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Table 1 Polynomially solvable PFB variants

From: Scheduling a proportionate flow shop of batching machines

Problem \(m=1\) \(m=2\) fixed \(m\ge 3\)
\(Fm \mid r_j, p_{ij} = p_i, p\text {-batch}, b_i \mid C_{\max }\) Ikura and Gimple (1986) Sung and Yoon (1997) Corollary 13/14
\(Fm \mid r_j, p_{ij} = p_i, p\text {-batch}, b_i \mid \sum C_j\) Ahmadi et al. (1992) Corollary 13 Corollary 13
\(Fm \mid p_{ij} = p_i, p\text {-batch}, b_i \mid \sum w_j C_j\) Baptiste (2000) Theorem 18 Theorem 18
\(Fm \mid p_{ij} = p_i, p\text {-batch}, b_i \mid L_{\max }\) Lee et al. (1992) Sung and Kim (2003) Theorem 18
\(Fm \mid p_{ij} = p_i, p\text {-batch}, b_i \mid \sum T_j\) Baptiste (2000) Sung and Kim (2003) Theorem 18
\(Fm \mid p_{ij} = p_i, p\text {-batch}, b_i \mid \sum U_j\) Lee et al. (1992) Sung and Kim (2003) Theorem 19
\(Fm \mid p_{ij} = p_i, p\text {-batch}, b_i \mid \sum w_j U_j\) Baptiste (2000) Theorem 19 Theorem 19
  1. Results achieved in this paper are given in bold
  2. To the best of our knowledge, each entry contains the first reference solving the corresponding problem or a generalization in polynomial time