CC BY
28
13. Richards. Shock waves on the highway. Operations Re- search, 4:42-51, 1956
14. Roess, W.R. McShane, and E.S. Prassas. Traffic Engineering: Second Edition. Prentice Hall, Upper Saddle River, NJ, 1998.
15. R. Scariza. Evaluation of Coordinated and Local Ramp Metering Algorithm Using Microscopic Traffic Simulation PhD thesis, Massachusetts Institute of Technology. Dept. of Civil and Environmental Engineering, 2003.
16. Smaragdis and M. Papageorgiou. Series of new local ramp metering strategies. Transportation Research Record: Journal of the Transportation Research Board, 1856:74-86, 2003.
17. J. Taylor, P. C. Young, A. Chotai, and J. Whittaker. Non- minimal state space approach to multivariable ramp metering control of motorway bottlenecks. IEEE Proceedings: Con- trol Theory and Applications, 145(5):568-574, 1998.
Boumediene Kamel
University of Artois - Technoparc Futura.
E-mail: kamel boume2002@yahoo.fr
62408 BETH U N E CE D E X
Laboratoire de Genie Informatique et d’Automatique de l’Artois (LGI2A E.A. 3926).
Amar Benasser
E-mail: amar benasser@univ-artois. fr
Daniel Jolly
E-mail: daniel.jolly@univ-artois.fr
УДК 71-57
Issam Nouaouri, J. Christophe Nicolas, Daniel Jolly SIZING OPERATING ROOMS IN CASE OF A DISASTER PLAN
In case of a disaster, the need for medical and surgical treatments overwhelms hospitals ’ capabilities with respect to standard operating procedures. In this paper, we deal with the preparation phase of the disaster management plan. We focus on the sizing activity of emergency resources, more precisely on operating rooms. So, we propose integer linear programming model. This model provides the optimal number of operating rooms that best respond to mass casualty events such that all victims are treated. Computational experiments performed by the Cplex solver show that a substantial aid is proposed by using this model in hospital disaster management.
Integer programming) Disaster plan, Sizing; operating rooms.
1. Introduction. The annual report of the International Federation of Red Cross and Red Crescent Societies proves that the national societies were impacted by 429 different disasters or crises in 2006. This shows an increase of 22 % from 2005 and 47 % from 2003 (Ghezail and al., 2007). Such an incident, affects hospitals of all sizes and geographic locations. Different countries require that their hospitals have plans for emergency preparation and disaster preparedness. For example, in the USA, the Joint Commission on the Accreditation of Healthcare Organizations requires US hospitals to have a disaster management plan (DMP). In other countries, like in France and in Tunisia, state requirements or laws impose each hospital to have a disaster plan so called white plan (Ministere de la sante et de la solidarite, 2006) (Ministere de la sante pub-lique, 2002).
Any emergency management plan must address the following phases: preparation, response, and recovery (Kimberly and al., 2003). The preparation phase is considered as the driving force behind a successful response. Indeed, it is vital to have a strong framework to activate in case of a disaster (Lipp and al., 1998). It includes all emergency preparedness activities such as defining medical and technical supplies, maintaining accu-
rate contact lists of the involved actors and conducting regular exercises with various disaster scenarios. This phase allows the hospital to be able to estimate the additional resources that may be needed in a given disaster situation, to keep adequate supplies on hand and to establish employees’ emergency planning. This is not an easy undertaking in any hospital setting (Kimberly and al., 2003) (Lipp and al., 1998).
