Flow-Shop Scheduling Problem Applied to the Planning of Repair and Maintenance of Electromedical Equipment in the Hospital Industry

Abstract

In the literature, several approaches have been proposed to integrate and optimize product supply and construction processes associated with demand management. However, in Industry 4.0, there needs to be more studies related to applying techniques that directly affect the programming and reprogramming process that integrates the industries at the operational level. This document proposes a flow-shop scheduling procedure to address the problem of planning the repair of medical equipment in public hospitals whose main objective is to eliminate downtime and minimize total production time. The research stems from the practical problem of responding to clinical users who make use of critical equipment, such as mechanical respirators, due to COVID-19, and the limited quantity of this equipment, which makes it necessary to have repair planning processes that seek to keep the equipment in operation for the most extended amount of time. The novelty of this study is that it was applied to a critical and real problem in the industry with a high economic and social impact, which has not been explored previously. The results show improvements in the overall planning and execution of electro-medical equipment repair. Several improvements to the applied methods were identified as future work, such as the need to consider work interruptions and psychosocial effects on workers due to the stricter planning of execution times.

1. Introduction

Task planning is considered one of the most critical problems of production systems in modern industries [1], as they are one of the greatest challenges exposed by Industry 4.0 [2,3,4] since various decisions affect the way resources are used for the production of goods and services. Planning is a stage that precedes the execution of any work and encompasses a specific objective and occurs in a dynamic context in which managers must adjust their plans according to the pace of change [5] and the criticality of the goods or services, such as the case of health equipment in the face of health contingencies such as the case of COVID-19 [6,7]. The planning must cover different time scopes, such as (i) long-term, e.g., planning for the construction of a new industrial plant; (ii) medium-term, e.g., the production of production units and sales plans; and (iii) short-term, e.g., a plan that is made for production schedules [8].

The task planning problem is “the problem of organizing the various resources in time to perform a certain number of tasks” [9]. Various productive sectors need to carry out a workload assignment with associated times. These tasks are assigned to workgroups of multiple sizes, dynamically formed by people and machinery, which can carry out said activities individually or jointly [10,11]. Each type of planning must be defined according to a specific need for information and control of the associated productive sector [12]. Both in the service area and the manufacturing area, the production systems establish the associated productive structures related to the process of converting inputs (raw materials and resources in general) into products (assets and services) satisfy the needs, requirements, and consumer expectations [9]. The implementations observed range from traditional techniques, for example, mass production and artisan production Campo [13], to the most modern hybridizations, such as a hybrid system in a flow-shop environment [14]. Within a set of production configurations, the flow-shop is categorized into the semi-repetitive type of production systems, where the production elements are extracted in line and adapted with production depending on the context in which it is applied [15].

Flow-shop models can be successfully applied to different production environments [16,17] both in theoretical and practical contexts. A specific sector of interest corresponds to logistics in public health, in which it is necessary to plan for the efficient use of resources that respond to the needs of the population [18,19]. It is possible to find examples of adaptations of preparing proposals in the domain of health logistics that cover emergency medical services [20], the routing and scheduling of home health care [21], or the scheduling of patient admissions [22].

The repair of medical equipment is a challenge that the medical industry must constantly face [23,24,25] due to strong demand and the limited availability of these instruments [26]. A case applied to the preceding is that artificial respirators, categorized as critical equipment, cannot be absent from in-hospital equipment due to their dependence and demand [27,28]. This is why any repair process must be as fast and efficient as possible to make the equipment available for active use. In the case of Chile, the country spends approximately 8.1% of its GDP (gross domestic product) on the health budget [29], which includes the management and repair of medical equipment. Based on the review, there are no proposals in the literature on the use of planning methods, such as the flow-shop; they are used in Chilean health contexts to manage public resources, specifically in the repair of medical equipment. As a contribution to this body of knowledge, this research proposes using a tabu search algorithm to repair medical equipment and electromedical devices for a public hospital in Chile. In this way, we hypothesized that the application of an algorithm by solving the flow-shop type planning problem for the repair of electromedical equipment improves the estimated times and their distribution of tasks; in the case of the null hypothesis, it was established that the application of an algorithm did not improve the times or the distribution of functions; thus the manual solution was more convenient than the automated one. Based on this, the following research questions were posed:

• Can an algorithm be applied to solve the flow-shop scheduling type problem applied to medical equipment repair scheduling?
• Does applying an algorithm to solve the flow-shop scheduling problem for medical equipment repair scheduling produce better schedules based on the total planned time and the distribution of tasks compared to traditional programs?
• Can a software application execute this algorithmic solution in a real hospital environment?

To validate the above, an application was built in which the input data were used to generate repair schedules and task assignments. The results obtained from the application were then compared with the programs used in the case study. In this way, the novelty of this study is that it applied traditional algorithmic techniques to a problem not addressed in the entire industry, which generates a high impact at a social and economic level, where good results were obtained. Thus, the contributions of this research are:

• It applied a metaheuristic algorithm in a real industry flow-shop scheduling type case study to optimize medical equipment repair planning.
• It defined a test scenario through the selection of data and practical cases in a real industrial environment with a high social and economic impact in Chile.
• It evaluated the impact of the differences in production times and costs associated with the case study in which the proposal was tested.

