installationandlocationforlevelofrepairanalysiswith...
TRANSCRIPT
INSTALLATION AND LOCATION FOR LEVEL OF REPAIR ANALYSIS WITH A MINIMUM COST FLOW MODEL
Leandro Gama dos Santos Ramos
Universidade Federal Fluminense (UFF) – Departamento de Engenharia de Produção Rua Passo da Pátria nº 156, sala 306, Bloco D, S. Domingos, Niterói -‐ RJ
Artur Alves Pessoa Universidade Federal Fluminense (UFF) – Departamento de Engenharia de Produção
Rua Passo da Pátria nº 156, sala 306, Bloco D, S. Domingos, Niterói – RJ [email protected]
Eduardo Siqueira Brick
Universidade Federal Fluminense (UFF) – Departamento de Engenharia de Produção Rua Passo da Pátria nº 156, sala 306, Bloco D, S. Domingos, Niterói -‐ RJ
RESUMO LORA é uma metodologia que tem como objetivo determinar a alocação ótima de
recursos e instalações para a manutenção de sistemas complexos e escolher a mais adequada política de manutenção. Esta pode ser entendida como uma decisão a priori sobre quais componentes defeituosos descartar ou reparar e onde essas ações devem ser executadas. Este trabalho propõe um novo modelo baseado em fluxo de custo mínimo e uma formulação de programação inteira mista, que determina a localização ideal de cada recurso para permitir a análise de falhas antes de executar a ação de reparo e escolhe a melhor ação a ser realizada dentre os diferentes tipos de ações de reparo possíveis de acordo com os tipos de escalão que foram instalados.
PALAVARAS CHAVE. Logística de Manutenção, Análise de Nível de Reparo. Fluxo de Custo Mínimo
Área principal -‐ PO em Serviços
ABSTRACT LORA is a methodology that aims to determine the optimal deployment of facilities and
resources for maintenance of complex systems components and choose the most appropriate repair policy. A repair policy may be understood as an a priori decision on what faulty components to discard or repair and where these actions should take place. This work proposes a new model based on minimum cost flow and a Mixed Integer Programming (MIP) formulation, which determines the optimal location of each resource allowing failure analysis before performing the repair action, and chooses the best action to be performed among different types of possible actions according to the types of echelons that were installed.
KEYWORDS. Maintenance Logistics. Level of Repair Analysis. Minimum Cost Flow.
Main area SE – PO em Serviços
1. Introduction Organizations dedicated to manufacture and supply of services often rely on a great
quantity of capital assets (facilities, machinery, tools, etc.). These assets are frequently subject to failures and cease to function properly, interrupting the production of goods or provision of the services. Thus, one of the main problems faced by these organizations is how to effectively guarantee the availability of these assets.
When addressing this problem there are two main concerns: the effectiveness of the actions taken, measured by the availability (or productive capacity) of the assets and the efficiency of system, measured by the life cycle cost or total ownership cost of these assets.
In order to achieve the desired availabilities, the main concern is the time needed to put the system back to the operating condition. One of the most used strategies to achieve high availabilities is to replace immediately, whenever possible, the faulty parts. This situation is shared by almost every sector of activity: transportation, defense, communications, energy, metallurgy, chemistry, information technology etc.
According to ABRAMAN (2013) maintenance costs about 4% of company’s total revenues and the requires about 25 % of theirs man power. So, the problem of designing effective and efficient maintenance structures is of great and general interest to all organization types. Nevertheless, the subject has received more attention in the defense sector, mainly represented by the Department of Defense (DoD) of the United States of America. This is easily explained by the huge quantity of complex assets owned by this organization and the need to deploy and operate them all over the world, normally in combat situations that subject them to extreme conditions that contributes to amplify the number of failures. In the international military jargon, these maintenance structures are known as logistic support systems.
There are two general assumptions shared by methodologies and models used to address logistic support systems related problems. The first one is that the assets (systems, equipment, etc.) may be broke down into several indenture levels in a tree like structure. The highest level represents the system or equipment and the lowest the more simple and indivisible components (bolts, nuts, transistors, coils, etc.).
There are many different criteria adopted to perform this system breakdown. The most used one considers the capability that exists in a particular site available resources to extract the part from its parent in the tree structure and replace it. A Line Replaceable Unit (LRU), for example, is a part that can be replaced using local resources at the site where the asset is located, whereas Shop Replaceable Unit (SRU) is a part that requires specialized facilities and tools to perform this action. An example of a system tree like structure is given in Figure 1.
