# Multi-objective multi-product process planning with reconfigurable machines: Exact and metaheuristic approaches

Abdelkader Mechaacha, Fayçal Belkaid, Nadjib Brahimi

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Abdelkader Mechaacha, Fayçal Belkaid, Nadjib Brahimi. Multi-objective multi-product process planning with reconfigurable machines: Exact and metaheuristic approaches. Expert Systems with Applications, 2026, 298, 10.1016/j.eswa.2025.129591. hal-05280876

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# Multi-objective multi-product process planning with reconfigurable machines: Exact and metaheuristic approaches

Abdelkader Mechaacha ID a,b, Faycal Belkaid ID a, Nadjib Brahimi ID b,*

## ARTICLE INFO

### Keywords:

Reconfigurable manufacturing system

Process planning

Product sequencing

Multi-objective optimization

Evolutionary metaheuristics

## ABSTRACT

Process planning in reconfigurable manufacturing systems usually considers a single product, this reduces the efficiency of the overall production plan when multiple products are combined. This paper tackles the Multi-Product Process Planning Problem (MPPP), optimizing both individual process plans and their sequencing. We propose a 0-1 LP model, the model is relaxed by fixing the product sequencing variables and implemented in a Normal-Boundary Intersection method (NBI-es), the method uses a function for iteratively updating $β$ values. Three metaheuristics are also developed: NSGA-II, and two variants of MOEA/D, one enhanced by Opposition-based learning (OBL). Computational experiments show that the update function enhances the performance of NBI over simple Normalized-Weighted Sum (NWS) method. Additionally, NBI-es performs better in HV metric for small size instances if it is given enough CPU times, while MOEA/D significantly outperforms NSGA-II on larger instances on most convergence and spread based metrics. OBL further enhances solution diversity for MOEA/D, albeit with less convergence. A special case of the MPPP is investigated, involving identical products: the Multi-Unit Process Planning (MUPP). An integrated approach was compared with a sequential separated approach. Results indicate that the integrated approach outperforms the separated method for smaller problem instances. Moreover, the analysis of high-quality MUPP solutions revealed a tendency towards diverse process plan combinations rather than repetitive identical ones.

## 1. Introduction

In today's dynamic market, characterized by a continuous demand for personalization, customers request a wide range of products to choose from. This makes the diversification of product portfolio an important attribute for companies aiming to gain and sustain a competitive edge (Andersen et al., 2016; Rothaermel et al., 2006). This introduces challenges both for the product's design phase and the manufacturing system design (Campos Sabioni et al., 2022). Underscoring the need for innovative solutions, such as Reconfigurable Manufacturing Systems (RMS), designed specifically to adapt to volatile markets (Koren et al., 1999).

Consequently, the challenges persist into the subsequent phases of production. Given its pivotal role in bridging product design and manufacturing, process planning is instrumental in realizing RMS's reconfiguration potential (Gu & Koren, 2018). Thus, the process planning problem

with RMS has been studied extensively in the literature (as detailed in Section 2), the majority of the papers has focused on single-product planning. An assumption that is not necessarily met in practice, especially that RMS was designed for managing a diverse product portfolio in the first place (Andersen et al., 2016).

Additionally, the cost and time associated with reconfiguring the system between successive products can represent a noticeable portion of the total production cost and time (Campos Sabioni et al., 2022). Optimizing product sequencing and their corresponding process plans is therefore crucial for achieving significant cost and time reductions.

In such a case, conventional process and production planning methods, can generate inadequately designed process and thus production plans (Musharavati & Hamouda, 2011). Furthermore, in RMS excessive reconfiguration and under-utilization of reconfiguration potential, negatively impacts product quality, increases costs, and extends production time (Khan et al., 2021). This work is motivated by Gu and Koren (2018)

<footnote>ª Manufacturing Engineering Laboratory of Tlemcen, University of Tlemcen, Tlemcen, Algeria</footnote>

<footnote>ᵇ Department of Supply Chain Management and IS, Rennes school of business, Rennes, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: abdelkader.mechaacha@univ-tlemcen.dz (A. Mechaacha), faycal.belkaid@univ-tlemcen.dz (F. Belkaid), nadjib.brahimi@rennes-sb.com (N. Brahimi).</footnote>
conclusion, stating that intelligent process planning must not only determine operation sequence for each product but also the order in which to produce these products.

In addressing this requirement, the present study deals with the Multi-product process planning problem (MPPP). The focus is on determining the process plans for multiple products and establishing the sequence for their processing within an RMS.

The concept of RMS, as described in Gu and Koren (2018), envisions a manufacturing system designed for mass customization. Koren et al. (1999) define it as manufacturing systems designed for the rapid adjustment of their production capacity and functionality across a product family. This is achieved through the rearrangement or change of components, both hardware and software, to adapt to various market situations. In contrast to the static dedicated and flexible manufacturing systems, RMS is a dynamic system with reconfigurable machine tools (RMTs) that can operate in different configurations thanks to their adjustable structures (Koren, 2010).

In manufacturing, process planning consists of converting design information into practical manufacturing steps and determining the sequence of operations. Essentially, process planning addresses two fundamental questions: “How?”-related to the execution and manufacturing methods of each operation, and “How long?”-concerning the duration required for executing operations and their sequence. The resulting process plan serves as a flexible input for different subsequent planning activities in industry, including scheduling, lot-sizing, layout planning, production planning, and other related tasks (Scallan, 2003).

However, when handling MPPP within the context of reconfigurability, a new set of challenges emerges. Specifically, understanding the interactions between different process plans in the RMS and assessing how these interconnections influence the overall performance of the production plan become critical considerations. Overall the complexity of RMS introduces an additional layer of difficulty in managing and optimizing the production process.

The MPPP problem arises in a number of contexts: In scenarios where process plans are executed as they are, typically in contexts of high variety and low quantity production. Additionally, it becomes crucial in situations where minimizing reconfiguration effort is important (Kant et al., 2020), including reconfigurations occurring during transitions from one process plan to another—a consideration often overlooked in single-product single-Unit process planning. Moreover, sequencing is important in cases where resource constraints oblige the sequential processing of products (Kant et al., 2020). Examples include limitations on operators (Single-manned workstation) or fixtures, which necessitate that only one product is processed at a time.

This paper aims at addressing the MPPP problem in a multi-objective framework, using exact and metaheuristic approaches. The exact approach is based on the Normal-Boundary Intersection (NBI-es) method involving a relaxation of a proposed 0–1 LP model, while the metaheuristics consist of three evolutionary algorithms, specifically, the Non-dominated Sorting Genetic Algorithm-II (NSGA-II), known for its effectiveness in solving process planning problems (Khan et al., 2022), and two variants of the Multi-Objective Evolutionary Algorithm based on De-composition (MOEA/D), one enhanced using Opposition-based learning (OBL) technique. MOEA/D is particularly noteworthy for its ability to operate without the computationally expensive calculation of crowding distance, a potential that warrants further investigation. Lastly, Insights into the structure of MUPP solutions obtained through the integrated approach are derived.

The rest of this paper is organized as follows: The next section presents a comprehensive literature review. Following that, we elaborate on the problem under consideration and present the mathematical model. Section 4 outlines the four proposed approaches for tackling the problem, with Section 5 being dedicated to describing the conducted experiments and interpreting the obtained results. Finally, in the last section, we provide some conclusions.

## 2. Literature review

In this section, we survey the current literature and position our work within its context. Initially, we conduct a review of various process planning papers. Subsequently, we narrow our scope to explore literature that specifically addresses process planning with RMS. Lastly, we examine the subset of literature dedicated to MPPP in the context of RMS.

### 2.1. Process planning

It is often said that two process planners will never come up with the same process plan for the same item. This inconsistency in human crafted plans, particularly affecting optimal efficiency or defined criteria, emphasizes the need for Computer Aided Process Planning (CAPP) as a solution to these issues (Chryssolouris, 2013). In their comprehensive review of CAPP, Xu et al. (2011) shed light on computerized process planning and its typical problems, which were classified into: *tool selection, setup planning, operations selection and sequencing, decision models, integration of process planning with a production system, and considerations for energy-conscious and energy-efficient CAPP*.

Luo et al. (2023) approached the problem from a mathematical standpoint, specifically addressing a flexible process planning problem. This problem involved alternative routes of operations for manufacturing a product. The planner is tasked with choosing a route, sequencing the selected operations, and assigning each operation to a machine, a tool, and a TAD. The proposed models demonstrated optimal solutions for small-scale problems and relatively good performances for larger ones.

### 2.2. RMS process planning

RMS represents a complex category of manufacturing systems, necessitating extensive expertise in production planning for efficient usage. The system's performance is inherently sensitive to changes in layout, configuration, tools, modules, and other factors, thereby impacting its overall stability. Therefore, practitioners might encounter difficulties when utilizing systems that rely on existing knowledge to navigate RMS (Khan et al., 2022).

Researchers have been looking into these new production planning challenges. Yelles-Chaouche et al. (2021) conducted a literature review on optimization problems related to RMS, including a section on process planning problems. Recognizing the crucial role of process planning in unleashing RMS potential, Khan et al. (2022) dedicated a literature review to process planning problems with RMS. Unlike traditional analyses, this study provides various classifications covering both theoretical and practical aspects. The theoretical part explores research themes and the inherent characteristics of RMS, while the practical part focuses on the type of problem and the approach used for its solution.

In their study, Youssef and ElMaraghy (2006) explored how the manufacturing system's configuration impacts the subsequent reconfiguration process. They introduced the term “reconfiguration smoothness” and presented a metric to evaluate it. This metric helps to better understand the activities associated with different levels of reconfiguration, enhancing our understanding of RMS dynamics. In a related work, Shabaka and ElMaraghy (2008) developed a model for the process planning problem. In this model, they addressed machine assignment, machine configuration, operation sequencing, operation cluster sequencing, and the allocation of tools and TADs to operations. The proposed model was solved using a genetic algorithm with the objective of minimizing the cost of machine usage, machine change, tool change and set-up change. This paper was one of the earliest works in RMS process planning with tools and TADs requirements, which are considered in our work.

Considering precedence constraints as well as TADs and tools constraints, Bensmaine et al. (2013) studied RMS process planning with machine selection. The work was among the first works that solved the
<table><caption>**Table 1**
Difference from literature.</caption><thead><tr><th>Article</th><th>Gao et al. (2025)</th><th>Stief et al. (2023)</th><th>Yazdani et al. (2022)</th><th>Gao et al. (2021)</th><th>Bensmaire et al. (2014)</th><th>Bensmaire et al. (2013)</th><th>Touzout and Benyoucef (2019)</th><th>Kazemisaboor et al. (2022)</th><th>Musharavati and Hamouda (2011)</th><th>This work</th></tr></thead><tbody><tr><td rowspan="3">Process planning combined with</td><td>Scheduling</td><td>✓</td><td>✓</td><td></td><td></td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Scheduling</td><td></td><td></td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Production planning</td><td></td><td></td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">Product type</td><td>System Design</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Single</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">Reconfiguration level</td><td>Multiple</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>System</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">Optimization</td><td>Machine</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Mono</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">Approach</td><td>Multi</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Integrated</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">Solution Method</td><td>Separated</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Exact</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>Approximate</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr></tbody></table>

problem in a multi-objective framework, where they utilized the NSGA-II algorithm, and the two objectives considered were: the total cost and total time. Building upon this work, Haddou Benderbal et al. (2018) extended the problem by introducing modular RMTs. They proposed an approach where each machine consisted of basic and auxiliary modules, allowing for diverse configurations with distinct functionalities. To further enhance the optimization process, they introduced a third objective, the maximization of the modularity index. The solution generation process involved the usage of the Archived Multi-Objective Simulated Annealing (AMOSA) metaheuristic, followed by ranking non-dominated solutions using the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS). These studies rely solely on metaheuristics and lack comparisons for instances where the true Pareto front is known, raising the challenge of efficient pareto front identification. In this paper, we propose an exact method to address this gap.

Solving the problem with exact solution approaches, Mechaacha et al. (2025) proposed a mathematical model and a mathematical based heuristic. The work compared a "one-main-decision-variable" (1MDV) model to an existing model from the literature and found that 1MDV is superior. The problem was studied in a Mono-objective context: the minimization of total production time. In this work, we extend the 1MDV model and adapt it to handle the additional complexities of MPPP and multi-objective optimization.

Quality considerations were brought into focus by Khan et al. (2021), who incorporated TADs, operations precedence, demand satisfaction, and constraints related to modular RMTs. Moreover, quality-related constraints were incorporated to evaluate the conformity and failure rates of products delivered by a given process plan. The researchers proposed a multi-objective mixed integer non-linear programming (MINLP) model, and a hybrid meta-heuristic that combined the strengths of NSGA-II and multi-objective particle swarm optimization (MOPSO).

In the same research stream, a recent work of Ameer and Dahane (2023) proposed an approach that addresses setup and process plan constraints, taking into account modular RMTs and fixtures. The approach operates in two stages: The first stage involves a heuristic for generating a setup plan for the given part, along with fixture assignments for each setup. In the second stage, a genetic algorithm (GA) is employed to identify the optimal process plan for executing the required operations, with the objective of minimizing the total cost.

The studies reviewed in this subsection all study Single-product Process Planning, reflecting the current trend in RMS process planning research. As noted by Khan et al. (2022), approximately 70% of reviewed articles address this type of the problem. However, process planning for multiple products offer a more realistic representation of manufacturing scenarios and has more potential in enhancing the efficiency of RMS. The next subsection delves into this research stream.

## 2.3. Process planning with multiple products and RMS

With the growing demand for diversified products, there is more focus on optimizing multi-product production. Therefore, the category of "integration of process planning with production systems" from Xu et al. (2011) classification (see Section 2.1), has gained increasing attention.

To address this type of problems, researchers have explored various planning issues. Similar problems to MPPP can be found in multi-product assembly lines-related works.

For instance, Kant et al. (2020) addressed the problem of forming reconfigurable machine cells, followed by sequencing the products on an assembly line. Their objective was to minimize total reconfiguration time and effort. Stief et al. (2023) addressed the simultaneous optimization of resource selection, operations and machine allocation, and sequencing. They proposed a Mixed-Integer Linear Programming (MILP) model for this problem and implemented it in a solver using the normalized weighted sum (NWS) method, which has two key limitations: it requires a pre-specified objective weights and fails to explore narrow regions of the search space with multiple non-dominated solutions, and it cannot adapt to changing computational times. The approaches presented in this paper overcome both issues and remain easy to implement, as will be demonstrated through comparison to NWS.

Table 1 summarizes the most relevant research on MPPP for RMS, along with their classifications. First it highlights the types of decisions integrated with process planning within the studied problem, in our case, product sequencing. It also considers products type, if it's identical across all planned products or varying. Reconfiguration level (system-wide or machine-specific), ours focuses on machine-level reconfiguration for RMTs. And the other three classifications are based on: optimization methods, solution approaches, and whether the methods are exact or approximate.

Several studies have explored the integration of process planning and scheduling (IPPS) problem. An IPPS problem with AGVs task assignment was solved with a variant of a genetic algorithm in Liu et al. (2023). The problem considered dedicated machines and was proven to be NP-Hard. When adding reconfigurable machines the problem becomes even more complex as additional decisions have to be made. Gao et al. (2021) proposed an integrated approach that combines IPPS decisions along with layout design for an RMS. Bensmaire et al. (2014) investigated the integrated process planning and scheduling problem with the goal of minimizing makespan. They compared integrated and separated approaches, finding that the integrated heuristic outperformed the separated method in terms of solution quality. Inspired by these findings, this research focuses on developing integrated approaches for the MPPP problem. Reconfigurable cells and human workers were both considered in IPPS by Guo et al. (2025), the authors proposed a genetic programming based heuristic. While IPPS related works are relevant to ours, the execution of operations of different products with no resource constraints makes a different type of problem.

Some researchers studied the process planning problem with product sequencing, labelling this problem as Multi-Unit Process Planning (MUPP) or Multi-part Process Planning (in the case of MUPP all the sequenced products are identical). This problem is relevant when resource constraint limits the capacity of resources to process operations of different products. The primary difference between IPPS and MPPP is that in the former, the processing of different parts is overlapped, thereby enabling the simultaneous processing of multiple parts within the system. In contrast, MPPP involves the processing of parts in a sequential order.

Touzout and Benyoucef (2019) addressed a multi-unit single-type process plan generation problem (MUPP) using three algorithms based on the NSGA-II metaheuristic employing separated product assembly lines-related works.
approaches, wherein process planning for each single product is performed independently, followed by sequencing the obtained process plans. The main limitation of these separated approaches is that they cannot guarantee the quality of the results, as no proper benchmark exists. In this work we propose a mathematical model that permits solving larger instances, and two exact approaches are proposed for identifying a maximum number of points from the true Pareto front given a fixed computational time. Additionally, integrated metaheuristic methods are proposed and validated against separated methods.

Others, such as Kazemisaboor et al. (2022), and Bensmaine et al. (2011), also adopted separated approaches to solve the MUPP problem, decomposing it into optimization and simulation phases. Using metaheuristics like NSGA-II, AMOSA, and MOPSO, Kazemisaboor et al. (2022) found NSGA-II to be superior, reinforcing its position as the most used solution approach in process planning with RMS (Khan et al., 2022). In Section 5.5 we compare separated and integrated approaches for MUPP and discuss the strengths and limitations of each. Additionally, as NSGA-II is the most adopted metaheuristic in process planning works a recent competitor has emerged, MOEA/D. Unlike NSGA-II, MOEA/D doesn't require calculating the crowding distances, making it a computationally attractive option (Zhang & Li, 2007). Investigating the effectiveness of MOEA/D for MPPP with RMS is performed in this work with comparison to NSGA-II.

In studying the multi-part flow line process planning problem, Musharavati and Hamouda (2011) proposed a modified genetic algorithm method. They considered the minimization of the total cost. The line is composed of stages organized in series and containing parallel reconfigurable machines. However, the constraints regarding the flow line studied limits the problem scope. For instance the flow line is unidirectional which limits the RMS. This limitation is further highlighted by Gu and Koren (2024), where the authors proposed using a return conveyor or a gantry on the flow line to allow for more flexibility in the parts flow. Thus appropriately representing and retrieving the potential of a true RMS. In the problem studied, we do not limit the flow of products in any direction.

Overall, MUPP works don't consider products of different types that belong to the same product family. While these works effectively represent the scalability characteristic of RMS considering multiple units, they fall short in highlighting its counterpart—convertibility, since the units are of exactly the same type. In this study, the units are of different types, thus better representing real world RMS planning problems.

The main contributions of the paper can be summarized as follows:

1. Regarding the problem specifics, to the best of our knowledge, this is the first work addressing the MPPP with RMS where products are of different types.

2. We propose efficient separated and integrated approaches, which permits meaningful comparison regarding process plan structure. Which is lacking in the literature.

3. As mentioned in Touzout and Benyoucef (2019) their MILP was not compared to metaheuristic methods because it can only solve microscopic instances. We propose an efficient 0-1 LP mathematical model which is implemented in NWS and NBI approached, along with adaptive β values update function. The methods permit the performance monitoring and validation of approximate methods for larger instances than usual works from literature.

4. While NSGA-II is the most implemented metaheuristic in process planning with RMS works (Khan et al., 2022), the method doesn't seem to be completely adequate especially due to the disruptive nature of the encoding procedure. Thus we compare NSGA-II with MOEA/D.

5. The capacity of the OBL strategy for solution diversification is tested as a replacement of the crossover as a genetic operator and for population initialization.

6. Most studies evaluating exact methods consider only cardinality and convergence metrics (C-metric and NRP). Here, we additionally assess spread-related metrics (SM and HV), with NBI and NBI-es demonstrating strong performance.

### 3. Problem description and formulation

In this section we describe the problem studied, then the mathematical model used to solve it.

#### 3.1. Problem description

We address the MPPP problem with reconfigurable machine tools (RMT). This problem involves a set of products, belonging to the same product family, to be manufactured on a predefined shop-floor. Each product comprises a distinct set of operations to be processed. The operations are linked through precedence constraints, coming from various technological and logical considerations, necessitating the completion of certain operations before others can start. Each operation within this framework requires a specific set of alternative tools and TADs that can be used for its execution.

A Tool Approach Direction (TAD) refers to the direction from which the tool approaches the workpiece during machining. There are six standard TADs (± x, ± y, ± z) which follow the Cartesian coordinate system. Additionally, two more TADs (± a) are considered for some special operations that require a 45-degree angle. Consequently, there are a total of eight distinct TADs. This TAD configuration has also been used in previous research, as seen in studies like Azab and ElMaraghy (2007), Li and McMahon (2007), Zhang and Wong (2016).

The shop-floor consists of a set of prepositioned RMTs. Each RMT has a set of usable tools in its tool magazine, and its structure is adjustable to obtain different machine configurations. In each configuration, the RMT provides one or multiple TAD for machining the workpiece. By considering the machine's TADs and available tools, we can determine the feasible operations for a given machine-configuration. Therefore, if a machine doesn't possess at least one of the alternative tools and the alternative TADs required for an operation, it cannot process that operation.

#### Notation

To formulate the problem, the following notations are used:

##### Indices

v Index of products

j Index of machines

ℓ Index of configurations

u Index of operations

q Index of tools

t Index of TADs

s Index of positions

As depicted in Fig. 1 the MPPP problem consists of two sub-problems: single product process planning, and product sequencing. At the process planning level the goal is to determine two aspects: i) the execution sequence of operations respecting the precedence constraints, and ii) the machine-configuration and the tool used for processing each operation. Meanwhile, the second sub-problem consists of sequencing the products (specifically their process plans). There are two minimization
<figure><img src="image_2.png" /><figcaption>Fig. 1. Multi-Product process planning explanation.</figcaption></figure>

objective functions: total production time and total production cost. The total production time, encompasses the processing time of the operations on the machines, the changeover time of the machines, the configuration changeover time, and the tool changeover time. Similarly, the total cost is composed of the processing cost, and the machine, configuration, and tool changeover costs.

### Parameters

<table><caption>Table 2
An MPPP Solution representation.</caption><tr><td>N</td><td>Number of products to manufacture</td></tr><tr><td>M</td><td>Number of available machines</td></tr><tr><td>T</td><td>Number of tools</td></tr><tr><td>NC<sub>j</sub></td><td>Number of available configurations for machine j</td></tr><tr><td>NOP<sub>v</sub></td><td>Number of operations of product v</td></tr><tr><td>TNOP</td><td>Total number of all operations of all products</td></tr><tr><td>MT<sub>j<sub>q</sub></sub></td><td>equals 1 if machine j can use tool q, 0 otherwise</td></tr><tr><td>CTAD<sub>j<sub>l</sub></sub></td><td>equals 1 if machine j in configuration l can machine the workpiece from TAD t, 0 otherwise</td></tr><tr><td>OPTAD<sub>v<sub>ut</sub></sub></td><td>equals 1 if operation u of product v can be executed from TAD t, 0 otherwise</td></tr><tr><td>OPT<sub>v<sub>uq</sub></sub></td><td>equals 1 if operation u of product v can be executed with tool q, 0 otherwise</td></tr><tr><td>OPP<sub>v<sub>uq</sub></sub></td><td>equals 1 if operation u' has to be processed after operation u of product v</td></tr><tr><td>CCTime<sub>j<sub>l</sub>l'</sub></td><td>Configuration change time for machine j from configuration l to l'</td></tr><tr><td>TCTime<sub>q<sub>q</sub></sub></td><td>Tool change time from tool q to q'</td></tr><tr><td>MCTime<sub>j<sub>j'</sub></sub></td><td>Machine change time from machine j to j'</td></tr><tr><td>PrTime<sub>v<sub>uq</sub>l</sub></td><td>Processing time for operation u of product v on machine j with configuration l</td></tr><tr><td>CCCost<sub>j<sub>l</sub>l'</sub></td><td>Configuration change cost per unit time for machine j from configuration l to l'</td></tr><tr><td>TCCost<sub>q<sub>q</sub></sub></td><td>Tool change cost per unit time from tool q to q'</td></tr><tr><td>MCCost<sub>j<sub>j'</sub></sub></td><td>Machine change cost per unit time from machine j to j'</td></tr><tr><td>PrCost<sub>v<sub>uq</sub></sub></td><td>Processing cost per unit time for operation u of product v on machine j</td></tr><tr><td>B</td><td>A big number</td></tr></table>

<table><thead><tr><th>Product</th><th>P2</th><th>P2</th><th>P2</th><th>P2</th><th>P1</th><th>P1</th><th>P1</th></tr></thead><tbody><tr><td>Operation</td><td>OP1</td><td>OP4</td><td>OP3</td><td>OP2</td><td>OP2</td><td>OP1</td><td>OP3</td></tr><tr><td>Machine</td><td>M2</td><td>M1</td><td>M2</td><td>M1</td><td>M2</td><td>M2</td><td>M1</td></tr><tr><td>Configuration</td><td>C3</td><td>C2</td><td>C2</td><td>C1</td><td>C2</td><td>C1</td><td>C1</td></tr><tr><td>Tool</td><td>T1</td><td>T1</td><td>T5</td><td>T3</td><td>T1</td><td>T2</td><td>T1</td></tr></tbody></table>

<table><thead><tr><th>Sets</th><th></th></tr></thead><tbody><tr><td>V</td><td>Set of all products</td></tr><tr><td>J</td><td>Set of all machines</td></tr><tr><td>L</td><td>Set of all configurations</td></tr><tr><td>U</td><td>Set of all operations</td></tr><tr><td>Q</td><td>Set of all tools</td></tr><tr><td>TAD</td><td>Set of all TADs</td></tr><tr><td>S</td><td>Set of all positions</td></tr></tbody></table>

An example solution is illustrated in Table 2. From this table we can retrieve the process plan of each product, as well as the products' sequence (P2 → P1). The table can be interpreted from left to right column by column, the first column indicates that operation OP1 of product P2 is processed first on machine M2 in configuration C3 with tool T1. Subsequently, operation OP4 is processed on machine M1, and so on.

From the same table, changeovers can be extracted; for example, there is a machine changeover between the first and second positions, switching from machine M2 to M1. Additionally, machine M2 changes both configuration and tool between positions 1 and 3, transitioning from configuration C3 to C2 and from tool T1 to T5.

## 3.2. Mathematical model

In this section, we present the problem formulation and the 0-1LP mathematical model used to solve it.

In addition to the classical indices usually found in SUPP works, we introduce the product index, employing a six-dimensional decision variable *P*. This variable is set to one if operation *u* of product *v* is processed
in position *s* by machine *j*, configuration *ℓ*, and using tool *q*. The first column of the previous example presented in Table 2 can be represented by $P_{1231}^{21} = 1$.

