An integrated automated guided vehicle design problem and preventive maintenance planning

Nowadays, automated guided vehicles (AGVs) play a key role in manufacturing systems because of improving system efficiency and lowering the cost of production. To increase the efficiency and stability of AGVs, it is crucial to consider maintenance planning for them. To the best of our knowledge, there are rarely found studies related to maintenance planning and AGVs’ design and control. Accordingly, in this paper, a new integrated nonlinear mathematical model is developed for optimizing the AGV design (including AGV fleet sizing and AGV assignment to workshops) and preventive maintenance policy. The proposed model aims to determine the preventive maintenance cycles, the optimal number of employed AGVs in manufacturing, and the optimal assignment of AGVs to manufacturing workshops so that the total cost is minimized. To solve this model, a genetic algorithm (GA) is developed, and its performance is compared with the global solver of LINGO software on 15 test problems, some of which are large dimensions. To tune the GA parameters, a Taguchi method is used. Moreover, a sensitivity analysis is performed to represent the validity of the model and solution approach. The results have demonstrated the effectiveness of GA in terms of computational time and solution quality.


Introduction
The AGVs have been used as an important part of material handling in inside and outside environments such as manufacturing systems, warehouses, cross-docking centers, and container terminals since their introduction (Vis 2006).They can realize intelligent manufacturing for the industrial 4.0 era (Hu et al. 2021;Lin et al. 2021).They can bring many benefits to any environment in which they are used, including reduced costs, improved safety as well as productivity, routing flexibility, and reduced material damage (Shneor et al. 2006).The issues which are addressed with the AGV system can be divided into two categories: (1) AGV design and (2) AGV control.The design concerns the flow path layout, the number of employed AGVs, the traffic flow pattern, buffer capacity for vehicles, the location of pickup and delivery stations, and AGV assignment to workshops.On the other hand, the control issues cover the dispatching, scheduling, and routing problems of AGVs.Two of the most important design issues are to determine the optimal number of employed AGVs and the optimal AGV assignment to workshops.In the first issue, the minimum number of AGVs required in the system has to be determined because the underestimated number of them does not guarantee that all jobs are performed within a desirable time, while an overestimated number leads to more congestion (Saidi-Mehrabad et al. 2019).The goal of the second issue is to assign the AGVs to manufacturing workshops so that only the AGV allocated to one workshop is responsible for transferring tasks from the same workshop to other workshops and even the warehouse.
Each AGV consists of several subsystems such as laser navigation, safety, battery, brake, and steering system.Each of these subsystems can be encountered with random failure and makes AGVs unavailable.Therefore, AGVs cannot perform their duties properly in such circumstances, and this influences the efficiency of the system.An appropriate maintenance policy can ensure the availability of AGVs and increase system efficiency and performance (Yan et al. 2018a).Corrective maintenance and preventive maintenance are two basic categories of maintenance.Maintenance that is regularly performed on each piece of equipment to decrease the probability of its failure and keep it up-to-date and functional is called preventive maintenance (PM).On the other hand, corrective maintenance (CM) involves any task that resolves the occurred failures with equipment and returns it to a proper operating state.In practice, some actions such as cleaning, oil changes, lubrication, repairs, adjustments, inspecting and replacing parts and partial or complete overhauls that are periodically scheduled are included in a PM schedule.In an AGV consisting of different subsystems, reliability and availability as two main factors for increasing the system efficiency can be achieved by replacing and repairing critical subsystems (Fazlollahtabar and Naini 2013).An unplanned stop resulting from the random failure of any subsystem can reduce the performance of AGVs, and therefore, the system cannot meet the consumer demand and faces higher costs.Thus, a PM schedule can reduce costs by increasing reliability and availability.There are many papers about AGV design problems that integrate the types of issues mentioned above.But, most of them assume that AGVs can act at their maximum capacity (nominal capacity), while this assumption is not real because in a real environment, an unexpected failure can influence system efficiency and capacity.In this state, AGVs work with a real capacity that is usually less than the nominal capacity.Although when a random failure takes place for AGVs, both cost and time for a PM plan are needed to maintain them and return to a suitable status and this follows fewer completed missions through the horizon planning, but on the other hand, a PM plan will generally reduce the total cost by increasing the availability and reliability of AGVs in manufacturing system.The purpose of this paper is to provide an integrated mathematical model of AGV design and PM scheduling.Also, the research questions are: 1. What kind of AGV is assigned to each workshop?2. How many AGVs are placed in each workshop?3. What is the size of the maintenance cycle for each selected AGV in the manufacturing system? 4. What is the effect of AGV's speed, AGV capacity, and workshop distance on the objective function value (overall cost)?
In this model, the optimal maintenance policy incorporating each kind of AGV, the optimal AGV fleet size, and the assignment of them to workshops is determined.
Generally, the main considered innovations in this paper are as follows: 1.A new mathematical model that considers the AGV design problem and maintenance planning simultaneously has been developed.2. A multiple AGV-based job shop with various AGVs in capacity and speed is considered in the proposed model.3.There is both a regular periodic inspection and a repair/ replacement action after random failure in a developed PM. 4. For solving the new proposed problem, a genetic algorithm as an efficient algorithm for large-size examples is applied for the first time.5. Sensitivity analysis is applied to investigate the effect of several important parameters on the objective function value.
The remainder of the paper is organized as follows.A review of the literature on the AGV design problem and integrated AGV and maintenance planning problem is included in Sect. 2. In Sect.3, the problem is introduced, and in Sect.4, the mathematical representation of the problem is developed.In Sect.5, a genetic algorithm is described in detail.In Sect.6, several examples are generated to illustrate the efficiency of the model and algorithm.Section 7 is associated with sensitivity analysis, and finally, the conclusions and future research are stated in Sect.8.

