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Answer the Following Questions According to their Specific Requirements.
1. Giapetto’s Woodcarving, Inc., manufactures two types of wooden toys: soldiers and trains. A soldier sells for $27 and uses $10 worth of raw materials. Each soldier that is manufactured increases Giapetto’s variable labor and overhead costs by $14. A train sells for $21 and uses
$ 9 worth of raw materials. Each train built increases Giapetto’s variable labor and overhead costs by $10. The manufacturer of wooden soldiers and trains requires two types of skilled labor: carpentry and finishing. A soldier requires 2 hours of finishing labor and 1 hour of carpentry labor. A train requires 1 hours of finishing labor and 1 hour of carpentry labor. Each week, Giapetto can obtain all the needed raw material but only 100 finishing hours and 80 carpentry hours. Demand for trains is unlimited, but at most 40 soldiers are bought each week. If Giapetto wants to maximize weekly profit, answer the questions bellow.
- Formulate the linear programing problem. (2 pts)
- Solve the problem using graphic method. (3 pts)
2. My diet requires that all the food that I eat come from one of the four basic food groups (chocolate cake, ice cream, soda, and cheesecake). At present, the following four foods are available for consumption: brownies, chocolate ice cream, cola, and pineapple cheese-cake. Each brownie costs $50, each scoop of chocolate ice cream costs $20, each bottle of coca cola costs $30, and each piece of pineapple cheese-cake $80. Each day, I must in-gest at least 500 calories, 6 oz of chocolate, 10 oz of sugar, and 8 oz of The nutritional content per unit of each food is shown in the following table.
Type of food | Calories | Chocolate
(Ounce) |
Sugar
(Ounce) |
Fat
(Ounce) |
Brownies | 400 | 3 | 2 | 2 |
Chocolate ice
cream (1 scoop) |
200 | 2 | 2 | 4 |
Cola (1 bottle) | 150 | 0 | 4 | 1 |
pineapple
cheese-cake (1 piece) |
500 |
0 |
4 |
5 |
- Formulate the linear Programing problem. (2 pts)
- Solve the problem using Simplex method. (5 pts)
- Find the dual form of this problem. (2 pts)
- Find the solution of the dual from the optimal tableau of the solution of the primal problem. (2 pts)
- Find the shadow prices and interpret them. (2 pts)
- For which values of the objective function coefficient of the non-basic variables would the optimal solution mix remain unchanged? (2 pts)
3. Powerco has three electric power plants that supply the needs of four cities. Each power plant can supply the following numbers of kilowatt-hours (kWh) of electricity: plant one 35 million; plant two 50 million; plant three 40 million as indicated in the table below. The peak power demands in these cities, which occur at the same time (2 pm), are as follows (in Kwh): city one 45million; city two 20 million; city three 30 million; city four 30 The costs of sending 1 million KWh of electricity from plant to city depend on the distance the electric city must travel.
From |
To | Supply (million kwh) | |||
City 1 | City 2 | City 3 | City 4 | ||
Plant 1 | 8 | 6 | 10 | 9 | 35 |
Plant 2 | 9 | 12 | 13 | 7 | 50 |
Plant 3 | 14 | 9 | 16 | 5 | 40 |
Demand (million kwh) | 45 | 20 | 30 | 30 |
- Find an initial feasible solution to the above transportation problem using Vogel’s Approximation method. (5 pts)
- Test the initial feasible solution you obtained using MODI method and find the optimal solution. (5 pts)
4. Five employees are available to perform four jobs. The time it takes each person to perform each job is given in the table below.
Person |
Time (hours) | |||
Job 1 | Job 2 | Job 3 | Job 4 | |
1 | 22 | 18 | 30 | 18 |
2 | 18 | - | 27 | 22 |
3 | 26 | 20 | 28 | 28 |
4 | 16 | 22 | - | 14 |
5 | 21 | - | 25 | 28 |
- Using Hungarian method, determine the assignment of employees to jobs that minimizes the total time required to perform the four jobs. (5 pts)
5. A toy manufacturer is considering a project of manufacturing a dancing doll with three different movement designs. The doll will be sold at an average of birr 10. The first movement design using ‘gears and levels’ will provide the lowest tooling and set up cost of birr 100,000 and birr 5 per unit of variable A second design with spring action will have a fixed cost of birr 160,000 and variable cost of birr 4 per unit. Yet another design with weights and pulleys will have a fixed cost of birr 300,000 and variable cost birr 3 per unit. The demand events that can occur for the doll and the probability of their occurrence is given below:
Demand (Units) | Probability | |
Light demand | 25, 000 | 0.1 |
Moderate Demand | 100, 000 | 0.7 |
Heavy Demand | 150, 000 | 0.2 |
- Construct a payoff table for the above project. (2 pts)
- Which is the optimum design using; EMV, EOL? (5 pts)
- How much can the decision maker afford to pay in order to obtain perfect information about the demand? (3 pts)
6. The following matrix gives the profit payoff of different strategies (alternatives) A, B, and C against states of nature W, X, Y, and Z. Identify the decision taken under the following approaches:
W | X | Y | Z | |
A | 4, 000 | -100 | 6, 000 | 18, 000 |
B | 30, 000 | 5, 000 | 400 | 0 |
C | 20, 000 | 15, 000 | -2, 000 | 1, 000 |
- Pessimistic, (3 pts)
- Optimistic, (3 pts)
- Equal probability, (3 pts)
- Regret, (3 pts)
- Hurwicz The decision maker’s criterion of realism (α) being 0.7 (3 pts)