### Decision variables

<table><tr><td>P<sup>vu</sup><sub>sjeq</sub></td><td>equals 1 if operation u of product v is in position s executed by machine j in the configuration ℓ using tool q, 0 otherwise</td></tr><tr><td>Z<sub>jss'</sub></td><td>equals 1 if machine j is used in positions s and s', 0 otherwise</td></tr><tr><td>W<sub>sjj'</sub></td><td>equals 1 if there is a change from machine j to j' between positions s and s + 1, 0 otherwise</td></tr><tr><td>X<sub>sjeℓℓ'</sub></td><td>equals 1 if there is a change for machine j from configuration ℓ to ℓ' in position s, 0 otherwise</td></tr><tr><td>Y<sub>sjqq'</sub></td><td>equals 1 if there is a change for machine j from tool q to q' in position s, 0 otherwise</td></tr></table>

### Objective functions

$f_time$ and $f_cost$ are respectively the total time and total cost, where:

$$ \text{Minimize } f\_time = \sum_{s \in V} \sum_{v \in V} \sum_{u \in V} \sum_{j \in J} \sum_{\ell \in \mathcal{L}} \sum_{q \in Q} P_{sjeq}^{vu} \times \text{PrTime}_{vuj} \\ + \sum_{s \in V} \sum_{v \in V} \sum_{u \in V} \sum_{j \in J} W_{sjj'} \times \text{MCTime}_{jj'} \\ + \sum_{s \in S} \sum_{v \in V} \sum_{j \in J} \sum_{\ell \in \mathcal{L}} \sum_{\ell' \in \mathcal{L}} X_{sje\ell\ell'} \times \text{CCTime}_{j\ell\ell'} \\ + \sum_{s \in S} \sum_{v \in V} \sum_{j \in J} \sum_{q \in Q} Y_{sjqq'} \times \text{TCTime}_{qq'} \quad (1) $$

$$ \text{Minimize } f\_cost = \sum_{s \in V} \sum_{v \in V} \sum_{u \in V} \sum_{j \in J} \sum_{\ell \in \mathcal{L}} \sum_{q \in Q} P_{sje\ell q}^{vu} \times \text{PrTime}_{vuj\ell} \\ \times \text{PrCost}_{vuj} \\ + \sum_{s \in V} \sum_{v \in V} \sum_{u \in V} \sum_{j \in J} W_{sjj'} \times \text{MCTime}_{jj'} \\ \times \text{MCCost}_{jj'} \\ + \sum_{s \in V} \sum_{v \in V} \sum_{j \in J} \sum_{\ell \in \mathcal{L}} \sum_{\ell' \in \mathcal{L}} X_{sje\ell\ell'} \times \text{CCTime}_{j\ell\ell'} \\ \times \text{CCCost}_{j\ell\ell'} \\ + \sum_{s \in V} \sum_{v \in V} \sum_{j \in J} \sum_{q \in Q} Y_{sjqq'} \times \text{TCTime}_{qq'} \\ \times \text{TCCost}_{qq'} \quad (2) $$

### Constraints

$$ Z_{jss'} \ge 1 - \sum_{s''=s+1}^{TNOP} \sum_{v \in V} \sum_{u \in V} \sum_{\ell \in \mathcal{L}} \sum_{q \in Q} P_{s''je\ell q}^{vu} \\ + \sum_{v \in V} \sum_{u \in V} \sum_{\ell \in \mathcal{L}} \sum_{q \in Q} (P_{sje\ell q}^{vu} + P_{s'je\ell q}^{vu}) - 2, \quad (3) $$

$$ \forall j \in \mathcal{J}, \quad s = 1, \dots, TNOP - 1, \quad s' = s + 1, \dots, TNOP. $$

$$ W_{sjj'} \ge \sum_{v \in V} \sum_{u \in V} \sum_{\ell \in \mathcal{L}} \sum_{q \in Q} P_{sje\ell q}^{vu} + P_{s+1j'\ell q}^{vu} - 1 \quad (4) $$

$$ \forall s = 1, \dots, TNOP - 1, \quad j, j' = 1, \dots, M. $$

$$ X_{sje\ell\ell'} \ge Z_{jss'} + \left( \sum_{v \in V} \sum_{u \in V} \sum_{\ell \in \mathcal{L}} P_{sje\ell q}^{vu} + P_{s'je\ell' q}^{vu} \right) - 2 \quad (5) $$

$$ \forall s, s' = 1, \dots, TNOP, \quad j = 1, \dots, M, \quad \ell, \ell' = 1, \dots, NC_j. $$

$$ Y_{sjqq'} \ge Z_{jss'} + \left( \sum_{v \in V} \sum_{u \in V} \sum_{\ell \in \mathcal{L}} P_{sje\ell q}^{vu} + P_{s'je\ell' q'}^{vu} \right) - 2 \quad (6) $$

$$ \forall s, s' = 1, \dots, TNOP, \quad j = 1, \dots, M, \quad q, q' = 1, \dots, T. $$

$$ \begin{gathered} \mathrm{OP}_{vuu'} + \sum_{j \in \mathcal{J}} \sum_{\ell \in \mathcal{L}} \sum_{q \in Q} P_{sje\ell q}^{vu} + P_{s'je\ell' q}^{vu'} \le 2 \\ \forall s = 2, \dots, TNOP, \quad s' = 1, \dots, s, \quad u, u' = 1, \dots, NOP_v, \quad v = 1, \dots, N. \end{gathered} \quad (7) $$

$$ \begin{gathered} P_{sje\ell q}^{vu} \le \sum_{\ell \in \mathcal{L}} \mathrm{CTAD}_{j\ell} \times \mathrm{OPTAD}_{vut} \\ \forall s \in S, \quad u \in V, \quad v \in V, \quad \ell \in \mathcal{L}, \quad q \in Q, \quad j \in \mathcal{J}. \end{gathered} \quad (8) $$

$$ \begin{gathered} P_{sje\ell q}^{vu} \le MT_{jq} \times OPT_{vjq} \\ \forall s \in S, \quad v \in V, \quad u \in V, \quad \ell \in \mathcal{L}, \quad q \in Q, \quad j \in \mathcal{J}, \quad u \in V. \end{gathered} \quad (9) $$

$$ \begin{gathered} \sum_{j \in \mathcal{J}} \sum_{\ell \in \mathcal{L}} \sum_{u \in V} \sum_{q \in Q} (P_{s-1j\ell q}^{vu} - P_{sje\ell q}^{vu} + 1) \times B \ge \sum_{j \in \mathcal{J}} \sum_{\ell \in \mathcal{L}} \sum_{u \in V} \sum_{q \in Q} \sum_{s'=1}^{s-2} P_{s'je\ell q}^{vu} \\ \forall s = 2, \dots, TNOP. \end{gathered} \quad (10) $$

$$ \sum_{s \in S} \sum_{j \in \mathcal{J}} \sum_{\ell \in \mathcal{L}} \sum_{q \in Q} P_{sje\ell q}^{vu} = 1 \quad \forall u \in V, \quad v \in V. \quad (11) $$

$$ \sum_{v \in V} \sum_{u \in V} \sum_{j \in J} \sum_{\ell \in \mathcal{L}} \sum_{q \in Q} P_{sje\ell q}^{vu} = 1 \quad \forall s \in S. \quad (12) $$

$$ \begin{gathered} P_{sje\ell q}^{vu}, Z_{jss'}, W_{sjj'}, X_{sje\ell\ell'}, Y_{sjqq'} \in \{0, 1\} \\ \forall s \in S, \quad v \in V, \quad u \in V, \quad \ell \in \mathcal{L}, \quad q \in Q, \quad j \in \mathcal{J}. \end{gathered} \quad (13) $$

The objective function (1) is the total production time of all the products which includes the processing times of operations, as well as the changeover times for machines, configurations, and tools. The second objective function (2) represents the total production cost which is calculated as the multiplication of times by costs per time. The total cost is the sum of the processing cost of operations of the products, as well as the changeover costs for machines, configurations, and tools.

Constraints (3) associate, through the Z variable, each machine usage with its immediate subsequent usage position $s'$. Constraints (4) indicate when there is a machine changeover between positions $s$ and $s+1$. Constraints (5) and (6) specify the occurrence of configuration and tool changeovers, respectively. Constraints (7) indicate that an operation is only processed when its predecessors have been processed before. Constraints (8) ensure that each machine configuration must possess at least one TAD required to process the operations assigned to it. Constraints (9) ensure that each machine should have the tool required for machining the operations assigned to it. Constraints (10) ensure that products are sequenced one after the other. It gathers operations of the same product together in the sequence so that no operation of a different product could be scheduled inside another product's operations sequence. Constraints (11) guarantee that every operation of every product $v$ is processed exactly once. Constraints (12) state that only one operation of a product is executed at a given time. Constraints (13) are integrality constraints.

As seen from the equations, the two objectives are related as the total cost is computed by multiplying the time by the cost per unit time, thereby establishing a direct relationship between the two objectives. This relationship between the two objectives suggests that there's no conflict between them and instead they are in harmony, meaning that an improvement in one objective leads also to an improvement in the other Chugh et al. (2023). However, the level of conflict between the two objectives is contingent on the values of the time and cost parameters. For instance if all cost parameters are equal, the difference between solutions depends solely on the first objective function (total production
<table><caption>**Table 3**
Details on the mathematical model size and runtime per instances sizes.</caption><thead><tr><td>Instance</td><td>Decision variables</td><td>Constraints</td><td>Runtime (sec)</td></tr></thead><tbody><tr><td>2.8_2</td><td>34,256</td><td>129,707</td><td>2.8</td></tr><tr><td>3.15_5</td><td>236,037</td><td>899,159</td><td>500</td></tr><tr><td>5.25_5</td><td>644,511</td><td>2,561,977</td><td>&gt;3600</td></tr></tbody></table>

time). In this scenario, there is no conflict. When the cost per time parameters are generated randomly with a positive variation (determined through experimental tests), the two objectives will be in conflict for most instances, which will be further discussed in the Section 5.

### 3.3. Model complexity

The MPPP with RMS is NP-hard since it is a general case of the single-product problem which is proven to be NP-hard (Khan et al., 2021; Mechaacha et al., 2025; Touzout & Benyoucef, 2019). The proof uses the reduction from the travelling salesman problem which is known to be NP-hard, as operations would be equivalent to cities that have to be visited only once, without consideration of configuration, tools, and TADs related constraints. The MPPP version adds an additional layer of difficulty by grouping operations into distinct products and requiring the sequencing of both operations and products.

In the model *P* is the main decision variable, while the remaining three are auxiliary variables. The constraints can be categorized into three types, Constraints (3)–(6) are used to compute the auxiliary variables, while constraints (7)–(9) impose the technical requirements that restrict each operation to eligible machines, configurations, and tools. Lastly, constraints (10) enforce the grouping of operations belonging to the same product in the sequence, without limiting the order in which the products can be sequenced.

Table 3 reports the total number of decision variables and constraints, as well as the computational time needed to solve each instance to optimality. The table quantifies how the size of the 0-1LP grows (and how long it takes to solve) as we move from small to larger test instances. For the smallest instance (2.8_2) composed of 2 products, 8 operations, and 2 machines, the model has about 34 000 decision variables and 130 000 constraints, and the solver reaches optimality in under 3 seconds. Increasing the instance to 3.15_5 pushes the model to a solve time of around 500 seconds. The largest instance (5.25_5) cannot be solved within a one-hour time limit. Therefore, exact solution approaches are not a good choice for solving large size instances of the problem and we have opted for heuristic and evolutionary approaches.

## 4. Solution approaches

This section describes the exact and approximate solution approaches proposed to solve the MPPP problem. In the first subsection, we present the Normal Boundary intersection based algorithm (NBI-es). Next, we present the coding procedure, and the three evolutionary metaheuristics NSGA-II, MOEA/D, MOEA/D-OBL. In the last subsection we present an illustrative example.

### 4.1. Normal-boundary intersection based algorithm (NBI-es)

The Normal Boundary Intersection (NBI) approach produces non-dominated solutions for the multi-objective problem through solving a set of single-objective sub-problems. NBI allows for searching for non-dominated solutions in different regions of the Pareto front by adjusting its parameters (β variables). We introduce two modifications: first, we use a function to update β variables in each iteration based on the available CPU time and the time taken to solve the previous sub-problems, and second, we use a strategy that we name Enumerate-and-solve, where we fix the products sequence to solve a relaxed version of the problem

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** The NBI-es algorithm flowchart.</figcaption></figure>

proposed in Section 3. The objective is to ensure that NBI searches in uniformly distributed regions as much as possible, thus efficiently use the available CPU time.

Introduced by Das and Dennis (1998) the NBI method demonstrated its independence from the relative scales of the objective functions, and proved to be efficient in generating an evenly distributed set of points in the Pareto set when provided with an evenly distributed set of parameters. This is a notable advantage compared to the popular weighted sum method, which lacks this property (Das & Dennis, 1998). Furthermore, the NBI method allows for an effective solution search control by adjusting the β vector, where varying values lead to exploration of different regions of the search space. This attribute makes it easy to implement and also to control the search process.

The method comprises three key phases: i) Initially, the ideal point is determined by minimizing each objective individually. ii) Subsequently, the ideal point is utilized to construct a payoff matrix. iii) Lastly, a new sub-problem is formulated and a new real variable *t* is introduced. The new objective becomes the maximization of *t*.

Fig. 2 depicts the NBI-es approach. Initially, the two anchor points (extremes) are determined by solving the two single objective sub-problems (minimizing total time and minimizing total cost). Then, we formulate the sub-problem and calculate the parameters $β_{1t}$ and $β_{2t}$ using Eqs. (14) and (15), with the initial values for the first iteration set
<figure><img src="image_4.png" /><figcaption>Fig. 3. Example of the decoding procedure for the vector of products.</figcaption></figure>

to $\beta_{10} = 0$ and $\beta_{20} = 1$. i is the index of iteration (for the first iteration i = 0).

$$ \beta_{1i} = \beta_{1i-1} + \frac{\beta_{2i-1}}{\left[ \frac{\text{time\_left} \times (i+2)}{\text{total\_time} - \text{time\_left}} \right] + 1} \quad (14) $$

$$ \beta_{2i} = 1 - \beta_{1i} \quad (15) $$

To determine the two anchor points, the mathematical model is implemented in a solver. And after updating $\beta$ variables all possible product sequences are enumerated, and for each sequence, the product sequence variables are fixed within the model by setting $\sum_{j \in J} \sum_{\ell \in L} \sum_{q \in Q} \sum_{u \in U} P_{1j\ell q}^{\nu u} = 1$ for product $u$ in the first position, a different product $u'$ in the second, and so on. The solver is then launched to identify the optimal process plans for that specific sequence. This process is repeated for all generated sequences. This method can be used only when the number of products is small, as $N!$ grows quickly with increasing $N$, making enumeration and solving for each sequence infeasible for large values of $N$.

## 4.2. Evolutionary metaheuristics for the MPPP

In this subsection we present the three evolutionary metaheuristics as well as the encoding procedure used to code solutions.

### 4.2.1. Chromosome encoding

The solution representation significantly impacts both the final solution quality and the approaches that can be used to obtain it. Similarly, the choice of the solution representation for a metaheuristic plays a crucial role, as it directly impacts its performance (Gendreau et al., 2010). This importance is amplified in our case due to the numerous solution alternatives, together with the difficulty of generating feasible ones without violating the constraints. For example, a simple permutation of two operations can lead to an infeasible solution due to violating precedence constraints. Additionally, implementing a repair heuristic for the generated infeasible solutions is costly in terms of CPU, and even designing genetic operators to ensure that generated neighbouring solutions are always feasible is not straightforward (Shabaka & ElMaraghy, 2008).

