The Sequence of Equilibrium Patterns in Problem II for ∆ < 0

As we have discussed all the conditions for ∆ > 0, we now consider the situation where

∆ < 0. The situation ∆ < 0 exerts a direct influence that the critical interval of Eq1 01is next below Eq1

11(by Proposition 2.3.3 and 2.3.7). Sequences of equilibrium patterns in Problem II are discovered and classified by the main characteristic values S_α, S_β, ∆ and the additional characteristic values θ, Cα, Cβ if necessary.

Lemma 3.4.1. If ∆ < 0 and S_α > 0, S_β > 0, the critical intervals in problem II are

. In this condition, several possible sequences of equilibrium patterns occur and the critical intervals, which are classified by additional characteristic value θ, C_α, C_β, are given below:

i) If θ > 0 and C_α> 0, the critical intervals are given below :

ii) If θ > 0 and Cα= 0. In this condition, the critical intervals are given below :

- d

iv) If θ < 0 and C_β > 0. In this condition, the critical intervals are given below :

- d

v) If θ < 0 and C_β = 0. In this condition, the critical intervals are given below :

vi) If θ < 0 and C_β < 0. In this condition, the critical intervals are given below :

- d

vii) If C_β = C_α > 0. In this condition, the critical intervals are given below :

- d First, the critical interval of Eq1

1

1always exists ath maxn Suppose θ > 0, only three kinds of critical interval assignments will take place in the next below of Eq1

⇒ The above result yields the determined intervals for Eq0

11and Eq⁰1

⇒ The above result yields the determined intervals for Eq0 11.

C_α < 0 ⇒ C_β > 0

⇒ The above result yields the determined intervals for Eq0

11and Eq⁰⁰ determined by h

0, minn

We enumerate some examples by the pattern in Lemma 3.4.1. All the cases with ∆ < 0 and S_α > 0, S_β > 0 are classified into seven categories.(see Example 3.4.1 - 3.4.7)

Example 3.4.1. Given the travel cost on each link as following :

f₁(u) = 1u + 30, f₂(u) = 10u + 40, f₃(u) = 10u + 50, f₄(u) = 1u + 10, f₅(u) = 1u + 0.

Example 3.4.2. Given the travel cost on each link as following :

f₁(u) = 1u + 30, f₂(u) = 10u + 40, f₃(u) = 10u + 50, f₄(u) = 1u + 10, f₅(u) = 1u + 20.

We compute the characteristic values that ∆ = −99 < 0, S_α = 100 > 0, S_β = 10 > 0, θ = 20 > 0 and Cα = 0. It satisfies the classification of Lemma 3.4.1-(ii). There is an interval [0, 5.4545] of Eq0

11, an interval [5.4545, ∞) of Eq¹1 1.

Example 3.4.3. Given the travel cost on each link as following :

f₁(u) = 5u + 20, f₂(u) = 10u + 40, f₃(u) = 10u + 15, f₄(u) = 5u + 10, f₅(u) = 1u + 0.

We compute the characteristic values that ∆ = −75 < 0, S_α = 275 > 0, S_β = 100 > 0, θ = 210 > 0 and C_α = −5 < 0. It satisfies the classification of Lemma 3.4.1-(iii).

There is an interval [0, 0.5000] of Eq0 0

1, an interval [0.5000, 5.3889] of Eq⁰¹

1, an interval [5.3889, ∞) of Eq1

11.

Example 3.4.4. Given the travel cost on each link as following :

f₁(u) = 5u + 10, f₂(u) = 10u + 50, f₃(u) = 10u + 40, f₄(u) = 5u + 30, f₅(u) = 1u + 0.

We compute the characteristic values that ∆ = −75 < 0, S_α = 350 > 0, S_β = 400 > 0, θ = −60 < 0 and Cβ = 20 > 0. It satisfies the classification of Lemma 3.4.1-(iv).

