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Question BANK ON Operations Research UNI
Operations management (MBA 706)
Lagos State University
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QUESTION BANK ON OPERATIONS RESEARCH
UNIT-1: Basics of operations research Q1. Discuss the origin and development of OR. Q2. How computer has helped in popularizing OR? Q3. What are the limitations of OR? Q4. Describe the various objectives of OR. Q5. What are the main characteristics of OR? Explain with suitable examples. Q6Give features of OR. Briefly discuss technique and tools of OR. Q7. What is the role of decision making in OR. Explain its scope. Q8. “OR is the application of scientific methods, technique and tool to problems involving the operation of a system so as to provide those in control of the system with optimum solution to the problems.” Q9. Discuss the significance and scope of OR in modern management. Q10. “Mathematics of OR is mathematics of optimization.” Discuss. Q11. “OR is an aid for the executive in making his decision by providing him with the needed quantitative information, based on the scientific method analysis.” Discuss the statement in detail, illustrating it with OR methods that you know. Q12. Discuss in brief the role of OR model in decision making.
UNIT-2: Linear Programming Q1. What are the essential characteristics of a linear programming model? Q2 is linear programming? Discuss the application of linear programming to managerial decision making. Q3 Discuss the assumption of proportionality, additivity, continuity, certainty and finite choices in the context of linear programming problems. Q4. Explain the meaning of linear programming problem stating its uses and give its limitations. Q5. Write at least five application areas of linear programming. Q6. A small manufacturer employs 5 skilled men and 10 semi skilled men and makes an article in two qualities, a deluxe model and an ordinary model the making of deluxe model requires 2 hours work by a skilled man and 2 hours work by a semi-skilled man. The ordinary model requires 1 hour work by a skilled man and 3 hour work by a semi- skilled man. By union rules no man can work more than 8 hours per day. The manufacturer’s clear profit of the deluxe model is Rs and of ordinary model Rs. Formulate the model of the problem. Q7. Old hens can be bought for Rs each but young one cost Rs each. The old hens lay 3 eggs per week, and young one 5 eggs per week, each egg being worth 30 paise. A hen cost Re per week to feed. If a person has only Rs to spend on hens, how many of each kind should he buy to get a profit of more than Rs per week assuming that he can’t house more than 20 hens? Q8. A firm manufactures three products A, B, and C. The profits are Rs, 2 and 4 respectively. The firm has two machines and required processing time in minutes for each machine on each product is given below: Product A B C Machine X 4 3 5 Y 2 2 4 Machine X and Y have 2000 and 1500 machine minutes respectively. The firm must manufacture 100 A’s, 200B;s and 50C’s but no more than 150 A’s. set up an Lp model to maximize the profit. Q9. The manager of an oil refinery has to decide upon the optimal mix of two possible blending processes, of which the input and output per production run are as follows:
Q15. Maximize subject to Z= -150x 1 -100 x2,+280000, 20≤ x 1 ≤60, 70≤x 2 ≤140, 12 0≤ x 1 +x 2 ≤140, x1, x 2 ≥ Use simplex method to solve the following problem: Q16. Maximize subject to Z= 3x 1 +2 x 2 + 5x 3 , x 1 + x 2 + x 3 ≤9, 2x 1 + 3x 2 + 5x 3 ≤30, 2 x 1 -x 2 -x 3 ≤8, x1, x2, x 3 ≥ Q17. Maximize subject to Z= 2x 1 +4x 2 + x 3 + x 4 , x 1 + 3x 2 + x 4 ≤4, 2x 1 + x 2 ≤3, x 2 +4x 3 +x 4 ≤3, x1, x2, x3, x 4 ≥ Q18. Maximize subject to Z= 2x 1 +3x 2 + x 3 +7 x 4 , 8x 1 + 3x 2 + 4x 3 + x 4 ≤6, 2x 1 +6x 2 + x 3 +5x 4 ≤3, x 1 +4x 2 +5x 3 +2x 4 ≤7, x1, x2, x3, x 4 ≥ Q19. Maximize subject to Z= 6x 1 +7x 2 + 9x 3 , 3x 1 + 7x 2 + 6x 3 ≤245, 5x 1 +8x 2 + 9x 3 ≤424, 11 x 1 +6x 2 +8x 3 ≤235, x1, x2, x3, ≥ Q20. Maximize subject to Z= 2x 1 +3x 2 + 4x 3 +x 4 , -x 1 - 5x 2 - 9x 3 + 6x 4 ≤2, 3x 1 -x 2 + x 3 +3x 4 ≤10, 2 x 1 +3x 2 -7x 3 +8x 4 ≤0, x1, x2, x3, x 4 ≥
