Alternative Optimal Solutions and Combinatorial Risk

Optimization in practice is usually not just about setting up a well-behaved model with an objective function and finding 'the optimal answer'. While that is an interesting exercise, the real 'value add' comes from the subsequent process of recognizing the business reality that a practical decision problem typically has many answers. Consequently, analyzing alternative optimal solutions in a way that is useful to the client is quite important. In other words, practical optimization is more often about analyzing feasible alternatives that initially appear to be equal to us, but in reality have vastly different qualities from our client's point of view (note: returning 'infeasible' is not really an option, and showing why our model returned 'infeasible' is only slightly more useful).  As we initiate a dialogue with our client to understand these differences and the context in which some alternatives are better than the others, we can see our lab model gradually transform into a useful business analytics tool.

Cutting across industries, I have not yet come across a single optimization problem deployed in production that does not have multiple objectives. Every seller loves to maximize profit, but not at the cost of losing out on volume or sales dollars in the process. Over time, the number and priority of such considerations change. For example, in the airline industry,  it is not uncommon for large-scale crew schedule planning problems to have hundreds of different goals and priorities. The richer the solution space, the more the number of goals it seems. In fact, optimizing just a single measure is risky because such a gain ("extreme point") almost always comes at the expense of other metrics that haven't been included within the analysis yet. Which leads us to:

Combinatorial Risk
This hidden problem of 'risk', even within a deterministic modeling context, is exacerbated in combinatorial (or global) optimization situations. Here, our model analyzes multiple inter-linked decisions that can produce solutions that radical differ from current practice and looks great numerically, but in reality, can potentially hurt our client's business if actually used. 'Locally optimal' does not always and automatically mean 'inefficient'. Like globalization, combinatorial or global optimization based holistic decision making can and does bring in more efficiency and profitability compared to that obtained by combining multiple locally optimal decisions when things go as per forecast. On the other hand, if the alternative (near-) optimal scenarios along with their corresponding risk of failure are not well mapped out and thought through, the resultant machine-generated combinatorial solution can cascade the risk of a bad decision through the system.

Part-3 here.