assignment problem matlab

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Solve linear assignment problem

Description

M = matchpairs( Cost , costUnmatched ) solves the linear assignment problem for the rows and columns of the matrix Cost . Each row is assigned to a column in such a way that the total cost is minimized. costUnmatched specifies the cost per row of not assigning each row, and also the cost per column of not having a row assigned to each column.

[ M , uR , uC ] = matchpairs( Cost , costUnmatched ) additionally returns indices for unmatched rows in uR and indices for unmatched columns in uC .

[ ___ ] = matchpairs( Cost , costUnmatched , goal ) specifies the goal of the optimization using any of the output argument combinations in previous syntaxes. goal can be 'min' or 'max' to produce matches that either minimize or maximize the total cost.

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Assign Flights with Minimal Cost

Assign salespeople to flights such that the total cost of transportation is minimized.

A company has four salespeople who need to travel to key cities around the country. The company must book their flights, and wants to spend as little money as possible. These salespeople are based in different parts of the country, so the cost for them to fly to each city varies.

This table shows the cost for each salesperson to fly to each key city.

Dallas Chicago New   York   City St.   Louis Fred $ 6 0 0 $ 6 7 0 $ 9 6 0 $ 5 6 0 Beth $ 9 0 0 $ 2 8 0 $ 9 7 0 $ 5 4 0 Sue $ 3 1 0 $ 3 5 0 $ 9 5 0 $ 8 2 0 Greg $ 3 2 5 $ 2 9 0 $ 6 0 0 $ 5 4 0

Each city represents a sales opportunity. If a city is missed, then the company loses out on an average revenue gain of $2,000.

Create a cost matrix to represent the cost of each salesperson flying to each city.

Use matchpairs to assign the salespeople to the cities with minimal cost. Specify the cost of unassignment as 1000, since the cost of unassignment is counted twice if a row and a column remain unmatched.

matchpairs calculates the least expensive way to get a salesperson to each city.

Dallas Chicago New   York   City St.   Louis Fred $ 6 0 0 $ 6 7 0 $ 9 6 0 $ 560 Beth $ 9 0 0 $ 280 $ 9 7 0 $ 5 4 0 Sue $ 310 $ 3 5 0 $ 9 5 0 $ 8 2 0 Greg $ 3 2 5 $ 2 9 0 $ 600 $ 5 4 0

Unequal Numbers of Rows and Columns

Match rows to columns when you have many more columns than rows in the cost matrix.

Create a 3-by-8 cost matrix. Since you have only three rows, matchpairs can produce at most three matches with the eight columns.

Use matchpairs to match the rows and columns of the cost matrix. To get the maximum number of matches, use a large cost of unassignment (relative to the magnitude of the entries in the cost matrix). Specify three outputs to return the indices of unmatched rows and columns.

Five of the columns in C are not matched with any rows.

Assign Taxis to Maximize Profit

Assign taxis to routes such that the profit is maximized.

A taxi company has several ride requests from across the city. The company wants to dispatch its limited number of taxis in a way that makes the most money.

This table shows the estimated taxi fare for each of five ride requests. Only three of the five ride requests can be filled.

Ride   1 Ride   2 Ride   3 Ride   4 Ride   5 Cab   A $ 5 . 7 0 $ 6 . 3 0 $ 3 . 1 0 $ 4 . 8 0 $ 3 . 5 0 Cab   B $ 5 . 8 0 $ 6 . 4 0 $ 3 . 3 0 $ 4 . 7 0 $ 3 . 2 0 Cab   C $ 5 . 7 0 $ 6 . 3 0 $ 3 . 2 0 $ 4 . 9 0 $ 3 . 4 0

Create a profits matrix to represent the profits of each taxi ride.

Use matchpairs to match the taxis to the most profitable rides. Specify three outputs to return any unmatched rows and columns, and the 'max' option to maximize the profits. Specify the cost of unassignment as zero, since the company makes no money from unfilled taxis or ride requests.

matchpairs calculates the most profitable rides to fill. The solution leaves ride requests 3 and 5 unfilled.

Calculate the total profits for the calculated solution. Since costUnmatched is zero, you only need to add together the profits from each match.

Track Point Positions over Time

Use matchpairs to track the movement of several points by minimizing the total changes in distance.

Plot a grid of points at time t = 0 in green. At time t = 1 , some of the points move a small amount in a random direction.

Use matchpairs to match the points at t = 0 with the points at t = 1 . To do this, first calculate a cost matrix where C(i,j) is the Euclidean distance from point i to point j .

Next, use matchpairs to match the rows and columns in the cost matrix. Specify the cost of unassignment as 1. With such a low cost of unassignment relative to the entries in the cost matrix, it is likely matchpairs will leave some points unmatched.

The values M(:,2) correspond to the original points ( x 0 , y 0 ) , while the values M(:,1) correspond to the moved points ( x 1 , y 1 ) .

Plot the matched pairs of points. The points that moved farther than 2*costUnmatched away from the original point remain unmatched.

Input Arguments

Cost — cost matrix matrix.

Cost matrix. Each entry Cost(i,j) specifies the cost of assigning row i to column j .

Data Types: single | double

costUnmatched — Cost of not matching scalar

Cost of not matching, specified as a scalar. matchpairs compares the value of 2*costUnmatched to the entries in Cost to determine whether it is more beneficial for a row or column to remain unmatched. Use this parameter to make matches more or less likely in the algorithm. For more information, see linear assignment problem .

