Rigid Constraints: Goal Programming Models

Goal Programming Models

SML304 Nikhil Sahu
2011CS10237

Goal Programming is a optimization methodology where there are multiple, probably conflicting goals that need to be achieved simultaneously.

Rigid Constraints
Goals

Goal programming formulations do not contains inequalities. Every constraint is written as an equation.
We introduce a extra non-negative variable to convert a inequality into a equality and that is called a slack or surplus variable.

Thus any linear programming problem can be converted into a standard form: Max c1x1 + c2x2 + ........ + cnxn subject to a11x1 + a12x2 + ........ + a1nxn = b1 .... .... .... am1x1 + am2 x2 + ........ +

Other constraints like a target profit of 800 or minimization of production are not so rigid and are the goals of our formulation

So the formulation of rigid constraints should be: 3x1 + 4x2 = type constraint pushes η in the objective function, while a =0

Now we will solve the problem using the traditional graphical approach to solve linear programming problems. We will plot the lines as indicated by excluding η and ρ from the equations and derive the feasible region as solution. The role ρ and η play here is that the indicate the direction in which the feasible region lie with respect to the line.
Say for (i), ρ will be increasing towards increasing y intercept of the line(keeping slope same) which is opposite to η.

The point that we will touch wil be (0,10) for (x1 , x2) with all parameters as 0 beside p5 which would be 2. This would lead our assumption to change to 10, which will make p5 to be 0 also giving a final feasible solution.

Resulting in the following formulations: 4x2 + 3y2 + η4 – ρ4 = 20 subject to minimization of ρ4

7x1 + 6x2 + 8y1 + 7y2 + η5 – ρ5 = 200 subject to minimization of η5

So the objective funtion becomes: minimize : [ (η1 + η2 + ρ3) , ρ4 , η5 ]

Now because, these assumed numbers are completely arbitrary, these estimates of parameters may be poor.

Now, if the objective function takes on the value 0, then it means the rigid constraints have a feasible solution set.

Now, if the estimate is poor, we will get a large positive value for the corresponding variable indicating that the target is unaceivable and needs to be revised.

However, when a variable takes a positive value, it means we are unable to proceed. Then we will not optimize further but evaluate rest of objectives further

Usually , if only the value of the last goal needs to be assumed, any arbitrary value an be assumed and proceed. However, if more than one goals needs assumed values, what is assigned to the lst but one decides whether it can be optimized further or not

We will solve this problem using the simplex