Reference no: EM132538749
Worksheet
Question 1. An individual investor has Birr 70,000 to divide among several investments. The alternative investments are municipal bonds with an 8.5% return, certificates of deposits with a 10% return, Treasury bill with a 6.5% return, and income bonds with a 13% return. The amount of time until maturity is the same for each alternative. However, each investment alternative has a different perceived risk to the investor; thus it is advisable to diversify. The investor wants to know how much to invest in each alternative in order to maximize the return. The following guidelines have been established for diversifying the investment and lessening the risk perceived by the investor.
1. No more than 20% of the total investment should be in an income bonds.
2. The amount invested in certificates of deposit should not exceed the amount invested in other three alternatives.
3. At least 30% of the investment should be in treasury bills and certificates of deposits.
4. The ratio of the amount invested in municipal bonds to the amount invested in treasury bills should not exceed one to three.
The investor wants to invest the entire Birr 70,000.
Required:
a. Formulate a LP model for the problem.
Question 2. The manager of a department store in Adama is attempting to decide on the types and amountsof advertising the store should use. He has invited representatives from the local radio station,television station, and newspaper to make presentations in which they describe their audiences.
a. The television station representative indicates that a TV commercial, which costs Birr 15,000, would reach 25,000 potential customers. The breakdown of the audience is as follows.
Male Female
Old 5000 5000
Young 5000 10000
b. The news paper representative claims to be able to provide an audience of 10,000 potential customers at a cost of Birr 4000 per ad. The breakdown of the audience is as follows.
Male Female
Old 4000 3000
Young 2000 1000
c. The radio station representative says that the audience for one of the station's commercials,which costs Birr 6000, is 15,000 customers. The breakdown of the audience is as follows.
Male Female
Old 1500 1500
Young 4500 7500
The store has the following advertising policy:
a. Use at least twice as many radio commercials as news paper ads.
b. Reach at least 100,000 customers
c. Reach at least twice as many young people as old people
d. Make sure that at least 30% of the audience is women.
Available space limits the number newspaper ads to 7. The store wants to know the optimal number of each type of advertising to purchase to minimize total cost.
Required:
a. Formulate appropriate linear programming model.
Question 3. A sale manager has to assign salesmen to four territories. He has four candidates of varying experience and capabilities and assesses the possible profit for each salesman in each territory as given below. Find the assignment which maximizes profit?
|
Salesmen
|
Territories
|
|
A
|
B
|
C
|
D
|
|
1
|
35
|
27
|
28
|
37
|
|
2
|
28
|
34
|
29
|
40
|
|
3
|
35
|
24
|
32
|
33
|
|
4
|
24
|
32
|
25
|
82
|
Question 4. Peniel House and Furniture manufacturer produces two products: Beds and Chairs. Each unit of Bed requires 3 hrs in molding unit, 4hrs in painting unit, and 1 hr in finishing. On the other hand, each unit of Chair requires 3 hrs in molding unit, 2 hrs in the paint shop and 2 hours in finishing. Each week, there are 210 hrs available in molding, 200hrs in painting, and 120 hrs in finishing unit. The demand for Beds cannot exceed 40 units per week. Each unit of Bed contributes Birr 20 to profit, while each unit of chair contributes Birr 30. Determine the number of units of each product per week to maximize the profit? (Use Graphic Method to solve the given LP problem).
Question 5. Construct the CPM network described by the following set of activities, compute the length of each path in the network, and indicate the critical path.
|
Activity
|
Time (months)
|
|
1 - 2
1 - 3
2 - 4
2 - 5
3 - 5
4 - 5
4 - 6
5 - 6
3 - 6
|
4
7
8
3
9
5
2
6
5
|
Question 6. A firm plans to purchase at least 200 quintals of scrap containing high metal quality metal X and low quality metal Y. It decides that the scrap to be purchased must contain at least 100 quintals of X-metal and not more than 35 quintals of Y-metal. The firm can purchase the scrap from two suppliers ( A and B) in unlimited quantities. The percentage of X and Y metals in terms of weight in the scrap supplied by A and B is given below.
