# Optimizing Rural Highway Maintenance Scheme with Mathematical Programming
###### Abstract
Therefore, an irrational highway repair strategy could lead to numerous traffic bottlenecks, which would be detrimental to system users. Additionally, one of the key factors limiting the highway maintenance timetable is the availability of funds. The operators of rural transportation systems must create optimal maintenance schedules within a limited budget, necessitating greater adaptability in mode selection. To address this problem, we propose using bi-level mathematical programming. At the lower level, the traffic flow generated by the maintenance scheme is determined, while, at the higher level, the road segment combination to be maintained is decided. This issue has been studied extensively and is known as the Discrete Network Design Problem (DNDP). The DNDP is an NP-hard problem. To balance the computing speed and precision, several methods have been developed [32; 33; 34; 35; 36]. Although large-scale networks present a  
Figure 1: (**a**) A pavement management system that detects pavement damage. (**b**) A typical rural pavement with crack damage. well-known NP-hard challenge, small-scale networks can still be solved. Furthermore, by incorporating binary variables, we reformulate the proposed bi-level mathematical programming into a single-level mathematical programming problem. This allows the DNDP to be solved using effective solvers like Gurobi ([[https://www.gurobi.com/](https://www.gurobi.com/)]([https://www.gurobi.com/](https://www.gurobi.com/))) and CPLEX ([[https://www.ibm.com/products/ilog-cplex-optimization-studio](https://www.ibm.com/products/ilog-cplex-optimization-studio)]([https://www.ibm.com/products/ilog-cplex-optimization-studio](https://www.ibm.com/products/ilog-cplex-optimization-studio))). We present a reliable model for the optimization of the rural transportation system. Rural road network administrators can use this model to determine the number of links that they wish to maintain concurrently while adhering to specific constraints, including financial plans, maintenance schedules, and other guidelines. This model enables the development of policies that minimize the total general cost of the rural transportation system. The following summarizes the primary findings of this study:  
1. We suggest using bi-level mathematical programming to optimize maintenance projects for rural roads, aiming to reduce the total costs and disruptions to traffic;
2. By incorporating binary variables, we transform the proposed bi-level mathematical programming into a single-level mathematical programming problem, allowing solvers like Gurobi and CPLEX to be used to solve the DNDP;
3. We validate our proposed models on four benchmark example networks--the Eastern Massachusetts network, the Sioux Falls network, the Nguyen-Dupuis network, and the Braess network--providing practical insights into model implementation. The remainder of this paper is organized as follows. In the second section, we present a mathematical model for the DNDP at two levels. In Section 3, we reformulate the suggested bi-level mathematical model into a single-level form that is amenable to commercial solvers. In Section 4, we validate the proposed model on four benchmark networks: the Eastern Massachusetts network, the Sioux Falls network, the Braess network, and the Nguyen-Dupuis network. Finally, we conclude in Section 5.