Fig. 1 - Example of s system tree structure
The second general assumption that is usually made is with regard to the maintenance structure itself. It is usually assumed that the support system is also organized in the format of a
tree like structure in which each level is known as an echelon. One to four echelons have been considered and the structure adopted by the DoD has four, as illustrated in Figure 2. The first echelon is located where the systems are deployed and operated. The forth echelon normally refers to maintenance performed at the original equipment or components manufacturers (OEM) facilities. Echelons are associated with different capabilities that perform maintenance actions. At the lowest level only simple actions, such as exchange of LRU, are possible. At the highest level all possible maintenance actions can be performed.
The DoD has sponsored the development and/or adopted a great number of methodologies dedicated to solve this problem. These are known as Logistics Support Analysis (LSA) tools, namely: fault tree analysis (FTA), failure mode, effect and criticality analysis (FMECA), maintenance task analysis (MTA), life cycle cost analysis (LCCA) and level of repair analysis (LORA). Other methodologies such as reliability centered maintenance analysis (RCMA) and the ones designed to define stock levels for spares or when and which part to replace, are also used.
Fig. 2 - Example of maintenance structure with four echelons
According to Crabtree & Sandell (1989, apud Barros, 1998) the US Army defines LORA as “not only the repair and discard location for the items that make up the system or equipment, but the extent of the maintenance permitted and the resources needed to support the maintenance process”. This definition is similar to the one given by Basten et al (2011, 2012). Given a product design and a repair network, a LORA decides upon:
• Which components should be repaired and which should be discarded when a failure occurs;
• Where in the repair network each failure mode will be repaired and; • Where the necessary resources for repair will be deployed.
A key activity in the process of identifying and repairing failures, and not considered in other works about LORA, is the failure analysis. When a failure occurs in a complex system, the first task, in an attempt to address the problem, is to perform an analysis to identify the cause of this failure (Murthy, Solem and Roren, 2004). Only after an analysis aimed at identifying the faulty component that has originated the failure is made, a corrective action may be taken.
In more sophisticated systems, there is a built-in-test-equipment (BITE). The BITE is characterized as fault management and diagnostics function incorporated in the equipment to support the maintenance process. Modern systems built-in-test (BIT) can be divided into two categories based on their operational methods: active and passive.
Active systems perform test routines during the operation and generate reports with the results presented to operators and maintainers. Passive systems monitor and record the performance of various platform subsystems. The results are examined after the operation for evidence of malfunction of the system and archived for historical trend data analysis. BITE is
widely used in transportation and even space weapons systems to operational controls, and as a central tool for fault isolation and rapid restoration of full functionality (Previsich, 2007).
The level of service (or the extent of maintenance permitted) for the failure solution is also an important factor for LORA. In the literature, this level of service is associated with the echelon.
1.1. Literature Review Alfredsson (1997) is one of the first to propose optimization models to solve LORA
problems. In the proposed model, the LORA problem is integrated into the spare parts inventory problem assuming some simplifications. Only one product indentation level and two echelons (operation and maintenance) are considered. Another assumption is that each component has its own test equipment that can be configured to repair other components.
Brick and Uchoa (2009) have generalized the definition of the LORA problem by extending it to include a fourth decision regarding where (geographical coordinates) to locate and build the maintenance facilities. Previous papers have considered the maintenance network as given (previously defined). To our knowledge this is the only paper so far that addresses repair policy decisions simultaneously with facility location ones.
The assumption that all components in each indentation level can share the same resource is seen in Barros (1998) and Barros and Riley (2001). Choosing the best action within a set of maintenance actions is discussed by Cassady (apud Basten et al. 2011a). Implications of these choices in the stock of spare parts for maintenance are observed in Fortuin and Martin (2004 apud Basten et al. 2011a).
Saranga and Dinesh Kumar (2006 apud Basten et al. 2011a) assume that the components do not share resources and each component needs just a single resource to be repaired.
Brick and Uchoa (2009) chose not to classify the repair facilities by echelons levels. A single echelon, in addition to the operating site, where only LRU replacements are allowed, forming a two echelon network. The problem is modeled as an integer programming problem that decides which facilities should be opened and where. In their model resources for repairs may be capacitated, installed in variable quantities and shared by different components. The fixed cost of resources, as well as the variable cost for maintenance actions may depend on the location. Transportation costs depend on component parameters (cost, weight, volume, etc.), transport modals, the distances between locations and even on origin and destination locations.