MetalsSupplier ASupplier B
X 25% 75%
Y 10% 20%
The price of A's scrap is Birr 200 per quintal and that of B is Birr 400 per quintal. The firm wants to determine the quantities that it should buy from the two suppliers so that total cost is minimized? (Use graphic approach)
Question 7. Six salesmen are to be allocated to six sales regions so that the cost of allocation of the job will be minimum. Each salesman is cable of doing the job at different costs in each region. The cost (in birr) matrix is given below.
|
Region
|
Salesmen
|
|
I
|
II
|
III
|
IV
|
V
|
VI
|
|
A
|
15
|
35
|
0
|
25
|
10
|
45
|
|
B
|
40
|
5
|
45
|
20
|
15
|
20
|
|
C
|
25
|
60
|
10
|
65
|
25
|
10
|
|
D
|
25
|
20
|
35
|
10
|
25
|
60
|
|
E
|
30
|
70
|
40
|
5
|
40
|
50
|
|
F
|
10
|
25
|
30
|
40
|
50
|
15
|
a. Find the allocation to give minimum cost. What is the minimum?
b. If the figures given in the above table represent the earning of each salesman at each region, then find an allocation so that the earning will be maximum. Also workout this maximum possible earning.
Question 8. A firm makes two products X and Y, and has a total production capacity of 9 tones per day, x and y requiring the same production capacity. The firm has a permanent contract to supply at least two tones of x and at least three tones of y per day to another company. Each tone of x requires 20 machine hours of production time and each tone of y requires 50 machine hours of production time. The daily maximum possible number of machine hours is 360. The entire firm's output can be sold and the cost per tone is 100 and 120 birr respectively and the selling price per tone of x and y is 180 and 240 Birr respectively. It is required to determine the production schedule for maximum profit using graphic approach? What is the maximum profit?
Question 9. The final simplex table for this model is as follows.
|
Cj
|
70
|
80
|
0
|
0
|
0
|
Quantity
|
|
|
X1
|
X2
|
S1
|
S2
|
S3
|
|
|
X1 70
|
70
|
1
|
0
|
2/3
|
-1/3
|
6
|
|
S2 0
|
0
|
0
|
0
|
-1/3
|
-1/3
|
1
|
|
X2 80
|
0
|
0
|
1
|
-1/3
|
2/3
|
7
|
|
Cj
|
70
|
80
|
20
|
0
|
30
|
980
|
|
Cj-Zj
|
0
|
0
|
-20
|
0
|
-30
|
|
a. Formulate the dual of this problem.
b. Define the dual variables and explain what they mean.
c. What do the dual variable equal?
d. Resolve the problem for its dual and see the differences in the solution if any
e. Interpret the Cj-Zj values in the above table
f. Allocate the profit among the resources to show how much each resource contributes to the total profit.
Question 10. Repi Soap Company wishes to make a new detergent from two of its current soaps: Soap A and soap B. The company wants the new detergent to contain at least 20 ounces of cleaning ingredient D. Each pound of soap A costs $0.2 and contains 5 ounces of D, whereas each pound of B costs 0.25 and one ounce of D. Furthermore, for ecological reasons, the amount of soap A must be no more than that of soap B. Finally, the new mixture must weight exactly 12 pounds.How many pounds of each soap should be used in the mixture in order to meet the above requirements at a minimum cost? What is the minimum cost? (Use graphic approach)
Question 11. A transistor radio Co., manufactures models A, B, and C which have profit contribution of Birr 16, Birr 30 and Birr 50 respectively. The weekly minimum production requirements are 20 for model A, 120 for model B, and 60 for model C. Each type of radio requires a certain amount of time for the manufacturing of component parts, assembly and packaging. Specifically a dozen unit of model A requires 3 hrs for manufacturing of component parts, 4b hrs for assembly and 1 hr for packaging. The corresponding figure for a dozen unit of model B is 3.5, 5 and 1.5 and for a dozen unit of C are 5, 8, and 3 hrs. During the forth coming week, the company has availability of 120 hrs of manufacturing, 160 hrs of assembly, and 45 hrs of packaging time.
Required:
a. Formulate linear programming problem?
b. Solve the problem using simplex method?