There is an interval [0, 3.3333] of Eq0

10, an interval [3.3333, 5.1111] of Eq¹1

0, an interval [5.1111, ∞) of Eq1

11.

Example 3.4.5. Given the travel cost on each link as following :

f₁(u) = 5u + 10, f₂(u) = 10u + 50, f₃(u) = 10u + 40, f₄(u) = 5u + 30, f₅(u) = 1u + 20.

We compute the characteristic values that ∆ = −75 < 0, S_α = 50 > 0, S_β = 100 > 0, θ = −60 < 0 and C_β = 0. It satisfies the classification of Lemma 3.4.1-(v). There is an interval [0, 1.7778] of Eq1

10, an interval [1.7778, ∞) of Eq¹¹

1.

Example 3.4.6. Given the travel cost on each link as following :

f₁(u) = 5u + 10, f₂(u) = 10u + 15, f₃(u) = 10u + 40, f₄(u) = 5u + 20, f₅(u) = 1u + 0.

We compute the characteristic values that ∆ = −75 < 0, S_α = 100 > 0, S_β = 275 > 0, θ = −210 < 0 and Cβ = −5 < 0. It satisfies the classification of Lemma 3.4.1-(vi).

There is an interval [0, 0.5000] of Eq1

00, an interval [0.5000, 5.3889] of Eq¹1

0, an interval [5.3889, ∞) of Eq1

11.

Example 3.4.7. Given the travel cost on each link as following :

f₁(u) = 1u + 10, f₂(u) = 10u + 20, f₃(u) = 10u + 20, f₄(u) = 1u + 10, f₅(u) = 1u + 5.

We compute the characteristic values that ∆ = −99 < 0, S_α = 55 > 0, S_β = 55 > 0 and θ = 0. It satisfies the classification of Lemma 3.4.1-(vii). There is an interval [0, 2.5000]

of Eq0

10, an interval [2.5000, ∞) of Eq¹¹

1.

Lemma 3.4.2. If ∆ < 0 and S_α > 0, S_β 6 0 , the critical intervals in problem II are determined by cr ⁰¹

1

1
. In this condition, one possible sequence of equilibrium patterns occurs and the critical intervals are given below:

- d

Proof. By Lemma 3.1.1 and Lemma 3.1.2, cr ¹1 0 In conclusion, the critical interval of Eq1

11is [cr ⁰¹

We enumerate some examples by the pattern in Lemma 3.4.2. All the cases with ∆ < 0 and S_α > 0, S_β 6 0 are classified into one categories.(see Example 3.4.8)

Example 3.4.8. Given the travel cost on each link as following :

f₁(u) = 1u + 25, f₂(u) = 10u + 40, f₃(u) = 10u + 10, f₄(u) = 1u + 20, f₅(u) = 1u + 10.

We compute the characteristic values that ∆ = −99 < 0, Sα = 75 > 0 and Sβ = −240 < 0.

It satisfies the classification of Lemma 3.4.2. There is an interval [0, 2.5000] of Eq0 01, an

interval [2.5000, 6.5909]of Eq0 1

1, an interval [6.5909, ∞)of Eq¹¹

1.

Lemma 3.4.3. If ∆ < 0 and S 6 0, S∗ > 0 , the critical intervals in problem II are determined by cr ¹1

0

1

11
, cr ¹0 0

1

10
. In this condition, one possible sequence of equilibrium patterns occurs and the critical intervals are given below:

- d

0 cr ¹0

0

1 1 0

cr ¹1

0

1 1 1

Eq1 00

 Eq1

10

 Eq1

11



Proof. Similarly to the proof of Lemma 3.4.9.

We enumerate some examples by the pattern in Lemma 3.4.3. All the cases with ∆ < 0 and S_α 6 0, Sβ > 0 are classified into one categories.(see Example 3.4.9)

Example 3.4.9. Given the travel cost on each link as following :

f₁(u) = 1u + 20, f₂(u) = 10u + 20, f₃(u) = 10u + 50, f₄(u) = 1u + 20, f₅(u) = 1u + 10.

