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Optimization of Preventive Maintenance Program for Imaging Equipment in Hospitals
Zdravko Kravanja, Miloš Bogataj (Editors), Proceedings of the 26th European Symposium on Computer Aided Process Engineering – ESCAPE 26 June 12th -15th, 2016, Portorož, Slovenia © 2016 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/B978-0-444-63428-3.50310-6
Optimization of Preventive Maintenance Program for Imaging Equipment in Hospitals Danahe Marmolejo-Correa,a* Rosa G. Juarez-Valdivia,b Alejandra RodriguezNavarrob a
Department of Physics Engineering, University of Guanajuato, Loma del Bosque 103, Colonia Lomas del Campestre, MX37150, León, Guanajuato, México. b
Department of Chemical, Electronics and Biomedical Engineering, University of Guanajuato, Loma del Bosque 103, Colonia Lomas del Campestre, MX37150, León, Guanajuato, México. [email protected]
Abstract The goal of this work is to obtain the optimal schedule for the distribution of human resources that provide Preventive Maintenance (PM) on Medical Equipment (ME) in thirteen Hospitals and Medical Imaging Centers (H&MICs). Only Computed Tomography and Magnetic Resonance Imaging Devices were selected for this study. The abovementioned types of ME are commonly found and used in General Hospitals in major cities. When the problem is stated as a Travelling Salesman Problem (TSP), and additionally combining two optimization techniques: the Shortest Route Method (SRM) and the Tabu Search (TS), the optimum schedule may be obtained. The thirteen hospitals are located in four different cities in the Center of Mexico. Keywords: Preventive Maintenance, Optimization, Shortest Route Method, Tabu Search, Computed Tomography, Nuclear Magnetic Resonance.
1. Introduction Mathematical Programming has shown important improvements to the total efficiency, quality assurance, and reduction to the Total Annual Cost (TAC) of numerous industrial processes, as well as solving numerous supply chain management problems. GonzalezSilva and Hernández (1996) state that in the health sector Biomedical Engineers (BE) are tasked with encouraging the financing and establishment of protocols for periodic Preventive Maintenance (PM) and replacement of Medical Equipment. Optimization techniques promote the correct and efficient management of Hospitals and Medical Imaging Centers (H&MIC). Nowadays, an efficient PM plan and schedule for Medical Equipment and Devices (MED) upkeep is very important. In Mexico, heuristic solutions are still favored in many labor sectors. The lack of information and diffusion of optimization techniques in small companies is due to their perception as large and unnecessary expenses. PM services are usually offered and carried out by three different actors in H&MICs:
1834
D. Marmolejo-Correa et al.
• The provider or manufacturer of the MEDs, when the device’s warrantee is still valid. • Independent companies (Outsourcing) hired for this specific service. • The Department of Biomedical Engineering of the Hospital or Centre. This paper is a case study of an outsourcing company, the second option in the list above. The outsourcing company provides services for preventive and corrective maintenance of MEDs in thirteen H&MICs. However, the company lacks protocols for optimal scheduling that minimize the time and costs involved in performing PM. Before the optimization study was performed, this company would establish its routes using only heuristic considerations. In general, the Total Annual Cost for providing PM is composed of three components: • Transport (vehicle and fuel use) • Salaries and meals (which depends on the number of engineers on the road) • Parts (replacement of pieces of the devices) The first two components are directly dependent on the distance between H&MICs. This study aims to minimize the cost of raising orders for PM. The cost is a function of the road-time of trip (distance/average velocity) and time for raising the order. The optimal schedule for the actual MP will be presented in subsequent work. 1.1. Problem Statement: A leader and pioneer company in the manufacture and maintenance of high-level medical technology has Biomedical Engineers (BE) trained for providing preventive and corrective maintenance to a variety of MEDs. The BE of this pioneer company have been given the task of making the first PM for Computed Tomography (CT) and Nuclear Magnetic Resonance (NMR) devices. These MEDs were purchased by thirteen H&MICs located in the Bajío Region in the Center of México. The first PM should be conducted two years after acquisition, within a maximum window of 2 months. The first stage of the PM is conducting a diagnostic of the equipment. The second stage is performing the actual maintenance. It is estimated that the maximum time for raising the order on these two kinds of equipment is close to 60 minutes providing it is done by two BEs, order time can be almost double if only one BE is assigned to the task. However, spending on the salaries of two skilled engineers means a significant increase in the labor cost, so many try and seek only one BE. Presently, the company is not interested in minimizing the number of days used for completing the tours. Thus, a problem statement that reflects the objective of the work presented in this paper can be written as follows: “Optimize the schedule of a team dedicated to raising orders for PM on N MEDs located in N different H&MICs by minimizing travel, fuel use, and labor costs”. The following section describes the methodology used to provide an optimal solution to the abovementioned problem.
