CYBERNETICS AND CONTROL THEORY

S. M. Shvartin

ON TRANSPORTATION OVER ROAD NETWORKS WITH LOSSES

(Presented by Academician A. I. Berg on 5 VI 1961)

Let \(V\) be a directed connected network \((^{1})\). Let \(G=\{g\}\) be the set of possible states of a vertex of the network \(V\). We shall call a mapping \(\mu(x)\) of the set of vertices of the network \(V\) into \(G\) a state of the network \(V\). We shall call a network \(V\) for which a mapping \(\mu(x)\) is specified a loaded network. Let \(\Xi=\{\mu(x)\}\) be the set of all possible states of the network \(V\). Let \(\mu_0(x)\in\Xi\) and \(\Xi_1\subset\Xi\) be given. We shall call the symbol \(W=(V,\Xi,\mu_0(x),\Xi_1)\) a definite loaded network, \(\mu_0(x)\) the initial state of the network, and \(\Xi_1\) the set of terminal states of the network. Let a class
\[ \mathcal{F}=\{F(g,g')=(\bar g,\bar g'):G^2\to G^2\} \]
of functions be given, called transfer functions. A function \(F\in\mathcal{F}\) and an edge \((x_i,x_j)_k\in W\) generate an elementary transfer operator \(A_k\) on the state of the network \(W\)

\[ A_k(\mu(x))=\mu_1(x)= \begin{cases} \bar g, & \text{if } x=x_i,\\ \bar g', & \text{if } x=x_j,\\ \mu(x), & \text{if } x\ne x_i \text{ and } x\ne x_j. \end{cases} \]

Let a class
\[ \Psi=\{\psi(g)=\bar g:G\to G\} \]
of functions be given, called expressed transfer functions. A function \(\psi\in\Psi\) and a vertex \(x_i\in W\) generate an elementary expressed transfer operator

\[ B_i(\mu(x))=\mu_1(x)= \begin{cases} \bar g, & \text{if } x=x_i,\\ \mu(x), & \text{if } x\ne x_i. \end{cases} \]

Let
\[ H_k=\{[(x,x')_k,F]\} \]
be the set of ordered pairs composed of one edge of the network and transfer functions from \(\mathcal{F}\). Let
\[ H=\bigcup_{k=1}^{\alpha_0} H_k. \]
Let
\[ N_i=\{[x_i,\psi]\} \]
be the set of ordered pairs composed of one vertex of the network and a function \(\psi\in\Psi\). Let
\[ N=\bigcup_{i=1}^{\alpha_1} N_i \]
(\(\alpha_0\) and \(\alpha_1\) are the numbers of edges and vertices of the network \(W\), respectively). We shall call the symbol \(S=(W;H,N)\) a fully specified loaded network (hereafter called simply a network). It is required to construct a sequence of operators, generated by ordered pairs from \(H\) and \(N\), that takes the network from the state \(\mu_0(x)\) into a state \(\mu(x)\in\Xi_1\). If the solution is not unique, then additional requirements of various kinds may be imposed on the sequence of operators. We shall call a network a network transmitting mass if all \(g\in G\) are nonnegative real numbers. We shall call a network a network transmitting information if all \(g\in G\) are ordered sequences of code symbols. In what follows—

networks transmitting mass are considered. We shall call the quantity
\(d_F=(\bar g+\bar g')-(g+g')(d_\psi=\bar g-g)\) the mass defect of the transmission function \(F(\psi)\). We shall call a network a network with losses if \(d_F<0\) and \(d_\psi=0\) for all \(F\) and \(\psi\) ordered pairs from \(H\) and \(N\), respectively. In what follows only networks with losses are considered.

We shall call a network special if

1) \(\Xi=\{\mu(x):\mu(x_i)\ge 0\}\).

2) \(\Xi_1=\{\mu(x):\mu(x_i)\ge a_i\}\) (\(a_i\ge0\) is a prescribed real number) \((i=1,2,\ldots,\alpha_1)\).

3) For every \(m\ge0\), in each \(H_k\in H\) there is an ordered pair
\([ (x,x')_k,\, F(g,g')=(\bar g,\bar g')]\) such that \(g-\bar g=m\).

4) For all \(F\) ordered pairs from \(\bar H_k\), \(d_F=f_k(g-\bar g)\); \(g\ge \bar g,\; g'\le \bar g'\).

