- Email: [email protected]
An algorithm for single shift scheduling of hierarchical workforce
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER
European Journal of Operational Research 96 (1996) 113-121
Theory and Methodology
An algorithm for single shift scheduling of hierarchical workforce Rangarajan Narasimhan * School of Business, Queen's Unioersity, Kingston, Ont., Canada K7L 3N6 Received April 1994; revised November 1995
Abstract This paper presents an optimal algorithm for single shift scheduling of hierarchical workforce in seven-day-a-week industries. There are many categories of workers; their capabilities are arranged hierarchically. The objective is to arrive at the most economical mix of categories of employees satisfying the pattern of demand for employees and desired work characteristics. The demand for employees varies from one category to the other and for each category the demand during weekdays is fixed and is different from that of weekends. The desired work characteristics are: (i) each employee must be given two days off every week including at least a proportion A out of every B weekends off and (ii) each employee can have at most 5 working shifts between any two offdays. The algorithm provides a computational scheme for arriving at the minimum workforce size comprising the most economical mix of categories of employees for this demand context and a new technique for characterizing the problem to achieve feasible assignment of employees to shifts.
Keywords: Scheduling theory; Manpower planning
O. Introduction Scheduling of hierarchical workforce using combinatorial methods has its beginning in the research of Emmons and Burns (1991). This class of problems arises in scheduling of health care personnel, job shop employees, maintenance crew and so on. In these problems there are many categories of employees to be scheduled. The categories of employees form a hierarchy in that the employees at a higher level can do the work of lower level employees but not vice versa. Prior to the work of Emmons and
* Correspondence to: Ranga Naras, The Fred C. Manning School of Business Administration, Acadia University, Wolfville, NS, Canada BOP 1X0. email: [email protected]
Burns (1991), the focus of research in this field has been on single shift and multiple shift scheduling of single category employees. Burns and Carter (1985) provide a comprehensive version of the single shift scheduling problem generalizing the earlier research of Brownell and Lowerre (1976), Lowerre (1977), Baker and Magazine (1977), Bums (1978), Baker, Burns and Carter (1979) and Emmons (1985). Bums and Koop (1987) provide a modular approach for solving workforce scheduling problems characterized by multiple shifts and single category of employees. Burns (1981) describes an iterative algorithm for the same problem. This paper focuses on single shift scheduling of hierarchical workforce and contributes to the progress of the research in this field by providing a computa-
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R. Narasimhan / European Journal of Operational Research 96 (1996) 113-12l
tional scheme for amving at the minimum size of the workforce for a pattern of demand not considered in the literature so far. As opposed to the uniform daily demand considered by Emmons and Bums (1991), a demand pattern for employees characterized by different constants for weekdays and weekends is considered in this paper. The computational time of the algorithm increases linearly with the increase in problem size. The remainder of the paper is divided into five sections. Section 1 describes the problem and introduces notation. Section 2 gives the computational scheme for arriving at the most economical workforce size using which Section 3 constructs a feasible schedule assigning employees to shifts. Section 4 illustrates the algorithm with a numerical example and Section 5 identifies the directions for further research in this field. The feasibility and the optimality of the algorithm are established in the Appendix.
workers required on Sa and Su and D k be the cumulative number of workers of categories 1 up to and including k required on Mo to Fr consisting of at least d k workers of category k, Vk ~ K. Let w k be the optimum number of people to be hired in category k and Wk the cumulative number of people to be hired in categories 1 up to and including k. Clearly,
1. Problem description
We compute three lower bounds for workforce size based on three demand characteristics: 1) the weekend demand for each category, 2) the total number of shifts required in a week for each category, and 3) the cumulative number of employees required up to and including category k ~ K during the weekdays. The most economical workforce size is determined by satisfying all the three bounds suitably. L o w e r bound based on weekend demand for each category: L~. The number of employees of category k available each weekend must be sufficient to meet the weekend demand. Since each employee has at least A out of every B weekends off, over a cycle of B weeks, we have
In this paper, we consider a facility that must be staffed seven days a week. There are many categories of workers to be scheduled; their capabilities are arranged hierarchically. The facility operates only one shift a day. The objective is to arrive at the most economical mix of categories of employees satisfying the pattern of demand for employees and the desired work characteristics. The demand for employees is of the following form: the weekday and the weekend demand are fixed with no particular relationship between them. The desired work characteristics are: (i) each employee must be given two days off every week including at least a proportion A out of every B weekends off and (ii) each employee works for at most 5 consecutive days. Let K = {1, 2, 3 . . . . . m} be the set of categories of employees. Assume that the categories are ranked with category 1 workers at the top as the most highly qualified so that a category r worker can be used in place of a worker of category k (for any k > r, r ~ m, and k, r ~ K ) but not vice versa. Thus we have a hierarchy of categories having downward substitutability during weekdays as explained in Emmons and Bums (1991). Let c k be the cost to employ a person of category k, k ~ K . Let n k be the number of category k
Dk-- D k - I >I d~ for k > 1 and k ~ K ,
and
D l = d I. There is no restriction on the relationship between d k and n k.
2. Determination of the most economical workforce size
( B - A) w k >1 Bn k or
wk/> L~, where L~ = [ B n k / ( B - A ) | and [ x | is the nearest integer equal to or above x. Lower bound based on total demand f o r category k employees: Lk2. The total number of employee days per week available from employees of all categories must be sufficient to meet the total weekly demand for that category. Since each employee is to be given
R. Narasimhan / European Journal of Operational Research 96 (1996) 113-121
two days off every week, s / h e works at most five days per week. Hence, 5 w k >: 5 d k + 2 n k or
wk >~ tk2,
where L~ = d~ + [0.4 nk]. The workforce size for category k is at least the maximum of the above two bounds Lkl and L~ and we have w k/> max{L~, L~}. L o w e r b o u n d b a s e d on the cumulative w o r k e r demand: Lk3. T h e cumulative number of employees
available from categories 1 to k should not only satisfy the cumulative number of employees required but also consist of the minimum number of category k employees for any k ~ K. The cumulative number of shifts available from employees of categories 1 to k should at least equal the total number of shifts required in the week for categories 1 to k, i.e., k
115
ber of employees required from categories 1 to m, W,~, is arrived at by equating it to the lower bound, L~', the total workforce size W ( = Wm) is the minimum required to satisfy the demand. To determine the most economical mix of categories of employees, we proceed as follows: Compute Sk = L~ - L~-1 - max(L~, L~) for k > 1 and k ~ K. Sk represents the additional number of employees required from categories I to k over and above the number of employees required exclusively from category k. Choose these Sk employees from the least cost category among the hierarchically substitutable categories, i.e., choose from category r such that r ~< k, r, k ~ K and c r = min( c I , c 2 , c 3 . . . . . ck).
