2351-9789 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of AHFE Conference doi: 10.1016/j.promfg.2015.07.455
Procedia Manufacturing 3 ( 2015 ) 4448 – 4455
ScienceDirect
6th International Conference on Applied Human Factors and Ergonomics (AHFE 2015) and the
Affiliated Conferences, AHFE 2015
Using augmented İ-constraint method for solving a multi-objective
operating theater scheduling
Amirhossein Najjarbashi*, Gino Lim
Department of Industrial Engineering, University of Houston, 4800 Calhoun Road, Houston, TX, 77204, USA
Abstract
Operating Theaters are among the most critical parts in hospitals. Generally, an Operating Theater consists of three stages: Public Health Unit (PHU), Operating Rooms, and Post-Anesthesia Care Unit (PACU). Operating Theater significantly affects annual costs and revenues of a hospital due to requiring skillful staff and expensive equipment to operate surgeries beside high service fees. In this paper, proposing an operating theater daily schedule to serve surgical demands is considered as a three-stage flow-shop scheduling problem. The problem is formulated as a multi-objective mixed integer programming model aiming to balance workload of operating rooms and surgeons, minimize waiting times and maximize human (surgeons) reliability. Weighted-sum
method and an augmentedİ-constraint method are applied to solve two test problems. Analysis of obtained solutions verifies that
both solution approaches are capable of providing appropriate schedules in a reasonable time. © 2015 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of AHFE Conference.
Keywords: Operating theater scheduling; Multi-objective mixed-integer programming; Human reliability;$XJPHQWHGİ-constraint method; Priority level
1. Introduction
According to Hans and Nieberg [1], around %70 of receptions in a hospital are requesting surgical cases. Thus, major proportion of hospital cost and revenue is related to operating theater. Besides, operating theaters have a wide variety of interactions with other sections within hospital such as, ICU, CCU, Emergency, and so more. Hence,
* Corresponding author. Tel.:+1-832-952-7496; fax: +1-713-743-4190.
E-mail address: [email protected]
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
improving operating theatre performance in terms of efficient utilization of available resources for surgical cases, and providing high quality services can increase patient satisfaction along with escalating hospital revenues.
Magerlein and Martin [2], Blake and Carter [3], Przasnyski [4] and Smith-Daniels et al. [5] were the first researchers to review the operating room planning and scheduling problems. The most recent categorization of operating room planning and scheduling is proposed by Cardoen et al. [6]. They grouped characteristics of an operating room planning and scheduling problem into six parts as follows:
x Patient characteristics: two main patient types are considered: elective and non-elective. Elective patients are divided into inpatients [7-9] and outpatients [10-12]. Non-elective patients are divided into urgent [13,14] and emergent [15,16] patients. Non-elective patients arrive at hospital randomly.
x Performance measures: many performance measures are considered as objectives of the problem. Makespan [17], waiting time [18], utilization [19], financial value, preferences [20], and throughput [21,22] are a number of objectives considered in the literature.
x Decision delineation: decisions are usually made on date [9,19] time [7,8,10,18], room [21], or capacity [11,23]. However, there are different levels for these decisions and they might be applied to a medical discipline, a surgeon [24], or a patient [8,22].
x Research methodology: refers to the chosen methodology and type of analysis performed e.g. Optimization, Scenario, etc. and utilized solution approaches e.g. Mathematical Programming, Simulation and Heuristics/Meta-heuristics.
x Uncertainty: existing uncertainty in this problem is due to two main sources: 1) Patient arrival [7,13,15,25], 2) Duration [15,18,26]. One of the most unpredictable processes is surgery duration.
x Applicability of research: it is very important to show that the proposed model and solution approach works well in real-world cases. Many authors evaluated applicability of their research using theoretic [8,15] or real data [7,10,11,12,22,27].
After a thorough review of the literature, it is noted that a few authors addressed the operating theater planning/scheduling problem by considering PHU, PACU, and/or ICU as a stage of service or a resource[10,25]. Nomenclature i, j patients k stage lk, l ’ k machine of stage k
l
k= 1, 2,…,L
k n, n’ surgeon tik process time suij setup time clij cleaning timeȜ1n non-critical error rate
Ȝ2n critical error rate
wi priority weight of patient i
Rn reliability of surgeon n
M a very large positive number
Xiklk binary decision variable for assigning patient i to machine lkin stage k
Siklk service start time for patient i in stage k
Yijklk binary decision variable for assigning patient j after patient i to machine lkin stage k
Vni binary decision variable for assigning surgeon n to operate patient i
Patients Step 1 - WARD Step 2 - ORs Step 3 - PACU Bed #1 Bed #2 Bed #m OR #1 OR #2 OR #n Bed #1 Bed #1 Bed #p
Fig. 1. Flow-shop structure of the operating theater scheduling problem.
