SHRI VISHNU ENGINEERING FOR WOMEN

OPE

UNIT – I

1. a) Explain various models in operations research. b) Use Simplex method to solve the following problem:

2.a) Explain necessity and scope of operations research. b) Food X contains 6 units of vitamins A per gram and 7 units of vitamin B per gram and costs 12 paise per gram. Food Y contains 8 units of vitamin A per gram and 12 units of vitamin B per gram and costs 20 paise per gram. The daily minimum requivitamin B is 100 units and 120 units respectively. Find the minimum cost of product mix by Big-M method.

3. a) Explain characteristics and phases of operations research. [8] b) Use the Two-phase Simplex method to

4. a) Explain the applications of Operations research in Mechanical Engineering. [6] b) Solve the following LPP graphically:

5. (a) Discuss the origin and development of OR with a suitable classification. (b) OR is a tool for decision support system. Justify.6. (a) Explain what is meant by degeneracy in LPP? How can these problems be solved? (b) Solve the following LP Problem by two phase method

7. (a) Describe the various steps involved in OR study. (b) Explain the scope and methodology of OR. [8+8. (a) What is the principle of duality in LPP? Explain its advantages. (b) Use duality to solve the LPP

SHRI VISHNU ENGINEERING FOR WOMEN

OPERATIONS RESEARCH-ASSIGNMENT

1. a) Explain various models in operations research. b) Use Simplex method to solve the following problem:

2.a) Explain necessity and scope of operations research.

b) Food X contains 6 units of vitamins A per gram and 7 units of vitamin B per gram and costs 12 paise per gram. Food Y contains 8 units of vitamin A per gram and 12 units of vitamin B per gram and costs 20 paise per gram. The daily minimum requirement of vitamin A and vitamin B is 100 units and 120 units respectively. Find the minimum cost of product mix by

3. a) Explain characteristics and phases of operations research. [8] phase Simplex method to

the applications of Operations research in Mechanical Engineering. [6]b) Solve the following LPP graphically:

5. (a) Discuss the origin and development of OR with a suitable classification. (b) OR is a tool for decision support system. Justify.

6. (a) Explain what is meant by degeneracy in LPP? How can these problems be solved?(b) Solve the following LP Problem by two phase method

7. (a) Describe the various steps involved in OR study. (b) Explain the scope and methodology of OR. [8+8]

8. (a) What is the principle of duality in LPP? Explain its advantages. (b) Use duality to solve the LPP

b) Food X contains 6 units of vitamins A per gram and 7 units of vitamin B per gram and costs 12 paise per gram. Food Y contains 8 units of vitamin A per gram and 12 units of vitamin B

rement of vitamin A and vitamin B is 100 units and 120 units respectively. Find the minimum cost of product mix by

the applications of Operations research in Mechanical Engineering. [6]

6. (a) Explain what is meant by degeneracy in LPP? How can these problems be solved?

UNIT – II

1. A company wishes to determine an investment strategy for each of the next four years.Five investment types have been selected, investment capital has been allocated for eachof the coming four years and maximum investment levels have been established for eachinvestment type. An assumption is that amounts invested in any year will remain invested until the end of the planning horizon of four years. The following table summarizes the data for this problem. The values in the body of the table represent net return on investment of one rupee up to the end of the planning horizon. For example, a rupee invested in investment of type B at the beginning of year I will grow to Rs. 1.90 by the end of the fourth year, yielding a net return of Re. 0.90.

The objective in this problem is to determine the amount to be invested at the beginning of each year in each investment to maximize the net rupee return for the four-year period. 2. a) Give three different example of sequencing problem from your daily life. Explain the process of solving sequencing problem. [6] b) Two major parts P1and P2 for a product require processing through six machine centers. The technological sequence of the parts on six machines and manufacturing times on each machine are

What would be the optimal scheduling to minimize the total processing time for these two parts? Find also the total elapsed time. For each machine specify the job that should be done first.

