Explanation Given Information: A company makes commercial heating system components and domestic furnaces in Monaca and Hamburg where in Monaca plant, not more than 1000 units per day can be produced and the number of commercial components cannot be 100 more than half the number of domestic furnaces and in the Hamburg plant, not more than 850 units per day can be made. Cost of every commercial components is $ 380 at the Monaca plant and $ 400 at the Hamburg plant. The order for 500 commercial components and 750 domestic furnaces is received. Formula used: The Simplex Method for mixed Constraints: Step 1: If the provided problem is to minimize f , then maximize − f . Step 2: Make the constraints ≤ by multiplying − 1 to both sides of any ≥ constraints. Step 3: Examine the upper portion of the last column for any negative entries in the simplex matrix. a. If there are no negative entries, apply the simplex method. b. Go to step 4 if there are negative entries. Step 4: Follow the following procedure if there are negative entries in the upper portion of the matrix: a. Select any negative entry in the same row and use its column as the pivot column. b. Compute all quotients for the entries above the line in the pivot column and determine the pivot form the smaller positive quotient. c. After completing the pivot operations, return to step 3. Calculation: Consider the statement, Let x be the number of heating components produced at Monaca and y be the number of domestic furnaces produced at Monaca. Total Needed Monaca Hamburg Heating Component 500 Produced x 500 − x Cost 380 x 400 ( 500 − x ) Domestic Furnaces 750 Produced y 750 − y Cost 200 y 185 ( 750 − y ) Then the objective function is Minimize: f = 380 x + 400 ( 5 − x ) + 200 y + 185 ( 750 − y ) = − 20 x + 15 y + 338750 Maximize: f = 20 x − 15 y − 338750 And the constraints are x + y ≤ 1000 x + y ≥ 400 x ≤ 100 + 1 2 y Simplifying the constraints, x + y ≤ 1000 − x − y ≤ − 400 2 x − y ≤ 200 Add the slack variables s , t and u as below, x + y + s = 1000 − x − y + t = − 400 2 x − y + u = 200 Step 2: Write the equations in an augmented matrix. The equations are, x + y + s = 1000 − x − y + t = − 400 2 x − y + u = 200 And, − f + 100 x + 40 y Write the equations in augmented matrix, [ x y s t u f s 1 1 1 0 0 0 1000 t − 1 − 1 0 1 0 0 − 400 u 2 − 1 0 0 1 0 200 − 20 15 0 0 0 1 − 338750 ] The upper portion of the last column has negative values. If values for the variables x , y , s , t , u are computed then t = − 400 are obtained. This violates the condition of simplex method that all the variables be non-negative. Choosing t = − 400 which is row 2. In this row, there is negative entry in column 1 and column 2. Choosing the pivot column as column 2. The pivot is determined from the quotients as, 200 − 1 = − 200 , − 400 − 1 = 400 and 1000 1 = 1000 Thus, the pivot is − 1 in column y

Mathematical Applications for the Management, Life, and Social Sciences

12th Edition

ISBN: 9781337625340

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Chapter 4.5, Problem 33E

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To calculate: The number commercial components and domestic furnaces be produced in Monaca and Hamburg plant at minimum cost and also find the minimum cost.

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Chapter 4.5, Problem 32E

Chapter 4.5, Problem 34E

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General Ford produces cars in Los Angeles and Detroit and has a warehouse in Atlanta. The company supplies cars to customers in Houston and Tampa. The costs of shipping a car between various points are listed as the table below. where a blank means that a shipment is not allowed. Los Angeles can produce up to 1600 cars, and Detroit can produce up to 3200 cars. Houston must receive 1800 cars, and Tampa must receive 2900 cars. 1. Determine how to minimize the cost of meeting demands in Houston and Tampa. 2. Modify the answer to part a if no shipments through Atlanta are allowed. From LA Detroit Atlanta Houston Tampa To LA Detroit $97 $94 $87 $129 $147 $100 $103 $145 Atlanta Houston $95 $125 $103 $110 $95 $88 $60 $103 Tampa $156 $148 $56 $100

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The number commercial components and domestic furnaces be produced in Monaca and Hamburg plant at minimum cost and also find the minimum cost.

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