Explanation Given Information: Provided information is as follows, “Jason’s House of Cheese offer two types of cheese and has 3600 ounces of cheddar, 1498 ounces of Brie and 2396 ounces of Stilton. The St. Nick assortment contains 10 ounces of cheddar, 5 ounces of Brie, and 6 ounces of Stilton whereas the Holly assortment contains 8 ounces of cheddar, 3 ounces of Brie, and 8 ounces of Stilton. The price of each St. Nick assortment is $ 16 and the price of each Holly assortment is $ 14 ”. Formula used: The Simplex Method to solve Problems in Standard Maximization Form 1. Introduce the slack variables, and state the problem in terms of a system of linear equations. 2. Construct the simplex table corresponding to the system obtained in step 1. 3. If the left part of the bottom row has negative entries, the solution corresponding to the tableau yields a maximum and the problem is solved. 4. If the left part of the bottom row contains negative entries, construct a new simplex tableau. (a) Choose the pivot column by inspecting the entries of the bottom row of the current tableau, excluding the right-hand entry. The pivot column is the one containing the most negative of these entries. (b) Choose the pivot element by computing ratios associated with the positive entries of the pivot column and the right column. The pivot element is the one corresponding to the smallest nonnegative ratio. (c) Construct the new simplex tableau by pivoting around the selected element. 5. Return to step 3 . Repeat steps 3 and 4 as many times as necessary to find a maximum. 6. Interpret the result. Calculation: Consider the provided information. Write it in the mathematical form as below, Maximize 16 x + 14 y subject to the constraints { 10 x + 8 y ≤ 3600 5 x + 3 y ≤ 1498 6 x + 8 y ≤ 2396 x ≥ 0 , y ≥ 0 Since, there are three non-trivial inequalities, the simplex method will have three slack variables. Denote the slack variables by u , v and w . So, the three inequalities become, 10 x + 8 y + u = 3600 5 x + 3 y + v = 1498 6 x + 8 y + w = 2396 Also, change the objective function ‘Maximize 16 x + 14 y ’ into an equation M = 16 x + 14 y or − 16 x − 14 y + M = 0 . Now, the above four equations in matrix form as below, x y u v w M u v w M [ 10 8 1 0 0 0 3600 5 3 0 1 0 0 1498 6 8 0 0 1 0 2396 − 16 − 14 0 0 0 1 0 ] The last row of the above table shows the coefficients of the objective function − 16 x − 14 y + M = 0 and the row 1 , row 2 and row 3 show the coefficient of the constraints or system of inequality. Now, find the pivot column and pivot row of the table, Select most negative value from the last row for the pivot column. From the above table, the most negative value is − 16 and it lies in the column 1 . So, the pivot column is column 1 Now, find the quotient to select the pivot row. Value of quotient for the row 1 is, 3600 10 = 360 Value of quotient for the row 2 is, 1498 5 = 299.6 Value of quotient for the row 3 is, 2396 6 = 399.33 The minimum of 360 , 299.6 , 399.33 is 299.6 , So, the pivot row is row 2 . So, the pivot element is 5 as shown in the below table, x y u v w M u x w M [ 10 8 1 0 0 0 3600 5 3 0 1 0 0 1498 6 8 0 0 1 0 2396 − 16 − 14 0 0 0 1 0 ] 360 ( 1498 / 5 ) ← ( 2396 / 6 ) ↑ Now, make the pivot element as a unit by applying the row operation R 2 → R 2 5 and then Make all elements 0 below and above the pivot element 1 by applying the row operations R 1 → R 1 − 10 R 2 , R 3 → R 3 − 6 R 2 , R 4 → R 4 + 16 R 2

Finite Mathematics & Its Applications (12th Edition)

12th Edition

ISBN: 9780134437767

Author: Larry J. Goldstein, David I. Schneider, Martha J. Siegel, Steven Hair

Publisher: PEARSON

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Chapter 4, Problem 3P

To determine

To calculate: The solution set of the linear programming,

Maximize 16x+14y subject to the constraints,

{10x+8y≤36005x+3y≤14986x+8y≤2396x≥0,y≥0

using simplex method.

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Chapter 4, Problem 2P

Chapter 4, Problem 4P

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Chapter 4 Solutions

Finite Mathematics & Its Applications (12th Edition)

Ch. 4.1 - 1. Determine by inspection a particular solution...Ch. 4.1 - Prob. 2CYU Ch. 4.1 - For each of the following linear programming...Ch. 4.1 - For each of the following linear programming...Ch. 4.1 - For each of the following linear programming...Ch. 4.1 - For each of the following linear programming...Ch. 4.1 - For each of the following linear programming...Ch. 4.1 - For each of the following linear programming...Ch. 4.1 - 712For each of the linear programming problems in...Ch. 4.1 - 7–12 For each of the linear programming problems...

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The solution set of the linear programming, Maximize 16 x + 14 y subject to the constraints, { 10 x + 8 y ≤ 3600 5 x + 3 y ≤ 1498 6 x + 8 y ≤ 2396 x ≥ 0 , y ≥ 0 using simplex method.

Solution Summary: The author explains the simplex method used to solve linear programming problems. The number of St. Nick and Holly assortments is 218 and 136 respectively.

Videos

To calculate: The solution set of the linear programming,

Maximize 16x+14y subject to the constraints,

{10x+8y≤36005x+3y≤14986x+8y≤2396x≥0,y≥0

using simplex method.

Want to see the full answer?

Chapter 4 Solutions

The solution set of the linear programming, Maximize 16 x + 14 y subject to the constraints, { 10 x + 8 y ≤ 3600 5 x + 3 y ≤ 1498 6 x + 8 y ≤ 2396 x ≥ 0 , y ≥ 0 using simplex method.

Solution Summary: The author explains the simplex method used to solve linear programming problems. The number of St. Nick and Holly assortments is 218 and 136 respectively.

Videos

To calculate: The solution set of the linear programming, Maximize 16x+14y subject to the constraints, {10x+8y≤36005x+3y≤14986x+8y≤2396x≥0,y≥0 using simplex method.

Want to see the full answer?

Chapter 4 Solutions

To calculate: The solution set of the linear programming,

Maximize 16x+14y subject to the constraints,

{10x+8y≤36005x+3y≤14986x+8y≤2396x≥0,y≥0

using simplex method.