Section 4: Linear Progamming applications (1)
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The document describes a linear programming model for a snack mix called 'zak' that combines dried fruit, nuts, and cereal, outlining constraints for calories, fat, and ingredient proportions. It provides a mathematical formulation to optimize the mixture while adhering to specified dietary limits and ingredient requirements. Additionally, it presents a media selection model for an ad campaign with constraints on budget and ad exposure.
Section 4: Linear Progamming applications (1)
- 1. By: Zakaria Elsayed Hasaneen Teaching Assistant at Faculty of Commerce Accounting Department (English Section) Kafrelsheikh University OR Section (4) Linear Progamming applications (1) Grade 4 –Term 2 Zakaria Hasaneen Zakaria Hasaneen Zakaria Hasaneen
- 2. ZIKO Foods manufactures a snack mix called ZAK by blending three ingredients: a dried fruit mixture, a nut mixture, and a cereal mixture. Information about the three ingredients (per ounce) is shown below: Blending Problem
- 3. The company needs to develop a linear programming model whose solution would tell them how many ounces of each mix to put into the ZAK blend. ZAK is packaged in boxes that will hold between three and four cups. The blend should contain no more than 1100 calories and at most 30 grams of fat. Dried fruit must be at least 30% of the volume of the mixture, and nuts must be no more than 25% of the weight of the mixture. Let:- x = amount (ounces) of dried fruit to put into ZAK. y= amount(ounces) of nut mix to put into ZAK. z= amount (ounces) of cereal mix to put into ZAK.
- 4. (1) What is the objective function? A) max0.4x+0.5y+0.2z B) max0.25x+0.5y+z C) min0.4x+0.5y+0.2z D) min0.25x+0.5y+z E) None of the above ANS: Given Information: We have to develop a model that meets the restrictions given to minimize the cost of the blend. The Objective function of Linear Programming Problem is given by: MinZ=0.4x+0.5y+0.2z Hence, option (C) is the correct answer.
- 5. (2) What is the constraint for calories? A)150x+400y+50z≥1100 B)150x+400y+50z≤1100 C) x+y+z≥1100 D) x+y+z≤1100 E) None of the above ANS: The blend should contain no more than 1100 calories that is the maximum amount of calories blend can contain is 1100 The constraint for calories is: 150x+400y+50z≤1100 Hence, option (B) is the correct answer.
- 6. (3) What is the constraint for fat? A) x+y+z≥30 B) x+y+z≤30 C)15y+3z≥30 D)15y+3z≤30 E) None of the above ANS: The blend should contain at most 30 grams of fat. The constraint for fat is: 0x+15y+3z≤30 15y+3z≤30 Hence, option (D) is the correct answer.
- 7. (4) What is the volume constraint for dried fruit? A) −0.7x+0.3y−0.7z≥0 B) 0.3x−0.7y+0.3z≥0 C) 0.21x−0.15y−0.3z≤0 D) 0.21x−0.15y−0.3z≥0 E) None of the above ANS: The Dried fruit must be at least 30% of the volume of the mixture. The volume constraint for dried fruit is: x≥0.3(0.3x+0.5y+1z) x≥0.09x+0.15y+0.3z 0.91x−0.15y−0.3z≥0 Hence, option (E) is the correct answer.
- 8. (5) What is the weight constraint for nut mix? A) 0.25x−y−z≤0 B) 0.75x−0.25y−0.25z≤0 C) x−0.25y−0.25z≤0 D)−0.25x+0.75y−0.25z≤0 E) None of the above ANS: Nuts must be no more than 25% of the weight of the mixture. y≤0.25(0.3x+0.5y+1z) 0.075x−0.8750y+0.25z≥0 Hence, option (E) is the correct answer.
- 9. An ad campaign for a new snack chip will be conducted in a limited geographical area and can use TV time, radio time, and newspaper ads. Information about each medium is shown below. If the number of TV ads cannot exceed the number of radio ads by more than 4, and if the advertising budget is $10000. Media selection Required: Develop the model that will maximize the number reached and achieve an exposure quality if at least 1000.
- 10. ANS: Let T = the number of TV ads Let R = the number of radio ads Let N = the number of newspaper ads Max 10000T + 3000R + 5000N s.t. 500T + 200R + 400N ≤ 10000 30T + 40R + 25N ≥ 1000 T − R ≤ 4 T, R, N ≥ 0