Solving the Multi-activity Shift Scheduling Problem using Variable

Neighbourhood Search

Yi Qu

and Timothy Curtois

Newcastle Business School, Northumbria University, Newcastle-upon-Tyne, NE1 8ST, U.K.

Staff Roster Solutions, Sir Colin Campbell Building, Nottingham, NG7 2TU, U.K.

Keywords: Multi-activity Shift Scheduling, Variable Neighbourhood Search, Benchmarks.

Abstract: This paper presents a set of benchmarks instances for the multi-activity shift scheduling problem and the

results produced using a variable neighbourhood search method. The data set is intended as a resource to

generate and verify novel research on an important and practical but challenging problem. The variable

neighbourhood search uses four different neighbourhood operators and can produce feasible solutions within

short computation times.

1 INTRODUCTION

The multi-activity shift scheduling problem is a

widely occurring, difficult optimisation problem. It is

commonly found in retail environments for example.

At shops and stores staff are required to work

different activities at different times throughout a day.

Each shop could have a different number of activities.

Staff may be required to operate tills, or assist

customers on shop floors, or manage stock in a

warehouse, or supervise other staff etc. The demand

for each activity may also fluctuate throughout the

day as busy periods arise for that activity. For

example, there could be times when more staff are

required in the warehouse when deliveries arrive. The

demand will also vary per day as some days are busier

than others. There are also constraints on the

employee’s individual schedule. They will have a

maximum number of contracted hours. They may

also have a minimum number of hours work they

must be assigned. There are constraints on their shift

lengths and when they can start. There are often

working directives on how much rest they must have

between shifts and so on. The objective of the

scheduler is often to minimise costs from overstaffing

whilst satisfying all other constraints.

One of the earliest attempts to solve the problem

is (Loucks & Jacobs, 1991). Until recently it was a

https://orcid.org/0000-0002-7773-8949

https://orcid.org/0000-0002-4743-416X

relatively under-studied problem. Possibly due to its

size and complexity most papers have focussed on

solving a single stage of the decomposed problem,

such as day off scheduling, shift scheduling or tour

scheduling. More recently mathematical

programming-based approaches have been used on

different variations of the problem (Gérard,

Clautiaux, & Sadykov, 2016; Restrepo, Gendron, &

Rousseau, 2016, 2018; Restrepo, Lozano, &

Medaglia, 2012; Salvagnin & Walsh, 2012) and also

neighbourhood search methods (Dahmen & Rekik,

2015; S Pan, Akplogan, Létocart, Touati, & Calvo,

2016; Stefania Pan et al., 2018; Quimper & Rousseau,

2010).

The benchmark data set captures the core features

of the problem. Instances of varying planning length,

numbers of staff and numbers of activities have been

produced. There are instances of length 7, 14 and 28

days. The number of staff varies from 10 to 150 and

the numbers of different activities varies from 1 up to

19 in the largest instances. There are 225 instances in

total. The instances and their characteristics are listed

in Table 1. Due to the way that the solutions were

created it is known that every instance does have a

feasible solution. When creating the instances, it was

also observed that at least three of the instances have

zero cost solutions (that means their total penalty

score is zero). The instances are available for

download from www.schedulingbenchmarks.org.

Qu, Y. and Curtois, T.

Solving the Multi-activity Shift Scheduling Problem using Variable Neighbourhood Search.

DOI: 10.5220/0009118102270232

In Proceedings of the 9th International Conference on Operations Research and Enterprise Systems (ICORES 2020), pages 227-232

ISBN: 978-989-758-396-4; ISSN: 2184-4372

227

2 PROBLEM DESCRIPTION

The problem requires the assignment of shifts to

employees and the assignment of activities within the

shifts.

The planning horizon is divided into 15 minute

intervals and shifts must start and finish at the

beginning of a 15 minute interval. For example,

09:00-17:15 would be a valid shift but 09:08-17:15

would not be, nor would 09:00-17:04.

Shifts consist of one or more activities. For

example, Figure 1. shows a representation of a shift

which starts at 09:00 and finishes at 17:00. Within

this shift an employee starts on Activity 1 (green) for

3.5 hours and then transfers to Activity 2 (grey) for

the remaining 4.5 hours of the shift.