Sizing problems are frequently encountered in hospitals, in order to optimize critical resources (operating rooms, medical staffs, beds,...) (Kusters and Groot, 1996) (Wang and al., 2008) and to satisfy health care requirements. Different works exist in the literature. They are based on four solving approaches: Markov chains, mathematical programming, queuing theory and simulation. Markov chains have been used by (Kao, 1974) to analyse the flows of patients among the different care units and therefore to assess their stay time in the hospital. The requirement on resources such as nurses’ staff and hospital beds are then determined. Several works based on mathematical programming models have been also proposed (Vissers, 1994). (Teow and Tan, 2007) presents a two-stage programming model. The first stage involves the sizing of beds by room configurations while the second stage assigns beds to physical locations with respect to operational considerations such as physical spanning and nurses’ training. (Lapierre and al., 1999) and (Murray, 2005) deal with bed sizing and planning according to the number of admitted patients. (Kusters and Groot, 1996) uses the dynamic programming for the sizing of hospital beds while taking into account the operating room planning, the availability of nurses’ staff, the arrival and exit patterns of patients. (Gorunescu and al., 2002) (Kao and Tung, 1981) and (Mackay, 2001) address the bed sizing problem with respect to different room configurations. The authors use queuing theory. Simulation has been employed in health care systems (Dumas, 1984) (Harper and Shahani, 2002) (Kim and Horowitz, 2002) (Ridge and al., 1998) with the purpose of a better understanding and analysis of care activities to evaluate the decisions that will be taken.
In this paper, we deal with resources optimisation in emergency preparedness. We focus more precisely on operating rooms and surgical staffs. Our purpose here is to find the optimal schedule of surgical cares that minimises the number of operating rooms needed to treat all persons injured by the disaster. To achieve this, we propose an integer-programming model.
The remainder of this paper is organized as follows. Section 2 describes the sizing problem we address. Section 3 details the integer-programming model we propose. Section 4 discusses the obtained numerical results. Section 5 concludes the paper and presents possible extensions of this work.
2. Problem description. In case of a disaster, victims are evacuated from the damaged zone to the nearby hospital. The triage allows classifying victims according to the urgency of the medical and/or surgical cares they need. In this paper, we consider victims that require surgical cares with predefined processing times and different ready dates in the operating theatre. Each victim is characterized by an emergency level, which is defined by the latest starting time of its surgical care. Therefore, the surgical care must be planned before the vital prognosis of the victim is being overtaken (Dhahri, 1999) (Ministere de la sante et de la solidarite, 2006).
Each hospital has a maximum of human resources that should be requested in a disaster situation. Therefore, surgical staffs, their number, configurations and ready dates in the operating theatre are detailed in a pre-established emergency planning. Each surgical staff (composed by a surgeon, an anaesthetist and nurses) is assigned to an operating room. In such a situation, all operating rooms are considered to be polyvalent. The hospital have to forecast the minimal number of operating rooms that ensure facing a given disaster situation.
The sizing problem we address is stated as follows: given a set of surgical staffs, find a schedule of surgical cares to be realized, so as all victims are treated, while minimizing the number of required operating rooms and satisfying some given constraints such as the ready dates of victims and surgical staffs.
3. Problem modelling. We propose a mathematical model. So, an integer linear programming model is developed using ILOG OPL 5.5 Studio.
Before presenting our model, we will first introduce the following notations:
N - Number of victims;
C - Number of staffs;
T - Time horizon;
p,. - Processing time of surgical care i; dlj - Latest starting time of surgical care i; dar, - Ready date of victim i;
rc - Ready date of surgical staff c with respect to the hospital emergency planning; M - Very big positive number.
Besides, we define the following decision variables: t, - Starting time of surgical care i;
f 1 if victim i is assigned to staff c at time t ’tc 10 otherwise.
1 if victim i is treated before victim j by staff c otherwise.
1 if staff c is assigned to an operating room
Уi]c I 0 otherwise.
NO„ .
[0 otherwise
Using the notations listed above, we propose the following integer linear programming:
C
Minimize^ NOc. (1)
c
X ^Xitc = 1 V/- g{L.n). (2)
t c
N
£x„ < 1 VeM Vcep.q. (3)
i
tt < dlt Vi e{1..N}. (4)
tt > dar Vi e{1..N}. (5)
■,-r.iv„ -«|1-X*Jг0, WeM. Vce|1..C}.
