Explanation Given Information: Minimize c = x − 3 y subject to the given following constraint 3 x + y ≥ 5 , 2 x − y ≥ 0 , x − 3 y ≤ 0 and x ≥ 0 , y ≥ 0 . Formula used: Draw a rectangle large enough that all the corner points are inside the rectangle. Shade the outside of the rectangle so as to define a new bounded feasible region and locate the new corner points. Obtain the optimal solutions using bounded feasible region. If the optimal solution occur at the one of the corner points, then LP problem has that corner point as optimal solution otherwise it has no optimal solution. Calculation: Consider the first constraint equation, 3 x + y = 5 Isolate the variable y , by subtracting 3 x from both sides, 3 x − 3 x + y = 5 − 3 x y = 5 − 3 x To plot the graph of the function, find the ordered pairs of the equation by substituting different values of x . For x = 0 , substitute 0 for x , y = 5 − 3 ⋅ 0 = 5 − 0 = 5 For x = 1 , substitute 1 for x , y = 5 − 3 ⋅ 1 = 5 − 3 = 2 For x = 2 , substitute 2 for x , y = 5 − 3 ⋅ 2 = 5 − 6 = − 1 Construct a table for the ordered pairs, x 0 1 2 y 5 2 -1 Test the first inequality by ( 0 , 0 ) test coordinates as; 3 x + y ≥ ? 5 3 ⋅ 0 + 0 ≥ ? 5 0 + 0 ≥ ? 5 0 ≥ ? 5 The inequality is not satisfied. If the test point does not satisfy the inequality then the set of solutions is the line plus the entire region on the other side of the line i.e. away from the test coordinate ( 0 , 0 ) . Consider the second constraint equation, 2 x − y = 0 Isolate the variable y , by subtracting 2 x from both sides, 2 x − 2 x − y = 0 − 2 x − y = − 2 x y = 2 x To plot the graph of the function, find the ordered pairs of the equation by substituting different values of x . For x = 0 , substitute 0 for x , y = 2 ⋅ 0 = 0 For x = 1 , substitute 1 for x , y = 2 ⋅ 1 = 2 For x = 2 , substitute 2 for x , y = 2 ⋅ 2 = 4 Construct a table for the ordered pairs, x 0 1 2 y 0 2 4 Test the second inequality by ( 1 , 1 ) test coordinates as; 2 x − y ≥ ? 0 2 ⋅ 1 − 1 ≥ ? 0 2 − 1 ≥ ? 0 1 ≥ ? 0 The inequality is satisfied. If the test point satisfies the inequality then the set of solutions is the line plus the entire region on the same side of the line as the test coordinate ( 1 , 1 ) . Consider the third constraint equation, x − 3 y = 0 Isolate the variable y , by subtracting x from both sides, x − x − 3 y = 0 − x − 3 y = − x y = x 3 To plot the graph of the function, find the ordered pairs of the equation by substituting different values of x

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Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 5.2, Problem 17E

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To calculate: Optimal solution of given linear programming problem to minimize c=x−3y subject to the given following constraints 3x+y≥5,2x−y≥0,x−3y≤0 and x≥0,y≥0.

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Chapter 5.2, Problem 16E

Chapter 5.2, Problem 18E

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Optimal solution of given linear programming problem to minimize c = x − 3 y subject to the given following constraints 3 x + y ≥ 5 , 2 x − y ≥ 0 , x − 3 y ≤ 0 and x ≥ 0 , y ≥ 0 .

Solution Summary: The author calculates the optimal solution of a linear programming problem to minimize c=x-3y subject to the following constraints.

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To calculate: Optimal solution of given linear programming problem to minimize c=x−3y subject to the given following constraints 3x+y≥5,2x−y≥0,x−3y≤0 and x≥0,y≥0.

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