Basten et al (2011a) and Basten et al (2011b) solve the problem using a minimum cost flow model that allows for repair, discard or move the component. In another work, Basten et al (2012) also evaluate occurrences of unsuccessful repair, disposal as the only option, no fault found, finite resource capacity and stock of spare parts.
It was not found in the literature models to solve LORA incorporating the analysis and identification of the faulty component in its formulation, nor examples in which the model itself determines how many echelons to install, based on demand and resources characteristics.
1.2. Contributions In this work, it is presented a new model for the LORA problem that considers a wider
range of alternative actions to be taken in the repair sites than the ones considered by Basten et al (2012). It also generalizes the model of Brick and Uchoa (2009) by adding the following decisions:
• Which set of resources, with different effectiveness and costs, to use for each maintenance action (this decision will lead automatically to the creation of additional echelons as natural solutions);
• Whether to include resources at the operating echelon level to perform analysis to identify more precisely the LRU where the failure has occurred (this may lead to lower costs by avoiding sending more expensive, voluminous and heavier parts to discard or repair, instead of only a part of it).
It is also proposed a new generic optimization model based on minimum cost flow that encompasses the previously mentioned LORA model, and provide MIP formulation for it. Then, the application of the MIP formulation to seven instances based on a realistic case is described, and an analysis of the solutions obtained through a MIP solver is presented, illustrating the usefulness of the approach.
This paper is organized into five sections. In section 1 the general definitions about LORA methodology, literature review, the insertion of this work in the literature and an overview of results achieved are presented. Section 2 details the concepts and characteristics of the problem, which are mathematically formalized in Section 3. Section 3 also contains the new generic model, the corresponding MIP formulation and the description of how to apply it to the original problem. Section 5 describes the realistic case used to generate the problem instances for the computational experiments. Section 6 presents the description of each tested instance and the computational results. Finally, Section 7 contains a discussion about the modeling gains evidenced from aspects that had not been considered in other studies and the performance of the algorithm.
2. Problem Description The problem considered in this paper consists in designing a logistics support system
network capable to attend the corrective maintenance demands generated by a set of different systems deployed in different geographical locations.
The systems to be maintained consist of a large quantity of components that can be modeled as a tree structure obeying a contain-contained (parent-child) relationship, associated with the practical possibility that a child can be removed from its parent and replaced by a spare, when a fault is present. Depending on the maintenance resources available at the places where the systems are deployed (maintenance first echelon), only a set of components can be removed and substituted at these locations. As mentioned before, they are called Line Replaceable Units (LRU). The components that can be removed and replaced only at places where special facilities exist are called Shop Replaceable Units (SRU).
When there is a functional failure, due to a component fault, for the system to continue operating the defective component should be immediately replaced with a spare. Defective components, which are removed from the system, generate a demand for maintenance actions in the support system.
The failure occurs in a location and the LORA policy determines to which site this defective component should be sent for maintenance. This policy also determines if the component should be repaired or discarded.
Resources, such as test instruments, tools, equipment, technical documentation, support software, special facilities, consumables and specialized human resources are required to perform each of these possible maintenance actions. There may exist more than one alternative of resources to perform a maintenance action, with different effectiveness and costs.
Some repairs can be made on sites belonging to third parties. In this case, it is not necessary to install facilities, but the unit cost per repair is generally higher (Brick and Uchoa, 2009).
The maintenance network is divided into echelons differentiated according to the effectiveness and capacity of the resources installed on them. The operation site is considered as the first echelon. It is assumed that system’s operators are able to change LRU without any additional maintenance resources. It is also assumed that the resources needed to perform an analysis to identify precisely in which component the fault that generated the failure has occurred, may be installed at these locations. Less complex service points that can perform only simple repairs, like in a small workshop, are classified as second echelon. Service points that perform complex services, such as the component manufacturer facilities, where there are expensive and specialized resources, are classified as third echelon. The first two levels are usually organic but the last levels can be organic or outsourced.
A failure is detected initially as an indication of system functionality loss. Normally it is
necessary to perform some testing to identify in which component the fault occurs. In many occasions it is not clear in which level of indentation the fault is localized and, thus, which LRU should be replaced in the field.
