The table below shows the number of hours required in each of two departments to make one unit of various products A, B and C. For example, product B requires 1 hour of time in department I and 3 hours in department II.
HOURS REQUIRED PER UNIT OF PRODUCT
DEPARTMENT 1
A=1
B=1
C=9
DEPARTMENT 2
A=1
B=3
C=7
Find the number of units of A, B, and C which could be made if department I has 75 hours available and Department II has 65 hours available. It is necessary that all of the available hours be used.
Dept. 1.
75/11h = 6 and 9 remainder.
A = 1h/u * 6 units = 6h.
B = 1h/u * 6 units = 6h.
C = 9h/u * 7 units = 63h.
Total Units = 6 + 6 + 7 = 19.
Total Hours = 6 + 6 + 63 = 75.
Dept #2.
A = 1h/u * 15 units = 15h.
B = 3h/u * 5 units = 15h.
C = 7h/u * 5 units = 35h.
Total Units = 15 + 5 + 5 = 25.
Total Hours = 15 + 15 + 35 = 65.
65/11h = 5, and 10 remainder.
To find the number of units of products A, B, and C that can be made using the available hours in departments I and II, we need to perform a linear programming problem.
First, let's define the variables:
Let x1 be the number of units of product A.
Let x2 be the number of units of product B.
Let x3 be the number of units of product C.
Now, let's set up the constraints based on the hours required per unit of each product:
In department I:
1x1 + 1x2 + 9x3 ≤ 75
In department II:
1x1 + 3x2 + 7x3 ≤ 65
These constraints represent the total number of hours used in each department, which should be less than or equal to the available hours in department I (75 hours) and department II (65 hours).
We also need to consider that the number of units cannot be negative, so we add the non-negativity constraints:
x1 ≥ 0
x2 ≥ 0
x3 ≥ 0
Now, we need to define the objective function, which is the quantity we want to maximize or minimize. In this case, we want to maximize the total number of units produced. Therefore, the objective function is:
Maximize x1 + x2 + x3
Now, we have defined the Linear Programming problem with the constraints and the objective function. To solve this problem, you can use any Linear Programming solver software or optimization tool such as Excel Solver, MATLAB, or Python's scipy.optimize.linprog function.
By plugging in the given data into the linear programming problem, you can find the maximum number of units of products A, B, and C that can be made using the available hours in departments I and II.