探索快遞物流網(wǎng)的離散數(shù)學(xué)模型

張明軍; 張玉婧; 楊見青; 姚兵

doi:10.11999/JEIT240767

探索快遞物流網(wǎng)的離散數(shù)學(xué)模型

doi: 10.11999/JEIT240767

張明軍^{1, 2},
張玉婧^1, ,,
楊見青¹,
姚兵³

1.
蘭州財經(jīng)大學(xué)信息工程與人工智能學(xué)院蘭州 730020
2.
甘肅省智慧商務(wù)重點實驗室蘭州 730020
3.
西北師范大學(xué)數(shù)學(xué)與統(tǒng)計學(xué)院蘭州 730070

基金項目: 國家自然科學(xué)基金(61363060, 61662066)，蘭州財經(jīng)大學(xué)科研資助項目(Lzufe2022B-002)，甘肅省自然科學(xué)基金(23JRRA1785, 25JRRA232)

詳細信息

作者簡介:
張明軍：男，教授，研究方向為圖論及其應(yīng)用，物聯(lián)網(wǎng)和供應(yīng)鏈，網(wǎng)絡(luò)信息安全等

張玉婧：女，碩士生，研究方向為物聯(lián)網(wǎng)和供應(yīng)鏈

楊見青：女，碩士生，研究方向為物聯(lián)網(wǎng)和供應(yīng)鏈

姚兵：男，教授，研究方向為拓撲編碼及其應(yīng)用，圖格和圖群，網(wǎng)絡(luò)信息安全

通訊作者:
張玉婧　zyjdm@qq.com

中圖分類號: TN911.7; O157.6
計量
- 文章訪問數(shù): 100
- HTML全文瀏覽量: 37
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- 被引次數(shù): 0
出版歷程
- 收稿日期: 2024-09-09
- 修回日期: 2025-02-20
- 網(wǎng)絡(luò)出版日期: 2025-03-05

Exploring The Discrete Mathematical Models of Express Logistics Networks

1.
School of Information Engineering and Artificial Intelligence, Lanzhou University of Finance and Economics, Lanzhou 730020, China
2.
Gansu Provincial Key Laboratory of Smart Commerce, Lanzhou 730020, China
3.
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Funds: The National Natural Science Foundation of China (61363060, 61662066), Lanzhou University of Finance and Economics Research Funding Programme (Lzufe2022B-002), The Natural Science Foundation of Gansu Province (23JRRA1785, 25JRRA232)