Figure 1: Example shift with one transfer.

Figure 2 shows another 8 hour shift but in this

example there are two activity changes. The

employee starts with Activity 1 and then switches to

Activity 2 after 3.5 hours and then transfers back to

Activity 1 for the final hour of the shift.

Figure 2: Example shift with two transfers.

There are no limits on the number of different

activities that a shift can contain or how many activity

changes can occur. However, every activity duration

must be at least one hour before the employee

changes to a different activity or the shift ends.

Activity changes must also occur at the beginning of

a 15 minute interval. For example, an activity change

could occur at 12:00 or 12:15 but not at any time

between 12:00-12:15.

Staff demand requirements (also called cover) are

provided for each activity for every 15-minute time

interval in the planning horizon e.g. 00:00-00:15,

00:15-00:30 … 23:30-23:45, 23:45-00:00.

In all the instances the planning horizon starts at

06:00 on the first day and finishes at 06:00 on the last

day. Therefore if the planning horizon is 7 days then

it runs from 06:00 on day 1 to 06:00 on day 8.

Shifts must be designed and then assigned such

that a minimum demand for each activity is satisfied

as well as the employees’ constraints and the

organisation’s work regulations. The other

constraints are as follows:

2.1 Constraints

C1: Maximum one shift per day - Employees cannot

start more than one shift on a day (where a day is

considered as starting at midnight and finishing 24

hours later).

C2: Maximum Five Consecutive Working Days -

The maximum number of consecutive working days

that can be assigned to an employee is 5. A day is

considered as a working day if a shift is started on that

day (where the day is considered as starting at

midnight and finishing 24 hours later). This constraint

always assumes that the last day of the previous

planning period was a day off and the first day of the

next planning period is a day off.

C3: Minimum Total Minutes - Each employee has

a minimum total time in minutes that must be assigned

over the whole planning horizon. This is specified in

the data file for each employee. The duration of each

shift is from the start time to the end time.

C4: Maximum Total Minutes – Each employee has

a maximum total time in minutes that can be assigned

over the whole planning horizon. This is specified in

the data file for each employee. The duration of each

shift is from the start time to the end time.

C5: Minimum Rest Time between Shifts - There

must be a minimum of 14 hours rest after every shift.

This means that after a shift finishes the employee

cannot start another shift until at least 14 hours later.

C6: Minimum Activity Duration - A shift can

contain any number of different activities, but each

activity duration must be at least 1 hour. This means

that a shift cannot contain an activity duration that is

less than one hour. For example, a shift which starts

at 09:00 and has the first 45 minutes assigned to

Activity 1 and then switches to Activity 2 at 09:45 is

not valid, but if the shift switches to Activity 2 at

10:00 then it would be valid.

C7: Minimum Shift Length - The minimum shift

duration is 6 hours.

C8: Maximum Shift Length - The maximum shift

duration is 10 hours.

C9: Valid Shift Start Times - Shifts can only start

between the following times each day: 06:00-10:00,

14:00-18:00 and 20:00-00:00. A shift cannot start

outside one of these intervals. For example, a shift

could start at 10:00 but not 10:15.

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

228

C10: Minimum Cover Requirements - The

minimum number of required staff for every time

interval and every activity must be satisfied. These

requirements are specified in the instance data files.

2.2 Objective Function

The objective is to minimise assigning more staff that

is required for each activity for every interval. It is

modelled as a quadratic function to ensure that over-

assignment is spread out over the planning horizon

rather than occurring in a small number of intervals,

which could happen with a linear function.

A maximum cover requirement is given for every

activity and every interval. If more than the maximum

required number of staff at the specified time interval

for the specified activity is assigned then a penalty

cost is added to the objective function value: If the

number assigned (x) is more than the maximum

required number then the penalty for that activity and

time interval is:

(x - max) * (x - max) * weight

The weight for all instances is 1.

The solution's total penalty is the sum of all the

penalties for every time interval and activity

requirement.

Figure 3: Screenshot of verifier.

2.3 Solution Verification

To ensure the accuracy of new computational results

a verifier has been made available (screenshot in

Figure 3). The solutions can be saved in a defined

XML format. These XML files can be read by and

opened using the verifier. The verifier will display the

objective function as well as displaying any errors or

constraint violations that may have been accidentally

introduced. This will help researchers to verify their

solutions and identify any errors if their solution does

not match the verifier’s calculated objective function.

