Models for Simulation & Optimization – An Introduction Yale Braunstein

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”)

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

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

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

Graphical Approach to Linear Programming (A standard optimization technique)

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).

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.

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

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.

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.

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

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

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

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

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

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

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

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

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

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.