formula sheet - quantitative analysis

7/23/2019 Formula Sheet - Quantitative Analysis

1/11

Formula Sheet for CPIT 603 (Quantitative Analysis)PROAI!IT"

Probability of any event: 0 P (event) 1 P(AorB) =P(A) +P(B) P(AandB)For Mutually exclusive events:

P(AorB) =P(A) +P(B)

P(AB) =P(A |B)P(B)

)(

)()(yProbabilitlConditiona

BB|A

P

ABPP =

nde!endent "vents:

P(AandB) =P(A)P(B)

P(A |B) =P(A)

#e!endent "vents:P($ and %) = P($) & P(% 'iven $)

P($ and % and C) = P($) & P(% 'iven $) & P(C

'iven $ and %)

%ayes *eore

)()()()(

)()()(

AA|BAA|B

AA|BB|A

+

=PPPP

PPP

A,B = any t-o events

A = co!leent ofA

"x!ected .alue

( ) ( )

)/(/000)/(/)/(/

///

11

1

nn

n

i

ii

PPP

PE

+++=

==

Xi = rando variables !ossible values

P(Xi) = !robability of eac* of t*e rando

variables !ossible values

=

n

i 1

= suation si'n indicatin' -e are addin'

all n!ossible values

E(X) = ex!ected value or ean of t*e rando

variable

)/()2/(/31

=

==n

i

iiPEVariance

Xi = rando variables !ossible values

E(X) = ex!ected value of t*e rando variable

3XiE(X)2 = difference bet-een eac* value oft*e rando variable and t*e ex!ected value

P(Xi) = !robability of eac* !ossible value of t*e

rando variable

.ariance#eviation4tandard ==Discrete Uniform Distribution

For a series of n values, f(x) = 15 nFor a ran'e t*at starts fro a and ends -it* b (a,

a+1, a+, 6, b) and a b

)( ab +=

1

1)1( +== abVariance

%inoial #istribution

rnrqprnr

n

=

)7(7

7n trialsinsuccessrofyProbabilit

"x!ected value (ean) = np

.ariance = np(1 p)

8eoetric #istribution

pp x 1)1(successfirstt*euntil

trialsofnuberofyProbabilit =

"x!ected value (ean) = 1/p

.ariance = (1 p)5!

Poisson #istribution

7)(

XX

=

eP

x

P(X) = !robability of exactlyXarrivals oroccurrences

= avera'e nuber of arrivals !er unit of

tie

(t*e ean arrival rate), !ronounced 9labdae= ;1


2/11

deviation,

=X

Z

X= value of t*e rando variable -e -ant to easure

= ean of t*e distribution

= standard deviation of t*e distribution

Z= nuber of standard deviations froXto t*e ean, m

e= ;1< (t*e base of natural lo'arit*s)

1=.ariance

tieservice$vera'e=1

=value"x!ected

tetP = 1)/(

for+ulaby t*e'ivenistti+etoe?ualort*anlessiscusto+eraserveto

re?uired(/)ti+eddistribute

llyex!onentiaany t*at!robabilit*e

>e'ative %inoial #istribution

( ) rrx ppr

xf

= 11

1)(x

= r5!

2= variance = r(1@!)5!

Continuous Uniform Distribution

For a series of n values, f(x) = 15 (b a)

For a ran'e t*at starts fro a and ends -it* b (a,

a+1, a+, 6, b) and a b

)( ba+=

1

)( ab== Variance

#$CISIO% A%A!"SISCriterion of Aealis

Bei'*ted avera'e = (best in ro-) + (1 )(-orst in

ro-)

For Miniiation:

Bei'*ted avera'e = (best in ro-) + (1 )(-orst in

ro-)

"x!ected Monetary .alue

)(=ative)"M.(altern iiPXXXi= !ayoff for t*e alternative in state of

nature i

P(Xi) =!robability of ac*ievin' !ayoffXi(ie,!robability of state of nature i)

