Mathematical Modelling And Simulation Rr321502

Set No. 1

Code No: RR321502

III B.Tech II Semester Supplimentary Examinations, Aug/Sep 2007 MATHEMATICAL MODELLING AND SIMULATION (Computer Science & Systems Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. A farm is engaged in breeding pigs. The pigs are fed on various products grown on the farm. In view of the need to ensure certain nutrient constituents (call them X, Y and Z), it is necessary to buy additional products, say, A and B. One unit of product A contains 36 units of X, 3 units of Y and 20 units of Z. One unit of product B contains 6 units of X, 12 units of Y and 10 units of Z. The maximum requirement of X, Y and Z is 108 units, 36 units and 100 units respectively. Product A costs Rs. 20 per unit and product B Rs. 40 per unit. Formulate the above as L. P. P. to minimize the total cost, and solve the problem by using graphic method. [16] 2. (a) Give the mathematical formulation of transportation problem (b) Solve the following transportation problem :

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D E F G Available A 11 13 17 14 250 B 16 18 14 10 300 C 21 24 13 10 400 Demand 200 225 275 250 3. (a) Derive the E. O. Q. formula for the manufacturing model with shortages [6] (b) A manufacturing firm has to supply 3,000 units annually to a customer who does not have enough space for storing the material. There is a contract that if the supplier fails to supply the material, a penalty of Rs. 40 per unit per month will be levied. The inventory holding cost amounts to Rs. 20 per unit per month and the setup cost is Rs. 400 per run. Find the expected number of shortages at the end of each scheduling period. [10] 4. (a) State various types of items in inventory control techniques.

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(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: [10] 1 2 4 9 75 4 25 3 6 13 2 4 12 30 100 2 7 40 15 55 1 11 15 8 19 1 20 1 3 5 1 of 2

Set No. 1

Code No: RR321502

5. Discuss various steady-state measures of performance for a generalized poission queuing model with C parallel servers. [16] 6. A company in the business of manufacturing equipments for chemical industry has taken up an order. The work to execute the order comprise following set of activities with time indicated against each. [16] Activity A B C D E F G H

Immediate preceeding activity A A C D B E,F,G

Activity time (week) 3 4 5 6 7 8 9 3

Estimate the time needed for completing the work or due date that can be promised to customer. 7. Discuss various factors to be considered during the selection of simulation software. [16] 8. What is the Quantile - Quantile plots differ from Histograms. ⋆⋆⋆⋆⋆

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Set No. 2

Code No: RR321502

III B.Tech II Semester Supplimentary Examinations, Aug/Sep 2007 MATHEMATICAL MODELLING AND SIMULATION (Computer Science & Systems Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Write a short essay on the definition and scope of operations research. (b) Solve the following L. P. Problem by graphical method: Max z = 2x1 + x2 subject to the constraints: x1 + 2x2 ≤ 10 x1 + x2 ≤ 6 x1 − x2 ≤ 2 x1 − 2x2 ≤ 1 and x1 , x2 ≥ 0.

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2. (a) Define constrained optimization problem in non - linear programming problem? [8] (b) Write a short note on Kuhn - Tucker conditions. 3. (a) Explain E. O. Q and sketch its graph.

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(b) Find the optimum order quantity for a product for which the price breaks are as follows: [10] Quantity Unit cost(Rs.) 0 ≤ q1 < 500 10.00 500 < q2 9.25 The monthly demand for a product is 200 units, the cost of storage is 2% of unit cost and the cost of ordering is Rs. 350. 4. (a) Explain ABC analysis.

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(b) What are its advantages and limitations, if any.

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5. Patients arrive at a clinic according to a poisson distribution at a rate of 30 patients per hour. The waiting room does not accommodate more than 14 patients. Examination time per patient is exponential with mean rate of 20 per hour. [16] (a) Find the effective arrival rate at the clinic (b) What is the probability that an arriving patient will not wait? (c) What is the expected waiting time until a patient is discharged form the clinic? 6. The following information is available about the various activities of a network.

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Set No. 2

Code No: RR321502 Activity 1-2 1-3 2-3

Normal Crash Duration (weeks) Cost (Rs.) Duration (weeks) Cost (Rs.) 4 4,000 3 7,000 8 5,000 7 8,000 5 8,000 3 10,000

Project overhead costs are at Rs.2,000 per week. Determine: (a) Direct cost duration relationship (b) Total cost duration relationship (c) Also draw the least cost network. 7. (a) Explain the role of state descriptor in discrete system simulation (b) Define the terms

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i. Discrete event ii. Simulation time iii. Clock time (c) Explain the representation of time in discrete system simulation. 8. List out the commonly used parameter estimators for various probability distributions. ⋆⋆⋆⋆⋆

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Set No. 3

Code No: RR321502

III B.Tech II Semester Supplimentary Examinations, Aug/Sep 2007 MATHEMATICAL MODELLING AND SIMULATION (Computer Science & Systems Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Write a short notes on the following:

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i. Analytical method ii. Heuristic method. (b) Compute all the basic feasible solutions of the L. P. problem: Maximize z =2x1 + 3x2 + 4x3 − 7x4 subject to the constraints: 2x1 + 3x2 − x3 + 4x4 = 8 x1 − 2x2 + 6x3 − 7x4 = −3 and choose that one which maximizes z.

