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December, 2009

Ms-51 : Operations Research

1. Solve the following problem through SIMPLEX method.

Minimize Z  = — 3x₁+ x₂+ x₃

Subject to

x₁ —2x₂ + x₃ < 11

—4x₁+ x₂ + 2x₃ > 3

— 2x₁       + x₃ = 1

x₁ > 0, x₂ > 0 and x₃ >0

2. (a) Discuss the queue parameters. How do you specify a querraing system ?

(b) A refuelling station is served by a single pump machine for providing service to the petrol vehicles. The arrival process has shown that the distribution of times between arrivals is negative exponential with a mean of 10 minutes. Similarly, service times were found to be adequately described by a negative exponential distribution with a mean of 6 minutes. Waiting space is unlimited. Determine:

(i) Probability that the customer has to wait.

(ii) Mean number of customer in the system.

(iii) Percentage utilisation of the service station.

(iv) Steady - state probability of having four customers in the system.

3. (a) Define Operations Research. Discuss its application in manufacturing and non - manufacturing sectors.

(b) State and explain Bellman's Principle of optimality.

4. (a) Write the general form of LP problem in matrix notation.

(b) The demand of an item is uniform at a rate of 25 units per month. The fixed cost is Rs. 30/- each time a production is made. The production cost is Rs. 2/- per item and rinventory carrying cost is 50 paisa per unit per month. If the shortage cost is Rs. 3/-per item per month, determine how often to make a production and of what size ?

5. Write short notes on any three of the following :

(a) GOAL Programming

(b) GOMORY'S cutting plane Algorithm

(c) Selective Inventory Control

(d) Pure and Mixed Strategy

(e) Sensitivity Analysis

(f) Monte Carlo Simulation

June, 2010

MS-51 : Operations Research

1. (a) "Operations Research (OR) is an aid for the executive in making his/her decisions by providing him/her with needed quantitative information based on the scientific method of analysis". Discuss the statement and give atleast two examples to illustrate how operations research is helpful in decision making.

(b) A manufacturing company is producing two products A and B. Each requires processing on two machines 1 and 2. Product A requires 03 hours of processing on machine 1 and 02 hours on machine 2. Product B requires 2 hours of processing on machine 1 and 6 hours on machine 2. The unit profits for product A and product B are Rs. 10 and Rs. 20 respectively. The available time in a given quarter on machine 1 and machine 2 are 1200 hrs and 1500 hrs respectively. The market survey has predicted that not more than 400 units of product A and not more than 250 units of product B can be sold in the given quarter.  The company wants to determine the product mix to maximize the profits. Formulate the problem as linear programming mathematical model and determine the profit.

2. (a) Why is the simplex method a better technique than a graphical approach for ost real cases ? Construct the dual to the following primal problem :

Max z = 6x₁ + 10x₂

Subject to the constraints

4x ₁ +12x₂ < 100

6x₁ + 4x₂ < 70

10x₁ — 6x <  20

x₂ <  20

x₁, > x₂ > 0

(b) In a grocery store, the daily demand of bread over a 100 days period has the following frequency distribution :

Using the above data, and random numbers  (27, 13, 80, 10, 54, 60, 49, 78, 66, 44), simulate a 10 - days sequence of the demand of bread.

3. (a) Suggest optimum assignment of 4 workers A, B, C and D to 4 Jobs I, II, III, and IV. The time taken by different workers in completing the different jobs is given below :

Also indicate the total time taken in completing the jobs.

(b) Find the initial solution of the following transportation problem by using Vogel Approximation Method :

Also find its optimal solution by using MODI method.

4. (a) What are the limitations of Game theory ? Show how a two persons zero - sum game problem can be formulated as a linear programming problem.

(b) For the following game, find the optimal strategies of A and B and the value of the game by using the principle of dominance :

5. (a) "Small variations in optimal order size will not change the total cost appreciably".  Do you agree with this statement ? Give Justification in support of your answer.

(b) A purchase manager places an order each time for a lot of 500 units of product A. From the available data, the following results are obtained :

Inventory carrying cost = 40% of purchase cost

Ordering cost per order            = Rs.600

Cost per unit                             = Rs.50

Annual demand                         = 1000

Find out the loss to the organization due to his ordering policy.

6. Write short notes on any three of the following :

(a) Travelling salesman problem

(b) ABC Analysis

(c) Non-linear programming

(d) Similarities between dynamic programming and linear programming

(e) Branch and bound method

June, 2011

MS-51 : Operations Research

1.  (a)   Discuss the historical background of Operations Research (O.R). Explain its significance and scope in Management Des ci s ion Ma king. Enumerate the limitations of O.R.

(b) Solve the following Linear Programming Problem graphically.

