Understanding DoCPLEX Multi Objective

Question

I am working on Pure LP problem with around 3 Million Constraints and I am currently using objective function with different weights. But to improve run time I am hoping to explore DoCPLEX multi- objective import "ObjectiveSense". Before implementation I wanted to understand how this solve works.

For Example:

My Objective Function is Maximize(-1000B -100C +10A -D -0.1E) Using Multi- Objective it is: ObjectiveSense.Maximize, [-B,A,-C,-D,-E], priorities=[5, 4, 3, 2, 1])

The one of the problem with weights is if 10A gets B1000 (which can be possible in few case) then A gets prioritized over B

Does Multi-Objective prevents this situation? Will it strictly prioritize B over all?

Answer 1

in https://developer.ibm.com/docloud/blog/2019/03/12/multiobjective-optimization-for-lp-and-mip-in-cplex/

you may read

Priority: an integer, default 0. Defines the order in which KPIs will be dealt with. If several sub-objectives have the same priority, they are blended together.

With priorities you get hierarchical KPIs so a less important KPI even with a huge value won t get more important than a less important KPI

Let me share a tiny example out of the zoo example:

from docplex.mp.model import Model

mdl = Model(name='buses')

nbbus50 = mdl.integer_var(name='nbBus50')
nbbus40 = mdl.integer_var(name='nbBus40')
nbbus30 = mdl.integer_var(name='nbBus30')

cost = mdl.continuous_var(name='cost')
co2emission = mdl.continuous_var(name='co2emission')

mdl.add_constraint(nbbus50*50+nbbus40*40 + nbbus30*30 >= 200, 'kids')
mdl.add_constraint(co2emission==nbbus50+nbbus40*1.1+nbbus30*1.2)
mdl.add_constraint(cost==nbbus40*500 + nbbus30*400+nbbus50*625)
                
sense="min"
exprs=[cost,co2emission]
priorities=[1,2]
weights=[1,1]
mdl.set_multi_objective(sense, exprs, priorities, weights, abstols=None, reltols=None, names=None)

mdl.solve(lex_mipgaps = [0.001, 0.05], log_output=True)

for v in mdl.iter_integer_vars():
    print(v," = ",v.solution_value)

print("The minimum cost is ",cost.solution_value);
print("CO2 emission is ",co2emission.solution_value);

which gives

nbBus50  =  4.0
nbBus40  =  0
nbBus30  =  0
The minimum cost is  2500.0
CO2 emission is  4.0

Answer 2

To add to Alex's commentary, a situation where multi-objective is helpful is when the coefficients in the objective function due to priorities, are orders of magnitude different. If the coefficients are at at least 1e6 different, this can lead to some answers being part of the rounding error and you can have numerical instability. In your case you are at 1e4 different, so unless there is another issue in the formulation, you are unlikely to see perfomance gains. If this is a LP, I suggest you start by changing the "lpmethod" parameter to 4 since you have such a large model.

Would you be able to comment on how things worked out?