摘要

摘要: 針對快遞物流網(wǎng)絡(luò)，該文研究：(1) 構(gòu)建全新的快遞物流網(wǎng)的離散數(shù)學(xué)模型 (又稱拓撲模型)；(2) 根據(jù)理論基礎(chǔ)從圖論學(xué)科的角度對快遞物流網(wǎng)絡(luò)拓撲模型進行定性分析，通過數(shù)學(xué)模型法結(jié)合參數(shù)統(tǒng)計、優(yōu)化算法等數(shù)學(xué)手段對模型進行定量分析。對拓撲模型中的邊賦予路長權(quán)重，并為快遞物流網(wǎng)拓撲模型設(shè)計了新的優(yōu)化算法(集散算法、控制集算法、預(yù)先指定子圖算法)；(3) 以蘭州市城關(guān)區(qū)主城區(qū)作為快遞物流網(wǎng)拓撲模型的應(yīng)用實例，實施了相應(yīng)的優(yōu)化算法。同時針對模型計算面臨的復(fù)雜度等困難提出了解決辦法，為進一步完善、優(yōu)化快遞物流網(wǎng)絡(luò)提供了一定參考。
- 快遞物流網(wǎng) /
- 拓撲模型 /
- 優(yōu)化算法 /
- 計算復(fù)雜度 /
- 賦權(quán)圖
Abstract: Objective With the rapid growth of e-commerce, express delivery volumes have surged, placing increased demands on existing logistics infrastructure and operational models. An efficient express logistics network can help reduce costs, improve transportation efficiency, and enhance logistics management. Therefore, analyzing the structure and operation of express logistics networks, as well as identifying ways to optimize these networks, has become a critical focus for logistics companies. The goal is to improve operational efficiency and support balanced regional economic development. Current research on express logistics networks involves constructing various models, such as mathematical optimization models, decision models, and network evaluation models, and applying algorithms like heuristic, genetic, and greedy algorithms, as well as those based on complex network theory, to optimize network structure, performance, and planning decisions. However, a limitation of existing studies is the lack of models closely aligned with the practical realities of express logistics, and the absence of effective new algorithms to address the complex, evolving challenges faced by express logistics networks. This study proposes a novel discrete mathematical model, also known as a topology model, for express logistics networks from the perspective of graph theory. The model comprises a road network (physical network), a topology network (mathematical model), and an information network (soft control system), providing a closer alignment with real-world express logistics scenarios. Through both qualitative and quantitative analyses of the model, along with the design of corresponding optimization algorithms, this research offers a reference for the in-depth study and scientific optimization of express logistics networks. Methods This study employs various methods: (1) Mathematical Model Construction: A new discrete mathematical model for express logistics networks is developed, accounting for the nonlinear, stochastic, and discrete characteristics of the network. The model integrates physical, topological, and informational networks. (2) Qualitative Analysis: The topology model of the express logistics network is qualitatively analyzed using graph theory concepts and algorithms, where the network topology model is represented as a weighted structure in graph theory. (3) Quantitative Analysis: The mathematical model is analyzed quantitatively using statistical parameters, optimization algorithms, and other mathematical techniques. The edges in the topological model are assigned route length weights, and new optimization algorithms—such as the distribution algorithm, control set algorithm, and pre-designated subgraph algorithm—are proposed to optimize the express logistics network topology. (4) Case Study and Optimization: The topology model is applied to the express logistics network in the central district of Lanzhou City (Chengguan District), where corresponding optimization algorithms are implemented. Solutions to challenges, such as the computational complexity of the model, are proposed. Results and Discussions The mathematical model in this study is a topological graph based on graph theory, where various matrices are used to input the express logistics network’s topology into the computer for subsequent calculations. Innovation 1: A topological model of the express logistics network is created. Innovation 2: The topological model of the express logistics network is optimized and quantitatively analyzed, and a minimum weight path m-control set algorithm (m ≥ 2) and a pre-designated subgraph control algorithm are developed(Algorithm 3, Algorithm 4). These models and algorithms are then applied to the study of the express logistics network in the Chengguan District of Lanzhou City. Innovation 3: In response to the large-scale data and the limitations in computer computing power, as well as the absence of a super-large computer at the author’s institution, the large-scale matrix calculation is divided into smaller regional matrices for optimized computation. Innovation 4: Different optimization algorithms are selected for different areas of the road network map of Lanzhou City’s Chengguan District (Fig. 4, Fig. 5). Multiple calculation results are integrated to obtain the minimum weight path of the pre-designated subgraph for the Chengguan District, validating the effectiveness of the model and algorithms. Conclusions This study addresses existing issues in express logistics network research through the aforementioned work and innovations. A new model and new algorithms, better suited to practical logistics scenarios, are developed. Based on the case study, new problems and methods are proposed, offering further possibilities for optimizing express logistics networks. With the rapid development of emerging technologies such as the Internet of Things, big data, and artificial intelligence, future research could focus on deeply integrating these technologies to enable real-time and accurate collection and analysis of logistics data. Access to high-quality and diverse data can further improve the accuracy of the model’s calculations and enhance intelligent decision-making capabilities. This study has only considered assigning route length weights to the edges in the topological model; future work may explore multi-objective, multi-weight optimization models for express logistics networks to meet the practical decision-making needs of different logistics service providers.
- Express logistics networks /
- topological model /
- optimization algorithm /
- computation complexity /
- weighted graph.

