COB 291 Final Exam
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