M.KUMARASAMY COLLEGE OF ENGINEERING, KARUR NUMERICAL METHODS.

UNIT I TWO MARKS

1. What is the order of convergence of Newton – Raphson method if the multiplicity of the root is one. Order of convergence of Newton – Raphson method is 2. 2. Newton – Raphson method is also known as the method of …… Answer: Iteration (Newton iterative method). 3. Derive Newton algorithm for finding the p th root of a number N. Solution: If x = N 1/p, then x p – N = 0 is the equation to be solved. Let f (x) = x p – N = 0 , f 1 (x) = p x p-1 By Newton – Raphson rule, if x r is the r th itrate f(xr) X r+1 = x r - ------ f1(xr) xr

p - N = xr - -------- Pxp-1

(p-1)xr

p + N = ----------------- Pxp-1

4. When should we not use Newton – Raphson method? If x1 is the exact root and x0 is its approximate value of

F(x0) the equation f(x) = 0. We know x1 = xo - -------. If f ’(x0) is small F ’(x0) F(x0) the error -------- will be large and the computation of the root by F ’(x0) this, method will be a slow process or may even by impossible. Hence the method should not be used in cases where the graph of the function when it crosses the X axis is nearly horizontal.

5. What is the rate of convergence in Newton – Raphson method? The rate of convergence in Newton – Raphson method is of order 2. 6. What is the criterion for the convergence (convergence condition) in Newton – Raphson method? |||| f(x) f”(x) |||| <<<< |||| f ’(x) ||||. 7. Write the iterative formula of Newton – Raphson method. f(xr) X r+1 = x r - ------ f ’(xr) 8. What are the merits of Newton’s method of iterative? Newton’s method is successfully used to improve the result obtained by other methods. It is applicable to the solution of equations involving algebraical functions as well as transcendental functions. 9. Say true or false: Newton’s method is useful when the graph of the function when it crosses the x – axis is nearly vertical. (True). 10. Say true or false: Newton’s method is useful in cases where the graph of the function when it crosses the x – axis is nearly horizontal. ( False ) . 11. Choose correct answer Newton’s method is convergent. (a) Linearly (b) Quadratically (c) cubically (d) Biquadratically Answer: (b).

12. What is condition for applying the fixed point iteration (successive approximation method) method to find the real root of the equation x=f(x) (or) If g(x) is continuous in [a, b], then under what conditions the iterative method x=g(x) has a unique solution in [a, b]? Let x=r be a root of x=g(x).Let I be an interval combining the point x=r. If ||||g’(x)|||| < 1 for all x in I, the sequence of approximation x1, x2, …. X n will converge to the root r, provided that the initial approximation x 0 is chosen in I. 13. What is the order of convergence for fixed point iteration? Soln: The convergence is linear and is of order 1.

14. Say true or false: ’iteration method’ is self correcting method Soln: True, iteration method is a self correcting method, since the round off error is smaller. 15. In case of fixed point iteration method, the convergence is a)Linear b)Quadratic c)Very slow d)h2 soln: linear. 16. What is the condition for the convergence of the iteration method for solving x=ΦΦΦΦ(x)? Soln: 1ΦΦΦΦ’(x)1 < 1 in the range. 17. True or False : ” Iteration method will always converge”. Soln: False. 18.Explain the term”pivot elements”. Soln: In an augmented matrix a11 a12 ..... a1n b1 a21 a22 ..... a2n b2 …. …. …. …. … an1 an2 ..... ann bn The elements a11,a22,…ann which have been assumed to be non-zero Are called pivot elements. 19. State the principle used in Gauss-Jordon method. Soln: Coefficient matrix is transformed into diagonal matrix. 20.For solving a linear system ,compare Gaussian elimination method and Gauss-Jordon method. Gaussian elimination method Gauss-Jordon method Coefficient matrix is transformed into upper triangle matrix. Direct method We obtain the solution by back substitution method

Coefficient matrix is transformed into diagonal matrix. Direct method No need of back substitution method

21. Give two indirect method to solve a system of linear equation. Soln: (i) Gauss Jocobi method (ii) Gauss-seidal method. 22.Compare Gauss – Jacobi and Gauss – Seidel methods. Gauss - Jacobi method Gauss- Seidel method 1 2 3

Convergence rate is slow Indirect method. Condition for convergence is the coefficient matrix is diagonally dominant.

The rate of convergence of Gauss – Seidel is roughly twice that of Gauss – Jacobi. Indirect method. Condition for convergence is the coefficient matrix is diagonally Dominant.

23. Is the iterative method, a self – correcting method always? In general, Iteration is a self correcting method, since the round off error is smaller. 24. Distinguish between direct and iterative(indirect) method of solving simultaneous equations. Direct method Indirect method 1 2

We get exact solution. Simple, take less time.

Approximate solution. Take consuming time.

25. What do you mean by ‘diagonally dominant’? A matrix is diagonally dominant if the numerical value of the leading diagonal element in each row is greater than or equal to the sum of the numerical values of the other value of the other element.

M.KUMARASAMY COLLEGE OF ENGINEERING, KARUR NUMERICAL METHODS.

