Chapter VI: Experimental Results
5.11. Case study
5.11.1. Sensitivity analysis
To do the sensitivity analysis, firstly, the sensitivity of second objective function with changes of flow (w), designed capacity of hubs (𝛤), designed capacity of connection links (𝜉), service rate of hubs (𝜇), and disruption related parameters including failure probability of complete disruption at hubs (q), disruption rate at hubs (f), retrieval time rate at hubs (r), disruption probability at hubs (𝜂), capacity disruption factor at hub (𝜃), disruption probability at connection links (𝜗), and capacity disruption factor at connection links (𝛿) is investigated. Next, the sensitivity of the first objective function is investigated with changes of parameters including w, 𝛤, 𝜉, q, 𝜂, 𝜃, 𝜗, and 𝛿. The results of the current network are presented in Tables 5.13 and 5.14, respectively for second and first objective functions, in which the results are shown based on changes according to the base scenario. In Tables 5.13 and 5.14, negative values relate to decrease in the value of objective function. It should be noted that the best values of objective functions in ℙ2 is considered as the base scenario with the values equal to 45.84 (Billion €) and 8.58 (h) for ℤ1 and ℤ2, respectively.
140
a) PF
3→5: Road; 3→24: Rail; 3→35: Road; 3→46: Road; 5→3:
Road; 5→24: Rail; 5→35: Air; 5→46: Rail; 24→3: Rail;
24→5: Road; 24→35: Road; 24→46: Road; 35→3: Rail;
35→5: Air; 35→24: Rail; 35→46: Road; 46→3: Road;
46→5: Road; 46→24: Rail; 46→35: Air
24→29: Road; 24→30: Air; 24→35: Rail; 24→46: Road;
29→24: Road; 29→30: Rail; 29→35: Rail; 29→46: Rail;
30→24: Rail; 30→29: Road; 30→35: Road; 30→46: Air;
35→24: Air; 35→29: Rail; 35→30: Rail; 35→46: Air;
46→24: Rail; 46→29: Rail; 46→30: Air; 46→35: Air;
24→25: Rail; 24→35: Air; 24→41: Air; 24→46: Rail; 25→24:
Rail; 25→35: Air; 25→41: Air; 25→46: Road; 35→24: Rail;
35→25: Air; 35→41: Rail; 35→46: Air; 41→24: Rail; 41→25:
Air; 41→35: Rail; 41→46: Rail; 46→24: Rail; 46→25: Rail;
46→35: Air; 46→41: Air;
b) Network structure for Sol. A c) Network structure for Sol. B d) Network structure for Sol. C
Figure 5.14. Results of ℙ1
0 10 20 30 40 50 60
45 50 55 60 65 70
ℤ2
ℤ1 (Billion €)
Failable hub level 1 Failable hub level 2 Failable hub level 3 Non-failable hub
46 35
24
5 3
46 30
35
24 29
46 35
24
25 41
A
B
A C
A
141
a) PF
10→24: Road; 10→27: Road; 10→30: Road; 10→35:
Rail; 10→46: Road; 24→10: Road; 24→27: Road;
24→30: Air; 24→35: Rail; 24→46: Rail; 27→10: Road;
27→24: Road; 27→30: Air; 27→35: Rail; 27→46: Road;
30→10: Road; 30→24: Rail; 30→27: Rail; 30→35: Road;
30→46: Road; 35→10: Road; 35→24: Rail; 35→27: Rail;
35→30: Road; 35→46: Rail; 46→10: Road; 46→24: Rail;
46→27: Road; 46→30: Rail; 46→35: Rail;
5→24: Rail; 5→29: Rail; 5→30: Air; 5→35: Air; 5→46:
Road; 24→5: Rail; 24→29: Rail; 24→30: Air; 24→35:
Rail; 24→46: Rail; 29→5: Rail; 29→24: Road; 29→30:
Rail; 29→35: Rail; 29→46: Rail; 30→5: Air; 30→24:
Rail; 30→29: Rail; 30→35: Rail; 30→46: Road; 35→5:
Air; 35→24: Rail; 35→29: Rail; 35→30: Road; 35→46:
Air; 46→5: Road; 46→24: Rail; 46→29: Rail; 46→30:
Rail; 46→35: Air;
24→25: Rail; 24→29: Rail; 24→30: Air; 24→35: Air; 24→46:
