Research on Optimization Method of Allocation of Equipment Maintenance Devices Based on Sensitivity Analysis

In order to scientifically and efficiently allocate maintenance devices to improve the military economic benefits of equipment maintenance, combined with the characteristics of equipment maintenance work, the optimization goals of minimizing the completion time of batch maintenance tasks, minimizing the average maintenance time of a single equipment, and minimizing the consumption of labor costs were proposed. Based on flexible scheduling of maintenance processes, an optimization model is established, and the HNSGSA algorithm is designed to solve the problem by combining the advantages of NSGA-II and SA. According to the sensitivity analysis of the objective function on maintenance devices allocation, an optimization method for maintenance devices configuration is proposed.


Introduction
The maintenance work is the main means to maintain and restore the technical performance of weapons and equipment, and plays an important role in ensuring the normal combat readiness and training of troops.With the continuous updating and increasing use of weapons and equipment, the task of equipment maintenance is becoming increasingly heavy.The maintenance of weapons and equipment is highly specialized and relies heavily on maintenance devices, which are important maintenance support resource.With the characteristics of a large number of types, large investment in procurement, and long usage time, the short quantity of maintenance devices allocation will affect the implementation of maintenance operations, and the excessive allocation will increase procurement and maintenance costs.Therefore, studying the optimal allocation of maintenance devices is of great significance for improving the efficiency of weapon equipment maintenance.
The allocation of maintenance devices is affected by many factors, such as the amount of maintenance tasks, the structural characteristics of equipment, the maintenance process standards, and the requirements for maintenance time limits.It is difficult to accurately calculate the types and quantities of maintenance devices.Currently, there are relatively few studies on the optimization of maintenance equipment allocation at home and abroad [1][2].The literature [3] established an optimization model for aircraft maintenance devices configuration with the target function of the wartime aircraft sortie rate; the literature [4,5] studied the maintenance devices configuration of ship weaponry and surface-to-air missiles respectively.The literature [6] studied the optimization configuration model of multiple types of support equipment under multiple constraints, applicable to a case in which multiple types of support equipment are required for various support activities.None of these documents studied the problem of maintenance devices configuration at the organizational level

Model Conditions
Let the disassembling time of the n-th equipment in a batch of equipment to be repaired be B n t , the assembling time be E n t , and the p-th professional maintenance have np J processes.The sequence constraint matrix between processes is: There are M maintenance devices, and the processes and devices applicability matrix is . When the value of element 1 {0,1} npjm b  is 1, it means that the m-th device is suitable for the j-th process of the p-th profession of the n-th equipment.When the value is 0, it means that it cannot be implemented.represents the time required for maintenance on this type of equipment.Maintenance labor are divided into 3 levels: high, medium, and low.The labor cost standard for the g-th level is g w .The quality of maintenance labor at level g in the p-th professional team for the n-th equipment is npg L .

Optimization Objectives
For each piece of equipment, the assembly is implemented after the all professional maintenance, and a batch maintenance task is completed after all equipment are repaired, so the objective function for minimizing the completion time of a batch maintenance task is: Each piece of equipment can be put into use after being repaired, then the objective function of minimizing the average maintenance time for each equipment is: The salary of maintenance personnel is the maintenance cost, and the objective function of minimizing the maintenance labor cost is:

Constraint Functions Let k npj
T represents the starting time of the j-th process under the k-th proper process order for the p-th professional maintenance of the n-th equipment, {0,1} npk   represents whether the k-th proper process order is implemented, 1 represents being implemented, 0 represents not being implemented, and m O represents the set of processes on device m E .The constraints of the optimization model are: ] 1 Equation (5) indicates that each professional maintenance process begins after the disassembly process is completed.Equation ( 6) means that each professional maintenance of each piece of equipment can only be repaired according to a selected process sequence.Equation (7) indicates that the next process will not begin until the previous process is over.Equation (8) means that if 2 processes of different equipment or different professional maintenance are scheduled to be implemented on the same device according to the selected process order, the completion of one process must be followed by the completion of the other process.

