Please check that this question paper contains_09_questions and _04_printed pages within first ten minutes. MORNING [Total No. of Questions: 09] [Total No. of Pages: 04] Uni. Roll No. ...esseeeseees O 1 Nak wis Program: B.Tech, (Batch 2018 onward) Semester: 5! Name of Subject: Operation Research Subject Code: HSMME-101 Paper ID: 16379 Time Allowed: 03 Hours Max. Marks: 60 NOTE: 1) Parts A and B are compulsory 2) Part C has Two Questions, Q8 and Q9. Both are compulsory but with internal choice. 3) Any missing data may be assumed appropriately Part—A [Marks: 02 each] a. (a) Discuss the scope and limitations of OR in real-life problems. (b) Solve the linear programming problem graphically Maximize Z = 5x + 2y Subject to xtysl0, x+y <4, xy>0 (©) State the common and distinguishing features of the assignment and the transportation models. (@)_ Write four limitations of game theory. (©) What is the difference between PERT and CPM? (What is degeneracy in the transportation problem Part-B [Marks: 04 each] Q2. Solve the following L.P. problem using the simplex method: Maximize Z = 10x1 + 15x2 + 20x3 Subject to 2x1 + Axo + 6x3 $24 3x1 + 9x2 + 6x3 $30 XI, X2,%3 20 Q3. Find the initial basic feasible solution to the following transportation problem by North ‘West Comer Rule and Vogel’s Approximation Method Page 1 of 4 P.T.O.Qs. Qs. 6. Q MORNING Ot Max 2026 ‘Warehouse WI W2 W3 W4 WS Available FL 7 6 4 5 9 40 Factory F2 8 5 6 7 8 30 FB 6 8 9 6 5 20 Fa 5 7 7 8 6 10 Required 30 30 15 20 s 100 (total) Four new machines M-1, M-2, M-3 and M-4, 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 M- 2 cannot be placed at C and M-3 cannot be placed at A. Ci, the assignment cost of machine i to place j in rupees is shown below. Find the optimal assignment schedule. A]B[C[DIE MI [4/6/10/5/6 M2[7[4|-[s]4]| D3 é[9 ela twa To [377 [273 ‘There are five jobs, each of yghich is to be processed through three machines A, B, and Cin the order ABC. Processing times in hours are Jobae Am | GBR |C i oma 7 2 ear aes! 9 3 7 [1 3 4 5 6 5 4 [3 | 10 Determine the optimum sequence for the five jobs and the minimum elapsed time. Also find the idle time for the three machines and waiting time for the jobs. A and B play a game in which each has three coins a 5p, a 10p and a 20p. Each player selects a coin without the knowledge of the other's choice. If the sum of the coins is an odd amount, A wins B's coin, if the sum is even, B wins A's coin, Find the best strategy for each player and the value of the game. A project schedule has the following characteristics: Activity | Time (weeks) | Activity | Time (weeks) 1-2 4 5-6 4 1-3 1 3-7 8 2-4 1 6-8 I 34 1 7-8 2 3-5 6 8-10 5 4-9 3 9-10 7 Page 2 of 4 PTO.a8. Q. MORNING ) Construct the network, (i) Compute E and L for each event, and OT MAK tues (iii) Find the eritical path. Part-C [Marks: 12 each] ‘A company has three media A, B and C available for advertising its product. The data collected-over the past years about the relationship between the sales and frequency of advertisement in the different media is as follows: Frequency/month_| Estimated sale (units) per month A B c 1 125 180 300 2 225 290 330 3 260 340 450 q 300 370 300 The advertisement cost is Rs. 5,000 in medium A, Rs 10,000 in medium B and Rs. 20,000 in medium C. The total budget allocated for advertising the product is Rs. 40,000. Determine the optimal combination of advertising media and frequency. OR The project schedule has the following characteristics: Activity | t0 | tm | tp | Activity | to | tm | tp 1-2 143 [3 [57 4 {5 |6 23 1 ]2 [3 [67 6 |7 {8 24 1 ]3 [5 |78 2 [4 {6 3-5 3 |4 |s [79 4 [6 |8 45 2 |3 4 |810 1/2 |3 4-6 3 | |7 [9-10 3 |5 [7 (a) Construct the project network. (b) Find expected duration and variance for each activity. (©) Find the critical path and expected project length. (@) What is the probability of completing the project in 30 days? A plant manufactures two products, A and B. The profit contribution of each product has been estimated as Rs. 20 for product A and Rs. 24 for product B. Each product passes through three departments of the plant. The time required for each product and total time available in each department are as follows: A company has a contract to supply at least 250 units of product B per month. Formulate the problem as a linear programming model and solve it Page 3 of 4 PTOMORNING 01 Man 202 Hour Required Available hours rtm Department | Product A| Product B | ioe month i 2 6 1500 2 3 2 1500 3 I i 600 oR The arrival rate of telephone calls at a telephone booth is according to Poisson distribution, with an average time of 9 minutes between two consecutive arrivals. The length of a telephone call is assumed to be exponentially distributed, with a mean of 3 minutes. a) Determine the probability that a person arriving at the booth will have to wait. b) Find the average queue length that is formed from time to time. ©) The telephone company will install a second booth when convinced that arrival would expect to wait af least four minutes for the phone. Fins the increase in flow rate of arrivals which will justify a second booth. 4) What probability will an arrival have to wait for more than 10 minutes before the phone is free? ©) 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? ) Find the fraction of a day that the phone will be in use. Page 4 of 4