Rr321502-mathematical-modelling--and--simulation

Set No. 1

Code No: RR321502

III B.Tech Supplimentary Examinations, Aug/Sep 2008 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 simplex? Write the steps used in the simplex method. (b) Express the following L. P. problem in standard from: Minimize z = x1 − 2x2 + x1 subject to the constraints: 2x1 + 3x2 + 4x3 ≥ −4 3x1 + 5x2 + 2x3 ≥ −7 x1 ≥ 0, x2 ≥ 0 and x3 is unrestricted in sign.

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2. (a) How the problem of degeneracy arises in a transportation? Explain how does one over come it? [4] (b) Consider the following unbalanced transportation problem? 1 2 3 Supply 1 5 1 7 10 2 6 4 6 80 3 3 2 5 15 Demand 75 20 50 Since there is not enough supply, some of the demands at these destinations may not be satisfied. Suppose there are penalty costs for every unsatisfied demand unit which are given by 5, 6 and 2 for destination 1, 2 and 3 respectively. Find the optimal solution. [12] 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. Make an ABC analysis for the following items in a store and construct the ABC analysis chart [16] 5. At a railway station, only one train is handled at a time. The railway yard is sufficient only for two trains to wait while others is given signal to leave the station. Trains arrive at the station at an average rate of 6 per hour and the railway station 1 of 2

Set No. 1

Code No: RR321502

can handle them on an average of 12 per hour. Assuming poisson arrivals and exponential service distribution, find the steady-state probabilities for the various number of trains in the system. Also find the average waiting time of a new train coming into the yard. [16] 6. A small maintenance project consists of the following 12 jobs with duration in days. Summarize the CPM calculations in standard tabular form calculating total, free and independent floats of the jobs. [16] Job Duration 1-2 2 3-4 3 5-8 5 7-9 4 2-3 7 3-5 5 6-7 8 8-9 1 2-4 3 4-6 3 6-10 4 9-10 7 7. (a) Generate a sequence for three two-digit random integers using linear congruential method [8] Let x0 = 27, a = 8, c = 47 and m = 100 (b) Generate a sequence of four three-digit random integers using multiplicative congruential method [8] Let x0 = 117, a = 43, and m = 1000 8. Explain the iterative process of calibrating a model with a schematic diagram. [16] ⋆⋆⋆⋆⋆

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Set No. 2

Code No: RR321502

III B.Tech Supplimentary Examinations, Aug/Sep 2008 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. 2. Explain briefly:

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[5+5+6]

(a) North - West corner rule (b) Minimum matrix method (c) Vogel?s approximation method, for finding an initial basic feasible solution for a transportation problem. 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) 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?

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Set No. 2

Code No: RR321502

6. (a) Explain PERT and its importance in network analysis. What are the requirements for applications of PERT techniques. [10] (b) List at the differences between PERT and CPM 7. Explain the execution of simulation algorithm in

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(a) SIM SCRIPT (b) GPSS 8. Explain the process of calibration and validation of simulation models. ⋆⋆⋆⋆⋆

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[16]

Set No. 3

Code No: RR321502

III B.Tech Supplimentary Examinations, Aug/Sep 2008 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 do you mean by primal and dual problems? Is the number of constraints in the primal and dual the same? [8] (b) Obtain the dual of the L. P. P. Minimize z = x1 + x2 + x3 subject to the constraints: x1 − 3x2 + 4x3 = 5 x1 + 2x2 ≤ 3 2x2 − x3 ≥ 4 x1 ≥ 0,x2 ≥ 0 and x3 is unrestricted in sign.

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2. A manufacturing company produces two products: Radios and T. V. sets. Sales price relationships for these two products are given below: product Radios T V sets

Quantity demanded Unit price 1,500-5 p1 p1 3,800-10 p2 p2

The total cost functions for these two products are given by 200x1 + 0.1x22 and 300x2 + 0.1x22 respectively. The production takes places on two assembly lines. Radio sets are assembled on assembly line I and TV sets assembled on Assembly line II. Because of the limitations of the assembly - line capacities, the daily production is limited to no more than 80 radios sets and 60 TV sets. The production of both types of products requires electronic components. The production of each of these sets requires five units and six units of electronic equipment components respectively. The electronic component are supplied by another manufacturer and the supply is limited to 600 units per day. The company has 160 employees, i.e., the labour supply amounts to 160 man - days. The production of one unit of radio sets requires 1 man - day of labour, whereas 2 man - days of labour are required for a TV set. How many units of radio and TV sets should the company produce in order to maximize the total profit? Formulate the problem as a non-linear programming problem. [16] 3. (a) Write a shorts on the following:

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i. General inventory ii. Deterministic inventory iii. Dynamic inventory. (b) For a fixed order quantity system, find out 1 of 2

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Set No. 3

Code No: RR321502

i. economic order quantity ii. optimum buffer stock iii. reorder level, for an item to the following data: Annual consumption D = 10,000 units, cost of one unit = Re. 1.00 C3 = Rs. 12.00 per production run, C1 = Re. 0.24 per unit. Past lead times: 15 days, 25 days, 13 days, 14 days, 30 days, 17 days. 4. Describe various selective inventory management techniques. Explain how these techniques can be combined to develop broad policy guidelines for selective control. [16] 5. With respect to queuing system, explain the following

[4x4=16]

(a) Balking (b) Jockeying (c) Traffic Intensity (d) Queue length 6. (a) Discuss in brief i. ii. iii. iv.

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Dummy activity Free float Independent float Total float

(b) What are the three estimates needed for PERT analysis? How do you use these estimates to compute the expected activity time and the variance in activity time? [8] 7. List and discuss various periods in the history of simulation software.

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8. List out the commonly used parameter estimators for various probability distributions.

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Set No. 4

Code No: RR321502

III B.Tech Supplimentary Examinations, Aug/Sep 2008 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 such that 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 lan the production programme so as to maximize the profit. The company may have its production capacity at any plant partly unused. 3. What are the costs associated with inventory? Distinguish between deterministic and stochastic models in inventory theory. [16] 4. What is the use of ABC, VED and other classifications to departments other than inventory control? What is the use of purchasing, for maintenance, for quality control? [16] 5. With respect to queuing theory, explain the following

[8+8=16]

(a) Cost models in queuing theory (b) Non-poisson queues. 6. (a) Discuss in brief

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i. Dummy activity ii. Free float iii. Independent float 1 of 2

Set No. 4

Code No: RR321502 iv. Total float

(b) What are the three estimates needed for PERT analysis? How do you use these estimates to compute the expected activity time and the variance in activity time? [8] 7. (a) What are the various tests used to ensure the desirable properties in random numbers. [6] (b) Generate a sequence of random numbers with x0 = 27, a = 17, c = 43 and m = 100 8. (a) Distinguish model verification and validation (b) Explain conceptual and operational model-building process. ⋆⋆⋆⋆⋆

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