Resoure Management Technique

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Resource Management Techniques MC9242 II MCA

MC9242 RESOURCE MANAGEMENT TECNI!UES

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UNIT I "INEAR #ROGRAMMING MO&E"S 9

Mathematical Formulation - Graphical Solution of linear programming models Simplexmethod Artificial variable Techniques- Variants of Simplex method

UNIT II TRANS#ORTATION AN& ASSIGNMENT MO&E"S 9

Mathematical formulation of transportation problem- Methods for finding initial basic

feasible solution optimum solution - degenerac Mathematical formulation of

assignment models !ungarian Algorithm Variants of the Assignment problem

UNIT III INTEGER #ROGRAMMING MO&E"S 9

Formulation Gomor"s #$$ method Gomor"s mixed integer method %ranch and

bound technique&

UNIT I' SCE&U"ING () #ERT AN& C#M 9

'et(or) *onstruction *ritical $ath Method $ro+ect ,valuation and evie(

Technique esource Analsis in 'et(or) Scheduling

UNIT ' !UEUEING MO&E"S 9*haracteristics of .ueuing Models $oisson .ueues - /M 0 M 0 12 3 /F#F4 0 5 0526/M 0 M 0 12 3 /F#F4 0 ' 0 526 /M 0 M 0 *2 3 /F#F4 0 5 0 526 /M 0 M 0 *2 3 /F#F4 0 ' 0 52

models&

Tota* No+ o, #erio-s . 4/

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RESOURCE MANAGEMENT TECNI!UES MC9242

#art0A !uestions an- Ans1ers

UNIT0I

+ 3hat is oerations research5

4perations research is a stud of optimi7ation techniques& #t is applied decision theor&

4 is the application of scientific methods6 techniques and tools to problems involvingthe operations of sstems so as to provide these in control of operations (ith optimum

solutions to the problem&

2+ "ist some a*ications o, OR+

4ptimal assignment of various +obs to different machines and different operators&

To find the (aiting time and number of customers (aiting in the queue and

sstem in queuing model

To find the mimimum transportation cost after allocating goods from differentorigins to various destinations in transportation model

8ecision theor problems in mar)eting6finance and production planning and

control&

$+ 3hat are the 6arious t7es o, mo-e*s in OR5

Models b function

i2 8escriptive model ii2 $redictive model iii2 'ormative model

Models b structure

i2 #conic model ii2 Analogue model iii2 Mathematical model

Models b nature of environment

i2 8eterministic model ii2 $robabilistic model

4+ 3hat are main characteristics o, OR5

,xamination of functional relationship from a sstem overvie(&

9tili7ation of planned approach

Adaptation of planned approach

9ncovering of ne( problems for stud

/+ Name some characteristics o, goo- mo-e*+

The number of assumptions made should be as fe( as possible

#t should be eas as possible to solve the problem

The number of variables used should be as fe( as possible&

#t should be more flexible to update the changes over a period of time (ithout

change in its frame(or)&

8+ 3hat are the -i,,erent hases o, OR5

Formulation of the problem

*onstruction of mathematical modeling

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8eriving the solution from the model

Validit of the model

,stablishing the control over the solution

#mplementation of the final solution&

+ "ist out the a-6antages o, OR5

4ptimum use of managers production factors #mproved qualit of decision

$reparation of future managers b improving their )no(ledge and s)ill

Modification of mathematical solution before its use&

:+ 3hat are the *imitations o, OR5

Mathematical model do not ta)e into account the intangible factors such as human

relations etc& can not be quantified&

Mathematical models are applicable to onl specific categories of problems&

equires huge calculations& All these calculations cannot be handled manuall

and require computers (hich bear heav cost&

9+ 3hat is *inear rogramming5

;inear programming is a technique used for determining optimum utili7ation of

limited resources to meet out the given ob+ectives& The ob+ective is to maximi7e the profit

or minimi7e the resources /men6 machine6 materials and mone2

%+ 3rite the genera* mathematica* ,ormu*ation o, "##+

1& 4b+ective functionMax or Min < = *1x1> *:x:> ?&&> *nxn

:& Sub+ect to the constraints

a11x1>a1:x:>????> a1nxn/@=2b1a:1x1>a::x:>????> a:nxn/@=2b:

??????????????????????&&

??????????????????????&&

am1x1>am:x:>????> amnxn/@=2bmB& 'on-negative constraints

x16x:6?&xm C

+ 3hat are the characteristic o, "##5

There must be a (ell defined ob+ective function&

There must be alternative course of action to choose&

%oth the ob+ective functions and the constraints must be linear equation orinequalities&

2+ 3hat are the characteristic o, stan-ar- ,orm o, "##5

The ob+ective function is of maximi7ation tpe&

All the constraint equation must be of equal tpe b adding slac) or surplus

variables

!S of the constraint equation must be positive tpe

All the decision variables are of positive tpe

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$+ 3hat are the characteristics o, canonica* ,orm o, "##5 ;NO' *e ,or not more than 8 hours 4% minutes 1hi*e machine M2 is a6ai*a>*e

,or % hours -uring an7 1or?ing -a7+ @ormu*ate the ro>*em as a "## so as to

maimiBe the ro,it+ ;MA) Bx:Sub+ect tot the constraints3

x1> x: @ DCC

:x1> x:@ ECCx16x: C

/+ A coman7 se**s t1o -i,,erent ro-ucts A an- ( ma?ing a ro,it o, Rs+4% an-

Rs+ $% er unit on themresecti6e*7+The7 are ro-uce- in a common ro-uction

rocess an- are so*- in t1o -i,,erent mar?ets the ro-uction rocess has a tota*

caacit7 o, $%%%% man0hours+ It ta?es three hours to ro-uce a unit o, A an- one

hour to ro-uce a unit o, (+ The mar?et has >een sur6e7e- an- coman7 o,,icia*

,ee* that the maimum num>er o, units o, A that can >e so*- is :%%% units an- that

o, ( is 2%%% units+ Su>Dect to these *imitations ro-ucts can >e so*- in an7

com>ination+ @ormu*ate the ro>*em as a "## so as to maimiBe the ro,it

Maximi7e 7 =DCx1>BCx:Sub+ect tot the constraints3

Bx1> x: @ BC6CCC x1 @ CCC

x: @ 1:CCC

x16x: C

8+ 3hat is ,easi>i*it7 region5 ;MA) i*it7 region in an "# ro>*em5 Is ti necessar7 that it shou*- a*1a7s

