x+2y=10

2x+y=14

Y

XO

UNIVERSITY OF ENGINEERING & MANAGEMENT, JAIPURQUESTION BANK

SUBJECT NAME: OPERATIONS RESEARCH, SUBJECT CODE: ME705C

B.TECH, 4TH YEAR, 7TH SEMESTER

GROUP A

(OBJECTIVE/ MULTIPLE TYPE QUESTIONS)

1. Let m be the number of sources and n be the number of destinations in a transportation problem. Then a feasible solution is called a basic feasible solution if the number of non-negative allocations is equal to (a) m – n+ 1 (b) m - n – 1 (c) m + n – 1 (d) None of the above

2. Which method gives the smallest transportation cost:(a) Vogal’s approximation (b) North-West Corner (c) Lowest cost entry (D) None

3. The maximum value of objective function c=2 x+3 y in the given feasible region, is

(a) 29(b) 18(c) 14(d) 15

4. Non-negativity condition is an important component of LPP, becausea. Variables are interrelated in terms of limited resources. b. Value of variables makes sense and corresponds to real world problems. c. Value of variables should remain under the control of decision-maker. d. None of the above.

5. Given an LPP to maximize subject to , and.Using graphical method, we have

a. No feasible solution. b. Unbounded solution. c. Unique optimum solution. d. Multiple optimum solutions.

6. The role of artificial variables in simplex method isa. To aid in finding initial basic feasible solution. b. To start phases of simplex method. c. To find shadow prices from the final simplex table. d. None of the above.

7. Given the first initial iteration of simplex table, the first entering and leaving variable will be? 6 4 0 0 0 -m

0 30 2 3 1 0 0 0

0 24 3 2 0 1 0 0

-m 3 1 1 0 0 -1 1

a. and

b. and

c. and

d. and 8. Given the final iteration of simplex method, the final optimal value of the objective function

will be?

-1 4 1 0 4/5 2/5 1/10 0

3 5 0 1 2/5 1/5 3/10 10

0 11 0 0 10 2/5 -1/2 1

0 0 12/5 1/5 8/10 0

a. -11b. 11c. 12d. -12

9. ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ are expressed is n the form of inequities or equations A. Constraints B. Objective Functions C. Both A and B D. None of the above 10. The objective functions and constraints are linear relationship between ‐‐‐‐‐‐‐‐‐‐‐‐‐ A. Variables B. Constraints C. Functions D. All of the above 11. If the feasible region of a LPP is empty, the solution is ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ A. Infeasible B. Unbounded C. Alternative D. None of the above 12. Any column or raw of a simplex table is called a ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ A. Vector B. Key column C. Key Raw D. None of the above13. If there are ‘m’ original variables and ‘n’ introduced variables, then there will be ‐‐‐‐‐‐‐‐‐‐‐‐‐

columns in the simplex table A. m + n B. m– n C. 3 +m + n D. m + n – 1 14. A minimization problem can be converted into a maximization problem by changing the sign

of coefficients in the ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐

A. Constraints B. Objective Functions C. Both A and B D. None of the above 15. If in a LPP, the solution of a variable can be made infinity large without violating the

constrains, the solution is ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ A. Infeasible B. Unbounded C. Alternative D. None of the above 16. In simplex method, we add ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ variables in the case of ‘=’ A. Slack Variable B. Surplus Variable C. Artificial Variable D. None of the above 17. In simplex method, if there is tie between a decision variable and a slack (or surplus)

variable, ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ should be selected A. Slack variable

