Model Question Paper - I - Or - 5th Sem 5k6r11

M.S. RAMAIAH INSTITUTE OF TECHNOLOGY, BANGALORE – 54 (Autonomous Institute, d to VTU) DEPARTMENT OF INDUSTRIAL ENGINEERING AND MANAGEMENT MODEL QUESTION PAPER – 1 OPERATIONS RESEARCH – IM504 Time: 3 hrs

Max. Marks: 100

NOTE: 1) PART A is Compulsory 2) PART B Answer five full questions choosing one from each unit PART – A 1) The mathematical technique for finding the best use of limited resources in an optimum manner is known as (a) Operations Research (b) Linear Programming (c) Network Analysis (d) Queuing Theory 2) A linear programming problem is called so, because in that problem (a) all the functions expressing the constraints are linear (b) the objective function should also be linear (c) both (a) & (b) above (d) none of the above. 3) Linear programming can be applied to (a) steel industry (b) oil industry (c) chemical industry (d) all of the above 4) In the graphical method of linear programming problem, every corner of the feasible polygon indicates (a) a basic feasible solution (b) optimum solution (c) both (a) & (b) above (d) none of the above. 5) In canonical form of linear programming model (a) all the inequalities indicating the constraints are in the same manner (b) the right-hand side of the inequation are constraints (c) all the variables are nonnegative (d) all of the above 6) The simplex method is the basic method for (a) value analysis (b) operations research (c) linear programming (d) model analysis 7) In the simplex method, the variables which have not been assigned the value zero, during an iteration, are called as

(a) basic variable (b) artificial variable (c) slack variable (d) none of the above. 8) A basic feasible solution in simplex method is one, when (a) all the decision variables are in the base (b) all the decision variables and surplus variables are assigned zero values (c) all the base variables are non – negative (d) all the base variables satisfy the constraint equations. 9) Graphical method, simplex method and transportation method are concerned with (a) value analysis (b) linear programming (c) break-even analysis (d) queuing theory 2 10) Artificial variable is introduced in the simplex method to (a) determine the IBFS, when surplus variable is present (b) convert the inequation with the sign greater than or equal to, in the form of an equation (c) apply Big- M method for solution to LPP (d) indicate the sensitivity of surplus variable. 11) Dual of the dual is the (a) dual (b) primal (c) either dual or primal (d) none of the above 12) At the optimum level (i.e., final iteration) the values of the slack variables in the primals are (a) positive (b) negative (c) zero (d) none of the above 13) In transportation problem, during an iteration, the total no. of allocation should not be more than (a) m + n (b) m + n + 1 (c) m + n – 1 (d) m – n – 1 14) The IBFS can be determined by the application of (a) northwest corner method (b) least cost method (c) vogel’s approximation method (d) all the above 15) Optimality is reached when all the index values are (a) zero (b) negative (c) positive (d) none of the above 16) In assignment model (a) degeneracy always present in all the problems (b) no. of resources is equal to no. of jobs (c) only one unit of the ith source can be assigned to any one of its destinations (d) all the above

17) In an n x n matrix of an assignment problem, the optimality is reached when the minimum number of straight lines scoring all the zeros is (a) n2 (b) 1 / n (c) n (d) none of the above 18) In pert network, some times dummy activities are introduced to (a) prevent cycling of network (b) maintain network logic of precedence (c) avoid crossing of activities in the network (d) have an activity with zero duration of time 19) A pert activity has an optimistic tie of 3 days, pessimistic time of 15 days and the expected time is 7 days. The most likely time of the activity is (a) 5 days (b) 6 days (c) 7 days (d) 9 days 20) The estimated duration of times for an activity, in the PERT network under the worst and best environment are 9 days and 3 days. The variance of this activity is (a) 6 days (b) 1 day (c) 2 days (d) none of the above 21) In a M/M/1 Queue, the service rate is (a) poisson (b) exponential (c) linear (d) none of the above 3 22) M/M/1 model is (a) a single channel & single service unit (b) a “First come, First serve basis” (c) a system having probability distribution independent of time (d) all of the above 23) Queue length (Lq) is equal to ς ς 1 ς 1 ς (a) 1 −ς (b) (c) λ (1 −ς ) (d) M (1 −ς ) 1 −ς 24) A situation is termed as competitive situation when (a) the action of one depends on the action of the other (b) the action of one does not depend on the action of the other (c) the action of one does not follow any logic (d) none of the above 2

25) A pay-off, which is an outcome of all combination of courses of action is (a) positive (b) negative (c) zero (d) all of the above PART – B UNIT – I 1 a) What is OR? Describe briefly its applications. 5

b) The manager of an oil refinery has to decide upon the optimal mix of 2 possible blending processes of which the inputs and outputs per production run are as follows 10 Process 1 2

