APIEMS2009
Dec. 14-16, Kitakyushu
Rostering Ambulance Services Yuan Li 1 and Erhan Kozan † 2 School of Mathematical Sciences, Queensland University of Technology Brisbane, Queensland, 4001, AUSTRALIA Email:
[email protected] [email protected] Abstract. This paper developed a model for rostering ambulance crew in order to maximise the coverage throughout a planning horizon and minimise the number of ambulance crew. Rostering Ambulance Services is a complex task, which considers a large number of conflicting rules related to various aspects such as limits on the number of consecutive work hours, the number of shifts worked by each ambulance staff and restrictions on the type of shifts assigned. The two-stage models are developed using nonlinear integer programming technique to determine the following sub-problems: the shift start times; the number of staff required to work for each shift; and a balanced schedule of ambulance staff. At the first stage, the first two sub-problems have been solved. At the second stage, the third sub-problem has been solved using the first stage outputs. Computational experiments with real data are conducted and the results of the models are presented. Keywords: rostering, ambulance services, scheduling
1. INTRODUCTION Ambulance Services are 24/7 services and require ambulance staff to work on night shifts, on weekends and even on public holidays. The major significance of the Ambulance Services is the increase in the level of demand for acute cases. This increasing demand has put pressure on resources and has led to deterioration in response time. Therefore, a good schedule for ambulance staff can help to reduce their fatigue levels and increase the efficiency of Ambulance Services performance. The objective of this paper is to develop analytical models to analyse staffing at Ambulance Services and determine the matching of personnel resources. The models are developed using nonlinear integer programming technique. The impact on the resources required in relation to a reduction or increase in the number of shifts is investigated as well. Firstly, a deterministic model is developed for determining shift start time and necessary number of ambulance staff to be assigned to each shift. Secondly, an allocation model is developed to assign all ambulance staff to proper shifts and so generate a monthly (four weeks) schedule. They are solved using optimisation software Lingo. The rostering models for Emergency Department have been paid more and more attention in the Operations Research area over the past half century. Ernst et al. (1999) described a number of network algorithms that were used to develop rosters for ambulance officers, such as shortest path algorithm and alternative network algorithm for cyclic
† : Corresponding Author
795
rosters. The most relevant research conducted into ambulance problems was conducted in Canada by Trudeau et al. (1988). A mathematical model was developed to simulate all emergency operations. Demand forecasting, staff scheduling and operations strategy were concentrated as well as the cost versus service problem. For demand forecasting they used a simple statistics with the day split into small time blocks to account for variations. Scheduling was then implemented using the above forecasts with segments reflecting differing demands. From this point the simulation model was created to account for the actual dispatch and service with the goal of minimising costs for a given level of service given socially acceptable constraints. They found several ‘good’ solutions yet an optimal could not be found. The most recent paper of scheduling ambulance crew is Erdogan et al. (2008). One ambulance location model and two Integer Programming models were constructed to find the maximum coverage. For solution method, a tabu search was employed and it produced better approaches than previous works in their literature. An investigation into the entire emergency dispatch and service model was recently conducted in Chile by Weintraub et al. (1999). A simulation model was developed of the city and its subsequent emergency services and emergency locations. They incorporated several features such as priority calls and made significant improvements to service time especially in adverse conditions when reports were highest. A research conducted by Felici and Gentile (2004)
APIEMS2009
Dec. 14-16, Kitakyushu
considered a very general weekly staff scheduling problem. A pure 0-1 model was transformed from the directly built up Integer Programming model after introducing a new binary variable to associate with the original one. This pure 0-1 model can fit to all the staff rostering problems that defined like this: in order to maximise the staff satisfaction, determine a weekly schedule for all the staff members that satisfies constraints derived from their work agreement and minimizes the daily demand. A Branch and Bound algorithm have been designed to solve the pure 0-1 model and an interesting discovery is the algorithm always providing good results where commercial solvers fail. Real world problems, however, are too complex to be optimized, many authors employ heuristics, e.g. Aickelin and Dowsland (2000), Bailey et al. (1997), Barnhart et al. (1998), Brusco and Jacobs (1993), Cai and Li (2000), Gendreau et al. (1997), Gutjahr and Rauner (2007) , Mason et al. (1998) and Monfroglio (1996). Ernst et al. (2004) summaries all the application areas of staff scheduling and rostering that have been studied before 2004 and their solution methods that have been applied.