Optimization of Preventive Maintenance Program for Imaging Equipment in Hospitals
1835
2. Methodology 2.1. Travelling Salesman Problem (TSP) The case study can be put in the form of the well-known TSP. Comprehensive reviews regarding TSP can be found in several papers such as, Laporte (1992) and Lenestra (1975). The main characteristics of the TSP are listed as follows: • The objective is to minimize the distance between cities visited. • All cities should be visited only once. • The tour is closed, i.e. the origin and destination are the same location. The mathematical program is defined by the Equation (1). n
n
Min z = dij xij i =1 j =1
subject to n
x
ij
= 1, ∀i = 1, 2,..., n
j =1 n
x
ij
i =1
(1)
= 1, ∀j = 1, 2,..., n
xij = {0,1}
where dij is the distance from city i to city j, dij =¥ when i = j and dij = dji. 2.2. Tabu Search Algorithm (TS) The TS is a Metaheuristic Algorithm designed to escape from a local optimum by allowing flexible movements. The TS selects a new search movement in such a way that temporally forbids the evaluation of previous solutions. The basic TS is composed by the next elements. • Tabu List: This is the instrument that lends a short-to-medium size memory to the algorithm. The List “remembers” and disables movements from previous searches. These disabled movements are referred to as Tabu Moves. • Tenancy Period: This is the time (i.e. number of iterations) that the Tabu List disables the Tabu Moves. Taha (2012) gives a comprehensive and general procedure for the TS Algorithm. • Step 0: Select an initial solution s0 ∈ S . Initialize the Tabu List L0 = ∅ and select a list tabu size. Establish k = 0 . • Step 1: Determine the neighborhood feasibility N ( sk ) that excludes inferior members of the tabu list Lk . • Step 2: Select the next movement sk +1 from N ( sk ) or Lk if there is a better solution and update Lk +1 . • Step 3: Stop if a condition of termination is reached, else, k = k + 1 and return to 1. 2.3. Location of the Hospitals and Imaging Medical Centers The H&MICs are located in the Center of Mexico, in the so-called Bajío region. The H&MICs selected for this case study are situated in four mid-sized cities: León, Irapuato, Salamanca and Celaya, all in the State of Guanajuato. Table 1 shows the actual distance between medical centers and Figure 1 shows the total network of the
1836
D. Marmolejo-Correa et e al.
Seven H&MIC Cs are located in León, the other o H&MICs in the abovemenntioned cities. S h two H&M MICs each. Thee central office (OFL) is loccated in León. The three cities have label for each h location is constructed by the letter H neext to the first letter of the k city where the ith h H&MIC is located.
