An approach to non convex/concave bi-level programming problems integrating Goal Programming with Satisfaction Function

An approach to non convex/concave bi-level
programming problems integrating Goal
Programming with Satisfaction Function

Nicolò Paternoster - Sergejs Pugacs

Outline
Introduction: multi-level programming

State of the art
Goal Programming
Satisfaction Function
Roghanian approach

Model proposal

Numerical examples

Results

Conclusions

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



LEADER

where y, for each value of x, is the solution of the lower level problem:

FOLLOWER

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

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

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.

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

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

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

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.

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.

Model proposal-2

the Follower.

First Strategy

g1

g2

Model proposal-2

the Follower.

First Strategy Second Strategy

g1 g1

ﬁx x treating it as a parameter
and then solve
g2 g2(x)

Model proposal - 3
Introducing Satisfaction Function

In order to refine results we can introduce the satisfaction function S in our model

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

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

Results -1

Satisfaction function

An approach to non convex/concave bi-level programming problems integrating Goal Programming with Satisfaction Function

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