last update: 23.09.99 (SK)         http://www.mathematik.uni-osnabrueck.de/research/OR/class

Shop problems with multiprocessor tasks

• maximal polymomially solvable:

  $FMPTm \vert r_i;p_{ij}=1 \vert C_{max}$	Brucker & Kraemer (1996) [5]
  $FMPTm \vert r_i;p_{ij}=1 \vert \sum C_i$	Brucker & Kraemer (1996) [5]
  $FMPTm \vert p_{ij}=1 \vert \sum w_iC_i$	Brucker & Kraemer (1996) [5]
  $FMPTm \vert p_{ij}=1 \vert \sum w_iU_i$	Brucker & Kraemer (1996) [5]
  $FMPTm \vert prec;r_i;n=k \vert \sum w_iU_i$	Kraemer (1995) [8]
  $FMPTm \vert p_{ij}=1 \vert \sum T_i$	Brucker & Kraemer (1996) [5]
  $FMPTm \vert prec;r_i;n=k \vert \sum w_iT_i$	Kraemer (1995) [8]
  $JMPT2 \vert n=k \vert C_{max}$	Brucker & Kraemer (1995) [4]
  $JMPT \vert n=2 \vert \sum w_iU_i$	Brucker (1995) [2]
  $JMPT \vert prec;r_i;p_{ij}=1;n=k \vert \sum w_iU_i$	Brucker & Kraemer (1996) [5]
  $JMPT \vert n=2 \vert \sum w_iT_i$	Brucker (1995) [2]
  $JMPT \vert prec;r_i;p_{ij}=1;n=k \vert \sum w_iT_i$	Brucker & Kraemer (1996) [5]
  $OMPT2 \vert \vert C_{max}$	Brucker & Kraemer (1995) [4]
  $OMPTm \vert r_i;p_{ij}=1 \vert C_{max}$	Brucker & Kraemer (1996) [5]
  $OMPTm \vert r_i;p_{ij}=1 \vert \sum C_i$	Brucker & Kraemer (1996) [5]
  $OMPTm \vert p_{ij}=1 \vert \sum w_iC_i$	Brucker & Kraemer (1996) [5]
  $OMPT \vert prec;r_i;p_{ij}=1;n=2 \vert \sum w_iU_i$	Kraemer (1995) [8]
  $OMPTm \vert p_{ij}=1 \vert \sum w_iU_i$	Brucker & Kraemer (1996) [5]
  $OMPTm \vert prec;r_i;n=k \vert \sum w_iU_i$	Kraemer (1995) [8]
  $OMPTm \vert p_{ij}=1 \vert \sum T_i$	Brucker & Kraemer (1996) [5]
  $OMPT \vert prec;r_i;p_{ij}=1;n=2 \vert \sum w_iT_i$	Kraemer (1995) [8]
  $OMPTm \vert prec;r_i;n=k \vert \sum w_iT_i$	Kraemer (1995) [8]

• minimal NP-hard:

  	$FMPT \vert n=3 \vert C_{max}$	Kraemer (1995) [8]
  $*$	$FMPT \vert prec;p_{ij}=1 \vert C_{max}$	Timkovsky (2003) [16]
  $*$	$FMPT \vert tree;r_i;p_{ij}=1 \vert C_{max}$	Brucker & Knust (1999) [3]
  $*$	$FMPT2 \vert \vert C_{max}$	Brucker & Kraemer (1995) [4]
  $*$	$FMPT \vert tree;p_{ij}=1 \vert L_{max}$	Brucker & Knust (1999) [3]
  $*$	$FMPT2 \vert \vert \sum C_i$	Garey et al. (1976) [6]
  $*$	$FMPT2 \vert chains;p_{ij}=1 \vert \sum w_iC_i$	Tanaev et al. (1994) [14]
  $*$	$FMPT2 \vert chains;p_{ij}=1 \vert \sum U_i$	Single-machine problem
  $*$	$FMPT2 \vert chains;p_{ij}=1 \vert \sum T_i$	Single-machine problem
  	$JMPT3 \vert n=3 \vert C_{max}$	Sotskov & Shakhlevich (1995) [13]
  $*$	$JMPT2 \vert p_{ij}=1 \vert C_{max}$	Brucker & Kraemer (1995) [4]
  	$JMPT2 \vert r_i;p_{ij}=1 \vert \sum C_i$	Timkovsky (1998) [15]
  	$JMPT3 \vert n=3 \vert \sum C_i$	Sotskov & Shakhlevich (1995) [13]
  $*$	$JMPT2 \vert chains;p_{ij}=1 \vert \sum C_i$	Timkovsky (1998) [15]
  $*$	$JMPT3 \vert p_{ij}=1 \vert \sum C_i$	Lenstra (-) [12]
  	$JMPT2 \vert p_{ij}=1 \vert \sum w_iC_i$	Timkovsky (1998) [15]
  $*$	$JMPT2 \vert r_i;p_{ij}=1 \vert \sum w_iC_i$	Timkovsky (1998) [15]
  	$OMPT \vert n=3 \vert C_{max}$	Gonzalez & Sahni (1976) [7]
  	$OMPT3 \vert \vert C_{max}$	Gonzalez & Sahni (1976) [7]
  $*$	$OMPT \vert \vert C_{max}$	Lenstra (-) [12]
  $*$	$OMPT \vert prec;p_{ij}=1 \vert C_{max}$	Timkovsky (2003) [16]
  $*$	$OMPT \vert tree;r_i;p_{ij}=1 \vert C_{max}$	Timkovsky (2003) [16]
  $*$	$OMPT2 \vert chains \vert C_{max}$	Tanaev et al. (1994) [14]
  $*$	$OMPT2 \vert r_i \vert C_{max}$	Lawler et al. (1981,1982) [10] [11]
  $*$	$OMPT \vert tree;p_{ij}=1 \vert L_{max}$	Timkovsky (2003) [16]
  $*$	$OMPT2 \vert \vert L_{max}$	Lawler et al. (1981,1982) [10] [11]
  $*$	$OMPT2 \vert \vert \sum C_i$	Achugbue & Chin (1982) [1]
  $*$	$OMPT2 \vert chains;p_{ij}=1 \vert \sum w_iC_i$	Timkovsky (2003) [16]
  $*$	$OMPT \vert r_i;p_{ij}=1 \vert \sum U_i$	Kravchenko (1999) [9]
  $*$	$OMPT2 \vert chains;p_{ij}=1 \vert \sum U_i$	Timkovsky (2003) [16]
  $*$	$OMPT2 \vert chains;p_{ij}=1 \vert \sum T_i$	Timkovsky (2003) [16]

