**4.2 Scheduling optimization criteria according to the prioritization by the method of lexicographic goal programming**

In this section we will make multicriteria optimization of six criteria identifies previously, in order of priority Realize by AHP. The problem is to find an optimal solution of scheduling in a cutting process by acting on all the criteria considered indispensable. We propose:

a1: cost per worker qualification;

a2: cost of training per worker safety culture;

a3: maintenance cost per machine;

a4: cost of extending the area of machine scheduling;

a5: cost of purchasing software programmable machine;

a6: cost of recycling machine.

Then, the goal to which the decision maker for each criterion to minimize the deviation are as follows (**Table 4**):


**Table 4.** *Permissible limit of criterion.*

In order of priority for the AHP method the objective functions of each criterion is as follows:

```
The objective function L6 is:
Max L = δ-6
Subject to:
800x1+ δ-1+ δ + 1 ≤ 1600;
400x2+ δ-2+ δ + 2 ≤ 2000;
50x3+ δ-3+ δ + 3 ≤ 5000;
200x3+ δ-4+ δ + 4 ≤ 1600;
50x4+ δ-5+ δ + 5 ≤ 35000;
20x5+ δ-6+ δ + 6 ≤ 500;
δ-i, δ + i ≥ 0 pour (i = 1,2, … ,6)
xj ≥ 0 pour (j = 1,2, … ,6)
```

```
The objective function L1 is:
Max L = δ-1
Subject to
800x1+ δ-1+ δ + 1 ≤ 1600;
400x2+ δ-2+ δ + 2 ≤ 2000;
50x3+ δ-3+ δ + 3 ≤ 5000;
200x3+ δ-4+ δ + 4 ≤ 1600;
50x4+ δ-5+ δ + 5 ≤ 35000;
20x5+ δ-6+ δ + 6 ≤ 500;
δ-i, δ + i ≥ 0 pour (i = 1,2, … ,6)
xj ≥ 0 pour (j = 1,2, … ,6)
```

```
The objective function L3 is:
Max L = δ-3
Subject to
800x1+ δ-1+ δ + 1 ≤ 1600;
400x2+ δ-2+ δ + 2 ≤ 2000;
50x3+ δ-3+ δ + 3 ≤ 5000;
200x3+ δ-4+ δ + 4 ≤ 1600;
50x4+ δ-5+ δ + 5 ≤ 35000;
20x5+ δ-6+ δ + 6 ≤ 500;
δ-i, δ + i ≥ 0 pour (i = 1,2, … ,6)
xj ≥ 0 pour (j = 1,2, … ,6)
```
The objective function L4 is: Max L = δ-4

*Optimization Multicriteria Scheduling Criteria through Analytical Hierarchy Process… DOI: http://dx.doi.org/10.5772/intechopen.96557*

```
Subject to
800x1+ δ-1+ δ + 1 ≤ 1600;
400x2+ δ-2+ δ + 2 ≤ 2000;
50x3+ δ-3+ δ + 3 ≤ 5000;
200x3+ δ-4+ δ + 4 ≤ 1600;
50x4+ δ-5+ δ + 5 ≤ 35000;
20x5+ δ-6+ δ + 6 ≤ 500;
δ-i, δ + i ≥ 0 pour (i = 1,2, … ,6)
xj ≥ 0 pour (j = 1,2, … ,6)
```

```
The objective function L2 is:
Max L = δ-2
Subject to
800x1+ δ-1+ δ + 1 ≤ 1600;
400x2+ δ-2+ δ + 2 ≤ 2000;
50x3+ δ-3+ δ + 3 ≤ 5000;
200x3+ δ-4+ δ + 4 ≤ 1600;
50x4+ δ-5+ δ + 5 ≤ 35000;
20x5+ δ-6+ δ + 6 ≤ 500;
δ-i, δ + i ≥ 0 pour (i = 1,2, … ,6)
xj ≥ 0 pour (j = 1,2, … ,6)
```

```
The objective function L5 is:
Max L = δ-5
Subject to
800x1+ δ-1+ δ + 1 ≤ 1600;
400x2+ δ-2+ δ + 2 ≤ 2000;
50x3+ δ-3+ δ + 3 ≤ 5000
200x3+ δ-4+ δ + 4 ≤ 1600;
50x4+ δ-5+ δ + 5 ≤ 35000;
20x5+ δ-6+ δ + 6 ≤ 500;
δ-i, δ + i ≥ 0 pour (i = 1,2, … ,6)
xj ≥ 0 pour (j = 1,2, … ,6)
```
By using the software LINDO, the ideal solution obtained for optimize the scheduling problem in a cutting process is (**Table 5**).

The solution is satisfactory with the support of the decision maker's preferences. It is obvious that the implementation of the prioritization criterion that the lexicographic goal programming, the solution is much improved. The level of satisfaction


achieved for the six objectives is 100%. Achieved this level of satisfaction implies that all specifications are met.
