B.E. Sixth Semester (Mechanical Engineering) (C.B.S.) Operations Research

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1 B.E. Sixth Semester (Mechanical Engineering) (C.B.S.) Operations Research P. Pages : 4 NKT/KS/1/39 Time : Three Hours *0561* Max. Marks : 80 Notes : 1. All questions carry marks as indicated.. Solve Question 1 Questions No.. 3. Solve Question 3 Questions No Solve Question 5 Questions No Solve Question Questions No Solve Question 9 Questions No Solve Question 11 Questions No Assume suitable data whenever necessary. 9. Use of normal distribution table is permitted. 10. Use of Random-number table is permitted. 1. a) Define operation research. Explain the importance of operation research in the decision making process. b) A firm makes two products X and Y and has a total production capacity of 9 tonnes per day. X and Y requiring the same production capacity. The firm has a permanent contract to supply at least tonnes of X and at least 3 tonnes of Y per day to another company. Each tonne of X requires 0 machine hours of production time and each tonnes of Y requires 50 machine hours of production time. The daily possible number of machine hours is 360. All the firm's output can be sold, and the profit made is Rs. 80 per tonne of X and Rs. 10 per tonnes of Y. It is required to determine the production schedule for maximum profit and calculate this profit.. Solve by simplex method Minimize Z 5x1 3x subject to constraints x x 5x x, x x x x A solicitors firm employ typist on hourly piece rate basis for their daily work. There are fine typists and their charges and speed are different. According to an earlier understanding only one job is given to one typist and he is paid for full hour even if he work for a fraction of an hour. Find the least cost allocation for the following data : Typist R/Hour No. of pages typed/hour Job No. of pages A 5 1 P 199 B 6 14 Q 15 C 3 8 R 145 D 4 10 S 98 E 4 11 T 18 NapurStudents.Org 4 9 NKT/KS/1/39 1 P.T.O

2 4. A manufacturer has distribution centers located at Agra, Allahabad, and Mumbai. These centers have 40, 0 and 40 units of product available with them. The retail outlets require following number of units A-5, B-10, C-0 and D-30. The transportation cost per unit in Rs. Between each centre and outlet is as follows. Find out the optional transportation program. Distribution Centre Retail outlets A B C D Agra Allahabad Mumbai a) An automobile manufacturer needs 50 piston per day in his assembly shop. The cost of ordering is Rs. 00/- order. The inventory holding cost is Rs. 0.15/unit/day. If the piston cost Rs. 160 and no shortages are permitted. i) Find optimal order quantity ii) What is optimal total expected system cost? iii) Find minimum cost procurement level if lead time is 4 days. iv) Sketch the inventory model. b) Write short notes on any two. 6 i) ABC analysis. ii) Cost Associated with Inventory. iii) Pure strategy, mixed strategy, saddle point. iv) Sequencing with proper example. 6. a) In a machine shop 8 different products are being manufactured, each requiring time on two machines M1 and M as given below : Machines Product I II III IV V VI VII VIII M1 (Time in min) M (Time in min) i) Decide the optimum sequence of processing of different products in order to minimize the total manufacturing time for all the products. ii) The total minimum elapsed time. iii) Idle time for machine M1 and Machine M. iv) Name the scheduling model used. NapurStudents.Org b) The labour contract between your management and the union will terminate in the near future. A new contract must be negotiated preferably before the old one expires. You are a member of management group charged with selecting a strategy for them during the coming negotiations. After a consideration of post experience, the group agrees that feasible strategies for the company and union are : i) All out attack, hard aggressive bargaining ii) A reasonably logical approach. iii) A legalistic strategy NKT/KS/1/39 8

3 iv) An agreeable conciliatory approach. The pay off table is given below : Union I II Strategies III IV I II Company Strategies III IV Find optimal strategy for the company. Determine the worth of your negotiation.. a) A small project consist of eight activities whose time estimates are given below : Activity Preceding Time Estimates (in weeks) Activity to tm tp A B C A D A E A.5 11 F B, C G D H E, F, G i) Draw a PERT Network for the project. ii) Determine the critical path iii) If a 30 week deadline is imposed what is the probability that the project will be finished within the time limit? iv) If the project manager want to be 99% sure that the project is completed on the scheduled date, how many week before that date should be start the project? 8. A small project consist of seven activities. The normal and crash cost and time is given in the table. If indirect cost is Rs. 160/- per day find optimum project duration. Activity Normal Crash Time (in days) Cost in (Rs.) Time (in days) Cost in (Rs.) NapurStudents.Org NKT/KS/1/39 3 P.T.O