After having chosen Sk employees from category r, denote this S k by Skr. Skr signifies the additional number of people to be hired from category r to satisfy the cumulative demand from categories 1 to k. Repeat this procedure for all k > 1 and k ~ K. The most economical combination of employees is
5 W k >1 5 D k + 2 ~ , n i i=1
wk = max(L~, L~) + ~ Sik. i=k
or
W k >~D k +
I04i1 1 ni
•
Also, the cumulative number of employees available from categories 1 to k should have at least the required number of employees of category k for k E K , i.e., Wk>~ {Wk_ ,
+ max(L~, L~)}
for k > 1, and WI = w l >/max(L~, L~) for category k = 1.
Thus, wt/> L~, where L~=max
( [o4i, ] D k+
n i , L~- ' + m a x ( L~, L
We compute Wk = E/k= i wi . The composition of the minimum total workforce size is such that each additional employee required to satisfy the cumulative demand is chosen from the least cost category among the hierarchically substitutable categories. Hence, the categorywise composition of workforce computed is the least cost combination of employees that satisfies the staffing requirements. In the next section, we describe an algorithm to construct a schedule for the most economical workforce size computed in this section. The schedule is such that it satisfies the pattern of demand for each category of employees and the desired work characteristics.
for category k for k > 1, and
L5 ; max(L , Li ).
3. Algorithm
We now have three lower bounds for the workforce size. Set Wm = L~'. Since the cumulative num-
The inputs to the algorithm are the values of the following variables: A, B, m, n,, d k, D k, Vk E K.
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R. Narasimhan / European Journal of Operational Research 96 (1996) 113-12 l
The algorithm computes the minimum total workforce size consisting of the most economical mix of categories of employees required and generates a feasible schedule for assignment of these employees to shifts. Step 1. Compute the minimum workforce size. Calculate the three bounds L~, Lk2 and L~ and arrive at the categorywise distribution of workforce, i.e., w k, V k ~ K . Step 2. Schedule the weekends off for each category k E K. Consider any category k. For convenience, assign a number from 1 to w k to each employee. Since n k people are required to work each weekend, the remaining (w k - n k) employees can have the weekend off. Assign the first weekend off to the first (w k - n k) employees. Assign the second weekend off to the next (w k - n k) employees. The process is continued cyclically with employee 1 being treated as the next employee after employee w k. Step 3. Determine the number of offday pairs required for each category k ~ K. In any week, employees will fall into one of four types depending on the weekends off at the beginning and at the end of the week as in Table 1. Since there are exactly n k employees working each weekend, IT3 I +IT41 = n k (from weekend i) and IT2 I + IT41 = n k (from weekend i + l ) . Hence IT2 1 = [T3I. As shown by Bums and Carter (1985), type T~ and type T4 employees cannot co-exist, i.e., either type TI may exist or type T4 may exist but not both.
Table 1 Employee types and offdays required weekend i
weekend i + 1
Type
Off Off On
Off On Off
T1 T2 T3
On
On
T4
No. of offdays required during the week
Step 4. Form the offday pairs for each category k ~ K. For each category k, 2n k offdays are required each week from Mo to Fr. The objective in this step is to form n k offday pairs for each category k ~ K from the offdays available. Form n k offday pairs going in a cyclical fashion over the five days of the week (Mo-Fr) for each category k ~ K. The starting day is Mo for category k = 1. We call the day at which the offday formation process ends as the stopping day for that category. The starting day for other categories (i.e., k > 1) is the day next to the stopping day for the previous category k - 1. Step 5. Assignment in week 1. The goal behind the offday assignment process is to ensure that an employee works no more than five consecutive days. Start with category k = 1. Assign pairs with adjacent offdays to T4 employees. Each remaining offday pair is assigned to a T 3 / T 2 employee pair such that T3 gets the earlier day of each pair and T2 gets the later day, In Lemma 1, we show that after the assignment of 2n k offdays for each category k ~ K, there will be D k workers from categories 1 up to and including k consisting of d k workers of category k available on each weekday (Mo-Fr). Repeat this step for all other categories, k ~ K. In Lemma 2, we show that the adjacent offday pairs available will be sufficient to cover all T4 employees. Step 6: Assignment of offdays in week i (i > 1). Assume that weeks 1 up to i - 1 have already been scheduled. For each employee who is of type T4 in week i and was of type T4 in week i - 1, assign the same pairs of offdays which they received in week i - 1. For each employee who is of type T4 in week i, was of type T2 in week i - 1 and was associated with adjacent pair of offdays for T 3 / T 2 in week i - 1, assign both days of that pair in week i. For each employee who is of type T4 in week i, was of type T2 in week i - 1 and was associated with the non-adjacent pair of offdays (Mo, Fr) for T 3 / T 2 in week i - 1, assign a new pair of offdays from an employee who was of type T4 in week i - 1 but is of type T3 in week i and associate the (Mo, Fr) pair with that T4. In Lemma 3, we show that this interchange is feasible. For each employee who is of type T 3 in week i, was of type T4 in week i - 1 and has the associated (Mo, Fr) offday pair, assign the earlier day of the pair t o T 3 employee and the later day to a T 2
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R. Narasimhan / European Journal of Operational Research 96 (1996) 113-121
Table 2 Parameters of the example problem. A = 2, B = 7 and c t > c2;
Table 3 Computations of L] and L~ bounds
C2 < C3 < C4
k
nt
1
2
3
2
3
2 3 4
3 2 7
5 3 10
6 6 5
6 6 10
k
nk
dk
Dk
1
2
1
1
2 3 4
3 2 7
4 5 2
7 13 21
e m p l o y e e in week i. F o r the other T 3 e m p l o y e e s in week i, associate the same pair o f offdays from which they received an offday in week i - 1. The type T 3 e m p l o y e e s get the earlier day of the associated pair while the type T 2 e m p l o y e e s get the later day o f the pairs. In L e m m a 4, we show that n o e m p l o y e e works for more than 5 consecutive days.