Given this context, (Shabaka & ElMaraghy, 2008) proposed a continuous domain variable representation, which can be decoded to generate feasible solutions. The proposed encoding system, has found application in various subsequent works such as Bensmale et al. (2013), Haddou Benderbal et al. (2018). However, the original system doesn't account for product indices, since it was designed for SUPP problems, therefore we enhance their encoding approach by adding the product's index. Hence, a new row is added to the matrix.

Table 4 illustrates a coded string representing a process plan. The solution is presented in a matrix of dimensions $5 \times TNOP$, where $TNOP$ is the sum of the number of operations for all products. This matrix is filled with real values ranging between 0 and 1. After decoding, the table is transformed into a feasible process plan resembling the one depicted in Table 2. The decoding process is done from left to right, line by line, employing a fundamental principle: all potential alternatives that can be put in a cell are enumerated in a vector. Subsequently, the

<table><caption>**Table 4**
A MPPP Solution representation with P2 processed before P1.</caption><thead><tr><th>Product</th><th>0.71</th><th>0.28</th><th>0.38</th><th>0.89</th><th>0.25</th><th>0.11</th><th>0.46</th></tr></thead><tbody><tr><td>Operation</td><td>0.16</td><td>0.92</td><td>0.66</td><td>0.28</td><td>0.47</td><td>0.42</td><td>0.23</td></tr><tr><td>Machine</td><td>0.65</td><td>0.52</td><td>0.17</td><td>0.51</td><td>0.92</td><td>0.10</td><td>0.42</td></tr><tr><td>Configuration</td><td>0.81</td><td>0.36</td><td>0.56</td><td>0.11</td><td>0.80</td><td>0.26</td><td>0.73</td></tr><tr><td>Tool</td><td>0.03</td><td>0.25</td><td>0.52</td><td>0.31</td><td>0.29</td><td>0.17</td><td>0.62</td></tr></tbody></table>

continuous value in the cell to be decoded is multiplied by the number of alternatives (the vector length). The resulting value is then rounded to the nearest higher integer, serving as an index to retrieve the corresponding element from the vector. This procedure is repeated for each subsequent cell.

Fig. 3 illustrates the decoding procedure for the first line of the example presented in Table 4. Possible alternatives are enumerated in a vector. In this example, products 1 and 2 can be placed in the first cell. With two possible alternatives, the value inside the cell, which is 0.71 in this case, is multiplied by the number of alternatives 2. The resulting number is then rounded to its ceiling. The rounded value serves as an index to extract the product to be placed in that cell, which, in this instance, is 2. Since product 2 has 4 operations, all the 4 subsequent cells will be filled with 2 to ensure that no operation from another product is processed before all operations of product 2 are processed.

This procedure is repeated through all lines of the matrix to extract the operations, machines, configurations, and tools.

### 4.2.2. Genetic operators

Before presenting the evolutionary metaheuristics, we first introduce the crossover and mutation operators, which are essential for balancing exploitation and exploration within the search space. As illustrated in Fig. 4, we employ three different operators: i) a one-point crossover: a random index is chosen as the crossover point, the parent chromosomes are split at this point, and offsprings are generated by combining the leading segment of one parent with the trailing segment of the other. ii) Opposite point: is a concept from opposition based learning (Mahdavi et al., 2018), where for each real-valued gene $x$, its “opposite” is computed as $x_{opp} = 1 - x$. iii) Mutation consists of randomly altering randomly selected genes of the parent to generate a child, the number of elements to change is set according to a predetermined mutation ratio. For selection, we use the tournament selection strategy where 4 randomly selected chromosomes are put in pairs and compared in a tournament style and the best is selected, this process is repeated to select the second parent. The OBL-based strategy is particularly well-suited for this problem because the encoding procedure permits the generation of an opposite and different solution, while keeping each gene within [0, 1], thus ensuring a feasible process plan.

### 4.2.3. Non-dominated sorting genetic algorithm (NSGA-II)

NSGA-II, developed by Deb et al. (2002), has been extensively used in solving multi-objective optimization problems. It has particularly proved its efficiency in the context of process planning with RMS
<figure><img src="image_5.png" /><figcaption>Fig. 4. Crossover, opposite point, and Mutation operators used in the evolutionary approaches.</figcaption></figure>

problems (Khan et al., 2022). The method involves a non-dominated sorting and crowding distance mechanism to produce a set of well-distributed non-dominated solutions, starting with an initial population. The flowchart for the implemented NSGA-II algorithm is presented in Fig. 5. The algorithm builds on the NSGA-II framework and incorporates the encoding/decoding procedure. This encoding both facilitates generating an initial feasible population and enables the use of crossover and mutation operators. However, in some cases the mutation operator may not generate a different child solution from the mutated parent. Specifically, if the selected cell for mutation has a single alternative (e.g., the last operation in the sequence) or the mutation's variation doesn't change the element selected from the alternatives vector. This causes the propagation of identical parent solutions within the population which limits the pool of population in parent selection and thus limits the diversity of solutions. To address this, a filter is employed to remove duplicate solutions from the child population as indicated in Fig. 5.

### 4.2.4. Multi-objective evolutionary algorithm based on decomposition (MOEA/D and MOEA/D-OBL)

Introduced by Zhang and Li (2007), MOEA/D has emerged as a powerful competitor to other evolutionary algorithms such as NSGA-II (Li & Zhang, 2008; Zhang & Li, 2007). This metaheuristic operates by decomposing the multi-objective problem into single-objective sub-problems, assigning different weights to the objectives for each solution in the population. It aims at simultaneously optimizing multiple conflicting objectives, generating a diverse set of non-dominated solutions that capture trade-offs between these objectives. Various decomposition approaches exist such as the weighted sum, Tchebychev and boundary intersection methods (see Zhang & Li, 2007). In MOEA/D algorithms, we adopt the Tchebychev decomposition, ranking solutions by their maximum normalized distance to a reference point.

The two MOEA/D variants are depicted in Fig. 6, with shared steps highlighted in red, MOEA/D-OBL in green and MOEA/D in blue. Both begin with distinct initialization strategies but employ NSGA-II's non-dominated sorting and crowding-distance procedures to ensure a diverse initial population, which is important for MOEA/D, where each parent represents a subproblem defined by a weight for each objective. Another important difference is in *T* which represents the number of solutions in a group of neighbours to a parent, from which we select parents for crossover to generate offsprings, and then the child is compared against all *T* neighboring parents (sub-problems) rather than only its own parent *j*, additionally, MOEA/D doesn't imply the K-way tournament selection because the limited parent pool (*T*) prevents the selection of certain

parents. Finally, as mutation can produce duplicate solutions, a filtering step is applied before updating the overall non-dominated solutions.

### 4.3. Illustrative example

We used the example presented in Youssef and ElMaraghy (2006) as a practical illustration. This example involves two parts, ANC-101 and its variant ANC-90, consisting of 20 and 12 operations, respectively. These operations can be processed on three machines: a reconfigurable horizontal milling machine (M1) with five configurations, a reconfigurable drilling press (M2) with two configurations, and a reconfigurable boring machine (M3) with a single configuration. Additionally, there are ten available tools for use. For more comprehensive information regarding this example, see Youssef and ElMaraghy (2006) (Fig. 7).

Table 5 presents the 7 obtained non-dominated solutions for the illustrative example. As can be seen, solution 1 has a total time of 1647.5 (time units) and a total cost of 898.92 (cost units). Only machine 1 was selected with configurations 2, 4, and 5. This solution achieves the best total time but other solutions offer better performance in terms of cost minimization. No machine changeovers occurred since only one machine was used, whereas the configuration of M1 has been changed 4 times, and tools were changed 17 times. In all 7 solutions, part ANC-101 is processed before part ANC-90. The operations sequences for both parts show minimal differences across the solutions.

The second column of Table 5 presents the machines and configurations used. Using machine M1 in configuration C5 is unavoidable because it is the only setup capable of machining features F11 and F12 of ANC-101. However, this configuration processes operations slowly. In contrast, configuration C4 is the fastest but also the most expensive due to its requirement for four spindles.

## 5. Numerical experiments

This section outlines first the experimental tests generation and the parameter calibration for the two proposed evolutionary metaheuristics. Next, we compare NBI with update $\beta$ function to NWS and NBI-es under different CPU time scenarios for small size instances, and we compare the performance of MOEA/D and NSGA-II within these scenarios. We then conduct a comparative analysis between the two metaheuristics, focusing on large size instances. Finally, we apply MOEA/D to Multi-Unit Process planning (MUPP), offering valuable insights.

All the algorithms in this section were coded in Java programming language and the NBI models are solved using IBM CPLEX version 12.6, the tests are conducted on a PC with a 2.9 GHz Intel Pentium processor and 2GB RAM.
<table><caption>**Table 5**
Results obtained for solving the illustrative example.</caption><thead><tr><th>N</th><th>Selected machines configurations</th><th>(Total time; Total cost)</th><th>Machine changeover</th><th>Configuration changeover</th><th>Tool changeover</th><th>Total changeovers</th></tr></thead><tbody><tr><td>1</td><td>M1C2 M1C4 M1C5</td><td>(1647.5; 898.92)</td><td>0</td><td>4</td><td>17</td><td>21</td></tr><tr><td>2</td><td>M1C2 M1C4 M1C5 M2C4</td><td>(1657; 892.43)</td><td>1</td><td>4</td><td>16</td><td>21</td></tr><tr><td>3</td><td>M1C2 M1C4 M1C5 M2C4</td><td>(1672; 884.83)</td><td>1</td><td>4</td><td>17</td><td>22</td></tr><tr><td>4</td><td>M1C2 M1C4 M1C5 M2C4</td><td>(1674; 722.98)</td><td>1</td><td>3</td><td>16</td><td>20</td></tr><tr><td>5</td><td>M1C2 M1C4 M1C5 M2C4</td><td>(1686; 714.39)</td><td>1</td><td>3</td><td>17</td><td>21</td></tr><tr><td>6</td><td>M1C2 M1C4 M1C5 M2C4</td><td>(1706; 693.56)</td><td>1</td><td>4</td><td>17</td><td>22</td></tr><tr><td>7</td><td>M1C2 M1C4 M1C5 M2C4 M3C1</td><td>(1738; 679.14)</td><td>3</td><td>4</td><td>16</td><td>23</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>**Fig. 5.** NSGA-II flowchart.</figcaption></figure>

## 5.1. Experimental setup

Because of the lack of benchmark instances in the literature on process plan generation problem with RMS, we conducted experiments using randomly generated instances. A configuration instance is identified by the total number of products *N*, number of machines *M*, and the total number of operations *TNOP*, denoted as $(N_{TNOP}, M)$. For each instance of a configuration, we randomly distribute the number of operations (*TNOP*) across the *N* products. For example, an instance of (2_20_3) involves two products (*N* = 2), 20 operations (*TNOP* = 20), and three machines (*M* = 3). A random distribution of *TNOP* can be: 8 operations for product 1 (*NOP*₁ = 8) and 12 operations for product 2 (*NOP*₂ = 12).

To ensure consistency with instances in other works (Had-dou Benderbal et al., 2018; Mahmoodjanloo et al., 2020; Shabaka & ElMaraghy, 2008; Vahedi-Nouri et al., 2022; Youssef & El-Maraghy, 2006), we generated the following parameters in a uniform distribution: *PrTime* ∈ [100, 800]; *TCTime* ∈ [15, 50]; *CCTime* ∈ [10, 50]; *MCTime* ∈ [5, 64]; *PrCost* ∈ [0.1, 0.8]; *TCCost* ∈ [0.01, 0.04]; *CCCost* ∈ [0.01, 0.04]; *MCCost* ∈ [0.05, 0.1]; *NC* ∈ [1, 5]; *T* = 10; Moreover, we set the average machine-configuration combinations capable of processing an operation as 20%.

## 5.2. Parameter tuning

The performance of metaheuristics depends on the values assigned to their parameters. Even a well-designed metaheuristic may underperform if its parameters are not setup correctly. Thus, in this subsection we calibrate the parameters of the two evolutionary metaheuristics.

The performance of developed algorithms in Multi-objective optimization can be measured using different performance indicators, Audet et al. (2021) classify these indicators in four categories: Cardinality indicators. Convergence-based indicators, Distribution and spread-based indicators, and metrics that combine both convergence and diversity (Audet et al., 2021). Here we use the Hypervolume (HV) as the performance indicator also named Hyperarea or S-metric, since it captures both convergence and diversity. It represents the area of the objective space covered between the solution points and a reference point *Z_ref*. In our case, we set *Z_ref* as *Z_max*, representing the maximum value of objective *i* from the reference set (Audet et al., 2021; Talbi, 2009).

Initially, we identified the parameters requiring calibration for each metaheuristic, as outlined in Table 6. The two metaheuristics share common parameters—population size, crossover probability, and mutation ratio—MOEA/D has an additional parameter, *T*. Each parameter has three distinct levels, as detailed in Table 6.

For NSGA-II, the number of possible parameter combinations is 27 (3³), while for MOEA/D, it is 81. Testing all these combinations is computationally infeasible, so we use the Taguchi method (Phadke, 1995). An orthogonal array *L*₉ is used to identify 9 tests with different parameter combinations for each metaheuristic. These tests were conducted on three generated instances (2_10_3, 2_15_2, and 3_15_5), and each test was repeated 10 times for reliability. To measure the impact of the parameters variation on HV we compute the Signal-to-Noise ratio (SNR) (Phadke, 1995). The results are depicted in Fig. 8. The most effective
<figure><img src="image_7.png" /><figcaption>Fig. 6. MOEA/D (in red and green) and MOEA/D-OBL (in red and blue) algorithms flowchart.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. The two example parts ANC-101 and its variant ANC-90 (Youssef & ElMaraghy, 2006).</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>(a): NSGA-II instance 2_10_3</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(d): MOEA/D instance 2_10_3</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(b): NSGA-II instance 2_15_2</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(e): MOEA/D instance 2_15_2</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(c): NSGA-II instance 3_15_5</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(f): MOEA/D instance 3_15_5</figcaption></figure>

<caption>**Fig. 8.** Plot of SNR variation as function of different parameters of NSGA-II and MOEA/D.</caption>

parameter settings are those associated with a high SNR. In subfigures a, b, and c of Fig. 8 a crossover probability of 0.5 yielded the highest SNR for the second and third instances, and ranking second in the first instance. Consequently, we opted for this value (0.5). Similarly, a mutation ratio of 0.5 proved to be good, and a population size of 40 demonstrated good SNR levels. The results of MOEA/D tests are illustrated in subfigures d, e, and f of Fig. 8. Best parameter values for MOEA/D were determined to be 20 for population size, 0.1 for crossover probability, 0.2 for the mutation ratio, and we set T at 5 as best parameters.

