Pitall 1: Problems in linear programming involving "whole things" like filing cabinets are a bit tricky to adapt. Usually this means checking that the coordinates of the intersection point of the main constraint equations are whole numbers.
Pitall 2: You must check that the final shaded area is convex (no star points). Otherwise there might not be a unique answer to your problem (more than one answer or no answer).
Luckily, we can easily check these with GeoGebra! Try it with the interactivities in the example below! Google doc Worksheet and Survey
Example with YouTube Mathcast and GeoGebra InterActivities:
Let's look again at Example 1: where the answer is numbers of filing cabinets. This means x and y can only be whole numbers.
Example 1:We need to put file cabinets in our office. File cabinet A costs 10 Euros, has a footprint of 0.6m2, and a storage capacity of 0.4m3. File cabinet B costs 20 Euros, has a footprint of 0.8m2, and a capacity of 0.6m3. We can spend a maximum of 140 Euros and we have a maximum floor space of 7.2 m2. How many cabinets of type A and type B should we purchase to maximize our storage capacity? Our constraints were: x ≥0 andy ≥0,10x+20y≤140 and 0.6x+0.8y≤7.2 and our solution was the whole numbers x=8 and y=3.
Pitall:Changing the numbers in the constraints 10x+20y≤140 and 0.6x+0.8y≤7.2 ...
We must check that the area is convex and that points A, E and D are gridpoints (or only E if we are in a hurry).
YouTube Mathcast
GeoGebra InterActivity 1: Just changing the totals (2 numbers)
GeoGebra InterActivity 2: Changing all the numbers
Pleasures for Educators
Pleasure 1: You can always change the numbers in the objective function as desired!
Pleasure 2: You can make GeoGebra worksheets with sliders to change the parameters of the problem. Then you or your students can make up good problems and have interesting discussions about what is happening. Watch the YouTube mathcast and then DIY in the GeoGebra interactivity below.
DIY Example with YouTube Mathcast and GeoGebra InterActivity:
YouTube Mathcast
DIY InterActivity with GeoGebra Directions for InterActivity
Remember - only gridpoints in the purple-shaded area satisfy our constraints.
• First practice with the interactivities in the example above. Our goal is to make these interactivities.
• The numbers in the objective function do NOT matter! Only the intersection points of the constraint equations.
• Quick: Make 2 sliders for the
== total cost ct from 0 to 200 increment 10 and == total footprint fp from.from 0 to 10 increment 0.1
• In the Algebra View, double-click on line a and change x+2y=14 to10*x+20y=ct and then on line c and change 7.2 to fp.
• Now slide sliders for ct and fp until E is a gridpoint. • More changes: Make 4 more sliders for the cost of x and cost of y: ctx and cty and for the footprints of x and y: ftx and fty.
• In the Algebra View, double-click on line a and change 10x toctx*x and 20y tocty*y and then on line c and change 7.2 to fp.
• Now slide all sliders until A, E and D are gridpoints.
How and Why
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