S$2$

S$24$wk$1$b: Problem $14$

$(1$ point$)$

$($Book: $3\mathrm{.}5$ problem $7)$ Each hour from $10$ A$.$M$.$ to $7$P$.$M$.,$ Big Bank receives checks and must process them. Its goal is to process all the checks the same day they are received. The bank has $13$ check$-$processing machines, each of which can process up to $500$ checks per hour. It takes one worker to operate each machine. Bank One hires both full$-$time and part$-$time workers. Full$-$time workers work $10$ A$.$M$.-6$ P$.$M$.,11$ A$.$M$.-7$ P$.$M$.,$ Noon$-8$ P$.$M$.$ and are paid $$160$ per day. Part$-$time workers work either $2$ P$.$M$.-7$ P$.$M$.$ or $3$ P$.$M$.-8$ P$.$M$.$ and are paid $$75$ per day. The number of checks received each hour is given in the Table below. In the interest of maintaining continuity, Big Bank requires it must have at least three full$-$time workers under contract. Set up the LP and solve to determine the minimal cost schedule that processes all checks by $8$P$.$M$.$

For simplicity we allow fractional numbers of employees.

The minimum labor cost subject to the conditions is just the number, no dollar sign.

$\backslash$table$[[$Time$,$checks$],[10$AM$,5000],[11$AM$,4000],[$Noon$,3000],[1$PM$,4000],[2$PM$,2500],[3PM,3000],[4PM,4000],[5PM,4500],[6$PM$,3500],[7$PM$,3000]]4$wk$1$b: Problem $14$

$(1$ point$)$

$($Book: $3\mathrm{.}5$ problem $7)$ Each hour from $10$ A$.$M$.$ to $7$P$.$M$.,$ Big Bank receives checks and must process them. Its goal is to process all the checks the same day they are received. The bank has $13$ check$-$processing machines, each of which can process up to $500$ checks per hour. It takes one worker to operate each machine. Bank One hires both full$-$time and part$-$time workers. Full$-$time workers work $10$ A$.$M$.-6$ P$.$M$.,11$ A$.$M$.-7$ P$.$M$.,$ Noon$-8$ P$.$M$.$ and are paid $$160$ per day. Part$-$time workers work either $2$ P$.$M$.-7$ P$.$M$.$ or $3$ P$.$M$.-8$ P$.$M$.$ and are paid $$75$ per day. The number of checks received each hour is given in the Table below. In the interest of maintaining continuity, Big Bank requires it must have at least three full$-$time workers under contract. Set up the LP and solve to determine the minimal cost schedule that processes all checks by $8$P$.$M$.$

For simplicity we allow fractional numbers of employees.

The minimum labor cost subject to the conditions is just the number, no dollar sign.

$\backslash$table$[[$Time$,$checks$],[10$AM$,5000],[11$AM$,4000],[$Noon$,3000],[1$PM$,4000],[2$PM$,2500],[3PM,3000],[4PM,4000],[5PM,4500],[6$PM$,3500],[7$PM$,3000]]$

This one needs a hint: the decision variables need to be at a minimum a variable for how many full$-$time employees start at each permitted starting time and similarly for part$-$time employees. This will already give your cost $($objective$)$ function. But, to more easily express the many constraints, it is helpful $($and perfectly allowed$)$ to introduce many other decision variable. For example, let $$ denote the number of checks that get processed in the hour starting at time $i.$ Also let $Ii($for inventory$)$ denote the number of checks that have arrived up to and including time i that have not yet been processed. We don't need $I10$ but notice that $I11=5000-P10$ would be one of the constraints, and for all $i,6500$ is a constraint. Another constraint is that the number of checks processed depends on how many employees are working.

Since it is pretty hard, here are my $10$AM and $11$AM constraints:

$c10a$: $111=5000-P10$;

$c10b$: $P10==6500$;

$c10c$: $P10={500}^{**}10$;

$c11a$: $112=4000+111-P11$;

$c11b$: $P116500$;

$c11c$: $P11500**(10+11)$;

etc. Note that we do not restrict ourselves to at most $13$ employees per hour because, although likely, it's not strictly required, so we let the solver work this out itself. It would be easy to find similar problems where you might have to have more than $13$ employees, e$.$g$.$ If there were only two shifts $10-3$ and $2-8,$ then we'd almost certainly have to have more than $13$ on "duty" from $2-3.$

also don't forget that in each hour you have checks left over from previous hour