2. TWO-STAGE SHIFT SCHEDULING MODELS
Objective function:
Constraints: The demand of ambulance staff at period p must be less than or equal to the number of staff who are working at period p. This can be formulated using the if-then constraints by summing the products of the shifts that cover period p and the number of ambulance staff assigned to that corresponding shift. If a shift works in period p then q is the starting period of that shift and finishes at the beginning of p+1 after periods can be formulated as follows:
where
The two-stage mathematical programming models were developed to solve the ambulance staff rostering problem. The first stage is to decide the shift start times and the second stage assigns all the ambulance staff to proper shifts so that all the requirements can be satisfied.
2.1 A Shift Start Times Determination Model
is a binary number and M is a large positive
number. There should be no gap between any two consecutive shifts, for example, for a hp hours three shifts per day problem, if there is one shift starts at time p, then there must be another shift starts before the end of this shift. And this constraint can also be formulated using the following If Then constraint:
The objective of this model is to minimize the volatility between ambulance supply and demand. Notations of the models is as follows given below: Notations: There must be enough shifts to be assigned to cover every day. The possible shift durations are bounded between 8 hours and 12 hours; this is why the number of shifts everyday can only be three or two.
The shift start time on each day will be the same.
The total working hours cannot exceed the existing resources.
796
APIEMS2009
where
Dec. 14-16, Kitakyushu
is the upper bound of the total
number of available working . The decision variable
Constraints: An ambulance staff needs to work (n/7) hours in a working period, where is the average working hours per week and n is the number of working days, thus
is binary variable that can
only take the values 0 or 1.
Note that there is a link between stage one and stage is the demand of ambulance crew at the end of time two. and it is provided by an Ambulance Services station. and
will be generated from stage one.
find the shifts’ start period
Hence, we could
To keep the fatigue level of ambulance staff at a low level, firstly, the ordinary time and likely overtime ( ) combined should were possible not exceed hours. Because the responsibility of the ambulance staff is lift saving, so the risk of the unreasonable consecutive hours working is patients’ lives. Therefore, Equation 10 ensures that the working hours is less than .
and the number of ambulance Equation 11 ensures that consecutive night shifts must not exceed .
staff should be assigned to be on duty for each shift. 2.2 An Ambulance Staff Allocation Model Due to stage one and two are built separately, the parameter notations are not related. Hence the parameters of stage two are not affected by the ones of stage one, the inner relation between stage one and two is the output of the first stage will be used as the input of stage two.
Equation 12 ensures that consecutive roster days should not exceed the consecutive days .
Notations:
Equation 13 ensures that maximum number of working hours worked continuously should not exceed hours.
The set of shifts; Days Set of ambulance crew. The number of hours of shift on day . Maximum consecutive roster days. Maximum continuously worked hours. Maximum overtime. Maximum consecutive night shifts.
The model has to make sure enough ambulance staff is assigned to each shift to satisfy the staff requirements.
Equation 15 ensures that ambulance staff cannot work more than one shift per day. Objective function: After we got the staff need to be assigned on each shift, we minimize the total number of working units at stage two in order to minimize the total cost.
797
APIEMS2009
Dec. 14-16, Kitakyushu
3. CASE STUDY
3.1 The Computational Results of the First Stage
Demand Profiles are designed to aid managers in assessing the effectiveness of staff placements and existing or proposed rostering arrangements. They may also assist in the preparation of business cases for either allocation of additional resources or transfer of resources from areas of declining workload to areas of greater need. The demand profile derived from a database of an ambulance station and summarized below: rosters must reflect average 40 hours/week over the roster cycle; ordinary time and likely overtime combined should were possible not exceed 12 hours total per shift; consecutive night shifts must not exceed 2; shift length for Nightshifts must not exceed 12 hours and considered as one shift; rostered days should not exceed 5 consecutive shifts; maximum number of hours worked continuously should not exceed 50 hours; and minimise administrative disruption during periods of annual leave and reduce prolonged periods of work. hourly demand profiles are known (because of the confidentiality, it will not be given); the minimum level of rostering at the station is derived from the demand profiles; the average dispatch-clear time; and the number of responses and the availability of units by hour of day during a year. In order to implement the two-stage models, it is necessary to generate the number of ambulance staff required at the end of the each period ( in stage one). Therefore, we can determine the number of ambulance staff should be assigned on to the duty for each shift and the shifts start times.
