Linear Programming

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1. Complete the following two problems.
Problem 1
Activity Mean Duration (Days) Std. Dev. (Days)
A 11 0.9
B 13 1.1
C 7 0.2
D 9 0.8
E 6 1
F 7 1.2
G 10 0.7
H 9 0.6
I 8 0.8
a) Calculate the project completion time.
b) Indicate the critical path activities.
c) What is the probability of completing this project between 38 and 40 days? d) What are the slack values for activities C and F? Interpret the meaning of their slack values?
Problem 2
A registered nurse is trying to develop a diet plan for patients. The required nutritional elements are the total daily requirements of each nutritional element are as indicated in table 2:
Table 2
(Required Nutritional Element Total and Daily Requirements)
Calories Not more than 2,700 calories
Carbohydrates Not less than 300 grams
Protein Not less than 250 grams
Vitamins Not less than 60 units
The nurse has four basic types to use when planning the menus. The units of nutritional elements per unit of food type are shown in the table below. Note than the cost associated with a unit of ingredient also appears at the bottom of table 3. Table 3

(Required Nutritional Element and Units of Nutritional Elements Per Unit of Food Type) Element Milk Chicken Bread Vegetables
Calories 160 210 120 150
Carbohydrates 110 130 110 120
Protein 90 190 90 130
Vitamins 50 50 75 70
Cost per unit $0.42 $0.68 $0.32 $0.17
Moreover, due to dietary restrictions, the following aspects should also be considered when the developing the diet plan:
i) The chicken food type should contribute at most 25% of the total calories intake that will result from the diet plan.
ii) The vegetable food type should provide at least 30% of the minimum daily requirements for vitamins.
Provide a linear programming formulation for the above case. (No need to solve the problem.)
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