A Data-Driven Methodology for Hierarchical Production Planning with LSTM-Q Network-Based Demand Forecast

Abstract

Mass customization makes it necessary to upgrade production planning systems to improve the flexibility and resilience of production planning in response to volatile demand. The ongoing development of digital twin technologies supports the upgrade of the production planning system. In this paper, we propose a data-driven methodology for Hierarchical Production Planning (HPP) that addresses the upgrade requests in the production management system of a fuel tank manufacturing workshop. The proposed methodology first introduces a novel hybrid neural network framework with symmetry that integrates a Long Short-Term Memory network and a Q-network (denoted as LSTM-Q network) for real-time iterative demand forecast. The symmetric framework balances the forward and backward flow of information, ensuring continuous extraction of historical order sequence information. Then, we develop two relax-and-fix (R&F) algorithms to solve the mathematical model for medium- and long-term planning. Finally, we use simulation and dispatching rules to realize real-time dynamic adjustment for short-term planning. The case study and numerical experiments demonstrate that the proposed methodology effectively achieves systematic optimization of production planning.

1. Introduction

As manufacturing technology and production capacity continue to advance, the demand for personalized vehicles is rapidly increasing with the diversification of products [1]. The fuel tank is a crucial component in the assembly of an automobile. Its production faces challenges in terms of production planning and management. On the one hand, the characteristics of the market environment include unpredictable demand, rapid product updates, and short lead time. These characteristics require that the production planning of enterprises has resilience in response to fluctuations in demand [2]. On the other hand, the growth in electric vehicles and the increasing competition from traditional fuel vehicles force companies to reduce costs [3]. The accelerating adoption of electric vehicles, driven by government incentives and emission regulations, is fundamentally reshaping automotive industry dynamics. As electric vehicle manufacturers innovate battery technologies, electric vehicles have an advantage over traditional fuel vehicles in terms of performance, cost, and environmental protection. In recent years, electric vehicles have crowded out a large portion of the market for traditional fuel vehicles, which has made the competition in the entire automotive industry particularly fierce. The increasing competition forces the traditional fuel vehicle companies to reduce costs [4]. Therefore, production plans should not only be adaptable to varying demands but also minimize costs wherever possible.

To address these challenges, enterprises must comprehensively upgrade their production management systems. Hierarchical production planning (HPP) provides a valuable perspective for achieving global optimization [5] through multi-level decision-making with specific objectives and planning horizons [6]. Many studies have explored the application of HPP [7,8]. However, in the context of Industry 4.0, the technology related to digital twins has developed rapidly [9], such as real-time data analytics and virtual simulations of production processes [10]. These technologies introduce new dynamics to the application and research of HPP. Specifically, the methods of HPP can be upgraded to data-driven methods through the integration of digital twin technology. The data-driven approach enables the collection of extensive system data to support improved decision-making and facilitates real-time monitoring and response based on digital twin models [11]. The symmetry between digital twin models and physical systems enables precise representation of uncertainties through bidirectional data mapping, where virtual and real-world parameters maintain geometric and temporal congruence [12]. However, this development introduces new challenges for HPP. The complexity of HPP systems under uncertain environments is significantly increased, because changes in a single parameter may have widespread effects on the entire HPP system. For example, an error in one process of the HPP system can lead to the failure of the overall solution generated by the HPP, rendering it unusable. This necessitates that the HPP has to be able to cope with the uncertainties in the system more precisely and rapidly.

Limited literature has discussed HPP in the context of digital twins, in particular with real-world case studies [13]. To fill this research gap and address the challenges of upgrading the planning system faced by the manufacturing workshop of fuel tanks, we propose a data-driven approach for HPP. The proposed data-driven methodology comprises three levels. In the first level, we introduce an LSTM-Q network for continuous iterative prediction of product demand based on real-time data. The second level formulates the production planning problem of the fuel tank workshop as a multi-item capacitated lot-sizing problem with inventory constraints (MCLSP-I). To solve this model, we propose two relax-and-optimize algorithms for the MCLSP-I. In the third level, we integrate simulation and dispatching rules to achieve real-time dynamic planning.

The main contributions of this study are (1) providing a data-driven approach for HPP based on a fuel tank manufacturing workshop; (2) designing a new LSTM-Q network-based network framework to predict the demand for HPP; (3) providing an improved relax-and-optimize heuristic for long-term planning, which is a multi-item capacitated lot-sizing problem with inventory constraints; (4) proposing a hybrid dispatching strategy at the short-term planning level that outperforms dispatching rules both based on time window and available capacity.

The following sections of this article are structured as follows: Section 2 reviews the related papers. Section 3 describes the problem description and introduces the solution framework for hierarchical production planning. Section 4 delineates the three levels in HPP, including demand forecast, long-term planning, and short-term planning. Section 5 provides the results and analyses. Finally, we summarize this article in Section 6 and outline future directions.

2. Literature Review

2.1. Hierarchical Production Planning

Hierarchical production planning (HPP) is a structured methodology for managing production processes by decomposing them into distinct decision levels, each with specific objectives and planning horizons [6]. This approach is particularly effective in complex manufacturing environments [8,14]. How to apply recursive production planning to the ever-changing modern manufacturing industry is a problem that deserves continued attention and research. Many studies have contributed valuable insights on this topic, such as bidirectional constraints between hierarchical levels and dynamic feedback loops [15,16]. However, integrating and coordinating hierarchical planning poses significant challenges due to its inherent complexity. One potential solution is reducing the problem complexity to enable integrated optimization across multiple planning levels. For example, one of the challenges in HPP is the lot-sizing problem, which involves determining when and how much to produce. Addressing this problem with a highly detailed solution over a long planning horizon can be complex and difficult to manage. However, by dividing the problem into a long-term plan and a short-term plan, it becomes feasible to approach. We can cope with it in a broader perspective for the long-term plan and consider the details more closely in the short-term plan. While the deterministic problem for HPP is already complex, extending this approach to uncertain environments presents even greater challenges [5,17,18,19].

In enterprises, the system architecture typically follows a hierarchical structure [20]. However, separate departments often manage production planning at different levels, leading to independent optimization of planning methods in the system. This independence creates silos between hierarchical levels, making coherent and systematic optimization difficult to achieve [21,22]. In extreme cases, the planning systems at different levels may operate entirely independently due to mismatched iteration speeds, effectively abandoning coordination between upper and lower levels [23]. Therefore, studying HPP is essential. With the ongoing advancements in technology, digital twins and data-driven methods have garnered increasing attention [11,24,25]. These technologies offer significant support for the implementation and enhancement of HPP [26,27].

Unlike traditional planning research, which often emphasizes theoretical models and algorithmic innovations, HPP focuses more on iterative improvement and integration of existing techniques [28,29]. This research needs to be closely tied to the specific production systems of individual companies, as the characteristics and concerns of production systems vary greatly across industries [7]. It is therefore essential to conduct industry-specific studies and validate HPP through real-world case studies [30,31,32]. This study contributes to the field by applying HPP to a fuel tank manufacturing workshop, thereby complementing existing studies and applications.

2.2. Demand Forecast

Long Short-Term Memory (LSTM) networks, a subtype of Recurrent Neural Networks (RNNs), have been pivotal in surmounting the challenges associated with learning long-term dependencies. Traditional RNNs encounter difficulties due to vanishing and exploding gradients, issues that LSTMs circumvent by implementing a complex system of gates—namely the input, forget, and output gates—that regulate the flow of information [33]. These gates determine the retention, updating, and discarding of information within the cell state, a crucial component that conveys relevant data across time steps [34]. LSTMs have shown considerable success in domains that require the processing of sequential data, such as natural language processing, speech recognition, and time series analysis, improving upon tasks like machine translation and language modeling [35]. Building upon the established efficacy of LSTM networks in handling sequential data, their applicability extends to the domain of time series forecasting, such as predicting future part order quantities based on historical ordering data. The inherent architecture of LSTMs, with their capacity to capture long-term temporal correlations, is particularly suited to address the challenges posed by the sequential nature and potential temporal lags present within inventory management datasets [36]. The strength of LSTMs in this context lies in their ability to learn and remember patterns over extended time frames, a critical requirement for accurate demand forecasting in supply chain operations. By discerning patterns and trends in historical data, such as seasonal fluctuations, cyclic demands, and irregular occurrences, LSTMs can make informed predictions about future order quantities [37]. Therefore, we developed a network framework based on LSTM-Q to forecast the demand quantity of all types of parts for the next weeks.

