QBA Exam 3 (Final)

Constraint function

- A function of the decision variables in the problem - Mathematically, expressed as equations with variables on LEFT and constant terms on RIGHT

(Change in right-hand side of a constraint) NO change is NOT within allowable range

- Cannot predict how the objective function will change using the shadow price. - Must solve the linear program using the updated constraint to find the new solution

(Changes in Obj. Function Coefficient) YES change within allowable range

- Optimal values of decision variables will not change - Must recalculate value of objective function using new value of the coefficient (Final value * change)

(Change in right-hand side of a constraint) YES change is within allowable range

- Use shadow price to predict how the objective function value will change (shadow price * change) - Must solve the model with the updated constraint to find the new values of the decision variables

Basic property of Linear Optimization Model (4/4)

We will add constraints to require nonnegative values

The Solver Sensitivity Report allows us to understand how the optimal objective value and optimal decision variables are affected by what 3 things?

1. Changes in objection function coefficients 2. changes in certain decision variables 3. changes in constraint resource limitation or requirements

4 basic steps used to develop any optimization model

1. ID decision variables 2. ID objective function 3. ID appropriate constraints 4. Write obj. function and constraints as mathematical expressions

Linear Optimization Model has the following basic properties (All)

1. The objective function and all constraints are linear functions of the decisions variables 2. All variables are continuous 3. Each function is a sum of terms. Each of which is a constant multiplied by a decision variable 4. We will add constraints to require nonnegative values

"Cannot exceed"

<=

"must contain exactly"

=

"at least"

>=

Basic property of Linear Optimization Model (2/4)

All variables are continuous

Feasible solution

Any solution that satisfies all constraints of an optimization problem

Basic property of Linear Optimization Model (3/4)

Each function is a sum of terms. Each of which is a constant multiplied by a decision variable

(Changes in Obj. Function Coefficient) NO change is NOT within allowable range

Must solve the model with the updated objective function to find the new optimal values of the decision variables

Basic property of Linear Optimization Model (1/4)

The objective function and all constraints are linear functions of the decisions variables

Optimization

The process of finding a set of values for decision variables that minimize or maximize some quantity of interest and the most important tool for prescriptive analytics

Objective function

The quantity that is to be minimized or maximized; minimizing or maximizing some quantity of interest by optimization

Decision Variables

the unknown values that an optimization model seeks to determine

Unique Optimal Soution

there is exactly one solution that will result in the maximum/minimum of the objective function

Binding constraints

a constraint for which the cell value is equal to the right-hand side of the value of the constraint

Reduced cost

a number that tells how much the objective function coefficient needs to be reduced for a nonnegative variable that is zero in the optimal solution to become positive

Negative shadow price means...

a one unit increase in the right-hand side of the associated constraint results in a DECREASE in the optimal objective function value

Positive shadow price means...

a one unit increase in the right-hand side value of the associated constraint results in an INCREASE in the optimal objective function value

Unbounded solution

a solution that has allowed the value of the objective function to be increased or decreased without bound and without violating any of the constraints

Slack

difference between the left and right-hand sides of the constraints for the optimal solution

Shadow price

how much the value of the objective function will change as the right-hand side of a constraint is increased by 1 unit

Negated shadow price used to...

indicate the amount by which the optimal objective function value changes given a one unit decrease in the right-hand side value of the constraint

Constraints

limitations, requirements, or other restrictions that are imposed on any solution, either from practical or technological considerations or by management policy

Infeasible problem

no feasible solution exists. There is no way to satisfy all the constraints in the problem simultaneously

Alternate optimal solutions

solutions that result in a max (or min) objective function value by more than one combination of decision variables