Unit I University Questions

1. (a) Using Gauss elimination method solve the equation X + 2 y + z = 3; 2 x + 3 y +3 z = 10; 3 x – y – 2 z = 13 (b) Solve the following equations by Gauss-seidel method 28 x + 4 y – z = 32; X + 3y + 10 z = 24; 2 x + 17 y + 4 z = 35. 2. (a) Find the root between ( 2 , 3) of x 3 –2 x - 5=0 by regular false method. (b) Compute real root of f( x , y ) = x 2 + y 2 – 4 = 0, g ( x ,y)= y+ex – 1 =0 correct to three decimal places using Newton Raphson method . 3. (a) Find the positive root of 3 x-√√√√ 1+ sinx = 0 by iteration method (b) Determine the real root of x e x – 3 = 0 correct to five decimal places, using the method of false position. 4. (a) Using Gauss Jacobin iteration method solve the equation X + 17 y - 2z = 48 30 x – 2 y + 3z = 75 2 x + 2 y + 18 z = 30. (b) Solve the following equations by Gauss Jordan method 10 x + y + z = 12 2 x + 10 y + z = 13 X + y + 5 z = 7. 5. (a) Solve the following system by gauss Seidal method x + y + 54 z =110; 27 x + 6y – z = 85; 6x + 15y + 2z = 72. 2 1 1 (b) Find the inverse matrix of by Jordan method. 3 2 3 1 4 9 6. (a) Find the solution of equation 4 x 2 + 2 x y + y 2 = 30 and 2 x 2 + 3 x y + y 2 = 3 correct to three Places of decimals, using Newtons Raphson method, given that x0 = - 3 and y0 = 2 . (b) Using iterative method, find the root of the equation cosx – x e x =0.

7. (a)Solve for positive root of the equation x-cos x=0, by Newton’s method. (b).Solve by Gauss elimination method, the system of equations 3 x – y + 2 z = 13 , X + 2 y + z = 3, 2 x + 3 y + 3 z = 10. 8. (a) Solve by Gauss seidel method, X + y + 54z = 110, 27 x + 6 y – z = 85, 6 x + 15 y + 3 z = 72. 2 2 3 (b).Using Gauss Jordan method,find inverse of the matrix 1 2 1 1 3 5 9. (a) Using Gauss Jacobin method, solve X + y + 5 z = 16, 2 x + 3 y + z = 4, 4 x + y – z = 4. (b).Using Gauss elimination method solves the following equations. 4 x + 4 y – 3 z = 4; 10 x + 8 y – 6 z = 5; 20 x – 4 y + 22 z = 7. 10.(a) Solve the equation x 6 – 5 x 2 + 136 = 0 and y 4 – 3 x 4 y + 80 = 0 assuming that ( 2 , 3 ) is an approximate solution. Obtain the solution correct to 2 decimals. (b). Solve X + 3 y + 3 z = 16, X + 4 y + 34 z = 18, X + 3 y + 4 z = 19, by Gauss Jordan method.

UNIT – 2

1. Define a Queue A flow of customers from finite or infinite population

towards the service facility forms a queue (waiting line) on account of lack of capacity to serve them all at a time.

2. State two characteristic of Queuing models. Basic elements of Queuing system are

(i) Input process (ii) Queue discipline (iii) Service mechanism (iv) Capacity of the system (v) The input describes the way in which the customers arrive

and join the system (vi) It is a rule according to which customers are selected for

service when a Queue has been formed. (vii) It is concerned with service time and service facilities (viii) The source from which the customers are generated may

be finite or infinite.

3. State some of the measures (or) operating character is of Queuing system.

(i) Expected number of customers in the system denoted by E(n) or Ls.

(ii) Expected no. of customers in the queue denoted by E(m) or L9

(iii) Expected waiting time in the system denoted by Ws (iv) Expected waiting time in the queue denoted by Wq (v) The server utilization factor denoted by ρρρρ=λλλλ / µµµµ.

4. Define arrival rate, service rate. The average number of customers arriving per unit of time is known as arrival rate. It is denoted by λλλλ. The average no. of customers completing service per unit of time is known as servicing rate. It is denoted by µµµµ.

5. Define Transient state of queuing system. A system is said to be intransient state when it’s operating characteristics are dependent on time.

6. Define Steady state. A system is said to be in steady state when the behaviors of the of the system is

Independent of time. 7. What do you understand by Kendall’s notation?

Kendal’s notation is used to represent Queuing models. Generally Queuing model may be completely specified in the following form (a/ b/c) : (d/c) where, a = Probability law for arrival. b = Probability law according to which customers are served. c = Number of channels ( or service station) d = Capacity of the system e = Queue discipline

8. Mention some of the Queue discipline with examples 1.FIFO - First in first out (OR) FCFS – First come First served eg : Person waiting to reserve ticket in the railway station 2.LIFO – Last in first out (OR) LCFS – Last come first served FCLS - First come last served eg : In the case of a big go down the items which came last are taken out first. 3.SIRO – Service in random order eg : Tools checked out from a crib in a machine shop. 9.Explain the term Balking and Reneging in the queue. Balking : It is one of the customers behaviors in the queue . A customer may have the Queue if there is no waiting space. Reneging : This occurs when the waiting customer leaves the queue due to Impatience. 10.Explain the term Priorities and Jockeying in the queue. Priorities : In certain applications some customers are served before others regardless of their order of arrival. Jockeying : Customers may jump from one waiting line to another for their personal gain.