This paper is structured as follows: Section 2 briefly explains the flow-shop problem as a general approach, which can be applied more in the public health field, and it is included and addresses related work from the literature, focusing on research proposals and industry examples. Section 3 mathematically formulates both the workflow and the domain of the problem considered in this investigation, and details on designing the solution to address the issue with the description of the case study, the proposed solution’s application, and the results. Section 4 presents a discussion of the results. Finally, in Section 5, several conclusions and future works are presented.

2. Flow-Shop Scheduling Problem

2.1. Theoretical Framework

2.1.1. General Description

Task scheduling encompasses various problems related to processing optimization or combinatorial optimization, which vary in complexity and structure [30]. The variables used are dependent and independent, and various elements within a process and control model exist. This work used a simplified scheme of a modern system [5]. In particular, the flow-shop scheduling configuration consists of the organization of n tasks broken down into m work items that must be processed in M machines to obtain a schedule that achieves the objectives of having the least number of devices in time-dead (no use) and its execution in the shortest possible time [31,32,33,34]. This task-planning model must follow the following essential characteristics [35]:

• Each machine performs a single task and performs this task for one work item at a time.
• Tasks require a single visit (run) to complete (if this job does not use the corresponding machine, the time is zero).
• The job goes through each machine only once.
• The order of the machines is always the same.

The objective of the problem is to order the sequence of work entries and meet the required times given the constraints for a particular series [36]. This schedule can be as simple as minimizing the time needed to complete all the work or as complex as finishing the series precisely on the due date; if the work is completed before or after the delivery date, there will be losses [37,38].

Flow-shop problems in the literature are classified as NP-Hard [30,31,32,33], which is understood as a highly complex non-deterministic polynomial problem in computational optimization. This is also reflected in the processing for N number of jobs carried out by M machines. All of its work must be performed for a task to be classified as complete. The total time invested in completing tasks or jobs is known as the makespan [34].

2.1.2. Mathematical Model

The mathematical model of flow-shop used for this research is presented in [35] and refers to the programming of binary integers, for which the objective function is to minimize the final makespan, where the final makespan is the total time in the processing of the works [36]. It is possible to define the following:

• n: number of the work to be performed ranges from 1 to N, where N represents the jobs in that range.
• m: the stages of the process of the work, in a range from 1 to M.
• SJ: the machine number in step j. The range is from 1 to S_m for each step m.

In addition, within the definition of variables, the following is established:

• R_n,M,s is assigned the value 1 in the case where the work n is performed on machine s of stage M; otherwise, it is given the value zero.
• T_n,M corresponds to the processing start time of the n referenced work, contextualized in step M, where:

  $$T_{nM}>0$$

  (1)
• Y_n,n’,M,s is assigned the value 1 in case the work n precedes the work n’, and all of this is included in the machines of stage M; otherwise, it is assigned the value zero. It is necessary to emphasize that this variable is used to determine the sequence in which the work passing through the machine s is processed.

For the definition of the mathematical model, it is necessary to consider a finite number of parameters:

• d_n,M,s represents the duration of the work n in the machine s of stage M. In particular cases, it is considered that this value is composed of two components: one that refers to the processing time (b_nms) and another that refers to the preparation times of the machines (a_nms). Finally, it is necessary to consider the time required to remove the work performed by the machine before placing the next one (c_nms). Therefore, it can be established that:

  $$d_{nms}=a_{nms}+b_{nms}+c_{nms}$$

  (2)
• S_M represents the number of machines in stage M.

The following is the objective function that is responsible for minimizing the final makespan:

$$Min \left(C_{max}\right)=Min \left({\displaystyle \sum }_{n=1}^N\left[T_{n,M}+{\displaystyle \sum }_{s=1}^{S_M}{d}_{n,M,s}{R}_{n,M,s}\right]\right)$$

(3)

Considering the previous model, the total residence time of all the works on the line is calculated, which is the sum of the times of the term of each work. Additionally, there are several restrictions that must be taken into account:

• All start and end times of the work to be processed must be expressed in integers greater than or equal to zero, where:

  $$\forall T_{n,M}, d_{n,M,s}>0 \in z$$

  (4)
• The term time must always be greater than the start time, and a job can only go through one machine in its lifeline. Therefore,

  $${\displaystyle \sum }_{n=1}^N{T}_{n,M} \le 1, T_{n,M}=\left(0,1\right)$$

  (5)
• A task must go through all the machines to determine that it is complete.

2.2. Theorical Applications

The first studies related to the subject followed a traditional line, such as that of [37] that used hybrid flow workshop tasks, or the use of different heuristic methods associated with the sequencing of tasks in a flow workshop, as is in [38], where the best performing alternative was selected and improved. Other studies added specific restrictions such as that the processed jobs must be the same on each machine and the total planning time must be minimized [39]. On the other hand, in [40], extensive work was conducted on two practical cases related to task scheduling. After multiple tests and comparisons carried out by the authors, it was possible to show a relevant improvement for the final programming times for several cases. There are also cases in which a configuration was used that incorporated a greedy adaptive random search procedure and NEH-RB heuristics [41], seeking to minimize processing times in a manufacturing environment. In a more innovative sense, ref. [42] applied greedy algorithms for the production line configuration, which generated permutation programs without delays in work. Furthermore, ordered flows were considered in this work, which were proportional to different speeds and configuration times. The results showed that the implemented algorithm worked well for a problem with a preferential order.

