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IGNOU AOR-01 - Operational Research

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IGNOU AOR-01 Code Details

  • University IGNOU (Indira Gandhi National Open University)
  • Title Operational Research
  • Language(s) English
  • Code AOR-01
  • Subject Mathematics
  • Degree(s) BA, B.COM, B.Sc., BSW
  • Course Application-Oriented Courses (AOC)

IGNOU AOR-01 English Topics Covered

Block 1 - Linear Programming and Applications

  • Unit 1 - Graphical Method
  • Unit 2 - The Simplex Method
  • Unit 3 - Duality and Applications

Block 2 - Integer and Dynamic Programming

  • Unit 1 - Special and Structured Linear Programming Problems
  • Unit 2 - Integer Linear Programming
  • Unit 3 - Dynamic Programming

Block 3 - Sequencing, Scheduling & Inventory

  • Unit 1 - Sequencing
  • Unit 2 - Project Scheduling
  • Unit 3 - Inventory Models

Block 4 - Queueing Models

  • Unit 1 - Basic Elements of Queueing Models
  • Unit 2 - Queueing Systems
  • Unit 3 - Simulation of Queueing Systems
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IGNOU AOR-01 (January 2023 - December 2023) Assignment Questions

1. Which of the following statements are true? Give reasons for your answer. a) In an LPP if all the co-efficients of the objective function are positive and all the constraints are of type, the problem has a bounded optimal solution. b) In any assignment problem, the optimal assignments are always along the main diagonal. c) An ABC analysis is used to determine the re-order point in an inventory problem. d) For a (M/M/1) queueing model, steady state solution exists. e) An LPP with 4 variables and3 constraints can have a maximum of 4 basic solutions. 2. a) A department has 5 jobs to be performed and there are five employees to perform these jobs. The time (in hours) each employee will take to perform each job is given in the following matrix: How should the jobs be allocated, one per employee, so as to minimize the total man-hours? b) i) Formulate the dual of the LPP ii) Check whether S1 = (0, 35/9, 20/9) is an optimal solution to the primal and S2 = (50/9, 8/9), is an optimal solution to the dual without solving the problem. 3. a) In a factory, the machines break down at an average rate of 10 machines per hour. Assume that the number of machines is fairly large. The idle time cost of a machine is estimated to be Rs. 20 per hour. The factory works 8 hours a day. The factory manager is considering 2 mechanics for repairing the machines. The first mechanic A takes about 5 minutes on an average to repair a machine and demands wages of Rs. 10 per hour. The second mechanic B takes about 4 minutes in repairing a machine and demands wages at the rate of Rs. 15 per hour. Assuming that the rate of machine breakdown is Poisson distributed and the repair rate is exponentially distributed, which of the two mechanics should be engaged? b) Solve the following LPP by simplex method: 4. a) A fertilizer company distributes its products by trucks loaded at its only loading station. Both, company trucks and contractor’s trucks are used for this purpose. It was found out that on an average every 5 minutes one truck arrived and the average loading time was 3 minutes. 40 percent of the trucks belong to the contractors. Making suitable assumptions, determine: i) The probability that a truck has to wait. ii) The waiting time of a truck that waits. iii) The expected waiting time of contractor’s trucks per day b) Find the shortest route from A to G using Bellman’s Optimality Principle. 