4. Example
The parameters o f the e x a m p l e p r o b l e m are presented in Table 2. L e t q = B / ( B - A) = 1.4. Step 1. C o m p u t e the m o s t e c o n o m i c a l worlcforce combination: See T a b l e s 3, 4 and 5. F r o m the c o m p u t a t i o n s in the tables, we get W =
W 4 = 27.
C o n c e r n i n g the c o m b i n a t i o n of employees: C o n sider k = 3. S 3 = 1, i.e., we need one additional
L~ :[qn k]
L~: d k + [0.4nk]
max(L], L~)
employee. Since of the first three categories, category 2 has the least cost, r = 2 and $32 = 1. Similarly $42 = 1. The most e c o n o m i c a l c o m b i n a t i o n o f employees and the c u m u l a t i v e n u m b e r of e m p l o y e e s required to be hired are given in the last two c o l u m n s of Table 5. Step 2. Schedule w e e k e n d s off." A week starts on S u n d a y and ends in Saturday. C o n s i d e r category k=l.
w k - n k = 1.
Therefore, one w e e k e n d off is rotated in the schedule and m a r k e d with ' × ' on Sa and Su in Table 7. Similar explanations hold good for other categories. Step 3. Identify the types o f employees: T h e types to which e m p l o y e e s b e l o n g according to their posi-
Table 4 Computation of L~ bound k
,n i
r~
Dk
D k + [0.4E~= ,n~l
L~ = max{Dk + [0.4E~= ,nil, Lk3- ' + max(L~. L~)} for k> land L~ = max(L',, L~)
1
2
1
2
3
2 3 4
5 7 14
7 13 21
9 16 27
9 16 27
Table 5 Determination of the most economical combination of employees k
Wk
S~.= W, - Wk_ 1 -max(L~, L~)
r
wk = max(L], L~) + Y~"__t Sik
W~
1
3
-
-
3
3
2 3 4
9 16 27
0 1 1
2 2
8 6 10
11 17 27
R. Narasimhan / European Journal of Operational Research 96 (1996) 113-121
118 Table 6 Formation o f offday pairs k
nk
Start day
Offday pairs (Mo, Tu), (We, T h )
1
2
Mo
2
3
Fr
(Mo, Fr), (Tu, We), (Th, Fr)
3 4
2 7
Mo Fr
(Mo, Tu), (We, Th) (Mo, Fr), (Tu, We), (Th, Fr), (Mo, Tu), (We, Th), (Mo, Fr), (Tu, W e )
tion in the rotating cycle of weekends off are marked for each week in Table 7 under the columns titled 'Type'. Step 4. Form offday pairs: See Table 6. Step 5. Assignment of offdays in week 1: See Table 7. Consider category 1. Assign the pair of consecutive offdays (Mo, Tu) to T4 employee (employee number 3). Assign We, the first day of the
offday pair (We, Th), to T3 employee (employee number 2) and the latter, Th to T2 employee (employee number 1). Proceed to allocate offdays in a similar fashion to all categories of employees in week 1 as in Table 7. Step 6. Assignment of offdays in week i, i > 1: Consider week i = 4 and category k = 4 for example without loss of generality. Assume weeks 2 and 3
Table 7 Final schedule assigning offdays to e m p l o y e e s No.
Sa
Su
Week 1 Type
x
Tu
We
Th
Fr
Sa
Su
Type
x
x
T4 T2 T3
1 2 3
x
4
x
x
TI
x
x
T2
5 6 7 8 9 10 11
x x x x
x x x x
TI T2 T2 T2 T3 T3 T3
x
x
T2 T3 T3 T3 TI T T2
12 13 14 15 16 17
x x x x
18 19 20 21 22 23 24 25 26 27
X x x
x x x x
X x x
T2 T3 T4
Week 2 Mo
TI TI T2 T2 T3 T3 T2 T2 T2 T3 T3 T3 T4 T4 T4 T4
x x x
x
x x x x x x
x x x
x x x
x x
x x
x x x
x x
x
x x
X x x x x x x
x x x x x
x
x x
x
x
x x x
T2 T2 T3 T3 T1 TI T4 T4 T4 T2 T2 T2 T3 T3 T3 T4
Mo
Tu
We
Th
Fr
Sa
Su
X
X
x x x
x x x x x
x x x x x
x x x x
x x x x
x
x
x
x
x
x
x x x x x
x
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R. Narasimhan / European Journal of Operational Research 96 (1996) 113-121
Fr), assign M o to T 3 in week 4 (employees 20 and 21) and F r to T2 in week 4 (employees 18 and 19). For other T3 employee in week 4 who was o f type T4 in week 3 and was associated with (Th, Fr), assign Th to T 3 in week 4 (employee 22) and Fr to T 2 in week 4 (employee 27). In similar fashion, assign offdays to employees in other categories.
have been scheduled already. O f the four T4 employees (employees 23 to 26) in week 4, three were of type T2 and one was of type T4 in week 3. F o r T4 employees in week 4 who were also o f type T4 in the previous week and received adjacent pair of offdays, assign the same pair in week 4. Employee 23 gets (Mo, Tu) in week 4 also. Employee 24 was associated with (We, Th) in week 3 and so assign (We, Th) in week 4. E m p l o y e e s 25 and 26 were associated with (Mo, Fr) in week 3 and so swap these pairs with adjacent pairs o f offdays received by T4 employees in week 3 (and who are o f type T3 in week 4), viz., (Tu, We), (Tu, W e ) of employees 20 and 21 in week 3. Assign these pairs to employees 25 and 26 in week 4. For T3 employees in week 4 who were of type T 4 in week 3 and were associated with (Mo,
Mo
Tu
T3 T4 T2
x
x x
T3 T3 T1 T1 T2 T2 T3x T3 T4 T4 T4 T4 T2 T2 T2 T3
In this paper, we have presented an algorithm for finding the optimal solution to single shift scheduling of hierarchical workforce on five-day workweeks to meet a particular pattern o f demand. The algorithm assigns adjacent pairs of offdays to employees
Week 4
Week 3 Type
T3 T3 TI TI T2 T2 T2 T3
5. Conclusion
We
Th
Fr
Sa
Su
x
x
T2
Type
Mo
Tu
T3 T4
x x x x x
x x x x
We
Th
X
X
TI T2 T2 T2 T3 T3
x
X
X
X X X
X
X
X
X
X
X
X
X
X
X
TI
X
X
TI TI T2 T2 T3x T3
X
X
X
X
T2 X
x
Su
x
X
X
Sa
T3
x
x
x
Fr
T2 T3x T3 T3 T4 T4 T4 T4 T2
X X X
X
X
X
120
R. Narasimhan / European Journal of Operational Research 96 (1996) 113-121
who work on successive weekends whenever A / B >t 1/6. However, if A / B < 1/6, the algorithm could assign the maximum possible number of adjacent pairs of offdays to employees who work on successive weekends. Further research is in progress for extending this algorithm to multiple shift contexts and for different pattern of demand for employees.