N
j *i
(б)
yjc < 1, Vie|1..N}, Vce|1..C}. (7a)
N
Ё y]ic < 1, ViЄМ, Vce|1-C} (7b)
j *i'
І І У-c = І ±X,IC -1, Vc (8)
i j*i i t
ti= t titX, Vi . (9)
lj > I, + JVA -M( 1 -^W. Vi.je|1..N}. Vce|1..C}. (10)
Xltc < NOc. Vie|1..N}, Vce{1..C}. Vte {0..T}. (11)
yjc e {0.1}. Vi.je|1..N}, Vce|1..C}. (12)
XUc e {0,1}, Vi, je{1..N}, Vt e{0..T}. Vce{1..C}. (13)
NOc e{0,l], Vce|1..C}. (14)
The objective function (1) minimises the number of operating rooms used in a given disaster situation. Constraints (2) ensure that each victim must be treated only once during the horizon T. In case these constraints can not be satisfied, so the available human resources are not sufficient enough to face the disaster. The hospital is then aware of this important information and should reinforce its staffs. Constraints (3) grantee that every staff makes one surgical care at most at each time t. Constraints (4) impose to satisfy the emergency degree of each victim. Equations (5) and (6) express the availability dates of respectively victims and staffs to begin surgical cares. Constraints (7a), (7b) and (8) are disjunctive precedence constraints. Equations (9) give the starting times of surgical cares. Constraints (10) impose no overlapping between two successive cares made by the same staff. Equations (11) guarantee that an operating room is used by a staff that is assigned at least one surgical care.
4. Computational experiments. In this section, we present the computational experiments that are performed using the Cplex solver 10.1 on a pentium® 4 of 3.00 GHz processor and 504 Mo RAM. We assess the performances of the proposed two-stage optimization model in different situations described on the following.
4.1. Problem tests. Different disaster situations are considered by varying the number of victims (N=25, 50 and 70) and the durations of surgical cares (given between 30 minutes and 2 hours). Moreover, 10 staffs are available with different ready dates (R = (rI,^rC), C = 10) according to the hospital emergency planning. The different data are reported in tables 1, 2 and 3 of the appendix. The instance label PN.R means the problem P involves N victims and ready dates R of staffs. For example P50.R1 denotes a problem of 50 victims and 10 staffs which ready dates in minutes are given by R1, thus r1 = 0, r2 = 0, r3 = 0, r4 = 30, r5 = 30, r6 = 30, r7 = 60, r8 = 60, r9 = 120, r10 = 120.
The computational experiments are performed while fixing the time horizon
T = max (dli + p ). Indeed, after this date, no victim can be treated.
i =1,...,N '
4.2. Results. The results presented in Tables 4 obtained by solving the proposed integer program. For each instance, we report the CPU time, the number of constraints (N.Cont), the number of variables (N.Var.), the number of iterations (N.Iter.) and the
optimal values of objective functions denoted by NOc.
Table 4
Results
Instances NO* CPU(s) N.Cont. N.Var.
P25.R1 3 3.4 бЗ 148 21G4
P25.R2 3 3.5 бЗ 148 21G4
P25.R3 3 5.87 бЗ 148 21G4
P25.R4 3 3.G4 бЗ 148 21G4
P25.R5 3 3.5G бЗ 148 21G4
P5G.R1 5 1б.2б 572845 7GG6
P5G.R2 5 32.2б 572845 7GG6
P5G.R3 5 29.G6 572845 7GG6
P5G.R4 5 29.7G 572845 7GG6
P5G.R5 5 33.42 572845 7GG6
P7G.R1 б 31G 14G9158 13G27
P7G.R2 б 1б7 14G9158 13G27
P7G.R3 б 1бЗ 14G9158 13G27
P7G.R4 б 185 14G9158 13G27
P7G.R5 б 1б2 14G9158 13G27
Table 4 shows the minimal number of operating rooms needed to treat all victims requiring surgical cares with respect to staffs’ ready dates given by the hospital emergency planning (Table 3). This sizing allows an optimal use of available resources. For example, to treat 70 victims in time with different staffs’ ready dates, we need six operating rooms. Therefore, if the hospital possesses a greater number of operating rooms, only six rooms will be used. Consequently, the remaining rooms can be kept on hand as safety rooms more particularly in case the disaster is much more important than forecasted. Otherwise, the minimal number of needed operating rooms exceeds the number of operating rooms possessed by the hospital. In this context, the proposed model allows the hospital to estimate how many modulated operating rooms are required to be kept on hand and to update the emergency planning of its employees. As a result, the hospital can extend, if possible, its resources, to avoid the evacuation of victims to farther hospitals, thus decreasing the risk of loss of lives.