Question 12. Consider the following transportation problem:
| To: |
|
|
|
|
| |
Store1 |
Store 2 |
Store 3 |
Supply |
| From: |
|
|
|
|
| |
12 |
20 |
15 |
|
| Warehouse A |
50 |
| |
9 |
11 |
4 |
|
| Warehouse B |
15 |
| |
|
| |
20 |
14 |
8 |
|
| Warehouse C |
55 |
| |
|
|
|
|
| Demand |
25 |
50 |
45 |
120 |
a. Develop the initial feasible solution using the northwest corner method. Compute the total cost for this solution.
b. Evaluate the solution using the stepping stone method?Is the solution is optimal? Explain?
c. Evaluate the solution using the MODI method and compare your cell evalutions to those obtained using the stepping stone method?
d. Develop the initial feasible solution using the least cost method. Compute the total cost for this solution? Evaluate your solution by stepping stone method&MODI method?
e. Develop the initial feasible solution using the voge's approximation method. Compute the total cost for this solution.Evaluate your solution by stepping stone method&MODI method?
Question 13. A company has two plants, each of which produces and supplies two products: A and B. The plants can each work up to 16 hrs a day. In plant 1, it takes 3 hrs to prepare and pack 1000 gallons of A and 1 hr to prepare and pack 1 quintal of B. In plant 2, it takes 2 hrs to prepare and pack 1000 gallons of A and 1.5 hrs to prepare and pack a quintal of B. In plant 1, it costs Birr15,000 to prepare and pack 1000 gallons of A and Birr 28,000 to prepare and pack a quintal of B, whereas these costs are Birr 18,000 and Birr 26,000 respectively in plant 2. The company is obliged to produce daily at least 10,000 gallons of A and 8 quintals of B.
Required:
a. Formulate linear programming problem?
b. Solve the problem using simplex method?
Question 14. The owner of the Burger Doodle Restaurant is considering two ways to expand operations:opening a drive-up window or serving breakfast. The increase in profits resulting from theseproposed expansions depends on whether a competitor opens a franchise down the street. Thepossible Profits from each expansion in operations given both future competitive situations areshown in the following payoff table.
Competitor
Decision Open Not Open
Drive-up window $-6,000 $20,000
Breakfast 4,000 8,000
Select the best decision using the following decision criteria.
a. Maximax
b. Maximin
c. Equally likely
d. Minimax regret
Question 15. Convert the following in to dual and solve it using simplex approach
Min Z=2x1 +8x2
Subject to:
5x1+ x2 > 10
2x1+ 2x2 > 14
X1+ 4x2 > 14
X1, x2 > 0
Question 16. An XYZ company is planning a production of 300 units of the product each day. The requirement of manufacture is purchase of two machines costing birr 30,000 and 15,000. The total capital investment possible is birr 300,000. The machine A can give an output of 25 products a day and machine B only 15 unites. The plot being small, there is restriction of floor space of only 200 sq. meters for installation of machines , each machines A needing 5 sq.meter and machine B needing 6 sq. meter.
Required
a. Determine the optimal number of each machine that should be purchased (use Gomory's cutting plane algorithm method)
Question 17. Use Gomory's cutting plane algorithm, solve the following integer linear programming
Max Z=3x1+4x2
Subject to
3x1+2x2 < 8
X1+4x2 > 10
X1,x2>0
18. An investor is considering investing in stock, real estate, or bonds under uncertain economic conditions. The payoff table of returns for the investor's decision situation is shown below
Economic Conditions
Investment Good Stable Poor
Stocks $ 5,000 $ 7,000 $ 3,000
Real estate -2,000 10,000 6,000
Bonds 4,000 4,000 4,000
a.What is Equal likely decision?
b.What is the Maximin decision?
c. what is the Maximax decision?
d. what is the Hurwicz decision ? Use ?= 0.3
e.What is the Minimax regret decision?
Question 19.A Manufacturing firm produces electric motors for washing machines and vacuum cleaners. The firm has resource constraints for production time, steel and wire. The LP model determining the number of washing machine motors (x1) and vacuum cleaner motors (x1) to produce has been formulated as follows.
Maximize Z = 70x1 + 80x2
Subject to:
2x1 + x2 <19 ( production, hrs)
X1 + x2 <14 (steel, lb)
X1 + 2x2 < 20 (wire, ft)
x1,x2 >0
Question 20. A university department head has five instructors to be assigned to four different courses.All of the instructors have taught the courses in the past and have been evaluated by the students. The rating for each instructor for each course is given in the following table (aperfect score is 100).
Course
InstructorA B CD
1 80 75 90 85
2 95 90 90 97
3 85 95 88 91
4 93 91 8084
5 91 92 93 88
The department head wants to know the optimal assignment of instructors to courses that will maximize the overall average evaluation. The instructor who is not assigned to teacha course will be assigned to grade exams. Solve this problem using the assignmentmethod.