Lemma 2. There exist enough adjacent offday pairs to cover all type T4 employees in Steps 5 and 6 of the algorithm. Consider any category k ~ K. When type T4 exists, as shown by Bums and Carter (1985) we have
ITz l = IT3] = w k - n k and
Appendix L e m m a 1. The offdays available at the Step 4 of the algorithm will be sufficient to meet the offdays assigned (= offdays required) i.e., after the assignment of 2n k offdays for each category k = 1, 2, 3 . . . . . m, there will be D k workers from categories 1 up to and including k consisting of d k category k workers available on each weekday (Mo-Fr). The algorithm always allocates 2n k offdays to each category k = 1, 2, 3 . . . . . m. We have to show that by doing so (i) we have d k type k employees are available for each day of the week from Mo to Fr and (ii) we have D k employees cumulatively for categories 1 to k for each day of the week from Mo to Fr. (i) The offday pair formation process of the algorithm goes over the five days of the week (Mo-Fr) in a cyclical fashion and this process distributes the total number of offdays for any category k, (2nk), uniformly on all the five days of the week (Mo-Fr) such that on any one day the number of shifts assigned off is at most [2nk/5]. The number of category k employees available for work on any weekday is w ~ - [2nk/51 which is >/d k by the L~ bound. (ii) Again, the cyclical offday pair formation goes over the five days of the week in a continuous fashion starting with Mo for category k = 1 and with the day next to the one at which the process ended for the previous category. Hence the number of shifts assigned off for category 1 up to k on any one day is at most [Ek~12ni/5]. The number of employees from categories 1 to k on any day from Mo to Fr is Wk - [E~= 12nil5], which is >/D k by the L~ bound.
IT4[ = 2 n k - w k. n k offday pairs are constructed going in a cyclical fashion from Mo to Fr. The following five standard offday pairs are formed: (Mo, Tu), (We, Th), (Mo, Fr), (Tu, We) and (Th, Fr). Therefore the minimum number of adjacent offday pairs in the above scheme is n k - Ink/5]. It is required to show that - rnk/51}
> / ( 2 n k - wk),
which is true if w~ >/n k + r nk/5]. By assumption A / B >~ 1 / 6 and this combined with the L{ bound implies w k i> [1.2 nk]. Hence the lemma. L e m m a 3. If a type T4 employee is associated with a non-adjacent offday pair, then there exists another employee who is associated with an adjacent offday pair with whom an interchange of offday pairs is possible in Step 6. Consider any category k ~ K. Since IT4[ is a constant, whenever a new type T4 employee occurs there must be some other employee who was of type T4 in the previous week but no longer. By construction, type T4 employees always have pairs of adjacent offdays. Therefore the employee who has ceased to be of type T4 is associated with a pair of adjacent offdays which can be relinquished to the employee who has become type T4. L e m m a 4. No employee works more than 5 consecutive days. Consider T3 employees in any week i (i > 1). They would have been of type T4 or T2 in the previous week i - 1. Since T3 employees in week i get the earlier offday from the pair which was asso-
R. Narasimhan / European Journal of Operational Research 96 (1996) 113-121
ciated with their types in the previous week, the maximum workstretch cannot exceed 5 days. Since T3 employees in week i have the weekend off at the end of week i and have one day off during the week i, the maximum workstretch cannot exceed 5 days regardless of their types in the next week i + 1. Consider T4 employees in week i. They were of type T 2 or T4 in the previous week i - 1. Since T4 employees are associated with the pair of offdays that they received in the previous week when they were of type T2 or T4, the maximum workstretch cannot exceed 5 days. T4 employees in week i remain T4 or become T3 in week i + 1. As argued for T3 employees, the workstretch cannot exceed 5 days. Consider 7"2 employees in week i. Since they have the weekend off at the beginning of the week and an offday during the week, regardless of their type in the previous week, the maximum workstretch cannot exceed 5 days. T2 employees in week i become T3 or T4 in week i + 1. As argued for T 3 / T 4 employees, the workstretch cannot exceed 5 days. Consider T~ employees who have both the weekends off in any week. So the workstretch cannot exceed 5 days regardless of their change of type.
Acknowledgements I would like to thank Dr. R.N. Burns of the School of Business, Queen's University, Kingston,
121
Canada for supervising the research and two anonymous referees for their valuable comments.
References Baker, K.R., and Magazine, M.J. (1977), "'Workforce scheduling with cyclic demands and day off constraints", Management Science 24, 161-167. Baker, K.R., Burns, R.N., and Carter, M.W. (1979), "Staff scheduling with day-off and workstretch constraints", AIIE Transactions 6/11,286-292. Brownell, W.D., and Lowerre, J.M., (1976), "'Scheduling of work forces required in continuous operations under alternative labour policies", Management Science 22, 597-605. Burns, R.N. (1978), "Manpower scheduling with variable demands and alternate weekends off" INFOR 16, 101-111. Burns, R.N. (1981), "An iterative approach to multiple shift scheduling", Manuscript, School of Business, Queen's University, Kingston, Ont., Canada. Burns, R.N., and Carter, M.W. (1985), "Work force size and schedules with variable demands", Management Science 31, 599-607. Burns, R.N., and Koop, G.J. (1987), " A modular approach to optimal multiple-shift manpower scheduling' ', Operations Research 35, 100-110. Emmons, H. (1985), "Workforce scheduling with cyclic requirements and constraints on days off, weekends off, and workstretch", lEE Transactions 17, 8-16. Emmons, H., and Burns, R.N. (1991), "Off-day scheduling with hierarchical worker categories", Operations Research 39, 484-495. Lowerre, J.M. (1977), "Workstretch properties for the scheduling of continuous operations under alternative labour policies", Management Science 23,963-971.
