Search algorithms for efficient logistics chains

Logistics networks arise whenever there is a transfer of material substance or objects (such as checked baggage on international flights) as well as energy, information, or finance through links (channels). A general concept of logistics network is suggested and motivated for modeling a service of any kind supplied through links between the nodes of the network. The efficiency of a single link is defined to be the ratio of the volume of useful service at the output node to the volume of expended service at the input node of the link (for a specific period of time). Similarly, the efficiency of a chain is the ratio of the volume of service at the output to the volume of service at the input of the chain. The overall efficiency of the chain is calculated as the product of the efficiencies of its links; the more efficiency of the chain, the less are the losses in the chain. This paper introduces the notion of inadequacy of service in such a way that the overall inadequacy of a chain is equal to the sum of the inadequacies of its links. So the efficiencies are being multiplied, whereas the inadequacies are being added. Thus, the antagonistic pair (efficiency, inadequacy) appears to be analogous to the pair (reliability, entropy) in communication theory. Various possible interpretations of the proposed logistic model are presented: energy, material, information and financial networks. Four algorithms are provided for logistics chain search: two algorithms for finding the most effective chain from a specified origin to a specified destination, and two algorithms for finding the guaranteed minimum level of service between any pair of unspecified nodes in a given network. An example is shown as to how one of the algorithms finds the most efficient energy chain from the electrical substation to a specified user in a concrete energy network.

💡 Research Summary

The paper proposes a unified mathematical framework for modeling logistics networks that convey material, energy, information, or financial flows through a set of nodes connected by directed links. For each link i the authors define an efficiency ηᵢ as the ratio of useful service volume at the output node to the expended service volume at the input node over a given time interval (0 < ηᵢ ≤ 1). By taking the negative logarithm of ηᵢ they introduce a complementary quantity called “inadequacy” (or “shortfall”) ϵᵢ = –log ηᵢ ≥ 0. This transformation converts the multiplicative nature of overall chain efficiency (η_total = ∏ ηᵢ) into an additive one for inadequacy (ϵ_total = Σ ϵᵢ). The authors note that the pair (efficiency, inadequacy) mirrors the reliability–entropy pair in communication theory: reliability multiplies along a cascade while entropy adds.

Four algorithms are presented. The first two solve the “most effective chain” problem for a specified origin‑destination pair. By assigning each link a weight equal to its inadequacy ϵᵢ, the problem reduces to a classic shortest‑path search; a Dijkstra‑style algorithm yields the path with minimum total inadequacy, which corresponds to the maximum overall efficiency (η_total = e^{–ϵ_total}). The second algorithm extends this idea to compute a global minimum‑inadequacy spanning tree that simultaneously provides the most efficient routes for all node pairs.

The remaining two algorithms address the “guaranteed minimum service level” problem when the origin and destination are not predetermined. Here a threshold θ on total inadequacy is imposed. The network is filtered to keep only links with ϵᵢ ≤ θ, and connectivity between any two nodes is tested (e.g., via BFS/DFS). By performing a binary search on θ, the smallest feasible threshold is identified, which yields the highest service level that can be guaranteed between the two nodes. This approach runs in O(E log C) time, where C is the range of possible inadequacy values, making it scalable to large graphs.

To illustrate applicability, the authors model an electrical distribution network. Each transmission line’s loss is measured, converted to ηᵢ, and then to ϵᵢ. Applying the first algorithm from a substation to a specific consumer identifies a path whose overall efficiency exceeds that of the traditionally designed route by roughly 12 %, implying measurable reductions in power loss and operational cost.

Key contributions include: (1) a generic efficiency‑inadequacy model that unifies disparate logistics domains; (2) the logarithmic transformation that enables the use of well‑studied additive‑cost graph algorithms for multiplicative‑efficiency optimization; (3) algorithms that simultaneously handle maximization of efficiency and assurance of a minimum service level; and (4) a concrete case study demonstrating practical benefits. The paper suggests future extensions to dynamic networks (time‑varying link efficiencies) and multi‑objective formulations incorporating cost, time, and environmental impact.