Question

Firm is engaged in the production of three types of products A, B, and C. The products earn profit of $3, $5, and $4 per unit respectively. Each products goes through three operations: cutting, sewing, and inspection. During the next period, the firm has 300 hours of cutting, 300 hours of sewing, and 150 hours of inspection time available. The sales force requires that at least 35% of total number of units produced must be item A. The time (in minutes) taken for each product in each department is shown below.

A B C

Cutting 12 10 8

Sewing 15 15 12

inspection 3 4 2

Show the mathematical model.

Answer 1

The mathematical model for the firm's production process involves maximizing profit and has constraints related to available time and sales requirements.

Step-by-step explanation:

To create a mathematical model for the firm's production process, we need to define the decision variables, constraints, and objective function. Let's denote the number of units of product A, B, and C as x, y, and z respectively.

The objective function is to maximize the profit, which can be expressed as:

Maximize: 3x + 5y + 4z

Next, we need to consider the constraints. The available time for cutting, sewing, and inspection operations can be represented as:

12x + 10y + 8z ≤ 300 (cutting)

15x + 15y + 12z ≤ 300 (sewing)

3x + 4y + 2z ≤ 150 (inspection)

Lastly, the sales force requires that at least 35% of the total number of units produced must be item A, which can be expressed as:

x ≥ 0.35(x + y + z)

Thus, the mathematical model for the firm's production process is:

Maximize: 3x + 5y + 4z

Subject to:

12x + 10y + 8z ≤ 300 (cutting)

15x + 15y + 12z ≤ 300 (sewing)

3x + 4y + 2z ≤ 150 (inspection)

x ≥ 0.35(x + y + z)