We produce two products: product 1 and product 2 on
two machines (machine 1 and machine 2). The number of
hours of machine time and labor depends on the machine
and the product as shown in Table 64.
The cost of producing a unit of each product is shown in
Table 65.
The number of labor hours and machine time available
this month are in Table 66.
This month, at least 200 units of product 1 and at least 240 units of product 2 must be produced. Also, at least half of product 1 must be made on machine 1, and at least half of product 2 must be made on machine 2. Determine how we can minimize the cost of meeting our monthly demands.
TAB L E 64
Product 1 Product 2 Product 1 Product 2
Machine 1 Machine 1 Machine 2 Machine 2
Machine time 0.75 0.75 0.8 0.9
Labor 0.75 0.75 1.2 1.9
TAB L E 65
Product 1 Product 2 Product 1 Product 2
Machine 1 Machine 1 Machine 2 Machine 2
$1.50 $0.40 $2.20 $4.00
TAB L E 66
Resource Hours Available
Machine 1 200
Machine 2 200
Labor 400
To determine how to minimize the cost of meeting the monthly demands, we can use linear programming. Linear programming is a mathematical optimization technique that helps find the best solution to a problem given certain constraints.
In this scenario, the monthly demands are:
- At least 200 units of product 1
- At least 240 units of product 2
We also have the following constraints:
1. Machine time available:
- Machine 1: 200 hours
- Machine 2: 200 hours
2. Labor hours available: 400 hours
3. Product allocation constraints:
- At least half of product 1 must be made on machine 1
- At least half of product 2 must be made on machine 2
To minimize the cost, we need to formulate an objective function. In this case, the objective is to minimize the total cost. The cost of producing a unit of each product is given in Table 65.
Let's denote the decision variables as:
- x1: Number of units of product 1 produced on machine 1
- x2: Number of units of product 1 produced on machine 2
- x3: Number of units of product 2 produced on machine 1
- x4: Number of units of product 2 produced on machine 2
With the given constraints and decision variables, we can set up the linear programming problem as follows:
Objective function: minimize 1.5x1 + 1.5x2 + 2.2x3 + 4x4
Subject to the following constraints:
1. Machine time constraints:
- 0.75x1 + 0.75x2 <= 200 (machine 1)
- 0.8x3 + 0.9x4 <= 200 (machine 2)
2. Labor hour constraint:
- 0.75x1 + 0.75x2 + 1.2x3 + 1.9x4 <= 400
3. Product allocation constraints:
- x1 + x2 >= 200 (at least 200 units of product 1)
- x3 + x4 >= 240 (at least 240 units of product 2)
- x1 >= 0.5(x1 + x2) (at least half of product 1 on machine 1)
- x4 >= 0.5(x3 + x4) (at least half of product 2 on machine 2)
To solve this linear programming problem, you can use software such as Excel Solver or specialized optimization libraries in programming languages like Python. By inputting the objective function, constraints, and variable bounds into the solver, it can find the optimal solution that minimizes the cost while meeting the given demands and constraints.