birt gs1 128

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birt gs1 128

birt ean 128

birt ean 128,

birt gs1 128,
birt ean 128,

birt ean 128,
birt gs1 128,

birt gs1 128,
birt gs1 128,
birt ean 128,
birt ean 128,
birt gs1 128,
birt ean 128,
birt gs1 128,
birt ean 128,
birt gs1 128,
birt gs1 128,
birt gs1 128,
birt gs1 128,
birt ean 128,
birt gs1 128,
birt gs1 128,
birt ean 128,

birt gs1 128,
birt gs1 128,
birt ean 128,
birt ean 128,
birt ean 128,
birt ean 128,
birt ean 128,
birt ean 128,
birt ean 128,
birt gs1 128,
birt ean 128,
birt ean 128,
birt gs1 128,
birt gs1 128,
birt ean 128,
birt ean 128,
birt ean 128,
birt gs1 128,
birt ean 128,
birt ean 128,
birt gs1 128,
birt ean 128,
birt gs1 128,
birt gs1 128,
birt gs1 128,
birt ean 128,
birt ean 128,
birt gs1 128,
birt gs1 128,
birt ean 128,
birt ean 128,
birt gs1 128,
birt gs1 128,
birt ean 128,
birt ean 128,
birt ean 128,
birt ean 128,
birt ean 128,
birt gs1 128,
birt ean 128,
birt ean 128,
birt ean 128,
birt ean 128,
birt ean 128,
birt gs1 128,
birt ean 128,
birt ean 128,
birt ean 128,

units of output, the rm should build and utilize plant 1 (given by SAC1). From 300 to 550 units of output, it should build the larger plant given by SAC2, etc. Note that the rm could produce an output of 400 with plant 1, but only at a higher cost than with plant 2. The irrelevant portions of the SAC curves are dashed. The remaining (undashed) portions form the LAC curve. By drawing many more SAC curves, we would get a smoother LAC curve. If in the long run we increase all factors used in production by a given proportion, there are three possible outcomes: (1) output increases in the same proportion, so that there are constant returns to scale or constant costs; (2) output increases by a greater proportion, giving increasing returns to scale or decreasing costs; and (3) output increases in a smaller proportion, giving decreasing returns to scale or increasing costs. Increasing returns to scale or economies of mass production may result because of division of labor and specialization in production. Beyond a certain size, however, management problems resulting in decreasing returns to scale may arise.

CHAP. 15]

WEIGHTED DIGRAPHS AND GRAPHS A weighted digraph is a pair (V, w) where V is a finite set of vertices and w is a function that assigns to each pair (x, y) of vertices either a positive integer or (infinity). The function w is called the weight function, and its value w(x, y) can be interpreted as the cost (or time or distance) for moving directly from x to y. The value w(x, y) = indicates that there is no edge from x to y. A weighted graph is a weighted digraph (V,w) whose weight function w is symmetric, that is, w(y,x) w(x,y) for all x,y V. Just as every digraph has an embedded graph, every weighted digraph has an embedded weighted graph (V, w) and an embedded (unweighted) digraph. The weight function for the embedded weighted graph can be defined as w (x, y) = min{w(x,y), w(y,x)}, where w is the weight function of the weighted digraph. The vertex set for the embedded digraph can be defined as E = {(x,y) : w(x,y) < }. The properties described above for digraphs and graphs apply to weighted digraphs and weighted graphs. In addition there are some extended properties that depend upon the underlying weight function in the obvious manner. For example, the weighted path length is the sum of the weights of the edges along the path. And the shortest distance from x to y would be the minimum weighted path length among all the paths from x to y. EXAMPLE 15.21 A Weighted Digraph and Its Embedded Structures

Figure 15.20 shows a weighted digraph together with its embedded weighted graph, its embedded digraph, and its embedded graph. The weights are shown on the edges.

In graph G1 the weighted path length of the path cabd is |cabd| = 2 + 3 + 2 = 7, and the shortest distance from c to d is 6 (along the path cad). But in graph G2 that shortest distance is 1 (along the path cd). Note that graph G3 is the same as that in Example 15.13 on page 292, and graph G4 is the same as that in Example 15.1 on page 285. Figure 15.21 shows the adjacency matrix, the incidence matrix, and the adjacency list for graph G1.

EULER PATHS AND HAMILTONIAN CYCLES An euler path in a graph is a walk that includes each edge exactly once. An euler cycle is a closed walk that includes each edge exactly once. An eulerian graph is a graph that has an euler cycle. Note that euler paths and cycles need not have distinct vertices, so they are not strict paths. EXAMPLE 15.22 Euler Paths and Cycles

The LAC curve derives its shape from the possible SAC curves that a rm has over ranges of outputs.

In the graph in Figure 15.22, the closed walk acedabefdbcfa is an euler cycle. So this is an eulerian graph. Note that every vertex in this graph has degree 4, and its 12 edges are partitioned into three circles. As the Theorem 15.8 reports, each of these two properties will always guarantee that the graph is eulerian.