So far we have focused on the algorithmic details of solving SCAP problems. Although these details are interesting from an academic perspective, the true motivation of this work is to aid policy makers in making decisions about disaster management. In this section we discuss several key applications of this algorithm that can provide decision support to policy makers and were directly requested by policy makers of the U.S. government.
2.6.1
Strategic Budget Decisions
Every year policy makers must decide how much money should be used for disaster preparedness and recovery operations. Without rigorous scientific study, the potential consequences and risks are
500000 1000000 1500000 0 20 40 60 80 100
Expected Demand Met
Budget ($)
Expected Demand Met (%)
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0% giving 25% giving 50% giving 75% giving 500000 1000000 1500000 0 20 40 60 80 100
Expected Last Delivery Time
Budget ($) Expected Time ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0% giving 25% giving 50% giving 75% giving
Figure 2.13: The Benefits of Corporate Giving with a Central Location.
unclear and as a result policy makers must resort to the philosophy,“the more money allocated to preparedness the better”. However, budgets are limited and decisions on preparedness allocation become increasingly difficult when funds must be taken from other public services such as infrastruc- ture maintenance and education programs. Fortunately, this SCAP algorithm can aid policy makers in making budgetary dissections by quantifying the quality of recovery operations under different budgets. Figure 2.12 demonstrates how to provide budgetary decision support to policy makers. In both graphs, the budget is varied on the x-axis and performance metrics are quantified on the y-axis. The left graph shows the percentage of demands that can be met. A policy maker can clearly see that at least a budget of about $900,000 is required if all demands are to be satisfied. The right graph shows the time of the last delivery. Once all demands can be satisfied, this graph indicates how additional budget will reduce the time of the last delivery. A policy maker can see that the delivery time can be reduced significantly up to a budget of $1,100,000, but additional funds bring few benefits. The SCAP algorithm for these kinds of decision support studies can be computed quickly and shows policy makers the quantifiable consequences of various budgetary decisions.
2.6.2
Effects of Corporate Giving
After a catastrophic disaster, corporations often make donations to aid federal organizations in relief efforts, so that a study of corporate giving is of particular interest to policy makers. Fortunately, the SCAP model can easily support analysis of corporate giving simply by modifying the problem input. Once these modifications are made, the SCAP algorithm can be used to understand how corporate giving affects disaster planning and response.
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Expected Demand Met
Budget ($)
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Expected Last Delivery Time
Budget ($) Expected Time ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0% giving 25% giving 50% giving 75% giving
Figure 2.14: The Benefits of Corporate Giving with a Distant Location.
We consider donations of 25%, 50%, and 75% of the total demand, and compare them with no donations (i.e., 0% of the demand). Figure 2.13 presents the behavior when the donation site is near the center of demand (e.g., a large retail store location). The left graph shows the demand met as the budget increases and the effect of corporate giving is what you would expect: it enables more demands to be met when the budget is too small. The right graph depicts the last delivery time as the budget increases. These results are quite interesting. Initially, for low budgets, the maximum delivery time increases as more of the demand is met; as the budget increases, the last delivery time decreases substantially because the giving location is placed close to the demand locations.
Figure 2.14 presents the behavior when the donation site is far away from the demands (e.g., a regional supply warehouse). The results are identical for the demand (left graph). However, the results for the latest delivery times are fundamentally different. Until the demand is met, the latest delivery times increase, as is normal. However, the increase is significantly larger than for a central location. When the budget is large, there is no benefit to corporate giving, even though it had significant benefits from a central location. It is only for budgets that meet all the demand but are not very large that corporate giving reduces the latest delivery time. This analysis quantifies the benefits of corporate giving in monetary terms and may help decision makers in influencing corporate giving.
2.6.3
Robust Optimization
So far the SCAP algorithm has considered minimization of the expected outcome of the stochastic set of disasters. Specifically, the Stochastic Storage Model considers the objective function
Minimize: X s∈S PsUnsatisfiedDemandss, X s∈S PsTravelTimes, StorageCosts
This is a natural formulation but policy makers may be interested in minimizing the worst-case outcome instead of the average-case outcome, as with the delivery-time objective in the routing aspect of the SCAP problem. This alternate objective is known as robust optimization. The objective above can be formulated as a robust optimization problem as:
Minimize: max
s∈S UnsatisfiedDemandss, maxs∈S TravelTimes, StorageCosts
The SSM is easily adapted for this objective in two steps. First we introduce auxiliary variables Md,Mt and add the constraints
Md ≥ UnsatisfiedDemandss ∀s
Mt≥ TravelTimes ∀s
then the objective becomes
Minimize: Md, Mt, StorageCosts
Together these modifications allow the SSM to solve a robust optimization variant of the SCAP problem. Both objective formulations are worth consideration and we allow the policy makers to choose the kind of objective formulation they prefer.
2.6.4
Recovery Practice in the Field
Due to the complexity of disaster recovery problems, it is often impossible to calculate a globally optimal solution. However, there is often some measure of uncertainty in the problem specification so a global optimal solution is not crucial in this case. In this context we shift our goal from finding an optimal solution to simply improving the best current field practice. In the rest of this section we discuss our approximation of the best field practices for the SCAP problem.
The SCAP problem can be viewed simply as storage decisions (first-stage variables) and routing decisions (second-stage variables). In developing an algorithm of best field practices, we replace each of these decisions with an algorithm similar to the ones currently used by policy makers. First we consider the storage decisions. Despite considerable effort, we have found no clear documentation on how policy makers make storage decisions for disaster preparation. In lieu of this information, we approximate the storage decisions by giving our best-practice algorithm the same storage decisions as the SCAP algorithm. This is highly optimistic in that, since that stage is solved to optimality, policy makers could not make better allocation decisions. Second, we consider the routing decisions. In the context of disaster recovery, truck drivers are assigned to specific trucks and possibly specific
areas and work more or less independently to deliver as much commodity as possible. We can simulate this behavior using an agent-based algorithm where each driver is an agent in the system and greedily makes pickups and deliveries as fast as possible. The agent decision process is as follows:
Greedy-Truck-Agent (GTA)()
1 while there exists some commodity to be picked up and demands to be met 2 do if I have some commodity
3 then drop it off at the nearest demand location
4 else pick up some commodity from the nearest warehouse 5 goto final destination
While this agent-based algorithm roughly approximates current relief delivery procedures, it is in fact optimistic because it assumes all the drivers have global and up-to-date information about the state of the stored commodities and demand locations. This is not an unreasonable as the recovery efforts can be orchestrated from a central office, but in practice perfect coordination of their efforts is unlikely. Together these algorithms permit a solution to the SCAP problem that is an optimistic approximation of current practices in the field. We call this approximation the GTA algorithm and it will serve as the basis of comparison for the evaluation of the SCAP algorithm in Section 2.7.