We shall call the function
\[ \varphi_k(g-\bar g)=(g-\bar g)+f_k(g-\bar g) \]
the transmission function of the edge \((x,x')_k\). In what follows special networks \(S\) are considered. Let
\(A=A_{k_1}, A_{k_2}, \ldots, A_{k_n}\) be a sequence of elementary operators on the state of the network \(S\). We shall call the functional
\[ P(A)=-\sum_{i=1}^{n} dA_{k_i} \]
the loss functional (\(dA_{k_i}\) is the mass defect of the operator \(A_{k_i}\in A\)). We shall call the sequence \(A\) optimal if it transfers the network \(S\) from the state \(\mu_0(x)\) to the state \(\mu(x)\in\Xi_1\) and makes the functional \(P(A)\) minimal. We shall call the loaded network \(\bar S\) a working subnet of the network \(S\) if it is obtained from the network \(S\) by deleting edges that do not generate elementary operators of the sequence \(A\), optimal for the network \(S\), and its state is the state of the network \(S\) obtained as a result of applying the sequence \(A\) to the network \(S\). We shall call a network a network with linear losses if
\(\varphi_k(g-\bar g)=p_k(g-\bar g)\) \((0\le p_k<1;\ k=1,2,\ldots,\alpha_0)\), and denote it by \(S_L\). Let \(A\) be an optimal sequence for the network \(S_L\); \(\bar S_L\) its working subnet, and \(\mu(x)\) the state of the working subnet \(\bar S_L\).

Lemma 1. For all vertices \(x_i\) of the network \(S_L\) such that \(\mu_0(x_i)\ge a_i\), \(\mu(x_i)\ge a_i\), and for all \(x_j\) such that \(\mu_0(x_j)\le a_j\), \(\mu(x_j)=a_j\).

Lemma 2. Let \((x_i,x_j)_{k_s}\) \((s=1,2,\ldots,n)\) be edges of the network \(S_L\); then the network \(\bar S_L\) contains no more than one of these edges.

Lemma 3. The network \(\bar S_L\) contains no loops.

Theorem 1. Let \((x_i,x_j)_k,\ (x_j,x_i)_s\) be edges of the network \(S_L\). The network \(\bar S_L\) contains no more than one of these edges, i.e. counter transmissions of masses are absent.

Theorem 2. The network \(\bar S_L\) contains no circuits and, consequently, is a tree*.

Let \(Z_i^0\) be the star of the vertex \(x_i\) of the network \(S_L\) (2). Attach to \(Z_i^0\) all vertices of the network \(S_L\) incident with its edges. We shall call these vertices the boundary vertices of the star. We shall call \(Z_i^0\), together with the boundary, a closed star and denote it by \(Z_i\). We shall say that the vertex \(x_i\) of the network \(S_L\) belongs to class \(K_1, K_2, K_3\) if \(\mu_0(x_i)>a_i\), \(\mu_0(x_i)<a_i\), \(\mu_0(x_i)=a_i\), respectively. We shall say that the star \(Z_i\) of the network \(S_L\) belongs to class \(K_{sp}, K_{spq}\) \((p<q)\), \(K_{s123}\), if \(x_i\in K_s\) and its boundary vertices belong only to classes \(K_p\); \(K_p\) and \(K_q\); \(K_1, K_2\), and \(K_3\), respectively \((s,p=1,2,3;\ q=2,3)\). Each star \(Z_i\) of the network \(S_L\) belongs to one of the 21 possible classes of stars.

* An analogous result for the case of transport without losses was obtained by M. L. Tsetlin (3).

Theorem 3. Every star \(Z_i\) of the network \(\overline S_L\) belongs to one of the 4 classes \(K_{13}, K_{31}, K_{33}, K_{313}\); in a star \(Z_i \in K_{13}\) each boundary vertex is the end of an edge of the star; in a star \(Z_i \in K_{313}\) there is only one boundary vertex of class \(K_1\). This vertex is always the beginning of an edge of the star \(Z_i\).

Let the network \(S_L\) be a tree. Let \(x_i \in K_2\) and be an end vertex of the network \(S_L\), and let \((x_i, x_j)_s\) be an edge of the network \(S_L\). Define the operator \(\mathfrak A^{-}\) on the state of the network:

\[ \mathfrak A^{-}(\mu(x))=\mu_1(x)= \begin{cases} a_i, & \text{if } x=x_i;\\[4pt] \mu(x_j)-\dfrac{a_i-\mu(x_i)}{p_s}, & \text{if } x=x_j;\\[8pt] \mu(x), & \text{if } x\ne x_i \text{ and } x\ne x_j. \end{cases} \]

Let \(x_i \in K_1\) and be an end vertex of the network \(S_L\). Let \((x_i, x_j)_k\) be an edge of the network. Define the operator \(\mathfrak A^{+}\) on the state of the network:

\[ \mathfrak A^{+}(\mu(x))=\mu_1(x)= \begin{cases} a_i, & \text{if } x=x_i;\\ \mu(x_j)+[\mu(x_i)-a_i]p_k, & \text{if } x=x_j;\\ \mu(x), & \text{if } x\ne x_i \text{ and } x\ne x_j. \end{cases} \]

