Question 1
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Activity
Mean duration
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
Table 1
Complete the following:
1. Calculate the project completion time.
2. Indicate the critical path activities.
3. What is the probability of completing this project between 38 and 40 days?
4. What are the slack values for activities C and F? Interpret the meaning of their slack values?
Question 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 as indicated in Table 2:
Required nutritional element total and daily requirements
Calories
Not more than 2,700 calories
Carbohydrates
Not more than 300 grams
Protein
Not less than 250 grams
Vitamins
Not less than 60 units
Table 2
The nurse has four basic types to use when planning the menus. The units of nutritional element per unit of food type are shown in Table 3 below. Note that the cost associated with a unit of ingredient also appears at the bottom of 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
Table 3
Moreover, due to dietary restrictions, the following aspects should also be considered when developing the diet plan:
1. The chicken food type should contribute at most 25% of the total caloric intake that will result from the diet plan.
2. The vegetable food type should provide at least 30% of the minimum daily requirements for vitamins.
Complete the following:
Provide a linear programming formulation for the above case. (You do not need to solve the problem.)
Project
A firm uses three machines in the manufacturing of three products:
· Each unit of product 1 requires three hours on machine 1, two hours on machine 2 and one hour on machine 3.
· Each unit of product 2 requires four hours on machine 1, one hour on machine 2 and three hours on machine 3.
· Each unit of product 3 requires two hours on machine 1, two hours on machine 2 and two hours on machine 3.
The contribution margin of the three products is £30, £40 and £35 per unit, respectively.
Available for scheduling are:
· 90 hours of machine 1 time;
· 54 hours of machine 2 time; and
· 93 hours of machine 3 time.
The linear programming formulation of this problem is as follows:
Maximise Z = 30X1 + 40X2 + 35X3
3X1 + 4X2 + 2X3 <= 90 2X1 + 1X2 + 2X3 <= 54 X1 + 3X2 + 2X3 <= 93 With X1, X2, X3 >= 0
Answer the following questions by looking at the solution. Submit your answers by the end of Day 7 (Wednesday).
1. What is the optimal production schedule for this firm? What is the profit contribution of each of these products?
2. What is the marginal value of an additional hour of time on machine 1? Over what range of time is this marginal value valid?
3. What is the opportunity cost associated with product 1? What interpretation should be given to this opportunity cost?
4. How many hours are used for machine 3 with the optimal solution?
5. How much can the contribution margin for product 2 change before the current optimal solution is no longer optimal?
