Models for Simulation & Optimization – An Introduction Yale Braunstein.

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

0

2

4

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14

0 2 4 6 8

Objective Contour Constraints

Construct axes to represent each of the outputs

Running shoes on vertical axis

Basketball shoes on horizontal axis

0

2

4

6

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14

0 2 4 6 8


Canvas

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

Requirements determine intercepts

0

2

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6

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14

0 2 4 6 8


Labor Hours

Canvas

Graph the second constraint: maximum labor time = 21,000 hours

Which is more of a constraint?

0

2

4

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0 2 4 6 8 10 12


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

0

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0 2 4 6 8 10 12


Labor Hours

Canvas

Machine Hours

0

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0 2 4 6 8 10 12


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

0

2

4

6

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14

0 2 4 6 8 10 12


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.

Models for Simulation & Optimization – An Introduction Yale Braunstein.

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