Chapter 5 Discussion and Concluding Remarks
5.2 Concluding Remarks
,...,
(u1 um must satisfy the following constraint:
1
1
∑
== m
u
θ θ .
Actually, the proposed method is not appropriate for some practice applications which do not meet the above constraint.
5.2 Concluding Remarks
This study develops the Superior Representation of Binary Variables (SRB) core technique.
It can effectively reduce the traditional binary variables number from O(m) to O(log2m) and easily applies in any mixed-integer problems; moreover, it has some directions for future research are described below:
(i) Task allocation problem (TAP): For example, a TAP with n agents and m tasks requires nm binary variables in existing deterministic approaches to obtain global solution while our method, based on SRB, only needs
⎡
log2nm⎤
binary variables.We try to find an efficient method to conquer the classic TAP. Furthermore, it is appropriate to solve the Multilevel Generalized Assignment Problem.
(ii) Haplotype inference: This is the optimal haplotype inference (OHI) problem as given a set of genotypes and a set of related haplotypes, find a minimum subset of haplotypes that can resolve all the genotypes in biology, which is also NP-hard and can be formulated as an integer quadratic programming (IQP) problem. We also want to find an efficient method through our SRB approach.
(iii) Integrating deterministic and heuristic approaches: It is an appropriate way to solve large scale optimal problems under the following strategy:
(a) Use the heuristic algorithms to find an initial solution to enhance the efficiency of finding the optimal solution.
(b) Then, use deterministic approaches are utilized to transcend the incumbent solution.
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