1 Chapter 10 The assignment model section

2 The general problem in this section is that we have people in our employ and we have tasks to be accomplished. Each person will work on a task, but each task needs only one person to do the work. How do we assign people to tasks? This is what we figure out. We typically want to minimize the cost for the company of getting all jobs completed. An example: Adams, Brown and Cooper having to complete projects 1, 2 and 3, but each person does only one job. The cost of each person, if they did a job is: Project 12 3 Adams Brown Cooper 9127

3 The brute force method The brute force method of solution would have you list every combination of assignments and pick the one with the minimum cost. This would entail something like the following (I just put the first letter of the person’s name from the example above) Pro 1Pro2 Pro 3 Total Costs ABC = 28 ACB = 34 BA C = 29 B C A = 26 C A B = 34 C B A = 25

4 Flood’s (he is Hungarian) technique The basic idea behind this approach to finding a solution is to think about each person in each job. If Brown is in project 1, that is the lowest cost job for him. But maybe that makes someone else take another project that is not their lowest cost job. Maybe Brown should be in a higher cost job for him and there will be a large reduction in cost for another person so that the total cost of all jobs combined is lowest.

5 QM for windows QM for Windows has a module to help us solve these problems. Go to Module on the toolbar and scroll down to assignment. The context of the problem will tell you to maximize or minimize. In this problem you would minimize. Type a title in and hit OK. Then a pop-up screen pops-up. Notice the rows are Jobs and columns are machines generically. You can just go into the cells and type over what is there. Put people in rows and projects in columns (although it really does not matter.)

6 After you type in the names put in the cost data in the appropriate cells. Save the file. Go to the green solve button in the upper right or go to file and solve. On the assignment screen you get the low cost solution. The assignment list screen has an even easier to read solution list. The MC screen is a little more crazy to understand.

7 The MC screen in QM for Windows project person Adams 0 Brown 1 9 Cooper 3 This table looks similar to the one in QM for Windows for the Marginal Costs in the transportation problem. But it should be interpreted differently. I will use it on the following screens to show you how to interpret what is happening.

8 project person Adams 0 x Brown 1 x 9 Cooper x 3 On this screen I put an x in each cell that represents the best solution. These cells have no numbers at all. If I add them up I get nothing. This means the marginal cost, or change in total cost, of this assignment from the best assignment is nothing. We expect this because this is the best assignment.

9 project person Adams x 0 Brown 1 x 9 Cooper x3 On this screen I put an x in each cell that represents an alternative to the best solution. If I add them up I get 3. This means the marginal cost, or change in total cost, of this assignment from the best assignment is 3. From the brute force screen you see this is assignment is 28, 3 more than 25.

10 project person Adams x0 Brown 1 x9 Cooper x 3 On this screen I put an x in each cell that represents an alternative to the best solution. If I add them up I get 9. This means the marginal cost, or change in total cost, of this assignment from the best assignment is 9. From the brute force screen you see this is assignment is 34, 9 more than 25.

11 Output

12 MC screen Note the best solution was adams project 3, brown project 2 and cooper project 1 for cost of 25. The MC screen helps in finding changes in Total cost from a different assignment. Essentially what we do on the MC screen is put x’s in cells of a different assignment and then add the values in those cells and this would represent the addition to cost of the different assignment.

13 Concern What if you have more jobs than people or more people than jobs? The solution is to create dummy jobs or people with values of 0 in the table. This means you create a job ( or person) to add to the table and then solve the problem. A person would get assigned to a “dummy” job. (Why not just fire the worker?:)

14 10 point assignment. Use data in problem pages 444 and What is the assignment of faculty to classes that maximizes ratings? 2. How would the overall rating change if the assignment was Anderson(He taught me stats!) – Stats, Sweeney - Management, Williams – Finance, McKinney – Economics? 3. Say that reliable research has indicated that if whomever teaches economics wears a fake nose and glasses the rating will increase 25 points. If all of these folks agree to do this if they get the econ course, will the assignment of teachers change? Just type the answers in a WEBCT mail message.