On the other hand, ref. [43] showed that each job’s main characters in each stage could be processed by one or more machines simultaneously; this approach allowed the relaxation of the classic problem of the flexible flow workshop, assuming that one or more devices can process a job at the same time. In the meantime, ref. [44] proposed an estimation of a distributed algorithm based on a memetic algorithm to answer the flow-shop problem in distributed assemblies. In [45], the following two scenarios were presented: the orders or batches of parts are divisible and can be manufactured in various instances of time, and batches of parts cannot be divided and must be manufactured in a single sample. This was addressed with four formulations of mathematical models, with variables that solved the multi-objective problem, concluding with computational tests that provided appreciable quantitative results. This was very similar to what was conducted by [46], with the difference being that flexible programming was considered with a preparation sequence that depended on the time that occurred for parts grouped in families.

There are also new proposals, such as the case of [47], who proposed a new heuristic based on Lagrangian relaxation; another possibility is [48], which offered a model that independently considered assembly times, sequences, orders, and machines; or [49], which proposed a bee colony algorithm, for which the convergence curves of the algorithm demonstrated not only a high processing speed but were also accompanied by competitive solutions concerning other techniques. Other studies are more avant-garde; for example, ref. [50] presented a hybrid algorithm that combined a colony of artificial bees and a tabu search by testing sets of instances with high workloads that aimed at a realistic production, obtaining results that corroborated the algorithm’s high efficiency. Another notable proposal is that of [51], which considered synchronous movements, where work moves between machines due to a single synchronous transport supported. Other reviews provided information related to task sequencing and setup times in the shop environment [52] or the study of exact solution methods, such as branching, delimitation, and restriction spreading, and the objectives were to minimize both the manufacturing time and mean process time [53].

More current studies are in the line of studying approaches that are novel for combinatorial analysis, as is the case of [54]. The objective was to determine the dominant properties between the permutation flow-shop (PFS) and non-permutation (NPFS) programs, developing a theoretical approach to graphics to describe the sets of operations that define the feasible duration PFS and NPFS programs. Another case is that of [55], in which, under the context of industrial control networks, it proposed a flow-shop scheduling generation model that guaranteed that no frames were lost during network updates, and experiments showed that the mechanism achieved zero frame loss without an updated overhead additional compared to existing methods. In [56], a multi-objective mixed integer programming model was proposed for the energy-efficient programming of the distributed weld flux shop based on three subproblems assigning jobs between factories, scheduling jobs in each factory, and determining the number of machines in each job; for the tests, diversified indicators were used to evaluate the proposed algorithm and the performance of others. Recent studies have also gone along the lines of improving multi-objective harmony search algorithms and a Gaussian mutation to solve flexible flow-shop scheduling problems with a sequence-based setup time and transport time, and possible rework, demonstrating the effectiveness of the proposal compared to existing algorithms for solving multi-objective problems [57]. Finally, other current published studies went along the lines of work with multi-objective mixed integers for the programming of the hybrid flow-shop, focusing on energy efficiency [58], enhanced hidden Markov models to formulate the problem and generate an optimal autonomous manufacturing task orchestration solution for a given process flow-shop [59], and artificial bee colony hybrid algorithms to solve a batch-distributed flow-shop problem with deteriorating jobs [60].

Finally, in [61], the author used a reinforcement learning algorithm based on a network of pointers that attacked the problem that the computation time in metaheuristic algorithms increases exponentially as the number of input objects increases, and computations are required separate optimization tests for each issue; the above was applied to the sequencing process to determine the order of incoming products to minimize the duration of a shipbuilding process. The results obtained showed an improved performance compared to heuristics and a similar performance compared to metaheuristics in terms of reducing makespan; in terms of computation time, the pointer network provided an immediate solution in real-time compared to metaheuristics, which take more time. Moreover, in [62], they address the problem of minimizing the total delay for the programming of two machines restricted to the publication dates of the works and the known periods of unavailability of the machines, considering, as in [61], that the majority of the existing studies that solve the problem are of a heuristic nature. There are no approximate algorithms or behavioral limits. They developed five metaheuristics, all of which used population-based approaches. Of the algorithms developed, the so-called “Imperialist Competitive Algorithm” proved to deliver better results in all evaluation contexts.

In this sense, the reviewed works give evidence that, regardless of the different techniques used, they generally find solutions that improve the final application times in benchmark-type comparisons, pending tests in real production environments and the estimation of the economic impact of executing these plans.

2.3. Applications in the Industry

In [14], a solution to the real problem of a self-adhesive label company was presented. The problem translated into solving a two-stage flexible flow-shop problem, for which the first stage consisted of a machine and a sequence that depended on set-up times, while in the second stage, there were two identical parallel machines. In this experiment, the results obtained with small samples were compared, and the actions to be taken for their future implementation in the company were proposed. Meanwhile, in [63], the multi-processor stream storage problem was addressed using a tabu search simulation-optimization approach. The stochastic modeling ability of the simulation of discrete events was presented as the local search algorithm’s tabu that applied the results. Finally, the work converged on a case study based on a multilayer ceramic capacitors’ manufacturer using a methodology proposed by the authors. The results obtained were promising at the practical level, leaving the stage open for future experiments.