Rail; 25→24: Rail; 25→29: Rail; 25→30: Air; 25→35: Air;
25→46: Rail; 29→24: Rail; 29→25: Rail; 29→30: Rail; 29→35:
Rail; 29→46: Rail; 30→24: Rail; 30→25: Air; 30→29: Rail;
30→35: Rail; 30→46: Rail; 35→24: Air; 35→25: Air; 35→29:
Rail; 35→30: Rail; 35→46: Air; 46→24: Rail; 46→25: Road;
46→29: Rail; 46→30: Air; 46→35: Air;
b) Network structure for Sol. A c) Network structure for Sol. B d) Network structure for Sol. C
Figure 5.15. Results of ℙ2
8,5 10,5 12,5 14,5 16,5 18,5 20,5 22,5 24,5
45 50 55 60 65
ℤ2 (h)
ℤ1 (Billion €)
46 30
35
24
27 10
46 30
35
24 29
5 46
30 35
24 29
25
A
B
A
C
A
142
a) PF
24→30: Road; 24→35: Rail; 30→24: Rail; 30→35: Rail;
35→24: Rail; 35→30: Road; 24→30: Rail; 24→35: Air; 30→24: Rail; 30→35: Rail;
35→24: Rail; 35→30: Rail; 1→30: Air; 1→35: Air; 30→1: Rail; 30→35: Rail; 35→1: Air;
35→30: Rail;
b) Network structure for Sol. A c) Network structure for Sol. B d) Network structure for Sol. C
Figure 5.16. Results of ℙ3
8 13 18 23 28 33
55 60 65 70 75 80
ℤ2 (h)
ℤ1 (Billion €)
30
35
24
30
35
24
30
35
1
A
B
A
C
A
143
a) PF
24→35: Rail; 24→46: Road; 24→47: Rail; 35→24: Road;
35→46: Road; 35→47: Road; 46→24: Road; 46→35:
Rail; 46→47: Rail; 47→24: Road; 47→35: Rail; 47→46:
Road;
24→27: Rail; 24→35: Rail; 24→41: Rail; 27→24: Rail;
27→35: Air; 27→41: Rail; 35→24: Rail; 35→27: Air;
35→41: Rail; 41→24: Rail; 41→27: Rail; 41→35: Road;
24→30: Air; 24→35: Air; 24→46: Rail; 30→24: Air; 30→35:
Rail; 30→46: Air; 35→24: Air; 35→30: Rail; 35→46: Air;
46→24: Rail; 46→30: Air; 46→35: Air;
b) Network structure for Sol. A c) Network structure for Sol. B d) Network structure for Sol. C
Figure 5.17. Results of ℙ4
5 10 15 20 25 30
55 60 65 70 75 80
ℤ2 (h)
ℤ1 (Billion €)
47 35
24
46
41
35
24
27
30
35
24
46
A
B
A
C
A
144
Table 5.13. Second objective function changes vs. changes in parameters
Parameter
Increase (%) Objective function value changes (%)
w Γ ξ μ q f r η θ 𝜗 δ
10 12 -5 -3 -6 4 8 6 5 9 3 6
15 18 -8 -5 -10 9 13 10 9 12 6 8
20 25 -16 -11 -18 14 18 15 14 21 11 16
25 31 -20 -16 -28 17 23 19 19 30 15 19
30 40 -28 -19 -34 21 28 23 25 41 19 19
35 48 -30 -19 -42 25 36 30 32 41 24 28
40 53 -30 -23 -45 34 41 35 32 41 27 34
45 65 -30 -25 -49 39 49 39 39 53 27 38
50 78 -30 -25 -53 46 58 45 42 58 35 43
55 90 -30 -25 -57 52 64 52 42 69 35 49
60 105 -30 -25 -60 56 72 60 50 76 42 54
Table 5.14. First objective function changes vs. Changes in parameters
Parameter
Increase (%) Objective function value changes (%)
w Γ ξ q η θ 𝜗 δ
10 12 -5 -3 4 5 9 3 6
15 18 -8 -5 9 9 12 6 8
20 25 -16 -11 14 14 21 11 16
25 31 -20 -16 17 19 30 15 19
30 40 -28 -19 21 25 41 19 19
35 48 -30 -19 25 32 41 24 28
40 53 -30 -23 34 32 41 27 34
45 65 -30 -25 39 39 53 27 38
50 78 -30 -25 46 42 58 35 43
55 90 -30 -25 52 42 69 35 49
60 105 -30 -25 56 50 76 42 54
Figure 20 illustrates the results of Table 13, while increase in flow (w) and increase in disruption probability at connection links (𝜗) have the highest and the lowest effect on increase of the second objective function, respectively; and increase in service rate of hubs (𝜇) and increase in designed capacity of connection links (𝜉) have the highest and the lowest effect on decrease in the second objective function.
