Explanation Given: The objective function to be minimized is C = − 3 x − 4 y . The constraints are, x + y ≤ 45 x + y ≥ 15 x ≤ 30 y ≤ 25 x ≥ 0 , y ≥ 0 Approach: (1) If all the entries are nonnegative, the optimal solution has been reached. (2) If there are one or more negative entries, the optimal solution has not reached. Calculation: Convert the minimization problem into standard maximization problem by multiplying with negative sign. P = − C P = − ( − 3 x − 4 y ) P = 3 x + 4 y The constraints are, x + y ≤ 45 x + y ≥ 15 x ≤ 30 y ≤ 25 x ≥ 0 , y ≥ 0 Introduce the slack variables, u , v , w , and n in the given conditions and rewrite the objective function in the standard form that gives the system of linear equation as follows, x + y + u = 45 − x − y + v = − 15 x + w = 30 y + n = 25 Construct the initial simplex table. x y u v w n P Constant 1 1 1 0 0 0 0 45 − 1 − 1 0 1 0 0 0 − 15 1 0 0 0 1 0 0 30 0 1 0 0 0 1 0 25 3 − 4 0 0 0 0 1 0 The constant in the last column are not positive. Calculation of pivot element, Let column 2 be the pivot column. x y u v w n P Constant 1 1 1 0 0 0 0 45 − 1 − 1 0 1 0 0 0 − 15 − 1 = 15 1 0 0 0 1 0 0 30 0 = undefine 0 1 0 0 0 1 0 25 − 3 − 4 0 0 0 0 1 0 The least ratio is 15 that lies in row 2 So the pivot row is the second row. Hence, the element corresponding to pivot column and the pivot row is − 1 . Apply the row operation R 2 → ( − 1 ) R 2 in the above matrix. x y u v w n P Constant 1 1 1 0 0 0 0 45 1 1 0 − 1 0 0 0 15 1 0 0 0 1 0 0 30 0 1 0 0 0 1 0 25 − 3 − 4 0 0 0 0 1 0 Apply operation R 1 → R 1 − R 2 , R 4 → R 4 − R 2 , and R 5 → R 5 + 4 R 2 x y u v w n P Constant 0 0 1 1 0 0 0 30 1 1 0 − 1 0 0 0 15 1 0 0 0 1 0 0 30 − 1 0 0 1 0 1 0 10 1 0 0

Finite Mathematics for the Managerial, Life, and Social Sciences, 11th Edition

11th Edition

ISBN: 9781285464657

Author: Soo T. Tan

Publisher: Brooks Cole

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Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

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Finite Mathematics for the Managerial, Life, and Social Sciences, 11th Edition

Ch. 4.1 - Give the three characteristics of a standard...Ch. 4.1 - a. When the initial simplex tableau is set up, how...Ch. 4.1 - In the simplex method, how is a pivoted column...Ch. 4.1 - In Exercises 1-6, a write the linear programming...Ch. 4.1 - In Exercises 1-6, a write the linear programming...Ch. 4.1 - In Exercises 1-6, a write the linear programming...Ch. 4.1 - In Exercises 1-6, a write the linear programming...Ch. 4.1 - In Exercises 1-6, a write the linear programming...Ch. 4.1 - Prob. 6E Ch. 4.1 - In Exercises 7-16, determine whether the given...

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To find:

The solution to the minimization problem.