Another study corresponded to [64], in which the authors considered a flow-shop scheduling problem with sequence-dependent setup times in an uncertain environment. The authors proposed two different approaches to deal with the input data variability, firstly working with robust optimization, and this was contrasted with fuzzy optimization. This work provided an exciting approach to an actual case study at the Tehran-Madar company, a producer of printed circuits and electronic equipment manufacturers. Finally, after obtaining the results, a comparison was made between the approaches to determine which is the most appropriate for the industry in which the case study is being developed. On the other hand, in [65], disruptions that were inherently stochastic were considered, similar to unexpected events that occur in real industries. For this reason, reprogramming methods were proposed so that outages did not appreciably affect the identified solution. Additionally, this work was accompanied by a case study in the petrochemical industry to show the approach’s proposed performance.

In [66], a computer system was approached with a server in the first stage and two identical processors in the second stage, with a “no waiting” restriction between stages and where all the work that went through the first stage had the same processing time. The objective function minimized the sum of the total work time. The authors proposed an algorithm for whole processing times that solved the problem in polynomial time, where the work was ordered by SPT (least processing time) and was assigned according to the FMA criterion (first available machine). Meanwhile, in [13], an application of a practical case on an industrial scale was presented, considering the computational processing times and the precision of the results. Therefore, a new hybrid algorithm was proposed to improve planning times for assembly operations. Hybridization between genetic algorithms and tabu-search was performed to apply dispatch rules later to work orders. In the implementation of [50], the problem of flow-shop scheduling was addressed in a hybrid workshop for the back-end assembly of semiconductors, considering supply and demand as limiting. This job considered that an order can be divided into separate jobs to be processed in parallel and then merged to be considered complete, ultimately reducing flow-shop time. Several elements were included in this work: a simulation model for performance evaluation, an optimization strategy with the application of a genetic algorithm, and an acceleration technique through an optimal computational budget allocation. Another case developed by [67] presented the Colorama case, solving the problem of an enterprise digital printing configuration with a processing scenario for different papers and packaging supported by machines that generate a response to the adopted production configuration.

Another industrial case corresponds to a flow-shop proposal for the agricultural industry aimed at improving post-harvest resource planning and applying it to a floriculture company [68]. Meanwhile, in [69], a flexible hybrid flow-shop production scheduling methodology was proposed by using genetic algorithms. Flow-shop applications can also be found in the education domain, as in the case of the Agustitiana University in Colombia, where a sequencing approach using Johnson’s algorithm was presented for a problem similar to a flow-shop programming system [70]. In this context, the classrooms were used as lively and active didactic cases that yielded results that revealed the teaching and learning processes’ discernment. In [71], a high-impact industrial application was presented, where the continuous scheduling of steelmaking under a hybrid schedule was considered. For this, controlled processing times were considered, with multiple objectives related to total waiting time, delays, anticipations, and adjustments in production costs. For this, a hybrid differential evolution algorithm combined with a variable neighborhood decomposition search was used, and the effectiveness of this two-layer optimization approach was validated by computational experiments on planning instances designed in the real world. Finally, there is a specific case study of the Chilean industry related to the scheduling of production in copper smelters for CODELCO Chile [72].

Currently, you can find studies such as that of [73], in which, in the context of an automobile manufacturing company, they studied the problems of programming and the sizing of lots of multiple levels and stages with the updating of the demand information to determine the quantities and production sequences in various stages of production to minimize total production and inventory costs. They made comparisons of the heuristics and selected the one that delivered the best results, giving a satisfactory answer to the proposed objective. In [74], by studying a hybrid flow-shop scheduling problem with limited dampers and two process paths that came from an engine hot test production line in a diesel engine assembly plant of an engine company, the effectiveness of the proposal was validated in three groups of instances and a real-world industrial case, giving excellent results. Finally, in [75], we worked with the stochastic programming of a robotic flow-shop cell of two machines with controllable inspection times, applying this to the machinery parts’ maintenance industry and organization.

In this way, it is important to highlight that the existing works applied to the industry are limited; most of them develop pilot plans in which the real production process is not intervened; therefore, there are fewer sources to compare. For this reason, this study greatly contributes to showing a real application in the industry.

3. Case Study

3.1. Context

A practical study case was applied to Biomédica Ingeniería Ltda., a business dedicated to electromedical equipment repair. This business is responsible for the repairs and maintenance of the medical equipment of Dr. Gustavo Fricke Hospital in Viña del Mar, Chile. The hospital has more than 1300 units of electromedical equipment of high, medium, and low complexity. It is projected that, within one year, the completion of construction of the new hospital at the same location will increase this type of equipment to approximately 4000 units [76]. The fitting out of the new hospital compound was accelerated due to the national contingency due to the health emergency caused by COVID-19. The principal problem elements are: (1) there is a finite number of equipment (designated work); (2) there are a limited number of technicians; (3) every piece of equipment is different and requires using the technicians for its repair; and (4) an estimation is made for the production of the equipment that requires repair or maintenance.

