MATH451-Unit 2 Intellipath - Blending

MATH451 – Unit 2 Intellipath Blending Blending is another application of linear programming model. In this case, you can mix several items to produce sometime product. You will consider a diet problem in this section. You will see how you can mix several different types of milk to create a special glass of milk to meet a particular physician prescription. Example You drink and eat all kinds of drinks and food or junk food as you desire. However, there are many people who cannot eat or drink anything they wish and the amount they desire due to heath issues. Consider a case a person has been prescribed by a physician. A person should mix three types of milk (whole, 2%, and skim) to obtain the required daily dietary in the following ways:  A full glass of whole milk containing 4 grams of fat, 10 grams of protein, 30% calcium, and 100 IU (International Unit) of vitamin D.  A full glass of 2% milk containing 2 grams of fat, 7 grams of protein, 45% calcium, and 120 IU of vitamin D.  A full glass of skim milk containing 1 gram of fat, 9 grams of protein, 50% calcium, and 120 IU of vitamin D. The patient needs to drink milk every day to obtain at least 10 grams of protein, 60% of calcium, 250 IU, and at most 4 grams of fat. The cost of whole, 2%, and skim milk are $0.90, $1.10, and $1.10 per glass, respectively. What percentage of a glass does a patient need to mix each type of milk to achieve his or her requirement? Setting Up the Linear Programming Model The objective function and the constraints are as follows: Minimizing the cost = $0.90W + $1.10T + $1.10S subject to 10W + 7T + 9S ≥ 10 grams of protein

0.30W + 0.45T + 0.50S ≥ 0.60 100W + 120T + 120S ≥ 250 4W + 2T + 1S ≤ 4 grams of fat W, T, and S ≥ 0 (nonnegativity) Figure 1 shows the Excel’s Solver program used to solve the linear programming model containing all of the required Excel functions. Figure 1: Excel’s Solver Program Used to Solve the Linear Programming Model Figure 2 shows the same program containing the numerical values of the decision- making variables and the amount of resources consumed. Figure 2: Excel’s Solver Program Used to Solve the Linear Programming Model Results

The patient needs to drink milk every day to obtain at least 10 grams of protein, 60% of calcium, 250 IU of vitamin D, and at most 4 grams of fat. The individual costs of whole, 2%, and skim milk are $0.90, $1.10, and $1.10 per glass, respectively. Suppose that, instead of 250 IU of vitamin D, you need 400 IU. How many glasses of skim milk are required? Round to the nearest hundredth. A: 3.16 Q: A patient must mix three types of milk (i.e., whole, 2%, and skim) to obtain the required daily dietary allowance, in the following ways:  A full glass of whole milk containing 4 grams of fat, 10 grams of protein, 30% calcium, and 100 IU (International Unit) of vitamin D  A full glass of 2% milk containing 2 grams of fat, 7 grams of protein, 45% calcium, and 120 IU of vitamin D  A full glass of skim milk containing 1 gram of fat, 9 grams of protein, 50% calcium, and 120 IU of vitamin D The patient needs to drink milk every day to obtain at least 10 grams of protein, 60% of calcium, 250 IU of vitamin D, and at most 4 grams of fat. The individual costs of whole, 2%, and skim milk are $0.90, $1.10, and $1.10 per glass, respectively. Suppose that the patient's doctor required 500 IU of vitamin D. How many glasses of skim milk are needed? Round to the nearest hundredth. A: The problem is not solvable Q: A full glass of whole milk contains 4 grams of fat, 10 grams of protein, 30% calcium, and 100 IU (International Unit) of vitamin D. A full glass of 2% milk contains 2 grams of fat, 7 grams of protein, 45% calcium, and 120 IU of vitamin D. Finally, a full glass of skim milk contains 1 gram of fat, 9 grams of protein, 50% calcium, and 120 IU of vitamin D. The patient needs to drink milk every day to obtain at least 15 grams of protein, 70% of calcium, 250 IU of vitamin D, and at most 5 grams of fat. The cost of whole, 2%, and skim milk are $0.85, $1.10, and $1.15 per glass, respectively. What is the objective function? A: Minimizing the cost = $0.85W + $1.10T + $1.15S Q: A full glass of whole milk contains 4 grams of fat, 10 grams of protein, 30% calcium, and 100 IU (International Unit) of vitamin D. A full glass of 2% milk contains 2 grams of fat, 7 grams of protein, 45% calcium, and 120 IU of vitamin D. Finally, a full glass of skim milk contains 1 gram of fat, 9 grams of protein, 50% calcium, and 120 IU of vitamin D. The patient needs to drink milk every day to obtain at least 15 grams of protein, 70%