The solution visualisation may also be useful in

designing and testing new algorithms. The verifier is

available at www.schedulingbenchmarks.org.

3 VARIABLE NEIGHBOURHOOD

The variable neighbourhood search (VNS) uses four

different neighbourhood operators. Each operator is

applied to the solution in an iterative process until the

solution is a local optimum with respect to all four

operators. Once the local optimum is reached, all

shifts are un-assigned and the process is repeated.

When the time limit is reached the solution with the

best local optimum is returned.

To make the search space more connected and to

assist the neighbourhood search to reach better local

optima, all constraints except C1, C5 and C9 are

relaxed and made soft constraints but with very high

weights. The search neighbourhood operators are as

follows:

N1: For each shift, try replacing it with a new

shift. When creating the new shift, every possible

start time and shift length is tried and each

combination of three different activities within the

shift is tried.

N2: For each shift, test making it up to 30 minutes

longer or shorter and up to 30 minutes earlier or later

and simultaneously making another shift for the same

employee on a different day up to 30 minutes longer

or shorter and up to 30 minutes earlier or later.

Similarly to neighbourhood 1, it will also test each

combination of three different activities within the

shift

N3: For each day, for each pair of employees, try

swapping the shifts assigned between the two

employees on that day. This is repeated for that day

but also simultaneously swapping over the next x

consecutive days where x is 1..4.

N4: For each shift, try moving it up to one hour

earlier or later and up to 15 minutes shorter or longer

and simultaneously moving another shift for another

employee on the same day up to one hour earlier or

later and up to 15 minutes shorter or longer. When

moving the first shift, it will also test different

combinations of activities within the shift. For the

second shift it will keep the activities within them the

same and the absolute times of the activity changes.

Solving the Multi-activity Shift Scheduling Problem using Variable Neighbourhood Search

229

4 RESULTS

The algorithm was run for 10 minutes on each

instance. The best solution found after 5 minutes and

10 minutes on each instance is listed in Table 1. If no

feasible solution was found in the time allowed, then

the row contains ‘-’. The experiments were conducted

on an Intel Core i5-4690K CPU 3.50GHz.

The VNS method can find feasible solutions for

most of the smaller instances within 5 minutes. An

extra five minutes further improves the solutions on

some instances and results in solutions for some other

instances that could not be solved within 5 minutes.

The larger instances with more staff, more activities

or longer planning horizons appear to be more

difficult to solve and often a feasible solution could

not be found within the time provided.

5 CONCLUSIONS

New benchmark instances have been introduced for

the multi-activity shift scheduling problem. They are

publicly available for download from

www.schedulingbenchmarks.org. A verifier with a

graphical user interface is also available to validate

new results and assist with development. A Variable

Neighbourhood Search that uses four different

neighbourhoods has also been presented. It can find

feasible solutions to the smaller and medium sized

instances in relatively short computation times.

Future research should focus on methods for solving

the larger instances. Data sets which also consider

break scheduling and task scheduling within the shifts

may also be introduced in the future.

REFERENCES

Dahmen, S., & Rekik, M. (2015). Solving multi-activity

multi-day shift scheduling problems with a hybrid

heuristic. Journal of Scheduling, 18(2), 207–223.

Gérard, M., Clautiaux, F., & Sadykov, R. (2016). Column

generation based approaches for a tour scheduling

problem with a multi-skill heterogeneous workforce.

European Journal of Operational Research, 252(3),

1019–1030.

Loucks, J. S., & Jacobs, F. R. (1991). Tour scheduling and

task assignment of a heterogeneous work force: A

heuristic approach. Decision Sciences, 22(4), 719–738.

Pan, S, Akplogan, M., Létocart, L., Touati, N., & Calvo, R.

W. (2016). Solving a Multi-Activity Shift Scheduling

Problem with a Tabu Search Heuristic. PATAT 2016:

Proceedings of the 11th International Conference of the

Practice and Theory of Automated Timetabling, 317–

326.

