3IntoductionWorkforce allocation & personnel scheduling deal with the arrangement of work schedules and the assignment of personnel to shits to cover the demand for resources that vary over time.In service environments the operations are often prolonged and irregular, and the staff requirements fluctuate over time.The schedules are typically subject to various constraints dictated by equipment requirements, union rules, ...
4Days-off schedulingAn elementary personnel assignment problem.The problem is to find the minimum number of employees to cover a 7-days-a-week operation so that the following constraints are satisfied:The demand per day, nj, j = 1, ..., 7 (Sunday to Saturday), is metEach employee is given k1 out of every k2 weekends offEach employee works exactly 5 out of 7 days (Sunday to Saturday)Each employee works no more than 6 consecutive days.k1/k2 = 1/3
5Days-off scheduling : lower bounds Weekend constraint(k2 – k1)W >= k2 max(n1, n7)Where W is the minimum size of the workforceTotal demand constraint:5W >= SnjMaximum daily demand constraint:W >= max (n1, ..., n7)k1/k2 = 1/3
6Days-off scheduling : Algorithm W = min workforce, n = max(n1, n7)Step1. (Schedule the weekends off)Assign the 1st weekend off to the first W-n employeesAssign the 2nd weekend off to the second W-n employeesThis process is continued cyclically.
7Days-off scheduling : Algorithm uj = W- nj, j= 2, ..., 6, uj = n – nj, j = 1, 7Step2. (Determine the additional off-day pairs)Construct a list of n pairs of off days, numbered 1 to n.Choose day k such that uk = max(u1, ..., u7)Choose day l (l k), such that ul > 0; if ul = 0 for all l k, set l = kAdd the pair (k, l) to the list and decrease uk and ul by 1.Repeat the process n times.(2, 1), Sunday-Monday(2, 2), Monday-Monday (non distinct pairs)
8Days-off scheduling : Algorithm Set i = 1Step3. (Categorize emplyees in week i)Type T1 : weekend i off, no days needed during week i, weekend i+1 offType T2 : weekend i off, 1 off day needed during week i, weekend i+1 onType T3 : weekend i on, 1 off day needed during week i, weekend i+1 offType T4 : weekend i on, 2 off days needed during week i, weekend i+1 on|T3| + |T4| = n, |T2| + |T4| = n (as n people working each weekend)Pair Each employee of T2 with one of T3Step 4 (Assign off-day pairs in week i)Assign the n pairs of days, starting from the top off the list as follows:First assign pairs of days to the employees of T4Then, to each employee of T3 and his companion of T2, assign the one of T3 the earliest day of the pair.Set i = i+1 and return to step 3.
9Days-off scheduling : Algorithm Week 1 : T2 = 1, T3 = 2, T4 = 3Week 2 : T2 = 2, T3 = 3, T4 = 1Week 3 : T2 = 3, T3 = 1, T4 = 2The schedule generated by the days-off scheduling algorithm is always feasible.
10Shift schedulingA cycle (one day, one or several weeks) is fixed.Each work assignment pattern over a cycle has its own cost.Problem:m time intervals/periods in the predetermined cyclebi personnel are required for period ib different shift patterns, and each employee is assigned to one and only one pattern(a1j, a2j, ..., amj) = shift pattern j with aij = 1 if period i is a work period.cj = cost of patern jDetermine the number of employees of each pattern in order to minimise the total cost.
13Shift schedulingThe integer programming formulation of the general personnel scheduling problem (with arbitrary 0-1 A matrix) is NP-hardThe special case with each column containing a contiguous set of ones is easy and the solution of the LP-relaxation is always integer.
14Cyclic staffing problem The objective is to minimise the cost ofassigning people to an m-period cyclic scheduleso thatsufficient workers are present during time period i, in order to meet requirement bi,and each person works a shift of k consecutive periods and is free the other m-k periods.Each column is a possible shift(5,7) cyclic staffing
15Cyclic staffing problem : algorithm Step 1. Solve the linear relaxation of the problem to obtain xi’If (xi’) are integer, STOPStep 2. Form two linear programs LP’ and LP’’ from the relaxation of the original problem by adding respectively the constraints:LP’’ has an optimal solution that is integerIf LP’ does not have a feaible solution, then the solution of LP’’ is the optimal solutionIf LP’ has a feasible solution, then it has an optimal solution that is integer and the best of LP’ and LP’’ solutions is the optimal solution.
17CREW SCHEDULINGCrew scheduling problems are very important in transportation especially in airline industryConsider a set of m jobs, or flight legs.A flight leg is characterized by a point of departure and a point of arrival, as well as an approximate time interval during which the flight has to take place.There is a set of n feasible and permissible combinations of flight legs that one crew can handle, round trips or tours.Each round trip j, has a cost cj.Crew schedule determines round trips to select in order to minimize the total cost under the constraint that each flight leg is covered exactly once by one and only one round trip.
18CREW SCHEDULINGEach column in the A matrix is a round trip, and each row is a flight leg that must be covered exactly once by one round trip.Set partitioning problem.