CYBERNETICS AND CONTROL THEORY

V. A. EVSTIGNEEV

THE TRANSPORTATION PROBLEM IN TIME IN GRAPH THEORY

(Presented by Academician S. L. Sobolev on 17 III 1964)

Let us call a finite connected graph \(G=(X,U)\) without loops or multiple arcs a transportation network \(G_{\|}=(x_0, X_n, z; U)\), if two vertices are distinguished in it: \(x_0\) \((\Gamma^{-1}x_0=\varnothing)\), the entrance of the network, and \(z\) \((\Gamma z=\varnothing)\), the exit, and with each arc \(u\in U\) there are associated two integers: \(l(u)\ge 0\), the length of the arc, and \(c(u)\ge 0\), the capacity of the arc.

The transportation problem in time consists in finding the minimum time required to transport a load \(P\) from \(x_0\) to \(z\), the transportation being carried out along the arcs of the network.

A function \(\varphi(u)\), defined on the set \(U\) and taking integer values, will be called a flow in the transportation network if:

1) \(0\le \varphi(u)\le c(u),\ u\in U\);

2)
\[ \sum_{u\in U^-_{x_i}}\varphi(u)-\sum_{u\in U^+_{x_i}}\varphi(u)=0 \qquad (x_i\in X_n,\ x_i\ne x_0,\ x_i\ne z). \]

The quantity
\[ \Phi=\sum_{u\in U^-_z}\varphi(u)=\sum_{u\in U^+_{x_0}}\varphi(u) \]
is called the value of the flow \(\varphi\). For a given transportation network the greatest value of the flow is equal to the least capacity of a cut (the Ford–Fulkerson theorem \((^{1,5})\)), i.e.,
\[ \max_{\varphi}\Phi=\min_{\substack{x_0\in A\\ z\notin A}} c(U^-_{\bar A}). \tag{1} \]

In the general case the flow in a transportation network is not defined uniquely \((^2)\). Uniqueness can be achieved by imposing an additional condition on the flow \(\varphi(u)\):
\[ \sum \varphi(u)\cdot l(u)\le \sum \varphi'(u)\cdot l(u), \tag{2} \]
where the summation is over all arcs entering a \(\varphi\)-cycle \((^2)\), and \(\varphi'(u)\) is a flow of the same value as the flow \(\varphi(u)\). The flow \(\varphi(u)\) must satisfy condition (2) on all \(\varphi\)-cycles.

Theorem 1. The minimum time for transporting a load \(P\) over a two-terminal transportation network is determined by the formula
\[ T=\left[\frac{P-1+\sum_{u\in U}\varphi(u)\cdot l(u)}{\Phi}\right], \tag{3} \]
where the square brackets denote taking the integer part.

For the proof of Theorem 1 let us consider the so-called parallel transportation networks, i.e., networks composed of parallel branches \(\mu\) (Fig. 1). Denote such a network by \(\widetilde G\). For this network a transportation plan ensures minimal transportation time if it satisfies the following condition.

Theorem 2. The time required to transport cargo \(P\) through the network \(G\) is minimal if and only if

\[ \max_{i,j}\left|T_i-T_j\right|=0 \text{ or } 1, \tag{4} \]

where \(T_i\) is the duration of transportation along the branch \(\mu_i\), \(i,j=1,2,\ldots,m\), where \(m\) is the number of branches. Transportation along all branches begins at the moment \(t=0\), and the size of the cargo portions, except possibly the last ones, is maximal.

From Theorem 2 there follows directly the solution of the transportation problem in time for parallel networks.

Theorem 3. The minimal time \(T\) for transporting cargo \(P\) through the network \(G\) is determined by the formula

\[ T=\left[ \frac{ P-1+\sum_{i=1}^{m}\varphi(\mu_i)\cdot l_i }{\Phi} \right], \tag{5} \]

Fig. 1

where \(\varphi(\mu_i)\) is the flow along the branch \(\mu_i\), and

\[ l_i=\sum_{u\in \mu_i} l(u). \]

To solve the transportation problem in time for arbitrary networks, we introduce the concept of equivalence of networks. Networks \(G_1\) and \(G_2\) are called equivalent if 1) the flow through the network \(G_1\) is equal in magnitude to the flow through \(G_2\), and 2) equal quantities of cargo are transported through the networks \(G_1\) and \(G_2\) in equal time.

Theorem 4. For any arbitrary transportation network there exists an equivalent parallel transportation network.

From Theorems 3 and 4, Theorem 1 follows directly.

The result obtained here—the formula (3)—admits generalizations both to the case of bipartite networks of a more complicated nature, for example \(n\)-partite transportation networks, and to the case of certain special kinds of multipartite transportation networks. It is necessary to note, however, that in the latter case the concept of flow differs from that accepted in graph theory.

Remark. The results obtained in the article are valid for

\[ P>\sum_{u\in U}\varphi(u)\cdot l(u). \]

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
12 II 1964

REFERENCES

1. K. Berge, Graph Theory and Its Applications, IL, 1962.
2. V. G. Vizing, DAN, 144, No. 6 (1962).
3. Hoang Tuy, Siberian Mathematical Journal, 4, No. 2 (1963).
4. M. A. Stein, DAN, 129, No. 4 (1959).
5. P. Elias, A. Feinstein, K. Shannon, “On the maximum flow through a network,” in: C. Shannon, Works on Information Theory and Cybernetics, IL, 1963, pp. 729–734.