Sparsity-Constrained Transportation Problem

Transcription

1 MSOM 204 Conference June 20, 204 Sparsity-Constrained Transportation Problem Annie I. Chen Stephen C. Graves Massachusetts Institute of Technology

2 Inventory Positioning for Online Retailers Large scale: Fulfillment centers (FCs): tens Customer demand zones: hundreds Items (SKUs): millions Integrated and centralized control: Customers served by FC not limited to geographic location Not all items have to be stocked everywhere 204/6/20 A. I. Chen and S. Graves (MIT) 2

3 Sparsity Constraints Limit the number of FCs that can carry an item There may be other types of sparsity constraints, e.g., limit the number of items in each FC. Benefits: Reduce operational complexity (fewer FCs/items to keep track of) Reduce fixed costs (e.g. transportation to FC) Reduce variability (risk pooling of demand zones) Trade-off: fulfillment costs 204/6/20 A. I. Chen and S. Graves (MIT) 3

4 Sparsity-Constrained Transportation Problem Given: Demand Extension: item affinity FC capacity Throughput, storage Fulfillment cost Distance, weight, time Item sparsity # FCs carrying the item Decision: Flow amount FC usage 3 Fulfillment Centers /3 /4 /3 A B C Demand Zones 2 204/6/20 A. I. Chen and S. Graves (MIT) 4

5 Sparsity-Constrained Transportation Problem Given: Demand Extension: item affinity FC capacity Throughput, storage Fulfillment cost Distance, weight, time Item sparsity # FCs carrying the item Decision: Flow amount FC usage??? Fulfillment Centers /3 /4 /3 A B C What if we only used 2 FCs? Demand Zones 2 204/6/20 A. I. Chen and S. Graves (MIT) 5

6 Sparsity-Constrained Transportation Problem Given: Demand Extension: item affinity FC capacity Throughput, storage Fulfillment cost Distance, weight, time Item sparsity # FCs carrying the item Decision: Flow amount FC usage A B A C B C /6/20 A. I. Chen and S. Graves (MIT) 6

7 Sparsity-Constrained Transportation Problem Given: Demand Extension: item affinity FC capacity Storage, throughput Fulfillment cost Distance, weight, time Item sparsity # FCs carrying the item Decision: Flow amount FC usage Fulfillment Centers... u Demand Zones... v 204/6/20 A. I. Chen and S. Graves (MIT) 7

8 MIP Formulation u v 204/6/20 A. I. Chen and S. Graves (MIT) 8

9 Solution Approaches. Sparsify-Improve Decompose sparsity constraint Conceptually similar to subgradient projection 2. Column Generation Branch-and-price for large scale MIP 3. Capacity Allocation Decompose by item Conceptually similar to primal decomposition 204/6/20 A. I. Chen and S. Graves (MIT) 9

10 Decompose sparsity constraint (Master Problem) 204/6/20 A. I. Chen and S. Graves (MIT) 0

11 The Sparsify-Improve Algorithm Initialize: start with all FCs active for all items Step : Sparsify Generate a sparse solution by progressively eliminating the least active FC-item pair Step 2: Improve Explore other sparse solutions by heuristically swapping FCs Output: sparse near-optimal solution for Master Problem (Master Problem) 204/6/20 A. I. Chen and S. Graves (MIT)

12 Sparsify-Improve in Pictures Sparsify Improve 204/6/20 A. I. Chen and S. Graves (MIT) 2

13 Numerical Experiment: Setup # of items:, 2, 4, 8, 6, 32, 64 For each item setting, 0 random graphs: 30 fulfillment centers (each with random capacity) 00 demand zones (each with random demand) Sparsity limit = 5 for all items Implemented in Python + Gurobi Comparison: MIP formulation Results: ~5% sub-optimal in 30-40% time 204/6/20 A. I. Chen and S. Graves (MIT) 3

14 Numerical Experiment: Results # of items:, 2, 4, 8, 6, 32, 64 For each item setting, 0 random graphs: 30 fulfillment centers (each with random capacity) 00 demand zones (each with random demand) Sparsity limit = 5 for all items Implemented in Python + Gurobi Comparison: MIP formulation Results: ~5% sub-optimal in 30-40% time 204/6/20 A. I. Chen and S. Graves (MIT) 4

15 Solution Approaches. Sparsify-Improve Decompose sparsity constraint Conceptually similar to subgradient projection 2. Column Generation Branch-and-price for large scale MIP 3. Capacity Allocation Decompose by item Conceptually similar to primal decomposition 204/6/20 A. I. Chen and S. Graves (MIT) 5

16 Column Generation Select best pattern from a discrete set Simplifying assumption: each demand zone served by only one FC Pattern: a vector specifying which FC serves each demand zone Master problem: choose best pattern for each item Restricted master problem: choose best pattern from a subset of patterns Pricing subproblem: generate new patterns (columns) and add to subset 204/6/20 A. I. Chen and S. Graves (MIT) 6

17 Solution Approaches. Sparsify-Improve Decompose sparsity constraint Conceptually similar to subgradient projection 2. Column Generation Branch-and-price for large scale MIP 3. Capacity Allocation Decompose by item Conceptually similar to primal decomposition 204/6/20 A. I. Chen and S. Graves (MIT) 7

18 Capacity Allocation Model a continuous spectrum of capacity allocations (rather than discrete usage patterns) View each item (and each FC) as a decisionmaking agent and decompose by item. FC subproblem: capacity allocation Capacity Marginal cost Item subproblem: min-cost flow 204/6/20 A. I. Chen and S. Graves (MIT) 8

19 Conclusion Sparsity is practical in online retail inventory positioning. We proposed algorithms that solve the sparsity-constrained transportation problem:. Sparsify-Improve: decomposes sparsity constraint; gives near-optimal solutions. 2. Column Generation Ongoing work 3. Capacity Allocation Related problems: What is the right sparsity level? (Sensitivity and cost-benefit analysis) Scaling up: Decomposition by items? Aggregation? Network design: flexibility and risk Questions? Comments? 204/6/20 A. I. Chen and S. Graves (MIT) 9

20 References Agatz, N. A., Fleischmann, M., and van Nunen, J. A. (2008). E-fulllment and multi-channel distribution-- a review. European Journal of Operational Research, 87: Swaminathan, J. and Tayur, S. (2003). Models for supply chains in e- business. Management Science, 49(0): Bahmani, S., Raj, B., and Boufounos, P. (203). Greed sparsity-constrained optimization. Journal of Machine Learning Research, 4: Beck, A. and Eldar, Y. C. (202). Sparsity constrained nonlinear optimization. Johnson, M. and Whang, S. (2002). E-business and supply chain management: An overview and framework. Production and Operations Management, (4): Barkhard, Johnson, Nemhauser, Savelsbergh, and Vance (998). Branchand-Price: Column Generation for Solving Huge Integer Programs. Operations Research, 46(3): /6/20 A. I. Chen and S. Graves (MIT) 20