I. $(20$ pts$.)$ Bonus Prohlem.

You wish to build a house and you have divided the process into a number of tasks, namely:

B$.$ excawation and building the foundation,

F$.$ raising the wooden frame,

F$.$ electrical wiring,

P$.$ indoor plumbing,

D$.$ dry walls and flooring, and

I. landscuping.

You estimate the following daration for each of the tasks $($in werks$)$:

Some of the taiks can only be started when some other tawks are conmpleted. For instanen,

you can only build the frume oner the foumdation has bexe completed, i$.$e$.$ F$\text{'}$ can surt only

after $13$ is completed. All the precolence conestraints are mammarized as follownc:

F$\text{'}$ ean otart only after $BI$ is completed,

L$.$ can ntart only after $B$ is completed,

$E$ ean ntart only after $F$ is completed,

$P$ can start only after $F$ is completed,

Dean start only after $R$ is completerl,

D can start only after $P$ is completind.

The goal is to schexdule the starting time of each tack such that the entire projost is onmpleted

as mon as powible.

As an example, here is a fesaitle schedfle with a completion time of ten weeks.

$($a$)$ Let $x_1$ represent the week on which task i is scheduled to start for any i $in\{B,F,E,P,D,L\}.$

Ilow can the constraint $"$F cau start only after $B$ is completed" be matheraatically en$-$

forcerd?

b$)$ Notice that the non$-$linear function y $=$ max$\{$z$1,$ z$2,...,$ zn$\}$ can be modeled with n linear

constraints: y $>=$ z$1,$ y $>=$ z$2,...,$ y $>=$ zn$.$ Use this information to formulate the problem

as an IP$.($Note: You do not need to solve this problem, only formulation is required$)$