Civil aviation engine under uncertain conditions of fleet long-term maintenance plan optimization method

Abstract

The present application relates to the civil aviation engine maintenance technical field, specifically speaking, a kind of civil aviation engine's long-term maintenance plan optimization method under uncertain condition capable of optimizing maintenance cost, under the condition of same scale fleet, the solution result of uncertain fleet maintenance plan model and the solution result of deterministic model are compared, good results are obtained in the number of missing days and the number of wasted days, there is little difference in equilibrium index, the absolute error of optimal value of solution is within the unit of digit, the model under uncertain condition considers engineering practice, triangular fuzzy number is proposed to process fleet engine original data, can be applied to practical engineering, which reflects the advancement and practicality of long-term maintenance plan model of fleet under uncertain condition.

Description

Technical fields:

[0001] This invention relates to the field of civil aviation engine maintenance technology, specifically a method for optimizing long-term fleet maintenance plans under uncertain conditions for civil aviation engines, which can optimize maintenance costs. Background technology:

[0002] Civil aircraft engines are large and complex pieces of equipment with a long service life, and a single engine will undergo multiple major overhauls throughout its entire life cycle. The timing and scope of each major overhaul affect subsequent maintenance time and scope, thus impacting the overall maintenance cost over the entire engine's life cycle. Therefore, in order to rationally plan fleet maintenance schedules, it is necessary to determine the major overhaul time and on-wing life between each major overhaul from the perspective of the entire life cycle of a single engine. On-wing life is measured in flight cycles, which can be obtained from civil aviation company data by converting the number of daily flight cycles into days.

[0003] After an existing engine is removed from the aircraft stand, a suitable spare engine needs to be selected and installed. If there are no spare engines available, there are generally five possible solutions. The first is to ground the aircraft, which is unacceptable for airlines as it reduces transport capacity, lowers operating revenue, and negatively impacts reputation. The second is to purchase new engines, a good option if the airline already has few spare engines, but unwise if the shortage is due to poor planning, such as a large number of engines requiring repair in a short period. The third is to lease engines, which involves transportation costs and daily rental fees higher than the airline's own engine rental rates. The fourth is to send engines for repair in advance, which, while potentially wasteful, is a viable option if it avoids engine shortages. The fifth is intermittent maintenance, where spare engines are stored on the ground with remaining service life, the wing engine is removed and sent for repair in advance, and the spare engine is installed on the aircraft stand, thus avoiding engine shortages. For chief engineers of airline engines, the maintenance plan with the fewest engine shortages and leased engine days is generally chosen. For example... Figure 1 As shown, this illustrates the full lifecycle management process for civil aviation engines.

[0004] In summary, the long-term maintenance plan for civil aviation engines can be described as follows: within the planning period, determining the ground storage time for each engine in the fleet, determining the total number of engine downtime days within the planning period, and determining the total number of wasted engine days. The following constraints and assumptions are made for the long-term maintenance plan for civil aviation engines.

[0005] 1) Assume that the engine's total on-wing life will not change between the two major overhauls due to its installation at different fuselage positions;

[0006] 2) The number of spare engines is fixed during the planning period, and it is assumed that the number of engines in the fleet remains unchanged during the planning period, with no additions or retirements of engines;

[0007] 3) The long queues during the planning period will remain unchanged;

[0008] 4) Each engine can only be installed in one machine position at a time;

[0009] 5) Only one engine can be installed on a single aircraft stand at a time;

[0010] 6) Assume that the replacement of the aircraft is completed instantly and does not require any time;

[0011] 7) During the planned period, the engine's on-wing life will not be unexpectedly interrupted or stopped due to external factors.

[0012] 8) Assuming that engines can be leased at any time, leasing engines is the only solution when the aircraft is short of engines. Summary of the Invention:

[0013] This invention addresses the problems existing in the prior art by proposing a maintenance plan for the entire fleet and its entire life cycle, and proposes a method for optimizing the long-term maintenance plan of civil aviation engines under uncertain conditions, which can optimize maintenance costs and obtain a better maintenance plan within constraints.

[0014] This invention achieves its purpose through the following measures:

[0015] A method for optimizing long-term fleet maintenance plans under uncertain conditions, characterized by comprising the following steps:

[0016] Step 1: Establish an optimization model:

[0017] Step 1-1: Obtain the number of engine positions M, engine number N, planning period length P, and on-wing life {Q} of the fleet. l i,j Q m i,j Q u i,j Repair time {MT} l i,j,MT m i,j ,MT u i,j};

[0018] Step 1-2: Calculate the maintenance schedule;

[0019] Steps 1-3: Calculate the repair time list This includes calculating the interval between each repair. Then, the maintenance balance index is obtained; and the maximum and minimum time that each engine can provide is calculated from the start point of the planning period to each maintenance time and the end point of the planning period.

[0020] Steps 1-4: Output the number of missing and wasted days, as well as the balance index and ground retention time;

[0021] Step 2: Solve the model using the improved whale optimization algorithm, specifically as follows:

[0022] Step 2-1: Set the number of aircraft positions M, the number of engines in the fleet N, the planning period length P, and input the engine life information and the number of decision variables N·D;

[0023] Step 2-2: Set the relevant parameters for the whale algorithm: number of whales n, maximum number of iterations T iter_max ;

[0024] Steps 2-3: Initialize the location of the whale population, calculate the fitness value of each whale, sort them according to the size of the fitness value, and select n whales as the initial population;

[0025] Steps 2-4: Calculate the fitness values ​​of the n individuals and find the position of the individual with the smallest fitness value as the optimal position;

[0026] Steps 2-5: Update the position of the next generation;

[0027] If the termination condition is met in step 2-6, output the optimal individual, which is the optimal solution found by the algorithm; otherwise, return to step 2-4.

[0028] This invention assumes that the number of engines remains constant during the planning period, with no engine retirements or new engine purchases, and that the fixed costs of the engines remain constant. Given the on-wing lifespan interval between two major overhauls throughout the entire lifespan, and the intervals between each engine disassembly, replacement, and maintenance cycle, and if ground storage time can also be obtained, the maintenance timing for each engine can be determined. Assuming the number of engine bays in the fleet is M, the number of aircraft engines is N, and the planning period for establishing the fleet model is P, to increase the model's flexibility, the on-wing lifespan is set as a triangular fuzzy number, varying within a range according to the membership function relationship, {Q}. l i,j Q m i,j Q u i,j} has been obtained from the single engine life maintenance decision, representing the on-wing life of the i-th engine from the (j-1)-th repair return to the j-th repair before the j-th repair, where Q l i,j Q represents m i,j Subtract the value of the left limit, Q u i,j This represents the right limit minus the middle number Q. m i,j The value of E(Q) i,j ) represents {Q l i,j Q m i,j Q u i,j The expected value of} represents the triangular fuzzy number {Q}. l i,j Q m i,j Q u i,j The most likely outcome is that the actual on-wing time from the (j-1)th repair return of engine i to the jth repair is less than or equal to Q. m i,j -Q l i,j Decision variables are SG. i,j Let represent the total ground storage time from the end of the (j-1)th repair of the i-th engine to the beginning of the j-th repair. To increase the flexibility of the model, the maintenance cycle is fuzzified into a triangular fuzzy number. The engine maintenance time is represented by {MT}. l i,j ,MT m i,j ,MT u i,j} represents the maintenance cycle from the end of the (j-1)th repair of the i-th engine to the beginning of the j-th repair, where MTl i,j MT represents the difference between the median repair time and the left limit. u i,j MT represents the difference between the right limit and the median. m i,j E(MT) is the average repair cycle in historical repair schedules. i,j ) represents {MT l i,j ,MT m i,j ,MT u i,j The expected value of} is {MT} l i,j ,MT m i,j ,MT u i,j The most likely number in};

[0029] Formula (1) indicates that the expected value of the sum of the time from when engine i returns from its j-th maintenance to when it is sent for maintenance (j+1) is less than or equal to the original expected value of the on-wing life from when engine i returns from its j-th maintenance to when it is sent for maintenance (j+1), because some lifespan is wasted when the engine is sent for maintenance prematurely. l i,j Q m i,j Q u i,j} originally represented the on-wing lifespan from the return of the j-th engine after its j-th maintenance request to the (j+1)-th maintenance request; if the (j+1)-th maintenance request exceeds the planned period, {Q l i,j Q m i,j Q u i,j} represents the total on-wing life from the time the i-th engine is sent for maintenance and returned j-th time until the end of the planned period, {E l i,j E m i,j E u i,j} represents the end time after engine i returns from its j-th repair, and is a triangular fuzzy number; {S l i,j ,S m i,j ,S u i,j} represents the start time after engine i returns from its j-th repair, and is a triangular fuzzy number, V. i This represents the total number of times engine i needs to be repaired during the planned period.

[0030]

[0031] Formula (2) indicates that the engine is sent for repair at the same time as the last time it was disassembled and started before being sent for repair, where {b l i,j ,b m i,j ,b u i,j} represents the time when engine i is sent for repair for the jth time, and is a triangular fuzzy number.

[0032]

[0033] Formula (3) indicates that the engine repair return time is equal to the sum of the engine repair time and the repair time, where {T} l i,j ,T m i,j ,T u i,j} represents the repair time of engine i during the j-th repair, which is a triangular fuzzy number. The result of addition and subtraction operations between triangular fuzzy numbers is also a triangular fuzzy number. l i,j ,r m i,j ,r u i,j} indicates the time when engine i is usable after its lth repair return.

[0034]

[0035] Formula (4) indicates that the time when the engine can be used after being sent back for repair is always less than the time when it is first installed on the machine after being sent back for repair. This is because the two numbers being compared are triangular fuzzy numbers. The two triangular fuzzy numbers are compared by comparing the magnitude of their expected values. The triangular fuzzy number with the larger expected value is larger.

[0036]

[0037] Formula (5) indicates that the start time of engine i after the jth repair is always less than the end time, and the end time of the jth use is always less than the start time of the (j+1)th use. Only after the jth use is completed can the next use be carried out. Here, the comparison is based on the triangular fuzzy numbers. The present invention uses the comparison method of comparing expected values ​​for comparison.

[0038]

[0039] The cost of the long-term maintenance plan model for the fleet under uncertain conditions can be expressed as shown in formula (6):

[0040] minC(s)=C rent +C waste

[0041]

[0042] In the formula, s represents the solution vector formed by the decision variables;

[0043] C rent —Due to a shortage of funds for rented engines;

[0044] C waste — Expenses wasted due to sending the item for repair ahead of schedule.

[0045] In this invention, the daily cost of rental and distribution is C. rend C waste This represents the daily wasted cost incurred due to the engine being sent for repair ahead of schedule. Therefore, the optimization objective 1 for the long-term maintenance plan of the civil aviation engine fleet can be summarized as: minimizing the cost of engine shortages and the wasted cost of sending the engine for repair ahead of schedule. The costs of sending the engine for repair ahead of schedule and the wasted cost are expressed by formula (7):

[0046] minD(s)=C rend ·D rend +C waste ·D waste

[0047]

[0048] D rend Indicates the total number of days of unpaid payments during the planned period; D waste This indicates the total number of wasted days during the planning period.

[0049] In this invention, the quality of balanced engine repair is measured by the standard deviation of the engine repair interval within the planned period. The standard deviation is calculated as shown in formula (8):

[0050] in The average interval between engine repairs during the planned period is calculated as shown in formula (9).

[0051]

[0052] In the formula, n represents the total number of times the engine needs to be repaired during the planned period;

[0053] X w —The scheduled time for the wth engine to be sent for maintenance during the planned period;

[0054] σ — Standard deviation of engine repair intervals during the planned period.

[0055] When optimizing the repair balance index, this invention uses the expected value of the triangular fuzzy number to compare the values ​​of the triangular fuzzy number. Therefore, the repair time in this model is also a triangular fuzzy number, denoted by {X}. l w ,X m w ,X u w} represents the target. Because the sample size is large, fuzzy simulation of the optimization objective is required. This invention uses... To optimize E(σ), let's first... The fuzzy simulation process will be introduced in the following steps:

[0056] Step 1: Set e = 0;

[0057] Step 2: Extract the fuzzy numbers from the set Θ = {Q} for all time intervals. l i,j Q m i,j Q u i,j Generate random numbers θ uniformly in the} k , such that Pos{θ k}≥ε, let ν k =Pos{θ k}, k = 1, 2, ..., N, where ε is a sufficiently small number, in fact, as long as θ k ∈{X l w -X l w-1 ,X m w -X m w-1 ,X u w -X u w-1}, Pos{θ k If ≥ε, then it holds true.

[0058] Step 3: Set a=f(ξ(θ1))∧···∧f(ξ(θ N )),b=f(ξ(θ1))∨···∨f(ξ(θ N ));in, n represents the number of on-wing lifetimes that meet the criteria. a is all f(ξ(θ) i The smallest value among them is b, which is f(ξ(θ)). i The largest median value;

[0059] Step 4: Generate r uniformly from [a,b];

[0060] Step 5: If r≥0, then e←e+Cr{f(ξ)≥r};

[0061] Step 6: If r < 0, then e ← e-Cr {f(ξ) ≤ r};

[0062] Step 7: Repeat steps 4 to 6 a total of N times; in this invention... During the fuzzy simulation, N = 1000;

[0063] Step 8: Finally, calculate E[f(ξ)]=a∨0+b∧0+e·(ba) / N, that is... Similarly, the variance is calculated using the same steps, and the square root of the variance is taken to obtain E(σ).

[0064] Under the same fleet size conditions, the solution results of the uncertain fleet maintenance planning model, compared with the solution results of the deterministic model, achieved better results in terms of engine downtime and wasted days, with very little difference in equilibrium indicators, and the absolute error of the optimal value was within single digits. Six sets of comparative experiments verified the correctness of the long-term maintenance planning model under uncertain conditions. However, compared with the overly idealistic model under deterministic conditions, the model under uncertain conditions considers engineering realities and proposes processing the raw engine data of the fleet using triangular fuzzy numbers, making it applicable to practical engineering. This reflects the advanced nature and practicality of the long-term fleet maintenance planning model under uncertain conditions. Attached image description:

[0065] Appendix Figure 1 This is a schematic diagram of the full life cycle management process for civil aviation engines.

[0066] Appendix Figure 2 This is a flowchart for establishing a long-term fleet planning model under uncertain conditions.

[0067] Appendix Figure 3 This is a schematic diagram of the whale intelligent optimization algorithm.

[0068] Appendix Figure 4 This is a flowchart of the improved whale intelligent optimization algorithm. Detailed implementation method:

[0069] The present invention will now be further described with reference to the accompanying drawings.

[0070] Long-term fleet maintenance planning aims to determine the optimal maintenance schedule for each engine in the fleet. This invention uses the ground storage time between engine maintenance appointments as a decision variable. Engine shortages lead to engine leasing costs, while engine surpluses result in waste costs. This invention also studies the equilibrium indicators crucial to fleet maintenance planning. Therefore, this invention establishes a long-term fleet maintenance planning model with ground storage time as the decision variable and shortage costs, waste costs, and equilibrium indicators as optimization objectives. This invention applies triangular fuzzy number processing to on-wing lifetime and maintenance cycle, making them triangular fuzzy numbers and increasing the model's flexibility.

[0071] Given the on-wing lifespan interval between two major overhauls throughout the entire lifespan, and the intervals between each engine disassembly, replacement, and maintenance cycle, and if ground storage time can also be obtained, then the maintenance timing for each engine can be determined. Assume the number of engine bays in the fleet is M, the number of aircraft engines is N, and the planning period for establishing the fleet model is P. To increase the flexibility of the model, this invention sets the on-wing lifespan as a triangular fuzzy number, which varies within a range based on the membership function relationship, {Q}. l i,j Q m i,j Q u i,j} has been obtained from the single engine life maintenance decision, representing the on-wing life of the i-th engine from the (j-1)-th repair return to the j-th repair before the j-th repair, where Q l i,j Q represents m i,j Subtract the value of the left limit, Q u i,j This represents the right limit minus the middle number Q. m i,j The value of E(Q). i,j ) represents {Q l i,j Q m i,j Q u i,j The expected value of} represents the triangular fuzzy number {Q}. l i,j Q m i,j Q u i,j The most likely outcome is as follows. Since this invention does not consider the impact of delayed repairs on long-term maintenance plans, the constraint that repairs cannot be delayed and must be advanced is that repairs cannot be delayed. Therefore, the actual on-wing time from the (j-1)th repair return of engine i to the jth repair is less than or equal to Q. mi,j -Q l i,j The decision variables in this invention are SG. i,j This represents the total ground storage time from the end of the (j-1)th repair of the i-th engine to the beginning of the j-th repair. To increase the model's flexibility, this invention fuzzifies the maintenance cycle into a triangular fuzzy number, and the engine maintenance time is represented by {MT}. l i,j ,MT m i,j ,MT u i,j} represents the maintenance cycle from the end of the (j-1)th repair of the i-th engine to the beginning of the j-th repair. Where MT l i,j MT represents the difference between the median repair time and the left limit. u i,j MT represents the difference between the right limit and the median. m i,j E(MT) is the average repair cycle in historical repair schedules. i,j ) represents {MT l i,j ,MT m i,j ,MT u i,j The expected value of} is {MT} l i,j ,MT m i,j ,MT u i,j The most likely number in}.

[0072] This invention assumes that the number of engines remains constant during the planning period, with no engine retirements or new engine purchases, and that the fixed costs of the engines remain constant. The optimization of the long-term fleet maintenance plan aims to determine the number of times each engine needs maintenance, the number of times it can be used between two maintenance visits, and the start and end times of each engine installation at its designated location, thereby minimizing maintenance costs during the planning period. Due to the inherent characteristics of civil aviation engines—high reliability, long lifespan, and high cost—specifically, the ability to be reused throughout its lifespan, the civil aviation engine fleet maintenance planning model requires additional constraints.

[0073] Formula (1) indicates that the expected value of the sum of the time from when engine i returns from its j-th maintenance repair to before its (j+1)-th maintenance repair is less than or equal to the original expected value of the on-wing lifespan from when engine i returns from its j-th maintenance repair to before its (j+1)-th maintenance repair. This is because some lifespan is wasted when the engine is sent for maintenance prematurely. {Q li,j Q m i,j Q u i,j} originally represented the on-wing lifespan from the return of the j-th engine after its j-th maintenance request to the (j+1)-th maintenance request; if the (j+1)-th maintenance request exceeds the planned period, {Q l i,j Q m i,j Q u i,j} represents the total on-wing life from the time the i-th engine is sent for maintenance and returned j-th time until the end of the planned period, {E l i,j E m i,j E u i,j} represents the end time after engine i returns from its j-th repair, and is a triangular fuzzy number; {S l i,j ,S m i,j ,S u i,j} represents the start time after engine i returns from its j-th repair, and is a triangular fuzzy number. V i This represents the total number of times engine i needs to be repaired during the planned period.

[0074]

[0075] Formula (2) indicates that the engine is sent for repair at the same time as the last time it was disassembled and started before being sent for repair, where {b l i,j ,b m i,j ,b u i,j} represents the time when engine i is sent for repair for the jth time, and is a triangular fuzzy number.

[0076]

[0077] Formula (3) indicates that the engine repair return time is equal to the sum of the engine repair time and the repair time. Where {T} l i,j ,T m i,j ,T u i,j} represents the repair time of engine i during its j-th repair, and is a triangular fuzzy number. The result of addition and subtraction operations between triangular fuzzy numbers is also a triangular fuzzy number. l i,j ,r m i,j ,ru i,j} indicates the time when engine i is usable after its lth repair return.

[0078]

[0079] Formula (4) indicates that the time when the engine can be used after being sent for repair and returned is always less than the time when the engine is first installed on the machine after being sent for repair and returned. This is because the two numbers being compared are triangular fuzzy numbers. In this invention, the expected value is compared to the two triangular fuzzy numbers. The triangular fuzzy number with the larger expected value is larger.

[0080]

[0081] Formula (5) indicates that the start time of engine i after the jth repair is always less than the end time, and the end time of the jth use is always less than the start time of the (j+1)th use. Only after the jth use is completed can the next use be carried out. Here, the comparison is based on the triangular fuzzy numbers. The present invention uses the comparison method of comparing expected values ​​for comparison.

[0082]

[0083] The cost of the long-term maintenance plan model for the fleet under uncertain conditions can be expressed as shown in formula (6):

[0084] minC(s)=C rent +C waste

[0085]

[0086] In the formula, s represents the solution vector formed by the decision variables;

[0087] C rent —Due to a shortage of funds for rented engines;

[0088] C waste — Expenses wasted due to sending the item for repair ahead of schedule.

[0089] To judge the merits of long-term maintenance plans, this invention uses the minimum number of missing and wasted days, as well as a balance index, to introduce a function about missing and wasted days and the balance index, and uses the minimum function value as the optimization objective.

