Improved approximation algorithms for two-stage flexible flow shop scheduling

A two-stage flexible flow shop scheduling is a manufacturing infrastructure designed to process a set of jobs, in which a single machine is available at the first stage and m parallel machines are available at the second stage. At the second stage, each task can be processed by multiple parallel machines. The objective is to minimize the maximum job completion time, i.e., the makespan. Sun et al. (J Softw 25:298–313, 2014) presented an $$O(n\log n)$$ O ( n log n ) -time 3-approximation algorithm for $$F2(1, Pm)~|~size_i~|~C_{\max }$$ F 2 ( 1 , P m ) | s i z e i | C max under some special conditions. Zhang et al. (J Comb Optim 39:1–14, 2020) presented a 2.5-approximation algorithm for $$F2(1, P2)~|~line_i~|~C_{\max }$$ F 2 ( 1 , P 2 ) | l i n e i | C max and a 2.67-approximation algorithm for $$F2(1, P3)~|~line_i~|~C_{\max }$$ F 2 ( 1 , P 3 ) | l i n e i | C max , which both run in linear time. In this paper, we achieved following improved results: for $$F2(1, P2)~|~line_i~|~C_{\max }$$ F 2 ( 1 , P 2 ) | l i n e i | C max , we present an $$O(n\log n)$$ O ( n log n ) -time 2.25-approximation algorithm, for $$F2(1, P3)~|~line_i~|~C_{\max }$$ F 2 ( 1 , P 3 ) | l i n e i | C max , we present an $$O(n\log n)$$ O ( n log n ) -time 7/3-approximation algorithm, for $$F2(1, Pm)~|~size_i~|~C_{\max }$$ F 2 ( 1 , P m ) | s i z e i | C max with the assumption $$ \mathop {\min }_{1 \le i \le n} \left\{ {{p_{1i}}} \right\} \ge \mathop {\max }_{1 \le i \le n} \left\{ {{p_{2i}}} \right\} $$ min 1 ≤ i ≤ n p 1 i ≥ max 1 ≤ i ≤ n p 2 i , we present a linear time optimal algorithm.