Literature review
Various papers have been published about AGV design and control after introducing them in 1955 (Muller 1983).Several studies are in the design field.For example, Nishi et al. (2020) discussed the guide path of AGV systems.Also, De Ryck et al. (2019a) designed a charging station into an AGV-based manufacturing system.Fransen et al. (Fransen et al. 2020) developed path planning for AGVs.Dehnavi-Arani et al. (2020) determined the optimal dwell location for AGVs in a cellular manufacturing system.Cheong and Lee (2018) investigated an AGV design problem through image detection and precise positioning.Lim et al. (2003) suggested a Q-learning technique for designing guide-path networks.Qiuyun et al. (2021) studied an AGV path planning problem of a one-line production line in the workshop.Gu et al. (2020) proposed a mathematical model and a data-driven intelligent algorithm for dynamic path design problem of AGVs.
Some other studies considered the control issue of AGVs.For instance, Vale et al. (2017) evaluated the navigation of AGVs in nuclear fusion facilities.Mahaleh and Mirroshandel (2018) addressed path detection for an AGVbased system.Wang et al. (2019) investigated a novel scheduling problem for AGV in workshop environments.Bae and Chung (2017) presented an AGV routing problem.Dehnavi-Arani et al. (2019) considered a scheduling problem in a cellular manufacturing system considering AGVS' movement.Abderrahim et al. (2020) considered battery management of AGVs in an AGV-based job shop manufacturing.Chen et al. (2020) addressed the AGV charging mechanisms in a manufacturing system and solved the proposed model by two-stage simulation optimization.Lopez et al. (2022) presented a flexible framework to simulate AGV-based transport systems.There are several comprehensive review studies in the literature about AGV design and control that the readers can refer to for a better understanding like (Vis 2006;Oyekanlu et al. 2020;De Ryck et al. 2019b;Le-Anh and Koster May 2006;Suparjon 2022;Balaji et al. 2022).In this section, initially, several studies on AGV fleet sizing and assignment are reviewed, and then studies on the integration of maintenance and AGV issues are reviewed.

