# PHPSimplex

## Optimizing resources with Linear Programming

### 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
X

PHPSimplex
Version 0.81