10. PLAN DE EMPRESA
10.4. ORGANIZACIÓN
Criteria n1 n2 … nj
Alternatives
m1 m2
… X
ij
mi
Figure 4-10 Decision matrix
All the original criteria receive tendency treatment. Usually the cost criteria are transformed into benefit criteria by the reciprocal ratio method as it shown in Equation (4.2). (García-cascales &
Lamata, 2012; L. Ren et al., 2007)
(4.2)
Step 2: Calculate the normalized decision matrix A. Since different criteria have different dimensions, the values in the decision matrix X are first transformed into normalized, non-dimensional values in order to convert the original attribute values within the interval [0, 1] under the following Equation:
(4.3)
where aij stands for the normalized value; i = 1, 2, …, m; j = 1, 2, …, n
Step 3: Coefficient vector of importance of the criteria. This step allows decision makers to assign weights of importance to a criterion relative to others. The weighted normalized matrix V is calculated by multiplying each value within the individual criterion in the normalized matrix A by the weight of this criterion:
(4.4) where wj stands for the weight of the individual criterion j; i = 1, 2, …, m; j = 1, 2, …, n.
Step 4: Determine the positive ideal and negative ideal solution from the matrix A. The ideal solution (A+) is the group of weighted normalized criteria values, which indicates the ideal criteria values (maximum value for benefit criteria and minimum value for cost criteria), and the non-ideal solution (A – ) is a group of weighted normalized criteria values, which indicates the negative ideal criteria values (minimum value for benefit criteria and maximum value for cost criteria):
(4.5) (4.6) Where J + = {i = 1, 2, …, m} when i is associated with benefit criteria ; J − = {i = 1, 2, …, m} when i is associated with cost criteria. j = 1, 2, …, n.
Step 5: Calculate Euclidean distance. Calculate the separation measures, using the n-dimensional Euclidean distance. (García-cascales & Lamata, 2012; Pinter & Pšunder, 2013)
(4.7)
(4.8)
For i = 1, 2, …, m.
Step 6: Calculate the relative closeness to the ideal solution. In M-TOPSIS, unlike TOPSIS, the positive ideal solution (Di+ ) and negative ideal solution (Di −) in finite planes are found at first; and then, the D+ D−-plane is constructed and set the optimized ideal reference point.
Finally, the relative distance from each evaluated alternative to the ideal reference point is calculated with (Lifeng Ren et al., 2007). Set the point A in Figure 4-11 [min (Di +
), max (Di−
)]
as the optimized ideal reference point because the aim is to have the lowest distance
4. Methods and tools for ecodesign: combining MOO, PCA and MCDM 119
Figure 4-11 Example D+ D− – plane of M-TOPSIS method (Lifeng Ren et al., 2007)
between the ideal criteria set values (A+) and get away as much as possible of non-ideal criteria set values (A – ). The ratio value of Ri is calculated as follows:
(4.9) Where i = 1, 2, …, m.
Step 7: Rank order. Rank alternatives in increasing order according to the ratio value of Ri. The best alternative is the one that having the M-TOPSIS coefficient Ri nearest to 0.
Example of application of M-TOPSIS method 4.4.2
The M-TOPSIS procedure described above is applied here on 15 points from a Pareto front obtained after a bi-objective optimization, each point representing a potential solution. The criteria involve the maximization Qout (kWh) and the minimization of EPBT (year). The different stages of the M-TOPSIS algorithm for this example are applied as follows:
From original data, the decision matrix is built (see Table 4-5). Because the EPBT criterion represents a cost criterion (minimization), it is transformed into benefit criterion (maximization) by Equation (4.2). The transformed values are displayed in Table 4-6.
Table 4-5 Decision data matrix
Criteria Criteria
Qout EPBT Qout EPBT
max min max min
Alternatives
1 2,286,757.98 1.753
Alternatives
9 1,424,028.60 1.701
2 1,072,808.71 1.699 10 2,088,618.68 1.735
3 2,005,066.69 1.730 11 716,057.32 1.692
4 1,710,340.98 1.711 12 2,040,111.49 1.731
5 1,933,294.35 1.727 13 358,578.79 1.691
6 2,183,467.41 1.747 14 2,076,489.16 1.732
7 2,253,731.29 1.749 15 1,760,111.18 1.718
8 716,068.22 1.692
Table 4-6 Transformed values matrix
Table 4-7 Normalized decision matrix
Criteria Criteria
Table 4-8 Weighted normalized matrix
Criteria Criteria
The normalized decision matrix A is obtained using Equation (4.3) (see Table 4-7). None of the criteria is preferred over the other, so the coefficient vector of importance W is equal to [1, 1]. The normalized weighted matrix is then represented in Table 4-8. The positive ideal (respectively negative ideal, i.e. non-ideal) solution is determined from the matrix A as well as the Euclidean distance matrix (Equations (4.5) to (4.8)). The obtained results are shown in Table 4-9 and Table 4-10. Considering
4. Methods and tools for ecodesign: combining MOO, PCA and MCDM 121
the point A [min (Di
+), max (Di
−)] as the optimized ideal reference point, the Figure 4-12 displays the position of all the alternatives in D+ D−–plane.
