1. Introduction
Scheduling methods are still important means to optimize production processes such as [
1,
2]. This significance is particularly evident in multi-site manufacturing systems [
3,
4,
5]. However, different from the mass production in which the scheduling is simpler and can be described as a disjunctive scheduling problem, the civil aircraft production system is typically a Discrete Manufacturing System (DMS), where the products are assembled from various materials through various production processes. Compared with other manufacturing industries, civil large aircraft production has characteristics, including larger scale, advanced technology, and longer production cycle, requiring a wider variety of resources in the production processes [
6]. Furthermore, it is expensive to purchase and stock some of the materials in civil aircraft production. Therefore, it is generally challenging to plan production based on the held material inventory. Instead, the production as well as material consumption is pulled by orders received from customers, typically from airline companies. In other words, the production environment of civil aircraft falls under the category of make-to-order (MTO). The production company must make a long-term production plan according to the received orders. Before the production of a certain flight is commenced, a scheduling scheme must be made in advance. By providing scheduling solutions with shorter production times, it is beneficial to improve production efficiency.
As shown in
Figure 1, the civil aviation industry is usually organized on a prime manufacturer–supplier structure, where suppliers are responsible for providing a variety of materials to the prime manufacturer undertaking the final assembly of the aircraft. Since the resources required by the civil aviation production are geographically dispersed, the factories of suppliers are also distributed in different places. Manufacturing companies commonly employ Material Requirements Planning (MRP) for production orders and management. However, MRP may have limitations when applied in the considered civil aircraft production environments. Based on our investigation on the companies, inventory shortages are commonly encountered in the production of the prime manufacturer of civil aircraft, and these shortages usually arise due to the limited production capacity of upstream suppliers. While other production industries can solve this issue by (1) adding more suppliers or (2) requesting suppliers to increase their production capacity, the unique characteristic of civil aircraft production presents challenges in implementing these solutions. The number of aviation suppliers that meet the necessary requirements is often limited, making (1) relatively difficult to achieve. Additionally, the implementation of (2) tends to be slow for aircraft production. As a result, it is necessary to schedule under the limited inventory.
By conducting on-site investigation, we conducted a comprehensive analysis of the factors that could affect civil aircraft production efficiency and discover that the factors constraining the production time mainly come from two aspects: precedence constraints and resource constraints. Precedence constraints include the certain time required for each production operation to complete, and the precedence relationship between each production operations, which can be represented by a directed acyclic graph in the form of activity-on-node (AON). Resource constraints are usually reflected in the limited amount of materials required for each production operation. Taking into consideration the existence of these two aspects of constraints and the distributed characteristic of civil aircraft manufacturing, the production activities of the work packages in aircraft production can be modeled by Resource-Constrained Project Scheduling Problem (RCPSP).
RCPSP and its variants have been widely practiced already in describing industrial scheduling problems, including civil aviation production in recent years. For example, ref. [
7] described an aircraft assembly scheduling problem using RCPSP, and attempted to solve it using two different coding methods based on a demand-driven GA. However, the types of resource it mainly considered were reusable human resources, with limited consideration on material inventory. Ref. [
8] described the paced aircraft assembly line rescheduling problem as an RCPSP, and proposed novel optimization criteria suitable for the aircraft manufacturing environment. However, the material resources were mainly not replenishable. Considering the situation where there were multiple stations sharing resources in an aircraft moving assembly line, ref. [
9] proposed a resource investment problem based on project splitting with time windows and a two-stage iterative loop algorithm was used to solve the corresponding optimization problem. The unpredictability of resource shortages was also considered, but its application scenario did not include material inventory.
The above examples of research on scheduling aircraft production with RCPSP modeling have insufficient consideration on inventory. In their models, the materials are not replenishable and limited by the application scenarios. In practical production, the material inventory would fluctuate rather than strictly being renewable or unrenewable during the production process, because of the continuous supply from upstream suppliers. We stress the importance of considering replenishable material because civil aircraft production enterprises are prone to production delays caused by inventory shortages in practical production. The shortages are usually caused by inflexible scheduling strategy.
Considering that the consumption of the material inventory is driven by orders, it is necessary to model the replenishable material inventory. Since the production capacity of the upstream suppliers is limited, it is unrealistic to stock all the required materials in advance when an order occurs. From the scheduler’s point of view, the material inventory is supplied periodically, and different processes may potentially compete for materials. Thus, it is necessary to consider the production as an RCPSP-IR.
In this article, we propose a GA-based scheduling method for civil aircraft distributed production, taking into consideration material inventory replenishment. First, we formulate the civil aircraft production scheduling problem as an RCPSP-IR. By analyzing the production process, we formulate the problem into a mixed integer programming (MIP) problem. Then, we introduce GA with resource constraints to solve RCPSP-IR so as to minimize the production time. To deal with the complex constraints including the inventory constraints, different constraint handling approaches and local optima preventing methods are used. Finally, to evaluate effectiveness of the proposed scheduling method, we conduct a case study of a jet twin-engine civil aircraft production of COMAC. The results of the case study show that the proposed method can give a nearly optimal scheduling strategy to be applied to actual civil aircraft product919ion.
