Network flow example

This transportation problem is an example of an important class of LPs called network flow problems Find a set of values for the flow of a single commodity on the arcs of a graph (or network) that satisfies both flow conservation constraints at each node (i.e., flow in equals flow out) and upper and lower limits on each flow, and maximize or minimize a linear objective (say, total cost). There are specified supplies of the commodity at some nodes and demands at others. Such problems have the important special property that, if all supplies, demands, and flow bounds are integers, then an optimal solution exists in which all flows are integers. In addition, special versions of the simplex method have been developed to solve network flow problems with hundreds of thousands of nodes and arcs very quickly, at least ten times faster than a general LP of comparable size. See Glover et al. (1992) for further information. 

In the usual network problem, the terminal pressures, line lengths, and line diameters are specified and the flow rates throughout are required to be found. The solution can be generalized, however, to determine other unknown quantities equal in number to the number of independent friction equations that describe the network. The procedure is illustrated with the network of Example 6.6. 

The efficacy of the proposed methodology is demonstrated with the two examples of a gasoline supply network. In Example 1, the combinatorially feasible solutions, i.e., networks, are identified via algorithms MSG and SSG [4-6]. The second columns of Tables 2 and 3 list the feasible networks determined via MIP for all combinatorially feasible networks contained in the first column of each table. Note that not all the combinatorally feasible networks are feasible moreover, the number of feasible networks identified varies according to the mass constraints imposed as discemable in Tables 2 and 3. Specifically, Table 1 contains the results obtained without any mass flow constraint, and Table 2 contains the results obtained when the minimum mass... 

The remainder of this chapter gives more information concerning the formulation, solution and apphcation of network flow models. Section 2 details some of the principal network models. Sections 3 and 4 discuss solution procedure concepts and give information concerning available software for solving network flow models. Section 5 discusses some examples of network applications. 

Notice that every variable appears in exactly two equations—once with a coefficient of +1 and once with a coefficient of —1. For example, the variable x,., appears in the first equation with a +1 coefficient and in the seventh equation with a — 1 coefficient. This special property for the variable coefficients holds for any transportation problem and more generally for ANY general minimum cost flow problem. This special structure is also the key to many of the specialized solution approaches for network flow problems. 

One of the most effective algorithms for these classes of network problems has been a specialized implementation of the simplex algorithm for linear programming. This type of approach uses special data structures to exploit the special properties of the network models and accelerate the steps of the simplex algorithm. For example, for these network flow models (and aU of the other related subclasses discussed in this chapter), the set of basic variables corresponds to a set of arcs that form a spanning tree for the underlying network. Computing such items as the current values for the dual multiphers is easily done with a specialized procedure that exploits the basis tree structure (Ahuja et al. 1993). 

This section gives further examples of appUcations that can be cast as network flow models. These examples also help Ulustrate the types of situations that can be formulated as network flow problems. In many other appUcations, the network model structure is often hidden and can require considerable ingenuity to identify and formulate. 

An Example of Dynamic Network Flow A Material-Handling System... 

Network flow models can represent temporal as weU as spatial relationships. Consider the four-node dynamic maximum flow network example depicted in Figure 4. For each arc the two labels represent... 

The most common of these exact cases are optimization problems that can be modeled as singlecommodity network flows (see Chapter 99). Equivalently, these are the (ILP)s that can be written so that for each variable Xj, at most one constraint coefficient A j equals 1, at most one A j equals —1, and all other equal 0. Such ILP) s are totally unimodular in that any submatrix formed by the Ajj associated with a collection of rows i and a like-sized collection of variables y has determinant 0, 1 or — 1. This is enough to ensure optimal basic solutions to (/LP) (produced, for example, by the simplex algorithm for linear programming) are integer whenever right-hand-side coefficients are all integer. 

Figure 3.10 Optimal stock levels and network flows for periods 1 to 8 of the MC-RTP example...

Oil and gas production systems, drainage networks, supply (delivery) networks, evacuation networks, flows of information on the Internet towards a common destination, wireless networks transferring information from Wi-Fi access points to a wired access point that coimect to the Internet, root network of plants, river basin systems, water distribution systems, the blood vessel system of animals, the morphology of the human limgs, certain data collection networks, etc. are all examples of flow networks with merging flows. In an oil and gas production system or a drainage system for example, the branches correspond to the components characterised by flow capacities while the nodes are notional, used to represent the topology of the network where the streams flow into another stream. 

Most of the reliability theory literature focuses on binary systems, i.e., systems with only two states functioning or failed. See e.g., (Barlow Proschan, 1981). In many real life applications, however, systems have more than two states. A typical example is a network flow system where the state of the system may be defined as the flow capacity of the system. Depending on the number of functioning links in the system, this capacity varies between full capacity and zero capacity, but with several intermediate states as well. 

There are two major approaches for the synthesis of crystallization-based separation. In one approach, the phase equilibrium diagram is used for the identification of separation schemes (For example Cisternas and Rudd, 1993 Berry et al., 1997). While these procedures are easy to understand, they are relatively simple to implement only for simple cases. For more complex systems, such as multicomponent systems and multiple temperatures of operation, the procedure is difficult to implement because the graphical representation is complex and because there are many alternatives to study. The second strategy is based on simultaneous optimization using mathematical programming based on a network flow model between feasible thermodynamic states (Cisternas and Swaney, 1998 Cisternas, 1999 Cisternas et al. 2001 Cisternas et al. 2003). 

Linear polymers are often thermoplastic, meaning they can flow at some temperature depending on the molecular weight. Polymers need not be simple chains but can be branched or networked. Cross-linked polymers can be gels that swell in a solvent or thermosets, which form three-dimensional networks for example, epoxy resins. This interconnectedness allows long-range coupling. 

One remaining possibility that is less costly from an energy point of view but needs to be carefully controlled is to incorporate additives called flow improvers. These materials favor the dispersion of the paraffin crystals and in doing so prevent them from forming the large networks which cause the filter plugging. The conventional flow improvers essentially change the CFPP and pour point, but not the cloud point. They are usually copolymers, produced, for example, from ethylene and vinyl acetate monomers ... 

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