Doulton - BDM Flashcards
1
Q
Probability:
P(A)
A
The probability of event A
2
Q
Probability:
P(A’)
A
The probability of everything which is not A
3
Q
Probability:
P(A and B)
or P(AB)
A
The intersection/joint probability:
The probability of event A and event B occurring at the same time
Equation Given
4
Q
Probability:
P(A or B)
A
The union:
The probability of either event A or event B occurring
Equation given
5
Q
Probability:
P(A / B)
A
Conditional Probability:
The probability of event A, given that B already occurred
Equation given
6
Q
Mutually Exclusive =
A
When two event cannot occur at the same time. (Not independent)
Ex: You cannot roll and 4 & 5 on a dice at the same time./
7
Q
Independence
A
Two events, A and B, are independent if the fact that A occurs DOESN’T affect the probability of B occurring
P(A and B)
P(A/B)
8
Q
Rule of Conditional Probability
A
P(A/B) = P(A and B)
__________
P(B)
9
Q
Bayes Theorem =
A
Describes the probability of an event, based on prior knowledge of conditions that might be related to the event .
Ex: performing well in class is related to having attended class (Bayes theorem involves assessing with more information)
Equation Given
10
Q
Binomial Distribution
A
The binomial distribution is a discrete distribution
It is used when we do the same thing ‘n’ times, and each time there is the same probability of success
11
Q
Poisson distribution
A
The Poisson distribution models the number of arrivals/occurences (X) which happens over a fixed interval (ex: hour) when you know the average number of arrivals (lagrange)
12
Q
Decision Making:
Decision Making under Certainty =
A
When the decision maker knows the payoff of each option/choice and can choose the option that maximizes their utility
Ex: Choosing to buy the same product online vs. in the store - You would choose the lowest price option.
13
Q
Decision Making:
Decision Making under Uncertainty =
A
When there are multiple outcomes for each choice, but the decision maker doesn’t know the probability of each outcome.
Ex: Which job makes you happiest in life
14
Q
Decision Making:
Decision Making under Risk =
A
When there are multiple outcomes for each choice, but the decision makers knows the probability of each outcome
- Most of our problems
Ex: Playing a card game or the lottery, the odds of winning can be calculated.
15
Q
Decision Making under Uncertainty:
MaxiMax (optimistic)
A
The option with the greatest potential payoff
16
Q
Decision Making under Uncertainty:
MaxiMin (pessimistic)
A
The option with the highest WORST option
17
Q
Decision Making under Uncertainty:
Weighted Average
A
Given probabilities are assigned to each outcome
(basically makes it Decision making under risk).
Choose the one which has the highest expected payoff (or lowest expected cost)
18
Q
Equally Likely (Laplace)
A
Simular to the Weighted Average, however equal probabilities are used (the amount of all options to choose from)
19
Q
MinMax Regret
A
The outcome with the minimum of all possible regrets
Use a table to calculate
Regret (opportunity loss) = Best pay off - payoff received
Find the Maximum regret for each alternative, and then choose the minimum from this quantities
20
Q
Decision Trees =
A
Decision Trees model sequential events. It captures decisions to be made, probabilities for each event and the outcome of each set of decisions and events
They are draw from start to finish, but solved backwards
21
Q
Decision Tree :
Square
A
Represents a decision
22
Q
Decision Tree :
Circle
A
Represents a probability event
23
Q
Decision Tree :
Triangle
A
The end of the set of events
24
Q
EMV (Decision making under risk)
A
Expected Monetary Value :
= The expected value of the payoffs. No perfect information
Equation Given
25
EVxPI (Decision making under risk)
Expected Value with Perfect Information:
= The expected value if we had all the information before the decision had been made
Equation Given
26
EVPI (Decision making under risk)
Expected Value of Perfect information:
= sets the maximum we would be willing to pay for the perfect information
Equation Given
27
EVSI (Decision making under risk)
Expected Value of Sample information:
= expected value of sample information is the increased expected value resulting from having sample information. The value of the sample information is the difference of payoff you would get with the sample and without the sample
28
Expected Utility
The utility of each event added up
29
Risk Neutral =
A risk neutral person has NO preference between outcomes which have the same expected payoff. (As long as it gets you the same in the end)
30
Risk Adverse/Risk Avoider
A risk adverse person prefers certainty over risk. it is a CONCAVE function.
