📘 linear programming

1. **State the problem:** We want to maximize the objective function $$z = 5x + 6y$$ subject to the constraints: $$y \geq 0$$

1. **Stating the problem:** We are given the objective function $$z = 5x + 3y$$ and the constraints:

1. **State the problem:** A farmer wants to buy bags of two types of animal feed, Type A and Type B, to meet minimum nutritional requirements and minimize cost. 2. **Define variabl

1. **State the problem:** We want to find the maximum value of the function $$z = 4x + 3y$$ subject to the constraints: $$x \geq 4$$

1. **State the problem:** We need to solve the Linear Programming Problem (LPP) using the Big-M penalty method.

1. **State the problem:** We want to maximize the objective function $$Z = 3x_1 - x_2$$ subject to the constraints:

1. **State the problem:** The farm owner has 140 acres of land to plant corn and soybeans. Each acre of corn yields a profit of 370 and requires 2 hours of labor. Each acre of soyb

1. **State the problem:** Jacob needs to buy meat and cheese to meet minimum weekly requirements of 12 units of carbohydrates and 8 units of protein.

1. **State the problem:** Jacob needs to buy meat and cheese to meet minimum weekly requirements of 12 units of carbohydrates and 8 units of protein. Meat contains 2 units carbs an

1. **State the problem:** We want to find the minimum and maximum values of the objective function $$z = 5x + 4y$$ subject to the constraints: $$2x + y \leq 20$$

1. **State the problem:** Maximize the objective function $$z = 3x + 7y$$ subject to the constraints: $$4x + 6y \leq 24$$

1. **Problem Statement:** We have a linear programming problem (P) with objective function $$w = 5x_1 + \alpha x_2 - x_3$$ and constraints:

1. **State the problem:** Minimize the objective function $$R(x,y) = x + 2y$$ subject to the constraints: $$x + y \geq 2$$

1. **Problem Statement:** We have a primal linear program (P): $$\max w = 5x_1 + 8x_2$$

1. **Stating the problem:** We are given a primal linear programming (LP) problem (P):

1. **State the problem:** We want to derive the dual problem of the given primal linear program:

1. **Problem Statement:** We want to maximize the objective function $$w = 3x_1 + x_2$$ subject to constraints:

1. **Problem Statement:** We have the system of inequalities:

1. **State the problem:** We have the linear programming (LP) problem:

1. **State the problem:** We want to maximize the profit from producing tables (x) and chairs (y) given constraints on available hours in two departments.

1. **State the problem:** We want to determine how many units of products A and B should be produced and sold to maximize total profit, given constraints on raw materials and labor