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LINEAR PROGRAMMING. These visual aids will assist in class participation for the section on Linear Programming. They have no new information and less detail than the Word Document. Concepts and geometric interpretations Basic assumptions and limitations Post-optimal, results analysis

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These visual aids will assist in class participation for the section on Linear Programming. They have no new information and less detail than the Word Document.

- Concepts and geometric interpretations
- Basic assumptions and limitations
- Post-optimal, results analysis
- Model formulations
- (Details in Notes, textbook, and references)

Multiple choice question: Why study Linear Programming?

1. The professor cannot formulate N-L models

2. Linear programming is the most frequently used optimization method.

3. Linear programming was developed in the 1940’s

4. Formulation and solution knowledge will also help when addressing non-linear models

5. Excellent software is available

Linear programming is formulated using the general problem statement. How do we satisfy the modelling limitations?

Objective function

Equality constraints

Inequality constraints

Variable Bounds

Linear programming is formulated using the general problem statement. We must learn some formulation “tricks”.

Objective function

Equality constraints

Inequality constraints

Variable Bounds

Linear programming requires inequality constraints. Remember that they require the lhs (lefthand side) to be less than or greater than or equal to the rhs (right hand side).

Corner Point: A solution for a linear program is a corner point that is located at the intersection of inequality constraints. A point is a corner point (p) if every line segment in the set (feasible region) containing p has p as an endpoint. When explaining linear programming, various references use the following terms, all having the same meaning: corner point, extreme point, and vertex.

The best corner-point solution must be the optimal value of the objective function! Thus,if the problem has one optimal solution, it must be a corner point (vertex); if it has multiple optimal solutions, at least two must be located at corner points (vertices).

Linear programming is a convex optimization problem; therefore, in linear programming a local optimum is a global optimum!

Using the Corner Point concept as the foundation for a solution method.

Let’s just calculate the objective function value at all corner points and select the best. With 20 variables and ten constraints, 185,000 corner points exist! And, that is a small problem.

Back to the drawing board!

Using the Corner Point concept as the foundation for a solution method.

We want to convert this general formulation to a system of linear equations - we know how to solve these!

Using the Corner Point concept as the foundation for a solution method.

We will add “slack” variables to all inequalities to convert them to equalities.

What do the slack variables measure?

Using the Corner Point concept as the foundation for a solution method.

We have the LP problem in “Standard Form”. Note that the system has more variables than equations.

Using the Corner Point concept as the foundation for a solution algorithm - the Simplex Tableau.

The Simplex method moves for among feasible corner points to find the best. But, it requires an initial feasible corner point. The “Big-M” method adds artificial variables.

Po = c1x1 + …. + cnxn the original objective

Pa = c1x1 + …. + cnxn + Mxa1 + ….+Mxam the modified objective

The artificial variables get us started, but they must not appear in the optimal solution.

The artificial variables give us an initial solution in canonical form.

Using the Corner Point concept as the foundation for a solution method - the Simplex Algorithm.

Start here

1. Are we optimal; if yes, we are done

2. If not optimum, move along an edge to an adjacent feasible corner point. Select the corner point that has the largest rate of change of the objective.

Using the Corner Point concept as the foundation for a solution method - the Simplex Algorithm.

Start here

1. Are we optimal;

2. Move along an edge to an adjacent corner point.

3. This adds one variable to the basis and removes another variable from the basis.

Iteration 3

Iteration 2

LINEAR PROGRAMMING

Using the Corner Point concept as the foundation for a solution method - the Simplex Algorithm.

The LP Simplex Algorithm Extensions

- Variables can be positive, zero, or negative
- x = x’ – x’’ with x’ 0 and x’’ 0
- Variables can have upper bounds
- (We can do this without adding constraints)
- Efficient Calculations
- - Do not recalculate a big matrix every time
- - Restart from a previous solution

The LP Simplex Algorithm Weird Events

- NO FEASIBLE SOLUTION
- Diagnosis -
- Remedial Action -

The LP Simplex Algorithm Weird Events

- NO FEASIBLE SOLUTION
- Diagnosis - At least one artificial variable in optimal basis - software reports this as infeasible.
- Remedial Action - reformulate, if appropriate

The LP Simplex Algorithm Weird Events

- UNBOUNDED SOLUTION
- Diagnosis -
- Remedial Action -

The LP Simplex Algorithm Weird Events

- UNBOUNDED SOLUTION
- Diagnosis - The distance to the next corner point is infinity - software will report.
- Remedial Action - Reformulate, which is always possible - realistic variables never go to

The LP Simplex Algorithm Weird Events

- ALTERNATIVE OPTIMA
- Diagnosis -
- Remedial Action -

The LP Simplex Algorithm Weird Events

- ALTERNATIVE OPTIMA
- Diagnosis 1 - The basis can change with no change in objective.
- One or more non-basic variables has a zero marginal cost.
- Software does not report warning

The LP Simplex Algorithm Weird Events

- ALTERNATIVE OPTIMA
- Diagnosis 2 - One or more active constraint rhs can be changed without affecting the objective.
- An active constraint has a zero marginal value.
- Software does not report warning

Constraint rhs can be changed with no change to OBJ

The LP Simplex Algorithm Weird Events

- ALTERNATIVE OPTIMA
- Remedial action- We have found the best value of the objective function!
- We likely prefer one of the different sets of x values. We would like to know all solutions and select the “best”, using other criteria.

