A Multi-Agents System for Solving Facility Layout Problem: Application to Operating Theater

3.1 Problem Description

Given a set of facilities, the dimensions and orientation of each facility, and the available area, the SOTLP seeks to determine the optimal placement of the set of facilities within the available space, subject to a set of constraints while optimizing an objective function. In this work, a bi-objective function is considered, which includes quantitative and qualitative measures related to the characteristics of the SOTLP under consideration.

To address the OTLP, the safety and hygiene considerations necessitate a layout organization with a multi-section structure to prevent contamination, to avoid accidents, and to facilitate the movement of material and different actors of the OT. The multi-section structure consists of separated zones (i.e. surgery zone, restricted zone, public zone, etc.), each one having all necessary facilities to operate.

The new variant is formulated with the objective of determining the locations of the sections in the final layout, and the location of facilities and corridors within each section that minimizes the total traveling costs and maximizes the total adjacency rating.

To formulate the mathematical model of the multi-section SOTLP, several assumptions have been considered:

• The facilities have rectangular shapes with unequal areas of known dimensions;

• The movement between facilities is modeled as a rectilinear centroid-to-centroid movement;

• The number of trips, the difficulty of movements, the desirable relationship values between facilities and the cost factors assigned to each entity types are given;

• The orientation of the OT layout is a given parameter;

• The maximum number of corridors and their orientations are given. In the solution, corridors are used to travel between two facilities in the same section, and to transit from one section to another;

• The location of corridors is a decision variable;

• The corridors are aligned according to the layout orientation to ensure the entrance to the OT unit, the movements between sections, and the exit from the OT unit;

• No relationships of facilities with the outside world are considered;

• A single facility cannot be split among multiple sections.

The main two objectives we aim to optimize in this work are to minimize the total traveling cost (quantitative) and to maximize the total adjacency rating (qualitative). We express these two objectives in the following objective function:

where N is the set of facilities, K is the set of actors in the OT, F_ijk is the traveling frequency from facility i to j by actor k (parameter), D_ij is the distance between facility i and j (decision variable), φ_ijk is the difficulty of the trip according to the provided effort and the requested resources (parameter), σ_k is the traveling cost corresponding to the actor k (parameter), R_ij is the desirability relationship value between facility i and j according to the AEIOUX rating system [21] (parameter), and μ_ij is the proximity coefficient according to the distance between facility i and j (decision variable).

The AEIOUX rating system represents the need for proximities between two facilities, where a strong positive relationship indicates a proximity rank of A. The AEIOUX rates are associated to values equal to A=16, E=8, I=4, O=2, U=0, and X=−2, respectively, and are defined according to the international standards in terms of hygiene or decision makers’ preferences.

The parameters, decision variables, constraints and results of this mathematical formulation are reported in detail in Ref. [3]. This paper is devoted to the MAS architecture and the new additions or alterations to the principal model.

3.2 MAS Approach

In our approach, we used cognitive and communicative agents. Thus, to communicate, agents need a developed language to be able to exchange messages. However, a simple model of message is not enough to tolerate conversations between agents and to allow language acts to fulfill their meaning. For this, we introduced the notion of protocol to support such conversations. The negotiation strategy is proposed based on the Contract Net protocol implemented using the JADE platform that aims to simplify the development of MAS while providing a complete set of services and agents complying with foundation for Intelligent Physical Agents specifications [7].

In this paper, the OT is divided into several departments according to their specialty, (e.g. orthopedics, gynecology, etc.), while each one contains all necessary facilities to be autonomous (e.g. operating rooms, induction room, post-anesthesia care unit, etc.).

The newly adopted approach is modeled in three levels of abstraction (see Figure 1). The first one consists of defining the dimensions and orientation of the plan layout or the initial construction site in which all departments and facilities will be implanted. In the second level, the MILP formulation is used to solve the layout problem for departments first, and to calculate the placement of each one of them in the initial plan layout. Finally, the third level considers the resolution of the MILP for each department in order to determine the optimal locations and orientation for the facilities within.

Figure 1:

Three-Level FLP Approach.

