last update: 06.03.06 (SK)         http://www.mathematik.uni-osnabrueck.de/research/OR/class

Flow-shop problems with a single server

• maximal polynomially solvable:

  $F2 \vert \vert C_{max}$	Johnson (1954) [3]
  $F2;S1 \vert p_{ij}=p;s_{ij}=s;r_i \vert C_{max}$	Brucker et al. (2005) [1]
  $F;S1 \vert p_{ij}=1;s_{ij}=s;r_i \vert C_{max}$	Brucker et al. (2005) [1]
  $F;S1 \vert p_{ij}=p;s_{ij}=s \vert C_{max}$	Brucker et al. (2005) [1]
  $F2;S1 \vert p_{ij}=p;s_{ij}=s;r_i \vert \sum C_i$	Brucker et al. (2005) [1]
  $F;S1 \vert p_{ij}=1;s_{ij}=s;r_i \vert \sum C_i$	Brucker et al. (2005) [1]
  $F \vert p_{ij}=p;r_i \vert \sum w_iC_i$	Single-machine problem
  $F2;S1 \vert p_{ij}=p;s_{ij}=1;r_i \vert \sum w_iC_i$	Single-machine problem
  $F \vert p_{ij}=p;r_i \vert \sum w_iU_i$	Single-machine problem
  $F2;S1 \vert p_{ij}=p;s_{ij}=1;r_i \vert \sum w_iU_i$	Single-machine problem
  $F2;S1 \vert p_{ij}=p;s_{ij}=s \vert \sum w_iU_i$	Assignment problem
  $F;S1 \vert p_{ij}=1;s_{ij}=s \vert \sum w_iU_i$	Assignment problem
  $F \vert p_{ij}=p;r_i \vert \sum T_i$	Single-machine problem
  $F2;S1 \vert p_{ij}=p;s_{ij}=1;r_i \vert \sum T_i$	Single-machine problem
  $F \vert p_{ij}=1;r_i \vert \sum w_iT_i$	Single-machine problem
  $F \vert p_{ij}=p \vert \sum w_iT_i$	Single-machine problem
  $F2;S1 \vert p_{ij}=p;s_{ij}=s \vert \sum w_iT_i$	Assignment problem
  $F;S1 \vert p_{ij}=1;s_{ij}=s \vert \sum w_iT_i$	Assignment problem

• minimal NP-hard:

  	$F2;S1 \vert p_{ij}=p \vert C_{max}$	Brucker et al. (2005) [1]
  $*$	$F2 \vert r_i \vert C_{max}$	Lenstra et al. (1977) [4]
  $*$	$F2;S1 \vert s_{ij}=s \vert C_{max}$	Brucker et al. (2005) [1]
  $*$	$F3 \vert \vert C_{max}$	Garey et al. (1976) [2]
  $*$	$F2 \vert \vert L_{max}$	Lenstra et al. (1977) [4]
  $*$	$F2;S1 \vert p_{ij}=1;r_i \vert L_{max}$	Brucker et al. (2005) [1]
  $*$	$F3;S1 \vert p_{ij}=1;r_i \vert L_{max}$	Brucker et al. (2005) [1]
  $*$	$F2 \vert \vert \sum C_i$	Garey et al. (1976) [2]
  $*$	$F2;S1 \vert p_{ij}=1;r_i \vert \sum C_i$	Brucker et al. (2005) [1]
  $*$	$F3;S1 \vert p_{ij}=1;r_i \vert \sum C_i$	Brucker et al. (2005) [1]
  	$F2;S1 \vert p_{ij}=1 \vert \sum w_iU_i$	Brucker et al. (2005) [1]
  	$F3;S1 \vert p_{ij}=1 \vert \sum w_iU_i$	Brucker et al. (2005) [1]
  	$F2;S1 \vert p_{ij}=1 \vert \sum T_i$	Brucker et al. (2005) [1]
  	$F3;S1 \vert p_{ij}=1 \vert \sum T_i$	Brucker et al. (2005) [1]
  $*$	$F2;S1 \vert p_{ij}=1 \vert \sum w_iT_i$	Brucker et al. (2005) [1]
  $*$	$F3;S1 \vert p_{ij}=1 \vert \sum w_iT_i$	Brucker et al. (2005) [1]

• minimal open:

  $F2;S1 \vert p_{ij}=1 \vert C_{max}$
  $F2;S1 \vert s_{ij}=1 \vert C_{max}$
  $F3;S1 \vert p_{ij}=1 \vert C_{max}$
  $F3;S1 \vert p_{ij}=p;s_{ij}=1;r_i \vert C_{max}$
  $F2;S1 \vert p_{ij}=1;s_{ij}=s;r_i \vert L_{max}$

  $F3;S1 \vert p_{ij}=1;s_{ij}=1;r_i \vert L_{max}$
  $F3;S1 \vert p_{ij}=p;s_{ij}=1 \vert L_{max}$
  $F2;S1 \vert p_{ij}=1 \vert \sum C_i$
  $F3;S1 \vert p_{ij}=1 \vert \sum C_i$
  $F3;S1 \vert p_{ij}=p;s_{ij}=1 \vert \sum C_i$

  $F2;S1 \vert p_{ij}=1;s_{ij}=s;r_i \vert \sum w_iC_i$
  $F3;S1 \vert p_{ij}=1;s_{ij}=1;r_i \vert \sum w_iC_i$
  $F2 \vert p_{ij}=p;r_i \vert \sum w_iT_i$
  $F2;S1 \vert p_{ij}=1;s_{ij}=1;r_i \vert \sum w_iT_i$
  $F3 \vert p_{ij}=p;r_i \vert \sum w_iT_i$

• maximal open:

  $F2;S1 \vert s_{ij}=1 \vert C_{max}$
  $F3;S1 \vert p_{ij}=p;r_i \vert C_{max}$
  $F;S1 \vert p_{ij}=1;r_i \vert C_{max}$
  $F;S1 \vert p_{ij}=p \vert \sum w_iC_i$

  $F3;S1 \vert p_{ij}=p \vert \sum U_i$
  $F;S1 \vert p_{ij}=1 \vert \sum U_i$
  $F;S1 \vert p_{ij}=p;s_{ij}=s;r_i \vert \sum w_iU_i$
  $F;S1 \vert p_{ij}=p;s_{ij}=s;r_i \vert \sum w_iT_i$