Ex: a Risk adverse person with a utility function of U(x) = (square root)x
31
Risk Loving/Seeeking
A risk loving/seeking person prefers the potential upside of risk of certainty. CONVEX function
32
Linear Programming =
The maximization or minimization of linear objective function subject to linear constraints using the algorithm of Simplex LP.
The solution is based on defining the 1) decision variables, 2) objective functions, 3) and constraints
33
Properties of Linear Programming
1. One objective function
2. One or more constraints
3. Alternative courses of action (i.e. the decision variables can take on different numbers)
4. Equations are linear - proportionality and divisibility
5. Certainty (coefficients you are using are not going to change)
6. Divisibility (solutions do not need to be whole numbers)
7. Nonnegative variables (you can't have a negative number of cups)
34
Components of Linear Programming Solution:
| 1. Decision Variables =
"Let X be ......."
What you control directly in the problem
Results of core decisions are constraints, and not decisions
35
Components of Linear Programming Solution:
| 2. Objective Function =
Profit, Revenue, or Cost
The value you are trying to minimize or maximize.
Must relate the decision variables to a single value.
Must be linear.
36
Components of Linear Programming Solution:
| 3. Constraints =
Constraints limit the objective function
37
Graphical Solutions for Linear Programming
Simple problems can be solved graphically (limited to two variables only) where the feasible region is found, and the objective function is pushed away from (0,0), if the maximization problem or the objective function is pushed towards (0,0) if a minimization problem.
38
Linear Programming Types of Problems:
```
Resource allocation
Inventory
Marketing
Investment
Transportation Problem
Blending Problems
```
39
Linear Programming Types of Problems: Resource Allocation
You decide how much to make/produce of each item. The recipe is given
40
Linear Programming Types of Problems: Inventory
You must order or produce each period and you have the option of holding inventory until the next period. There may be reference to holding costs or the cost of producing or ordering the product changes period to period
Decision Variable:
Xi - the quantity of widgets to produce in period I
Ii - the quantity of widgets to hold in inventory at the end of month i
Objective Function:
min Cost = Production costs + Holding costs
S.t.:
Amount left over this period = Amount started with + in - out
41
Linear Programming Types of Problems: Marketing
You are deciding how many ads to run in each market or type
42
Linear Programming Types of Problems: Investment
Decision variables:
| the number of shares/stocks to buy or the dollar amount to invest
43
Linear Programming Types of Problems: Transportation
Decision Variables:
Decide how much to ship along each route, typically xii variable
"Let Xii represent the amount shipped from location i to location j
Objective:
Usually minimize the cost of shipping
Constraints:
Supply: what leaves a supply node < what is available at the node
Demand: what arrives to a demand node > what is required at that node
44
Linear Programming Types of Problems: Blending Problems
Decision Variables:
the amount of each ingredient in each project
Let xij be the quantity of input i which goes into product j
Objective Function:
max Profit = Revenue from products - cost of inputs
Constraints
Common constraints include supply and demand and can be treated similarly as outlines in the transportation section.
45
Sensitivity Analysis
The Sensitivity Analysis gives us information on the impact of changing:
1) the coefficients of the objective function, and
2) the right hand side of the constraint.
46
Sensitivity Analysis
Analyzing the Variable Cells (in sensitivity analysis report)
"Reduced costs":
How much final value will drop (or increase) if the variable cell was to increase by 1.
Alternative - have to change before we would consider making the product.
"Objective Coefficient"
Optimal quantity of decision variables X change in coefficient
- multiple number of final values
"Allowable increase/decrease"
= how much the coefficient can increase (or decrease) without having to change the entire system.
47
Sensitivity Analysis
Change in objective function formula
= optimal quantity of decision variable x Change in coefficient
48
Sensitivity Analysis
Analyzing the Constraints table
Contraint
"Allowable increase/decrease"
= how much the constraint can increase (or decrease) without having to change the entire system.
"Shadow price"
= the shadow price exists ONLY if the constraint is binding (otherwise it is zero).
it is the change in the objective function value for a 1 unit change in the right-hand-side of the constraint.