BITE information is relevant to decide which maintenance action will be taken. But BITES do not cover all possible faults and in many situations they can indicate faults only on a parent of the faulty component (a lower indentation component). Thus, for a fault that is pre-diagnosed in a low indentation level component, probably it will be interesting to perform an analysis to identify the more precisely faulty part before setting the repair or disposal of the component. Nevertheless, if pre-diagnosis indicates a failure in a high indentation level component, the analysis will probably not be necessary.
In addition to analysis, maintenance locations can also perform repair and disposal actions. Depending on the resource installed on the site, i.e. the echelon level, repair effectiveness can be higher or lower. An unsuccessful repair means that a proportion of component failures that are serviced in this workshop cannot be repaired and will have to be reworked (another repair or disposal).
3. New model and Mixed Integer Programming Formulation Consider an organization with 𝐾 locations where the systems are located and generate
demand actions, and 𝐽 facilities where the repair actions may take place. K may be a subset of J. At each site 𝑘 ∈ 𝐾 there are systems that consist of components 𝑞 ∈ 𝑄 that can be
distributed in up to 𝑁 indentation levels. When in operation, these items may fail. Each failure is initially presented as a fault indication (𝑡 ∈ 𝑇) in the defective component itself or in a component that is contained by it (a component with indentation level higher than the level of the indicated one). Thus, when there is a failure indication one can only know the probability 𝛽 that it corresponds to a failure (𝑡 ∈ 𝑇) in the indicated component or in one of its subcomponents. That is, there is an uncertainty in the correct faulty component identification that is considered by the model.
Resources can be allocated to a maintenance site (𝑗 ∈ 𝐽) to perform the necessary repair actions to restore the component functionality, or to discard it. Each resource 𝑟 ∈ 𝑅 is composed of maintenance site 𝑗 and resource type h. Also in the maintenance sites (𝑗), each failure indication or failure (𝑡) undergoes an action (𝑎 ∈ 𝐴). It will be possible to choose between two different sets of resources, with different effectiveness and costs for each action.
Each triple (k, t, i), where k is a site, t is a failure indication or failure, and i is repair attempt index, forms a demand node 𝑙 ∈ 𝐿. The repair attempt indexes are:
(i=1) Indicates that the component has not undergone any attempted repair.
(i=2) Indicates that the component has undergone a unique attempt to repair with the Resource Set 1;
(i=3) Indicates that the component has undergone a unique attempt to repair with the Resource Set 2;
(i=4) Indicates that the component undergone two repair attempts.
Each triple (j, t, a), where j is a site, t is an failure indication or failure, and a is an action, determines a maintenance or action node 𝑣 ∈ 𝑉. The following actions are possible in maintenance site (𝑗):
(a=1) Analysis: Action taken on a fault indication. After the analysis, a new demand failure is created according to 𝛽;
(a=2) Disposal: Action taken on failure indications and failures. It discards the component completely;
(a=3) Repair with Resources Set 1 (first or third attempt): Action trying to repair the fault indication or failure. It is possible that the repair can be unsuccessful and generate rework. Thus, a new demand arises in this site with the same fault
according to 𝛽.
(a=4) Repair with Resources Set 2 (first or third attempt): Similar to the previous action but with a different set of resources that can be installed in the same location for repair the component.
(a=5) Repair with Resources Set 1 (second attempt): Action trying to repair the fault indication or failure. It is possible that the repair can be unsuccessful again and generate rework. Thus, a new demand arises in this site with the same fault according to 𝛽.
(a=6) Repair with Resources Set 2 (second attempt): Similar to the previous action but with a different set of resources that can be installed in the same location for repair the component.
It is not possible that the same repair action be executed twice in the same place on the same defective component, i.e., when suffering an unsuccessful repair the same node cannot be chosen for the new maintenance action. The third attempt to repair occurs only in nodes that have resource set with 100% effectiveness. Hence, only up to 3 maintenance attempts are possible to repair a faulty component.
A problem graph is then built using the previously defined two sets of nodes such that 1. 𝐺 = (𝐿 ∪ 𝑉,𝐴! ∪ 𝐴!), 𝐿 ∩ 𝑉 = ∅, 2. 𝑙, 𝑣 ∈ 𝐴! ⇒ 𝑙 ∈ 𝐿, 𝑣 ∈ 𝑉, 3. 𝑣, 𝑙 ∈ 𝐴! ⇒ 𝑣 ∈ 𝑉, 𝑙 ∈ 𝐿.