The remainder of this paper is organized as follows:
Section 2 introduced the related work; In
Section 3, the aircraft production process is summarized and the corresponding mathematical model of RCPSP-IR is described; In
Section 4, the model is solved by GA considering the complex constraints;
Section 5 gives the computational result of a case study; Finally,
Section 7 summarizes the paper.
3. RCPSP-IR Modeling for the Distributed Production Scheduling of Civil Aircraft
3.1. Introduction of Civil Aircraft Production Process
The manufacturing of civil aircrafts is a typical DMS where its production process can be decomposed into many processing tasks to complete. Its production consists of several large, isolated work packages. These work packages are completed by stations on the actual production line. The production efficiency of those stations on the main production line directly affects the total aircraft production time. These stations include the installation and docking of the front, middle, and rear sections of the fuselage, as well as the assembly of the front and rear wings. The large work packages can be further divided into sub work packages, corresponding to the substations inside the station as shown in
Figure 2. The substations involve many working procedures. It takes a certain amount of time and consumes a certain quantity of materials including structural parts, standard parts, and raw materials to carry out a procedure, and if the work package’s production design does not change, its consumption quantity usually does not change greatly.
There exists a precedence relationship between the procedures, where each certain procedure can only start after its predecessor procedures have all been completed. Each procedure progresses according to this order, until the final procedure is conducted. By regarding each procedure as a node and the temporal precedence relationship between them as an edge, we can represent the precedence relationship as an Activity-on-Node (AON) structured graph as shown in
Figure 3. In the AON graph, a node is connected to the nodes representing its predecessor procedures through the incoming edges, and the outgoing edges are linked to the nodes representing its successor procedures. All paths in the graph do not always lead to the same final node, which represents a branch that occurs in the production process. The overall graph is a directed acyclic graph (DAG) for there is no cycle in the production, and the graph must be a connected graph.
In addition to the precedence relationship mentioned above, inventory is also a major factor restricting the production. It is worth noting that the materials required by each procedure is listed by its technological document. Before the procedure is started, all materials required must be available and transported from the inventory to the shop. The types and numbers of these materials would be checked. Only after that, the procedure can begin. All these materials are stored in the warehouse of the prime manufacturer. These materials mainly come from the orders from upstream suppliers. Generally, according to the technological document, a procedure requires several types of materials, and the amount of each type of material required is different. Both the types and amount required would be different among the procedures. The inventory replenishment periods and replenishment quantities for each type of material are also different. Materials with low complexity and technology such as standard parts and raw materials have large output and can be produced and supplemented at a stable rate, while the structural parts produced by strategic suppliers have limited output and longer replenishment cycle.
From the above, it can be seen that for the prime manufacturer, the production capacity is constrained by the capacity of stations on the main line, and the factors restricting the capacity of stations mainly come from two aspects: (1) precedence constraints; (2) resource constraints. In terms of precedence constraints, each procedure needs a certain amount of time to complete, and the execution of each procedure requires the completion of some other procedures as a precondition.
The factors of resource constraints should be emphasized. In terms of resource constraints, each procedure consumes a certain type and quantity of materials. Restricted by the limited production capacity of upstream suppliers and the consumption by production, the main manufacturer may not have enough materials in the inventory to simultaneously conduct all procedures that meet time constraints. On the other hand, some procedures may require the same types of material, which will inevitably lead to the competition among procedures for materials. Because of these two aspects of constraints, the scheduling solution needs to be responsible for both the procedure scheduling and resource scheduling.
These two aspects of constraints make the problem fit the characteristics of RCPSP, and the production activities in the stations can be modeled in the framework of RCPSP. Additionally, compared with the classic RCPSP, the replenishable inventory should be considered, because the total amount of resources in many RCPSP models are initially given at the beginning of the project and are renewable as the project processes. The main difference between our aircraft production problem and traditional RCPSPs is that the inventory of material can be replenished by upstream suppliers and consumed by the procedures over time. Therefore, it should be defined as an RCPSP with inventory replenishment (RCPSP-IR).
Table 1 compares difference between RCPSP and our model considering inventory constraints.
3.2. Formulization of RCPSP-IR
Considering that civil aircraft production is DMS, we assume the whole production process is decomposed into many stations and procedures, and the precedence relationship of the procedures in a station can be represented by a connected AON which is a DAG. In addition, the aircraft production environment is MTO, and inventory is limited by the amount and period of material supplied by upstream suppliers. So, the production plan must be scheduled in advance because of the inventory constraints for a specific production link. According to the historical inventory data, we assume that upstream suppliers and manufacturing departments provide materials regularly and quantitatively.