"Constraint"
Change in objective function = shadow price X RHS of strait+- cost
49
Sensitivity Analysis
| Analyzing changes in the constraints: Change in objective function formula
= Shadow price x RHS of constraints+/- costs
50
Why do we have inventory? (5 reasons)
1. Buffer between manufacturing and other processes (decoupling)
2. Storing resources
3. Irregular supply and demand
4. Quantity discounts
5. Avoiding stock outs and shortages
51
Inventory control models: Inventory decisions
1. How much to order (Q)
| 2. When to order (ROP)
52
Inventory control models: Economic Order Quantity:
| Assumptions of the model (6) =
1. Demand is known and constant overtime
2. The lead time is known and constant (time between placing the order and getting it)
3. Order is received all at once, instantaneously
4. The purchase cost per unit is constant (quantity discounts are not possible)
5. The only variable costs are the costs of placing an order (order cost) and the cost of holding inventory over time. The holding costs per unit per year and ordering costs must be constant
6. Orders are placed so stockout and shortages are avoided completely
53
Inventory control models: Total Cost formula
= annual ordering cost + annual holding cost + purchase cost
```
Q = number of units in an order
EOQ = Q* = optimal number of units to order
D = annual demand (total)
d = daily demand (really demand per unit time AKA not total)
Co= Ordering cost of each other
Cs = set up cost for each production run
Ch = holding cost per unit per year
C = cost per unit
L = lead time
```
54
Inventory control models:
| EOG = Q*
The quantity which minimizes cost occurs when annual holding cost is equal to annual ordering cost (also known as taking the derivative of the total cost function and setting it equal to zero
Formula Given
55
Time between Orders (Cycle time
T = Q/D
56
Number of orders
1/T = D/Q
57
Average Inventory
Max inventory/2 = Q/2
58
ROP
Re-order point (Lead time)
59
```
Inventory control models:
Bulk Discounts (steps)
```
1. Find the optimal order quantity for each price per unit, C, using "EOG with ch expressed as percentage of unit cost" (given in formula sheet)
2. If the EOG is not in the acceptable range for Q, use the closest acceptable order quantity
3. For each price point caulcuite the total cost
4. Choose the lower total cost
60
Production Run model
when inventory is not received instantaneously. Instead of a ordering cost there is usually.a set up cost for initiating the production run
61
Safety Stock
= When demand or the lead time is uncertain, safety stock can be used to help prevent stockout.
safety stock is found by adjusting the re-order point (ROP).
Safety stock increases the total holding cost but does not impact the optimal order quantity.
The greater the lead time, or the greater the variability, the more safety stock that will be needed
Equation of sheet
62
Determining Safety Stock
Service Level = 1 - probability of stick out
63
Linear Programming Types of Problems: Max Flow problem
```
Decision variable (how much can get from start to finish):
Xij: the quantity travelling along arc ij or the quantity travelling from node i to node j
Helper variable: Uij: the maximum flow along ij
```
Objective Function:
Maximize Flow
```
Constraint:
Node Balance (i.e. Quantity out - Quantity in)
```
64
Linear Programming Types of Problems: Shortest Path (Do I want to take it or not)
The equipment replacement problem (when to repair vs. replace a piece of equipment) is a type of shortest path problem.
65
Linear Programming Types of Problems: Shortest Path
"Connect the Dots" - to make the smallest distance possible. Once all connected, add the up.
66
EV with SI
Apart of the EVSI equation
= expected value with sample information
67
EV without SI
Apart of the EVSI equation
= expected value without sample information (ie. the EMV if no sample was taken)
68
Decision variables:
one variable for each type of item you are making/selling
69
Objective function:
Usually max profit or revenue
70
Contraints
one constraints for each resource, ie/ amt labour used < amt. available
71
Optimal production Quantity
Used for Production Run model.
