Asynchronous optimization of part logistics routing problem

Abstract

The solution to the capacitated vehicle routing problem (CVRP) is vital for optimizing logistics. However, the transformation of real-world logistics problems into the CVRP involves diverse constraints, interactions between various routes, and a balance between optimization performance and computation load. In this study, we propose a systematic model originating from the part logistic routing problem (PLRP), which is a two-dimensional loading capacitated pickup-and-delivery problem that considers time windows, multiple uses of vehicles, queuing, transit, and heterogeneous vehicles. The newly introduced queuing and transit complicate the problem, and to the best of our knowledge, it cannot be solved using existing methods or the standard commercial optimizer. Hence, this problem has caused the existing research to develop, generalize, and extend into the two-dimensional CVRP (2L-CVRP). To solve this problem, we provide a framework that decouples the combination of 2L-CVRP and global optimization engineering and derives an efficient and realistic solver that integrates diverse types of intelligent algorithms. These algorithms include: (1) a heuristic algorithm for initializing feasible solutions by imitating manual planning, (2) asynchronous simulated annealing (SA) and Tabu search (TS) algorithms to accelerate the optimization of global routes based on novel bundling mechanics, (3) dynamic programming for routing, (4) heuristic algorithms for packing, (5) simulators to review associated time-related constraints, and (6) truck-saving processes to promote the optimal solution and reduce the number of trucks. Moreover, the performances of the SA and TS solver algorithms are compared in terms of various size scales of data to obtain an empirical recommendation for selection. The proposed model successfully established an intelligent management system that can provide systematic solutions for logistics planning, resulting in higher performance and lower costs compared to that of manual planning.

Appendices

Appendices

Hyperparameter combinations in robust test

The hyperparameter in SA is $\theta _1$ and $\theta _2$ in Eq. (29), and the initial temperature $t_{\max }$ and $B_{s}$ in Algorithm 2. The hyperparameters used in this study are listed in Table 7, and 24 unique combinations were generated from them.

Table 7 SA Hyperparameter

The hyperparameter in TS is the length of Tabu list $|{\mathcal {T}}|$ and the unbundling threshold $B_t$. The values are listed in Table 8

Table 8 TS Hyperparameter

Genetic programming for 2D strip packing problem

We strictly followed [9] for GP training and utilization. The GP approach includes two parts: a GP tree with given functions and terminals based on the property of loading states, loading position, and column attributes, which represents a heuristic function and evaluates the goodness of placing a column in a loading position. Thus, a packing scheme could be generated by repeated selection of the best column-position pair. The heuristic function was trained by an evolution algorithm aiming to minimize the fitness, e.g., the overall length of the complete loading area. We adopted the same GP tree functions and terminal as well as the training method, except for the following conditions.

• The B2 constraint was considered to ensure that the column to be unloaded will not be blocked, regardless of the high value of the heuristic function.

• Fitness was altered as the maximum length of the loading area of all stations.

Manual heuristic function design

$$\begin{aligned} H_l(m, u_{0m}, u_{1m}; \Upsilon )= & {} \frac{(1 + \text {(Fit)}) \times \text {(Free area)} \times \text {(Rate)} \times \delta _{0m} \times \delta _{1m} }{(\text {(Wasted area)} + 0.1) ((\text {Gap}) + 0.1) } , \end{aligned}$$

where (Top Length) represents the top-loading area of, (Fit) $\in \{0, 1, 2\}$ evaluates the possibility of a column to perfectly fit a position. (Free Area) determines the free area above the top-loading surface. (Wasted Area) calculates the area below the edge surface unoccupied by the columns. (Gap) indicates the distance from the right-most edge to the truck boundary. (Rate) is a soft form of the objective function of the 2D strip packing problem [23]. These are defined by:

$$\begin{aligned} {\tilde{\Upsilon }}&= \Upsilon \cup \{(m, u_{0m}, u_{1m})\},\\ \text {(Top length)}&= \max _{(m', u_{0m'}, u_{1m'}) \in {\tilde{\Upsilon }}} (u_{0m'} + \delta _{0m'}),\\ \text {(Occupied)}&= \\&\int _{0}^{u_{1}} \max _{(m', u_{0m'}, u_{1m'}) \in {\tilde{\Upsilon }}} \left\{ \left[ u_{1m'} \le u_1 \le u_{1m'} + \delta _{1m'}\right] (u_{0m'} + \delta _{0m'}) \right\} \text {d}u_{1},\\ \text {(Free area)}&= D_{0k}D_{1k} - \text {(Occupied)},\\ \text {(Wasted area)}&= \text {(Occupied)} - \sum _{(m', u_{0m'}, u_{1m'}) \in {\tilde{\Upsilon }}}\delta _{0m'}\delta _{1m'},\\ \text {(Gap)}&= D_{1k} - \max _{(m', u_{0m'}, u_{1m'}) \in {\tilde{\Upsilon }}} \left\{ (u_{1m'} + \delta _{1m'}) \right\} ,\\ \text {(Fit)}&= [u_{1m} + \delta _{1m} = D_{1k}]+ [u_{0m} = D_{0k}],\\ \text {(Rate)}&= \frac{1}{1 + e^{\text {(Top length)}/D_{0k}}}. \end{aligned}$$

Details of result as a universal solver

The most used dataset in 2L-CVRP is generated by [20]. It is for 2L-CVRP without TW, PD, HV, MU, queuing, and transit constraint. We use this dataset only to show that our algorithm could be used as a universal 2L-CVRP solver comparing to the existing methodology. It is not our intention to outperform state-of-the-art algorithms for the original 2L-CVRP. We used only the instances reporting the optimal solution. Each line in Table 9 represents the average of three runs executing the ASA for at most 10 s.

Table 9 Results of universal solver

Initialization with queuing and transit

Due to the Time-Window (TW) constraints accompanied by the limited number of docks for loading/unloading, which leads to a first-come-first-served queuing principle corresponds to A2, A3, we argue that it is not trivial to find a feasible solution at the initialization. For instance, assuming there are 24 shipments in a station, of which the load equals a truck capacity and the unloading/loading time is 1 hour, there might be no feasible solution.

To illustrate this viewpoint, we tried the following two experiments of initializations on the 311 shipments dataset to illustrate the queuing problem due to the limits of the docks at each unloading sites:

• In the first experiment, we firstly assigned all shipments within each cluster without considering the queuing process, equivalently assuming the number of the docks is unconstrained, by the Tabu search and validate the queuing process. Then, we collect all violated shipments and insert/assign them by RS. After the each-cluster optimizing, the mileage sum is 864.8 km, with one route containing 9 shipments violating the TW if considering queuing. After a complete ergodic search on insertions of these shipments, we fail to find a feasible solution.

• In the second experiment, we conduct initialization and iteratively optimize each cluster. For each cluster, we add the A2 constraint to the previous clusters at each iteration. However, we still fail to find a feasible by this way at the 4th cluster (out of 6), where the routes from previous clusters prevent feasibility at this cluster. The reason might be that some previous routes share first-few stations with the current cluster, and any new trucks in these stations will cause queuing and delay previous routes.

Our initialization process is to mimic manual planning in the SAIC, which is proved to find a good feasible initial solution in most realistic situations. However, as we argued above, there must exist cases that we cannot identify a feasible solution by this method. In general, many papers that concern TWs [17] initialize solutions by assuming infinite vehicles and assigning shipments each. It might not work in our case because the combinations of each feasible route may produce an infeasible solution due to queuing.

So, seeking a better initialization is truly worthy of further investigation and will be our future orient of research.