D = suation sybol

"M. (alternative i) = (!ayoff of first state of nature) x

(!robability of first state of nature) + (!ayoff of second

state of nature) x (!robability of second state of nature) +6 + (!ayoff of last state of nature) x (!robability of last

state of nature)

"x!ected .alue -it* Perfect nforation

".-P = D(best !ayoff in state of nature i)

(!robability of state of nature i)".-P = (best !ayoff for first state of nature) x

(!robability of first state of nature) + (best

!ayoff for second state of nature) x (!robability

of second state of nature) + 6 + (best !ayoff forlast state of nature) x (!robability of last state of

nature)

"x!ected .alue of Perfect nforation

".P = ".-P %est "M.

"x!ected .alue of 4a!le nforation

".4 = (". -it* 4 + cost) (". -it*out 4)

100E

".P

".4=ninforatiosa!leof"fficiency

tility of ot*er outcoe = (p)(utility of bestoutcoe,-*ic* is 1) + (1 p)(utility of t*e worstoutcoe, -*ic*

is 0)


3/11

R$&R$SSIO% 'O#$!S

errorrando

linere'ressiont*eofslo!e

0)/-*enGof(valueinterce!t

y)ex!lanatoror(!redictort variableinde!enden/

(res!onse)variablede!endentG

1

0

10

=

===

==

++=

XY

resultssa!leonbased,ofestiate

resultssa!leonbased,ofestiate

Gofvalue!redictedH

H

11

00

10

===

+=

b

b

bb

Y

XY

"rror = ($ctual value) (Predicted value)

YY H=e

XY

XX

YYXX

YY

Y

XX

X

10

1 )(

))((

valuesof(ean)avera'e

valuesof(ean)avera'e

bb

b

n

n

=

=

==

==

)(44otal4?uaresof4u+ == YY

44"+44A44=

=== )H(44""rror4?uaresof4u+ YYe

==)H(44AAe'ression4?uaresof4u+ YY

44

44"1

44

44Aion#eterinatoftCoefficien === r

Correlation Coefficient = rr =1

44"M4"

===kn

sErrorSquaredMean

M4"== sEstimateofErrorStandard ++= XYModelLinearGeneric 10

odelin t*est variableinde!endenofnuberI

44AM4A

=

=k M4"

M4A=F:StatisticF

de'rees of freedo for t*e nuerator = df1= k

de'rees of freedo for t*e denoinator = df= n

k 1

0:

0:

11

10

=

H

HTestHypothesis

1

ifAeJect

1

,, 1

==

>

knk

FFff!a"!#"ate

dfdf

=

value@ifAeJect

)statistictestcalculated(value@

p

FPp

$= 0+ 1X1+ X+ 6 + kXk+

$= de!endent variable (res!onse variable)Xi= i

thinde!endent variable (!redictor or

ex!lanatory variable)

0= interce!t (value of $-*en allXi= 0)

i= coefficient of t*e ithinde!endent variable

k= nuber of inde!endent variables

= rando error

kkbbbb XXXY ++++= H 110

YH = !redicted value of $b0= sa!le interce!t (an estiate of %)

bi= sa!le coefficient of t*e i t* variable (an

estiate of i)

)15(44

)15(44"1$dJusted

=n

knr

FOR$CASTI%&

n

= errorforecast(M$#)#eviation$bsoluteMeann

=)error(

(M4")"rror4?uaredMean


4/11

100actual

error

(M$P")"rrorPercent$bsoluteMeann

=

nn

nttt

t

11

1

!eriodsn!reviousindeandsofsuForecast$vera'eMean

++

+++===

YYYF

ttt

ww

wwi

++++++

== +

)Bei'*ts(

)!eriodinvalue$ctual)(!eriodinBei'*t(

1

111

YYF:AveraeMovin!ei"ted

forecast)s!eriodKastdeandactuals!eriodKast(forecasts!eriodKastforecast>e-

)(1

+=+=+ tttt FYFF:Smoot"inlEx#onentia

111

11

1

)(

)(

:rend-it*4oot*in'l"x!onentia

+++

++

+

+=+=+=

ttt

tttt

tttt

$FF%$

F%$F$$

F%$YF%$F

n),=,,1,/(ie,!eriodti+e/

linet*eofslo!eb1

interce!tb0

value!redictedH-*ere

H10

==

=

=

=

+=

Y

XY bb

LL11H XXXXY bbbba ++++=

M$#

error)(forecast

M$#

A4F"si'nalracIin'

