this opportunity is funded by the united states department of education. award # 2010-38422-19963 -...

This opportunity is funded by the United States Department of Education.

Award # 2010-38422-19963 - WEEK 2 – 6/17/13 – 6/21/13

INTRODUCTION TO OPTIMIZATION

INTRODUCTION TO OPTIMIZATION

Why teach optimization? What is it? How does it fit into

engineering? Why is it

important?

We have lots of excellent choices to cover during this second week


Award # 2010-38422-19963 - WEEK 2 – 6/17/13 – 6/21/13

WHAT IS OPTIMIZATION?

ENGINEERS AND OPTIMIZATIONEngineers use optimization all the time !!!

• Production planning• Product distribution• Logistics network design• Telecommunication networks• Scheduling• Product design• Asset allocation• Portfolio optimization• Vehicle routing• Manufacturing processes• Product configuration• Health care systems• Etc…

HOW DO WE OPTIMIZE?

Mathematical optimization – (model-based)

Two options exist

Empirical optimization

• We follow the optimization process

• What is the objective?

• What type of variables?

• Are there any constraints?

• We follow our intuition

• We use brute-force

SOLUTION !

MANY TYPES OF OPTIMIZATION TECHNIQUES

The challenge is to put the numbers in a 3x3 grid so that any line (horizontal, vertical or diagonal) of three numbers in the grid equals the same amount.

Mini optimization challenge

WHAT YOU WILL LEARN THIS WEEK

• What optimization is and the role it plays in Engineering

• The optimization process

• Some basic algorithms Linear programming Integer programming Combinatorial

optimization Network optimization

• Real-life optimization applications

WHAT IS OPTIMIZATION?

Optimization is a scientific approach to analyzing problems and making decisions.

Much of this work is done using analytical and numerical techniques to develop a mathematical model to organize systems that

involve people, machines and

procedures.

This opportunity if funded by the United States Department of Education.

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OPERATIONS RESEARCH

Is a discipline of optimization that deals with the application of advanced analytical methods to help make better decisions.


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OPERATIONS RESEARCH

The use of operations research expanded beyond the military to include both private companies and other governmental organizations.


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GRAPH THEORY

The field of graph theory dates back more than 250 years to the Swiss mathematician Leonhard Euler (1707-1783).

• Operations researchers use graphs to represent and solve practical routing problems This opportunity if funded by the United States Department of Education.

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GRAPH THEORY

An example is drawing an star, is it possible to draw the star without lifting the pencil or going over the same line twice?


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GRAPH THEORY

An example of Graph Theory problems include the routing of a certain travel delivery


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NETWORK REPRESENTATION

A network consists of a set of points, called nodes, which are connected by arcs


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ARC

Arcs can be undirected or directed

Sometimes the arcs can be labeled with numeric values representing distance, travel time, or cost.

23


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SHORTEST DISTANCE

Do you use always the same path when going to work?

What shortest mean to you?


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CASE STUDY 1


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ACTIVITY 1: TRAVELING SALESMAN PROBLEM

Sandy’s grandmother lives in an old one-story house. There are many connecting doors between the rooms. One day, Sandy wanted to find a route that would take her though each door exactly once. Help Sandy find a route.


Award # 2010-38422-19963 - WEEK 2 – 6/17/13 – 6/21/13

TSP SOLUTION

If a room has an odd number of doors, you must either begin in that room or end in that room


Award # 2010-38422-19963 - WEEK 2 – 6/17/13 – 6/21/13

COMPLETING A TRIP

Lets discuss the best way to complete a trip, fulfilling the following list of errands:

Mailing letters at the post officeMaking a deposit at the bankRenting a movie at the video storePurchasing items at the grocery

store


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TRAVELING SALESMAN PROBLEM

Given a number of cities and the costs of traveling from any city to any other city, what is the cheapest round-trip route that visits each city exactly once and then returns to the starting city


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CASE STUDY 2


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SHORT CIRCUIT TRAVEL AGENCY

Steve Isaac works in the human resources department of an engineering company in the Washington, D.C., area.

He plans to visit 4 universities in 4 days and then return to Washington


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SHORT CIRCUIT TRAVEL AGENCYCircuit #

Start

1 2 3 4 Return

Circuit Sequence

Total Cost

1 W P C S A W WPCSAW 74+98+65+149+76=462

2 W P C A S W WPCASW

3 W P S A C W WPSACW

4 W P S C A W WPSCAW

5 W P A S C W WPASCW

6 W P A C S W WPACSW

7 W C P S A W WCPSAW

8 W C W

9 W C W

10 W C W

11 W C W

…..


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How many possible circuits are there?

Among the circuits that start WC, you should have identified circuit WCSAPW. If you travel this circuit in reverse, what circuit would it be?


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How many unique circuits remain?

How does the number of unique circuits compare to the total number of circuits?

