Bachelor s Degree Programme Operations Research (Valid from 1st January, 2012 to 30th November, 2012.)

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1 AOR-01 ASSIGNMENT BOOKLET Bachelor s Degree Programme Operations Research (Valid from 1st January, 2012 to 30th November, 2012.) It is compulsory to submit the assignment before filling in the exam form. School of Sciences Indira Gandhi National Open University Maidan Garhi, New Delhi (For January, 2012 cycle)

2 Dear Student, Please read the section on assignments in the Programme Guide for elective Courses that we sent you after your enrolment. As you may know already from the programme guide, the continuous evaluation component has 30% weightage. This assignment is for the continuous evaluation component of the course. Instructions for Formating Your Assignments Before attempting the assignment please read the following instructions carefully. 1) On top of the first page of your answer sheet, please write the details exactly in the following format: ROLL NO : NAME : ADDRESS : COURSE CODE : COURSE TITLE : STUDY CENTRE : DATE PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND TO AVOID DELAY. 2) Use only foolscap size writing paper (but not of very thin variety) for writing your answers. 3) Leave a 4 cm margin on the left, top and bottom of your answer sheet. 4) Your answers should be precise. ) While solving problems, clearly indicate which part of which question is being solved. ) This assignment is valid only up to November, If you fail in this assignment or fail to submit it by November, 2012, then you need to get the assignment for the January 2013 cycle and submit it as per the instructions given in the Programme Guide. We strongly suggest that you retain a copy of your answer sheets. Wish you good luck. 2

3 ASSIGNMENT 1) Which of the following statements are true? Give reasons for your answer. Course Code: AOR-01 Assignment Code: AOR-01/TMA/2012 Maximum Marks: 100 i) In an optimal solution (x 1,x 2 ) to the LPP Max 4x 1 + 3x 2 x 1 + x 2 = 2 x 1,x 2 0 both x 1, x 2 cannot be positive. ii) iii) iv) For a transportation problem with 3 origins and 4 destinations, the maximum number of basic cells is. If the LP relaxation of an integer linear programming problem has a feasible solution, the integer linear programming problem also has a feasible solution. In queueing theory, if the arrivals are according to a Poisson distribution with parameter λ, the inter-arrival time is according to an exponential distribution with parameter e λ. v) In a project network diagram, the critical path is the longest among all paths. (10) 2) a) Find all the basic solutions of the following system of equations: x 1 + 2x 2 + 3x 3 = 2 2x 1 + x 2 + 3x 3 = Which of these solutions are basic feasible solutions? (4) b) Solve graphically the following linear programming problem: Max 3x 1 + x 2 2x 1 x 2 4 2x 1 + 3x 2 12 x 2 3 x 1,x 2 0 () 3) a) A paint factory produces both interior and exterior house paints for wholseale distribution. Two basic raw materials, A and B, are used to manufacture the paints. The maximum availability of A is tonnes a day and that of B is 8 tonnes a day. The requirements of raw materials per tonne of interior and exterior paints are summarised in Table 1 on the following page. A market survey has established that the daily demand for interior paint cannot exceed that of exterior paint by more than one 1 tonne. The survey also shows that the maximum demand for interior paint is limited to 2 tonnes daily. The whole sale price per tonne is Rs. 3000/- for exterior paint and Rs. 2000/- for the interior paint. How much of interior and exterior paints should the company produce daily to maximise the gross income? Formulate the problem as an LPP. () 3

4 Table 1: Availability of Raw Material. Tonnes of raw material per ton of Paint Exterior Interior A 1 2 B Raw Material Maximum Availability (Tonnes) b) A manager wants to appoint 4 sales-persons to 4 different cities. If the expected profit when different persons are appointed to different cities is as given in Table 2, find the assignment that will maximise the profit. () Table 2: Table for problem 3b) Sales-persons I II III IV Cities A B 3 14 C D ) Solve the following LPP by two Phase simplex method. Max 3x 1 + x 2 + x 3 x 1 + x 2 + x 3 2 2x 1 + x 2 + 2x 3 10 x 1 + 2x 2 + 2x 3 1 x 1,x 2,x 3 0 (10) ) a) Consider the transportation problem with the cost and requirements as given in Table 3. Find an Table 3: Availability, demand and transportation costs. Market P Q R S Supply Warehouse A B C Demand initial basic feasible solution using Vogel s method and hence find the optimum transportation schedule. () b) Solve the Stage Coach problem in Fig. 1 on the next page using Bellman s optimality principle. () 4

5 Figure 1: Stage coach problem. ) Solve the following integer linear programming problem: Max 2x 1 + x 2 + x 3 x 1 + x 2 + x 3 2x 1 + 3x 2 + x 3 x 1,x 2,x 3 0 where x 1, x 2 and x 3 are integers. (10) ) a) A bakery shop keeps stock of a popular brand of cake. Previous experience shows that the daily demand for the item, with associated probabilities, is as given in Table 4. Use the following Table 4: Table for problem (a) Daily Demand Probability sequence of random numbers to simulate the demand for the next 10 days. Random numbers: 40,19,83,3,84,29,09,02,20 Also, estimate the daily average demand for the cakes on the basis of simulated data. () b) Consider an 8-job sequencing problem on a single machine with processing time requirements as in Table. Find the optimal sequence of jobs that minimises the average job flow times. Table : Processing times. Job Processing time Compute the value of average flow times for the optimal sequence. () 8) a) A grocery shop sell milk packets containing half a litre of milk. It makes bulk purchase of the same from the nearby city. Every time a purchase is made, it incurs a cost of Rs. 0 towards

6 transportation. The daily demand of packets is about 20. The holding cost is Rs for a packet per day to preserve it in a refrigerator. Determine the optimum quantity of a bulk purchase and the cycle time. (4) b) The construction of a bridge involves major activities A, B, C,..., G. Table gives the immediate predecessor(s) of each activity. Durations(in days) of the activities are also provided, where t 0 = optimistic time, t m = most likely time and t p = pessimistic time. Prepare a network Table : Process activities and times. Activity Immediate Duration predecessor(s) t 0 t m t p A 1 3 B 3 4 C A, B 4 D B 3 E D 13 F C, E 4 10 G D 8 10 diagram for the project. Also, find expectation and variance of the shortest completion time for the project. () 9) a) A warehouse has only one loading dock manned by a three person crew. Trucks arrive at the loading dock at an average rate of trucks per hour and the arrival rate is Poisson distributed. The loading of a truck takes 10 minutes on an average and can be assumed to be exponentially distributed. The operating cost of a truck is Rs. 20 per hour and the members of the loading crew are paid Rs. 8 each per hour. Would you like to advise the truck owner to add another crew of 3 persons? () b) Determine the optimal sequence of jobs that minimises the total elapsed time based on the information in Table. Processing time on machines is given in hours and passing is not allowed. () Table : Machine processing times Job Machine (M1) Machine (M2) Machine (M3) A 3 4 B 8 3 C 2 D 4 11 E 9 1 F 8 4 G ) a) In a railway yard, goods trains arrive at the rate of 30 trains per day. Assuming that the inter-arrival time follows an exponential distribution and the service time distribution is also exponential with an average of 3 minutes, calculate the following: i) The average number of trains in the queue. ii) The probability that the queue size is greater than or equal to 10. ()

7 b) The information regarding a project with activities is given in Table 8. Table 8: Project information. Activity Immediate Duration Predecessor(s) (days) A 9 B 8 C A D C 3 E D 2 F B, E 1 G E, F 14 i) Construct a corresponding network diagram. ii) Identify a critical path and find the total project duration. ()

M.Sc.(Mathematics with Applications in Computer Science) Probability and Statistics

MMT-008 Assignment Booklet M.Sc.(Mathematics with Applications in Computer Science) Probability and Statistics (Valid from 1 st July, 013 to 31 st May, 014) It is compulsory to submit the assignment before