A manufacturer of bicycles builds racing, touring, and mountain models. The bicy
ID: 3173852 • Letter: A
Question
A manufacturer of bicycles builds racing, touring, and mountain models. The bicycles are made of both steel and aluminum. The company has available 91,800 units of steel and 50,400 units of aluminum. The racing, touring, and mountain models need 17, 27, and 34 units of steel, and 12, 21, and 18 units of aluminum, respectively.
Complete parts (a) through (c) below.
(a) How many of each type of bicycle should be made in order to maximize profit if the company makes $7 per racing bike, $11 per touring bike, and $22 per mountain bike?
Let x 1 be the number of racing bikes, let x 2 be the number touring bikes, and let x 3 be the number of mountain bikes.
What is the objective function?
To maximize profit, the company should produce ____ racing bike(s), __ touring bike(s), and ___ mountain bike(s).
(b) What is the maximum possible profit?
The maximum profit is $___ (c) Does it require all of the available units of steel and aluminum to build the bicycles that produce the maximum profit? If not, how much of each material is left over?
Compare any leftover to the value of the relevant slack variable.
No. Since s1=___ and s2= ___ in the optimal solution, there is/are ___ unit(s) of steel and ___ unit(s) of aluminum, respectively, left over.
Explanation / Answer
(a) The objective function is Maximize P = 7x1 + 11x2 + 22x3
The constraints are
17x1 + 27x2 + 34x3 91800
12x1 + 21x2 + 18x3 50400
x1, x2, x3 0
Tableau #1
x1 x2 x3 s1 s2 p
17 27 34 1 0 0 91800
12 21 18 0 1 0 50400
-7 -11 -22 0 0 1 0
Tableau #2
x1 x2 x3 s1 s2 p
0.5 0.794118 1 0.0294118 0 0 2700
3 6.70588 0 -0.529412 1 0 1800
4 6.47059 0 0.647059 0 1 59400
The optimum solution is x1 = 0, x2 = 0, x3 = 2700, P = $59400
(b) The maximum profit is $59400
(c) Since s1 = 0 and s2 = 1800 in the objective function, there are 0 units of steel and 1800 units of aluminum respectively left over
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