4 9. A computer has a large number of electronic tubes. They are subjected to a mortality as given below : Period Age of failure (Internal time period) Probability of failure If the tubes are group replaced, the cost of replacement is Rs. 15 per tube. Group replacement can be done at a fixed internal in the night shift when the computer is not normally used. Replacement of individual tube which fail in service cost Rs. 60/- tube. How frequently should the tube be replaced? 10. A mini truck owner estimates that the running cost and salvage value of truck for various years as tabulated below. If purchase price of truck is Rs. 1,60,000, estimate optimal replacement age. Assume a rate of return on capital invested to 5% per year. Also assume running cost incurred in the beginning of the year. Year (k) Running cost (Rk) in Rs. Salvage value (Sk) in Rs a) Define simulation. Give one application area where this technique used in practice. 4 b) A company manufactures around 00 vehicles. Depending upon the availability of raw materials and other conditions, the daily production has been varying from 196 vehicles to 04 vehicles whose probability distribution is as given below : Production per day Probability The finished vehicles are transported in a specially designed three storeyed lorry that can accommodate only 00 vehicles. Using the given 15 random numbers, 8, 89, 8, 4, 53, 61, 18, 45, 04, 3, 50,,, 54, 10. Simulate the process to find out. i) What will be the average number of vehicles waiting in the factory? ii) What will be the average number of empty space on the lorry? 1. a) Patron arrive at a reception counter at an average inter arrival rate of min. The receptionist in duty takes an average of one minute per patron. i) What is the chance that a patron will straight way meet the receptionist. ii) For what portion of time the receptionist is busy? iii) What is the average queue length? iv) What is the average number of patrons in the system? v) What is the average waiting time of a patron? vi) What average time patron spends in the system? NapurStudents.Org b) What is queuing theory? What information can be obtained by analyzing a queuing system? 4 *********** 9 9 NKT/KS/1/39 4

5 B.E. Sixth Semester (Mechanical Engineering) (C.B.S.) Operations Research P. Pages : 4 KNT/KW/16/39 Time : Three Hours *016* Max. Marks : 80 Notes : 1. All questions carry marks as indicated.. Solve Question 1 Questions No.. 3. Solve Question 3 Questions No Solve Question 5 Questions No Solve Question Questions No Solve Question 9 Questions No Solve Question 11 Questions No Assume suitable data whenever necessary. 9. Illustrate your answers whenever necessary with the help of neat sketches. 10. Use of non programmable calculator is permitted. 11. Used of Random Number Table & Normal distribution chart is permitted. 1. a) Define operation research? Explain the scope and application of operation research? 6 b) XYZ company produces an automobile spare part. The contract that it has signed with large manufacturer calls for the following 4-month shipping schedule. Month No. of parts to be shipped January 3,000 February 4,000 March 5,000 April 5,000 The company can manufacture 3,000 parts per month on a regular time basis and,000 parts per month on an overtime basis. Its production cost is Rs. 15,000 for a part produced in regular time and 5,000 for a part produced in overtime. Its monthly inventory holding cost is Rs Formulate this problem as an LP model to minimize the overall cost.. a) Use Penalty (Big M) method to solve the following LP problem. Maximize Z x1 x 3x3 x4 Subject to constraints. x 3x 15 x1 3 x1 x 5x3 0 x1 x x3 x4 10 x1, x, x3 & 0 KNT/KW/16/39 1 P.T.O b) Solve following LP problem with graphical method. Max Z x 1 x Subject to x1 x 0 3x1 x 3 & x1, x

6 3. a) Differentiate between transportation and Transshipment model. 3 b) Consider a firm having two factories. The firm is to ship its products from the factories to three retail stores. The number of units available at factories X & Y are 00 and 300, respectively, while those demanded at retail stores A, B and C are 100, 150 and 50 respectively. Rather than shipping directly from factories to retail stores, it is asked to investigate the possibility of trans-shipment. The transportation cost (in rupees) per unit is given in table. 10 Factory Retail stores X 0 8 Factory Y 6 0 A Retail stores B 1 5 C 8 9 Find optimal shipping schedule. KNT/KW/16/39 X Y 4. An airline that operates seven days a week has the timetable which is given below. The crew must have a minimum layover time of five hours between flights. Flight No. Delhi - Jaipur Dep. Arrival A Flight No. B C Jaipur - Delhi Dep. Arrival am 8.00 am am 9.15 am am 9.00 am am 9.45 am pm.30 pm noon 1.15 pm pm.30 pm pm 6.45 pm Obtain the pairing of flights that minimizes layover time away from home. For any given pairing, the crew will be based at the city that results in the smaller layout. For each pair also mention the town where the crew should be based. 5. Explain the principle of dominance in game theory and solve the following. Player A A1 A A3 Player B B B B a) Find the sequence that minimizes the total time required in performing the following jobs on three machines in the order ABC. Processing times (in hours) are given in the following table. Job : MachineA : MachineB: MachineC : b) The production department for a company requires 3,600 kg of raw material for manufacturing a particular item per year. It has been estimated that the cost of placing an order is Rs.36 and the cost of carrying inventory is 5% of the investment in inventories. The price is Rs.10 per kg. The purchase manager wishes to determine an ordering policy for raw material. 6