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圖 1 一個具有權(quán)重的區(qū)域道路圖G

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圖 2 解釋第k個集散過程及其算法模塊

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圖 3 蘭州市城關(guān)區(qū)道路圖(摘自高德地圖)

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圖 4 基于圖G的兩個最小權(quán)重森林

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圖 5 基于圖G的最小權(quán)重支撐樹

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1 無約束快遞集散 CD-算法

輸入：集裝$C(k) = \{ {A_{k,1}},{A_{k,2}}, \cdots ,{A_{k,n(k)}}\} $，其中${A_{k,j}}$是貨物，且有${A_{k,j}} \ne {A_{k,t}}$, $j \ne t$
輸出：分散$D(k + 1)$
在一般的快遞物流網(wǎng)中，不會為一個貨物進行式(1)的無約束快遞集散，而是進行下面的多個快遞貨物的集裝和分散：
步驟1　集裝$C(k)$被分散，形成分散$D(k)$：
(1) 分散集裝$C(k)$的貨物，其中有貨物${{\mathrm{ED}}_k} = \{ {A_{{k_i},1}},{A_{{k_i},2}}, \cdots ,{A_{{k_i},a({k_i})}}\} $被發(fā)送到客戶手中；
(2) 又有新的貨物${{\mathrm{NB}}_k} = \{ {B_{k,1}},{B_{k,2}}, \cdots ,{B_{k,m(k)}}\} $補充進來。
步驟2　形成集裝$C(k + 1)$：將集合$(C(k)\backslash {{\mathrm{ED}}_k}) \cup {{\mathrm{NB}}_k}$中的貨物組裝成小集裝模塊${C_{k + 1,1}},{C_{k + 1,2}}, \cdots ,{C_{k + 1,n(k + 1)}}$，其中　$n(k + 1) \le n(k) - a({k_i}) + m(k)$，使得集裝$C(k + 1)$由這$n(k + 1)$個集裝模塊構(gòu)成。
步驟3 由集裝$C(k + 1)$返回分散$D(k + 1)$。

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2 最小權(quán)重路控制集算法

輸入：權(quán)重路拓撲模型 ${G_{\rm{weight}}}(t)$
輸出：權(quán)重路拓撲模型 ${G_{\rm{weight}}}(t)$ 的一個最小權(quán)重路控制集
步驟1　選擇節(jié)點 ${u_1} \in V({G_{\rm{weight}}}(t))$ 以及與節(jié)點 ${u_1}$ 相鄰的節(jié)點 ${w_1} \in {N_{\rm{ei}}}({u_1})$，使得路 $P({u_1},{w_1})$ 的權(quán)重之和為最小。得到節(jié)點集　合： ${U_1} \leftarrow \{ {u_1}\} $, ${W_1} \leftarrow \{ {w_1}\} $, ${V_1} = V({G_{\rm{weight}}}(t))\backslash ({U_1} \cup {W_1})$。
步驟2　選擇節(jié)點 ${u_k} \in {V_k}$ 以及與節(jié)點 ${u_k}$ 相鄰的節(jié)點 ${w_k} \in {N_{\rm{ei}}}({u_k})$，使得路 $P({u_k},{w_k})$ 的權(quán)重之和是最小的，且 ${w_k}\not \in {U_{k - 1}}$。得到　節(jié)點集合：${U_k} \leftarrow {U_{k - 1}} \cup \{ {u_k}\} $, ${W_k} \leftarrow {W_{k - 1}} \cup \{ {w_k}\} $, ${V_k} = V({G_{\rm{weight}}}(t))\backslash ({U_k} \cup {W_k})$。轉(zhuǎn)到步驟3。
步驟3　如果有節(jié)點 $x \in {V_k}$ 使得路 $P(x,y)$ ($y \in {N_{\rm{ei}}}(x)$) 的權(quán)重之和達到最小值，但是 $y\not \in {W_k}$，則令 ${W_k} \leftarrow {W_{k - 1}} \cup \{ x\} $，以及　${U_k} \leftarrow {U_{k - 1}} \cup \{ y\} $, ${V_k} = V({G_{\rm{weight}}}(t))\backslash ({U_k} \cup {W_k})$。轉(zhuǎn)到步驟4。
步驟4　如果有 $\|{V_k}\| \ge 2$，轉(zhuǎn)到步驟2。否則，轉(zhuǎn)到步驟5。
步驟5　如果 ${V_k} = \{ z\} $，且有節(jié)點$ z' \in {N_{\rm{ei}}}(z) $，使得路 $ P(z,z') $ 的權(quán)重之和達到最小。當(dāng)節(jié)點 $ z' \in {U_k} $，則令 ${W_{k + 1}} \leftarrow {W_k} \cup \{ z\} $, 　${U_{k + 1}} \leftarrow {U_k}$；當(dāng)節(jié)點 $ z' \in {W_k} $，則令 ${U_{k + 1}} \leftarrow {U_k} \cup \{ z\} $, ${W_{k + 1}} \leftarrow {W_k}$, ${V_{k + 1}} = V({G_{\rm{weight}}}(t))\backslash ({U_{k + 1}} \cup {W_{k + 1}}) = \varnothing $。轉(zhuǎn)到步驟6。
步驟6　返回權(quán)重路拓撲模型 ${G_{\rm{weight}}}(t)$ 的最小權(quán)重路控制集 ${W_{k + 1}}$。

