Stochastic Programming for Dynamic Temperature Control of Refrigerated Road Transport

TL;DR

The paper tackles maintaining product quality in refrigerated road transport under uncertainties in door openings and initial product temperatures by integrating a high-fidelity thermodynamic trailer model into a multistage stochastic programming framework solved with stochastic dual dynamic programming (SDDP). It compares four policies—SP, rolling lookahead (RLP), and two myopic on/off heuristics (H1, H2)—on a detailed case study with four real routes, demonstrating that SP substantially reduces temperature violations (about 35% on average) and can cut fuel use by up to 40% under excursion constraints, with precooling and product-temperature monitoring identified as key drivers. The study provides managerial insights, notably the value of monitoring pallet-level product temperatures, the importance of knowing stop durations, and the effectiveness of preemptive cooling; it also shows SP closely approaches a perfect-information benchmark, highlighting the practical viability of sophisticated stochastic cooling policies. The findings offer a clear path toward more reliable cold-chain operations, lower energy consumption, and reduced carbon footprints, while suggesting directions for extending the model to broader routing/inventory problems and more detailed environmental factors.

Abstract

Temperature control in refrigerated delivery vehicles is critical for preserving product quality, yet existing approaches neglect critical operational uncertainties, such as stochastic door opening durations and heterogeneous initial product temperatures. We propose a framework to optimize cooling policies for refrigerated trucks on fixed routes by explicitly modeling these uncertainties while capturing all relevant thermodynamic interactions in the trailer. To this end, we integrate high-fidelity thermodynamic modeling with a multistage stochastic programming formulation and solve the resulting problem using stochastic dual dynamic programming. In cooperation with industry partners and based on real-world data, we set up computational experiments that demonstrate that our stochastic policy consistently outperforms the best deterministic benchmark by 35% on average while being computationally tractable. In a separate analysis, we show that by fixing the duration of temperature violations, our policy operates with up to $40$\% less fuel than deterministic policies. Our results demonstrate that pallet-level thermal status information is the single most crucial information in the problem and can be used to significantly reduce temperature violations. Knowledge of the timing and length of customer stops is the second most important factor and, together with detailed modeling of thermodynamic interactions, can be used to further significantly reduce violations. Our analysis of the optimal stochastic cooling policy reveals that preemptive cooling before a stop is the key element of an optimal policy. These findings highlight the value of sophisticated control strategies in maintaining the quality of perishable products while reducing the carbon footprint of the industry and improving operational efficiency.

Figures (6)

• Figure 1: Temperature profiles of the air and products inside the trailer during a trip with a generic stop at a customer. The left panel shows a pre-emptive cooling strategy that cools the air and, consequently, the products in anticipation of the stop, while the right panel shows a purely reactive policy that starts cooling once the air temperature leaves a certain predefined range.
• Figure 2: The time structure of the multi-stage stochastic program: the truck starts at the depot at time $0$ and then loads/unloads refrigerated goods at $S$ stops. The time between the stops are the stages of the stochastic program, each consisting of a driving phase and all but the first of a loading/unloading phase. Within each stage, there is a finer temporal structure used to model thermodynamic interactions and decisions about cooling.
• Figure 3: Logarithmic scale representation of the average performance of each policy [⋅], varying route (left), refrigeration unit capacity [k ] (center), and heat transfer coefficient [WmK] (right).
• Figure 4: Evolution of average product temperatures for all out-of-sample scenarios for route R1 with unit capacity $12$ and $h=4$. The $x$ axis represents actual clock time, which results in different trip lengths due to the random door opening duration. Green periods correspond to handling operations with open doors, while blue periods represent driving.
• Figure 5: Effect of random components on average violations [⋅] obtained by SP (left), value added associated with the information available to each policy, expressed as a percentage of average H1 violations (right).