A company has reserved 5 candidates to occupy 4 work jobs. The work jobs consist in driving 4 different hardwares (one worker for each hardware). The company tested the 5 workers at the 4 machines, doing the same work every worker at every machine, Obtaining the following times:

Machine 1 | Machine 2 | Machine 3 | Machine 4 | |

Candidate A | 10 | 6 | 6 | 5 |

Candidate B | 8 | 7 | 6 | 6 |

Candidate C | 8 | 6 | 5 | 6 |

Candidate D | 9 | 7 | 7 | 6 |

Candidate E | 8 | 7 | 6 | 5 |

Determining wich candidates must choose the company and to wich machine must be assigned.

Determining decision variables and expressing them algebraically. In this case:

- Xij: action that the i worker is assigned to j machine (0 points that worker has not been assigned and 1 yes)

Determining the restrictions and expressing them as equations or inequalities in function of the decision variables.

- Each worcker must be assigned only to one machine or none if is not choosen:

- XA1 + XA2 + XA3 + XA4 ≤ 1
- XB1 + XB2 + XB3 + XB4 ≤ 1
- XC1 + XC2 + XC3 + XC4 ≤ 1
- XD1 + XD2 + XD3 + XD4 ≤ 1
- XE1 + XE2 + XE3 + XE4 ≤ 1

- In each machine must be a worker:

- XA1 + XB1 + XC1 + XD1 + XE1 = 1
- XA2 + XB2 + XC2 + XD2 + XE2 = 1
- XA3 + XB3 + XC3 + XD3 + XE3 = 1
- XA4 + XB4 + XC4 + XD4 + XE4 = 1

Expressing all implicit conditions established by the origin of variables: negativeness, integer, only a few allowed values... . In this case, the restrictions are that workpeople's assignments to hardware must be boolean (0 not assigned, 1 yes), and so, can't be negatives:

- Xij ≥ 0
- Xij are boolean

Determining objective function:

- Minimize Z = 10·XA1 + 8·XB1 + 8·XC1 + 9·XD1 + 8·XE1 + 6·XA2 + 7·XB2 + 6·XC2 + 7·XD2 + 7·XE2 + 6·XA3 + 6·XB3 + 5·XC3 + 7·XD3 + 6·XE3 + 5·XA4 + 6·XB4 + 6·XC4 + 6·XD4 + 5·XE4

Perform a change of variable with the following correspondence:

XA1 | XA2 | XA3 | XA4 | XB1 | XB2 | XB3 | XB4 | XC1 | XC2 |

X1 | X2 | X3 | X4 | X5 | X6 | X7 | X8 | X9 | X10 |

XC3 | XC4 | XD1 | XD2 | XD3 | XD4 | XE1 | XE2 | XE3 | XE4 |

X11 | X12 | X13 | X14 | X15 | X16 | X17 | X18 | X19 | X20 |

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Copyright ©2006-2023. All rights reserved.

Developed by:

Daniel Izquierdo Granja

Juan José Ruiz Ruiz

English translation by:

Luciano Miguel Tobaria

French translation by:

Ester Rute Ruiz

Portuguese translation by:

Rosane Bujes