# Many Optimal Solutions

## Math Can’t Do Everything

When I was in college, working on a research project for a class, I had an awakening of sorts.  I was shocked at how close a mathematical representation of a business problem — one that I worked through on a chalk board one evening — came to mimicking the behavior of the operation I was studying. With some more math, I could optimize the resourcing of the business and greatly improve the business’ performance. Math that I made up could actually perfect a real-world operation and save them a lot of money. That awakening changed my course of study and led me on a 30-year career.

I became a real believer in mathematical modeling.  But for those of you who have read this column before (Hi, mom!), you might be shocked to hear me say this:  math and modeling can’t always solve all problems.

What I mean by this is that there is often nuance associated with many business problems that cannot be captured by math or algorithms.  And this nuance is best left to humans and intuition.

## Optimality (Again)

Mathematical optimality has a very specific definition: to say something is “mathematically optimal” means it is the absolute, very best.  It cannot be improved upon, and you can prove, with math, that an optimal solution is the very best.

For example, if I am putting together agent schedules and I use Integer Programming, a mathematical optimization algorithm, I can very likely find a solution that is mathematically optimal for my objective (say, minimize staff hours), given my constraints (say, hit service standards and don’t violate work rules).  My solution will hit my goals and not violate any of my business rules, and be the set of schedules that has the absolute fewest staff hours.

Optimality gives a business many advantages in addition to the obvious one: lowest cost or highest profit.  An optimal solution is a benchmark against other possible solutions and a starting point to understanding the costs of additional business rules. For example, let’s say that human resources wanted to require all agents to be trained two hours a week, every month, on some sort of procedure or practice.  If we had optimal schedules, we could add this training constraint to our math model and see the actual dollar-cost of layering onto our schedules this single work rule.

Many years ago, we were working with a business that had a rule to simplify the hiring and training process: all new hires must start on the first day of the month.  Because we used optimization models to determine their hiring plans, we could, in minutes, compare the unconstrained optimal staff plan to one with hiring plans constrained to start on the first of the month. We could run these two models side by side to determine the cost of this specific hiring policy.  If this policy cost \$200,000,

we would be able to make a value judgement on its appropriateness. (Note: The company ditched the policy.)

## Alternative Optimal Solutions

But there is a problem with finding optimal solutions in that there is sometimes more than one optimal solution for a specific problem.  In these situations, the business problem is said to have Alternative Optimal Solutions, meaning there may be two or a hundred other solutions with the same objective value that meets all constraints. This can be an embarrassment of riches of sorts, in that it can be difficult to choose between several solutions.  For instance, with agent scheduling, there are many nuances of scheduling that are not captured through a single objective metric. Week over week, it might be good for agents to work similar schedules, and two equally optimal costing schedule sets could possibly differ on schedule consistency. Week over week consistency could be a tie breaker.

If your business model returns many competing optimal solutions, the question is whether decision makers care which solution (e.g., which set of schedules) they choose.  If one optimal solution is as good as the next, then the analyst can choose any one of them and move on. If there are qualities of the solution that makes one better than the next, on the margin, then the goal becomes to see whether there is a marginally better choice between these already great alternatives.

## Ways to Choose Between Alternative Optimal Solutions

The most basic way to choose between one optimal solution and the next may be to simply look at them, or to present them to the decision makers. If the number of solutions is small, or metrics about each solution can be graphed or tabulated, then it may be a simple choice, and an expert can make the call.

Similarly, different solutions may be judged by the value of specific constraints.  For example, if one set of schedules has more weekends off, even though all solution sets have more than the bare minimum weekends required, that solution with the most weekends off, may be the obvious best choice of the optimal alternatives.

Often, the idea of secondary objectives, usually relegated to a constraint, can be elevated to part of the objective when evaluating solutions.  Meaning we might be able to construct our model with two objectives, say minimize the cost of our schedules AND maximize the number of weekends off our agents get.  In this way, our algorithms help us winnow down the number of possible competing solutions. It will solve for the best of both objectives, which should minimize the number of competing optimal solutions.

Are there aspects of the solution set that measure risk to the business?  For instance, there may be time periods where any understaffing will lead to dramatic drops in performance (when economies-of-scale and volumes are low).  Schedule sets that leave a small overstaffing at these critical time periods can help us avoid a service catastrophe if too many agents call in sick. Are there solutions that simply have too many undesired shifts?  That solution may be a risk to agent attrition. Even though solutions may be equally cost-optimal, these other considerations — even if they are not explicit constraints — may be the tie breaker.

Optimality is cool, it saves us money, maximizes service consistency, and serves as a benchmark on the cost of our business rules.  Having more than one optimal solution to our business problems affords us the opportunity to improve upon solutions that are near perfect.

Ric Kosiba, Ph.D. is a charter member of SWPP and vice president of Genesys’ Workforce Systems. He can be reached at Ric.Kosiba@Genesys.com or (410) 224-9883.