11.Under (M/M/S): (∞∞∞∞/FCFS) model , µµµµn will be equal to what? µµµµn = Sµµµµ if n ≥≥≥≥ S µµµµn = nµµµµ if n ≤≤≤≤ S. 12.The utilization factor under multiserver model is (M/M/S): (∞∞∞∞/FCFS) is what? ρρρρ = λλλλ/Sµµµµ 13.What is the formula for ρρρρn under (M/M/1): (N/FCFS) ?. Pn = (1- ρρρρ) / (1- ρρρρ N +1 ) where ρρρρ = λλλλ/µµµµ. 14.Given λλλλ = 10| hr , µµµµ = 3|hr ,S = 4, Po = 0.0213,Find the length of the system. Ls = λλλλ/µµµµ + Lq , λλλλ/µµµµ = 3.3333 λλλλ/Sµµµµ (λλλλ/µµµµ) S * Po

Where Lq = 1 - λλλλ/Sµµµµ S!!!! = 3.287 Ls = 3.3333 + 3.287 = 6.6. 15.What is the distribution for the service time? Exponential distribution or Erlang distribution. 16. What do you mean by queue discipline? It is the rule according to which customers are selected for service when a queue has been formed . Examples: FCFS – First come First second LCFS – Last come First served SIRO - Serviced in random order

17. What is the traffic intensity? If traffic intensity of M/M/1 system is given to be 0.76, What percent of time the system would be idle? The server utilization factor or busy period or traffic intensity (ρρρρ =λλλλ/µµµµ) is the proportion Of customers. Here λλλλ stands for the average no. of customers arriving per unit of time , and µµµµ stands for the average no. of customers completing per unit of time Given that ρρρρ =0.76

Expected idle time =1-ρρρρ=1-o.76=0.24 Percent of time the system would be idle = .24x100=24%

18. Write little’s formula ? Relations between average Queue length and Average waiting time is called Little’s formula. WKT for (M/M/1) : (∞∞∞∞/FCFS) model, Ls =λλλλ/(µµµµ-λλλλ) Lq= λλλλ2/(µµµµ (µµµµ-λλλλ )) ,Wq= λλλλ/(µµµµ (µµµµ-λλλλ)) , Ws=1/(µµµµ-λλλλ) Using these expression , we get

Ls=λλλλ Ws, Lq=λλλλ Wq, Ws= Wq+1/µµµµ. 19. Describe the queuing models (M/M/1) and (M/M/1/k)

(M/M/1) -This is a queuing model with single server and infinite capacity of the system First symbol M represents that arrival rate follows poisson distribution Second symbol M represents that service time follows Exponential distribution (M/M/1/k) -This is a queuing model with single server and finite capacity of the system . First symbol M represents that arrival rate follows poisson distribution Second symbol M represents that service time follows Exponential distribution

20. Given example for an application of( M/M/c ) (N/FCFS) queuing system . A barbershop has two barbers and three chairs for customers. Assume that the Customers arrive in poisson fashion at a rate of 5 per hour and that each barber services Customers according to an

exponential distribution with mean of 15 minutes. Further if the arrives and there are no empty chairs in the shop, he will leave. 21.Give the formula for Poisson distribution and exponential distribution formula. Poisson distribution p(x) = (e-λλλλx λλλλx) /x! Exponential distribution f(x)= = λλλλ e-λλλλx ,x>0

QUEING THEORY 12 MARK QUESTIONS 1. Customers arrive at a watch repair shop according to a

Poisson process at a rate of one per every 10 minutes , and the service time is an exponential random variable with mean 8 minutes. i. Find the average number of customers L s in the shop . ii. Find the average time a customer spends in the shop Ws . iii. Find the average number of customers in the queue L q . iv. What is the probability that the server is idle .

2. Automatic car wash facility operators with only one bay .Cars arrive according to a Poisson process at the rate 4 cars per hour and may wait in the facility ‘s parking lot if yhe bay is busy . If the service time for all cars is constant and equal to 10 minutes , determine i. Mean number of customers in the system L s . ii. Mean number of customers in the queue L q . iii. Mean waiting time of a customer in the system W s . iv. Mean waiting time of a customer in the queue W s .

3. Find the average number of customers L s in the M/M/1 queuing system when λ=µ .

4. A bank has two tellers working on savings account. The first teller handles withdrawals only. The second teller handles deposits only . It has been found that the service time distributions for both deposits and withdrawals are exponential with mean service time of three minutes per customer .Depositors are found to arrive in a Poisson fashion throughout the day with mean arrival rate of 16 per hour .What would be the effect on the average waiting time for the customers if each teller could handle both withdrawals and deposits.

5. There are three typists in an office .Each typist can type an average of 6 letters per hour. If letters arrive for being typed at the rate of 15 letters per hour , what fraction of time all the typists will be busy? What is the average number of letters waiting to be typed?.