2. Problem description
This paper studies an operating theater scheduling problem. The operating theater consists of PHU, operating rooms, and PACU. Figure 1 illustrates the three-stage operating theater scheduling problem. First, each patient is assigned to an available bed in PHU to receive necessary pre-surgical services and get prepared for operation. Then, he/she is assigned to the first available OR to be operated. Finally, the operated patient is assigned to an available bed in PACU to make recovery. It should be noted that when no bed/OR is available, the patient have to wait until a bed/OR become idle.
2.1. Assumptions
Availability of surgeons is an important factor in providing appropriate schedules. Surgeons are assigned to surgical cases at the beginning of the second stage. Then scheduling and sequencing of patients in operating rooms will be carried out. Limited available resources are PHU and PACU beds, Operating Rooms, and Surgeons. Patients belong to several surgical groups with different priority levels. After each operation, operating room must be disinfected and set up for the next surgery that might belong to a different surgical group. Cleaning times and set up times are assumed to be sequence dependent. Another novel concept considered in this research is human reliability. According to B. S. Dhillon [28], there are three models for human reliability and error in healthcare: human reliability in normal work environment, human reliability in fluctuating work environment, and human reliability with critical and non-critical human errors in normal work environment. The third model fits well to the considered problem in this paper since human errors during operation can be either non-critical or critical in terms of how much they pose a threat to patient’s health. The reliability of surgeon s at time t is given by
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Equation (2) tries to balance workload of operating rooms and surgeons using equations (5-7). Equation (3) calculates total patients’ waiting time during their stay at operating theater. Equation (4) tries to minimize surgeons working duration using equations (8-11) which consequently increase human reliability function. Equation (12) ensures that only one assignment for each patient in each stage is made. Equation (13) states that start time in each stage must be greater than start time plus process time in the previous stage. Two patients are allowed to be sequenced successively to a bed/OR only when both of them have been assigned to that specific bed/OR according to Equation (14). Equations (15) and (16) confine start times of successive patients in each stage. Equation (17) is similar to equation (14) but deals with assigning patients to a surgeon. Equation (18) limits start times of successive patients assigned to a surgeon. Equations (19-21) states capacity constraints of PHU and PACU, operating rooms and surgeons. Binary and non-negativity constraints for decision variables are indicated by equations (22-24). 3. Solution approach
Methodology of this paper in dealing with the operating theater scheduling problem is multi-objective optimization. Hwang and Masud [29] classified the solution methods for multi-objective mathematical programming problems into three categories regarding the stage in which the decision maker implement his/her preferences in the decision making process: 1) the a-priori methods e.g. weighted-sum method; 2) the interactive methods, and 3) the generation (a-posteriori) methods e.g. İ-constraint method. Although the a-priori methods are the most popular due to much faster computation process, the interactive and the generation methods convey much more information to the decision maker and reinforce the decision maker’s confidence on the final decision. In this paper, we applied weighted-sum method [30] and an augmented İ-constraint method (AUGMENCON) by Mavrotas [31] to solve the problem.
4. Numerical examples
In this section, two numerical examples are solved using the weighted-sum method and the İ-constraint method. In the following of this section, test problems are introduced, and obtained solutions are indicated and analyzed. Then, performance of the solution approaches are compared in the final subsection.
4.1. Test problem #1
There are five patients from a single surgical type in this example. Operating theater resources are three PHU beds, one OR, and three PACU beds. PHU is open from 8:00 to 6:00, OR from 8:00 to 6:00, and PACU from 8:0 to 6:00 during weekdays. There are two surgeons, one experienced and one intermediate with different working shifts.
The problem is first solved using weighted-sum method for three different sets of importance weights and the obtained results are indicated in Table 1.
Table 1. Obtained solutions for test problem #1 using the weighted-sum method.
Set number w1 w2 w3 Workload balance (hr) Waiting time (hr) Surgeons’ working
time (hr) Solution time (sec) 1 0.333 0.333 0.333 0.75 18.25 12.75 2 2 0.4 0.2 0.4 1.25 20 12.75 2 3 0.35 0.45 0.2 0.75 20 13 3
Then, augmented İ-constraint method is applied to solve the problem by considering two sets of grid points and expected number of solutions. Larger values for grid points raise density of solutions and facilitate decision making process, but increase computational effort. The problem is solved for and obtained results are shown Table 2. Table 2. Obtained solutions for test problem XVLQJWKHİ-constraint method.
