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.

/pdf/an-exact-algorithm-for-constrained-k-cardinality-unbalanced-mua0fqk4.pdf

An exact algorithm for constrained k-cardinality unbalanced assignment problem

. thors; licensee Growing Science, Canada 2018 by the au ©

/pdf/an-open-close-multiple-travelling-salesman-problem-with-13lekmvx08.pdf

An open close multiple travelling salesman problem with single depot

Brain pathologies often manifest as partial or complete loss of tissue. The goal of many neuroimaging studies is to capture the location and amount of tissue changes with respect to a clinical variable of interest, such as disease progression. Morphometric analysis approaches capture local differences in the distribution of tissue or other quantities of interest in relation to a clinical variable. We propose to augment morphometric analysis with an additional feature extraction step based on unbalanced optimal transport. The optimal transport feature extraction step increases statistical power for pathologies that cause spatially dispersed tissue loss, minimizes sensitivity to shifts due to spatial misalignment or differences in brain topology, and separates changes due to volume differences from changes due to tissue location. We demonstrate the proposed optimal transport feature extraction step in the context of a volumetric morphometric analysis of the OASIS-1 study for Alzheimer's disease. The results demonstrate that the proposed approach can identify tissue changes and differences that are not otherwise measurable.

Optimal Transport Features for Morphometric Population Analysis

Novel optimization method for unbalanced assignment problems with multiple jobs: The Dhouib-Matrix-AP2

The $k$-cardinality assignment problem asks for finding a maximal (minimal) weight of a matching of cardinality $k$ in a weighted bipartite graph $K_{n,n}$, $k \leq n$. The algorithm of Gassner and Klinz from 2010 for the parametric assignment problem computes in time $O(n^3)$ the set of $k$-cardinality assignments for those integers $k \leq n$ which refer to "essential" terms of a corresponding maxpolynomial. We show here that one can extend this algorithm and compute in a second stage the other "semi-essential" terms in time $O(n^2)$, which results in a time complexity of $O(n^3)$ for the whole sequence of $k=1,...,n$-cardinality assignments. The more there are assignments left to be computed at the second stage the faster the two-stage algorithm runs. In general, however, there is no benefit for this two-stage algorithm on the existing algorithms, e.g. the simpler network flow algorithm based on the successive shortest path algorithm which also computes all the $k$-cardinality assignments in time $O(n^3)$.

Computing the sequence of k-cardinality assignments.

In this paper the “Generalized Time-dependent Travelling Salesman Problem” is a three dimensional travelling salesman problem where the cost matrix C (i, j, k) is the cost of the salesman visiting from city i to city j at time (facility) k. The cost matrix C(i,j,k) [i,j = 1,2,3,…..,n;k = 1,2,3,…..,m] is given. There are n cities and N = {1, 2, 3,…, n}. We are given a partition of cities into groups and we require to find a minimum length tour that includes at least one city from each group. The problem is to find a feasible tour for m(<n) cities for the salesman with minimum total cost with given number of cities from each group with the restriction that he visits only one pair of cities at each point of time(facility) i.e., at a particular time he has to travel from one city to another city. We propose a Lexi Search algorithm based on Pattern Recognition Technique for solving “Generalized Time-dependent Traveling Salesman Problem” with an illustrative example. For this problem a computer program is developed for the algorithm and is tested. It is observed that it takes less time for solving fairly higher dimension problem also.

U. Balakrishna

Papers

An exact algorithm for constrained k-cardinality unbalanced assignment problem

A pattern recognition lexi search approach to generalized time-dependent travelling salesman problem