Optimal Combination of Aircraft Maintenance Tasks by a Novel Simplex Optimization Method

Combiningmaintenance tasks into work packages is not only necessary for arrangingmaintenance activities, but also critical for the reduction of maintenance cost. In order to optimize the combination of maintenance tasks by fuzzy C-means clustering algorithm, an improved fuzzy C-means clustering model is introduced in this paper. In order to reduce the dimension, variables representing clustering centers are eliminated in the improved cluster model. So the improved clustering model can be directly solved by the optimization method. To optimize the clustering model, a novel nonlinear simplex optimization method is also proposed in this paper.The novel method searches along all rays emitting from the center to each vertex, and those search directions are rightly n+1 positive basis. The algorithm has both theoretical convergence and good experimental effect. Taking the optimal combination of some maintenance tasks of a certain aircraft as an instance, the novel simplex optimization method and the clustering model both exhibit excellent performance.


Introduction
Combining maintenance tasks into work packages is not only necessary for arranging maintenance activities, but also critical for the reduction of maintenance cost.In the field of civil aircraft maintenance, in order to reduce the maintenance cost, the number of character check is constantly increased.For example, the number of  check is increased from 64 to 128.Recently, character checks have been completely cancelled, instead, the maintenance tasks executed in character check are combined into several small maintenance work packages and are equally allocated to line maintenance, so as to balance the maintenance and reduce the maintenance cost.However, it is still inevitable to optimally combine maintenance tasks into work packages no matter whether character check is utilized or not.As combinatorial optimizations usually belong to NP-hard problems that are difficult to be exactly solved, only optimal approximation solutions are obtained on the basis of fuzzy clustering of maintenance tasks in the paper.First of all, a modification is made on the fuzzy C-means clustering model to reduce the dimension.Then, a novel nonlinear simplex algorithm is put forward.Lastly, the optimal combination of maintenance tasks is transformed into a clustering problem which can be solved by a novel simplex algorithm.
In the traditional fuzzy C-means (FCM) clustering method, not only the degree of fuzzy membership is taken as an independent variable but also the center of every class is unknown.For this reason, the FCM model is a high-dimension optimization problem that is difficult to be directly solved.Usually, only its local optimal solution can be found out by the alternative optimization algorithm.Therefore, the traditional FCM may present high error rate and poor adaptability.In this paper, class centers are regarded as a function of fuzzy membership degree and samples; thus, an improved FCM model is developed.The improved model does not explicitly include clustering centers, so the number of unknown variables is reduced, which not only results in (

Improvement of Fuzzy C-Means Clustering
The key to the traditional FCM algorithm is to solve the class center matrix [  ] × and the fuzzy membership matrix [  ] × .However, clustering is a NP problem, and (1) is usually a high-dimension optimization problem with numerous local minimum values [8].Traditionally, Model (1) is alternatively solved by the steepest descent method.Firstly, assuming that the current clustering center   is known, and the stationary point of Model (1) with respect to   is taken as the next   .Then, the new   is supposed to be constant, and the optimum   is achieved with the stationary point of Model (1) with respect to   , as shown in In the whole course of computation, the unknown   ,   ,  = 1, . . ., ,  = 1, . . .,  are not simultaneously computed as a whole, and also, the correlation between   and   is ignored.Therefore, only the local minimum point can be found out.The traditional FCM model must repeatedly compute the distance between each sample and all centers of class at each iteration, which consumes a large amount of computation.

Improved Model of FCM.
Actually, there is a close correlation between   and   .According to (2), if the class center   is regarded as the function of   and   , and then   can be eliminated and the number of variables will be reduced.From the property of norm and inner product, Different from ( 1), (3) does not contain class center   .Instead, it is composed of inner products among samples.If they can be computed and stored in advance, the computed amount of Model (3) can be reduced especially when sample size is large.Compared with (1), (3) contains much less unknown variables, which makes it possible to directly solve (3).As (3) has numerical stationary points with respect to   , the optimization method which needs derivative is inappropriate for (3).Inspired by the traditional simplex algorithm, a novel direct optimization method that can be utilized to solve nonlinear programming, namely, a novel simplex optimization method based on center ray, is presented to solve (3) in the paper.