Q21. Minimize subject to Z= 4x 1 -3x 2 + 7x 3 -x 4 , 7x 1 + 3x 2 ≤400, 5x 1 + 4x 3 ≥250, x 1 +x 4 =43, x 1 ,x 2 ,x 3 ,and x 4 are non negative and none is below 20. Solve by Big M-method Q22. Maximize subject to Z= x 1 +2x 2 + 3x 3 -x 4 , x 1 +2x 2 +3x 3 =15, 2x 1 +x 2 + 5x 3 =20, x 1 +2x 2 +x 3 +x 4 =10, x1, x2, x3, x 4 ≥ Q23. Maximize subject to Z= x 1 +2x 2 + 3x 3 , x 1 -x 2 +x 3 ≥4, x 1 +x 2 + 2x 3 ≤8, x 1 -x 3 ≥2, x1, x2, x3, ≥ Q24. A manufacturer produces three products A, B, and C. Each product requires processing on two machines I & II. The time required to produce one unit of each product on a machine is: Product Time to produce one unit (hrs.) Machine I Machine II A 0 0. B 0 0. C 0 1. There are 850 hrs are available on each machine. The operating cost is Rs/hr. for machine I and Rs/hr. for machine II. The market requirements are at least 90 units of A, at least 80 units of B and at least 60 units of C. The manufacturer wishes to meet the requirement at minimum cost. Solve the problem by simplex method. Q25. A factory has decided to diversify its activities. The data collected by sales and production department is summarized below. Potential demand exist for three products A,B and C. market can take any amount of A and C. whereas the share of B for this organization is expected to be not more than 400 units a month.
UNIT-3a: Transportation Model Q1. Explain the following in the context of transportation problem. a) Degenerate transportation problem b) Modified distribution method. Q2. What is degeneracy in transportation problem? How it can be resolved? Q3. What are the conditions for the application of optimality test in case of transportation problem? Briefly explain as to why these conditions should be satisfied? Q4. Find the feasible solution of the following transportation problem using North West corner method. W 1 W 2 W 3 W 4 Supply Factory F 1 14 25 45 5 6 F 2 65 25 35 55 8 F 3 35 3 65 15 16 Requirement 4 7 6 13 30 (Total) Q5. Find the initial basic feasible solution of the following transportation problem using Vogel’s approximation method. W 1 W 2 W 3 W 4 Capacity Factory F 1 19 30 50 10 7 F 2 70 30 40 60 9 F 3 40 8 70 20 18 Requirement 5 8 7 14 34 (Total)
Q6. Find the initial basic feasible solution of the following transportation problem using North West corner method and Vogel’s approximation method. A 1 B 1 C 1 D 1 E 1 Supply Origin A 2 11 10 3 7 4 B 1 4 7 2 1 8 C 3 9 4 8 12 9 Requirement 3 3 4 5 6 Q7. Solve the transportation problem for which the cost, origin, availability and destinations requirements are given below: D 1 D 2 D 3 D 4 D 5 D 6 ai O 1 1 2 1 4 5 2 30 O 2 3 3 2 1 4 3 50 O 3 4 2 5 9 6 2 75 O 4 3 1 7 3 4 6 20 bj 20 40 30 10 50 25 175 (Total) Q8. A departmental store wishes to purchase the following quantities of ladies’ dresses: Dress type A B C D Quantity 150 100 75 250 Tenders are submitted by three different manufacturers who undertake to supply not more than the quantities below: Manufacturer W X Y Total Quantity 350 250 150
From To G H I J A 13 25 12 21 B 18 23 14 9 C 23 15 12 16 What is the optimal distribution plan to minimize the moving cost? Q11. Solve the following transportation problem: From To A B C D E Supply 1 20 19 14 21 16 40 60 90
2 15 15 19 16
3 18 20 18 20
Demand 30 40 70 40 60 Q12. A manufacturing company has three factories F1, F2, F 3 with monthly manufacturing capacities of 7000,4000,10000 units of a product. The product is to be supplied to seven stores. The manufacturing costs of these factories are slightly different but the important factor is the shipping cost from each factory to a particular store. Following table represent the factory capacities, store requirement and unit cost in rupees of shipping from each factory to each store and slack. Here slack is difference between total factory capacity and total store requirement. S 1 S 2 S 3 S 4 S 5 S 6 S 7 Slack Factory capacity Factory F 1 5 6 4 3 7 5 4 0 7000 F 2 9 4 3 4 3 2 1 0 4000 F 3 8 4 2 5 4 8 3 0 10000 Store requirement
1000 2000 4500 4000 2000 3500 3000 1000
Work out a transportation plan so as to minimize the transportation cost. Q13. General electrode is a big electrode manufacturing company. It has two factories and three main distribution centers in three cities. The supply and demand conditions for units of electrode are given below along with unit cost of production. How should the trips be scheduled so that cost of production is minimized?