B. Surplus variable C. Decision variable D. None of the above

18. A BFS of a LPP is said to be ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ if at least one of the basic variable is zero A. Degenerate B. Non‐degenerate C. Infeasible D. Unbounded 19. The term linearity implies ‐‐‐‐‐‐‐‐‐‐‐ among the relevant variables: A. Straight line B. Proportional relationships C. Linear lines D. Both A and B 20. Which of the following O.R. problems cannot be expressed as a network flow problems?(i) An assignment problem(ii) A transportation problem(iii) A queuing problem(iv) None of the above21. Which of the following situations cannot be modeled as a network flow problem?(i) Determining the shortest route between two cities(ii) Finding the maximum capacity of water flowing in a pipe lines(iii) Determining the optimal stock to keep for cattle(iv) Finding the most scenic driving route for the weekend22. Which of the following statements is not correct?(i) A directed network allows flows in one direction only(ii) The network algorithms are more efficient than the Simplex method(iii) A connected network always has at least one path(iv) A cycle in a network may be directed or undirected23. Dijkstra’s algorithm is designed to determine the shortest path between(i) Any two nodes of the network(ii) The source node and any other node(iii) Any node and the destination node(iv) Any pair of nodes of a unidirectional network24. A transportation problem can be represented as a network flow problem where(i) Origins represent sinks and destinations the sources (ii) Origins represent sources and destinations the sinks

(iii) Objective is to maximize the network flow(iv) Per unit transportation costs become irrelevant25. The general linear programming problem is an standard form, if

1. The constraints are strict equations 2. The constraints are inequalities of ‘≤’ type3. The constraints are inequalities of ‘≥’ type4. The decision variables are unrestricted in sign

26. Which of the following statement is wrong?1. Slack variables are used to convert the inequalities of the type ‘≤’ into equations2. surplus variables are used to convert the inequalities of the type ‘≥’ into equations 3. An LPP with all its constraints are of the type ‘≥’ is said to be in standard form4. An LPP with all its constraints are of the type ‘≤’ is said to be in canonical form

27. A necessary and sufficient condition for a basic feasible solution to a maximization LPP to be an optimum is that (for all j) :1. z j−c j ≥02. z j−c j ≤03. z j−c j=04. z j−c j>0∨z j−c j<0

28. If two constraints do not intersect in the positive quadrant of the graph, then 1. The solution is feasible2. The solution is infeasible3. The solution is unbounded 4. None of these29. A Basic feasible solution of a LPP is said to be ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ if at least one of the basic

variable is zero a. Degenerate b. Non‐degenerate c. Infeasible d. Unbounded

30. If in a LPP, the solution of a variable can be made infinity large without violating the constrains, the solution is ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐

a. Infeasible b. Unbounded c. Alternative d. None of the above

31. Which technique is used in finding a solution for optimizing a given objective, such as profit maximization or cost minimization under certain constraints?

A. Quailing Theory B. Waiting Line C. Both A and B D. Linear Programming32. Assignment problem is solved by

a. Stepping Stone methodb. Two phase methodc. Hungarian methodd. Simplex method

33. The full form of CPM isa. Critical path managementb. Critical path methodc. Crash project managementd. None of these

34. When all constraints are of type in primal, the dual will havea. All constraints are of typeb. All constraints are of typec. Some constraints are of typed. None of these

35. A competitive situation is known as a ‘game’ if it has given characteristicsa. No of players is finiteb. Every game results in a payoffc. Both a and bd. None of these

36. If in a game, the minimax value= maximin value= 0, thena. The game is fairb. The game is strictly determinablec. Saddle points do not existd. The game is not fair

37. In an assignment problem involving four workers and three jobs, the numbers of possible assignments

a. 4b. 3c. 7d. 12

38. The coefficients for slack variables are alwaysa. Zerob. Positivec. Negatived. –M

39. Traffic intensity is given by

a.

b.

c.d. None of these

40. If the dual has unbounded solution, then primal hasa. Unbounded solutionb. Infeasible solutionc. Feasible solutiond. None of these

41. What is the method used to solve an LPP involving artificial variables?a. Simplex b. VAMc. Big Md. None of these

42. The shortest route between any two nodes in a network id determined by the following:a. Dijkastra’s algorithm b. Floyd’s algorithm c. Critical path methodd. None of these

43. Consider the constraint . Find the value of the slack variable is associated to

this constraint for the point .a. 8

b. 6c. 0d. -1

44. Which of the following accurately describes steps of the northwest corner rule, after making the initial allocation of units in the northwest cell?

a. Move down first, and then move rightb. Move right first, and then move downc. Move right or down first, depending on whether the demand requirement or the supply

capacity, respectively, is exhausted firstd. Move right or down first, depending on whether the supply capacity or the demand

Requirement, respectively, is exhausted45. What is the overall objective in applying the transportation method to the facility location

problem?a. minimize the distance traveledb. maximize the value of items shippedc. minimize the cost of the distribution systemd. minimize the number of items shipped

46. As the no. of persons is the same as the number of jobs, ……………is said to be balanceda. Assignment problemb. hungarian methodc. L.P.Pd. Transportation problem

47. The canonical form of LPP if the objective function is of minimization then all the constraints other than non-negativity conditions are ___________.a. greater than type. b. lesser than type. c. greater than or equal to type. d. lesser than or equal to type.