3. a) State the common and distinguishing features of the assignment and the transportation Models. b) A company has four territories open and four salesmen available for assignment. The territories are not equally rich in their sales potential. It is estimated that typical salesman operating in each territory would bring in the following annual sales:

The four salesmen are also considered to differ in ability; it is estimated that working under the same conditions, their yearly sales would be proportionately as follows:

If the criterion is maximum expected total sales, the intuitive answer is to assign the best salesman to the richest territory, the next best salesman to the second richest territory and so on. Verify this answer by the assignment method. 4. a) Why is not the Simplex method not used to solve Transportation problems? b) Find the initial basic feasible solution of the following transportation problem by North-West Corner rule. Also find the optimal transportation plan. [10]

5. ABC Limited has three production shops supply a product to five warehouse. The cost of production varies from shop to shop and cost of transportation from one shop to a warehouse also varies. Each shop has a specific production capacity and each warehouse has certain amount of requirement. The costs of transportation are given below:

The cost of manufacturing the product at different production shops is

Find the optimum quantity to be supplied from each shop to different warehouses at minimum total cost.

6. (a) Give the mathematical formulation of transportation problem. How does it differ from an assignment problem. (b) Solve the assignment problem given in figure 2.

7. (a) What is an unbalanced Assignment problem. (b) A department head has four subordinates and four tasks to be performed. The subordinates differ in efficiency and the tasks differ in their intrinsic difficulty. His estimates of the times that each man would take to perform each task is given in figure 2b.

8. (a) What do you understand by the following terms in the context of sequence of jobs: i. Job arrival pattern ii. Number of machines iii. The flow pattern in the shop. (b) Determine the optimal sequence of jobs that minimize the total elapsed time based on the following information. Processing time on machines is given in hours and passing is not allowed.

UNIT – III

1. a) Explain how the theory of replacement is used in replacement of items whose maintenance cost varies with time. b) Machine A costs Rs. 9000. Annual operating costs are Rs. 200 for the first year, and then increase by Rs.2000 every year. Determine the best age at which to replace the machine. If the optimum replacement policy is followed, what will be the average yearly cost of owning and operating the machine? Assume that the machine has no resale value when replaced and that future costs are not discounted. Machine B costs Rs. 10000. Annual operating costs are Rs. 400 for the first year and then increase by Rs. 800 every year. You have now a machine of type A

and which is one year old. Should you replace it with B, and if so, when? Suppose you are just ready to replace machine A with another machine of the same type,when you hear that machine B will become available in a year. What would you do? 2. A manufacturer is offered two machines A and B. A has cost price of Rs. 2500, its running cost is Rs. 400 for each of the first 5 years and increases by Rs. 100 every subsequent year. Machine B, having the same capacity as A, costs Rs. 1250, has running cost of Rs. 600 for 6 years, increasing by Rs. 100 per year thereafter. If money is worth 10% per year, which machine should be purchased? Scrap value of both machines is negligibly small. 3. A manual stamper currently valued at Rs. 1000 is expected to last 2 years and cost Rs. 4000 per year to operate. An automatic stamper which can be purchased for Rs. 3000 will last 4 years and can be operated at an annual cost of Rs. 3000. If money carries the rate of interest 10% per annum, determine which stamper should be purchased. 4. Find the cost per period of individual replacement policy of an installation of 300 light bulbs, given the following: i) Cost of replacing an individual bulb is Rs. 2. ii) Conditional probability of failure is given below:

Also calculate the number of light bulbs that would fail during each of the four weeks. 5. It has been suggested by a data processing firm that a company adopts the policy of periodically replacing all the tubes in a certain piece of equipment. A given type of tube is known to have mortality distribution as shown in the following table:

The cost of replacing the tubes on an individual basis is estimated to be Rs 1.00 per tube and the cost of group replacement policy average Re 0.30 per tube. Compare the preventive replacement with that of remedial replacement. 6. (a) State some of the simple replacement policies and give the average cost function for the same explaining your notations. (b) A factor has a large number of bulbs, all of which must be in working condition. The mortality of bulbs is given in the following table :

If a bulb fails in service, it cost Rs. 3.50 to replace; but if all the bulbs are replaced at a time it costs Rs. 1.20 each. Find the optimum group replacement policy. 7. (a) Discuss the treatment of unequal service life in replacement analysis. Discuss the minimum cost replacement model. Under what conditions is the minimum cost replacement model applicable?