Input Crude A 5 4

Output Gasoline X Gasoline Y

Crude B 3 5

5 4

8 4

The maximum amounts available of Crude A & B are 200 units and 150 units respectively. Market requirements shows that at least 100 units of gasoline X and 80 units of gasoline must be produced. Profits per production run from process 1 & process 2 are Rs.30 & Rs.40 respectively. Formulate the LPP & solve it graphically. 2 a) List and explain the important phases of OR study. 6 b) Solve the following LPP graphically 9 Maximize Z = 2x1 + x2 Subject to x2 ≤ 10 2x1 + x2 ≤ 60 x1 + x2 ≤ 18 3x1 + x2 ≤ 44 x1, x2 ≥ 0

4 UNIT – II 3) Solve the following LPP by Simplex method 15 Maximize Z = 5x1 + 2x2 + 10x3 Subject to x1 – x3 ≤ 10 x2 – 3x3 ≥ 10 x1 + x2 + x3 ≤ 10 x1, x2, x3 ≥ 0 4) Solve the following LPP by using the dual simplex method 15 Maximize Z = 6x1 + 7x2 + 3x3 + 5x4 Subject to 5x1 + 6x2 – 3x3 + 4x4 ≥ 12 x2 + 5x3 - 6x4 ≥ 10 2x1 + 5x2 + x3 + x4 ≥ 8 x1, x2, x3, x4 ≥ 0

UNIT – III 5 a) What are the rules to determine degeneracy? 3 b) Freshwater is supplied through 3 reservoirs with daily supply capacities of 15, 20 & 25 million ltrs of water respectively. On each day supply must be provided for 4 cities A, B, C & D whose demands are 8, 10, 12 & 15 million litres respectively whose costs are given below: 12 Cities

Reservoirs

1 2

A 2 3

B 3 2

C 4 5

D 5 2

3

4

1

5

2

Use the transportation algorithm to determine the cheapest pumping schedule, if excess water can be disposed of at no cost. 6) A firm produces a component and distributes them to 5 wholesalers at a fixed price of Rs. 10/unit. Sales forecast indicate that monthly demand will be 3,000, 3,000, 1,000, 5,000 and 4,000 units at wholesale dealers a, b, c, d & e respectively. The monthly production capacities are 5,000, 1,000 & 10,000 at plants A, B & C respectively. The production costs are Rs. 2, Rs. 1 & Rs. 3 at plants A, B & C respectively. The unit transportation cost in rupees between the plants and wholesalers are given in the following table 15 Plants

Wholesales a b c

d

e

A 0.5 0.5 1.0 1.5 1.5 B 1.0 0.5 1.0 1.0 1.5 C 1.0 1.0 0.5 1.5 1.0 Determine the transportation schedule between plants and wholesalers in order to maximize the total profit per month. Use VAM to obtain the IBFS. 5 UNIT – IV 7 a) The owner of a machine shop has 4 machinists available to assign to jobs for that day. Five

jobs are offered with expected profit for each machinist on each job as follows 9

1 2 3 4

A

B

C

D

E

60 69 85 46

76 82 90 62

48 59 109 85

99 71 69 75

80 57 79 78

Determine the optimum assignment schedule. Which job should be declined? b) The following table gives the activities of a construction project and other relevant information 10 Activity

1–2 1–3

Duration

20

2–3 25

2–4

10

12

3–4 6

4–5

10

(i) Draw the network for the project (ii) Find critical path 8) The following table lists the jobs of a network along with their time estimates 15 Duration in days Jobs Optimistic Most likely Pessimistic 1 2 2 3 3 5 3 4 (i) (ii) (iii)

– – – – – – – –

2 3 4 4 5 6 6 6

3 6 5 3 1 2 4 2

6

15 12 11 9 4

5

30 17 27 7 8

19 5

28 14

Draw the network and calculate the length and variance of critical path What is the probability that jobs on critical path will be completed by the due date of 40 days? What is your estimate of the probability that the entire project will be completed by the due date?

6 UNIT – V 9 a) Explain Kendal – lee notation for M/M/1 Model 3 b) The arrival of customers at a banking counter follows poisson distribution with a mean of 45 per hour 12 (i) What is the probability of having zero customer in the system (Po)? (ii) What is the probability of having 5 customers in the system? Probability of having 10 customers in the system. (iii) Determine the steady state performance statistics, namely, Ls, Lq, Ws & Wq? 10 a) Define the following 3 (a) Saddle Point (b) Fair game b) The following matrix represents the payoff to P1 in a rectangular game between two persons P1 & P2 12 8 19 0

15 15 20

-4 17 15

-2 16 5

By the notion of dominance, reduce the game to 2 x 4 game & then solve it graphically.