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Session 3b Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

Overview More Network Flow Models Assignment Model Traveling Salesman Model Decision Models Prof. Juran Decision Models Prof. Juran

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**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 Decision Models Prof. Juran

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**Professor Preferences**

Decision Models Prof. Juran Decision Models Prof. Juran

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**Managerial Problem Definition**

Determine an assignment of professors to sections that maximizes the total satisfaction of the professors. Decision Models Prof. Juran Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

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 Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

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 Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

Formulation Decision Models Prof. Juran Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

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 Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

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 Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

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 Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

Solution Methodology Decision Models Prof. Juran Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

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**Decision Models -- Prof. Juran**

Optimal Solution Decision Models Prof. Juran Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

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 Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

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 Decision Models Prof. Juran

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**Network Representation**

Prof 1 Prof 2 Prof 3 9:00 10:00 11:00 1:00 2:00 3:00 Decision Models Prof. Juran Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

Optimal Solution Prof 1 Prof 2 Prof 3 8 6 6 9 9 8 9:00 10:00 11:00 1:00 2:00 3:00 Decision Models Prof. Juran Decision Models Prof. Juran

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**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 Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

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 Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

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**Decision Models -- Prof. Juran**

1 Decision Models Prof. Juran Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

Trouble! Each source city is own destination. We’ll use the old “big cost” trick: Decision Models Prof. Juran Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

2 Decision Models Prof. Juran Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

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 Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

3 Decision Models Prof. Juran Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

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 Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

4 Decision Models Prof. Juran Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

5 Decision Models Prof. Juran Decision Models Prof. Juran

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**Decision Models -- Prof. Juran**

Summary More Network Flow Models Assignment Model Traveling Salesman Model Decision Models Prof. Juran Decision Models Prof. Juran

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Math443/543 Mathematical Modeling and Optimization

Math443/543 Mathematical Modeling and Optimization

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