Chapter 3 Workforce scheduling. Plan. Introduction Days-off scheduling Shift scheduling Cyclic staffing probme Crew scheduling. Intoduction.
k1/k2 = 1/3
(k2 – k1)W >= k2 max(n1, n7)
Where W is the minimum size of the workforce
Total demand constraint:
5W >= Snj
Maximum daily demand constraint:
W >= max (n1, ..., n7)
k1/k2 = 1/3
W = min workforce, n = max(n1, n7)
Step1. (Schedule the weekends off)
Assign the 1st weekend off to the first W-n employees
Assign the 2nd weekend off to the second W-n employees
This process is continued cyclically.
uj = W- nj, j= 2, ..., 6, uj = n – nj, j = 1, 7
Step2. (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 (lk), such that ul > 0; if ul = 0 for all lk, set l = k
Add 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)
Set i = 1
Step3. (Categorize emplyees in week i)
Type T1 : weekend i off, no days needed during week i, weekend i+1 off
Type T2 : weekend i off, 1 off day needed during week i, weekend i+1 on
Type T3 : weekend i on, 1 off day needed during week i, weekend i+1 off
Type 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 T3
Step 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 T4
Then, 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.
Week 1 : T2 = 1, T3 = 2, T4 = 3
Week 2 : T2 = 2, T3 = 3, T4 = 1
Week 3 : T2 = 3, T3 = 1, T4 = 2
The schedule generated by the days-off scheduling algorithm is always feasible.
A cycle (one day, one or several weeks) is fixed.
Each work assignment pattern over a cycle has its own cost.
m time intervals/periods in the predetermined cycle
bi personnel are required for period i
b 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 j
Determine the number of employees of each pattern in order to minimise the total cost.
What if overtime is allowed?
The objective is to minimise the cost of
assigning people to an m-period cyclic schedule
sufficient 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
Step 1. Solve the linear relaxation of the problem to obtain xi’
If (xi’) are integer, STOP
Step 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 integer
If LP’ does not have a feaible solution, then the solution of LP’’ is the optimal solution
If 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.
Truck routing network