The sets of arcs AL and AV are built as follows. The arcs AL connect demand nodes to maintenance nodes with the same failure or fault indication (𝑡), from each site 𝑗 to any site 𝑘. However, only demand nodes with 𝑡 corresponding to a fault indication are connected to maintenance nodes with 𝑎 = 1 (analysis action). Demand nodes with 𝑖 = 1 and any value of 𝑡 are connected to maintenance nodes with 𝑎 = 2 (discard action), 𝑎 = 3 and 𝑎 = 4 (repair actions first attempt), demand nodes with 𝑖 = 2 or 𝑖 = 3 are connected to demand nodes with 𝑎 = 2, 𝑎 = 5 and 𝑎 = 6 (repair actions second attempt) and, finally, demand nodes with 𝑖 = 4 are connected to maintenance nodes with 𝑎 = 2, 𝑎 = 3 and 𝑎 = 4 (repair actions third attempt), provided they have 100% efficiency repair action.
The AV arcs connect maintenance nodes to demand nodes. These arcs are responsible for creating new demands that have to be reworked. Thus, maintenance nodes are only connected to demand nodes when the corresponding sites 𝑗 and 𝑘 are the same. Maintenance nodes with 𝑎 = 1 (analysis of action) are linked to demand nodes with 𝑖 = 1 and 𝑡 value equivalent to a failure identified from a fault indication analyzed in the initial maintenance node. Maintenance nodes with 𝑎 = 3 are connected to demand nodes with 𝑖 = 2, maintenance nodes with 𝑎 = 4 are connected to demand nodes with 𝑖 = 3, maintenance nodes with 𝑎 = 5 or 𝑎 = 6 are connected to maintenance nodes with 𝑖 = 4. Maintenance nodes with 𝑎 = 2 are not connected to demand nodes.
The quantity of flow through an arc (𝑙, 𝑣) is defined as xlv. The cost to send flow through the arc (𝑙, 𝑣) is given by 𝑐!", which consists of the costs of transporting the component q from node 𝑙 to node 𝑣, perform the maintenance action in 𝑣 and return the component to the node 𝑙. The number units of a resource type h that needs to be installed in a site j is defined by 𝑦!, where ℎ, 𝑗 = 𝑟 ∈ 𝑅. The cost to deploy this resource is 𝑓!. The decision variables are xlv and 𝑦!, while the other quantities are given.
Thus, the MIP formulation is presented starting from its objective function:
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑐!"𝑥!"!,! ∈!!
+ 𝑓!𝑦!! ∈ !
In a site k, the amount of primary occurrences of a failure indication or failure 𝑡 is
named 𝑑(𝑙), where l is defined by the triple (k, t, i). Note that 𝑑 𝑙 = 0 if 𝑖 ≠ 1. Every demand or
amount of flow that reaches a node l should be directed to a node v for a maintenance action to be performed. This leads to the following flow conservation constraint:
𝑥!"!: !,! ∈!!
= 𝑥!"!: !,! ∈!!
+ 𝑑 𝑙 , ∀ 𝑙 ∈ 𝐿 (1)
From the failures originated in the node 𝑙 ∈ 𝐿 that have undergone an action in the node
𝑣 ∈ 𝑉, the percentage that will generate new demands in the node 𝑤 ∈ 𝐿 is defined by 𝛽!"#. Note that the sites where the new demands are generated are the same as where actions are performed. As these new demands should also be sent to an action node, the following constraint is also added:
𝑥!" = 𝛽!"# . 𝑥!"!: !,! ∈!
, ∀ 𝑣,𝑤 ∈ 𝐴! (2)
The maximum flow permitted in an arc (𝑙, 𝑣) is defined as 𝑈!". As one of the model's
objectives is to indicate an inventory policy with plausible implementation, it is important that only one action option be allowed for the same failure indication or failure in a same location. To model this feature, a binary decision variable zlv is defined so that a non-zero flow is allowed to pass through the arc (l, v) when zlv = 1, and such flow is forbid when zlv = 0. The following constraints ensure this property, and use it avoid more than one non-zero flow leaving the same demand node l ∈ L.
𝑥!" ≤ 𝑧!" .𝑈!" , ∀ 𝑙, 𝑣 ∈ 𝐴! (3)
𝑧!"!: !,! ∈!