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3 最小權(quán)重路 m-控制集算法 (m≥2)

輸入：在最小權(quán)重路控制集算法2中，最小權(quán)重路控制集 ${W_{k + 1}}$ 導(dǎo)出權(quán)重路拓撲圖${H_1} = G[{W_{k + 1}}]$，它是權(quán)重路拓撲模型 ${G_{\rm{weight}}}(t)$ 的一　個子圖
輸出：權(quán)重路拓撲圖 ${H_m}$ 的最小權(quán)重路 m-控制集算法 (m≥2)
步驟1　對圖${H_1} = G[{W_{k + 1}}]$運用最小權(quán)重路控制集算法，得到權(quán)重路拓撲圖${H_1}$的一個最小權(quán)重路控制集${S_1}$，它是權(quán)重路拓撲模型　${G_{\rm{weight}}}(t)$的一個最小權(quán)重路 2-控制集，使得權(quán)重路拓撲模型${G_{\rm{weight}}}(t)$的任何一個節(jié)點$a$滿足：$a \in {S_1}$，或者有節(jié)點$b \in {S_1}$，使得路　$P(a,b)$的權(quán)重之和達到最小值，且有邊集$\|E(P(a,b))\| \le 2$。
步驟2　節(jié)點集${S_1}$導(dǎo)出權(quán)重路拓撲圖${H_2} = G[{S_1}]$。實施最小權(quán)重路控制集算法2，得到權(quán)重路拓撲圖${H_1}$的一個最小權(quán)重路控制集${S_2}$，它　是權(quán)重路拓撲模型${G_{\rm{weight}}}(t)$的一個最小權(quán)重路 3-控制集。
步驟$(m - 1)$　節(jié)點集${S_{m - 2}}$ 導(dǎo)出權(quán)重路拓撲圖${H_{m - 1}} = G[{S_{m - 2}}]$。實施最小權(quán)重路控制集算法2后，得到權(quán)重路拓撲圖${H_m}$的一個最　小權(quán)重路控制集${S_{m - 1}}$，它是權(quán)重路拓撲模型${G_{\rm{weight}}}(t)$的一個最小權(quán)重路m-控制集。