6. A super market has 2 girls attending to sales at the counters . If the service time for each customer is exponential with mean 4 mins and if people arrive in Poisson fashion at the rate of 10 per hour , i . What is the probability that a customer has to wait for service? ii.What is the expected percentage of idle time for each girl?

7. Customers arrive at a one man barber shop according to a Poisson process with a mean arrival time of 12 min.Customers spend an average of 10 min in the barbers chair? i. What is the expected no of customers in the barber show

and in the queue? ii. What is the probability that more than 3 customers are in

the system? 8. A petrol pump station has 2 pumps . The service time

follow the exponential distribution with mean of 4 minutes and cars arrive for service is a Poisson Process at the rate of 10 cars per hour . Find the probability that a customer has to

wait for the service.What is the probability that the pumps remain idle?.

9. Obtain the steady state probabilites for M/M/1 / N / FCFS queueing model.

10. In a given M/M/1 queueing system , the average arrivals is 4 customers per minute : ρ = 0.7 . What are i. Mean number of customers L s in the system ii. Mean number of customers L q in the queue iii. Probability that the server is idle iv. Mean waiting time W s in the system.

11. Customer arrive at a one man barber shop according to a Poisson process with a mean inter arrival time of 20 mins. Customers spend an average of 15 mins in the barber chair . If an hour is used as a unit of time , then i. What is the probability that a Customer need not wait for

a hair cut? ii. What is the expected no of customers in the barber show

and in the queue? iii. How much time can a customer expect to spend in the

barber shop? iv. Find the average time that the Customer spend in the

queue. v. What is the probability that there will be 6 or more

customers waiting for queue?.

12. Derive the formula for the average number of customers in the queue and the probability that an arrival has to wait for (M/M/C) with infinite capacity .Also derive for the same model the average waiting time of a customer in the queue as well as in the system.

UNIT – 3 1.Define a string A string (or) A word over an alphabet ΣΣΣΣ is a finite ordered list of symbols chosen from the set ΣΣΣΣ.

Ex: (i) 1001, abab 2.Define empty string or null string The string having no symbol is called the empty string. It is denoted by ‘εεεε’or ‘λλλλ’, and we have εεεε = λλλλ=0. 3.Define kleene star (or) closure Let ΣΣΣΣ be an alphabet set. The kleene star ΣΣΣΣ denotes the set of all strings including εεεε the set of all strings the alphabet set ΣΣΣΣ. Here ΣΣΣΣ+ = ΣΣΣΣ*-{ εεεε }. 4. Concalsenation-Define it. Let u ad v be any 2 string of alphabet ΣΣΣΣ, a cancalsensation of u and v denoted by uv is a new string obtained by u followed v Ex: u=abc v=123 uv=abc123 5.Define language Let ΣΣΣΣ be an alphabet set and ΣΣΣΣ* be the set of all strings over ΣΣΣΣ. Consider the subset of ΣΣΣΣ*. this subset is language over ΣΣΣΣ and it is denoted by (ie) L⊆⊆⊆⊆ΣΣΣΣ*

6.Define phrase structure grammar The phrase structure grammar is defined by G={N,T,P,S} where N - Non-terminal symbols T- Terminal symbols P – set of production rules. S – Starting symbol Ex: If G = ⟨⟨⟨⟨{s,B,c}{a,b}p,s⟩⟩⟩⟩ P consist of following elements , S→→→→ asb S→→→→ aB B→→→→ ab

7.State types of grammar , types of language.

(i) Type-0 grammar (ii) Type-1 grammar (iii) Type-2 grammar(context free) (iv) Type-0 grammar (regular) Similarly (i) Type-0 language (ii) Type-1 language (iii) Type-2 language (iv) Type-3 language

8. Define type-0 grammar. A grammar that has no restrictions on its production rule is called type-0 grammar. 9. Define type -1 grammar. A grammar in which every production is of the form w1 →→→→w2 , where w1≤≤≤≤ w2 or of the form w1 →→→→λλλλ , is called a type -1 grammar. Eg: If G contains P = {aB →→→→abb , B →→→→ εεεε},then G is called Type -1. 10.Define type -2 grammar. A grammar in which every production is of the form w1 →→→→w2 , where ‘w1’ is a single non-terminal is called Type – 2. Ex: If G contains P={ A→→→→ab, S→→→→εεεε , B→→→→εεεε},then G is called Type-2. 11.Define type -3 grammar. A grammar in which every production is of the form w1 →→→→w2 , where ‘w1’ is a single non-terminal and‘w2’ is a single terminal or terminal followed by a non-terminar or of the form S→→→→λλλλ is called Type – 3 or regular grammar. Ex: If G contains P={ A→→→→a, S→→→→BA , B→→→→bC}. 12.Define Context sensitive.