The Verification of Improved Model of FCM.
To solve the constrained optimization Model (3), ( 3) is firstly transformed into the unconstrained Model (4) by the exterior penalty function method.Then, the novel optimization method proposed in the paper is employed to solve the unconstrained problem, namely, (4): If   is the center of the th class of simulation sample and   is the random vector obeying certain distribution, then   = {  +  ,  = 1, . . ., |  |} can be regarded as the set of the th class simulation sample.In the experiment,   , respectively, follows normal distribution and uniform distribution, in other words, simulate, respectively, under   ∼ (0, (( +1 −   )/2) 2 ) and   ∼ (−(( +1 −   )/2), ( +1 −   )/2) circumstance.Three classes of eight-dimensional random vectors are randomly generated, which comprise sample set { 1 ,  2 ,  3 }, where, respectively, set  1 = 1,  2 = 2, and  3 = 3.Before the sample is generated, the number of every class sample is randomly determined by |  | = ⌊  + (  − 0.5)⌋, where,  = 5,   ∼ (0, 1) and , and the error rate of classification is utilized to evaluate the effect of the classification and compare the performance of the clustering model.Model ( 4) is solved by Algorithm 5.In the experiment, numerical simulation is repeated 300 times under the condition of same experimental parameters and some average results are totaled, which are tabulated in Table 1.  in Table 1 is the practical optimum value of the clustering object function, which can indicate whether the clustering model has advantages and disadvantages.In the experiment, the main parameters of Algorithm 5 are as follows: ... After analyzing the results in Table 1, the following conclusion is obtained.  in Model (3) is always smaller than that in Model (1), which implies the result of Model (3) is better than that of Model (1) from the view of mathematical optimization of object function.  ,   of Model (3) are also better than those of Model (1), which shows that, after clustering of Model (3), the interior of each class becomes more compact and the separability and dissimilarity between classes are more significant apparent.Besides, in terms of the classification error rate, Model (3) is also more accurate than Model (1).However, if   and   are taken into consideration, Model (1) seems to be better, indicating that makingdecisions of Model (1) are clearer and more arbitrary while the membership degree of Model ( 3) is more uniform than that of Model (1).Thus, compared with traditional FCM, Model (3) is stronger in adaptability and higher in recognition accuracy.from which the degeneration of the simplex can be judged.

