Chapter 2: Business Efficiency Lesson Plan

1. For All Practical Purposes Chapter 2: Business EfficiencyLesson Plan • Business Efficiency • Visiting Vertices-Graph Theory Problem • Hamiltonian Circuits • Vacation Planning Problem • Minimum Cost-Hamiltonian Circuit • Method of Trees • Fundamental Principle of Counting • Traveling Salesman Problem • Helping Traveling Salesmen • Nearest Neighbor and Sorted Edges Algorithms • Minimum-Cost Spanning Trees • Kruskal’s Algorithm • Critical-Path Analysis Mathematical Literacy in Today’s World, 8th ed.

2. Chapter 2: Business EfficiencyBusiness Efficiency • Visiting Vertices • In some graph theory problems, it is only necessary to visit specific locations (using the travel routes, or streets available). • Problem: Find an efficient route along distinct edges of a graph that visits each vertex only once in a simple circuit. Applications: • Salesman visiting particular cities • Delivering mail to drop-off boxes • Route taken by a snowplow • Pharmaceutical representative visiting doctors

3. Chapter 2: Business EfficiencyHamiltonian Circuit • Hamiltonian Circuit • A tour that starts and ends at the same vertex (circuit definition). • Visits each vertex once. (Vertices cannot be reused or revisited.) • Circuits can start at any location. • Use wiggly edges to show the circuit.

4. Chapter 2: Business EfficiencyBusiness Efficiency Starting at vertex A, the tour can be written as ABDGIHFECA, or starting at E, it would be EFHIGDBACE.

5. Chapter 2: Business EfficiencyBusiness Efficiency A different circuit visiting each vertex once and only once would be CDBIGFEHAC (starting at vertex C).

6. Chapter 2: Business EfficiencyHamiltonian Circuit vs. Euler Circuits Euler circuit – A circuit that traverses each edge of a graph exactly once and starts and stops at the same point.

7. Chapter 2: Business EfficiencyHamiltonian Circuit vs. Euler Circuits Hamiltonian circuit– A tour (showed by wiggly edges) that starts at a vertex of a graph and visits each vertex once and only once, returning to where it started.

8. Chapter 2: Business EfficiencyHamiltonian Circuit vs. Euler Circuits Hamiltonian vs. Euler Circuits • Similarities • Both forbid re-use. • Hamiltonian do not reuse vertices. • Euler do not reuse edges.

9. Chapter 2: Business EfficiencyHamiltonian Circuit vs. Euler Circuits • Differences • Hamiltonian is a circuit of vertices. • Euler is a circuit of edges. • Euler graphs are easy to spot (connectedness and even valence). • Hamiltonian circuits are NOT as easy to determine upon inspection. • Some certain family of graphs can be known to have or not have Hamiltonian circuits.

10. Chapter 2: Business EfficiencyHamiltonian Circuits Example 1 Vacation Planning Problem

11. Chapter 2: Business EfficiencyHamiltonian Circuits Let’s imagine that you are a college student in Chicago. During spring break you and a group of friends have decided to take a car trip to visit other friends in Minneapolis, Cleveland, and St. Louis. There are many choices as to the order of visiting cities and returning to Chicago, but you want to design a route that minimizes the distance you have to travel. This will also cut costs because of the cost of gasoline for the trip.

12. Chapter 2: Business EfficiencyHamiltonian Circuits Similar problems with complications would arise for bus, railroad, or airplane trips. Now the local automobile club has provided you with the inter-city driving distances between Chicago, Minneapolis, Cleveland, and St. Louis. We can construct a graph model with this information, representing each city by a vertex and the legs of the journey between cities by edges joining the vertices. To complete the model, we add a number called a weight to each graph edge.

13. Chapter 2: Business EfficiencyHamiltonian Circuits Road mileage between four cities

14. Chapter 2: Business EfficiencyHamiltonian Circuits • Hamiltonian circuit concept is used to find the best route that minimizes the total distance traveled to visit friends in different cities. (assume less mileage less gas minimizes costs)

15. Chapter 2: Business EfficiencyHamiltonian Circuits Hamiltonian circuit with weighted edges • Edges of the graph are given weights, or in this case mileage or distance between cities. • As you travel from vertex to vertex, add the numbers (mileage in this case). • Each Hamiltonian circuit will produce a particular sum.

16. Chapter 2: Business EfficiencyHamiltonian Circuit • Minimum-Cost Hamiltonian Circuit • A Hamiltonian circuit with the lowest possible sum of the weights of its edges. Algorithm is a step-by-step description of how to solve a problem.

17. Chapter 2: Business EfficiencyHamiltonian Circuit • Algorithm (step-by-step process) for Solving This Problem • Generate all possible Hamiltonian tours (starting with Chicago). • Add up the distances on the edges of each tour. • Choose the tour of minimum distance.

18. Chapter 2: Business EfficiencyHamiltonian Circuit Steps 2 and 3 of the algorithm are straightforward. Thus, we need worry only about Step 1, generating all the possible Hamiltonian circuits in a systematic way. To find the Hamiltonian tours, we will use the method of trees. In this step we disregard the distances.

