2. 1 ENM 503 Block 3 Lesson 12 – Discrete Optimization Models Combinatorial Problems and their Solutions An Introduction to Discrete Optimization Narrator: Charles Ebeling University of Dayton

3. 2 The Lineup The Assignment Problem The Knapsack Problem The Traveling Salesman Problem The Postman problem

4. 3 Combinatorial Optimization Problems Optimization problem having a finite number of discrete solutions. One solution approach is to explicitly generate and evaluate all possible solutions Explicit enumeration Consider a problem with 100 variables where x j = 0, 1, 2, …, 50; j = 1,2, …, 100 number of possible solutions = 51 100 Explicit enumeration may not be possible Implicit enumeration – attempt to account for all possible solution without enumerating all of them Problem dependent

5. 4 The Assignment Problem Assign n workers to n tasks A Problem in Permutations I heard that this is really good!

6. 5 A problem Three workers are to be assigned to one of 4 distinct tasks. Each worker can perform each task in a different time. The objective is to minimize the total time to complete the assigned tasks. worker task times We need to add a dummy worker. timetask1task2task3task4 w112131011 w210121410 w314111512

7. Complete Enumeration 6 timetask1task2task3task4 w112131011 w210121410 w314111512 (4)(3)(2) = 24 possible solutions w1w2w3time w1w2w3time 1task1task2task33513task1task4task337 2task1task3task23714task1task3task438 3task2task1task33815task4task1task336 4task2task3task14116task4task3task139 5task3task1task23117task3task1task432 6task3task2task13618task3task4task134 7task4task2task33819task1task2task436 8task4task3task23620task1task4task233 9task2task4task33821task2task1task435 10task2task3task43922task2task4task137 11task3task4task23523task4task1task232 12task3task2task43424task4task2task137

8. A Bigger Problem What if there were 20 workers and 25 tasks to be completed? 25 P 20 = 29,260,083,694,425,000,000,000 What if it takes 1 second to evaluate each assignment? 29,260,083,694,425,000,000,000 sec / (60 sec/min X 60 min/hr x 24 hr/day x 365 days/yr = 4,098,810,365,754,210 years 7

9. 8 Some Applications workers to tasks jobs to machines facilities to locations Truck drivers to customer pick-up points Umpire crews to baseball games Judges to court dockets State inspectors to construction sites Weapons to targets These are some terrific applications.

10. The Knapsack Problem A total of m items whose weights are w 1, w 2, …, w m are available for packing a knapsack. The total weight to be packed cannot exceed W total. The objective is to pack as many items as possible. 9

11. The Knapsack Problem - a binary problem 10

12. New & Improved Knapsack Problem 11 It is comforting to know my load is optimal! A thief robbing a store can carry a maximum weight of w. There are n items and i th item weighs w i and is worth v i dollars. What items should thief take?

13. A Knapsack Problem to Solve Johnny has five friends who would like to go on a fishing trip with him to the remote island of Chichagof located 70 miles northwest of Sitka, Alaska. However, Johnny can only carry an additional 319 pounds of weight in his seaplane – his only means of transportation. Johnny has assigned a “pleasure” index to each friend – the larger the value the more Johnny would enjoy the friend’s company. Which friend(s) should Johnny take with him? 12

14. The Data 13 FriendJaneJoanJudyJoyJim Pleasure Index869103 Weight (lb.)130180140125185 2 5 = 32 alternatives

15. The Results 14 FriendJaneJoanJudyJoyJim Pleasure Index869103 Weight (lb.)130180140125185valueweight 11111367601 11110335752 11101266353 11011276204 10111305805 01111286306 11100234507 11001174958 10011214409 001112245010 110102443511 101012045512 010111949013 011102544514 011011850515 101102739516 feasible110001431017 feasible100011131518 feasible000111331019 feasible101001727020 01001936521 feasible010101630522 011001532023 feasible001101926524 feasible100101825525 001011232526 feasible10000813027 feasible01000618028 feasible00100914029 feasible000101012530 feasible00001318531 feasible000000032 319 lb.

16. 15 Let’s count the number of solutions… Dad, how many solutions are there for the knapsack problem if I have 30 items to consider? You should know that Johnny. It is 2 30 or 1,073,741,824 solutions if you include the infeasible ones as well.

17. Other Binary Selection Problems Menu selection Select items from a menu to maximize protein Calorie and carbohydrate or cost constraints Cargo loading on trucks or aircraft Which crates or pallets to load – maximize value Volume and weight constraints Project selection – n potential engineering projects to fund Maximize expected profit Cost or resource constraint 16

18. 17 The Traveling Salesman Problem There once was a farmer’s daughter… You got to know the territory!

19. 18 Traveling Salesman Problem A salesman must visit each of n cities once and only once returning to his starting city. What route should be followed so that the total distance (cost or time) traveled is minimized? (n-1)! possible routes Itinerary Dayton Cincinnati Denver New York Atlanta Chicago Boston San Francisco Wapakoneta My secretary always finds for me, the minimum distance itinerary.

