Formulating a Mixed Integer Programming Problem to Improve Solvability

Transcription

1 Formulating a Mixed Integer Programming Problem to Improve Solvability Barnhart et al Japhet Niyobuhungiro June 16, 2015 Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

2 A standard formulation of a real-world distribution problem could not be solved, even for a good solution, by a commercial mixed integer programming code Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

3 A standard formulation of a real-world distribution problem could not be solved, even for a good solution, by a commercial mixed integer programming code Reformulating it by reducing the number of 0-1 variables and tightening the linear programming relaxation An optimal solution could be found efficiently. Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

4 A standard formulation of a real-world distribution problem could not be solved, even for a good solution, by a commercial mixed integer programming code Reformulating it by reducing the number of 0-1 variables and tightening the linear programming relaxation An optimal solution could be found efficiently. Purpose of the paper Demonstrate, with a real application, the practical importance of the need for good formulations in solving MIP problems Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

5 A standard formulation of a real-world distribution problem could not be solved, even for a good solution, by a commercial mixed integer programming code Reformulating it by reducing the number of 0-1 variables and tightening the linear programming relaxation An optimal solution could be found efficiently. Purpose of the paper Demonstrate, with a real application, the practical importance of the need for good formulations in solving MIP problems Company (Baxter Healthcare Corporation, Distribution Service Division) uses Short term studies: analyze how existing product flows/mode choices should change to obtain cost reduction Manufacturing initiated studies: determine how existing product flows should change to respond to production site changes Strategic network design: decide where new replenishment centers should be located given future production and distribution center network plans Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

6 Company s hierarchical distribution system gew in size and complexity. Distribution model became nearly impossible to solve Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

7 Company s hierarchical distribution system gew in size and complexity. Distribution model became nearly impossible to solve New formulation The original formulation, running on a commercial MIP code did not yield a feasible solution after > 100 hours of CPU time on a mainframe The new formulation yielded a provably optimal solution in about 10 min on a workstation Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

8 Problem Description Distribution network Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

9 Old model Purpose: Meet demand at each DC (Distribution Center) while minimizing shipping, handling, and inventory costs Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

10 Old model Purpose: Meet demand at each DC (Distribution Center) while minimizing shipping, handling, and inventory costs Several other constraints must be met Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

11 Old model Purpose: Meet demand at each DC (Distribution Center) while minimizing shipping, handling, and inventory costs Several other constraints must be met Weigh-out/cube-out constraints: The weight and volume capacity of a shipping container cannot be violated DCs must receive shipments with some minimum frequency RCs (Replenishment Center(s)) have finite inventory capacities that cannot be violated Most PGs (Product Group(s)) must be single sourced Only one transportation mode may be used on a given RC-DC lane A DC must receive all of its shipments of a PG from the same RC Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

12 Old model Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

13 Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

14 Product group NPG + 1 represents air-partially filled containers may have to be shipped to meet the minimum frequency! Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

15 Old model Two major weaknesses 1 Constraints are not as tight as they could be Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

16 Old model Two major weaknesses 1 Constraints are not as tight as they could be 2 There are more 0 1 variables than are necessary Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

17 New formulation c jkn = cost of shipping a container on mode k from RC j to DC n Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

18 Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

19 Computational results Two versions of the new model 1 The disaggregated model Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

20 Computational results Two versions of the new model 1 The disaggregated model 2 The aggregated model. Aggregate constraint set (N8) [ ] x ijkn a in y jkn for all j, k, n. i i Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

21 Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

22 Conclusion It is (sometimes!) far more efficient to perform simple preprocessing tasks, such as disaggregation, manually rather than depending on IP preprocessing Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14

23 Conclusion It is (sometimes!) far more efficient to perform simple preprocessing tasks, such as disaggregation, manually rather than depending on IP preprocessing For Solvability of Mixed Integer Programming Problems, (Strong) Formulation is key and very important Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, / 14