The LP Simplex Algorithm Weird Events

- CONSTRAINT
- DEGENERACY
- Diagnosis -
- Remedial Action -

The LP Simplex Algorithm Weird Events

- CONSTRAINT
- DEGENERACY
- Diagnosis - The marginal value of an active constraint rhs depends upon the direction of change.
- The range of the (non-zero) marginal value is 0.0 in one direction.

The LP Simplex Algorithm Weird Events

- CONSTRAINT
- DEGENERACY
- Remedial action- The solution is correct.
- The sensitivity information is not reliable!
- If you need sensitivity information, introduce the change (rhs, cost, etc.) and rerun the optimization.

The LP Simplex Algorithm - Great News - Sensitivity

We always want to answer “what if” questions. Much valuable information is available with the optimal solution!

1. Sensitivity = z/ with all x = constant

2. Sensitivity = z/ with all basic variables, xB, allowed to change so the results represent an optimal solution, i.e., corner point, for the modified problem.

The sensitivity is reported using the optimal basis and evaluates the range and effects of parameter changes within the basis, i.e., without requiring a basis change.

The LP Simplex Algorithm - Great News - Sensitivity

Change to rhs, e.g., max production rate, min flow, etc.

One-at-a-time changes to a rhs parameter

0

Resulting change in basic variables; non-basic are not changed if bi small

The LP Simplex Algorithm - Great News - Sensitivity

One-at-a-time changes to a rhs parameter

How much can we change constraint 1 without changing the basis?

The LP Simplex Algorithm - Great News - Sensitivity

One-at-a-time changes to a rhs parameter - Software products report the value and range of the sensitivity for every constraint! Outside of the range, the basis changes; what do we do?

Remember to check for constraint degeneracy!

Check the units!!!!!

The LP Simplex Algorithm - Great News - Sensitivity

Sensitivity to one change in cost - basic variable

Formulating Models for LPs

Formulating Models for LPs

Straightforward model represents effects of dominant variables and ignores all other effects.

Formulating Models for LPs

Base-Delta models include the effects of secondary variables over a limited range

Fi = (F) + (T-T0) + (P-P0)

is the “yield” at T0 and P0

Formulating Models for LPs

Disjunctive Programming provides alternative models for the same system.

Formulating Models for LPs

Separable Programming extends the range over which the linear models are accurate by using piecewise linear models.

Formulating Models for LPs

Goal Programming provides a method for “getting close” to desired conditions that are not possible. This is a penalty function.

We want to achieve three properties and the total flow by mixing two flows.

What do we do?

Formulating Models for LPs

Goal Programming

Extra variables are added to ensure “math” feasibility; they measure the amount of infeasibility related to the original problem.

They are penalized to minimize infeasibility.

Formulating Models for LPs

Linearizing Transformations - Some properties can be transformed to behave linearly.

Formulating Models for LPs

Flow-property relationships- When properties combine linearly and the properties do not depend on the variables (flows), a special formulation can be used.

Non-linear

Linear

Formulating Models for LPs

Mini-Max Problem- Often, we have a range of outcomes possible from alternative systems.

These could systems could be

- Alternative decisions (e.g., investments)

- possible outcomes from an uncertain system

Here, we would like to consider every system and minimize the worst case, i.e., the maximum of the minimums.

Formulating Models for LPs

Mini-Max Problem- Minimize the worst case, i.e., the maximum of the minimums from all systems.

fi = a set of linear equations and inequalities with the parameters i yielding an objective function fi

i = The parameters associated with outcome I

x = The optimization variables, which are used in every model I

- The LP solution method offers the following:
- whether the feasible region is unbounded (giving an objective value of ).
- whether feasible region does not exist (no solution).
- the unique optimum value of the objective at a corner point, or the existence of multiple optima.

- The LP solution method offers the following:
- excellent sensitivity information regarding the effects of changes in selected parameters.
- Excellent software
- Extremely fast solutions

We will seek to formulate an optimization problem as an LP, when the method provides adequate accuracy for the problem being solved.

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