Calculating a Linear Program

The process for calculating a linear program is fairly straightforward:  Specify the constraints as inequalities (formulas), and solve them simultaneously (substitute one into the other to find their common solution, or—alternatively—plot the equations on a common graph to find their intersection).  The trick, of course is to convert the information at hand into formulas that will be useful.

Consider an example:

1.      Suppose a city department has 54 employees—36 office staff and 18 field workers.  The department has 2 functions, planning and maintenance, each of which requires some effort from both groups.  The labor requirements are as follows:

 Maintenance (x) Planning (y) Office 1 2 Field 1 .5

2.      And, since production cannot be negative,

a.       Maintenance > 0

b.      Planning > 0

3.      Ideally, how much should the department provide of each service?

Therefore,

·        From 1, office labor requirements are                            x + 2y > 36

·        From 1, field labor requirements are                              x + .5y > 18

To solve a family of equations, you need as many equations as you have unknowns.  Converting the first two to equalities, and substituting:

x + 2y = 36, so        y = 18 - .5x

x + .5y = 18, so       x + .5 (18 - .5x) = 18     so     x + 9 - .25x = 18

so   x =  .75 (9)  =  12

and   12 + 2y = 36, so      y = (36 – 12) / 2 = 12

Returning these values into the formulas, we find that the ideal distribution of office labor will be 12 in maintenance and 24 in planning; field labor will be 12 in maintenance and 6 in planning.

The same result could have been obtained graphically: