ASSIGNMENT BOOKLET Bachelor's Degree Programme (BS.c./B.A/B.Com) OPERATIONS RESEARCH

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1 PHE-0 AOR-0 ASSIGNMENT BOOKLET Bachelor's Degree Programme (BS.c./B.A/B.Com) OPERATIONS RESEARCH Valid from st January to st December 0 It is compulsory to submit the Assignment before filling in the Term-End Examination Form. It is mandatory to register for a course before appearing in the Term- End Examination of the course. Otherwise, your result will not be declared. For B.Sc. Students Only You can take electives ( or credits) from a minimum of TWO and a maximum of FOUR science disciplines, viz. Physics, Chemistry, Life Sciences and Mathematics. You can opt for elective courses worth a MINIMUM OF 8 CREDITS and a MAXIMUM OF 8 CREDITS from any of these four disciplines. At least % of the total credits that you register for in the elective courses from Life Sciences, Chemistry and Physics disciplines must be from the laboratory courses. For example, if you opt for a total of credits of electives in these disciplines, then at least credits out of those credits should be from lab courses. School of Sciences Indira Gandhi National Open University Maidan Garhi, New Delhi-008 (0)

2 Dear Student, Please read the section on assignments in the Programme Guide for Elective Courses that we sent you after your enrolment. A weightage of 0 per cent, as you are aware, has been earmarked for continuous evaluation. For this you need to answer one tutor-marked assignment for this course, which is in this booklet. Instructions for Formating Your Assignments Before attempting the assignment please read the following instructions carefully: ) On top of the first page of your answer sheet, please write the details exactly in the following format : ROLL NO.:. NAME:. COURSE CODE:. ADDRESS: COURSE TITLE:. ASSIGNMENT NO.: STUDY CENTRE:... DATE:.. PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND TO AVOID DELAY. ) Use only foolscap size writing paper (but not of very thin variety) for writing your answers. ) Leave a cm. margin on the left, top and bottom of your answer sheet. ) Your answers should be precise. ) While solving problems, clearly indicate which part of which question is being solved. ) The assignment response is to be submitted to your Study Centre Coordinator by st December, 0, not later. Please retain a copy of your answer sheets. Wish you good luck!

3 TUTOR MARKED ASSIGNMENT Course Code: AOR-0 Assignment Code: AOR-0/TMA/0 Maximum Marks: 00 Q. Which of the following statements are true and which are false? Give reasons for your answers: i) The optimal solution to an integer linear programming problem can be found by rounding off the optimal solution of its LP relaxation. ii) The mean and variance of then Poison distribution are equal. iii) The total number of possible sequences for processing jobs on machines is. iv) A Two-dimensional linear programming problem can have at most optimal solutions. v), x, x is a basic feasible solution for the system of equations vi) vii) x x x x x x x The assignment problem can be solved using simplex method. The optimal values of the objectives functions of the primal and dual problems are not same. viii) For each cell (i, j) in an optimal solution to a transportation problem the value u v x 0. i j c ij ιj ix) For a queuing model M/M/ : GD/ /, if the service rate µ increases, the expected number of customers in the system decreases. x) In the inventory model with finite replenishment rate, if the replenishment rate is equal to consumption rate, the holding cost is 0. (0) Q. (a) Solve, by simplex method, the following linear programming problem: subject to: Minimize Z x x x x x x x x, x x x x x 0 7 8x 0 A book binder has one printing press, one binding machine and manuscripts of seven different books. The time required for performing printing and binding operations for different printing and binding operations for different books are shown below: ()

4 Book: 7 Printing Time (Days) Binding Time (Days) Find the optimum sequence of processing of the jobs that minimises the total time required. Also compute the optimal time required. Q. (a) The time taken in hours by four different machines for completing four different jobs is given below: Machines A B C D I 0 Jobs II 8 9 III 0 IV 0 8 () Find the optimal assignment and the optimal time taken to finish the jobs. () Customers arrive at a single window server according to a Poisson distribution with mean 0 minutes, and the service time is exponential with mean minutes per customer. Find the following: (i) expected number of customers in the system; (ii) expected number of customers in the queue; (iii) variance of the queue length () Q. (a) A particular item has demand of 000 units per year. The cost of one procurement is Rs. 00 and the holding cost per unit is Rs..0 per year. The replenishment is instantaneous and no shortages are allowed. Determine: (i) the economic order quantity (ii) the number of orders per year (iii) the time between orders and (iv) the optimum cost per year () Write the dual of the linear programming problem: Min x x subject to:

5 x x x x x x x x (c) x, x, x 0 () The jobs, predecessors of the jobs and duration of the jobs, in a project as given below: E C F D 8 Draw the network diagram, and hence find the critical path and the optimal duration for completing the project. () Q. (a) Solve the ILPP given below by graphical method: Job Immediate Predecessor (s) Duration (in days) A - B A 0 C A D B Maximize x x subject to x x x x x, x are non- negative integers. () Find, an initial basic feasible solution using Vogel s approximation method and hence find an optimum transportation schedule for the following data: Destination A B C Capacity (in units) I Sources II 9 III Demand (units) 0 Q. (a) An unbiased die is thrown. Let X be the random variable denoting the number appearing on the throw of the die. Find the random variate X corresponding to 0.. () Using the mixed generator rn rn (modulo 7), () Calculate r when r o =. Q7. (a) A diet for a sick person must contain at least 000 units of vitamins, 0 units of minerals and 00 calories. Two foods A and B are available at a cost of Rs. and Rs. per unit,

6 respectively. If one unit of Food A contains 00 units of vitamins, unit of mineral and 0 calories, and one unit of Food B contains 00 units of vitamins, units of minerals and 0 calories. Find, by graphical method, what combination of foods A and B be used to minimise the cost. () In an automobile company, the daily production of scooters varies from to, depending upon the availability of raw material and other conditions, as shown in the chart below: Production per day Probability Use the following random numbers 80, 70,, 8,,, 7 To simulate the process to find out the average number of scooters manufactured daily. () Q.8 (a) Solve the following LPP using graphical method. (c) Maximize x x Subject to x x x x x x x x x 0 (), Solve the following profit maximization assignment problem. A 8 0 Machine B 7 7 C 0 9 D 8 9 () A project schedule has the following characteristics: 9 Activity Time Activity Time

7 (i) Construct the network diagram. (ii) Compute earliest and latest occurrence times for each event. () Q9. (a) Find the sequence that minimizes the total time required to complete the following tasks on two machines: Task I II III IV V VI Machine A 8 9 Machine B (c) Find the total time taken when using this sequence. Also, find the idle time for each machine. The production department for a company requires 700kg of raw material for manufacturing particular item per year. It has been estimated that the cost of placing an order is A 7 and the cost of carrying inventory is percent of the investment in the inventories. The price of raw material is A 0 per kg. Find the economic order quantity (EOQ), the total cost per unit time and the cycle time. () Use the dual simplex method to solve the following LPP: Minimize z x y z () Subject to x y z 0 x y z 0 x, y,z 0 () 7