## 5.3. Comparing the solution approaches for small size instances with different CPU times

In this subsection, we evaluate the performance of the two evolutionary metaheuristics NSGA-II and MOEA/D with their ideal parameter values and NBI-es with two simple approaches that execute the mathematical model presented in Section 3 directly, namely NBI with update function and NWS, under different computational times for small size instances.

<table><caption>**Table 6**
Table of parameters and their levels.</caption><thead><tr><td>Criteria</td><td>Crossover Probability</td><td>Mutation Ratio</td><td>Population size</td><td>T</td></tr></thead><tbody><tr><td>Level 1</td><td>0.9</td><td>0.5</td><td>100</td><td>20</td></tr><tr><td>Level 2</td><td>0.5</td><td>0.2</td><td>40</td><td>10</td></tr><tr><td>Level 3</td><td>0.1</td><td>0.05</td><td>20</td><td>5</td></tr></tbody></table>

The NWS method is implemented as described in Stief et al. (2023), with the definition of 12 combinations of weights of the two objectives. As for NBI and NBI-es, the method starts by defining the two anchor points to be able to proceed with the normalized values. For NBI, the method does not need a pre-specified number of points and uses the update function defined in Section 4.1 to update the values of $\beta$.

We propose a formula for computing CPU time that ensures a fair comparison between instances sizes and between solution methods. The computational time is computed as: $TNOP \times M \times C_{cpu}$ (in seconds), where $C_{cpu}$ is a constant integer. By varying the values of $C_{cpu}$
<caption>**Table 7**</caption>

The Hypervolume results for the proposed approaches (HV gap percentages to the reference front).

<table><thead><tr><th rowspan="2">Configuration</th><th rowspan="2">C<sub>cpu</sub></th><th rowspan="2">NWS Gap</th><th rowspan="2">NBI Gap</th><th rowspan="2">NBI-es Gap</th><th colspan="2">NSGA-II</th><th colspan="2">MOEA/D</th></tr><tr><th>Gap</th><th>CV</th><th>Gap</th><th>CV</th></tr></thead><tbody><tr><td rowspan="6">2_8_2</td><td>1</td><td>8.59</td><td>8.59</td><td><strong>7.41</strong></td><td>15.43</td><td>0.15</td><td>14.71</td><td>0.18</td></tr><tr><td>2</td><td>5.97</td><td>3.57</td><td><strong>1.57</strong></td><td>8.47</td><td>0.12</td><td>15.37</td><td>0.22</td></tr><tr><td>3</td><td>4.80</td><td>3.57</td><td><strong>1.48</strong></td><td>7.64</td><td>0.08</td><td>14.52</td><td>0.17</td></tr><tr><td>6</td><td>4.80</td><td>2.98</td><td><strong>0.03</strong></td><td>4.70</td><td>0.06</td><td>11.54</td><td>0.14</td></tr><tr><td>12</td><td>4.80</td><td>1.39</td><td><strong>0.04</strong></td><td>3.98</td><td>0.06</td><td>10.47</td><td>0.16</td></tr><tr><td>1</td><td>27.33</td><td>27.33</td><td>22.99</td><td>14.82</td><td>0.15</td><td><strong>13.11</strong></td><td>0.11</td></tr><tr><td rowspan="6">2_10_2</td><td>2</td><td>11.46</td><td>11.06</td><td><strong>0.69</strong></td><td>7.81</td><td>0.08</td><td>9.93</td><td>0.08</td></tr><tr><td>3</td><td>1.87</td><td>1.83</td><td><strong>0.14</strong></td><td>9.09</td><td>0.09</td><td>9.76</td><td>0.09</td></tr><tr><td>6</td><td>0.92</td><td>0.73</td><td><strong>0.10</strong></td><td>7.74</td><td>0.08</td><td>8.31</td><td>0.07</td></tr><tr><td>12</td><td>0.92</td><td>0.05</td><td><strong>0.04</strong></td><td>5.94</td><td>0.07</td><td>6.64</td><td>0.08</td></tr><tr><td>1</td><td>29.22</td><td>19.41</td><td>19.33</td><td><strong>1.87</strong></td><td>0.03</td><td>5.83</td><td>0.10</td></tr><tr><td>2</td><td>18.26</td><td>18.26</td><td>19.34</td><td><strong>2.52</strong></td><td>0.06</td><td>6.10</td><td>0.09</td></tr><tr><td rowspan="6">2_10_3</td><td>3</td><td><strong>0.13</strong></td><td>0.46</td><td>0.17</td><td>2.04</td><td>0.06</td><td>6.56</td><td>0.10</td></tr><tr><td>6</td><td>0.13</td><td><strong>0.03</strong></td><td>0.17</td><td>1.96</td><td>0.06</td><td>3.14</td><td>0.07</td></tr><tr><td>12</td><td>0.13</td><td><strong>0.00</strong></td><td>0.15</td><td>0.18</td><td>0.00</td><td>1.67</td><td>0.04</td></tr><tr><td>1</td><td>73.16</td><td>73.16</td><td>63.47</td><td>31.63</td><td>0.15</td><td><strong>16.07</strong></td><td>0.07</td></tr><tr><td>2</td><td>67.54</td><td>67.54</td><td>60.19</td><td>25.60</td><td>0.11</td><td><strong>13.57</strong></td><td>0.08</td></tr><tr><td>3</td><td>62.78</td><td>62.78</td><td>58.70</td><td>23.15</td><td>0.14</td><td><strong>13.01</strong></td><td>0.07</td></tr><tr><td rowspan="6">2_15_2</td><td>6</td><td>62.21</td><td>62.21</td><td>58.52</td><td>20.72</td><td>0.13</td><td><strong>11.29</strong></td><td>0.07</td></tr><tr><td>12</td><td>61.74</td><td>61.74</td><td>57.64</td><td>16.26</td><td>0.08</td><td><strong>8.97</strong></td><td>0.07</td></tr></tbody></table>

we obtain different scenarios with different available computational times.

We conducted tests on small instances under 5 scenarios for the values of $C_{cpu}$: 1, 2, 3, 6, and 12. The tested instance configurations ($N_T$,$T$)$_{NOP}$,$M$) are 2_8_2, 2_10_2, 2_10_3, and 2_15_2. For each configuration 5 random instances were generated. In contrast to the deterministic nature of the NBI approaches, metaheuristics exhibit stochastic behaviour. Therefore, we performed 10 runs for each instance and each metaheuristic, the average result was recorded.

To evaluate the quality of the obtained results, we compute the percentage covered of the Hypervolume of a reference set, as in Kuk et al. (2021), Zapotecas-Martínez et al. (2024), which combines all the obtained points and saves only overall non-dominated solution in a single reference front, representing the Pareto front approximation. Thus, we compare the percentage of the reference HV cover by HV of the obtained solution.

In addition to HV percentages, the results were compared on a spread-based and a cardinality metric: Spread Metric (SM) and the number of solutions in the reference points (NRP).

The Spacing metric (SM), reflects the diversity of solutions within an obtained front. A lower SM value indicates a low variation of the distance between points of the Pareto front approximation, suggesting good diversity (Audet et al., 2021). The Spacing metric is calculated using the following formula:

$$SM = \sqrt{\frac{\sum_{i=0}^{|A|} (d_i - \bar{d})^2}{|A|}}$$

Where $d_i$ is the normalized rectilinear distance between the point $i$ and the closest point to it, and $\bar{d}$ is the mean value of all distances.

Table 7 presents the obtained results across the 5 instances for each configuration, with different $C_{cpu}$ values. As the two metaheuristics were repeated 10 times for each instance, the table also includes the average Coefficient of variation (CV) of the five instances. The best gap for each configuration for each $C_{cpu}$ value are highlighted in bold.

Table 7 shows that NBI consistently outperforms NWS across all instances and $C_{cpu}$ values, except for instance 2_10_3 at $C_{cpu} = 3$ where NWS performs better (0.13%). This gap in performance is the largest at $C_{cpu} = 6$ and $C_{cpu} = 12$ where NBI seems to find solutions in the pareto front that NWS can't find. For the largest configuration (2_15_2), both

methods give the same results, as they begin by computing the anchor points that could not be determined within the allowed time.

Comparing NBI and NBI-es, it can be seen that NBI-es strategy improves the average HV compared to NBI, this suggests that the Enumerate-and-solve strategy boosts the solver's performance. It should be noted, however, that the NBI approach performs better for some configurations when the allowed CPU is small ($C_{cpu} = 3$ or 2).

For instances (2_10_3) and small CPU times ($C_{cpu} = 1,2$), NSGA-II outperforms all solution approaches. However for the largest instances (2_15_2) MOEA/D largely outperforms the other approaches regardless of the amount of given CPU time, in such instances, the average gaps for NBI and NBI-es are, respectively 61.74% and 57.64% when $C_{cpu} = 12$. These gaps are even larger than that of the metaheuristics with smallest CPU time ($C_{cpu} = 1$), 31.63% for NSGA-II and 16.07% for MOEA/D.

Both NSGA-II and MOEA/D exhibit improved stability (lower CV) with increased computational times. NSGA-II demonstrates a sharper decrease in CV, suggesting it might converge more consistently. This is further supported by its superior performance, for the three smallest configurations, achieving a 0.18% average gap with a 0 average CV for 2_10_3 (compared with MOEA/D's 1.67% gap and 0.04 CV). Interestingly however, the trend seems to reverse for the largest configuration (2_15_2). MOEA/D appears to handle its complexity better, achieving a low average gap of 16.07% with the minimum $C_{cpu}$ outperforming NSGA-II even for the maximum $C_{cpu}$ (16.26% gap). This suggests that MOEA/D might perform better than NSGA-II for large size configurations, which is tested in the next subsection.

The 2_10_3 configuration exhibits an interesting trend where both metaheuristics achieve very good approximate Pareto fronts for $C_{cpu} = 12$. This is in contrast to the other two smaller configurations' results where the NBI approaches generally outperform the metaheuristics. The resulting gaps for NBI, NBI-es, NSGA-II, and MOEA/D are 0%, 0.15%, 0.18%, and 1.67%, respectively.

A closer examination of individual instances within the 2_10_3 configuration reveals that 2 out of 5 instances demonstrate no conflict between the two objectives. Comparing the average performance of the two metaheuristics across instances of 2_10_3 with conflicting and non-conflicting objectives, we observe that the metaheuristics perform better in conflicting-objectives instances. For non-conflicting objectives, both algorithms struggled to find high-quality solutions for all $C_{cpu}$ values except for the largest ($C_{cpu} = 12$). Notably, for NSGA-II, the gap in performance didn't improve for $C_{cpu}$ between 1 and 6, and even
<figure><img src="image_15.png" /><figcaption>**Fig. 9.** Comparing average HV gaps, SM, and Number of solutions of NSGA-II, MOEA/D, NWS, NBI, and NBI-es for the three smallest configurations 2_8_2, 2_10_2, and 2_10_3 (N_TNOP_M).</figcaption></figure>

deteriorated. MOEA/D exhibited worse HV gaps than NSGA-II and a decline in performance between 1 and 3 $C_{cpu}$.

This emphasis the importance of considering the level of conflict between objectives when selecting optimization algorithms. Furthermore, the superior performance of NSGA-II in handling conflicting objectives suggests a stronger exploitation capability compared to MOEA/D.

Overall, for all methods, increasing the computational time leads to a decrease in both the average gap (closeness to the optimal solution) and CV for metaheuristics. This is intuitive, as more processing time allows for more refined search and greater consistency in results. However, a key trade-off exists between exact and approximate methods, since exact solution approaches perform poorly in limited computational times. Although metaheuristics often outperform exact approaches in scenarios with limited computational time, they can't guarantee optimal solutions even when large computational time is available. This underscores the classic decision-making challenge: balancing solution quality against computational time. Fig. 9 visually illustrates this trade-off.

Fig. 9 presents the average results for the three smallest configurations. Sub-figure (a) presents the average HV gap percentage to the reference front per different $C_{cpu}$ values, whereas (b) and (c) present SM and the average number of non-dominated points in the generated fronts, respectively.

From Fig. 9 and as already discussed in Table 7, we can see that NBI outperforms NWS in terms of HV gap and it achieves better SM values for $C_{cpu} \le 3$. Additionally, for $C_{cpu} \ge 6$ NBI generates a greater number of solutions compared to NWS. These results indicate that NBI, equipped with the update function, explores the search space more effectively within the available CPU time and achieves broader coverage of HV. In contrast, NWS tends to stagnate beyond $C_{cpu} = 3$, failing to find additional solutions.

However, a notable trend appears in subfigure (b), where NWS achieves the best SM value among all methods. This is primarily because NWS, due to its mechanism of using pre-specified fixed search

points, fails to generate additional solutions. As a result, the few solutions it produces are more evenly spaced. In comparison, other methods find more points of the Pareto front, but due to the front's structure, this often leads to a less uniform distribution of solutions.

Figure (a) also shows that NWS, NBI, and NBI-es approaches tend to perform worse than the two metaheuristics for low computational times ($C_{cpu} = 1$). However, it surpasses them for larger times, with NBI and NBI-es achieving almost 0% gap for $C_{cpu}$ larger than 6. For the two metaheuristics a similar behaviour can be observed, with NSGA-II consistently outperforming MOEA/D. At the largest computational time ($C_{cpu} = 12$) NSGA-II achieves an average gap of 4.79%, while MOEA/D reaches 7.65%.

Interestingly, NBI and NBI-es experience a sharper decline in the HV gap between $C_{cpu} = 1$ and $C_{cpu} = 3$, dropping from 18.83% to 1.31% for NBI and from 16.70% to 0.56% for NBI-es. This is due to the NBI method procedure with initial generation of the two extreme (anchor) points that significantly contribute to the HV value. Subsequent points are iteratively generated, prioritizing those furthest from existing points. This suggests that the prioritization results in points generated in later iterations being closer to existing points, thus contributing less to the overall HV. This is evident in the minimal decrease in the HV gap for NBI and NBI-es when $C_{cpu}$ doubles from 6 to 12.