The present cost of transportation is around Rs/month. What can be the maximum saving by proper scheduling? Centers A B C Requirement 50 50 150 Cost per trip from X plant
25 35 10
Cost per trip from Y plant
20 5 80
Capacity of plant X 150 units of electrodes Capacity of plant Y 100 units of electrodes Q14. A company has three plants at location A,B,C which supply warehouses located at D,E,F,G,H.. Monthly plant capacities are 800,500,900 units resp. Monthly warehouse requirements are 400,350, 250,900. the unit transportation cost in rupees given below: From To D E F G H A 8 8 9 4 3 B 5 8 5 11 6 C 8 9 7 3 3 Determine an optimum distribution for the company in order to minimize the total transportation cost. How much is the cost? Q15. Priyanshu enterprise has three auditors. Each auditor can work up to 160 hours during the next month, during which time 3 projects must be completed. Project I will take 130 hours, project II will take 140 hours and III will take 160 hours. The amount in rupee per hour that can be billed for assigning each auditor to each project is given below: Project 1 Project 2 Project 3 Auditor 1 1200 1500 1900 2 1400 1300 1200 3 1600 1400 1500 Find the optimal solution. Also find the maximum total billing during the next month. Q16. Four suppliers have submitted sealed bids that quote the price per case of hairnets delivered to four regional stores of the army. The bids are summarized in the following table. The regional stores requirements as well as supplying capacities of suppliers are also shown. Supplier 4 has quoted for only region 1. Because of previous contractual obligations, region 3 will have to get a minimum of 200 cases from supplier 2.
UNIT-3b: Assignment Model Q19. Explain the following in the context of assignment problem: a) Balanced assignment problem b) The Hungarian method c) An infeasible assignment Q20. Show that the assignment model is a special case of the transportation model. Q21. Six machines M1, M2, M3, M4, M5, M 6 are to be located in six places P1, P2, P3, P4, P5, P 6. Cij the cost of locating machine Mi at place Pj is given in the matrix below: P 1 P 2 P 3 P 4 P 5 P 6 M 1 20 23 18 10 16 20 M 2 50 20 17 16 15 11 M 3 60 30 40 55 8 7 M 4 6 7 10 20 25 9 M 5 18 19 28 17 60 70 M 6 9 10 20 30 40 55 Formulate an Lp model to determine an optimal assignment. Write the objective function and the constraints in detail. Define any symbol used. Find an optimal layout by assignment technique of linear programming. Q22. Five new machines are to be located in a machine shop there are five possible locations in which the machine can be located. Cij the cost of placing machine i in place of j is given in the table below: Place 1 2 3 4 5 Machine 1 20 23 18 10 16 2 50 20 17 16 15 3 60 30 40 55 8 4 6 7 10 20 25 5 18 19 28 17 60 It is required to place the machine at suitable places so as to minimize the total cost. a) Formulate an L model to find an optimal assignment. b) Solve the problem by assignment technique of L
Q23. Solve the following assignment problem for minimum optimal cost: From city To city
1 2 3 4 5 6
A 12 10 15 22 18 8
B 10 18 25 15 16 12
C 11 10 3 8 5 9
D 6 14 10 13 13 12
E 8 12 11 7 13 10
Q24. A department has four subordinates and four tasks to be performed. The subordinates differ in efficiency and task differs in their intrinsic difficulty. The estimates of the profit in rupees each man would earn is given in the effectiveness matrix. How should the task be allocated, one to each man, so as to maximize the total earning? Subordinate Task 5 6 7 8 1 5 40 20 5 2 25 35 30 25 3 15 25 20 10 4 15 5 30 15 Q25. Xyz airlines operating 7 days a week has given the following time table. Crew must have a minimum layover of 5 hours between flights. Obtain the pairing of flights that minimize layover time away from home. For any given pairing the crew will be based at the city that results in the smaller layover. Chennai-Mumbai Mumbai-Chennai Flight no. Departure Arrival Flight no. Departure Arrival A 1 6 A. 8 A. B 1 8 A 10A. A 2 8 A. 10A. B 2 9 A 11A. A 3 2 P. 4 P. B 3 2 P 4 P. A 4 8 P. 10 P. B 4 7 P 9 P.