Question 21. The Bunker manufacturing firm has five employees and six machines, and wants to assign the employees to the machines so as to minimize cost. A cost table showing the cost incurred by each employee on each machine is given below.
Machine
Employe A B C D E F
1. $12 7 20 14 8 10
2. 10 14 13 20 9 11
3. 5 3 6 9 7 10
4. 9 11 7 16 9 10
5. 10 6 14 8 10 12
Because of union rules regarding departmental transfers, employee 3 cannot be assignedto machine E and employee 4 cannot be assigned to machine B. Solve this problem,indicate the optimal assignment, and compute total minimum cost
Question 22. A Manufacturing firm produces electric motors for washing machines and vacuum cleaners. The firm has resource constraints for production time, steel and wire. The LP model determining the number of washing machine motors (x1) and vacuum cleaner motors (x1) to produce has been formulated as follows.
Maximize Z = 70x1 + 80x2
Subject to:
2x1 + x2 <19 ( production, hrs)
X1 + x2 <14 (steel, lb)
X1 + 2x2 < 20 (wire, ft)
x1,x2 >0
Question 23. Given the following information,
Estimate(months)
Activity a m b
1--->2 4 8 12
1--->3 6 10 15
1--->4 2 10 14
2--->5 3 6 9
2--->6 1 4 13
3--->5 3 6 18
4--->5 0 0 0
4--->7 2 8 12
5--->8 9 15 22
5--->7 5 12 21
7--->8 5 6 12
6--->9 7 20 25
8--->9 3 8 20
Determine the following:
a. Expected activity times
b. Earliest event times
c. Latest event times
d. Activity slack
e. Critical path
f. Expected project duration and variance
Question 24. Two players A and B match coins. If the coins match, then A wins one unit of value; if the coins do not mach, B wins one unit of value. Determine optimum strategies for the players and the value of game.
Question 25. Solve the transportation problem for which the cost, origin availabilities and destination requirements are give below:
| To: |
|
|
|
|
|
|
Supply |
| |
D1 |
D2 |
D3 |
D4 |
D5 |
D6 |
| From: |
|
|
|
|
|
|
| O1 |
1 |
2 |
1 |
4 |
5 |
2 |
30 |
| O2 |
3 |
3 |
2 |
1 |
4 |
3 |
50 |
| O3 |
4 |
2 |
5 |
9 |
6 |
2 |
75 |
| O4 |
3 |
1 |
7 |
3 |
4 |
6 |
20 |
| Demand |
20 |
40 |
30 |
10 |
50 |
25 |
|
Question 26. A large steel manufacturing company has three options with regard to production. (i) Produce commercially (ii) build pilot plant (iii) stop producing steel. The management has estimated that their pilot plant, if built, has 0.8 chance of high yield and 0.2 chance of low-yield . if the pilot plant does show a high yield , management assigns a probability of 0.75 that the commercial plant will also have a high yield. If the pilot plant shows a low yield, there is only a 0.1 chance that the commercial plant will show a high yield. Finally management best assessment of the yield on a commercial-size plant without building a pilot plant first has a 0.6 chance of high yield. A pilot plant will cost birr 300,000. The profit earned under high and low yield conditions are birr 12000,000 and birr 1200,000 respectively. Find the optimum decision for the company?
Question 27. What is the role of decision making in operation research? Define scientific decision-making and explain it affects operation research decisions.
Question 28. Explain how and why operation research methods have been valuable in aiding executive decision?
Question 29. Explain approximation in operation research model. What are advantage and characteristics of a good model?
Question 30. What are the situations where operation research techniques will be applicable?
Question 31. For what type of business problems might game theory helpful?
Question 32. A business man has two independent investments A and B available to him, but he lacks the capital to undertake both of them simultaneously. He can choose to take A first and then stop, or if, A is successful, then take B or vice-versa. The probability of success on A is 0.7, while for B, it is 0.4. Bothinvestments require an initial capital outlay of birr 2000 and both return nothing, if the venture is unsuccessful. Successful completion of A will return birr 3000 (overcost) whereas successful completion of B will return birr 5000(overcost).
Required: draw the decision tree and determine the best strategy?
Attachment:- Worksheet.rar