48. In the simplex method, the slack, surplus and artificial variables are restricted to be

a) multipliedb) negativec) non-negatived) divide

49. The third requirement of simplex method is that all the variables are restricted to include

a) negative even valuesb) odd valuesc) even valuesd) non-negative value

50. In simplex method basic solution set as (n-m), all the variables other than basic are classified as

a) constant variableb) non positive variablesc) basic variablesd) non-basic variable

GROUP B

(SHORT ANSWER TYPE QUESTIONS)

1. Find the basic feasible solution of the following transportation problem by North-West Corner Method

2. Draw the feasible region of the following LPP and find the optimal solution.

.3. Draw the feasible region of the following LPP and find the optimal solution.

.4. Draw the feasible region of the following LPP and find the optimal solution.

.5. Draw the feasible region of the following LPP and find the optimal solution.

.6. Draw the feasible region of the following LPP and find the optimal solution.

.7. Write the Dual of the following LPP:

D1 D2 D3 D4 SupplyS1 3 7 6 4 5S2 2 4 3 2 2S3 4 3 8 5 3Demand 3 3 2 2

.8. Write the dual of the following linear programming problem.

Maximize

Subject to

and .9. Write the dual of the following linear programming problem.

Minimize

Subject to

and .10. Find the basic feasible solution of the following transportation problem by Lowest Cost Entry

Method

11. Solve the following Assignment problem for least working hours:

I II III IVA 8 26 17 11B 13 28 4 26C 38 19 18 15D 19 26 24 10

12. Find the basic feasible solution of the following transportation problem by Vogal’s Approxiation Method

Supply

6 4 1 5 14

8 9 2 7 16

4 3 6 2 5

Demand 6 10 15 413. Write the dual of the following linear programming problem.

Minimize

Subject to

D E F G Available

A 11 13 17 14 250B 16 18 14 10 300C 21 24 13 10 400Demand

200

225

275

250

and .14. Write the dual of the following linear programming problem.

Minimize

Subject to

and .15. Solve the following assignment problem:

9 22 58 11 19 2743 78 72 50 63 4841 28 91 37 46 3374 42 27 49 39 3236 11 57 22 25 183 56 53 31 17 28

16. We have to purchase two articles A and B of cost Rs. 45 and Rs. 25 respectively. I can purchase total article maximum of Rs. 1000. After selling the articles A and B, the profit per unit is Rs. 5 and 3 respectively. If I purchase the x and y numbers of articles A and B respectively, then write the mathematical formulation of the problem.

17. Defined the followings:a. Mathematical formulation of the Transportation problem.b. Degenerate BFS and Unbalanced Transportation problem.c. Mathematical formulation of the Assignment problem.

18. Determine which of the following two person zero sum games are strictly determinable and fair Player A

Player B1 14 -3

Player A

Player B-5 2-7 -4

19. Find the range of the values of and which will render the entry a saddle point for the game

Player A

Player B2 4 510 7 q4 p 6

20. Find the saddle point/points and value of the following game Player A

Player BI II III IV V

I 9 3 1 8 0II 6 5 4 6 7III 2 4 4 3 8IV 5 6 2 2 1

21. Determine the optimal strategies and value of the following gamePlayer A

Player B5 13 4

22. The payoff matrix of a game is given. Find the solution of the game to the player A and B. Play Player B

er A I II III IV VI -2 0 0 5 3II 3 2 1 2 2III -4 -3 0 -2 6IV 5 3 -4 2 -6

23. The payoff matrix of a game is given. Find the solution of the game to the player A and B. Player A

Player BI II III IV V

I 3 -1 4 6 7II -1 8 2 4 12III 16 8 6 14 12IV 1 11 -4 2 1

24. Two players A and B match coins. If the coins match, the A wins two units of value. If coins do not match, the B wins two units of value. Determine the optimum strategies for the players and the value of the game.