20. 19 A From-to Matrix From / To City ACity BCity CCity D City A253012 City B251723 City C301737 City D122337 distances in miles

21. 20 Traveling on a network 2 5 5 34 6 7 8 11 A B C D E F G 6! = 720 routes A solution is a permutation!

22. 21 A Traveling Salesman Problem P. Rose, currently unemployed, has hit upon the following scheme for making some money. He will guide a group of people on a tour of all National League baseball parks. The tour will start and end in Cincinnati. What route should he follow in order to minimize total distances (costs)?

23. 22 Distances between parks in miles ATL CHI CIN HOU LA MON NY PHI PIT STL SD SF ATL 702 454 842 2396 1196 864 772 714 554 2363 2679 CHI 702 324 1093 2136 764 845 764 459 294 2184 2187 CIN 454 324 1137 2180 798 664 572 284 338 2228 2463 HOU 842 1093 1137 1616 1857 1706 1614 1421 799 1521 2021 LA 2396 2136 2180 1616 2900 2844 2752 2464 1842 95 405 MON 1196 764 798 1857 2900 396 424 514 1058 2948 2951 NYK 864 845 664 1706 2844 396 92 386 1002 2892 3032 PHI 772 764 572 1614 2752 424 92 305 910 2800 2951 PIT 714 459 284 1421 2464 514 386 305 622 2512 2646 STL 554 294 338 799 1842 1058 1002 910 622 1890 2125 SD 2363 2184 2228 1521 95 2948 2892 2800 2512 1890 500 SF 2679 2187 2463 2021 405 2951 3032 2951 2646 2125 500 11! = 39,916,800 alternatives

24. 23 A Heuristics Algorithm I need a good heuristic to solve this problem. Joe EngineerP. Rose Heuristic: A procedure for solving problems by an intuitive approach in which the structure of the problem can be interpreted and exploited intelligently to obtain a reasonable solution.

25. 24 Nearest Neighbor Heuristic – a greedy heuristic algorithm ATL CHI CIN HOU LA MON NY PHI PIT STL SD SF ATL 702 454 842 2396 1196 864 772 714 554 2363 2679 CHI 702 324 1093 2136 764 845 764 459 294 2184 2187 CIN 454 324 1137 2180 798 664 572 284 338 2228 2463 HOU 842 1093 1137 1616 1857 1706 1614 1421 799 1521 2021 LA 2396 2136 2180 1616 2900 2844 2752 2464 1842 95 405 MON 1196 764 798 1857 2900 396 424 514 1058 2948 2951 NYK 864 845 664 1706 2844 396 92 386 1002 2892 3032 PHI 772 764 572 1614 2752 424 92 305 910 2800 2951 PIT 714 459 284 1421 2464 514 386 305 622 2512 2646 STL 554 294 338 799 1842 1058 1002 910 622 1890 2125 SD 2363 2184 2228 1521 95 2948 2892 2800 2512 1890 500 SF 2679 2187 2463 2021 405 2951 3032 2951 2646 2125 500

26. 25 LAXSNFCIN MON SNDATLHOUCHIPITSTLPHINYK DISTANCE = 8015 MILES

27. 26 LAXSNFCIN MON SNDATLHOUCHIPITSTLPHINYK DISTANCE = 8015 MILES X X OPTIMUM = 7577 MILES

28. 27 Other Applications manufacture of circuit boards assembly line reconfiguration mixing paints colors in vats searching for stars

29. 28 The Chinese Postman Problem A study in the optimal delivery of the postal mail

30. 29 The problem defined The Chinese postman problem concerns a postman who has to deliver mail to houses along each of the streets in a particular housing district and wants to minimize the distance that has to be walked. The problem was first considered by the Chinese mathematician Mei-ko Kwan in the 1960s.

31. 30 More of the Chinese Postman Problem Given a network with distances assigned to each arc, find the minimum distance walk that walks each arc at least once and returns to the starting node. 2 5 5 34 6 7 8 11 A B C D E F G I get tired walking this route. It is too long.

32. Solve by Trial and Error 31 A – B – C – D – F – G – E – D –E –B –E – G – A - 67 miles Start and end At A Minimum possible distance = 51 miles 2 5 5 34 6 7 8 11 A B C D E F G A – B – C – D – F – G – E – D – E – B – A – G – A - 63 miles What if I need to walk every street once in each direction?

33. 32 The Postman Problem Model Given a network with distances assigned to each arc, find the minimum distance walk that walks each arc at least once and returns to the starting node.

34. 33 Other Applications trash pickup newspaper delivery and door-to-door soliciting railroad track inspections and maintenance snow plowing, salting streets, and street cleaning meter reading school bus routes

35. 34 Play ball! Now it’s your turn. Engineering Management & Systems has proudly presented Discrete Systems Methods and Models