Column Generation and its applications

Transcription

1 Column Generation and its applications Murat Firat, dept. IE&IS, TU/e BPI Cluster meeting

2 Outline Some real-life decision problems Standard formulations Basics of Column Generation Master formulations Case: Shift scheduling of airport workers Problem description Master formulation Reduced cost and pricing problem Column Generation overview Towards an integer solution Using data science in the CG approach PAGE 2

3 Outline Some real-life decision problems Standard formulations Basics of Column Generation Master formulations Case: Shift scheduling of airport workers Problem description Master formulation Reduced cost and pricing problem Column Generation overview Towards an integer solution Using data science in the CG approach PAGE 3

4 Dierent industry sectors, dierent problems Telecommunication: Stable workforce assignments. PAGE 4

5 Dierent industry sectors, dierent problems Telecommunication: Stable workforce assignments. Telecommunication: Designing FTTH network. PAGE 5

6 Dierent industry sectors, dierent problems Telecommunication: Stable workforce assignments. Telecommunication: Designing FTTH network. Airport ground operations: Shift scheduling of workers. PAGE 6

7 Dierent industry sectors, dierent problems Telecommunication: Stable workforce assignments. Telecommunication: Designing FTTH network. Airport ground operations: Shift scheduling of workers. Logistics: Planning routes of vehicles. PAGE 7

8 Dierent industry sectors, dierent problems Telecommunication: Stable workforce assignments. Telecommunication: Designing FTTH network. Airport ground operations: Shift scheduling of workers. Logistics: Planning routes of vehicles. Machine learning: Constructing max-accuracy decision trees. PAGE 8

9 Instance sizes Stable workforce assignments: 100 workers, 25 jobs, 15 skills. PAGE 9

10 Instance sizes Stable workforce assignments: 100 workers, 25 jobs, 15 skills. Designing FTTH network: Local areas of 5-10K citizens, PAGE 10

11 Instance sizes Stable workforce assignments: 100 workers, 25 jobs, 15 skills. Designing FTTH network: Local areas of 5-10K citizens, Shift scheduling of airport workers: 200 workers, 4 weeks, 5 skills. PAGE 11

12 Instance sizes Stable workforce assignments: 100 workers, 25 jobs, 15 skills. Designing FTTH network: Local areas of 5-10K citizens, Shift scheduling of airport workers: 200 workers, 4 weeks, 5 skills. Planning routes of vehicles: 100-1K customers, 50 vehicles. PAGE 12

13 Instance sizes Stable workforce assignments: 100 workers, 25 jobs, 15 skills. Designing FTTH network: Local areas of 5-10K citizens, Shift scheduling of airport workers: 200 workers, 4 weeks, 5 skills. Planning routes of vehicles: 100-1K customers, 50 vehicles. Decision trees: 5K data instances, 50 attributes. PAGE 13

14 Common properties of the problems Strict feasibility requirements. PAGE 14

15 Common properties of the problems Strict feasibility requirements. Worst cases: NP-Hard. PAGE 15

16 Common properties of the problems Strict feasibility requirements. Worst cases: NP-Hard. Exponentially many feasible solutions PAGE 16

17 Common properties of the problems Strict feasibility requirements. Worst cases: NP-Hard. Exponentially many feasible solutions Many local optimal points. PAGE 17

18 Standard formulations One big formulation, low-quality bound. Stable workforce assignments: Worker-job decisions. PAGE 18

19 Standard formulations One big formulation, low-quality bound. Stable workforce assignments: Worker-job decisions. Shift scheduling of airport workers: Worker-shift decisions. PAGE 19

20 Standard formulations One big formulation, low-quality bound. Stable workforce assignments: Worker-job decisions. Shift scheduling of airport workers: Worker-shift decisions. Planning routes of vehicles: Vehicle-customer decisions. PAGE 20

21 Standard formulations One big formulation, low-quality bound. Stable workforce assignments: Worker-job decisions. Shift scheduling of airport workers: Worker-shift decisions. Planning routes of vehicles: Vehicle-customer decisions. Decision trees: Data row-tree node decisions. PAGE 21