≤ 1, ∀ 𝑙 ∈ 𝐿 (4)
The maximum number of resource units of a given type h that can be installed in a maintenance site j is given by 𝑀!, where r = (h, j). The installed resources have a limited capacity. The percentage of the capacity of a resource r used by each flow unit passing through the node 𝑣 ∈ 𝑉! is defined by 𝑢!, where 𝑉! is the set of nodes v that use the resource r. The following constraints ensure that the capacity usage in each site does not exceed the installed capacity.
𝑢! . 𝑥!"!: !,! ∈!!! ∈ !!
≤ 𝑦! , ∀ 𝑟 ∈ 𝑅 (5)
Finally, the domains of all decision variables are determined by the following constraints. 𝑥!" ≥ 0, ∀ 𝑙, 𝑣 ∈ 𝐴! (6) 𝑦! ∈ {1, 2,… ,𝑀!} (7) 𝑧!" ∈ {0,1} (8)
4. A realistic case The formulation defined in the previous section was applied to a realistic case with the
following characteristics to develop a 10-year maintenance policy:
Table 1 - Overview of the instances Parameter Cardinality Comments
Systems |S| = 3 - Operation Sites |K| = 10 -
Maintenance sites |J| = 15 - Components |Q| = 9 Only LRU
Failures |T| = 22 9 failure indications and 13 failures
Resources Types |H| = 6 -
The three systems have the following indentation structures, failure indications and
failures:
Fig. 3 - Tree structure of the components, failure indication and failure
The quantities of each system deployed at each operating location are shown in Table 2.
Table 2 - Quantity of assets in each location Location (k) Asset Quantity Manaus (1) A 25 Manaus (1) B 150 Belém (2) A 25 Belém (2) B 100
Fortaleza (3) C 200 Salvador (4) C 125 Cuiabá (5) B 25 Cuiabá (5) C 50 Brasília (6) A 25
São Paulo (7) B 75 Rio de Janeiro (8) C 100
Curitiba (9) A 50 Porto Alegre (10) A 100
Each site/component has the following quantities of failures:
Table 3 - Amount failure by component in each demand node for ten years
k q t Amount Failure k q t Amount
Failure k q t Amount Failure
1 1 10 3,75 4 2 13 3,75 7 1 10 0,25 11 3 4 15 8,25 11 1,75
12 9,5 5 16 6,25 12 4,25 2 13 3,5 7 18 13,5 3 14 6,25 3 14 5,5 19 12,5 6 17 10,25 5 16 7,75 9
21 18,75 8 20 13,5 6 17 11,5 22 16,75
8
2 13 4 8 20 12,5
5
1 10 0,75 4 15 8,5
2
1 10 3,5 11 1,5 5 16 8 11 0,5 12 4,25 7 18 12 12 7,75 2 13 2,75 19 12,75
2 13 4,25 3 14 5,25 9 21 15 3 14 5 4 15 6,25 22 19,75 5 16 8,25 5 16 9,25
9 1
10 0,25 6 17 13 6 17 12,25 11 2 8 20 13,25 7
18 11,25 12 5
3
2 13 5 19 12,75 2 13 4,25 4 15 8,5 8 20 12,5 5 16 9,75 5 16 8,5 9 21 15,5
10 1
10 1,5
7 18 10 22 17,5 11 1,25 19 13,5
6 1
10 2,25 12 4,75
9 21 20,75 11 0,5 2 13 4,75 22 17 12 4,75 5 16 7,5
2 13 3,5 5 16 6,25
In the drawn up instances, the operating sites may be also maintenance sites (J ∩ K = K).
In addition to the 10 sites above, there are also possible maintenance sites on the cities of Porto Velho, Campo Grande, Recife, Brasília and Florianopolis, thus completing 15 possible maintenance sites.