M-TOPSIS coefficient Ri is calculated for each alternative by Equation (4.9) and the ranking is presented in Table 4-11. The Pareto front EPBT-Qout in Figure 4-13 indicates the position of the three best alternatives after applying the M-TOPSIS method. The best alternative selected by M-TOPSIS method is alternative 1.
Table 4-9 A+ and A– values Criteria Qout EPBT A+ 0.3338 0.2627 A – 0.0523 0.2534
Table 4-10 Euclidean distance matrix (Di+ and Di−) Di+
Di− Di+
Di−
Alternatives
1 0.0093 0.2815
Alternatives
9 0.0842 0.1975
2 0.1772 0.1046 10 0.0297 0.2526
3 0.0415 0.2404 11 0.2293 0.0530
4 0.0842 0.1974 12 0.0365 0.2455
5 0.0519 0.2299 13 0.2815 0.0093
6 0.0173 0.2664 14 0.0313 0.2508
7 0.0100 0.2767 15 0.0770 0.2047
8 0.2293 0.0530
Figure 4-12 D+ D—plane for M-TOPSIS example
Table 4-11 Rank alternatives by M-TOPSIS coefficient Ri
Ri Rank Ri Rank
Alternatives
1 0.0000 1
Alternatives
9 0.1125 10
2 0.2439 12 10 0.0354 4
3 0.0522 7 11 0.3172 14
4 0.1126 11 12 0.0451 6
5 0.0669 8 13 0.3849 15
6 0.0171 3 14 0.0378 5
7 0.0049 2 15 0.1024 9
8 0.3172 13
Figure 4-13 Pareto front EPBT-Qout with top 3 ranked alternatives
As it can be seen in Figure 4-13, the best alternatives are located at the upper corner of the curve representing the Pareto front. If these three alternatives are compared with some of the alternatives that are in the knee of the curve, e.g. alternative 4 as shown in Figure 4-14, although EPBT is reduced, the energy produced is also strongly reduced. EPBT reduction is approximately 0.05 year while annual energy produced suffers a reduction of about 30%. The result provided by M-TOPSIS indicates that the best compromise that can be found at equal weight to both objectives in this example is to produce the maximum amount of energy because the difference between the growth in EPBT value is minimal as compared to the gain of Qout.
4.5 Conclusion
In this chapter, three methods to be applied for sizing PVGCS were presented. The number of objectives and the characteristics of the model developed in Chapter 3 make attractive the use of a GA to obtain the best alternatives embodied through a Pareto front. A variant of NSGA-II, embedded in MULTIGEN library, is selected. It must be emphasized that most of the works reported for PVGCS sizing through AG only consider economic or technical aspects. The main contribution of this work will be to integrate the environmental aspect from earlier design stage and not at end-of-pipe stage as
4. Methods and tools for ecodesign: combining MOO, PCA and MCDM 123
Figure 4-14 Pareto front EPBT-Qout with top 3 ranked alternatives and alternative 4
currently carried out.
The use of PCA is particularly attractive to reduce the number of environmental categories that are generally involved in LCA impact methods as described in Chapter 2. The reduction of intermediate impact categories to be evaluated will save AG computational time and provide a better interpretation of the results. Finally, a post-optimization analysis by use of a MCDM method based on m-TOPSIS is implemented to search for the best configuration among the alternatives represented in the Pareto front.
Figure 4-15 summarizes how the three methods will be integrated and applied for PVGCS, which constitutes the core of the following chapter.
Obtaining optimal Pareto front Optimization loop
(NSGA II)
Reduction of objectives (PCA)
Decision-making (M-TOPSIS) Multi-objective decision
problem
Figure 4-15 Integration of NSGA-II, PCA method and M-TOPSIS method