Table $29\mathrm{.}22$ A company must complete three jobs. The amounts of

processing time $($in minutes$)$ required are shown in Table

A job cannot be processed on machine $j$ unless for all

ijji the job has completed its processing $on$ machine $i.$

Once a job begins its processing $on$ machine $j,$ the job

cannot $be$ preempted $on$ machine $j.$ The flow time for a job

$is$ the difference between its completion time and the time

$at$ which the job begins its first stage $of$ processing.

Formulate $anIP$ whose solution can $be$ used $to$ minimize the

average flow time $of$ the three jobs. $(Hint$: Two types $of$

constraints will $be$ needed: Constraint type $1$ ensures that $a$

job cannot begin $tobe$ processed $on$ a machine until all

earlier portions $of$ the job are completed. You will need five

constraints $of$ this type. Constraint type $2$ ensures that only

one job will occupy a machine $at$ any given time. For

example, $on$ machine $1,$ either job $1is$ completed before job

$2$ begins, $or$ job $2is$ completed before job $1$ begins.$)$

Formulate an LP whose solution will tell LIMECO how

to maximize their daily profit.

$11$ Chemco produces three products: A$,$ B$,$ and C$.$ They

can sell up to $30$ pounds of each product at the following

prices $($per pound$)$: product $A,$$$10$; product $B,$$$12$; product

$C,$$$20.$ Chemco purchases raw material at $$\frac{5}{l}b.$ Each pound

of raw material can be used to produce either $1lb$ of $A$ or

$1lb$ of $B.$ For a cost of $$\frac{3}{l}b$ processed, product A can be

converted to $\mathrm{.}6lb$ of product $B$ and $\mathrm{.}4lb$ of product $C.$ For

a cost of $$\frac{2}{1}b$ processed, product $B$ can be converted to

$\mathrm{.}8lb$ of product C$.$ Formulate an LP whose solution will tell

Chemco how to maximize their profit.

$12$ Chemco produces $3$ chemicals: B$,$ C$,$ and D$.$ They begin

by purchasing chemical A for a cost of $$\frac{6}{100}$ liters. For an

TABLE $28$