1. Introduction to optimization
Hello! Welcome to this introduction about optimization techniques in decision-modeling.
2. Introduction
Optimization involves finding the best solution to a problem from a set of possibilities.
By maximizing or minimizing criteria such as:
3. Introduction
Minimal cost
4. Introduction
Minimal time
5. Introduction
and highest efficiency
6. Introduction
Applying optimization techniques to decision models helps decision-makers make the best choices.
Let's further explore optimization in decision-making.
7. Optimizing decisions
Decisions can be segmented into objectives, alternatives, uncertainties and constraints.
8. Optimizing decisions
In optimizations, the objective of the decision is the main factor; referring to "what" is being optimized.
9. Optimizing decisions
The alternatives, are the choices available to optimize the objective.
10. Optimizing decisions
Lastly, constraints are the requirements that must always be met.
Next, let's see the steps to optimize a decision.
11. The optimization journey
Step 1 is formulating the problem by identifying the objective, alternatives and constraints.
12. The optimization journey
Next, a mathematical model is built using the factors identified in Step 1.
13. The optimization journey
Step 3 is applying optimization techniques to find the best solution for the model.
14. The optimization journey
Finally, step 4 is the implementation and monitoring of the optimal solution.
Let's see these steps in one example:
15. Optimizing a farmer's land
A farmer has 1,000 ft of fence to enclose a rectangular field, with one side bordering a river.
Since the river side doesn't require a fence, he only needs to fence the three sides represented in the illustration by "Xs" and "Y".
16. Optimizing a farmer's land
What should be the dimensions of the fence to maximize the area of the field?
17. Identifying the decision factors
We'll start with the objective function:
Maximizing the area of the field which is "x" multiplied by "y".
18. Identifying the decision factors
The variables "x" and "y" represent the. alternatives available to find the optimal area.
19. Identifying the decision factors
The constraint is the 1000 ft of fence to enclose the sides of the field. This can be mathematically represented as 2(x)+y =1000.
20. Building a mathematical model
These are the objective and the constraint functions which we'll manipulate next.
21. Building a mathematical model
Let's isolate y by rewriting the constraint function as y=1000-2x.
22. Building a mathematical model
Now, replacing "y" in the objective function for 1000-2x...
23. Building a mathematical model
we update the objective function as "x times 1000 - two x".
Now we can solve the model finding the optimal value for "x".
But before that, let's investigate x further.
24. Building a mathematical model
x can have a minimum value of 0. In this case, y would be 1000. The fence would be placed straight along the river and the area would be 0.
25. Building a mathematical model
Inversely, "x" can have a maximum value of 500 with "y" being zero. Here, each lateral side would have 500 ft and the area would also be zero.
26. Building a mathematical model
So, we conclude that 'x' ranges from 0 to 500 ft. Now, we need to find the value of 'x' that maximizes the objective function.
27. Finding the optimal solution
The final model corresponds to "Area = x times 100 minus 2x" and x ranges from 0 to 500.
We can actually solve this model graphically.
28. Finding the optimal solution
This parabola shows the objective function with x varying from 0 to 500.
The function reaches its highest value when x = 250. This is the optimal point!
29. Finding the optimal solution
So,the two lateral sides of the fence must have 250 ft.
30. Finding the optimal solution
Considering the whole fence measures 1000 ft. The value of "y" must be 500 ft. With these measures, the farmer can place the fence maximizing the area of his land!
31. Let's practice!
Great job! Let's practice a bit more.