>e a con6e set5A region in (hich all the constraints are satisfied is called feasible region& The

feasible region of an ;$$ is al(as convex set&

:+ &e,ine so*ution

A set of variables x16x:?&xn (hich satisfies the constraints of ;$$ is called a

solution&

9+ &e,ine ,easi>*e so*ution5 ;MA)


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An solution to a ;$$ (hich satisfies the non negativit restrictions of ;$$"s called thefeasible solution

2%+ &e,ine otima* so*ution o, "##+ ;MA)


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2:+ &e,ine (asic so*ution5

Given a sstem of m linear equations (ith n variables/mIn2&The solution obtained

b setting /n-m2 variables equal to 7ero and solving for the remaining m variables iscalled a basic solution&

29&&e,ine non &egenerate (asic ,easi>*e so*ution5

The basic solution is said to be a non degenerate basic solution if 'one of thebasic variables is 7ero&

$%+ &e,ine -egenerate >asic so*ution5

A basic solution is said to be a degenerate basic solution if one or more of the

basic variables are 7ero&

$+ 3hat is the ,unction o, minimum ratioL

To determine the basic variable to leave

To determine the maximum increase in basic variable

To maintain the feasibilit of follo(ing solution

$2+ @rom the otimum sim*e ta>*e ho1 -o 7ou i-enti,7 that "## has un>oun-e-

so*ution5

To find the leaving variables the ratio is computed& The ratio is I=C

then there is an unbounded solution to the given ;$$&

$$+ @rom the otimum sim*e ta>*e ho1 -o 7ou i-enti,7 that the "## has no

so*ution5

#f atleast one artificial variable appears in the basis at 7ero level (ith a >ve value

in the b column and the optimalit condition is satisfiedthen the original problem has no feasible solution&

$4+ o1 -o 7ou i-enti,7 that "## has no so*ution in a t1o hase metho-5

#f all 7 -egenerac75The concept of obtaining a degenerate basic feasible solution in ;$$ is )no(n as

degenerac& This ma occur in the initial stage (hen atleast one basic variable is 7ero in

the initial basic feasible solution&

$8+ 3rite the stan-ar- ,orm o, "## in the matri notation5

#n matrix notation the canonical form of ;$$ can be expressed as Maximi7e < = */ob+ fn&2

Sub to A I= b/constraints2 and J= C /non negative restrictions2

here * = /*16*:6?&&*n26

A = a11 a1: ?&& a1n = x1 b = b1

a:1 a::?&& a:n 6 x: 6 b:

& & & & & &

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am1 am:?& amn xn bn

$+ &e,ine >asic 6aria>*e an- non0>asic 6aria>*e in *inear rogramming+

A basic solution to the set of constraints is a solution obtained b setting an nvariables equal to 7ero and solving for remaining m variables not equal to 7ero& Such m

variables are called basic variables and remaining n 7ero variables are called non-basic

variables+

$:+So*6e the ,o**o1ing "# ro>*em >7 grahica* metho-+ ;MA) Dect tot the constraints.

F 2 /

2H :

2H %

$9+ &e,ine unrestricte- 6aria>*e an- arti,icia* 6aria>*e+ ;NO' *em+

#t is a special tpe of linear programming model in (hich the goods are shipped

from various origins to different destinations& The ob+ective is to find the best possible

allocation of goods from various origins to different destinations such that the totaltransportation cost is minimum&

$+ &e,ine the ,o**o1ing. @easi>*e so*ution

A set of non-negative decision values xi+ /i=16:6?&mN +=16:?n2 satisfies theconstraint equations is called a feasible solution&

4+ &e,ine the ,o**o1ing. >asic ,easi>*e so*ution

A basic feasible solution is said to be basic if the number of positive allocations

are m>n-1&/ m-origin and n-destination2f the number of allocations are less than /m>n-

12 it is called degenerate basic feasible solution&

/+ &e,ine otima* so*ution in transortation ro>*em

A feasible solution is said to be optimal6 if it minimi7es the total transportationcost&

8+ 3hat are the metho-s use- in transortation ro>*em to o>tain the initia* >asic

,easi>*e so*ution+

'orth-(est corner rule

;o(est cost entr method or matrix minima method

Vogel"s approximation method

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+ 3rite -o1n the >asic stes in6o*6e- in so*6ing a transortation ro>*em+

To find the initial basic feasible solution

To find an optimal solution b ma)ing successive improvements from the initial

basic feasible solution&

:+ 3hat -o 7ou un-erstan- >7 -egenerac7 in a transortation ro>*emL /'4V "CO2

#f the number of occupied cells in a m x n transportation problem is less than

/ m>n-12 then the problem is said to be degenerate&

9+ 3hat is >a*ance- transortation ro>*em un>a*ance- transortation ro>*em5

hen the sum of suppl is equal to demands6 then the problem is said to be

balanced transportation problem&A transportation problem is said to be unbalanced if the total suppl is not equal

to the total demand&

%+ o1 -o 7ou con6ert an un>a*ance- transortation ro>*em into a >a*ance- one5

The unbalanced transportation problem is converted into a balanced one b

adding a dumm ro( /source2 or dumm column /destination2 (hichever is necessar&

The unit transportation cost of the dumm ro(0 column elements are assigned to 7ero&Then the problem is solved b the usual procedure&

