Steps to Linear Programming

On the quiz and final you will be asked to formulate a linear programming problem. Here is Professor Burgiel's interpretation of the problem formulation instructions on pages 248-250 of the textbook.
  1. Understand the problem.
    The goal of a linear programming problems is to find a way to get the most, or least, of some quantity -- often profit or expenses. This quantity is called your objective.

    The answer should depend on how much of some decision variables you choose. Your options for how much will be limited by constraints stated in the problem.

  2. Describe the objective.
    What are you trying to optimize? Are you trying to minimize costs? Maximize production quantities?

    You may have constraints like "you can't spend more than $1000" or "you mus't ship at least 50 tons of product C". These are limits but don't necessarily reflect your employer's top priorities; don't mistake these for suggestions that you minimize costs or maximize production.

  3. Define the decision variables.
    The answer to a linear programming problem is always "how much" of some things. What are those things? Choose variables to represent how much of each of those things. For example:
    L = number of leadership training programs offered
    P = number of problem solving programs offered.

  4. Write the objective function.
    Use the variables you just chose to write down an algebraic expression that describes the amount you're trying to minimize. Do not use an = sign. Do not use < or > signs.

  5. Describe the constraints.
    What are the limits on "how much" your decision variables can be? Look for words like "at least", "no more than", "two thirds of of", "we must fill orders for", etc.

  6. Write the constraints in terms of the decision variables.
    For each constraint such as "at least $500" or "no more than 29" write an inequality using the decision variables. For example: 2.5 L > 500
    P + L < 29

  7. Add the nonnegativity constraints.
    Don't forget to include non-negativity constraints like P >= 0. These are worth a quick two points on the quiz.

  8. Write it up pretty.
    Your formulation should look something like:

    Maximize 4U + 3H
    Subject to:U < 70
    H < 60
    2U- H < 100
    U > 0
    H > 0