The Shamrock Transportation Company has four terminals: A$,$ B$,$ C$,$ and D$.$ At the start of a particular day, there are $8,8,6,$ and $3$ tractors available at those terminals, respectively. During the previous night, trailers were loaded at plants R$,$ S$,$ T$,$ and U$.$ The number of trailers at each plant is $2,10,5,$ and $6,$ respectively. The company dispatcher has determined the distances between each terminal and each plant, as follows.

Shipments

R

S

T

U

A

$21$

$44$

$25$

$40$

B

$42$

$24$

$45$

$22$

C

$78$

$32$

$42$

$55$

D

$44$

$42$

$32$

$30$

a$.$ Is the problem balanced?

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$.$ If not, where is the unbalance and the number of units?

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b$.$ How many tractors should be dispatched from terminal D to plant U$?$

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c$.$ What is the optimal solution?

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d$.$ If the demand for tractors at plant R now increased to $6,$ what impact does it have on the overall miles?

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Are all demand levels at each plant met?

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If not, which plant will not have it's demand levels met?

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e$.$ Based on your solution to letter d$.,$ if one tractor is required to be sent from terminal C to plant T$,$ what impact would it have on the total number of miles traveled?

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