Figure (a) also reveals that NBI-es consistently outperforms NBI in terms of HV for computational times of 1, 2, and 3. This superiority is further evidenced by Figure (c), which demonstrates that NBI-es generates a larger number of non-dominated solutions than NBI across all $C_{cpu}$ values. The better performance of NBI-es can be attributed to its ability to identify more than one non-dominated solution within each subproblem. This suggests that NBI-es is more effective in exploring the solution space compared to NBI.

Figure (b) reveals that the NBI approaches, particularly NBI-es, generate Pareto fronts with a more evenly distributed spread of solutions compared to the metaheuristics. This is most evident for the highest
<figure><img src="image_16.png" /><figcaption>**Fig. 10.** The non-dominated solutions generated by the three approaches under different computational times for the instance 5 of configuration 2_8_2 (N_TNOP_M).</figcaption></figure>

computational time ($C_{cpu} = 12$) where NBI-es achieves the lowest SM value of 0.66.

As mentioned earlier, for $C_{cpu}$ of 6 and 12, the two NBI methods has almost 0 HV gaps. This is despite the increase in the number of non-dominated solutions for these $C_{cpu}$ values, especially for NBI-es, as evident in Figure (c). This suggests that the additional solutions discovered by NBI-es are located in regions of the Pareto front with minimal impact on the overall HV, for example near the already discovered non-dominated points. However, Figure (b) reveals a different trend. The SM remains relatively stable for NBI-es between $C_{cpu}$ 6 and 12, suggesting that the additional solutions are distributed uniformly in the existing Pareto front, with minimal impact on the overall solution spread.

This phenomenon is likely due to the staircase structure of the Pareto front, this is further confirmed by the Pareto fronts presented in Fig. 10, where the additional points found for $C_{cpu} \ge 3$ align with the HV space limit and thus contributing marginally to the overall HV.

Fig. 10 depicts the Pareto fronts obtained by the NBI algorithm for various $C_{cpu}$ values for instance 5 of configuration 2_8_2, and the best overall solutions found by the metaheuristics. This figure emphasizes the importance of having a good representation of the true Pareto front for solution evaluation. As the NBI algorithm progressively reveals the true Pareto front, the quality of obtained solutions can be better evaluated, allowing for easier identification of poor solutions or fronts.

Fig. 10 indicates that the NBI method produces distinct Pareto fronts for varying $C_{cpu}$ values. This is because a larger computational time allows the generation of more non-dominated solutions with a more even distribution. On the other hand NWS discovers the front progressively from bottom right to the top left, and still fails to find most of the pareto optimal solutions.

In Fig. 10(a) ($C_{cpu} = 1$), the NBI method identifies the two anchor points and one additional solution, While NWS finds only the two extremes. As the computational time increases (Fig. 10(b) and (c)), a fourth
Pareto point is found. Interestingly, although the three Pareto points from Fig. 10(a) reappear in Fig. 10(b) and (c) the additional point is different between the two Fig. 10(b) and (c). This is due to the NBI method's ability to adapt to different computational times by exploring different regions of the search space and promoting solution diversity. In Fig. 10(d) with $C_{cpu} = 6$, 8 non-dominated solutions were generated. Finally, Fig. 10(e) shows the most diverse Pareto front with 13 solutions generated for the largest computational time ($C_{cpu} = 12$). NWS discovers only 4 of the 13 solutions.

The variability in the required CPU time for each iteration poses a challenge that NBI effectively addresses, unlike NWS that searches the same regions regardless of the available time. For instance, generating anchor points for this example involves solving two problems. The first problem (minimization of total production time) is solved in 9 seconds while the second one (minimization of total production cost) is solved in 5 seconds. Despite the difficulty in predicting the solver's computational time in each iteration, our NBI approach consistently demonstrates a relatively uniform spacing among the generated solutions for all $C_{cpu}$ scenarios.

Fig. 10 also showcases the ability of both NSGA-II and MOEA/D to generate solutions close to the true Pareto front. The red and blue points represent the best solutions found by each metaheuristic based on the HV metric of the ten runs. The obtained HV values are 11,449 and 11,206 for NSGA-II and MOEA/D, respectively. In comparison, the NBI solution in sub-figure (a) has an HV of 8715. However, the NBI solution in Fig. 10(e), obtained with the largest computational time ($C_{cpu}$), achieves an HV of 11644, surpassing the best solutions from both metaheuristics.

Analysis of the staircase structure of the Pareto front reveals insights into the decision-making process. Solutions belonging to the same plateau within this structure share identical operation-to-machine assignments. Consequently, variations in product and operation sequences, along with tool and configuration selection, have a minimal impact on total cost. Conversely, modifying operation-to-machine assignments results in transitions between plateaus, significantly affecting total cost.

This observation suggests that in a hierarchical planning system, performing operation-to-machine assignments first would result in prioritizing the minimization of the total cost. Such a framework can be explored further, building upon the heuristic framework proposed in Mechaacha et al. (2025).

From the results presented in this section, we can conclude that, first the two NBI based methods exhibit good performance for smaller instances, with NBI-es performing better. Second, NBI and NBI-es achieve higher HVs than the metaheuristics for the smallest instances when allowed larger computational time ($C_{cpu} \ge 3$). Third, the metaheuristics' performance is validated against the exact solution approaches, demonstrating close convergence with gaps of almost 4% and 7% from the reference front HV for the three smallest configurations. Since mathematical modelling is not applicable to medium or large instances, the next subsection will focus solely on evaluating the two metaheuristics' performance for these larger problem sizes.

## 5.4. Comparing two metaheuristics

The behaviour and performance of Multi-Objective Evolutionary Algorithms (MOEAs) are primarily studied through experimentation due to the inherent complexities in theoretical analysis (Li & Zhang, 2008). This subsection presents the results obtained for testing NSGA-II, MOEA/D, and MOEA/D-OBL on medium to large size instances. Based on the performances in Figure (c), we set $C_{cpu}$ value to 2. This choice balances solution quality with computational efficiency. Increasing $C_{cpu}$ beyond 2 appears to offer minimal improvement in solution quality, while significantly increasing computation time.

In addition to HV and SM, two other metrics are used to evaluate the performance of the metaheuristics: C-metric, and Number of solutions in the reference points (NRP).

C-metric, denoted as C(A,B), is a quantitative measure used to assess the dominance relationship between the two sets of solutions A and B (He et al., 2022). It calculates the proportion of solutions in set B that are dominated by at least one element within set A. Its value ranges from 0 to 1 (inclusive) where:

* A value of C(A, B) equal to 0 implies that no solution in B is dominated by any solution in A (ideal scenario for set B meaning that B completely dominates A).

* A value of 1 signifies that all solutions in B are dominated by solutions in A (worst case for set B).

It should be noted that C(A, B) does not necessarily equal C(B, A).

Number of solutions in the reference points (NRP) represents the number of points found in A that belong to the reference set, a higher value of NRP indicates that more solutions found are non-dominated. This metric is similar to the ratio of reference points (Audet et al., 2021).

Five random instances for each of the 10 configurations were generated. Each instance was run 10 times for both metaheuristics. Table 8 presents the average values of the four metrics (HV, C-metric, SM, and NRP) across all runs for each instance for NSGA-II, MOEA/D, and MOEA/D-ONL. Additionally, the table shows the average values for each configuration across all instances.

To assess the statistical significance of the observed differences between the results of the two metaheuristics, we employed the Wilcoxon signed-rank test (Wilcoxon, 1947) with a significance level of 0.05. Two comparisons were conducted, between NSGA-II and MOEA/D, and between MOEA/D and MOEA/D-OBL. Cases where one metaheuristic achieved a statistically significant difference are marked with an underline.

Table 8 reveals that MOEA/D and MOEA/D-OBL perform significantly better than NSGA-II across most metrics and configurations. MOEA/D consistently achieves higher HV values for all configurations averages. This is statistically significant against NSGA-II in 45 out of 50 instances, and in 5 against MOEA/D-OBL. For the largest configuration (10_100_20), MOEA/D's average HV across the five instances is 0.68, more than double NSGA-II's average 0.32, and better than MOEA/D-OBL 0.59. This suggests that MOEA/D's Pareto fronts cover a significantly larger area from the reference points compared to NSGA-II. However, as instances sizes increase, it can be seen that MOEA/D covers a lesser average portion of the reference set HV, this implies that MOEA/D solution quality in terms of stability also decreases.

MOEA/D is generally better than NSGA-II in terms of C-metric values, indicating better solution dominance relationships. This dominance is statistically significant according to the Wilcoxon test in 46 out of 50 instances. The remaining four instances did not show a statistically significant difference. While MOEA/D and MOEA/D-OBL HV are relatively similar, C-metric results indicate that MOEA/D performs much better than MOEA/D-OBL with 15 instances with significant difference. This suggests that, while the OBL technique decreases the convergence of the pareto front approximation, it increases the spread of the front and thus maintains a good HV. This is further confirmed with SM results, as MOEA/D-OBL achieves a better spread in 26 instances and 1 being significant, while MOEA/D is better in 17 instances with 1 significant. While the observed differences in SM are less evident compared to HV and C-metric, MOEA/D and MOEA/D-OBL still achieves better average SM values than NSGA-II in all configurations except one (4_20_5). MOEA/D exhibits statistically significant differences to NSGA-II in 8 instances, while NSGA-II shows a significant difference in only 1 instance. This suggests a generally better distribution of solutions within MOEA/D's approximate Pareto fronts.

As NRP is a cardinality and convergence metric, MOEA/D also performs better than MOEA/D-OBL, however none of the differences are significant. MOEA/D generates more reference points than NSGA-II in all 50 instances except for 3, and the difference is significant in 2 instances where MOEA/D generates significantly more NRP points than NSGA-II. For instances of configuration 3_20_3 NSGA-II doesn't generate any NRP indicating complete superiority of MOEA/D
<caption>**Table 8**</caption>

Comparing the performance of the two metaheuristics NSGA-II and MOEA/D for different metrics.

