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Sankalchand Patel College of Engineering, Visnagar

Assignment Subject: Modelling Simulation and Operation Research (161601)

Branch & Sem.: VI – IT

1. For the following project: (i) Draw an arrow diagram. (ii) Identify critical path and calculate the total

float and free float for each activity:

Activity: A B C D E F G

Immediate

predecessors:

- - A A, B C, D B, D E, F

Time (days): 2 1 3 2 1 3 1

2. Explain the basic difference between CPM and PERT.

3. Explain the following:

a. Optimistic time

b. Pessimistic time

c. Most likely time

d. Expected activity time

4. Explain the following queue discipline:

a. First come first served

b. Last come, first served

c. Serve in Random order

d. Priority service

5. On an average 6 customers reach a beauty shop every hour. Determine the probability that exactly 2

customers will reach in a 30 minute period, assuming that the arrivals follow Poisson distribution.

6. Cost matrix for players A and B are as below. Find the optimum strategy for player – A:

B’s

I II III IV

A’s I 20 15 12 35

II 25 14 8 10

III 40 2 10 5

IV -5 4 11 0

7. Explain in detail about Dominance rules.

8. Using Dijkstra’s algorithm, solve the following shortest route problem:

9. Jobs arrive in a machine shop following a Poisson process with a mean rate of one per hour, If the

average time to complete the job is 30 minutes; what is the probability that at least two jobs will be

waiting always?

10. Explain the difference between linear and dynamic programming and also define return function.

Define and explain the following terms:

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a. Free float

b. Interference float

c. Total float

11. Explain the Monte-Carlo simulation technique with its steps.

12. Determine minimum spanning trees and their lengths for following network:

13. Explain the infeasibility and unboundedness in graphical approach of LPP.

14. Solve the following assignment problem:

Worker

Time (Minutes)

Job-1 Job-2 Job-3

A 4 2 7

B 8 5 3

C 4 5 6

15. Solve the following transportation problem:

From

To

1 2 3 4 Supply

1 8 8 5 12 7

2 6 9 11 9 7

3 10 15 6 13 10

4 6 8 7 8 6

5 11 10 11 13 5

6 8 14 5 12 6

Demand 9 10 8 14

16. Determine the critical path for given project and also determine the total time required for the project:

Activity 1-2 1-3 2-4 3-4 3-5 4-6 5-6

Duration 2 8 4 1 2 5 6

Also calculate latest and earliest occurrence time.

17. A department store has a single cashier. During the rush hours, customers arrive at the rate of 20

customers per hour. The average number of customers that can be processed by the cashier is 24 per

hour. Assume that the conditions for use of single-channel queuing model apply. What is the (i)

probability that cashier is idle? (ii) Average number of customers in the queuing system (iii) average

time a customer spends in the system.

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18. The city health department is having a computer to record birth in a locality, Where it is estimated that,

on an average there is a birth in every two hours. Determine (a) How many records have to be stored –

annually? (b) What is probability that no birth – takes place in a day, in this locality?

19. What is simulation? Explain. Also state the application of simulation.

20. Briefly explain the major applications of linear programming in business.

21. Solve by Revised simplex method

𝑀𝑖𝑛𝑚𝑖𝑧𝑒 ∶ 𝑍 = 2𝑥1 + 𝑥2

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 3𝑥1 + 4𝑥2 ≤ 6

6𝑥1 + 𝑥2 ≤ 3

𝑥1, 𝑥2 ≥ 0

22. Solve the following problem using simplex

method:

𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒:𝑍 = 4𝑥1 + 3𝑥2

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 2𝑥1 + 𝑥2 ≤ 1000

𝑥1 + 𝑥2 ≤ 800

𝑥1 ≤ 400

𝑥2 ≤ 700

𝑥1, 𝑥2 ≥ 0

23. Solve the problem by Simplex method:

𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒:𝑍 = 4𝑥1 + 10𝑥2

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 2𝑥1 + 𝑥2 ≤ 10

2𝑥1 + 5𝑥2 ≤ 20

2𝑥1 + 3𝑥2 ≤ 18

𝑥1, 𝑥2 ≥ 0

24. Using two phase method, solve the following

problem:

