Mid$-$West Trucking is a medium$-$sized TL carrier based in Des Moines, Iowa. Like many other carriers based in Iowa and surrounding states, Mid$-$West Trucking faces much demand for eastbound shipments $($because in most cases productions of goods take place in Midwest areas and consumptions take place in large cities in East Coast regions$).$ Since not much demand exists for westbound shipments $($from East Coast cites to midwestern areas$),$ the carrier had struggled with finding backhaul customers for many years. Historical data showed that only about $20$ percent of the time the carrier could find backhaul customers, meaning that about $80$ percent of the time the carrier had to return its trucks to Iowa empty $($no revenue load$).$ For this reason, Mid$-$West Trucking now provides a large price discount for the westbound shipments to capture the customer demands that would otherwise not exist $($needless to say, generating a little money on backhaul is better than generating no money on backhaul$).$

The actual cost of moving a truck from a Midwest city to an East Coast city $($and vice versa$)$ is typically around $$1,000.$ Because of the uncertainty the company faces for finding backhaul customers at East Coast cities, Mid$-$West Trucking typically charges $$2,000$ for an eastbound shipment. The extra $$1,000($beyond the cost$)$ reflects the cost of backhaul $($which is less than $$1,000$ in the long run, as the company can sometimes find backhaul customers$),$ as well as the profit for the company. On the other hand, Mid$-$West Trucking charges roughly $$700$ for a westbound shipment, which means that the company does not even charge enough to cover the cost for westbound shipments. Nevertheless, Mid$-$West Trucking is willing to use this discount because the company knows from experiences that the probability of finding backhaul customers increases drastically from $20$ to $80$ percent with this discount. By using this strategy, Mid$-$West Trucking has been making a fair, but small, amount of profit over the past several years. The management has been happy with what they have been doing so far.

One day, James Black, the marketing manager of Mid$-$West Trucking, was meeting with David White, one of his clients who tenders many eastbound shipments to Mid$-$West Trucking. During this meeting, David expressed concerns about the large difference in pricing between eastbound and westbound shipments. Specifically, David said the following to James:

$$I am not happy with what you are doing for westbound shipments. I know what you charge does not even cover the cost. This means that you are making money from shippers like me$,$ who tender eastbound shipments, and losing money from westbound shipments. This is not fair, because it is like we are subsidizing the cost of shipments for your westbound clients. I don$$t want to subsidize anyone whom I don$$t even know. You will have to stop doing this immediately. You must charge the same price for both eastbound and westbound shipments. Otherwise it is not fair.$$

James ponders how he should respond to this complaint made by David. James wants to make sure that all of his clients are happy, but at the same time, he also wants to make sure that Mid$-$West Trucking minimizes the cost of backhauling by finding westbound customers via large price discounts.

$1.$ Calculate the profit which Mid$-$West Trucking makes under two scenarios, namely, with and without discount for west$-$bound shipments. Use the following table to compute the profit.

With no discount:

Shipping cost $($one way$)-1000$

East bound charge $-2000$

West bound charge $-2000$

Probability of finding BH customer $-\mathrm{.}2(20\%)$

Expected profit per shipment $-?$

With discount:

Shipping cost $($one way$)-1000$

East bound charge $-2000$

West bound charge $-700$

Probability of finding BH customer $-\mathrm{.}8(80\%)$

Expected profit per shipment $-?$

$2.$ You should have found that if Mid$-$West Trucking cannot use discount, the company$$s profit will decrease. Now, if Mid$-$West Trucking follows the request by David White and stops providing discounts on westbound freights $($westbound shipments must be charged $$2,000),$ how much does the company have to charge for each headhaul $($eastbound$)$ shipment to make the same profit as before $($scenario with discount$)?$

$3.$ Do you agree with David White? Why or why not?