+ E*ain ho1 the ro,it maimiBation transortation ro>*em can >e con6erte-

to an equi6a*ent cost minimiBation transortation ro>*em+ ;MA)


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performs onl one operation6 the overall ob+ective is to maximi7e the total profit orminimi7e the overall cost of the given assignment&

8+ E*ain the -i,,erence >et1een transortation an- assignment ro>*ems5

Transortation ro>*ems Assignment ro>*ems

12 suppl at an source ma be a Suppl at an source (illan positive quantit& be 1&

:2 8emand at an destination ma 8emand at an destinationbe a positive quantit& (ill be 1&

B2 4ne or more source to an number 4ne source one destination&of destination&

+ &e,ine un>oun-e- assignment ro>*em an- -escri>e the stes in6o*6e- in

so*6ing it5

#f the no& of ro(s is not equal to the no& of column in the given costmatrix theproblem is said to be unbalanced& #t is converted to a balanced one b adding dumm ro(

or dumm column (ith 7ero cost&

:+ E*ain ho1 a maimiBation ro>*em is so*6e- using assignment mo-e*5

The maximi7ation problems are converted to a minimi7ation one of the follo(ingmethod&

/i2 Since max 7 = min/-72

/ii2 Subtract all the cost elements all of the cost matrix from the!ighest cost element in that cost matrix&

9+ 3hat -o 7ou un-erstan- >7 restricte- assignment5 E*ain ho1 7ou shou*- o6ercome it5

The assignment technique6 it ma not be possible to assign a particulartas) to a

particular facilit due to technical difficulties or other restrictions& This can be overcome

b assigning a ver high processing time or cost /it can be 52 to the corresponding cell&

2%+ o1 -o 7ou i-enti,7 a*ternati6e so*ution in assignment ro>*em5

Sometimes a final cost matrix contains more than required number of 7eroes atthe independent position& This implies that there is more than one optimal solution (ith

some optimum assignment cost&

2+ 3hat is a tra6e*ing sa*esman ro>*em5A salesman normall must visit a number of cities starting from his head quarters&

The distance bet(een ever pair of cities are assumed to be )no(n& The problem of

finding the shortest distance if the salesman starts from his head quarters and passesthrough each cit exactl once and returns to the headquarters is called Traveling

Salesman problem&

22+ &e,ine route con-ition5

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The salesman starts from his headquarters and passes through each cit exactlonce&

2$+ Gi6e the areas o, oerations o, assignment ro>*ems5

Assigning +obs to machines&

Allocating men to +obs0machines&

oute scheduling for a traveling salesman&

24+ o1 -o 7ou con6ert the un>a*ance- assignment ro>*em into a >a*ance- one5

;MA) *em ;I##=5 ;&EC *em5

#n ;$$ the values for the variables are real in the optimal solution& !o(ever incertain problems this assumption is unrealistic& For example if a problem has a solution

of 10: cars to be produced in a manufacturing compan is meaningless& These tpes of

problems require integer values for the decision variables& Therefore #$$ is necessar to

round off the fractional values&

$+ "ist out some o, the a*ications o, I##5 ;MA)


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Mixed #$$$ure #$$

/+ 3hat is ure I##5

#n a linear programming problem6 if all the variables in the optimal solution are

restricted to assume non-negative integer values6 then it is called the pure /all2 #$$&

8+ 3hat is Mie- I##5#n a linear programming problem6 if onl some of the variables in the optimal

solution are restricted to assume non-negative integer values6 (hile the remaining

variables are free to ta)e an non-negative values6 then it is called A Mixed #$$&

+ 3hat is Jero0one ro>*em5

#f all the variables in the optimum solution are allo(ed to ta)e values either C or 1as in Qdo" or Qnot to do" tpe decisions6 then the problem is called et1een #ure integer rogramming mie- integer

integer rogramming&hen an optimi7ation problem6 if all the decision variables are restricted to ta)e

integer values6 then it is referred as pure integer programming& #f some of the variablesare allo(ed to ta)e integer values6 then it is referred as mixed integer integer

programming&

9+ E*ain the imortance o, Integer #rogramming5

#n linear programming problem6 all the decision variables allo(ed to ta)e an

non-negative real values6 as it is quite possible and appropriate to have fractional valuesin man situations& !o(ever in man situations6 especiall in business and industr6

these decision variables ma)e sense onl if the have integer values in the optimal

solution& !ence a ne( procedure has been developed in this direction for the case of ;$$sub+ected to additional restriction that the decision variables must have integer values&

%+ 3h7 not roun- o,, the otimum 6a*ues in stea- o, resorting to I#5 ;MA)


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$+ 3hat is search metho-5

#t is an enumeration method in (hich all feasible integer points are enumerated&

The (idel used search method is the %ranch and %ound Technique& #t also starts (iththe continuous optimum6 but sstematicall partitions the solution space into sub

problems that eliminate parts that contain no feasible integer solution& #t (as originall

developed b A&!&;and and A&G&8oig&

4+ E*ain the concet o, (ranch an- (oun- Technique5

The (idel used search method is the %ranch and %ound Technique& #t starts

(ith the continuous optimum6 but sstematicall partitions the solution space into subproblems that eliminate parts that contain no feasible integer solution& #t (as originall

developed b A&!&;and and A&G&8oig&

/+ Gi6e the genera* ,ormat o, I##5

The general #$$ is given b

Maximi7e < = *Sub+ect to the constraints

A @ b6 C and some or all variables are integer&

8+ 3rite an a*gorithm ,or Gomor7a>7 -o**s ma?es t1o t7es o, -o**s -o** K an- -o** )+

