R05321502-mathematical-modeling-and-simulation

Set No. 1

Code No: R05321502

III B.Tech Supplimentary Examinations, Aug/Sep 2008 MATHEMATICAL MODELING AND SIMULATION (Computer Science & Systems Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) What is the role of decision variables in linear programming model? (b) What is the standard form of linear programming model and explain. (c) What is the use of functional constraints in linear programming model?[4+6+6] 2. Discuss the streamlined simplex method for the transportation Problem.

[16]

3. (a) Discuss stochastic multi period model with batch orders and no setup cost. (b) Draw the graph for the above.

[12+4]

4. (a) A company purchases three items A,B,C Their annual demand and prices are given in the following table. Items A B C

Annual Demand Unit Price 1,00,000 3 80,000 2 600 96

If the company wants to place 40 orders per year for all the three items, what is the optimal number of orders for each item? (b) The following thirty numbers represent the annual value in thousand of rupees of some thirty items of materials selected at random. Carry out an ABC analysis and list out the values of ‘A’ items only. [6+10] 1 2 4 9 75 4 3 6 13 2 4 12 100 2 7 40 15 55 1 11 25 15 8 10 1 20 30 1 3 5 5. (a) Explain the meaning of a ‘Queue’ with suitable examples. (b) Customers arrive at a box office window being manned by a single individual according to a Poisson input process with a mean rate of 30 per hour. The time required to serve a customer has an exponential distribution with a mean of 90 seconds. Find average waiting time of a customer. Also determine the average number of customers in the system and average queue length. [7+9]

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Set No. 1

Code No: R05321502

6. A company manufacturing plant and equipment for chemical processing is in the process of quoting a tender called by a public sector undertaking. Delivery data once promised is crucial and penalty clause is applicable. Project manager has listed down the activities in the project as under: Activity A B C D E F G H

Immediate Predecessor A A C D B E,F,G

Optimistic 1 2 3 5 5 6 7 2

Activity Time (weeh) Most Likely Pessimistic 3 5 4 6 5 7 6 7 7 9 8 10 9 11 3 4

Using PERT: (a) find out the delivery week from the date of commencement of the project, and (b) total float and free float for each of the non-critical activities.

[16]

7. A sample of 100 arrivals of customers at a retail sales depot is according to the following distribution: Time between arrivals (minutes) Frequency

0.5 1 15 2 2 6 10 25

2.5 20

3 3.5 4 14 10 7

4.5 4

5 2

A study of the time required to service customers by adding up the bills, receiving payments, placing packages, etc. yields the following distribution. Service time (minutes) I: 0.5 1 1.5 Frequency 12 21 36

2 2.5 19 7

3 5

Estimate the average percentage of customer waiting time and average percentage idle time of the server by simulation for the next 10 arrivals. Assume any suitable random numbers. [16] 8. Distinguish verification and validation of simulation models with appropriate examples. [16] ⋆⋆⋆⋆⋆

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Set No. 2

Code No: R05321502

III B.Tech Supplimentary Examinations, Aug/Sep 2008 MATHEMATICAL MODELING AND SIMULATION (Computer Science & Systems Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. A television company has three major departments for the manufacture of its two models A and B. The monthly capacities are given as follows: [16] Per unit Hours available this month time requirement (hours) Model A Model B Dept - I 4.0 2.0 1600 Dept - II 2.5 1.0 1200 Dept - III 4.5 1.5 1600 The marginal profit per unit from of model A is Rs 400 and that of model B us Rs100. Assuming that the company can sell any quantity of either product to do favorable market conditions, Determine the optimum output for both the models, the highest possible profit for this month and the slack time in the three departments. 2. A pure Binary Integer Programming example with the constraint, 2x1 +3x2 ≤ 4, was used to illustrate the procedure for tightening constraints. Show that applying the procedure for generating cutting planes to this constraint yields the same new constraint, x1 +x2 ≤ 1. [16] 3. (a) Describe the basic characteristics of an inventory system? (b) Explain with suitable examples.

