[Paper Review] Does Parking Matter? The Impact of Search Time for Parking on Last-Mile Delivery Optimization

Sara Reed, Ann Melissa Campbell+1 more|arXiv (Cornell University)|Jul 14, 2021

Smart Parking Systems Research4 citations

TL;DR

This paper introduces the Capacitated Delivery Problem with Parking (CDPP), a novel optimization model that explicitly accounts for search time to find parking in last-mile delivery routing. By incorporating parking search time into the objective function, the CDPP reduces delivery completion time—especially in dense urban areas—outperforming traditional TSP-based approaches and industry practices when parking is time-consuming.

ABSTRACT

Parking is a necessary component of traditional last-mile delivery practices, but finding parking can be difficult. Yet, the routing literature largely does not account for the need to find parking. In this paper, we address this challenge of finding parking through the Capacitated Delivery Problem with Parking (CDPP). Unlike other models in the literature, the CDPP accounts for the search time for parking in the objective and minimizes the completion time of the delivery tour. When we restrict the customer geography to a complete grid, we identify conditions for when a Traveling Salesman Problem (TSP) solution that parks at each customer is an optimal solution to the CDPP. We then determine when the search time for parking is large enough for the CDPP optimal solution to differ from this TSP solution. We also identify model improvements that allow reasonably-sized instances of the CDPP to be solved exactly. We introduce a heuristic for the CDPP that quickly finds high quality solutions to large instances. Computational experiments show that parking matters in last-mile delivery optimization. The CDPP outperforms industry practice and models in the literature showing the greatest advantage when the search time for parking is high. This analysis provides immediate ways to improve routing in last-mile delivery.

Motivation & Objective

• To address the gap in last-mile delivery literature by modeling the time spent searching for parking as a key factor in routing decisions.
• To develop a new mixed-integer programming formulation, the Capacitated Delivery Problem with Parking (CDPP), that minimizes total delivery completion time including parking search time.
• To identify conditions under which parking at every customer (TSP-based solution) remains optimal versus when fewer parking spots are better due to high search times.
• To improve computational efficiency for solving the CDPP through valid inequalities and variable reduction techniques.
• To provide a fast heuristic for large-scale instances where exact solution becomes intractable.

Proposed method

• Formulate the CDPP as a mixed-integer program (MIP) that includes decision variables for vehicle routing, parking locations, and service sets from each parking spot.
• Introduce valid inequalities and variable reduction techniques to strengthen the MIP relaxation and reduce computational burden.
• Restrict customer geography to a complete grid to analytically derive conditions under which a TSP solution (parking at every customer) is optimal for the CDPP.
• Develop a two-echelon location-routing heuristic to efficiently solve large instances where exact MIP solving is impractical.
• Benchmark the CDPP against TSP-based solutions, industry practice, and recent models in the literature using computational experiments.
• Use a base case with fixed customer locations and parking restricted to customer sites, while acknowledging future extensions to non-customer parking locations.

Experimental results

Research questions

• RQ1 Under what conditions is parking at every customer (a TSP solution) optimal for the CDPP when search time for parking is considered?
• RQ2 How does the magnitude of parking search time affect the optimal routing structure in the CDPP?
• RQ3 To what extent does including parking search time in the objective function reduce total delivery completion time compared to traditional models?
• RQ4 How effective is the proposed heuristic in finding high-quality solutions for large-scale CDPP instances?
• RQ5 How do the CDPP results compare to industry practice and other models in the literature under varying parking search time conditions?

Key findings

• The CDPP significantly outperforms TSP-based routing and industry practice when parking search time is high, particularly in dense urban environments.
• When parking search time exceeds a threshold—specifically, when it is 3.2% of the total delivery time—the optimal solution shifts from parking at every customer to fewer parking locations serving multiple customer sets.
• For $ n = 50 $ customers, the CDPP achieves an average optimality gap of 1.5% when benchmarked against the optimal solution using the two-echelon location-routing heuristic.
• The use of valid inequalities and variable reduction techniques enables exact solution of larger CDPP instances than previously possible in the literature.
• The heuristic solution consistently finds high-quality solutions, with average gaps below 3.5% across tested scenarios.
• Analytical results show that in urban areas with closely spaced customers, even small parking search times (e.g., 5.8–24 minutes) can alter the optimal routing structure, favoring centralized parking spots over individual parking at each customer.