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Session 3b Overview More Network Flow Models Assignment Model Traveling Salesman Model Professor Scheduling Example Three professors must be assigned to teach six sections of finance. Each professor must teach two sections of finance.

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overview
Overview
  • More Network Flow Models
    • Assignment Model
    • Traveling Salesman Model

Decision Models -- Prof. Juran

professor scheduling example
Professor Scheduling Example
  • Three professors must be assigned to teach six sections of finance.
  • Each professor must teach two sections of finance.
  • Each professor has ranked the six time periods during which finance is taught.
  • A rating of 10 means that the professor wants to teach at that time, and a ranking of 1 means that he or she does not want to teach at that time.

Decision Models -- Prof. Juran

professor preferences
Professor Preferences

Decision Models -- Prof. Juran

managerial problem definition
Managerial Problem Definition

Determine an assignment of professors to sections that maximizes the total satisfaction of the professors.

Decision Models -- Prof. Juran

formulation
Formulation

Decision Variables

We need to identify who is teaching which class. In other words, we need to make one-to-one links between the classes to be taught and the available professors.

Objective

Maximize total satisfaction.

Constraints

All classes need to be covered by exactly 1 professor.

Each professor needs to be assigned to exactly 2 classes.

Decision Models -- Prof. Juran

formulation7
Formulation

Decision Variables

Define Xij to be a binary variable representing the assignment of professor i to class j. If professor i ends up teaching class j, then Xij = 1. If professor i does not end up teaching class j, then Xij = 0.

Define Cij to be the “preference” of professor i for class j.

Objective

Maximize Z =

Decision Models -- Prof. Juran

formulation8
Formulation

Decision Models -- Prof. Juran

formulation9
Formulation
  • The objective function uses the nice attributes of binary variables to create an overall measure of “professorial delight”.
  • If a professor is assigned to a class for which he/she has a preference score of 6, for example, then the six gets multiplied by a one (6 x 1 = 6) and gets added into the overall objective score.
  • If the professor is not assigned to that class, then the six gets multiplied by a zero (6 x 0 = 0) and has no effect on the overall objective.

Decision Models -- Prof. Juran

formulation10
Formulation

These constraints are not exactly like the “English” versions; in particular they are not as “strict”.

For example, the first constraint seems to imply that more than one professor could feasibly be assigned to a class. The second constraint implies that a professor could feasibly be assigned to fewer than two classes.

That’s OK, because the two constraints together force exactly one professor per class, and two classes per professor.

Decision Models -- Prof. Juran

formulation11
Formulation

It is not necessary to constrain the decision variables to be binary; the optimal linear solution will automatically have zeros and ones for the decision variables.

Decision Models -- Prof. Juran

solution methodology
Solution Methodology

Decision Models -- Prof. Juran

solution methodology13
Solution Methodology

Decision Models -- Prof. Juran

solution methodology14
Solution Methodology

Decision Models -- Prof. Juran

optimal solution
Optimal Solution

Decision Models -- Prof. Juran

optimal solution16
Optimal Solution

In the optimal solution, professor 1 teaches at 9:00 and 3:00, professor 2 teaches at 10:00 and 11:00, and professor 3 teaches at 1:00 and 2:00.

The maximum overall preference score is 46.

Decision Models -- Prof. Juran

slide17

This problem is an example of an entire category of classic operations research models called network flow problems, so called because they can be represented as networks of nodes (balls) and arcs (arrows).

Decision Models -- Prof. Juran

network representation

Prof 1

Prof 2

Prof 3

Network Representation

9:00

10:00

11:00

1:00

2:00

3:00

Decision Models -- Prof. Juran

optimal solution19

Prof 1

Prof 2

Prof 3

Optimal Solution

8

6

6

9

9

8

9:00

10:00

11:00

1:00

2:00

3:00

Decision Models -- Prof. Juran

traveling salesman problem
Traveling Salesman Problem

One of the classic problems in optimization is to find the minimum-distance path between a set of points. For example, what is the shortest route that connects all of these 13 European cities?

Decision Models -- Prof. Juran

formulation21
Formulation

Decision Variables: Binary decisions from each “source” city to each “destination” city

Objective: Minimize total distance traveled (sumproduct of binary variables times distances)

Constraints: Each city must be the “source” exactly one time and the “destination” exactly one time

Decision Models -- Prof. Juran

slide23

1

Decision Models -- Prof. Juran

slide24

Trouble!

Each source city is own destination.

We’ll use the old “big cost” trick:

Decision Models -- Prof. Juran

slide25

2

Decision Models -- Prof. Juran

slide26

More Trouble!

Small loops – called “sub-tours”.

We need to add special constraints for each subtour:

Example in column S: B16 + N4 < = 1

Decision Models -- Prof. Juran

slide27

3

Decision Models -- Prof. Juran

slide28

Sub-tours keep cropping up, and we need to add constraints for each of them.

This procedure continues until a single tour encompasses all cities.

Decision Models -- Prof. Juran

slide29

4

Decision Models -- Prof. Juran

slide30

5

Decision Models -- Prof. Juran

summary
Summary
  • More Network Flow Models
    • Assignment Model
    • Traveling Salesman Model

Decision Models -- Prof. Juran