5. a) An insurance company has decided to modernize and refit one of its branch offices. Some of the existing office equipments will be disposed of but the remaining will be returned to the branch on completion of the renovation work. Tenders are invited from a number of selected contractors. The contractors will be responsible for all the activities in connection with the renovation work excepting the prior removal of the old equipment and its subsequent replacement. The major elements of the project have been identified as follows along with their durations and immediately preceding element: i) Draw the network diagram showing the inter-relations between the various activities of the project. ii) Calculate the minimum time that the renovation can take from the design stage. iii) Find the effect on the overall duration of the project if the estimates or tenders can be obtained in 2 weeks from the contractors by reducing their numbers. iv) Calculate the ‘independent float’ that is associated with the non-critical activities in the network diagram. b) The final table in Phase I of a Linear Programming Problem is given below: Here x a1 is the artificial variable. Does this LPP have a feasible solution? Justify your answer. 6. a) ABC limited has 3 production shops supplying a product to 5 warehouses. Each shop has a specific production capacity and each warehouse has certain requirements. The costs of transportation, capacities and requirements are given below: Find a basic feasible solution by Vogel’s method and check its optimality. b) Find the sequence that minimizes the total elapsed time required to complete the following tasks. Each job is processed in the order ABC. 7. a) A super bazaar in a city daily needs between 20 to 35 workers. The rush hours are between noon and 2 pm. The requirement of the workers at various hours is given in the following table: The super bazaar can call up to 24 full time workers in a day. Others can be hired on contract for 4 hours at a stretch. The part-time workers can be employed beginning at 10 AM, 12 noon or 1 PM. Full time workers are given a lunch break of 1 hour, half of them taking their lunch from 12 noon to 1 PM and the other half between 1 PM to 2 PM. Full time workers are paid Rs. 90 per day and part-time workers are paid Rs. 40 for 4 hours. Formulate the problem of finding the number of full time workers to be employed and the number of part-time workers to be called at 10 AM, 12 noon and 1 PM as a cost minimizing LPP. b) In a factory a valve is consumed at the rate of 64 units per month. The inventory carrying costs are 20 paise per valve per month. It takes Rs. 250 to place and order and the lead time for receiving an order is assumed to be 0. Determine: i) EOQ ii) Total cost per unit time iii) Cycle time. 8. a) The details of a project consisting of 10 activities are given below: Draw the network and find the project completion time. b) Solve the following integer programming problem using branch and bound method: 9. a) Use the dual simplex method to solve the following LPP: b) Three custom officers separately check the luggages of the passengers at an airport. The passengers arrive t an average rate of five per hour. The time a custom officer spends with a passenger is exponentially distributed, with mean service time 24 minutes. Find the probability that all the custom officers are idle. Also, find the probability that there are exactly 2 customers in the queue. 10. a) A factory manufacturers two articles A and B. To manufacture an article A, a certain machine has to work for 1.5 hours and in addition a craftsman has to work for 2 hours. To manufacture an article B, the machine has to work for 2.5 hours and in addition the craftsman has to work for 1.5 hours. In a week the factory can avail 80 hours of machine time and 70 hours of craftsman’s time. The profit on each article A is ₹ 5 and on each article B is ₹ 4. If all articles produced are sold away, find how many of each type of articles should be produced to earn the maximum profit per week using LPP. b) i) Define M/M/1 queuing model. ii) A bank plans to open a single server desk in banking facility at a particular centre. It is estimated that on an average 28 customers re arriving each hour and it requires 2 minutes to process a customer’s transaction. Determine: 1. The proportion of time that the system is idle. 2. Average time a customer will have to wait before reaching the server.