European Journal of Operational Research 96 (1996) 113-121
Theory and Methodology
An algorithm for single shift scheduling of hierarchical workforce Rangarajan Narasimhan * School of Business, Queen's Unioersity, Kingston, Ont., Canada K7L 3N6 Received April 1994; revised November 1995
Abstract This paper presents an optimal algorithm for single shift scheduling of hierarchical workforce in seven-day-a-week industries. There are many categories of workers; their capabilities are arranged hierarchically. The objective is to arrive at the most economical mix of categories of employees satisfying the pattern of demand for employees and desired work characteristics. The demand for employees varies from one category to the other and for each category the demand during weekdays is fixed and is different from that of weekends. The desired work characteristics are: (i) each employee must be given two days off every week including at least a proportion A out of every B weekends off and (ii) each employee can have at most 5 working shifts between any two offdays. The algorithm provides a computational scheme for arriving at the minimum workforce size comprising the most economical mix of categories of employees for this demand context and a new technique for characterizing the problem to achieve feasible assignment of employees to shifts.
Keywords: Scheduling theory; Manpower planning
O. Introduction Scheduling of hierarchical workforce using combinatorial methods has its beginning in the research of Emmons and Burns (1991). This class of problems arises in scheduling of health care personnel, job shop employees, maintenance crew and so on. In these problems there are many categories of employees to be scheduled. The categories of employees form a hierarchy in that the employees at a higher level can do the work of lower level employees but not vice versa. Prior to the work of Emmons and
* Correspondence to: Ranga Naras, The Fred C. Manning School of Business Administration, Acadia University, Wolfville, NS, Canada BOP 1X0. email: [email protected]
Burns (1991), the focus of research in this field has been on single shift and multiple shift scheduling of single category employees. Burns and Carter (1985) provide a comprehensive version of the single shift scheduling problem generalizing the earlier research of Brownell and Lowerre (1976), Lowerre (1977), Baker and Magazine (1977), Bums (1978), Baker, Burns and Carter (1979) and Emmons (1985). Bums and Koop (1987) provide a modular approach for solving workforce scheduling problems characterized by multiple shifts and single category of employees. Burns (1981) describes an iterative algorithm for the same problem. This paper focuses on single shift scheduling of hierarchical workforce and contributes to the progress of the research in this field by providing a computa-
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R. Narasimhan / European Journal of Operational Research 96 (1996) 113-12l
tional scheme for amving at the minimum size of the workforce for a pattern of demand not considered in the literature so far. As opposed to the uniform daily demand considered by Emmons and Bums (1991), a demand pattern for employees characterized by different constants for weekdays and weekends is considered in this paper. The computational time of the algorithm increases linearly with the increase in problem size. The remainder of the paper is divided into five sections. Section 1 describes the problem and introduces notation. Section 2 gives the computational scheme for arriving at the most economical workforce size using which Section 3 constructs a feasible schedule assigning employees to shifts. Section 4 illustrates the algorithm with a numerical example and Section 5 identifies the directions for further research in this field. The feasibility and the optimality of the algorithm are established in the Appendix.
workers required on Sa and Su and D k be the cumulative number of workers of categories 1 up to and including k required on Mo to Fr consisting of at least d k workers of category k, Vk ~ K. Let w k be the optimum number of people to be hired in category k and Wk the cumulative number of people to be hired in categories 1 up to and including k. Clearly,
1. Problem description
We compute three lower bounds for workforce size based on three demand characteristics: 1) the weekend demand for each category, 2) the total number of shifts required in a week for each category, and 3) the cumulative number of employees required up to and including category k ~ K during the weekdays. The most economical workforce size is determined by satisfying all the three bounds suitably. L o w e r bound based on weekend demand for each category: L~. The number of employees of category k available each weekend must be sufficient to meet the weekend demand. Since each employee has at least A out of every B weekends off, over a cycle of B weeks, we have
In this paper, we consider a facility that must be staffed seven days a week. There are many categories of workers to be scheduled; their capabilities are arranged hierarchically. The facility operates only one shift a day. The objective is to arrive at the most economical mix of categories of employees satisfying the pattern of demand for employees and the desired work characteristics. The demand for employees is of the following form: the weekday and the weekend demand are fixed with no particular relationship between them. The desired work characteristics are: (i) each employee must be given two days off every week including at least a proportion A out of every B weekends off and (ii) each employee works for at most 5 consecutive days. Let K = {1, 2, 3 . . . . . m} be the set of categories of employees. Assume that the categories are ranked with category 1 workers at the top as the most highly qualified so that a category r worker can be used in place of a worker of category k (for any k > r, r ~ m, and k, r ~ K ) but not vice versa. Thus we have a hierarchy of categories having downward substitutability during weekdays as explained in Emmons and Bums (1991). Let c k be the cost to employ a person of category k, k ~ K . Let n k be the number of category k
Dk-- D k - I >I d~ for k > 1 and k ~ K ,
and
D l = d I. There is no restriction on the relationship between d k and n k.
2. Determination of the most economical workforce size
( B - A) w k >1 Bn k or
wk/> L~, where L~ = [ B n k / ( B - A ) | and [ x | is the nearest integer equal to or above x. Lower bound based on total demand f o r category k employees: Lk2. The total number of employee days per week available from employees of all categories must be sufficient to meet the total weekly demand for that category. Since each employee is to be given
R. Narasimhan / European Journal of Operational Research 96 (1996) 113-121
two days off every week, s / h e works at most five days per week. Hence, 5 w k >: 5 d k + 2 n k or
wk >~ tk2,
where L~ = d~ + [0.4 nk]. The workforce size for category k is at least the maximum of the above two bounds Lkl and L~ and we have w k/> max{L~, L~}. L o w e r b o u n d b a s e d on the cumulative w o r k e r demand: Lk3. T h e cumulative number of employees
available from categories 1 to k should not only satisfy the cumulative number of employees required but also consist of the minimum number of category k employees for any k ~ K. The cumulative number of shifts available from employees of categories 1 to k should at least equal the total number of shifts required in the week for categories 1 to k, i.e., k
115
ber of employees required from categories 1 to m, W,~, is arrived at by equating it to the lower bound, L~', the total workforce size W ( = Wm) is the minimum required to satisfy the demand. To determine the most economical mix of categories of employees, we proceed as follows: Compute Sk = L~ - L~-1 - max(L~, L~) for k > 1 and k ~ K. Sk represents the additional number of employees required from categories I to k over and above the number of employees required exclusively from category k. Choose these Sk employees from the least cost category among the hierarchically substitutable categories, i.e., choose from category r such that r ~< k, r, k ~ K and c r = min( c I , c 2 , c 3 . . . . . ck).
