An attempt of solving optimization problem combining two methodologies shows good performance in a controlled environment.

1. An approach to non convex/concave bi-level
programming problems integrating Goal
Programming with Satisfaction Function
Nicolò Paternoster - Sergejs Pugacs

2. Outline

3. Outline
Introduction: multi-level programming
State of the art
Goal Programming
Satisfaction Function
Roghanian approach
Model proposal
Numerical examples
Results
Conclusions

4. Bi-level Programming (BLP)
“bilevel optimization problems are mathematical programs which have a sub-set of their
variables constrained to be an optimal solution of other programs parameterized by their
remaining variables”
By definition the bilevel programming problem is defined as

5. Bi-level Programming (BLP)
“bilevel optimization problems are mathematical programs which have a sub-set of their
variables constrained to be an optimal solution of other programs parameterized by their
remaining variables”
By definition the bilevel programming problem is defined as
LEADER

6. Bi-level Programming (BLP)
“bilevel optimization problems are mathematical programs which have a sub-set of their
variables constrained to be an optimal solution of other programs parameterized by their
remaining variables”
By definition the bilevel programming problem is defined as
LEADER
where y, for each value of x, is the solution of the lower level problem:
FOLLOWER

7. Goal Programming - 1
This model allows to take into account simultaneously several objectives in a problem for choosing the
most satisfactory solution within a set of feasible solutions.
When dealing with a multi criteria optimization problem the decision maker can choose a goal he wants
to achieve for for each objective function
OBJECTIVE FUNCTIONS

8. Goal Programming - 1
This model allows to take into account simultaneously several objectives in a problem for choosing the
most satisfactory solution within a set of feasible solutions.
When dealing with a multi criteria optimization problem the decision maker can choose a goal he wants
to achieve for for each objective function
OBJECTIVE FUNCTIONS
GOALS

9. Goal Programming - 2
Using the GP Model formulation the problem becomes
Instead of minimizing the objective function, using this approach we try to minimize the deviations
between goals and the achieved level.

10. Satisfaction Function - 1
Through the satisfaction functions, the DM can explicitly express his preferences for any deviation of
the achievement from the aspiration level of each objective
An general shape of SF can be

11. Satisfaction Function - 2
After defining the analytical expression for a general satisfaction function F we can write the model
where we try to maximize the satisfaction level for each goal :
OSS: there could be a different SF for each goal

12. Roghanian Approach
S. S. E. Roghanian, M.B. Aryanezhad, Integrating goal programming, khun-tucker conditions, and penalty
function approaches to solve linear bi-level programming problems, Applied Mathematics and Computa-
tion, 2008.
An approach to solve linear Bilevel Problems
They proposed to replace the follower's problem with its (KKT) conditions and append the resulting
system to the leaders problem as a constraint .
They point out that the optimal values of the Leader and Follower relaxed
problem are the lower bounds for the optimal values of F(x,y),f(x,y) ,respectively

13. Model proposal - 1
We propose an approach that can be extended to non-convex/concave functions
We replace the BLP with the multi-criteria single level problem and we use GP to solve the
multi-criteria problem.

14. Model proposal-2
Strategies for finding goals
We propose two different strategies which can help the DM in finding the goals for the Leader and
the Follower.

15. Model proposal-2
Strategies for finding goals
We propose two different strategies which can help the DM in finding the goals for the Leader and
the Follower.
First Strategy
g1
g2

16. Model proposal-2
Strategies for finding goals
We propose two different strategies which can help the DM in finding the goals for the Leader and
the Follower.
First Strategy Second Strategy
g1 g1
fix x treating it as a parameter
and then solve
g2 g2(x)

17. Model proposal - 3
Introducing Satisfaction Function
In order to refine results we can introduce the satisfaction function S in our model

18. Numerical Example - 1
Non-convex/concave follower’s function

19. Numerical Example - 2
This example is more complex as the goal for the follower problem depends on x

20. Results -1

21. Results -1
Satisfaction function

22. Results -2

23. THE END