22 Outline Some real-life decision problems Standard formulations Basics of Column Generation Master formulations Case: Shift scheduling of airport workers Problem description Master formulation Reduced cost and pricing problem Column Generation overview Towards an integer solution Using data science in the CG approach PAGE 22

23 Master formulations Stable workforce assignments: Team-job decisions. PAGE 23

24 Master formulations Stable workforce assignments: Team-job decisions. Shift scheduling of airport workers: Worker-schedule decisions. PAGE 24

25 Master formulations Stable workforce assignments: Team-job decisions. Shift scheduling of airport workers: Worker-schedule decisions. Planning routes of vehicles: Vehicle-route decisions. PAGE 25

26 Master formulations Stable workforce assignments: Team-job decisions. Shift scheduling of airport workers: Worker-schedule decisions. Planning routes of vehicles: Vehicle-route decisions. Decision trees: Data segment-tree node decisions. PAGE 26

27 Master formulation Linear programming model: PAGE 27

28 Master formulation Linear programming model: More complicated objects as decision variables. PAGE 28

29 Master formulation Linear programming model: More complicated objects as decision variables. Object set PAGE 29

30 Master formulation Linear programming model: More complicated objects as decision variables. Object set has exponentially many items. PAGE 30

31 Master formulation Linear programming model: More complicated objects as decision variables. Object set has exponentially many items. initially is empty or has few items. PAGE 31

32 Master formulation Linear programming model: More complicated objects as decision variables. Object set has exponentially many items. initially is empty or has few items. Iteratively nd promising columns and add them to object set. PAGE 32

33 Master formulation Linear programming model: More complicated objects as decision variables. Object set has exponentially many items. initially is empty or has few items. Iteratively nd promising columns and add them to object set. When no promising column exists: certicate for optimality. PAGE 33

34 Main loop of Column Generation Reduced cost of a column is PAGE 34

35 Main loop of Column Generation Reduced cost of a column is violation amount of the corresponding dual constraint. PAGE 35

36 Main loop of Column Generation Reduced cost of a column is violation amount of the corresponding dual constraint. estimated cost change per unit increase in its value. PAGE 36

37 Main loop of Column Generation Reduced cost of a column is violation amount of the corresponding dual constraint. estimated cost change per unit increase in its value. Adding promising columns PAGE 37

38 Main loop of Column Generation Reduced cost of a column is violation amount of the corresponding dual constraint. estimated cost change per unit increase in its value. Adding promising columns completes dual feasibility. PAGE 38

39 Main loop of Column Generation Reduced cost of a column is violation amount of the corresponding dual constraint. estimated cost change per unit increase in its value. Adding promising columns completes dual feasibility. improves primal objective. PAGE 39

40 Main loop of Column Generation Reduced cost of a column is violation amount of the corresponding dual constraint. estimated cost change per unit increase in its value. Adding promising columns completes dual feasibility. improves primal objective. No dual feasibility improvement primal optimality PAGE 40

41 Outline Some real-life decision problems Standard formulations Basics of Column Generation Master formulations Case: Shift scheduling of airport workers Problem description Master formulation Reduced cost and pricing problem Column Generation overview Towards an integer solution Using data science in the CG approach PAGE 41

42 Problem denition: Shift scheduling We are given: multi-skilled workers with availability info. PAGE 42

43 Problem denition: Shift scheduling We are given: multi-skilled workers with availability info. service demand within a planning horizon PAGE 43

44 Problem denition: Shift scheduling We are given: multi-skilled workers with availability info. service demand within a planning horizon labor regulations about shifts: PAGE 44

45 Problem denition: Shift scheduling We are given: multi-skilled workers with availability info. service demand within a planning horizon labor regulations about shifts: minimum resting time between shifts, PAGE 45

46 Problem denition: Shift scheduling We are given: multi-skilled workers with availability info. service demand within a planning horizon labor regulations about shifts: minimum resting time between shifts, maximum working time due to contracts, PAGE 46

47 Problem denition: Shift scheduling We are given: multi-skilled workers with availability info. service demand within a planning horizon labor regulations about shifts: minimum resting time between shifts, maximum working time due to contracts, night shifts: longer resting times. PAGE 47