[0090] In civil aviation, fleet maintenance optimization plans play a crucial role in cost, therefore, it's essential to clearly define the costs required for each stage. When optimizing for engine shortages and wasted days, we also optimize through cost. First, we calculate the fleet cost. Within the cost components, engine leasing is necessary due to shortages, and the daily cost of engine leasing is C. rend Cwaste This refers to the wasted cost incurred each day due to the engine being sent for repair ahead of schedule.

[0091] Therefore, the optimization objective 1 of the long-term maintenance plan for civil aviation engine fleet can be summarized as: minimizing the cost of engine shortage and the cost of premature repair waste. The cost of premature repair and the cost of waste are expressed by formula (7):

[0092] minD(s)=C rend ·D rend +C waste ·D waste

[0093]

[0094] In the formula D rend —This indicates the total number of days of unpaid payments within the planned period;

[0095] D waste — This indicates the total number of wasted days during the planning period.

[0096] In addition to optimizing costs, long-term fleet maintenance plans must also ensure reasonable engine maintenance schedules. If a large number of engines are sent for maintenance consecutively within a short period, it can lead to insufficient spare engines, or even zero spare engines. If an unexpected situation arises requiring spare engines, the lack of spare engines could ground the aircraft, which is unacceptable. Therefore, achieving a balanced maintenance schedule is a key aspect of this invention.

[0097] The optimization objective 2 of the long-term maintenance plan model for civil aviation engine fleets is the engine maintenance balance index. The quality of maintenance balance is measured by the standard deviation of the engine maintenance interval during the planning period. The standard deviation is calculated as shown in formula (8):

[0098]

[0099] in The average interval between engine repairs during the planned period is calculated as shown in formula (9).

[0100]

[0101] In the formula, n represents the total number of times the engine needs to be repaired during the planned period;

[0102] X w —The scheduled time for the wth engine to be sent for maintenance during the planned period;

[0103] σ — Standard deviation of engine repair intervals during the planned period.

[0104] Here, because the present invention has a wing lifespan {Q} l i,j Q mi,j Q u i,j Since {X} is a triangular fuzzy number, this invention uses the expected value of the triangular fuzzy number to compare its values ​​when optimizing the repair balance index. Therefore, the repair time in this model is also a triangular fuzzy number, denoted as {X}. l w ,X m w ,X u w} represents the target. Because the sample size is large, this invention requires fuzzy simulation of the optimization objective. This invention uses... And E(σ) are used for optimization. First, this invention will first optimize E(σ). The fuzzy simulation process will be introduced in the following steps:

[0105] Step 1: Set e = 0;

[0106] Step 2: Extract the fuzzy numbers from the set Θ = {Q} for all time intervals. l i,j Q m i,j Q u i,j Generate random numbers θ uniformly in the} k , such that Pos{θ k}≥ε, let ν k =Pos{θ k}, k = 1, 2, ..., N, where ε is a sufficiently small number, in fact, as long as θ k ∈{X l w -X l w-1 ,X m w -X m w-1 ,X u w -X u w-1}, Pos{θ k If ≥ε, then it holds true.

[0107] Step 3: Set a=f(ξ(θ1))∧···∧f(ξ(θ N )),b=f(ξ(θ1))∨···∨f(ξ(θ N ));in, n represents the number of on-wing lifetimes that meet the criteria. a is all f(ξ(θ) i The smallest value among them is b, which is f(ξ(θ)). i The one with the largest median value.

[0108] Step 4: Generate r uniformly from [a,b];

[0109] Step 5: If r≥0, then e←e+Cr{f(ξ)≥r};

[0110] Step 6: If r < 0, then e ← e-Cr {f(ξ) ≤ r};

[0111] Step 7: Repeat steps 4 to 6 a total of N times; in this invention... During the fuzzy simulation, N = 1000;

[0112] Step 8: Finally, calculate E[f(ξ)] = a∨0 + b∧0 + e·(ba) / N. That is... Similarly, the variance is calculated using the same steps, and the square root of the variance is taken to obtain E(σ).

[0113] Thus, this invention establishes a model with the optimization objectives of engine shortage and waste costs caused by premature repairs, as well as the repair balance index. That is, after obtaining some necessary information about the fleet's engines, the repair time of the fleet's engines can be calculated.

[0114] Currently, there are two main approaches to solving multi-objective optimization problems: direct methods and indirect methods. Direct methods, as the name suggests, do not process the individual optimization objectives; they directly obtain a solution that simultaneously satisfies the multi-objective function values ​​and all constraints. This solution is known as a Pareto solution. In contrast, indirect methods require preprocessing the optimization objectives. There are many preprocessing methods, the most common being the weighting coefficient method and the hierarchical method. The weighting coefficient method assigns a weight coefficient to each optimization objective, effectively dedimensionalizing them. This allows the objectives to be weighted and summed, resulting in a new optimization objective. This transforms the multi-objective optimization problem into a single-objective one, simplifying the problem. The hierarchical method analyzes the importance of each optimization objective, ranking them. The most important objective is optimized separately, without considering other objectives, yielding its solution set, which is denoted as Ω1. Next, the second most important optimization objective is optimized separately, but its search domain must be within Ω1. Let the solution set of the second most important optimization objective be Ω2. Then, the third most important optimization objective is optimized separately, but its search domain must be within Ω1 and Ω2. This process is repeated to find the solution sets for each optimization objective. It can be seen that the hierarchical processing method has a complex operation process, and if the optimal solution set cannot be found in the process of solving a certain optimization objective, the previous work becomes meaningless, and subsequent work becomes difficult to continue, resulting in an inability to solve the problem. From the above description, it is not difficult to find that among the methods for solving multi-objective optimization problems, the three methods, from simplest to most complex, are: the weighting coefficient method, the direct method, and the hierarchical processing method.

