The nature of simple LP• One, single decision criterion
– Either maximize or minimize something– Usually, profit or cost
• No consideration of probability• Usually only one best answer
• Elaborations of LP allow for probabilities, multiple criteria, integer-only solutions, and more.– We’ll look at some of these elaborations later in the
semester.
Introducing LP via example
• Suppose we manufacture furniture.
• We must decide how many tables and how many chairs to make this week.
• We assume we can sell all that we make.– At a profit of $4/table and $3/chair.
Obviously we can’t make an infinite number of tables or chairs.
• Each table requires 4 hours processing in machine (or ‘department’) A...– ... and 2 hours in machine (dept.) B.
• Each chair requires 2 hours in dept. A...– ... and 4 hours in B.
• We have capacity constraints:– A total of 60 hours/week in dept. A, and– A total of 48 hours/week in dept. B.
Let’s write all this in algebraic form
• Let x1 be how many tables we make.• Let x2 be how many chairs we make.• We want to maximize
4x1 + 3x2
• Subject to our capacity constraints:4x1 + 2x2 < 60
2x1 + 4x2 < 48
• We also require that x1 > 0 and x2 > 0
Because your professor cleverly drew an example with only two
variables x1 and x2,• We can solve the
exercise using a 2D graph.
• This is not possible for realistic problems which may have several thousand variables.
The best (“optimal”) solution is always at a vertex!
(12,6) 4x+2y = 60; 2x+4y = 48 66 Maximum
(15,0) 4x+2y = 60; y = 0 60
(0,12) 2x+4y = 48; x = 0 36
(0,0) x = 0; y = 0 0
Vertex Lines Through Vertex Value of Objective
(Graph and solution generatedby http://www.zweigmedia.com/RealWorld/LPGrapher/lpg.html .The site uses x for tables and y for chairs.)
Another site will solve problems with many variables.• http://www.zweigmedia.com/RealWorld/simplex.html
Now let’s see how to solve these using Excel
• You need the “Solver” plug-in.
• Watch http://www.youtube.com/watch?v=0KPHmyyghew
Now you try it. Set up this model, and solve it in Excel.
A customer of the Regal Corporation needs 1,000 pounds of a chemical mixture consisting of three raw materials. Cost for each of these is as follows: • X1 = $2 per pound • X2 = $3 per pound • X3 = $4 per pound
The customer requires the mixture to meet these conditions: • The mix must contain at least 200 pounds of X2. • The mix cannot contain more than 400 pounds of X1. • The mix must contain at least 100 pounds of X3.
Determine the least-cost mixture for the batch of 1,000 pounds which will satisfy the customer's requirements.
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