The resources available for installation have different costs according to the region in which they are installed. Below are the basic costs of resources and the multiplication factors according to the installation region. Table 4 – Resources data
Resource
Capacity (Hours
Available during 10
years)
10 years ownership
cost
Maximum Quantity
Installable Region
Resource Multiplying Factor
Equipment Worker Super 30.240 R$ 8.400.000 Infinite Extra 129.600 R$ 3.600.000 Infinite North 1,3 1,4
Regular 30.240 R$ 2.800.000 Infinite Northeast 1 1 Test Kit 30.240 R$ 70.000 Infinite Midwest 1,2 1,2
Worker 1 30.240 R$ 715.000 Infinite Southeast 1 1,2 Worker 2 45.360 R$ 1.001.000 Infinite South 1 1,1
Resource usage varies with the action performed and the failure indication or failure. A
component with the highest level of indentation will use a smaller amount of resource time to repair it. Analysis actions uses between 2 and 11 hours of the Test Kit resource, between 2 and 11 hours of the Worker resource and, if the analysis is over a SRU, 5 hours of SRU Kit. Disposal Actions uses between 6 and 12 hours of the Worker resource. Repair actions uses between 3 and
9 hours of the Equipment resource between 0 and 6 hours of the Test Kit resource and between 1 and 9 hours of the Worker resource.
Repair actions that use the Super resource have 100% effectiveness and, when using the Regular and Extra resources, have effectiveness of 75%, that is, 25% of the demand processed on that site should be repaired or discarded in a new try. The analysis actions and disposal have 100% effectiveness.
Maintenance action costs depend on the node where they are performed. Analysis actions cost between 5% and 10% of component value. Disposal actions cost between 120% and 150% of component value. Repair actions cost between 20% and 40% of component value.
The cost of transporting the components between the locations is determined according to the component’s attributes volume, weight and value and the distance between the origin and destination sites:
𝑡𝑟𝑎𝑛𝑝𝑜𝑟𝑡𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑠𝑡 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒×(10 ∗ 𝑣𝑜𝑙𝑢𝑚𝑒 + 10 ∗ 𝑤𝑒𝑖𝑔ℎ𝑡 + 0,08 ∗ 𝑣𝑎𝑙𝑢𝑒)
100+ 60
5. Experiments The formulation was implemented in MS Excel by VBA language and solved with the
help of UFFLP library (Pessoa and Uchoa, 2011). Currently this library supports solvers like CPLEX and COIN CBC. For this application the CPLEX was used.
The following instances were used in the experiments: § Instance 1: Analysis actions are not considered. Only disposal and repair
actions, with 100% effectiveness, are allowed. § Instance 2: Analysis actions are not considered. Only disposal and repair
actions, with 75% effectiveness, are allowed. § Instance 3: Analysis actions and disposal and repair actions with 100%
effectiveness are considered. § Instance 4: Analysis actions and disposal and repair actions with 75%
effectiveness are considered. § Instance 5: Analysis actions are not considered. Disposal and repair actions with
75% effectiveness and repair actions with 100% effectiveness are allowed. § Instance 6: Analysis actions and disposal and repair actions with 75%
effectiveness and repair actions with 100% effectiveness are considered. § Instance 7: Analysis actions are not considered. Disposal and repair actions with
75% effectiveness are considered. The model can then choose to install resources with regular or extra capacity. Both types of resources perform repairs with 75% effectiveness.
In the instances where analysis actions are not enabled, failure indications have been treated as failures. Without the analysis actions and not allowing the installation of different resources in the same site, the model is applicable and the results may be compared to most cases seen in the literature.
The model was able to solve all instances, with a GAP tolerance of 1%. A PC with Intel Core i5 processor and 4GB of memory was used to run the solver, and the resolution times are shown in Table 5.
Table 5 - Resolution times
Instance Total Cost Solution Gap Time 1 R$ 164.191.172 0% 0,9 second 2 R$ 193.195.690 0% 27,5 seconds 3 R$ 151.719.503 0% 178,4 seconds 4 R$ 167.901.582 0% 1.122,7 seconds 5 R$ 161.409.682 1% 126,7 seconds 6 R$ 149.408.796 1% 4.035,9 seconds
7 R$ 165.533.972 0% 328,9 seconds It is important to note the effect of analysis action on instances 3, 4 and 6. In
comparison with the instances 1, 2 e 5, respectively, the total cost of the maintenance system was reduced about 9,4%.
It is also relevant to observe that the model is able to automatically generate a maintenance structure of three echelons, as seen in the Figure 4 depicting the results of solution of instance 6. The filled circles represent operation/maintenance sites and unfilled circles are maintenance sites. The arrows indicate the components transported to undergo maintenance action. Larger the arrowheads represent higher flows of transported components.
