Formulating a Mixed Integer Programming Problem toImprove Solvability
Barnhart et al. 1993
Japhet Niyobuhungiro
June 16, 2015
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 1 / 14
A standard formulation of a real-world distribution problem could notbe solved, even for a good solution, by a commercial mixed integerprogramming code
Reformulating it by reducing the number of 0-1 variables andtightening the linear programming relaxation
An optimal solution could be found efficiently.
Purpose of the paper
Demonstrate, with a real application, the practical importance of theneed for good formulations in solving MIP problems
Company (Baxter Healthcare Corporation, Distribution ServiceDivision) uses
Short term studies: analyze how existing product flows/mode choicesshould change to obtain cost reductionManufacturing initiated studies: determine how existing product flowsshould change to respond to production site changesStrategic network design: decide where new replenishment centersshould be located given future production and distribution centernetwork plans
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 2 / 14
A standard formulation of a real-world distribution problem could notbe solved, even for a good solution, by a commercial mixed integerprogramming code
Reformulating it by reducing the number of 0-1 variables andtightening the linear programming relaxation
An optimal solution could be found efficiently.
Purpose of the paper
Demonstrate, with a real application, the practical importance of theneed for good formulations in solving MIP problems
Company (Baxter Healthcare Corporation, Distribution ServiceDivision) uses
Short term studies: analyze how existing product flows/mode choicesshould change to obtain cost reductionManufacturing initiated studies: determine how existing product flowsshould change to respond to production site changesStrategic network design: decide where new replenishment centersshould be located given future production and distribution centernetwork plans
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 2 / 14
A standard formulation of a real-world distribution problem could notbe solved, even for a good solution, by a commercial mixed integerprogramming code
Reformulating it by reducing the number of 0-1 variables andtightening the linear programming relaxation
An optimal solution could be found efficiently.
Purpose of the paper
Demonstrate, with a real application, the practical importance of theneed for good formulations in solving MIP problems
Company (Baxter Healthcare Corporation, Distribution ServiceDivision) uses
Short term studies: analyze how existing product flows/mode choicesshould change to obtain cost reductionManufacturing initiated studies: determine how existing product flowsshould change to respond to production site changesStrategic network design: decide where new replenishment centersshould be located given future production and distribution centernetwork plans
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 2 / 14
A standard formulation of a real-world distribution problem could notbe solved, even for a good solution, by a commercial mixed integerprogramming code
Reformulating it by reducing the number of 0-1 variables andtightening the linear programming relaxation
An optimal solution could be found efficiently.
Purpose of the paper
Demonstrate, with a real application, the practical importance of theneed for good formulations in solving MIP problems
Company (Baxter Healthcare Corporation, Distribution ServiceDivision) uses
Short term studies: analyze how existing product flows/mode choicesshould change to obtain cost reductionManufacturing initiated studies: determine how existing product flowsshould change to respond to production site changesStrategic network design: decide where new replenishment centersshould be located given future production and distribution centernetwork plans
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 2 / 14
Company’s hierarchical distribution system gew in size andcomplexity. Distribution model became nearly impossible to solve
New formulation
The original formulation, running on a commercial MIP code did notyield a feasible solution after > 100 hours of CPU time on a mainframeThe new formulation yielded a provably optimal solution in about 10min on a workstation
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 3 / 14
Company’s hierarchical distribution system gew in size andcomplexity. Distribution model became nearly impossible to solve
New formulation
The original formulation, running on a commercial MIP code did notyield a feasible solution after > 100 hours of CPU time on a mainframeThe new formulation yielded a provably optimal solution in about 10min on a workstation
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 3 / 14
Problem Description
Distribution network
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 4 / 14
Old model
Purpose: Meet demand at each DC (Distribution Center) whileminimizing shipping, handling, and inventory costs
Several other constraints must be met
Weigh-out/cube-out constraints: The weight and volume capacity of ashipping container cannot be violatedDCs must receive shipments with some minimum frequencyRCs (Replenishment Center(s)) have finite inventory capacities thatcannot be violatedMost PGs (Product Group(s)) must be single sourcedOnly one transportation mode may be used on a given RC-DC laneA DC must receive all of its shipments of a PG from the same RC
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 5 / 14
Old model
Purpose: Meet demand at each DC (Distribution Center) whileminimizing shipping, handling, and inventory costs
Several other constraints must be met
Weigh-out/cube-out constraints: The weight and volume capacity of ashipping container cannot be violatedDCs must receive shipments with some minimum frequencyRCs (Replenishment Center(s)) have finite inventory capacities thatcannot be violatedMost PGs (Product Group(s)) must be single sourcedOnly one transportation mode may be used on a given RC-DC laneA DC must receive all of its shipments of a PG from the same RC
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 5 / 14
Old model
Purpose: Meet demand at each DC (Distribution Center) whileminimizing shipping, handling, and inventory costs
Several other constraints must be met
Weigh-out/cube-out constraints: The weight and volume capacity of ashipping container cannot be violatedDCs must receive shipments with some minimum frequencyRCs (Replenishment Center(s)) have finite inventory capacities thatcannot be violatedMost PGs (Product Group(s)) must be single sourcedOnly one transportation mode may be used on a given RC-DC laneA DC must receive all of its shipments of a PG from the same RC
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 5 / 14
Old model
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 6 / 14
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 7 / 14
Product group NPG + 1 represents air-partially filled containers mayhave to be shipped to meet the minimum frequency!
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 8 / 14
Old model
Two major weaknesses1 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, 2015 9 / 14
Old model
Two major weaknesses1 Constraints are not as tight as they could be2 There are more 0 − 1 variables than are necessary
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 9 / 14
New formulation
cjkn = cost of shipping a container on mode k from RCj to DCn
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 10 / 14
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 11 / 14
Computational results
Two versions of the new model1 The disaggregated model
2 The aggregated model. Aggregate constraint set (N8)
∑i
xijkn ≤
[∑i
ain
]yjkn for all j , k, n.
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 12 / 14
Computational results
Two versions of the new model1 The disaggregated model2 The aggregated model. Aggregate constraint set (N8)
∑i
xijkn ≤
[∑i
ain
]yjkn for all j , k, n.
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 12 / 14
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 13 / 14
Conclusion
It is (sometimes!) far more efficient to perform simple preprocessingtasks, such as disaggregation, manually rather than depending on IPpreprocessing
For Solvability of Mixed Integer Programming Problems, (Strong)Formulation is key and very important
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 14 / 14
Conclusion
It is (sometimes!) far more efficient to perform simple preprocessingtasks, such as disaggregation, manually rather than depending on IPpreprocessing
For Solvability of Mixed Integer Programming Problems, (Strong)Formulation is key and very important
Japhet Niyobuhungiro (LiU) Discrete Optimization June 16, 2015 14 / 14
