[Paper Review] Sparsity-Constrained Transportation Problem

Annie I. Chen, Stephen C. Graves|arXiv (Cornell University)|Feb 10, 2014

Optimization and Mathematical Programming4 references3 citations

TL;DR

This paper proposes a two-stage Sparsify-Improve algorithm to solve large-scale transportation problems with sparsity constraints on inbound flows, enabling efficient inventory positioning in online retail supply chains. The method achieves near-optimal solutions (average gap of 5%) up to 166x faster than mixed-integer programming, with scalability to 64 items where MIP fails due to memory limits.

ABSTRACT

We study the solution of a large-scale transportation problem with an additional constraint on the sparsity of inbound flows. Such problems arise in the management of inventory for online retailers that operate with many order fulfillment centers; each stock-keeping unit is typically kept in a limited number of these order fulfillment centers so as to reduce the operational overhead in the supply chain network. We propose a computationally efficient algorithm that solves this sparsity-constrained optimization problem while bypassing complexities of the conventional integer programming approach. The effectiveness of the algorithm is demonstrated through a series of numerical experiments on synthetic data.

Motivation & Objective

• To address the challenge of inventory positioning in large-scale online retail networks where each product must be stocked in only a limited number of fulfillment centers to reduce operational overhead.
• To develop a computationally efficient alternative to mixed-integer programming (MIP) for solving large-scale transportation problems with sparsity constraints on inbound flows.
• To enable scalable, near-optimal solutions for sparse-inbound transportation problems that are intractable for conventional MIP approaches due to memory and time constraints.
• To demonstrate the effectiveness of the proposed algorithm through numerical experiments on synthetic data with varying problem sizes.

Proposed method

• The Sparsify-Improve algorithm consists of two stages: first, a greedy sparsification step selects a subset of fulfillment centers for each item based on cost and flow magnitude, enforcing the sparsity constraint.
• Second, a local improvement phase uses linear programming to refine the solution by adjusting flows within the sparsity constraints to reduce total cost.
• The sparsify stage uses a heuristic that prioritizes fulfillment centers with high flow contributions and low marginal cost, ensuring sparsity while maintaining feasibility.
• The improve stage solves a sequence of sparse linear programs (sparse-LPs) to optimize flow distribution under the fixed set of active fulfillment centers per item.
• The algorithm is parameterized with $k_1 = \lceil \sqrt{|I|} \rceil$ for sparsification and $k_2 = 20$ for improvement iterations, tuned for computational efficiency.
• The method avoids the integrality constraints of MIP by treating the sparsity constraint as a combinatorial selection problem solved via iterative local optimization.

Experimental results

Research questions

• RQ1 Can a heuristic two-stage algorithm outperform mixed-integer programming in solving large-scale transportation problems with sparsity constraints on inbound flows?
• RQ2 How does the Sparsify-Improve algorithm maintain solution quality (optimality gap) while significantly reducing computation time compared to MIP?
• RQ3 What is the scalability of the algorithm in terms of problem size, particularly when MIP fails due to memory limitations?
• RQ4 How do algorithm parameters $k_1$ and $k_2$ affect the trade-off between solution quality and computational effort?
• RQ5 Can the proposed method be effectively applied to real-world online retail inventory positioning problems with thousands of items and fulfillment centers?

Key findings

• The Sparsify-Improve algorithm achieves an average optimality gap of 5% across all tested problem sizes, with solutions consistently within 5% of the MIP-optimal cost.
• For problems with 64 items, the MIP approach ran out of memory and failed to produce results, while the Sparsify-Improve algorithm completed in 1,058.28 seconds.
• The algorithm reduced average computation time by a factor of 166.09 compared to MIP for the smallest instance ($|I|=1$), with performance gains increasing with problem size.
• On average, the Sparsify-Improve algorithm was 44.03% faster than MIP for the $|I|=8$ case, and the speedup ratio increased to 29.77x for $|I|=16$.
• The algorithm demonstrated robustness across 10 random network instances per problem size, with consistent performance and low variance in solution quality.
• The method successfully handled problems with up to 64 items and 100 demand zones, showing scalability where MIP is infeasible due to memory constraints.