Blending Problems: This is another large class of problems in which linear programming is applied heavily. Blending is concemed with mixing different materials called the constituents of the mixture (these may be chemicals, gasoline, fuels, solids, colors, foods, etc.) so that the mixture conforms to specifications on several properties or characteristics. To model a blending problem as an LP, the linearity assumptions must hold. This implies that the value for a characteristic for the constituents in the mixtures are the weighted average of the values of that characteristic for the constituents in the mixture; the weights being the proportions of the constituents. As an example, consider a mixture consisting of 4 barrels of fuel 1 and 6 barrels of fuel 2, and suppose the characteristic of interest is the octane rating (Oc.R). If linearity assumptions hold, the Oc.R of the mixture must be equal to (4 times the Oc.R of fuel 1 + 6 times the Oc.R of fuel 2)/(4 +6). These linearity assumptions hold to a reasonable degree of precision for many important characteristics of blends of gasolines, of crude oils, of paints, of foods, etc. That's why linear programming is used extensively in optimizing gasoline blending, in the manufacture of paints, cattle feeds, beverages, etc. The decision variables in a blending problem are usually either the quantities or the proportions of the constituents. Problem: Three liquid mixtures are to be designed as provided in the table below to contain a special chemical called "Optim" Availability [gallons] 800 Cost ($/gallon 20 Optim [%] 45 35 65 30 Material 1000 700 1500 15 30 25 Minimum Optim (%)Optim [%] 25 30 40 Required Amount Igallons] 600 1200 Maximum Selling Price ($/gallon) 70 Mixture 45 50 65 105 140 Y 900 Formulate an LP model to determine the production plan that maximizes profit. 1234
Blending Problems: This is another large class of problems in which linear programming is applied heavily. Blending is concemed with mixing different materials called the constituents of the mixture (these may be chemicals, gasoline, fuels, solids, colors, foods, etc.) so that the mixture conforms to specifications on several properties or characteristics. To model a blending problem as an LP, the linearity assumptions must hold. This implies that the value for a characteristic for the constituents in the mixtures are the weighted average of the values of that characteristic for the constituents in the mixture; the weights being the proportions of the constituents. As an example, consider a mixture consisting of 4 barrels of fuel 1 and 6 barrels of fuel 2, and suppose the characteristic of interest is the octane rating (Oc.R). If linearity assumptions hold, the Oc.R of the mixture must be equal to (4 times the Oc.R of fuel 1 + 6 times the Oc.R of fuel 2)/(4 +6). These linearity assumptions hold to a reasonable degree of precision for many important characteristics of blends of gasolines, of crude oils, of paints, of foods, etc. That's why linear programming is used extensively in optimizing gasoline blending, in the manufacture of paints, cattle feeds, beverages, etc. The decision variables in a blending problem are usually either the quantities or the proportions of the constituents. Problem: Three liquid mixtures are to be designed as provided in the table below to contain a special chemical called "Optim" Availability [gallons] 800 Cost ($/gallon 20 Optim [%] 45 35 65 30 Material 1000 700 1500 15 30 25 Minimum Optim (%)Optim [%] 25 30 40 Required Amount Igallons] 600 1200 Maximum Selling Price ($/gallon) 70 Mixture 45 50 65 105 140 Y 900 Formulate an LP model to determine the production plan that maximizes profit. 1234
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![Blending Problems: This is another large class of problems in which linear programming is applied heavily. Blending is concerned with mixing different
materials called the constituents of the mixture (these may be chemicals, gasoline, fuels, solids, colors, foods, etc.) so that the mixture conforms to
specifications on several properties or characteristics. To model a blending problem as an LP, the linearity assumptions must hold. This implies that the
value for a characteristic for the constituents in the mixtures are the weighted average of the values of that characteristic for the constituents in the mixture;
the weights being the proportions of the constituents. As an example, consider a mixture consisting of 4 barrels of fuel 1 and 6 barrels of fuel 2, and suppose
the characteristic of interest is the octane rating (Oc.R). If linearity assumptions hold, the Oc.R of the mixture must be equal to (4 times the Oc.R of fuel 1
+ 6 times the Oc.R of fuel 2)/(4 +6). These linearity assumptions hold to a reasonable degree of precision for many important characteristics of blends of
gasolines, of crude oils, of paints, of foods, etc. That's why linear programming is used extensively in optimizing gasoline blending, in the manufacture of
paints, cattle feeds, beverages, etc. The decision variables in a blending problem are usually either the quantities or the proportions of the constituents.
Problem:
Three liquid mixtures are to be designed as provided in the table below to contain a special chemical called "Optim"
Optim [%]
45
35
65
30
Availability [gallons]
800
1000
Cost [$/gallon]
20
15
30
25
Material
700
1500
Minimum
Optim [%] Optim [%]
25
30
40
Required Amount
Igallons]
600
1200
Maximum
Selling Price ($/gallon)
70
Mixture
45
50
65
105
140
Y
900
Formulate an LP model to determine the production plan that maximizes profit.
1234](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F388a9fab-648d-450d-a1a5-3932044f3fea%2F314a6a1d-6655-463b-8532-501b72fa5aaa%2Fdvw1zo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Blending Problems: This is another large class of problems in which linear programming is applied heavily. Blending is concerned with mixing different
materials called the constituents of the mixture (these may be chemicals, gasoline, fuels, solids, colors, foods, etc.) so that the mixture conforms to
specifications on several properties or characteristics. To model a blending problem as an LP, the linearity assumptions must hold. This implies that the
value for a characteristic for the constituents in the mixtures are the weighted average of the values of that characteristic for the constituents in the mixture;
the weights being the proportions of the constituents. As an example, consider a mixture consisting of 4 barrels of fuel 1 and 6 barrels of fuel 2, and suppose
the characteristic of interest is the octane rating (Oc.R). If linearity assumptions hold, the Oc.R of the mixture must be equal to (4 times the Oc.R of fuel 1
+ 6 times the Oc.R of fuel 2)/(4 +6). These linearity assumptions hold to a reasonable degree of precision for many important characteristics of blends of
gasolines, of crude oils, of paints, of foods, etc. That's why linear programming is used extensively in optimizing gasoline blending, in the manufacture of
paints, cattle feeds, beverages, etc. The decision variables in a blending problem are usually either the quantities or the proportions of the constituents.
Problem:
Three liquid mixtures are to be designed as provided in the table below to contain a special chemical called "Optim"
Optim [%]
45
35
65
30
Availability [gallons]
800
1000
Cost [$/gallon]
20
15
30
25
Material
700
1500
Minimum
Optim [%] Optim [%]
25
30
40
Required Amount
Igallons]
600
1200
Maximum
Selling Price ($/gallon)
70
Mixture
45
50
65
105
140
Y
900
Formulate an LP model to determine the production plan that maximizes profit.
1234
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