📘 linear programming

1. **State the problem:** We want to maximize the objective function $$z = 3x_1 + 4x_2$$ subject to the constraints:

1. The problem involves maximizing profit given constraints on acres for crops A, B, and C. 2. The profit function and constraints are given, and we need to verify the maximum prof

1. **Restate the problem:** We need to determine how the 100 acres should be used for planting crops A, B, and C by applying the vertex method to the feasible region defined by the

1. مسئله: دامنه تغییرات $b_1$ (محدودیت اول) را در مسئله برنامه‌ریزی خطی داده شده بیابید. 2. فرمول و قواعد مهم:

1. **Menyatakan masalah:** Diberikan dua jenis layanan bus: Ekspres ($x_1$) dan Reguler ($x_2$). Tujuan adalah memaksimalkan keuntungan dengan batasan kapasitas.

1. **Stating the problem:** We have two types of bus services: Ekspres (x) and Reguler (y). We want to maximize profit given constraints on Terminal A, Depo BBM, and pool/bengkel c

1. **Problem Statement:** Minimize total travel time for passengers from the main terminal to a business area using three transportation modes: regular bus, BRT, and feeder angkot.

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