PHPSimplex

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:

	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

Solve with PHPSimplex.