mathematical modeling and operational research

Question

I have a variable TE which 0=<TE<=180

Also, I have a parameters L(k) and U(K), showing lower bound and upper bound for interval K (k=1,2,3)

The value of these parameters are as follows: L(1)=0 K(1)=60 , L(2)=60 K(2)=120, L(3)=120 K(3)=180

I need to find the related range of the variable TE. For example,

  if `TE=12`,  then `K=1`

  if `TE=150`, then `k=3` 

Which mathematical constraint(s) do this?

Answer 1

Depends on how you prefer writing down your answer. Boolean logic is a perfectly valid way of writing mathematical expressions. For example, the logical "box" function, taking only the values 0 or 1 can be defined as follows:

$BOX(TE, L, U) = (TE > L)\; \& \;(TE < U)$

Then your entire choice can be written as

$k(TE) = 1 \cdot BOX(TE, L1, U1) + 2 \cdot BOX(TE, L2, U2) + 3 \cdot BOX(TE, L3, U3)$

which will always work as long as the intervals do not overlap. One can also use the step-function to define the box-function. Defining a piecewise function, as you did in the description is also mathematically perfectly valid. Take your pick