#rocessing o, these -o**s is -one on t1o machines A an- (+ &o** K requires 2 hours

on machine A an- 8 hours on Machine (+ &o** ) requires / hours on machine A an-

/ hours on Machine (+ There are 8 hours o, time er -a7 a6ai*a>*e on machine A

an- $% hours on machine (+ The ro,it is gaine- on >oth the -o**s is same+ @ormat

this as I##5

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;et the manufacturer decide to manufacture x1 the number of doll and x:number of doll R so as to maximi7e the profit& The complete formulation of the #$$ is

given b

Maximi7e < = x1>x:Sub+ect to : x1> K x:@1E

E x1> K x:@BC

and C and are integers&

9+ E*ain Gomor7ranche- the origina*ro>*em5

Geometricall it means that the branching process eliminates portion of thefeasible region that contains no feasible-integer solution& ,ach of the sub-problems

solved separatel as a ;$$&

2+ 3hat is stan-ar- -iscrete rogramming ro>*em5

#f all the variables in the optimum solution are allo(ed to ta)e values either C or 1

as in Qdo" or Qnot to do" tpe decisions6 then the problem is called standard discreteprogramming problem&

22+ 3hat is the -isa-6antage o, >ranche- or ortione- metho-5#t requires the optimum solution of each sub problem& #n large problems this could

be ver tedious +ob&

2$+ o1 can 7ou imro6e the e,,icienc7 o, ortione- metho-5

The computational efficienc of portioned method is increased b using the

concept of bounding& % this concept (henever the continuous optimum solution of a sub

problem ields a value of the ob+ective function lo(er than that of the best availableinteger solution it is useless to explore the problem an further consideration& Thus once

a feasible integer solution is obtained6 its associative ob+ective function can be ta)en as a

lo(er bound to delete inferior sub-problems& !ence efficienc of a branch and bound

method depends upon ho( soon the successive sub-problems are fathomed&

UNIT0I'

+ 3hat -o 7ou mean >7 roDect5

A pro+ect is defined as a combination on inter related activities (ith limitedresources namel men6 machines materials6 mone and time all of (hich must be

executed in a defined order for its completion+

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2+ 3hat are the three main hases o, roDect5

$lanning6 Scheduling and *ontrol

$+ 3hat are the t1o >asic *anning an- contro**ing techniques in a net1or?

ana*7sis5

*ritical $ath Method /*$M2

$rogramme ,valuation and evie( Technique /$,T=

4+ 3hat are the a-6antages o, C#M an- #ERT techniques5

#t encourages a logical discipline in planning6 scheduling and control of pro+ects

#t helps to effect considerable reduction of pro+ect times and the cost

#t helps better utili7ation of resources li)e men6machines6materials and mone

(ith reference to time

#t measures the effect of delas on the pro+ect and procedural changes on the

overall schedule&

/+ 3hat is the -i,,erence C#M an- #ERT

*$M

'et(or) is built on the basis of activit 8eterministic nature

4ne time estimation

$,T

An event oriented net(or)

$robabilistic nature

Three time estimation

8+ 3hat is net1or?5

A net(or) is a graphical representation of a pro+ect"s operation and is composedof all the events and activities in sequence along (ith their inter relationship and inter

dependencies&

+ 3hat is E6ent in a net1or? -iagram5

An event is specific instant of time (hich mar)s the starts and end of an activit&

#t neither consumes time nor resources& #t is represented b a circle&

:+ &e,ine acti6it75

A pro+ect consists of a number of +ob operations (hich are called activities& #t is

the element of the pro+ect and it ma be a process6 material handling6 procurement ccleetc&

9+ &e,ine Critica* Acti6ities5

#n a 'et(or) diagram critical activities are those (hose if consumer more than

estimated time the pro+ect (ill be delaed&

%+ &e,ine non critica* acti6ities5

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Activities (hich have a provision such that the event if the consume a specifiedtime over and above the estimated time the pro+ect (ill not be delaed are termed as non

critical activities&

+ &e,ine &umm7 Acti6ities5

hen t(o activities start at a same time6 the head event are +oined b a dotted

arro( )no(n as dumm activit (hich ma be critical or non critical&

2+ &e,ine -uration5

#t is the estimated or the actual time required to complete a trade or an activit&

$+ &e,ine tota* roDect time5

#t is time ta)en to complete to complete a pro+ect and +ust found from thesequence of critical activities& #n other (ords it is the duration of the critical path&

4+ &e,ine Critica* #ath5

#t is the sequence of activities (hich decides the total pro+ect duration& #t is

formed b critical activities and consumes maximum resources and time&

/+ &e,ine ,*oat or s*ac?5 ;MA)


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2%+ &e,ine Otimistic5

4ptimistic time estimate is the duration of an activit (hen everthing goes on

ver (ell during the pro+ect&

2+ &e,ine #essimistic5

$essimistic time estimate is the duration of an activit (hen almosteverthing

goes against our (ill and a lot of difficulties is faced (hile doing a pro+ect&

22+ &e,ine most *i?e*7 time estimation5

Most li)el time estimate is the duration of an activit (hen sometimes thing goon ver (ell6 sometimes things go on ver bad (hile doing the pro+ect&

24+ 3hat is a ara**e* critica* ath5

hen critical activities are crashed and the duration is reduced otherpaths ma

also become critical such critical paths are called parallel critical path&

2/+ 3hat is stan-ar- -e6iation an- 6ariance in #ERT net1or?5 ;NO' et1een -irect cost an- in-irect cost5 ;NO'


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ca*cu*ate the customerution+

3hat is the ro>a>i*it7 that the queue *ength ecee-s /5

Arrival rate= K0BC min

Service rate=:0Pmin$robabilit that the queue length exceeds K = /2n>:

= /&OK2 O=C&1BB

$+ E*ain !ueue -isci*ine an- its 6arious ,orms+ /i2 F#F4 or F*FS - First #n First 4ut or First *ome First Served&

