車輛路徑問題 (Vehicle Routing Problem，VRP) 是一個在理論研究與實際應用都得到了廣泛討論的組合優化問題。城市配送環境中，時變的交通路況對配送規劃產生了顯著的影響。不同出發時刻所面臨的路況變化直接影響到所需的配送時間。因此，越來越多的學者將時變因素 (Time-Varying) 納入了路徑規劃問題的討論中。
依時車輛路徑問題 (Time-Dependent Vehicle Routing Problem，TDVRP) 是帶時間窗約束車輛路徑問題 (Vehicle Routing Problem with Time Window，VRPTW) 的延伸與變形，除了需要在固定的時間窗 (Time Window) 要求下完成對客戶的配送服務，還要考慮到實際配送過程中，尖峰時刻、道路擁堵等時變因素的影響。
依時車輛路徑問題的討論中，不同的學者分別採用不同的模型代表不同出發時間的影響，如「配送時間時變函數」，「配送速度時變函數」等。少有學者用「連續速度時變函數」來體現不同出發時間影響。本研究採用「連續速度時變函數」模型設定來討論不同出發時間對配送過程的影響，以更好的模擬實際配送過程，避免離散函數模型會帶來突變點 (Breakpoint) 的影響(Ehmke, Campbell, & Thomas, 2018)。
在實際路徑規劃中，物流公司會採用不同運載體積、不同運載能力、不同配送成本的異質車型配送的方式來降低擁堵路況的影響，從而降低配送成本。本研究採用「連續速度時變函數」來體現時變因素，同時採用異質不定車型配送的方式進行配送。以包含固定配送成本和動態配送成本的最小配送成本為模型目標，實現車輛路徑規劃最佳化及車輛選擇最佳化。
車輛路徑問題被證明是NP-hard問題，模型中引入「連續速度時變函數」與異質不定車型設定能更符合實際配送過程，但也增加了求解複雜度。「連續速度時變函數」模型設定下，配送時間與配送距離為二次函數關係，為應對求解複雜性，本研究採用查表法來降低計算時間，結果表明，查表法所需計算時間為二次函數直接計算時間的0.046%。同時實驗也驗證了異質不定車型設定的必要性，與單一車型配送相比，異質不定車型設定下配送成本僅為單一車型最低配送成本的96.401%。
鑑於車輛路徑問題的求解複雜性，精確算法通常只適用於小型問題實例。因此，本研究具有創新性地將車輛路徑問題和網絡可靠度問題結合，為車輛路徑問題的求解提供了一個新的視角。基於在網絡可靠度問題求解中表現出色的二元累加樹演算法，提出了移動二元累加樹演算法，以解決容量限制車輛路徑問題，獲得唯一的真實最佳解，並將其作為其他演算法的比較基準。
透過先分群再排序的方法，利用隱式二元累加樹演算法求得近似解，並將這種思路沿用至二元累加樹演算法對其他元啟發演算法的改進中。提出二元累加樹改進簡化群體演算法，並與常被用於車輛路徑問題求解的基因遺傳演算法、粒子群演算法、模擬退火法、變動鄰域搜尋法、禁忌搜尋法進行比較驗證。本研究也討論了二元累加樹演算法對基因遺傳演算法、粒子群演算法、模擬退火法的改善。
實驗結果表明在所提「連續速度時變函數異質車型車輛路徑問題」，UK10數據集貼合編碼，隨機編碼，插板離散編碼三種編碼方式求解比較中，利用二元累加樹改進元啟發演算法可使得目標函數值改善比例依次為0.966%，0.461%，0.175%，改善均值為0.534%。在UK15數據集目標函數值改善比例依次為，2.174%，1.112%，1.956%，改善均值為1.747%。除目標函數值求解改善，二元累加樹演算法提供了更好的局域搜尋能力，大幅的縮短計算求解時間。在UK10數據集求解時間改善比例依次為35.058%，53.795%，35.085%，改善均值為41.312%。在UK15數據集求解時間改善比例依次為44.868%，46.390%，72.299%，改善均值為51.519%。
綜合比較下，所提二元累加樹改進簡化群體演算法在隨機編碼在中小規模 (10配送節點，15配送節點，20配送節點，25配送節點) 表現優異。在三種規模 (10至50配送節點) 配送問題求解上，二元累加樹改進簡化群體演算法隨機編碼所求得目標函數值為其他演算法所求目標函數值的77.405%。在大規模 (50配送節點) 配送問題求解中，二元累加樹改進簡化群體演算法的貼合編碼方式表現最為優異，所求目標函數值為其他演算法所求目標函數值的77.184%，證明了二元累加樹改進簡化群體演算法在路徑規劃問題求解的有效性。

Vehicle Routing Problem (VRP) is a combinatorial optimization problem that has been widely discussed in both theoretical research and practical applications. The surge in industrialization and urbanization has attracted a growing number of scholars to delve into VRP.
