University Linear Programming Assignment Solution - Optimization

This assignment solution addresses a linear programming problem faced by an agriculture business, focusing on maximizing profit while adhering to resource constraints. The student formulates a linear programming model, defining the objective function, constraints (farm land and pesticide usage), and non-negativity constraints. The solution employs the graphical method to determine the optimal combination of maize and vegetable cultivation acreage. By analyzing extreme points within the feasible region, the student identifies the combination that maximizes the firm's net profit. The optimized profit is then calculated, demonstrating the practical application of linear programming principles in business decision-making. The solution provides a step-by-step approach to solving the problem, making it a valuable resource for students studying quantitative methods and operations research.

Running Head: LINEAR PROGRAMMING ASSIGNMENT
Linear Programming Assignment
Name of the Student
Name of the University
Student ID

1LINEAR PROGRAMMING ASSIGNMENT
Answer 7
Part A
The given problem will be solved by using the graphical technique. The following method
described is followed in solving the problem using graphical technique (Mahato 2015). Let x1 be
the number of acres of maize crop and x2 be the number of acres of vegetables that has been
cultivated in the 80 acres of farm land.
The objective function of this problem can be given as follows:
Z=5000 x1 +3000 x2
The total acre of farm land present is 80 acres. Thus, the constraint related to firm land is given
by the following inequality:
x1+ x2 ≤ 80
The total pesticide usage is limited to 100 gallons. Maize plantation requires 2 gallons of
pesticide per acre and vegetable plantation requires 1 gallon of pesticide per acre. Thus, the
pesticide constraint can be given as follows:
2 x1 + x2 ≤ 100
Since, the number of acres of land cannot be negative, the non-negativity constraints are given as
follows:
x1 ≥ 0 , x2 ≥ 0
Part B
Let the equation corresponding to the first constraint be

2LINEAR PROGRAMMING ASSIGNMENT
L1: x1+ x2=80
The origin O (0, 0) satisfies (0 + 0) = 0 < 80, which satisfies the farm land constraint. Thus, the
inequality satisfies all points on L1 and on the origin side of L1.
Let the equation corresponding to the second constraint be
L2: 2 x1 + x2=100
The origin O (0, 0) satisfies (2 * 0 + 0) = 0 < 100, which satisfies the pesticide constraint. Thus,
the inequality satisfies all points on L2 and on the origin side of L2.
At some of the extreme points obtained in the feasible region, the optimal solution will exist.

3LINEAR PROGRAMMING ASSIGNMENT
Extreme Points Co-Ordinates Value of z
A (0, 80) (5000 * 0) + (3000
* 80) = 240000
B (20, 60) (5000 * 20) +
(3000 * 60) =
280000
C (50, 0) (5000 * 50) +
(3000 * 0) =
250000
O (0, 0) (5000 * 0) + (3000
* 0) = 0
The objective function is maximized at B (20, 60). Thus, 20 acres of maize and 60 acres of
vegetables has to be cultivated in order to maximize the profit.
Part C
The optimized net profit of the farm is ($5,000 * 20) + ($3,000 * 60) = 280,000

4LINEAR PROGRAMMING ASSIGNMENT
References
Mahato, D. (2015). Essentials of Operations Research. [online] researchgate.net. Available at:
https://www.researchgate.net/publication/298178045_Linear_Programming_Graphical_Method
[Accessed 25 Jul. 2018].