Last time I wrote about multi-objective programming. Today, I would like to follow up with a few more details and more generally a discussion of the practical use of it in humanitarian logistics.
In the paper, Rath and Gutjahr use a variant of the bounded objective method called the adaptive epsilon constraint method. Unfortunately, for anything beyond a bi-objective formulation, this method becomes quite nasty and complicated. Suffice it to say, where the epsilon constraint method chooses predetermined bounds on the constrained objective, the AECM uses knowledge of the objective space from previous solutions (We are solving for the Pareto frontier) to create better constraint values for future searches. Hence, it adapts its constraints based on new data and is able to significantly decrease the total search time over the more naïve ECM.
Using these constraints, Rath and Gutjahr then seek to find the optimal objective value by narrowing an upper and lower bounds on the solution until they achieve equality and hence optimality. From here, the primary contribution of the paper is in developing heuristics to aid in the computation of these bounds. In each iteration, they add new constraints to the upper bound to make a better, but more computationally intense relaxation. Further, they use basic heuristics to find a feasible lower bound.
All this effort is with the goal of finding a faster, if not quite as accurate solution. So the practical question then is whether this still provides a useful solution and moreover, whether it is even necessary.
I think it would be hard to argue that even an approximate answer is not reasonably acceptable where an exact answer would have been useful. This comes simply from the assumption of perfect knowledge. Of all the assumptions, this is probably the worst. As Sam pointed out in class, do we even know that all of the network connections (roads) are still in place? Even if this is accounted for, isn’t it possible that they may be restored over time making our solution suboptimal? Further, our knowledge of demand is limited to what information we have gathered so far- information that is itself inexact.
So the ability to solve the exact solution is of little use. We are merely looking for guidance and for that a heuristic is just fine. In fact, I would argue that in a real aid deployment, a heuristic is perhaps more useful, being faster and more understandable to non-practitioners while producing what is arguably an equally useful solution.
The other question is whether this complicated model is even worth solving. As was discussed last week, finding the Pareto frontier is not an insignificant problem and perhaps best avoided if we are reasonably able. Here, I think is the crux of the matter. At least for the problem that they are considering, I question the usefulness of the multi-objective criteria and the necessity to minimize cost and maximize service at the same time. Either the organization is working under a given budget or it can be changed based on increased donations. In either case, it makes as much sense to solve the Pareto optimal solution for the current budget, and resolve later if necessary. The lack of perfect information would necessitate these resolves in any case.
The only real benefit in knowing the entire Pareto Frontier is to get a good idea for the trade-offs. However, even this is of so much value given our lack of knowledge and the lag time between when an organization makes a decision to contribute resources and when those resources are actually deployed. In that time not only the knowledge, but the actual situation on the ground and hence the Pareto frontier would likely have changed.
So I see the benefit of models like this being more prominent in domestic disasters where we have better knowledge, faster response, and more control over the budget of the operation.