Operations Research Assignment: LPP Solutions and Analysis

This assignment presents solutions to three linear programming problems (LPP) using both graphical and simplex methods. The first problem aims to maximize the objective function, with the optimal solution found at x1=3, x2=4, and max Z = 18. The second problem also maximizes the objective function, utilizing the simplex method and Excel solver to determine the optimal solution as x1 = 0, x2 = 1.33, x3 = 1.33, and max Z = 14.67. The third problem focuses on minimizing the objective function, with the optimal solution at x1=2, x2=0, and min Z = 2. The document provides step-by-step explanations and the final solutions for each problem, demonstrating the application of linear programming techniques to achieve optimal results. The solutions demonstrate how to use graphical and simplex methods and Excel solver to solve LPP problems.

OPERATIONS
[DATE]

Question 1
Linear programming problem
The aim here is to maximize the objective function and find the optimal solution of the LPP.
Objective function
Way 1: Graphical Method
1

2

The maximum value of the objective function is observed as 18 at points C (3,4). Hence, the
optimal solution of the given LPP would be at x1=3, x2=4 and the min Z = 18.
Way 2: Simplex Method
3

As all the constraints are less than or equal to type and hence, slack variables (Artificial
variables), S1, S2, S3 and S4 are added to the corresponding constraints.
4

5

Finally, it can be concluded that in both the methods, the maximum value of the objective
function is 18 and the optimal solution of the decision variables x1 = 3, x2=4.
Question 2
Linear programming problem
6

(A) The aim here is to maximize the objective function and find the optimal solution of the
LPP.
Objective function
STEP1
STEP 2
STEP 3
7

STEP 4
Here, x1 = 0, x2 = 1.33, x3 = 1.33 and Max Z = 14.67
(B) Simplex method
8

9

(C) Excel solver model has been formed to find the optimal solution of the LPP.
Solver add-ins model
10

Answer Report
Sensitivity Report
11

Limits Report
Optimal solution
x1 = 0, x2 = 1.33, x3 = 1.33 and Max Z = 14.67
Question 3
Linear programming problem
The aim here is to minimize the objective function and find the optimal solution of the LPP.
Objective function
12

13

14

15

The minimum value of the objective function has occurred at the extreme point (2,0) and the
value is z = 2. Hence, the optimal solution of the given LPP would be at x1=2, x2=0 and the
min Z = 2.
16