On the basis of the production process, the following assumptions are made to solve and verify the model:
Upstream suppliers and manufacturing departments provide materials regularly and quantitatively;
Each type of material has its own supply quantity and supply cycle;
The whole production process of a civil aircraft can be decomposed into many procedures;
Each procedure must only be started after the completion of its predecessor procedure, and takes a certain amount of time to complete;
No cycle during the production. In other words, the AON graph is a DAG;
The AON graph is a connected graph;
Supply disruptions or production breakdown caused by external factors, like quality issues, human error, and sudden fault, are not considered;
Working hour is the time unit of this model.
In this article, the scheduling problem of civil aircraft production is solved through the idea of optimization. The objective of the established optimization model is to minimize the overall production time of each station. The decision variable is the start time of each procedure. As previously mentioned, time factors and resource factors, including inventory, are mainly considered. The variables related to time factors should include the procedure precedence constraints and execution time, etc. The variables related to resource factors encompass the quantity of materials required for each procedure, as well as the inventory replenishment quantities and periods. Since the procedures and material consumption are closely coupled, scheduling material resources can also be achieved by scheduling the start time of the procedures. Depending on the replenishment quantities and periods, three kinds of materials are considered: raw materials, standard parts, and structural parts. The symbols and their definitions appearing in this paper are shown in
Table 2, including the symbols related to GA used in this article.
The constraints presented in the form of equations and inequalities are as follows.
In our consideration, the whole production process starts at time 0 so all variables representing time should be non-negative. For all
,
The precedence constraints contain a series of inequalities, stating that only after all the predecessors are finished can a procedure start. “
” represents that the succeed procedure of procedure
is procedure
.
According to the definition of
we have the following:
By combining the previous Formulas (4) and (5), we have the following:
The current inventory depends on the delivery volume from suppliers and consume volume caused by the production activities. Taking structural part
for example (the situation for raw materials and standard parts are the same),
Here,
and
are the Boolean functions, respectively describing whether there is inventory replenishment of part
happening at moment
, and whether procedure
starts at moment
:
The value of depends on the order strategy of the factory and in our assumption so there should be a time sequence indicating the date list when there are parts arriving. As mentioned above, we assume here that they arrive at constant intervals, so represents whether is in .
In the AON representing the manufacturing process, there is no acyclic, so every job should and can only be conducted once. For a pair of fixed
and
,
The inventory of each kind of material should have a lower bound, or at least it should not be a negative value. The lower bound can be fixed to another value if it is necessary to set a safety inventory. In the following case study, we set the lower bound of inventory to 0 and directly consider the relative value of the current inventory compared to the safety inventory.
4. RCPSP-IR Solving Based on GA
The main challenge to solve RCPSP-IR by GA is how to deal with the constraints associated with the problem. The scale of the problem and the complexity of the constraints seriously restrict the efficiency of computation. There is a large number of constraints in the model. Firstly, precedence constraints exist between two adjacent procedures. Secondly, at every moment there is an inventory lower bound constraint for each type of material. Moreover, there is a strong coupling relationship among these constraints, making the feasible region quite narrow. If GA is left to evolve and iterate from random population without intervention, it may not reach the feasible region in acceptable runtime, let alone find a feasible optimal solution. Thus, the two aspects of constraints need to be handled accordingly.
4.1. Precedence Constraints
Precedence constraints can be solved by variable substitutions. The precedence constraints are easily violated because it is hard to generate a scheduling policy satisfying the constraint between a pair of procedures without violating the others. Since the GA does not strictly impose requirement on whether the optimization problem is linear or not, inequality (3) can be transformed into (10):
The inequalities can then be translated into equalities to reduce the difficulty of solving by introducing variable
as shown in (11) and (12). The main decision object
becomes the array composed of
for each procedure
as shown in (13).
4.2. Resource Constraints Considering Inventory Replenishment
For the resource constraints, the inventory fluctuates every moment and cannot be easily eliminated. Generally, when solving combinatorial optimization problems, GA often employs the penalty method and the correction algorithm to handle constraints. Since the correction algorithm depends on specially designed heuristics for complex problems, it is not as concise and flexible as the penalty function method while they achieve similar effects. So, the penalty function method applied to solve the inventory constraints. The penalty method generally adds a penalty function term to solutions that violate the constraints in order to reduce their fitness, so that they are less fit and easier to be eliminated during the iteration. represents the degree of constraint violation by solution , and its value depends on the selected penalty method.
It should be noted that the feasible regions of different problems have different characteristics, and there is no universal penalty function that can be applied. In addition, some penalty functions also involve coefficient adjustment such as static penalty, and the performance of the penalty functions may also be related to the hyperparameter configuration of GA, so it is necessary to compare experiments to find a penalty function that can deal with the constraints of the problem in this paper. Three penalty function methods are potentially helpful in handling these complex constraints.
