Kane Manufacturing has a division that produces two models of hibachis, model A and model B. To produce each model-A hibachi requires 3 lb of cast iron and 6 min of labor. To produce each model-B hibachi requires 4 lb of cast iron and 3 min of labor. The profit for each model-A hibachi is $2, and the profit for each model-B hibachi is $1.50. There are 1000 lb of cast iron and 22 labor-hours available for the production of hibachis each day.
How many hibachis of each model should the division produce to maximize Kane's profit?
model A
______hibachis
model B
______hibachis
What is the largest profit the company can realize?
$_____
Do this graphically:
Let the y axis be Model A, x axis Model B.
Now consider cast iron. On the Y axis, mark the point 0,333 (the point of all model A and all the iron). then on the x axis, mark 250,0 (all cast iron used on B). connect the points, that is the IRON constraint line.
Now consider manhours. On the Y axis, plot (0,220) the point if all labor hours were making ModelA. Then plot the point (440,0), connect the points, that is the labour hour constraint.
Now all the points within x,y axis, and below any of the lines is the area of possible solutions. There is a nice theorem that tells us the max and min will be somewhere on the boundries at a endpoint. So examine for profit at 0,220, and 250,0, and the crossing point of the iron/labour lines. Compute profit at each of those points
Profit=x*1.50+y*2
I didn't make a accurate graph, just a mental sketch, but in my head it looks like the vicinity of x= about 150 (model b) and y= about 90. Work that out accurately.
To determine the number of hibachis of each model that should be produced to maximize Kane's profit, we can use the concept of linear programming.
Let's define the decision variables:
Let x be the number of model-A hibachis to be produced.
Let y be the number of model-B hibachis to be produced.
We need to set up the objective and constraints based on the given information.
Objective:
We want to maximize the profit. The total profit can be calculated as:
Total Profit = (Profit per model-A hibachi * Number of model-A hibachis) + (Profit per model-B hibachi * Number of model-B hibachis)
Total Profit = 2x + 1.5y
Constraints:
1. The amount of cast iron used should not exceed the available 1000 lb:
3x + 4y <= 1000 lb
2. The labor hours used should not exceed the available 22 hours:
6x + 3y <= 22 hours
3. The number of hibachis produced cannot be negative:
x ≥ 0
y ≥ 0
Now, we can solve this linear programming problem using graphical or algebraic methods.
Graphically, you can plot the constraints on a graph and find the feasible region (the region of intersection of all constraints). Then, you can calculate the total profit at each corner point of the feasible region and determine the maximum profit.
Algebraically, you can use the simplex method or the graphical method to solve the linear programming problem. The simplex method involves setting up a tableau and iteratively finding the optimal solution.
Once you have determined the optimal solution (i.e., the values of x and y that maximize the profit), you can substitute these values into the objective function to calculate the largest profit the company can realize.
Note: Since the question did not specify any restrictions on producing fractional hibachis, we will assume that fractional hibachis are allowed in the solution.
Please calculate the values of x and y that maximize Kane's profit using either method mentioned above, and then substitute them into the objective function to find the largest profit the company can realize.