[8+8]

4. (a) “Purchase manager should shoulder special responsibility for A-items and the ‘A’ items should not be handled on any routine procurement policy.” Discuss. (b) Determine the Re-order level with uncertain demand.

[10+6]

5. (a) Discuss the fields of application for queuing theory? (b) Telephone users arrive at a booth following a poison distribution with an average time of 5 min between one arrival and the next. The time taken for telephone call is on an average 3 min. and it follows an exponential distribution. What is the probability that the booth is busy? How many more booths should be established to reduce the waiting time to less than equal to half of the present waiting time? [6+10] 6. A company has decided to market a new product for the consumer market. The problems of how to plan and control the various phases of this project-sales promotion, training of salespeople, pricing, packaging, advertising, and manufacturing-are 1 of 2

Set No. 2

Code No: R05321502

obvious to the management of this firm. They have asked you to guide then through this difficult venture using CPM, since time is of the essence. The first firm to market this type of product will reap substantial profits and will enhance its image by marketing such a revolutionary product. A list of the activities, with the expected time duration for each, is given in the table in terms of weeks: Activity Precedents Time(weeks) A -1 B A 1 C B 3 D C 3 E B 3 F B 8 G E 2 H G 4 I P 1 J F 0 K D,J,H,I,T 3 L K 7 M B,S 2 N M 4 O N 5 P O 1 Q K,S 2 R Q 3 S B 2 T S 3 Draw network diagram, identify tthe critical part and determine its length.

[16]

7. Discuss in detail various applications of simulations in real life with examples. [16] 8. Discuss need and scope of mathematical modeling with illustrations. ⋆⋆⋆⋆⋆

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[16]

Set No. 3

Code No: R05321502

III B.Tech Supplimentary Examinations, Aug/Sep 2008 MATHEMATICAL MODELING AND SIMULATION (Computer Science & Systems Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. A farmer has 1000 acres of land on which he can grow corn, wheat or soyabeans. Each acre of corn costs Rs100 for preparation, requires 7 men/day of work and yields a profit of Rs30. An acre of wheat costs Rs120 to prepare, require 10 men/day of work and yields a profit of Rs20. If the former has Rs1,00,000 for preparation and can count on 8000 men/day of work, determine how many acres should be allocated to each crop to max’m profits. [16] 2. (a) What is Initial solution in transportation problem. (b) Discuss the following methods for binding initial solution. i. North-west corner method ii. Least cost Method iii. VAM.

[4+4+4+4]

3. Discuss the basic EOQ model in deterministic continuous review models.

[16]

4. Draw the graph for ABC classification of inventory items and discuss the steps in ABC analysis with example. [16] 5. Arrival rate of telephone calls at a telephone booth is according to Poisson distribution, with an average time of 9 minutes between the consecutive arrivals. The length of telephone call is assumed to be exponentially distributed with mean 3 minutes. (a) Determine the probability that a person arriving at the booth will have to wait. (b) Find the average queue length that forms time to time. (c) The telephone company will install a second booth when convinced that an arrival would expect to have to wait at least four minutes for the phone. (d) What is the probability that an arrival will have to wait for more than 10 minutes before the phone is free? (e) What is the probability that he will have to wait for more than 10 minutes before the phone is available and the call is also complete? (f) Find the fraction of a day that the phone will be in use.

[16]

6. Listed in the table are the activities and sequencing requirements necessary for the completion of a research report.

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Set No. 3

Code No: R05321502 Activity A B C D E F G H I J K L M

Description Literature search Formulation of hypothesis Preliminary feasibility study B Formal proposal C Field analysis A,D Progress report D Formal research A, D Data collection E Data analysis G,H Conclusions I Rough draft G Final copy J,K Preparation of oral presentation L

Precedence Duration (weeks) 6 5 2 2 2 1 6 5 6 2 4 3 1

(a) Draw a network diagram for this project. (b) Find the critical path. What is its length? (c) Find the total float and the free float for each non-critical activity.