• minimal open:

  $FMPT \vert chains;n=2 \vert C_{max}$
  $FMPT \vert p_{ij}=1 \vert C_{max}$
  $FMPT \vert r_i;n=2 \vert C_{max}$
  $FMPT2 \vert chains;p_{ij}=1 \vert C_{max}$
  $FMPT2 \vert r_i;p_{ij}=1 \vert L_{max}$
  $FMPT \vert chains;n=2 \vert \sum C_i$
  $FMPT \vert n=3 \vert \sum C_i$
  $FMPT \vert p_{ij}=1 \vert \sum C_i$
  $FMPT \vert r_i;n=2 \vert \sum C_i$
  $FMPT2 \vert chains;p_{ij}=1 \vert \sum C_i$
  $FMPT2 \vert r_i;p_{ij}=1 \vert \sum w_iC_i$
  $FMPT2 \vert p_{ij}=1 \vert \sum w_iT_i$
  $JMPT2 \vert chains;n=2 \vert C_{max}$
  $JMPT2 \vert r_i;n=2 \vert C_{max}$
  $JMPT2 \vert n=3 \vert L_{max}$

  $JMPT2 \vert chains;n=2 \vert \sum C_i$
  $JMPT2 \vert n=3 \vert \sum C_i$
  $JMPT2 \vert p_{ij}=1 \vert \sum C_i$
  $JMPT2 \vert r_i;n=2 \vert \sum C_i$
  $OMPT \vert n=2 \vert C_{max}$
  $OMPT \vert p_{ij}=1;n=3 \vert C_{max}$
  $OMPT2 \vert chains;p_{ij}=1 \vert C_{max}$
  $OMPT2 \vert r_i;p_{ij}=1 \vert L_{max}$
  $OMPT \vert n=2 \vert \sum C_i$
  $OMPT \vert p_{ij}=1;n=3 \vert \sum C_i$
  $OMPT2 \vert chains;p_{ij}=1 \vert \sum C_i$
  $OMPT2 \vert r_i;p_{ij}=1 \vert \sum w_iC_i$
  $OMPT2 \vert p_{ij}=1 \vert \sum w_iT_i$

• maximal open:

  $FMPT \vert tree;p_{ij}=1 \vert C_{max}$
  $FMPT \vert chains;r_i;p_{ij}=1 \vert L_{max}$
  $FMPTm \vert prec;r_i;p_{ij}=1 \vert L_{max}$
  $FMPT \vert prec;r_i;p_{ij}=1 \vert \sum C_i$
  $FMPT \vert prec;r_i;n=k \vert \sum w_iC_i$
  $FMPT \vert r_i;p_{ij}=1 \vert \sum w_iU_i$
  $FMPT \vert r_i;p_{ij}=1 \vert \sum w_iT_i$
  $JMPT2 \vert p_{ij}=1 \vert \sum C_i$
  $JMPT \vert prec;r_i;n=2 \vert \sum w_iU_i$
  $JMPT2 \vert prec;r_i;n=k \vert \sum w_iU_i$
  $JMPT \vert prec;r_i;n=2 \vert \sum w_iT_i$
  $JMPT2 \vert prec;r_i;n=k \vert \sum w_iT_i$

  $OMPT \vert tree;p_{ij}=1 \vert C_{max}$
  $OMPT \vert chains;r_i;p_{ij}=1 \vert L_{max}$
  $OMPTm \vert prec;r_i;p_{ij}=1 \vert L_{max}$
  $OMPT \vert prec;r_i;p_{ij}=1 \vert \sum C_i$
  $OMPT \vert prec;r_i;n=k \vert \sum w_iC_i$
  $OMPT \vert p_{ij}=1 \vert \sum w_iU_i$
  $OMPT \vert prec;r_i;n=2 \vert \sum w_iU_i$
  $OMPT \vert prec;r_i;p_{ij}=1;n=k \vert \sum w_iU_i$
  $OMPTm \vert r_i;p_{ij}=1 \vert \sum w_iU_i$
  $OMPT \vert prec;r_i;n=2 \vert \sum w_iT_i$
  $OMPT \vert prec;r_i;p_{ij}=1;n=k \vert \sum w_iT_i$
  $OMPT \vert r_i;p_{ij}=1 \vert \sum w_iT_i$