Therefore, this business’s purpose is to evaluate using a risk-oriented maintenance plan consistent with the repair and maintenance priority assignments, starting with a critical evaluation of each piece of equipment [77]. There is equipment that have a low-risk level that were not included in the maintenance plan and were serviced at the user’s request or only when they failed (corrective maintenance). It was indicated that, if maintenance is limited to critical equipment, it becomes unmanageable or inefficient; therefore, preventive maintenance inclusion criteria are recommended, based on risk-oriented criteria, using the Fennigkoh and Smith criterion [78]. The medical equipment repair process, which is the MEU’s (Medical Equipment Unit) responsibility, fulfilled a process series linked to the equipment’s life:

• Failure analysis: This process corresponds to a set of activities that the technician/engineer must perform to determine the failure or problem that the equipment presents when it is sent to MEU.
• Sanitation: As an equipment repair process, this process may be a component or be linked to another relevant equipment repair process.
• Discharged equipment: This process may result from sanitation, because the repair may not be possible for several business reasons. In addition, it may be a failure analysis result related to complexity or a critical piece that the provider no longer manufactures. Therefore, the equipment must be discharged. It is relevant to note that there are cases where the equipment is released but may still be working for an indeterminate period in a health service clinical unit.
• Preventive maintenance by opportunity (PMO): This type of maintenance is performed for the equipment when it arrives at UEM or is repaired. It consists of checking the functionality and all the equipment parts. It is known as preventive by opportunity due to its preventative maintenance that is not planned the day the equipment is received, but due to the high demand and low amount of electromedical equipment in the country’s public health sector, this maintenance is performed mostly in later cases, because functioning equipment cannot be stopped to perform the normal maintenance.
• Sending for purchase parts or external analysis: This process corresponds to repairs that depend on an external factor, whereby it cannot be charged to the MEU repair or compliance planning, which are committed. In this case, MEU acts as a technical counterpart between a repair service provider or spare part seller and the hospital.
• Spare installation: When a spare change is required, this process considers removing the spare, replacing it with a new one, and testing the repaired equipment. Later, the process entails a PMO.
• Chilean Health Ministry (MINSAL) registration guide service: After all the activities are performed, without regard to operational results, these activities performed on the equipment must be registered in a consolidated statement.

The situation for the biomedical engineering study entails the preparation of the corresponding weekly work planning; this planning is realized considering the reason for service demanded in a determined timeframe. The process starts when the equipment is received to be repaired, and it is a requirement to have the reception and registry of equipment that enters the MEU for a process that was not considered. Some subprocesses that are considered at this stage are (1) physical equipment reception; (2) work order reception/filling; and (3) processing and/or assignment of one or more service guide(s).

Then, it becomes necessary to establish the pre-assignment failure analysis, which is conducted for a work request indication (work order), initiated from the hospital clinical unit. With this, it proceeds to generate the weekly planning activities to perform, depending on equipment needs or requirements and considering that the stages for equipment are always the same and have the same order. In addition, some can have zero duration, e.g., buying spare parts. The work order flow-shop storage is detailed in Figure 1.

The secretary considers the following records in the flows, always considering that this is a case in which all the flow occurred correctly with no mishaps.

At the moment the secretary or MEU technician are satisfied with the work order formality, they would perform and create at that time the following records, entailing the entry of the preliminary information, in their two available information systems: (1) the secretary or technician must write his name, his last name, and the reception date in the work order; (2) workbook “Administrative clinical units and services work order entry and assignment”; and (3) MS-Excel spreadsheet: Clinical unit and services WO (Work Order) entry.

In addition, the UEM secretary would make the following physical records (Book “Administrative clinical units and services work order entry and assignment”) and virtual records (MS-EXCEL spreadsheet).

The different flow-shop problem elements and how they are reflected in the application are: (1) technician: analogous to a machine, in charge of processing a task in a determined time; (2) work: equipment composed of different tasks; (3) task: composed of a time, and it has to be processed by a technician in the assigned time; and (4) planning: resource assignments that define what a machine must process and which task is performed in a determined time instance.

3.2. Domain of the Problem

The planning problem’s implementation was applied to the programming of electromedical equipment repair times in a Chilean hospital. The variables that were part of the problem domain are defined below. The secretary considered the following records in the flows, always considering that this is a case in which all the flow occurred correctly with no mishaps. The work orders flow storage is detailed later in the case study application. The next equation shows that, adding the final times minus the initial times of every single work (subtasks) that composed a repair and adding the delta defined by every activity implicitly to all the defined positions of human resources, established the final repair time to the corresponding planning.

$$t_{repair}={\displaystyle \sum }_{p=1}^k\left({\displaystyle \sum }_{j=1}^{m}t{\left(f\right)}_{pj}-t{\left(i\right)}_{pj}\right)+\Delta$$

(6)

where:

• t(f)_pj corresponds to the final time for a sub repair task of a medical team.
• t(i)_pj corresponds to the initial time for a sub repair task of a medical team.
• $\Delta$ corresponds to the time not occupied in repairs, which in some cases is negligible, calculated as:

  $$\Delta ={\displaystyle \sum }_{f=1}^n{t}_f$$

  (7)
• t_f represents the time that is used in other tasks that are not considered proper repair.

Considering all the above, it is possible to calculate the executed production planning corresponding times for the hospital.

3.3. Proposed Solution

3.3.1. General Overview

The implemented system has the functionality to receive equipment, to repair as input data, and to provide a planning schedule that is as close as possible to the optimum. The generated functional elements inside the solution flow are shown in Figure 2.

The generated functional elements inside the solution flow are: (1) equipment to be repaired (inputs); (2) Johnson algorithm application (initial solution); (2) tabu search application (search of the optimal planning); and (3) planning (outputs).

In the development, technical aspects were implemented in Java 8 in a Windows environment, with an Intel I3 2, 8 GHz processor, and 8 RAM of memory. It was considered a record on a flat text file (*.txt) with the values to be used to solve the problem. These data are: (1) technicians; (2) task; and (3) times.

