Solve a Minimization Problem Using Linear Programming

By Mary Jane Sterling

Finite math teaches you how to use basic mathematic processes to solve problems in business and finance. For example, you can use linear programming to stay within a budget.

When you’re dealing with money, you want a maximum value if you’re receiving cash. But if you’re on a tight budget and have to watch those pennies, then you’re concerned with minimizing your expenses. The following is a minimization problem dealing with saving money on supplements.

You’re on a special diet and know that your daily requirement of five nutrients is 60 milligrams of vitamin C, 1,000 milligrams of calcium, 18 milligrams of iron, 20 milligrams of niacin, and 360 milligrams of magnesium. You have two supplements to choose from: Vega Vita and Happy Health. Vega Vita costs 20 cents per tablet, and Happy Health costs 30 cents per tablet. Vega Vita contains 20 milligrams of vitamin C, 500 milligrams of calcium, 9 milligrams of iron, 2 milligrams of niacin, and 60 milligrams of magnesium. Happy Health contains 30 milligrams of vitamin C, 250 milligrams of calcium, 2 milligrams of iron, 10 milligrams of niacin, and 90 milligrams of magnesium. How many of each tablet should you take each day to meet your minimum requirements while spending the least amount of money?

A good way to organize this is to make a chart or table listing the requirements, costs, and amount of nutrients in each tablet.

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With all the information organized into the table, it’s time to solve for the number of tablets that will minimize your cost using linear programming.

  1. Choose variables to represent the quantities involved.
    The quantities here are the number of tablets. Let a tablet of Vega Vita be represented by v and a tablet of Happy Health be represented by h.
  2. Write an expression for the objective function using the variables.
    The goal is to spend the smallest amount of money necessary (so you want the minimum). Vega Vita costs 20 cents per tablet, and Happy Health costs 30 cents per tablet.
    Minimize: $0.20v + $0.30h
  3. Write constraints in terms of inequalities using the variables.
    The constraints are all in terms of meeting the daily requirements. Each requirement has at least in its form, so you use the greater than or equal to symbol in your equations.
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    It makes no sense to have a negative number of tablets, so you add the two constraints
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    This keeps the graph in the first quadrant.
  4. Graph the feasible region using the constraint statements.
    The graph will be completely in the first quadrant and will be boundless; the solution area extends forever upward and to the right, as shown in the figure. You’re more concerned with the points closer to the axes, though.

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    Graphing the combinations of nutrients in the two tablets.
  5. Identify the corner points of the feasible region.
    There are four corner points. Two of the corner points are where lines corresponding to the constraints intersect, and the other two lie on the axes.

    • Intersection of 9v + 2h = 18 and 60v + 90h = 360:
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    • Intersection of 60v + 90h = 360 and 2v + 10h = 20:
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    • Intercept on vertical axis: (0, 9)
    • Intercept on horizontal axis: (10, 0)
      The following figure shows the feasible with the corner points labeled.

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      Some of the intersections involve fractions.
  6. Find and compare the values at the corner points to determine the solution.
    You see that two of the intersections contain fractions of tablets. Yes, you can buy one of those handy-dandy pill cutters, but these are pretty strange fractions. It’s best to just round the numbers up to a whole number (a whole tablet). So the corner points you’ll consider are (2, 4), (5, 2), (0, 9), and (10, 0).
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    It appears that you have two choices. You can either take two Vega Vita and four Happy Health each day or five Vega Vita and two Happy Health each day. They cost the same amount. Of course, the first choice is fewer pills, but the Vega Vita may be easier to swallow. It’s up to you.