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 Sciences Department University of Wisconsin-Madison

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

3 Rockwell Model One CDC (Central Distribution Center) distributors 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 binary variable [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) Solve via branch & cut (CPLEX)

9 Algebraic Model

10 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) Modeling Notes

11 Hub&Spoke Scenarios Scenario Maximum Mileage (hub to spoke) maxHubs

12 CPLEX Results ScenarioTotal CostGap Solution Time(sec) Run Time (sec) 1$21,048,155 $657,155 (3.12%) 25010,000 2$17,216,529$ $18,004,390$ $19,099,000 $130,000 (0.68%) ,000 5$17,772,604$0510 6$16,332,781$0170

13 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 Features of Branch & Cut

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

15 CPLEX / Nested Partitions Hybrid (CPLEX/NP)

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

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 complex NP backtracking strategies Develop more sophisticated supply chain heuristics for NP Combine LP and heuristics to obtain node estimates