The arrowheads (triangles) without lines indicate components that suffered maintenance at the same site where the demand was generated. This happens, for example in Manaus, Fortaleza and Salvador. Cuiaba works as first echelon of attendance for demands originated in Brasilia. Fortaleza is the second echelon facility to demands originated in Belém that worked as the first echelon facility in this maintenance flow.
São Paulo attends the several demands of regions Midwest, Southeast and South. A depot level was created in São Paulo to service demands originated in Porto Alegre, which is a second echelon facility to demands originated from the first echelon site in Curitiba. The black arrow that connects Porto Alegre to São Paulo indicates a third attempt of repair.
Fig. 4 – Action Flow Map of Instance 6
Besides São Paulo, Manaus, Fortaleza and Salvador also receive the Super equipment in instance 6. That is why they do not have flow outing this sites. The Super equipment had an average utilization of 97% and the Regular equipment an average utilization of 70%. Since the super equipment is much more expensive, it is expected that a good solution uses this resource more efficiently.
6. Conclusions This paper proposes a new model for LORA that was applied to test instances
constructed from parameters obtained with a specialist. The cases may be defined as medium size. However, with more time and computing resources available, the model is able to solve large-scale problems optimally.
The constructed model is very comprehensive, can incorporate most of the models seen in the literature about the topic and further includes new choices:
• Failure Analysis: This action usually occurs in the real world, but had not been modeled in previous LORA models.
• Installation of different resources (capacity and effectiveness) in a location: this evolution allows that multiple echelons are modeled and defined in optimization.
It is important to point out that this model can be used to analyze health systems, defining the location of health centers, medium and large hospitals and reference centers for various diseases.
Despite the developments obtained, the model still has limitations: • Do not consider that the component failure rates may vary throughout the system
lifetime. • Do not consider the time of transport and the action processing times.
References ABRAMAN. Associação Brasileira de Manutenção e Gestão de Ativos, Documento Nacional 2013. (http://www.abraman.org.br/Arquivos/403/403.pdf/, 4, 2015) Alfredsson, P. (1997), Optimization of multi-echelon repairable item inventory systems with simultaneous location of repair facilities, European Journal of Operational Research 99, 584–595. Barros, L.L. (1998), The optimization of repair decisions using life-cycle cost parameters. IMA Journal of Mathematics Applied in Business and Industry 9, 403–413. Barros, L., Riley, M. (2001), A combinatorial approach to level of repair analysis, European Journal of Operational Research 129, 242–251. Barros, L., Riley, M., Brown, D. (2001), Special millennium issue of the EJOR: A global view of industrial logistic, European Journal of Operational Research 129, 231–234. Basten, R.J.I., Schutten, J.M.J, van der Heijden, M.C. (2009), An efficient model formulation for level of repair analysis, Ann Oper Res 172: 119–142. Basten, R.J.I., van der Heijden, M.C., Schutten, J.M.J. (2011a), A minimum cost flow model for level of repair analysis. International Journal of Production Economics 133 (1), 233–242. Basten, R.J.I., van der Heijden, M.C., Schutten, J.M.J. (2011b), Practical extensions to a minimum cost flow model for level of repair analysis. European Journal of Operational Research 211, 333–342. Basten, R.J.I., van der Heijden, M.C., Schutten, J.M.J. (2012), Joint optimization of level of repair analysis and spare parts stocks. European Journal of Operational Research 222, 374–483. Brick, E.S., Uchoa, E. (2009), A facility location and installation or resources model for level of repair analysis, European Journal of Operational Research 192 (2), 479–486. Crabtree, J.W, Sandel, B.C. Army Level of Repair Analysis (1989). Logistic Spectrum, Summer, 27-31 Gapso (2015), Integrando Programação Inteira Mista e Planilhas de Cálculo de forma simples e acessível, Website Gapso, (http://www.gapso.com.br/ufflp/, 4, 2015). Murthy, D.N.P., Solem, O., Roren, T. (2004), Product warranty logistics: Issues and challenges, European Journal of Operational Research 156, 110-126. Pessoa, A. A., Uchoa, E. (2011), Minicurso UFFLP: Integrando Programação Inteira Mista e Planilhas de Cálculo, Simpósio Brasileiro de Pesquisa Operacional. Previsich, N. (2007), Built-in-test Equipment for Integrated Weapons Systems: Achieving Utility and User Acceptance, Master’s Thesis of Science in Systems Engineering, Naval Postgraduate School, Monterey, CA, EUA.