An exact algorithm for constrained k-cardinality unbalanced assignment problem

An assignment problem (AP) usually deals with how a set of persons/tasks can be assigned to a set of tasks/persons on a one-to-one basis in an optimal manner. It has been observed that balancing among the persons and jobs in several real-world situations is very hard, thus such scenarios can be seen as unbalanced assignment models (UAP) being a lack of workforce. The solution techniques presented in the literature for solving UAP’s depend on the assumption to allocate some of the tasks to fictitious persons; those tasks assigned to dummy persons are ignored at the end. However, some situations in which it is inevitable to assign more tasks to a single person. This paper addresses a practical variant of UAP called k-cardinality unbalanced assignment problem (k-UAP), in which only of persons are asked to perform jobs and all the persons should perform at least one and at most jobs. The k-UAP aims to determine the optimal assignment between persons and jobs. To tackle this problem optimally, an enumerative Lexi-search algorithm (LSA) is proposed. A comparative study is carried out to measure the efficiency of the proposed algorithm. The computational results indicate that the suggested LSA is having the great capability of solving the smaller and moderate instances optimally.

On the sum of positive divisors functions

Properties of divisor functions $$\sigma _k(n)$$ σ k ( n ) , defined as sums of k-th powers of all divisors of n, are studied through the analysis of Ramanujan’s differential equations. This system of three differential equations is singular at $$x=0$$ x = 0 . Solution techniques suitable to tackle this singularity are developed and the problem is transformed into an analysis of a dynamical system. Number theoretical consequences of the presented dynamical system analysis are then discussed, including recursive formulas for divisor functions.