$(25$ Points$)$ Alabama Atlantic is a lumber company that has four sources of wood and

five markets to be supplied. The annual availability of wood at sources Thomasville, Demopolis,

Monroeville, and Tuscaloosa is $15,20,15,$ and $18$ million board feet, respectively. The amount

that can be sold annually at markets Birmingham, Jackson, Atlanta, Gulfport, and Biloxi is $15,18,$

$12,13,$ and $10$ million board feet, respectively. In the past the company has shipped the wood by

train. However, because shipping costs have been increasing, the alternative of using ships to

make some of the deliveries is being investigated. This alternative would require the company to

invest in some ships. Except for these investment costs, the shipping costs in thousands of dollars

per million board feet by rail and by water $($when feasible$)$ would be the following for each route:

Unit Cost by Rail $(1000$s of $$)$

Market

Source Birmingham Jackson Atlanta Gulfport Biloxi

Thomasville $6172455566$

Demopolis $6978604956$

Monroeville $5966636147$

Tuscaloosa $5187456646$

Unit Cost by Ship $(1000$s of $$)$

Market

Source Birmingham Jackson Atlanta Gulfport Biloxi

Thomasville $31382435$

Demopolis $3643282431$

Monroeville $33363226$

Tuscaloosa $272637$

The capital investment $($in thousands of dollars$)$ in ships required for each million board feet to

be transported annually by ship along each route is given as follows:

Investment for Ships $(1000$s of $$)$

Market

Source Birmingham Jackson Atlanta Gulfport Biloxi

Thomasville $275303238285$

Demopolis $293318270250265$

Monroeville $283275268240$

Tuscaloosa $264281243$

Considering the expected useful life of the ships and the time value of money, the equivalent

uniform annual cost of these investments is one$-$tenth the amount given in the table. The

objective is to determine the overall shipping plan that minimizes the total equivalent uniform

annual cost $($including shipping costs$)$ and satisfy the demand as much as possible. You are

asked to analyze this problem and make suggestions to your supervisor by considering the

following three options:

Option $1$: Continue shipping exclusively by rail.

Option $2$: Switch to shipping exclusively by water $($except where only rail is feasible$).$

Option $3$: Ship by either rail or water, depending on which is less expensive for the

particular route.

a$)$ By using the templates provided, find the optimal shipping plan and the optimal cost for

each option. State each solution explicitly in tabular form.

b$)$ Which option is better and why? State the major differences in the shipment plan between

each option in no more than five sentences