Pan, Stefania, Akplogan, M., Touati, N., Létocart, L.,

Calvo, R. W., & Rousseau, L.-M. (2018). A hybrid

heuristic for the multi-activity tour scheduling problem.

Electronic Notes in Discrete Mathematics, 69, 333–

340.

Quimper, C.-G., & Rousseau, L.-M. (2010). A large

neighbourhood search approach to the multi-activity

shift scheduling problem. Journal of Heuristics, 16(3),

373–392.

Restrepo, M. I., Gendron, B., & Rousseau, L.-M. (2016).

Branch-and-price for personalized multiactivity tour

scheduling. INFORMS Journal on Computing, 28(2),

334–350.

Restrepo, M. I., Gendron, B., & Rousseau, L.-M. (2018).

Combining Benders decomposition and column

generation for multi-activity tour scheduling.

Computers & Operations Research, 93, 151–165.

Restrepo, M. I., Lozano, L., & Medaglia, A. L. (2012).

Constrained network-based column generation for the

multi-activity shift scheduling problem. International

Journal of Production Economics, 140(1), 466–472.

Salvagnin, D., & Walsh, T. (2012). A hybrid MIP/CP

approach for multi-activity shift scheduling.

International Conference on Principles and Practice of

Constraint Programming, 633–646. Springer.

APPENDIX

Table 1: Results.

Instance Days Staff Tasks 5 mins 10 mins

1 7 10 1 387 387

2 7 10 1 192 176

3 7 10 1 317 317

4 7 10 1 330 328

5 7 10 2 115 115

6 7 20 1 900 900

7 7 20 1 879 818

8 7 20 2 884 884

9 7 20 2 513 500

10 7 20 3 274 268

11 7 30 1 909 844

12 7 30 2 1541 1541

13 7 30 2 1440 1440

14 7 30 3 1476 1469

15 7 30 4 553 553

16 7 40 2 1946 1883

17 7 40 2 1831 1831

18 7 40 3 1737 1737

19 7 40 4 1437 1437

20 7 40 5 990 955

21 7 50 2 1740 1740

22 7 50 3 2646 2646

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

230

Table 1: Results (continue).