Another observation stemming from Tables 4 is that if the number of victims gets more important, the computational time increases.
5. Conclusion. In this paper, we have addressed an emergency preparedness activity tied to the dimensioning of operating rooms. The suggested approach is based on integer linear programming model.
This model yields the minimal number of operating rooms needed to treat all injured persons, while satisfying the availability of surgical staffs in the operating theatre once they are requested by the hospital. The proposed approach allows the hospital to estimate the additional supplies to be kept on hand by updating their number and configuration. Another interesting advantage is that it proposes a scheduling program for the involved surgical cares with respect to the obtained optimal sizing.
This approach has been tested on various disaster situations for which we have shown that we can find the optimal solutions. Future research works should consider this important issue and deal with auxiliary services as well as sharing critical resources.
Acknowledgments. The authors thank the university group and medical partners of the project ARC-IR (France) as well as professionals of the university hospital Charles Nicole (Tunisia) and the General Direction of the military health (Tunisia). We are particularly grateful to Dr. Mondher Gabouj, Dr. Naoufel Somrani, Dr Henda Chebbi, Pr. Eric Wiel and Dr. Cedric Goze.
The authors also want to thank the Nord Pas-de-Calais region, the European Regional Development Fund (ERDF) and The European Doctoral College that sustain these works financially.
REFERENCES
1. Dhahri M.A. (1999) “Organisation des secours medicaux dans les situations d’exception en Tunisie”, D.E.S.S de medecine d’urgence et de catastrophe, Departement d’Anesthesie et de reanimation, Hopital Militaire de Tunis.
2. Dumas M.B. (1984) “Simulation modelling of hospital bed planning”. Simulation, Vol. 43,
- P. 69-78.
3. Ghezail F., Hajri-Gabouj S. and Pierreval H. (2007) "Analyzing the consequences of disruptions on schedules: A scheduling graph", 4th International Federation of Automatic Control Conference on Management and Control of Production and Logistics, Sibiu, Romania.
4. Gorunescu F., McClean S. and Millard P.H. (2002) “A queuing model for bed-occupancy management and planning of hospitals” Journal of the operational Research society. Vol.53.
- P. 19-24.
5. HarperP.R., Shahani A.K. (2002) “Modelling for the planning and management of bed capacities in hospitals”. Journal of the operational Research society. Vol. 53. - P. 11-18.
6. Kao E.P.C. (1974) “Modelling the movement of coronary patients within a hospital by a semi-markov processes”, Operations Research. Vol. 22. - P. 683-699.
7. Kao E.P.C. and Tung G. (1981) “Bed allocation in a public health care hospital delivery system” Management science. Vol.27. - P. 507-520.
8. Kim S.C., Horowitz L. (2002) “Scheduling hospital services: the efficacy of elective-surgery quotas” Omegaint, journal Manager. Vol.18. - P. 427-443.
9. Kimberly A. Cyganik, RN, MS, ACNP, CEN (2003) “Disaster Preparedness in Virginia Hospital Center-Arlington after Sept 11, 2001” Disaster Management & Response/Cyganik, Section Editor: Ted Cieslak, MD. Vol.1. - P. 80-86.
10. Kusters R.J. and Groot P.M.A. (1996) “Modelling resources availability in general hospitals. Design and implementation of a decision support model”, European Journal Operational Research. Vol.88. - P. 428-445.
11. Lapierre S.D, Goldsman D., Cochran R. and DuBow J. (1999) “Bed allocation techniques based on census data” Socio Economic Planning Sciences. Vol.33. - P. 25-38.
12. Lipp M., Paschen H., Daublander M., Bickel-Pettrup R. and Dick W. (1998) “Disaster management in hospitals”, Current Anaesthesia and Critical Care. Vol.9. - P.78-85.
13. Mackay M. (2001) “Practical experience with bed occupancy management and planning systems: an Australian view” Health care management science. Vol.4. - P. 47-56.
14. Ministere de la sante et de la solidarite (2006), Plan blanc et gestion de crise, Edition 2006, annexe a la circulaire n°DHOS / CGR / 2006 / 401. France.
