Background

Based on information documented in Style Seat $2022($online publication that caters to the hair salon industry in the U$.$S$.),$ there are a number factors that determine how much is charged for typical female hair cuts and styles. These factors range from the stylist's experience and skill level, the state of the client's hair and hair history, to the local cost of living.

The average revenue generated by a Cut and Style for a woman in the U$.$S$.$ is about $$120$ per head and the average revenue generated for a manicure and color is about $$21\mathrm{.}34.$

You decide to help a friend with a hair and nail salon shop that she started about two years ago. You want to help her optimize her revenue streams. The rent per month for the shop is $$1095$ per month $($this includes utilities$).$ The shop operates on a $40$ hour a week work schedule and a lady comes in two days a week $($Thursday and Friday's$)$ to do nails. The hair portion of the shop contributes $$929\mathrm{.}65$ towards the rent a month and the nails portion of the shop contributes $$165\mathrm{.}34$ a month towards rent per month. The shop has no more than $$1100$ every month to pay rent. Also, we represent the number of heads serviced as $H$ and the number of nail manicures as $N.$

It takes your friend about $2412$ minutes a month to do hair and typically has about $36$ heads a week. The nails lady normally takes about $296$ minutes a month to do nails. Because your friend also has another part time job, the shop has no more than $2708$ minutes a month to do both hair and nails.

Case Project

Contruct a linear program of the hair salon's case

Solve for the maximum values for both hair and nails

Solve for optimal revenue