23 7 50 4 2446 2446

24 7 50 5 1795 1795

25 7 50 7 - 1344

26 7 60 2 1734 1734

27 7 60 3 2904 2904

28 7 60 4 3276 3248

29 7 60 6 2463 2463

30 7 60 8 - -

31 7 70 3 2574 2574

32 7 70 4 3288 3288

33 7 70 5 3170 3170

34 7 70 7 - -

35 7 70 9 - -

36 7 80 3 2858 2709

37 7 80 4 3335 3335

38 7 80 6 3894 3894

39 7 80 8 - -

40 7 80 10 - -

41 7 90 3 2575 2575

42 7 90 5 4317 4317

43 7 90 6 4971 4877

44 7 90 9 - -

45 7 90 12 - -

46 7 100 4 3471 3471

47 7 100 5 5139 4837

48 7 100 7 5492 5302

49 7 100 10 - -

50 7 100 13 - -

51 7 110 4 3338 3338

52 7 110 6 5084 5084

53 7 110 8 6291 6237

54 7 110 11 - -

55 7 110 14 - -

56 7 120 4 3486 3486

57 7 120 6 5991 5991

58 7 120 8 - 6749

59 7 120 12 - -

60 7 120 15 - -

61 7 130 5 4932 4932

62 7 130 7 - 6720

63 7 130 9 - 7086

64 7 130 13 - -

65 7 130 17 - -

66 7 140 5 4057 4057

67 7 140 7 6009 6009

68 7 140 10 - -

69 7 140 14 - -

70 7 140 18 - -

71 7 150 5 4063 4063

72 7 150 8 - 7590

73 7 150 10 - -

74 7 150 15 - -

75 7 150 19 - -

76 14 10 1 604 598

77 14 10 1 814 814

78 14 10 1 634 634

79 14 10 1 607 607

80 14 10 2 292 292

81 14 20 1 1659 1659

82 14 20 1 1643 1643

83 14 20 2 1387 1387

84 14 20 2 1279 1168

85 14 20 3 520 520

86 14 30 1 1738 1738

87 14 30 2 2672 2672

88 14 30 2 2780 2780

89 14 30 3 2551 2551

90 14 30 4 - -

91 14 40 2 3514 3514

92 14 40 2 3767 3767

93 14 40 3 3942 3820

94 14 40 4 - 3980

95 14 40 5 - -

96 14 50 2 3666 3666

97 14 50 3 4921 4921

98 14 50 4 4802 4802

99 14 50 5 - -

100 14 50 7 - -

101 14 60 2 3419 3419

102 14 60 3 5473 5473

103 14 60 4 5942 5942

104 14 60 6 - 5620

105 14 60 8 - -

106 14 70 3 5137 5137

107 14 70 4 7208 6892

108 14 70 5 - -

109 14 70 7 - -

110 14 70 9 - -

111 14 80 3 - 5510

112 14 80 4 6748 6748

113 14 80 6 - 8124

114 14 80 8 - -

115 14 80 10 - -

116 14 90 3 5715 5598

117 14 90 5 - 8818

118 14 90 6 - -

119 14 90 9 - -

120 14 90 12 - -

121 14 100 4 - -

122 14 100 5 - -

123 14 100 7 - -

124 14 100 10 - -

125 14 100 13 - -

Solving the Multi-activity Shift Scheduling Problem using Variable Neighbourhood Search

231

Table 1: Results (continue).

126 14 110 4 - 7573

127 14 110 6 - -

128 14 110 8 - -

129 14 110 11 - -

130 14 110 14 - -

131 14 120 4 - 7475

132 14 120 6 - -

133 14 120 8 - -

134 14 120 12 - -

135 14 120 15 - -

136 14 130 5 - -

137 14 130 7 - -

138 14 130 9 - -

139 14 130 13 - -

140 14 130 17 - -

141 14 140 5 - 8013

142 14 140 7 - -

143 14 140 10 - -

144 14 140 14 - -

145 14 140 18 - -

146 14 150 5 - -

147 14 150 8 - -

148 14 150 10 - -

149 14 150 15 - -

150 14 150 19 - -

151 28 10 1 1677 1677

152 28 10 1 1509 1509

153 28 10 1 1729 1729

154 28 10 1 1535 1535

155 28 10 2 - -

156 28 20 1 3766 3766

157 28 20 1 3541 3523

158 28 20 2 3327 3327

159 28 20 2 2989 2989

160 28 20 3 - 1803

161 28 30 1 3513 3505

162 28 30 2 - 6551

163 28 30 2 6209 6209

164 28 30 3 - -

165 28 30 4 - -

166 28 40 2 - 7613

167 28 40 2 7317 7317

168 28 40 3 - 8270

169 28 40 4 - -

170 28 40 5 - -

171 28 50 2 7000 6843

172 28 50 3 - -

173 28 50 4 - -

174 28 50 5 - -

175 28 50 7 - -

176 28 60 2 7179 7179

177 28 60 3 - -

178 28 60 4 - -

179 28 60 6 - -

180 28 60 8 - -

181 28 70 3 - -

182 28 70 4 - -

183 28 70 5 - -

184 28 70 7 - -

185 28 70 9 - -

186 28 80 3 - 11181

187 28 80 4 - -

188 28 80 6 - -

189 28 80 8 - -

190 28 80 10 - -

191 28 90 3 - -

192 28 90 5 - -

193 28 90 6 - -

194 28 90 9 - -

195 28 90 12 - -

196 28 100 4 - -

197 28 100 5 - -

198 28 100 7 - -

199 28 100 10 - -

200 28 100 13 - -

201 28 110 4 - -

202 28 110 6 - -

203 28 110 8 - -

204 28 110 11 - -

205 28 110 14 - -

206 28 120 4 - -

207 28 120 6 - -

208 28 120 8 - -

209 28 120 12 - -

210 28 120 15 - -

211 28 130 5 - -

212 28 130 7 - -

213 28 130 9 - -

214 28 130 13 - -

215 28 130 17 - -

216 28 140 5 - -

217 28 140 7 - -

218 28 140 10 - -

219 28 140 14 - -

220 28 140 18 - -

221 28 150 5 - -

222 28 150 8 - -

223 28 150 10 - -

224 28 150 15 - -

225 28 150 19 - -

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

232