19. Chapter 2: Business EfficiencyHamiltonian Circuit 1st Stage This stage is where you select a starting vertex, say Chicago, and making a tree-diagram showing the next possible locations.

20. Chapter 2: Business EfficiencyHamiltonian Circuit

21. Chapter 2: Business EfficiencyHamiltonian Circuit 2nd Stage At each subsequent stage down, there will be one less choice .

22. Chapter 2: Business EfficiencyHamiltonian Circuit 22

23. Chapter 2: Business EfficiencyHamiltonian Circuit 3rd Stage At each subsequent stage down, there will be one less choice .

24. Chapter 2: Business EfficiencyHamiltonian Circuits

25. Chapter 2: Business EfficiencyHamiltonian Circuits Name the 6 tours from the tree. C-M-S-CL-C C-M-CL-S-C C-S-M-CL-C C-S-CL-M-C C-CL-M-S-C C-CL-S-M-C Notice that 1 and 6 are the same tour, 2 and 4 are the same tour, and 3 and 5 are the same tour. So there are only three are unique circuits. Find the total distance for each unique circuit.

26. Chapter 2: Business EfficiencyHamiltonian Circuits The optimal tour is Chicago, Minneapolis, St. Louis, Cleveland, Chicago. These graphs are called complete graphs. The three Hamiltonian circuits’ sums of the tours

27. 2nd Day Chapter 2: Business EfficiencyHamiltonian Circuits

28. Chapter 2: Business EfficiencyHamiltonian Circuits Complete graph – A graph in which every pair of vertices is joined by an edge.

29. Chapter 2: Business EfficiencyHamiltonian Circuits Suppose instead of going to 4 cities you had to plan a minimum cost circuit for 100 cities. Making a tree would take an incredible amount of time. There is a better way.

30. Chapter 2: Business EfficiencyHamiltonian Circuits Fundamental Principle of Counting If there are “a”ways of choosing one thing, “b”ways of choosing a second after the first is chosen, “c”ways of choosing a third after the second is chosen…, and so on…, and “z”ways of choosing the last item after the earlier choices, then the total number of choice patterns is a×b × c × … × z. Example: Jack has 9 shirts and 4 pairs of pants. He can wear 9 × 4 = 36 shirt-pant outfits.

31. Chapter 2: Business EfficiencyHamiltonian Circuits • In a restaurant there are 4 kinds of soup, 12 entrees, 6 desserts, and 3 drinks. How many different four-course meals can a patron choose? 4 x 12 x 6 x 3 = 864 possible meals

32. Chapter 2: Business EfficiencyHamiltonian Circuits • In a state lottery a contestant gets to pick a four-digit number that does not contain a zero followed by an uppercase or lowercase letter. How many such sequences of digits and a letter are there? Each of the four digits can be chosen 9 ways (that is, 1, 2, …, 9), and the letter can be chosen in 52 ways (that is, A, B, …, Z plus a, b, …, z). So 9 x 9 x 9 x 9 x 52 = 341, 172

33. Chapter 2: Business EfficiencyHamiltonian Circuits • A corporation is planning a musical logo consisting of four different ordered notes from the scale C, D, E, F, G, A, and B. • How many logos are there to choose from if no letter can be reused? The 1st note can be chosen in 7 ways, but because reuse is not allowed, the next note can be chosen in only 6 ways. The remaining two notes can be chosen in 5 and 4 ways, respectively. So 7 x 6 x 5 x 4 = 840 musical logos

34. Chapter 2: Business EfficiencyHamiltonian Circuits • How many logos are there to choose from if letters can be reused? 7 x 7 x 7 x 7 = 2401 musical logos

35. Chapter 2: Business EfficiencyHamiltonian Circuits Let’s use the Fundamental Counting Principal on the spring break trip. Stage 1 = 3 cities Stage 2 = 2 cities Stage 3 = 1 city

36. Chapter 2: Business EfficiencyHamiltonian Circuits So 3 x 2 x 1 = 6 tours C-M-S-CL-C C-M-CL-S-C C-S-M-CL-C C-S-CL-M-C C-CL-M-S-C C-CL-S-M-C Notice that 1 and 6 are the same tour, 2 and 4 are the same tour, and 3 and 5 are the same tour. So there are only three are unique circuits.

37. Chapter 2: Business EfficiencyHamiltonian Circuit • Principle of Counting for Hamiltonian Circuits • For a complete graph of n vertices, there are (n - 1)! possible routes. • Half of these routes are repeats, the result is: Possible unique Hamiltonian circuits are (n - 1)! / 2

38. Chapter 2: Business EfficiencyHamiltonian Circuits There are 4 vertices. So there are (4 – 1)! possibilities. 3! = (3)(2)(1) = 6 3!/2 = 6/2 = 3 circuits Road mileage between four cities

39. Chapter 2: Business EfficiencyHamiltonian Circuit Examples: • How many different tours could there be with six cities? (6 – 1)!/2 = 5!/2 = 60 different tours • How many different tours could there be with ten cities. (10 – 1)!/2 = 9!/2 = 181,440 different tours

40. Chapter 2: Business EfficiencyHamiltonian Circuit This method is good for small circuits but as you saw the number of different tours can rather large very quickly. That is why this method is called a brute force method.