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4 預(yù)先指定子圖控制算法

輸入：連通邊賦權(quán)圖$G(t)$，指定連通圖$G(t)$的一個真子圖$H(t)$
輸出：連通圖$G(t)$的一個子圖$L(t)$，使得$H(t) \subset L(t)$，每個節(jié)點$u \in V(L(t))$到真子圖$H(t)$的距離最小
步驟1　選擇最大的節(jié)點子集合${V_1} \subset {V^} = V(G(t))\backslash V(H(t))$，使對每一個節(jié)點 $x \in {V_1}$，有節(jié)點$y \in V(H(t))$ 與x 相鄰，并且邊　$xy \in {E^} = E(G)\backslash E(H(t))$ 的權(quán)重最小，${V_1}$叫做 1-層節(jié)點集。做：${E_1} \leftarrow E(H(t))$, ${E_2} \leftarrow ({E_1} \cup E({V_1},V(H(t)))$，其中邊集合　$E({V_1},V(H(t)))$中的邊$ xy $權(quán)重最小，且滿足$x \in {V_1}$和$y \in V(H(t))$。
步驟2　選擇最大的節(jié)點子集合${V_2} \subseteq {V^}\backslash {V_1}$，使對每一個節(jié)點$x \in {V_2}$，有節(jié)點$y \in {V^}\backslash {V_1}$與x相鄰，并且邊$xy \in {E^*}\backslash E({V_1},V(H(t)))$ 的　權(quán)重最小，${V_2}$叫做 2-層節(jié)點集。做：${E_3} \leftarrow {E_2} \cup E({V_2},{V_1})$，其中邊集合$E({V_2},{V_1})$中的邊$ xy $權(quán)重最小，且滿足$x \in {V_1}$和$y \in {V_2}$。
步驟3　如果$k$-層節(jié)點集${V_k} = {V^*}\backslash \cup _{i = 1}^{k - 1}{V_i} \ne \varnothing $，轉(zhuǎn)到步驟2，否則轉(zhuǎn)到步驟4。
步驟4　如果$k$-層節(jié)點集${V_k} = {V^*}\backslash \cup _{i = 1}^{k - 1}{V_i} = \varnothing $，做：$V(L(t)) \leftarrow V(G(t))$, $E(L(t)) \leftarrow {E_2} \cup E({V_2},{V_1}) \cup E({V_3},{V_2}) \cup \cdots \cup E({V_{k - 1}},{V_{k - 2}})$ 　到步驟5
步驟5　返回連通圖$G(t)$的子圖$L(t)$，每個節(jié)點 $u \in V(L(t))$到真子圖$H(t)$的距離最小。

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表 1 相關(guān)算法比較和評估

算法	核心思想	性能比較及適用范圍
最小權(quán)重路控制集算法	通過貪心選擇和迭代構(gòu)建的方式，逐步逼近最優(yōu)解，即在滿足控制集條件下使權(quán)重之和最小。	計算過程較為直觀，其時間、空間復(fù)雜度與圖的規(guī)模和迭代次數(shù)有關(guān)，可能在大規(guī)模復(fù)雜網(wǎng)絡(luò)中效率較低。適用于規(guī)模較小或結(jié)構(gòu)簡單、連通性較強的網(wǎng)絡(luò)。
最小權(quán)重路 m-控制集(m≥2)算法	通過基于子圖的迭代操作，利用算法 2作為基礎(chǔ)步驟，逐步構(gòu)建出權(quán)重路拓撲模型的最小權(quán)重m-控制集。這種方法通過分解問題為多個子問題來解決尋找特定階數(shù)最小權(quán)重控制集的復(fù)雜問題。	在計算上比最小權(quán)重路控制集算法更復(fù)雜，但通過多次迭代和基于子圖的操作，其求解精度相對算法 2會更高。適用于多層網(wǎng)絡(luò)的層次化資源管理等，特別是當(dāng)問題的求解依賴于前一階控制集的結(jié)果，并且需要不斷細化和擴展控制集以滿足更復(fù)雜的條件時。
預(yù)先指定子圖控制算法	算法通過層次選擇節(jié)點集逐步找到最優(yōu)路徑。通過最大化節(jié)點集和最小化邊權(quán)重來保證每個節(jié)點都與指定的真子圖保持最小距離。	在優(yōu)化子圖和控制集的過程中，能更好地處理特定拓撲結(jié)構(gòu)，計算復(fù)雜度也是較高的。適合處理較為特殊的拓撲結(jié)構(gòu)，如環(huán)型、樹型網(wǎng)等，尤其適合實際應(yīng)用中先給定子圖的情況。

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