This three-step decision-making approach can best be handled through an effective and efficient cooperation among multiple agents. To deal with this, we adopt a master/slave architecture where the master processor (MP) is the coordinator of the optimization process, while the slave processor (SP) is the worker executing tasks coming from the MP. The main task of an MP is to dispatch parameters and orders to each generated SP, which, in parallel, solves the sub-problems and sends back the results with objective value (see Figure 2).

Figure 2:

Master/Slave Architecture.

The used algorithm in Figure 1 can be described as follows:

This study on MAS allowed us to define how the capacity of cognitive agents can be used in a decision-making system, namely their intelligence, cooperation and learning.

In addition to the MILP formulation used in Ref. [3] to solve the layout problem for each department, we need to add some parameters, variables and constraints related to the department in order to define their optimal location and orientation in the OT layout. To make our MILP suitable for the three-level approach, these constraints are considered in order to calculate the total interdepartmental traveled distance:

where Xp_ij and Yp_ij are variables representing, respectively, the horizontal and vertical distances between two departments i and j; x_i/x_j and y_i/y_j are the used variables to determine the location of a department; x_c/y_c are the variables used to detrmine the location of a corridor; and OrD_d refers to each department orientation, which takes 1 if the length of department d is parallel to the y-axis (vertical orientation) and 0 otherwise. We can obtain it by using the following equations:

where

• ϑ_d : length of department d (parameter);

• θ_d : width of department d (parameter);

• L_d : X-length of department d depending on whether ϑ_d or θ_d is parallel on the x-axis (decision variable);

• W_d : Y-length of department d depending on whether ϑ_d or θ_d is parallel on the y-axis (decision variable).

The objective function in the second level is formulated in Eq. (10), where F_lhk, D_lh, φ_lhk, R_lh and μ_lh are, respectively, the inter-departments’ traveling frequency (see Appendix A), traveling distance, traveling difficulty (see Appendix A), relationship, and adjacency coefficient (see Appendix A):

4.1 Problem Description

To the best of our knowledge, hospital layout planning is still being addressed as SFLP, where material handling flows stay invariant over a long time. As far as we know, hospitals and specifically OT are volatile. Under such conditions, some parameters such as patients demand are not stable and the facilities must be adaptive to demand change requirements. For these reasons, a static layout analysis would not be sufficient.

To overcome the disadvantages of the SFLP, we provide an adaptive formulation for OTLP (AOTLP), with the objective of minimizing the interdepartmental travel costs among facilities, maximizing the adjacency rating, and minimizing the rearrangement costs [Eq. (11)]. The AOTLP is solved by studying an individual layout for each distinctive period based on patient demands, subject to a set of distance constraints, available areas, and non-overlapping facilities according to international medical standards and specifications.

In order to model the OTLP in an uncertain environment, two different solutions can be proposed. The first approach consists of finding in one single decision a robust OT layout that is the optimal layout for all periods and that could not be rearranged in later periods. The second one is an adaptive solution, which consists of generating a layout plan for each period of the planning horizon. Here, the layout will be rearranged in a way that it satisfies the patient demands of each period. Table 1 summarizes the differences between the robust approach of Chraibi et al. [4] and our adaptive approach.

Traditional FLP approaches consider a planning horizon generally divided into periods of weeks, months or years. Choosing the period length has an impact on the final results. In fact, in a short period, the flows are fairly constant and there will not have a need for layout rearrangement and the use of adaptive layout analysis may not be justified. In contrast, a long period will lead to prohibitive rearrangement costs and this goes against the objective of cost minimization. This choice should be made according to the studied system based on demand and the delivered asset and service.

In the case of our MAS approach, this paper will focus on the AOTLP and present just the adopted architecture and novelties compared to the developed MILP for the problem in Ref. [4]. The developed MILP aims to minimize the traveling costs for the considered entities (doctors, patients, and medical and non-medical staff) for each period in the planning horizon, to maximize the adjacency rating between dependent facilities and to minimize rearrangement costs generated by moving appropriated equipment for each room specialty [Eq. (11)]. This model provides an optimal solution for problems with <17 facilities, which represents the size of one simple department in large hospitals.

where A_ij is a binary parameter equals to 1 if facilities i and j can be exchanged, and 0 otherwise; C_ij is the relocation cost of facility i if exchanged with j (parameter); and r_ijt is a binary variable equals to 1 if facility i is exchanged with j in period t, and 0 otherwise.

