# Principles Of Operations Research 2006-07

Principles Of Operations Research

Note : (1) Attempt all questions.

(2)    All questions carry equal marks.

(3) The choice of questions is internal as indicated in each question.

(4) Graph papers will be provided on demand.

1. Attempt any four of the following :

(a) Solve the following LPP by graphical method :

Maximize Z = 3xx + 4x2

Subject to 4Xj + 2x2 < 80, 2x^ + 5x2 < 180 and x1, x2 > 0

(b)    Explain clearly the following terms used in LPP :

(1) Objective function

(2) Decision variables,

(3) Slack variables

(4) Surplus variables and

(5) Redundant constraints.

(c)    Convert the following LPP to the standard form : Maximize Z = 5xx + 7x2 + 9xg

Subject to 4x1 – 6x2 < 5 3xx + 2x2 + 7xg > 12 4xx + 3xg < 2 and xx, x2 > 0

(d) Write the dual of the following LPP :

Minimize Z = 3xx – 2x2 + 6xg Subject to 4xx + 5x2 + 3xx > 7 3xx + x2 + 6xg > 5 7xx – 2x2 – 3xg < 10 xx – 2x2 + 5xg > 3 4xx + 7x2 – 9xg > 2 and xx, x2, xg > 0

(e) Solve the following LPP by simplex method : Minimize : Z = xx – 3x2 + 3xg

Subject to 3xx – x2 + 2xg < 72xx +4x2 > -12 -4xx + 3x2 + 8xg <10 and xx, x2, xg > 0

(f) Fill in the blanks so that the following statements are correct :

(1) A constraint which does not affect the solution of an LPP, is called

(2) Any solution to an LPP which satisfies the non-negativity condition is called

(3) The feasible solution of an LPP is      of the objective function.

(4) Iso-profit lines on a graph of an LPP would always be to each other.

(5)    A constraint 5xx + 3x2 <12 on an LPP is replaced by the constraint 5xx + 3x2 < 7 . This would make the LPP more restrictive in nature. The constraint 5xx + 3x2 <12 now becomes

2. Attempt any four of the following :

(b) What is a transportation problem ? Give the mathematical formulation of the transportation problem.

(b) What is a trans-shipment problem ? Explain how it can be formulated and solved as a transportation problem ?

(c)    Find the initial basic feasible solution for the following transportation problem by YAM :

Factory

Destination

 Dl D2 Ds F1 11 13 17 14 250 F2 16 18 14 10 300 FS 21 24 13 10 400 Demand 200 225 275 250 950

Supply

(d) A company has five jobs to be done on five machines. Any job can be done on any machine. The cost of doing the jobs in different machines are given below :

 Jobs Machines Mi m2 m3 M4 M5 13 8 16 18 19 9 15 24 9 12 12 9 4 4 4 6 12 10 8 13 15 17 18 12 20

Assign the jobs for different machines so as to minimize the total cost ?

(e) Solve the following integer programming problem using branch and bound method :

Maximize Z = 3x1 + 2.5x2 Subject to Xj + 2x2 > 20, 3XJ + 2x2 > 50 and Xp x2 > 0 and both are integers.

(f)  Give the various steps involved in Hungarian method to solve an assignment problem.

3. Attempt any two of the following :

(a) Calculate the variance and expected activity time for the activities of the network shown in the following figure :

For each activity, the three estimates tQ, tm and t

are given along the arrows in the       tp order.

Enter calculations in the tabular form.

(b) Consider PERT network given in the following figur 1-2-3 Find the float of each activity and identify the critical path if the scheduled completion time for the project is 20 weeks. Also, identify the sub-critical path.

(c) There are five jobs, each of which has go through the two machines M1 and M2 in the order M^M2. Processing times are given in the following table :

 Processing times in hours for the job Machines 1 2 3 4 5 Machine – M1 5 1 9 3 10 Machine – M2 2 6 7 8 4

Determine a sequence of the five jobs that will minimize the total elapsed time T. Calculate the total idle time for the machines in this period.

4. Attempt any two of the following :

(a) A manufacturer has to supply his customer with 600 units of his product per year. Shortages are not allowed and the storage cost amount to Re. 0.60 per unit per year. The set up cost per run is Rs. 80. Find (i) The optimum run size, (ii) Optimum scheduling period and (ii) The minimum average yearly cost.

(b) A machine owner finds from his past records that the costs per year of maintanining (i.e. operations) a machine whose purchase price is Rs. 6000, are as given below :

 Year 1 2 3 4 5 6 7 8 Operating Cost (in Rs.) 1000 1200 1400 1800 2300 2800 3400 4000 Resale Value (in Rs.) 3000 1500 750 375 200 200 200 200

Determine at what age, the replacement of the machine is due ?

(c) The following failure rates have been observed for a certain type of light bulbs :

 Week 1 2 3 4 5 Percent failing by the end the week 10 25 50 80 100

There are 1000 bulbs in use and it costs Rs. 10 to replace an individual bulb which has burnt out. If all the bulbs were replaced simultaneously, it would cost Rs. 4 per bulb. It is proposed to replace all bulbs at fixed intervals of time, whether or not they have burnt out and to continue replacing burnt out bulbs as and when they fail. At what intervals, should all the bulbs be replaced ? At what group replacement price per bulb would a policy of strictly individual replacement become preferable to the adopted policy ?

5. Attempt any two of the following :

(a) What do you understand by inventory ? Give the merits and demerits of inventory ? What is inventory control ? What are the main objectives of inventory control ?

(b) Use dynamic programming to solve the following LPP :

Maximize Z = xl + 9x2 Subject to 2x1 + x2 < 25, x2 < 11 and x1, x2 > 0

(c) State Bellman’s principle of optimality in reference to dynamic programming problem. Give the basic characteristics of dynamic programming problem.