2.3. Relax-And-Fix Heuristic

Solving NP-hard problems like the lot-sizing problem within acceptable computational time, particularly for large-scale instances, remains a fundamental challenge in operations research [38]. In recent years, mixed-integer programming (MIP)-based heuristics (e.g., relax-and-fix, fix-and-optimize) have received much attention from scholars [39]. MIP-based heuristics effectively balance solution quality and computational efficiency, bridging the gap between exact algorithms and heuristic approaches. Their compatibility with commercial solvers (e.g., Gurobi, CPLEX) and straightforward implementation have further driven their widespread industrial adoption [40,41].

Pochet and Wolsey [42] introduced the R&F approach and applied it in production planning. Its core principle involves decomposing the original complex problem into smaller sequential subproblems. This dimensional reduction enables efficient resolution of subproblems using commercial solvers or established algorithms. Compared to developing new algorithmic frameworks from the beginning, R&F significantly reduces implementation costs while maintaining solution quality. It is particularly advantageous for industrial applications with time-sensitive and computational resource constraints.

Since its introduction, the R&F algorithm has been widely applied in planning and scheduling across diverse industries. Many researchers have been motivated to expand its range of applications and enhance its performance. As a critical process, the decomposition method must be tailored to the specific characteristics (e.g., decision variables, constraints) of each problem during implementation. Table 1 lists notable R&F studies, analyzing their problem type, objective functions, industrial applications, and decomposition methods.

Table 1 summarizes the decomposition methods explored in each study, with the optimal methods based on experimental results highlighted in bold. Existing research indicates that the time-based decomposition method is currently the most effective and stable for R&F algorithms [43,44]. R&F algorithms are commonly applied to solve lot-sizing problems for production planning, and they have also been applied to address scheduling, inventory balancing, vehicle routing, and production routing [43,44,45,46,47]. Although some extensions of the basic lot-sizing problem have incorporated factors such as shortage and setup, limited literature on R&F algorithms considers overtime and its associated costs. Our study supplements it by applying the R&F algorithm in the capacitated lot-sizing problems with shortage, setup, and overtime costs. Moreover, although many general studies apply R&F algorithms to lot-sizing problems, there is a notable gap in industry-specific case studies. To the best of our knowledge, no studies have specifically addressed the application of R&F algorithms in the fuel tank production industry. Our research fills this gap by providing an analysis of R&F algorithm applications in this context.

Table 1. Literature review of R&F algorithms.

Paper	Level	Capacity Constraints		Objective Costs	Decomposition Method	Case Type
		Production/ Transportation	Inventory
Araujo et al. [43]	Single	Yes	Yes	Inventory, Storage, Production, Setup	Time, Product, Machine, Problem- independent	Personal care consumer goods industry
Clark and Clark [45]	Single	Yes	–	Inventory, Storage	Time	–
Wang et al. [46]	Multi	–	Yes	Transportation, Pulling-off, Packing, Out-of-stock, Inventory	Time	Clothes industry
Joncour et al. [44]	Single	Yes	–	Transportation, Inventory, Production	Time, Problem- independent	–
Friske et al. [48]	Multi	Yes	Yes	Revenue, Traveling, Operation, Inventory Violation	Time	Maritime transport
Brahimi and Aouam [47]	Multi	Yes	Yes	Inventory, Storage, Production, Transportation, Service	Time	–

Table 1. Literature review of R&F algorithms.

Paper	Level	Capacity Constraints		Objective Costs	Decomposition Method	Case Type
		Production/ Transportation	Inventory
Araujo et al. [43]	Single	Yes	Yes	Inventory, Storage, Production, Setup	Time, Product, Machine, Problem- independent	Personal care consumer goods industry
Clark and Clark [45]	Single	Yes	–	Inventory, Storage	Time	–
Wang et al. [46]	Multi	–	Yes	Transportation, Pulling-off, Packing, Out-of-stock, Inventory	Time	Clothes industry
Joncour et al. [44]	Single	Yes	–	Transportation, Inventory, Production	Time, Problem- independent	–
Friske et al. [48]	Multi	Yes	Yes	Revenue, Traveling, Operation, Inventory Violation	Time	Maritime transport
Brahimi and Aouam [47]	Multi	Yes	Yes	Inventory, Storage, Production, Transportation, Service	Time	–

2.4. Simulation Methods and Dispatching Rules

The application of simulation methods and dispatching rules in planning and scheduling is a dynamic and evolving field, especially in the context of Industry 4.0 [49,50]. These methods play a crucial role in complex production systems, enabling real-time decision-making and improving scheduling efficiency [51,52]. Conventional scheduling methods based on mathematical models and heuristic algorithms face inherent computational challenges due to the NP-hard nature of scheduling problems [53]. During the formulation of a mathematical model, the necessary abstraction and simplification of real-world systems for managing the complexity inevitably lead to information loss. This information loss makes it difficult for the mathematical model to accurately reflect the real-world system [54,55]. The discrete-event simulation methods address these limitations by capturing detailed operational characteristics of the real system and reflecting uncertain environments in the real system. Therefore, simulation models provide a more accurate representation of real systems compared to mathematical models [56,57]. Furthermore, solving mathematical models for scheduling often requires substantial computational resources or the development of heuristic algorithms [58,59,60]. In contrast, dispatching rules can be implemented quickly, enabling rapid decision-making and real-time scheduling even in complex environments [61,62,63].

Simulation models simulate the system’s current state and project future scenarios, while dispatching rules guide dynamic strategy selection through continuous state evaluation. This dual mechanism achieves rapid adaptation to fluctuations in the system without complex pre-planning [64,65]. Due to hardware and software limitations, simulations have historically been primarily off-line and used mainly in the early planning stages. However, with the development of digital twin technology, real-time online simulation has gradually become popular in enterprises. This allows for continuous updates of production data and enables automatic adjustments of the simulation model based on changes in the production system facilities [66,67,68]. Thus, the integration of simulation and dispatching rules can be applied more widely and effectively in practical production management. In addition, the integration of simulation and dispatch rules has advantages from the implementation point of view. Its implementation can use existing simulation software and rule libraries, requiring neither complex algorithms nor extensive computational resources, thus providing high practicality and scalability [65,69,70,71].

This paper focuses on developing data-driven hierarchical production planning in the context of upgrading factories to digital twins. Specifically for short-term planning, we employ simulation and dispatching rules to optimize order sequencing and batch processing and evaluate the performance of various dispatching rules through simulation.

3. Problem Description and Solution Framework

This study aims to address production planning challenges in a fuel tank manufacturing factory, which can be viewed as a hierarchical production planning problem in a flow shop. The challenges faced by this factory include dynamic demand fluctuations, short delivery lead times, production capacity constraints, and inventory capacity constraints. Inventory capacity constraints in the fuel tank manufacturing factory are mainly due to the standing process of semi-finished products, where both semi-finished and finished products take up the inventory space and containers. In addition, the factory can obtain additional production capacity beyond the regular limited capacity by flexibly arranging personnel and production lines, which is defined here as overtime. However, overtime shifts must be planned in the long-term planning phase. These constraints necessitate a production planning system that simultaneously (1) ensures timely demand fulfillment under limited capacity, (2) effectively schedules overtime shifts, (3) maintains inventory levels within physical storage limits, and (4) minimizes total costs. To address these requirements, we propose a data-driven methodology for HPP in fuel tank manufacturing, as illustrated in Figure 1.

The data-driven methodology for HPP comprises three levels:

• The first level is data-driven demand forecasting. It uses an LSTM-Q network to iteratively update demand forecasts with historical and real-time demand data. The forecast period at this level is in weeks.
• The second level focuses on medium- and long-term planning. This level uses both known and predicted demand data to create plans for monthly and quarterly. The planning period at this level is in weeks. We generate production plans through a mixed-integer linear programming (MILP) model and employ the relax-and-fix algorithm to solve the MILP model.
• The third level is short-term planning. It refines the weekly production plans by specifying order release times and production sequences based on the results of the upper-level plans and supported by discrete-event simulation and dispatching rules.

4. Data-Driven Methodology for HPP

This section sequentially presents the three levels of the data-driven methodology for hierarchical production planning: demand forecasting, long-term planning, and short-term planning.