==

I%$%TOR" CO%TRO! 'O#$!S

=levelinventory$vera'e&

oo '&

('

ordereac"inunitsof)um*er

(emandAnnual

order)!ercost(rderin'

y!er!lacedordersof>ubercostorderin'$nnual

==

=

h'&

year!erunit!ercost(Carryin'

?uantityrder

year)!erunit!ercost(Carryin'nventory$vera'ecost*oldin'$nnual

=

=

= "conoic rder Nuantity

$nnual orderin' cost = $nnual *oldin' cost

"o '+

&'

&

(=

"

o

'

('&

"N & ==

otal cost (C) = rder cost + Ooldin' cost

"o '&

'&

($'

+=

Cost of storin' one unit of inventory for one year = &h

'&( -*ere & is t*e unit !rice or cost of an inventory iteand is $nnual inventory *oldin' c*ar'e as a !ercenta'

of unit !rice or cost

%'('& o& =

AP -it*out 4afety 4tocI:

Aeorder Point (AP) = #eand !er day x Keadtie for a ne- order in days

=)

nventory !osition = nventory on *and +nventory on order

EOQ without instantaneous receipt assumption

Maxiu inventory level =(otal !roduced durin' t*e

!roduction run) (otal used durin' t*e !roduction run

=(#aily !roduction rate)(>uber of days !roduction)

(#aily deand)(>uber of days !roduction)

=(pt) (t)

L


5/11

===

#

d&

#

&d

#

dt#t 1

otal !roduced *=pt

=

#

d&1

inventory$vera'e "'

#

d&

= 1

cost*oldin'$nnual

s'&

(=costsetu!$nnual o'

&

(=costorderin'$nnual

+ = t*e annual deand in units

* =nuber of !ieces !er order, or !roduction run

Production Run Model: EOQ without

instantaneous receipt assumption

$nnual *oldin' cost =$nnual setu! cost

s" '&

('

#

d&=

1

=

#

d'

('&

"

s

1

&

Quantity Discount Model

%'

('o"N=

f "N Miniu for discount, adJust t*e ?uantity to= Miniu for discount

otal cost =Material cost + rderin' cost + Ooldin' co

"o '&

'&

(('

++costotal =

Ooldin' cost !er unit is based on cost, so &h='&B*ere'= *oldin' cost as a !ercenta'e of t*e unit cost

(&)

Safety Stoc

AP = $vera'e deand durin' lead tie + 4afety4tocI

4ervice level = 1 Probability of a stocIout

Probability of a stocIout = 1 4ervice level

Safety Stoc with !ormal Distribution

AP = ($vera'e deand durin' lead tie) +ZsKZ = nuber of standard deviations for a 'iven

service level

K = standard deviation of deand durin' t*e lead

tie

4afety stocI =ZK

#eand is variable but lead tie is constant

( LZLd +=AP

daysintielead

deanddailyofdeviationstandard

deanddailyavera'e

===

L

d

#eand is constant but lead tie is variable

( ))dZLd +=AMP

deanddaily

tieleadofdeviationstandard

tieleadavera'e

===

d

L

)

%ot* deand and lead tie are variableAP ) dLZLd ++=

otal $nnual Ooldin' Cost -it* 4afety 4tocI

otal $nnual Ooldin' Cost = Ooldin' cost of re'ularinventory + Ooldin' cost of safety stocI