Example: M = matchpairs(C,10) specifies a cost of 10 for not matching a row or column of C .

goal — Optimization goal 'min' (default) | 'max'

Optimization goal, specified as either 'min' or 'max' . The optimization goal specifies whether the total cost should be minimized or maximized.

Example: M = matchpairs(Cost,costUnmatched,'max') specifies that the rows and columns of Cost should be matched together to maximize the total cost.

Output Arguments

M — matches matrix.

Matches, returned as a matrix. M is a p -by- 2 matrix, where M(i,1) and M(i,2) are the row and column indices of a matched pair in the cost matrix. The rows of M are sorted with the second column in ascending order.

Each row and column can be matched a single time only, so each M(i,1) value and each M(i,2) value is unique.

M contains p matches, and p is less than or equal to the maximum number of matches min(size(Cost)) .

The cost of the matches in M is sum([Cost(M(1,1),M(1,2)), Cost(M(2,1),M(2,2)), ..., Cost(M(p,1),M(p,2))]) .

uR — Unassigned rows column vector

Unassigned rows, returned as a column vector of indices. The entries in uR indicate which rows in Cost are unassigned. Each entry in uR and uC contributes to the total cost of the solution according to costUnassigned .

uC — Unassigned columns column vector

Unassigned columns, returned as a column vector of indices. The entries in uC indicate which columns in Cost are unassigned. Each entry in uR and uC contributes to the total cost of the solution according to costUnassigned .

Linear Assignment Problem

The linear assignment problem is a way of assigning rows to columns such that each row is assigned to a column and the total cost of the assignments is minimized (or maximized). The cost of assigning each row to each column is captured in a cost matrix . The entry Cost(i,j) is the cost of assigning row i to column j .

The cost of unassignment assigns a cost to any row or column that is not matched. This practice allows for minimum-cost solutions that do not assign all rows or columns. If a row and column are not matched, this increases the total cost by 2*costUnmatched .

The total cost of a solution M is the sum of the cost of all matched pairs added to the cost of all unmatched pairs:

T C = ∑ i = 1 p Cost ( M ( i , 1 ) , M ( i , 2 ) ) + costUnmatched   ⋅   ( m + n − 2 p )

In code the total cost is

Cost is an m -by- n matrix.

M is a p -by- 2 matrix, where M(i,1) and M(i,2) are the row and column of a matched pair.

(m+n-2*p) is the total number of unmatched rows and columns.

[1] Duff, I.S. and J. Koster. "On Algorithms For Permuting Large Entries to the Diagonal of a Sparse Matrix." SIAM J. Matrix Anal. & Appl. 22(4), 2001. pp 973–996.

Extended Capabilities

C/c++ code generation generate c and c++ code using matlab® coder™..

Usage notes and limitations:

Code generation does not support sparse matrix inputs for this function.

Version History

Introduced in R2019a

equilibrate | sprank | dmperm

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New Approach to Solve Assignment Problem using MATLAB

Assignment problem is a special case of transportation problem, in which the objective is to minimized total cost by assigning ‘m’ jobs to ‘n’ machines. By using MATLAB coding and some modification in ROA method, optimal solution can be trace for assignment problem within seconds. MATLAB coding result has given for various orders of illustrations.

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Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)

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• For each row of the matrix, find the smallest element and subtract it from every element in its row.
• Do the same (as step 1) for all columns.
• Cover all zeros in the matrix using minimum number of horizontal and vertical lines.
• Test for Optimality: If the minimum number of covering lines is n, an optimal assignment is possible and we are finished. Else if lines are lesser than n, we haven’t found the optimal assignment, and must proceed to step 5.
• Determine the smallest entry not covered by any line. Subtract this entry from each uncovered row, and then add it to each covered column. Return to step 3.

Try it before moving to see the solution

Explanation for above simple example:

  An example that doesn’t lead to optimal value in first attempt: In the above example, the first check for optimality did give us solution. What if we the number covering lines is less than n.

Time complexity : O(n^3), where n is the number of workers and jobs. This is because the algorithm implements the Hungarian algorithm, which is known to have a time complexity of O(n^3).

Space complexity :   O(n^2), where n is the number of workers and jobs. This is because the algorithm uses a 2D cost matrix of size n x n to store the costs of assigning each worker to a job, and additional arrays of size n to store the labels, matches, and auxiliary information needed for the algorithm.

In the next post, we will be discussing implementation of the above algorithm. The implementation requires more steps as we need to find minimum number of lines to cover all 0’s using a program. References: http://www.math.harvard.edu/archive/20_spring_05/handouts/assignment_overheads.pdf https://www.youtube.com/watch?v=dQDZNHwuuOY

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Title: multi-auv kinematic task assignment based on self-organizing map neural network and dubins path generator.

Abstract: To deal with the task assignment problem of multi-AUV systems under kinematic constraints, which means steering capability constraints for underactuated AUVs or other vehicles likely, an improved task assignment algorithm is proposed combining the Dubins Path algorithm with improved SOM neural network algorithm. At first, the aimed tasks are assigned to the AUVs by improved SOM neural network method based on workload balance and neighborhood function. When there exists kinematic constraints or obstacles which may cause failure of trajectory planning, task re-assignment will be implemented by change the weights of SOM neurals, until the AUVs can have paths to reach all the targets. Then, the Dubins paths are generated in several limited cases. AUV's yaw angle is limited, which result in new assignments to the targets. Computation flow is designed so that the algorithm in MATLAB and Python can realizes the path planning to multiple targets. Finally, simulation results prove that the proposed algorithm can effectively accomplish the task assignment task for multi-AUV system.

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