After having chosen Sk employees from category r, denote this S k by Skr. Skr signifies the additional number of people to be hired from category r to satisfy the cumulative demand from categories 1 to k. Repeat this procedure for all k > 1 and k ~ K. The most economical combination of employees is
5 W k >1 5 D k + 2 ~ , n i i=1
wk = max(L~, L~) + ~ Sik. i=k
or
W k >~D k +
I04i1 1 ni
•
Also, the cumulative number of employees available from categories 1 to k should have at least the required number of employees of category k for k E K , i.e., Wk>~ {Wk_ ,
+ max(L~, L~)}
for k > 1, and WI = w l >/max(L~, L~) for category k = 1.
Thus, wt/> L~, where L~=max
( [o4i, ] D k+
n i , L~- ' + m a x ( L~, L
We compute Wk = E/k= i wi . The composition of the minimum total workforce size is such that each additional employee required to satisfy the cumulative demand is chosen from the least cost category among the hierarchically substitutable categories. Hence, the categorywise composition of workforce computed is the least cost combination of employees that satisfies the staffing requirements. In the next section, we describe an algorithm to construct a schedule for the most economical workforce size computed in this section. The schedule is such that it satisfies the pattern of demand for each category of employees and the desired work characteristics.
for category k for k > 1, and
L5 ; max(L , Li ).
3. Algorithm
We now have three lower bounds for the workforce size. Set Wm = L~'. Since the cumulative num-
The inputs to the algorithm are the values of the following variables: A, B, m, n,, d k, D k, Vk E K.
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R. Narasimhan / European Journal of Operational Research 96 (1996) 113-12 l
The algorithm computes the minimum total workforce size consisting of the most economical mix of categories of employees required and generates a feasible schedule for assignment of these employees to shifts. Step 1. Compute the minimum workforce size. Calculate the three bounds L~, Lk2 and L~ and arrive at the categorywise distribution of workforce, i.e., w k, V k ~ K . Step 2. Schedule the weekends off for each category k E K. Consider any category k. For convenience, assign a number from 1 to w k to each employee. Since n k people are required to work each weekend, the remaining (w k - n k) employees can have the weekend off. Assign the first weekend off to the first (w k - n k) employees. Assign the second weekend off to the next (w k - n k) employees. The process is continued cyclically with employee 1 being treated as the next employee after employee w k. Step 3. Determine the number of offday pairs required for each category k ~ K. In any week, employees will fall into one of four types depending on the weekends off at the beginning and at the end of the week as in Table 1. Since there are exactly n k employees working each weekend, IT3 I +IT41 = n k (from weekend i) and IT2 I + IT41 = n k (from weekend i + l ) . Hence IT2 1 = [T3I. As shown by Bums and Carter (1985), type T~ and type T4 employees cannot co-exist, i.e., either type TI may exist or type T4 may exist but not both.
Table 1 Employee types and offdays required weekend i
weekend i + 1
Type
Off Off On
Off On Off
T1 T2 T3
On
On
T4
No. of offdays required during the week
Step 4. Form the offday pairs for each category k ~ K. For each category k, 2n k offdays are required each week from Mo to Fr. The objective in this step is to form n k offday pairs for each category k ~ K from the offdays available. Form n k offday pairs going in a cyclical fashion over the five days of the week (Mo-Fr) for each category k ~ K. The starting day is Mo for category k = 1. We call the day at which the offday formation process ends as the stopping day for that category. The starting day for other categories (i.e., k > 1) is the day next to the stopping day for the previous category k - 1. Step 5. Assignment in week 1. The goal behind the offday assignment process is to ensure that an employee works no more than five consecutive days. Start with category k = 1. Assign pairs with adjacent offdays to T4 employees. Each remaining offday pair is assigned to a T 3 / T 2 employee pair such that T3 gets the earlier day of each pair and T2 gets the later day, In Lemma 1, we show that after the assignment of 2n k offdays for each category k ~ K, there will be D k workers from categories 1 up to and including k consisting of d k workers of category k available on each weekday (Mo-Fr). Repeat this step for all other categories, k ~ K. In Lemma 2, we show that the adjacent offday pairs available will be sufficient to cover all T4 employees. Step 6: Assignment of offdays in week i (i > 1). Assume that weeks 1 up to i - 1 have already been scheduled. For each employee who is of type T4 in week i and was of type T4 in week i - 1, assign the same pairs of offdays which they received in week i - 1. For each employee who is of type T4 in week i, was of type T2 in week i - 1 and was associated with adjacent pair of offdays for T 3 / T 2 in week i - 1, assign both days of that pair in week i. For each employee who is of type T4 in week i, was of type T2 in week i - 1 and was associated with the non-adjacent pair of offdays (Mo, Fr) for T 3 / T 2 in week i - 1, assign a new pair of offdays from an employee who was of type T4 in week i - 1 but is of type T3 in week i and associate the (Mo, Fr) pair with that T4. In Lemma 3, we show that this interchange is feasible. For each employee who is of type T 3 in week i, was of type T4 in week i - 1 and has the associated (Mo, Fr) offday pair, assign the earlier day of the pair t o T 3 employee and the later day to a T 2
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R. Narasimhan / European Journal of Operational Research 96 (1996) 113-121
Table 2 Parameters of the example problem. A = 2, B = 7 and c t > c2;
Table 3 Computations of L] and L~ bounds
C2 < C3 < C4
k
nt
1
2
3
2
3
2 3 4
3 2 7
5 3 10
6 6 5
6 6 10
k
nk
dk
Dk
1
2
1
1
2 3 4
3 2 7
4 5 2
7 13 21
e m p l o y e e in week i. F o r the other T 3 e m p l o y e e s in week i, associate the same pair o f offdays from which they received an offday in week i - 1. The type T 3 e m p l o y e e s get the earlier day of the associated pair while the type T 2 e m p l o y e e s get the later day o f the pairs. In L e m m a 4, we show that n o e m p l o y e e works for more than 5 consecutive days.
