Explanation Given Information: There are 50 computers in stock in Boston and 80 in stock in Chicago. The dealer in Detroit orders 40 computers, and the dealer in Fletcher orders 50 computers. The cost of shipping one computer from each warehouse to each dealer is as follows: Detroit Fletcher Boston $125 $180 Chicago $100 $160 Formula used: The following steps are used for Problems in Nonstandard Form: (1) All inequalities (except x ≥ 0 , y ≥ 0 , ... ) should be in the form, [Linear expression] ≤ [Constant] (2) Remove any negative number present in the upper part of the last column of the simplex table by pivoting. (a) Select any one negative entry in its row. The corresponding column will be the pivot column. (b) Find the pivot element by evaluating the least of the positive ratios corresponding to entries in the pivot column. (3) Repeat step 2 till we get all the entries positive in the upper part of the right-hand column of the simplex table. Also, if this process does remove all negative entries from the upper part of the right-hand column, then the linear programming problem does not have an optimal solution. (4) Apply the simplex method for tableaux in standard maximization form. Calculation: Consider that there are 50 computers in stock in Boston and 80 in stock in Chicago. The dealer in Detroit orders 40 computers, and the dealer in Fletcher orders 50 computers. Also, the cost of shipping one computer from each warehouse to each dealer is as follows: Detroit Fletcher Boston $125 $180 Chicago $100 $160 Let x be the number of computers shipped from Chicago to Detroit and y be the number of computers shipped from Chicago to Fletcher. Then ( 40 – x ) equals the number of computers shipped from Boston to Detroit and ( 50 – y ) represents the number of computers shipped from Boston to Fletcher. Thus, simplex problem is, Minimize 125 ( 40 − x ) + 100 x + 180 ( 50 − y ) + 160 y = 14000 − 25 x − 20 y Subject to constraint: { x + y ≤ 80 x + y ≥ 40 x ≤ 40 y ≤ 50 x ≥ 0 , y ≥ 0 First, transform the problem so that the first two constraints are in ≤ form: { x + y ≤ 80 − x − y ≤ − 40 x ≤ 40 y ≤ 50 x ≥ 0 , y ≥ 0 Now write the linear programming equation as a linear system: From the above linear system, the simplex tableau is: x y u v w t M u v w t M [ 1 1 1 0 0 0 0 80 − 1 − 1 0 1 0 0 0 − 40 1 0 0 0 1 0 0 40 0 1 0 0 0 1 0 50 − 25 − 20 0 0 0 0 1 − 14000 ] The right-side column has a negative number − 40 . This means that the initial value for y is − 40 , which violates the condition that all variables be ≥ 0 . First put the tableau into standard maximization form by pivoting to remove the negative entry in the right column. To choose the pivot element - Look along the left side of the − 40 row of the tableau, and locate any negative entry. There are two − 1 . Use the column containing the − 1 as the pivot column (that is, use column 1). Now, compute ratios. Note, however, that in this circumstance, we compute ratios corresponding to both positive and negative entries (except the last) in the pivot column, considering further only positive ratios. x y u v w t M u v w t M [ 1 1 1 0 0 0 0 80 − 1 − 1 0 1 0 0 0 − 40 1 0 0 0 1 0 0 40 0 1 0 0 0 1 0 50 − 25 − 20 0 0 0 0 1 − 10400 ] 80 1 = 80 − 40 − 1 = 40 40 1 = 40 50 0 = undefined Now we use simplex method for tableaux in standard form maximization form. x y u v w t M u x w t M [ 0 0 1 1 0 0 0 40 1 1 0 − 1 0 0 0 40 0 − 1 0 1 1 0 0 0 0 1 0 0 0 1 0 50 0 5 0 − 25 0 0 1 − 13000 ] The left part of the bottom row has a negative entry ( − 25 )

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 4.3, Problem 22E

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To calculate: The minimum cost with shipping schedule when there are 50 computers in stock in Boston and 80 in stock in Chicago and the cost of shipping one computer from each warehouse to each dealer is as follows:

Detroit Fletcher

Boston $125 $180

Chicago $100 $160

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Chapter 4.3, Problem 21E

Chapter 4.3, Problem 23E

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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 minimum cost with shipping schedule when there are 50 computers in stock in Boston and 80 in stock in Chicago and the cost of shipping one computer from each warehouse to each dealer is as follows: Detroit Fletcher Boston $125 $180 Chicago $100 $160

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To calculate: The minimum cost with shipping schedule when there are 50 computers in stock in Boston and 80 in stock in Chicago and the cost of shipping one computer from each warehouse to each dealer is as follows: Detroit Fletcher Boston $125 $180 Chicago $100 $160

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To calculate: The minimum cost with shipping schedule when there are 50 computers in stock in Boston and 80 in stock in Chicago and the cost of shipping one computer from each warehouse to each dealer is as follows:

Detroit Fletcher

Boston $125 $180

Chicago $100 $160