7 . Consider a project having following activities & their time estimates as shown in table 1 a) Draw PERT network diagram for the project. b) Compute the expected project completion time. c) What should be the due date to have 0.90 probability of completion? d) Find the total float & free float for all non critical activities? Table 1 Activity Immediate Activity Time (weeks) Predecessor Most Optimistic Most likely Most Pessimistic A B C B D A, C E B F D, E G D, E H D, E I G 4 5 J F, I 8 10 K G L H 8 9 M J, K, L The following is a table showing details of a project. 14 Activity Immediate Normal Crash Predecessor Time (weeks) Cost (Rs'000) Time (weeks) Cost (Rs'000) A B C B D B E B F E G A, D, C Indirect cost is Rs. 400 per day. Find the optimum duration and the associated minimum project cost. 9. The data collected in running a machine, the cost of which is Rs. 60,000 are given below. Year : Resale value (Rs.) : 4,000 30,000 0,400 14,400 9,650 Cost of Spare (Rs.) : 4,000 4,0 4,880 5,00 6,800 Cost of labour (Rs.) : 14,000 16,000 18,000 1,000 5,000 Determine the optimum period for replacement of machine. 14 KNT/KW/16/39 3 P.T.O

8 10. A computer has a large number of electronic tubes. They are subject to mortality as given below. If the tubes are group replaced, the cost of replacement is Rs. 15 per tube Group replacement can be done at fixed intervals in the night shift when computer is not normally used. Replacement of individual tubes which fail in service cost Rs. 60 per tube. How frequently should the tubes be replaced? Period Age of failure (hrs) Probability of failures Arrivals at telephone booth are considered to be Poisson with an average time of 10 minutes between one arrival and the next. The length of phone call is assumed to be distributed exponentially, with mean 3 minutes. a) What is the probability that a person arriving at the booth will have to wait? b) The telephone department will install a second booth when convinced that an arrival would expect waiting for at least 3 minutes for a phone call. By how much should the flow of arrivals increase in order to justify a second booth? c) What is average length of queue that forms from time to time? d) What is the probability that it will take him more than 10 minutes altogether to wait for the phone and complete his call? 1. A dentist schedules all his patients for 30 minute appointments. Some of the patients take more or less than 30 minutes depending on the type of dental work to be done. The following summary shows the various categories of work, their probabilities and time actually needed to complete the work. Category of service Time required (minutes) Probability of category Filling Crown Cleaning Extraction Checkup Simulate the dentist's clinic for four hours and determine the average waiting time for patients as well as idleness of doctor. Assume that all the patients show up at the clinic at exactly their scheduled arrival time starting at 8.00 am. Use following random numbers for handling, the above problem ************ KNT/KW/16/39 4

