📘 linear programming

1. **بيان المشكلة:** يريد المستثمر استثمار مبلغ 150000 وحدة نقدية في ثلاثة بدائل استثمارية: شراء منازل صغيرة، قطع أراضي، وأسهم شركات، بحيث يكون العائد المتوقع في نهاية العام أكبر م

1. **State the problem:** We want to maximize the expression $a_1 + t_1, t_4$ subject to the constraints:

1. **Problem Statement:** A firm produces two products, X and Y. Each kilogram of X requires 8 hours of machine time (T) and 10 MJ of energy (K). Each kilogram of Y requires 12 hou

1. **Stating the problem:** Alea Bakery makes two types of cakes: brownies and bika ambon. Brownies require 4 kg of wheat flour and 2 hours to make. Bika ambon requires 4 kg of whe

1. **State the problem:** We need to determine the feasible region and vertices for the system of inequalities: $$\begin{cases} x + 2y < 24 \\ 2x + 4y > 16 \\ x > 0, y > 0 \end{cas

1. **State the problem:** Maximize the objective function $$Z = 40X_1 + 32X_2$$

1. **State the problem:** We want to maximize

1. **State the problem:** We want to maximize the objective function $$Z = -4x_1 + 6x_2 - 18x_3$$

1. **Problem statement:** We have the feasible region $$S = \{(x_1,x_2) \in \mathbb{R}^2 : -x_1 - x_2 \leq 1, -x_1 + 2x_2 \geq 2, x_1 - x_2 \leq -1, x_1 \leq 0, x_2 \geq 0\}$$

1. **Problem statement:** We have the feasible region $S = \{(x_1,x_2) \in \mathbb{R}^2 : -x_1 - x_2 \leq 1, -x_1 + 2x_2 \geq 2, x_1 - x_2 \leq -1, x_1 \leq 0, x_2 \geq 0\}$ and wa

1. The problem is to understand what "basic feasible" means in the context of linear programming. 2. In linear programming, a "feasible solution" is any solution that satisfies all

1. **Problem Statement:** We have two optimization problems: (a) a maximization problem and (b) a minimization problem, each with objective functions and constraints.

1. **Problem Statement:** We want to determine how many REGULAR and SUPER tents to manufacture weekly to maximize profit, given labor hour constraints and demand limits.

1. **Problem Statement:** We want to determine how many REGULAR tents ($x$) and SUPER tents ($y$) to manufacture weekly to maximize profit, given labor hour constraints and demand

1. **Problem Statement:** A farmer wants to minimize the daily cost of chicken feed using two types of feed, A and B.

1. **State the problem:** Minimize $$z = 2x_1 + 3x_2$$

1. **State the problem:** We want to minimize $$z = 2x_1 + 3x_2$$ subject to constraints:

1. **State the problem:** We want to maximize the objective function $$Z = 2x_1 + 10x_2$$

1. **Problem Statement:** A firm manufactures two products with profits of 3 and 5 per unit respectively. Each product requires processing time in two departments D1 and D2. Produc

1. **Problem Statement:** Minimize $$z = 4x_1 + 4x_2 + x_3$$ subject to: $$2x_1 + x_2 + 3x_3 \leq 2$$

1. **State the problem:** We need to minimize the cost function $$\text{Cost} = 1.80S + 2.20T$$ subject to the constraints: $$5S + 8T \geq 200$$