To implement the two-stage models for the case study, 18 shift patterns that start from 7am, 8am,…,24am have selected. It is required that there must be no shift starts at 1am, 2am,…, or 6am. Also, the shifts preferred not to be finished at 0am, 1am,…,5am. The shift end time, however, are dependent on the duration of each shift, i.e. the working hours. In this case study the possible duration of a shift is 8, 10 or 12 hours based on the current rosters of the Ambulance Station. We consider some cases with different working hours, the case involving the minimum number of ambulance staff is the recommended case. Result of a case for γ = 7 is provided in Table 1 and Table 2.
In stage 2, the number of ambulance staff on duty for each shift from Table 2 is used to generate a monthly roster for the Ambulance station for alternative cases. One of these results is presented in Table 3 as an example. As shown in Table 3 the monthly schedule for 50 ambulance crew worked for an Ambulance Station has been established. The duration of shift time of this schedule used the one in case one, i.e. 10 hours and 12 hours combination shifts. All the schedules for the other cases can easily simply change the requirement of units for each shift on each day. Table 3 is as an example here to show what the schedule should look like. The computational time of this programming running in Lingo is around five minutes which is reasonably shorter than the schedule generated manually. The running time is depending on the constraints we considered in the programming.
Table 2: Minimum daily staff requirements
Table 1: Alternative roster starting and finishing times Cases
3.2 The Computational Results of the Second Stage
Morning
Afternoon
Night
Cases
Mon
Tue
Wed
Thu
Fri
Sat
Sun
8 hrs
7:00~15:00
15:00~23:00
23:00~7:00
8 hrs
25
26
27
26
28
25
24
10 hrs
7:00~17:00
13:00~23:00
22:00~8:00
10 hrs
25
25
26
25
26
25
24
12 hrs
7:00~19:00
na
19:00~7:00
12 hrs
18
19
18
17
20
18
16
8&10hrs
8:00~16:00
15:00~23:00
23:00~9:00
8&10hrs
25
25
26
25
27
25
24
8&12hrs
8:00~16:00
15:00~23:00
20:00~8:00
8&12hrs
25
25
26
25
27
25
24
10&12 hrs
8:00~18:00
10:00~20:00
20:00~8:00
10&12 hrs
26
25
25
23
23
22
22
na: not applicable
798
APIEMS2009
Dec. 14-16, Kitakyushu
Table 3: An Example Schedule for ambulance staff Staff
Week1 1
1
2
A
2
A
3
N
4
M
M
5
M
A
6 7
3
M
8
Week2
4
5
N
N
7
1
2
M
M
M
A
N
M A A
M
M
A
N
M
M
A M
A
A
M
M
M
10
M
M
M
M
M
11
A
M
M
M
12
M
M
M
M
A
13
M
M
A
14
M
M
M
M
A
M M
M A
M
M
A M
16
M
M
17
M
M
M
M
M
A
18
M
M
M
M
M
A
19
M
N
20
M
M
M
M
21
M
M
M
22
M
M
M
23
M
M M
24
M
A
M
A
A
M
M
M
M
M
M
26
M
M
27
M
28
M
29
A
30
M
A
A
N
31
M
M
M
A
32
M
M
N
33
N
N
M
N M
M
A
N
M
M
M
M M N
A
N
M
A
N
A
N
M
M
35
M
M
M
36
A
M
A
37
N
M
A
M
M
M
N
N
41
M
42
N
43
M
M
44
M
N
A
N
45
N
N
M
A
46
N
47 48
N
49
A
50
A
N
N
N
A
M
M
M
N
A N
A
A
N
N
N
M
M
N
N
A A
M
N M
N
N
M
A
M
M
M
M
M
A
M
A
A
M
M
A
M
M
M
M M
M
M
M
A
M
M
A
A
M
M
M
A
M
M
A
M
A
M
M
M
M
A
M
M
A
A
M
M
M
N
M
M
A
M
M
N
A
A
N
N
N
N
A
N
N
M
A
M
N
M
A
N
A
A
M
N M
A
M
N N
N N
A
M
M
N
N
N
N
N A
M
M
M
N
N
N
A M
M
M: Morning shift; A: Afternoon shift; N: Night shift; Empty cell: off period.