Let \(x_i \in K_3\) be an end vertex of the network \(S_L\), and let \((x_i, x_j)_k\) and \((x_j, x_i)_s\) be edges of the network \(S_L\). Denote by \(\mathfrak A\) the operator that excludes these edges from the network \(S_L\). Let \(Z_i\) be the closed star of the vertex \(x_i\) of the network \(S_L\). We shall call folding the network \(S_L\) into the star \(Z_i\) the operation consisting in applying the operators \(\mathfrak A, \mathfrak A^{+}\), and \(\mathfrak A^{-}\) to the network \(S_L\) as long as this is possible under the prohibition on considering boundary vertices of the star as end vertices. Let \(S'_L\) be the closed star \(Z_i\) of the vertex \(x_i\). We shall call the star balanced if the operators \(\mathfrak A^{+}\) and \(\mathfrak A^{-}\) carry this star from its state into one of the final states. Otherwise the star \(Z_i\) is called unbalanced. Let \(Z_i\) be the closed star of the vertex \(x_i\) of the network \(S_L\). Then the following hold:

Theorem 4. Let the network \(S_L\) be folded into the star \(Z_i\), and let the star \(Z_i\) be unbalanced. Then every star \(Z_k\) of the network \(S_L\) is unbalanced after folding the network \(S_L\) into it.

Theorem 5. For the existence of an optimal sequence of operators on the network \(S_L\), it is necessary and sufficient that it contain a closed star which becomes balanced after folding the network \(S_L\) into it.

Let \(S_p\) be a special network of arbitrary nature with nondecreasing concave transfer functions of the edges (i.e.

\[ \varphi_k\!\left(\frac{x_1+x_2}{2}\right)\ge \frac{1}{2}\,[\varphi_k(x_1)+\varphi_k(x_2)] \quad \text{and} \quad 0\le \varphi_k(x)\le x. \]

Theorem 6. For the network \(S_p\), there is no optimal sequence of operators.

Let \(S'_L\) be a network with linear losses obtained from the network \(S_p\) by replacing the transfer functions \(\varphi_k(g-\overline g)\) of the edges of the network \(S_p\) by the linear transfer functions

\[ \left.\frac{d\varphi_k}{d(g-\overline g)}\right|_{(0+0)} \cdot (g-\overline g). \]

Let \(A'_L\) be an optimal sequence of operators on the network \(S'_L\), and let \(A_p\) be an arbitrary sequence of operators on the state of the network \(S_p\) that carries it from the initial state to one of the final states. Then the following holds:

Theorem 7. \(P(A_L) \leqslant P(A_p)\), and for every \(\varepsilon>0\) there exists a sequence of operators \(A_p\) on the network \(S_p\) that transfers it from the initial state to the state \(\mu(x)\in \Xi_1\), such that \(P(A_p)-P(A_L)<\varepsilon\).

Let \(S_q\) be a special network of arbitrary nature with nondecreasing convex transfer functions of the edges (i.e.,
\[ \varphi_k\left(\frac{x_1+x_2}{2}\right)\leqslant \frac{1}{2}\left[\varphi_k(x_1)+\varphi_k(x_2)\right] \]
and \(0\leqslant \varphi_k(x)\leqslant x\).

For such networks the optimal sequence may contain counter-transfers, and the working subnet may contain contours. The property determined by Lemma 1 does not hold. Let \(\ldots, t_{-n}, \ldots, t_{-1}, t_0, t_1,\ldots,t_n,\ldots\) be a discrete time scale. Let \(S\) be an arbitrary network. We shall call the symbol \((S,t_0)\) a network with initial state \(\mu_0(x)\) at time \(t_0\). We shall assume that an elementary operator \(A_k^t(\mu(x))=\mu_1(x)\), acting on the network \(S\) at time \(t\), transfers it to the state \(\mu_1(x)\) at time \(t+1\).

Here the following problems are considered:

1) Given a time instant \(t_1>t_0\). It is required to construct a sequence of operators \(A_k^t\) that transfers the network \((S,t_0)\) from the state \(\mu_0(x)\) at time \(t_0\) to a state \(\mu(x)\in \Xi_1\) at a time \(t\leqslant t_1\) and minimizes the loss functional.

2) It is required to construct a sequence of operators that transfers the network \((S,t_0)\) to a state \(\mu(x)\in \Xi_1\) in the least possible time.

Under these conditions the optimal sequences of operators may contain counter-transfers, and the working subnets may contain contours. These properties are determined not by the nature of the transfer functions of the edges, but exclusively by the topology of the network and by the boundedness of time.

Received
11 V 1961

CITED LITERATURE

1. S. V. Yablonskii, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 51, 270 (1958).
2. L. D. Kudryavtsev, UMN, 3, no. 4 (26) (1948).
3. M. L. Tsetlin, DAN, 129, no. 4 (1959).