# A Manufacturing-as-a-Service Scheduling Problem

Cristian Duran-Mateluna, Valeria Borodin, Alexandre Dolgui, Simon Thevenin

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Cristian Duran-Mateluna, Valeria Borodin, Alexandre Dolgui, Simon Thevenin. A Manufacturing-as-a-Service Scheduling Problem. ROADEF 2025: 26ème congrès annuel de la Société Française de Recherche Opérationnelle et d'Aide à la Décision, École Nationale des Ponts et Chaussées, Feb 2025, Champs Sur Marne, France. hal-04979779

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# A Manufacturing-as-a-Service Scheduling Problem

Cristian Duran-Mateluna<fnref n="1" />,², Valeria Borodin<fnref n="1" />, Alexandre Dolgui<fnref n="1" />, Simon Thevenin<fnref n="1" />

¹ IMT Atlantique, LS2N-CNRS, Nantes, France.

`{cristian.duran-mateluna, valeria.borodin, alexandre.dolgui, simon.thevenin}@imt-atlantique.fr`

² University of Santiago of Chile (USACH), Faculty of Engineering, Industrial Engineering Department, Santiago, Chile.

**Keywords:** *Manufacturing-as-a-Service*, *Scheduling Problem*, *Tardiness*, *Eligibility level*.

## 1 Context and motivation

Manufacturing-as-a-Service (MaaS)<fnref n="1" /> is a transformative trend in modern manufacturing, that relies on the servitization of manufacturing resources enabled by cloud manufacturing [2]. The development of MaaS applications can contribute significantly to increasing the resilience and sustainability of supply chains due to the flexibility, responsiveness, and scalability they offer [2]. Distributed by definition, a MaaS system includes three decision makers: (i) customers of services, (ii) service providers, and (iii) the MaaS framework that centralizes and coordinates the provisioning and release of services [1]. To support manufacturers in sharing and using manufacturing resources, let us consider a MaaS framework that manages the scheduling of requested tasks on shared unrelated parallel machines in a centralized way. In this work, we extend a state-of-the-art problem [3] by incorporating task-resource eligibility scores, generated by a matchmaking process, as input to the scheduling problem and propose a list of Key Performance Indicators (KPIs) to analyze the quality of solutions proposed by the MaaS framework from the perspective of resource providers and customers.

## 2 Problem statement and mathematical formulation

Consider the problem of [3] for the unrelated parallel machine scheduling with eligibility constraints and delivery times defined as follows. Given a set of $n$ service orders (tasks) $j \in \mathcal{J}$ and a set of $m$ shared resources (machines) $i \in \mathcal{M}$, the platform assigns tasks to eligible machines and sequences the tasks on machines. Each task $j$ has a due date $d_j$, a weight $w_j$, a processing time $p_{ij}$, and a delivery time $q_{ij}$ for each machine $i$. All tasks are available at time zero, each task $j$ must be processed by only one machine without interruption, and each machine $i$ can handle at most one task at a time. The objective is to minimize the total weighted tardiness $\sum w_j T_j$, where $T_j = \max\{C_j - d_j, 0\}$ and $C_j$ is the completion time defined as the time a task reaches the customer. Let binary variables $y_{ij}$ indicate if task $j$ is assigned to machine $i$.

In [3], the problem takes as input eligible machine sets $M_j$ for each task $j$. To represent and model both customers' and providers' requirements, MaaS frameworks are typically endowed with matchmaking capabilities (see e.g., [4]), allowing them to create these eligible sets or, even more accurately, to associate weights to express the eligibility between tasks and machines. Therefore, the eligibility scores $e_{ij} \in [0, 1]$ between machine $i \in \mathcal{M}$ and task $j \in \mathcal{J}$ allow us to consider explicitly a minimal eligibility level $\theta$ either in an individual or joint way, as follows:

$$ \sum_{i \in \mathcal{M}} e_{ij} y_{ij} \geq \theta \quad \forall j \in \mathcal{J} \qquad (1) $$

$$ \sum_{j \in \mathcal{J}} \sum_{i \in \mathcal{M}} e_{ij} y_{ij} / n \geq \theta \qquad (2) $$

<footnote n="1">¹https://www.horizon-europe.gouv.fr/manufacturing-service-technologies</footnote>
# 3 Numerical experiments and main findings

We analyze several KPIs related to the perspectives of decision makers characterizing eligibility levels of task assignment on machines, tardiness, and machine active/idle time. First experiments consider small instances with 4 machines and 10 tasks, $w_j = 1$, and $e_{ij} \sim U(0, 1.0)$, generated in a manner similar to [3]. Instances are characterized by two parameters: tardiness factor T, and relative range of due dates R. 15 instance samples have been generated for several pairs of (T,R). The program described in Section 2 subject to Constraints (1) (resp. Constraint (2)) has been firstly solved with a minimal eligibility level $\theta = 0$. Then $\theta$ is iteratively increased by 10% from $\theta^*$ until a feasible solution is not found, where $\theta^*$ corresponds to the effective eligibility level associated with the obtained solution.

<figure><img src="image_2.png" /><figcaption>FIG. 1: Overview of KPIs values: Model [3] s.t. Constraints (1)</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>FIG. 2: Overview of KPIs values: Model [3] s.t. Constraint (2)</figcaption></figure>

Figures 1-2 summarize a selection of KPIs. As expected, Constraint (2) has less impact on total tardiness and machine idle time than Constraints (1) due to the smoothing effect of the mean. This comes at a cost paid by certain customers, who are delivered later and with a lower eligibility level than other ones. To solve this issue and better align with real-life settings, this work will be extended in terms of both optimization criteria (e.g., equitable level of service) and constraints (e.g., available time slots of machines, availability of tasks at different starting times). In addition, the problem under study will be embedded in a more realistic dynamic and collaborative framework to explicitly consider the distributed nature of MaaS.

# References

[1] Wenchong Chen, Pengwei Feng, Xinggang Luo, and Libing Nie. Task-service matching problem for platform-driven manufacturing-as-a-service: A one-leader and multi-follower Stackelberg game with multiple objectives. *Omega*, 129:103157, 2024.

[2] Dmitry Ivanov, Alexandre Dolgui, and Boris Sokolov. Cloud supply chain: Integrating Industry 4.0 and digital platforms in the “Supply Chain-as-a-Service”. *Transportation Research Part E: Logistics and Transportation Review*, 160:102676, 2022.

[3] Söhnke Maecker, Liji Shen, and Lars Mönch. Unrelated parallel machine scheduling with eligibility constraints and delivery times to minimize total weighted tardiness. *Computers & Operations Research*, 149:105999, 2023.

[4] Frauke Schuseil, Michael Hertwig, Joachim Lentes, Nikolas Zimmermann, and Katharina Hölzle. A semantic matchmaking approach to empower human decision-making in manufacturing-as-a-service scenarios [unpublished manuscript, accepted]. In *AHFE International Conference 2024 - Hawaii Edition*, 2024.