$.$ Calculation Question $$ Transportation Problem

The Metal Company has a plant in Chicago, IL$,$ that ships truckloads of product to customers

daily. Shipments from Metal are dependent on customer demands and follow the following

distribution:

Trucks Demanded Probability

$00\mathrm{.}2$

$10\mathrm{.}3$

$20\mathrm{.}3$

$30\mathrm{.}2$

Fast is a trucking company that supplies trucks to Metal. To provide a truck to Metal in a given

period, Fast has to plan for truck availability the previous period. It costs Fast $$200$ per truck that

it plans to make available to Metal. Metal pays Fast $$400$ per truck used. If Metal needs a truck

that Fast does not have, Fast pays Metal a penalty per truck of $$200.$ Metal then rents a truck and

driver for a period at a cost of $$800$ per truck.

a$.$ Provide the planned trucks at Fast that will maximize Fast$$s profits. Also provide the

associated service level that Metal will observe, given this number of planned trucks by

Fast.

b$.$ Given Fast$$s planned trucks, provide Metal$$s expected costs.

c$.$ An executive at Metal offers Fast the following deal: Metal will guarantee Fast usage of

one truck every period, i$.$e$.,$ Metal will pay Fast for one truck whether they need it or not,

but in return Fast should provide $100\%$ service level. What would be the impact of this

deal on Metal$$s expected costs and Fast$$s expected profits? Should Fast accept this offer?