The total cost of circuit 1 is $462, calculate the cost for each of the remaining unique circuits and record it in the table


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Circuit #

Start

1 2 3 4 Return Circuit Sequence

Total Cost

1 W P C S A W WPCSAW 74+98+65+149+76=462

2 W P C A S W WPCASW 74+98+104+149+105=530

3 W P S A C W WPSACW 74+165+149+104+114=606

4 W P S C A W WPSCAW 74+165+65+104+114=484

5 W P A S C W WPASCW 74+113+149+65+114=515

6 W P A C S W WPACSW 74+113+104+65+105=461

7 W C P S A W WCPSAW 114+98+165+149+76=602

8 W C P A S W WCPASW 114+98+113+149+105=579

9 W C S P A W WCSPAW 114+65+165+113+76=533

10 W C S A P W WCSAPW

11 W C A S P W WCASPW

12 W C A P S W WCAPSW 114+104+113+165+105=601This opportunity if funded by the United States Department of Education.

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Circuit #

Start

1 2 3 4 Return

Circuit Sequence

Total Cost

13 W S C P A W WSCPAW 105+65+98+113+76=457

14 W S C A P W WSCAPW

15 W S P C A W WSPCAW 105+165+98+104+76=548

16 W S P A C W WSPACW

17 W S A P C W WSAPCW

18 W S A C P W WSACPW

19 W A P C S W WAPCSW

20 W A P S C W WAPSCW

21 W A C P S W WASPSW

22 W A C S P W WACSPW

23 W A S P C W WASPCW

24 W A S C P W WASCPW


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Which circuit is the cheapest? What is the cost?


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THE BRUTE-FORCE METHOD

The method we just used is sometimes called the “brute-force” method because it involves trying every unique circuit

How many nodes can you travel to directly from W?


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THE BRUTE-FORCE METHOD

After you have chosen the second node in your circuit, how many choices are there for the third node?

And for the fourth node and the fifth node?

3

2 and then 1


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WRITING A FORMULA

Lets write a general formula in terms of n for the total number of circuits that can be created starting at point W.

Just remember that some of the circuits were duplicates


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WRITING A FORMULA

How many unique circuits will be when there are only 6 cities, and for 7, what about 21 cities?

Can you solve a problem of 21 cities using the brute-force method?


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21 CITIES PROBLEM

The fastest computers can do approximately 1 trillion (1x1012) computations per second.

Assuming that the construction of a 21-node path requires 100 computations. Using the brute-force method, how long it would take to a computer to solve this problem?

(1.22x10^18)(100comp.)(= more than 1400 days


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EXPAND THE CIRCUIT

Steve just added a visit to Columbus, Ohio.


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THE NEAREST-NEIGHBOR ALGORITHM

I. Choose a node as your starting point

II. From the starting node, travel to the node for which the fare is the cheapest. We call this node the “nearest neighbor”. If there is a tie, choose one arbitrarily.

III. Repeat the process, one node at a time, traveling to nodes that have not yet been visited. Continue this process until all nodes have been visited.

IV. Complete the Hamiltonian circuit by returning to the starting point.

V. Calculate the cost of the circuit.This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 6 : 6/25/2012


Award # 2010-38422-19963 - WEEK 2 – 6/17/13 – 6/21/13



Award # 2010-38422-19963 - DAY 6 : 6/25/2012

What is the cost of the route? What is the new route?$531.00; WPCoCSAW


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Why does using the nearest-neighbor algorithm make more sense than using the brute-force method in this case?

Will the nearest-neighbor algorithm always give a good route? Why or why not?


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REPETITIVE NEAREST-NEIGHBOR ALGORITHM

I. Select any node as a starting point. Apply the nearest-neighbor algorithm from that node.

II. Calculate the cost of that circuit.

III.Repeat the process using each of the other nodes as the starting point.

IV.Choose the “best” Hamiltonian circuit.


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REPETITIVE NEAREST-NEIGHBOR : SOLUTION1.Start at P: PWACSCoP = 74+76+104+65+110+79= $508

2.WACSCoPW; its cost is the same, $508. A circuit goes into and out of each city. Therefore, the starting point within a given circuit has not effect on the total cost.

3.Start at C: CSWPCoAC = 65+105+74+79+121+104 =$548

4.Start at A: AWPCoCSA = 76+74+79+88+65+149 = $531

5.Start at S: SCCoPWAS = 65+88+79+74+76+149 = $531

6.Start at Co: CoPWACSCo = 79+74+76+104+65+110 = $508. This is identical to the circuit found by starting the nearest –neighbor algorithm at Pittsburgh. The cheapest circuit found by starting the algorithm at either Pittsburgh or Columbus translates to WACSCoPW and costs $508.


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ACTIVITY #2

Instructions:• Using the poster board and

the pins, place the different characteristics under the correct method


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SOLUTION


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this opportunity is funded by the united states department of education. award # 2010-38422-19963 -...

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