A Novel Simplex Optimization Algorithm
The whole iteration of the traditional nonlinear simplex algorithm mainly consists of four steps: reflection, expansion, contraction, and shrinkage.In contrast, the novel algorithm in the paper primarily solves problems by generating a simplex, raying outward from the simplex center and reflecting operations.Taking the unconstrained optimization problem min () as an example, the novel simplex algorithm can be described as below.
3.1.1.Generation of a Simplex.Due to the tendency of deformation and degeneration in the process of the simplex's iteration, it is usually better that the initial simplex is constructed as an orthogonal one.When one side of the orthogonal simplex gets close to the negative gradient direction −  , the simplex will search much rapidly.If   = ( 1 , . . .,   ) and  is any of solution of the linear equations     = 0, then  will be orthogonal to  1 , . . .,   .Firstly, we set  1 = −  and  1 = ( 1 ).If , that is, any of the solutions of     = 0, is taken as the last column of   to generate a new matrix  +1 , then the orthogonal matrix   = ( 1 , . . .,   ) can be obtained after repeatedly solving the linear equations −1 times, where,  1 , . . .,   represent  orthogonal vectors.Subsequently, we choose the point whose function value is better from {  + ℎ    ,   − ℎ    } as the vertex of the simplex.In this paper, the flow of simplex generation is shown as Algorithm 1.
Step 3. Solve the linear equations     = 0 and choose any of the solutions as  and let Step 4. When  <  − 1, then  =  + 1 and go to Step 2; otherwise exit.
Following the procedure, the simplex   is generated at   .Edges connecting with   are orthogonal vectors whose lengths are all ℎ  .Besides, the first edge is parallel to the negative gradient direction −  .
The center ray operation searches along the direction of   ,  = 0, . . .,  by comparing () and (  ),  = 0, . . ., .Firstly, as shown in (7), the first search is along the direction   ,  = 0, . . ., , where,   must satisfy (8), and it is advised to calculate   by (9) or (10).To be specific, when () > (  ), we let   =  outer > 1, which means search outside the simplex.If () ≤ (  ), it is reasonable we assume better points locate between  and   , so we let   =  inner ∈ (0, 1) in order to search inside the simplex: Next, in order to look for a better point, the second detection is attempted according to ( x  ), indicated in (11).The second detection point is located by parameter   whose scope is determined by comparing function values of x  ,   , , as indicated in (12): If x  , which is found out by the first search, is the minimum point, then the direction is supposed as the descent direction.Naturally, a better point may be acquired after extending forward along the same direction.So let   > 1 in the case.If expansion point x  is better than x  , x  will be accepted; or else, x  will be accepted.When (  ) ≤ (x  ) ≤ () ∨ () < (  ), better points are supposed to locate between   and x .Thus, set  −1  <   < 1, and the algorithm goes back for a certain distance to look for a better point x  .Then,   is replaced with the better point from x  and x  , as a result, a new simplex will be formed.When (  ) < () ≤ ( x ), assumption that better points locate between   and  is reasonable; thus, let Otherwise, vertex   still retains in the new simplex.Through the comparison between (  ) and (), center ray operation can search both inside and outside the simplex, so the algorithm can get close to the extreme point no matter it is inside or outside the simplex.Some illustrations of the center ray operation under 2-dimensional circumstance are shown in Figure 1.
In the process of the center ray, the simplex can become larger or smaller.The adaptive change can reduce calculated amount and accelerate the convergence.If search direction is appropriate, the second search can expand outwards, which can enlarge the simplex and speed up the search.If the extreme point is inside the simplex, the simplex can be automatically downsized by the comparison of function values among the center, the vertex and the detected point.Consequently, the convergence will be accelerated.The search of traditional simplex method relies mainly on reflection operations; unfortunately, its search is only along one direction and only one vertex is updated.Computation of the traditional method is slow and its effect is poor in solving high-dimension problems.In comparison, the center ray operation searches along directions of +1 positive bases, and one operation may update several vertexes.As a result, not only the computation speed is improved but also the omission of solution is avoided.

Reflection Operation.
In the simplex, the center of all vertexes except the worst vertex   is denoted as   , as shown (13).Firstly, like the traditional simplex method,   is reflected about   .As indicated in (14), x , namely, the reflection point of   about   , is computed.Different from the traditional simplex method, the algorithm in the paper can search along two inverse reflection directions by comparison between (  ) and (  ), as shown in (15).Besides, like center ray operation, there are two attempts in reflection operation, as shown in ( 14) and ( 18), respectively: The reflection coefficient  can be computed by either (16) or (17).As shown in (16), the reflection coefficient  is usually constant.When  decreases with iterations as shown in (17), the computation speed can be accelerated.In (17),  max indicates the preset maximum iterations,  forward init ,  reverse init are the reflection coefficients when  = 0; and  forward end and  reverse end are the reflection coefficients when  =  max : After that, by comparison among ( x ), (  ), (  ), x is computed by (18) to the second search.When (  ) ≥ (  ), coefficient   must satisfy (19); when (  ) < (  ),   must satisfy (20).The final reflection point is denoted as   , as shown in (21).If (  ) < (  ),   is replaced with   to form a new simplex.Otherwise, the reflection will fail.Figure 2 illustrates some of the process of reflection operations under 2-dimensional circumstance:

Mathematical Problems in Engineering
Other than the traditional simplex method, both the center ray and the reflection in the paper twice searches along a same direction, likewise, object function is also computed twice.If results of the first detection are better, the algorithm outward expands along the same direction with a longer step to start the second search.Thus, computation speed will be accelerated.Otherwise, the algorithm will try to make certain contraction on the basis of the first search or search along inverse direction.This will improve the accuracy, and what is more important, this also can cause adaptive changes in both the size and the shape of the simplex in the search process.It means to automatically and continuously change the direction and step of the search, which enhance adaptability of the algorithm.One operation in the improved algorithm can achieve effects of several operations of the traditional algorithm, including reflection, expansion, contraction, and  shrinkage.As long as the appropriate parameters are chosen, the computation efficiency can be greatly improved.