A grammar in which every production is of the form αααα w1 ββββ→→→→ αααα w2 ββββ is called a Context sensitive grammar. 13.For the following production identify the corresponding grammar

(i) S→→→→ aAB ; S →→→→ AB ; A →→→→a;B→→→→b. (ii) S→→→→ aB ; B→→→→ AB; aA→→→→b; A→→→→a;B→→→→b. (iii) S→→→→aAB; AB→→→→bB; B→→→→b; A→→→→aB. (iv) S→→→→AB; B→→→→bB; B→→→→bA; A→→→→a; B→→→→b. (v) S→→→→A; S→→→→AAB; Aa→→→→Aba; A→→→→aa; Bb→→→→Abb; AB→→→→ABB; B→→→→b. (vi) <S> : : = b<S>| a<A>|a ;

<A>: : = a<S> | b<B> ; <B> : : = b<A> | a<S> | b;

Ans: (i) Type – 2(ii) Type –0 (iii) Type – 1 (iv) Type – 3 (v) ontext sensitive (vi) Type – 3.

14. What is derivation tree? It is a graphical representation for the derivation of the given production rules for a given context free grammar . It has the following properties

(i) the root node is always a node that indicating a starting symbol. (ii) The derivation is read from left to right (iii) The leaf node is always a terminal node (iv) The interior nodes is always non- terminal nodes

15. What is left most derivation? The left most derivation is the derivation in which the left most non-terminal is replaced first from the sentential form. 16. What is right most derivation? The right most derivation is the derivation in which the right most non-terminal is replaced first from the sentential form. 17. Define :- ambiguous.

The grammar that can be derived in either left most or right most derivation but if there exist that one parse tree for a given grammar . that mean there could be more than one left most derivation or right most derivation possible and then that grammar is said to be an ambiguous grammar. Otherwise it is called unambiguous. 18.Define finite state automata Afinite state automata can be defined as 5 tuples M = { Q , ΣΣΣΣ ,δδδδ , q o , F }, Q = Non empty finite set of states , ΣΣΣΣ = Non empty finite set of symbols, q o = Initial state , F = set of finite states, δδδδ = Transition function Q ××××ΣΣΣΣ →→→→ 2 θθθθ where 2 θθθθ is the power set of Q. 22.Define Regular expression Let ΣΣΣΣ be an alphabet which is used to denote the input set . the Regular expression over ΣΣΣΣ can be defined as follows 1. ϕϕϕϕ is a regular expression which denotes the empty set 2.εεεε is a regular expression and denotes the setn {εεεε} 3.For each a ∈∈∈∈ΣΣΣΣ , a is a regular expression and denotes the set{a} 4.If r and s are regular expression denoting the languages L1 and L2 then r+s = L1 ∪∪∪∪ L2 ( union ) rs = L1 L2 ( cancatenation) r* = L1

* ( closure ) 23.Write the regular expression for the following (i) Accepting all combinations of a’s over the set {a}: R = a* (ii) Starting with 1 and ending with 0 : R = 1(0+1)*0 (iii) starting or ending with either 00 or 11 : R = [(00+11)(0+1)*]+[(0+1)*(00+11)].

UNIT -4

1. What is a project network? A project network is a graphical diagram consisting of a certain configuration of arrows and nodes for showing the logical sequence of various tasks (or activities) of a project to be performed to achieve project objectives.

2. What do you mean by an activity of a project?

Activity is a work to be done in a project that consumes resources like time, money, labour etc.

3. What are the three main phases of a project? 1. Planning 2. Scheduling 3. Control

4. What is meant by a project? A project is a collection of interrelated activities with each activity consuming time and resources.

5. Name the two important network scheduling techniques. a. Critical Path method (CPM) b. Programme Evaluation and Review Technique (PERT)

6. What is a dummy activity? It is an imaginary activity which does not consume any resource and is used to maintain precedence relationship in the network diagram.

7. Define critical path of a project network. The sequence of critical activities in a network called the critical path. The critical path is the longest path in the network from the starting event to ending event.

8. What is a critical activity? / Define Total float

The difference between the latest start time and earliest start time of an activity will indicate the amount of time by which the activity can be delayed without affecting the total project duration. This difference is usually called as total float. “Activities with zero total float are known as critical activities”

9. Write the formula for free float. Free float (FFij) = Total float – Head slack = Total Float – (Lj – Ej)

10. Distinguish the slack time and Float time Slack Float

a. It is used for events a. It is used for the activities

b. There are two slack times b. There are three floats

i. Head slack ((Lj – Ej) i. Total float (TF) ii. Tail Slack (Li – Ei) ii. Free Float (FF)

iii. Independent Float (IF)

11. What is the formula for the Head and Tail slack? Head slack = (LFj – ESj) Tail Slack = (LFi – ESi)

12.Write the drawbacks / demerits of CPM ?

(1) Baseed on the assumption of precise known time for each of the activity in the project which may not be true in real life situations.

(2) Does not incorporate statistical analysis in determining the times estimates.

