In industries such as mining that are characterized by heavy use of their equipment and machinery, an adequate stock of spare parts and critical components is essential. Insufficient inventories affect the overall performance of physical assets, and lack of spare parts can lead to penalties, reduced availability, or increased operational risks.
On the other hand, large inventories lead to inefficient use of capital expenditures and can have serious financial consequences for companies.
This paper provides a methodology to determine the optimal quantities of inventory of major repaired components from the analysis and reliability indicators of the critical equipment of the company, who is in charge of the Supply and Maintenance areas. The work is concentrated only on critical components in the area of high-cost mobile mine equipment, which are subject to failures with operational consequences, where coverage is required for unpredictable failures, regularly with significant costs and associated with inventory failures (stock- out).
The determination of the optimal quantities of inventories is done through a variation of the periodic review model (model R, S) with service level criteria. The proposed model indicates that no component orders are placed after the period R passes, to reach its target quantity S, because each component removed from the equipment returns to inventory after a logistic time T at; repair, warranty return, or purchase in replacement of the components canceled in the process, with this it is achieved that the inventory does not reach a stockout, which is ratified with the construction of a simulation model.
Additionally, indicators of interest are determined for the maintenance and supply area, as well as the optimization of logistics times T at and its consequences in determining the optimal quantity of components in stock, obtaining a decrease in repairable assets in the warehouse up to by 50%.
1. INTRODUCTION
Inventories represent approximately one third of all assets in a typical company (Díaz and Fu, 1997). Of these, spare parts and components are of particular importance for industries characterized by relatively expensive equipment that is heavily used in their production process. The present work aims to generate an inventory methodology that determines the optimal quantities of inventory of major repaired components based on the analysis and reliability indicators of critical equipment. This methodology differs from traditional inventory control techniques primarily in that it is assumed that there are no infinite populations, therefore the rate of demand for parts depends on the number of units currently in operation in the operation.
In most industries, expensive or hard-to-reach parts need to be in stock to protect the stock-out operation, and inventory value has been found to be concentrated in these low-consumption, high-unit-price parts. This type of spare part is maintained as an insurance against the costs of failure that would be incurred due to its unavailability in the event of being required. The “lost sales” rule, which is common in final product inventory problems, is not applicable in this case, because if parts are not available, equipment downtime is too expensive.
There are several problems when deciding how many of these parts to have. One of them is to estimate the demand rate and its associated distribution: long intervals of time are required to estimate it. This is in contrast to high-turnover parts where short-time interval records suffice. Another difficulty in managing low-turnover parts is their inflexibility. For example, over-stocking of fast-moving parts can be quickly remedied by natural consumption, which is not the case with low-turnover parts. Excess purchases at the start of operation (or after start-up) can only be remedied slowly, particularly if the part is unique, making it impossible to sell it to other companies.Another difference appears with the variation in logistics delivery or repair times. High-turnover spare parts stocks can be adjusted quickly by changing these delays. In the case of low-turnover spare parts, by adjusting the levels for transient variations in logistics times, overstock can be easily reached when stationary conditions are reached in logistics times.
As with high-turnover parts, space constraints or, more often, budget constraints apply in some cases. This generally entails suboptimization in decision-making. In the case of profit-oriented organizations (such as mining, gas - oil, etc.), the strategy for low-turnover spare parts must take the cost of failure as a parameter, which in general can be difficult to estimate.
Low turnover spare parts can be classified according to the degree of planning of their consumption (Pascual, R. (2008)):
- Spare Parts with Planned Use: These items are purchased for use on a specific date, for example, a major plant overhaul. As long as the supplier is given sufficient notice, there is no reason to dispose of these parts beyond the time it takes to inspect them before use. Spare Parts with Adequate Warning: they have defects that can be admitted for a longer time interval than the logistic supply time. They must be acquired as needed. Backup Spare Parts: this type of spare part does not give warning before failure, or very little against lead time. It is considered that it is better to have them due to the high cost of failure that can be incurred in case of their unavailability.A subclassification that can be performed considers whether the failures are random (constant failure rate) or old age (increasing failure rate).
2. DEFINITION OF TERMS
In determining the number of components necessary to protect the productive operation from costly stock-outs, it is first necessary to define under what criteria the stock level is "optimal." Of course, the criteria will not be the same for every application, although in industrial practice cost minimization is typically preferred. This imposes the need to generate reliable estimates for the costs associated with operating parts. The latter is not always an easy task, as the shortage of spare parts could have complex consequences that are rarely quantifiable in monetary terms. When shortage costs are unknown, the optimization criteria is usually shifted towards an inventory performance measure,such as the probability of having a part on hand when a demand is generated. The user may also be interested in maximizing the availability of the equipment supported by the inventory (Louit et al, 2005).
