The solution obtained by arbitrarily assigning values to some variables and then solving for the remaining variables is called the basic solution associated with the tableau. So the above solution is the basic solution associated with the initial simplex tableau. We can label the basic solution variable in the right of the last column as shown in the table below. x1

x2

y1

y2

Z

1

1

1

0

0

12

y1

2

1

0

1

0

16

y2

– 40

– 30

0

0

1

0

Z

4. The most negative entry in the bottom row identifies a column. The most negative entry in the bottom row is –40, therefore the column 1 is identified. x2

y1

y2

Z

x1 Q4. What is the significance of duality theory of linear y1 programming? Describe the general rules for writing the dual of a linear programming problem. Ans.Linear programming (LP) is a mathematical method for 2 1 0 1 0 16 y2 determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear – – relationships. Linear programming is a specific case 40 30 0 0 1 0 Z of mathematical programming. More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Given a polytope and a real-valued affine function defined on this polytope, a linear programming method will find a point on the polytope where this function has the smallest (or largest) value if such point exists, by searching through the polytope vertices. Linear programs are problems that can be expressed in canonical form: where x represents the vector of variables (to be determined), c and b are vectors of (known) coefficients and A is a (known) matrix of coefficients. The expression to be maximized or minimized is called theobjective function (cTx in this case). The equations Ax ≤ b are the constraints which specify a convex 1

1

1

0

0

12

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