Optimization problems, in their most general form, involve finding the values of *variables* which maximize the value of some *objective function*, subject to a set of *constraints*.

Problems are classed as either *continuous* or *discrete*, according to the type of variables they contain. The former are often solved by calculus; the latter often require enumeration of possible cases.

Many optimization problems include no random component: find the shortest path through a network; find the smallest set of coins necessary to make any needed amount of change (the answer is 10, incidentally: 3 quarters, 2 dimes, 1 nickel, and 4 pennies), or find the set of set of U.S. states with the smallest population that still add up to 270 electoral votes.

When a problem includes a random component, there may be more than one reasonable choice of objective function: maximize a mean or median value; minimize variability, while holding the mean fixed; ensure that some threshold will be exceeded only a fixed percentage of the time. An airline that wishes to overbook a particular flight, for instance, might seek to minimize (expected number of empty seats) x (airfare) + (expected number of bumped passengers) x (cost of a hotel room).

Large or small, statistical or deterministic, we can help you formulate the optimization problem your business needs to solve, and then help you solve it. Contact us for a free consultation, or read about problems we’ve solved before in these areas: