Transportation Timet. Outline Workforce Scheduling DMP204 SCHEDULING, TIMETABLING AND ROUTING 1. Transportation Timetabling Lecture 23 2. Workforce Scheduling Workforce Scheduling Crew Scheduling and Rostering Employee Timetabling Shift Scheduling Nurse Scheduling Marco Chiarandini 2 Transportation Timet. Transportation Timet. Outline Periodic Event Scheduling Problem Workforce Scheduling Workforce Scheduling 1. Transportation Timetabling Blackboard 2. Workforce Scheduling Crew Scheduling and Rostering Employee Timetabling Shift Scheduling Nurse Scheduling 3 4
Workforce Scheduling Transportation Timet. Crew Scheduling and Rostering Transportation Timet. Crew Scheduling and Rostering Outline Workforce Scheduling Employee Timetabling Workforce Scheduling Employee Timetabling Overview A note on terminology 1. Transportation Timetabling Shift: consecutive working hours Roster: shift and rest day patterns over a fixed period of time (a week or a month) 2. Workforce Scheduling Two main approaches: Crew Scheduling and Rostering coordinate the design of the rosters and the assignment of the shifts Employee Timetabling to the employees, and solve it as a single problem. Shift Scheduling Nurse Scheduling consider the scheduling of the actual employees only after the rosters are designed, solve two problems in series. Features to consider: rest periods, days off, preferences, availabilities, skills. 5 6 Workforce Scheduling Workforce Scheduling Transportation Timet. Crew Scheduling and Rostering Transportation Timet. Crew Scheduling and Rostering Workforce Scheduling Employee Timetabling Workforce Scheduling Employee Timetabling Overview Overview 2. Employee timetabling (aka labor scheduling) is the operation of Workforce Scheduling: assigning employees to tasks in a set of shifts during a fixed period 1. Crew Scheduling and Rostering of time, typically a week. 2. Employee Timetabling Examples of employee timetabling problems include: assignment of nurses to shifts in a hospital, 1. Crew Scheduling and Rostering is workforce scheduling applied in assignment of workers to cash registers in a large store the transportation and logistics sector for enterprises such as airlines, assignment of phone operators to shifts and stations in a railways, mass transit companies and bus companies (pilots, service-oriented call-center attendants, ground staff, guards, drivers, etc.) Differences with Crew scheduling: The peculiarity is finding logistically feasible assignments. no need to travel to perform tasks in locations start and finish time not predetermined 7 8
Transportation Timet. Crew Scheduling and Rostering Transportation Timet. Crew Scheduling and Rostering Crew Scheduling Shift Scheduling Workforce Scheduling Employee Timetabling Workforce Scheduling Employee Timetabling Input: A set of flight legs (departure, arrival, duration) A set of crews Creating daily shifts: Output: A subset of flights feasible for each crew roster made of m time intervals not necessarily identical during each period, b i personnel is required How do we solve it? n different shift patterns (columns of matrix A ) Set partitioning or set covering?? c T x min Often treated as set covering because: Ax ≥ b st its linear programming relaxation is numerically more stable and thus easier to solve x ≥ 0 and integer it is trivial to construct a feasible integer solution from a solution to the linear programming relaxation it makes possible to restrict to only rosters of maximal length 10 12 Transportation Timet. Crew Scheduling and Rostering Transportation Timet. Crew Scheduling and Rostering ( k, m ) -cyclic Staffing Problem Workforce Scheduling Employee Timetabling Workforce Scheduling Employee Timetabling Assign persons to an m -period cyclic schedule so that: requirements b i are met each person works a shift of k consecutive periods and is free for the Recall: Totally Unimodular Matrices other m − k periods. (periods 1 and m are consecutive) and the cost of the assignment is minimized. Definition: A matrix A is totally unimodular (TU) if every square submatrix of A has determinant +1, -1 or 0. min cx Proposition 1: The linear program max { cx : Ax ≤ b, x ∈ R m + } has an integral optimal solution for all integer vectors b for which it has a finite 1 0 0 1 1 1 1 optimal value if and only if A is totally unimodular 1 1 0 0 1 1 1 1 1 1 0 0 1 1 Recognizing total unimodularity can be done in polynomial time (see st 1 1 1 1 0 0 1 x ≥ b (P) [Schrijver, 1986] ) 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 x ≥ 0 and integer 13 14
Total Unimodular Matrices Total Unimodular Matrices Transportation Timet. Crew Scheduling and Rostering Transportation Timet. Crew Scheduling and Rostering Workforce Scheduling Employee Timetabling Workforce Scheduling Employee Timetabling Resume’ Resume’ Basic examples: All totally unimodular matrices arise by certain compositions from Theorem network matrices and from certain 5 × 5 matrices [Seymour, 1980] . This decomposition can be tested in polynomial time. The V × E -incidence matrix of a graph G = ( V, E ) is totally unimodular if and only if G is bipartite Definition Theorem A (0 , 1) –matrix B has the consecutive 1’s property if for any column j , b ij = b i ′ j = 1 with i < i ′ implies b lj = 1 for i < l < i ′ . That is, if there The V × A -incidence matrix of a directed graph D = ( V, A ) is totally is a permutation of the rows such that the 1’s in each column appear unimodular consecutively. Theorem Let D = ( V, A ) be a directed graph and let T = ( V, A 0 ) be a directed tree on Whether a matrix has the consecutive 1’s property can be determined in V . Let M be the A 0 × A matrix defined by, for a = ( v, w ) ∈ A and a ′ ∈ A 0 polynomial time [ D. R. Fulkerson and O. A. Gross; Incidence matrices if the unique v − w -path in T passes through a ′ forwardly; and interval graphs. 1965 Pacific J. Math. 15(3) 835-855.] := +1 M a ′ ,a if the unique v − w -path in T passes through a ′ backwardly; − 1 A matrix with consecutive 1’s property is called an interval matrix and if the unique v − w -path in T does not pass through a ′ 0 they can be shown to be network matrices by taking a directed path for M is called network matrix and is totally unimodular. the directed tree T 16 17 Transportation Timet. Crew Scheduling and Rostering Transportation Timet. Crew Scheduling and Rostering Workforce Scheduling Employee Timetabling Workforce Scheduling Employee Timetabling What about this matrix? Integer programs where the constraint matrix A have the circular 1’s 1 0 0 1 1 1 1 property for rows can be solved efficiently as follows: 1 1 0 0 1 1 1 1 1 1 0 0 1 1 Step 1 Solve the linear relaxation of (P) to obtain x ′ 1 , . . . , x ′ n . If 1 1 1 1 0 0 1 x ′ 1 , . . . , x ′ n are integer, then it is optimal for (P) and 1 1 1 1 1 0 0 STOP. Otherwise go to Step 2. 0 1 1 1 1 1 0 Step 2 Form two linear programs LP1 and LP2 from the 0 0 1 1 1 1 1 relaxation of the original problem by adding respectively the constraints Definition A (0 , 1) -matrix B has the circular 1’s property for rows (resp. for columns) if the columns of B can be permuted so that the 1 ’s in each x 1 + . . . + x n = ⌊ x ′ 1 + . . . + x ′ n ⌋ (LP1) row are circular, that is, appear in a circularly consecutive fashion and The circular 1’s property for columns does not imply circular 1’s property for rows. x 1 + . . . + x n = ⌈ x ′ 1 + . . . + x ′ n ⌉ (LP2) Whether a matrix has the circular 1’s property for rows (resp. columns) From LP1 and LP2 an integral solution certainly arises (P) can be determined in O ( m 2 n ) time [A. Tucker, Matrix characterizations of circular-arc graphs. (1971) Pacific J. Math. 39(2) 535-545] 18 19
Transportation Timet. Crew Scheduling and Rostering Cyclic Staffing with Overtime Workforce Scheduling Employee Timetabling Hourly requirements b i Basic work shift 8 hours Days-Off Scheduling Overtime of up to additional 8 hours possible Guarantee two days-off each week, including every other weekend. IP with matrix A : 21 Transportation Timet. Crew Scheduling and Rostering Transportation Timet. Crew Scheduling and Rostering Workforce Scheduling Employee Timetabling Workforce Scheduling Employee Timetabling Cyclic Staffing with Part-Time Workers Cyclic Staffing with Linear Penalties for Understaffing and Overstaffing Columns of A describe the work-shifts demands are not rigid Part-time employees can be hired for each time period i at cost c ′ i a cost c ′ i for understaffing and a cost c ′′ i for overstaffing per worker cx + c ′ x ′ + c ′′ ( b − Ax − x ′ ) min min cx + c ′ x ′ Ax + Ix ′ ≥ b st Ax + Ix ′ ≥ b st x, x ′ ≥ 0 and integer x, x ′ ≥ 0 and integer 22 23
Recommend
More recommend
