Q$3)$ A marine operator presently offers service between ports A and B$,6,000$ miles apart,

utilizing a single ship. At an equivalent land speed $20$ miles per hour, the total sailing time

for a one$-$way voyage is $300$ hours or $12\mathrm{.}5$ days. The average time spend loading or unloading

cargo in either port is two days, so the average one$-$way trip time for the ocean voyage for a

cargo is $12\mathrm{.}5+2=14\mathrm{.}5$ days. The round round$-$trip time is thus $29$ days. Far a working year

of $330$ days, the effective frequency is $\frac{330}{29}=11\mathrm{.}4$ round trips per year. The ship's capacity

is $15,000$ tons of cargo; the rate presently charged is $$25$ per ton. The average cost of a 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

$V=Z_0-*t_{iv}-\frac{}{Q}-*c$

Where

$V=$ round$-$trip volume in tons per year

$Z_0=$ market size factor

$t_{iv}=$ one$-$way trip time

$Q=$ frequency in round trips per year

$c=$ freight rate in dollars per ton

$a,,Y=$ parameters

His estimates of the parameters are $a=19,500,=3\mathrm{.}1{10}^6,=8,000$ and $Z_0=982,680.$

The average volume per voyage is $20,000$ tons.

a$.$ Are his estimates of the parameters consistent with this volume? 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 operator is considering replacing the present ship with a newer vessel, which would

have a speed of $24$ miles per hour, thus cutting the one$-$way sailing time to $10\mathrm{.}4$ days. The

cost per voyage of this newer, faster ship would be $$250,000$ per one$-$way trip. The capacity

and loading$/$unloading times would be the same. Would this ship be more attractive to the

operator: $($i$)$ if frequency and rate remained the same? $($ii$)$ if the frequency were increased to

take advantage of the increased speed but the rate remained the same? $($iii$)$ if the frequency

were increased and the rate increased to $$30$ per ton? Summarize and discuss your results:

What consequences can a change in vehicle speed have? Discuss the significance of the

parameters $a,$ and $.$

c$.$ After doing the analysis, operator suddenly realizes that he has 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. He estimates that an

average shipment spends about six days in the land portion of the trip, for a total travel time

of $20\mathrm{.}5$ days. How might this affect his estimates? Discuss qualitatively.