Explanation Given: The objective function to be minimized is C = x + 2 y . The constraints are, 3 x + y ≥ 12 x + 4 y ≥ 16 x ≥ 0 , y ≥ 0 Approach: (1) If all the entries are nonnegative, the optimal solution has been reached. (2) If there is one or more negative entries, the optimal solution has not been reached. Calculation: Convert the minimization problem into standard maximization problem by multiplying with negative sign. P = − C = − ( x + 2 y ) = − x − 2 y The constraint are now as, − 3 x − y ≤ − 12 − x − 4 y ≤ − 16 Introduce the slack variables, u , v in the given conditions and rewrite the objective function in the standard form that gives the system of linear equation as follows, − 3 x − y + u = − 12 − x − 4 y + v = − 16 P + x + 2 y = 0 Construct the initial simplex table. x y u v P Constant − 3 − 1 1 0 0 − 12 − 1 − 4 0 1 0 − 16 1 2 0 0 1 0 The constant in the last column are not positive. Calculate the pivot element. Let column 2 be the pivot column. x y u v P Constant Ratio − 3 − 1 1 0 0 − 12 − 12 − 1 = 12 − 1 − 4 0 1 0 − 16 − 16 − 4 = 4 1 2 0 0 1 0 The least ratio is 4 that lies in row 2 . So the pivot row is the second row. So, the element corresponding to pivot column and the pivot row is − 4 . Apply the row operation R 2 → − 1 4 R 2 in the above matrix. x y u v P Constant − 3 − 1 1 0 0 − 12 1 4 1 0 − 1 4 0 4 1 2 0 0 1 0 Apply operation R 1 → R 1 + R 2 and R 3 → R 3 − 2 R 2 . x y u v P Constant − 11 4 0 1 − 1 4 0 − 8 1 4 1 0 − 1 4 0 4 1 2 0 0 1 2 1 − 8 The element in the constant term is still not positive

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Author: Tan

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Chapter 4.CRE, Problem 12CRE

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The solution to the minimization problem having constraints,

3x+y≥12x+4y≥16x≥0,y≥0.

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Chapter 4.CRE, Problem 11CRE

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The solution to the minimization problem having constraints, 3 x + y ≥ 12 x + 4 y ≥ 16 x ≥ 0 , y ≥ 0 .

Solution Summary: The author explains how to find the solution to the minimization problem by multiplying with negative sign.

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To find:

The solution to the minimization problem having constraints,

3x+y≥12x+4y≥16x≥0,y≥0.

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Chapter 4 Solutions

The solution to the minimization problem having constraints, 3 x + y ≥ 12 x + 4 y ≥ 16 x ≥ 0 , y ≥ 0 .

Solution Summary: The author explains how to find the solution to the minimization problem by multiplying with negative sign.

Videos

To find: The solution to the minimization problem having constraints, 3x+y≥12x+4y≥16x≥0,y≥0.

Want to see the full answer?

Chapter 4 Solutions

To find:

The solution to the minimization problem having constraints,

3x+y≥12x+4y≥16x≥0,y≥0.