Estimation of Gradient with Simplex.
As derivative information is not utilized in the traditional simplex algorithm, the search is slow.Function values of at least  + 1 points are known in the simplex of -dimensional problems.Therefore, the gradient even the Hessian matrix at each iteration can be estimated by these  + 1 points.When one side of the simplex is in accordance with the negative gradient, the simplex may search along the descent direction.
If the th side vectors   −  0 ( = 1, . . ., ) connecting with the optimum point  0 is denoted as V  , then V  = ‖V  ‖ ⃗ V  , where ‖V  ‖ = ‖  − 0 ‖ is its side length and ⃗ V  is its direction of unit vector, as shown in (22).Since the directional derivative of () in the direction of ⃗ Suppose the gradient of () at  0 is ∇( 0 ); then, V  ]  and the vector   = [ 1 ,  2 , . . .,   ]  ; then,   ∇( 0 ) ≈   , especially, when the simplex is relatively small.As ⃗ V 1 , ⃗ V 2 , . . ., ⃗ V  are linearly independent, (24) can be regarded as the estimated value of the gradient:  In (24), the gradient can be accurately estimated without additionally calculating the function value.Especially, when the simplex is small or approximate to the extreme point, estimation is very accurate.Furthermore, Hessian matrix, Newton direction as well as the conjugate gradient at each iteration can be approximately estimated according to recent two successive gradients in the process of iterations.