799
N
N N
N
N
A
M M
N
M
A
M
A
A
A
A
A
N
M M
A
A M
N
A M M
M
A N
N
A
A M
A
N
A
A
M
M
N
M
A
M
A
M
N
M
M
A
N
M
N
A
A
A
N
M
A
M
A
A
A N
A
M
N
M N
M
A
M
M
M N
N
M
N
A
A
A
N
A
M
M
M
M
A M
N
A
A
N
M
A
A
M
M
M
A
M M
M
M
M A
N M
A
N
M N
A A
N
A
M
M
N
N
N N
M
A
A
A
A
A
A
A
M
M
A
A
M
M
M
N
M
M
A
M
M
M
A
M
N
M
M
M
M
M
A
M
M
M
A
A A
M A
M
A
M
M M
M
N
M M
N
N
N
M M
M
A
M
M
N M
M M
M
M
A
M
M
M
A
M
M
7
M M
M
M
A
N
A
N
M
N
A
M
6
N
A M
A
5
a
M
M
N
A N
M
N
A
M
M
M
N
M
A
M
M
A
M
M
M
M
A
M
M
M
A
A
M
A
A N
M
M
M M
A
M
A
M
N
M
M
A
A
A
N
A
N
M
M
A
M
N M
M
M
M
N
A M
M
M
N
M
A
A
M
M
A
M
M
N M
M N
A
N
A
40
A
M
N
A
N M
M
M
A
N
N
M
A
N
A
34
M
A
M
A
A
A
A
A
M
A
M
M
A
M
39
N
A
A
A M
N
N
A
N
M
A
M
M
N
A
M
M
M
A
M
M
M
N
M
M
M
N
M
M
N
A A
M
M
M M
N
M
A M
M
N
M
M
A
N
A
M
A
4
M N
M
N
M
3
M
M
A
M
2
N
M
N
M N
1
M
M
M
A
7
M
M
A
N
6
M
M
A
A M
Week4 5
A
M
N
4
A
M
M
3
A
A
A
A
A
N
A
A
M
A
A
M
M
2
M
M
A
38
M
1
A
A
25
M
M
N
N N
M A
A
M
A
M M
M
7
A
A
M
6 N
M
M
M
Week3 5 M
A
M
M
A
M
M
4
N
M
M
A
3
M M
9
15
6
A
M
A
M
N
N
A
N
N
N
N
A N
A
N M
M N
M
N M
N
M
A A
N
M N
N
M A
M A
M
A
A
M
A
N
M
M
M
N N
N
M M
M N
N N
M
A N
APIEMS2009
Dec. 14-16, Kitakyushu
4. CONCLUSIONS The challenge here is to integrate the separate steps in the rostering process into one single problem. And consider how to adjust a new rostering problem to a well developed problem. The problem will become NP-hard when we merge the two stages. In addition to this, the stochastic nature of the problem, e.g. sickness of the staff, delay of the services etc will make the so more complex. Another improvement is generalisation of models and techniques. At the current stage, it is difficult to transfer models or algorithms of one application area to another one. It is desirable that new formulated models are more flexible to accommodate individual workplace practices. None of the existing rostering softwares is flexible enough for all the users due to the inflexibility of rostering algorithms. In order to make a model more accessible most of the users, the effort should be put on the way to generate general algorithms. Most of the rostering problems did not consider personal preferences. But like our rostering Ambulance Services problem, adding personal preferences will change the problem to be an NP-hard problem. Only a roster that considers the rules of institution or organisation and the individual preferences is a complete roster. Like we mentioned in the introduction section, many authors employed heuristics methods to get near optimal solutions. This is also our future effort to make schedules we generated so far become smooth schedules.