3.3.2. Initial Solution

The Johnson algorithm was used for an initial solution generation to which tabu search was applied. Generally, this algorithm is represented as:

$$p_i^k={\displaystyle \sum }_{j=1}^k{p}_{ij} i=\mathrm{1..}n, k=\mathrm{1..}m-1$$

(8)

The Johnson algorithm is composed of a limited number of the following steps:

• Step 1: Make a list with all the tasks and two more (one for each machine). The first machine’s list is filled from left to right, and the second one is filled from right to left.
• Step 2: Find the shortest processing time task (p). Draws can be randomly broken.
• Step 3: If the time corresponds to the first machine, put the task on the first list. If it corresponds to the second machine, put the task on the second machine list.
• Step 4: Repeat until the task list is empty.

The algorithm identifies the machines that can be stopped in the shortest time. The obtained sequence processes the tasks in machine 1 that must go through machine 2 and then those that only have to go through machine 1. At the same time, machine 2 processes those tasks that only have to go through machine 2 and, later, only those that have an operation in machine 1. An optimal sequence is achieved by concatenating the first list and second list tasks. It is relevant to perform an analogous comparison between the machines and the technicians with the similar activities of machines to medical equipment repair. The work is sequentially read, and it is decided how it directly depends on each task specialty that is appropriately assigned to each technician. In this case, the selection and combinatorial operation only determine the task order for the realization and sequential repair of the electromedical equipment.

3.3.3. Tabu Search

Tabu search is a heuristic technique that may be used in combination with another searching method to solve combinatorial optimization problems [79] that have a high level of difficulty. This study only referenced the use of tabu search defined by [80] as a heuristic goal to improve the result (final makespan) provided by the planning schedule to find a result as close as possible to the optimal solution. This is because tabu search allowed us to move between solutions even if it is not better than the current one; in this way, we could get out of local optima and continue with the solution search execution.

Therefore, it used the operators subsequently defined to promote better solutions that complied with the objective of minimizing makespan C. In addition, it complied with the corresponding restrictions. It provided a size seven general tabu list, in which the movements used were saved with anteriority, and the Johnson algorithm’s initial solution was improved.

Table 1. Tabu search algorithm [67].

{n is number of iterations; t size tabu list; s number of electromedical equipment} BEGIN
while i <= n then
while j <= s then
search_best_solution (electromedical_equipment_j)
compare_solutions (list_of_solutions)
if (solution_is_better) then
if (solution_is_factable) then
select_solution (electromedical_equipment_j)
end_if
end_if
j = j + 1
end_while
apply_solution (electromedical_equipment_j)
add_movement (list_tabu)
i = i + 1
end_while
end_Begin

shows a tabu search algorithm used to improve the initial solution, taking into account an electromedical equipment repair case study.

The tabu search central factor is located in the process called short-term memory, and many strategic considerations of its fundamental aspects reappear in an amplified way but without big modifications in processes called long-term memory. The tabu search is a short-term memory component that is an aggressive exploratory procedure that tries to achieve the best possible movement, and it is subject to the restrictions imposed by the problem [79].

The different restrictions on the problem were used to avoid certain movements, and certain activities could become forbidden, e.g., technical load limitations, breakdown of repair work on the same equipment, or criticality of repair. It is relevant to know that the main objective of this movement was to ensure that this searching technique surpassed local optimal solutions while simultaneously considering the identification of better solutions in every iteration. If the above description does not apply, the solution moves to a point outside of the local optimum but risks moving to the same proposal point described as the solution movement later.

The best possible movement selection process is composed of many stages. First, all movements inside the feasible movement’s list are evaluated, and these usually have the restriction that the candidate movements must improve the solution makespan because the planning delta calculation is performed before applying the movement, and the result applies the candidate movement. The use of those positive deltas is considered, and those that generate a greater compression in the total processing time are prioritized.

The process of deciding the best movement, as shown in Figure 3, was more complex than a simple evaluation. In particular, it required performing a previously described estimation of all the possible combinations, which were simultaneously coupled to its execution feasibility, makespan improvement, and forbidden movement evaluation, which were generated in different iterations. These forbidden movements are called the tabu list [79].

In this approach, the next movement analysis can be included in the feasible movement selection. However, in certain cases, if the tabu list restrictions and selection criteria do not comply, then an allowance is made to select movements that contribute to reducing the total time of the makespan. In the analysis of the work performed in this case, the solution that was not feasible was not used for movements applied to the solution. For this, each solution belonging to the current iteration was temporarily stored, in order to later compare them all and verify which of them achieved a feasible and greater improvement to the final makespan.

3.3.4. Operators

The operators referred to the movements or the sub-algorithms that enable tabu search (TS) to find a solution that improves the obtained initial solution for makespan. It must be considered to apply to any operator as a sequence to which it will always be referred. Firstly, it was necessary to identify the sequences that correspond to the technicians who repair the equipment and that the sub-programming assigned to the machine. Then, movements were defined as interchanges that occur between work entities belonging to the same machine.

Two movement types, MI(a,b), correspond to realized movements, where M_a work is interchanged to be in front of an M_b work; this movement is defined as a left movement. Likewise, Md(a,b) is defined as a movement that takes mama work after mbmb work; this movement is defined as a right movement.