4.1. Demand Forecast

To forecast the requisite inventory levels for component parts over a forthcoming bi-weekly period, a novel hybrid neural network framework that integrated a Long Short-Term Memory (LSTM) network [33] and a Q-network, denoted as LSTM-Q network, is designed. The LSTM-Q is tailored to capture the temporal dependencies inherent in the historical data of part quantities spanning t consecutive weeks. Moreover, the LSTM-Q framework is designed to extract and integrate complex patterns and relationships from the time-series data, thereby facilitating a more robust and accurate prediction of part quantities necessary for the imminent two-week interval. The data information of continuous t weeks is represented as state vector $s_t$, as shown in Equation (1).

$$s_t=[X_0,X_1,\dots ,X_t]$$

(1)

In Equation (8), $X_t$ denotes the number of all parts in the tth week. Thus, we employ the LSTM to transform the variable-length state vector $s_t$ into a fixed-length vector $h_t$.

As shown in Figure 2, the data information of all parts in $s_t$ is stored in the cell state $c_t$. The cell state serves as a conveyor belt, transmitting pertinent information along the chain. Additionally, the cell generates a hidden state $h_t$ based on the current cell state. The forward propagation of the LSTM can be represented by Equations (2)–(6).

$$f_t=\sigma (W_f·[h_{t-1}\left|\right|s_t]+b_f)$$

(2)

$$i_k=\sigma (W_i·[h_{t-1}\left|\right|s_t]+b_i)$$

(3)

$$c_t=f_t·c_{t-1}+i_k·tanh(W_c·[h_{t-1}\left|\right|s_t]+b_c)$$

(4)

$$o_t=\sigma (W_o·[h_{t-1}\left|\right|s_t]+b_o)$$

(5)

$$h_t=o_t·tanh\left(c_t\right)$$

(6)

In Equations (2)–(6), $W_f$, $W_i$, $W_c$, $W_o$ and $b_f$, $b_i$, $b_c$, $b_o$ represent the weight matrices and bias vector parameters. $\sigma$ is the fully connected layer with activation function. $f_t$ is the forget gates. $i_k$ is the input gates. $o_t$ is the output gates.

The LSTM network can incorporate the data information $X_t$ of each week into the cell state. Therefore, the final hidden state $h_t$ captures all the data information belonging to the state $s_t$, i.e., $[X_0,X_1,\dots ,X_t]$. The fixed-length vector $h_t$ is then connected to a deep Q network, which outputs the predicted number of all parts in next two weeks. The parameters of the LSTM-Q network used in this paper are shown in

Table 2. Parameters of LSTM-Q Networks.

Parameters	Value
LSTM input size	12
LSTM number of hidden layers	2
LSTM hidden layer size	64
Q network input size	64
Q network number of hidden layers	2
Q network hidden layer size	64
Q network output layer size	53
Q network output layer activation function	Sigmoid
LSTM and Q network hidden layer activation function	Relu

.

4.2. Long-Term Planning

The primary goal of long-term planning is to generate a plan that specifies weekly product types, production quantities, overtime shift schedules, and inventory levels. We present a mathematical model constructed for long-term planning in Section 4.2.1, followed by a heuristic algorithm for solving the model in Section 4.2.2.

4.2.1. Mathematical Model

As described above, the fuel tank production workshop can be viewed as a flow shop, characterized by fast processing, high automation, and a simple production line layout. Each production line can process all product types, allowing the combined capacity of multiple lines to be treated as a single resource. We simplify the model by converting the production capacity of all production lines to the total production time in each planning period. The period duration is set longer than the maximum processing time for any individual product, meaning that, if production capacity allows, the entire product can be completed within one period. Overtime production is possible but capacitated, as the planner can reallocate workers and production lines from other workshops to address capacity shortages. However, reallocating workers and production lines from other workshops implies additional overtime costs, requiring a trade-off between overtime and shortage. In the mathematical model, we manage overtime by adding overtime shift constraints and assigning high overtime costs to avoid excessive overtime. In addition, the following basic assumptions underlie the long-term production planning scenarios.

• Assumptions:

• Each product is produced in a single production lot during a single production cycle.
• It should be noted that split batches of the same product in the same cycle are not considered.
• The relevant materials are sufficient in number and quality, and the production process is be disrupted by accidents.
• The production process is characterized by a 100% yield rate, indicating no wastage.
• The processing time represents the average total time per product, comprising not only the value-added processing time in machines but also other non-value-added activities, such as queuing and material handling.
• For the capacity constraints, production lines and workers are pre-configured, without consideration for splitting.
• The production capacity can be fully used, that is, the equipment is not in a state of disrepair or inoperable.
• The costs associated with inventory and shortage depend on the batch sizes of products.
• The setup costs are determined by whether a product type is scheduled for production in a specific period, regardless of the production quantity.
• The costs associated with overtime are only dependent on the shift schedules and number of shifts, not dependent on the specific product type or batch size.
• The model does not consider production costs, but rather the additional costs associated with inventory, setup, shortage, and overtime.

The parameters and symbols used to construct the mathematical model for the long-term production planning are defined as follows.

• Parameters and Symbols Definition:

Symbol	Description
Index:
i	Index representing product number, $i=1,2,3,\cdots ,I$
t	Index representing the period number, $t=1,2,3,\cdots ,T$
n	Index representing the overtime shift number, $n=1,2,3,\cdots ,N$
Parameter:
I	Total number of product categories.
T	Total number of planning periods.
N	Total number of overtime shifts during the unit period.
M	A big enough number.
$d_{it}$	Predicted demand for product i in period t.
$a_i$	Unit inventory cost for product i.
$l_i$	Unit shortage cost for product i.
$u_i$	The setup cost for all product i in per period.
R	Regular production capacity.
$O_{tn}$	Overtime production capacity for period t and shift n in hours.
$p_i$	Unit processing time for product i.
$I_{max}$	Inventory capacity limit.
${\beta }_{tn}$	The unit cost for overtime shift $tn$ in period t.
${\mathbb{Z}}^{+}$	The set of positive integers.
Integer variables:
$Q_{it}$	The production quantity of product i in period t.
$I_{it}$	Inventory level for product i at period t.
$S{H}_{it}$	Actual shortage quantity for product i at period t.
Binary variables:
$y_{tn}$	In period t, if the n th overtime shift is scheduled, the value of $y_{tn}$ is 1; otherwise, the value of $y_{tn}$ is 0.
$z_{it}$	In period t, if the product i in period t is scheduled, the value of $z_{it}$ is 1; otherwise, the value of $y_{tn}$ is 0. In other words, a setup operation occurs if a product is scheduled to be produced.

The problem we considered in long-term production planning belongs to multi-item capacitated lot-sizing problems with inventory constrains (MCLSP-I). The MCLSP-I can be formulated as a mixed integer linear programming model (denoted by ‘MILP model’):

• MILP Model:

$$\mathrm{Minimize}TC=\sum _{i=1}^I\sum _{t=1}^T{a}_i{I}_{it}+\sum _{i=1}^I\sum _{t=1}^T{l}_i{SH}_{it}+\sum _{i=1}^I\sum _{t=1}^T{u}_i{z}_{it}+\sum _{t=1}^T\sum _{n=1}^N{\beta }_{tn}{y}_{tn}$$

(7)

subject to

$$I_{it}-{SH}_{it}=I_{i,t-1}-S{H}_{i,t-1}+Q_{it}-d_{it}\forall i,t$$

(8)

$$\sum _{i=1}^I{p}_i{Q}_{it}\le \sum _{n=1}^N{O}_{tn}{y}_{tn}+R\forall t$$

(9)

$$\sum _{t=1}^T{Q}_{it}\ge \underset{t\in \{1,\dots ,T\}}{max}{d}_{it}\forall i$$

(10)

$$Q_{it}-M\times z_{it}\le 0\forall i\in I,t\in T$$

(11)

$$\sum _{i=1}^I{I}_{it}\le I_{\max }\forall t$$

(12)

$$\sum _{n=1}^N{y}_{tn}\le N\forall t$$

(13)

$$Q_{it},I_{it},S{H}_{it}\in {\mathbb{Z}}^{+}\forall i,t$$

(14)

$$y_{tn}\in \{0,1\}\forall t,n$$

(15)

$$z_{it}\in \{0,1\}\forall i,t$$

(16)

