For each problem, your responsibility is to formulate the decision environment a
ID: 353510 • Letter: F
Question
For each problem, your responsibility is to formulate the decision environment as a mathematical model. You are not required to solve the problems! Simply write up the appropriate models. This, you should recall, will include complete descriptions of the decision variables, a statement of the objective function, and all relevant constraints. (Note that constraints are not to contain “IF” statements or multiplication or division of variables.) Your models should be written in a form that I can interpret, not one that is meant for Excel.
Mane Event Stables is a horse boarding facility. Each month they are faced with varying needs in terms of the nutritional requirements of the horses they are boarding. They also face variable availability of mixers that they use do make the feed for the horses. Below are the data on four different mixers that they will use to make the feed for the horses they are boarding this month. (The fourth mixer is actually “pure” vitamin C.)
ProteinFatVitamin C$/poundLbs. Available
Mixer 1 29% 8% 2% .81 2000
2 12% 24% 11% . 46 800
3 20% 12% 4% .66 600
4 0% 0% 90% 13.15 30
Mane Event has determined that they will need to mix three distinct feeds to meet the nutritional needs of the horses they are boarding this month. Feed A will need a minimum of 24% protein in the blend, a minimum of 14% fat, and minimum of 9% vitamin C. They will need 1400 pounds of Feed A. Feed B will need a minimum of 20% protein, a maximum of 9% fat, and a minimum of 12% vitamin C. They will need 600 pounds of Feed B. Feed C will need a minimum of 15% protein, a minimum of 13% fat, and a minimum of 6% vitamin C. They expect to need 1000 pounds of Feed C.
You are to construct a linear program to determine the most cost efficient blending of the four input mixers, keeping in mind the nutritional requirements and demand for each of the three feeds.
Explanation / Answer
Let Xij be the amount (lb.) of Mixer-i present in Feed-j; i=1,2,3,4 and j=A,B,C
Objective function: Minimize Z = 0.81*(X1A + X1B + X1C) + 0.46*(X2A + X2B + X2C) + 0.66*(X3A + X3B + X3C) + 13.15*(X4A + X4B + X4C)
Subject to,
Constraints for availability of the Mixers
X1A + X1B + X1C <= 2000 (Mixer 1)
X2A + X2B + X2C <= 800 (Mixer 2)
X3A + X3B + X3C <= 600 (Mixer 3)
X4A + X4B + X4C <= 30 (Mixer 4)
Constraints for demand for the Feeds
X1A + X2A + X3A + X4A = 1400 (Feed A)
X1B + X2B + X3B + X4B = 600 (Feed B)
X1C + X2C + X3C + X4C = 1000 (Feed C)
Constraints for Protein in Feeds
0.29X1A + 0.12X2A + 0.2X3A + 0X4A >= 0.24(X1A + X2A + X3A + X4A)
or, 0.05X1A - 0.12X2A - 0.04X3A - 0.24X4A >= 0 (Feed A)
0.29X1B + 0.12X2B + 0.2X3B + 0X4B >= 0.20(X1B + X2B + X3B + X4B)
or, 0.09X1B - 0.08X2B + 0X3B - 0.2X4B >= 0 (Feed B)
0.29X1C + 0.12X2C + 0.2X3C + 0X4C >= 0.15(X1C + X2C + X3C + X4C)
or, 0.14X1C - 0.03X2C + 0.05X3C - 0.15X4C >= 0 (Feed C)
Constraints for Fat in Feeds
0.08X1A + 0.24X2A + 0.12X3A + 0X4A >= 0.14(X1A + X2A + X3A + X4A)
or, -0.06X1A + 0.10X2A - 0.02X3A - 0.14X4A >= 0 (Feed A)
0.08X1B + 0.24X2B + 0.12X3B + 0X4B <= 0.09(X1B + X2B + X3B + X4B)
or, -0.01X1B + 0.15X2B + 0.03X3B - 0.09X4B <= 0 (Feed B)
0.08X1C + 0.24X2C + 0.12X3C + 0X4C >= 0.13(X1C + X2C + X3C + X4C)
or, -0.05X1C + 0.11X2C - 0.01X3C - 0.13X4C >= 0 (Feed C)
Constraints for Vitamin C in Feeds
0.02X1A + 0.11X2A + 0.04X3A + 0.9X4A >= 0.09(X1A + X2A + X3A + X4A)
or, -0.07X1A + 0.02X2A - 0.05X3A + 0.81X4A >= 0 (Feed A)
0.02X1B + 0.11X2B + 0.04X3B + 0.9X4B >= 0.12(X1B + X2B + X3B + X4B)
or, -0.1X1B - 0.01X2B - 0.08X3B + 0.78X4B >= 0 (Feed B)
0.02X1C + 0.11X2C + 0.04X3C + 0.9X4C >= 0.06(X1C + X2C + X3C + X4C)
or, -0.04X1C + 0.05X2C - 0.02X3C + 0.84X4C >= 0 (Feed C)
Xij >= 0; i=1,2,3,4 and j=A,B,C
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