Optimization Algorithm for Equipment Maintenance Devices allocation
The No-dominated Sorting Genetic Algorithm (NSGA-II) [7,8] uses elite retention strategies and crowding comparison operators to ensure population diversity and uniform distribution of solution sets.It is an effective algorithm for maintenance optimization, but it has strong dependence on initial parameters, low search efficiency, poor local search performance, and is prone to premature convergence [9,10].The Simulated Annealing (SA) algorithm is a global optimization method [11], which has the characteristic of probabilistic mutation.It can find the global optimum point of the objective function using random search techniques in a probabilistic sense.As it accepts bad solutions to a limited extent, it expands the search range, allowing the algorithm to escape local optima and thus more easily find the global optimum solution [12].This paper integrates the NSGA-II and the SA, and proposes a new HNSGSA (Hybrid NSGA-II and SA) algorithm to comprehensively improve global and local search capabilities.
The algorithm framework of HNSGSA is shown in figure 1, and the steps are as follows.
Step 1: Set the population scale Popsize, evolution numbers GN, crossover chance Pc, mutation chance Pm, NS individuals bettered by SA algorithm, the option thresholds of individual transform manner NC1 and NC2, the incipient temperature Tr, temperature transform rate , the outer cycles number R and inner cycles number L.
Step 2: Encode process sequence, equipment selection, and process scheduling, randomly generate chromosomes.
Step 3: Using Equations ( 2), ( 3) and ( 4), calculate the objective function values 1 2 3 ,, Step 4: Using simulated annealing algorithm to randomly improve the quality of X chromosomes and using the M criterion to determine if it should be accepted.
Step 5: Implement fast non dominant sorting to chromosomes, calculate the Petro level of each chromosome [13].Step 6: Using equation ( 9), calculate the crowded distance of each chromosome. where indicates the function value of the -1 i chromosome.
Step 7: Based on the criteria of low Petro level and high crowded distance of chromosomes, using the roulette wheel method, select populations Parent1 and Parent2, each with a size of Popsize/2.
Step 8: Merge Parent1 and Parent2 into Parent, cross and mutate Parent to obtain Children, which size is Popsize.
Step 9: Merge Parent and Children into X , which size is 2Popsize .Calculate the non- domination level and crowded distance of each chromosome.Make a set with chromosomes of same to form a series of non-dominated sets , The number of each set in which is

Simulation Result Analysis
In the 200th iteration, the best chromosome is obtained, the objective functions of which are , and . The Gantt chart of this scheduling scheme is drawn, as shown in figure 2. The task arrangement of the scheme meets the requirements of all constraints, verifying the feasibility of the maintenance process scheduling model and the accuracy of the HNSGSA algorithm.

Equipment Maintenance Devices Allocation Based on Sensitivity Analysis
Increasing the number of devices can reduce the waiting time for maintenance processes and thus improve maintenance capabilities.Below, based on the case in Section 4.1, three sets of simulation are conducted to increase the number of specialized devices, the number of general devices, and the both, respectively, to study the sensitivity of the objective function to devices allocation and quantitatively investigate devices allocation optimization methods.Continue to add the specialized device, chassis profession (No. 6), fire control profession (No. 9), communication profession (No. 14), and optoelectronic profession (No. 17), one more of the same type, respectively numbered 36, 38, 40, and 42.The maintenance process uses the same time as the original device.Keeping other parameters unchanged, using the HNSGSA algorithm to iterate 300 times, an optimal process scheduling scheme is obtained, as shown in figure 3, the objective function values are 1 f =312, 2 f =278.75, 3 f =1165510.From the change in the objective function values, it can be seen that after adding one more specialized device for each profession, the task time, or overall maintenance progress, has decreased from 323 hours to 313 hours, with a significant effect.After adding one specialized device for each profession, although the average time and labor cost have decreased, the task time has not changed much.Observing the process scheduling diagram 3, after adding two more specialized devices, there has been a lot of idle state.However, the general devices No. 20 and No. 21 are basically fully utilized, becoming a bottleneck that restricts the overall maintenance progress.

Simulation2: Increasing the Number of General Devices.
Based on the the experiment in Section 3.1, add the general device No. 20 and No. 21, one more of the same type, respectively numbered 35, 37.The maintenance process uses the same time as the original device.Keeping other parameters unchanged, using the HNSGSA algorithm to iterate 300 times, the optimal process scheduling scheme is obtained, the objective function values are 1 f =305, 2 f =285.5, 3 f =1175650.
Continue to add the general device, No. 20 and No. 21, one more of the same type, respectively numbered 36, 38.The maintenance process uses the same time as the original device.Keeping other parameters unchanged, using the HNSGSA algorithm to iterate 300 times, an optimal process scheduling scheme is obtained, as shown in figure 4, the objective function values are 1 f =295, 2 f =282.5, 3 f =1149440.Although the objective function values decrease when adding two general devices compared to adding one, the scheduling diagram reveals that the increase of device 35 and 36 results in excessive idle time.Device 35 is used once and device 36 is used twice, resulting in an excessively high idle rate.In practice, such scarce general devices are often expensive, such as truss hoists and comprehensive testing platforms, which require large investment and complex installation and debugging.This device allocation is clearly uneconomical and undesirable.Keeping other parameters unchanged, using the HNSGSA algorithm to iterate 300 times, the optimal process scheduling scheme is obtained, the objective function values are 1 f =298, 2 f =271.25, 3 f =1121850.It can be seen that the comprehensive plan of adding specialized devices and general devices, with moderate increase and appropriate investment, has significantly decreased the value of each objective function, which has a significant effect on improving overall maintenance efficiency, and has a high device utilization rate, with obvious advantages.

