Explanation Given Information: A builder wants to construct cottages, ranch houses and McMansions. The construction of a cottage requires 1 4 acres of land and $ 100 , 000 , the construction of a ranch house requires 1 2 acres of land and $ 400 , 000 and the construction of a McMansion requires 1 acre of land and $ 200 , 000 . The builder has a total of $ 6 million and 12 acres of land. Also, a cottage gives a profit of $ 6000 , a ranch house gives a profit of $ 13 , 000 and a McMansion gives a profit of $ 14 , 000 . Formula used: Step 1: Introduce slack variables and write the linear programming problem as a system of linear equations. Step 2: Construct the corresponding simplex table. Step 3: Check if the left part of the bottom row contains negative entries. If no negative entry is present, the solution corresponding to the table gives the maximum value of the objective function. Step 4: If the left part of the bottom row contains negative entries, construct a new simplex table as below: Firstly, choose the pivot column by checking which column of the bottom row except the right-hand entry contains the most negative element. Secondly, choose the pivot element by computing ratios of right column to entries of the pivot column. The pivot element is the one corresponding to the smallest nonnegative ratio. Now, construct a new simplex table by pivoting around the selected pivot element Step 5: Repeat step 3 and step 4 till there is no negative entry left in the left part of the bottom row. Step 6: Write the result obtained. Calculation: Let the number of cottages to be x , the number of ranch houses be y and the number of McMansions be z . As, the construction of a cottage requires 1 4 acres of land, the construction of a ranch house requires 1 2 acres of land and the construction of a McMansion requires 1 acre of land, the total land required is 1 4 x + 1 2 y + z . Also, the maximum land available is 12 acres. So, the first constraint is 1 4 x + 1 2 y + z ≤ 12 . Now, the construction of a cottage requires $ 100 , 000 , the construction of a ranch house requires $ 400 , 000 and the construction of a McMansion requires $ 200 , 000 . The total money required is 100000 x + 400000 y + 200000 z . Also, maximum money available is $ 6 million. So, the second constraint is 100000 x + 400000 y + 200000 z ≤ 6000000 . Also, a cottage gives a profit of $ 6000 , a ranch house gives a profit of $ 13 , 000 and a McMansion gives a profit of $ 14 , 000 . Thus, the objective function is, Maximize 6000 x + 13000 y + 14000 z . Hence, the linear programming problem corresponding to the provided problem is, Maximize 6000 x + 13000 y + 14000 z Subject to the constraints { 1 4 x + 1 2 y + z ≤ 12 100000 x + 400000 y + 200000 z ≤ 6000000 x ≥ 0 , y ≥ 0 , z ≥ 0 Now, to solve the above problem, follow the steps shown below: Step 1: Introduce the slack variables and write the problem in terms of a system of linear equations. The two constraints 1 4 x + 1 2 y + z ≤ 12 and 100000 x + 400000 y + 200000 z ≤ 6000000 give the equations, 1 4 x + 1 2 y + z + u = 12 100000 x + 400000 y + 200000 z + v = 6000000 Where, u and v are slack variables with u ≥ 0 and v ≥ 0 . The objective function 6000 x + 13000 y + 14000 z gives the equation, M = 6000 x + 13000 y + 14000 z . So, the corresponding system of linear equation with slack variables is, { 1 4 x + 1 2 y + z + u = 12 100000 x + 400000 y + 200000 z + v = 6000000 − 6000 x − 13000 y − 14000 z + M = 0 Where, x ≥ 0 , y ≥ 0 , z ≥ 0 , u ≥ 0 , w ≥ 0 and M is as large as possible. Step 2: The initial simplex table corresponding to above system is: x y z u v M u 1 4 1 2 1 1 0 0 12 v 100000 400000 200000 0 1 0 6000000 M − 6000 − 13000 − 14000 0 0 1 0 Step 3: As the left part of the bottom row has negative entries − 6000 , − 13000 and − 14000 , the above table does not give the maximum value of the objective function. So, create a new simplex table by pivoting. Step 4: First, determine the pivot column as below: Since, − 14000 is the most negative entry in the last row, the third column becomes the pivot column. Now, to determine the pivot element, compute ratios as below: x y z u v M Ratios u 1 4 1 2 1 1 0 0 12 12 1 = 12 v 100000 400000 200000 0 1 0 6000000 6000000 200000 = 30 M − 6000 − 13000 − 14000 0 0 1 0 ↑ Since, the smallest non-negative ratio is 12, so the pivot element is 1

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 4.2, Problem 30E

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To calculate: The number of cottages, ranch houses and McMansions to be built so as to maximize the profit if the construction of one cottage requires 14 acres of land and $100,000, the construction of ranch house requires 12 acres of land and $400,000 and the construction of McMansion requires 1 acre of land and $200,000.

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Chapter 4.2, Problem 29E

Chapter 4.2, Problem 31E

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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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Solution Summary: The author calculates the number of cottages, ranch houses, and McMansions to be built so as to maximize the profit.

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