AGV fleet sizing and assignment
The goal of the fleet sizing model is to determine the number of employed AGVs in a manufacturing system.In the literature, both analytical models and simulation methods are mostly utilized to determine the optimum number of AGVs.For example, Valmiki et al. (2018) determined the estimation of the fleet size of AGVs in a flexible manufacturing system by simulation.The objective of their method was to minimize travel time or overall cost.Tao et al. (2010) developed analytical and simulation methods simultaneously to find the optimum number of AGVs in a flexible manufacturing system.Pjevcevic et al. (2017) designed a decision-making approach based on data envelopment analysis to determine the fleet size of AGVs at a port container terminal.They used the simulation as a solution approach in their paper.Chawla et al. (2018) developed a mathematical model to determine the number of AGVs in a flexible manufacturing system.The solution approach was the gray wolf optimization algorithm in their article.Choobineh et al. (2012) also used an analytical model based on queuing networks to find the optimal AGV requirement.Koo et al. (2004) used a queuing model for the fleet sizing procedure.Liu and Ioannou (2002) minimized the number of AGVs for achieving zero idle time in AGV-based job shops.They used heuristic and Petri net theory to solve this analytical model.Chang et al. (2014) represented also a simulation-based framework for AGV fleet size.Some papers are associated with AGV assignments, such as Angeloudis and Bell (2010) that they studied the AGV assignment problem in a container terminal under uncertainty.They also used a simulation method for earning the desirable factors.In another paper, Huang et al. (2017) optimized the AGV assignment for dynamic demand for transportation on a shop floor in an uncertain environment.Mohamad et al. (2018) focused on a multi-load AGV assignment in a flexible manufacturing system.In this paper, each operation of the job should be allocated to AGV.Weyns et al. (2006) studied task assignments for AGVs transportation systems through a field-based approach.Recently, the requirement for the use of autonomous mobile robots in in-patient care has been developed by Kriegel et al. (2021).Fu et al. (2021) determined the vehicle requirement of the AGV system based on discrete event simulation and response surface methodology.Other papers related to AGV assignments can be found in Hafidz and System (2011), Hafidz et al. (2013), Sabattini et al. (2017), and Lu and Moreira (2016).

Integrated preventive maintenance and AGV issues model
Maintenance has been an essential sector in the context of Industry 4.0 (Fernandes et al. 2021).Many papers show this importance, such as Tran et al. (2019); Mihai et al. (2021); and Wang et al. (2021).In this way, the maintenance of AGVs as widely used vehicles in smart manufacturing considered in Industry 4.0 is a crucial problem.The issues of combining PM and AGV design/control are extremely rare in the literature.As an example, Yan et al. (2017) investigated the reliability of AGVs in their paper.They analyzed the reliability by fault tree analysis and evaluated the vehicle mission reliability via the Petri net method.Fazlollahtabar and Naini (2013) proposed a Markovian model for flexible manufacturing systems.They considered the reliability of machines and AGVs simultaneously.Yan et al. (2018b) modeled the corrective and preventive maintenance of failed AGVs using colored Petri nets.To optimize the model, a genetic algorithm has been proposed for the results of Petri nets.They modeled the problem to achieve both the location selection of the maintenance site and the maintenance strategies.Fazlollahtabar and Saidi-Mehrabad (2013) developed a multiobjective mathematical model so that the objectives are to maximize the total reliability of machines and AGVs and minimize the total repair cost.They used fuzzy goal programming to change the multi-objective model to singleobjective one.Tavana et al. (2014) developed a bi-objective stochastic programming model with AGV reliability considerations.In that paper, the first and second objectives were to minimize the total cost of production and the total production time, respectively.The authors considered a job shop environment equipped with an AGV.In order to transform the bi-objective model to a single objective, they applied a perceptron neural network.Yu et al. (2021) considered a reliability-based AGV online scheduling and conflict-free routing in warehouse systems where AGVs quickly sort a large number of packages.The main difference between our proposed model and other approaches in other papers is that the proposed model considers PM as an AGV capacity reducer and also the probability of an AGV failure reducer in a multiple AGV system.Moreover, our model integrates the AGV fleet sizing and AGV assignment to shops as two important issues in the AGV design and maintenance model simultaneously.