Equation (7) represents the objective function of the mathematical model, comprising inventory costs, shortage costs, setup costs, and overtime costs. The objective is to minimize the total cost of the manufacturing system over the planning horizon. In this model, we focus more on inventory and shortage costs to streamline the mathematical formulation, thus ignoring production costs. Equation (8) represents the inventory balance constraints. At the end of each period, the quantity of the product remaining after fulfilling customer service demand is the inventory for that period. In the event of unfilled customer demand, the number of products not delivered on time is the shortage quantity. Equation (9) outlines the production capacity constraints. For any period, if there is no overtime, the total production work hours must not exceed the regular production capacity of the system. If overtime is engaged, the total production work hours cannot exceed the sum of the configured production capacity and overtime work hours. Equation (10) specifies the minimum production batch constraint, which states that for any product, the total production quantity over the planning horizon must be greater than or equal to the maximum expected demand across all periods. This prevents scenarios where a product has only a small demand in a certain horizon, potentially minimizing costs despite shortages, leading to ongoing out-of-stock status. Equation (11) guarantees that if no setup occurs during period t, the production quantity becomes zero. Equation (12) delineates the inventory capacity constraints, which limit the total warehouse inventory at the end of a period not to exceed the total warehouse capacity. Equation (13) defines the overtime shift constraints, indicating that the number of overtime shifts in a period must not exceed the upper limit of overtime production capacity. Equations (14)–(16) represent the fundamental constraints on the decision variables.

4.2.2. Relax-And-Fix Heuristic

The planning problem considered in this paper is a multi-item capacitated lot-sizing problem with inventory constraints, which is classified as strongly NP-hard [72]. The efficient resolution of such problems, particularly in large-scale scenarios, remains a significant challenge due to their computational complexity. To address this, we propose a relax-and-fix heuristic algorithm adapted to this class of problems. The methodology comprises three key steps: first, a pre-processing phase to generate a feasible solution rapidly; second, a comprehensive implementation of the R&F procedure; and third, the integration of heuristic cuts to accelerate the solution process further.

(1) 

Pre-processing for finding feasible solutions

Providing the R&F algorithm with rapid and feasible solutions at an early phase can facilitate efficient guidance for the subsequent R&F approach. The binary variables are the key to solving mixed-integer programming models. Consequently, when seeking a feasible solution, our attention is directed towards the values of the binary variables $y_{tn}$ and $z_{it}$. The variable $z_{it}$ indicates whether product i is scheduled for production in period t. While scheduling all products for production in every period is suboptimal, it always results in a feasible solution. The physical meaning of $y_{tn}$ is whether or not the nth shift in period t is scheduled for overtime. It is evident that if we do not consider overtime in the schedule even regular production shifts cannot cope with overtime, there are always feasible solutions. In such a scenario, customer demand that is not delivered on time would be transformed into a shortage. In other words, we let $y_{tn}=0\forall t,n$ and $z_{it}=1\forall t,i$; in this case, the total overtime cost is zero, ${\sum }_{t=1}^T{\sum }_{n=1}^N{\beta }_{tn}{y}_{tn}=0$, while the setup cost is fixed at ${\sum }_{i=1}^I{u}_i\times T$. The remaining decision variables can be obtained by solving the following integer linear programming model (denoted by ’ILP model’).

ILP model:

$$\mathrm{Minimize}Z=\sum _{i=1}^I\sum _{t=1}^T{h}_i{I}_{it}+\sum _{i=1}^I\sum _{t=1}^T{l}_i{S}_{it}+\sum _{i=1}^I{u}_i\times T$$

(17)

subject to Constraints (8), (10), (12), (14)

$$\sum _{i=1}^I{p}_i{Q}_{it}\le R\forall t$$

(18)

The objective Function (17) minimizes the total cost, which is the total inventory holding cost and shortage cost. Constraints (18) replace Constraints (9), with $y_{tn}=0\forall t,n$. Note that Constraints (13) and (15) in the MILP model are redundant for the ILP model due to $y_{tn}=0\forall t,n$; therefore, they are removed in the ILP model. Compared to the MILP model, the ILP model is an integer linear programming model and can be solved faster. In the pre-processing stage, we use a commercial solverto obtain feasible solutions for the ILP model as the initial solution of the subsequent R&F algorithm to guide the algorithm.

(2) 

Relax-and-fix approach

The R&F approach of this study is based on the works of Friske et al. in 2022 [48] and Wang et al. in 2023 [46]. The core idea of the R&F approach is to decompose the original problem into a series of subproblems, each solved iteratively. The decomposition focuses on binary variables $y_{tn}$ and $z_{it}$ as the primary objects of optimization. Considering the studies mentioned above, this research employs a period-based decomposition method. We let K denote the total number of subproblems, corresponding to the number of iterations.

Each iteration comprises three blocks: the fixed block, the integer block, and the relaxed block. In each subproblem, the optimization targets the binary variables $y_{tn}$ and $z_{it}$ in the integer block, which are maintained as 0–1 variables. The binary variables $y_{tn}$ and $z_{it}$ in the fixed block are set to the values obtained from the previous iteration, while the remaining time intervals form the relaxed block, where binary variables $y_{tn}$ and $z_{it}$ are relaxed to continuous variables in the range $[0,1]$. The time interval of the integer block for each subproblem is defined as $T_{\mathrm{len}}=\lfloor T/k\rfloor$. If T is not an integer multiple of k, the length of the final subproblem interval is $T_{\mathrm{len}}^{\mathrm{end}}=T-(k-1)\times T_{\mathrm{len}}$. The subproblems are solved using the CPLEX solver.

Figure 3 uses a small instance with $T=10,k=3$ to illustrate the fundamental logic of the iterative solution of the R&F method. As illustrated in Figure 3, this study employs a period-based decomposition method. The planning horizon is first divided into K time intervals. The integer block of each subproblem corresponds to one time interval, where the binary variables $y_{tn}$ and $z_{it}$ within this interval are optimized. In the instance shown in Figure 3, with $T=10$ and $K=3$, the time interval for each subproblem is three periods, while the last subproblem encompasses four periods. Once the decomposition is completed, an iterative solving process is performed. In the first iteration, since no results are available from prior iterations, this subproblem only comprises an integer block and a relaxed block, with no fixed block. In this iteration, $y_{tn}$ and $z_{it}$ are treated as binary variables in the time interval $T\in [0,3]$, while $y_{tn}$ and $z_{it}$ in the interval $T\in [3,10]$ are treated as continuous variables. Solving this subproblem yields the solutions for $y_{tn}$ and $z_{it}$ in the interval $T\in [0,3]$. In the second iteration, $y_{tn}$ and $z_{it}$ in the interval $T\in [0,3]$ are fixed to the solutions from the first iteration. $y_{tn}$ and $z_{it}$ in the interval $T\in [3,6]$ are treated as binary variables, while $y_{tn}$ and $z_{it}$ in $T\in [6,10]$ remain continuous. Solving the second subproblem provides solutions for $y_{tn}$ and $z_{it}$ in $T\in [3,6]$. In the third and final iteration, $y_{tn}$ and $z_{it}$ in the fixed block $T\in [0,3]$ and $T\in [3,6]$ are fixed to the solutions of the first and second iteration. The binary variables $y_{tn}$ and $z_{it}$ in the interval $T\in [6,10]$ of the integer block are then optimized, without the the relaxed block. After completing the third iteration, all binary variables are optimized and the final solutions are recorded. Pseudo-code Algorithm 1 provides a detailed explanation of the R&F algorithm logic.

Algorithm 1 The procedure of the R&F method.

