use (7) and (19) …”