25. Determine the optimal strategies and value of the following gamePlayer A

Player B4 -4-4 4

26. Derive expected numbers of customers in the system . 27. Derive the probability of customers in the system at steady state.28. Derive the expected waiting time of a customer in the queue for .29. A T.V. repairman finds that the time spent on his job has an exponential distribution with mean

30 minutes. If he repair sets in the order in which they come in, and if the arrival of sets is poissoinan with an average rate = 10/8 per hour, what is the repairman’s idle time each day? How many jobs are ahead of the average set just brought in?

30. Derive expected numbers of customers in the queue for . 31. Derive expected length of a non empty queue for .32. Derive the expected waiting time of a customer in the system for .33. Derive expected numbers of customers in the system . 34. Derive expected numbers of customers in the queue for .

35. Define convex set. Show that is a convex set.36. Four machines are to be installed in five factory locations. The cost of placing a particular

machine in a particular location is given in the following cost matrix.Location

Machine L1 L2 L3 L4 L5M1 9 11 15 10 11M2 12 9 - 10 9M3 - 11 14 11 7M4 14 8 12 7 8

Find the minimum cost assignment using Hungarian method.37. For what value of the following game, whose payoff matrix is given below, is strictly

determinable.Player B

Player A B1 B2 B3A1 6 2A2 -1 0A3 -2 4

38. Draw a network diagram corresponding to the following activity- predecessor table.

Activity A B C D E F G H IPredecessor

- - A, B

B B A,B F,D F,D C,G

39. Draw a network diagram corresponding to the following activity- predecessor table.Activity A B C D E F G HPredecessor

- - A B C,D D E F

40. Compare and contrast PERT and CPM, stating the circumstances where CPM is a better technique of project management than PERT.

41. Briefly mention the difficulties encountered in using network techniques.42. What is float? Discuss in brief different types of float, explaining their uses in the network.43. Draw a network diagram corresponding to the following activity- predecessor table.

Activity A B C D E F G H I J K L M N OPredecessor

- A A C B C D,E

G H F I,J

K L J M,N

44. What do you mean by network techniques? Also describe the areas of application of the network techniques.

45. What are the different phases of project management? Define event, activity, and looping.46. The annual demand of an item is 3200 units. The unit cost is Rs. 6 and inventory carrying

charges 25% per annum. If the cost of one procurement is Rs. 150. Determine.(i) EOQ(ii) Number of orders per year(iii) Time between two consecutive order(iv) The optimal cost

47. A contractor has to supply 10,000 bearings per day to an automobile manufacture. He finds that when he starts production run he can produce 25000 bearings per day. The holding cost of a bearing in stock is Rs. 0.02 per year. Set up cost of a production is Rs. 18. How frequently should production run be made?

48. An item is produced at the rate of 50 per day. The demand occurs at the rate of 25 items per day. If the set up cost is Rs. 100 per run and the holdings cost is Rs. 0.01 per unit of item per day. Find the economic lot size for one run assuming the shortages are not permitted. Also find the time of the cycle and minimum cost for one run.

49. A company has a demand of 12,000 units per year for an item and it can produce 2000 such items per month. The cost of one set up is Rs. 400 and the holding cost per unit per month is Rs. 0.015. Find the optimum lot size, max. Inventory, manufacturing time and the total time.

50. The annual requirements for a particular raw material are 2000 units, costing Rs. 1 each to manufacture. The ordering cost is Rs. 10 per order and the carrying cost 16% per annum of the average inventory value. Find the EOQ and the total inventory cost per annum.

GROUP C

(LONG ANSWER TYPE QUESTIONS)

1. A company produces two types of leather belts say of type and . Belt is of a superior quality and belt is lower quality. Profits on the two types of belts are 40 and 30 paisa per belt respectively. Each belt of type requires twice as much time as required by a belt of type . If all belts were of type , the company could produce 1000 belts per day. But the supply of leather is sufficient only for 800 belts per day. Maximum fancy buckles available for belt are 400 and for are 700 respectively. Find graphically how the company should manufacture the two types of belts in order to have maximum overall profit.

