1. Models for Simulation & Optimization – An Introduction
Yale Braunstein

2. Models are Abstractions
Capture some aspects of reality
Tradeoff between realism and tractability
Can give useful insights
Cover well-studied areas
Two basic categories
Equilibrium (steady-state)
Optimization (constrained “what’s best”)

3. Specific topics to be covered
Queuing theory (waiting lines)
Linear optimization
Assignment
Transportation
Linear programming
Maybe others
Scheduling, EOQ, repair/replace, etc.

4. The ABC’s of optimization problems
What can you adjust?
What do you mean by best?
What constraints must be obeyed?

5. General comments on optimization problems
Non-linear: not covered
Unconstrained: not interesting
Therefore, we look at linear, constrained problems
Assignment
Transportation
Linear programming

6. Graphical Approach to Linear Programming
(A standard optimization technique)

7. The “SHOE” problem
We want to use standard inputs--canvas, labor, machine time, and rubber--to make a mix of shoes for the highly competitive (and profitable!) sport shoe market.
However, the quantities of each of the inputs is limited.
We will limit this example to two styles of shoes (solely because I can only draw in two dimensions).

8. What can you adjust?
We want to determine the optimal levels of each style of shoe to produce.
These are the decision variables of the model.

9. What do you mean by best?
Our objective in this problem is to maximize profit.
For this problem, the profit per shoe is fixed.

10. What constraints must be obeyed?
First, the quantities must be non-negative.
Second, the quantities used of each of the inputs can not be greater than the quantities available.
Note that each of these constraints can be represented by an inequality.

11. Overview of our approach
Construct axes to represent each of the outputs.
Graph each of the constraints.
[Optional] Evaluate the profit at each of the corners.
Graph the objective function and seek the highest profit.

12. Detailed problem statement
We can make two types of shoes:
basketball shoes at $10 per pair profit
running shoes at $9 per pair profit
Resources are limited:
canvas………………….12,000
labor hours……………..21,000
machine hours…….……19,500
rubber…………………. 16,500

13. Resource requirements
Resources Basketball Running
Canvas 2 1
Labor hours 4 2
Machine hours 2 3
Rubber 2 1

14. Construct axes to represent each of the outputs
Running shoes on vertical axis
Construct axes to represent each of the outputs
Basketball shoes on horizontal axis

15. Graph the first constraint: maximum amount of canvas = 12,000
Requirements determine intercepts

16. Graph the second constraint: maximum labor time = 21,000 hours
Which is more of a constraint?

17. Graph the third constraint: maximum machine time = 19,500 hours
Why can we ignore the last constraint?

18. The set of values that satisfy all constraints is known as the feasible region

19. Optionally, evaluate the profit for each of the feasible corners.
(0,6500) = $58.5K
Optionally, evaluate the profit for each of the feasible corners.
(0,0) = $0
(5250,0) = $52.5K

20. Graph the objective function and seek the highest feasible profit.
(3000,4500) = $70.5K

21. In closing: two theorems
The number of binding constraints equals the number of decision variables in the objective function.
If a linear problem has an optimal solution, there will always be one in a corner.