Source node

From the topographical viewpoint illustrated in Figure El4.6a the process comprises a set of reactor-separator sections that connect a set of component feeds (specified as source nodes) to component products (specified as destination nodes). Each section is a prescribed sequence of reactors and associated separation units, and sev-... 

For the reactor-separator section, all source nodes must have a destination... 

Multiplier Sum Summing junction Ground Sinusoidal voltage source Node label... 

In practical terms, such a comprehensive analysis is hardly possible. However, as shown in the examples given below, the comparison of label flux to a small number of metabolites or even to different substructure components of a given metabolite can provide more detailed information than the measurement of flux from a single source node to a single sink metabolite. 

Single-layer feedforward networks include input layer of source nodes that projects onto an output layer of neurons (computation nodes), but not vice versa. They are also called feedforward networks. Since the computation takes place only on the output layer nodes, the input layer does not count as a layer (Figure 3.5(a)). 

Multi-layer feedforward networks contain an input layer connected to one or more layers of hidden neurons (hidden units) and an output layer (Figure 3.5(b)). The hidden units internally transform the data representation to extract higher-order statistics. The input signals are applied to the neurons in the first hidden layer, the output signals of that layer are used as inputs to the next layer, and so on for the rest of the network. The output signals of the neurons in the output layer reflect the overall response of the network to the activation pattern supplied by the source nodes in the input layer. This type of network is especially useful for pattern association (i.e., mapping input vectors to output vectors). 

Input layer of Layer of hidden source nodes neurons... 

Figure 1 gives an example of an eight node network that represents a distribution system between factories and retailers. Nodes 1 and 2 are factories whose production capacities (net supplies) are 6 and 9. Nodes 3, 4, and 5 are warehouses (intermediate nodes) whose net supplies are zero. Nodes 6, 7, and 8 correspond to the retailers, whose net supplies are —3, —5, and —7. Each arc is labeled with its unit transportation cost and its upper bound. In this example, there are only arcs between source nodes and intermediate nodes and tetween intermediate nodes and sink nodes. In a general minimum cost flow model there can be arcs between any two nodes in the network. 

In this subclass of problems there are only source and demand sites (no intermediate nodes are included), and shipments can only be made directly between source and demand sites. The source node i has a supply of s, and demand node j has demand of dj (or a net supply of —dj). The transportation problem can be formulated as the foUowing linear programming model ... 

This type of planning problem can be formulated as a transportation model, and the corresponding network is given in Figure 2. Each production shift corresponds to a source node, and the node supply represents the production shift capacity. There are six source nodes since each quarter i can have a regular shift (denoted by node and an overtime shift (denoted by o. For each quarter j there is a demand node j, and finally, there is a dummy demand node DUMMY, which is included to ensure that the total demand equals the total supply. The cost coefficient for arc (g, h) represents the unit cost of supplying demand node h from supply node g. For example, the unit cost of supplying demand node 3 from node Oj, the overtime shift in quarter 1, is ( 12.00 + (2 X 0.50) = 13.00). Note that the costs of aU arcs incident to the DUMMY node are zero since these flows do not represent any actual production. 

A special case of the transportation problem is the assignment problem where there are exactly n source nodes that all have a supply of 1 and n demand nodes that all have a demand of 1 (net supply of -1). Thus, each source node must be assigned to a unique demand node. The cost of assigning source node i to demand node j is and aU arcs are uncapacitated. The assignment problem can be represented in the following way ... 

The assignment problem can arise in a manufacturing system when we view the source nodes as jobs and the demand nodes as machines that can perform the jobs. In the next time cycle, each job must be assigned to a unique machine. The coefficient represents the cost of assigning job i to machine j during this time cycle. The optimal solution assigns the jobs to machines so that the total cost during the next time cycle is minimized. 

The typical maximum flow problem has a network where there are arc upper bounds and a designated source node s and a designated sink node t. The objective is to find the flow pattern that maximizes... 

One application of this model is in the area of emergency evacuation of a facility (such as a building or subway station). The source node represents the location of workers in the facihty and the sink node represents a safety area. The arcs can correspond to the various links from one part of the facility to another (stairways, corridors, etc.) and the arc capacity indicates the maximum number of people who could traverse a link per unit time. The maximum flow represents the maximum rate at which people could be evacuated from the facility (see Chalmet et al. 1982 for a more elaborate model). 

The assignment of personnel to jobs is an often-cited application area for network flow models. Consider the classical assignment problem where the model represents the assignment of n people to n jobs. There is one source node for each person available and one demand node for each job. One possible objective function would be to assign people to jobs in order to minimize the relocation... 

In order to solve the shortest path problem for the double weighting network, a new weight for graph G is constructed with the target weight and limit weight and which is expressed as w = aw, -H (1 - a) w and Wj = (k,W2 + fcj (0 < a < 1). If there is a feasible path p from the specified source node s to the destination node t, a function can be established as follows ... 

Assume that the optimal path p from the source node s to the target node t exists, a e [u, v] (wherein 0 < M, V < 1), from Conclusion 1 and 2 we know that if path p and p that satisfy min/(a) exist when a=u and a=v, and//m) < / /)(v) > / then p can be determined within interval [u, v] by the previously constructed auxiliary functions. The interval [m, v] is referred to as the approximate range of the optimal path. And path p is an approximate solution of the optimal path, referred to as p. If the path that satisfies min f[a) exists and is obtained when a = cf i.e. = / then the path obtained is the optimal solution p that we mean to find. 

If F2(Pi) /(, p"Dijkstra algorithm is called to seek the pathp, that satisfies min/(a) from the source node s to the destination node t else if FJip > /, there is no solution, then the algorithm ends else if Fjjpo) = l, p" p , the algorithm ends else... 

A permanent marking point is made according to the Dijkstra algorithm from the source node also called the positive direction, stored in a collection S. Then the adjacent temporary marker point is modified. 

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