Environmental Engineering Reference
In-Depth Information
support, or re-initialised by setting a different set of aspiration levels. Actually, in such a case
it could seem appropriate to revise some of the constraints that specify X ea , in particular
relaxing some of the acceptability conditions. Finally, let U k the set of indexes of the
objectives whose level is considered not satisfying and Sk the complementary set of indexes
of the objectives considered satisfying at the k-th iteration. The procedure aims at identifying
a trade-off in an implicit way asking the DM to indicate for at least one of the objective j∈S k
an increase (recall that a minimization is considered)
k
j
q Δ that the DM is willing to accept in
order to possibly improve the objectives in U k .
The procedure then computes a new reference point from the objective levels of the
current efficient solution as
k
j
+1
k
j
k
j
q
=
q
+
Δ
q
j S kTO S k , where S kTO is the set of the
k
j
+1
k
j
q
=
q
objectives in Sk for which the DM is willing to accept an implicit trade-off, and
j J\S kTO .
Then, a new candidate efficient and acceptable solution (x k+1 , q k+1 ) is found by
maximizing the order consistent achievement function as follows
k
+
max
σ
(
q
,
q
)
(3.6)
k
x
X
being X k =X k-1 ∪{
k
j
k
j
+≤ ,∀ j S kTO }. In this way the new reference points are taken
into account in (3.6) and a relevant set of new constraints are added which impose the
maximum worsening level accepted by the DM. This interaction continues till the DM is
satisfied for all the objectives. As recent approaches to MODM have pointed out [Wierzbicki
et al., 2002], information provided during the decision making process (also called
“progressive” information), generally lead to identify decisions that are easily recognized to
be consistent with the DM's preference and then finally accepted. In addition, the use of
progressive information does not require that the DM expresses definitive and accurate
preference judgements only once, but lets the DM free to revise the preference at each step of
the decision process, taking into account the current solution point at which the judgements
previously provided have led to.
q
q
Δ
q
j
3.4. The Formalization of the MODM Decision Problem
The primary decision variables correspond to the flows of materials and represent the
components of the decision vector x. The following decision variables are
considered:
α i
( =
i
1
,...,
11
)
ψ ,
ψ ,
ψ ,
λ ,
λ
θ
M
γ
,
(see figure 3.1).
I
L
I
M
3.4.1.Objectives
Four objective functions are considered: minimizing economic costs, minimizing
unrecycled waste, minimizing waste sent to landfill, and minimizing incinerator emissions.
For brevity, the complete formalization of these functions is not reported. Further information
can be found in [Fiorucci et al., 2003, Costi et al., 2004]. The first objective function f1(x) is
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