Allocative personnel's problem

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:

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:
  
  
  • X11 + X12 + X13 + X14 ≤ 1
  • X21 + X22 + X23 + X24 ≤ 1
  • X31 + X32 + X33 + X34 ≤ 1
  • X41 + X42 + X43 + X44 ≤ 1
  • X51 + X52 + X53 + X54 ≤ 1
  
  
• In each machine must be a worker:
  
  
  • X11 + X21 + X31 + X41 + X51 = 1
  • X12 + X22 + X32 + X42 + X52 = 1
  • X13 + X23 + X33 + X43 + X53 = 1
  • X14 + X24 + X34 + X44 + X54 = 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 booleans (0 not assigned, 1 yes), and so, can't be negatives:

• Xij ≥ 0
• Xij are booleans

Determining objective function:

• Minimize Z = 10·X11 + 8·X21 + 8·X31 + 9·X41 + 8·X51 + 6·X12 + 7·X22 + 6·X32 + 7·X42 + 7·X52 + 6·X13 + 6·X23 + 5·X33 + 7·X43 + 6·X53 + 5·X14 + 6·X24 + 6·X34 + 6·X44 + 5·X54