Figure 1. Co omplete Networrk Superstructuree for the H&MIC Cs Table 1 Shorteest distances betw ween Medical C Centers (103 m). HL1 HL2 HL3 HL4 H HL5 HL L6 HL7
HI2 HS1 H HS2
HC1
HC2
9.3
1 12.0
9.9
12.0
18.0 75.0 73.0 93.0 92.0 9
137.0
1 132.0
7.0
7.0
1 10.0
5.0
5.0
10.3 67.3 65.3 85.3 84.3
129.3
1 124.3
¥
1.0
2.0
2.0
4.0
9.4 66.4 64.4 84.4 83.4
128.4
1 123.4
¥
2.4
0.5
3.0
8.7 65.7 63.7 83.7 82.7
127.7
1 122.7
¥
2.9
5.0
6.0 63.0 61.0 81.0 80.0
125.0
1 120.0
¥
2.1
8.1 65.1 63.1 83.1 82.1
127.1
1 122.1
¥
6.0 63.0 61.0 81.0 80.0
125.0
1 120.0
¥ 61.0 59.0 79.0 81.0
123.0
1 117.0
72.0
66.0
OFL
7.7
8.6
HL1
¥
HL2 HL3 HL4 HL5 HL6 HL7 HI1
HI1
¥
2.0 27.0 28.0 2
HI2
¥ 30.0 31.0 3
79.0
74.0
HS1
¥
2.0
46.0
41.0
¥
47.0
42.0
¥
6.0
HS2 HC1
The company y usually has two t types of sservices: (a) Lo ocal and (b) R Regional. The local l service focuss on the H&M MICs located inn León and reggional service attend a the meddical centres in thee other three sm maller cities. For F both servicces, the followiing estimationss are used: 1. Time for lifting l the ordeer: 2 h if only oone BE is hiredd, 1 h with two BEs. 2. Salary forr the complete service (all 13 H&MICs): 133,000 Mexican Pesos (MXN). 3. Working hours h per day: 8 h. Overtime is paid doublee. 4. Fuel pricee 14 MXN per liter. 5. Vehicle peerformance and average veloocities: (a) in ciities: 13.7 km/L L and 30 km/hh and (b) in highhways: 19.2 km m/L and 80 km m/h.
Optimization n of Preventive Maintenance Program P for Im maging Equipm ment in Hospitals
1 1837
Thee actual cost is close to the 344 MMXN (withh extra workingg hours) and it is 6. distributed ov ver 3 days (seee Table 2). The optimal sequence and scenario s for liffting the PM orrder is sought. A soft restrictiion nal model (Eq. (2)). Eq. (3) regarding thee working hourrs must be addeed to the origin shows the rou ute time ti , j froom i and j as a function of the average veloocities v and thhe time it takes to finish one activity a (lifting of the order), ai = 1 h . The asssigned cost too each distancee is calculated by Eq. (4). In tthe case of oveertime, Eq. (2) is not includedd in the model, teex is the amounnt of extra workking hours in th hat tour while cex is the associated coost to the extra hours. n
n
t
i, j
xi , j ≤ 8
(22)
i =1 j =1
(33)
ti , j = vdi , j + ai
ci , j =
di , j ⋅ F p
+ ( tex ⋅ cex )
i≠ j
(44)
w hours are allowed, then t a weekly schedule shouuld be establisshed. If no extra working The completee algorithm waas implementedd in Wolfram Mathematica M vv.10.2.
3. Results For an initiaal approximatioon or solution,, one could co onsider that alll the H&MICss are visited in on ne tour. Thus, when there is no penalty co ost for extra working w hours,, the minimum disstance, cost, and a time are 3003.9 km, 14,24 45.8 MXP, 300.7 h, respectiv vely. This result im mplies that duriing 33 hours thhe BE is continnuously workinng, a condition that is not recom mmendable. Theere are 25 extrra hours that should be paid double (i.e. 2,000 MXN/h). Thhus, the total cost is 58,245.88 MXP. Figuree 2 shows the resulting pathh for minimum tottal cost. Accorrding to the preevious result, it i can be estim mated that at leaast 4 days of 8 wo orking hours arre needed for completing c the tour. Table 4 shows some off the Tabu Searchh iterations. Thhe minimum ccost is then 15,773.0 MXN N for the sequeence when the H& &MICs are orddered by cities which is also the safest soluution, since driv ving along highwaays is minimizzed as well. It should be meentioned that thhe TS was useed to heuristically rearrange the H&MICs H in thhe tours. The sh hortest route method m was useed to minimize thee distance in each tour. Thhe savings of using a 5 dayy-tour sequencce is 17,654.8 MX XN, close to 533% of the actuaal cost.
F Figure 2. Shortest tour: initial soolution.
