# Assignment Problem In Operational Research Efficient FORTRAN implementations for the case of complete and sparse matrices are given.

An assignment problem can be easily solved by applying Hungarian method which consists of two phases.

In this paper, we consider a truck dock assignment problem with operational time constraint in crossdocks where the number of trucks exceeds the number of docks available.

The objective is to find an optimal assignment of trucks that minimizes the operational cost of the cargo shipments and the total number of unfulfilled shipments.  In the second phase, the solution is optimized on iterative basis.In a given problem, if the number of rows is not equal to the number of columns and vice versa, then add a dummy row or a dummy column.The assignment costs for dummy cells are always assigned as zero.Reduce the matrix by selecting the smallest value in each row and subtracting from other values in that corresponding row.In row A, the smallest value is 13, row B is 15, row C is 17 and row D is 12.This paper analyzes the most efficient algorithms for the Linear Min-Sum Assignment Problem and shows that they derive from a common basic procedure.For each algorithm, we evaluate the computational complexity and the average performance on randomly-generated test problems.If there is no single zero allocation, it means multiple numbers of solutions exist.But the cost will remain the same for different sets of allocations.In column 1, the smallest value is 0, column 2 is 4, column 3 is 3 and column 4 is 0.The column-wise reduction matrix is shown in the following table.

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