"" ''&

(44)

OC +=

*e ex!ected ar'inal !rofit =P(MP)

*e ex!ected ar'inal loss = (1 P)(MK)*e o!tial decision rule

4tocI t*e additional unit ifP(MP) Q (1 P)MKP(MP) Q MK P(MK)

P(MP) +P(MK) Q MK

P(MP + MK) Q MK

MP+MK

MK,

R


6/11

PRO$CT 'A%A&$'$%T

"x!ected $ctivity ieS

+L+=

*mat

S

=.ariance

a*

"arliest finis* tie = "arliest start tie + "x!ected

activity tie"F = "4 + t

"arliest start =Kar'est of t*e earliest finis* ties of

iediate !redecessors"4 = Kar'est "F of iediate !redecessors

Katest start tie = Katest finis* tie "x!ectedactivity tie

K4 = KF t

Katest finis* tie = 4allest of latest start ties forfollo-in' activities

KF = 4allest K4 of follo-in' activities

4lacI = K4 "4, or 4lacI = KF "F ProJect .ariance = su of variances of activities on t*

critical !at*

rianceProJect vadeviationstandardProJect == $

$

co!letionofdate"x!ecteddate#ue =Z

.alue of -orI co!leted = (Percenta'e of -orIco!lete) x (otal activity bud'et)

$ctivity difference = $ctual cost .alue of -orIco!leted

Cras* tietie>oral

Cost>oralcostCras*Periodcost5ieCras*

=

*AITI%& !I%$S A%# Q+$+I%& T,$OR" 'O#$!SSin"le#Channel Model$ Poisson %rri&als$

E'ponential Ser&ice (imes )M*M*+,

= ean nuber of arrivals !er tie !eriod (arrival

rate)

= ean nuber of custoers or units served !er

tie !eriod (service rate)*e avera'e nuber of custoers or units in t*e

syste,)

=L

*e avera'e tie a custoer s!ends in t*e syste,

,

= 1!

*e avera'e nuber of custoers in t*e ?ueue,)q

)(

=qL

*e avera'e tie a custoer s!ends -aitin' in t*e

?ueue, ,q

)(

=q!

*e utiliation factor for t*e syste, -rho., t*e

!robability t*e service facility is bein' used

=

*e !ercent idle tie,P0, or t*e !robability no one

is in t*e syste

Multichannel Model$ Poisson %rri&als$ E'ponentia

Ser&ice (imes )M*M*m,

m= nuber of c*annels o!en

= avera'e arrival rate

= avera'e service rate at eac* c*annel

*e !robability t*at t*ere are ero custoers in t*esyste

>

+

= m

m

m

mn

, m-mn

n

n for

7

1

7

11

1=

0=

0

*e avera'e nuber of custoers or units in t*e syste

+

= 0

)()71(

)5(,

mmL

m

*e avera'e tie a unit s!ends in t*e -aitin' line or

bein' served, in t*e syste

L,

mm!

m

=+

=1

)()71(

)5(0

*e avera'e nuber of custoers or units in line-aitin' for service

= LLq

*e avera'e nuber of custoers or units in line

-aitin' for service

q

q

L!! ==

1

*e avera'e nuber of custoers or units in line

S


7/11

=10,

*e !robability t*at t*e nuber of custoers in t*e

syste is 'reater t*an k,PnTk1+

=.

./n,

-aitin' for service (tiliation rate)

m=

-inite Population Model

)M*M*+ with -inite Source,= ean arrival rate

= ean service rate

= sie of t*e !o!ulation

Probability t*at t*e syste is e!ty

=

=

)

n

n

n-)

),

0

0

)7(

7

1

$vera'e len't* of t*e ?ueue

( )01 ,)Lq

+=

$vera'e nuber of custoers (units) in t*e syste( )01 ,LL q+=

$vera'e -aitin' tie in t*e ?ueue

)( L-)

L!

q

q=

$vera'e tie in t*e syste

1+= q!!