1838
D. Marmolejo-Correa et al.
Table 2. Some results of the Tabu Search Algorithm Sequence
Dist. (km)
Time (h)
Total Cost (MXP)
OFL, HI2, HI1, HS2, HS1, HC1, HC2, HL7, HL1, HL6, HL5, HL3, HL4, HL2, OFL
303.9
30.7
58,653.4
{OFL, HI2, HI1, HS2 OFL},{ OFL, HS1, HC1, HC2, OFL},{ OLF,HL1, HL2, HL3, HL4, HL5, HL6, HL7 OFL}
522.9
34.0
33,427.8
{OFL, HI2, HI1, HL7, OFL}{OFL, HS1, HS2, HL1, OFL}{OLF, HC1, HC2, HL6, OFL}{OLF, HL2, HL3, HL4, HL5, OFL}
673.7
36.6
22,759.6
{OFL, HI2, HI1, OFL}{OFL, HS1, HS2, OFL}{OLF, HC1, HC2, OFL}{OLF,HL7, HL1, HL6, HL5, OFL}{OLF, HL3, HL4, HL2,OFL}
681.6
36.8
15,773.0*
{OFL, HC2, HC1, OFL}{OFL, HS1, HI1,HL7, OFL}{OFL, HS2, HI2,HL1, OFL}{OLF, HL3, HL4, HL2, OFL}{OLF, HI5, HL6, OFL}
734.7
37.8
18,919.4
*Optimum solution
4. Conclusions The optimum sequence of tours was found with a hybrid method of two methods: the shortest route method, using the Traveler Salesman Problem as basis, and the Tabu Search Algorithm for finding the best arrangement. According to the obtained results, the best sequence of tours should be scheduled over five days. The savings found are close 53% compared to the actual Preventive Maintenance Program. There is no need to hire a second Biomedical Engineer, however, less qualified help at a lower pay scale could be evaluated. The proposed methods can be used in various engineering fields, as well as more application in the field of biomedical engineering, implemented for optimizing hospital resources and the rotation and distribution of personnel and activities within health institutions.
References C.A. Gonzalez-Silva and Hernandez A., 1996, Maintanece Manual for Health Services: Equipment and Buildings, HSP-UNI/Operating Manuals, PALTEX, Vol. 2, Issue. 6, (In Spanish) G. Laporte, 1992, The Traveling Salesman Problem: An Overview of Exact and Aproximate Algorithms, The European Journal of Operational Research, Vol. 59, Issue. 2, pp 231-247. J. Lenstra, 1975, Some Simple Applications of the Traveling Salesman Problem, Vol. 26, Issue 4, pp 717-733. H. Taha, 2011, Heuristic Programming, Operations Research: An Introduction, Nineth Edition, Prentice Hall, Boston.
Optimization of Preventive Maintenance Program for Imaging Equipment in Hospitals Danahe Marmolejo-Correa,a* Rosa G. Juarez-Valdivia,b Alejandra RodriguezNavarrob a
Department of Physics Engineering, University of Guanajuato, Loma del Bosque 103, Colonia Lomas del Campestre, MX37150, León, Guanajuato, México. b
Department of Chemical, Electronics and Biomedical Engineering, University of Guanajuato, Loma del Bosque 103, Colonia Lomas del Campestre, MX37150, León, Guanajuato, México. [email protected]
Abstract The goal of this work is to obtain the optimal schedule for the distribution of human resources that provide Preventive Maintenance (PM) on Medical Equipment (ME) in thirteen Hospitals and Medical Imaging Centers (H&MICs). Only Computed Tomography and Magnetic Resonance Imaging Devices were selected for this study. The abovementioned types of ME are commonly found and used in General Hospitals in major cities. When the problem is stated as a Travelling Salesman Problem (TSP), and additionally combining two optimization techniques: the Shortest Route Method (SRM) and the Tabu Search (TS), the optimum schedule may be obtained. The thirteen hospitals are located in four different cities in the Center of Mexico. Keywords: Preventive Maintenance, Optimization, Shortest Route Method, Tabu Search, Computed Tomography, Nuclear Magnetic Resonance.
