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M. Antoniou and P. Korošec
reaction of the followers along with the decision results given by their counterparts
at the upper level.
A bilevel decision problem may involve multiple decision makers at the lower
level, formulating a bilevel multifollower problem. These followers may have
different reactions for a possible decision made by the leader. These followers may
share decision variables, objectives and/or constraints. The reactions of individual
followers and their relationships will affect the leader’s decision [27].
A definition of the multifollower problem is [17]: For x ∈ X ⊂ R p , y i ∈ Y i ⊂
R q i , i = 1, 2, .., K, in which one leader and K followers are involved
min
x∈X
F (x, y)
subject to
G(x, y) ≤ 0,
where for each x given by the upper level, y = (y 1 , y 2 , . . . , y K ) solves the i-th
follower’s problem
min
y i ∈Y i
f i (x, y)
subject to
g i (x, y) ≤ 0,
where x and y i are the decision variables of the leader and the i-th follower,
respectively. F, f i : R p × R q 1 × . . . × R q K → R 1 are the objective functions of the
leader and the i-th follower, respectively and G : R p × R q 1 × . . . × R qk → R m ,
g i : R p ×R q 1 ×. . .×R q K → R n i the constraint conditions of the leader and the i-th
follower, respectively. In this problem, the followers have to take into consideration
the decision results of their counterparts when making their individual decisions,
after knowing the leaders’ decision.
In literature, these problems have been accounted for problems with several
leaders and one follower, one leader and more followers and combination with more
than one leaders and followers. Papers tackling these problems are [28–30].
9.5.3 Bilevel Optimisation Problem Under Uncertainty
Decision variables of bilevel optimisation problems can present uncertainties at
both levels. Each level has different importance to the overall result of a bilevel
optimisation problem. Therefore, the impact of the uncertainties of upper and lower
level variables on the final robust solution of the problem is expected to be different,
M. Antoniou and P. Korošec
reaction of the followers along with the decision results given by their counterparts
at the upper level.
A bilevel decision problem may involve multiple decision makers at the lower
level, formulating a bilevel multifollower problem. These followers may have
different reactions for a possible decision made by the leader. These followers may
share decision variables, objectives and/or constraints. The reactions of individual
followers and their relationships will affect the leader’s decision [27].
A definition of the multifollower problem is [17]: For x ∈ X ⊂ R p , y i ∈ Y i ⊂
R q i , i = 1, 2, .., K, in which one leader and K followers are involved
min
x∈X
F (x, y)
subject to
G(x, y) ≤ 0,
where for each x given by the upper level, y = (y 1 , y 2 , . . . , y K ) solves the i-th
follower’s problem
min
y i ∈Y i
f i (x, y)
subject to
g i (x, y) ≤ 0,
where x and y i are the decision variables of the leader and the i-th follower,
respectively. F, f i : R p × R q 1 × . . . × R q K → R 1 are the objective functions of the
leader and the i-th follower, respectively and G : R p × R q 1 × . . . × R qk → R m ,
g i : R p ×R q 1 ×. . .×R q K → R n i the constraint conditions of the leader and the i-th
follower, respectively. In this problem, the followers have to take into consideration
the decision results of their counterparts when making their individual decisions,
after knowing the leaders’ decision.
In literature, these problems have been accounted for problems with several
leaders and one follower, one leader and more followers and combination with more
than one leaders and followers. Papers tackling these problems are [28–30].
9.5.3 Bilevel Optimisation Problem Under Uncertainty
Decision variables of bilevel optimisation problems can present uncertainties at
both levels. Each level has different importance to the overall result of a bilevel
optimisation problem. Therefore, the impact of the uncertainties of upper and lower
level variables on the final robust solution of the problem is expected to be different,