In urban distribution settings, it is essential to consider the substantial influence of time-varying traffic conditions when scheduling logistics operations. The Time-Dependent Vehicle Routing Rroblem (TDVRP) is an extension of the classical Vehicle Routing Problem with Time Windows (VRPTW). In addition to the need to complete the delivery service to customers within the time window requirements, it is also necessary to consider the impact of time-varying factors such as peak hours and road congestion during the actual delivery process.
Different scholars have used different models to represent the effects of different departure times, such as the " time-varying delivery time functions" and " time-varying distribution speed functions". However, few scholars have used the " time-varying continuous speed functions" to represent the effect of different departure times. In this study, the " time-varying continuous speed function" model is used to discuss the impact of different departure times on the distribution process to better simulate the actual distribution process and avoid the effect of discrete time-varying functions models that may bring about breakpoints.
In practical route planning, logistics companies adopt heterogeneous vehicles with different carrying capacities and different distribution costs to reduce the impact of congestion and reduce distribution costs. In this study, the " time-varying continuous speed function" is used to specify the time-varying factors, and the fleet size and mix vehicle type is used for distribution route planning which is called Heterogeneous Fleet Vehicle Routing Problem (HFVRP). The objective of the model is to optimize vehicle routing and composition of the fleet by minimizing the distribution costs, which contained fixed costs of acquiring and variable fuel costs.
The VRP is recognized as an NP-hard challenge. The Heterogeneous Fleet Vehicle Routing Problem with Time-Varying Continuous Speed Function can better match the actual distribution process, but increases the complexity of solving it. Under the model setting of " time-varying continuous speed function", the distribution time and distribution distance are quadratic functions, in order to cope with the complexity of the solution, this study adopts the look-up table method to reduce the calculation time, and the results show that the calculation time required by the look-up table method is 0.046% of the time required by the quadratic function direct calculation. Meanwhile, the experiment also verifies the necessity of Heterogeneous Fleet and compared with the single type vehicle distribution, the distribution cost under the Heterogeneous Fleet is only 96.401% of the lowest distribution cost of the single type vehicle.
Given the complexity of solving the VRP, the exact algorithm is usually only applicable to small problem instances. Therefore, this study innovatively combines the VRP with the Network Reliability Problem (NRP) to provide a new perspective for solving the VRP. Based on the BAT, which performs well in solving the NRP, this study proposes a Move Binary Addition Tree Algorithm (Move BAT) to solve the Capacitated Vehicle Routing Problem (CVRP) to obtain the unique true optimal solution, and to use it as a baseline for comparison with other algorithms.
An Implicit BAT is used to find an approximate solution for CVRP by first grouping and then sorting, and this idea is carried over to improve Meta-Heuristic algorithms with the BAT. The BAT-improved Simplified Swarm Optimization (BAT-SSO), and compared with the Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Simulated Annealing (SA), Variable Neighborhood Search (VNS) and Tabu Search (TS), which are often used for solving the vehicle path problem, for validation. The improvement of BAT on GA, PSO and SA method is also discussed.
The experimental results show that in the proposed " The Heterogeneous Fleet Vehicle Routing Problem with Time-Varying Continuous Speed Function ", UK10 dataset Combine coding, random key coding, Cut Discrete coding three coding methods of solving the comparison, the use of BAT to improve the Meta-Heuristic algorithms can make the object value improvement ratio of 0.966%, 0.461%, 0.175%, the improvement of the average value is 0.534%. In the UK15 dataset, the improvement ratio of object value is 2.174%, 1.112%, 1.956%, and the mean value of the improvement is 1.747%.
In addition to the improvement of object, the BAT provides a better local search capability and can significantly reduce the computation solving time, and the improvement ratios of the solving time in the UK10 dataset are 35.058%, 53.795%, and 35.085% in the order of the mean value of the improvement is 41.312%. The improvement ratio of solution time in UK15 dataset is 44.868%, 46.390%, 72.299% in that order, and the mean value of improvement is 51.519%.
In a comprehensive comparative analysis, the proposed BAT-SSO exhibited strong performance, particularly when employing random key coding for small and medium-scale scenarios involving 10, 15, 20, and 25 distribution nodes. In the medium-scale distribution problem-solving scenarios with 20 and 25 distribution nodes, the BAT-SSO utilizing 隨機 coding, achieves target function values equivalent to 77.405% of those achieved by other algorithms.
When addressing large-scale distribution problems, the BAT-SSO with Combine coding. It achieves an objective function value equivalent to 77.184% of that achieved by other algorithms. This outcome underscores the efficacy of the BAT-SSO in solving complex VRP.

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