Then, using neighborhood trees, a simulation of all possible movements was performed, relating them to their makespan. In addition, it must be considered that sequences always have to maintain precedence between work entities that belong to a task. Finally, complying with these rules, the best movement selection was applied to the selected sequence (those that mostly minimize the makespan), and the tabu list was updated.

Table 2. Solution selection [67].

{ n is the number of sequences or machines } BEGIN
/ * iteration of SBP * /
while i <= n then
s = searchAttachment (machine_i)
/ * GLS iteration * /
while (exists_solution)
solution = possible_tree_solutions(machine_i)
if solution is feasible then
solucionfactible = solution
end_while
if makespan(solution_power)<=makespan(machine_i) then
apply solution to machine
i = i + 1
end_while
end_Begin

presents the solutions selection algorithm.

3.3.5. Software Data

The results were expressed as output in a plain text format (*.txt). The total initial time of planning was provided, which entails the test execution and assignments of each technician. The parameters were defined within the built software application context, and provided, within reason, a planning solution to be tested, considering: (1) work number to which a task belongs; (2) processing order must have a task; and (3) consumed time (final time minus initial time).

In addition to the application’s general use, the considered parameters are: (1) iterations number for tabu search algorithm; (2) number of technicians; and (3) number of tasks by work to perform (these must match the total processing orders).

The size, number, quantity, and equipment types that UEM handles and the entire equipment registered by the hospital inventory were obtained, corresponding to 1537 pieces of electromedical equipment (used inventory in: http://bit.ly/3H4xqd2, last update: 1 October 2022.).

4. Results and Discussion

4.1. Configuration

Simulations were executed in the Java 8 environment, using plain text input data from every planning schedule, as presented in [67]. For the performed tests, the implementation was performed for the available equipment selection for repair, resulting in an equipment volume of 60 units, considering the following:

• The entire inventory, from which a sample was selected and obtained for the equipment used in the study case.
• MEU planning was performed for a series of equipment, using these same series for the test.
• Had a time estimation for every piece of equipment and its tasks.
• The technician did not notify the next task technician of the consumed time during the execution.
• All equipment presenting as failed must be repaired; this ensured that there was time to perform test data selection.

It is important to consider that, to perform the tests, the effectiveness of the implemented solution was measured for improving the initial planning generated by the biomedical engineer, and it was not evaluated based on their computational performance at the processor time and on the hardware.

4.2. Performed Tests

This work performed six test sets that were grouped in pairs (

Table 3. Set of entry and expected times.

Test ID	Initial Makespan (Hours)	Quantity of Equipment	Task to Plan	Technicians Available
1	15	5	20	4
2	23	5	20	4
3	45	10	40	4
4	47	10	40	4
5	48	15	60	4
6	55	15	60	4

). The pairs corresponded to the same equipment quantity but with different equipment between them, generating a combinatorial approach between the available equipment to repair. In addition, for each test set, a planning schedule that was developed by the biomedical engineer to the hospital was relied on, which was generated with past Microsoft Excel planning methods. The quantity of the equipment used by every test and the initial makespan prepared by the hospital are detailed in Table 3. In all input schedules, a major performance improvement was expected due to the initial planning that was performed with basic formulas based on the delayed history. The distribution and quantity of tasks to assign (a breakdown of steps that equipment must pass) and the technicians used for the tests are detailed in Table 3.

4.3. Results and Analysis

In

Table 4. Results and summary of improvements obtained.

Test ID	Initial Makespan (Hours)	Final Makespan (Hours)	Improvement
1	15	13.4	10.7%
2	23	14	39.1%
3	45	24.8	44.9%
4	47	36	23.4%
5	48	39	18.8%
6	55	37.6	31.6%

, the obtained results in the list are shown, using data sets previously discussed. The first column shows the applied tests, and the expected time is shown in the second column. The last column shows the objective times after using the planning provided by the application.

In comparison with other studies where the flow-shop problem was solved, it can be mentioned that the results are mixed, where there are cases such as in [81], in which the improvement ranged from 10–20% in comparison with the improvement in the makespan of our study, which in some cases reached 45%. Despite this, and considering the different contexts, there are other studies such as [82], where the aim was to improve delivery and payment dates, for which a biased random heuristic is applied, and then a metaheuristic with a variable neighborhood was used to carry it out finally a semi-heuristic through the incorporation of Monte Carlo simulations, demonstrating that the results were conclusive, exceeding a 40% improvement in the case of application and tests. In general, most of the applied techniques and algorithms manage to improve the initial makespan, highlighting studies such as that of [83,84] where techniques such as neural networks, multi-objective evolutionary algorithms with heuristic decoding, and graph theory were applied. In some cases, there are more discrete results, such as those of [85], in which artificial neural networks were used, compared to those obtained in our study.

Results showed that the planning generated through the algorithm implementation obtained final times that were lower than those of MEU planning generated using historical time records. Thus, the results generally provided a notable improvement in planning resources that the hospital must invest in for electromedical equipment repair or maintenance.

As seen in Figure 4, the graphical representation of a minor makespan compared to the initial was obtained in all the cases. The cases with major work levels were related to major difference because those tests had a significant equipment quantity. Therefore, the possibility of performing better combinatorial approaches, which could not be performed in the two initial difficulties due to the lack of a task quantity to organize and thus the limited possible movement quantity, led to considering the tabu list.

