Numerical example of an airport island in the North

A computer program has been written to solve Equations 7.3 through an iterative procedure. It turns out that both g > i and g < i can be processed y = a ■ gx + b

wi ■ pi,i + w2 ■ pi,2 = wi ■ p2,i + w2 ■ p2,2

without any difficulty. As an example, let us consider the case of an airport island in the North Sea as discussed in Chapter 15. The distance d from the shore is divided into two parts:

dminimal distance the island should be from the shoreline anyway; dextra extra distance from the shore to reduce hindrance.

For dextra we assume an exponential curve for the prices to be paid for it. For the first few kilometres we are prepared to pay more than for the more remote kilometres. We then can conduct both linear and non-linear simulations of this example.

The LP model can be formulated as:

subject to the restrictions:

I = Io + CIf (F - 6); T < tc; D > dc; F > fc; v ■ T = do + D; D - Dx = dmin. Linear:

Exponential:

Id = a ■ bD + p ^ ln(|IId - pi) = ln(|a|) + ln(|b|) ■ D Iid < IIdc ^ ln(|IId - pi) < ln(|IIdc - pi)

where:

Id = a ■ bD + p ^ ln(|IId - pi) = ln(|a|) + ln(|b|) ■ D Iid < IIdc ^ ln(|IId - pi) < ln(|IIdc - pi)

Io

= investment to build an island for 600 k flight movements

per year at a distance of 10 km from the shore line

hd

= investment to build a tunnel longer then 10 km from the

shore line

IIdc

= maximum investment for a tunnel longer then 10 km

from the shore line

Cif

= increase of required investment per 100 k flight

movements over 600 k

Cid

= increase of required investment per km more distance

from the shore than 10 km

F

= number of flight movements (x 100 k)

dmin

= minimum distance shuttle train travels from the shore

line

do

= distance shuttle train travels over land

v

= average speed of shuttle train

D

= total distance shuttle train

Dx = extra distance shuttle train travels from the shore line above dmin tc = maximum travelling time in the shuttle fc = minimum number of flight movements ( x 100 k)

dc = minimum length of tunnel a = scale factor b = base of the exponent p = intercept of the logarithmic function

The parameters

used in our simulation are:

Io =

€ 30 billion

JIdc =

€ 4 billion

CIf =

€ 1 billion per 100 k flight movements

CId =

€ 0.1 billion per km

dmin =

10 km

do =

30 km

v =

100 km per hr

tc =

0.58 hr (35 minutes)

fc =

8 100 k flight movements

dc =

30 km

The values of a,

b and p differ per example.

Linear

Investment for tunnel depends on its length: Ijd = Cjd ■ D (distance d from shore); maximum investment for extra € 4 billion: Ijd < 4. The maximum extra length of the tunnel is 38 km, constrained by the maximum travelling time of 35 minutes.

Exponential Example 1:

• From 0 to 10 extra km is € 2.5 billion available;

• From 10 to 50 extra km is € 5 billion available.

a ■ b0 + p = 0 a ■ b10 + p = 2.5 a ■ b50 + p = 5

Then:

Table 7.1 Results of simulations for an airport island in the North Sea

Variable

Linear

Exponential

Exponential

Costs extra tunnel 1 10 km [bil.]

1.0

2.0

2.5

Costs extra tunnel 10 50 km [bil.]

5.0

5.0

5.0

Flight movements [100k per yr]

8

8

8

Time [hr]

0.58

0.48

0.42

Length extra tunnel [km]

38

28.25

22.40

Costs extra tunnel [bil. ]

3.8

4

4

The maximum length of the tunnel is 22.40 km, constrained by the maximum budget for the 'extra' tunnel.

Example 2:

• From 0 to 10 extra km is € 2 billion available;

• From 10 to 50 extra km is € 5 billion available.

a • b0 + p = 0 a • b10 + p = 2 a • b50 + p = 5 (7.7)

Then:

The maximum length of the tunnel is 28.25 km, constrained by the maximum budget for the 'extra' tunnel.

Table 7.1 summarises the results of the different simulations for an airport island in the North Sea. The conclusion is that the non-linearity of the cost/distance ratio has a significant effect on the outcome, i.e. the optimum distance from the shore. The example shows how to deal with the difficulty that an exponential variable cannot be mixed with a linear one. We simply introduce two variables: a price per distance as a constraint in the LP model and an extra price per extra distance, which is assumed to vary exponentially.

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