Louit (2005), makes a classification of the objectives of spare parts management according to:
- Instantaneous Service Level: which corresponds to the probability that a spare part is available at any moment. This is equivalent to the fraction of demands that can be satisfied immediately with the available stock (Stock On Hand). Service Level in an Interval (or mission): which corresponds to the probability of not running out of stock at any time over a specific interval of time. This criterion is more demanding than the instantaneous availability of stock. Global Cost: it is the most used criterion. Includes: Acquisition Costs, that is, the cost of placing purchase orders. o Spare Parts Intervention Costs, proportional to the number of items, although it may be influenced by discount programs offered by suppliers. Ownership or storage costs: unearned interest, insurance,etc. Failure costs, due to unavailability and its effect on production. Availability of the Supported System: it is the fraction of the time in which a system or equipment is in service due to the availability of spare parts.
3. METHODOLOGY
The main steps carried out in the development of this work are shown below:
3.1 Major Component Repair Process
At present, the repair of major components is carried out according to the needs presented, generated by scheduled maintenance (Component Change Plan) and / or failures. Once the component is removed, technical maintenance personnel verify if it belongs to the group of serialized components or not of the company. This is because the serialized components keep a record by the suppliers who are responsible for the repair.
The component already withdrawn, its serialization verified and placed on the pale, it is observed in the records of the Component Change Plan if the component failure was premature, if so, it is sent as a guarantee to the supplier. Once received by the supplier, the technical evaluation (and failure report) and the repair quote are carried out, in which it is informed if the repair of the component is under warranty. If not, it is evaluated by the administration if the component is repaired or not, according to economic criteria established by the company, if the repair is not approved, the component is canceled and the process of purchasing a component begins. new for replacement.
Once the repair has been approved, the purchase order is generated to proceed with the work. Once the component is repaired, it is sent to the company and entered into the warehouse as a spare in stock (repaired) to solve maintenance plans and future failures of a specific equipment. A summary of the component repair management process is shown in Figure 1.
3.2 Model
3.2.1 Inventory Model
According to the behavior observed by the repaired components of the mine equipment, a variation of the periodic review model (Model R, S) with service level criteria will be used, assuming a Gaussian, Poisson or Exponential distribution (depending on the historical behavior of the spare parts) of its parameters of failure rate and standard deviation of the selected component (Pascual, R. (2008), Meruane, V., Espinoza, F). The proposed model indicates that a component order will not be placed every certain period of time R, in order to maintain the target quantity S, due to the fact that in current practice components enter the inventory with a certain repair time or delivery period T at(logistics time, Turnaround Time), different for each component according to when it was removed from the equipment and sent for repair. So the inventory is replenished with components once the logistics time T at for each component has passed.
For this we propose the following formulation:
∗ = () + (1)
| Where: | |
| Q * | : Optimum quantity of spare to have in inventory. |
| E t () | : Average demand time (scheduled maintenance events and failures) of components. |
| T at | : Logistic repair time. |
| β | : Safety factor based on the service level of the component. |
| σ | : Standard deviation of demand (scheduled maintenance events and |
failures) according to the probability distribution.
Observation 1: It is used to take into account that the standard deviation is given for a unit of time and not T at units of time.
The proposed model (1) is sure of not running out of inventories according to the service level defined for the components, and where the optimal quantity must be corrected in case of significant variations in the logistic repair times T at. For which, average over time period for management.
3.2.2 Repairable Spare Parts Model
We will assume that the failures are independent, and that cannibalization of components and spare parts does not apply in the company's maintenance area. We consider that the system requires I types of repairable spare parts and that it belongs to a fleet of equipment. The repair of these components is carried out in a workshop (s) and takes a certain logistical time of repair or delivery time (Turnaround Time, T at (ut)).
At any moment, a repairable part may be available (Stock On Hand, OH t), being sued for scheduled maintenance or failures (D t), being repaired (Due In, DI t), or being purchased (C t), at replacement of components that are derecognized (CB t) or due to the increase in stock defined by the administration or by this model, and finally in the Guarantee process (G t) according to the component purchase contracts with The manufacturers.
According to the above, the mass balance model is as:
α () s = (3)
which corresponds to the expected unavailability of each component of the fleet.