𝑀𝑎𝑥𝑖𝑚𝑖𝑠𝑒: 150𝑥1 + 150𝑥2 + 100𝑥3

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 2𝑥1 + 3𝑥2 + 𝑥3 ≥ 4

3𝑥1 + 2𝑥2 + 𝑥3 ≥ 3

𝑥1, 𝑥2, 𝑥3 ≥ 0

25. Using two phase method, solve the following

LPP:

𝑀𝑎𝑥𝑖𝑚𝑖𝑠𝑒:𝑍 − 3𝑥1 − 𝑥2

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 2𝑥1 + 𝑥2 ≥ 2

𝑥1 + 3𝑥2 ≤ 2

𝑥2 ≤ 4

𝑥1, 𝑥2 ≥ 0

26. Using two phase method, solve the following

LPP:

𝑀𝑎𝑥𝑖𝑚𝑖𝑠𝑒:𝑍 = 5𝑥1 + 3𝑥2

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 2𝑥1 + 𝑥2 ≤ 1

𝑥1 + 4𝑥2 ≥ 6

𝑥1, 𝑥2 ≥ 0

27. Solve the following by B-g-M method:

𝑀𝑎𝑥𝑖𝑚𝑖𝑠𝑒:𝑍 = 2𝑥1 + 𝑥2 + 3𝑥3

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 𝑥1 + 𝑥2 ≤ +2𝑥3 ≤ 5

2𝑥1 + 3𝑥2 + 4𝑥3 ≤ 12

𝑥1, 𝑥2, 𝑥3 ≥ 0

28. Solve the following by B-g-M method:

𝑀𝑖𝑛𝑖𝑚𝑖𝑠𝑒:𝑍 = 3𝑥1 − 𝑥2

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 2𝑥1 + 𝑥2 ≥ 2

𝑥1 + 3𝑥2 ≤ 3

𝑥2 ≥ 4

𝑥1, 𝑥2 ≥ 0

29. Solve by Big-M method

𝑀𝑖𝑛𝑖𝑚𝑖𝑠𝑒:𝑍 = 2𝑦1 + 3𝑦2

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 𝑦1 + 𝑦2 ≥ 5

𝑦1 + 2𝑦2 ≥ 6

𝑦1,𝑦2 ≥ 0

30. Obtain an optimal solution, if any , to the

following primal LP problem:

𝑀𝑎𝑥𝑖𝑚𝑖𝑠𝑒:𝑍 = 3𝑥1 + 2𝑥2

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 2𝑥1 + 𝑥2 ≤ 5

𝑥1 + 𝑥2 ≤ 3

𝑥1, 𝑥2 ≥ 0

31. Solve by dual simplex method

𝑀𝑎𝑥𝑖𝑚𝑖𝑠𝑒:𝑍 = −3𝑥1 − 2𝑥2

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜:𝑥1 + 𝑥2 ≥ 1

𝑥1 + 𝑥2 ≤ 7

𝑥1 + 2𝑥2 ≥ 10

𝑥2 ≤ 3

𝑥1, 𝑥2 ≥ 0

32. Solve the problem by mixed integral

programming by using the cutting plan method:

𝑀𝑎𝑥𝑖𝑚𝑖𝑠𝑒:𝑍 = 𝑥1 + 𝑥2

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 𝑥1 + 2𝑥2 ≤ 4

6𝑥1 + 2𝑥2 ≤ 9

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𝑥1, 𝑥2 ≥ 0 𝑎𝑛𝑑 𝑎𝑟𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠.

