## 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:

 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.

## Real Estate Essentials

Tap into the secrets of the top investors… Discover The Untold Real Estate Investing Secrets Used By The World’s Top Millionaires To Generate Massive Amounts Of Passive Incomes To Feed Their Families For Decades! Finally You Can Fully Equip Yourself With These “Must Have” Investing Tools For Creating Financial Freedom And Living A Life Of Luxury!

Get My Free Ebook