considers the problem of building railway tracks under the assumption that pieces fit exactly with no slack. Now consider the real problem, in which pieces don’t fit exactly but allow for up to 10 degrees of rotation to either side of the "proper" alignment. Explain how to formulate the problem so it could be solved by simulated annealing

By using simulated annealing we will sample the next state, evaluate and take the next state according to the probability e^Δv

Value function ( V ) = ( a * number of gaps ) + ( b * number of misconnected pieces ) + ( c * sum of sizes of gaps )

a,b,c = adjustable

Explanation:

In order to solve this problem by simulated annealing

First condition : assuming that pieces of the railways tracks fit exactly with no slack

Assume a state configuration of 32 pieces, use of discrete operations whose function is to remove pieces and reconnect it somewhere else without slack , we will also consider a continuous operations to help change angles to real values

Second condition : considering a real problem

This condition can be considered to be a closed loop because when one joint is moved all other joints are moved, here we will consider using a heuristic function

By using simulated annealing we will sample the next state, evaluate and take the next state according to the probability e^Δv

Value function ( V ) = ( a * number of gaps ) + ( b * number of misconnected pieces ) + ( c * sum of sizes of gaps )

a,b,c = adjustable

you will have $2593.74 in 10 years if you set aside $1,000 at 10%

you will have $9321.91 in 20 years if you set aside $2000 at 8%

you will have $1469.32 in 5 years if you set aside $1000 at 8%

you will have $1296.87 in 10 years if you set aside $500 at 10%

you will have $239.66 in 15 years if you set aside $100 at 6%

you will have $4660.96 in 20 years if you set aside $1000 at 8%