<table><thead><tr><th rowspan="2">Configuration</th><th rowspan="2">Instance</th><th colspan="3">HV</th><th colspan="4">C-metric</th><th colspan="3">SM</th><th colspan="3">NRP</th></tr><tr><th>NSGA-II</th><th>MOEA/D</th><th>MOEA/D-OBL</th><th>C(NSGA-II, MOEA/D)</th><th>C(MOEA/D, NSGA-II)</th><th>C(MOEA/D-OBL)</th><th>C(MOEA/D-OBL, MOEA/D)</th><th>NSGA-II</th><th>MOEA/D</th><th>MOEA/D-OBL</th><th>NSGA-II</th><th>MOEA/D</th><th>MOEA/D-OBL</th></tr></thead><tbody><tr><td rowspan="5">3_15_5</td><td>1</td><td>0.71</td><td><u>0.93</u></td><td><u>0.96</u></td><td>0.03</td><td><u>0.69</u></td><td>0.20</td><td><u>0.53</u></td><td>0.74</td><td>0.63</td><td><u>0.54</u></td><td>0.00</td><td>0.30</td><td><u>0.90</u></td></tr><tr><td>2</td><td><u>0.97</u></td><td>0.96</td><td>0.96</td><td><u>0.42</u></td><td>0.40</td><td><u>0.54</u></td><td>0.36</td><td><u>0.69</u></td><td>0.70</td><td>0.70</td><td>1.50</td><td><u>1.90</u></td><td>0.40</td></tr><tr><td>3</td><td><u>0.99</u></td><td>0.97</td><td>0.96</td><td><u>0.48</u></td><td>0.42</td><td><u>0.52</u></td><td>0.37</td><td>0.66</td><td><u>0.63</u></td><td>0.67</td><td>0.30</td><td>0.60</td><td><u>0.90</u></td></tr><tr><td>4</td><td>0.92</td><td><u>0.94</u></td><td><u>0.94</u></td><td>0.16</td><td><u>0.60</u></td><td><u>0.41</u></td><td>0.36</td><td><u>0.57</u></td><td>0.59</td><td>0.63</td><td><u>1.20</u></td><td>0.80</td><td>0.60</td></tr><tr><td>5</td><td>0.91</td><td>0.93</td><td><u>0.95</u></td><td>0.37</td><td><u>0.45</u></td><td>0.32</td><td><u>0.56</u></td><td><u>0.61</u></td><td>0.67</td><td>0.74</td><td>0.50</td><td>0.70</td><td><u>1.00</u></td></tr><tr><td rowspan="5">3_20_3</td><td>AVG</td><td>0.90</td><td><u>0.95</u></td><td><u>0.95</u></td><td>0.29</td><td><u>0.51</u></td><td>0.40</td><td><u>0.44</u></td><td>0.65</td><td><u>0.64</u></td><td>0.66</td><td>0.70</td><td><u>0.86</u></td><td>0.76</td></tr><tr><td>1</td><td>0.75</td><td><u>0.92</u></td><td><u>0.92</u></td><td>0.05</td><td><u>0.90</u></td><td>0.40</td><td><u>0.49</u></td><td>0.76</td><td><u>0.69</u></td><td>0.71</td><td>0.00</td><td><u>0.50</u></td><td>0.20</td></tr><tr><td>2</td><td>0.71</td><td><u>0.89</u></td><td><u>0.90</u></td><td>0.05</td><td><u>0.87</u></td><td>0.39</td><td><u>0.40</u></td><td><u>0.56</u></td><td>0.73</td><td>0.69</td><td>0.00</td><td><u>1.50</u></td><td>0.50</td></tr><tr><td>3</td><td>0.39</td><td><u>0.88</u></td><td>0.85</td><td>0.02</td><td><u>0.95</u></td><td><u>0.46</u></td><td>0.30</td><td>0.69</td><td><u>0.50</u></td><td>0.54</td><td>0.00</td><td><u>1.90</u></td><td>0.80</td></tr><tr><td>4</td><td>0.86</td><td><u>0.92</u></td><td>0.91</td><td>0.11</td><td><u>0.78</u></td><td><u>0.47</u></td><td>0.40</td><td>0.67</td><td><u>0.51</u></td><td>0.54</td><td>0.00</td><td>0.30</td><td><u>0.40</u></td></tr><tr><td rowspan="5">4_20_5</td><td>AVG</td><td>0.64</td><td><u>0.88</u></td><td>0.85</td><td>0.03</td><td><u>0.91</u></td><td><u>0.56</u></td><td>0.21</td><td>0.65</td><td><u>0.56</u></td><td>0.59</td><td>0.00</td><td><u>1.30</u></td><td>1.00</td></tr><tr><td>1</td><td>0.87</td><td><u>0.96</u></td><td>0.95</td><td>0.06</td><td><u>0.87</u></td><td><u>0.45</u></td><td>0.36</td><td>0.67</td><td><u>0.60</u></td><td>0.61</td><td>0.00</td><td><u>1.10</u></td><td>0.58</td></tr><tr><td>2</td><td>0.87</td><td><u>0.95</u></td><td>0.94</td><td>0.03</td><td><u>0.82</u></td><td><u>0.56</u></td><td>0.26</td><td>0.64</td><td><u>0.57</u></td><td><u>0.54</u></td><td>0.00</td><td><u>1.30</u></td><td>0.10</td></tr><tr><td>3</td><td>0.57</td><td><u>0.90</u></td><td><u>0.91</u></td><td>0.02</td><td><u>0.93</u></td><td>0.32</td><td><u>0.37</u></td><td>0.46</td><td><u>0.61</u></td><td>0.55</td><td>0.10</td><td><u>1.30</u></td><td>0.80</td></tr><tr><td>4</td><td>0.46</td><td><u>0.90</u></td><td><u>0.92</u></td><td>0.00</td><td><u>0.98</u></td><td>0.21</td><td><u>0.54</u></td><td>0.55</td><td><u>0.54</u></td><td><u>0.53</u></td><td>0.00</td><td>0.20</td><td><u>0.90</u></td></tr><tr><td rowspan="5">4_25_3</td><td>AVG</td><td>0.56</td><td><u>0.88</u></td><td>0.83</td><td>0.03</td><td><u>0.89</u></td><td><u>0.56</u></td><td>0.28</td><td><u>0.54</u></td><td>0.67</td><td>0.64</td><td>0.00</td><td><u>1.30</u></td><td>0.50</td></tr><tr><td>1</td><td>0.30</td><td><u>0.82</u></td><td>0.80</td><td>0.00</td><td><u>0.97</u></td><td>0.37</td><td><u>0.41</u></td><td><u>0.54</u></td><td>0.58</td><td>0.55</td><td>0.02</td><td><u>1.00</u></td><td>0.66</td></tr><tr><td>2</td><td>0.80</td><td><u>0.91</u></td><td>0.90</td><td>0.13</td><td><u>0.74</u></td><td>0.52</td><td>0.36</td><td>0.61</td><td>0.60</td><td><u>0.59</u></td><td>0.20</td><td><u>2.00</u></td><td>0.20</td></tr><tr><td>3</td><td>0.64</td><td><u>0.93</u></td><td><u>0.94</u></td><td>0.02</td><td><u>0.97</u></td><td>0.42</td><td><u>0.47</u></td><td>0.57</td><td>0.52</td><td><u>0.48</u></td><td>0.00</td><td>0.40</td><td><u>0.50</u></td></tr><tr><td>4</td><td>0.26</td><td><u>0.78</u></td><td>0.69</td><td>0.00</td><td><u>1.00</u></td><td><u>1.00</u></td><td>0.00</td><td>0.51</td><td><u>0.44</u></td><td>0.49</td><td>0.00</td><td><u>0.70</u></td><td>0.00</td></tr><tr><td rowspan="5">5_25_5</td><td>AVG</td><td>0.49</td><td><u>0.83</u></td><td><u>0.87</u></td><td>0.01</td><td><u>0.85</u></td><td>0.38</td><td><u>0.40</u></td><td>0.59</td><td>0.59</td><td><u>0.57</u></td><td>0.00</td><td>0.90</td><td><u>1.30</u></td></tr><tr><td>1</td><td>0.48</td><td><u>0.89</u></td><td>0.86</td><td>0.00</td><td><u>1.00</u></td><td><u>0.55</u></td><td>0.27</td><td>0.57</td><td>0.52</td><td><u>0.51</u></td><td>0.04</td><td><u>0.94</u></td><td>0.50</td></tr><tr><td>2</td><td>0.61</td><td><u>0.91</u></td><td><u>0.93</u></td><td>0.01</td><td><u>0.93</u></td><td>0.31</td><td><u>0.52</u></td><td><u>0.51</u></td><td><u>0.56</u></td><td>0.67</td><td>0.00</td><td>0.10</td><td><u>1.70</u></td></tr><tr><td>3</td><td>0.71</td><td><u>0.94</u></td><td>0.93</td><td>0.04</td><td><u>0.89</u></td><td><u>0.44</u></td><td>0.38</td><td>0.57</td><td>0.56</td><td><u>0.52</u></td><td>0.10</td><td><u>3.20</u></td><td>1.10</td></tr><tr><td>4</td><td>0.53</td><td><u>0.96</u></td><td>0.93</td><td>0.00</td><td><u>1.00</u></td><td><u>0.63</u></td><td>0.20</td><td>0.65</td><td>0.52</td><td><u>0.49</u></td><td>0.00</td><td><u>1.60</u></td><td>0.20</td></tr><tr><td rowspan="5">5_50_5</td><td>AVG</td><td>0.53</td><td><u>0.86</u></td><td><u>0.87</u></td><td>0.02</td><td><u>0.92</u></td><td>0.33</td><td><u>0.48</u></td><td>0.66</td><td>0.58</td><td><u>0.51</u></td><td>0.00</td><td><u>1.90</u></td><td>1.50</td></tr><tr><td>1</td><td>0.53</td><td><u>0.73</u></td><td>0.67</td><td>0.11</td><td><u>0.63</u></td><td><u>0.63</u></td><td>0.26</td><td>0.50</td><td>0.50</td><td><u>0.46</u></td><td>0.50</td><td><u>0.80</u></td><td>0.00</td></tr><tr><td>2</td><td>0.39</td><td><u>0.75</u></td><td>0.61</td><td>0.07</td><td><u>0.89</u></td><td><u>0.64</u></td><td>0.20</td><td>0.64</td><td><u>0.41</u></td><td><u>0.38</u></td><td>0.00</td><td><u>0.30</u></td><td>0.00</td></tr><tr><td>3</td><td>0.44</td><td><u>0.70</u></td><td>0.60</td><td>0.16</td><td><u>0.62</u></td><td><u>0.63</u></td><td>0.26</td><td>0.50</td><td>0.43</td><td><u>0.40</u></td><td>0.40</td><td><u>0.50</u></td><td>0.00</td></tr><tr><td>4</td><td>0.39</td><td><u>0.66</u></td><td>0.58</td><td>0.12</td><td><u>0.82</u></td><td><u>0.62</u></td><td>0.23</td><td>0.68</td><td><u>0.44</u></td><td><u>0.38</u></td><td><u>0.50</u></td><td>0.10</td><td>0.10</td></tr><tr><td rowspan="5">5_30_10</td><td>AVG</td><td>0.36</td><td><u>0.58</u></td><td>0.55</td><td>0.14</td><td><u>0.73</u></td><td><u>0.53</u></td><td>0.33</td><td>0.67</td><td><u>0.35</u></td><td>0.44</td><td>0.20</td><td><u>0.40</u></td><td>0.20</td></tr><tr><td>1</td><td>0.39</td><td><u>0.86</u></td><td>0.81</td><td>0.02</td><td><u>0.96</u></td><td><u>0.61</u></td><td>0.26</td><td>0.59</td><td><u>0.48</u></td><td>0.49</td><td>0.00</td><td><u>1.30</u></td><td>0.00</td></tr><tr><td>2</td><td>0.56</td><td><u>0.89</u></td><td>0.82</td><td>0.01</td><td><u>0.95</u></td><td><u>0.65</u></td><td>0.18</td><td>0.54</td><td><u>0.52</u></td><td>0.53</td><td>0.00</td><td><u>1.40</u></td><td>0.90</td></tr><tr><td>3</td><td>0.37</td><td><u>0.78</u></td><td>0.76</td><td>0.01</td><td><u>0.83</u></td><td><u>0.52</u></td><td>0.32</td><td>0.51</td><td>0.49</td><td><u>0.45</u></td><td>0.00</td><td><u>1.10</u></td><td>0.50</td></tr><tr><td>4</td><td>0.58</td><td><u>0.85</u></td><td><u>0.86</u></td><td>0.01</td><td><u>0.90</u></td><td>0.40</td><td><u>0.47</u></td><td>0.56</td><td><u>0.51</u></td><td>0.56</td><td>0.60</td><td>0.70</td><td><u>0.90</u></td></tr><tr><td rowspan="5">10_50_10</td><td>AVG</td><td>0.44</td><td><u>0.89</u></td><td><u>0.90</u></td><td>0.00</td><td><u>1.00</u></td><td>0.36</td><td><u>0.38</u></td><td>0.54</td><td>0.44</td><td><u>0.40</u></td><td>0.00</td><td><u>0.60</u></td><td><u>0.60</u></td></tr><tr><td>1</td><td>0.18</td><td><u>0.75</u></td><td>0.74</td><td>0.00</td><td><u>1.00</u></td><td>0.39</td><td>0.39</td><td>0.69</td><td><u>0.41</u></td><td>0.61</td><td>0.00</td><td><u>0.70</u></td><td>0.20</td></tr><tr><td>2</td><td>0.28</td><td><u>0.71</u></td><td><u>0.73</u></td><td>0.01</td><td><u>0.91</u></td><td>0.39</td><td><u>0.46</u></td><td>0.49</td><td><u>0.45</u></td><td>0.54</td><td>0.00</td><td><u>0.30</u></td><td>0.20</td></tr><tr><td>3</td><td>0.24</td><td><u>0.68</u></td><td>0.57</td><td>0.02</td><td><u>0.91</u></td><td><u>0.59</u></td><td>0.28</td><td>0.63</td><td><u>0.45</u></td><td>0.49</td><td>0.00</td><td><u>0.40</u></td><td>0.00</td></tr><tr><td>4</td><td>0.28</td><td><u>0.82</u></td><td>0.76</td><td>0.00</td><td><u>1.00</u></td><td><u>0.51</u></td><td>0.23</td><td>0.47</td><td>0.46</td><td><u>0.34</u></td><td>0.00</td><td>0.50</td><td><u>0.70</u></td></tr><tr><td rowspan="5">10_75_10</td><td>AVG</td><td>0.38</td><td><u>0.68</u></td><td><u>0.70</u></td><td>0.11</td><td><u>0.72</u></td><td>0.39</td><td><u>0.49</u></td><td>0.65</td><td><u>0.48</u></td><td>0.44</td><td>0.20</td><td>0.20</td><td><u>0.40</u></td></tr><tr><td>1</td><td>0.21</td><td><u>0.62</u></td><td>0.48</td><td>0.00</td><td><u>0.98</u></td><td>0.65</td><td>0.19</td><td>0.58</td><td>0.57</td><td><u>0.45</u></td><td>0.00</td><td><u>0.40</u></td><td>0.00</td></tr><tr><td>2</td><td>0.33</td><td><u>0.71</u></td><td>0.61</td><td>0.06</td><td><u>0.82</u></td><td>0.56</td><td>0.27</td><td>0.53</td><td>0.52</td><td><u>0.42</u></td><td>0.10</td><td><u>0.70</u></td><td>0.00</td></tr><tr><td>3</td><td>0.36</td><td><u>0.65</u></td><td><u>0.66</u></td><td>0.11</td><td><u>0.70</u></td><td>0.34</td><td><u>0.54</u></td><td>0.53</td><td>0.45</td><td><u>0.44</u></td><td>0.00</td><td><u>0.50</u></td><td>0.20</td></tr><tr><td>4</td><td>0.31</td><td><u>0.68</u></td><td>0.60</td><td>0.05</td><td><u>0.83</u></td><td><u>0.53</u></td><td>0.30</td><td>0.63</td><td>0.44</td><td><u>0.41</u></td><td>0.00</td><td><u>0.40</u></td><td>0.00</td></tr><tr><td rowspan="5">10_100_20</td><td>AVG</td><td>0.27</td><td><u>0.73</u></td><td>0.70</td><td>0.03</td><td><u>0.91</u></td><td><u>0.45</u></td><td>0.37</td><td>0.59</td><td><u>0.45</u></td><td>0.48</td><td>0.04</td><td><u>0.42</u></td><td>0.30</td></tr><tr><td>1</td><td>0.34</td><td><u>0.80</u></td><td>0.71</td><td>0.03</td><td><u>0.85</u></td><td><u>0.53</u></td><td>0.27</td><td><u>0.42</u></td><td>0.42</td><td>0.52</td><td>0.00</td><td><u>0.50</u></td><td>0.00</td></tr><tr><td>2</td><td>0.22</td><td><u>0.66</u></td><td>0.61</td><td>0.02</td><td><u>0.89</u></td><td><u>0.53</u></td><td>0.34</td><td>0.57</td><td>0.47</td><td><u>0.35</u></td><td>0.00</td><td><u>0.90</u></td><td>0.10</td></tr><tr><td>3</td><td>0.37</td><td><u>0.67</u></td><td>0.47</td><td>0.11</td><td><u>0.66</u></td><td><u>0.77</u></td><td>0.08</td><td>0.55</td><td><u>0.40</u></td><td>0.42</td><td>0.10</td><td><u>0.50</u></td><td>0.00</td></tr><tr><td>4</td><td>0.21</td><td><u>0.65</u></td><td>0.60</td><td>0.02</td><td><u>0.90</u></td><td><u>0.49</u></td><td>0.37</td><td>0.52</td><td>0.43</td><td><u>0.39</u></td><td>0.00</td><td><u>0.50</u></td><td>0.20</td></tr><tr><td rowspan="2">5</td><td>AVG</td><td>0.47</td><td><u>0.63</u></td><td>0.57</td><td>0.25</td><td><u>0.48</u></td><td><u>0.51</u></td><td>0.33</td><td>0.55</td><td><u>0.31</u></td><td>0.37</td><td><u>0.50</u></td><td>0.20</td><td>0.20</td></tr><tr><td>1</td><td>0.32</td><td><u>0.68</u></td><td>0.59</td><td>0.09</td><td><u>0.76</u></td><td><u>0.57</u></td><td>0.28</td><td>0.52</td><td><u>0.41</u></td><td>0.41</td><td>0.12</td><td><u>0.52</u></td><td>0.10</td></tr></tbody></table>

with 1.1 points per run, and MOEA/D-OBL with 0.58 points per run.