1: Build MILP model 2: Set total time limit $ϵ$ 3: Let $y_{tn}=0\forall t,n$ and $z_{it}=1\forall t,i$, obtain the ILP model 4: Obtain an initial solution by solving the ILP model with CPLEX 5: Record the initial solution of objective value $Z_{init}$ and relevant variable values 6: Divide the planning horizon into K time intervals. 7: Build MILP models for each subproblem 8: for $k\in \{1,2,\dots ,K\}$ do 9:    if $k=1$ then 10:      Input the initial solution 11:      Solve the 1st subproblem to optimality within the time limit $ϵ/K$ 12:      if $Z_{init}>Z$ then 13:         Record the solution $Z_{best}\leftarrow Z$, and relevant variable values 14:      else 15:         $Z_{best}\leftarrow Z_{init}$ 16:      end if 17:    else 18:      Input last iteration solutions as initial solutions 19:      Solve the kth subproblem to optimality within the time limit $ϵ/K$ 20:      if $Z_{best}>Z$ then 21:         Record the solution $Z_{best}\leftarrow Z$, and relevant variable values 22:         Report the solution 23:      else 24:         Do not update the solution 25:      end if 26:    end if 27: end for

(3) 

Method for accelerating computations

To accelerate the computation, we tight the constraints (11) in this LP-relaxation by defining M as follows:

$$M=\underset{\forall t}{max}\sum _{i\in I}{D}_{it}$$

(19)

To enhance the computational efficiency of the R&F algorithm, we extend the relaxation of variables. In the relaxed block of the subproblem, we not only allow binary variables to take on continuous values between 0 and 1 but also relax the integer variable $Q_{it}$ to a continuous variable. Throughout the iterative process, $Q_{it}$ is progressively constrained back to integer values. Since both inventory $I_{it}$ and shortages $S{H}_{it}$ are calculated from $Q_{it}$, we can also relax $I_{it}$ and $S{H}_{it}$ to continuous variables. As $Q_{it}$ becomes bounded to integer values during the iterations, $I_{it}$ and $S{H}_{it}$ are automatically converted to integers as well. This approach effectively accelerates the solution of the subproblem. We define the R&F algorithm in Pseudo-code Algorithm 1 without the acceleration method as R&F-1, and the R&F with the acceleration method as R&F-2.

4.3. Short-Term Planning Based on Simulation and Dispatching Rules

The main task of short-term planning is to refine production planning by batching orders and determining their release time and sequence. This process is guided by the planned products, quantities, and overtime shifts established in the long-term production planning. Unlike long-term planning, which treats all production lines as a single resource, short-term planning must consider more detailed constraints.

Specifically, the fuel tank workshop contains several production lines, requiring equipment setup changes when switching between different products. Thus, the plan should avoid frequent line changeovers. In addition, material transfer between machines on a production line is based on trolleys. This means that we need to consider the resource constraints of trolleys in short-term planning. To optimize resource utilization, production batch sizes should avoid being too small and ideally match multiples of the trolley capacity.

This study introduces an approach based on simulation and dispatching rules for short-term planning. First, orders are scheduled based on dispatching rules. Second, they are simulated to verify the operational feasibility of short-term plans through key performance indicators (KPIs) such as order completion times. The simulation results can provide decision support for plan adjustments. Furthermore, the simulation validation experiment is conducted only at the beginning of each week. If an emergency order arrives during the week, only the short-term plan is updated according to the dispatching rules. Based on existing dispatching rules, we propose two new dispatching rules that incorporate the earliest due date (EDD) rule and develop a hybrid dispatching strategy based on these two rules. The core ideas and main steps of these three approaches are outlined below.

4.3.1. Dispatching Rules Based on Time-Window

The dispatching rules based on time-window balance on-time delivery and production efficiency by aggregating orders of the same type with similar due dates. The core mechanism groups orders into batches according to predefined time windows and adjusts batch sizes through splitting/merging orders to meet production line constraints, e.g., trolley capacity and changeover number. While prioritizing on-time delivery, the approach requires balancing time window granularity with economies of scale. If the time window is too large, this method may create oversized batches leading to excessive WIP inventory, whereas overly small windows increase changeover frequency. The key steps of this rule are as follows:

(1)

Order encoding and initial sorting: Map due date to time-window identifiers, convert part numbers into product codes, and sort orders in ascending order of due dates to generate an initial order queue.

(2)

Dynamic batching based on time window: Perform batching sequentially in each time window. First, combine orders for the same product in the same time window. Then, evaluate batch sizes: split oversized batches into smaller ones based on trolley capacity and defer undersized batches to the next time domain for further combination.

(3)

Resource allocation: Recalculate the due date for each batch, which is determined by the latest due date among its constituent orders. Allocate decoded orders to available production lines using the EDD rule, prioritizing lines with no changeovers.

The key parameter of this method is the length of the time window. Adjusting the time-window length optimizes the balance between batch economics and resource constraints. For short-term planning with a weekly planning horizon, we consider four rule variants with time-window lengths of 4, 6, 8, and 12 h, labeled as TW-4, TW-6, TW-8, and TW-12, respectively.

4.3.2. Dispatching Rules Based on Available Capacity

The dispatch rule based on the available capacity minimizes the changeover of production lines by aggregating orders of the same type based on a fixed capacity threshold. The core mechanism groups orders by product type and forms production batches constrained by a preset capacity threshold (e.g., 2-h production capacity). This approach reduces the number of changeovers by centralizing production and relying on finished product inventories to buffer due date variances.

(1)

Order encoding and initial sorting: Extract the product type and due date of orders, convert them into product codes, and sort orders in ascending order of due dates to generate an initial order queue.

(2)

Dynamic batching based on available capacity: Traverse the orders cyclically, dynamically fill the batch with the capacity threshold for each product type. When the capacity threshold is exceeded, generate a new batch.

(3)

Resource allocation: This step is the same as Step 3 of dispatching rules based on time-window.

The key parameter of this method is the size of the capacity threshold, which can be adjusted to optimize the balance between batch economics and resource constraints. The short-term schedule is based on a weekly planning cycle, and since the regular working time of each production line is 8 h per day, we consider four rule variants with capacity threshold sizes of 2/4/6/8 h, which are defined as AC-2, AC-4, AC-6, AC-8.

4.3.3. Hybrid Dispatching Strategy

The hybrid strategy integrates two dispatching rules: those based on the time window and based on the available capacity, with a particular focus on trolley resource constraints. In the workshop, limited trolley resources represent a bottleneck. So this strategy prioritizes the production of products that consume more bottleneck resources, specifically products with smaller single-trolley capacities.

(1)

Order encoding and initial sorting: This step is the same as Step 1 of dispatching rules based on available capacity.

(2)

Dynamic batching based on the hybrid strategy: First, divide the order queue by the time window. In each time window, aggregate half products with smaller single-trolley capacities. For the remaining half products, dynamically fill batches based on the threshold of available capacity ignoring the time window.

(3)

Resource allocation: This step is the same as Step 3 of dispatching rules based on time-window.

The hybrid strategy requires parametrization of both two dispatching rules. In the implementation, we first test which variants of the two basic rules perform optimally and then apply the settings from these variants in the hybrid strategy.

5. Experiments and Results

This section comprises four main components. First, Section 5.1, Section 5.2 and Section 5.3 respectively present the performance test results for the three levels of the HPP system. Finally, Section 5.4 discusses the overall performance of the system as a whole.

The experiments in this section are run on a personal computer equipped with an Intel Core i5-10310U CPU 1.70 GHz/2.21 GHz and 16 GB of RAM. The proposed methods are implemented with Python 3.9.4, CPLEX Studio 22.1.0, and Anylogic 8.5. CPLEX Studio 22.1.0 is used for algorithm implementation and verification in Section 5.2, and Anylogic 8.5 is used for simulation in Section 5.3.

5.1. Results of Demand Forecast

5.1.1. Training Process

During training, datasets are production data obtained from a company. The training process employs the parameters outlined in

Table 3. The training parameters of LSTM-Q network.

Parameter	Value
Total number of training epochs (L)	10,000
Mini-batch size for gradient descent (B)	8
Learning rate ($\alpha$)	0.001
Optimizer	Adam

. Figure 4 shows the loss values for each training epoch on the dataset.

Figure 4 illustrates a gradual decrease in the loss value throughout the training process, indicating that the proposed LSTM-Q network framework achieves high-quality predict performance.

5.1.2. Validation of the Proposed LSTM-Q Network Framework

To verify the efficacy of the LSTM-Q network, a predictive analysis was conducted utilizing an actual production dataset to forecast the order quantities for 53 different parts over the upcoming two-week period. The comparison between the forecasted quantities and the actual order quantities is illustrated in Figure 5. As depicted, the trend lines for the predicted order quantities and the actual order quantities of the 53 parts exhibit a high degree of alignment (with the forecasted values closely mirroring the actual values), thereby demonstrating the LSTM-Q network’s proficient predictive performance.