(b) A truck-owner finds from his past experience that the maintenance costs are Rs 200 for the first year and then increase by Rs 2,000 every year. The cost of Truck Type A is Rs 9,000. Determine the best age at which to replace the truck. If the optimum replacement is followed what will be the average yearly cost of operating the truck? Truck Type B costs Rs 20,000. Annual operating costs are Rs. 400 for the first year and then increase by Rs 800 every year. The truck owner has now the Truck Type A, which is one year old. Should it be replaced by B type, and if so, when? 8. A company is considering to replace grinder X presently of worth Rs 10,000 by a new grinder Y of Rs 20,000 but will be economic in running expenditures. The expected life of grinder X is 5 years with running expenditures of Rs 4,000 in fist year and then additional increase of Rs 400 per year for next four years. For the new grinder, the annual running cost is Rs 1,000 per year and scrap value of Rs 2,000. As an advisor to the company, find (a) The present value of the cost of old and new grinders considering 12 percent normal rate interest. (b) Suggest whether the old grinder be replaced by the new grinder, assuming the life of new grinder to be 5 years.

UNIT – IV

1. a) Describe the role of theory of games for scientific decision-making. b) In a certain game, player A has three possible choices L, M, N, while player B has two possible choices P and Q. Payments are to be made according to the choices made.

What are the best strategies for player A and B in this game? What is the value ofthe game for A and B?

2. a) Explain the assumptions underlying game theory. b) The two armies are at war. Army A has two airbases, one of which is thrice as valuable as the other. Army B can destroy an undefended airbase, but it can destroy only one of them. Army A can also defend only one of them. Find the best strategy for A to minimize its losses. 3. a) Discuss the basic concepts of game theory giving examples. b) Solve the game given in the table below by the graphical method.

4. a) Discuss the algebraic method for solving 2 X 2 games by taking a suitable example. b) Solve the following 2 X 4 game by graphical method:

5. (a) Define: i. competitive game, ii. payoff matrix, iii. pure and mixed strategies, iv. saddle point, v. optimal strategies, and vi. rectangular (or two-person zero-sum) game (b) What is a game in game theory? What are the properties of a game? Explain the best strategy’ on the basis of mini-max criterion of optimality. 6. (a) Define ‘saddle point’. Is it necessary that a game should always possess a saddle point? (b) Use dominance rules to reduce the size of the payoff matrix given in figure 4b to (2 × 2) size and hence, find the optimal strategies and value of the game.

7. (a) What are the assumptions made in the theory of games? (b) Obtain the optimal strategies for both players and the value of the game for two-person zero-sum game whose payoff matrix is given in figure 4b. [8+8]

8. A game has the payoff matrix

Show that E(x,y)=1-2x(y-1/2) and deduce that in solution of the game the first player follows a pure strategy while the second has infinite number of mixed strate-

UNIT – V

1. a) Explain characteristics and classification of queuing models. b) Arrival rate of telephone calls at a telephone booth are according to Poisson distribution, with an average time of 9 minutes between two consecutive arrivals. The length of telephone call is assumed to be exponentially distributed, with mean 3 minutes. (a) Determine the probability that a person arriving at the booth will have to wait. (b) Find the average queue length that is formed from time to time. (c) Thetelephone company will install a second booth when convinced that an arrival would expect to have to wait at least four minutes for the phone. Find the increase in flow rate of arrivals which will justify asecond booth. (d) What is the probability that an arrival will have to wait for more than 10 minutes before the phone is free? (e) What is the probability that he will have to wait for more than 10 minutes before the phone is available and the call is also complete? (f) Find the fraction of a day that the phone will be in use. 2. a) “The assumptions in queuing theory are so restrictive as to render behavior prediction of queuingsystem practically worthless”. Discuss. b) Customers arrive at the first class ticket counter of a theatre at the rate of 12 per hour. There is one clerk serving the customers at the rate of 30 per hour. (i) What is the probability that there is no customer in the counter (i.e. that the system is idle)? (ii) What is the probability that there are more than 2 customers in the counter? (iii) What is probability that there is no customer waiting to beserved? (iv) What is the probability that a customer is being served and nobody is waiting? 3. a) Prove that if arrivals occur at random in time, then the number of arrivals occurring in a fixed time interval follows Poisson’s distribution. b) The mean rate of arrival of planes at an airport during the peak period is 20/hour asper Poisson distribution. During congestion, the planes are forced to fly over the field in the stack awaiting the landing of other planes that had arrived earlier. (a) How many planes would be flying in the stack during good and in bad weather? (b) How long a plane would be in the stack and in the process of landing in good and in bad weather? (c) How much stack and landing time to allow so that priority to land out of order would have to be requested only once in 20 times?Assume µ=60 planes/hour in good weather and 30 planes/hour in bad weather. 4. a) Write a note on various assumptions made in single-channel queuing theory

