LP: Summary thus far

• Requirements

• Graphical solutions

• Excel

• Sensitivity Analysis

Requirements of a Linear Programming Problem

• Objective Function: LP problems seek to maximize or minimize some quantity .

• Constraints: restrictions that limit the degree to which we can pursue our objective.

• Alternative courses of action.

• Linear equations or inequalities.

Graphical Solutions for LP

• Must be no more than 2 decision variables.

• Plot the limits of the constraints.

• Find the feasible region.

• Determine the optimized solution (using corner point method or iso-profit line method).

Graphical Solutions for LP: Corner Point Method

• Theory behind LP states that the optimal solution to any problem will lie at a corner point, or extreme point of the feasible region.

• Find result for each corner of the feasible region, pick the one that best satisfies the goal (within reason).

Graphical Solutions for LP: Iso-Profit Line Method

• Find the slope of the optimization equation:

• For a given profit level, there will be a (infinite) # combinations of X1 & X2, described as a line.

• An iso-profit line that is farther from the origin (0,0) will have a greater profit level.

• Find the intersection of the iso-profit slope with the farthest point of the feasible region.

Spreadsheet Solutions for LP: MS Excel

• Formulate the optimization equation and the constraints.

• Build spreadsheet with formulas showing the relationships between the various elements.

• Using MS Excel Solver, define the cell to optimize, and the constraints on it.

Using MS Excel for LP: Answer Report

• Binding & non-binding constraints

• Slack

Using MS Excel for LP: Sensitivity Analysis

• Reduced Cost: how much a coefficient would need to change to affect the optimal solution. (Ignore sign)

• Allowable increase & decrease: how much a coefficient can change without changing the optimal combination for the solution (but it can change the actual optimized value)

Using MS Excel for LP: Sensitivity Analysis

• Shadow Price: How much, per unit, the constraint affects the optimized value

• Allowable increase & decrease: In what range does the shadow price hold true.

Using MS Excel for LP: Sensitivity Analysis

• All data in the sensitivity analysis are based on changing ONE THING AT A TIME.

Using MS Excel for LP: Troubleshooting

• Infeasibility: two possibilities: – An answer cannot be found that satisfies all

constraints.– Part of the LP was input incorrectly. (More likely)

• Unboundedness: – An answer has no limit to its goal.– Part of the LP was input incorrectly. (See above)

Using MS Excel for LP: Example

• James Beerd and his cakes. GO TO

Using MS Excel for LP: Group Exercise

• Fill in the formulas• Detail the constraints, etc.• GO TO SPREADSHEET

Using MS Excel for LP: Homework

• pg. 55 #5: solve w/ MS Excel, discuss answer report & sensitivity report.

• pg. 59 #14, part A: solve w/ MS Excel, discuss answer report & sensitivity analysis.

Turn in printout of solved spreadsheet, and email or hand in on diskette the Excel files by BEGINNING OF CLASS Thursday, Sept. 10