Probability of nunits in t*e syste

( ) )n,n-)),

n

n 0,1,,for7

7 0 ==

otal service cost = (>uber of c*annels) x (Cost !er

c*annel)otal service cost = m&sm = nuber of c*annels

&s = service cost (labor cost) of eac* c*annel

otal -aitin' cost = (otal tie s!ent -aitin' by all

arrivals) x (Cost of -aitin')

= (>uber of arrivals) x ($vera'e -ait !er arrival)&w

= (,)&w

otal -aitin' cost (based on tie in ?ueue) = (,q)&

otal cost = otal service cost + otal -aitin' cost

otal cost = m&s+ ,&w

otal cost (based on tie in ?ueue) = m&s+ ,q&w

Constant Ser&ice (ime Model )M*D*+,

$vera'e len't* of t*e ?ueue

)(

=qL

$vera'e -aitin' tie in t*e ?ueue

)(

=q!

$vera'e nuber of custoers in t*e syste

+= qLL

$vera'e tie in t*e syste

1+= q!!

.ittle/s -low E0uations

)= , (or ,=)5)

)q= ,q (or ,q=)q5)

$vera'e tie in syste = avera'e tie in ?ueue +

avera'e tie receivin' service

,= ,q+ 15

'AR-O A%A!"SIS

;


8/11

(i) = vector of state !robabilities for !eriod

i

= (1, , , 6 , n)

-*ere

n = nuber of states

1, , 6 , n = !robability of bein' in state 1,

state , 6, state n

Pi0= conditional !robability of bein' in state0in t*efuture 'iven t*e current state of i

=

mnmm

n

n

PPP

PPP

PPP

1

1

1111

,

For any !eriod n-e can co!ute t*e state!robabilities for !eriod n+ 1

(n+ 1) = (n)P

"?uilibriu condition= P

Fundaental MatrixF= ('B)1

nverse of Matrix

=

=

=

ra

r!

r

b

r

!

ba

!

ba

1

01,

,

r a 2 b!

M re!resent t*e aount of oney t*at is in eac* of t*enonabsorbin' states

3= (31,3,3, 6 ,3n)

n = nuber of nonabsorbin' states31 = aount in t*e first state or cate'ory

3 = aount in t*e second state or cate'ory

3n = aount in t*e nt* state or cate'ory

Partition of Matrix for absorbin' states

=

BA

4',

= identity atrix

= a atrix -it* all 0s

Co!utin' labda and t*e consistency index

1C

=n

n

Consistency Aatio

A

CCA=


9/11

STATISTICA! Q+A!IT" CO%TRO!

x

x

5x

5x

=+=

(KCK)liitcontrolKo-er

(CK)liitcontrol!!er

x = ean of t*e sa!le eans5= nuber of noral standard deviations ( for

URRE confidence, for UU;E)

x = standard deviation of t*e sa!lin' distribution

of t*e sa!le eans =n

x

6Ax

6Ax

x

x

KCK

CK

=

+=

6 = avera'e of t*e sa!les

A= Mean factorx = ean of t*e sa!le eans

6+

6+

6

6

L

KCK

CK

=

=

CK6 = u!!er control c*art liit for t*e ran'eKCK6 = lo-er control c*art liit for t*e ran'e

+Land+= !!er ran'e and lo-er ran'e

!@c*arts

pp

pp

5p

5p

=

+=

KCK

CK

p = ean !ro!ortion or fraction defective in t*e sa!

exainedrecordsofnuberotal

errorsofnuberotal=p

5= nuber of standard deviations

p = standard deviation of t*e sa!lin' distributionp is estiated by pH

"stiated standard deviation of a binoial distribution

n

ppp

)1(H

=

-*ere nis t*e sie of eac* sa!le

c@c*arts*e ean is ! and t*e standard deviation is e?ual to!

o co!ute t*e control liits -e use !! = ( is

used for UU;E and is used for URRE)

!!

