## 3 Problem Description  
The last-mile logistics problem we consider in this work is a problem arising in this modern day. Each day drivers must distribute commodities to several locations. Delivery locations change daily depending on demands, and the list of locations to visit is available approximately one hour before the job assignment procedure starts. Our goal was to create a plan to utilize \(m\) vehicles to visit \(n\) locations but those vehicles should return to the depot within the time limit in order to avoid drivers to work overtime. In addition, vehicle times for all vehicles should be balanced which means the plan with lower vehicle time differences is more preferable. Finally, the plan should assign drivers to their original coverage areas as much as possible because the drivers have their own expertise in their responsible area. The drivers know shortcuts and detour routes which become very convenient when there are road incidents such as road blocks and extreme traffic congestion. Additionally, drivers gain familiarity to delivery protocols when visiting the same locations many times. Traffic information must be considered when creating a logistics plan. The traffic information, determining a vehicle speed, can be presented by the time-dependent traffic information and can be obtained by map APIs such as Longdo Map (Thailand map and road traffic application) [32], or Google Map. These APIs allow us to have a better travel time estimation between locations as the map providers use historical traffic information to predict road traffic congestion. For Longdo Map, the main source of the time-dependent traffic information is the data from Thai public data and GPS probes which are collected by the Thai Intelligent Traffic Information Center (iTIC) [33]. Time estimation obtained from the application is known to be one of the best traffic estimation tools in Thailand. To obtain traffic data, the map APIs require locations or points on Earth as a pair of longitude-latitude coordinates. Providing a request with two locations to the map APIs, we obtain travel duration of a _trip_ between the two locations. A trip starting at a specific time is referred to as a _trip instance_, and is a trip with the fastest route for the time being. Collecting a trip instance for every location pair generates a travel time matrix given that a vehicle departs at time instance \(t\). A collection of travel times for all locations for every time instance in the horizon is called the _time-dependent matrix_. We consider here a logistical planning problem with the assumption that the business has existing operations. This existing urban logistics plan provides city coverage for each driver. In this way, drivers will gain more area knowledge (e.g., the local road system, traffic issues) as they get assigned to the same coverage area repeatedly. The issue with the existing plan is the fact that there are significant differences of daily workload between drivers, and the visiting locations can vary depending on the daily orders. In such cases, some drivers may have to drop visits to some locations, or work overtime. Designing route for the last-mile logistics problem is known to be an NP-hard problem. Applying time-dependent travel matrices increases problem complexity [34]. It is almost impossible to provide the optimal solution for a real-world problem in a very limited computation time. Another computational challenge is the time taken for making requests for travel distances and times from the map API. Normally, times to enquire the distance and travel time between two locations at a time \(t\) is less than a second. However, the API request time grows at the order of \(O(n^{2})\) where \(n\) is the number of locations. The connection times can be reduced slightly if the API has an endpoint to collect information as a batch query or a matrix query.