What fraction of available capacity is utilized per voyage

Solve the below problem:

Q: A port operation presently offers service between ports X and Y, 6,000 miles apart, with a single vessel. At a land speed of 20 miles per hour (normally expressed in knots or nautical miles), one-way ship travel time is 300 hours (12.5 days).

The average loading or unloading cargo time in port is two days,therefore the average one-way trip  time for ocean voyage for a cargo is 14.5 days; 12.5 + 0.5(2+2).

Round-trip time is 29 days. For a "working year" of 330 days (allowing time for periodic maintenance),

the effective frequency is 11.4 , (= 330/29) round trips per year.

Ship's capacity is 15,000 tons of cargo;

the rate presently charged is $25 per ton.

The average cost of one-way trip is $225,000 per voyage, or ($15 per ton of available capacity).

The operator estimates that the demand function in this market is

V= Z0 - a tiv - b/Q - gc


V = round-trip volume in tons per year

Z0 = market size factor

tiv = one-way trip time

Q = frequency in round trips per year

c = freight rate in dollars per ton

a, b, g = parameters

The operation's estimates of the parameters are:

a = 19,500, b = 3.1x106, g = 8,000, and Z0 = 982,680. The average volume per voyage is 20,000 tons.

A- Are the operation's parameter estimates consistent with the average volume per voyage?

What fraction of available capacity is utilized per voyage?

What is his gross revenue (receipts from rates paid), total cost, and net revenue (gross revenue less total cost) per voyage?

B- The operation is considering replacing the current vessel with a newer one, which would have a speed of 24 miles per hour, thus cutting the one-way sailing time to 10.4 days. The cost per voyage of this newer, faster vessel is $250,000 per one-way trip. The capacity and loading/unloading times would be the same. Would this ship be more attractive to the operation:

If frequency and rate remained the same?

If the frequency were increased to take advantage of the increased speed but the rate remained the same?

If the frequency were increased and the rate increased to $30.00 per ton?

C- Summarize and discuss your results: What consequence can a change in vehicle speed have? Discuss the significance of the parameters -- a,b,g.

After doing this analysis, the operation realizes that they have ignored a fundamental principle: the time that will influence shipper's decisions (that is, the demand function) should be the total door-to-door time, not just the trip time on the ocean leg. They estimate that an average shipment spends six days in the land portion of the trip, for a total travel time of 20.5 days.
How does this affect their estimates? (qualitatively.)

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Operation Management: What fraction of available capacity is utilized per voyage
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