Given on formula sheet
72
Answer report
Objective Cell table
Maximize contribution = profit
"FInal Value":
= optimal solution output
73
Answer report
Variable Cell table
Amount each variable is producing
74
Answer report
Constraints table
"Cell value":
= ending value of how many you use from constraint
"Status"
binding = max constraints
non binding = not using all constant
"Slack":
whats left over from "status"
75
cycle time for ordering
Q/D
76
inventory cost models:
| Max in inventory
= Q
77
Inventory cost models:
d
= D/days
| total demanded/number of days -----> to find demand per day
78
Weighted Average
a probability is assigned to each outcome (basically makes it decision making under risk, but probabilities aren't calculated or truly known)
79
HUrwicz Criterion/Certerion of Realism
uses only the best and worst outcomes weithed by alpha (coefficient of realism
The closer to alpha is to one, the more optimistic the decision maker is
a(best outcome) + (1-a) (worst outcome)
80
Integer Programming 2 variables
Integer variables
| Binary Variables
81
Integer Variabels
Integer variables: variables which can take on WHOLE numbers
ex: number of employees, number of contracts
Deos not change the fundamental structure of the linear program
requires simplex algorithm (technique called branch and bound)
we add an additional constraint to indicate that the variable is an integer and when entering the model into excel we would indicate the variable is an integer (where K is an integer)
82
Binary Variables
variables which can only take on the value of 0 or 1. they capture "yes" or "no", on or off relationships in models
also used the branch and bound technique
we add an additional constraint to indicate that the variable is an integer and when entering the model into excel we would indicate the variable is an Binary
83
Binary Constraints
Logical upper bound
| Logical Lower bound
84
logical upper bound
used when a binary variable being zero forces another variable to be zero
ex: if we do not open the factory, we cannot produce any widgets in the factory
LUB: Y < MX where M is an arbitrarily large number (X as 1 if we open factory, 0 otherwise. and Y be the number of widgets produced
85
logical lower bound
used when we have a minimum with alternative to do nothing at all
LLB will not force the non-binary to be zero when the binary is zero (so must also have a LUB when there is a LLB)
ex: if I go to stauff library to study, I will go for at least 2 hours
86
Types of Simulation
Monte Carlo Simulation
Operational Gaming Simulation
Systems Simulation
87
Monte Carlo Simulation
= generates values for the uncertain variables making up the model being studied
ex: daily demand, LT, # of parts which fail, number of packages that arrive
Steps to Monte Carlo simulation
1. establish probability distribution for the important variables
2. create cumulative probability table for each variable
3. establish the interval of random numbers which applies to each variable
4. generate the random numbers
5. Simulate a series of trails using the random numbers to determine the value for each uncertain variable
88
Operation Gaming Simulation
- has two or more competitive players (common in business and military applications)
- focuses on the players decision making skills in hypothetical situations
- new information is provided each round, players must respond to it
ex: caps in 1st year was an operational gaming simulation
89
Systems Simulation
- tests the effects of policies on broad systems
- ex: model the economy and show the impact of new tax measures or minimum wage
- sim city is a system simulation (water, hydro, population growth)
90
Advantages and Disadvantages of Simulation
advantages
- relatively straightforward and flexible
- advances in technology to make simulations model easy
- can be used to analyze large and complex real - word simulations
- always "what-if?" type questions
- does not interfere with the real world system
- enables study of interactions between components
- enables time compression
- enables the inclusion of real-world complications
91
Disadvantages
- often expensive, may require a long complicated process to develop the model
- does not general optimal solution (trail and error approach, you must generate the options)
- requires managers to generate all conditions and constrains of real-world problem
- each model is unique and the solutions and interferences are not usually transferrable to other problems
92
TABULAR method of simulations
BY hand
one set of calculations of a simulation (proof of concept)
- use random numbers to generate the unknown variables
93
Computer based simulation
YASAI
- runs 1000's of calculations for each simulation
- Use random functions (GEN functions) to generate the unknown variables (like demand)
- We can run many different scenarios as long as the parameters of those scenarios are known (ie simulation is not an open ended problem)
94
Simulation Problem types
Inventory
random variables : demand, LT
decisions: ROP, Order quantity
Queing Theory
random variables: time between arrivals or number of arrivals, time to provide service or number able to service
decisions: how many service providers to have
95
Common calculations for inventory simulation
Daily order cost:
cost of placing one order X avg. # of orders places
Daily Holding cost:
cost of holding one unit X avg. inventory (per day)
Daily Stock out cost:
cost per lost sale X average # of lost sales per day
96
Common calculations for Queing theory simulation
cost of waiting per hour:
$ per hour lost from waiting X avg. # waiting at any time
Cost of providing service:
$ per hour paid for server X # of servers
97
Key YASAI formulas
Gen functions = the genfunctions generate a random variable according to the specified distribution
GENTABLE:
function replaces the random number interval table of the previous section
- list of outcomes, list of probabilities
- when you are provided with set of outcomes which each occur at a varying probability. these will appear CHART form
GENUNIFOR (min, max)
- equal probability of falling between two numbers (min and max)
- won't say uniform, but will imply it
- continuous variable (ie result will be a decimal answer)
- ex: price could be anything from $5-$10
GENPOISSON
- wil be given avg. # of arrives in a set time period
- the mean must match the time interval of your problem
- result is the # of occurrences (whole number)
- ex: on avg. 17 customers enter the store per day
GENNORMAL (mean, st. dev)
- will say normal distribution or will be the sum of many variables with a given mean and st. dev
- continuous variable (result will be a decimal)
- ex: length of a ruler is 15 cm on avg. with st. dev of 0.2 cm
GENBINOMIAL (# of trails, probability of success)
- models a yes/no or success/failure relationship
- a fixed number of trails (n)
- each trial has same probability of success (p)
- discrete (returns a whole number, ie. the number of successes)
- ex: 14 cars in a parking lot. on any given day there Is a 2% change that a car will get damaged
- ex: roll a 6 sided die
98
Other important functions in YASAI
MIN (A,B) = returns the smallest number
MAX (A,B) = returns the largest number
INT (A) = returns only integer portion of the number (whole number)
99
Quantative analysis: Types of Analytics
Descriptive analytics
- using historical data to describe a situation (ex: mean, median, standard deviation)
predictive analytics
- focuses on predicting future outcomes based on patterns in past information (ex: regression, decision trees, simulation)
Prescriptive analysis
uses optimization to better determine a better or new way to operate based on a specified objective (ex: EOQ, linear programming, trans models (anything you can use solver for)
100
Quantative analysis: Types of models
physical model:
- a physical, usually 3D, replica of a situation example (miniature car)
schematic model
- is a representation of the elements of a system using abstract, graphic symbols rather than realistic pictures, (Flow chart)
mathematical model
- uses a set of mathematical expression to portray relationships (linear programming)
101
deterministic model
the values to the inputs are known (linear programming)
102
probabilistic model
risk or uncertainty are a fundamental part of the model (decision trees, simulation)
103
Multifactor decision making
= allows the decision maker to consider many different factors in a quantitative fashion
104
Multifactor evalution process (MFEP)
= all factors are given appropriate weight, and each altnerative is evaluated in terms of each factor
105
Analytic Hierarchy process (AHP)
= uses pairwise compaisons and then computes the weighting factors and evaluations for us. Uses an arithmetic mean and normalizes each weighted factor. Sum of the factors is 1
106
Brazilai, Cook and Golany (BCG)
= uses pairwise comparisons like AHP but uses a geometric mean to compute the final evaluation. Does not normalize the alternative factors. The factors multiply to 1. This process is immune to rank reversal
107
Multifactor Evaluation process: decision criteria
each decision criteria is given an importance on a scale of 0 to 1 (sum to 1)
108
Multifactor Evaluation process: Options
rate each option from 0 to 1 for each criteria (do not need to sum to one)
109
number of comparisons formula
= n(n-1) / 2
110
ranking and meaning
```
1 - are of equal importance
3 - a little more important
5 - is more important
7 - is much more important
9 - very much more important
```
(ratio's are the opposites of above)
111
CI (consistency index)
= can be calculated for any set of pairwise comparisons (ie for each pairwise comparison matrix)
lamba: the average of the numbers found when diving the new terms bny their respective weights s
112
consistency ratio
CR of less than 10% means the decision maker was consistent in their decision making
113
Rank Reversal
the fundamental issue with AHP. if you have two identical best options, they will no tie for first but instead they reverse rank with the second place option
ex: A,B, C have final rankings (0.33, 0.55, 0.15). If we introduce D with exact weighting as B, A will now be first choice and B and D will tie for second
114
BCG approach
BCG prevents rank reversal
- uses geometric avg. instead of the arithmetic mean
- the final alternative weights from a BCG approach multiple to one (whereas the weights from AHP add to 1)
115
single period inventory modelling: Marginal loss
= the loss which occurs if you stock a unit which you don't sell (usually purchase cost - salvage value)
116
single period inventory modelling: Marginal Profit
= the profit if you stock a unit and it sells (usually the retail price - purchase cost)
117
single period inventory modelling: P
the probability demand will be greater than or equal to the given amount