2. Old hens can be bought at Rs. 2 each and young ones at Rs. 5 each. The old hens lay 3 eggs per week and young ones lay 5 eggs per week, each egg being worth 30 paisa. A hen costs Rs. 1 per week to be fed there are only Rs. 80 available to spend on purchasing the hens and it is

not possible to house more than 20 hens at a time. Formulate the LPP and solve it by the graphical method to find how many of each kind of hens should be bought in order to have a maximum profit per week.

3. Find the optimal solution of the following unbalanced transportation problem for maximum cost:

4. Find the optimal solution of the following transportation problems for least cost:

5. Find the dual of the following

problem and then solve by simplex method:

Min

s.t.

and .6. Find the dual of the following problem and then solve by simplex method:

Min

s.t.

and .7. A departmental head has four subordinates, and four tasks to be performed. The subordinates

differ in efficiency, and the tasks differ in their intrinsic difficulty. His estimate, of the time each man would take to perform each task, is given in the table below:

Tasks MenE F G H

A 18 26 17 11B 13 28 14 26C 38 19 18 15D 19 26 24 10

How should the tasks be allocated, one to a man, so as to minimize the total man-hours by using Hungarian assignment method? 8. A pharmaceutical company is producing a single product and is selling it through five

agencies located in different cities. All of a sudden, there is a demand for the product in another five cities not having agency of the company. The company is faced with the problem

Plan

t

CustomerD1 D2 D3 D4 Supply

S1 40 25 22 33 100S2 44 35 30 30 30S3 38 38 28 30 70Demand 40 20 60 30 150, 200

Plan

t

CustomerD1 D2 D3 D4 D5 Supply

S1 4 1 3 4 4 60S2 2 3 2 2 3 35S3 3 5 2 4 4 40Demand 22 45 20 18 30 135

of deciding on how to assign the existing agencies to dispatch the product to needy cities in such a way that the travelling distance is minimized. The distance between the surplus and deficit cities (in Km) is given in the following table:

Deficit cities

Surp

lus

citi

esa b c d E

A 85 75 65 125 75B 90 78 66 132 78C 75 66 57 114 69D 80 72 60 120 72E 76 64 56 112 68

Determine the optimum assignment schedule.9. Solve the following LPP by using Simplex method:

.10. Solve the following Assignment problem by Hungarian method for least working hours:

I II III IV VA 1 3 2 3 6B 2 4 3 1 5C 5 6 3 4 6D 3 1 4 2 2E 1 5 6 5 4

11. Draw the graph and then solve the following game Player A

Player B1 25 4-7 9-4 -32 1

12. Using Dominance property, solve the following game Player A

Player B1 7 26 2 75 1 6

13. Two companies A and B are competing for the same product. Their different strategies are given in the following payoff matrix.

Company A

Company B4 -3 3-3 1 -1

Determine the best strategies for the two companies (Graphical method).14. Solve the following game problem graphically

Player A

Player B3 -3 4-1 1 -3

15. Using Dominance property, solve the following game A

B2 -2 4 16 1 12 3-3 2 0 6

2 -3 7 716. Solve the game without saddle point approach

A

B1 7 3 45 6 4 57 2 0 3

17. At a one man barber shop, customers arrive according to the Poisson distribution with a mean arrival rate of 4 per hour and their hair cutting time is exponentially distributed with an average hair cut taking 12 minutes. There is no restriction in queue length. Calculate the followings:(i). Expected time in minutes that a customer has to spend in the queue.(ii). Probability that there are at least 5 customers in the system.(iii). Percentage of time the barber is idle in 8 hour day.

18. At a railway station, only one train is handled at a time. The railway yard is sufficient for two trains to wait while other is given signal to leave the station. Trains arrive at the station at an average of 6 per hour and the railway station can handle them on an average of 12 per hour. Assuming Poisson arrivals and exponential service distribution. Find the steady state probabilities for the various numbers of trains in the system. Also find the average waiting time of a new train coming into the yard. If the handling rate is halved, how will the above results be modified?

