Suppose that a machine shop has two different types of machines; machine 1 and machine 2, which can be used to make a single product .These machines vary in the amount of product produced per hr., in the amount of labor used and in the cost of operation. Assume that at least a certain amount of product must be produced and that we would like to utilize at least the regular labor force. How much should we utilize each machine in order to utilize total costs and still meets the requirement?
Resources used
Machine 1
(X1)
Machine 2
(X2)
Minimum Required hrs.
Product produced/Hr
20
15
100
Labor/Hr
2
3
15
Operation Cost
$25
$30
To determine how much each machine should be utilized in order to minimize total costs while still meeting the requirement of producing at least a certain amount of product and utilizing the regular labor force, we can set up the following mathematical optimization problem:
Let X1 be the number of hours that machine 1 is utilized, and X2 be the number of hours that machine 2 is utilized.
We want to minimize the total costs, which is given by the objective function:
Total Costs = (X1 * 20 * 2 * $25) + (X2 * 15 * 3 * $30)
Subject to the following constraints:
1) Product produced >= 100
This constraint can be written as: (X1 * 20) + (X2 * 15) >= 100
2) Labor utilized <= 15
This constraint can be written as: (X1 * 2) + (X2 * 3) <= 15
We also have the additional constraints: X1 >= 0 and X2 >= 0, since we can't have negative hours of machine utilization.
By solving this optimization problem, we will find the values of X1 and X2 that minimize the total costs while still meeting the requirements.
Note: The specific method for solving this optimization problem (e.g., linear programming) will depend on the specific software or mathematical tools you have access to.