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2003 Course Operations Research

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Total No. of Questions : 12] [Total No. of Pages : 5 [3664] - 156 P937 B.E. PRODUCTION OPERATIONS RESEARCH (2003 Course) Time : 3 Hours] [Max. Marks : 100 Instructions to the candidates : 1) Solve one question from every unit in each section. 2) Answers to the two sections should be written in separate books. 3) Neat diagrams must be drawn wherever necessary. 4) Figures to the right indicate full marks. 5) Use of logarithmic tables, electronic pocket calculator is allowed. 6) Assume suitable data, if necessary. SECTION - I UNIT - I Q1) a) b) What is the significance of duality theory in L.P.P.? Dual of a dual is primal . Explain in brief. [6] Find the optimal solution by simplex method. Minimize z = x2 + 3x3 + 2x5 Subject to 3x2 x3 + 2x5 7 [10] 2x2 + 4x3 12 i) ii) 4x2 + 3x3 + 8x5 10 x2, x3, x5 0 Formulate the dual problem of the primal and solve by simplex method for values of dual variables. Write the values of the primal variables. OR Q2) a) b) What do you understand by the term sensitivity analysis? Discuss briefly the effect of variation of cost coefficients. [6] A firm produces three items A, B and C and requires two types of resources - man hours and raw material as given below in the table. Determine the optimum production schedule that maximizes profit. P.T.O. Resources Products A B C i) Man hours ii) Raw material 6 3 3 4 Available 5 5 45 30 Profit/unit Rs. 3 1 5 Also find the range on the unit profit of product A. What is the effect if profit / unit of A is Rs. 4? [10] UNIT - II Q3) a) Explain the similarities and differences between the transportation problem and transshipment problem. [6] b) In a flood situation, military have to supply the food in flood affected zone. Three aircrafts are available to drop the food to five villages. Find the number of trips each aircraft should make to each village so as to maximize total quantity of food. [10] Aircraft Villages Trips / day 1 10 5 2 8 3 3 6 8 4 9 4 5 12 10 75 135 7 9 6 20 4 90 100 80 70 40 20 A1 A2 A3 Trips/day to village OR Discuss assignment model. Indicate a method of solving a travelling salesman problem. [6] b) A department has four subordinates and four tasks are to be performed. The estimates of the profit in rupees each man would earn is given in the effectiveness matrix. How should the tasks be allocated one to each man, so as to maximize the total earnings? [10] Tasks Subordinate Q4) a) [3664] - 156 1 2 3 4 A 5 25 15 15 B 40 35 25 5 C 20 30 20 30 D 5 25 10 15 2 UNIT - III Q5) a) b) State the Bellman s principle of optimality and explain how it is used for solving dynamic programming problem. [8] A confectioner sells the confectionery items of which past data of demand per week in kg is : [10] Demand /week : 0 5 10 15 20 25 Frequency : 2 11 8 21 5 3 Using the following sequence of random numbers, generate the demand for next 15 weeks. Also find out the average demand per week. Use random numbers : 35, 52, 90, 13, 23, 73, 34, 83, 94, 57, 35, 56, 67, 66, 60. OR Q6) a) Write a note on i) ii) b) [8] Parametric programming. Nonlinear programming. Solve the following problem by branch and bound method. Maximize z = x1 + x2 Subject to 3x1 + 2x2 20 [10] 6x1 + 5x2 25 x1 + 3x2 10 x1 and x2 are non-negative integers. SECTION - II UNIT - IV Q7) a) b) Write a note on goal programming. [6] The following mortality rates have been observed for a certain type of fuses : [10] End of week 1 2 3 4 5 % of failing to 5 15 35 75 100 There are 1000 fuses in use and it costs Rs. 5 to replace an individual fuse. If all the fuses were replaced simultaneously, it would cost Rs. 1.25 per fuse. It is now proposed to replace all the fuses at fixed intervals of time irrespective of their state and to continue replacing burnt out fuses as they fail. At what intervals group replacement should be made? OR [3664] - 156 3 Q8) a) b) Explain the concept of geometric programming. [6] A firm is thinking of replacing a particular machine whose cost price is Rs. 12,200. The scrap price of this machine is only Rs. 200. The maintenance costs are found to be as follows : [10] Year Maintenance Cost Rs. 1 2 3 4 5 220 500 800 1200 1800 6 7 8 2500 3200 4000 Determine when the firm should go for m/c replacement. UNIT - V Q9) a) b) Explain in brief the characteristics of queuing models. [6] Two items of worth Rs. 100 and Rs. 150 each are to be auctioned at a public sale. There are only two bidders A and B. Bidder A has Rs. 125 and the bidder B has Rs. 155 with him. If each bidder wants to maximize his own return, what should be his strategy? [10] OR Q10)a) b) Explain the Kendal s notation for representing queuing models. Consider a game having the following payoff. Player B Player A i) ii) A1 B1 2 [6] [10] B2 6 2 A2 Show that whatever the value of may be, the game is strictly deterministic. Solve the game. UNIT - VI Q11)a) Explain the following terms in networks. [8] i) Earliest time ii) Latest time iii) Slack b) iv) Event. Table below shows jobs, normal and crash time and cost for a project. Indirect cost for project is Rs. 300 per day. [3664] - 156 4 Job i-j Normal Time (days) Cost (Rs.) Crash Time (days) Cost (Rs.) 1-2 1-3 6 8 1400 2000 4 5 1900 2800 2-3 2-4 3-4 4 3 Dummy 1100 800 - 2 2 - 1500 1400 - 2-5 4-6 6 10 900 2500 3 6 1600 3500 5-6 3 500 i) Draw network of the project. 2 800 ii) What is the normal duration and cost of the project? iii) If all activities are crashed, what will be the cost? [10] OR Q12)a) Define the physical meaning of the terms : total float, free float and independent float. [6] b) A small project consists of seven activities, whose time estimates are given below. [12] Activity to : : 1-2 3 1-3 2 1-4 2 2-5 2 3-5 1 4-6 4 5-6 1 tm tp : : 6 9 5 8 4 6 3 10 3 11 6 8 5 15 Where to is time optimistic, tm is time most likely and tp is time pessimistic. i) Draw the network diagram for the project. ii) Determine the cirtical path. iii) Calculate the variance and standard deviation. iv) What is the probability that the project will be completed by 14 weeks. z : Probability : [3664] - 156 0 0.5 0.5 1.0 1.5 2.0 0.692 0.841 0.933 0.977 5

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