How many optimal basic feasible solutions does the following LP have? max z = 2x1 + 2x2 s.t. X1 + x2 < 6 2x1 + x2 < 13 All x; 2 0
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
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![**Linear Programming Problem Analysis**
**Objective:**
Determine how many optimal basic feasible solutions exist for the given linear programming (LP) problem.
**LP Formulation:**
Maximize:
\[ z = 2x_1 + 2x_2 \]
Subject to constraints:
\[
x_1 + x_2 \leq 6
\]
\[
2x_1 + x_2 \leq 13
\]
Non-negativity constraint:
\[
x_i \geq 0 \quad \text{for all } x_i
\]
**Explanation of Constraints:**
The constraints define a feasible region in the first quadrant of the Cartesian plane where \(x_1\) and \(x_2\) are non-negative. The objective function \(z = 2x_1 + 2x_2\) aims to maximize the value of \(z\) within the feasible region defined by the constraints.
**Graphical Representation:**
A graph of the constraints would consist of:
- A line representing \(x_1 + x_2 = 6\).
- A line representing \(2x_1 + x_2 = 13\).
The area bounded by these lines and the axes represents the feasible region. The corners of this region, also known as vertices, are potential candidates for the optimal solution. The maximum value of \(z\) is sought at these points because linear programming problems achieve their optimal solution at the vertices of the feasible region.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2d45e0e1-27ec-43c7-8595-f17aba038019%2Fc83dc996-3f80-4d87-a9a0-d32448966002%2Feznkb4_processed.png&w=3840&q=75)
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