1. Introduction Mathematical Programming has shown important improvements to the total efficiency, quality assurance, and reduction to the Total Annual Cost (TAC) of numerous industrial processes, as well as solving numerous supply chain management problems. GonzalezSilva and Hernández (1996) state that in the health sector Biomedical Engineers (BE) are tasked with encouraging the financing and establishment of protocols for periodic Preventive Maintenance (PM) and replacement of Medical Equipment. Optimization techniques promote the correct and efficient management of Hospitals and Medical Imaging Centers (H&MIC). Nowadays, an efficient PM plan and schedule for Medical Equipment and Devices (MED) upkeep is very important. In Mexico, heuristic solutions are still favored in many labor sectors. The lack of information and diffusion of optimization techniques in small companies is due to their perception as large and unnecessary expenses. PM services are usually offered and carried out by three different actors in H&MICs:
1834
D. Marmolejo-Correa et al.
• The provider or manufacturer of the MEDs, when the device’s warrantee is still valid. • Independent companies (Outsourcing) hired for this specific service. • The Department of Biomedical Engineering of the Hospital or Centre. This paper is a case study of an outsourcing company, the second option in the list above. The outsourcing company provides services for preventive and corrective maintenance of MEDs in thirteen H&MICs. However, the company lacks protocols for optimal scheduling that minimize the time and costs involved in performing PM. Before the optimization study was performed, this company would establish its routes using only heuristic considerations. In general, the Total Annual Cost for providing PM is composed of three components: • Transport (vehicle and fuel use) • Salaries and meals (which depends on the number of engineers on the road) • Parts (replacement of pieces of the devices) The first two components are directly dependent on the distance between H&MICs. This study aims to minimize the cost of raising orders for PM. The cost is a function of the road-time of trip (distance/average velocity) and time for raising the order. The optimal schedule for the actual MP will be presented in subsequent work. 1.1. Problem Statement: A leader and pioneer company in the manufacture and maintenance of high-level medical technology has Biomedical Engineers (BE) trained for providing preventive and corrective maintenance to a variety of MEDs. The BE of this pioneer company have been given the task of making the first PM for Computed Tomography (CT) and Nuclear Magnetic Resonance (NMR) devices. These MEDs were purchased by thirteen H&MICs located in the Bajío Region in the Center of México. The first PM should be conducted two years after acquisition, within a maximum window of 2 months. The first stage of the PM is conducting a diagnostic of the equipment. The second stage is performing the actual maintenance. It is estimated that the maximum time for raising the order on these two kinds of equipment is close to 60 minutes providing it is done by two BEs, order time can be almost double if only one BE is assigned to the task. However, spending on the salaries of two skilled engineers means a significant increase in the labor cost, so many try and seek only one BE. Presently, the company is not interested in minimizing the number of days used for completing the tours. Thus, a problem statement that reflects the objective of the work presented in this paper can be written as follows: “Optimize the schedule of a team dedicated to raising orders for PM on N MEDs located in N different H&MICs by minimizing travel, fuel use, and labor costs”. The following section describes the methodology used to provide an optimal solution to the abovementioned problem.
Optimization of Preventive Maintenance Program for Imaging Equipment in Hospitals
1835
2. Methodology 2.1. Travelling Salesman Problem (TSP) The case study can be put in the form of the well-known TSP. Comprehensive reviews regarding TSP can be found in several papers such as, Laporte (1992) and Lenestra (1975). The main characteristics of the TSP are listed as follows: • The objective is to minimize the distance between cities visited. • All cities should be visited only once. • The tour is closed, i.e. the origin and destination are the same location. The mathematical program is defined by the Equation (1). n
n
Min z = dij xij i =1 j =1
subject to n
x
ij
= 1, ∀i = 1, 2,..., n
j =1 n
x
ij
i =1
(1)
= 1, ∀j = 1, 2,..., n
xij = {0,1}
where dij is the distance from city i to city j, dij =¥ when i = j and dij = dji. 2.2. Tabu Search Algorithm (TS) The TS is a Metaheuristic Algorithm designed to escape from a local optimum by allowing flexible movements. The TS selects a new search movement in such a way that temporally forbids the evaluation of previous solutions. The basic TS is composed by the next elements. • Tabu List: This is the instrument that lends a short-to-medium size memory to the algorithm. The List “remembers” and disables movements from previous searches. These disabled movements are referred to as Tabu Moves. • Tenancy Period: This is the time (i.e. number of iterations) that the Tabu List disables the Tabu Moves. Taha (2012) gives a comprehensive and general procedure for the TS Algorithm. • Step 0: Select an initial solution s0 ∈ S . Initialize the Tabu List L0 = ∅ and select a list tabu size. Establish k = 0 . • Step 1: Determine the neighborhood feasibility N ( sk ) that excludes inferior members of the tabu list Lk . • Step 2: Select the next movement sk +1 from N ( sk ) or Lk if there is a better solution and update Lk +1 . • Step 3: Stop if a condition of termination is reached, else, k = k + 1 and return to 1. 2.3. Location of the Hospitals and Imaging Medical Centers The H&MICs are located in the Center of Mexico, in the so-called Bajío region. The H&MICs selected for this case study are situated in four mid-sized cities: León, Irapuato, Salamanca and Celaya, all in the State of Guanajuato. Table 1 shows the actual distance between medical centers and Figure 1 shows the total network of the