33. Determine the dual of the following LPP:

𝑀𝑖𝑛𝑖𝑚𝑖𝑠𝑒:𝑍 = 3𝑥1 − 2𝑥2 + 4𝑥3

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 3𝑥1 + 5𝑥2 + 4𝑥3 ≥ 7

6𝑥1 + 𝑥2 + 3𝑥3 ≥ 4

7𝑥1 − 2𝑥2 − 𝑥3 ≤ 10

𝑥1 − 2𝑥2 + 5𝑥3 ≥ 3

4𝑥1 + 7𝑥2 − 2𝑥3 ≥ 2

𝑥1, 𝑥2, 𝑥3 ≥ 0

34. Discuss on Limitation of graphical method, used for LPP.

35. What is dummy activity? Why is it used?

36. Explain the forward pass calculation for determining earliest start and earliest finish time.

37. Explain the following structure of service system:

a. Multiple parallel service facilities with single queue

b. Service facilities in a series

38. Explain the minimax-maximin principle.

39. Determine minimum spanning trees and their lengths for the network given below:

40. Using Dijkstra’s algorithm, solve the following shortest route problem.

41. Describe about following characteristics of dynamic programming problem:

a. Stage

b. State

c. Return function

42. Explain in brief about rules of network construction.

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43. Obtain the optimal strategies for both players and value of game for two person zero sum game whose

pay off matrix is given as

Player B

Player A 1 3 11

8 5 2

44. State and explain the three phase of operation research.

45. Solve the following transportation problem for minimum cost by taking initial feasible solution by

V0gel’s approximation method:

Origin Destination Availibility

1 2 3 4

1 10 8 11 7 20

2 9 12 14 6 40

3 8 9 12 10 35

Requirement 16 18 31 30 95

46. A car rental company has one car at each of five depots a, b, c, d and e. A customer in each of five towns

A, B, C, D and E requires a car. The distance in kilometre between the depots and towns where the

customers are is given in distance matrix:

Depots

a b c d E

Town A 160 130 175 190 200

B 135 120 130 160 175

C 140 110 155 170 185

D 50 50 90 80 110

E 55 35 70 80 105

How should the cars be assigned to customers so as to minimize the distance travelled?

47. Determine the initial basic feasible solution to following transportation problem:

Destination Supply

A B C D

Source S1 1 5 3 3 34

S2 3 3 1 2 15

S3 0 2 2 4 12

S4 2 7 2 4 19

Demand 21 25 17 17

48. Define and explain:

a. I conic model

b. Analogue model

c. Verbal model

d. Mathematical model

49. Explain the following terms in context of

LPP:

a. Infeasibility

b. Unboundedness

50. An office equipment manufacturer produces two kinds of products: Computer covers and floppy boxes.

Production of rather a computer cover or a floppy box requires 2 hour of production capacity in the

factory. The factory has a maximum production capacity of 8 hours per day. Because of the limited sales

capacity, the maximum numbers of computer covers and floppy boxes that can be sold are 8 and 12 per

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day, respectively. The gross margin from the sale of computer cover is Rs. 60 and Rs. 40 for 4 floppy

boxes. The over time hour should not exceed 2 hours per day. The plant manager has set the following

goals arranged in order of importance:

a. To avoid any underutilization of production capacity.

b. TO limit the overtime hours to 2 hours.

c. To sell as many computer covers and floppy boxes as possible. Since the gross margin from the

sale of a computer cover is set at twice the amount of profit from a floppy box, he has twice as

much desire to achieve the sales go for computer covers as for the floppy boxes.

d. To minimize the over time operation of the plant as much as possible.

Develop a goal programming model for this problem.

51. Arepair man is to be hired to repair machine that break down following a Poisson’s process, with an

average rate of 4 per hour. The cost of non productive machine is Rs. 9 per hour. The company has the

options of choosing either a fast or slow repairman. The Fast man charges Rs. 6 per hour and will repair

at an average rate of 7 per hour. The slow man charges Rs. 3 per hour and with repair at an average rate

5 per hour. Find which repairman should be hired.

52. Discuss application of dynamic programming with example and it’s formulation.

53. Explain how and why or methods have been valuable in aiding executive decisions.

54. Simulation is an initiation of reality justify.

55. PERT is non deterministic network justify.

56. A manufacture produces two different models X and Y of the same product. The raw materials r1 and r2

are required for production. At least 18kg of r1 and 12kg of r2 must be used daily. Also at most 34 hours

of labour are two be utilised. 2kg of r1 are needed for each model X and 1kg of r1 for each model Y. For

each model of X and Y, 1kg of r2 is required. It takes 3 hours to manufacture a model X and 2 hours to

manufacture a model Y. The profit is Rs. 50 for each model X and Rs. 30 for each model Y. How many

units of each model should be produced to maximize the profit? Solve the LP problem by graphical

method.

