Environmental Engineering Reference
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capacity is rapidly decreasing. For the sake of brevity the data relevant to the case under
concern are not reported here, but they can be found in Fiorucci et al. [2003], where the MSW
management problem was faced without introducing a multi-objective formulation. The
preliminary step that must be performed in order to apply the MODM approach to the MSW
case study is to identify for each objective function fj both the utopia qju, and the nadir qjn
solutions and to normalise the function with respect to the interval [qju, qjn]. Table 3.1
reports the four objective functions considered together with their dimensions and the
computed utopia and nadir solutions.
The case has been analyzed by two different decision makers, DM1 and DM2, showing
different attitudes in selecting the initial reference solution and in interacting with the
methodology. The first decision maker, DM1, is not able to initially identify a feasible
satisfying reference point. So, DM1 simply accepts to start the method from the (unfeasible)
utopia point. Then, the method computes the first solution from this reference and presents it
to DM1.
Table 3.1. Utopia and Nadir computation
f
j
Dim.
q
ju
q
jn
f1
M€
45.732
64.027
f2
Tons
376.616
880.750
f3
Tons
0.020
0.100
f4
Mg/m3
3.392
10.000
Table 3.2. Iteration sequence for DM1
Reference Point
Objective Values
k
q
q
q
q
q
q
q
q
2
4
2
4
1
0
0
0
0
0.34
0.34
0.34
0.16
2
0.5
0
0
0
0.65
0.15
0.05
0.00
3
0.5
0.45
0
0
0.56
0.23
0.06
0.00
4
0.5
0.45
0.06
0
0.52
0.30
0.08
0.00
Table 3.2 reports the iteration sequence characterizing DM1. The first column of the table
denotes the iteration, the other eight columns respectively reports first the references used and
the objective values obtained by the method from such references. Note that the values
reported for both the references and the objective have been normalized with respect to the
interval [q
ju
, q
jn
] for
j=1,...,4
.
After having analyzed the results obtained in the first iteration, DM1 is willing to accept a
worsening for the satisfying objective 1, accepting costs that are in the middle between the
normalized utopia and nadir (
q
1
=
, see table 2), with the aim to achieve a possible
improvement in at least one of the not satisfying objectives. Then, DM1 proceeds with the
other iterations as summarized in table 2. The objective values obtained at the iteration 2 are
quite good for
q
j
, j
=2,3,4, but the projection on the efficient frontier provided by minimizing
σ(
0.5
q
,
) leads to a cost value that is considered too high. Then, instead of reducing the
reference variation introduced for the objective 1, i.e., performing again the first iteration,
q
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