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.
Can you help me with this one?
To find the number of units of products A, B, and C that can be made using the available hours in each department, we need to apply a linear programming approach called the "production allocation problem."
Step 1: Define the decision variables:
Let's represent the number of units of product A, B, and C as x, y, and z, respectively.
Step 2: Formulate the objective function:
The objective is to maximize the number of units produced, which can be expressed as:
Maximize: x + y + z
Step 3: Define the constraints:
The constraints are based on the available hours in each department. We have the following constraints:
In Department I:
1x + 1y + 9z ≤ 75 (Restriction for available hours in Department I)
In Department II:
1x + 3y + 7z ≤ 65 (Restriction for available hours in Department II)
Additionally, since we cannot produce a negative number of units, we add the non-negativity constraints:
x ≥ 0, y ≥ 0, z ≥ 0
Step 4: Solve the linear programming problem:
Using any optimization software or method, we can solve this linear programming problem to find the values of x, y, and z that maximize the objective function while satisfying all the constraints.
The solution will give us the optimal number of units of products A, B, and C that can be produced using the available hours in each department.