📘 linear programming

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

1. **Problem Statement:** KIA Motors Limited (KML) wants to maximize total sales by sending promotions to two customer groups: current customers and new customers.

1. **State the problem:** We want to maximize the weekly profit from producing two toy guns, Acer ($8 profit per dozen) and Bulls-I ($5 profit per dozen), subject to resource and m

1. The problem is to determine whether to plot the constraint $$y - x \leq 350$$ given that all other constraints intersect in the upper right quadrant except this one. 2. The cons

1. **State the problem:** We want to maximize the weekly profit from producing two toy guns: Acer and Bulls-I.

1. **Stating the problem:** We want to maximize the objective function $$Z = -3x_1 - 2x_2$$

1. Задача: Минимизировать функцию цели $$Z(X) = -2x_1 + x_2 + 3x_3 - 2x_4$$ при условиях: $$\begin{cases} 3x_1 - x_2 - 4x_3 + x_4 = 2, \\ 5x_1 - x_2 - 7x_3 + 2x_4 = 6, \\ x_j \geq

1. Задача: Максимизировать функцию цели $$Z(X) = 3x_1 - x_2$$ при ограничениях: $$-3x_1 + 2x_2 \leq 6,$$

1. **State the problem:** Minimize the objective function $$Z = 4x + 3y$$ subject to the constraints: $$2x + y \geq 10$$

1. **Problem Statement:** Angela and Zooey want to decide how many fish dinners ($x$) and beef dinners ($y$) to prepare each night to maximize profit, given constraints on total me

1. **State the problem:** We want to maximize the objective function $$Z = x + 2y$$ subject to the constraints:

1. **State the problem:** Reduce the given linear programming problem to its standard form.

1. **State the problem:** We want to maximize the profit from products A and B given constraints on sales, availability, and raw material usage.

1. **Problem Statement:** An investor wants to invest in two companies: Company 1 (extractive) and Company 2 (tech). Prices per share are $40$ for Company 1 and $25$ for Company 2.

1. **Stating the problem:** Maximize $$Z = 4x_1 + 3x_2 + 2x_3$$