Problem statement
Here, a manufacturing system, including several workshops, parts, and AGVs, is considered.It is assumed that the process route for each job is known.AGVs are employed to transfer parts among different workshops.
Each AGV can be assigned to one or more workshops to pick up parts from these workshops and deliver them to destination workshops, while each workshop can select only one AGV type to transfer their processed parts.
There are various kinds of AGVs in terms of capacity, speed of movement, purchase price, fixed cost, operational cost, and failure rate.In reality, AGVS may be failed subject to uncertain fluctuations which cause the downtime of the production system.In such a situation, AGVs cannot work with maximum capacity through horizon planning.Therefore, failure of AGVs impacts on the number of completed emissions by them.A PM must be done to restore AGVs to ''as good as new'' status and increase the AGV capacity and decrease the failure rate, and corrective maintenance is used to restore AGVs to ''as good as old'' status and restart the AGVs again.Of course, it should be noted that maintenance is costly.The purpose of the model is to determine the PM cycle so that the needed completed emission for horizon planning is met, and the average total cost is minimized.Notably, the maintenance plan and cost for each kind of AGV in the manufacturing system are assumed the same.
An example of the above-mentioned manufacturing system is represented in Fig. 1.In this figure, there are five workshops and four AGV types.As can be seen, four AGVs type 3, three AGVs type 1, two AGVs type 1, two AGVs type 2, and one AGV type 4 have been assigned to workshops 1-5, respectively.The routes of AGVs assigned to workshops 1-5 have been represented with red, green, orange, blue, and yellow colors.Moreover, one of the AGVs type 1 in workshop 3 and AGV type 4 in workshop 5 are not available because they are under PM and CM, respectively.

The mathematical model
The integrated model of the problem under study is described in this section.The considered sets, parameters, and decision variables are as follows: 4.1 Sets j Index for AGV types (j ¼ 1:. ..:J).n:m:k:m 0 ð Þ Index for workshops (n:m:k:m 0 ¼ 1::N).t Index for periods (t ¼ 1:. ..:H).

Parameters fc j
The constant cost of AGV type j (e.g., purchase cost, installation cost, etc.).vc j The variable cost of AGV type j. nc nm The cost of missions that they do not complete between workshops n and m in horizon planning.
The distance workshops n and m. s j The speed of AGV type j. cap j The capacity of AGV type j.Rm nm The required volume of the parts to be transferred between workshops n and m. lt jn The loading time for AGV type j in workshop n. ut jm The unloading time for AGV type j in workshop m. pr nm If there is a processing route between workshops n and m equal to 1; otherwise 0.

C jt
The maintenance cost of AGV type j when they have the size of maintenance cycle t.AV jtt 0 The available time for each AGV type j in the length of period t if maintenance cycle t 0 is selected.Z A big number.

N jn
The fleet size of AGV type j assigned to workshop n.

N j
The fleet size of AGV type j.

Tm j
The total movement time for AGV type j.

Dc nm
The desirable level of missions between workshops n and m by AGV.Cm jnm The number of completed missions between workshops n and m by AGV type j.Cm jnmt The number of completed missions between workshops n and m by AGV type j in the length of period t.

A jnm
If AGV type j has been assigned to route between workshops n and m 1; otherwise 0.

T jnm
The time lasting for AGV type j from workshops n to m.

Tt jnm
The total time that AGV type j needs from workshops n to m to complete its mission.
If AGV type j select the size of maintenance cycle t equal to 1; otherwise 0.

Objective function
The objective function of the proposed model is as Eq. ( 1): The first term of the objective function is a constant cost for all of the AGVs used in the production system.This cost is obtained by the sum of the product of the fleet size of AGV type j and their associated costs such as purchase cost and installation cost.The second term shows the variable cost of employed AGVs.This cost is the sum of the product of the total time of movement for AGV type j (i.e., the AGV workload) and their associated cost.The third term calculates the total cost for uncompleted missions through horizon planning.It is the sum of the product of a maximum between zero and deviation of completed mission and a desirable level of completed mission from workshops n to m, and their associated cost.The final term is related to the total maintenance cost, which is the sum of the product of maintenance cost if AGV type j selects the size of the maintenance cycle t.The maintenance cost C jt is obtained as follows.

Maintenance cost
In this paper, the developed maintenance model by Yalaoui et al. (2014) is used.It is assumed that the failure rate has a probabilistic function as Eq.(2): where f j t ð Þ is the probability function and F j t ð Þ is the cumulative distribution function of time's failure of each component (i.e., component means AGV in this paper).The horizon planning is divided into H periods with a length of s.It is assumed that the PM of each AGV is performed in the first period of each maintenance cycle.Also, each maintenance cycle for each AGV is MC j which consists of g j periods (MC j ¼ g j sÞ.Then, the PM is done at the beginning of the 1, ( The maintenance cost for any AGV has a nonlinear function of both PM and CM.For each AGV type j in each cycle period of MC j , the cost of PM is fixed and equal to pc j , and the cost of CM is estimated based on the average number of failures r and unit cost CM that is equal to cc j . The PM and CM plan and costs are depicted in Fig. 2.  2014), the maintenance cost of AGV type j when they have the size of maintenance cycle t can be computed as Eq. ( 3): Equation ( 3) can be simplified as Eq. ( 4): Now, based on all of the candidate cycles, all of the expected maintenance costs are calculated.

