CWMOIP

We first implement CWMOIP in Matlab for proof of concept (here is the Matlab code and input for the case WebPortal Rand2). Then we implement CWMOIP in Java with CPLEX. Note that for the same case with the same input feature attribute values, the Matlab implementation finds 763 non-dominant solutions, and the CPLEX solution finds 768 non-dominant solutions. After inspection, we found that Matlab implementation missed 5 non-dominant solutions due to the precision floating issue in Matlab (only rounded to 4 decimals in Matlab, but round to 16 decimals for Java Double data type). Note, we find Matlab is about 10 times slower than CPLEX using their by-default configuration.

CWMOIP（Constraint Weighted Multi-Objective Integer Programming）proposed by Özlen et al. [17] is an objective reduction technique for MOIP. This method is based on epsilon-constraint method, and it aims to generate all the non-dominant solutions for a given MOIP. Our study on this (like SolRep) later borrows the idea of constrainted weighted for objective-reduction, which helps to avoid the generation of dominant solutions.

It improves the epsilon-constraint method by two steps:

1） Accurate upper/lower bound calculation for each objective

for each objective, the lower bound is not 0 and the upper bound is not the sum of feature attributes revelent to that objective. To be precise, BIP is applied

to get the true lower and upper bounds for each objective separately, subject to the conjunction of constraints for the specific problem.

2） Objective-reduction

Objective-reduction is implemented via the constraint weight method, to avoid generating dominant solutions. The k-objective problem is first reduced to that of k -1, then k - 2, iteratively, until the only last objective is left.

For a multi-objective integer programming (MOIP) problem with m objectives：

, where X denotes the vector of desicion varibles.

After the m-th objective is constraint weighted, this problem is reduced to a MOIP problem with m-1 objectives (CW(m-1)OIP):

, where we have

The algorithm presentation that is given in the original paper (Özlen^[17], page 7) is as follow:

The above algorithm is a bit high-level, and we provide the algorithm of CWMOIP at a more detailed implementation level.