A percentage summary of the improvement obtained for MEU planning represents the time used for the improvement. It is possible to affirm that there is a relevant difference, considering that the unit of measurement is men-work-hours; thus, it is observed that:

• In general, some tasks were overestimated in the hospital MEU.
• To decompose tasks and emulate their distribution between different technicians, they produced a major flexibility for planning the work of a particular technician.
• The preventive maintenance time by opportunity was underestimated by MEU, given that this was used as an indicator by MINSAL.

It is important to mention that chaos exists, in which the difference between the expected versus the resultant approaches 45%, indicating a very high value for a labor environment. In addition, it can be seen that the ten equipment test sets accumulated a major difference between the expected versus the resultant times, while less difference was shown for the five equipment tests. On the other hand, as shown in Figure 5, in those work entities that had less possible quantity movements, for instance, those with less equipment quantity to plan, there was a major percentage improvement compared to the total result percentage improvement in those planning schedules that had more equipment and, hence, a major combinatorial and iteration quantity product of the possibility of interchanges to perform. It is important to note that, about the performed implementation, it is a highly used algorithmic technique to answer the flow-shop type of planning problems, and it was contextualized in a real study case that considered poorly elaborated production planning with strategic level deficiencies inside the hospital.

4.4. Impacts on Production Costs

About production costs, currently, in the medical equipment unit, the value of the person-hour paid on average to each technician is USD 13.3.

Table 5. The apportionment of repair costs.

Test ID	Initial Cost (USD)	Final Cost Post Experiment (USD)
1	199.5	178.22
2	305.9	186.2
3	598.5	329.84
4	625.1	478.8
5	638.4	518.7
6	731.5	500.08
Total	3098.9	2191.84

presents the apportionment of repair costs in each test applied previously.

From Table 5, it can be seen that there was a reduction in the costs involved in using the technique presented in this work in all experiments. Therefore, at a productive level, the company would save USD 907.06 in man-hour costs, which, if extrapolated to the monthly work and the number of technicians hired, it can be established that:

• By applying the model proposed in this paper, you can save between 10% and 44% of production costs depending on the case.
• Considering the monthly load of the technicians, their production can increase on average by 28%, which means a saving of USD 595.84 for each one of the technicians.
• Knowing that in the unit of medical equipment of the hospital used in this work there are on average seven technicians hired, the savings that could be generated in repairing the same amount of medical equipment would be USD 4170.88.
• In male hours, with the savings generated, you can hire two extra technicians to support the work of the MEU.

Finally, it is also important to point out that all the savings in production costs that could be generated in a Chilean hospital have a great effect on the use of resources and the attention given to users, which is why the work presented may infer a social impact that may be the reason for study in another work.

It is in this way that the proposed solution is in line with other applications, as in the case of [86], where the generation of solutions and the calculation of the objective function for the two-dimensional guillotined shear problem were facilitated, where, in our case, the solution obtained through tabu search managed to be a facilitator to obtain the production planning for the repair of medical equipment. Moreover, this improvement is also evident at the time level resulting from the algorithm.

5. Conclusions and Future Work

For planning and production control problems, managing the course of events and their core elements is recognized as complex a problem inside organizations, particularly when facing scarce resource assignments needed to achieve client satisfaction. Therefore, it is important to develop methodologies that allow the improvement of these processes. Within the essential characteristics of this production system, the flow-shop approach became a problem representation approach that helped to manage the maintenance of serial production elements, in which products are diverse and stations that must be used for production process completion are previously defined. Thus, flow-shop production provided the necessary steps to achieve a product that met the requirements determined by the client, knowing that this compliance must also be compatible with other key resources, such as time.

This work presented a practical approach to flow-shop application for public health planning, including a state-of-the-art review of diverse applications in the industry. As a solving method considering the problem’s complexity, tabu search was implemented through a Java-built application. Finally, a series of experiments were performed based on real data provided by the selected study case with its respective analysis. The obtained results demonstrated a reduction of total work time, minimizing production costs and further allowing the better schedule and assignment of new tasks, improving the distribution of estimated labor charges for technicians. It was imperative to analyze repair assigned times for which large differences were seen. These results also showed a real opportunity for MEU-generated planning improvement. They motivated better electromedical equipment planning, resulting in an immediate increase in repaired equipment availability at the health service.

The presented solution has some limitations and further applications that can be considered for future work. For example, interruptions were not considered, which could occur in equipment failure situations in which it is necessary to contact the manufacturer or that may not be directly detected. Including these elements should require reviewing the mathematical model and the implementation itself and to explore other problem variants, such as flexible flow-shop, which is closer to a real health domain. On the other hand, results indicate that the application of planning algorithm techniques in lesser-explored businesses, particularly in Chile, such as medical equipment repair, could be conveniently extrapolated to other health systems or industrial areas. One should consider scenarios with poor distribution problems regarding product and service provision under diverse conditions. Other domains include automotive repair, industrial machinery, or initial flow-shop-based production systems.

Finally, other non-technical issues should be considered for these study cases. There could exist psychosocial effects on workers as a result of stricter planning schedules of execution times, which could be an influential factor for the occupational health of operators. Exploring different labor charge rolling techniques is possible, which are not measured variables in this work, but they could impact the performance of equipment repair. Finally, it is also relevant how people who will lead productive processes are getting used to these technological approaches. It is important to provide proper introductions and an overall perspective regarding how these innovations support their leading role and could impact productivity indicators for their teams.