/ii2 ;#F4 or ;*FS - ;ast #n First 4ut or ;ast *ome First Served&

/iii2 S#4 - Selection for service in random order& /iv2 $# - $riorit in selection

4+ &istinguish >et1een transient an- stea-7 state queuing s7stem+

A sstem is said to be in transient state (hen its operating characteristics are dependent

on time& A stead state sstem is one in (hich the behavior of the sstem is independent

of time&

/&&e,ine stea-7 state5A sstem is said to be in stead state (hen the behavior of the sstem

independent of time& ;et pn/t2 denote the prob that there are Qn" units in the sstem attime t& then in stead state=J lim pn/ t 2=C

tU5

8+ 3rite -o1n the *itt*e ,ormu*a5

;s=;q>0W

here ;s= the average no& of customers in the sstem

;q= the average no& of customers in the queue

+ I, tra,,ic intensit7 o, MMI s7stem is gi6en to >e %+8 1hat ercent o, time the

s7stem 1ou*- >e i-*e5Traffic intensit = C&OE /bus time2

Sstem to be idle = 1-C&OE =C&:D

:+ 3hat are the >asic e*ements o, queuing s7stem5

Sstem consists of the arrival of customers6 (aiting in queue6 pic) up for service

according to certain discipline6 actual service and departure of customer&

9+ 3hat is meant >7 queue -isci*ine5

The manner in (hich service is provided or a customer is selected for service is

defined as the queue discipline+

%+ 3hat are the c*assi,ications o, queuing mo-e*s5

m X m X # X5 m X m X # Xn m X m X cX5 m X m X c Xn

+ 3hat are the characteristic o, queuing rocess5

Arrival pattern of customers6 service pattern of servers6 queue discipline6

sstem capacit6 no& of service channels6 no& of service stage&

2+ &e,ine #oisson rocess5

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The $oisson process is a continuous parameter discrete state process /ie2 a goodmodel for man practical situations& if /t2represents the no& of 4ccurrences of a certain

in /C6 t2 then the discrete random process Y /t2Z is called the $oisson process& if it

satisfies the follo(ing postulates

#& $[1 occurrence in /t6t>\t2] =\t > 4/\t2

##& $[C occurrence in /t6t>\t2] =1-\t > 4/\t2

###& $[: or more occurrence in /t6t>\t2] =4/\t2#V& /t2 is independent of the number of occurrences of the event in an

interval prior /or2 after the interval/C6t2

V& The prob that the event occurs a specified number of times in /t C6 t C>t2depends onl on t but not on t C&

$+Gi6en an7 t1o eam*es o, #oisson rocess51& The number of incoming telephone calls received in a particular time

:& The arrival of customer at a ban) in a da

4+ 3hat are the roerties o, #oisson rocess5

1& The $oisson process is a mar)ov process&:& Sum of t(o independent poissen processes is a poisson process&

B& 8ifference of t(o independent poisson processes is not poisson process&D& The inter arrival time of a poisson process has an exponential distribution

(ith mean 10&

/+ Customer arri6es at a one0man >ar>er sho accor-ing to a #oisson rocess 1ith

an mean inter arri6a* time o, 2 minutes+ Customers sen- a a6erage o, % minutes

in the >ar>erar>er sho an-

in the queue5

Given mean arrival rate 10 = 1:&

Therefore = 101: per minute& Mean service rate 10W = 1C&

Therefore W = 101C per minute&

,xpected number of customers in the sstem

;s = 0W- = 101:0101C-101: = K customers&

8&&e,ine ure >irth rocess5 #f the death rates W) = C for all ) = 16 :?? (e have a pure birth process&

1O+ 3rite -o1n the ostu*ates o, >irth an- -eath rocess5

12 p [1 birth /t6 t > \t2] = n/t2\t > C/\t2 :2 p [C birth in /t6 t > \t2] = 1 - n/t2\t > C/\t2&

B2 p [1 death in /t6 t > \t2] = Wn/t2\t > C/\t2

D2 p [C death in /t6 t > \t2] = 1 - Wn/t2\t > C/\t2&

:+ 3hat is the ,ormu*a ,or the ro>*em ,or a customer to 1ait in the queue un-er

;mm N@C@S=

s = ;s0&

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9+ 3hat is the a6erage num>er o, customers in the s7stem un-er

;mme. @C@S=5

W /0W2c 0 /c-12^/cW - 2: > 0W&

2%+ 3hat is the -i,,erence >et1een ro>a>i*istic -eterministic an- mie- mo-e*s5$robabilistic3 hen there is uncertaint in both arrivals rate and service rate are

assumed to be random variables&

8eterministic3 %oth arrival rate and service rate are constants&Mixed3 hen either the arrival rate or the service rate is exactl )no(n and the other is

not )no(n&

2+ 3hat are the assumtions in mm mo-e*5

/i2 ,xponential distribution of inter arrival times or poisson distribution

of arrival rate& /ii2 .ueue discipline is first come6 first serve&

/iii2 Single (aiting line (ith no restriction no length of queue&/iv2 Single server (ith exponential distribution of service times&

22+ #eo*e arri6e at a theatre tic?et >ooth in oisson -istri>ute- arri6a* rate o,

2/hour+ Ser6ice time is constant at 2 minutes+ Ca*cu*ate the mean5

= :K0hr W = /_2EC = BC per hour& = 0W = :K0BC = K0E = C&BB

;q = : 0 1- = /&BB2:0 1 - &BB = D&1KKC:

Mean (aiting time=;q0 = D0:K = D0:K ` EC = P&E minutes&

8 Mar?s Unit

1. (a) Solve the LPP maximize Z=5x1-2x2+3x3

Subject to

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Resource Management Techniques MC9242 II MCA2x1+ 2x2 X3= 2