We compute the characteristic values that ∆ = −99 < 0, S_α = −80 < 0 and S_β = 190 > 0.

It satisfies the classification of Lemma 3.4.3. There is an interval [0, 1.0000] of Eq1 00, an

interval [1.0000, 11.3636] of Eq1 1

0, an interval [11.3636, ∞) of Eq¹¹

1.

Lemma 3.4.4. If ∆ < 0 and S 6 0, S∗ 6 0 , the critical intervals in problem II are determined by cr ¹0

1 ↔ ¹1

01
. In this condition, the critical intervals are given below:

i) If C_β > C_α . In this condition, the critical intervals are given below :

- d

ii) If C_β < C_α . In this condition, the critical intervals are given below :

- d

iii) If C_β = C_α . In this condition, the critical intervals are given below :

- d

Suppose d < cr ¹⁰

1 ↔ ¹¹

1
, the equilibrium solution y_γ < 0, it means that the interval of Eq1

01is h maxn

cr ⁰0 1

1

01
, cr ¹⁰

0

1 01

o , cr ¹0

1 ↔ ¹¹

1

i

,and the lower bound depends on C_β−C_αbeing positive,negative or zero. Either the critical interval of Eq0

0

1or Eq¹⁰

0exists if C_β 6= C_α.

We enumerate some examples by the pattern in Lemma 3.4.4. All the cases with ∆ < 0 and S_α 6 0, Sβ 6 0 are classified into one categories.(see Example 3.4.10 - 3.4.12)

Example 3.4.10. Given the travel cost on each link as following :

f₁(u) = 1u + 20, f₂(u) = 10u + 50, f₃(u) = 10u + 20, f₄(u) = 1u + 10, f₅(u) = 1u + 45.

We compute the characteristic values that ∆ = −99 < 0, S_α = −95 < 0, S_β = −455 < 0 and C_β − C_α = (−5) − (−45) = 40 > 0. It satisfies the classification of Lemma 3.4.4-(i).

There is an interval [0, 3.6364] of Eq0

01, an interval [3.6364, 5.5556] of Eq¹⁰

1, an interval [5.5556, ∞) of Eq1

11.

Example 3.4.11. Given the travel cost on each link as following :

f₁(u) = 1u + 10, f₂(u) = 10u + 20, f₃(u) = 10u + 50, f₄(u) = 1u + 20, f₅(u) = 1u + 45.

We compute the characteristic values that ∆ = −99 < 0, S_α = −455 < 0, S_β = −95 < 0 and C_β− C_α = (−45) − (−5) = −40 < 0. It satisfies the classification of Lemma 3.4.4-(ii).

There is an interval [0, 3.6364] of Eq1 0

0, an interval [3.6364, 5.5556] of Eq¹⁰

1, an interval [5.5556, ∞) of Eq1

11.

Example 3.4.12. Given the travel cost on each link as following :

f₁(u) = 1u + 10, f₂(u) = 10u + 20, f₃(u) = 10u + 20, f₄(u) = 1u + 10, f₅(u) = 1u + 40.

We compute the characteristic values that ∆ = −99 < 0, S_α = −330 < 0, S_β = −330 < 0 and Cβ − Cα = (−30) − (−30) = 0. It satisfies the classification of Lemma 3.4.4-(iii).

There is an interval [0, 6.6667] of Eq1

01, an interval [6.6667, ∞) of Eq¹1 1.

Once we fix the linear travel time function on each link, then the relation between equilibrium patterns is determined by total flow d. The relation between the patterns in detail is discussed in Lemma 3.4.1 - 3.4.4. With the increasing of the amounts of total flow from zero to infinity, the order of the patterns is shown in a sequence. Figure 3.6 lists all the possible sequence in Problem II for ∆ < 0.

Eq1

Fig. 3.6: Relation Between Equilibrium Conditions in Problem II for ∆ < 0

在文檔中 Braess運輸問題的均衡解 (頁 55-67)