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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.
A What can you adjust?
B What do you mean by best?
C What constraints must be obeyed?
The ABC’s of optimization problems
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)
• 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).
The “SHOE” problem
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
Resources Basketball Running
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Objective Contour Constraints
Construct axes to represent each of the outputs
Running shoes on vertical axis
Basketball shoes on horizontal axis
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0 2 4 6 8
Objective Contour Constraints
Canvas
Graph the first constraint: maximum amount of canvas = 12,000
Requirements determine intercepts
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0 2 4 6 8
Objective Contour Constraints
Labor Hours
Canvas
Graph the second constraint: maximum labor time = 21,000 hours
Which is more of a constraint?
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0 2 4 6 8 10 12
Objective Contour Constraints
Labor Hours
Canvas
Machine Hours
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
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Objective Contour Constraints
Labor Hours
Canvas
Machine Hours
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Objective Contour Constraints
Labor Hours
Canvas
Machine Hours
Optionally, evaluate the profit for each of the feasible corners.
Profit @ (0,0) = $0
Profit @ (0,6500) = $58.5K
Profit @ (5250,0) = $52.5K
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Objective Contour Constraints
Labor Hours
Canvas
Machine Hours
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.
