(a) Explanation Given Information: The city council has offered to subsidize the hotel development and at least 2 hotels with total capacity of at least 1 , 400 , you are thinking of 3 prototypes a convention style hotel with 500 rooms costing $ 100 million, a vacation style hotel with 200 rooms costing $ 20 million and a small motel with 50 rooms costing $ 4 million and the council will approve your plans if you build at least one convention style hotel and no more than 2 small hotels. Formula used: To compute solution of general linear programming problems, use slack and surplus variables to covert inequalities into equations and star the rows for which the active variables have negative value. If there are starred rows, use Phase I to proceed. Phase I: To remove the stars that is to make active variables have positive values. Start from the first starred row and locate the largest positive number and pivot the column, use test ratios and the smallest test ratio corresponds to a pivot and clear the pivot column. Check again and star the rows for which the active variables have negative value and repeat the process until there are no starred rows and move to Phase II. Phase II: Use the simplex method to solve the obtained standard maximization tableau. Choose the smallest negative number on the left side of the bottom row and pivot the column. Select the pivot which is the smallest of the test ratios a b , where a is the entry in the right most column and b is the corresponding entry in the pivot column. Clear the pivot column using the pivot and label the pivot row with the label of the pivot column. Repeat till all the entries in the bottom row are non negative. If there are no negative entries in the bottom row of the standard maximization tableau, then the obtained tableau represents the solution to the problem. Use row operations to clear the column of the augmented matrix. Calculation: Let x represent number of convention style hotel, y represents the number of vacation style hotel and z represents number of small motel. Thus, the minimum cost is given by c = 100 x + 20 y + 4 z . And, the constraints are 500 x + 200 y + 50 z ≥ 1400 x ≥ 1 z ≤ 2 x ≥ 0 , y ≥ 0 , z ≥ 0 Consider the maximisation problem. Maximize p = − 100 x − 20 y − 4 z subject to constraints, 500 x + 200 y + 50 z ≥ 1400 x ≥ 1 z ≤ 2 x ≥ 0 , y ≥ 0 , z ≥ 0 Recall that to compute solution of general linear programming problems, use slack and surplus variables to covert inequalities into equations and star the rows for which the active variables have negative value. Subtract and add the slack and the surplus variable s t and u . 500 x + 200 y + 50 z − s = 1400 x − t = 1 z + u = 2 Thus, the equations are 500 x + 200 y + 50 z − s = 1400 x − t = 1 z + u = 2 − 100 x − 20 y − 4 z + p = 0 Write the equations in augmented matrix as; x y z s t u p s 500 200 50 − 1 0 0 0 1400 t 1 0 0 0 − 1 0 0 1 u 0 0 1 0 0 1 0 2 p 100 20 4 0 0 0 1 0 Since, the value of s and t is negative, star the row. Thus, the tableau is; x y z s t u p * s 500 200 50 − 1 0 0 0 1400 * t 1 0 0 0 − 1 0 0 1 u 0 0 1 0 0 1 0 2 p 100 20 4 0 0 0 1 0 Since, there are starred rows, use Phase I to proceed. Phase I: To remove the stars that is to make active variables have positive values. Start from the first starred row and locate the largest positive number and pivot the column, use test ratios and the smallest test ratio corresponds to a pivot and clear the pivot column. Check again and star the rows for which the active variables have negative value and repeat the process until there are no starred rows and move to Phase II. The first starred row is s and the largest positive number in the s row is 500 . Thus, pivot the x column. The test ratios are, 1400 500 and 1 1 . Since, 1 1 is the smallest, pivot the entry 1 in x column as; x y z s t u p * s 500 200 50 − 1 0 0 0 1400 * t 1 0 0 0 − 1 0 0 1 u 0 0 1 0 0 1 0 2 p 100 20 4 0 0 0 1 0 Use row operations R 1 − 500 R 2 and R 4 − 100 R 2 to clear the pivot column of the augmented matrix as; x y z s t u p * s 0 200 50 − 1 500 0 0 900 x 1 0 0 0 − 1 0 0 1 u 0 0 1 0 0 1 0 2 p 0 20 4 0 100 0 1 − 100 Since, the value of s is negative, star the row. Thus, the tableau is; x y z s t u p * s 0 200 50 − 1 500 0 0 900 x 1 0 0 0 − 1 0 0 1 u 0 0 1 0 0 1 0 2 p 0 20 4 0 100 0 1 − 100 The first starred row is s and the largest positive number in the s row is 500 . Thus, pivot the t column. The test ratios are 900 500 . Thus, pivot the entry 500 in t column as; x y z s t u p * s 0 200 50 − 1 500 0 0 900 x 1 0 0 0 − 1 0 0 1 u 0 0 1 0 0 1 0 2 p 0 20 4 0 100 0 1 − 100 Use row operations 500 R 2 + R 1 , 5 R 4 − R 1 to clear the pivot column of the augmented matrix as; x y z s t u p t 0 200 50 − 1 500 0 0 900 x 500 200 0 − 1 0 0 0 1400 u 0 0 1 0 0 1 0 2 p 0 − 100 − 30 1 0 0 5 − 1400 Since, all the values are positive, there are no starred rows (b) To determine Whether the city council’s $ 50 million subsidy will be sufficient to cover 20 % of the total costs.

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Chapter 5.4, Problem 43E

(a)

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To calculate: The number of each type of hotel should you build to satisfy the city council’s wishes and stipulations while minimizing the cost.

(b)

To determine

Whether the city council’s $50 million subsidy will be sufficient to cover 20% of the total costs.

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Chapter 5.4, Problem 42E

Chapter 5.4, Problem 44E

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The number of each type of hotel should you build to satisfy the city council’s wishes and stipulations while minimizing the cost.

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To calculate: The number of each type of hotel should you build to satisfy the city council’s wishes and stipulations while minimizing the cost.

Whether the city council’s $50 million subsidy will be sufficient to cover 20% of the total costs.

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Chapter 5 Solutions