Mathematical modelling and solution approaches to tackle challenges in real-life routing problems
Creators: name: Pahlevani, Delaram
Publisher: name: RMIT University
Year: 2024
Subjects: classificationCode: 490108
subject: Operations research
classificationCode: 490404
subject: Combinatorics and discrete mathematics (excl. physical combinatorics)

description: This thesis aims to develop an optimisation models to tackle real-life routing and scheduling problems. Routing and scheduling problems are categorised as a special variant of vehicle routing problems. Operations research offers a variety of techniques to utilise mathematical tools for different practical problems. We concentrate on disaster management with a particular focus on wildﬁres and scheduling for home health care providers. These two problems share many features in common, such as the need to to provide services with varying capacities and capabilities in a timely fashion.

Australia is prone to natural disasters, and wildfires are inevitable events. Studies of previous wildfire disasters in Australia show that an accurate response plan is a key to effectively battling wildfires and protecting community assets. However, quickly and efficiently conducting a response plan for a disaster is challenging. Many things must be considered, such as the limited number of resources. Fire propagates quickly; therefore, the mission is to find a plan to minimise damage to community assets. The problem is known as Asset Protection Problem (APP), which got scholars’ attention in the past few years. While several studies were found in the literate to utilise optimisation techniques, the presented work is not quite the same as found in an actual operation. After a deep investigation of the problem-specific, we realised the capacity of vehicles is a challenging constraint to running operations. A vehicle with an empty tank is no longer able to protect assets. We introduced Capacitated Asset Protection Problem (CAPP) and considered the reloading capability for vehicles when the amount of water is not sufficient to save the remaining assets. Considering capacity and possibility for a vehicle to refill makes the problem more similar to the real operation and leads to a more accurate operations plan.

This study first presents a deterministic model for CAPP. The problem contains a set of nodes and a set of vehicles. A reward is a value associated with each asset. The model depends on time windows determining the earliest and the latest time that asset protection can be started. Each asset is labelled with a protection vector indicating an asset's requirements. The ultimate goal is maximising the total value of protected assets. Extensive computational experiments have been conducted to assess the performance of the model. The results suggest the response plan could be a valuable decision-aid for a co-ordinating manager.

In addition to the deterministic CAPP, we realised the necessity of stochastic optimisation for this problem. The previous works for bushfires reveal that it is likely to observe a change in wind direction. The research has been expanded to address uncertainty in the timing of wind change. It bridges the gap between the model and practice by keeping the capacity and reloading possibility as practical constraints. A change in wind direction will directly affect the time windows of each asset that was initially determined. Our model can handle various scenarios regarding the timing of wind change over the planning horizon. The model considers the probability of each scenario and proposes a plan to maximise the total rewards collected. This thesis conducts numerous computational experiments to evaluate the performance of the developed model under different settings. The results suggest that the presented model incorporates realistic constraints and generates a solution in a reasonable computational time. 

This thesis also studies Home Health Care (HHC) planning which shows several similarities with wildfire management. HHC refers to delivering social, medical and paramedical services to people in their homes. Caregivers are assigned and routed to perform various tasks such as personal care and household chores at the client's homes. Minimising the total cost and satisfying the client's requirements and preferences are critical in HHC. In this thesis, we present a mixed-integer linear programming model for HHC routing and scheduling problems, which considers fair and balanced workload allocation of caregivers while minimising the total cost and addressing the client's needs.

We realised that Australian Home Care Providers (HCP) suffer from an inefficient plan and must pay a significant amount of money to workers for travel reimbursement. In HHC optimisation, a set of workers must be assigned to clients despite many challenging constraints. Time windows which indicate the earliest and the latest time to visit a client, play an essential role in this model. Each client may require a specific caregiver, which forces a scheduler to find a skilled worker. Expectation from a client is not restricted to skills requirements, and they may have other preferences related to their caregiver (language-wise, gender-wise). Furthermore, each worker can specify the type of activities or gender of clients that she/he wants to serve. Time windows for workers must be specified in the input sample. The problem also contains many practical constraints. For instance, the model must be able to suggest a route for each worker in which the distance between any of two locations is less than 15 km. A home care provider must pay double travel costs to a worker if this is not the case. 

Our developed model has been tested using commercial solvers. While they are helpful for small-sized instances, they do not show any success in generating a solution for large-sized instances. To address this issue, we introduced a multi-step clustering algorithm that aims to subgroup the entire sample to find the most similar clients in terms of preferences, requirements and geographic location but with the least similarity in visiting time. Defining clusters in which the members are required to be visited simultaneously increases the need for the number of workers. The final solution finds the minimum travel distance and suggests the route for each worker. Disagree The performance of our algorithm was tested using various datasets. A comprehensive comparison between the proposed and current home care plans was conducted for real datasets. The results demonstrated an impressive enhancement for the entire operation.