Question

A company is making two products A and B. The cost of producing one unit of product A and B is $ 60 and $ 80 respectively. As per the agreement, the company has to supply at least 200 units of product B to its regular customers. One unit of product A requires one machine hours whereas product B has machine hours available abundantly within the company. Total machine hours available for product A are 400 hours. One unit of each product A and B requires one labour hour each and total of 500 labour hours are available. The company wants to minimize the cost of production by satisfying the given requirement.

Formulate the problem as a linear programming problem.
Compute the solution using Graphic method
Solve the following problem by simplex and Graphic method
Maximise Z = 30x1 + 40x2
Subject to, 60x1 + 120x2 < 12,000
8x1 + 5x2 < 600
3x1 + 4x2 < 500
x1, x2 > 0
Dorian makes luxury cars and jeeps for high-income men and women. It wishes to advertise with 1 minute spots in comedy shows and football games. Each comedy spot costs $50K and is seen by 7M high-income women and 2M high-income men. Each football spot costs $100K and is seen by 2M high-income women and 12M high-income men. How can Dorian reach 28M high-income women and 24M highincome men at the least cost?
A. Formulate the problem as a linear programming problem.
B. Compute the solution using Graphic method
A company which manufactures canoes employs 120 employees, each of whom working 30 hours per week. Half of them work in the carpenter department, 20 persons in the plastics department, and the rest of them at the completion department. The company manufactures the simple canoes with net unit profit € 7 and the luxury canoes with corresponding profit € 10. A simple canoe requires 4.5 hours in the carpenter department and two hours in each of the other two departments. The working hours for each luxury canoe are 5, 1 and 4 at the carpenter department, plastics department and completion department respectively. Marketing calculations have shown that not less than 1/3 and not more than 2/3 of the total number of the canoes should be luxurious.
Required:
Formulate the above problem as a linear programing model
Solve the problem using graphical method/ simplex method.
Solve the following LLP by graphical method (5pt)
Z max = 40X1 + 30X2
Subject to:
3x1 + x2 ≤ 12
x1 + x2 ≤ 6
5x1 + 3x2 ≤ 27
X1 , X2( 0
Solve the following LLP by graphical method (5pt)
Z min. : 50X1 + 80X2
Subject to
20X1 + 30X2 (1400
10X1 + 40X2 (1200
X1 , X2 ( 0

Answer 1

The labor-leisure budget constraint represents the trade-off between labor and leisure time, and helps individuals and families determine the utility-maximizing balance between work and leisure. Cost-related decisions for production methods are driven by the relative costs of labor and capital, with the preference for the least costly method based on current prices.

Step-by-step explanation:

The concept explored here is the labor-leisure budget constraint in economics, which illustrates the trade-off between working hours (and thus income) and leisure time. The budget constraint shows all possible combinations of labor and leisure that a person can achieve given their wage rate and available time.

For instance, Vivian can choose between 70 hours of leisure and no income or up to 40 hours of work for a weekly income of $400, which represents her highest total utility.

For single-parent families like the one mentioned, the labor-leisure trade-off becomes crucial for maximizing utility, which means striking the best balance between earning income and spending time on family responsibilities.

The budget constraint line shows that at a wage rate of $8 per hour, working 2,000 hours a year would generate an income of $16,000, which could be the most preferable option for the single mother.

The choice of production method depends on the relative costs of labor and capital. Initially, if labor costs $100 per unit and capital costs $400, the method with the least total cost would be selected. If labor costs increase to $200 per unit, it may prompt a shift to a method utilizing more capital and less labor.