This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13 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 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

WHAT IS OPTIMIZATION?

ENGINEERS AND OPTIMIZATION Engineers 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. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

OPERATIONS RESEARCH Is a discipline of optimization that deals with the application of advanced analytical methods to help make better decisions. This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

OPERATIONS RESEARCH The use of operations research expanded beyond the military to include both private companies and other governmental organizations. This opportunity if funded by the United States Department of Education. Award # DAY 6: 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

GRAPH THEORY The field of graph theory dates back more than 250 years to the Swiss mathematician Leonhard Euler ( ). Operations researchers use graphs to represent and solve practical routing problems This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

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? This opportunity if funded by the United States Department of Education. Award # DAY 6: 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

GRAPH THEORY An example of Graph Theory problems include the routing of a certain travel delivery This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

NETWORK REPRESENTATION A network consists of a set of points, called nodes, which are connected by arcs This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

ARC Arcs can be undirected or directed Sometimes the arcs can be labeled with numeric values representing distance, travel time, or cost. 23 This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

SHORTEST DISTANCE Do you use always the same path when going to work? What shortest mean to you? This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

CASE STUDY 1 This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

CASE STUDY 1 This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

CASE STUDY 1 This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

CASE STUDY 1 This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

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. This opportunity is funded by the United States Department of Education. Award # 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 This opportunity is funded by the United States Department of Education. Award # 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 office  Making a deposit at the bank  Renting a movie at the video store  Purchasing items at the grocery store This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

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 This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/ This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

CASE STUDY 2 This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

CASE STUDY 2 This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

CASE STUDY 2 This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

CASE STUDY 2 This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

CASE STUDY 2 This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity if funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

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 This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

SHORT CIRCUIT TRAVEL AGENCY This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

SHORT CIRCUIT TRAVEL AGENCY Circuit #Start1234ReturnCircuit SequenceTotal Cost 1WPCSAWWPCSAW =462 2WPCASWWPCASW 3WPSACWWPSACW 4WPSCAWWPSCAW 5WPASCWWPASCW 6WPACSWWPACSW 7WCPSAWWCPSAW 8WCW 9WCW 10WCW 11WCW ….. This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

SHORT CIRCUIT TRAVEL AGENCY 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? This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

SHORT CIRCUIT TRAVEL AGENCY 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 This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

SHORT CIRCUIT TRAVEL AGENCY Circuit #Start1234ReturnCircuit SequenceTotal Cost 1WPCSAWWPCSAW =462 2WPCASWWPCASW =530 3WPSACWWPSACW =606 4WPSCAWWPSCAW =484 5WPASCWWPASCW =515 6WPACSWWPACSW =461 7WCPSAWWCPSAW =602 8WCPASWWCPASW =579 9WCSPAWWCSPAW =533 10WCSAPWWCSAPW 11WCASPWWCASPW 12WCAPSWWCAPSW =601 This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

SHORT CIRCUIT TRAVEL AGENCY Circuit #Start1234ReturnCircuit SequenceTotal Cost 13WSCPAWWSCPAW =457 14WSCAPWWSCAPW 15WSPCAWWSPCAW =548 16WSPACWWSPACW 17WSAPCWWSAPCW 18WSACPWWSACPW 19WAPCSWWAPCSW 20WAPSCWWAPSCW 21WACPSWWASPSW 22WACSPWWACSPW 23WASPCWWASPCW 24WASCPWWASCPW This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

SHORT CIRCUIT TRAVEL AGENCY Which circuit is the cheapest? What is the cost? This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

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? This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

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 This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

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 This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

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? This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

21 CITIES PROBLEM The fastest computers can do approximately 1 trillion (1x10 12 ) 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? This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

EXPAND THE CIRCUIT Steve just added a visit to Columbus, Ohio. This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

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 # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

THE NEAREST-NEIGHBOR ALGORITHM This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 What is the cost of the route? What is the new route? $531.00; WPCoCSAW This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

THE NEAREST-NEIGHBOR ALGORITHM 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? This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

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. This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

REPETITIVE NEAREST-NEIGHBOR : SOLUTION 1.Start at P: PWACSCoP = = $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 = =$548 4.Start at A: AWPCoCSA = = $531 5.Start at S: SCCoPWAS = = $531 6.Start at Co: CoPWACSCo = = $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. This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

ACTIVITY #2 Instructions: Using the poster board and the pins, place the different characteristics under the correct method This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13

SOLUTION This opportunity if funded by the United States Department of Education. Award # DAY 6 : 6/25/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 2 – 6/17/13 – 6/21/13