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
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
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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?
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.
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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