Explanation Given Information: For each grade mine and the number of tons per week required to fill orders are given in the table. I II III Required Tons/Week A 10 10 10 90 Ore Grades B 10 10 10 50 C 10 0 10 60 Cost/day $ 6000 $ 8000 $ 12000 Calculation: Consider the table: I II III Required Tons/Week A 10 10 10 90 Ore Grades B 10 10 10 50 C 10 0 10 60 Cost/day $ 6000 $ 8000 $ 12000 Let x = Mine I , y = Mine II and z = Mine III . The minimization problem will become: M i n i m i z e : C = 6000 x + 8000 y + 12000 z Subject to constraints 10 x + 10 y + 10 z ≥ 90 10 y + 10 z ≥ 50 10 x + 10 z ≥ 60 The constraints can be written in matrix form as follows: [ 10 10 10 90 0 10 10 50 10 0 10 60 6000 8000 12000 g ] Now transpose the given matrix. This will convert into: [ 10 0 10 6000 10 10 0 8000 10 10 10 12000 90 50 60 g ] Introduce new slack variables x 1 , x 2 , x 3 . Now the matrix with slack variables will become: [ 10 0 10 1 0 0 0 6000 10 10 0 0 1 0 0 8000 10 10 10 0 0 1 0 12000 − 90 − 50 − 60 0 0 0 1 0 ] As − 90 is the most negative entry, column 1 is the pivot column. Now, find the quotient to select the pivot row. Value of quotient for the row 1 is, 6000 10 = 600 Value of quotient for the row 2 is, 8000 10 = 800 Value of quotient for the row 3 is, 12000 10 = 1200 The minimum of 600 , 800 , 1200 is 600 , so, the pivot row is row 1. And so, pivot element is 10. [ ( 10 ) 0 10 1 0 0 0 6000 10 10 0 0 1 0 0 8000 10 10 10 0 0 1 0 12000 − 90 − 50 − 60 0 0 0 1 0 ] Now apply the row operations to solve this matrix: 1 10 R 1 → R 1 − 10 R 1 + R 2 → R 2 − 10 R 1 + R 3 → R 3 90 R 1 + R 4 → R 4 The matrix will become: [ 1 0 1 0

Mathematical Applications for the Management, Life, and Social Sciences

11th Edition

ISBN: 9781305108042

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Chapter 4.4, Problem 30E

To determine

To calculate: The number of days the company should operate each mine to fill the orders at minimum cost and the minimum cost if The James MacGregor Mining Company owns three mines I, II, III. For each grade mine and the number of tons per week required to fill orders are given in the table.

IIIIIIRequired Tons/WeekA10101090Ore GradesB10101050C1001060Cost/day$6000$8000$12000

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Chapter 4.4, Problem 29E

Chapter 4.4, Problem 31E

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Solution Summary: The author calculates the number of days the company should operate each mine to fill the orders at minimum cost and the minimum costs if the James MacGregor Mining Company owns three mines I, II, III.

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