COMBUSTIBLES Y LUBRICANTES

Propositions 2–2 and 2–3 provide ample evidence that the simple pruning rule is flawed. But, we still desire a decision rule for pruning some new edge branches in an IE tree. In order to develop a pruning rule that can incrementally add defenses one at a time, we must first introduce some additional terminology.

Define excess supply as a source node that has more resources available than it can deliver to all destination nodes that are connected to it. A source node is connected to a destination node by one or more paths of existing edges or previously added edges. This definition allows for the possibility that the network could be connected or disconnected

into multiple subnetworks. If an entire network is connected, excess supply means that the sum of all capacities of all source nodes is strictly greater than the sum of all demands of all destination nodes. Similarly for disconnected subnetworks, excess supply means the sum of all capacities of all source nodes in a subnetwork is strictly greater than the sum of all demands from all destination nodes in that subnetwork. Without excess supply, the situation in the previous proposition could occur.

Incentive is defined to mean that the cost to transport resources over a network

from a source node to a destination node is less than the penalty cost for failing to fulfill demand at a destination node. Incentive assumes that a demand exists for a resource at a destination node. When incentive exists, the optimal solution will transport resources from source to destination because the cost of doing so is less than the penalty for not doing so. If incentive does not exist, then no reason to improve a network to transport supply to demand exists either.

Proposition 2–4: New edges must have an incentive to be built in order to be considered as defenses in an IE algorithm.

Proof of proposition 2–4. Assume to the contrary that edges did not need to have incentive to be considered in IE. For example, Figure 11 shows a supply node with more resources than demanded at nodes 2 and 3. The edge that connects nodes 1 and 2 has a cost less than the demand, so an incentive exists for resources to be transported to node 2. But, the new edge that connects nodes 2 and 3 would create a path from node 1 to node 3 that would have a traversal cost of eight units, which is more than the penalty of five units for not satisfying demand at node 3. In this bizarre case, the optimal solution would be for the resource to remain at the supply node and not travel to demand node 3, because it would cost more to transport the resource than to leave the destination unsatisfied. Optimally, edge (2, 3) would not be added to the network. The situation in Figure 11 can be avoided if the maximum cost of any acyclic path with new edges between source and destination nodes is less than the penalty at a destination node. If all edges have a unit cost, then the graph theoretic measure of “distance” between source and destination must be less than the penalty for failing to fulfill demand (Chartrand & Zhang, 2012, p. 15).▪

Figure 11. No Incentive Exists for New Edge to Demand Node

For the next proposition, consider Figure 12. Figure 12 contains three subnetworks that are each connected with existing edges. Subnetwork A only has demand nodes without any supply, so unmet demand exists. Subnetwork B has supply and demand nodes, but no excess supply exists. Subnetwork C has supply and demand nodes, and there is more supply available than demand. The interconnection edges between the subnetworks that are shown by the dashed lines between nodes 1 and 2 and 3 and 4 do not currently exist, but could be added as extra defenses. The penalty for unmet demand is significantly higher than the cost of transporting resources over the longest possible path that could interconnect any source and destination node across subnetworks. There is sufficient defense budget to add both new edges. Subnetwork A is a connected, but it penalizes the overall network objective function because of unmet demand. Subnetwork B is connected, and there is enough supply to satisfy demand, but nothing in excess. Subnetwork C is connected, and there is significant excess supply.

Figure 12. Disconnected Subnetworks with Unbuilt Connecting Edges

Proposition 2–5: Source nodes must have excess supply and new edges must have incentive in order to consider the incremental addition of new edges individually in an enumeration strategy.

Proof of proposition 2–5: Proof by counterexample. Assume to the contrary that network source nodes did not need to have excess supply and new edges did not needed incentive and consider Figure 12. Suppose new edges were considered as network defenses individually for addition. In the first round of selecting defenses incrementally, new edge (1, 2) would not be added to the network because adding it alone does not have an effect on the network. Incentive exists for resources to travel to subnetwork A over edge (1, 2), but no extra resources exist in subnetwork B to utilize the new edge (1, 2). The lack of excess supply in subnetwork B would mean that if edge (1, 2) was utilized, then penalties would be incurred in subnetwork B to satisfy unmet demand in subnetwork A. The travel cost of resources over new edge (1, 2) would make the addition of edge (1, 2) by itself a suboptimal solution. Also in the first round of selecting defenses, new edge (3, 4) would not be added to the network by itself because it would have no effect either. Excess supply exists in subnetwork C, but no incentive exists over edge (3, 4) for the extra resources to only travel to subnetwork B. All demand in subnetwork B is already satisfied. In the second round of selecting defenses incrementally, edges (1, 2) and (3, 4) again would be considered individually and neither would be added for the same reasons as in the first round. The contradiction occurs because the optimal solution would be to add both edges to the network so that the excess supply in subnetwork C could travel over both new edges to satisfy the unmet demand in subnetwork A. Adding both edges simultaneously is optimal because the high penalty cost for unmet demand creates incentive for the excess supply in network C to be able to reach the unmet demand in subnetwork A. Since the optimal solution is a contradiction, the proposition must be true. Excess supply does not exist in subnetwork B and that makes the addition of either new edge appear to be inconsequential when they are considered individually.

The result from the example in Figure 12 is that when disconnected excess supply components and disconnected excess demand components are involved in a network, an incremental approach to enumeration can only be considered when a maximum distance of one potential edge can exist to connect the components. Figure 12 proves that if there are intermediate connections between excess supply subnetworks and unmet demand

subnetworks, then an incremental approach toward adding new edge defenses will not work.

The situation in Figure 12 could be avoided in at least two different ways. First, if the entire network was connected with other existing edges, then the situation would not arise. Second, if the source nodes in subnetwork B had excess supply, then incremental addition would be possible. Then, the new edge (1, 2) would be added to the network in the first round of enumeration. If the excess supply in subnetwork B was not enough to completely satisfy the unmet demand in subnetwork A, then the second round of enumeration would add the new edge (3, 4) to the network.