[16]

7. A certain maintenance facility is responsible for the upkeep of five machines. The machines, which fail frequently, must be repaired as soon as possible to maintain as high a productive capacity of the production system as possible. Management is concerned about the average down time per machine and is considering an increase in the capacity of maintenance facility. The following distributions have been developed from historical data: Time between breakdown (days) Probability Repair time Probability 2 0.05 1 0.4 3 0.10 2 0.5 4 0.15 3 0.1 5 0040 4 0.2 6 0.20 5 0.3 7 0.10 6 0.6 Simulate the failure and repair of 10 machines. Begin by determining the time of the first breakdown by each of the 5 machines. Sequence the machines through the repair facility on first come first served basis. If there is more than one machine waiting to be repaired, arbitrarily choose one to repair the next. After a machine has been repaired, determine its next time of break down and continue until you have repaired 10. Assume any suitable random numbers. [16] 8. Discuss various definitions of modeling mentioning its appropriate applications.[16] ⋆⋆⋆⋆⋆

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Set No. 4

Code No: R05321502

III B.Tech Supplimentary Examinations, Aug/Sep 2008 MATHEMATICAL MODELING AND SIMULATION (Computer Science & Systems Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Explain about the levels of abstraction in model development? (b) Discuss the simplex method procedure for solving on Linear Programming Problem? [4+12] 2. (a) What is assignment problem? Discuss in detail. (b) Discuss the assignment problem solution procedure.

[8+8]

3. (a) Purchase manager has decided to place order to a minimum quantity of 500 member of a particular item in order to get a discount of 10%. Form the past records it was found out that in the last year, 8 orders each of size 200 units were placed. Given the ordering cost is Rs.500 per order, inventory carrying cost of 40% of the inventory value and the price of the item of Rs.400 per unit. Is the purchase manager justified in his decision? What is the effect of his decision to the company? (b) The annual demand for a product is 64,000 units (or 1280 units per week). The buying cost per order is Rs.10 and the estimated cost of carrying one unit in stock for a year is 20%. The normal price of the product is Rs.10 per unit. however, the supplier offer a quantity discount of 2% on an order of at least 1000 units of a time, and a discount of 5% if the order is for at least 5,000 units. Suggest the most economic purchase quantity per order. [8+8] 4. (a) Classify the material in A,B,C classification for the following information. This information is known about a group of items. Model Number Annual Consumption pieces 501 502 503 504 505 506 507 508 509 510

30,000 2,80,000 3,000 1,10,000 4,000 2,20,000 15,000 80,000 60,000 8,000

Unit Price (in paise) 10 15 10 5 5 10 5 5 15 10

(b) Draw the graph for cumulative % of items (q) and cumulative % of usage value for the above table. [10+6] 1 of 2

Set No. 4

Code No: R05321502

5. (a) State and prove Markovian Property of inter arrival times. (b) A repair shop attended by a single mechanic has an average of four customers an hour who bring small appliances for repair. The mechanic inspects them for defects and takes six minutes on an average. Arrivals are Poisson and service rate has the exponential distribution. You are required to i. ii. iii. iv.

Find the proportion of time during which there is no customer in the shop. Find the probability of finding at least one customer in the shop. What is the average number of customers in the system? Find the average time spent by a customer in the shop including service. [6+10]

6. A company manufacturing plant and equipment for chemical processing is in the process of quoting a tender called by a public sector undertaking. Delivery data once promised is crucial and penalty clause is applicable. Project manager has listed down the activities in the project as under: Activity A B C D E F G H

Immediate Predecessor A A C D B E,F,G

Optimistic 1 2 3 5 5 6 7 2

Activity Time (weeh) Most Likely Pessimistic 3 5 4 6 5 7 6 7 7 9 8 10 9 11 3 4

Using PERT: (a) find out the delivery week from the date of commencement of the project, and (b) total float and free float for each of the non-critical activities.

[16]

7. Why would an analyst prefer a general purpose language such as FORTRAN or BASIC in simulation? Explain anyone application area of simulation with illustration. [16] 8. Why would an analyst ever prefer a general purpose language such as FORTRAN or BASIC in a simulation when there are advantages of using special purpose languages such as GPSS or SIMSCRIPT? Illustrate with examples. [16] ⋆⋆⋆⋆⋆

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