1. Computational Methods for Management and Economics Carla Gomes
Module 8c
The Assignment Problem

2. The assignment problem
Special type of linear programming problem where assignees are assigned to tasks.
Assignees and tasks  people, machines, plants, vehicles, locations, etc
Given a set of tasks to be performed and a set of assignees who are available to perform these tasks, the problem is to determine which assignee should be assigned to each task.

3. The assignment problem
To fit the model for an assignment problem, the following assumptions need to be satisfied:
The number of assignees and the number of tasks are the same.
Each assignee is to be assigned to exactly one task.
Each task is to be performed by exactly one assignee.
There is a cost associated with each combination of an assignee performing a task.
The objective is to determine how all the assignments should be made to minimize the total cost.
Note: These assumptions appear quite restrictive. However by using e.g., dummy assignees and/or tasks we can apply it to several problems.

4. Sellmore Company Assignment Problem
The marketing manager of Sellmore Company will be holding the company’s annual sales conference soon.
He is hiring four temporary employees:
Ann; Ian; Joan; Sean
Each will handle one of the following four tasks:
Word processing of written presentations
Computer graphics for both oral and written presentations
Preparation of conference packets, including copying and organizing materials
Handling of advance and on-site registration for the conference
Question: Which person should be assigned to which task?

5. Data for the Sellmore Problem
Required Time per Task (Hours)
Temporary Employee
Word Processing
Graphics
Packets
Registrations
Hourly Wage
Ann 35 41 27 40
$14
Ian 47 45 32 51 12
Joan 39 56 36 43 13
Sean 25 46 15
Table Data for the Sellmore Company problem.

6. The Network Representation
Figure The network representation of the Sellmore Co. assignment problem shows all the possible assignments and their costs graphically.

7. Job Shop (Assigning Machines to Locations)
The Job Shop Company has purchased three new machines of different types.
There are five available locations where the machine could be installed.
Some of these locations are more desirable for particular machines because of their proximity to work centers that will have a heavy work flow to these machines.
Question: How should the machines be assigned to locations?

8. Materials-Handling Cost Data
Cost per Hour
Location: 1 2 3 4 5
Machine
$13
$16
$12
$14
$15 15 — 13 20 16 7 10 6
Table Materials-handling cost data for the Job Shop Co. problem.

9. The Assignment Problem Model
Decision variables:
1 if assignee i performs task j
xij
0 otherwise
i,j = 1,2 …, n
Each xij is a binary variables – it takes only two values: 0 or 1.

10. The Assignment Problem Model
Each task is performed
by exactly one assignee.
Each assignee
performs
exactly one task.

11. The Assignment Problem Model
The Assignment Problem Model is a particular type of integer programming problem
If we remove the constraints that state that xij is a binary variable (i.e., xij is 0 or 1):
we get a linear programming problem.
we get a transportation problem
Number source = number of destinations
Every supply sj = 1 .
Every demand dj = 1 .
Because of its special structure, like in transportation problems, we have the guarantee that the simplex method (the standard or the transportation simplex method) will produce BF integer solution (therefore we don’t have to worry about the binary constraints).
Note: in general, with integer programming problems, we are not that lucky  we have no guarantees of “automatically” finding an integer solution and therefore we cannot ignore integrality constraints.

12. The Assignment Problem Model – solution procedures
We can use several solution procedures:
Standard simplex (*)
Transportation simplex (faster)(*)
Specialized algorithms that further exploit the special structure of the Assignment Problem (e.g. the Hungarian Algorithm) and therefore are considerably faster than Transportation simplex .
(*) Because all the RHS are 1 and the matrix A only has 0 and 1 elements, the solutions
for the assignment problem are highly degenerate, which causes wasted iterations
where nothing changes except for the labeling of which allocations of zero correspond
to degenerate variables rather than non-basic variables.

13. Algebraic Formulation
Let xij = 1 if swimmer i swims stroke j; 0 otherwise tij = best time of swimmer i in stroke j Minimize Time = ∑ i ∑ j tij xij subject to each stroke swum: ∑ i xij = 1 for each stroke j each swimmer swims 1: ∑ j xij ≤ 1 for each swimmer i and xij ≥ 0 for all i and j.

14. Bidding for Classes
In the MBA program at a prestigious university in the Pacific Northwest, students bid for electives in the second year of their program.
Each of the 10 students has 100 points to bid (total) and must take two electives.
There are four electives available:
Quantitative Methods
Finance
Operations Management
Accounting
Each class is limited to 5 students.
Question: How should students be assigned to the classes?
Discuss: what is the right objective?

15. Points Bid for Electives
Student
Quantitative Methods
Finance
Operations Management
Accounting
George 60 10 20
Fred 40
Ann 45 5
Eric 50 25
Susan 30
Liz Ed 70
David 35 15
Tony
Jennifer

16. Better Products (Assigning Plants to Products)
The Better Products Company has decided to initiate the production of four new products, using three plants that currently have excess capacity.
Unit Cost
Product: 1 2 3 4
Capacity Available
Plant
$41
$27
$28
$24 75 40 29 — 23 37 30 27 21 45
Required production 20
Table 6.6 Data for the Better Products Co. problem.
Question: How to split up the production of the products among plants?
Management wants to consider two options: option1- permit product splitting – the same product can be produced in more than one plant; option2 – prohibit product splitting;

17. Better Products (Assigning Plants to Products) Option 1- Transportation Formulation
Option 1 – Permit product splitting  Transportation Formulation
Unit Cost
Destination (Product): 1 2 3 4
5(D)
Supply
Source(Plant) 41 27 28 24 75 40 29 M 23 37 30 21 45
Demand 20
Table 6.6 Data for the Better Products Co. problem.
Optimal solution: x12 =30; x13 = 30; x15 = 15; x25=60;x31=20;x34=25 Z*=$3260 (cost a day)
Plant 1 – produces all of products 2 and 3; Plant 2 produces 37.5% of product 4; Plant 3:
Produces 62.5% of product 4 and all of product 1;

18. Better Products (Assigning Plants to Products) Option 2
Option 2 – Prohibit product splitting  Assignment Formulation(*)
Unit Cost
Destination (Product): 1 2 3 4
5(D)
Source(Plant) 1a
820
810
840
960 1b 2a
800
870 M
920 2b
740
900
Demand 20 30 40 75
(**)
Table 6.6 Data for the Better Products Co. problem.
(*) we can also use the transportation simplex algorithm to solve the assignement model.
(**) $41 * 20 = $820; the other entries are obtained in a similar way;
Optimal solution:
Plant 1 – produces products 2 and 3; Plant 2 produces product 1; Plant 3:
Produces product 4; Z*=$3290 (cost a day) – slightly more, but there are hidden costs of splitting …

19. Summary
Linear programming model – general framework that includes a wide variety of specific types of problems.
Some specific types have special structures which allow for streamlined algorithms much more efficient than the simplex method – e.g., the transportation problem and the assignment problem.
Both the transportation problem and the assignment problem have a number of common applications  it is important to recognize them when they arise and to use the best available algorithm.