Constraints
Constraint (5) says that if there is a processing route between two workshops, an AGV type is assigned to that route; otherwise, no AGV is allocated.Constraint (6) states that all AGV types that transfer parts from one particular workshop to other workshops are the same.In other words, only one AGV type can be assigned to each workshop, and this AGV transfers the output of that workshop to another.Constraint (7) determines the number of AGVs assigned to each workshop.This number should be equal to or less than the number of AGVs assigned to routes that went out from the workshop.Constraint (8) assures that if an AGV is

Maintenance data:
Other data: The second stage

Integrated AGV design and maintenance planning:
The third stage

Final solution:
Determine the maintenance cost and available capacity under each maintenance cycle Determine the manufacturing system configuration, the required costs and times Use the input data and formulate the nonlinear mathematical model according to equations 1-17 Solve the proposed model by LINGO and GA in order to determine the optimal maintenance cycle, AGV fleet size and AGV assignment Fig. 3 An overview of the proposed model in this paper Fig. 4 An example of a considered chromosome in developed GA assigned to a route from workshop n toward any destination, the number of that AGV in that workshop should be greater than or equal to one.Constraint (9) calculates the total employed AGV type j in the manufacturing system.Constraint (10) computes the total movement time for AGV type j.This movement time is considered as AGV workload.Constraint (11) ensures that if no AGV is assigned to the route between workshops n and m, the completed missions on that route will be zero.Constraint (12) ensures that the total completed missions on each route are equal to the sum of completed missions on that route in the thorough horizon planning.Constraint (13) calculates the desirable completed mission between workshops n and m.This level is obtained by dividing the required volume of the parts to be transferred between workshops n and m to the capacity of the assigned AGV to workshop n.Constraints ( 14) and ( 15) compute the total time required for the movement of each AGV from workshop n into workshop m.Constraint ( 16) is related to the available time capacity for each AGV between workshops n and m.A detailed explanation of this constraint is given as follows.Constraint (17) indicates that each AGV type can select only one maintenance cycle time.Eventually,constraint (18) shows the kind of decision variable in the model.

Available capacity
It is now time to discuss exactly how the time capacities of each route are calculated for each period of H.According to reference [63], if the AGV type j is maintained over a cycle period g j , the available capacity values are identically distributed over all maintenance cycles.So, it is sufficient that the calculations for periods of the first cycle are obtained, and other cycles behave similarly to the first cycle.The capacities for periods of the first cycle based on [63] are as follows (Eq.( 19)): where NC j is the nominal capacity of the jth AGV in each period, h jp and h jr are the amount of reduction in nominal value by PM and CM, respectively, and are h jp ¼ xNC j and h jr ¼ uNC j in which x; u 2 0; 1 ½ :

Solution approach
The nonlinear above-mentioned mathematical model has been coded in LINGO 18 software to solve numerical examples, especially in small-sized dimensions.For medium-and large-sized, a genetic algorithm (GA) is utilized to find optimal or near-optimal solutions for example.Generally, in the first stage, the cost of each AGV for each maintenance cycle and the capacity of each maintenance period are determined under candidate maintenance cycles.Also, in this stage, other parameters are entered into the model.In the second stage, the integrated model of AGV design and maintenance planning is formulated.Finally, Lingo and GA are used to determine the optimal/near-optimal obtained solutions.A schematic overview of the proposed integrated model is shown in Fig. 3.