1836
D. Marmolejo-Correa et e al.
Seven H&MIC Cs are located in León, the other o H&MICs in the abovemenntioned cities. S h two H&M MICs each. Thee central office (OFL) is loccated in León. The three cities have label for each h location is constructed by the letter H neext to the first letter of the k city where the ith h H&MIC is located.
Figure 1. Co omplete Networrk Superstructuree for the H&MIC Cs Table 1 Shorteest distances betw ween Medical C Centers (103 m). HL1 HL2 HL3 HL4 H HL5 HL L6 HL7
HI2 HS1 H HS2
HC1
HC2
9.3
1 12.0
9.9
12.0
18.0 75.0 73.0 93.0 92.0 9
137.0
1 132.0
7.0
7.0
1 10.0
5.0
5.0
10.3 67.3 65.3 85.3 84.3
129.3
1 124.3
¥
1.0
2.0
2.0
4.0
9.4 66.4 64.4 84.4 83.4
128.4
1 123.4
¥
2.4
0.5
3.0
8.7 65.7 63.7 83.7 82.7
127.7
1 122.7
¥
2.9
5.0
6.0 63.0 61.0 81.0 80.0
125.0
1 120.0
¥
2.1
8.1 65.1 63.1 83.1 82.1
127.1
1 122.1
¥
6.0 63.0 61.0 81.0 80.0
125.0
1 120.0
¥ 61.0 59.0 79.0 81.0
123.0
1 117.0
72.0
66.0
OFL
7.7
8.6
HL1
¥
HL2 HL3 HL4 HL5 HL6 HL7 HI1
HI1
¥
2.0 27.0 28.0 2
HI2
¥ 30.0 31.0 3
79.0
74.0
HS1
¥
2.0
46.0
41.0
¥
47.0
42.0
¥
6.0
HS2 HC1
The company y usually has two t types of sservices: (a) Lo ocal and (b) R Regional. The local l service focuss on the H&M MICs located inn León and reggional service attend a the meddical centres in thee other three sm maller cities. For F both servicces, the followiing estimationss are used: 1. Time for lifting l the ordeer: 2 h if only oone BE is hiredd, 1 h with two BEs. 2. Salary forr the complete service (all 13 H&MICs): 133,000 Mexican Pesos (MXN). 3. Working hours h per day: 8 h. Overtime is paid doublee. 4. Fuel pricee 14 MXN per liter. 5. Vehicle peerformance and average veloocities: (a) in ciities: 13.7 km/L L and 30 km/hh and (b) in highhways: 19.2 km m/L and 80 km m/h.
Optimization n of Preventive Maintenance Program P for Im maging Equipm ment in Hospitals
1 1837
Thee actual cost is close to the 344 MMXN (withh extra workingg hours) and it is 6. distributed ov ver 3 days (seee Table 2). The optimal sequence and scenario s for liffting the PM orrder is sought. A soft restrictiion nal model (Eq. (2)). Eq. (3) regarding thee working hourrs must be addeed to the origin shows the rou ute time ti , j froom i and j as a function of the average veloocities v and thhe time it takes to finish one activity a (lifting of the order), ai = 1 h . The asssigned cost too each distancee is calculated by Eq. (4). In tthe case of oveertime, Eq. (2) is not includedd in the model, teex is the amounnt of extra workking hours in th hat tour while cex is the associated coost to the extra hours. n
n
t
i, j
xi , j ≤ 8
(22)
i =1 j =1
(33)
ti , j = vdi , j + ai
ci , j =
di , j ⋅ F p
+ ( tex ⋅ cex )
i≠ j
(44)
w hours are allowed, then t a weekly schedule shouuld be establisshed. If no extra working The completee algorithm waas implementedd in Wolfram Mathematica M vv.10.2.
