Hub Location under Uncertainty: a Minimax Regret Model for the Capacitated Problem with Multiple Allocations

In this paper the capacitated hub location problem is formulated by a minimax regret model, which takes into account uncertain setup cost and demand. We focus on hub location with multiple allocations as a strategic problem requiring one definite solution. Investigating how deterministic models may lead to sub-optimal solutions, we provide an efficient formulation method for the problem. A computational analysis is performed to investigate the impact of uncertainty on the location of hubs. The suggested model is also compared with an alternative method, seasonal optimization, in terms of efficiency and practicability. The results indicate the importance of incorporating stochasticity and variability of parameters in solving practical hub location problems. Applying our method to a case study derived from an industrial food production company, we solve a logistical problem involving seasonal demand and uncertainty. The solution yields a definite hub network configuration to be implemented throughout the planning horizon.

💡 Research Summary

The paper addresses the strategic design of hub‑and‑spoke logistics networks under uncertainty, focusing on a capacitated hub location problem (HLP) with multiple allocations. Traditional hub location models assume deterministic parameters, which can lead to sub‑optimal network configurations when real‑world costs and demands fluctuate. To overcome this limitation, the authors formulate a robust optimization model based on the minimax regret criterion.

The deterministic foundation of the model follows Campbell’s (1994) linear formulation. Decision variables include xᵢⱼᵏᵐ, the fraction of flow from origin i to destination j that passes through hub k and then hub m, and yₖ, a binary variable indicating whether node k is opened as a hub (1) or remains a spoke (0). The cost structure consists of a fixed hub setup cost Fₖ and three transportation components: collection cost (β·dᵢₖ), transfer cost (α·dₖₘ), and distribution cost (δ·dₘⱼ). The total transportation cost for a given i‑j‑k‑m combination is Cᵢⱼᵏᵐ = βdᵢₖ + αdₖₘ + δdₘⱼ. Constraints enforce flow continuity through selected hubs, respect hub capacity limits Γₖ, and maintain variable domains (binary yₖ, 0 ≤ xᵢⱼᵏᵐ ≤ 1).

Uncertainty is captured through two scenario sets: S⁰ for hub setup cost variations and Sʷ for demand fluctuations. For each cost scenario s⁰∈S⁰, the setup cost becomes Fₖ^{s⁰}; for each demand scenario s∈Sʷ, the origin‑destination demand is Wᵢⱼ^{s}. The regret of a solution under scenario s is defined as the difference between the solution’s total cost Z(s) and the optimal cost Z⁎(s) that could be achieved if the scenario were known in advance. The minimax regret model introduces an auxiliary variable R and adds constraints R ≥ Z(s) − Z⁎(s) for all scenarios. The objective is to minimize R, thereby limiting the worst‑case regret across all plausible realizations of cost and demand. Because the underlying problem is linear, the regret formulation remains a mixed‑integer linear program (MILP) that can be solved with commercial solvers.

The authors conduct computational experiments to compare three approaches: (1) a deterministic model that uses average parameter values, (2) a seasonal‑optimization approach that solves a separate deterministic model for each season and aggregates the results, and (3) the proposed minimax regret model. Results show that the deterministic model can suffer large cost overruns in adverse scenarios, especially when demand spikes or setup costs increase. Seasonal optimization improves performance but still leaves significant variability because each season is optimized in isolation. In contrast, the minimax regret model delivers a single hub configuration that keeps the maximum regret low, leading to a more stable total cost across all scenarios. The benefit is most pronounced when hub capacities are tight; the regret‑based solution tends to locate hubs in positions that can accommodate peak demand, avoiding capacity violations.

A real‑world case study involves an Iranian food‑production company that experiences pronounced seasonal demand and volatile land/equipment costs. Using historical demand data and cost estimates, the authors generate a set of demand and cost scenarios, then solve the minimax regret model. Compared with the company’s existing hub network, the new configuration reduces total logistics and setup cost by approximately 7 % and eliminates bottlenecks during peak seasons. Importantly, the model provides a single, definitive hub layout that can be implemented for the entire planning horizon, satisfying the strategic nature of the decision.

Key contributions of the paper are:

1. Integration of capacity constraints and multiple allocations into a linear hub location framework, preserving tractability.
2. Introduction of a scenario‑based minimax regret formulation that handles both setup‑cost and demand uncertainty without requiring probability distributions.
3. Empirical evidence that the regret‑based solution outperforms both naive deterministic planning and season‑by‑season optimization in terms of cost stability and overall savings.
4. Demonstration of practical applicability through a detailed industrial case study, delivering a robust, implementable hub network.

The authors suggest several avenues for future research: extending the model to incorporate additional sources of uncertainty such as hub failure or re‑configuration costs, developing a multi‑period dynamic version that captures the timing of hub openings and closures, and exploring decomposition techniques (e.g., Benders or column generation) to solve larger‑scale instances. Overall, the study showcases how robust optimization, specifically minimax regret, can bridge the gap between theoretical hub location models and the uncertain realities faced by logistics managers.