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Question $6$A:

A large sporting goods store is placing an order for bicycles with its supplier. Four models can be ordered: the adult Open Trail, the adult Cityscape, the girl's Sea Sprite, and the boy's Trail Blazer. It is assumed that every bike ordered will be sold, and their profits, respectively, are $30,25,22,$ and $20.$ The LP model should maximize profit. There are several conditions that the store needs to worry about. One of these is space to hold the inventory. An adult's bike needs two feet, but a child's bike needs only one foot. The store has $500$ feet of space. There are $1200$ hours of assembly time available. The child's bike need $4$ hours of assembly time; the Open Trail needs $5$ hours and the Cityscape needs $6$ hours. The store would like to place an order for at least $275$ bikes.

a$.$Formulate a model for this problem.

b$.$Solve your model with any computer package available to you.

c$.$How many of each kind of bike should be ordered and what will the profit be$?$

d$.$What would the profit be if the store had $100$ more feet of storage space?

e$.$If the profit on the Cityscape increases to $$35,$ will any of the Cityscape bikes be ordered?

f$.$Over what range of assembly hours is the dual price applicable?

g$.$If we require $5$ more bikes in inventory, what will happen to the value of the optimal solution?

h$.$Which resource should the company work to increase, inventory space or assembly time?

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