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Decision Making via Linear Programming: A simple introduction

Decision Making via Linear Programming: A simple introduction. Fred Phillips fred.phillips@stonybrook.edu. 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

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Decision Making via Linear Programming: A simple introduction

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  1. Decision Making via Linear Programming:A simple introduction Fred Phillips fred.phillips@stonybrook.edu

  2. 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.

  3. 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.

  4. 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.

  5. 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

  6. 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.

  7. The best (“optimal”) solution is always at a vertex! (Graph and solution generated by http://www.zweigmedia.com /RealWorld/LPGrapher/lpg.html . The site uses x for tables and y for chairs.)

  8. Another site will solve problems with many variables. • http://www.zweigmedia.com/RealWorld/simplex.html

  9. Now let’s see how to solve these using Excel • You need the “Solver” plug-in. • Watch http://www.youtube.com/watch?v=0KPHmyyghew

  10. 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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