Simulation Conclusion
The objective function values of the optimal scheduling schemes under various devices allocations are compared, as well as the reduction rate of the objective function values compared to the initial devices allocation in the simulation in Section 3.1, as shown in table 4. Based on table 4 and the process scheduling scheme under various maintenance device allocation, the method for optimizing the allocation of maintenance deviecs can be derived: At first, priority should be given to devices with strong versatility.Observing the optimal process scheduling diagram 2 under the initial equipment condition in Section 3.1, the utilization levels of general devices No. 20 and No. 21 are roughly equivalent to those of specialized devices for each profession, but comparing scheme 3 with schemes 1 and 2, the objective function values of the scheme with two additional general devices are significantly better than those of the scheme with four additional specialized devices, and the task time is even better than that of the scheme with eight additional specialized devices, mainly because the increase of general devices has a boosting effect on each professional maintenance operations, and it has a greater impact on overall maintenance efficiency.Secondly, increasing appropriately.The increase of devices is not the more the better.Comparing schemes 1 and 2, after adding eight specialized devices, some are seriously idle, and the task time function value decreases slightly, indicating that the benefit of the scheme is poor.Therefore, when allocating maintenance devices, the benefit should be scientifically calculated, and blind implementation should not be allowed.The third, focus on comprehensive allocation.Comparing schemes 2, 4, and 5: After adding four general devices, the task time and labor cost decrease significantly, but the average time decreases slightly; after adding two general devices and four specialized devices, the objective function values decrease significantly.General devices generally require large investment, so from the perspective of both economic and maintenance efficiency, general devices and specialized devices should be coordinated when allocation devices.

Conclusion
Based on the organizational process of military equipment maintenance work, this article studies the optimization of equipment maintenance process scheduling under resource constraints, analyzes the maintenance process, establishes a multi-objective equipment maintenance process optimization scheduling model that comprehensively considers military economic benefits, integrates the advantages of NSGA-II algorithm and simulated annealing algorithm, proposes the HNSGSA algorithm, and verifies the model and algorithm using actual data cases.By adding specialized equipment, general equipment, and comprehensive equipment separately, the sensitivity of the objective function to various types of equipment is calculated, and the optimization strategy for maintenance equipment matching is obtained.The research in this paper has certain guiding significance for strengthening the construction of maintenance support capabilities.
the chromosomes of each set into the empty population Y until the size of the population exceeds Popsize after n Z is added.According to the from the bigest to the smallest, rank the chromosomes in n Z , and place the first into Y, the size of which reaches Popsize, then the new generation population Y generates.Step 10: Repeat steps 3-9 until the iteration is over.The chromosomes of the 1 Z in Y are the optimal option set.

Figure 2 .
Figure 2. The Gantt chart of the scheduling scheme based on the HNSGSA algorithm.

5. 1 . 1 .
Simulation1: Increasing the Number of Specialized Devices.Add the specialized device, chassis profession (No. 5), fire control profession (No. 8), communication profession (No. 13), and optoelectronic profession (No. 16), one more of the same type, respectively numbered 35, 37, 39, and 41.The maintenance process uses the same time as the original device.For example, in the experiment in Section 3.1, the time of the chassis process No. 3 of equipment No. 1 on device No. 5 is 25 hours.In this experiment, it applies the device No. 5 and No. 35 for 25 hours.Keeping other parameters unchanged, using the HNSGSA algorithm to iterate 300 times, the optimal process scheduling scheme is obtained, the objective function values are 1 f =313, 2 f =289.75, 3 f =1194440.

Figure 3 .
Figure 3.The Gantt chart of the optimal scheduling scheme after each profession increasing two special devices.

Figure 4 .
Figure 4.The Gantt chart of the optimal scheduling scheme after each general machine increases two.5.1.3.Simulation3: Both Increasing in Special and General Devices.Based on the the experiment in Section 3.1, add the specialized device, chassis profession (No. 5), fire control profession (No. 8), communication profession (No. 13), and optoelectronic profession (No. 16), the general device No. 20 and No. 21, one more of the same type, respectively numbered 35, 36, 37, 38, 39, 40.The maintenance process uses the same time as the original device.Keeping other parameters unchanged, using the HNSGSA algorithm to iterate 300 times, the optimal process scheduling scheme is obtained, the objective function values are 1 f =298, 2 f =271.25, 3 f =1121850.It can be seen that the comprehensive plan of adding specialized devices and general devices, with moderate increase and appropriate investment, has significantly decreased the value of each objective function, which has a significant effect on improving overall maintenance efficiency, and has a high device utilization rate, with obvious advantages.

Table 1 .
Simulation data setting range.

Table 2 .
The data of application between process and device and allocation of maintenance labor.

Table 3 .
The constraint matrices of the professional maintenance processes.

Table 4 .
The objective functions value of optimal scheduling plan under 5 device improvement schemes.