3. Results For an initiaal approximatioon or solution,, one could co onsider that alll the H&MICss are visited in on ne tour. Thus, when there is no penalty co ost for extra working w hours,, the minimum disstance, cost, and a time are 3003.9 km, 14,24 45.8 MXP, 300.7 h, respectiv vely. This result im mplies that duriing 33 hours thhe BE is continnuously workinng, a condition that is not recom mmendable. Theere are 25 extrra hours that should be paid double (i.e. 2,000 MXN/h). Thhus, the total cost is 58,245.88 MXP. Figuree 2 shows the resulting pathh for minimum tottal cost. Accorrding to the preevious result, it i can be estim mated that at leaast 4 days of 8 wo orking hours arre needed for completing c the tour. Table 4 shows some off the Tabu Searchh iterations. Thhe minimum ccost is then 15,773.0 MXN N for the sequeence when the H& &MICs are orddered by cities which is also the safest soluution, since driv ving along highwaays is minimizzed as well. It should be meentioned that thhe TS was useed to heuristically rearrange the H&MICs H in thhe tours. The sh hortest route method m was useed to minimize thee distance in each tour. Thhe savings of using a 5 dayy-tour sequencce is 17,654.8 MX XN, close to 533% of the actuaal cost.
F Figure 2. Shortest tour: initial soolution.
1838
D. Marmolejo-Correa et al.
Table 2. Some results of the Tabu Search Algorithm Sequence
Dist. (km)
Time (h)
Total Cost (MXP)
OFL, HI2, HI1, HS2, HS1, HC1, HC2, HL7, HL1, HL6, HL5, HL3, HL4, HL2, OFL
303.9
30.7
58,653.4
{OFL, HI2, HI1, HS2 OFL},{ OFL, HS1, HC1, HC2, OFL},{ OLF,HL1, HL2, HL3, HL4, HL5, HL6, HL7 OFL}
522.9
34.0
33,427.8
{OFL, HI2, HI1, HL7, OFL}{OFL, HS1, HS2, HL1, OFL}{OLF, HC1, HC2, HL6, OFL}{OLF, HL2, HL3, HL4, HL5, OFL}
673.7
36.6
22,759.6
{OFL, HI2, HI1, OFL}{OFL, HS1, HS2, OFL}{OLF, HC1, HC2, OFL}{OLF,HL7, HL1, HL6, HL5, OFL}{OLF, HL3, HL4, HL2,OFL}
681.6
36.8
15,773.0*
{OFL, HC2, HC1, OFL}{OFL, HS1, HI1,HL7, OFL}{OFL, HS2, HI2,HL1, OFL}{OLF, HL3, HL4, HL2, OFL}{OLF, HI5, HL6, OFL}
734.7
37.8
18,919.4
*Optimum solution
4. Conclusions The optimum sequence of tours was found with a hybrid method of two methods: the shortest route method, using the Traveler Salesman Problem as basis, and the Tabu Search Algorithm for finding the best arrangement. According to the obtained results, the best sequence of tours should be scheduled over five days. The savings found are close 53% compared to the actual Preventive Maintenance Program. There is no need to hire a second Biomedical Engineer, however, less qualified help at a lower pay scale could be evaluated. The proposed methods can be used in various engineering fields, as well as more application in the field of biomedical engineering, implemented for optimizing hospital resources and the rotation and distribution of personnel and activities within health institutions.
References C.A. Gonzalez-Silva and Hernandez A., 1996, Maintanece Manual for Health Services: Equipment and Buildings, HSP-UNI/Operating Manuals, PALTEX, Vol. 2, Issue. 6, (In Spanish) G. Laporte, 1992, The Traveling Salesman Problem: An Overview of Exact and Aproximate Algorithms, The European Journal of Operational Research, Vol. 59, Issue. 2, pp 231-247. J. Lenstra, 1975, Some Simple Applications of the Traveling Salesman Problem, Vol. 26, Issue 4, pp 717-733. H. Taha, 2011, Heuristic Programming, Operations Research: An Introduction, Nineth Edition, Prentice Hall, Boston.