3x1 x2= 3

X2+ 3x3= 5

!"# x1$ x2$ x3= %

2& ! com'a" 'o#uce* to #i,,ee"t 'o#uct* ! a"# & .he com'a" ma/e* a

'o,it o, 0*& % a"# 0*&3% 'e u"it o" ! a"# e*'ectivel& .he 'o#uctio"'oce** ha* a ca'acit o, 3%$%%% ma" hou*& t ta/e* 3 hou* to 'o#uce o"e

u"it o, ! a"# o"e hou to 'o#uce o"e u"it o, & .he ma/et *uve i"#icate*

that the maximum "umbe o, u"it* ! that ca" be *ol# i* $%%% a"# tho* o, i*

12$%%% u"it*& omulate the 'oblem a"# *olve it b 4a'hical metho# to 4et

maximum 'o,it&

(ii) ite the al4oithm ,o Sim'lex 6etho#

3&(a)7*e ati,icial vaiable tech"i8ue to *olve the ,olloi"4 LP 'oblem&

6aximize Z = X1 + 2X2 + 3X3 XSubject to X1 + 2X2 + 3X3 = 15

2X1 + X2 + 5X3 = 2%X1 + 2X2 + X3 + X = 1%

X1 $ X2 $ X3 $ X $ =%&& ! ,im 'la"* to 'ucha*e atlea*t 2%% 8ui"tal* o, *ca' co"tai"i"4 hi4h 8ualit metalX a"# Lo 8ualit metal 9& it #eci#e# that the *ca' to be 'ucha*e# mu*t co"tai"atlea*t 1%% 8ui"tal* o, X-6etal a"# "o moe tha" 35 8ui"tal* o, 9 6etal& .he ,im ca"'ucha*e the *ca' ,om 2 *u''lie* ( ! a"# %i" u"limite# 8ua"titie*& .he : o, X a"#

9 metal* i" tem* o, ei4ht i" the *ca' *u''lie* b ! a"# i* 4ive" belo&

Meta*s Su*ier A Su*ier (

K 2/P /P

) %P 2%P

.he 'ice o, !;* *ca' i* 0*& 2%% 'e 8ui"tal a"# that o, ;* 0*& %% 'e 8ui"tal&


21/34


2 44 $/ $% $% $%

$ $: $: 2: $% %

&eman- 4% 2% 8% $%

2+ 3rite the a*gorithm ,or MO&I metho-

3& (i) ite the al4oithm ,o u"4aia" metho#(ii) ,ive o/e* ae available to o/ ith the machi"e* a"# the e*'ective co*t*a**ociate# ith each o/e-machi"e a**i4"me"t i* 4ive" belo& ! *ixth machi"e* i*available to e'lace o"e o, the exi*ti"4 machi"e* a"# the a**ociate# co*t* ae al*o4ive" belo&

o/e*

6achi"e*

61 62 63 6 65 6>

1 12 3 > - 5

2 11 - 5 - 3

3 2 1% ? @ 5

- @ > 12 1%5 5 ? > -

&etermine 1hether the ne1 machine can >e accete-+ &etermine a*so otima*

assignment an- the associate- sa6ing in the cost+

&(a)! 'o#uct i* ma"u,actue b ,actoie* !$$A$< .he u"it 'o#uctio" co*t* i"them ae 0*& 2 0*&3 a"# 0*&1 a"# 0*&5 e*'ectivel& .hei 'o#uctio" ca'acitie* ae5%$ @%$ 3%$ a"# 5% u"it* e*'ectivel the*e ,actoie* *u''l the 'o#uct to *toe*$#ema"#* o, hich ae 25$35$1%5 a"# 2% u"it* e*'ectivel& 7"it ta"*'otatio" co*t i"u'ee* ,om each ,acto to each *toe i* 4ive" i" the table beloB

Stores

2 $ 4

A 2 4 8

( % : /

C $ $ 9 2

& 4 8 : $


22/34


Cost o, Reairs ;Rs+ "a?h=

R R2 R$ R4

C 9 4 9 /

C2 2% 9

C$ 9 : 2 :

C4 % 2 : 9

C/ % / 2 8

i& ,i"# the be*t a o, a**i4"i"4 the e'ai o/ to the co"tact a"# theco*t*&

ii& hich o, the ,ive co"tacto* ill be u"*ucce**,ul i" hi* bi#&

8 Mar?s Unit $

1& 7*e ba"ch a"# bou"# metho# to *olve the ,olloi"4 i"te4e 'o4ammi"4'oblem

6ax Z = 3x1+ x2Subject to@x1+ 1>x2= 523x1 2x2= 1

a"# x1$ x2 = a"# i"te4e&

2& Solve the ,olloi"4 mixe# i"te4e 'o4ammi"4 'oblem

6ax Z x1+ >x2+ 2x3 x1= x2= 5-X1+ >X2= 5

-X1+ X2+ X3= 5

3&Solve the ,olloi"4 mixe#-i"te4e 'o4ammi"4 'oblem b u*i"4 Domo;* cutti"4'la"e metho#&

6aximize z = X1 + X2Subject to the co"*tai"t*B

3X1 + 2X2 = 5X2 = 2

!"# X1 X2 = %E i* a" i"te4e&

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23/34

Resource Management Techniques MC9242 II MCA& Solve the ,olloi"4 all-"te4e 'o4ammi"4 'oblem u*i"4 the ba"ch a"# bou"#metho#&

