*3.4. Problem-Solving Approach with Lexicographer*

One of the methods employed to solve goal programming problems when objectives are prioritized is lexicographers. In fact, the lexicographer uses a set of methods that an analyzer needs to get basic information from the decision-maker. The difference between lexicographers and the weighting method is receiving weights from the decision-maker. In the weighting method the weights taken from the decision-maker reveal the importance of the objective functions. However, in the lexicographer's method, the only order of priorities or preferences is determined by the decision-maker.

The lexicographer's approach is to select an objective function with the highest priority and the problem is optimized in the target space with the highest priority. Afterwards, if the found point is unique, it is considered the optimum. If multiple solutions are found, the point which satisfies the less prior objective functions as well.

This method consists of the following steps:

Step 1: Prioritize the desired goals;

Step 2: The goal has the highest priority from which the deviation is more important to the decision-maker, so in this step select the most important goal and minimize the related deviation; and

Step 3: The problem will be solved only for the highest prior function and its optimum will be searched. Two situations could happen at this stage:

Mode 1: If the solution is unique, the answer is found; and

Mode 2: If some points are found, lower prior functions determine which point is superior.

Steps above are iterated until reaching a unique answer.