Expected availability of the system (each truck, spade drill, etc.) A s, ie, the fraction of time that the equipment can operate because there is spare available:
1 I
- ∑ EBO i () I i N i = 1
(8)
2 ∑ n (2 i −1)
Where:
n: Number of data
fx (): Theoretical probability distribution function
F () x: Empirical distribution function
3.5 Simulation
To study the behavior of the process of repaired components of the company, determined by the Markov Chain of Figure 2, a simulation tool will be used with the support of the ARENA software (Rockwell Automation, Basogain, X and Olabe, MA). The model built in this application is illustrated in Figure 3. The objective in this stage of the study is to determine a simulation model that represents the real behavior of the process, in relation to the elements included in it, the level of detail, the constraints and input and output elements.
For the construction of the simulation model we will consider the following conditions (Pascual, R. (2008)):
- The mean intervals between failures follow a probability distribution, which must be adjusted for the analysis of the model. The components are considered repairable. The components do not fail or are decommissioned when they are in the warehouse. The simulation period T corresponds to at least a life cycle of the components considered for the study, in order for the model to enter into service. T at logistics times for normal component repair are the same as those for warranty repair.
Considering these conditions, the simulation model is built that outlines the behavior of the repaired components. With this model we proceed as follows:
- We obtain the behavior of the time between demands of the component under study, chronologically ordering the failures or scheduled events. We adjust a probability distribution to the times between demand events. Subsequently, we enter the Stock box of the model (see Figure 3), the distribution determined in the previous step, we enter the input variables into the model:
o Number of components in current operation in existing equipment o Number of components in inventory (determined by equation (1))
- We enter the probability that the component will fail prematurely (component sent to the warranty process) We enter the probability that the component will be accepted as warranty We enter the probability that the component will be withdrawn from the warranty process or from the normal repair.
The construction of this simulation model manages to obtain the behavior of the demands that will be produced in the future in the components, as well as to predict if the repairs carried out are due to premature failures (guarantee process) or due to normal failure (repair process). It is also possible to observe the quantity of components sent for repair that are decommissioned, driving the purchase process for their replacement (Kelton, Sadowski & Sturrock). Additionally, it is possible to observe in the simulation model if the optimal quantity suggested according to expression (1) has a good behavior in the face of the demands produced for the component avoiding the Stock-out in the inventory.
Figure 3: Simulation model in ARENA software
At the same time, it is possible to observe variations in the behavior of the components in the simulation model, modifying the variables of the model. For example:
- Modify the distribution of demand probabilities, according to the ranking provided by the Easy Fit system. This modification shows the variation in the behavior of demand, producing stock-out or surpluses in the inventory. Varying the optimal quantity of inventory (Q *), with this it is possible to see the behavior of the values currently suggested by the company based on experience Modify logistics time T at, this modification shows the impact it has on the optimal quantity of inventory. Modify the probabilities in the events of the Markov Chain (Figure 2), that is, for the repair, warranty or low components process. This modification varies the quantities that go through the events of the chain, obtaining from the simulation model that the only action that can modify the quantity of the inventory is the action of removing components, as long as the purchase time is less than the logistics time and the number of decommissioned components are greater than that of repaired.
4 RESULT AND DISCUSSION
As an example, to illustrate the methodology, let's consider the largest component with code
Oracle 1092096R corresponding to a CUMMINS brand QSK60 engine from the Komatsu 830 E and 830 AC truck fleet, out of a total of 15 engines installed in the fleet, and with a stock of components for change and repair of 3 units, mainly determined by the experience. The following table shows the demand behavior that this component has experienced from January 2012 to December 2013.
Figure 4: Quantity of demands of the Cummins QSK60 Engine component, between 2012 and 2013
Figure 5: Goodness-of-fit test
Figure 6: Histogram of the sample
Subsequently, through the Statgraphics software, the historical data in Figure 4 are analyzed, to check that they correspond to a simple random sample, in order to rule out that there is a particular cause associated with the demands of this component. According to the analysis carried out, it is verified that the sample has a random behavior with 95% confidence. Next, using the EasyFit software, a probability distribution is fitted, as shown in Figure 6. Applying the AndersonDarling goodness-of-fit test, the distribution behaves exponentially, being the one that best adapts to the data (see Figure 5).
Obtained the probability distribution and its parameters, we can replace in equation (1) to obtain the optimal quantity of components to have in stock, resulting in Q * = 6 units for this example, with a service level of 97%, these is: β = F - 1 (0.97).
Sorted by ranking the probability distributions provided by the EasyFit system, after adjusting the data, Table 1 shows the possible quantities of optimal stock that we can have according to its distribution and its level of service.