6i"imize Z = 3X1 + 2&5X2Subject to the co"*tai"t*

X1 + 2X2 = 2%3X1 + 2X2 = 5%

8 Mar?s Unit 4

1 Aalculate the total ,loat$ ,ee ,loat a"# i"#e'e"#e"t ,lot ,o the 'oject ho*eactivitie* ae 4ive" belo&

Acti6it7 02 0$ 0/ 20$ 204 $04 $0/ $08 408 /08

&uration :Q 2 4 % $ / % 4

@in- the critica* ath a*so+

:&! mai"te"a"ce ,oema" ha* 4ive" the ,olloi"4 e*timate o, time* a"# co*t o, job*i" a mai"te"a"ce 'oject

o> #re-ecessor Norma* Crash

Time Cost Time Cost

A 0 : :% 8 %%

( A 4% 4 94C A 2 %% / :4

& A 9 % / %2

E (+C+& 8 /% 8 /%

Cvehea# co*t i* 0*&25 'e houi"#(i) the "omal #uatio" o, the 'oject a"# the a**ociate# co*t(ii) Aa*h **tematicall a"# ,i"# a**ociate# co*t(iii) .he mi"imum #uatio" o, the 'oject a"# it* co*t(iv) , all the activitie* ae ca*he# hat ill be the 'oject #uatio" a"# thecoe*'o"#i"4 co*t

3& ! 'oject ha* the ,olloi"4 activitie* a"# othe chaactei*tic*B

Estimate- -uration ;in 1ee?s=

Acti6it7

Otimistic Most *i?e*7 #essimistic

;i D =

02

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24/34


0$ 4

04 2 2 :

20/

$0/ 2 / 4

408 2 / :

/08 $ 8 /

i& hat i* the ex'ecte# 'oject le"4thF

ii& hat i* the 'obabilit that the 'oject ill be com'lete# "omoe tha" ee/* late tha" ex'ecte# timeF

&Li*te# i" the table ace the activitie* a"# *e8ue"ci"4 e8uieme"t* "ece**a ,o thecom'letio" o, e*each e'ot

8 Mar?s Unit /

1&(i) Cbtai" the **tem o, *tea# *tate e8uatio"* a"# he"ce ,i"# the value o, P i"u*ual "otatio"* hee "G * a"# = * (ii) ! 4e"eal i"*ua"ce com'a" ha* thee claim a#ju*te* i" it* ba"ch o,,ice&Peo'le ith claim* a4ai"*t the com'a" ae ,ou"# to aive i" 'oi**o" ,a*hio" at a"avea4e ate o, 2% 'e hou #a& .he amou"t o, ti"e that a" a#ju*te *'e"#* ith a

claima"t i* ,ou"# to have "e4ative ex'o"e"tial #i*tibutio" ith mea" *evice time %mi"ute*& Alaima"t* ae 'oce**e# i" the o#e o, thei a''eaa"ce& o ma" hou* aee/ ca" a" a#ju*te ex'ect to *'e"# ith claima"t* a"# ho much time$ o" theavea4e #oe* claima"t *'e"# i" the ba"ch o,,ice

2&(i) Hx'lai" vaiou* Iueuei"4 ith exam'le

(ii) ! ba"ch o, a "atio"al ba"/ ha* o"l o"e t'i*t$ Si"ce the t'i"4 o/ vaie* i"

le"4th$ the t'i"4 ate i* a"#oml #i*tibute# a''oximati"4 'oi**o" #i*tibutio"

ith mea" ate o, lette* 'e hou& .he lette aive at a ate o, 5 'e hou #ui"4

the e"tie hou o/ #a& , the t'eite i* value# at 0*& 1&5% 'e hou&


25/34

Resource Management Techniques MC9242 II MCA ii& .he tele'ho"e #e'atme"t ill i"*tall a *eco"# booth he"

co"vi"ce# that a" aival oul# ex'ect aiti"4 ,o atlea*t 3 mi"ute* ,o a 'ho"e call& ho much *houl# the ,lo o, aival* i"cea*e i" o#e to ju*ti, a *eco"# boothF

iii& hat i* the avea4e le"4th o, the 8ueue that ,om ,om time totimeF

D& i& hat i* the 'obabilit that a 'e*o" aivi"4 at the booth ill have to ait

ii& .he tele'ho"e #e'atme"t ill i"*tall a *eco"# booth he"co"vi"ce# that a" aival oul# ex'ect aiti"4 ,o atlea*t 3 mi"ute* ,o a 'ho"e call& ho much *houl# the ,lo o, aival* i"cea*e i" o#e to ju*ti, a *eco"# boothF

iii& hat i* the avea4e le"4th o, the 8ueue that ,om ,om time totimeF

- :K -


26/34


- :E -


27/34


Other 8 Mar?s

+ a= i= 4ld hens can be bought at s& : each and oung ones at s K each& The old hens

la B eggs per (ee) and the oung one la K eggs per (ee)6 each egg being (orth BC

paise& A hen costs s 1 per (ee) to feed& A person has onl s C to spend for hens&

!o( man of each )ind should he bu to give a profit of more than s&E per (ee)6

assuming that he cannot house more than :C hensL Formulate this ;&$&$ H solve b

graphical methodL =

ii=,xplain the follo(ing (ith respect to graphical method

'o Feasible solution6 9nbounded solution6 Alternate optimal solution

>=9sing simplex algorithm

Min - : x1- x:Sub+ect to the constraints

x1> x: :

x1> x: @ D and

x16x: C

2 a=A product is produced b four factories A6 %6 *6 and 8& The unit production costs

in them

are s :6 s B6 s&1 and s& K respectivel& Their production capacities are3

Factor A- KC

units6 %- OC units6 *- BC units and 8- KC units& These factories suppl the product

to four

stores6 demands of (hich are :K6 BK6 1CK and :C units respectivel& 9nit

transportation cost

in rupees from each factor to each store is given in the table belo(&

Stores

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28/34


@actories

8etermine the extent of deliveries from each of the factories to each of the stores

so that the

total production and transportation cost is minimum& 9se VAM for finding the

initial basic

feasible solution&

2 >= i= #n the modification of the plant laout of a factor6 four ne( machines6 M16 M:6