5.1.3. Computational Experiments and Analysis

In the realm of predictive analytics, the performance of neural networks is often benchmarked by the proximity of their forecasted outputs to actual data. For LSTM-Q, Gated Recurrent Unit (GRU) networks, and Recurrent Neural Networks (RNNs), this proximity is quantifiably measured by the ‘gap’ value—the difference between predicted order quantities and actual order quantities. A smaller gap value signifies superior predictive performance of the network.

Table 4. Comparison results of LSTM-Q, GRU, and RNN networks (mean/std).

Weeks	LSTM-Q		GRU		RNN
	Mean	Std	Mean	Std	Mean	Std
1–2	0.009995	0.000238	0.015385	0.002458	0.030377	0.004613
2–3	0.011961	0.000725	0.018611	0.003010	0.072564	0.004535
3–4	0.006272	0.001007	0.011313	0.002559	0.019162	0.011007
4–5	0.005579	0.001251	0.011230	0.002358	0.041516	0.012276
5–6	0.010553	0.003427	0.015776	0.003044	0.061825	0.001746
6–7	0.013302	0.009164	0.016156	0.003700	0.098373	0.085105
7–8	0.019472	0.017844	0.022133	0.007164	0.116105	0.168153
8–9	0.010753	0.006660	0.008158	0.000490	0.056897	0.177908

below summarizes the mean and standard deviation of the gap values for both LSTM and GRU networks, where each network was run 20 times on each test case to ensure robust statistical assessment. The mean gap value (mean) represents the average difference over the multiple runs, offering insight into the overall forecasting accuracy of the networks. Meanwhile, the standard deviation (std) provides a measure of the variability in the predictive performance across the runs, with a smaller standard deviation indicating more consistent performance.

The proposed LSTM-Q network exhibits remarkable predictive performance compared to existing methodologies. Table 4 reports the comparison results, and the ones marked in bold in this table are the optimal results. As presented in Table 4, the LSTM-Q network achieves an impressive average prediction accuracy of 98.9%, surpassing the RNN with 93.8% accuracy and GRU with 98.5%. Specifically, RNNs employed for forecasting France’s day-ahead and year-ahead load demand have demonstrated prediction accuracies ranging from 98.51% to 98.92% [73]. Additionally, Honjo et al. [74] developed a Gated Recurrent Unit (GRU) network model that achieved a prediction accuracy of 96%, representing a 15% increase over conventional forecasting methods.

Comparative analysis reveals that the LSTM-Q network consistently maintains a smaller mean prediction gap across the majority of forecasting instances, indicating its superior ability to capture intricate temporal dependencies and non-linear demand patterns inherent in order data. While the prediction stability of the LSTM-Q, GRU, and RNN models shows negligible differences, all three models demonstrate commendable robustness. Nevertheless, the LSTM-Q network’s higher average accuracy underscores its effectiveness in delivering more precise and reliable demand forecasts.

To evaluate the computational efficiency of the proposed algorithms, we measured the prediction times of LSTM-Q, GRU, and RNN across different week intervals ranging from 1–2 weeks to 8–9 weeks. As illustrated in

Table 5. Time cost comparison of neural networks (seconds).

Weeks	Time Cost
	LSTM-Q	GRU	RNN
1–2	0.002001	0.000000	0.000997
2–3	0.003008	0.002013	0.000000
3–4	0.002816	0.001998	0.000000
4–5	0.003497	0.001001	0.000000
5–6	0.001995	0.000998	0.000000
6–7	0.002508	0.000997	0.001047
7–8	0.003502	0.001508	0.000000
8–9	0.001994	0.001508	0.000000

(the best results are in bold), all three models exhibit minimal time costs, with GRU and RNN generally outperforming LSTM-Q in terms of speed. This low computational burden ensures that the models are highly suitable for real-time scenarios, enabling swift and accurate predictions without significant delays.

5.2. Results of Long-Term Planning

This section evaluates the performance of the proposed R&F algorithms in solving long-term planning models. The experiments comprises two parts. First, we randomly generate some instances of different sizes to test the performance of the proposed R&F algorithms. We also perform a sensitivity analysis to examine how the number of sub-problems affects solution quality and solving speed. Second, the proposed R&F algorithm is applied to a real-world case in a fuel tank manufacturing factory.

5.2.1. Experiments with Randomly Generated Instances

(1) 

Instance Generation

Table 6. Parameters for randomly generated instances.

Parameters	Values	Parameters	Values
Length of the planning horizon	10, 20	Time between orders	2
Number of products	5, 10, 15	Regular time production capacity (h)	720
Demand quantity	U [100, 2000], U [2000, 4000]	Overtime production capacity per shift (h)	24
Process time (s)	U [36, 72], U [72, 120]	Number of overtime shifts per period	6
Initial inventory	U [50, 400]	Unit cost of holding inventory	1
Inventory capacity	8000	Unit cost per overtime shift	1
Overtime cost coefficient (for six shifts per period)	[200, 200, 400, 400, 400, 400]	Unit cost of shortage	5

presents the key parameters for generating the instance dataset. The primary parameters include the number of products, the planning horizon, demand, process time, and initial inventory. We define three different values for the number of products to represent three problem scales, including small, medium, and large. Demand quantity, process time, and initial inventory are generated randomly using a uniform distribution (denoted by U in Table 6). We classify demand into high demand (HD) and low demand (LD), and process time into long process time (LP) and short process time (SP). Using a full factorial design, we generated 10 instances for each combination of these five parameters, resulting in a total of 240 instances (2 × 3 × 2 × 2 × 1 × 10).

In addition to the key parameters, other parameters are set based on the real case, while cost-related parameters are primarily referenced from existing literature [46,75]. The setup costs are generated as follows:

$$u_i={\displaystyle \frac{{\sum }_{i\in I}{a}_i\times Ave{D}_i\times {\theta }^2}{2}}$$

(20)

where $Ave{D}_i$ represents the average demand of product i, and $\theta$ is the time between orders (TBO).

(2) 

Performance Analysis

The performance evaluation experiments aim to validate the effectiveness of the proposed algorithms and compare the performance of the two R&F variants. We benchmark the proposed algorithms against CPLEX, with a time limit of 2 h for CPLEX and 10 min for the R&F algorithms. The number of subproblems K is set to three based on the findings in existing studies [46,76].

Table 7. The experiment results with comparisons for the problem instances randomly generated.

I	T	DT	PT	CPLEX		R&F-1			R&F-2
				Obj	CT (s)	Ave.Gap	Max.Gap	CT (s)	Ave.Gap	Max.Gap	CT (s)
5	10	LD	LP	76,405	1.78	0.89%	1.28%	4.37	0.43%	0.84%	0.63
			SP	71,525	1.71	0.81%	1.10%	5.46	0.31%	0.79%	0.74
		HD	LP	231,967	4.19	0.79%	1.00%	8.15	0.38%	0.51%	0.83
			SP	229,432	6.35	0.84%	0.95%	6.18	0.41%	0.51%	0.89
	20	LD	LP	142,430	32.24	0.90%	1.06%	115.81	0.68%	0.82%	1.62
			SP	144,425	16.36	0.86%	1.01%	65.66	0.71%	0.92%	1.62
		HD	LP	460,752	299.63	0.86%	0.93%	149.57	0.62%	0.68%	3.18
			SP	451,796	210.05	0.83%	0.88%	188.42	0.61%	0.68%	3.34
10	10	LD	LP	146,616	5.04	0.78%	0.92%	22.96	0.38%	0.54%	1.44
			SP	146,512	4.84	0.81%	0.95%	23.98	0.32%	0.61%	1.43
		HD	LP	560,553	324.68	0.70%	0.84%	238.29	0.30%	0.64%	5.78
			SP	506,798	103.32	0.88%	0.90%	54.44	0.51%	0.59%	1.92
	20	LD	LP	289,246	132.50	0.82%	0.88%	421.79	0.64%	0.71%	9.22
			SP	282,475	87.68	0.84%	0.98%	418.81	0.63%	0.78%	9.22
		HD	LP	1,008,977	7200.24	8.10%	8.98%	600.67	4.99%	6.00%	68.03
			SP	998,132	7200.22	8.97%	9.33%	600.61	6.92%	7.33%	65.01
15	10	LD	LP	217,033	15.84	0.93%	0.93%	135.10	0.32%	0.51%	2.73
			SP	220,951	13.57	0.84%	0.84%	69.34	0.29%	0.47%	3.13
		HD	LP	2,946,513	7200.44	9.37%	13.29%	436.42	9.37%	12.92%	10.57
			SP	800,897	7200.58	9.00%	9.16%	247.20	6.19%	6.54%	14.41
	20	LD	LP	429,148	7200.28	8.68%	10.57%	588.44	6.43%	8.20%	27.28
			SP	441,966	7200.30	8.51%	9.56%	572.19	6.25%	6.97%	33.43
		HD	LP	10,722,417	7200.28	1.91%	1.56%	600.90	2.68%	5.69%	238.78
			SP	1,587,924	7200.31	8.79%	9.41%	600.94	7.74%	7.96%	535.90