9 Scanned by CamScanner Special Thanks to Abhijith Alolickal

10 Scanned by CamScanner Special Thanks to Abhijith Alolickal

11 Scanned by CamScanner Special Thanks to Abhijith Alolickal

12 Scanned by CamScanner Special Thanks to Abhijith Alolickal

13 B.E. (Mechanical Engineering) Sixth Semester (C.B.S.) Operations Research P. Pages : 4 NJR/KS/18/4536 Time : Three Hours *083* Max. Marks : 80 Notes : 1. All questions carry marks as indicated.. Solve Question 1 Questions No.. 3. Solve Question 3 Questions No Solve Question 5 Questions No Solve Question Questions No Solve Question 9 Questions No Solve Question 11 Questions No Due credit will be given to neatness and adequate dimensions. 9. Assume suitable data whenever necessary. 10. Use of non programmable calculator is permitted. 11. Use of normal distribution table is permitted. 1. Use of Random Tables is permitted. 1. a) A firm makes two products X and Y and has a total production capacity of a tonnes per day. X and Y require the same production capacity. The firm has a permanant contract to supply atleast tonnes of X and at least 3 tonnes of Y per day to another company. Each tonne of X requires 0 machine hours of production time and each tonne of Y requires 50 machine hours of production time. The daily possible number of machine hours is 360. All the forms output can be sold and the profit made is Rs. 80 per tonne of X and Rs. 10 per tonne of Y. It is required to determine the production schedule for maximum profit and calculate this profit. Use graphical method. b) Write dual of the following LPP Maximize Z 40 x1 10 x Subject to x1x 8 3x 5x x1 44x 660 x, x 0 1. A chemical company produces two components A & B. The following table gives the unit of ingredients C and D per kg of compound A and B as well as minimum requirements of C & D and cost per kg of A and B. Using simplex method find the quantities of A and B which would give a supply of C & D at a minimum cost. Compound Minimum A B requirement Ingredient C 1 80 D Cost / Kg NJR/KS/18/ P.T.O

14 3. A salesman wants to visit cities A, B, C, D and E. He does not want to visit any city twice before completing his tour of all cities and wishes to return to the point of starting journey. Cost of going from one city to another (in rupees) is shown in table. find the least cost route? 4. Consider the following unbalanced problem. To From A B C D E A B C D E To D E F Supply From A B C Demand 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, 3 and for D, E and F respectively. Find out the optimal solution. 5. a) Compute EOQ, optimum number of orders, time between two consecutive orders and optimum inventory cost considering EOQ. Demand rate units, unit price Rs. 30, ordering cost Rs. 16, storage rate % per annum, interest 10% per annum and obsolescence rate 8% per annum. b) A m/c operator has to perform three operations turning, threading and knurling on a number of different jobs. The time required to perform these operations in minutes for each job is known. Determine the order in which jobs should be processed in order to minimize the total time required to turn out all the jobs. also find the idle time for the three operations. Job (s) Turning (Time in minutes) Threading (Time in minutes) Knurling (Time in minutes) a) Reduce the following game by dominance and find the game value. Strategy of Rakesh Strategy of NAVED A B C A B C NJR/KS/18/4536

15 b) Write short notes on : i) Significance of ii) Various costs associated with Inventory control.. Project has following seven activities from A to G. The time estimate in days for different activities are given in the following table. i) Draw PERT Network for the Project. ii) Determine the critical path. iii) Find out the probability of completing the project in 19 days. Activity Depends on t o t m t p A B C -- 8 D A E B 5 14 F C 5 8 G D, E The following is a table showing the details of a project. Find the optimum duration and associated minimum project cost. Normal Crash Activity Depends on Time (weeks) Cost (Rs' 000) Time (weeks) Cost (Rs' 000) A B C B D B E B F E G A, D, C Indirect cost is Rs. 400 per week. 9. A SPM two years ago was brought for Rs. 1,60,000/- This machine A can be replaced by m/c. B. The operational and maintenance cost for m/c A is Rs and Rs for first two years and then increases by Rs per year subsequently. The optimum present net worth cost associated with m/c B is Rs. 3,50,000/- at 0% rate of interest. Decide whether m/c A is to be replaced by m/c B or not. 10. Following rates have been observed for certain type of electric bulbs. There are 1000 bulbs in use. If the cost is Rs. to replace the bulb individually at Rs to replace the bulb in group. Find out optimum replacement policy. What should be group replacement cost if individual policy is adopted End of Week % Probability of failure to date NJR/KS/18/ P.T.O

16 11. Workers come to tool store room to receive special tools (required by them) for accomplishing a particular project assigned to them. The average time between two arrivals is 60 seconds and the arrivals are assumed to be Poisson distribution. The average service time (of tool room attendant) is 40 seconds. Determine : i) Average number of workers in the system. ii) Average number of workers in the Queue. iii) Average time a worker spends in a system. iv) Average time workers wait in Queue. v) Average length of non empty Queue. vi) Utilization factor. vii) Probability that there are 6 workers in system. 1. An airline has 15 flights leaving a base per day each with an air hostess. The airline keeps three airhostesses in reserve, so that they may be called in case the scheduled air hostess for a flight reports sick. The probability distribution for the number of sick hostesses is as below. Number of sick air hostesses Probability Use Monte Carlo simulation technique to estimate the utilisation of reserved air hostesses and also the probability that atleast one flight is cancelled in a day because of unavailability of air hostess. *********** NJR/KS/18/4536 4