Flow of the Novel Simplex Method.
For each simplex, the algorithm looks for a better point along rays emitting from the center  to the vertex   to replace the current vertex   .By continuously replacing the former simplex with a new one, iterations will be repeated until the evolution of the simplex stagnates.Once no better points are worked out or the simplex collapses, a new smaller simplex will be generated at the current optimal point in line with the current estimated descent direction and resume iterations again.The flow of the algorithm is shown in Figure 3.
Step 2.1.Generate the orthogonal simplex with the side length ℎ  at   in accordance with Algorithm 1.
Step 3. Determine whether   has degenerated.In case of degeneration, go to Step 2; or else, go to Step 4.
Step 5. Estimate the gradient   of the simplex   .
Step 6. Generate simplex  after center ray operations of simplex   .
Step 6.6.Sort vertexes of simplex  in ascending order by their function values.
Step The center ray only searches along  + 1 fixed directions.If the direction is not descending, the center ray may fail, and the search will stop for the simplex tends to beinfinitely small.For some ill-conditioned problems, even though search directions are descent, it is theoretically feasible to find optimum value; however, this can be realized only when the step is extremely small.Since such extremely small simplex is likely to collapse or the computation is too slow, it is difficult to realize in practice.Although Algorithm 3 is theoretically convergent, sometimes, it actually tends to converge to saddle point and nonoptimum point in case of complex functions.The performance only can be improved through continuous changes of searching directions.A new simplex will be forced to generate when the number of successive execution of ray operations exceeds certain threshold, which is equivalent to changes of the search direction to some degree.Whereby Algorithm 4 is formed with improvements of Algorithm 3, as shown in Figure 4.
Step 2.1.Generate the orthogonal simplex with the side length ℎ  at   in accordance with Algorithm 1.
Step 3. If  <  max , go to Step 4; otherwise, go to Step 9.
Step 4. Determine whether   has degenerated.In case of degeneration, go to Step 2; or else, go to Step 5.
Step 6. Estimate the gradient   of the simplex   .
Step 6.1.Compute each side vector ⃗ V  by ( 22) and create the matrix   .
Step 7. Generate simplex  after the application of center ray operations of simplex   .
Step 7.6.Sort vertexes of simplex  in ascending order by their function values.
Different from Algorithm 3, Algorithm 4 makes restrictions on the times of the repetitive execution of ray operations, denoted as , which is not allowed to surpass the maximum value  max .If  ≥  max , even though ray operations are successful, a new simplex at the current optimal point   is generated according to the current   so as to change the search direction.
Considering the better performance of reflection operations in the searching process of traditional simplex, in order to further improve the search abilities, a reflection operation will be performed in case of failure of ray operations.Consequently, search directions are more abundant and Algorithm 5 is formed, as shown in Figure 5.
Step 2.1.Generate the orthogonal simplex with the side length ℎ  at   in accordance with Algorithm 1.
Step 3. If  <  max , go to Step 4; otherwise, go to Step 11.
Step 4. Determine whether   has degenerated.In case of degeneration, go to Step 2; or else, go to Step 5.
Step 6. Estimate the gradient   of the simplex   .
Step 7. Generate simplex  1 after the application of center ray operations of simplex   .
Step 9. Form novel simplex  2 after the reflection of simplex  1 .
Step 9.4.Determine reflection point   by (21).If (  ) < (  ), then   is replaced with   to form new simplex  2 and sort vertexes of  2 in ascending order by function value.Algorithms 3, 4, and 5 all have two layers of loops.The sequence {  } is composed of optimal points of simplex   .The th execution of the inner loop generates the finite sequence {  },   ≤  <  +1 , which satisfies (  ) ≤ ( −1 ) −   < ( −1 ).The outer loop will be repeatedly executed, which may generate the infinite sequence {  } satisfying ( +1 ) ≤ (  ).That is to say, the sequence (  ) generated by the algorithm must be nonincreasing.The sequence {  } is composed of best vertexes of the simplex when both ray operations and reflection operations fail, which satisfies ( +1 ) ≤ (  ).Upon the failure of ray and reflection operations, a new smaller orthogonal simplex will be generated.The algorithm can ensure the current minimum point to be one of the vertexes of the new simplex.Besides, it also ensures that the current descent direction is parallel to an edge of the new simplex.The generation of a new smaller simplex achieves better effect than shrinkage operations, which can reduce the possibility of the simplex degeneration.  indicates   which can update several vertexes at each iteration by center ray operations, is effective and efficient.
With respect to the performance, generally, Algorithm 5 is the best, while Algorithm 4 is better than Algorithm 3.For both Rosenbrock function and Schwefel function, Algorithm 5 can obtain the smallest Ω 1 , Ω 2 and the largest Ω 3 , Ω 4 , Ω 5 , Ω 6 ; this implies that increasing of search directions can greatly improve the efficiency.Ω 1 , Ω 2 of Algorithm 4 are smaller than those of Algorithm 3, which indicates that the former costs less in computation.In general, Ω 3 , Ω 4 , Ω 5 , Ω 6 of Algorithm 3 is smaller than that of Algorithm 4, which reflects that efficiency and speed of Algorithm 4 is higher than that of Algorithm 3.
The parameters of the algorithm, especially the parameters controlling changes of ℎ  have great influence on the performance of the algorithm.According to the experiment,