MB and MD are to installed in a machine shop& There are five vacant places A6 %6 *6 8

and , available& %ecause of limited space6 machine M: cannot be placed at * and MB

cannot be placed at A& The cost of placing a machine at a particular is given belo(6 in

rupees& Find the optimal assignment schedule&

"ocation

A ( C & E

M P 11 1K 1C 11

M2 1: P - 1C P

Machine M$ - 11 1D 11 O

M4 1D 1: O

ii=rite an algorithm to find an optimal solution for assignment problem

- : -

2 $ 4

A : D E 11

( 1C O K

C 1B B P 1:

& D E B


29/34


$ a=Solve the follo(ing problem b the Gomor"s cutting plane method&

Max < = x1>x:

Sub+ect to

Bx1> :x: @ K

x: @ :

and x16x: C and integers

$ >=9se branch and bound method to solve the follo(ing3

Max < = :x1>:x:

Sub+ect to

Kx1> Bx: @

x1> : x: @ D


4 a= i= 8ra( the net(or) and determine the critical path for the follo(ing data& Also

calculate the

floats3

o>s. 1-: 1-B :-D B-D B-K D-K D-E K-E

&uration ;&a7s= E K 1C B D E : P

ii=*onstruct the net(or) for the pro+ect (hose precedence relationships are given belo(3

% I ,6 F N * I G6; N , 6 G I ! N ; 6! I # N ; I MN !6 M I ' N A I N #6 I $N $ I

.&

4 >= i=Assuming that the expected times are normall distributed6 find the probabilit

of meeting

the schedule date as given for the net(or)&

Acti6it7. 1-: 1-B :-D B-D D-K B-K

a : P K : E

m K 1: 1D K E 1O

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30/34


> 1D 1K 1O 1: 1: :C

Scheduled pro+ect completion date is BC das& Also find the date on (hich the

pro+ect manager can complete the pro+ect (ith a probabilit of C&PC or PC

ii=8istinguish bet(een $,T and *$M&

/ a=4btain the sstem of stead state equations and hence find the value of p nin usual

notation

(here i2 n I s ii2 n s&

/ >= Ships arrive at a port at the rate of one in ever D hours (ith exponential

distribution of inter arrival times& The time a ship occupies a berth for unloading

has exponential distribution (ith an average of 1C hours& #f the average dela of

ships (aiting for berths is to )ept belo( 1D hours6 ho( man berths should be

provided at the portL

a= A farmer has 1CCC acres of land on (hich he can gro( corn6 (heat and soabeans&

,ach acre of corn costs s 1CC per preparation6 O man das of (or) and ields a profit of

s BC&An acre of (heat costs 1:CN requires 1C man-das (or)6 and ields a profit of

s&DC&An acre of soabeans cost s 1OC to prepare6 requires man-das (or)6 and ields

a profit of s :Cf the former has s& 1 la)h for preparation and can count on CCC man-

das (or)& !o( man acres should be allocated to each crop to maximi7e profitL

Formulate this problem and solve it using the Simplex method&;8=

>=9se t(o-phase simplex method to

;8=

- BC -


31/34


Minimi7e

xB :

x16 x:6 xB C

2 a= i=Solve the follo(ing transportation problem&

;%=

&estination

# ! R S Su*7

A :1 1E :K 1B 11

Source ( 1O 1 1D :B 1B

C B: :O 1 D1 1P

&eman- E 1C 1: 1K

ii=rite M48# method to solve the transportation problem&

;%8=

Or

2 >= The o(ner of a small machine shop has four mechanics available to assign +obs for

the da& Five +obs are offered (ith expected profit for each mechanic on each +ob (hich

are as follo(s3

o>

A ( C & E

E: O KC 111 :

Mechanic 2 O1 D E1 OB KP

$ O P: 111 O1 1

4 D ED O OO C

Find b using the assignment method6 the assignment of mechanics to the +ob that (ill

result in a maximum profit& hich +ob should be declinedL

;8=

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32/34


$ a=Find the optimum integer solution using cutting plane algorithm for the follo(ing3

;8=

Maximum < = x1> : x:

Sub+ect to

x1> x: @ O

: x1@ D

: x: @ O

x16x: C and integers&

Or

$ >=9se branch and bound method to solve the follo(ing3

;8=

Max < =B x1>:x:

Sub+ect to the constraints

:x1> :x:@ O

x1@ :

x:@ :


4 a= The follo(ing table sho(s the +obs of a pro+ect (ith their duration in das& 8ra(

the net(or) and determine the critical path& Also calculate all the floats&

;8=

obs 1-: 1-B 1-D :-K B-O D-E K-O K-

8uration 1C P 1E O O O

obs E-O E-P O-1C -1C P-1C 1C-11 11-1:

8uration K 1: 1C 1K K

Or

4 >=A maintenance foreman has given the follo(ing estimate of times and cost of +obs

in a

maintenance pro+ect&

;8=

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33/34


ob $redecessor 'ormal Time 'ormal *ost *rash Time *rash

*ost

A - C E 1CC

% A O DC D PD

* A 1: 1CC K 1D

8 A P OC K 1C:

, %6 *6 8 E KC E KC

4verhead cost is s& :K per hour

i2 The normal duration of the pro+ect and the associated cost

ii2 The minimum duration of the pro+ect and associated cost

iii2 #f all the activities are crashed (hat (ill be the pro+ect duration and the

corresponding cost&

/ a= i=For the queuing model/M0M0S23 /50F#F426 derive the formula for average

(aiting time of

a customer in the sstem

;%8=

ii=For the model /M0M0123 /'0 F#F426 derive the formula for average number of

customers in

the sstem&

;%8=

iii=,xplain the various characteristics of queuing sstems in detail&

;%4=

Or

>= *ustomers arrive at one-man barber shop according to a $oisson process (ith a mean

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