(3) It is difficult to use CPM as controlling device for the simple reason that entire evaluation of the project is to be repeated and a new critical path is to be found out each time when certain changes are introduced in the network

(4) CPM was initially developed as a static planning model and not as a dynamic controlling device

13. write the assumptions made on a PERT calculation the two main assumptions made on PERT calculation are

a. the activity duration independent the time required to complete an activity will be have no bearing on the completion time of any other activity of the project .

b. the activity duration follows β –distribution whose , standard deviation σ = tp-t0/ 6 and mean te = t0+4tm+tp/6

14. Distinguish between CPM and PERT

CPM 1 A deterministic model with well known activity , times based upon the past experience . it therefore does not deal with uncertainty in time. 2 It is activity oriented as the results of calculations are considered in terms of activites or operations of the project. 3 The use of dummy activity is not necessary. 4 CPM is commonly used for those projects which are repetitive in nature and where one has prior experience of handling similar projects. 5 It is used for construction projects and business problems PERT A probability model with uncertainty in activity duration the duration of each activity is normally computed from multiple time estimates with a view to take into account time uncertainty. 2 It is said to be event oriented as the results of analysis are expressed in terms of events or distinct points in time in dilative of progress. 3 The use of dummy activity is required sequencing.

4 PERT in generally used for those projects where time required to complete various activities is not known a priori. 5 it is applied for widely planning and scheduling reason programmes and developing projects.

15. With t o ,t m , t p write the formula for expected duration and expected variance of

each activity. Expected duration of each activity te = (to +4 tm

+tp)/6 PERT AND CPM 12 MARKS

1. From the data given below for the network , draw the network diagram.Crash the project ti its minimum completion time . What is the completion time and cost for the crashed network?

Activity Pre requisite Time (days) Cost( Rs.) Activities Normal Crash Normal Crash A _ 40 35 12,000 16,000 B A 20 10 300 600 C A 30 15 500 800 D B , C 50 40 600 1,000 E C 30 25 1,000 2,000 F D, E 0 0 _ _

2. For the following project draw the network and find the critial

path.Also calculate the earliest , latest and slacktimes for all events. Activity : 1 – 2 2 – 3 2 – 4 2 – 5 3 – 5 4 – 6 5 – 7 6 – 7 Performance time: 5 8 7 5 2 9 6 3

3. A small maintenance profect consists of the following jobs where precedence relationships is given below :

Job : 1 – 2 1 – 3 2 – 3 2 – 5 3 – 4 3 – 6 4 – 5 4 – 6 5 – 6 6 – 7 Duration : 1 5 15 3 5 8 12 1 14 3 14 (i) Draw an arrow diagram representing the project. (ii) Find the total float for each activity. (iii) Find the critial path and total project duration.

4. A small project composed of 7 activities whose time estimates are listed in the table below.Activities are identified by their beginning (i) and ending (j) node numbers :

Activity Expected duration ( weeks ) i - j Optimistic most likely pessimistic 1 – 2 1 1 7 1 – 3 1 4 7 1 – 4 2 2 8 2 – 5 1 1 1 3 – 5 2 5 14 4 – 6 2 5 8 5 – 6 3 6 15 (i) Draw the network , find the critical path and project time

duration. (ii) What is the probability that the project will be completed

atleast four weeks earlier than expected?. 5. A project consists of the following activities and time

estimates: Activity : 1 – 2 2 – 3 1 – 4 2 – 5 2 – 6 3 – 6 4 – 7 5 – 7 6 -7

Least time(days) :3 2 6 2 5 3 3 1 2 Greatest time(days) : 15 14 30 8 17 15 27 7 8 Most likely time(days): 6 5 12 5 11 6 9 4 5 (i) Draw the network , find the critical path and project time

duration. (ii) What is the probability that the project will be completed

in 27 days? (iii) What due date has about 95% chance of being met?.

6. Distinguish between PERT and CPM .

7. A maintenance foreman has given the following estimates of times and cost of jobs in a maintenance project :

Job predecessor Normal Crash Time(hours Cost(Rs) Time(hours) Cost(Rs) A - 8 80 6 100 B A 7 40 4 94 C A 12 100 5 184 D A 9 70 5 102 E B , C , D 6 50 6 50 Overhead cost is Rs. 25 per hour . Find the optimum duration of the project and associated cost.

8. A project schedule has the following data : Activity : 1 – 2 1 – 3 1 – 4 2 – 5 3 – 5 5 – 6 4 – 6 Time : 4 6 5 3 4 2 3 Find the critical path . 9. Bring out the difference between CPM and PERT . Also discuss the various steps involved in developing PERT network analysis.

10. A project consists of the following activities: Activity : 1 – 2 1 – 3 1 – 4 2 – 5 2 – 6 3 – 6 4 – 7 5 – 7 6 -7 Least time(days) :5 18 26 16 15 6 7 7 3 Greatest time(days) : 10 22 40 20 25 12 12 9 5 Most likely time(days): 8 20 33 18 20 9 10 8 4 Perform the PERT network analysis and identify any one critical path. 11. A small project is composed of 7 activities whose time estimates are given as below: Activity : 1 – 2 1 – 3 1 – 4 2 – 5 3 – 5 4 – 6 5 – 6 Optimistic time(weeks) : 1 1 2 1 2 2 3 Most likely time : 1 4 2 1 5 5 6 Pessimistic time : 4 7 8 1 14 8 15 (i). Draw the project network . (ii). What is the expected project length? (iii). What is the probability that the project will be completed atleast 4 weeks earlier than expected time?(Given P(0≤ z ≤ 1.33) = 0.4082). 12. Listed in the table are the activities and sequencing requirements necessary for the completion of a research report. Activity Description Precedence Duration A Literature search _ 6 B Formulation of hypothesis _ 5 C Preliminary feasibility study B 2 D Formal proposal C 2 E Field analysis A,D 2 F Progress report D 1 G Formal research A,D 6 H Data collectio E 5 I Data analysis G,H 6 J Conclusion I 2 K Rough draft G 4 L Final copy J,K 3 M Preparation of oral presentation L 1 i) Draw a network diagram for this project. (5)

ii) Find the critical path. What is a lenght? (2) iii) Find the Earliest strat time, Earliest finish time , Latest start time and latest finish time (6) iv) Find the total float and the free float for activity. (3)

