L.P. notes

Optimal solution

A feasible solution that results in the largest possible value when maximizing (or the smallest when minimizing)

all integer linear program (ILP)

An LP in which all the variables are restricted to be integers

Feasible Solution

Any values of the decision variables that satisfies all of the problems constraints (i.e. within the feasible region)

mixed integer linear program (MILP)

If only a subset of the variables are restricted to be integers

Non binding constraints

If the slack or surplus is greater zero at the optimal solution. Geometrically they do not pass through the optimal solution.

Binding constraints

If the slack or surplus is zero at the optimal solution. If LHS=RHS then it is a binding constraint. Geometrically, this will pass through the optimal solution.

Linear Programming

Involves choosing a course of action when the mathematical model of the problem only contains linear functions (objective functions, constraints) with respect to the decision variables.

Constraints

Limit the overall objective function value and the value the decision variables can take

Slack

Represent the difference between the LHS & the RHS hand side of the constraints. IN less than or equal to constraints slack =RHS-LHS of the constraint and indicates unused capacity.

Surplus

Represent the difference between the LHS & the RHS hand side of the constraints. Surplus=LHS-RHS and usually represents the amount over some minimum required level

LP relaxation

The LP that results from dropping the integer requirements is called the LP Relaxation of the ILP. Solving the problem as a linear program ignoring the integer constraints, the optimal solution to the linear program gives fractional values for our decision variables.

Extreme points

The corners or vertices of the feasible region. When looking for the optimal solution you only have to consider the extreme points.

Objective

The maximization or minimization of some quantity

L.P. problem

When the objective function and the constraints are linear (separate and to the power of one)

relevant cost

a resource cost in which the amount is paid for it is dependent on the amount of the resource used based upon decision variables. They are reflected in the OF.

sunk cost

a resource cost that must be paid regardless of the amount of resources actually used based on the DV and are not reflected in the OF.

binary variables

are variables whose values are restricted to be 0 or 1. These variables will be often be used as logical switches (No/Yes; Off/On) - in our case, typically whether a project is undertaken (=1) or not (=0) (or whether an operation is performed or not, etc.).

unbounded

has an objective function that can be increased without bound

non degenerate optimal solution

is one at which the number of positive-valued decision, slack, and surplus variables = the number of constraints (excluding non-negativity conditions).

allowable decrease

make up the range of feasibility of the shadow price and help determine if the shadow price is still valid or operative.

allowable increase

make up the range of feasibility of the shadow price and help determine if the shadow price is still valid or operative.

alternative (multiple) optimal solution

multiple solutions exist where all of these solutions result in the identical objective function value

degenerate optimal solution

n is one at which the number of positive-valued decision, slack, and surplus variables < the number of constraints (excluding non-negativity conditions).

infeasible optimal solution

no solution to the LP problem satisfies all constraints , including the nonnegativity condition. Reasons for this include: calculation error, too many restrictions on the problem, or the feasible region does not exist.

unique optimal solution

one solution (values of the dv) that satisfies all of the constraints and provides the best value (maximum in a maximization and minimum in a minimization of the objective function

range of optimality aka sensitivity range of an objective coefficient)

provides the range of values over which the current solution will remain optimal. the range of optimality for the current OFC is the range given by this minimum and maximum change over the current objective function values.

reduced cost

the amount by which the OFC that DV must be improved (min or max increase or decrease) to have an optimal solution where that dv has a positive value.

shadow price

the rate of change in the optimal value of the objective function per unit increase in the constraints RHS. The rate of change is operative for an increase in the RHS of up to but not surpassing the allowable increase. The shadow price for a nonbinding constraint is zero

sensitivity analysis (aka post optimality analysis)

used to see how the optimal solution is affected by changes, within specified ranges, in: the objective functions coefficients and RHS constraints

If both variables are mutually exclusive...

xj+xi is less than or equal to one. Now my options are limited to do either 1 or the other of project 'i' nor project 'j' (xj=1, xi=0 or xj=0, xi=1) or I can opt to do neither xj=0, xi=0)

If a variable is conditional upon another variable...

xj-xi is less than or equal to 0. Can only undertake project 'j' if I do project 'i' i.e. xj cannot be 1 unless xi is 1. However, I can still opt to do project i but not project j (xj=0, xi=1) or I can choose to do neither 'i' or 'j' (xj=0, xi=0). Easier to say at the beginning that xj is less than or equal to xi (which can be confirmed in the constraint above).

If a variable is a corequisite for another variable...

xj-xi=0 (or equivalently xj=xi). If I want to undertake project "j" I must do project i and vice versa. I either do both or do neither. My options are more restricted than the conditional constraint. Now xj=1, xi=1 OR xj=0, xi= 0