Question:

$Z = 7x + y$, subject to $5x + y \ge 5, x + y \ge 3, x \ge 0, y \ge 0$. The minimum value of Z occurs at

Updated On: Jul 7, 2022
  • $(3, 0)$
  • $\left(\frac{1}{2}, \frac{5}{2}\right)$
  • $(7, 0)$
  • $(0, 5)$
Hide Solution
collegedunia
Verified By Collegedunia

The Correct Option is D

Solution and Explanation

We have, maximize $Z = 7x + y$, Subject to : $5x + y \ge 5, x + y \ge 3, x, y \ge 0$. Let $\ell_{1} : 5x + y = 5$ $\ell _{2} : x + y = 3$ $\ell _{3} : x = 0$ and $\ell _{4} : y = 0$ Shaded portion is the feasible region, Where $A\left(3, 0\right), \,B\left(\frac{1}{2}, \frac{5}{2}\right), C\left(0, 5\right)$
Solving $\ell _{1}$ and $\ell _{2}$, we get $B\left(\frac{1}{2}, \frac{5}{2}\right)$ Now maximize $Z = 7x + y$ Z at $A\left(3, 0\right) = 7\left(3\right) + 0 = 21$ Z at $B\left(\frac{1}{2}, \frac{5}{2}\right) = 7\left(\frac{1}{2}\right) + \frac{5}{2} = 6$ Z at $C\left(0, 5\right) = 7\left(0\right) + 5 = 5$ Thus Z, is minimized at $C\left(0, 5\right)$ and its minimum value is $5$
Was this answer helpful?
0
0

Top Questions on Linear Programming Problem

View More Questions

Notes on Linear Programming Problem

Concepts Used:

Linear Programming

Linear programming is a mathematical technique for increasing the efficiency and effectiveness of operations under specific constraints. The main determination of linear programming is to optimize or minimize a numerical value. It is built of linear functions with linear equations or inequalities restricting variables.

Characteristics of Linear Programming:

  • Decision Variables: This is the first step that will determine the output. It provides the final solution to the problem.
  • Constraints: The mathematical form in which drawbacks are expressed, regarding the resource.
  • Data: They are placeholders for known numbers to make writing complex models simple. They are constituted by upper-case letters.
  • Objective Functions: Mathematically, the objective function should be quantitatively defined.
  • Linearity: The function's relation between two or more variables must be straight. It indicates that the variable's degree is one.
  • Finiteness: Input and output numbers must be finite and infinite. The best solution is not possible if the function consists infinite components.
  • Non-negativity: The value of the variable should be either positive (+ve) or 0. It can't be a negative (-ve) number.