Mathematical Problems in Engineering
Maintenance interval is interval-valued.It is necessary to redefine its distance and inner product to adapt to model (3) [15,16].Supposing that set  is composed of interval-valued [, ], ,  ∈   , and and the scalar multiplication is defined as it can be proved that  is the linear space on  and the zero element is [0, 0].In addition, if the inner product is defined as is proved to be the inner product space.Hence, the distance between  1 and  2 can be defined as which is in consistent with the definition of the city block distance [17,18].Obviously, it also meets the requirements of metric distance.Consequently, the distance between Period  and Period  is defined as  Supposing the th maintenance work package is Package  , man-hours of the package can be computed by (28) if maintenance tasks are executed in sequence.Its man-hours shall be computed by (29) if maintenance tasks are performed concurrently.To realize the successful execution of maintenance tasks, the workload of the maintenance work package must meet the requirement of actual maintenance capability, namely, PH  < PH max , where PH max indicates the upper limit of every work package man-hours, which is required by maintenance conditions: To satisfy requirements of the reliability and economy, when those work packages are executed, maintenance intervals of work packages should be acceptable to all maintenance tasks in the package.Namely, there shall be a public interval among the intersection of all tasks in the package.If the maintenance interval   of the th maintenance work package Package  is defined as (30), then P −  ̃ ≥ 0, ∀ is required: Correspondingly, the combinatorial optimization model of the maintenance task will be   [20].Its detailed steps are described in Algorithm 7.
Step 1. Initialize the parameter and set  = 2.
Step 5. Analyze and compare maintenance cost between before and after the combination.
Step 2 to Step 4 form an inner loop, in which the preset number of classes is increased constantly until a combination of maintenance task satisfies the constraints.
Suppose costs of every maintenance includes four parts: labor costs, device costs, halt costs, and fixed costs, which are, respectively, represented as Cost  , Cost  , Cost ℎ , and Cost  .Specifically, the fixed cost typically refers to the depreciation of maintenance tools and fixed assets and the management fees.Cost ℎ and Cost  are always the same for any repair.Cost  , which is determined by maintenance items, is the total device costs of all maintenance items repaired and is irrelative with the way of combination.Thereby, maintenance cost before the combination is Cost  =  labour ∑ 17 =1 Hour  + ∑ 17 =1 Device  + 17Cost ℎ + 17Cost  . labour refers to the labor costs per unit time.Device  indicates the device cost of the th maintenance item.After the combination, the maintenance cost will be Cost  =  labour ∑ 13 =1 PH  + ∑ 17 =1 Device  + 13Cost ℎ + 13Cost  .The maintenance cost is compared between before and after combination is tabulated in Table 12.As Cost  − Cost  = 4Cost ℎ + 4Cost  , it can be concluded that the combination helps saving cost and improving efficiency.Halt cost will cause great cost, which accounts for a large proportion in the total maintenance cost.Therefore, optimal combination of maintenance tasks can greatly increase profits of the airlines.

Conclusions and Discussions
Firstly, the improved fuzzy C-means clustering model is presented, which can be directly solved.Considering FCM model has many stationary points, the optimization method which needs derivatives is not applicable to solve it.Based on the positive bases theory, center ray operation is introduced and a novel nonlinear simplex optimization method is developed.Besides, its convergences are roughly analyzed.As a direct optimization method, it can be utilized to solve the clustering model effectively.Finally, the improved FCM model and the novel optimization method are employed to optimally combine maintenance tasks and exhibit good effects.However, there are still several problems to be further discussed.(1) The number of independent variables contained in the improved clustering model is still large.In case of a large number of samples, it is still inconvenient to solve FCM directly; while the clustering model, which is easier to be solved, needs to be further studied.(2) The simplex algorithm proposed in the paper is much more effective than the traditional algorithm.However, sometimes it is likely that the improved method fails to achieve good result because the searching direction can not be adjusted timely.From this view, the algorithm has to be improved.The most prominent point is that a certain rotation operation should be taken in the simplex algorithm to change the search directions.(3) ℎ  has great influences on the performance of the improved algorithm proposed in the paper.Although it is proved to be feasible that ℎ  is taken as a decreasing exponential function, the efficiency is still to be improved.Various factors can be associated to find out other way to control changes of ℎ  so as to improve the performance.(4) The experimental results show that the algorithm proposed in the paper can

Figure 2 :
Figure 2: Illustration of the reflection operation under 2-dimensional circumstance.
as a new vertex of simplex  to replace   ; otherwise, choose the best point from   , x  as a new vertex of simplex .
6.4.If () > ( x ), choose the best point from x  , x . If () > ( x ), choose the best point from x  , x  as a new vertex of simplex  to replace   ; otherwise, choose the best point from   , x as a new vertex of .
If () > ( x ), choose the best point from x  , x  as a new vertex of simplex  1 to replace   ; otherwise, choose the best point from   , x  as a new vertex of simplex  1 .Step 7.5.If  < , then set  = +1 and go to Step 7.2; otherwise, go to Step 7.6.

Table 2 :
Comparison between the algorithm proposed in the paper and Price.