Genetic algorithm
The genetic algorithm (GA) is a powerful method for combinatorial optimization problems which is proposed by Holland [58].In GA, each problem is encoded by chromosomes in which each gene represents a feature of the considered problem.GA usually consists of the five following steps: Step 1: The population of chromosomes is initialized.
Step 2: The fitness of each chromosome is evaluated.
Step 3: New chromosomes are created by applying crossover and mutation to the current chromosomes.
Step 4: The fitness of the new population of chromosomes is evaluated.
Step 5: Stop when the termination condition is satisfied, and the best chromosome is returned; otherwise, go to Step 3.
In general, each GA method has several mechanisms that should be determined, including representation, initial population, selection, operators, fitness function, and the termination condition.These mechanisms, in the proposed GA, for the problem under this study are as follows:

Representation
To design a proper chromosome for the solution structure coding is the first and the most important mechanism of GA.The chromosome was designed based on the model's variables and constraints.Here, each chromosome consists of the following genes: 1.The gene related to the number of AGVs assigned to each workshop and maintenance cycle which is a matrix X ½ n 1;k n ÂJ where k n ¼ P N m¼1 pr nm 8n 2 N, 1 is the number of rows and k n Â J is the number of the column.The alleles of the matrix are limited to 0 and 1. 2. The gene associated with the number of completed missions between workshops which is the matrix ½Y H 1;k where k ¼ P N m¼1 P N n¼1 pr nm , 1 is the number of rows and k is the number of columns.The alleles of the matrix are limited to integer numbers and follow Eq. ( 20) by considering Eq. ( 17): There are other variables such as A jnm :N jn :Tt jnm that they are obtained based on the defined chromosome, and there is no need to define them into chromosomes separately.As an example, it is assumed that there are five workshops, three AGV types, and a horizon, including two periods as well as the jobs which should be transferred from a workshop toward all of the other workshops.Hence, the chromosome shown in Fig. 4 can be a solution for this example.This solution states that workshop 1 has four AGVs type 2 with 123 a maintenance cycle 2, workshop 2 has three AGVs type 1 with a maintenance cycle 2, and workshop 3 has two AGVs type 3 with a maintenance cycle 1.Also, the number of completed missions between workshops 1 and 2 is 20 in period 1 and 18 in period 2; the number of completed missions between workshops 1 and 3 is 11 in period 1 and 9 in period 2, and so on.

Initial population
The initial population is a subset of chromosomes.In order to create an initial population, several feasible solutions as shown in Fig. 4 are generated.

Selection
The selection is needed because it provides the opportunity to transfer the genes of a good solution to the next generation.The various selection methods are described in the literature.In this study, the roulette wheel selection is used.

Genetic operators
Three kinds of crossover operators are considered in this paper: XjY ½ -level, ½X-level, and ½Y-level.The XjY ½ -level crossover is to randomly select two genes ½X or ½Y from parents and swap corresponding matrices.In ½X-level, one cut point is selected randomly on the only matrix ½X in the vertical or horizontal direction, then the partial matrices from two parents are swapped together.½Y-level crossover is a two-child arithmetic crossover.In the two-child arithmetic crossover, two offspring by linear combining two selective parents are obtained.This crossover is based on Eqs. ( 21) and ( 22).An example of kinds of crossover is drawn in Fig. 5a-c.
Mutation has two levels: ½X-level mutation and ½Ylevel mutation.The mutation used in ½X-level is so that a value of greater than 0 changes to 0 and other randomly selected genes except genes related to a value of greater than 0 change to a random number of greater than 0. On the other hand, the mutation used in ½Y-level is an arithmetic mutation where the value of a selected gene is reduced by the amount of D, and then it is added to another selective gene.It should be noted that D should be selected so that the value of the gene does not exceed its acceptable value.Figure 6a, b represents two kinds of mutation, respectively.

Fitness function
The fitness function is the same as the objective function Eq. (1) in Sect.3.

Stoppage condition
The stoppage condition is to reach an upper limit on the number of generations (i.e., a maximum iteration.