Table 1: Optimal quantities according to service level and probability distribution
| Distribution | Weather
Logistic (days) |
Standard deviation
(unit / month) |
85% | 90% | 97% | |||
| Security factor | Q * Optimal (Unit) | Security factor | Q * Optimal (Unit) | Security factor | Q * Optimal (Unit) | |||
| Exponential | 154 | 0.58333 | 1.1067 | 5 | 1.3432 | 5 | 2,0455 | 6 |
| Gamma | 154 | 0.65386 | 1.1458 | 5 | 1.4202 | 6 | 2,2501 | 7 |
| Weibull | 154 | 0.56942 | 1.4106 | 5 | 1.6081 | 6 | 2.1377 | 6 |
| Normal | 154 | 0.65386 | 1,261 | 5 | 1.4213 | 6 | 1.8131 | 6 |
| Uniform | 154 | 0.65386 | 1.3761 | 5 | 1.4894 | 6 | 1,6479 | 6 |
With this optimal quantity of 6 units, the behavior of the inventory is observed with the history of existing demands of the component in the years 2012 and 2013 (see figure 7), and its future behavior for the years 2014 to 2016 according to the Change Plan of Components projected for the Budget / LOM (see figure 8).
Figure 7: Inventory behavior according to the optimal quantity proposed
Figure 8: Behavior of the inventory according to the optimal quantity proposed in the future
Now we carry out the simulation process considering the projection of component failures plus scheduled maintenance, to obtain a more real behavior in the future. The results obtained by varying the probability distribution are those shown in Table 2.
Table 2: Behavior distribution of probabilities according to their mean time between demands.
| Distribution | Parameters | Stock-Out | Excess | ||
| if not | times | if not | quantity | ||
| Gamma | α = 0.99138 β = 48.099 | yes | two | no | - |
| Exponential | λ = 0.02097 | yes | one | no | - |
| Weibull | α = 1.0349 β = 43.042 | no | - | yes | one |
| Normal | σ = 47.892 µ = 47.685 | yes | one | no | - |
| Triangular | m = 6.9163 a = 6.9163 b = 157.74 | no | - | no | - |
| Uniform | a = -35.266 b = 130.64 | yes | one | no | - |
| Erlang | No setting | - | - | - | - |
Once the exponential probability distribution has been selected (as an example or test), the simulation model is entered as well as the number of components in operation, plus the optimal quantity to have in stock and the percentages of repair by warranty and given components low. With these, it is possible to observe the future behavior of the inventory of repaired components, see Figure 9.
Figure 9: Simulation of the stabilized model, for three years.
As we can see from Figure 9, only one Stock-out is reached, in a short period during the three years that the simulation was carried out. Therefore, the value of the optimal amount suggested according to the service level of 97% of the component is acceptable. This checks the efficiency of the optimal quantity proposed to keep in inventory in the company.
5 CONCLUSION
The equation (1) proposed for the determination of the optimal quantity of inventory can be sensitized by varying the level of service in the Supply area, which is shown in Table 3. As we can see, the lower the level of service, the greater the quantity of days in stock-out we can have.
Table 3: Modification of the service level.
| Weather
Logistics (Days) |
Service level % | Security factor | Optimal Quantity | ||
| Q * Obtained (Units) | Q * Rounding (Unit) | Amount of
Days of Stock-Out |
|||
| 155 | 99 | 2.6863 | 6,5099 | 7 | 9 |
| 155 | 97 | 2,0453 | 5.6659 | 6 | 27 |
| 155 | 90 | 1.3432 | 4.7413 | 5 | 89 |
| 155 | 85 | 1.1067 | 4.4299 | 5 | 134 |
We see that it is possible to eliminate the Stock-out of the inventory according to the optimal quantity, by reducing the logistic time T at (see Table 4), and keeping the other variables constant. Furthermore, it is possible not to have Stock-out, reducing internal management time, mainly in the areas in charge of packaging the components to be sent for repair and the generation time of the purchase order in the Supply area.
