Question

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 needs 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 260 bikes.

a.	Formulate a LP model for this problem.
b.	Solve your model using Solver.
	How many of each kind of bike should be ordered and what will the profit be?
c.	What would the profit be if the store had 100 more feet of storage space?
d.	If the profit on the Cityscape increases to $35, will any of the Cityscape bikes be ordered?
e.	Over what range of assembly hours is the dual price applicable?
f.	If we require 5 more bikes in inventory, what will happen to the value of the optimal solution?

Answer 1

QUESTION – A:

Decision Variables:

Let,

Number of adult Open Trail = x1

Number of adult Cityscape = x2

Number of girl’s Sea Sprite = x3

Number of boy’s Trail Blazer = x4

Objective Function:

Here we should maximize the profit.

Profit from adult Open Trail = 30

Profit from adult Cityscape = 25

Profit from girl’s Sea Sprite = 22

Profit from boy’s Trail Blazer = 20

So, the objective function is:

30*x1 + 25*x2 + 22*x3 + 20*x4

Constraints:

Adult’s bike needs 2 feet and child’s bike needs 1 foot to store. Available space is 500 feet. So the total space that can be used should not exceed the available space.

2*x1 + 2*x2 + x3 + x4 <= 500

Child’s bike needs 4 hours of assembly time. Open Trail needs 5 hours and the Cityscape needs 6 hours. Available assembly time is 1200 hours. The used assembly time should not exceed the available assembly time.

5*x1 + 6*x2 + 4*x3 + 4*x4 <= 1200

Minimum requirement is 260 bikes

x1 + x2 + x3 + x4 >= 260

No bikes can be ordered in negative quantity.

x1, x2, x3, x4 >= 0

***

QUESTION – B:

Solving using Excel:

Solver Parameters:

Optimal Solution:

Adult Open Trail	Adult Cityscape	girl's Sea Sprite	boy's Trail Blazer
x1	x2	x3	x4
160	0	100	0

Adult Open Trail = 160

Adult Cityscape = 0

Girl’s Sea Sprite = 100

Boy’s Trail Blazer = 0

Profit = $7,000

***

The sensitivity Report is:

We will use the sensitivity report to answer the rest parts

QUESTION – C:

The allowable increase of the space constraint is infinite and shadow price is 0. So, if we increase the space by any amount, the optimal solution and profit will remain same.

So, the profit would be = $7,000

***

QUESTION – D:

The allowable increase of Cityscape is $13, means till the profit is $38, the optimal solution wont change. Hence, Number of Cityscape bikes will be 0.

No Cityscape bikes will be ordered.

***

QUESTION – E:

The allowable increase in assembly hour is 80 and allowable decrease is 160,

The dual price is applicable over (1200 – 80) and (1200 + 160)

The dual price is applicable over 1120 and 1360.

***

QUESTION – F:

The allowable increase in number of bikes in inventory is 40. So it can accommodate 5 more bikes. The shadow price of this constraint is (-10). So with each additional bike, the profit will reduce by $10.

So here,

The value of optimal solution will decrease by $50

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