Maximize Z = 4x₁  + 6x₂

Subject to constraints

x₁+x₂ =5

x₁  ≥ 2

x₂ ≤ 0

x_1, x₂ ≥  0 

2.(a) Derive the equation for Economic Batch Quantity (EBQ) for simultaneous production and consumption

(b) The annual demand for an item is 3200 units.  The unit cost is Rs. 6/- and inventory carrying cost is 25% per annum.  If the cost of one procurement is Rs. 150/- find out.

(i)  Economic Order Quantity

(ii)  No. of orders per year

(iii)  Time between two consecutive orders

(iv)  The optimal cost Mention assumptions made, if any.

3.(a) Discuss the application of dynamic programming in decision making. How is this different from linear programming ?

(b) An organization has three consultants. Each consultant can work upto 160 hours during next month during which three projects roust be completed. Project 1 will take 130 hours, Project 2 will take 140 hours and Project 3 will take 160 hours. The mount (l's.) per hour that can be billed for assigning each consultant to each project is given below:-

Formulate this as a transportation problem and find the optimal solution. What is the

maximum total billing for next month ?

4.  (a) Explain the meaning of Dominance Principle in Game Theory. Illustrate with a small example.

(b)  A bakery keeps stock of a popular brand of cakes. Previous experience shows the daily pattern for the item with associated probabilities as given :

Use the following sequence of random numbers to simulate the demand for next 10 days. Also find the average demand per day. Random Nos. 25, 39, 65, 76, 12, 05, 73, 89, 19, 49.

5.  (a)  Discuss the parameters of Queuing Problem.

(b)  A self - service store employs one cashier at its counter. Nine customers arrive on an average every 5 minutes while the cashier can serve 10 customers in 5 minutes. Assuming Poisson distribution for arrival rate and exponential distribution for service rate, find

(i)  Average number of customers in the system.

(ii)  Average number of customers in queue or average queue length.

(iii)  Average time a customer spends in the system.

(iv)  Average time a customer waits before being served.

6.  Write short notes on any four of the following :

(a)  Branch and bound algorithm

(b)  Goal Programming

(c)  Non - linear Programming

(d)  Assignment Problem

(e)  Dual Linear Programming Problem

(f)  Traveling salesman problem 

June, 2011

MS-51 : Operations Research

1.  (a)   Discuss the historical background of Operations Research (O.R). Explain its significance and scope in Management Des ci s ion Ma king. Enumerate the limitations of O.R.

(b) Solve the following Linear Programming Problem graphically.

Maximize Z = 4x₁  + 6x₂

Subject to constraints

x₁+x₂ =5

x₁  ≥ 2

x₂ ≤ 0

x_1, x₂ ≥  0 

2.(a) Derive the equation for Economic Batch Quantity (EBQ) for simultaneous production and consumption

(b) The annual demand for an item is 3200 units.  The unit cost is Rs. 6/- and inventory carrying cost is 25% per annum.  If the cost of one procurement is Rs. 150/- find out.

(i)  Economic Order Quantity

(ii)  No. of orders per year

(iii)  Time between two consecutive orders

(iv)  The optimal cost Mention assumptions made, if any.

3.(a) Discuss the application of dynamic programming in decision making. How is this different from linear programming ?

(b) An organization has three consultants. Each consultant can work upto 160 hours during next month during which three projects roust be completed. Project 1 will take 130 hours, Project 2 will take 140 hours and Project 3 will take 160 hours. The mount (l's.) per hour that can be billed for assigning each consultant to each project is given below:-

Formulate this as a transportation problem and find the optimal solution. What is the

maximum total billing for next month ?

4.  (a) Explain the meaning of Dominance Principle in Game Theory. Illustrate with a small example.

(b)  A bakery keeps stock of a popular brand of cakes. Previous experience shows the daily pattern for the item with associated probabilities as given :

Use the following sequence of random numbers to simulate the demand for next 10 days. Also find the average demand per day. Random Nos. 25, 39, 65, 76, 12, 05, 73, 89, 19, 49.

5.  (a)  Discuss the parameters of Queuing Problem.

(b)  A self - service store employs one cashier at its counter. Nine customers arrive on an average every 5 minutes while the cashier can serve 10 customers in 5 minutes. Assuming Poisson distribution for arrival rate and exponential distribution for service rate, find

(i)  Average number of customers in the system.

(ii)  Average number of customers in queue or average queue length.

(iii)  Average time a customer spends in the system.

(iv)  Average time a customer waits before being served.

6.  Write short notes on any four of the following :

(a)  Branch and bound algorithm

(b)  Goal Programming

(c)  Non - linear Programming

(d)  Assignment Problem

(e)  Dual Linear Programming Problem

(f)  Traveling salesman problem