4. Example
The parameters o f the e x a m p l e p r o b l e m are presented in Table 2. L e t q = B / ( B - A) = 1.4. Step 1. C o m p u t e the m o s t e c o n o m i c a l worlcforce combination: See T a b l e s 3, 4 and 5. F r o m the c o m p u t a t i o n s in the tables, we get W =
W 4 = 27.
C o n c e r n i n g the c o m b i n a t i o n of employees: C o n sider k = 3. S 3 = 1, i.e., we need one additional
L~ :[qn k]
L~: d k + [0.4nk]
max(L], L~)
employee. Since of the first three categories, category 2 has the least cost, r = 2 and $32 = 1. Similarly $42 = 1. The most e c o n o m i c a l c o m b i n a t i o n o f employees and the c u m u l a t i v e n u m b e r of e m p l o y e e s required to be hired are given in the last two c o l u m n s of Table 5. Step 2. Schedule w e e k e n d s off." A week starts on S u n d a y and ends in Saturday. C o n s i d e r category k=l.
w k - n k = 1.
Therefore, one w e e k e n d off is rotated in the schedule and m a r k e d with ' × ' on Sa and Su in Table 7. Similar explanations hold good for other categories. Step 3. Identify the types o f employees: T h e types to which e m p l o y e e s b e l o n g according to their posi-
Table 4 Computation of L~ bound k
,n i
r~
Dk
D k + [0.4E~= ,n~l
L~ = max{Dk + [0.4E~= ,nil, Lk3- ' + max(L~. L~)} for k> land L~ = max(L',, L~)
1
2
1
2
3
2 3 4
5 7 14
7 13 21
9 16 27
9 16 27
Table 5 Determination of the most economical combination of employees k
Wk
S~.= W, - Wk_ 1 -max(L~, L~)
r
wk = max(L], L~) + Y~"__t Sik
W~
1
3
-
-
3
3
2 3 4
9 16 27
0 1 1
2 2
8 6 10
11 17 27
R. Narasimhan / European Journal of Operational Research 96 (1996) 113-121
118 Table 6 Formation o f offday pairs k
nk
Start day
Offday pairs (Mo, Tu), (We, T h )
1
2
Mo
2
3
Fr
(Mo, Fr), (Tu, We), (Th, Fr)
3 4
2 7
Mo Fr
(Mo, Tu), (We, Th) (Mo, Fr), (Tu, We), (Th, Fr), (Mo, Tu), (We, Th), (Mo, Fr), (Tu, W e )
tion in the rotating cycle of weekends off are marked for each week in Table 7 under the columns titled 'Type'. Step 4. Form offday pairs: See Table 6. Step 5. Assignment of offdays in week 1: See Table 7. Consider category 1. Assign the pair of consecutive offdays (Mo, Tu) to T4 employee (employee number 3). Assign We, the first day of the
offday pair (We, Th), to T3 employee (employee number 2) and the latter, Th to T2 employee (employee number 1). Proceed to allocate offdays in a similar fashion to all categories of employees in week 1 as in Table 7. Step 6. Assignment of offdays in week i, i > 1: Consider week i = 4 and category k = 4 for example without loss of generality. Assume weeks 2 and 3
Table 7 Final schedule assigning offdays to e m p l o y e e s No.
Sa
Su
Week 1 Type
x
Tu
We
Th
Fr
Sa
Su
Type
x
x
T4 T2 T3
1 2 3
x
4
x
x
TI
x
x
T2
5 6 7 8 9 10 11
x x x x
x x x x
TI T2 T2 T2 T3 T3 T3
x
x
T2 T3 T3 T3 TI T T2
12 13 14 15 16 17
x x x x
18 19 20 21 22 23 24 25 26 27
X x x
x x x x
X x x
T2 T3 T4
Week 2 Mo
TI TI T2 T2 T3 T3 T2 T2 T2 T3 T3 T3 T4 T4 T4 T4
x x x
x
x x x x x x
x x x
x x x
x x
x x
x x x
x x
x
x x
X x x x x x x
x x x x x
x
x x
x
x
x x x
T2 T2 T3 T3 T1 TI T4 T4 T4 T2 T2 T2 T3 T3 T3 T4
Mo
Tu
We
Th
Fr
Sa
Su
X
X
x x x
x x x x x
x x x x x
x x x x
x x x x
x
x
x
x
x
x
x x x x x
x
119
R. Narasimhan / European Journal of Operational Research 96 (1996) 113-121
Fr), assign M o to T 3 in week 4 (employees 20 and 21) and F r to T2 in week 4 (employees 18 and 19). For other T3 employee in week 4 who was o f type T4 in week 3 and was associated with (Th, Fr), assign Th to T 3 in week 4 (employee 22) and Fr to T 2 in week 4 (employee 27). In similar fashion, assign offdays to employees in other categories.
have been scheduled already. O f the four T4 employees (employees 23 to 26) in week 4, three were of type T2 and one was of type T4 in week 3. F o r T4 employees in week 4 who were also o f type T4 in the previous week and received adjacent pair of offdays, assign the same pair in week 4. Employee 23 gets (Mo, Tu) in week 4 also. Employee 24 was associated with (We, Th) in week 3 and so assign (We, Th) in week 4. E m p l o y e e s 25 and 26 were associated with (Mo, Fr) in week 3 and so swap these pairs with adjacent pairs o f offdays received by T4 employees in week 3 (and who are o f type T3 in week 4), viz., (Tu, We), (Tu, W e ) of employees 20 and 21 in week 3. Assign these pairs to employees 25 and 26 in week 4. For T3 employees in week 4 who were of type T 4 in week 3 and were associated with (Mo,
Mo
Tu
T3 T4 T2
x
x x
T3 T3 T1 T1 T2 T2 T3x T3 T4 T4 T4 T4 T2 T2 T2 T3
In this paper, we have presented an algorithm for finding the optimal solution to single shift scheduling of hierarchical workforce on five-day workweeks to meet a particular pattern o f demand. The algorithm assigns adjacent pairs of offdays to employees
Week 4
Week 3 Type
T3 T3 TI TI T2 T2 T2 T3
5. Conclusion
We
Th
Fr
Sa
Su
x
x
T2
Type
Mo
Tu
T3 T4
x x x x x
x x x x
We
Th
X
X
TI T2 T2 T2 T3 T3
x
X
X
X X X
X
X
X
X
X
X
X
X
X
X
TI
X
X
TI TI T2 T2 T3x T3
X
X
X
X
T2 X
x
Su
x
X
X
Sa
T3
x
x
x
Fr
T2 T3x T3 T3 T4 T4 T4 T4 T2
X X X
X
X
X
120
R. Narasimhan / European Journal of Operational Research 96 (1996) 113-121
who work on successive weekends whenever A / B >t 1/6. However, if A / B < 1/6, the algorithm could assign the maximum possible number of adjacent pairs of offdays to employees who work on successive weekends. Further research is in progress for extending this algorithm to multiple shift contexts and for different pattern of demand for employees.