4.2.1. Static Penalty
In our consideration, since the inventory quantity has a lower bound, the inventory constraint violation term of a certain material at a specific time should be defined as (15). In the static penalty method (denoted by SP), the total constraint violation
is the weighted sum of violations of all constraint terms shown in (16), where
,
are constant values and
can be 1 or 2, similar to the method mentioned in [
22]. Obviously, results of SP will be influenced by the values of these constant coefficients.
4.2.2. Dynamic Penalty
In the dynamic penalty method (denoted by DP), the violation coefficient values depend on the current iteration of GA, as shown in (17), where
are constant, and
is the current iteration. In this way, a larger penalty can be imposed on infeasible solutions when the number of iteration rounds is relatively large, and theoretically it is expected to avoid the algorithm from making too many useless explorations in the infeasible domain.
4.2.3. Adaptive Penalty
We adopt two adaptive penalty methods for potentially better algorithm performance. In the first adaptive penalty method (denoted by AP1), the factor of penalty term is computed dynamically according to the individuals in the population at the current generation. As shown in (18) and (19), theoretically it is expected to automatically expand its search range if the past
k generations are all feasible and narrowing its search area to obtain a feasible solution if the past
k generation are not feasible, same as [
22].
Different from the method above, the second penalty adaptive method (denoted by AP2) normalized the objective function of populations and constraint violations of individuals separately. Similar with [
23], when there were fewer feasible solutions in the population, the weight of the penalty for violating the constraints is increased to obtain more feasible solutions. When there are more feasible solutions, the weight of the optimization goal is increased to obtain the optimal solution. Where
is the normalized optimization goal,
is the normalized violation of the individual, and
is the proportion of feasible solutions in
.
4.3. Prevention of Local Optima
Another concern is the local optima resulting from premature convergence, which is quite common when applying GA. According to [
24], although it is hard to totally avoid falling into local optima and verify the current optimal solution’s optimality, increasing the individual diversity of the population is still useful to reduce local optima. This can be achieved by properly increasing the crossover rate or mutate rate and increasing the magnitude of mutation.
One of the characteristics of our problem is that the feasible regions are distributed in isolation if the range of the optimization goal is restricted. Some penalty methods, such as DP and AP1, trend to assign a large penalty to the infeasible individuals, which would lead the individuals to fall into a nearby feasible region where there is no optimal solution in a hurry.
An extinction mechanism is introduced to solve local optima. If the optimal feasible solution is not being improved for generations, where is the countdown, all optimal feasible individuals would be removed from the population and replaced by randomly generated new individuals, because that potentially means it may be trapped in a local optimum.
By adopting the methods above, the algorithm can be expressed as Algorithm 1.
Algorithm 1 Overview of GA in this article |
1: | Define the selection mechanism, the crossover mechanism, the mutation mechanism. |
2: | Define the extinction countdown . |
3: | Initialize the population P and maximum iteration episodes . |
4: | For each individual in , calculate the fitness . |
5: | for : |
| (a) | Select from to obtain a new population according to the selection mechanism. |
| (b) | for each pair of individuals in , with a crossover probability: Crossover to obtain offspring according to the crossover mechanism. Replace with . end for |
| (c) | for each individual in , with a mutation probability: Mutate to get solution according to the mutation mechanism Replace with . end for |
| (d) | Recalculate the fitness of each individual in . |
| (e) | Obtain the current optimal feasible solution from . If the fitness of the current optimal feasible solution is not getting improved for generations, remove all optimal feasible individuals from , and recalculate the fitness of each individual in . |
| (f) | Replace with . |
| end for |
6: | Return the optimal feasible solution O in population P. |
6. Discussion
The RCPSP-IR is proposed to formulate the scheduling problem of civil aircraft production. Despite the narrow feasible regions inherent in this combinatorial optimization problem, the obtained results demonstrate the feasibility and practicality of finding near-optimal solutions, forming an effective scheduling scheme for civil aircraft production.
In terms of convergence speed, AP2 outperforms DP and AP1 because AP2 can adaptively adjust the penalty terms, while DP and AP2 easily degrade to the death penalty situation. Moreover, from the point of view of the scheduling goal value that it finally converges to, AP2 shows a lower likelihood to fall into local optima for similar reasons. Additionally, compared with SP, AP2 also takes less effort to fine-tune parameters. By additionally importing extinction, local optimality can be further avoided. These efforts make the scheduler obtain faster scheduling schemes.
The purpose of our method is to schedule production activities in advance when order demands occur, providing a reference for the layout of production. As an approximate optimal scheduling scheme has been generated, it is possible to serve as guidance for production assembly. The results of this case study have already been used to guide the aircraft production assembly in COMAC.