Assignment Problem IP

Task Assignee a b c d A 85 95 82 86 B 92 87 83 90 C 73 78 79 80 D 90 95 90 88
Assignment problem
Solution
1 x ij = 0
Define if person Pi is assigned to perform ta T j sk ( i = 1, 2 ,3, 4; j = 1, 2 ,3, 4 ) otherwise
Solution：This problem satisfies the conditions of Hungarian ： algorithm, so we use Hungarian to solve it. Step 1 Row Reduction.
m in 58 75 65 82 69 180 260 58 50 150 230 50 ⇒ 70 170 250 65 55 200 280 55
Assignment problem
Summary of the Hungarian Algorithm
Step 1 Subtract the smallest number in each row from every number in the row. (This is called row reduction). Enter the result in a new table. Step 2 Subtract the smallest number in each column of the new table from every number in the column. (This is called column reduction). Enter the result in another table. Step 3 Test whether an optimal assignment can be made. You do this by determining the minimum number of lines needed to cover (i. e., cross out) all zeros. If the number of lines equals the number of rows, an optimal set of assignment is possible. In that case, go to step 6. Otherwise go on to step 4.
m Z = 85x11 + 92x12 + 73x13 + 90x14 + 95x21 + 87x22 + ax The mathematical + 78x23 + 95x24 + 82x31 + 83x32 + 79x33 + 90x34 + model： ： + 86x41 + 90x42 + 80x43 + 88x44 x11 + x12 + x13 + x14 = 1 x11 + x21 + x31 + x41 = 1 x + x + x + x = 1 s.t. x21 + x22 + x23 + x24 = 1 12 22 32 42 x31 + x32 + x33 + x34 = 1 x13 + x23 + x33 + x43 = 1 x41 + x42 + x43 + x44 = 1 x14 + x24 + x34 + x44 = 1
Assignment problem
Step 4 If the number of lines is less than the number of rows, modify the table in the following way: a. Subtract the smallest uncovered number from every uncovered number in the table. b. Add the smallest uncovered number to the number at intersections of covering lines. c. Numbers crossed out but not at the intersection of cross-out lines carry over unchanged to the next table. Repeat step 3 and 4 until an optimal set of assignment is possible.
Table 2 Product 1 Product 2 Product 3 Product 4 Plant1 Plant2 Plant3 Plant4 58 75 65 82 69 50 70 55 180 150 170 200 260 230 250 280
Assignment problem
The condition required by the Hungarian Algorithm: (1) The assignment problem is a minimization problem; (2) The number of assignees and tasks are the same; (3) The element in the efficiency matrix is nonnegative.
m in(m ax)Z = ∑∑cij xij
i= 1 j= 1
m i =1 L m , , ∑xij =1 j =1 m j =1 L m , , ∑xij =1 i=1 xij = 0 or 1 i, j =1 L , m
Assignment problem
2 The Hungarian Algorithm for the Assignment problem
xij = 0or1 i、 = 1 , j ,2,3,4
Assignment problem
In general, we assume that each assignee is to be assigned exactly one task
and there is a cost cij≥0 associated with assignee i performing task j (i, j =1, …,m), the efficiency matrix is [cij] given in the Table 1，and try to find a solution to ， optimize the efficiency （min or max). The mathematical model is m m
Assignment problem
Theorem 2
If the elements in matrix A can be cut into two parts : “zero” and “ non-zero” , then the minimum number of lines used to cover “0” element is the maximum number of “0” elements which are in both the different row and column ( called independent element) . If the minimum number of lines is m，then there are m independent “0” ， elements, so the corresponding xij =1, the value of other variables is 0，at ， this time, the objective value is 0, and we get the optimal solution.
11 22 22 0 0 0 5 5 5 0 45 0
0 25 0 27 11 122 202 0 100 180 5 105 185 0 145 225
Assignment problem
Step 2
Column Reduction.
0 25 0 27
11 122 202 0 25 0 100 180 ⇒ 0 5 105 185 0 145 225 27
Assignment problem
Step 5 Make the assignments one at a time in positions that have zero elements. Begin with rows or column that have only one zero. Since each row and each column needs to receive exactly one assignment, cross out both the row and the column involved after each assignment is made. Then move on to the rows and columns that are not yet crossed out to select the next assignment, with preference again given to any such row or column that has only one zero that is not crossed out. Continue until every row and every column has exactly on assignment and so has been crossed out.