7 Introduction to Optimisation
243
• Goal attainment: it is a combination of the previous two techniques. Objectives
goals are assigned as before, together with relative under or over attainment
weight coefficients. The problem becomes
min
x∈Ω
α
subject to c(x) ≤ 0
f i (x) ≤ T i + αw i , i = 1, . . . , n obj ,
where α ∈ R and the weights w i ≥ 0 are normalised so that
n obj
i=1
w i = 1.
It is possible to prove that the Pareto front can be covered varying the weight
coefficients and the methodology is able to deal also with non-convex problems
[49].
• The ε constraint method: the objectives are minimised one at a time, constraining the others below a certain level
min
x∈Ω
f j (x)
subject to c(x) ≤ 0
f i (x) ≤ ε, i = 1, . . . , n obj , i = j.
The main weaknesses of the approach are the same as listed above, computational
efficiency, and a necessary a priori knowledge of the problem for covering the
global Pareto front.
• Lexicographic order: the objectives are sorted by user intervention. The
optimisation problem is divided in n obj subproblems solved sequentially with
a pre-established order and with additional constraints for not violating the
satisfaction of the minimum values of the former subproblems. Assuming that
{f 1 (x), f 2 (x), . . . , f n obj (x)} are the ordered objectives and f ∗
i the minimum
value achieved for the i-th objective. Then the i-th subproblem is defined as
min
x∈Ω
f i (x)
subject to c(x) ≤ 0
f j (x) = f ∗
j , j = 1, . . . , i − 1.
To cover the Pareto front, different optimisation runs with different sequences of
objectives must be performed, heavily increasing the overall computational time.
• Game theory: a ‘player’ is assigned to each objective function. The player
has the goal to minimise its objective. Assuming that the players are playing
a non-cooperative game (i.e. the players make decisions independently), the
243
• Goal attainment: it is a combination of the previous two techniques. Objectives
goals are assigned as before, together with relative under or over attainment
weight coefficients. The problem becomes
min
x∈Ω
α
subject to c(x) ≤ 0
f i (x) ≤ T i + αw i , i = 1, . . . , n obj ,
where α ∈ R and the weights w i ≥ 0 are normalised so that
n obj
i=1
w i = 1.
It is possible to prove that the Pareto front can be covered varying the weight
coefficients and the methodology is able to deal also with non-convex problems
[49].
• The ε constraint method: the objectives are minimised one at a time, constraining the others below a certain level
min
x∈Ω
f j (x)
subject to c(x) ≤ 0
f i (x) ≤ ε, i = 1, . . . , n obj , i = j.
The main weaknesses of the approach are the same as listed above, computational
efficiency, and a necessary a priori knowledge of the problem for covering the
global Pareto front.
• Lexicographic order: the objectives are sorted by user intervention. The
optimisation problem is divided in n obj subproblems solved sequentially with
a pre-established order and with additional constraints for not violating the
satisfaction of the minimum values of the former subproblems. Assuming that
{f 1 (x), f 2 (x), . . . , f n obj (x)} are the ordered objectives and f ∗
i the minimum
value achieved for the i-th objective. Then the i-th subproblem is defined as
min
x∈Ω
f i (x)
subject to c(x) ≤ 0
f j (x) = f ∗
j , j = 1, . . . , i − 1.
To cover the Pareto front, different optimisation runs with different sequences of
objectives must be performed, heavily increasing the overall computational time.
• Game theory: a ‘player’ is assigned to each objective function. The player
has the goal to minimise its objective. Assuming that the players are playing
a non-cooperative game (i.e. the players make decisions independently), the