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Global Meta-Hybrids for Large-Scale Combinatorial Optimization

Global Meta-Hybrids for Large-Scale Combinatorial Optimization. Professor Leyuan Shi Department of Industrial Engineering University of Wisconsin-Madison Dan S. Ludwikoski Rockwell Automation Professor Robert Meyer Computer Sciences Department University of Wisconsin-Madison.

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Global Meta-Hybrids for Large-Scale Combinatorial Optimization

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  1. Global Meta-Hybrids for Large-Scale Combinatorial Optimization Professor Leyuan Shi Department of Industrial Engineering University of Wisconsin-Madison Dan S. Ludwikoski Rockwell Automation Professor Robert Meyer Computer SciencesDepartment University of Wisconsin-Madison

  2. Distributors Suppliers CDC Assembly Cell DMI Customers Electronic Pull Lead Time Manufacturing Inventory True Demand Finished Goods Inventory 1 3 2 Supply Chain Network

  3. Rockwell Model • One CDC (Central Distribution Center) • 544distributors • All product families aggregated into one family • Multi-level transportation costs • No Manufacturing Assembly considered • 961 constraints • 4572 variables

  4. Hub & Spoke Model Identify the best locations (distributors to be used as hubs) while meeting each distributor’s demand and minimizing total cost. A hard problem for large supply chains.

  5. Hub & Spoke

  6. Parameters d(i,j) = distance in miles from distributor i to distributor j K(i) = Facility cost at distributor i D(i) = Demand in lbs at distributor i c(0, i) = shipping cost from CDC to distributor i c(i, j) = shipping cost from distributor i to distributor j h(j) = handling cost maxHubs = maximum number of hubs allowed in the system maxDist = maximum range in miles that a Hub can serve

  7. Variables z(0, i) = amount shipped from CDC to distributor i z(i, j) = amount shipped from distributor i to distributor j x(i) = hub&spoke binaryvariable [x(i)=1 if distributor i chosen to be a hub, x(i)=0 if distributor i is a spoke]

  8. Traditional Approach • Model the problem as a MIP (Mixed Integer Program) • Solvevia branch & cut(CPLEX)

  9. Algebraic Model 1 2 3 4

  10. Modeling Notes Objective is sum of shipping, handling, location costs Net flow into distributor i must equal demand Total number of hubs in supply chain <maxHubs If distributor i receives shipments from the CDC, then it is a hub (and its location cost is charged in objective) 1 2 3 4

  11. Hub&Spoke Scenarios

  12. CPLEX Results

  13. Features of Branch & Cut • For easy problems, Branch&Cut is fast. • However, Branch & Cut algorithms often fail to provide quality solutions within acceptable time frames. • Additional disadvantages of Branch & Cut : • They do not employ/allow problem-specific heuristics • Data from known feasible solutions not very helpful • Tree size can be huge, leading to memory problems

  14. CPLEX / NP Hybrid • Start with problem specific preprocessing • Run CPLEX for a short period, obtaining a feasible solution • Utilize feasible solution from CPLEX to initialize the NP approach • Using the result from NP to provide a cutoff, restart CPLEX

  15. CPLEX / Nested Partitions Hybrid (CPLEX/NP)

  16. Interface: CPLEX/NP Solver User Initial solution via CPLEX Partitioning Sampling Promising Index (CPLEX) Backtracking Input interface NP Manual / Excel Graphical solution interface User

  17. CPLEX/NP vs. CPLEX Best objective obtained on hub/spoke model: CPLEX (using NP’s pre-processing):$21,197,214 Gap: $701,214 (total run time : 2950 sec.) CPLEX (default):$21,048,155 Gap: $657,155 (total run time: 172,000 sec) CPLEX/NP:$20,769,200 Gap: $273,200 (total run time : 2950 sec.)

  18. Conclusions • CPLEX/NP hybrid significantly outperforms CPLEX • CPLEX provides starting point and LB • NP able to obtain a higher quality solution • NP runs quickly • Hybrid approach much more powerful than CPLEX or NP • as stand-alone methods

  19. Future Research • Extend model to more product families and customers • Consider more complexNP backtracking strategies • Develop more sophisticated supply chain heuristics for NP • Combine LP and heuristics to obtain node estimates

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