.b) Goods trucks arrive randomly at a stockyard with a mean of 8 trucks/hour. A crew of four operatives can unload a truck in 6 minutes. Trucks waiting in queue to be unloaded are paid a waiting charge at the rate of Rs. 60 per hour. Operatives are paid a wage rate of Rs. 20 per hour. It is possible to augment the crew strength to 2 or 3 (of four operatives per crew) when the unloading time will be 4 minutes or 3 minutes respectively per truck. Find the optimal crew size. 5. The tool room company’s quality control department is manned by a single clerk who takes an average of 5 minutes in checking parts of each of the machine coming for inspection. The machines arrive once in every 8 minutes on the average. One hour of the machine is valued at Rs.15 and a clerk’s time is valued at Rs.4 per hour. What are the average hourly queuing system costs associated with the quality control department? 6. (a) Explain the characteristics of waiting line models. (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. 7. A T.V. repairman finds that the time spent on his jobs have an exponential distribution with mean of 30 minutes. If he repairs sets in the order in which they come in, and if the arrival of sets is approximately Poisson distribution with an average rate of 10 per 8 hour day, what is repairmen’s expected idle time each day? How many jobs are ahead of the average set just brought in? 8. A branch of National Bank has only one typist. Since the typing work varies in length (number of pages to be typed), the typing rate is randomly distributed approximating a Poisson distribution with mean service rate of 8 letters per hour. The letters arrive at a rate of 5 per hour during the entire 8-hour work day. If the type writer is valued at Rs. 1.50 per hour, determine (a) Equipment Utilization. (b) The per cent time an arriving letter has to wait. (c) Average system time. (d) Average idle time cost of the typewriter per day.

UNIT – VI

1. a) Explain the necessity for maintaining inventories. b) Find the optimal order quantity for a product when the annual demand for the product is 500 units, the cost of storage per unit per year is 10% of the unit cost and ordering cost per order is Rs. 180. The unit costs are given below:

2. a) Give the general classification of inventories. b) A stockiest has to supply 400 units of a product every Monday to his customers. He gets the product at Rs. 50 per unit from the manufacture. The cost of ordering and transportation from the manufacturer is Rs. 75 per order. The cost of carrying inventory is 7.5% per year of the cost of the product. Find (i) The economic order quantity (ii) The total optimal cost (including the capital cost) (iii) The total weekly profit if the item is sold for Rs. 55 per unit. 3. a) State the assumptions while deriving EOQ formula. b) The purchasing manager of a distillery company is considering three sources of supply for oak barrels. The first supplier offers any quantity of barrels at Rs. 150 each. The second supplier offers barrels in lots of 150 or more at Rs. 125 per barrel. The third supplier offers barrels in lots of 250 or more at Rs. 100 each. The distillery uses 1500 barrels a year at a constant rate. Carrying costs are 40%, and its costs the purchasing agent Rs. 400 to place an order. Calculate the total annual cost for the orders placed to the probable suppliers and find out the supplier to whom the orders should be placed? 4. a) How do you control the inventories of a manufacturing organization? Discuss various inventory costs associated with any organization. b) A company uses an item at a uniform rate of 2000 units per year. Delivery is instantaneous and no shortages are permitted. The ordering, receiving and hauling cost is Rs. 13 per order, while inspection cost is Rs. 12 per order. Interest costs Rs. 0.056 and deterioration and obsolescence costs Rs. 0.004 respectively per year for each item actually held in inventory plus Rs. 0.02 per year based on the maximum number of units in inventory. Calculate the EOQ if lead time is 25 days, find reorder level. 5. 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 per setup and holding cost is 10 per 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 minimum total cost for one run. 6. 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 setup cost is Rs.100 per setup and holding cost is Re.0.01 per unit of item per unit of item per day. Find the economic lot size for one run, assuming that shortages are not permitted. Also find the time of cycle and minimum total cost for one run.