UNIT-V

1.Let S be the sample space associated with an experiment. To each event A , there is a real number P(A) associated, called the probability of A satisfying the following axioms: (i) P(A) ≥≥≥≥0 (ii) P(S) = 1 (iii) If A1,A2, ………An are mutually exclusive events, then

P(A1∪∪∪∪A2∪∪∪∪ ………∪∪∪∪An) = P(A1)+ P(A2) + ….+P(An) 2. Define mutually exclusive events. Events are called mutually exclusive when no two of them can occur simultaneously. 3.Define equally likely events. Events are called equally likely when none of them can be performed rather than the other. 4. Define independent event. A set of events is said to be independent if the occurance of any one of them does not depend on the occurance of the other. 5. What is the chance that a leap year selected at random will contain 53 Sundays? In aleap year there are 366 days that is 52 weeks and two days remaining can be {(s,m),(m,t),(t,w),(w,th),(th,f),(f,sa),(sa,s)}. For 53 Sundays out of remaining two days one must Sunday.

Therefore number of favourable cases = 2 Total number of cases = 7 Required probability = 2/7. 6.A and B are events such that P(A Ụ B)=3/4 , P(A∩ B)=1/4 ,P(Ac) 2/3 Find P(Ac/B). By definition , P(Ac ∩ B) P(Ac/B) = P(B) we know that P(Ac ∩ B) = P(B) - P(A∩ B) 3/4 =1/3+ P(B) -1/4 Therefore P(B) = 2/3 Implies P(Ac ∩ B)= 2/3 -1/4 =5/12 Therefore P(Ac/B) = 5/8. 7.If you twice flip a coin ,what is the probability that getting at least one head? S={ HH ,HT,TH,TT} Let E= { Getting atleast one head}={ HH ,HT,TH} Therefore P(E) = n(E)/n(S) = 3/4 . 8.Define conditional probability. The conditional probability of an event B , assuming that event A has happened,is denoted by P(B/A) = n(A∩ B)/n(A) = P(A∩ B)/P(A), provided P(A) ≠ 0.

9.Define Multiplication theorem of probability.

For any two events A and B , P(A∩ B) = P(A) P(B/A) provided P(A) ≠ 0. = P(B)P(A/B) provided P(B) ≠ 0. 10. Define Addition theorem of probability. For any two events A and B , P(A Ụ B) = P(A)+P(B) – P(A∩ B).

PROBABILITY 12 MARK QUESTIONS

1. An urn contains five balls . Two balls are drawn and found to be white . What is the probability of all the balls being white? 2. There are two boxes .Box1 contains 2 white ,1black balls and Box2 contains 1 white & 5black balls . A ball is drawn from box1 and is placed into box2 and hence a ball is drawn.It happens to be white.What is the probability that the transformed ball is white given that the white ball is drawn?. 3. A condence of Box1 , Box2& Box3 are as follows.2 white , 3black , 4red balls ; 3 white , 2black ,2red balls; 4 white,1black ,3 red balls; and box is choosen at random and 2 balls are drawn .They happen to be white and red.What is the probability that they came from Box1, Box2& Box3. 4. An ice cream seller has to decide whether to order more stoke for the week end , he estimate based on the past experience that if weather is hot ,he has a 90% chance of selling his all stoke .If it is cloudy his chance is 60% & if it is rainy his chance is only 20% according to weather forecast the probability of sunshine is 0.45, the probability of cloudy is 0.3 & the probability of rain is 0.25 .Determine the probability that the salesman sell all his stoke. 5. A box contains 5 red ,4 white balls,2 balls are drawn successively from the box without replacement and its is noted that second one is white . What is the probability that first is also white?. 6. In a shooting test the probability of hitting is 1/2 for A , 2/3 for B , 3/4 for C if all of them fire at the target.Find the probability that (i) None of them hit the target (ii) Atleast one of them hit the target. 7. a )From a well suffled pack one card is drawn .What is the probability that a card being red or king? b )What is the chance that (i) a non-leap year should have 53 Sundays (ii) a leap year having53 Sundays. 8. State and prove Baye ‘ s theorem. 9. State and prove addition and multiplication theorem on probability.

