# Linear Assignment Problem

The assignment problem is specified by the cost matrix cij for minimization problems or the benefit matrix aij for maximization problems.Those matrices describe the cost or benefit of assigning object j to person i.

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Visit Stack Exchange I wonder if there is any literature on the following problem $$\begin \underset & \displaystyle\sum_ C_ X_\\ \text & \displaystyle\sum_ X_ = \displaystyle\sum_ X_ = 1\\ & X_ \geq 0\end$$ The closest related problem might be Rectangular Linear Assignment Problem (RLAP)$^\dagger$, as RLAP further constrains $X_ \in \$.

The LSAP is also used as subproblems in a number of network flow and applied combinatorial optimization problems such as the quadratic assignment problem (QAP), the traveling salesman problem (TSP) and the transportation problem.

The assignment problem is also used as a preprocessing step for pivoting methods.

So, do the relaxed RLAP share the same optimum as RLAP? The converted problem is a standard assignment problem whose optimal objective cost is the same as that of the Rectangular Assignment Problem.

Per the Integrality Theorem, the Assignment Problem, without integrality constraints, has an optimal solution consisting of all integers (which must be 0 and 1).Due to the dynamic nature of many applications, as well as the expansion of problem sizes, where a solution needs to be found under tight time constraints, heuristics that give rise to solutions that are close to optimal solution are sought.Our efforts in that regards lead us to the Deep Greedy Switching (DGS) algorithm.The auction algorithm proved to be unsuitable for our application.Even attempts of a parallelized version of the auction algorithm did not meet our requirements.Even as the auction is considered one of the fastest algorithms, for large-scale and complex instances of the assignment problem the auction algorithm can take a lot of time to find the optimal solution. discussed five types of the assignment problem instances which are Geometric, Fixed-Cost, High-Cost, Low-Cost and Uniformly Random.It was shown that for the first two problem types the auction algorithm performs poorly in terms of the running time in comparison with the other three problem types.Trick applied a greedy heuristic approach for the generalized assignment problem, where it was shown that some randomization to the greedy approach is required to reach better results than a completely greedy approach.Another heuristic approach is the greedy randomized adaptive search procedure (GRASP, which consist of a multi-start random initial solutions and a local search.approximate dual projective algorithm by Ramakrishnan et al.) and algorithms that use forest construction as dual forest algorithm by Achatz et al.The auction algorithm is considered one of the fastest algorithm that can find the optimal solution for the assignment problem.

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In the general linear programming model of the assignment problem, one agent is assigned to one and only one task. The assignment problem is a special case of the…

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Techniques, many of which built on linear programming for generating a global view of large, complex optimization problems 5. 2. Mathemtical LP Model for assignment problem Some linear programming models for the assignment problem is presented is assumed that the cost or time for every machine is known denoting that C…

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The assignment problem is a special case of the transportation problem, and the transportation problem is a special case of a linear programming problem. Both assignment and transportation problems may be solved using LPP methods, like the simplex method, although there are more specific methods for these problems, as mentioned in other replies.…

• ###### Scipy.optimize.linear_sum_assignment — SciPy v1.3.0 Reference Guide

The linear sum assignment problem is also known as minimum weight matching in bipartite graphs. A problem instance is described by a matrix C, where each Ci,j is the cost of matching vertex i of the first partite set a “worker” and vertex j of the second set a “job”. The goal is to find a complete assignment of workers to jobs of minimal cost.…

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Assignment Problem Linear Programming The assignment problem is a special type of transportation problem, where the objective is to minimize the cost or time of completing a number of jobs by a number of persons.…

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