7. (a) Define the terms i. Lead time. ii. Reorder point. iii. Buffer stock. (b) A company uses 24,000 units of a raw material which costs Rs.12.50 per unit. Placing each order costs Rs.22.50 and the carrying cost is 5.4% per year of the average inventory. Find the economic order quantity and the total inventory cost (Including the cost of the material). 8. (a) Discuss various costs involved in inventory. (b) The production department for a company requires 36,000kg of raw material for manufacturing a particular item per year. It has been estimated that the cost of placing an order is Rs.36 and the cost of carrying inventory is 25% of investment an the inventories. The prince is Rs.10 per kg. The purchase manager wishes to determine the operating doctrine for raw materials.

UNIT –VII

1. A manufacturing company has three sections producing automobile parts, bicycle partsand sewing machine parts respectively. The management has allocated Rs. 20000 for expanding the production facilities. In the auto parts and bicycle parts sections, the production can be increased either by adding new machines or by replacing some old inefficient machines by automatic machines. The sewing machine parts section was started only a few years back and thus the additional amount can be invested only by adding new machines to the section. The cost of adding and replacing the machines, along with the associated expected returns in the different sections is given in table below. Select a set of expansion plans which may yield the maximum return.

2. A dealer has to dispose of certain goods within 5 weeks time. The market prices are fluctuating from week to week. It is estimated that the chances of getting Rs. 2000 for the whole stock are 45%, chances of getting Rs. 2500 are 35% and chances for getting Rs. 3000 are 20%. If the goods are not sold in the first 4 weeks, then they will have to be disposed of in the fifth week at the prevailing market price in that week. When should the stocks be sold? 3. Use Bellman’s principle of optimality,

4. Use dynamic programming to solve the following LPP:

5. An oil company has 8 units of money available for exploration of three sites. If oil is present at a site, the probability of finding it depends upon the amount allocated for exploiting the site, as given below:

The probability that oil exists at the sites 1, 2 and 3 is 0.4, 0.3 and 0.2 respectively.Find the optimal allocation of money. 6. Seven units of capital can be invested in four activities with the return from each activity given in the accompanying table. Find the allocation of capital to each activity that will maximize the total return.

7. An oil company has 8 units of money available for exploration of three sites. If oil is present at a site, the probability of finding it depends upon the amount allocated for exploiting the site, as given below:

The probability that oil exists at the sites 1, 2 and 3 is 0.4, 0.3 and 0.2 respectively. Find the optimal allocation of money. 8. In 18th century, when transportation systems were not developed, a family wanted to travel from their home to reach a friend’s home in other part of the country. They had a choice of alternate routes and haltages between their home and the destination. The costs of travel from each point to the other point on route, based on such factors as distance, difficulty, mode of transportation, etc. are given below

Find the safest route so that the total traveling cost is minimum

UNIT – VIII

1. a) Define simulation. “When it becomes difficult to use an optimization technique for solving a problem, one has to resort to simulation”. Discuss. b) Write the advantages and disadvantages of simulation. c) Name different simulation languages. What are their distinguishing features? 2. A company trading in motor vehicle spares wishes to determine the level of stock it should carry for the items in its range. Demand is not certain and there is lead time for stock replenishment. For one item X, the following information is obtained:

Carrying cost per unit per day=20 Paise, ordering cost per order=Rs. 5/-, Lead time for replenishment=3days. Stock in hand at the beginning of the simulation exercise was 20 units. You are required to carry out a simulation run over a period of 10 days with the objective of evaluating the following inventory rule: Order 15 units when present inventory plus any outstanding order falls below 15units. The sequence of random numbers used is 0, 9, 1, 1, 5, 1, 8, 6, 3, 5, 7, 1, 2, 9 using the first number for day one. Your calculation should include the total cost of operating this inventory rule for 10 days. 3. Two persons X and Y work on a two-station assembly line. The distributions of activity times at their stations are

s i) Simulate operation of the line for eight items.

ii) Assuming Y must wait until X completes the first item before starting work, will he have to wait to process any of the other seven items? What is the average waiting time of items for Y. Use the following random numbers:

iii) Determine the inventory of items between the two stations. iv) What is the average production rate? 4. Find the value of experimentally by simulation. 5. Discuss about various types of simulation models. 6. Define simulation. Explain utility of simulation to solve inventory problems 7. What is the importance of simulation and modeling? Explain utility of simulation to solve inventory problems. 8. How can simulation help to optimize the inventory?