19. A visitor parking at a college is limited to five spaces only. Cars making use of this space arrive at the rate of 6 cars per hour. Parking time is exponentially distributed with a mean of 30 minutes. Visitors who cannot find an empty space on arrival may temporarily wait inside the lot until a parked car leaves. That temporary space can hold only three cars other cars that cannot park or can find a temporary waiting space must go elsewhere. Determine the following.

(i) The probability, , of n cars in the lot(ii) The effective arrival rate for cars that actually use the lot.(iii) The average number of cars in the lot.(iv) The average time a car waits for a parking space inside the lot

20. Automatic car wash facility operates with only one bay. Cars arrive according to a Poisson process, with mean of 4 cars per hour and may wait in the facility’s parking lot if the bay is busy. If the service time for all cars is constant and equal to 10 minutes, determine

.21. A Xerox machine is maintained in an office and operated by a secretary who does

other job also. The service rate is Poisson distributed with a mean service rate of 10 jobs per hour. Generally, the requirements for use are random over the entire 8-hour working day but arrive at a rate of 5 jobs per hour. Several people have noted that a waiting line develops occasionally and have questioned the office policy of maintaining only Xerox machine. If the time of a secretary is valued at Rs.10 per hour. Find the following.i. Utilization of the Xerox machineii. The probability that an arrival has to wait.iii. The mean number of job of the systemiv. The average waiting time of a job in system.

22. The arrival rates of telephone calls at telephone booth are according to Poisson distribution with an average time of 12 minutes between arrivals of consecutive calls. The length of telephone call is assumed to exponentially distribute with mean 4 minutes. (i) Determine the probability that the person arriving at the booth will have to wait.(ii) Find the average queue length that formed from time to time.(iii) The telephone company will install the second’s booth when convinced that an arrival

would expect to wait at least 5 minutes for the phone. Find the increase in flows of arrivals which will justify the second booth.

(iv) What is the probability that an arrival will have to wait for more than 15 minutes before the phone is free?

23. In the city airport, flights arrive at rate of 24 flights per day. It is known that the inter arrival time follows an exponential distribution and the service time distribution is also exponential with an average of 30 minutes. Finds follows:i. The Probability that the system will be idle.ii. The Mean queue size.iii. The average number of flights in the queue.iv. The probability that the queue size exceeds 7.v. If the input of flights increases to an average 30 flights per day, what will be the

changes in (i) - (iv).24. The daily demand of biscuit packets in a manufacturing company follows a discrete

distribution as follows:The purchase price of each packet is Rs. 8. The selling price per packet is Rs. 11. If the biscuit packets are not sold within the daily of purchase, then they are sold at Rs. 4 per packet to shops. Find the optimal order size of biscuits.

Observation

1 2 3 4 5 6 7 8 9 10 11

Demand (Di)

25 26 27 28 29 30 31 32 33 34 35

Probability (Pi)

0.2 0.11 0.10 0.09 0.08 0.12 0.14 0.05 0.04 0.04 0.03

25. The production department for a company requires 3800 kg 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 the inventory is 25% of the investment in the inventories. The price is Rs. 10 per kg. find out the optimal order size, the order interval and the total annual inventory cost.

26. The following table shows the jobs of a network along with their time estimates. The time estimates are in days.

Activity 1-2 1-3 2-3 2-5 3-4 3-6 4-5 4-6 5-6 6-7Duration (Days)

15 15 3 5 8 12 1 14 3 14

(i) Draw the arrow diagram.(ii) Find the total float and free float for each activity.(iii) Find the critical path and total project duration.

27. A manufacturing company purchases 10000 parts of a machine for its annual requirements. Ordering one month’s requirements at a time. Each part costs Rs. 15. The ordering cost per order is Rs. 10 and the carrying cost is 16% of the average inventory per year. You have been assigned to suggest a more economical purchasing policy for the company. What advice would you suggest and how much would it save the company per year.

28. Consider the following data for activities in a given project:Activity A B C D E FPredecessor - A - B,C C D,EDuration (Days)

5 4 7 3 4 2

Draw the network diagram. Compute the earliest and latest event times. Find the critical path. What is the minimum project completion time?

