Question

A small factory has two products (A and B). In order to keep the factory running, at least 20+k units of products must be produced each day where k is the largest digit of your student number. Product A costs $200 to produce, while Product B costs $100 to produce. The total costs must be less than $8000 as the budget each day. In addition, the market regulations require that the number of Product B cannot exceed twice the number of Product A, but must be at least more than half the number of Product A. If each Product A gives a profit of $40 and each Product B gives a profit of $60, design a simple daily production portfolio (in terms of the numbers of Products A and B to be produced) so that the total profit is maximized. • Write this problem as a simple linear programming problem with the objective and all appropriate constraints. • Find its optimal solution by solving it using two different methods (e.g., graphical method and Excel Solver/Matlab)?

Answer 1

The question involves creating a linear programming model to optimize the production of two products, A and B, maximizing profit within budget and market constraints. The objective function is to maximize Z = 40x + 60y. Solving the inequalities graphically or with computational tools will yield the optimal solution.

Step-by-step explanation:

The subject of the question is the application of linear programming to determine an optimal production strategy for two products, with a specific focus on maximizing profits under a set of given constraints. To formulate this as a linear programming problem, let x represent the quantity of Product A and y represent the quantity of Product B.

Objective Function

The objective is to maximize the total profit, which can be represented by the equation:

Maximize Z = 40x + 60y

Constraints

Given the constraints, the linear programming model will have the following inequalities:

To find the optimal solution, one can use the graphical method, plotting the inequalities on a graph and finding the vertices of the feasible region, or computational tools such as Excel Solver or Matlab.

Production Example

A simple example would be to check if producing more of the higher profit item, Product B, within the acceptable range would maximize the profit or if a balanced approach would be more beneficial considering total budget constraints.