Lemma 2. There exist enough adjacent offday pairs to cover all type T4 employees in Steps 5 and 6 of the algorithm. Consider any category k ~ K. When type T4 exists, as shown by Bums and Carter (1985) we have
ITz l = IT3] = w k - n k and
Appendix L e m m a 1. The offdays available at the Step 4 of the algorithm will be sufficient to meet the offdays assigned (= offdays required) i.e., after the assignment of 2n k offdays for each category k = 1, 2, 3 . . . . . m, there will be D k workers from categories 1 up to and including k consisting of d k category k workers available on each weekday (Mo-Fr). The algorithm always allocates 2n k offdays to each category k = 1, 2, 3 . . . . . m. We have to show that by doing so (i) we have d k type k employees are available for each day of the week from Mo to Fr and (ii) we have D k employees cumulatively for categories 1 to k for each day of the week from Mo to Fr. (i) The offday pair formation process of the algorithm goes over the five days of the week (Mo-Fr) in a cyclical fashion and this process distributes the total number of offdays for any category k, (2nk), uniformly on all the five days of the week (Mo-Fr) such that on any one day the number of shifts assigned off is at most [2nk/5]. The number of category k employees available for work on any weekday is w ~ - [2nk/51 which is >/d k by the L~ bound. (ii) Again, the cyclical offday pair formation goes over the five days of the week in a continuous fashion starting with Mo for category k = 1 and with the day next to the one at which the process ended for the previous category. Hence the number of shifts assigned off for category 1 up to k on any one day is at most [Ek~12ni/5]. The number of employees from categories 1 to k on any day from Mo to Fr is Wk - [E~= 12nil5], which is >/D k by the L~ bound.
IT4[ = 2 n k - w k. n k offday pairs are constructed going in a cyclical fashion from Mo to Fr. The following five standard offday pairs are formed: (Mo, Tu), (We, Th), (Mo, Fr), (Tu, We) and (Th, Fr). Therefore the minimum number of adjacent offday pairs in the above scheme is n k - Ink/5]. It is required to show that - rnk/51}
> / ( 2 n k - wk),
which is true if w~ >/n k + r nk/5]. By assumption A / B >~ 1 / 6 and this combined with the L{ bound implies w k i> [1.2 nk]. Hence the lemma. L e m m a 3. If a type T4 employee is associated with a non-adjacent offday pair, then there exists another employee who is associated with an adjacent offday pair with whom an interchange of offday pairs is possible in Step 6. Consider any category k ~ K. Since IT4[ is a constant, whenever a new type T4 employee occurs there must be some other employee who was of type T4 in the previous week but no longer. By construction, type T4 employees always have pairs of adjacent offdays. Therefore the employee who has ceased to be of type T4 is associated with a pair of adjacent offdays which can be relinquished to the employee who has become type T4. L e m m a 4. No employee works more than 5 consecutive days. Consider T3 employees in any week i (i > 1). They would have been of type T4 or T2 in the previous week i - 1. Since T3 employees in week i get the earlier offday from the pair which was asso-
R. Narasimhan / European Journal of Operational Research 96 (1996) 113-121
ciated with their types in the previous week, the maximum workstretch cannot exceed 5 days. Since T3 employees in week i have the weekend off at the end of week i and have one day off during the week i, the maximum workstretch cannot exceed 5 days regardless of their types in the next week i + 1. Consider T4 employees in week i. They were of type T 2 or T4 in the previous week i - 1. Since T4 employees are associated with the pair of offdays that they received in the previous week when they were of type T2 or T4, the maximum workstretch cannot exceed 5 days. T4 employees in week i remain T4 or become T3 in week i + 1. As argued for T3 employees, the workstretch cannot exceed 5 days. Consider 7"2 employees in week i. Since they have the weekend off at the beginning of the week and an offday during the week, regardless of their type in the previous week, the maximum workstretch cannot exceed 5 days. T2 employees in week i become T3 or T4 in week i + 1. As argued for T 3 / T 4 employees, the workstretch cannot exceed 5 days. Consider T~ employees who have both the weekends off in any week. So the workstretch cannot exceed 5 days regardless of their change of type.
Acknowledgements I would like to thank Dr. R.N. Burns of the School of Business, Queen's University, Kingston,
121
Canada for supervising the research and two anonymous referees for their valuable comments.
References Baker, K.R., and Magazine, M.J. (1977), "'Workforce scheduling with cyclic demands and day off constraints", Management Science 24, 161-167. Baker, K.R., Burns, R.N., and Carter, M.W. (1979), "Staff scheduling with day-off and workstretch constraints", AIIE Transactions 6/11,286-292. Brownell, W.D., and Lowerre, J.M., (1976), "'Scheduling of work forces required in continuous operations under alternative labour policies", Management Science 22, 597-605. Burns, R.N. (1978), "Manpower scheduling with variable demands and alternate weekends off" INFOR 16, 101-111. Burns, R.N. (1981), "An iterative approach to multiple shift scheduling", Manuscript, School of Business, Queen's University, Kingston, Ont., Canada. Burns, R.N., and Carter, M.W. (1985), "Work force size and schedules with variable demands", Management Science 31, 599-607. Burns, R.N., and Koop, G.J. (1987), " A modular approach to optimal multiple-shift manpower scheduling' ', Operations Research 35, 100-110. Emmons, H. (1985), "Workforce scheduling with cyclic requirements and constraints on days off, weekends off, and workstretch", lEE Transactions 17, 8-16. Emmons, H., and Burns, R.N. (1991), "Off-day scheduling with hierarchical worker categories", Operations Research 39, 484-495. Lowerre, J.M. (1977), "Workstretch properties for the scheduling of continuous operations under alternative labour policies", Management Science 23,963-971.