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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)
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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 Canvas 2 1 Labor hours 4 2 Machine hours 2 3 Rubber 2 1 ResourcesBasketballRunning
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
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.