UNIT-4: Game theory Q1. For what type of business problems might game theory be helpful? Q2. Describe the role of theory of games for scientific decision making. Q3. Explain the assumptions underlying game theory. Q4. What do you understand by zero sum and non zero sum games? What do you mean by strategy, dominance and saddle point? Q5. Explain the following: a) minimax and maximin principles b) pure and mixed strategy c) two person zero sum game Q6. State the four properties which a competitive situation should have if it is to be called a competitive game. Q7. How is the concept of dominance used in simplifying the solution of a rectangular game? Q8. Show how a game can be formulated as an L.P? Q9. For a two person zero sum game , the payoff matrix for player A is a 11 b 12 a 21 b 22 With no saddle point. Obtain the optimal strategies (x1, x2) and (y1, y2). Q10. The following games have saddle point solutions. Determine the saddle point and optimum strategies for each player. a) Y X 4 6 4 2 10 0 b) Y X 2 -1 3 4 5 6
Q11. Determine the optimum strategies and the value of the following games: a) A
B
-3 4 2 9
7 8 6 10
6 2 4 -
b) A
B
-1 9 6 8
-2 10 4 6
5 3 0 7
7 -2 8 4
c) A
B
10 4 2 9 1
7 6 5 7 8
3 5 4 4 9
6 7 3 3 2
Q12. Find the value of the games shown below. Also indicate whether they are fair or strictly determinable. a) A
B
1 9 6 0
2 3 8 -
-5 -2 10 -
7 4 -2 -
b) A
B
6 -2 -3 8
-1 -2 -7 0
8 9 -6 -
9 5 -7 7
Q13. Find the range of values of p and q so that the entry (2,2) is a saddle point in the following games:
Q18. Find the optimum strategies for Y and the value of game. X
Y
4 -1 4 -1 2
2 2 3 -4 2
1 -3 1 0 -
Q19. Solve the following game: Player A Player B 10 81 32 43 93 59 63 39 69 73 71 20 05 27 84 34 14 44 44 69 Q20. Reduce the following game to 2×2 game by using dominance and modified dominance property and then solve the game. A
B
B 1 B 2 B 3 B 4
A 1 1 2 -1 2
A 2 3 1 2 3
A 3 -1 3 2 1
A 4 -2 2 0 -
Q21. Solve by using dominance property, the following game: A
B
I II III IV V VI
1 4 2 0 2 1 1
2 4 3 1 3 2 2
3 4 3 7 -5 1 2
4 4 3 4 -1 2 2
5 4 3 2 -2 2 2
Q22. Reduce the following game to 2×2 game by using graphical method: a) b) A
B
0 4 -8 -5 1
1 5 8 -4 0
c) A
B
2 3
6 7
-6 10
-3 -
3 2
d) A
B
-4 3
-7 1
-2 -
-5 -
-1 -
f) A
B
I II III IV
I 2 2 3 -
II 4 3 2 6
Q23. Solve by using the method of matrices, the following game: A
B
1 0 2
3 0 0
0 2 1
A
B
3 0 6 -1 7
-1 5 -2 2 1
Question BANK ON Operations Research UNI
Course: Operations management (MBA 706)
University: Lagos State University
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