[0115] In summary, the optimization method for the long-term fleet maintenance planning problem based on fuzzy simulation in this invention is the weighted coefficient method. The weighted coefficients corresponding to the two optimization objectives are shown in Table 1:

[0116] Table 1. Weight coefficients corresponding to the optimization objectives

[0117]

[0118]

[0119] Based on the weight coefficients in the table, this invention performs a weighted merging of the two optimization objectives, combining them into one optimization objective, as shown in formula (10):

[0120] minF(s)=λD(s)+(1-λ)E(σ) (10)

[0121] The purpose of the long-term maintenance plan for the fleet is to determine the maintenance schedule for each engine. Once the maintenance schedule is determined, the value of optimization objective 2 can be obtained through calculation. However, optimization objective 1 needs to be optimized. This invention provides a series of calculation rules for optimization objective 1 and optimization objective 2, as shown below:

[0122] Step 1: Obtain the number of aircraft engines (M), the number of engines in the fleet (N), and the length of the planning period (P). Input the lifespan information of the N engines and the on-wing lifespan (Q) of each engine between two major overhauls. l i,j Q m i,j Q u i,j}, through the planned period {Q l i,j Q m i,j Q u i,j The quantity of} can be used to obtain the quantity of decision variables, i.e., the quantity of ground storage time, because the quantities of the two are equal, let the quantity be D. The engine overhaul time is {MT}. l i,j ,MT m i,j ,MT u i,j}

[0123] Step 2: Calculate the maintenance time list based on D obtained in Step 1, and set the current time... Given {0, 0, 0}, iterate through the on-wing lifetime of each engine, first setting {Q l i,j Q m i,j Q u i,j} and {0,SG i,j Adding {0} together, the sum is {Q}. l i,j +0,Q m i,j +X i,j Q u i,j Let S be the integer '+0', then... Add the value of S to Will Compared with P, if Then retain the triangular fuzzy number. Time list obtains triangular fuzzy number With {MT l i,j ,MT mi,j ,MT u i,j Add them together and assign the result to the given value. Otherwise, increment j+1 to proceed to the next iteration. Finally, sort the triangular fuzzy numbers in the time list according to their expected values.

[0124] Step 3: Calculate the repair time list, remove duplicates from the previously sorted sequence, and then... middle Items with three identical numbers are eliminated, and the remaining items are the repair times, which are then sorted according to their expected values, denoted as . Because we need to consider the maximum and minimum time each engine can provide at each time it is sent for maintenance, we need to remove duplicate items.

[0125] Step 4: Based on the list of previous repair requests Calculate the intervals between each repair request; the first interval is the first item in the repair time list. The remaining items have a value of Recorded as

[0126] Step 5: Based on the intervals between previous repairs The variance is solved using fuzzy simulation, and the standard deviation is obtained by taking the square root of the variance, denoted as V. This is the value of the optimization objective 2.

[0127] Step 6 calculates the maximum and minimum time each engine can provide from the start of the planning period to each repair request time and the end of the planning period. The repair request list is obtained from Step 3 at different times. At each time point, calculate the maximum and minimum on-wing time that the engine can provide. The minimum on-wing time is calculated by consuming all ground-based storage time before consuming the remaining on-wing lifetime. The maximum on-wing lifetime is the on-wing lifetime the engine can provide up to the time of service, excluding ground-based storage time. Determine the minimum on-wing time that each engine can provide at each time point. We take the expected value for each time. Similarly, the maximum time can be obtained.

[0128] Step 7: Solve for missing data and wasted days. Based on the results obtained in Step 3... Before each time point, calculate and if This will result in waste, with D days wasted. waste for

[0129] if This will result in missed payouts, with the number of days of missed payouts being D. short for

[0130] Flowchart as follows Figure 2 As shown in the preceding explanation, the solution space of this invention is extremely large. Using a traversal method for the aforementioned algorithm would consume a significant amount of time and resources. The long-term maintenance plan problem for civil aviation engine fleets is essentially a combinatorial optimization problem. Currently, algorithms for solving combinatorial optimization problems mainly include intelligent optimization algorithms such as genetic algorithms, gray wolf algorithms, particle swarm optimization algorithms, and whale algorithms, or heuristic algorithms for specific problems. The whale algorithm is widely used due to its simplicity, minimal parameter adjustments, and strong ability to escape local optima. Therefore, this invention employs the whale intelligent optimization algorithm to solve the model.

[0131] The Whale Optimization Algorithm is a novel heuristic optimization algorithm that mimics the hunting behavior of humpback whales. In the Whale Optimization Algorithm, the position of each whale represents a feasible solution. In the ocean, whales employ a unique hunting method called the bubble-net predation strategy. Their hunting behavior is as follows: Figure 3 As shown

[0132] The whale algorithm achieves its search objective through processes such as whale swarm searching, surrounding, chasing, and attacking prey, providing mathematical models for surrounding prey, spiral bubbles, and prey locating. At the initial position, each whale's position is X = (x1, x2, ..., x...). D () represents a feasible solution. Through the two stages of exploration and development, the optimal position, i.e. the optimal solution, is gradually found.