Number of grid points
Number of expected solutions
Pareto solution # Workload balance (hr) Waiting time (hr) Surgeons’ working time (hr) Solution time (sec) 5 20 1 0.5 17 15 30 2 1 17 13.75 3 1.5 17 12.25 10 50 1 0.5 19.5 14.5 180 2 0.5 20.5 14 3 1 19.5 13 4.2. Test problem #2
There are 15 patients from two different surgical types with 10 and five patients, respectively. Operating theater resources are six PHU beds, three ORs, and five PACU beds. PHU is open from 8:00 a.m. to 6:00 p.m., ORs from 8:00 a.m. to 8:00 p.m., and PACU from 8:00 a.m. to 9:00 p.m. every day. Three surgeons, including two experienced and one intermediate, operate surgeries cases of type I, and two other experienced surgeons operate surgical cases of type II. Surgeons have different working shifts varying from eight-hour to 12-hour shifts. Patients from surgical type I have priority value of five, while priority value of patients from surgical type II is equal to three. Obtained solutions from solving the problem by each method are indicated in the following tables.
Table 3. Obtained solutions for test problem #2 using the weighted-sum method.
Set number w1 w2 w3 Workload balance (hr) Waiting time (hr) Surgeons’ working
time (hr) Solution time (sec) 1 0.333 0.333 0.333 4.25 191.75 30.5 860 2 0.4 0.2 0.4 1.5 224.25 34.25 910 3 0.35 0.45 0.2 3.75 222 38 880
Due to high complexity of the problem, a slight increase in problem parameters such as number of patients, beds, and/or operating rooms leads to an exponential increase in solution time. Despite the first test problem that was solved after a few seconds, it takes about 15 minutes for the weighted-sum method to obtain each solution indicated
in table 3. Similarly, the İ-constraint method solved test problem #2 in approximately 6 hours with 9 grid points and 100 expected solutions.
Table 4. Obtained solutions for test problem #2 using the İ-constraint method. Number of grid
points
Number of expected solutions
Pareto solution # Workload balance (hr) Waiting time (hr) Surgeons’ working time (hr) Solution time (sec) 9 100 1 0 196.5 38.5 20940 2 0 204.75 39 3 0.25 213.25 40 4 0.25 230.25 42 5 0.25 232.75 45.25 6 0.25 245.5 42 7 0.25 262 41.5 4.3. Numerical analysis
Test problem #1 was solved several times using the weighted-sum method and the İ-constraint method. Both methods solved the problem within reasonable times from two to three seconds to 10 minutes. However, the computation time increased significantly for test problem #2 due to NP-hardness of the problem. Obtained solutions for test problems are indicated and compared in table 5.
Table 5. Analysis of obtained results by the two solution methods.
Criterion Test problem #1 Test problem #2
Weighted-sum İ-constraint Weighted-sum İ-constraint Workload balance
(hr)
Surgeons 1.75 0.6 2 0
ORs 2.25 1 3 0.5
Waiting time (hr) Priority I 2 1.5 2 1
Priority II 4 5
Reliability (%) – working time (hr)
Type I 95-6 85-6.5 90-6 90-6
Type II 85-7 80-7 85-8 80-8.5
Solution time (sec) 3 105 875 20940
According to table 5, the weighted-sum method is able to obtain high quality solutions in a reasonable time for both test problems. On the other hand, the İ-constraint method provides a set of high quality solutions for both test problems but lacks time efficiency for test problem #2. The İ-constraint method outperforms the former method in terms of workload balance. Both solution methods perform quite similar in terms of waiting time. Less waiting time for patients of priority level I is another positive point about performance of the applied solution methods. Both solution methods perform well in producing reliable and safe schedules i.e. with at least 80% reliability.
5. Summary and conclusion
In this paper, and operating theater scheduling problem has been studied and considered as a three- stage flow-shop scheduling problem. Sequence-dependent set-up times and cleaning times are considered in this paper along with considering different priority levels for surgical types. The problem is formulated as a multi-objective mixed-integer linear programming model. Two numerical examples are solved using the weighted-sum method, and the İ-constraint method. Analysis of the obtained solutions proves ability of both methods in providing high quality solutions. The weighted-sum method is time efficient for both test problems while computation time for İ-constraint method increase significantly for the second test problem. Incorporating major uncertainties in the real-world problem such as uncertain operation times and emergency patient’s arrival could be helpful contributions to the present research.
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