MATH451-Unit 2 Intellipath - Blending

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Jan 9, 2024

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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

As can be seen from the results, the patient needs to mix only 60.5% of a glass of whole milk and 1.579 of a glass of skim milk. 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, 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. Subject to the following constraints: 1. 10W + 7T + 9S ≥ 10 grams of protein 2. 0.30W + 0.45T + 0.50S ≥ 0.60 3. 100W + 120T + 120S ≥ 250 4. 4W + 2T + 1S ≤ 4 grams of fat 5. W, T, and S ≥ 0 (non-negativity) Using the Excel function =sumproduct(), you can compute each of the expressions and set up a single constraint using cell ranges to enter items 1–3. Item 5 can be set by a check box in the Solver dialog or by using another range constraint. Item 4 looks different from the rest and requires a separate constraint line. It is not a big, but it is ugly. How can you include item 4 as well with items 1 through 3? A: Negate the fat values, and require that the sumproduct for fat ( ) be greater than or equal to . 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, 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%

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