[0133] The Whale Intelligent Optimization Algorithm Process is as follows:

[0134] 1. Initialization parameters: namely, the number of whales n in the whale population, and the maximum number of iterations T. iter_max ;

[0135] 2. Initialize the location of the whale population;

[0136] 3. Calculate the fitness value of each whale, sort them according to the size of the fitness value, and select n as the initial population;

[0137] 4. Calculate the fitness values ​​of n individuals and find the position of the individual with the smallest fitness value as the optimal position;

[0138] 5. Update the position of the next generation;

[0139] 6. If the termination condition is met, output the optimal individual, which is the optimal solution found by the algorithm; otherwise, return to step 4.

[0140] The flowchart of the improved whale intelligent optimization algorithm is as follows: Figure 4 The style.

[0141] The proposed model is validated and evaluated below based on six groups of maintenance schedules for aircraft engine fleets of different sizes generated from survey data. The long-term fleet maintenance planning model based on triangular fuzzy numbers is compared with a deterministic plan to illustrate the effectiveness of this model. The six fleet sizes are shown in Table 2.

[0142] Table 2 Fleet Size Information

[0143]

[0144] The whale intelligent optimization algorithm was used to solve the uncertain model and the deterministic model of the same scale, and the solution results are shown in Table 3.

[0145] Table 3 shows the experimental results.

[0146]

[0147] The experimental results in Table 3 show that, under the same fleet size conditions, the solution results of the uncertain fleet maintenance plan model are better than those of the deterministic model in terms of engine downtime and wasted days, with very little difference in the equilibrium index, and the absolute error of the optimal value is within single digits. Six sets of comparative experiments verify the correctness of the long-term maintenance plan model under uncertain conditions. However, compared to the overly idealistic model under deterministic conditions, the model under uncertain conditions considers engineering realities and proposes processing the raw fleet engine data with triangular fuzzy numbers, making it applicable to practical engineering. This reflects the advanced nature and practicality of the long-term fleet maintenance plan model under uncertain conditions.

Claims

1. A method for optimizing long-term maintenance planning of a fleet under uncertain conditions, characterized in that, The method comprises the following steps: Step 1: establishing an optimization model: Step 1-1: Obtain the number of fleet engine positions M, the number of engines N, the planning period length P, the in-flight life Q l i,j ,Q m i,j ,Q u i,j , the maintenance time MT l i,j ,MT m i,j ,MT u i,j ; Step 1-2: calculating a maintenance time list; Step 1-3: Calculate the list of repair times including calculating the interval between each repair then obtaining the repair balance index; and calculating the maximum and minimum time each engine can provide from the start of the planning period to each repair time and the end of the planning period Step 1-4: outputting missing and waste days and balance indexes and ground storage time; Step 2: using an improved whale optimization algorithm to solve the model, specifically: Step 2-1: setting the number of aircraft parking spaces M, the number of engines in the aircraft fleet N, the planned period length P, inputting engine life information and the number of decision variables N·D; Step 2-2: Set the whale algorithm related parameters: the number of whales n, the maximum number of iterations T iter_max ; Step 2-3: initializing the position of the whale population, calculating the corresponding fitness value of each whale, sorting according to the size of the fitness value, and selecting n as the initial population; Step 2-4: calculating the size of the fitness value of the n individuals, and finding the position of the individual with the minimum fitness value as the optimal position; Step 2-5: updating the position of the next generation; Step 2-6 if the termination condition is reached, the optimal individual is output, that is, the optimal solution found by the algorithm; otherwise, return to step 2-4.

