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## Introduction to Operations Research - Part 1

“Both graphical and normal simplex methods can be used to solve linear programming that consists of two decision variables.” Based on your understanding, discuss whether the given statement is true by giving justification.

To analyse the given statement we can say that when we have two variables we can solve it with graphical method, but when we have three or more variables we have the option to solve large LP problems using Simplex Method. The simplex system uses a highly efficient approach. It does not determine the value of the objective function at any point; it starts first with a corner of the viability area in which all the key variables are negative. Then the value of the objective function is gradually shifted from the corner point to the corner point. The method moves on to determine the best solution.

In the resolution of an LP problem with one or two decision variables, the graphical approach is constrained. It gives a simple description of the positions and vertical portions of both feasible and non-feasible areas. Visual interpretation of the dilemma leads to a more logical way of thinking. We learnt, for example, that when a linear programme has a non-empty, confined area, the optimal solution is always one of its viable vertices. It is then important to locate all the crossing points, vertices, and then analyse which vertices are the ideal solution across all feasible vertices.

## Introduction to Operations Research - Part 2

1. We use a graphical linear programming approach for the resolution of the problems by locating the intersection between the object and the feasible area at the limit or bottom of the diagram.
2. Many real-life problems can be described as problems of optimization which face many restrictions. How far would I go if I plan to be 60 percent on the highway and 40 percent in the city without stopping at the gas station? What is the most enjoyable I get with 10 dollars of chocolate bars because I want at least one butterfinger strip, but twice as much as Snickers? How will I achieve the highest GPA in the schools, in terms of the ranking system for each class and just 2 more days before finals can be studied? This approach uses a slack variables, tableaux and crucial variables as a way to address linear programming models by hand to find the right solution to an optimisation problem. A linear scheme is a way of obtaining the best result considering the linear restriction of a maximum or minimum equation. Alternatively, the Simplex approach is a technique of treating linear programmes using an online solver such as MatLab.
3. The key principle of the simple method is to begin at a single vertex and try to locate an adjacent vertex which will maximise the objective function (in case of maximisation). In addition, this process is started. The query now is how to pick the beginning vertex. Typically, zero selects the beginning vertex. However, zero is not necessarily a realistic option. This is where the two-phase approach is to be used. Fundamentally, to try a viable alternative or to say that your LP is unfeasible, you create another LP. You have a vertex before you find it and should begin the latter process.
4. Big M is a method used to solve problems of linear programming. The methods used for the solution of LP must generally start from BAS (BFS)-(0, 0). The presumption now extends to LESS THAN and EQUAL TO (< =). The simplex algorithm is widely used for these restrictions. When an LP is restricted > or =, it cannot be quickly accessible to provide a simple solution. The Big M method is a variant of Simplex Algorithms, which, by adding "artificial" variables, first finds a feasible solution to the problem of artificial areas and then moves to real regions to boost the objective solution. The Big M benefit is that only one goal feature is needed, and we can trace the artificial variable M as to whether or not the problem has a solution. However, if the problem is solved on the optical computer, they present a challenge.
5. The dual simplex approach can switch from a dual dictionary, which can be done, to a dual dictionary, which can be done. It is called the Dual Simplex Method since the normal Simplex Solution on the dual linear problem is the same one. This describes also the word "dual possible": for the basic, any dictionary has a corresponding dictionary for the dual dictionary, and the primal dictionary is dual precisely when (in the ordinary sense) the corresponding dual dictionary is feasible. However, we will not even use this correspondence: instead of explaining how these dual swivel pivots can be achieved directly using a dual vocabulary, we will not directly speak about the dual LP.

## Introduction to Operations Research - Part 3

In order to apply the dual simplex method, convert all ≥ constraint to ≤ constraint by multiply -1.

Problem is

Max Z =x1-2x2

Subject to

-2x1-2 x2≤-6

x1-x2 ≤-5

and x1,x2≥0;

The problem is converted to canonical form by adding slack, surplus and artificial variables as appropiate

1. As the constraint-1 is of type '≤' we should add slack variable S1
2. As the constraint-2 is of type '≤' we should add slack variable S2

After introducing slack variables

Max Z =x1-2x2+0S1+0S2

subject to

-2x1-2x2+S1= -6

x1-x2+S2=-5

And x1,x2,S1,S2≥0

Here not all Zj-Cj≥0. (BecauseZ1-C1=-1)

Hence, method fails to get optimal basic feasible solution.

Remember, at the center of any academic work, lies clarity and evidence. Should you need further assistance, do look up to our Management Assignment Help

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