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2003 Course Operations Research & Management

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Total No. of Questions : 6] [Total No. of Pages : 6 [3864]-191 P1354 B.E. (Prod./SW) OPERATIONS RESEARCH & MANAGEMENT (2003 Course) (411126) Time : 3 Hours] [Max. Marks : 100 Instructions to the candidates : 1) Answers to the two sections should be written in separate books. 2) Neat diagrams must be drawn wherever necessary. 3) Figures to the right indicate full marks. 4) Use of logarithmic tables, slide rule, Mollier charts, electronic pocket calculator and steam tables is allowed. 5) Assume suitable data, if necessary & state it clearly. 6) All questions are compulsory. SECTION - I Q1) a) Solve the following L.P.P. by Simplex method. [11] Max. Z = 3x1 + 5x2, s.t. x1 4, 2x2 12, 3x1 + 2x2 18 & x1, x2 0. b) Form the dual of above & write values of dual decision variables from primal final table. [5] c) Do the RHS ranging for 3rd constraint of primal. [2] OR a) Mathematics of OR is mathematics of optimization - Explain. [3] b) Explain Big-M method of solving L.P.P. [6] c) Explain sensitivity analysis in L.P.P. w.r.t. addition of new variable. [2] d) Define the following w.r.t. L.P.P. i) Feasible region ii) [4] Redundant constraint iii) Degenerate solution iv) Slack variable e) Sketch the graphical solution for alternate & unbounded solution. [3] P.T.O. Q2) a) Solve the following transportation problem for minimization. To ai 2 4 5 3 1 8 5 4 7 7 1 6 2 14 7 bj 7 3 From [10] 9 18 b) Write the L.P. form of Transportation problem. [3] c) What is a meaning of sub-optimal solution for travelling salesman problem ? [3] OR a) The Captain of a cricket team has to allot five middle batting positions. The average runs scored by each batsman at these positions are as follows. Batsman II Batting Positions III IV V VI P 40 40 35 25 50 Q 42 30 16 25 27 R 50 48 40 60 50 S 20 19 20 18 25 T 58 60 59 55 53 i) Find the assignment of batsmen to positions which would give the maximum number of Re. runs. [5] ii) If it is decided to give 4th position to P , how the decision is going to change? [5] b) What are trans-shipment problems? How they are solved? [4] c) Write mathematical expression for assignment problem. [2] [3864]-191 -2- Q3) a) Find the sequence that minimises the total time required in performing the following jobs on three machines in order ABC. Processing time in minutes is given below: [7] Jobs I II III IV V M/C A 8 10 6 7 11 B 5 6 2 3 4 C 4 9 8 6 5 b) Derive the basic EOQ formula with inventory cycle sketch. [6] c) Define: [3] i) Reorder level ii) Buffer stock iii) Average inventory level. OR a) Mention any six optimality criteria in sequencing problems. [6] b) Discuss the concept of service level & its use in probabilistic inventory model. [3] c) Find the optimal order quantity for a product for which the price breaks are as follows. [7] Q 0 < Q < 600 Unit cost (Rs.) Rs. 10/- 600 < Q < 750 750 < Q Rs. 9.25/Rs. 8.75/SECTION - II Q4) a) State the advantages of simulation. [4] b) A dentist schedules all his patients for 30 min. appointments. Some of the patients take more or less time than 30 min. depending on the type of dental work to be done. The following summary shows the various categories of work, their probabilities and time needed to complete the work. Simulate the dentist s clinic for 4 hours starting from 8.00 am & [12] determine average waiting time for patients as well as dentist. [3864]-191 -3- Random nos. 40, 82, 11, 34, 25, 66, 17, 19 Category Time required (min.) Probability of category Filling 45 0.40 Crowning 60 0.15 Cleaning 15 0.15 Extraction 45 0.10 Check up 15 0.20 OR a) Discuss the replacement policy for items that fail suddenly. [4] b) State the limitations of simulation. [3] c) Two functionally identical machines P & Q are available in market with no scrap value. Machine P costs Rs. 12,000/- & its annual cost is Rs.400/- in 1st year, progressively increases by Rs.100/- in next 2 years, then by Rs.200/- in next 2 years & finally by Rs.300, 400, 600 & 800 in subsequent years. Machine Q costs Rs. 13,000/- & annual cost is Rs. 200, 350, 550, 750, 1000, 1300, 1800, 2400 & 3000 from year to years. If worth of money remains constant, which is better choice, P or Q? And in that case, what should be the replacement policy? [9] Q5) a) On an average 96 patients per 24 hours day require the service of an emergency clinic. Also on the average, a patient requires 10 min. of active attention. Assume that the fascility can handle only one emergency at a time. Suppose it costs clinic Rs. 100/- per patient treated to obtain an average servicing time of 10 min. and that each minute of decrease in this average time would cost the clinic Rs. 10/- per patient treated. How much would have to be budgeted by the clinic of decrease the average size of queue from present 4/3 patients to 1/2 patients? [9] b) Mention the costs involved in queuing theory. Sketch cost-service level tradeoff. [3] c) Discuss Column & Row Dominance. [3864]-191 -4- [4] OR a) Write LP form of the following game from view point of player B. [6] B 4 3 3 1 6 3 A 1 4 2 b) North & South Korea armies are at war. N Korea has 2 air bases, one of which is 3 times valuable than other. S Korea can destroy an undefended air base, but it can destroy only one of them. N Korea can also defend only one of them. Find the best strategy for N Korea to minimizes it s losses. [6] c) Explain Kendall s notations used in queuing models. [4] Q6) a) A small project is composed as follows Activity 1-2 1-3 1-4 2-5 3-5 4-6 5-6 Lowest time Medium " 1 1 1 4 2 2 1 1 2 5 2 5 3 6 Highest " 7 7 8 1 14 8 15 i) Draw the Network & show C.P. [3] ii) Find expected duration & variance of each activity. [4] iii) What is relative probability (compare to expected duration of project) of project completion two days latter than expected? [2] b) A small project has following data. Activity Time Normal (Week) Crash Cost penalty/ Week (Rs.) [3864]-191 1-2 1-3 1-4 2-4 2-5 3-6 4-6 5-6 6 4 8 4 5 3 3 3 5 3 12 8 8 5 6 6 2400 2700 1500 1200 6000 1500 -5- The cost of completing all activities in normal time is Rs.2,00,000/without overheads or indirect cost, which is Rs. 4,900/- per week. i) Identify critical path, find out normal duration & normal cost for project. [4] ii) Find out optimum duration & minimum cost for project. iii) Calculate total float for activity 2-5. [3] [2] OR a) A project data is given below. Activity 1-2 1-3 1-4 2-5 2-6 3-7 4-8 5-9 6-9 7-8 8-9 Duration 2 2 0 2 5 4 5 6 3 4 6 Man Power requirement 5 4 0 2 3 6 2 8 7 4 3 There are 11 persons employed on this project. Carry out approximate man-power levelling so that the fluctuations of work force requirement from day to day is as small as possible. [15] b) Discuss Fulkerson s rules for numbering the events. rrr [3864]-191 -6- [3]

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