48 Basic notation Let S be set of all schedules st. a schedule s S PAGE 48

49 Basic notation Let S be set of all schedules st. a schedule s S covers time interval i N if s i = 1, not otherwise. PAGE 49

50 Basic notation Let S be set of all schedules st. a schedule s S covers time interval i N if s i = 1, not otherwise. should comply all regulations PAGE 50

51 Basic notation Let S be set of all schedules st. a schedule s S covers time interval i N if s i = 1, not otherwise. should comply all regulations should have length bet. 3-9 hrs. PAGE 51

52 Basic notation Let S be set of all schedules st. a schedule s S covers time interval i N if s i = 1, not otherwise. should comply all regulations should have length bet. 3-9 hrs. For worker w W PAGE 52

53 Basic notation Let S be set of all schedules st. a schedule s S covers time interval i N if s i = 1, not otherwise. should comply all regulations should have length bet. 3-9 hrs. For worker w W Sk d w indicates if worker w is skilled in skill type d D. PAGE 53

54 Basic notation Let S be set of all schedules st. a schedule s S covers time interval i N if s i = 1, not otherwise. should comply all regulations should have length bet. 3-9 hrs. For worker w W Sk d w indicates if worker w is skilled in skill type d D. Service demand Ri,d at time i in skill type d, PAGE 54

55 Master IP formulation Min subject to w W s S c swx sw Skws d i x sw R i,d, d D, i N w W x sw = 1, w W s S x sw {0, 1}, s S, w W PAGE 55

56 LP Relaxation of Master formulation Min subject to c swx sw w W s S Skws d i x sw R i,d, d D, i N w W s S x sw = 1, w W 0 x sw 1, s S, w W Note: Restricted set S S PAGE 56

57 LP Relaxation of Master formulation Min subject to c swx sw w W s S Duals Skws d i x sw R i,d, d D, i N π i,d w W x sw s S = 1, w W θ w 0 x sw 1, s S, w W Note: Restricted set S S PAGE 57

58 Reduced cost of column x sw Dual constraint of column x sw: Skws d i πi,d + θw c sw (1) Reduced cost of column x sw: d D i N c sw = c sw d D Skws d i πi,d θw (2) Case c sw < 0: (1) Estimated objective decrease (why?), (2) dual feasibility violation. i N PAGE 58

59 Pricing problem: Objective Pricing problem: Find the most promising column (schedule) with the objective min s S,w W = max s S,w W { c sw } Skws d i πi,d θw (3) d D i N { ( )} s i Skwπ d i,d c iw θw (4) i N d D PAGE 59

60 Pricing problem: Modeling Dene a graph, reds: shifts, blues: resting times, blacks: unavailability. PAGE 60

61 Pricing problem: Modeling Dene a graph, reds: shifts, blues: resting times, blacks: unavailability. labor regulations: arc structure and side constraints PAGE 61

62 Pricing problem: Modeling Dene a graph, reds: shifts, blues: resting times, blacks: unavailability. labor regulations: arc structure and side constraints Solve max w a x a subject to a PAGE 62

63 Pricing problem: Modeling Dene a graph, reds: shifts, blues: resting times, blacks: unavailability. labor regulations: arc structure and side constraints Solve max w a x a subject to a conservations, resting after shifts, passing unavailability arcs. PAGE 63

64 Pricing problem: Modeling Dene a graph, reds: shifts, blues: resting times, blacks: unavailability. labor regulations: arc structure and side constraints Solve max w a x a subject to a conservations, resting after shifts, passing unavailability arcs. Pricing: nd the constrained 0 N "Longest Path"! PAGE 64

65 Pricing problem: Modeling A schedule s on the graph looks like: PAGE 65

66 Column Generation overview Initialization: PAGE 66

67 Column Generation overview Initialization: Formulate Master ILP model, obtain the Restricted Master Problem (RMP). Express the reduced costs, formulate pricing. PAGE 67

68 Column Generation overview Initialization: Formulate Master ILP model, obtain the Restricted Master Problem (RMP). Express the reduced costs, formulate pricing. Warm up: Find several initial columns for a warm start of RMP. PAGE 68