of calcium, 250 IU of vitamin D, and at most 5 grams of fat. The cost of whole, 2%, and skim milk are $0.85, $1.10, and $1.15 per glass, respectively. What is the value for the objective function after running the problem through Excel’s Solver? A: $2.27 Q: A full glass of whole milk contains 4 grams of fat, 10 grams of protein, 30% calcium, and 100 IU (international Unit) of vitamin D. A full glass of 2% milk contains 2 grams of fat, 7 grams of protein, 45% calcium, and 120 IU of vitamin D. Finally, a full glass of skim milk contains 1 gram of fat, 9 grams of protein, 50% calcium, and 120 IU of vitamin D. The patient needs to drink milk every day to obtain at least 15 grams of protein, 70% of calcium, 250 IU of vitamin D, and at most 5 grams of fat. The cost of whole, 2%, and skim milk are $0.85, $1.10, and $1.15 per glass, respectively. What would the value of the objective function be if the price of whole milk decreased to $0.75 per glass after running the problem through Excel’s Solver? A: $2.20 Q: A full glass of whole milk contains 4 grams of fat, 10 grams of protein, 30% calcium, and 100 IU (International Unit) of vitamin D. A full glass of 2% milk contains 2 grams of fat, 7 grams of protein, 45% calcium, and 120 IU of vitamin D. Finally, a full glass of skim milk contains 1 gram of fat, 9 grams of protein, 50% calcium, and 120 IU of vitamin D. The patient needs to drink milk every day to obtain at least 15 grams of protein, 70% of calcium, 250 IU of vitamin D, and at most 5 grams of fat. The cost of whole, 2%, and skim milk are $0.85, $1.10, and $1.15 per glass, respectively. What would the value of the objective function be if the price of whole milk increased to $1.00 per glass after running the problem through Excel’s Solver? A: $2.30 Q: A full glass of whole milk contains 4 grams of fat, 10 grams of protein, 30% calcium, and 100 IU (International Unit) of vitamin D. A full glass of 2% milk contains 2 grams of fat, 7 grams of protein, 45% calcium, and 120 IU of vitamin D. Finally, a full glass of skim milk contains 1 gram of fat, 9 grams of protein, 50% calcium, and 120 IU of vitamin D. The patient needs to drink milk every day to obtain at least 15 grams of protein, 70% of calcium, 250 IU of vitamin D, and at most 5 grams of fat. The cost of whole, 2%, and skim milk are $0.85, $1.10, and $1.15 per glass, respectively. What is the fat constraint?

D. The patient needs to drink milk every day to obtain at least 15 grams of protein, 70% of calcium, 250 IU of vitamin D, and at most 5 grams of fat. The cost of whole, 2%, and skim milk are $0.85, $1.10, and $1.15 per glass, respectively. To what amount the price of a glass of skim milk should be changed to force the patient to buy skim milk after running the problem through Excel’s Solver? A: $1.12 Q: A full glass of whole milk contains 4 grams of fat, 10 grams of protein, 30% calcium, and 100 IU (International Unit) of vitamin D. A full glass of 2% milk contains 2 grams of fat, 7 grams of protein, 45% calcium, and 120 IU of vitamin D. Finally, a full glass of skim milk contains 1 gram of fat, 9 grams of protein, 50% calcium, and 120 IU of vitamin D. The patient needs to drink milk every day to obtain at least 15 grams of protein, 70% of calcium, 250 IU of vitamin D, and at most 5 grams of fat. The cost of whole, 2%, and skim milk are $0.85, $1.10, and $1.15 per glass, respectively. What type of milk does the patient have to buy if all brands of milk had the same price per glass after running the problem through Excel’s Solver? A: Skim Q: Rather than work with real products and real nutrients, you can work with abstract products and components. Determine the minimum cost way to mix products I, II, III, and IV to achieve the required amounts. Component Cost Product a b c d e I 6.52 1.44 6.32 9.05 7.38 56 II 7.48 8.79 4.92 3.73 7.96 24 III 0.24 8.64 5.44 8.98 8.73 35 IV 1.58 6.71 0.22 3.24 1.03 66 Required 57.76 89.523 88.262 50.028 71.98 What is the cost of the lowest cost solution? Round to the nearest thousandth. A: None of the offered choices is correct Q: Rather than work with real products and real nutrients, you can work with abstract products and components. Determine the minimum cost way to mix products I, II, III, and IV to achieve the required amounts. Component Cost Product a b c d I 9.73 1.16 0.52 0.836 64.85

II 1.23 4.35 1.216 1.648 72.48 III 0.948 0.44 9.68 0.892 76.46 IV 1.438 1.258 1.544 5.43 72.78 Required 88.057 42.201 63.641 79.225 What is the cost of the minimum cost solution? Round to the nearest thousandth. A: 1,866.099 Q: Rather than work with real products and real nutrients, you can work with abstract products and components. Determine the minimum cost way to mix products I, II, III, and IV to achieve the required amounts. Component Cost Product a b c d e I 6.52 1.44 6.32 9.05 7.38 56 II 7.48 8.79 4.92 3.73 7.96 24 III 0.24 8.64 5.44 8.98 8.73 35 IV 1.58 6.71 0.22 3.24 1.03 66 Required 57.76 89.523 88.262 50.028 71.98 How much of product I is used in the lowest cost solution? Round to the nearest thousandth. A: 8.530 Q: Rather than work with real products and real nutrients, you can work with abstract products and components. Determine the minimum cost way to mix products I, II, and III to achieve the required amounts. Component Cost Product a b c d I 0.064 0.487 0.845 0.57 26 II 0.196 0.344 0.015 0.092 74 III 0.527 0.499 0.02 0.091 28 Required 2.425 3.323 1.754 2.078 How much of product I will be used? Round to the nearest thousandth. A: 2.969 Q: Rather than work with real products and real nutrients, you can work with abstract products and components. Determine the minimum cost way to mix products I, II, III, and IV to achieve the required amounts. Component Cost