The Theory of Graphs and Its Applications

Burger, Edward B.; Berge, Claude; Doig, Alison

doi:10.2307/1911908

Cited by 315 publications

(114 citation statements)

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“…To illustrate, suppose the network structure is as shown in Figure 4, below. Also suppose the hypotheses of Theorem 23 are satisfied for each of the following pairs of facilities: (2,6) (3,4). Then it follows from Theorem 23 that ------Y2 = Ye, 73 = Y4 = Y7 = Y8, y5 = yg, so we only need to determine TI, Jrz, y3, T5.…”

mentioning

confidence: 99%

Optimal policy for a dynamic multi‐echelon inventory model

Bessler

Veinott

1966

Naval Research Logistics

113

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A general multiperiod multi-echelon supply system consisting of n facilities each stocking a single product is studied. At the beginning of a period each facility may order stock f r o m an exogenous source with no delivery lag and proportional ordering costs. During the period the (random) demands at the facilities are satisfied according to a given supply policy that determines to what extent stock may be redistributed fromfacilities with excess stockto those experiencing shortages. There a r e storage, shortage, and transportation costs. An ordering policy that minimizes expected costs is sought. If the initial stock is sufficiently small and certain other conditions a r e fulfilled, i t is optimal t o order up t o a certain base stock level at each facility.The special supply policy in which each facility except facility 1 passes its shortages on to a given lower numbered facility called i t s direct supplier is examined in some detail. Bounds on the base stock levels a r e obtained. It i s also shown that if the demand distribution at facility j is stochastically smaller ("spread" l e s s ) than that at another facility k having the same direct supplier and i f certain other conditions a r e fulfilled, then the optimal base stock level ("virtual" stockout probability) at j is less than (greater than) o r equal to that at facility k.

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confidence: 99%

Optimal policy for a dynamic multi‐echelon inventory model

Bessler

Veinott

1966

Naval Research Logistics

113

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“…It may also be remarked here that for a given digraph F(A, E), and its corresponding topology (A, TE), C1(i), lEA, is equivalent to Berge's "transitive closure" of the point i, in terms of digraphs, [3].…”

Section: K(~) --Cl(d) a Kl(d)mentioning

confidence: 97%

On topological spaces associated with digraphs

Bhargava¹,

Ahlborn²

1968

Acta Mathematica Academiae Scientiarum Hungaricae

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“…The length of any path from node i to node j is the sum of the weights of the edges in that path; the distance da between nodes i and j is the length of the shortest path from i to j, with dti taken as zero. The matrix whose elements are da is the distance matrix D. This development is limited to graphs which are trees [2], It is convenient to introduce the path matrix of a tree. If an arbitrary vertex r is taken as a reference vertex, then the path matrix P is defined as follows.…”

Section: Properties Of the Distance Matrix Ofmentioning

confidence: 99%

Properties of the distance matrix of a tree

Boesch¹

1969

Quart. Appl. Math.

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In a recent paper, Hakimi and Yau [1] have given the necessary and sufficient conditions for a matrix D to be the distance matrix of a linear graph. They also give a computational technique for determining a tree realization if it exists. The purpose of this note is to exhibit certain interesting algebraic properties of the distance matrix of a tree. Two algorithms for tree realization emerge from these properties. However, these algorithms are not elaborated on as there are many computational techniques which are faster than the original method proposed by Hakimi and Yau.

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The Theory of Graphs and Its Applications

Cited by 315 publications

References 0 publications

Optimal policy for a dynamic multi‐echelon inventory model

Optimal policy for a dynamic multi‐echelon inventory model

On topological spaces associated with digraphs

Properties of the distance matrix of a tree

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