Models for Simulation

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Models for Simulation Optimization An
Introduction

• Yale Braunstein

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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 whats best)

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Specific topics to be covered

• Queuing theory (waiting lines)
• Linear optimization
• Assignment
• Transportation
• Linear programming
• Maybe others
• Scheduling, EOQ, repair/replace, etc.

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The ABCs of optimization problems

• What can you adjust?
• What do you mean by best?
• What constraints must be obeyed?

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General comments on optimization problems

• Non-linear not covered
• Unconstrained not interesting
• Therefore, we look at linear, constrained
  problems
• Assignment
• Transportation
• Linear programming

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

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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.

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What do you mean by best?

• Our objective in this problem is to maximize
  profit.
• For this problem, the profit per shoe is fixed.

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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.

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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.

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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

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Resource requirements
Resources Basketball Running

• Canvas 2 1
• Labor hours 4 2
• Machine hours 2 3
• Rubber 2 1

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Running shoes on vertical axis
Construct axes to represent each of the outputs
Basketball shoes on horizontal axis
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Graph the first constraint maximum amount of
canvas 12,000
Requirements determine intercepts
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Graph the second constraint maximum labor time
21,000 hours
Which is more of a constraint?
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Graph the third constraint maximum machine time
19,500 hours
Why can we ignore the last constraint?
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The set of values that satisfy all constraints is
known as the feasible region
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Profit _at_ (0,6500) 58.5K
Optionally, evaluate the profit for each of the
feasible corners.
Profit _at_ (0,0) 0
Profit _at_ (5250,0) 52.5K
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Graph the objective function and seek the highest
feasible profit.
Profit _at_ (3000,4500) 70.5K
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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.