Use appropriate big$-M$ values wherever needed.

$($a$)$ Use binary variables $x_t$ for $t=1,$dots,$T$ to denote whether we have

a transaction in period $t(x_t=1)$ or not $(x_t=0).$ Write down

a linear constraint to enforce that if we have a transaction in the

first period than we can have transactions in at most half of the

remaining periods. You can assume $T$ is an odd number. Justify

your formulation.

$($b$)x_i$ for $i=1,2,3$ are continuous nonnegative variables with values in

the range $0,3.$ Write a set of constraints to model the requirement

that:

$|2x_1-x_2-x_3|2$

by introducing an additional binary variable. Justify your formula$-$

tion.

$($c$)x_1$ and $x_2$ are integer variables whose values are restricted to be in

$0,10.$ Write a set of constraints to model the requirement that:

either $x_1+x_{2}10$ or $2x_1-x_{2}5$ but not both

by introducing an additional binary variable.