REFERENCES Aickelin, U. and K. A. Dowsland. 2000. Exploiting problem structure in a genetic algorithm approach to a nurse rostering problem. Journal of Scheduling: 139-153. Bailey, R. N., K. M. Garner and M. F. Hobbs. 1997. Using simulated annealing and genetic algorithms to solve staff-scheduling problems. Asia-Pacific Journal of Operational Research: 27-43. Barnhart, C. E., L. Johnson, G. L. Nemhauser, M. W. P. Sacerlsbergh and P. H. Vance. 1998. Branch-and-Price: Column generation for solving huge integer programs. Operations Research 46(3): 316-329. Brusco, M. J. and L. W. Jacobs. 1993. A simulated annealing approach to the solution of flexible labour scheduling problems. The Journal of the Operational Research Society: 1191-1200. Cai, X. and K. N. Li. 2000. Theory and methodology: A genetic algorithm for scheduling staff of mixed skills under multi-criteria. European Journal of Operations Research: 359-369. Erdogan, G., E. Erkut, A. Ingolfsson and G. Laporte. 2008. Scheduling ambulance crews for maximum coverage. Journal of the Operations Research Society.
800
Ernst, A. T., P. Hourigan, M. Krishnamoorithy, G. Mills, H. Nott and D. Sier. 1999. Rostering ambulance officers. Proceeding of the 15th National Conference of the Australian Society for Operations Research: 470-481. Ernst, A. T., H. Jiang, M. Krishnamoorthy and D. Sier. 2004. Staff scheduling and rostering: A review of applications, methods and models. European Journal of Operational Research, 3-27. Felici, G. and C. Gentile. 2004. A polyhedral approach for the staff rostering problem. Management Science 50(3): 381-393. Gendreau, M., G. Laporte and F. Semet. 1997. Solving an ambulance location model by tabu search. Location Science 5(2): 75-88. Gutjahr, W. J. and M. S. Rauner. 2007. An ACO algorithm for a dynamic regional nurse-scheduling problem in Austria. Computers and Operations Research: 642-666. Mason, A. J., D. M. Ryan, and D. M. Panton. 1998. Integrated simulation, heuristic and optimisation approaches to staff scheduling. Operations Research 46(2): 161-175. Monfroglio, A. 1996. Hybird genetic algorithms for a rostering problem. Software-Practive and Experience: 851-862. Trudeau, P., J. M. Rousseau, J. A. Ferland and J. Choquette. 1998. An operations research approach for the planning and operation of an ambulance service. INFOR. 27: 95-114. Weintraub, A., J. Aboud, C. Fernandez, G. Laporte and E. Ramirez. 1999. An emergency vehicle dispatching system for electric utility in Chile. Journal of the Operations Research Society 50: 690-696.
APIEMS2009
Dec. 14-16, Kitakyushu
AUTHOR BIOGRAPHIES Erhan Kozan is a Chair Professor of Operations Research, in the School of Mathematical Sciences at Queensland University of Technology, Australia. He has had 35 years industrial, managerial, teaching and research experience in the areas of Operations Research. He worked with the World Bank Group and the United Nations Development Program. He is the National President of the Australian Society for Operations Research and the President of the Asia Pacific Industrial Engineering and Management Society. Professor Kozan has acted as principal investigator for over 26 long-term industrial projects, and 18competitive national and international research grants in the area of health, finance, production, railways and seaports transportation. He is the author of a book, nine softwares and over 165 articles. He is the editor and associate editor of eight journals. He has supervised over 33 postgraduate research students. He is currently supervising six PhD students in the health and transportation area. Yuan Li is a PhD student supervised by Professor Kozan. She received a Bachelor of Mathematics Degree and a Bachelor of Applied Science (Honours) Degree from the School of Mathematical Sciences at Queensland University of Technology, Australia in 2007 and 2008. Now she is studying in the health area with Professor Kozan.
801
QUT Digital Repository: http://eprints.qut.edu.au/
Li, Yuan and Kozan, Erhan (2009) Rostering ambulance services. In: Industrial Engineering and Management Society, 14-16 December 2009, Kitakyushu International Conference Center, Kitakyushu, Japan.
©
Copyright 2009 The Korean Institute of Industrial Engineers