15. Ministere de la sante publique (2002), Guide de mise en place d’une strategie d’accueil dans les hopitaux en cas d’afflux massif des victimes, Circulaire 2002/50, Tunisie.
16. Murray C.J. (2005) “A note on bed allocation technique based on census data”. Socio Economic Planning Sciences. Vol.39. - P. 183-192.
17. Ridge J.C., Jones S.K., Nielsen M.S. and Shahani A.K. (1998) “Capacity planning for intensive care units”. European journal of operational research, 105,346-355.
18. Teow K.L. and Tan W.S. (2007) “Conference on Industrial Engineering and Systems Management (IESM), Beijing, China”.
19. Vissers J.M.H. (1994) “Patients flow-based allocation of hospital resource”. These Eindhoven university of technology, Eindhoven, Holland.
20. Wang T., Belaidi A., Guinet A. and Besombes B. (2008) “Modeling and simulation of emergency services with ARIS and ARENA”, Conference on Industrial Engineering and Systems Management (IESM), Beijing, China.
Appendix
Table 1. Data about victims Table 2. Victims of the different instances
І Pi (mn) dh (mn) darl (mn) Ї Pi (mn) dh (mn) darl (mn)
1 60 510 0 36 30 330 180
2 30 270 0 37 90 330 120
3 30 420 240 38 60 810 240
4 30 150 30 39 30 450 300
5 90 630 120 40 60 570 330
6 30 750 420 41 60 600 420
7 30 300 120 42 30 450 360
8 30 630 450 43 30 600 270
9 60 450 60 44 60 900 450
10 60 480 150 45 30 660 240
11 60 660 210 46 90 630 480
12 120 630 390 47 60 480 150
13 30 660 240 48 30 630 180
14 30 630 300 49 90 600 270
15 90 630 30 50 30 510 210
16 30 660 330 51 90 510 420
17 30 600 330 52 30 450 270
18 30 660 300 53 60 60 0
19 60 630 420 54 60 720 360
20 30 660 60 55 30 480 120
21 30 270 30 56 120 330 30
22 120 270 180 57 30 450 360
23 30 270 30 58 30 540 180
24 90 180 90 59 60 510 0
25 120 540 0 60 120 240 120
26 60 510 270 61 30 30 0
27 60 600 300 62 90 450 210
28 60 480 120 63 30 330 180
29 90 630 450 64 60 120 0
30 30 540 300 65 30 450 270
31 120 630 450 66 30 240 120
32 60 420 360 67 60 360 30
33 30 360 30 68 30 540 300
34 90 570 330 69 60 360 180
35 120 510 420 70 30 270 150
N Victims
2: 3:5:8: 10:12: 15: 16: 20:30:
25 33 35; 37: 39: 41: 43: 46: 48:
50 52:54:58:61:65:68
IS 23:25:2:3:27: 29:5:8:31:
36 3S: 10: 12: 15: 16; 20: 69:
30 33: 35: 37: 39: 40: 41: 44:
47 49: 51; 53: 55: 43: 46: 48:
50 56: 57; 59: 60: 52: 54: 62:
63 64:58:61:65:66:67:68
70 1 to 70
Table 3. Ready dates of surgical staffs according to hospital emergency planning
R (nm) 1*2 (ЯШ) la (mn) І*4(ят) 1*5 (ЯШ) fg (nm) fj (mn) /g (mn) #9 (mn) /*10<mn)
Ri 0 0 0 30 30 30 60 60 120 120
r2 0 0 0 30 30 30 60 60 60 120
Rs 0 0 0 30 30 60 60 60 120 120
R4 0 0 30 30 30 60 60 60 120 120
Rs 0 0 30 30 60 60 60 120 120 120
mn : mmutes.
Issam Nouaouri
Univ Lille Nord de France, F-59000 Lille, France, UArtois, LGI2A, F-62400 Bethune, France.
E-mail: issam.nouaouri@fsa.univ-artois.fr.
J. Christophe Nicolas
E-mail: jchristophe.nicolas@univ-artois.fr.
Daniel Jolly
E-mail: daniel.jolly@univ-artois.fr.
