The Assignment Problem

Transcription

1 I heard that this is really good! The Assignment Problem A special case of the transportation problem which is a special case of a linear program which is a special case of a mathematical program. Chapter 5.4 1

2 The Problem There are n workers and n tasks to be performed. The time it takes worker i to perform task j is c ij. Which task should be assigned to which worker? I want the easy task. 2

3 The Model-1 Min z st.. = n i= 1 n i= 1 n j= 1 j= 1 x n c x ij ij x = 1 j = 12,,..., n ij x = 1 i = 12,,..., n ij ij = 01, for all i, j 3

4 The Model-2 Min z st.. = n i= 1 n i= 1 n j= 1 j= 1 x n c x ij ij x = 1 j = 12,,..., n ij x = 1 i = 12,,..., n ij ij = 01, for all i, j This is just the transportation problem with the right hand side values equal to one! 4

5 Some Applications These are terrific applications. workers to tasks jobs to machines facilities to locations Truck drivers to customer pick-up points Umpire crews to baseball games Judges to court dockets State inspectors to construction sites Weapons to targets 5

6 A combinatorial problem If there are n workers and n tasks there are n! (factorial) possible assignments. Example: Workers are Al, Art, Alice, and Ann. There are four tasks: 1,2,3, & = 4! Al Art Alice Ann If n = 10 then 10! = 3,628,800 6

7 An Example - assign a construction project (building) to a contractor Contractor A Building B C D Bids (in $10,000) 7

8 The Algorithm (Flood s or the Hungarian Method) 1. Subtract the smallest cost element in each row from every element in that row. 2. Subtract the smallest cost element in each column from every element in that column. 3. Test for optimality by drawing the minimum number of lines that will cover every zero cell (no diagonal lines). If the minimum = n, a feasible assignment involving only zero cells is possible. Go to step Select the smallest element not having a line through it. Subtract this amount from all elements not covered by a line;and add this amount to all elements at the intersection of lines.go to step Solution is optimum; make assignments using zero cells so that all constraints are satisfied. 8

9 I bet it works by magic. z = n i= 1 n j= 1 cij xij Some Magic Subtract a constant a from row k: n n n n n n ij ij kj kj ij ij i= 1 j= 1 j= 1 i= 1 j= 1 j= 1 i k z' = c x + ( c a) x = c x a x kj n n = cx a = z a i= 1 j= 1 ij ij since n j= 1 x kj = 1 animated 9

10 Let s solve the problem-1! A B C D

11 Let s solve the problem-2! Subtract 44 from row A B C D

12 Let s solve the problem-3! Subtract 56 from row A B C D

13 Let s solve the problem-4! Subtract 85 from row 3 and 42 from row A B C D

14 Let s solve the problem-5a! Subtract 2 from column 2 and 4 from column 3 Is there a feasible solution using the zero cells? A B C D

15 Let s solve the problem-5b! Subtract 2 from column 2 and 4 from column 3 Is there a feasible solution using the zero cells? A B C D = 4 15

16 Let s solve the problem-6! Minimum uncovered element A B C D

17 Let s solve the problem-7! A 4-1=3 2-1=1 2-1=1 0 B =13 C 11-1=10 7-1=6 1-1=0 0 D =5 17

18 Let s solve the problem-8a! A B C D

19 Let s solve the problem-8b! A B C D Optimal assignment is now possible using the zero cells! 19

20 Let s solve the problem-9b! A B X C D X X X 20

21 The Final Cost A B C D Cost = = 234 or $2,340,000 21

22 Another Example The Match Maker, a computerized dating service that attempts to bring two compatible people together, has to match the following individuals: Mandy, Mollie, and Martha with Bill, Bob, Bruno, and Bruce. The ladies have ranked each man on a scale of 1 to 10 with the higher number being the more preferred. fine print: A really good problem that illustrates the algorithm when maximizing with unbalanced rows and columns. 22

23 The Rankings-1 Bill Bob Bruno Bruce Mandy Mollie Martha But there are not enough women! 23

24 The Rankings-2 Bill Bob Bruno Bruce Mandy Mollie Martha Dummy

25 Converting from Max to Min Max f = {9, 5, 12, 8, 15, 20, 7} = 20 Min f = {-9, -5, -12, -8, -15, -20, -7} = -20 Max f = - Min -f n n n n Max z = cx = Min zwhere z= cx ij ij ij ij i= 1 j= 1 i= 1 j= 1 25

26 More Converting from Max to Min Ymax X* Ymax = Max f(x) -Ymin Ymin = Min -f(x) 26

27 Convert to a minimization problem Bill Bob Bruno Bruce Mandy Mollie Martha Dummy

28 Generate a zero in every row Bill Bob Bruno Bruce Mandy 10-7=3 10-5= =0 10-6=4 Mollie 9-8=1 9-6=3 9-9=0 9-5=4 Martha 9-9=0 9-7=2 9-8=1 9-4=5 Dummy

29 Test for optimality-1a Bill Bob Bruno Bruce Mandy Mollie Martha Dummy

30 Test for optimality-1b Bill Bob Bruno Bruce Mandy Mollie Martha Dummy

31 Test for optimality-2 Bill Bob Bruno Bruce Mandy Mollie Martha Dummy Minimum uncovered cell 31

32 Test for optimality-3 Bill Bob Bruno Bruce Mandy 3-1=2 5-1= =3 Mollie 1-1=0 3-1= =3 Martha =2 5 Dummy =1 0 32

33 Test for optimality-4a Bill Bob Bruno Bruce Mandy Mollie Martha Dummy

34 Test for optimality-4b Bill Bob Bruno Bruce Mandy Mollie Martha Dummy Minimum uncovered cell 34

35 Test for optimality-5 Bill Bob Bruno Bruce Mandy 2 4-2= =1 Mollie 0 2-2= =1 Martha 0 2-2= =3 Dummy 0+2= =3 0 35

36 Test for optimality-6a Bill Bob Bruno Bruce Mandy Mollie Martha Dummy

37 Test for optimality-6b Bill Bob Bruno Bruce Mandy Mollie Martha Dummy

38 Test for optimality-7a Bill Bob Bruno Bruce Mandy Mollie Martha Dummy

39 Test for optimality-7b Bill Bob Bruno Bruce Mandy Mollie X Martha X Dummy X X 39

40 The End Bill Bob Bruno Bruce Mandy Mollie Martha Dummy

41 An Interesting Application Need to assign swimmers to a 200 meter medley relay race. Below are the 5 fastest swimmers on the Pink Dauphins swim team. Which swimmers are assigned which strokes? Times given are individual best times (in seconds) of each swimmer. Stroke Jane Jean Jessica Jackie Jenifer Backstroke Breaststroke Butterfly Freestyle

42 The Assignment Problem This has been another delightful OR experience that both invigorates and excites the mind. Don t you agree? 42