Even though network modeling can present system complexity, it also provides assurances through investment planning, system design, emergency planning, source optimization, operational strategy development, improved customer service, reductions in service interruptions, reductions in responses times for information, and reductions in the source output, demand management, and pressure control (Elton &Schulte, 1994). The theory of network modeling can be applied in a wide range of areas as it provides the blueprint for the manner in which the nodes on the network can be utilized for maximum efficiency.
The shortest route technique emerges from the shortest route problem in which associated costs are minimized through the identification of the shortest route between nodes on a grid (Sargent & Stachurski, 2014). Presented in mathematics and computer science, the shortest route problem has applications in economics, operations research and transportation, robotics and artificial intelligence, and network design and routing, among numerous others while variations of the solutions available are observed through applications such as Google maps and routing packets that are transmitted via the internet (Sargent & Stachurski, 2014). A systemic solution is applied to determine the least-cost solution for the shortest route problem as it applies to large graphs. In order to determine the least-cost paths from larger graphs, J(v) denotes the minimum cost-to-go from node v, which represents the total cost from v when the most efficient route is identified (Sargent & Stachurski, 2014). Assuming that J(G) = 0, the most efficient route can be identified by starting at point A and then moving to any node from node v that provides the solution to min/w∈Fv{c(v,w)+J(w)} where Fv identifies the set of nodes attainable from v in one step and c(v, w) represents the cost associated with traveling from v to w (Sargent & Stachurski, 2014). This necessitates identifying the function J through the application of the Bellman equation through a standard algorithm consisting of J0(v)=M if v≠ destination, else J0(v)=0 with M representing a large number (Sargent & Stachurski, 2014). Next, n = 0, then Jn+1(v)=minw∈Fv{c(v,w)+Jn(w)} for all v (Sargent & Stachurski, 2014). However, if Jn+1 and Jn are not equal, apply increment n and reapply the last algorithm(Sargent & Stachurski, 2014).
The maximal-flow technique is applied in response to the maximal-flow problem which addresses the issue of increasing, or maximizing, the amount of flow of items from the point of origin to the point of destination (Taylor, 2006). The types of items addressed through maximal-flow techniques can consist of liquids such as water, gas, or oil that traverses through a network of pipelines, traffic as it travels through a network of roads, or products as they move along a production line system; however, each type of flow has the commonality of fluctuations within the flow capacities throughout the branches of the network, resulting in assessments of the network to obtaining the most effective and efficient method of obtaining the maximum flow from the system (Taylor, 2006). The solution for the maximal-flow problem entails the arbitrary selection of any path in the network that extends from the point of origin to the point of destination, apply adjustments to the capacities that are present at each node by subtracting the maximal flow for the previously selected path, and add the maximal flow as it exists along the path in the opposite directions of each node, repeating each step until there are no paths with available flow capacity (Taylor, 2006).
Network modeling has numerous practical uses which have increased as linear programming has led to the development of several software packages to enhance and simplify the process. One such example is observed through the efforts of the World Bank in 2005 to assist in establishing a partnership strategy in Ghana (Many donors, n.d.). The research applied in this area presented a survey to detail the types of support international donors were currently and planning to provide. The data obtained was then assimilated into a two-mode network matrix providing insight into the objectives presented as aspects of the planned strategy in Ghana and the support being pledged or provided by the originating country of donors (Many donors, n.d.). The significance of the matrix is dependent on the intentions and interests of the actors consisting of donors, government ministries and agencies, and NGOs who may attempt to influence the objectives and implementation of the proposed policies (Many donors, n.d.). The matrix can then be expanded through the addition of nodes to provide further details into areas of concern, such as donor harmonization activities to clarify who needs to work with whom on the agreed approaches to policy specificities as well as highlighting areas in which variations are present in the level of support needed and provided (Many donors, n.d.). The network structure can be treated as an independent variable which can be used to gauge the effects of changes, as a mediating variable which presents the same questions multiple times with the assumption that the influence of the network will be minimal, and as a dependent variable to gauge the changes that will occur throughout the network structure in the event of successful policy changes (Many donors, n.d.). In addition, a network model allows the ability to apply variations in approaches by being opportunistic in which any notable change in policies might be beneficial even though it is not the optimal change or by being committed through the pursuit of issues that are identified as areas of concern while noting the amount of support in these areas as opposed to other issues (Many donors, n.d.).
- Elton, A. & Schulte, A.M. (1994). Network modeling: advances at a major British utility. American Water Works Association.
- Many donors, many objectives. (n.d.). Developing network models of development projects: An introduction. Retrieved from http://www.mande.co.uk/donorobjectives.htm
- Sargent, T.J. & Stachurski, J. (2014). Shortest Paths. Quantitative Economics. Retrieved from http://lectures.quantecon.org/py/short_path.html
- Taylor, B.W. (2006). The Maximal Flow Problem. Introduction to Management Science, Ninth Edition. Retrieved from http://flylib.com/books/en/3.287.1.99/1/