!!

!

!

KCK

CK

=

+=

Aan'e of t*e sa!le = /ax@ /in

U


10/11

OT,$RSCo!utin' labda and t*e consistency index

1C

=n

n

Consistency Aatio

A

CCA=

*e in!ut to one sta'e is also t*e out!ut fro

anot*er sta'esn21 = ut!ut fro sta'e n

*e transforation function

tn= ransforation function at sta'e n

8eneral forula to ove fro one sta'e toanot*er usin' t*e transforation function

sn1= tn(sn, n)*e total return at any sta'e

fn= otal return at sta'e n

ransforation Functions

( ) ( ) nnnnnn !bsas ++=1Aeturn "?uations

( ) ( )nnnnnn!bsar ++=

7s

f

=

=cost5unit.ariablePrice5unit

costFixed(units)!ointeven@%reaI

Probability of breaIin' even

=

!ointeven@breaI

Z

P(loss) =P(deand breaI@even)

P(!rofit) =P(deand T breaI@even)

costsFixed

dea(Meanunit

cost.ariable

unit

Price"M.

=

>=

=%"P/V0for

%"//)for!ointeven@W(breaIKossy!!ortunit

-*ere

8= loss !er unit -*en sales are belo- t*e breaI@even

!oint

X= sales in units

sin' t*e unit normal loss inte"ral,"K can b

co!uted usin'

"K =8(+)

"K = ex!ected o!!ortunity loss

8 = loss !er unit -*en sales are belo- t*e breaI

even !oint= standard deviation of t*e distribution

(+) =value for t*e unit noral loss inte'ral fo

'iven value of+

!ointevenbreaI=(

( ) 'AB =

=

=

!e

be

ae

!

b

a

e

!

b

a

( ) ( )!fbea

f

e

!ba ++=

++++

=

h!f9!e

bhafb9ae

h9

fe

!

ba

!

ba

#eterinant .alue = (a)() (!)(b)

ih9

fe

!ba

#eterinant .alue = aei+ bf9+ !h9e! hfaib

deterinrdenoinatoofvalue>uerical

deterinanueratorofvalue>uerical=X

10


11/11

=

=

=

=

a!

b

ab

!

!ba

!

ba

atrixt*eof$dJoint

cofactorsofMatrix

atrixori'inaloft value#eterinan

atrixri'inal

=

!ba

a

!ba

!

!ba

b

!ba

!

ba 1

"?uation for a line$= a+ bX

-*ere bis t*e slo!e of t*e line

8iven any t-o !oints (X1, $1) and (X, $)

1

1

inC*an'e

inC*an'e

XX

YY

X

Y

X

Y* =

==

For t*e >onlinear function$ =X LX + S

Find t*es"opeusin' t-o !oints and t*is e?uatio

1

1

inC*an'e

inC*an'e

XX

YY

X

Y

X

Y* =

==

!ba

!ba

++++=

++=

)()(

/

1

XXXXY

XY

1 )()()( XXXXYYY ++== !ab

XXX

XXX

XXXXXY

++=

++=

++=

!ab!ab

X

!ab

)(

)()()(

)()(

)()(

1

xhx9

xhx9

!

n

n

n

=+=

=

==

=

Y

Y

XY

XY

XY

'Y

)()(

)()(

0

1

1

1

xhx9

xhx9

n

!n

n

n

n

n

=+=

=

=

==

+

Y

Y

XY

XY

XY

Y

otal cost = (otal orderin' cost) + (otal *oldin' cost)

+ (otal !urc*ase cost)

(''&

'&

($' "o +=

+

* = order ?uantity+ annual deand

&o = orderin' cost !er order

&h = *oldin' cost !er unit !er year& = !urc*ase (aterial) cost !er unit

"conoic rder Nuantity

o "'

&

('

&

$' +=

"

o

'

('&

=

&

('

&

$' o=

11

formula sheet - quantitative analysis

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