Table 3 :
Parts of results of solving 90-dimensional Schwefel function by Algorithm 3.

Table 4 :
Parts of results of solving 45-dimensional Rosenbrock function by Algorithm 3.

Table 5 :
Parts of results of solving 90-dimensional Schwefel function by Algorithm 4.

Table 6 :
Parts of results of solving 45-dimensional Rosenbrock function by Algorithm 4.

Table 7 :
Parts of results of solving 90-dimensional Schwefel function by Algorithm 5.

Table 8 :
Parts of results of solving 45-dimensional Rosenbrock function by Algorithm 5.The basic properties of the th preventive maintenance task of aircraft includes ATA code, maintenance interval, zone, check and, man-hours, respectively, denoted as ATA  , Period  , Zone  , Check  , and Hour  .ATA code is a sixdigit code indicating the system, subsystem and components of the aircraft.The last two digits of ATA code are not uniformly standardized, so only the preceding four codes are adopted in the paper.Being interval-valued, maintenance interval refers to the floating range of the maintenance cycle and is denoted as Period  = [  ,   ].In terms of ATA standards, the structure and the space of a civil aircraft are usually divided into several zones.Every task is executed in some zones.There are great similarities among maintenance tasks in adjacent zones.In line with MSG-3, there are five types of checks, namely, LU/SV, OP/VC, IN/FC, RS, and DS.If we define set I = {LU/SV, OP/VC, IN/FC, RS, DS}, then Check  ∈ I. Man-hours represents how long time execution of the maintenance task need, which can be utilized to measure the size of the work package.ATA  , Period  , Zone  , and Check  are utilized to measure similarities between maintenance tasks.However, roles and influences of the four properties on similarity evaluation are very different.Weights  period ,  zone ,  check , and  ATA are, respectively, quantified those differences.
[19]iod  ,Period  = (  −   ) 2 + (  −   )2.Check merely involves several discrete values.So the measurement of the similarity or distance is usually specified subjectively from expertise.After detailed investigations and comprehensive researches, similarities of those checks are summarized in Table9.Table9develops the similarity matrix [I  ] 5×5 of Check  and Check  ; thus, the distance matrix can be obtained as 2Check  ,Check  = 1 − I [19].Zones are symbolized by digital symbols from 100 to 800.These zone codes can be regarded as digits to measure similarities although they are not numeric values in the real sense.Estimating the distance between two zones by the common numerical computation can approximately meet the requirements of engineering.Suppose that the th task associates with |Zone  | zones, denoted as Zone  = {CZ ,1 , . .., CZ ,|Zone  | }, and the th task involves |Zone  | zones, expressed as Zone  = {CZ ,1 , . .., CZ ,|Zone  | }; then, the distance between Zone  and Zone  is defined as  2 Zone  ,Zone  = (1/ |Zone  ||Zone  |) ∑ CZ , − CZ , ) 2 .Similar to zone codes, it is acceptable in practice to evaluate similarities of ATA codes by numerical computations if they are treated as numerical values.Thus, the distance between ATA  and ATA  is defined as  2 ATA  ,ATA  = ⟨ATA  − ATA  , ATA  − ATA  ⟩. (

Table 9 :
Similarities of checks.Based on the clustering model, Model (31) is supplemented with two additional constraints.Besides, the constraints are difficult to be explicitly represented by   ,   .It is not suitable to directly solve the maintenance tasks combination problem as a normal clustering problem.However, maintenance tasks can be clustered first, and then the result of clustering should be examined or checked whether clustering results satisfy constraints.If clustering results do not meet the requirements, we increase the number of clustering and recomputation until the results satisfy the constraint When   () < , go to Step 3. Otherwise, set  +1 =   ,  =  + 1 and go to Step 2.3.Tasks in the same class are taken as same maintenance work package.The man-hours PH  of the th maintenance work package Package  is totaled by (28) or (29).Summarize the maintenance interval   = [ ̃, P ] by (30).

Table 11 :
Optimal work package after the combination of maintenance tasks.

Table 12 :
Optimal work package after the combination of maintenance tasks.Before combination After combination Cost saved Labor cost  labour ∑