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2. (a) Distinguish between transportation model and assignment model.

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(b) Four new machines M1 , M2 and M3 and M4 are to be installed in a machine shop. There are five vacant places A, B, C, D and E available. Because of limited space, machine M2 cannot be placed at C and M3 cannot be placed at A. Cij , the assignment cost of machine i to be place j in rupees is shown below:

M1 M2 M3 M4

A 4 7 9

B 6 4 6 3

C D E 10 5 6 - 5 4 9 6 2 7 2 3

Find the optimal assignment schedule. 3. (a) Describe the basic characteristics of inventory system.

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(b) A company has a demand of 12,000 units/year for an item and it can produce 2000 such items per month. The cost of one setup is Rs. 400 and the holding cost / unit/ month is Rs. 0.15. Find the optimum lot size and the total cost per year, assuming the cost of 1 unit as Rs. 4. Also, find the maximum inventory. [12] 4. (a) Explain ABC analysis.

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(b) What are its advantages and limitations, if any.

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5. Patients arrive at a clinic according to a poisson distribution at a rate of 30 patients per hour. The waiting room does not accommodate more than 14 patients. Examination time per patient is exponential with mean rate of 20 per hour. [16] 1 of 2

Set No. 3

Code No: RR321502 (a) Find the effective arrival rate at the clinic

(b) What is the probability that an arriving patient will not wait? (c) What is the expected waiting time until a patient is discharged form the clinic? 6. A PERT network has the following activities with their time estimates given below: Activity 0-1 0-2 0-3 1-2 1-5 2-4 3-4 3-5 4-5

Optimistic (days) Most likely (days) Pessimistic (days) 2 3.5 8 3 3.75 6 1 2.5 7 3 7.5 9 4 5.5 10 2 5 8 2 2.75 5 3 6 9 2 5 8

(a) Construct a network and find the expected completion time of the project. (b) Find the probability of completing the project 3 days ahead of the expected schedule. [16] 7. Name three of the principal entities, attributes and activities to be considered if one simulates the operation of [4x4=16] (a) Cafeteria (b) Barber Shop (c) Software development centre (d) A hospital emergency room 8. Discuss the steps in the development of a useful model of input data with suitable example. [16] ⋆⋆⋆⋆⋆

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Set No. 4

Code No: RR321502

III B.Tech II Semester Supplimentary Examinations, Aug/Sep 2007 MATHEMATICAL MODELLING 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 a model? Discuss various classification schemes of models. (b) Find all basic solutions for the problem Max z = x1 + 2x2 subjected to x1 + x2 ≤ 10 2x1 − x2 ≤ 40 and x1 , x2 ≥ 0.

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2. A company manufacturing air - coolers has two plants located at Mumbai and Kolkata with capacities of 200 Units and 100 units per week respectively. The company supplies the air coolers to its four show rooms situated at Ranchi, Delhi, Lucknow and Kanapur which have a maximum demand of 75, 100, 100 and 30 units respectively. Due to the difference in raw material cost and transportation cost, the profit per unit in rupees differs which is shown in the table below: [16] Ranchi Delhi Lucknow Kanpur Mumbai 90 90 100 100 Kolkata 50 70 130 85 Find the production programme so as to maximize the profit. The company may have its production capacity at any plant partly unused. 3. (a) What are the types of inventory? Why they are maintained?

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(b) A particular item has a demand of 9,000 units/year. The cost of one procurement is Rs. 100 and the holding cost per unit is Rs. 2.40 per year. The replacement is instaneous and no shortages are allowed determine. [10] i. ii. iii. iv.

the economic lot size the number of orders per year the time between orders total cost per year if the cost of one unit is Rs. 1.

4. Explain M.R.P. with a suitable example. Write a short notes on optimal Replenishment system. [16] 5. Patients arrive at a clinic according to a poisson distribution at a rate of 30 patients per hour. The waiting room does not accommodate more than 14 patients. Examination time per patient is exponential with mean rate of 20 per hour. [16] 1 of 2

Set No. 4

Code No: RR321502 (a) Find the effective arrival rate at the clinic

(b) What is the probability that an arriving patient will not wait? (c) What is the expected waiting time until a patient is discharged form the clinic? 6. By using the data given below, draw the PERT network and find the critical path. With in how many days would you expect the project to be completed with 99% chances. [16] Activity A B C D E F G H I J K L

Predecessor activity A A B B C,D E C,D G,H F J,K

t0 tm tp (days) (days) (days) 2 2 2 1 3 7 4 7 8 3 5 7 2 6 9 5 9 11 3 6 8 2 6 9 3 5 8 1 3 4 4 8 11 2 5 7

7. (a) Explain the role of state descriptor in discrete system simulation (b) Define the terms

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i. Discrete event ii. Simulation time iii. Clock time (c) Explain the representation of time in discrete system simulation. 8. What is the Quantile - Quantile plots differ from Histograms. ⋆⋆⋆⋆⋆

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