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a) Write the objective function of this LP. b) Write the constraint of the basic element Y. c) Write the constraint that expresses "the factional contributions from each mine must add up to 1."

a) Write the objective function of this LP. b) Write the constraint of the basic element Y. c) Write the constraint that expresses "the factional contributions from each mine must add up to 1."

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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Exercise 4

 

a) Write the objective function of this LP.
b) Write the constraint of the basic element Y.
c) Write the constraint that expresses
"the factional contributions from each mine must add up to 1."
d) What is the optimal fraction of a ton to be chosen from mine 4?
e) To make the optimal fraction of a ton to be chosen from mine 4 non-zero, how should the cost per ton of ore from mine 4 change?
f) If the minimum requirement amounts of basic element X increases to 7 tons, how much does the cost of a feasible blend (i.e., the value of the objective function) become?
g) If the minimum requirement amount of basic element Z decreases to 25 tons, how much does the cost of a feasible blend (i.e., the value of the objective function) become?

Copper ore from four mines will be blended to make a new product. Analysis has shown that to produce a blend with suitable qualities, minimum requirements must be met on three basic elements, denoted for simplicity as X, Y, and Z. Each ton of copper ore must contain at least 5 pounds of basic element X, at least 100 pounds of basic element Y, and at least 30 pounds of basic element Z. The ore from the four different mines possesses each of the three basic elements but in different amounts. These compositions, in pounds per ton, are given in the following table. Composition from Each Mine Mine (pounds per ton of each element) 1 2 3 4 Basic Element X Y 10 3 8 2 Z 90 45 150 25 75 20 175 37 Since the ore from each mine has a different cost, different blends will also have different costs. The cost data are given in the following table. Cost of Ore from Each Mine Mine Dollar Cost Per Ton of Ore 1 800 2 400 3 600 4 500 The objective of management in this company is to minimize the cost of a feasible blend to make each of three basic elements (i.e., elements X, Y, and Z) from ore from each of the four different mines (i.e., mines 1, 2, 3, and 4). Hence, decision variables are set up as follows: Si= fraction of a ton to be chosen from mine i, where i = 1, 2, 3, and 4. Since there are no other contributions to the 1 ton besides the four mines, the factional contributions from each mine must add up to 1.
Transcribed Image Text:Copper ore from four mines will be blended to make a new product. Analysis has shown that to produce a blend with suitable qualities, minimum requirements must be met on three basic elements, denoted for simplicity as X, Y, and Z. Each ton of copper ore must contain at least 5 pounds of basic element X, at least 100 pounds of basic element Y, and at least 30 pounds of basic element Z. The ore from the four different mines possesses each of the three basic elements but in different amounts. These compositions, in pounds per ton, are given in the following table. Composition from Each Mine Mine (pounds per ton of each element) 1 2 3 4 Basic Element X Y 10 3 8 2 Z 90 45 150 25 75 20 175 37 Since the ore from each mine has a different cost, different blends will also have different costs. The cost data are given in the following table. Cost of Ore from Each Mine Mine Dollar Cost Per Ton of Ore 1 800 2 400 3 600 4 500 The objective of management in this company is to minimize the cost of a feasible blend to make each of three basic elements (i.e., elements X, Y, and Z) from ore from each of the four different mines (i.e., mines 1, 2, 3, and 4). Hence, decision variables are set up as follows: Si= fraction of a ton to be chosen from mine i, where i = 1, 2, 3, and 4. Since there are no other contributions to the 1 ton besides the four mines, the factional contributions from each mine must add up to 1.
Variable Cells Cell Name $B$3 Ton Fractions S₁ $C$3 Ton Fractions S2 $D$3 Ton Fractions S3 $E$3 Ton Fractions S4 Constraints Cell $F$6 X Total Name $F$7 Y Total $F$8 Z Total $F$9 Balance Total Final Value 0.259 0.7037 0.0370 0 Final Value 5 131.67 30 1 Reduced Objective Allowable Allowable Coefficient Increase Decrease Cost 223.636 66.848 85.714 1E+30 0 0 0 91.11 800 400 600 500 44.44 0 4.44 155.56 Shadow Constraint Allowable Allowable Price R.H. Side Increase Decrease 5 100 30 1 120 300 2.375 31.667 0.714 0.25 118.269 91.11 0.25 1E+30 7 0.043
Transcribed Image Text:Variable Cells Cell Name $B$3 Ton Fractions S₁ $C$3 Ton Fractions S2 $D$3 Ton Fractions S3 $E$3 Ton Fractions S4 Constraints Cell $F$6 X Total Name $F$7 Y Total $F$8 Z Total $F$9 Balance Total Final Value 0.259 0.7037 0.0370 0 Final Value 5 131.67 30 1 Reduced Objective Allowable Allowable Coefficient Increase Decrease Cost 223.636 66.848 85.714 1E+30 0 0 0 91.11 800 400 600 500 44.44 0 4.44 155.56 Shadow Constraint Allowable Allowable Price R.H. Side Increase Decrease 5 100 30 1 120 300 2.375 31.667 0.714 0.25 118.269 91.11 0.25 1E+30 7 0.043
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