A segment of automatically controlled highway is being equipped with sensors spa

Question

A segment of automatically controlled highway is being equipped with sensors spaced equally along length The maximum error in estimating traffic volume, which occurs halfway between any two sensors, can be expressed as (d/s)2, where d is the length of the segment and s is the number of sensors. Each sensor costs p dollars, and designers want to reduce the maximum error as much as possible within a budget for sensors of b dollars. Identify each of the following for this design problem. The decision variable The input parameters The objective function The constraints

Explanation / Answer

since there are 's' sensors and total length of segment is 'd'

Total cost is s*p

available budget is b

so for project feasibility cost should be less than or equal to budget.

b >= s*p ------(1)

Maximum error = (d/s) ^2

s = d/sqrt(maximum error)

substitute in (1)

b >= d*p/sqrt(maximum error)

Maximum error >= (d * p/b)^2

so

for maximum error to be minimum,

Max error = (d * P/b)^2 -----(2)

but max error = (d/s)^2

so

d/s = d*P/b

s = b/p --------(3)

Decision variable is 's'

Input parameters are 'p' , 'b','d'

objective fuction is (d * P/b)^2

Constraints is

s >= b/p

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