Genetic parameter setting
A Taguchi method is applied for tuning the parameters of GA and finding the optimum combination of effective parameters on the performance of GA.The Taguchi method is based on a signal/noise (S/N) ratio which means a ratio of an average standard deviation.A higher ratio indicates better performance for parameters.The S/N ratio for the developed problem can be calculated as Eq. ( 23): where n is the number of observations in the experiment and of i is the same as the objective function value.The important parameters which should be tuned for a GA usually consist of many chromosomes in a population (NPop), the number of iterations to reach the best result (MaxIt), crossover rate (CrR), and mutation rate (MuR).
In summary, the developed GA algorithm is shown in Fig. 7.
To compare and study the efficiency of solutions obtained by LINGO and GA, 15 test problems were generated and solved with both the exact global solver of the LINGO and GA method mentioned in Sect.4.1 for 10 runs on MATLAB.Furthermore, the mean of the objective function and CPU time, the standard deviation, and the mean of the gap% are represented in Table 13.Also, Figs.11 and 12 show the performance of GA against LINGO.As observed, the maximum deviation GA from LINGO in terms of the objective function is 7.4%.On the other hand, the computational times for LINGO are much longer than GA for small-or medium-sized examples, and for large-sized ones, Lingo is not able to find the response even within 24 h (86,400 s).As a result, the developed GA is more efficient than LINGO and can find justifiable and

Sensitivity analysis
In this section, variations of three parameters including the AGV speed, the AGV capacity, and workshop distance have been investigated on values of the objective function for example 3 given in Table 3.As shown in Fig. 13, the objective function value has an oscillating behavior subject to the speed of AGVs as well as the distance between workshops.This behavior is absolutely logical because these two parameters influence both the cost of total movement time for AGVs and the cost of uncompleted missions through the horizon planning in the objective function at the same time.On the other hand, the objective function value has a descending trend toward increasing the capacity of AGVs.It is obvious that the greater the capacity of AGVs, the fewer number of AGVs, the less movement time, and the less uncompleted mission is needed, and therefore, this leads to fewer costs.

Conclusion
In this paper, a nonlinear mathematical model for an automated guided vehicle (AGV) design problem considering the preventive maintenance policy of AGVs has been developed.In other words, the main goal of this paper is to find the optimal fleet sizing of AGVs and assign them to workshops together with determining the optimal preventive maintenance cycle.Our proposed model minimizes the constant and variable costs of AGVs, uncompleted mission costs, and maintenance costs.The performance of the model is verified by a numerical example solved in LINGO software.Since the considered problem is a hard problem to solve, a genetic algorithm (GA) together with the Taguchi method for tuning the GA parameters was represented.In order to check the efficiency of GA, 15 examples were generated and solved by both GA and global solver in LINGO software.By comparing the obtained objective function value for all examples, it is concluded that GA has a maximum 7.4% percentage gap in terms of the objective function value.Also, GA solves these examples at much lower times (on average about 50 times smaller).These results show the efficiency of the proposed GA for solving the problems, especially for large-sized ones where Lingo is not able to achieve a feasible solution even in 24 h.Finally, a sensitivity analysis was performed on some key parameters and investigated the effects of those parameters on objective value.
The future extension of this study could be to consider the job shop scheduling together with the above problem, development of the multi-objective model and solution approach, apply the uncertainties in parameters, and integrate other AGV design or control problems such as battery management, navigation strategies, AGV routing, and dispatching.
Author contributions All authors contributed to the study's conception and design.Material preparation, data collection, and analysis were performed by Saeed Dehnavi-Arani and Aliakbar Hasani.
Funding The authors did not receive support from any organization for the submitted work.

Fig. 1
Fig. 1 A schematic view from the manufacturing understudy

Fig. 2
Fig. 2 Considered PM plan and cost in this study Fig. 5 a XjY ½ -level crossover; b ½X-level crossover; c ½Y-level crossover

Fig. 8
Fig. 8 Optimal manufacturing system view obtained by LINGO

Fig. 9
Fig. 9 Signal-to-noise ratio in the Taguchi method

Fig. 12
Fig. 11 Objective value obtained by LINGO and GA

Table 1
Parameters of AGVs

Table 2
Parameters of workshops

Table 4
Available capacity of AGV type 1 according to the candidate maintenance cycle

Table 5
Available capacity of AGV type 2 according to the candidate maintenance cycle

Table 6
Available capacity of AGV type 3 according to the candidate maintenance cycle

Table 7
Optimal number of AGVs assigned to workshops, total movement times, and preventive maintenance cycles obtained by LINGO Dc nm À Cm jnm ÞÞ

Table 9
GA parameters and their levels

Table 11
Optimal number of AGVs assigned to workshops, total movement times, and preventive maintenance cycles were obtained by GA in MATLAB

Table 13
Comparison of the objective function obtained by LINGO and GA in MATLAB (10 runs)