summarizes the experimental results, where LD and HD represent demand ranges of U [100, 2000] and U [2000, 4000], respectively, while SP and LP correspond to processing time ranges of U [36, 72] and U [72, 120], respectively. Performance metrics include the optimal objective value (denoted as Obj), which is the best value found within the time limit, computation time (CT), and the average gap (denoted as Ave.Gap) and maximum gap (denoted as Max.Gap) relative to CPLEX. The gap is calculated as follows:

$$Gap={\displaystyle \frac{Ob{j}_{CPLEX}-Ob{j}_{R\&F}}{Ob{j}_{CPLEX}}}$$

(21)

where $Ob{j}_{CPLEX}$ and $Ob{j}_{R\&F}$ are the objective values obtained by solving the MILP model with CPLEX directly and by using the R&F methods, respectively.

The experimental results in Table 7 are analyzed under two scenarios, and the best results are marked in bold. In the first scenario, CPLEX can find the optimal solution within the time limit (i.e., $CT<7200$ for CPLEX). In this case, the gap of the R&F algorithms measures their deviation from the optimal solution, with smaller gaps indicating better performance. Both proposed R&F algorithms achieve an average gap of less than 1% and a maximum gap below 2%, outperforming CPLEX in computation time. Notably, the R&F-2 algorithm demonstrates higher stability, with an average gap below 0.8% and a maximum gap under 1%.

In the second scenario, CPLEX fails to find the optimal solution within the time limit (i.e., $CT\ge 7200$ for CPLEX) and returns only feasible solutions. In this case, a positive gap indicates that the R&F algorithms obtain better solutions than CPLEX, corresponding to lower-cost production plans. Experimental results show that both R&F-1 and R&F-2 algorithms consistently outperform CPLEX in solution quality and computation time. While R&F-1 occasionally produces slightly better solutions than R&F-2, the difference is marginal. However, R&F-2 significantly outperforms R&F-1 in computation speed.

In summary, both proposed R&F algorithms efficiently solve the MILP model, generating high-quality long-term production plans. R&F-2 exhibits superior performance in terms of computation speed and stability compared to R&F-1.

(3) 

Sensitivity analysis

The existing literature demonstrates that the number of subproblems K significantly impacts both solution quality and computational efficiency in R&F algorithms [76]. Therefore, we conduct a sensitivity analysis on the number of subproblems K for the R&F-2 algorithm, which was shown to be the best-performing variant in Section 5.2.1. For the 240 instances in the test dataset, we vary K from 2 to 10 with a step size of 1, running each instance 9 times with different K values. Note that $K=1$ corresponds to solving the original problem directly using CPLEX without decomposition. The analysis is performed across three instance scales: small ($I=5$), medium ($I=10$), and large ($I=15$).

Figure 5 and Figure 6 compare the performance of the R&F-2 algorithm under different K values for various problem scales. Performance is evaluated using two metrics: (1) the average value of objective gap $Gap$ computed by Equation (21) and (2) the average value of computation time gap ${Gap}_{CT}$, calculated as

$$Ga{p}_{CT}={\displaystyle \frac{{CT}_{CPLEX}-{CT}_{R\&F}}{C{T}_{CPLEX}}}$$

(22)

where $C{T}_{CPLEX}$ and $C{T}_{R\&F}$ are the computation time spent to solve the MILP model with CPLEX directly and by using the R&F-2 method, respectively.

A positive $Ga{p}_{CT}$ indicates that R&F-2 is faster than CPLEX, with larger values representing better performance. For small-scale instances, a smaller $Gap$ indicates that the R&F-2 solution is closer to optimal. Figure 6 and Figure 7 show that R&F-2 achieves the best balance between solution quality and computation speed at $K=2$. For medium- and large-scale instances, where CPLEX sometimes fails to find optimal solutions within the time limit, a larger average $Gap$ indicates better performance. In these cases, R&F-2 performs optimally at $K=3$. Additionally, Figure 7 reveals that the computation time of the R&F-2 algorithm increases with the number of subproblems K, suggesting that excessive decomposition should be avoided. The optimal K value depends on the problem scale and should be determined based on specific instance characteristics.

5.2.2. Experiments with Real-World Instances

This case study implements the R&F-2 algorithm in a real-world fuel tank manufacturing plant. The production system comprises three parallel lines producing 53 different products. The production capacity of these lines is represented by the total available hours, which accounts for the plant’s limited capacity.

Table 8. Parameter configuration.

Parameters	Values
Number of products	53
Length of the planning horizon	4, 8, 12, 52
Unit cost of holding inventory	1
Unit cost of shortage	10
Unit cost of setup	Calculated in Equation (20)
Overtime cost coefficient	[200, 200, 400, 400, 400, 400]
Regular time production capacity (h)	720
Overtime production capacity per shift (h)	108
Inventory capacity	8000

details the parameter configuration in our mathematical model where cost-related parameters are adjusted based on experimental tuning to meet the real planning requirements.

We evaluate the performance of the R&F-2 algorithm under different planning horizons, including monthly (T = 4), two-month (T = 8), quarterly (T = 12), and annual (T = 52) plans. Each period corresponds to one week in the production planning. First, we compare the R&F-2 algorithm with the Genetic algorithm (GA) implemented in the original system. The original system employs GA with two selection strategies: roulette wheel selection (defined as GA_R) and tournament selection (defined as GA_T). The parameters for the genetic algorithm are set as follows: Population size is 100, mutation probability is 0.1, and crossover probability is 0.8. For different planning horizons (T = 4, 8, 12, 52), the maximum generations are set to [1500, 3000, 3000, 10,000], respectively. We compare the R&F-2 algorithm with the two GAs based on two metrics, including computational time (CT) and the comparison coefficient (CC) with R&F-2, calculated as the ratio of the optimal results obtained by the GAs to those obtained by the R&F-2 algorithm.

Table 9. Comparison results.

I	T	GA_R		GA_T		R&F-2
		CC	CT (s)	CC	CT (s)	CC	CT (s)
53	4	1.08	34.63	1.13	21.78	1	0.33
53	8	1.11	99.75	1.12	104.33	1	2.25
53	12	1.31	135.73	1.24	136.55	1	13.99
53	52	1.67	1589.15	1.33	1619.80	1	604.34

reports the comparison results. The higher values of the comparison coefficient for GAs indicate superior solution quality of R&F-2. The experimental results in Table 9 indicate that R&F-2 outperforms GAs in both solution quality and speed, particularly when handling large-scale problems, such as quarterly and annual planning horizons.

Second, we carefuly evaluate the performance of R&F-2 in

Table 10. Implementation results of R&F-2 algorithm for different planning horizons.

I	T	CT (s)	Ave.Inv	Ave.OT.Util	Ave.Cap.Util	OTD
53	4	0.33	868.00	27.87%	91.07%	99.61%
53	8	2.25	3352.00	66.67%	91.35%	99.31%
53	12	13.99	5877.17	79.17%	90.27%	99.88%
53	52	604.34	6989.56	77.44%	90.30%	99.87%

. Our implementation optimizes the total cost while monitoring five key performance indicators (KPIs):

(1)

The computation time $CT$ measures the efficiency of the algorithm execution. Notably, when implementing the R&F-2 algorithm, the time limit is set to 600 s based on the planning requirements of the real factory.