69 Column Generation overview Initialization: Formulate Master ILP model, obtain the Restricted Master Problem (RMP). Express the reduced costs, formulate pricing. Warm up: Find several initial columns for a warm start of RMP. Step 1: Solve the Restricted Master Problem, pass duals to pricing. PAGE 69

70 Column Generation overview Initialization: Formulate Master ILP model, obtain the Restricted Master Problem (RMP). Express the reduced costs, formulate pricing. Warm up: Find several initial columns for a warm start of RMP. Step 1: Solve the Restricted Master Problem, pass duals to pricing. Step 2: Solve pricing: PAGE 70

71 Column Generation overview Initialization: Formulate Master ILP model, obtain the Restricted Master Problem (RMP). Express the reduced costs, formulate pricing. Warm up: Find several initial columns for a warm start of RMP. Step 1: Solve the Restricted Master Problem, pass duals to pricing. Step 2: Solve pricing: If i : c i < 0: update S, go to Step 1. If i : c i 0: RMP is solved to optimality, go to Step 3. PAGE 71

72 Column Generation overview Initialization: Formulate Master ILP model, obtain the Restricted Master Problem (RMP). Express the reduced costs, formulate pricing. Warm up: Find several initial columns for a warm start of RMP. Step 1: Solve the Restricted Master Problem, pass duals to pricing. Step 2: Solve pricing: If i : c i < 0: update S, go to Step 1. If i : c i 0: RMP is solved to optimality, go to Step 3. Step 3: Output the RMP solution. PAGE 72

73 Outline Some real-life decision problems Standard formulations Basics of Column Generation Master formulations Case: Shift scheduling of airport workers Problem description Master formulation Reduced cost and pricing problem Column Generation overview Towards an integer solution Using data science in the CG approach PAGE 73

74 Towards integer solutions Having fractional RMP optimal solution, we have two choices: Use rounding heuristics: PAGE 74

75 Towards integer solutions Having fractional RMP optimal solution, we have two choices: Use rounding heuristics: Use meta heuristics to nd a feasible solution quickly (hopefully) PAGE 75

76 Towards integer solutions Having fractional RMP optimal solution, we have two choices: Use rounding heuristics: Use meta heuristics to nd a feasible solution quickly (hopefully) Make decisions how to (smartly) round the fractional solution PAGE 76

77 Towards integer solutions Having fractional RMP optimal solution, we have two choices: Use rounding heuristics: Use meta heuristics to nd a feasible solution quickly (hopefully) Make decisions how to (smartly) round the fractional solution Start a smart enumeration, e.g. Branch-and-Price, either. PAGE 77

78 Towards integer solutions Having fractional RMP optimal solution, we have two choices: Use rounding heuristics: Use meta heuristics to nd a feasible solution quickly (hopefully) Make decisions how to (smartly) round the fractional solution Start a smart enumeration, e.g. Branch-and-Price, either. to obtain "optimal" integer sol'n. to output "best-found" integer sol'n with quality measure in a time limit. PAGE 78

79 Outline Some real-life decision problems Standard formulations Basics of Column Generation Master formulations Case: Shift scheduling of airport workers Problem description Master formulation Reduced cost and pricing problem Column Generation overview Towards an integer solution Using data science in the CG approach PAGE 79

80 Using data science: Advantage or waste? Consider a data of instance history and their solutions. Find "similar" instances and round the current fractional solution towards corresponding solutions. PAGE 80

81 Using data science: Advantage or waste? Consider a data of instance history and their solutions. Find "similar" instances and round the current fractional solution towards corresponding solutions. Simply nd "similar" previous solutions to the current fractional solution and round accordingly. PAGE 81

82 Using data science: Advantage or waste? Consider a data of instance history and their solutions. Find "similar" instances and round the current fractional solution towards corresponding solutions. Simply nd "similar" previous solutions to the current fractional solution and round accordingly. Cluster instances and analyze the commonalities (patterns) in the cluster solutions. PAGE 82

83 THANKS PAGE 83