4.2 MAS Approach

To solve the AOTLP, we introduce a new three-level approach, which is illustrated in Figure 3. The first step defines the dimensions and orientation of the plan layout or the initial construction site in which all departments and facilities will be implanted and divides it into equal-sized departments corresponding to number of OT specialties. For each department, the second level consists of solving the adaptive layout problem for all periods of the planning horizon using the objective function in Eq. (11) and the MILP in Ref. [4] in order to determine the optimal location and orientation for each facility within the department. Considering these solutions, the optimal OT layout in a period t is addressed by using all resulting departments in period t. For this, departments are assigned to the defined locations – in the first step – according to the total relations rate (TRR) calculated in Eq. (12). To assign departments, the quadratic assignment problem (QAP) formulation in Eqs. (13)–(15) is adopted to determine the location of each department in the final layout while minimizing the traveling costs and maximizing the adjacency of the departments.

Figure 3:

Three-Level Adaptive FLP Approach.

where

• D is the set of departments.

• r_ijhk is the relationship value that expresses the need for proximity between the facility i in the department h and the facility j in the department k by using the AEIOUX rating system.

Subject to

∑k=1DXkm=1        m=1, …,  D;(15)

• X_hl : a binary variable equal to 1 if department h is assigned to location l, and 0 otherwise;

• μ_hk : the adjacency coefficient that expresses the desirability of locating adjacent departments next to each other. It will be 1 if two departments are fully adjacent, 0.5 if they belong to the same section, or 0 if they are located in different sections;

• C_hk : the traveling cost between department h and k;

• D_hk : the distance between department h and k.

Constraints in Eqs. (14) and (15) are used to ensure that each department is assigned to only one location and that each location contains only one department.

This three-step decision-making approach can be best handled through an effective and efficient cooperation of multiple agents in the second level to find the best layout, and a negotiation protocol in the third level to allow each agent to occupy the best location in the OT layout. For this, we adopt a master/slave/sub-slave architecture, where the MP is the coordinator of the optimization process, the SP is a worker executing tasks coming from MP and at the same time a coordinator of sub-SPs (Sub_SPs) attached to him, while the Sub_SP is a worker executing tasks coming from the SP (see Figure 4).

Figure 4:

Master/Slave/Sub-slave Architecture.

The main task of an MP is first to generate as many SPs as departments (i.e. specialties) and then to instruct each SP to solve a department layout problem for all the planning horizon periods. The main task of each SP is to generate as many Sub_SPs as periods and then to instruct each Sub_SP to solve the FLP for each period of the planning horizon. At the bottom of the hierarchy, Sub_SPs receive all required data to solve the layout problem and start the optimization process. Each group of Sub_SPs (i.e. attached to the same SP) compare their solutions in order to count the number of facility rearrangements (i.e. facilities having exchanged their locations from a period to another) in the whole planning horizon to be calculated in the objective function in Eq. (11). When all departments’ layouts are solved, the SPs start a negotiation phase to assign the departments to the available locations in the final OT layout for each period. Finally, the MP addresses all the obtained layouts (i.e. a layout per period) and presents the solutions to decision makers.

The developed interaction between agents to solve AOTLP can be described in the following algorithm:

In this formulation, we used two agent communication protocols. The first is cooperation, where agents try to find a solution that minimizes the total rearrangement cost in a way that it satisfies the all periods’ surgery demand. The second protocol is negotiation, in which each agent competes to find a location that satisfies its own benefit. This allows us to exploit agent properties to find efficient solutions in a reasonable time.

5.1 SOTLP

Experiments have been conducted on the SOTLP by configuring OT instances with a combination of departments of generally 16 facilities and 13 facilities. Several OT configurations have been constituted with medium sizes (e.g. 29 and 54 facilities) and large sizes (e.g. 87, 116, 145 and 174 facilities). They all were solved using MAS of one MP and as many departments as SPs. An example of larger-sized OTLP with 12 departments with 174 facilities is solved using 13 agents (i.e. 12 SPs and 1 MP). The number of facilities is no longer a limitation as the sub-tasks are feasible MILP for each SP. We decided to stop our experiments at OT with 174 facilities to deal with realistic OT sizes. Table 2 presents a description of the rooms list in each instance size.