Table 4: Modification of logistics time keeping the other variables constant
| Logistics Time | Quantity
Optimal (Pc) |
Repaired (Unit) | Warranty (Unit) | Unsubscribed components
I buy (Pc) |
State | ||||
| Weather
Management Internal (Days) |
Weather
Repair (Days) |
Weather
Total (Days) |
Stock-Out | Excess | Quantity | ||||
| 111 | 44 | 155 | 6 | 19 | two | one | yes | no | one |
| 100 | 44 | 144 | 6 | 19 | two | one | yes | no | one |
| 90 | 44 | 134 | 6 | 19 | two | one | yes | no | one |
| 80 | 44 | 124 | 6 | 19 | two | one | yes | no | one |
| 70 | 44 | 114 | 6 | 19 | two | one | yes | no | one |
| 60 | 44 | 104 | 6 | 19 | two | one | no | no | 0 |
| fifty | 44 | 94 | 6 | 19 | two | one | no | no | 0 |
| 40 | 44 | 84 | 6 | 19 | two | one | no | no | 0 |
| 30 | 44 | 74 | 6 | 19 | two | one | no | no | 0 |
| twenty | 44 | 64 | 6 | 19 | two | one | no | no | 0 |
| 10 | 44 | 54 | 6 | 19 | two | one | no | yes | two |
As can be seen from Table 4, reducing the logistic time T at to 104 days it is possible to eliminate the
Stock-out of inventory, and if we further reduce the logistics time to 54 days, we reach an excess of components in inventory, which allows us to reduce the proposed inventory by two units. On the other hand, reducing the logistics time T at in conjunction with the optimal quantity to have in inventory (see Table 5), variations according to the service level of 97% are obtained. Some reductions in logistics time, obtain the same optimal quantity, but the time in which the inventory is kept in stock-out is less as the logistics time is decreased (See Figure 10).
Figure 10: Comparison of Stock-out time reduction.
Another variable that can be sensitized is the probability that a component will be removed (see Table 6). We observe that by increasing the probability to 30% of the decommissioned components, the stock-out is eliminated from the inventory. Now, if we increase to probability values greater than or equal to 35%, it is possible to have an excess of components which makes it possible to adjust to the optimal amount of inventory. In the case of 35%, it is possible to optimize the optimal quantity of repaired components from 6 units to 5 units. This phenomenon is mainly due to the fact that when a component is canceled, the process of buying a new one starts automatically, with the arrival time of the component in replacement being less than the logistics time T at of repair.
Table 5: Modification of the logistic time and the optimal quantity, keeping the other variables.
| LOGISTICAL time | Optimal Quantity | Units in Process
Repaired |
Units in
Warranty Process |
Units in Process
Low and / or Purchase |
Stock-Out | |||
| weather
Management Internal (Days) |
Weather
Repair (Days) |
Weather
Total (Days) |
Q * Rounding (Unit / approx) | Exists | Number of times | |||
| 111 | 44 | 155 | 6 | 19 | two | one | yes | one |
| 90 | 44 | 134 | 6 | 19 | two | one | yes | one |
| 80 | 44 | 124 | 5 | 19 | two | one | yes | one |
| 70 | 44 | 114 | 5 | 19 | two | one | yes | one |
| 60 | 44 | 104 | 5 | 19 | two | one | yes | one |
| fifty | 44 | 94 | 4 | 19 | two | one | yes | 3 |
| 40 | 44 | 84 | 4 | 19 | two | one | yes | two |
| 30 | 44 | 74 | 4 | 19 | two | one | yes | one |
| twenty | 44 | 64 | 3 | 19 | two | one | yes | 3 |
| 10 | 44 | 54 | 3 | 19 | two | one | yes | 4 |
Table 6: Modification of the probabilities of removing a component
| Probability of deregistering components from: | Quantity
Optima (Pc) |
Weather
logistic (Days) |
State | Amount of
Components Casualty Dice (Unit) |
|||
| Process of
Repair Normal (%) |
Guarantee Process (%) | Stock-out | Excess | Quantity | |||
| 12.5 | 16.6 | 6 | 155 | yes | no | one | one |
| twenty | twenty | 6 | 155 | yes | no | one | two |
| 25 | 25 | 6 | 155 | yes | no | one | two |
| 30 | 30 | 6 | 155 | no | no | 0 | 5 |
| 35 | 35 | 6 | 155 | no | yes | one | 7 |
| Four. Five | Four. Five | 6 | 155 | no | yes | one | 12 |
| 55 | 55 | 6 | 155 | no | yes | one | 13 |
| 65 | 65 | 6 | 155 | no | yes | one | fifteen |
| 75 | 75 | 6 | 155 | no | yes | one | 16 |
| 85 | 85 | 6 | 155 | no | yes | one | 18 |
| 95 | 95 | 6 | 155 | no | yes | one | twenty-one |
| 100 | 100 | 6 | 155 | no | yes | one | 22 |
This sensitivity analysis to the model variables (equation (1) and (2)) proposed in this work shows that it is possible to minimize the company's warehouse costs for assets in inventory, since it is possible to reduce the number of components list or improve inventory response (service level). In the example developed, it is achieved that from 6 components, it is possible to go down to 3 units, reducing its logistics repair time, and contributing to a decrease in assets in the warehouse by up to 50%, with a service level of 97 %.
6 REFERENCES
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