COB 291 Final Exam

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Moving Direction of Graphical solutions

* Min: move toward the origin • Max: move far from the origin

Binding and non binding variables slack and surplus

-If binding the optimal solution needs to use all of the two resources and the slack variables of these constraints are zero -If nonbinding we expect a positive slack or positive surplus variables for these constraints.

<= constraint (tighten/extend)

-Increase the RHS will extend(improve) the feasible region -Decrease the RHS will tighten(worsen) the feasible region

>= constraint (tighten/extend)

-Increasing the RHS may tighten the feasible region -Decrease the RHS may extend the feasible region

Mathematical Functions are considered linear if

1. Each variable appears in a separate term 2. Each variable is raised to the first power

Calculate the optimal solution point

1. Identify the constraint equations that intersect at the optimal point 2. Solve these equations simultaneously to find the optimal values for x, and y

Redundant Constraint

A constraint that can be removed from the model without changing the feasible region.

Objective Function of an LP

A function of the decision variables that the decision-maker wants to either Maximize or Minimize.

Feasible Solution

A solution point that satisfies all the constraints simultaneously.

Integer Programming

All decision variables are integers Feasible points are only integer points, and the optimal point is one of them If the number of integer points is limited, we can calculate the objective function value for the integer points and find the optimal point.

Alternative Optimal Solutions

All feasible points (more than one) that maximize (or minimize) the value of the objective function.

Infeasible

An LP problem is infeasible if there is no way to satisfy all the constraints in the problem simultaneously.

Seasonal Index

Average of the Actual/Trend for the previous seasons

EVwoPI

Best EV found on the side of the decision tree without the information

What types of constraints have shadow prices

Binding

Expected Value of Perfect Information (EVPI)

EVPI = EVwPI - EVwoPI • If possible, use the payoff table to calculate the EVwPI, and EVwoPI

Degenerate solution

If an allowable increase or allowable decrease of the RHS value of a constraint is zero

Unbounded Solution

If the objective function of an LP problem can be made infinitely large (in the case of a maximization problem) or infinitely small (in the case of a minimization problem), the LP model has unbounded solutions.

EVwPI

Multiple the Prior probabilities by the best option for that state of nature • Find the summation of all states of nature

Exponential Smoothing

Prediction 2+1= Prediction 2+ alpha(Actual 2- Prediction 2)

Weighted moving average

Previous X period averages times there respected weights

Reject Ho means?

Statistically significant

Standard Error

The standard error measures the scatter in the actual data around the estimate regression line

What are the decision variables for an LP

They are the controllable inputs

Non-negativity constraints Graphed

They go on the X and Y axis and point towards the first quadrant

Identify the feasible region

to identify the feasible region of a constraint plug in a test point (0,0) and see if it fits the constraint

Rolling back/ Solving decision trees

• Circle nodes = Event Nodes - Hint: sum (payoff * Probability) • Square nodes = Decision Nodes - Hint: choose the best option

Change in the right-hand side of a constraint • If the change is within the allowable range

• The constraint is nonbinding • The optimal solution (𝑥∗) will not change, and The shadow price is zero, and• The optimal objective function value (𝑂𝐹𝑉∗) will not change The constraint is binding • The optimal solution (𝑥∗) will change • The change in the optimal objective function (𝑂𝐹𝑉∗)value can be exactly calculated(Change in 𝑂𝐹𝑉∗ = shadow price *change 𝑖𝑛 𝑅𝐻𝑆)

Change in Coefficients of the objective function -Change in the coefficient is beyond the allowable range

• The optimal solution (x*) will change • The optimal objective function value (OFV*) will change too

Change in Coefficients of the objective function -Change in the coefficient is within the allowable range

• The optimal solution (x*) will not change • The optimal objective function value (OFV*) will change(change in 𝑂𝐹𝑉∗ = change in 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡. 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑥∗)

Change in the right-hand side of a constraint • If the change is beyond the allowable range

• The optimal solution will change, • The exact change cannot be calculated • The general pattern of the change in the optimal objective function value can be predicted • Relaxing a constraint => the optimal objective function value (𝑂𝐹𝑉∗) will be improved or not changed • Tightening a constraint => the 𝑂𝐹𝑉∗ will worsen or not change

A decision tree can start with?

• a circle node (chance/event node) • a square node (decision node).

95% prediction interval

𝑌 ± 2𝑆𝑒

Higher confidence creates a wider or slimmer interval?

Wider

Break even point

X= F/(Q-v) F=Fixed costs Q= Unit selling price V= unit variable costs

Mean square error

average of (Actual - Prediction)^2

Mean absolute percent error

average of (absolute error / Actual) *100

Mean absolute deviation

average of Absolute value of actual-prediction

slack and surplus variables

in a ≤ constraint represent the slacks (unused)capacities associated with the constraint, and surplus variables in a ≥ constraint represent the surplus (extra) capacities associated with the constraint. 𝐿𝐻𝑆 ≤ 𝑅𝐻𝑆 → 𝐿𝐻𝑆 + 𝑆𝑙𝑎𝑐𝑘 = 𝑅𝐻𝑆 𝐿𝐻𝑆 ≥ 𝑅𝐻𝑆 → 𝐿𝐻𝑆 − 𝑆𝑢𝑟𝑝𝑙𝑢𝑠 = 𝑅𝐻𝑆

R-Square

measures the percentage of the variation in Y around its mean that is accounted for by the estimated regression equation

Shadow Price

the amount of change in the objective function value when the RHS of the constraint increases by one unit, assuming all other coefficients remain constant. - Shadow Price is only valid within the allowable range


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