9 Multilevel Optimisation
319
For more information on the bilevel multiobjective problems, the reader can
refer to [23, 24]. An example of a bilevel multiobjective problem can be found in
[25], where it was applied in a transportation system planning problem and solved
using genetic algorithm. Another example can be found in [26] where a ‘probabilistic bilevel linear multiobjective programming problem’ and its application in
enterprise-wide supply chain planning problem was approached and solved with a
fuzzy programming technique.
9.5.2 Bilevel Multileader and/or Multifollower Optimisation
Problems
In many applications, a bilevel optimisation problem may involve multiple decision
entities (decision makers) on the upper level. The bilevel problem, in this case, has
many leaders that may have their individual decision variables, objective functions
and/or constraint conditions. This kind of bilevel decision problem is called a bilevel
multileader problem [27]. A general definition of the bilevel multileader problem is
[17]:
For x i ∈ X i ⊂ R p i , y ∈ Y ⊂ R q , i = 1, 2, .., L, in which L is the number of
leaders and one follower
min
x i ∈X i
F i (x, y)
subject to
G i (x, y) ≤ 0,
where for each x = (x 1 , x 2 , . . . , x L ) given by the first level, y solves
min
y∈Y
f (x, y)
subject to
g(x, y) ≤ 0,
where L is the number of leaders, while x i and y are the decision variables of the
i-th leader and the follower, respectively. F i , f : R p 1 × . . . × R p L × R q → R 1
are the objective functions of the i-th leader and the follower, respectively. G i :
R p 1 × . . . × R p L × R q → R m i , g : R p 1 × . . . × R p L × R q → R n are the constraint
conditions of the i-th leader and the follower, respectively. The sets X i and Y can
induce upper and lower bounds on the decision variables, restricting the problem
even more. In general, in this problem the leaders have to take into consideration the
319
For more information on the bilevel multiobjective problems, the reader can
refer to [23, 24]. An example of a bilevel multiobjective problem can be found in
[25], where it was applied in a transportation system planning problem and solved
using genetic algorithm. Another example can be found in [26] where a ‘probabilistic bilevel linear multiobjective programming problem’ and its application in
enterprise-wide supply chain planning problem was approached and solved with a
fuzzy programming technique.
9.5.2 Bilevel Multileader and/or Multifollower Optimisation
Problems
In many applications, a bilevel optimisation problem may involve multiple decision
entities (decision makers) on the upper level. The bilevel problem, in this case, has
many leaders that may have their individual decision variables, objective functions
and/or constraint conditions. This kind of bilevel decision problem is called a bilevel
multileader problem [27]. A general definition of the bilevel multileader problem is
[17]:
For x i ∈ X i ⊂ R p i , y ∈ Y ⊂ R q , i = 1, 2, .., L, in which L is the number of
leaders and one follower
min
x i ∈X i
F i (x, y)
subject to
G i (x, y) ≤ 0,
where for each x = (x 1 , x 2 , . . . , x L ) given by the first level, y solves
min
y∈Y
f (x, y)
subject to
g(x, y) ≤ 0,
where L is the number of leaders, while x i and y are the decision variables of the
i-th leader and the follower, respectively. F i , f : R p 1 × . . . × R p L × R q → R 1
are the objective functions of the i-th leader and the follower, respectively. G i :
R p 1 × . . . × R p L × R q → R m i , g : R p 1 × . . . × R p L × R q → R n are the constraint
conditions of the i-th leader and the follower, respectively. The sets X i and Y can
induce upper and lower bounds on the decision variables, restricting the problem
even more. In general, in this problem the leaders have to take into consideration the