2. The method for optimizing long-term maintenance planning of a fleet under uncertain conditions according to claim 1, characterized in that, The fixed cost of engine possession is a fixed value, which remains unchanged during the planning period, provided that the number of engines remains unchanged, neither the retirement of engines nor the purchase of new engines, the in-flight life time interval between two overhauls in the known life cycle, and the engine disassembly, replacement and repair cycle time interval, if the ground storage time can be obtained, the engine maintenance time can be determined, assuming that the number of aircraft engine positions in the fleet is M, the number of aircraft engines is N, and the planning period of the fleet model is P. In order to increase the flexibility of the model, the in-flight life is set as a triangular fuzzy number, which varies within the range according to the membership function relationship. l i,j , m i,j , u i,j} is obtained by single engine life maintenance decision, which represents the in-flight life of the ith engine before the jth repair after the (j-1)th repair, wherein l i,j represents the left limit of Q m i,j , the value of Q u i,j represents the value of Q m i,j , the right limit minus the middle number i , j represents the expected value of {Q l i,j , m i,j , u i,j} is the most likely result in the triangular fuzzy number {Q l i,j , m i,j , u i,j} is the actual in-flight time of the ith engine from the (j-1)th repair to the jth repair, which is less than or equal to Q m i,j -Q l i,j , the decision variable is represented by SG i,j , which represents the total ground storage time of the ith engine from the end of the (j-1)th repair to the jth repair, in order to increase the flexibility of the model, the repair cycle is fuzzified as a triangular fuzzy number, and the engine repair time is represented by {MT l i,j , MT m i,j , MT u i,j} represents the maintenance cycle from the end of the (j-1)th repair of the i-th engine to the beginning of the j-th repair, where MT l i,j MT represents the difference between the median repair time and the left limit. u i,j MT represents the difference between the right limit and the median. m i,j E(MT) is the average repair cycle in historical repair schedules. i,j ) represents {MT l i,j ,MT m i,j ,MT u i,j The expected value of} is {MT} l i,j ,MT m i,j ,MT u i,j The most likely number in}; Equation (1) indicates that the expected value of the sum of the time that engine i is actually installed on the airplane engine site from the return of the jth repair to the (j+1)th repair is less than or equal to the expected value of the on-wing life from the return of the jth repair to the (j+1)th repair of the original engine i, because some life is wasted when the engine is repaired early, l i,j , m i,j , u i,j the original indicates the on-wing life of the ith engine from the return of the jth repair to the (j+1)th repair; if the time of the (j+1)th repair exceeds the planned period, l i,j , m i,j , u i,j then indicates the total on-wing life of the ith engine from the return of the jth repair to the end of the planned period, l i,j , m i,j , u i,j indicates the end time of the return of the jth repair of engine i, which is a triangular fuzzy number, l i,j , m i,j , u i,j indicates the start time of the return of the jth repair of engine i, which is a triangular fuzzy number, and i indicates the total repair times of engine i in the planned period, Equation (1) represents the engine repair time as the last disassembly time before the engine repair, where {b l i,j ,b m i,j ,b u i,j} represents the time of the jth repair time of engine i, which is a triangular fuzzy number, Equation (3) represents the engine return-to-service time as the sum of the engine service time and the repair time, where {T l i,j m i,j u i,j represents the repair time for the jth service of engine i, and is a triangular fuzzy number. The addition and subtraction of triangular fuzzy numbers results in a triangular fuzzy number. {r l i,j m i,j u i,j represents the time at which engine i can be used after the lth service.​​​​ Formula (4) indicates that the time when the engine is returned after repair can be used is always less than the time when the engine is first installed on the parking space after repair, because the two numbers compared are triangular fuzzy numbers, the size of the expected value is compared to compare two triangular fuzzy numbers, and the triangular fuzzy number with the larger expected value is larger, Formula (5) indicates that the start time of the engine i after the jth repair return is always less than the end time, and the end time of the jth use is always less than the start time of the j+1 use, only after the jth use is ended, the next use can be performed, and here, the comparison method of comparing the expected values of triangular fuzzy numbers is adopted for comparison; The cost of the long-term maintenance plan model of the aircraft fleet under uncertain conditions can be represented as formula (6): min C(s) = C rent +C waste In the formula, s is a solution vector composed of decision variables; C rent — the cost of a missing lease engine; C waste Waste of costs due to premature repair.

3. The method of claim 1, wherein, The daily cost of the lease is C rend , C waste represents the waste cost of the engine due to the advance repair, and thus, the optimization objective 1 of the long-term maintenance plan of the civil aviation engine fleet can be arranged: the minimum of the lack cost and the advance repair waste cost. The advance repair cost and the waste cost are represented by formula (7): min D(s) = C rend • D rend + C waste • D waste D rend represents the total number of days lost during the planning period; D waste represents the total number of days wasted during the planning period; In the application, the repair balance is measured by the standard deviation of the engine repair interval in the optimal use plan period, and the standard deviation is calculated according to formula (8): where is the average value of engine repair intervals during the planning period, calculated as shown in equation (9) In the formula, n is the total number of engine repairs in the planned period; X w - the wth engine service time within the planning period; σ is the standard deviation of the engine repair interval in the planned period.

4. The method of claim 1, wherein, The expected value of triangular fuzzy number is used to compare the value of triangular fuzzy number, so the repair time in the model is also a triangular fuzzy number, which is represented by {X l w ,X m w ,X u w} ; because the sample is large, the fuzzy simulation of the optimization target is needed, and the present application uses and E (sigma) to optimize, and the fuzzy simulation process of is introduced as follows: Step 1: set e=0; Step 2: Generate random numbers θ l i,j ,Q m i,j ,Q u i,j} from all time interval fuzzy number sets Θ = {Q k , respectively, such that Pos{θ k} ≥ ε, let v k = Pos{θ k}, k = 1, 2, ···, N, where ε is a sufficiently small number, and in fact, as long as θ k ∈ {X l w -X l w-1 ,X m w -X m w-1 ,X u w -X u w-1}, Pos{θ k} ≥ ε holds; Step 3: Set a = f(ξ(θ1)) Λ ··· Λ f(ξ(θ N )), b = f(ξ(θ1)) V ··· V f(ξ(θ N )); where, n is the number of wings that meet the condition in the life of the wing, a is the minimum value of all f(ξ(θ i )), and b is the maximum value of f(ξ(θ i )). Step 4: generate r uniformly from [a,b]; Step 5: if r>=0, then e <- e+Cr{f(ξ) >= r}; Step 6: if r < 0, then e <- e-Cr{f(ξ) <= r}; Step 7: end. Step 7: Repeat steps 4 through 6 for N times; in the present application, N = 1000 is taken in the fuzzy simulation process. Step 7: Repeat steps 4 through 6 for N times; in the present application, N = 1000 is taken in the fuzzy simulation process. Step 8: Finally, compute E[f(ξ)] = a ∨ 0 + b ∧ 0 + e · (b - a) / N, i.e. Similarly, the variance is computed in the same way and the square root of the variance is computed to obtain E(σ).

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