Figure 21 illustrates the results of Table 14, while increase in failure probability of complete disruption at hubs (q) and increase in disruption probability at connection links (𝜗) have the highest and the lowest effect on increase of the first objective function, respectively; and increase in designed capacity of hubs (𝛤) and increase in designed capacity of connection links (𝜉) have the highest and the lowest effect on decrease in the second objective function.
145
Figure 5.18. Current transportation network in France
Figure 5.19. Current network vs. non-dominated Pareto fronts
Figure 5.20. Increase in ℤ2 vs. Increase in parameters
41 35
24
0 10 20 30 40 50 60
45 50 55 60 65 70 75 80
ℤ2 (h)
ℤ1 (Billion €)
Tight Capacity, P=5 Tight Capacity, P=6 Excess Capacity, P=3 Excess Capacity, P=4
Current Transportation Network
-60 -45 -30 -15 0 15 30 45 60 75 90 105
10 20 30 40 50 60
Increase in ℤ2 (%)
Increase in parameters (%)
w q f r η θ θ δ Γ ξ μ
146
Figure 5.21. Increase in ℤ1 vs. Increase in parameters
Figure 5.22. Increase of Γ, ξ and μ vs. decrease in ℤ1 and ℤ2
In Figure 5.20, since w, 𝜃, f, r, and μ have the highest effect on the second objective function, it can be easily demonstrated that congestion in hubs strongly affects the travel time between each pair of O-D nodes. This result highlights the importance of analyzing congestion in HLP. On the other hand and based on Figure 5.21, since q, 𝜃, and Γ have the highest effect on the first objective function, the high importance of complete and partial disruption can be easily shown. Remarkably, analyzing congestion and considering complete and partial disruptions in the hub network are important issues that this thesis tried to address.
Figure 5.22 shows the general sensitivity analysis and illustrates the effect of increase in Γ, ξ and μ on decrease of objective functions. Accordingly, in the first objective function, increase in Γ has the highest effect, while in the second objective function, μ plays this role. Regarding to those parameters that their increase will increase the value of objective function, Figure 5.23 depicts the general sensitivity of objective functions respecting to parameters w, q, f, r, η, θ, 𝜗, and δ. In Figure 5.23, it can be easily shown that flow (w) and disruption factors (q and θ) have the highest
-40 -20 0 20 40 60 80
10 20 30 40 50 60
Increase in ℤ1 (%)
Increase in parameters (%)
w q η θ θ δ Γ ξ
30%
57% 23%
43% 47%
0 20 40 60 80 100
Z2 Z1
Parameters effect (%) μ
ξ Γ
147
effect on both objective functions. Figures 5.22 and 5.23 again demonstrate the importance of studying disruption and congestion in HLPs.
Figure 5.23. Increase in parameters vs. Increase in ℤ1 and ℤ2