57. A company has one surplus truck in each of the cities A, B, C, D and E and the deficit truck in each of the

cities 1, 2, 3, 4, 5, 6. The distance between the cities in kilometres shown in matrix below. Find the

assignment of truck from cities in surplus to cities in deficit so that total distance covered by vehicles is

minimized.

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

58. A company manufacturing air-coolers has two plants located at Bombay and Calcutta with a capacity of

200 units and 100 units per week respectively. The company supplies the air coolers to its four show

rooms situated at Ranchi, Delhi, Luckhnow and Kanpur which have max. Demand of 75, 100, 100 and

30 units respectively. Due to differences in rawmaterial cost and transportation cost, the profit per unit

in rupees differs which is shown in the table below:

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Ranchi Delhi Lucknow Kanpur

Bombay 90 90 100 110

Calcutta 50 70 130 85

Plan the production programme so as to maximize the profit. The company may have its production

capacity at both plants partly or wholly unused.

59. Explain following terms for game theory:

a. Payoff matrix

b. Strategy

c. Saddle point

d. Two person zero sum game

60. A company has five jobs to be done. The following matrix shows the return in rupees of assigning (i+4)

machine (i=1,2,...5) to the (j+4) job (j=1,2,... 5). Assign the five jobs to the five machines so as to

maximize the total expected profit:

1 2 3 4 5

1 5 11 10 12 4

2 2 4 6 3 5

3 3 12 5 14 6

4 6 14 4 11 7

5 7 9 8 12 5

61. Consider the details of a profit as shown in table:

Activity A B C D E F G H I J K L M N O P Q

Immediate

Predecessor

(s)

- - - A A B B C C D E F G H I J,

K,

L

M,N,O

Duration

(months)

4 8 5 4 5 7 4 8 3 6 5 4 12 7 10 5 8

Draw network diagram and Identity critical path. Also calculate project duration.

62. Explain Kendall’s notation in queuing systems.

63. Advantage and disadvantage of simulation. Write a short note on it.

64. The arrival rate of customer at banking counter follows Poisson distribution with a mean of 45 per

hour. The service rate of the counter clerk also follows Poisson distribution with a mean of 60 per hour.

a. What is the probability of having 0 customers in the system (P0).

b. What is the probability of having 10 customers in the system (P10)?

c. Find Ls, Lq, Ws and Wq.

65. Find the optimum strategies of the players in the following games:

a.

B

1 2 3

A 1 30 20 40

2 55 50 60

3 60 30 40

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b.

B

1 2 3

A 1 10 5 20

2 65 50 40

3 55 25 30

66. What are the differences between linear programming method and goal programming method?

67. A super market has two girls managing sales at the counters. If the service time for each customer is

exponential with mean 4 minutes and if people arrive in a Poisson fashion at the rate of 10 per hour.

Then calculate the

a. Probability of having to wait for service.

b. Expected percentage of idle time for each girl.

c. If a customer has to wait, what is the expected length of this waiting time?

68. Explain – Monte Carlo simulation

69. List out applications of queuing system

70. Explain dominance property in game theory.

71. A production manager is faced with the problem of job allocation to his two production reams. The

production rate of mean 1 is 8 units per hour; while the production rate of team 2 is 5 units per hour.

The normal working hours for each of the team is 40 hours/week. The production manager has

prioritized the following goals for the coming week:

P1 = Avoid under achievement of the desired production level of 550 units.

P2 = Overtime operation of team 1 is limited to 5 hours.

P3 = the total overtime for both teams should be minimized.

Formulate this problem as a goal programming model.

72. A distance network consists of eight nodes which are distributed as shown in following fig. Find the

shortest path from node 1 to node 8 and also the corresponding distances.

73. Solve above problem using dynamic programming method.

Faculty Name: Shreekant S. Tatosaniya