(2)

The average inventory level $Ave.Inv$ allows planners to evaluate the inventory status, guiding the setup of safety stock and efficient management of storage space, calculated as

$$Ave.Inv={\displaystyle \frac{{\sum }_{t=1}^T{\sum }_{i=1}^I{I}_{it}}{T}}$$

(23)

(3)

The average utilization of production capacity $Ave.Cap.Util$ measures the ratio of planned production capacity to the total available capacity, helping to assess whether the production system is appropriately loaded. $Ave.Cap.Util$ is calculated as

$$Ave.Cap.Util={\displaystyle \frac{{\sum }_{t=1}^T{\sum }_{i=1}^I{p}_i{Q}_{it}}{{\sum }_{t=1}^T{\sum }_{n=1}^N(O_{tn}{y}_{tn}+R)}}$$

(24)

(4)

The average utilization of overtime shifts $Ave.OT.Util$ reflects the proportion of overtime shifts scheduled relative to the total number of shifts available. This metric assists planners in optimizing overtime schedules and determining whether additional production lines and workers are necessary to enhance factory capacity from a long-term perspective. $Ave.OT.Util$ is calculated as follows:

$$Ave.OT.Util={\displaystyle \frac{{\sum }_{t=1}^T{\sum }_{n=1}^N{y}_{tn}}{N\times T}}$$

(25)

(5)

The average rate of on-time delivery $Ave.OTD$ indicates the proportion of orders delivered on time to the total number of orders during the whole planning horizon, and it is an important indicator to measure the service level. $Ave.OTD$ is calculated as follows:

$$Ave.OTD={\displaystyle \frac{{\sum }_{t=1}^T{\sum }_{i=1}^I{\gamma }_{it}}{{\sum }_{t=1}^T{\sum }_{i=1}^I{D}_{it}}}$$

(26)

where ${\gamma }_{it}$ presents the number of on-time delivery orders during period t.

Table 10 demonstrates that the R&F-2 algorithm efficiently generates optimized production plans for both medium-term (1 to 2 months) and long-term (quarterly, semi-annual, and annual) horizons. The computation speed is impressive, especially for production planning of less than six months. In contrast, CPLEX is already very difficult to handle for a two-month program, and the suboptimal feasible solutions obtained within the two-hour time limit are worse than solutions obtained by the R&F-2 algorithm.

The experimental results reveal the average rate of on-time delivery $Ave.OTD$ below 100%, mainly due to inventory capacity constraints. During demand fluctuation peaks, even with sufficient production capacity, inventory space limitations prevent 100% fulfillment of customer demands. Nevertheless, the achieved service level exceeds 99%, surpassing the target threshold of 95%.

Table 4 reports that the average utilization of overtime shifts $Ave.OT.Util$ mostly exceeds 50%, and the average utilization of production capacity $Ave.Cap.Util$ is above 90%. These metrics indicate inherent production capacity limitations in the current workshop configuration. Although the current workshop can handle the existing demand for products, it relies heavily on overtime shifts and support from other workshops. This indicates that the system lacks resilience to accommodate sudden demand surges or new products. From a long-term perspective, it is necessary to consider expanding production lines and workers or exploring task redistribution to other workshops to avoid additional costs due to transfer and overtime shifts.

5.3. Results of Short-Term Planning

The experiments for short-term planning use data derived from the long-term production plan of the fuel tank workshop. Specifically, the dataset consists of actual orders for a specific week, matching the planned production quantities and product types. The order data contains 2506 orders for 11 product types. The workshop comprises three parallel production lines, and products are transferred between machines on the production lines using specific trolley resources. Different product types require different types of trolleys with varying capacities.

Table 11. Product types and their corresponding trolley capacities.

Product Type	Trolley Type	Capacity
P1	Trolley 1	6
P2	Trolley 1	6
P3	Trolley 1	6
P4	Trolley 2	10
P5	Trolley 2	10
P6	Trolley 3	10
P7	Trolley 4	6
P8	Trolley 4	6
P9	Trolley 5	10
P10	Trolley 5	10
P11	Trolley 6	8

presents detailed information about the products and their corresponding trolley types.

We develop a simulation model for the fuel tank workshop, as illustrated in Figure 8, and perform simulation experiments to analyze the dispatching rules. The key evaluation metrics include the average utilization of equipments, the number of changeovers, and average inventory levels. Figure 9 reports the results of these experiments.

We first implement the variants of two basic dispatching rules, including TW-4, TW-6, TW-8, and TW-12 for rules based on time window, and AC-2, AC-4, AC-6, and AC-8 for rules based on available capacity. The results in Figure 9 demonstrate that the dispatching rules based on the time window effectively reduce the average inventory level, with the lowest inventory achieved when the time window is set to 4 h. On the other hand, the dispatching rules based on the available capacity significantly reduce the number of changeovers, with the fewest changeovers observed when the capacity threshold is set to 8 h. However, this setting results in a higher average inventory level compared to the inventory capacity of 8000.

Based on the experimental results of basic dispatching rules, we employ a hybrid dispatching strategy with the time window of 4 h and the capacity threshold of 2 h, denoted as HR-4-2. Figure 9 also presents the experimental results for this hybrid strategy. The experiments demonstrate that the hybrid strategy yields optimal results by reducing the number of changeovers while maintaining a low inventory level.

In addition, we compare the hybrid dispatching strategy HR-4-2 with the original system’s dispatching approach. In the original system, the short-term planning is generated by planners based on EDD rules and their practical field experience.

Table 12. Comparison before and after optimization.

Metric	Before Optimization	After Optimization
Average Inventory	8864	7918
Number of Changeovers	70	64
Average Utilization	72.6%	74.8%

presents the results before and after optimization. The comparison demonstrates the effectiveness of the proposed method HR-4-2, which maintains the average inventory below the inventory capacity, significantly reduces the number of changeovers, and enhances resource utilization.

5.4. Discussion of System Efficiency

Because the HPP is a whole system, we provide a discussion on the overall operational efficiency of the HPP.

Table 13. Efficiency analysis of HPP system.

Level	Ave.CT (s)	Time Consumption Details (s)
Demand Forecasting	0.002664	–
Long-term Planning	–	Monthly	Quarterly	Annual
		0.582384	17.97031	605.8001
Short-term Planning	32.4151	Dispatching Rule	Simulation Model
		0.1851	32.23

presents the average time required for updating the demand forecast, long-term plan, and short-term plan components of the proposed HPP system. In the demand forecasting layer, the forecasting process is rapid enough to allow for updates as soon as a new order arrives. This timely adjustment can enhance the iterative learning of the LSTM-Q algorithm. However, to prevent data redundancy, enterprises may choose to establish an update cycle based on actual requirements. In the long-term planning level, we detail the time spent updating the monthly, quarterly, and annual plans. The quarterly and annual plans do not require frequent updates; therefore, the monthly plan is typically revised in response to changes in demand, with updates occurring weekly. Consequently, the average time for a single execution of the entire HPP system is approximately 32.999 s.

Furthermore, the validation of the simulation primarily focuses on the short-term plan. The simulation validation experiment is conducted only at the beginning of each week, coinciding with the development of the short-term plan for the week. If demand or system conditions change during the week, only the short-term plan is updated according to the dispatching rules. Thus, without running the simulation, the processing time for the HPP system is approximately 0.770 s. It indicates that the updates for the demand forecast, long-term plan, and short-term plan can be completed in under one second.

6. Conclusions and Perspective

This paper proposes a data-driven methodology for hierarchical production planning specific to fuel tank workshops. First, we propose an LSTM-Q network for demand forecast. Experimental results demonstrate that the LSTM-Q network can effectively forecast demand in the real-world case and outperform the gated recurrent unit network. Second, we develop two R&F algorithms to solve the MCLSP-I for long-term planning based on the demand data obtained from the first level. Both R&F algorithms can efficiently generate high-quality production planning, and the R&F-2 algorithm exhibits better performance in computation speed and stability compared to the R&F-1 algorithm. Third, short-term planning uses long-term plans as input and applies simulation and dispatching rules to generate plans. The experiments compare the three proposed dispatching rules. The results indicate that the hybrid dispatching rule is optimal for the fuel tank workshop and can effectively balance the inventory level and changeover frequency.

Based on the limitations of this study, future research directions can focus on the following three points. The first direction is to enhance the demand forecast towards stochastic scenario trees. The proposed approach to demand prediction generates only a single quantity of demand for a single product and period. However, this approach can not handle more complex demand fluctuations. Future research could explore generating demand forecasts in the form of scenario trees to support stochastic programming for production planning. The second direction is to extend the deterministic mathematical model of MCLSP-I to a stochastic programming model that can generate production plans to accommodate various demand scenarios. The third direction is to enrich the dispatching rules. This paper evaluates limited dispatching rules. Future research could build a comprehensive dispatching rule library and use deep reinforcement learning to support real-time dynamic decision-making.