ISE 362 EXAM 1 Cheat Sheet (DRAFT) by musikdr

Rseries = Rⁿ
Rparallel = 1-(1-R)ⁿ

2 Parallel Components in Series = (1-(1-R)²)²
2 Series Components in Parallel = 1-(1-R²)²
Legitimate Values of R 0<=­R<=1

Consider f(R) = R(2-R) where does f(R) achieve it's max and what' the max value?
g(R) = R(2-R0 = 2R-R² taking it's derivative and setting equal to 0 we get
dg(R)/dR = 2-2R=0 Its max is achieved when R=1
Plugging this in for R in g(R) yields 2-1=1>0

k out of N Redundancy - system that operates if at least k out of N components function properly.
R=Sum(I=k to n) of R^I(1-R)^n-i (ex Sum(I=2 to 3(for ito 3)(.0.9)ⁱ(0.1)^3-i)

How much would you need to invest on September 27th, 2019 to have $10,000 on September 27th, 2027 given an interest rate of 5%?
PV = FV(P/F­,i,­n)=­10k­(P/­F,5,7)

How much would you need to deposit on January 1, 2020 in a fund that yields 5% annually in order to draw out $250.00 at the end of each year starting December 31, 2020 for 7 years, leaving nothing in the fund at the end?
PV=250­(P/­A,5%,7)

Dominate - when one of the outcome paths is clearly the better choice for every case

P34 = P[In State 4|Current State] - Take Pmatrix to the 4th power and look at position 34.

EMV(i) = SUM(j=1 to N) Pj*rij
How much does P have to change before another altern­ative is best. Add a z to one state and subtract z from another state, then set the equations = to each other and solve for z. If z is (-) then it's not better.

EV of Perfect inform­ation means you take the largest value at each state and multiply it by it's probab­ility.

Expected Loss of Sales cost (When D exceeds inventory)
E[lost sales cost | Demand­=d,­Inv=I]
MAX(d-i,0)

To remove a condition
E[lost sales] = SUM(d=0 to nd)SUM(I=0 to ni)max(d-i,0)Pd(d)*­Pi(i)

Can also use PMF (PI)

Markov Case Stochastic process where we only take into account the present to predict the future. That is the probab­ility of going to state j(future) from state I(present)

-If there is one (and only one) closed, aperiodic commun­icating class, the process is ergodic.

Pi(n) = P[In state after transi­tions]
Pi(1) = Pi1(0)p11 + Pi2(0)P21+...+ Pin(0)*Pn1

Haircuts take exactly 15min, we have
r.v. Y = # customers who arrive during a 15m interval
Haircuts cost 10; each customer consumes $1 of snacks every 15min

Pi(n) = Pi(n-1)*P
=Pi(0)*pⁿ

Q1 - We have 3 customers in the shop, what's the Probab­ility there will be 4 customers in the shop 4 periods from now? A - P⁴34

What is the probab­ility in the long run, a customer comes to the shop but leaves because there is no seat available?
P[A]=S­UM(I=0 to 4]P(A|­Sta­tei­)*P­(St­atei)

What is the Long run expected number of customers that come to the shop but leaves? E[A] = SUM(I=1 to n) E[A|B}­*P[Bi]
E[a|st­ate=0] = 0(.2) +0(.7) ......