29. The following table shows the jobs for a network along with their estimateJob 1-2 1-6 2-3 2-4 3-5 4-5 6-7 5-8 7-8t0 1 2 2 2 7 5 5 3 8tm 7 5 14 5 10 5 8 3 17tp 13 14 26 8 19 17 29 9 32

Draw the project network and find the probability of the project completing in 40 days.

30. Find the optimal solution of the following transportation problems for least cost:

31. Find the optimal solution for the following Transportation problem to minimize the overall transportation cost.

W1 W2 W3 W4 Availability

F1 2 3 5 1 7F2 7 3 4 6 9F3 4 1 7 2 18Demand 5 8 7 14 34

32. Solve the following LPP by using Simplex method:

.33. Solve the following LPP by using Simplex method:

.34. Solve the following LPP by using Simplex method:

.35. Solve the following LPP by using Simplex method:

.36. Using Dominance property, solve the following game.

B1 B2 B3 B4 B5A1 4 4 2 -4 -6A2 8 6 8 -4 0A3 10 2 4 0 12

37. Using Dominance property, solve the following game.B1 B2 B3 B4

A1 3 2 4 0A2 3 4 2 4

D1 D2 D3 D4 SupplyS1 19 30 50 10 7 S2 70 30 40 60 9 S3 40 8 70 20 18Demand 5 8 7 14

A3 4 2 4 0A4 0 4 0 8

38. Solve the following game problem graphically.

39. Solve the following game problem graphically.

40. Solve the following game.

41. Defined the followings:(i) Basic solution of an LPP and its types.(ii) Basic feasible solution of an LPP and its types.(iii) Optimal solution of an LPP.(iv) Convex set

42. Prove the followings:

(i) is a convex set.

(ii) is a convex set.43. Find the dual of the following problem and then solve by simplex method:

Max

s.t.

and .44. Find the dual of the following problem and then solve by simplex method:

Min

s.t.

and .45. Find the dual of the following problem and then solve by simplex method:

Min

s.t.

and .46. A farmer has a supply of chemical fertilizer of Type I which contains 10% nitrogen and 6%

phosphoric acid and Type II fertilizer which contains 5% nitrogen and 10 % phosphoric acid. After testing the soil conditions of a field, it is found that at least 14 kg of nitrogen and 14 kg of phosphoric acid is required for a good crop. The fertilizer type I cost Rs. 2 per kg and the Type II costs Rs. 3 per kg. How many kilograms of each fertilizer should be used to meet the requirement and the cost be minimum?

47. A firm manufactures two products A and B on which the profits earned per unit are Rs. 3 and Rs. 4 respectively. Each product is processed on two machines M1 and M2. Product A requires one minute of processing time on M1 and two minutes on M2

while B requires one minute on M1 and one minute on M2. Machine M1 is available for more than 7 hours while machine M2 is available for 10 hours during any working day. Find the number of unit’s products A and B to be manufactured to get maximum profit.

48. Patrons arrive at a reception counter at an average inter arrival rate of 3 minutes. The receptionist in duty takes an average of two minutes per patron. Calculate

a. Probability that the cashier is idle b. Average number of customer in the system c. Average time a customer spends in the systemd. Average time a customer spends in the queue

49. Consider a single server queuing system with Poisson input, exponential service times. Suppose the mean arrival rate is 3 calling units per hour, the expected service time is .25 hour and the maximum permissible number calling units in the system is two.

a. Probability that the cashier is idleb. Average number of customer in the systemc. Average number of customer in the queued. Average time a customer spends in the system

50. A farm is engaged in breeding pigs. The pigs are fed on various products grown on the farm in view of the need to ensure certain nutrients (call them X, Y and Z), it is necessary to buy two additional products, say A and B. One unit of product A contains 36 units of X, 3 units of Y and 20 units of Z. One unit of product B contains 6 units of X, 12 units of Y and 10 units of Z. The minimum requirement of X, Y and Z are 108 units, 36 units and 100 units respectively. Product A costs Rs. 20 per unit and product B Rs. 40 per unit.Formulate the above as a linear programming problem to minimize the total cost, and get the feasible region by using graphic method.