10. A and B throw alternatively with a pair of ordinary dice .A wins if he throws 6 before B throws 7 and Bwins if he throws 7 before A throws 6.If A begins , find probability of their respective winning. 11. A,B and C in order toss a coin . The first one to throw a head wins .What are their respective chances of winning assumming that the game may continue indefinitely?. 12. a )Show that P(AB) ≤ P(A) ≤P(A+B) ≤P(A)+P(B) for any two events A and B. b )If A and B are independent events then (i) A and Bc are independent (ii) Ac and B are independent and (iii) A c and Bc are independent.

PERT AND CPM 12 MARKS

13. From the data given below for the network , draw the network diagram.Crash the project ti its minimum completion time . What is the completion time and cost for the crashed network?

Activity Pre requisite Time (days) Cost( Rs.) Activities Normal Crash Normal Crash A _ 40 35 12,000 16,000 B A 20 10 300 600 C A 30 15 500 800 D B , C 50 40 600 1,000 E C 30 25 1,000 2,000 F D, E 0 0 _ _

14. For the following project draw the network and find the critial

path.Also calculate the earliest , latest and slacktimes for all events. Activity : 1 – 2 2 – 3 2 – 4 2 – 5 3 – 5 4 – 6 5 – 7 6 – 7

Performance time: 5 8 7 5 2 9 6 3

15. A small maintenance profect consists of the following jobs where precedence relationships is given below :

Job : 1 – 2 1 – 3 2 – 3 2 – 5 3 – 4 3 – 6 4 – 5 4 – 6 5 – 6 6 – 7 Duration : 1 5 15 3 5 8 12 1 14 3 14 (iv) Draw an arrow diagram representing the project. (v) Find the total float for each activity. (vi) Find the critial path and total project duration.

16. A small project composed of 7 activities whose time estimates are listed in the table below.Activities are identified by their beginning (i) and ending (j) node numbers :

Activity Expected duration ( weeks ) i - j Optimistic most likely pessimistic 1 – 2 1 1 7 1 – 3 1 4 7 1 – 4 2 2 8 2 – 5 1 1 1 3 – 5 2 5 14 4 – 6 2 5 8 5 – 6 3 6 15 (iii) Draw the network , find the critical path and project time

duration. (iv) What is the probability that the project will be completed

atleast four weeks earlier than expected?. 17. A project consists of the following activities and time

estimates: Activity : 1 – 2 2 – 3 1 – 4 2 – 5 2 – 6 3 – 6 4 – 7 5 – 7 6 -7 Least time(days) :3 2 6 2 5 3 3 1 2 Greatest time(days) : 15 14 30 8 17 15 27 7 8

Most likely time(days): 6 5 12 5 11 6 9 4 5 (iv) Draw the network , find the critical path and project time

duration. (v) What is the probability that the project will be completed

in 27 days? (vi) What due date has about 95% chance of being met?.

18. Distinguish between PERT and CPM .

19. A maintenance foreman has given the following estimates of times and cost of jobs in a maintenance project :

Job predecessor Normal Crash Time(hours) Cost(Rs) Time(hours) Cost(Rs) A - 8 80 6 100 B A 7 40 4 94 C A 12 100 5 184 D A 9 70 5 102 E B , C , D 6 50 6 50 Overhead cost is Rs. 25 per hour . Find the optimum duration of the project and associated cost.

20. A project schedule has the following data : Activity : 1 – 2 1 – 3 1 – 4 2 – 5 3 – 5 5 – 6 4 – 6 Time : 4 6 5 3 4 2 3 Find the critical path .

21. Bring out the difference between CPM and PERT . Also discuss the various steps involved in developing PERT network analysis.

22. A project consists of the following activities: Activity : 1 – 2 1 – 3 1 – 4 2 – 5 2 – 6 3 – 6 4 – 7 5 – 7 6 -7 Least time(days) :5 18 26 16 15 6 7 7 3 Greatest time(days) : 10 22 40 20 25 12 12 9 5 Most likely time(days): 8 20 33 18 20 9 10 8 4 Perform the PERT network analysis and identify any one critical path.

23. A small project is composed of 7 activities whose time estimates are given as below:

Activity : 1 – 2 1 – 3 1 – 4 2 – 5 3 – 5 4 – 6 5 – 6 Optimistic time(weeks) : 1 1 2 1 2 2 3 Most likely time : 1 4 2 1 5 5 6 Pessimistic time : 4 7 8 1 14 8 15 (i). Draw the project network . (ii). What is the expected project length? (iii). What is the probability that the project will be completed atleast 4 weeks earlier than expected time?(Given P(0≤ z ≤ 1.33) = 0.4082).

24. Listed in the table are the activities and sequencing requirements necessary for the completion of a research report.

Activity Description Precedence Duration A Literature search _ 6 B Formulation of hypothesis _ 5 C Preliminary feasibility study B 2

D Formal proposal C 2 E Field analysis A,D 2 F Progress report D 1 G Formal research A,D 6 H Data collectio E 5 I Data analysis G,H 6 J Conclusion I 2 K Rough draft G 4 L Final copy J,K 3 M Preparation of oral presentation L 1 v) Draw a network diagram for this project.

(5) vi) Find the critical path. What is a lenght?

(2) vii) Find the Earliest strat time, Earliest finish time , Latest

start time and latest finish time (6)

viii) Find the total float and the free float for activity. (3)