Fig. 11 depicts the HV and SM averages per instance size for each metaheuristic. For all metaheuristics, as the instance size increases the HV and SM decrease. This implies that the solution quality decreases and the spread is improved since the search space will not be concentrated on specific regions. The figure also highlights the clear superiority of MOEA/D and MOEA/D-OBL over NSGA-II. From Fig. 11(b)

it can be seen that MOEA/D-OBL achieves less SM than MOEA/D for most instances, indicating an improvement in distribution while covering less HV.

The difference in performance of NSGA-II and the two MOEA/D approaches, as observed in this and previous sections (Sections 5.3 and 5.2), can be attributed to their distinct exploration-exploitation strategies employed. NSGA-II's larger population size (40), and high crossover probability (0.5) and mutation ratio (0.5), promotes a broader
<figure><img src="image_17.png" /><figcaption>Fig. 11. HV and SM averages across the 10 different instance configurations.</figcaption></figure>

exploration of the search space. Conversely, MOEA/D's smaller population (20) and lower crossover probability (0.1) and mutation ratio (0.2) indicate a more focused exploitation of promising regions within the solution space.

Despite both being evolutionary algorithms, the different strategies of NSGA-II and MOEA/D are understandable given their computational requirements. NSGA-II's computationally expensive crowding distance and large population, limits the number of iterations (generations). On the other hand, MOEA/D's smaller population and simpler mechanisms allow for more iterations, enabling a more focused exploitation of the search space.

This can justify NSGA-II's superior performance on small size instances and those with non-conflicting objectives. In such scenarios, the search space is relatively small, and MOEA/D's minor genes modification strategy have minimal to no impact on the solutions after decoding due to the encoding mechanism. This leads to mutation being ineffective for multiple cells and duplicate solutions may appear in the child population, which are subsequently filtered out. However, as problem size increases, NSGA-II's performance deteriorates due to its limited number of iterations. This suggests that MOEA/D's relative advantage in larger instances may be partially due to NSGA-II's limitations rather than solely by MOEA/D's superior capabilities.

These results suggest that while the OBL strategy improves the distribution of solutions thanks to a better exploration, it decreases the convergence toward the Pareto optimal front. This can be attributed to the generation of opposite point being more computationally expensive than simple crossover, which permits the execution of more generations and it generates offsprings that dominate the parents population. However, OBL strategy improves the distribution as it diversifies the non-dominated solutions obtained.

## 5.5. Insights on solution structure in the case of same product type

We consider here the special case of MPPP in which all units are identical, termed Multi-Unit Process Planning (MUPP). A Single-Unit Process

Plan (SUPP) is a process plan for one unit, and a Multi-Unit Process Plan (MUPP) is a sequence of SUPPs. The work of Touzout and Benyoucef (2019) explored MUPPs structure, but due to limitations in used solution approaches, their study relied on microscopic instances and developed heuristics based on three observations about the potential structure of an optimal MUPP solution, which can be:

1. **RSUPP**: repeat a single Pareto-optimal SUPP for all units.

2. **ILSSUPP**: sequence different Pareto-optimal SUPPs.

3. **ABILS**: sequence neighbours of Pareto-optimal SUPPs to trade off objectives.

Their validity on larger instances remains unverified. However in MUPP, the separated approach (See Section 2.3 for more details about this approach) requires solving only one SUPP—then repeating it *N* times—allowing direct comparison to the RSUPP strategy.

To test the RSUPP observation, we tested two MUPP configurations (five instances each). For the integrated approach, we used MOEA/D and stored the best approximate fronts (highest HV). For RSUPP, we applied NBI with *N* = 1 to enumerate Pareto-optimal SUPPs, then duplicated each solution *N* times. CPU time was set to *TNOP* × *M* × *C<sub>cpu</sub>* with *C<sub>cpu</sub>* = 6. Performance was measured by HV and C-metric, using a combined reference set of all non-dominated solutions. Results are given in Table 9.

To better understand the structure of the MUPPs, each MUPP solution from MOEA/D is decomposed into its constituent SUPPs (denoted SUPP_ON) and compared to the Pareto-optimal SUPPs obtained via NBI in the RSUPP first step. We compute for every SUPP_ON the Euclidean distance to its closest Pareto-optimal neighbour (generational distance, GD) on the SUPP front, then aggregate these distances by their mean and standard deviation per MUPP. To allow comparisons across instances, we normalize each mean distance by the distance from the origin (0,0) to the Pareto front, so that a normalized value of 1 indicates a distance equal to the distance to the front's origin distance. Table 9 reports these normalized mean distances and their coefficients of variation for all instances, as well as the overall averages.
<table><caption>**Table 9**
The two approaches integrated vs separated HV and C-metric results and GD for single process plans for each instance.</caption><thead><tr><th rowspan="2">Configuration</th><th rowspan="2">Instance</th><th colspan="2">HV</th><th colspan="2">C-metric</th><th rowspan="2">GD</th><th rowspan="2">Norm. GD</th><th rowspan="2">CV</th></tr><tr><th>RSUPP</th><th>MOEA/D</th><th>C(RSUPP,MOEA/D)</th><th>C(MOEA/D,RSUPP)</th></tr></thead><tbody><tr><td rowspan="6">3_12_3</td><td>1</td><td>0.47</td><td>0.90</td><td>0.20</td><td>0.40</td><td>72.7</td><td>0.05</td><td>0.88</td></tr><tr><td>2</td><td>0.76</td><td>0.97</td><td>0.18</td><td>0.31</td><td>12.7</td><td>0.01</td><td>0.63</td></tr><tr><td>3</td><td>0.29</td><td>0.77</td><td>0.00</td><td>0.43</td><td>16.8</td><td>0.02</td><td>0.50</td></tr><tr><td>4</td><td>0.31</td><td>0.97</td><td>0.04</td><td>0.86</td><td>47.1</td><td>0.05</td><td>0.94</td></tr><tr><td>5</td><td>0.39</td><td>0.89</td><td>0.05</td><td>0.56</td><td>65.2</td><td>0.06</td><td>0.65</td></tr><tr><td>AVG</td><td>0.44</td><td>0.90</td><td>0.09</td><td>0.51</td><td>42.9</td><td>0.04</td><td>0.72</td></tr><tr><td rowspan="6">4_16_2</td><td>1</td><td>0.84</td><td>0.49</td><td>0.44</td><td>0.00</td><td>445.8</td><td>0.56</td><td>0.43</td></tr><tr><td>2</td><td>1.00</td><td>0.36</td><td>1.00</td><td>0.00</td><td>177.2</td><td>0.14</td><td>0.83</td></tr><tr><td>3</td><td>0.21</td><td>0.82</td><td>0.15</td><td>0.00</td><td>250.7</td><td>0.28</td><td>0.89</td></tr><tr><td>4</td><td>0.03</td><td>0.97</td><td>0.00</td><td>1.00</td><td>41.54</td><td>0.02</td><td>0.63</td></tr><tr><td>5</td><td>0.77</td><td>0.71</td><td>0.82</td><td>0.35</td><td>144.3</td><td>0.11</td><td>0.90</td></tr><tr><td>AVG</td><td>0.57</td><td>0.67</td><td>0.48</td><td>0.27</td><td>211.9</td><td>0.22</td><td>0.74</td></tr></tbody></table>

RSUPP = separated approach represented by NBI;

For the smallest configuration, MOEA/D significantly outperforms RSUPP, achieving an HV value of more than double that of RSUPP. Additionally, 51% of RSUPP solutions are dominated by at least one integrated approach solution, while only 9% are dominated by at least one RSUPP solution. The second configuration has a lesser difference between the two approaches however the integrated approach still achieves a better average HV.

The CV values in Table 9 ranging from 0.5 to 0.9 indicate that the distances between SUPP_ON and their nearest Pareto-optimal neighbours vary within each MUPP solution. This challenges the previously mentioned observations 1 and 2 (repetition or sequence of Pareto-optimal SUPPs) because a CV of 0, or close to 0, would be expected in those scenarios. Since the integrated approach performed better in the first configuration 3_12_3, (Table 9), and from the observed CV range, the optimal MUPPs consist of a mix of SUPPs with varying distances to the Pareto front, with a standard deviation between 50% and 90% of the average distance, rather than SUPPs belonging to the same front.

For the average distances and normalized distances we can say that the optimal GD depend on the parameters of the instance as it varies between instances of the same size. However, the general trend from Table 9 suggests that good quality MUPP solutions might exhibit an average normalized distance between 0.01 and 0.5. Further investigation is still needed to solidify this connection. It should be noted that the conclusions of these section are only valid for the instance sizes tested and their generalization still has to be verified.

## 6. Conclusion

In this paper, we study the multi-product process planning (MPPP) problem with a reconfigurable manufacturing system (RMS), it consists of: i) Process planning: Assigning operations to machines, configurations, and tools while considering tool and TAD requirements and sequencing the operations. ii) Product sequencing: Sequencing the process plans of the different products to ensure a minimal changeover time and cost from one product to the other. The problem was studied in a Multi-Objective framework where two objectives were considered: The minimization of the total production time and total cost. An exact solution approach was proposed based on the Normal Boundary Intersection (NBI-es) method with a function to iteratively update the NBI parameters β values. The NBI-es approach integrates a relaxed version of the proposed 0-1 LP model based on an Enumerate-and-solve strategy, which was compared to two scalarization techniques NBI and NWS that execute the model directly. The two approximate solution methods consist of two evolutionary metaheuristics: NSGA-II (Non-dominated Sorting Genetic Algorithm II) and MOEA/D (Multi-Objective Evolutionary Algorithm based on Decomposition) with another enhancement using OBL technique (MOEA/D-OBL).

Our findings demonstrate the effectiveness of all three approaches in solving small size instances, with NBI surpassing NWS thanks to its

update function, and NBI-es outperforming both methods. Metaheuristics performed better for small CPU times. Conversely, the NBI-es approach excelled when given larger CPU time both in terms of convergence (HV) and spread (SM), proving that the adaptive β values update formulas maintain well distributed fronts for computational times scenarios. For large instances MOEA/D and MOEA/D-OBL outperformed NSGA-II on most metrics, suggesting its potential as a strong competitor in process planning optimization. Additionally, for convergence-based metrics MOEA/D outperformed MOEA/D-OBL, while MOEA/D-OBL performed better in spread related metric suggesting that OBL strategy improves distribution of non-dominated points across Pareto front approximation. The results also indicate that the total cost is more sensible to operations assignment decisions than sequencing decisions. Suggesting that a hierarchical decision-making heuristic can provide good quality solutions. Such an approach could be developed as an extension of the heuristic framework proposed in Mechaacha et al. (2025).

The disruptive nature of the encoding procedure and genetic operators provides MOEA/D with an advantage over NSGA-II. Therefore, to enhance the performance of NSGA-II and other evolutionary metaheuristics, alternative encoding schemes such as a vector-based solution representation can be investigated, where each product and each component (operation, machine, configuration, and tool) is encoded in a vector and genetic operators can be executed on each vector individually. The results also suggest that a combination of simple one-point crossover and opposite point as a genetic operator can further enhance the performance of metaheuristics. Other crossover operators can be used such as uniform crossover or a designed crossover operator that take certain decisions from each parent, for example maintaining the operations assignment from parent 1 while taking operations sequence from parent 2.

The analysis of Multi-Unit Process Planning (MUPP) solutions indicates that high-quality solutions often comprise a combination of near-optimal Single-Unit Process Plans (SUPPs) alongside suboptimal SUPPs. This result can help design future solution methods for MUPP problem, where Pareto optimal SUPPs are generated first, then these process plans are adjusted to form good quality combinations of MUPPs.

To explore the potential of other metaheuristic paradigms, swarm intelligence-based approaches such as Multi-Objective Ant Colony Optimization could be investigated where the solution is constructed one column after the other, starting from the first column. Additionally, a comparative study between the proposed operations position-based model and a sequence-based one, would provide valuable insights into the efficiency of different mathematical formulations in terms of model complexity and number of decision variables and constraints.

The proposed approaches can integrate into an RMS model (André & Cardin, 2023), where a dedicated software module is designed for process planning and scheduling. This package considers various factors related to machine configurations and operational requirements. The availability of multiple approaches allows for the selection of the most
suitable model, taking into account factors such as available CPU time and the weight of each objective, providing decision makers with multiple trade-off solutions for optimal decision-making in RMS operations. However, to optimize algorithm selection, the tests have demonstrated that it is essential to consider the anticipated level of objective conflict. Therefore, developing a mechanism to predict the level of conflict based on problem instance parameters values would be an interesting contribution.

Additionally, as RMS has been recognized as a facilitator of sustainable manufacturing (Skärin et al., 2022), incorporating environmental objectives, such as minimizing greenhouse gas emissions and energy consumption, into the MPPP optimization problem can be a promising direction. Additionally, considering objectives related to reconfigurability and flexibility could enhance the ability to adapt to changing market demands.

With the emergence of human worker considerations, workforce planning with RMS have gained increased attention (Dolgui et al., 2019). Thus, another future extension can incorporate these aspects, introducing worker assignment to operations. The operation-worker assignment would depend on different factors such as operation difficulty and worker proficiency. This can take use of the results of the data-driven analysis presented in Lee et al. (2023).

Furthermore, a sensitivity analysis studying the impact of varying the processing and changeover times/costs on solution structure and objective function values, similar to Zhang and Wong (2016), but considering MPPP, would be an interesting work and provide valuable insights into the problem's behaviour.

## CRediT authorship contribution statement

**Abdelkader Mechaacha:** Conceptualization, Methodology, Implementation, Writing – original draft; **Fayçal Belkaid:** Writing – review & editing, Supervision, Analysis; **Nadjib Brahimi:** Writing – review & Editing, Supervision, Project administration.

## Data availability

Data will be made available on request.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgement

The authors would like to thank Dr. Boufelouh Radhwane for his valuable comments and suggestions for enhancing the quality of the work.

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