Table 3 provides some computation results obtained using the MILP model in Ref. [3]. The limit of this MILP is 18 facilities; beyond this value, optimality is not guaranteed. Compared to these obtained results, our MAS gives an efficient solution for larger-sized problems in a shorter time. For example, a problem with 16 facilities took 623 s with simple MILP, while MAS solved in the same duration a problem with 16 facilities. This shows the performance of this MA approach in terms of timeliness and capacity.

To be feasible, each agent is constrained to respect the limitation of the MILP model (i.e. 18 facilities); otherwise, this agent will not be able to return a solution. Table 4 provides the results of solving multi-department OTLP using MAS with different instance sizes. The first column gives the number of facilities in the OT, the second represents the number of used SPs to solve this problem size (number of SPs=number of departments), while the third one gives the processing time in different running iterations to verify the ability of agents to learn from their previous experiences. The MAS has been executed several times on the same instance to examine the agents’ characteristics and properties in each iteration. We observed that the MAS took less time for solving the problem than in the previous run, while the objective value remained fixed. This improvement on the execution time is due to the agents’ learning capacity. Figure 6 shows that this execution time regresses while repeating calculation for each problem size, while Figure 7 presents the final results for an OT problem with 87 facilities assigned to six departments.

Figure 6:

Processing Time Regression.

Figure 7:

SOTLP: Final OT Layout.

5.2 AOTLP

Experiments have been conducted on the AOTLP by configuring OT instances with a combination of departments of generally 13 facilities and 11 facilities. Several OT configurations have been constituted with medium sizes (e.g. 52 facilities) and large sizes (e.g. 78 and 104 facilities). They were all solved using MAS of one MP, as many departments as SPs, and as many periods as Sub_SPs under each SP (e.g. the final number of agents for an OT of four departments in a planning horizon of three periods is 1+4*3=13).

The adaptive approach consists of exchanging rooms’ or departments’ specialty from a period to another while minimizing rearrangement costs in a way that two departments can exchange their specialty and move the needed equipment and materials from one to the other to satisfy patient demand. Assumptions used in this formulation were approved by a senior manager of the OT of the hospital in Roanne city (in France).

The proposed MILP model solved small instances with 17 facilities as maximum value with a planning horizon of three and six periods. Beyond these values, an optimal solution will not be guaranteed. Table 5 presents a part of the obtained solutions using just the MILP model.

In this work, we consider a multi-disciplinary OT, which means that each department is affected to a surgical specialty. To rearrange the OT layout, we adopt the exchange of the surgical specialty between departments and the surgery operations’ affectation between operating rooms, as restructuration of the whole OT after each period can be very expensive. In this case, the OT will keep the first layout configuration and room specialty could be exchanged at the end of each period to satisfy patient demands.

Compared to the obtained results in Table 5, our MAS gives an efficient solution for larger-sized problems in a shorter time. For example, a problem with 17 facilities took 3935.89 s with a simple MILP, while MAS solved problems with larger sizes in less than this duration. This shows the performance of this MAS approach in terms of timeliness and capacity. In addition to this, the MILP capacity is limited to a planning horizon of six periods, while in MAS we investigate problems with 9 and 12 periods in a reasonable execution time.

To be feasible, each agent is constrained to respect the limitation of the MILP model (i.e. 17 facilities); otherwise, this agent will not be able to return a solution. Table 6 provides the results of solving AOTLP using MAS with different instance sizes and different planning horizon repartitions. The first and second columns give the number of departments and the number of facilities in the OT, whereas the third represents the number of used agents to solve this problem size according to the number of periods in the fourth column. The last column presents the processing time in different running iterations. We observed running time to also regress in each execution due to the agents’ learning capacity. As in the SOTLP, the number of facilities we tested here is not a limitation. We can make the size >104 by increasing the number of agents. Figure 8 presents the results for an OT problem with 52 facilities assigned to four departments in a planning horizon of three periods. From the three obtained layouts, we can observe the reallocation of facilities within the OT at different periods to ensure optimality and adequacy for various surgical demands. These assumptions and results were viewed and approved by a senior manager of the OT of the hospital in Roanne city (in France).

Figure 8:

AOTLP: Final OT Layout.