Probability Distributions: is < or > then 2 column DECISIONS WITH UNCERTAINTY Optimal decision under risk = D1
Uniform: each result has same probability Anderson-Darling Laplace (assume equally likely) EOL(D1) = .6(0) + .2(0) +.2(9000) = 1800
(rolling a single dice) Modification of K-S, placing greater weight MAXIMAX/MINIMAX (optimistic) Opportunity loss matrix
Normal: z is the number of stan dev’s away on the tails of the distribution, value < 1.5 MAXIMIN / MINIMAX (conservative)
from the mean = z = value – mean / st dev. indicates good fit MINIMAX REGRET
S1 S2 S3
Binomial: (put formula here) If a decision is dominated in all D1 0 0 9000
Triangular: has mode (most likely), max and outcomes, then you can remove it. D2 2000 500 4000
min. LAPLACE (choose higher
Poisson: a discrete distribution (a count) average/total payoff) D3 5000 1500 0
number of successes in an interval (λ) x − λ
depends on the success λ e
rate and the interval length.x) = f(x) =
Exponential: μ is average time to complete a
task, or interval between events. It is a
continuous distribution. If poisson rate
applies, then interval between is exponential.
P x≤x =1−e ) (−x0μ ) Compare the KS / AD values to the chart or MAXIMAX (MAX the MAXimum Payoff)
Above formula will give0you percentage that ‘good fit’ indicator. Take the Maximum value from all the maxs UTILITY
the task will be completed within x mins of Utilities are used when the decision
the average.Turn the interval (μ) into a rate Validation and Verification criteria must be based on more than just
λ = 1/μ – f(x) = λe λx Verification done by expert, validation expected monetary values.
can be done by using past data in Utility measures of the value of a
Simulation is the process of designing a particular outcome, reflecting the
simulation and seeing if results are correct.
mathematical or logical model of a real Steps in Validation: decision maker’s attitude towards a
system and then conducting computer-based 1. develop model that looks reasonable to collection of factors.
experiments with the model to describe, those that understand system MAXIMAX (MAX the Minimum Payoff) These factors may be profit, loss, and risk.
explain, and predict the behavior of the real 2. Validate assumption Take the greatest value from all the mins Deriving Utility
system. Simulation is the most widely 3. Validate output Assign 100 utiles and 0 utiles to best and
used Decision Analysis technique used to You are looking for the long run steady worst case payoffs.
gain knowledge about outcomes to assess Take value in middle, and give them that
state operation of a system.
risk for certain or the gamble to get best or
Continuous Simulation: based on math eqn’s QUEUING worst. The probability of getting the best
and used for simulating continuous values. Size of population can be infinite or finite that makes these 2 options equal is the
utility of the middle value.
(time spend waiting in line) Behaviour of arrivals (patient, balk, renege)
Discrete Simulation: simulation specific Waiting line can have limited or unlimited
values or points (number waiting in queue) capacity MINIMAX Regret (minimize Opport Cost)
Advan: leads to better understanding, can Take the oppourtunity loss of each cell, and RISK AVERSE
model any assumptions, easier to explain, M – Markov Distribution (Poss / expon) take the decision which has the least highest
provides a more realistic replication. D – Deterministic: constant regret.
Disadvan: no guarantee it will provide good G – General Distribution with known Regret States of Nature Maximum
Tnable s s s Regret
results, no way to prove reliable, time mean/variance i 1 2 3
consuming, it’s random based may be less λ = average arrival rate c d1 0 0 9,000 9,000 RISK LOVERS
accurate. 1/λ = average time between arrivals e
Static vs Dynamic μ = average service rate for each server D d 2 2,000 500 4,000 4,000
Does time have a role in model? Dynamic 1/ μ = average service time d3 5,000 1,500 0 5,000
simulation is time driven. Arrival Rate Distribution Expected Utility Approach
Continuous change vs Discrete change Question: What is the probability that no For each alternative, multiply the utility
orders are received within a 15- These are for PROFIT, the opposite are for by the probability for that state. Sum for
Can system change continuously or only at minute period? (mean is 5 in a 15
discrete points in time COST questions. each alternative, and take highest
Deterministic vs Probabilistic x - interval) alternative.
Is everything certain, or is there uncertainty?x) = (λ e )/x! DECISIONS under RISK Neutral Approach uses Normal EV
P(0) = (5 e )/0!
Use Monte Carlo technique for uncertainty. 3 options: 1) avoid, 2) manage (insurance), 3)
Doing probabilistic and taking averages over Service Time Distribution Embrace (casinos)
time is better then doing best case/worst casQuestion: What % of orders will take Probabilities known for outcomes.
(determ) Use Expected Value Approach and take
less than one minute to
Most operational models process? (mean = 30/hr, 2 min highest EV.
Dynamic, Discrete change, Probabilistic an order):
P(T ≤ t) = 1 - e = 1 - e0(1/60)
Mapping this Utility Chart:
The Monte Carlo technique is defined as a = 1 - .6065
technique for selecting numbers randomly