IGNOU AOR-01 (January 2022 - December 2022) Assignment Questions

1. Are the following statements true or false? Give reasons for your answers. a) When the ordering quantity is the sa,e as EOQ, the ordering cost is equal to the holding cost. b) x 1 = 1, x2 = 2 and x3 = 1 is a basic feasible solution for the system of equations x 1 + x 2 + x 3 = 5 2x 1 + x 2 + x 3 = 5 c) In an assignment problem, the optimal solution is always along the main diagonal. d) In order to shorten a project completion time, we must reduce the duration of noncritical activities. e) Little’s formula relates the waiting time of a customer and the number of customers present in a service facility. 2. a) A farm is engaged in breeding horses. The horses are fed on various products grown on the farm. Because of the need to ensure that certain important nutrients α, and β γ, are present in the meal, it is necessary to buy the products A and/or B. The amount of each nutrient available per unit of either product is given below, along with the minimum requirement of each nutrient Formulate the problem of deciding the amount of the products that should be purchased in order to meet the minimum requirements of nutrient at lowest cost, as an LP problem. b) A petrol station sells 4000 litres of petrol every month. The parent company, whenever it refills the station’s tank, charges the station Rs.50 besides the cost of petrol. The annual cost of holding a litre of petrol is Rs 0.30. Find out the economic order quantity 3. a) A factory has four machines. Four jobs are required to be processed on them. Each machine must be assigned exactly one job. The time (set-up and processing) requirement of each machine to complete any job is shown below. How should the jobs be assigned to the machines so that the total time needed to complete the jobs is minimized? What is the total machine time for the optimal assignment? b) Write down the dual (D) of the LPP, (P), given by he dual of the following LPP: Maximize 80x 1 +120x 2 Subject to Further, without actually solving the LPPs check whether ( x1 = 20, x2 = 10) is an optimal solution for the primal (P) and (y 1 = 20, y2 = 10, y 3 = 5) is an optimal solution for the dual, D. 4. a) Consider a 4/2/F/Fmax problem with the data: i) Obtain an optimal sequence of jobs. ii) Find the total idle time of M2 and the value of Fmax for the optimal sequence. b) For rail booking there are two reservation counters for customers who arrive according to a Poisson distribution at an average rate of 10 per hour. The service time for booking clerks at both the counters are exponentially distributed with a mean time of 5 minutes. The counters remain open for 12 hours per day. i) Find the hours of the day for which all the clerks are busy. ii) Find the probability that both the clerks are idle. iii) Find the probability that one clerk is idle. iv) Find the expected waiting time of customers in the queue. 5. a) We have a project with the activities: A ,B,……F. The following table gives the immediate predecessor(s) and duration of each activity: i) Draw the corresponding network diagram. ii) Find the critical path. iii) What is the shortest completion time of the project? b) At a certain iteration, the simplex table of a maximization problem looks like this: Determine the leaving and entering variables and hence find the values of the new basic variables. 6. a) Suppose that an average of 10 customers per hour arrive at a single-server bank teller. The average service time for each customer is 4 minutes. Assume that both, inter-arrival times and service times, are exponentially distributed. i) What is the probability that the teller is idle? ii) Find the average number of waiting customers. iii) Determine the average amount of time a customer spends in the bank (including time in service). b) Consider the following integer linear programming problem: where x1, x2 and x3 are integers. The final table for the LP relaxation is given below: Use the branch and bound algorithm to find the optimal solution of the integer linear programming problem. 7. a) The proportion of defective pieces of a certain product in a manufacturing process is 0⋅10. Call a piece D (for defective) if it is defective, G (for good) otherwise. Simulate the outcomes (D or G) of 10 pieces from the given process using the random numbers: 0⋅57, 0⋅76, 0⋅49, 0⋅09, 0⋅95, 0⋅35, 0⋅68, 0⋅ 22, 0⋅86, 0⋅54. b) The following table shows all the necessary information on the availability of supply to each warehouse, the requirement of each market and unit transportation cost (in Rs.) from each warehouse for each market: The shipping clerk has worked out the following schedule from experience: 12 units from A to Q, 1 unit from A to R, 9 units from A to S, 15 units from B to R, 7 units from C to P and 1 unit from C to R. Check if the clerk has the optimal schedule. If the schedule is not optimal, find the optimal schedule and minimum transportation cost. 8. a) Use Vogel’s approximate method, to solve the following transportation problem of minimization: b) A company manufactures around 200 mobiles. The daily production varies from 196 mobiles to 204 mobiles with the following probability distribution: The finished mobiles are transported in box with a capacity of only 200. Using the random numbers 82, 89, 78, 24, 53, 61, 18 and 45, simulate the mobiles which could not be transported. Determine the average number of such mobiles. 9. a) Use dual simplex method to solve the following LPP: Minimize z = x1 + x2 Subject to 2x1 + x2 ≥ 4, x1 + 7x ≥ 7, x1 , x2 ≥ 0. b) A T.V. repairman finds that the time spent on his jobs has an exponential distribution with mean 30 minutes. He repairs sets in the order in which they arrive. If the arrival of sets has Poisson distribution with an average rate of 10 per day, (taking 8 hours in a day), then what is the repairman’s expected idle time each day? How many jobs are in the queue when a new set is brought for repair? 10. a) Use two-phase simplex method to solve the following LPP: Maximize z = 3x1 + 2x2 Subject to 2x1 + x2 ≤ 2, 3x1 + 4x2 ≥ 12, x1 , x2 ≥ 0. b) An item is produced at the rate of 50 items per day. The demand occurs at the rate of 25 items per day. If the set-up cost is Rs.100, and the holding cost is Rs.0.01 per unit of item per day, find the economic lot size for one run, assuming that the shortages are not permitted. Also, find the time of cycle and total minimum cost for one run
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