Spring 2023 11:40 Decision Analysis

How do you find an activity's mean completion time?

(a+4m + b)/6

How do you find the variance?

(x-mean)^2

What are the different types of special cases?

- Alternative Optimal Solutions: when the objective function line is parallel to a binding constraint all points on the line are optimal - Infeasibility: There are no solutions that satisfy all constraints. - Unbounded: Improper formulation causes infinite solutions.

What are the different types of data patterns?

- Horizontal: Data fluctuates around a constant mean - Trend: gradual shifts over time - Seasonal: highs and lows are repeated over the same time periods Trend and Seasonal combined - Cyclical: Alternating sequence of points below and above the trend line lasting over a year

What is the shadow price for a non binding constraint?

0

How do you graph the objective function line?

1. Set the objective function equal to the product of the coefficients 2. Determine the coefficients 3. Graph

When are quantitative methods used to forecast?

When historical data is available

When does the optimal solution change when within the allowable increase/decrease?

When the RHS of the binding constraint changes.

When should the production quantity be increased?

When the profit surpasses than the allowable increase.

When are constraints binding?

When the touch the optimal solution and when they do not have any slack or surplus

Probability

A number that describes how likely an event will occur

What is the multiplication law?

AND

How do you calculate the variance of a project?

Add the critical path variances

How do you assign a probability?

All assigned probabilities must be between 0-1 and add up to 1 - Classical: data is based on equally likely events - Relative Frequency: probabilities are based on historical or experimental data - Subjective: data is subjective

What is the union of an event?

All sample points that those events contain

Experiment

Any process that generates well-defined outcomes. An outcome is a sample point

Blending Models

Blend several resources or materials together to create one or more products corresponding to a demand. Ex. Diaz Coffee Company, Strength Aroma etc.

What is the minimax approach?

Create a regret table, of the differences of each state of nature. Choose the decision alternative with the lowest amount of regret.

Exponential Probability Distribution

Describes how the length of the interval between occurrences, often time or distance. Example: Mean time between arrivals is three minutes. What's the probability that the time between two arrivals will be two minutes? p(x<=2) = 1 - e^-2/3 = .4866

Poisson Distribution

Describes the amount of occurrences per interval. F(amount of occurrences) = average amount of occurrences within that time period^amount of occurrences (e^-average amount of occurrences within that time period) / amount of occurrences! For example: Given 10 arrivals in 15 minutes find the probability of 5 arrivals in 15 minutes f(5) = 10^5 e^-10 / 5!

How do you find the optimal solution graphically?

Determine which extreme point of the feasible region touches the objective function line in the direction of improvement first. If the slope of the objective function is steeper than that of the binding constraint, than the extreme point to the left is the optimal solution and Vice versa.

What direction do you shade >= for maximization problems?

Down

Uniform Probability Distribution

Example: Given 120<=x<=140 1/140-120 = 1/20 What is the probability of 135<=x<=140? 140 - 135 = 5 1/20 x 5 = 1/4 There can never be an exact number

How do you find the Efficiency of Sample Information?

Expected value of sample information / Expected value of perfect information EVwSI - EVwoSI / EVwPI - EVwoSI

What is the expected value of perfect information?

Expected value with perfect information - expected value without perfect information

How do you find the range of optimality graphically? Or What are the values for which x or y can rake on before another set of constraints become binding?

Find the slopes of the binding constraints and objective function by -x/y Take the inverse when x is negative and solve lower binding constraint slope <= objective function slope <= higher binding constraint slope

Normal Probability Distribution

Find the z = x - mean / SD Find p < .05, find .95 on the z table (z-table value)(SD) + mean

What is the expected value without perfect information?

For each payoff: (state of nature probability)(payoff) Add this up for each decision alternative and choose the highest

How do you find the absolute error percentage?

Forecast error / sales Move two decimals to the right

Dummy Variables

Given that there is an equation with three seasons, that means that there are four total seasons Input dummy variables, all zeroes besides a single one, in each season with the fourth season having all zeroes.

How would you graph x <= 6?

Graph a straight line going upwards from 6 on the x axis

Allocation Models

Have a maximization objective which is usually profit, that is subject to less than constraints on capacity. Ex. Veerman's Furniture and their production departments

What is the conservative approach?

Highest of the smallest states of nature for each decision variable

What are mutually exclusive events?

If one event occurs, then the other event cannot occur. Either one or the other can occur, but not both.

How do you determine a new optimal solution if a single products profit increases?

If the increased profit is within the range of optimality, the optimal solution doesn't change. If it is not, than we know that it would be at least the allowed increase.

How do you find the objective function value?

Input the optimal solution into the objective function and solve.

How do you find the coordinates for minimization problems?

LHS should add up to RHS

What is the expected value with perfect information?

Making the decision after the state of nature. The highest payoff from each state of nature multiplied by the according probability an added together.

Covering Models

Minimize the objective function, which is usually costs, and is subject to greater than constraints on required coverage. Ex. the amount of ingredients in a trail mix based on their nutrients

Random Variables

Numerical description of the outcome of an experiment - Discrete Random Variables: either a set amount of values or an infinite sequence of values. This includes Poisson Distribution. - Continuous Random Variables: A number within a set interval or collection of intervals often using <= x <=. They include uniform, normal and exponential distributions.

How do you find the new objective function value when the unit profit changes?

OF value + (change)(final value)

How do you find the new objective function value when the RHS of a constraint changes?

OF value + (change)(shadow price)

How do you graph expected values?

On two parallel y-axis graph the values of the decision alternative.

What is the addition law?

P(A or B) = P(A) + P(B) - P(A and B) OR

How do you determine whether two events are independent?

P(A) x P(B) = P(A and B) When the probability of event A is not affected by the occurrence of event B those events are independent.

How do you find the range of optimality using sensitivity analysis?

Sensitivity Report: In the variable cells, objective coefficient +/- the allowable increase and decrease

What are Joint Probability tables?

Tables that display the probabilities of each event occurring

Range of feasibility

The RHS values for which the shadow price won't change Constraint R.H. Side +/- the allowable increase and decrease

What is the range of optimality?

The amount by which the product objective coefficients can change before the optimal solution changes

What is the shadow price?

The amount of money that you would be willing to pay for an additional unit

What is surplus?

The amount of remaining resource in greater than or equal to constraints

What is slack?

The amount of remaining resource in less than or equal to constraint

How do you find the shadow price for a binding constraint graphically?

The change in the objective function value from a change in the RHS of a binding constraint

What is the optimistic approach?

The decision alternative with the highest possible payoff

What is the Expected Value without Sample Information?

The highest payoff of the option that does not include further information.

What is time series analysis?

The use of historical sales data to find sales patterns

What are decision variables?

They show the amount to produce, invest, purchase and hire etc.

What direction do you share <= for maximization problems?

Up

What is the expected value with sample information?

Updated information from further research or experimentation. Multiply the highest payoffs from each possibility by the according probabilities and add together.

Network Models

Uses diagrams to formulate problems Transportation network models have capacities, demand locations and the unit costs of transportation between supply demand pairs. xij, i = from and j = to.

Moving Averages Forecasting Method

Uses the most recent data averages for horizontal trends

Naive Forecasting Method

Using the most recent data to predict future data

What is the standard deviation?

the positive square root of the variance. Denoted by an o symbol.

How do you find an activity's completion time variance?

[(b-a)/b]^2

Event

a collection of sample points whereas the sum is the probability

Exponential Smoothing Forecasting Method

predicts the next value will be a weighted average between the last realized value and the old forecast

What is the complement of an event?

the complement of A is all the sample points that are NOT in A