Game Theory Models to Support Manufacturing Systems: History
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Paolo Renna

Game theory is a branch of mathematics that studies strategic decision making in situations where multiple agents interact with each other. It provides a framework for analyzing and predicting the behavior of players in competitive and cooperative scenarios. Game theory is a powerful tool for analyzing strategic decision making in various contexts.

  • game theory
  • manufacturing systems
  • cooperation
  • decision making
  • network

1. Introduction

The digital transformation, driven by the principles of Industry 4.0, offers unprecedented opportunities for industrial companies to revolutionize efficiency and customer satisfaction [1][2][3]. This evolution hinges on interoperable physical and cyber systems, decentralization, and real-time data analytics. These advancements empower companies to establish geographically dispersed multi-factory supply chains, enhancing flexibility while reducing labor and logistics costs [4]. Companies achieve this through the strategic distribution of production capacity or by forging collaborative multi-entity supply chains. Industrial applications of multi-site production planning and scheduling abound, spanning semiconductor manufacturing [5], automotive [6], pharmaceutical [7], and TFT-LCD [8]. However, this paradigm shift introduces complexity into planning and scheduling models, requiring solutions that can be found quickly and efficiently. Metaheuristic approaches have shown promise [4], and game theory, in particular, offers a powerful analytical tool that is well-suited to address interactions among multiple decision makers engaged in multi-objective optimization. Game theory is a branch of mathematics that studies strategic decision making in situations where multiple agents interact with each other. It provides a framework for analyzing and predicting the behavior of players in competitive and cooperative scenarios. Game theory is a powerful tool for analyzing strategic decision making in various contexts. Game-theory-based approaches have been successfully applied to solve various complex engineering problems, including power systems, collaborative product design, and production planning, enhancing solving efficiency [9]. A notable advantage of game theory is that it solves distributed algorithms in less time and with less computations compared to heuristic-based approaches [10].

2. Production and Capacity Planning

This section addresses the challenges of production and capacity planning in distributed geographic networks. These networks enable enterprises to pool capacity, services, and technology, enhancing efficiency, responsiveness, and competitiveness. By leveraging this network, enterprises can effectively respond to unforeseen events such as demand fluctuations, machine breakdowns, rush orders, and supplier delays.
Argoneto and Renna [11] proposed a model to support capacity sharing for a set of independent firms that were geographically distributed, combining their resources and predicting demand to improve production and cost efficiency. The partners of the network are independent and share partial information. A multi-agent architecture has been developed to support cooperation activities in this context. The coordination model uses the Gale–Shapley algorithm to find a stable matching among plants in the network, including the information that each partner decides to share using the preferences function.
Krenczyk and Olender [12] studied the problem of production planning in a virtual manufacturing network with geographically distributed manufacturers. The objectives were to minimize cycle time and production costs. The proposed approach uses a multi-agent system that solves the problem with a non-cooperative game. The selection of alternative routes for a set of production orders is modeled as a non-cooperative game f-player non-zero-sum game with complete information. The model proposed is a framework of a potential application but any numerical test is provided.
Yin et al. [13] proposed a non-cooperative model to allocate production to multi-suppliers from one manufacturer. The model considers quality and demand variations. The proposed approach is a non-cooperative game based on the Stackelberg equilibrium, where the manufacturer is regarded as a leader and the suppliers as followers. As argued by the authors, the model needs to be studied on a larger scale to evaluate its application in real industrial cases.
Olender and Krenczyk [14] proposed the use of a game theory approach to support the production planning problem in a virtual manufacturing network. The objectives were the minimization of production and transport costs using a non-cooperative game. A very limited numerical case was discussed.
Hafezalkotob et al. [15] addressed the approach of coalitions of production plants for cooperative production planning problems. They proposed several methods of cooperative game theory, including the Shapley value. The numerical results highlight how cooperation ensures the satisfaction of production plants, reducing total costs.
Bigdeli et al. [16] proposed a game theory model to support a production planning problem with fuzzy variables. Duality theory in the single-objective and weighted sum methods in multi-objective games is proposed to obtain the payoffs of the players.
Renna [17] studied capacity and resource allocation in flexible production networks. The objective is to obtain a trade-off between the costs and flexibility of the network to satisfy the customer demand. A dynamic allocation of the flexibility is proposed based on the game theory approach using the Gale–Shapley algorithm. The proposed model allows the performance of the network to improve compared to the long-chain approaches proposed in the literature.
Nishizaki et al. [18] addressed two-stage stochastic linear production planning with partial cooperation, involving resource pooling, technology transfer, and product transshipment. Manufacturers determine production levels individually in the first stage, and then collaborate to produce products using pooled resources in the second stage. Additional profits from cooperative game theory are distributed among all manufacturers.
Table 1 summarizes the primary contributions of recent research on production and capacity planning, outlining key characteristics such as the addressed problem (capacity or production planning), the development of multi-agent System (MAS) architecture to support activities, the level of cooperation (non-cooperative or coalition), and the specific algorithm proposed (Gale–Shapley or Shapley Value). Notably, recent studies show that while one study utilized a coalition approach, the majority focused on non-cooperative models. Additionally, capacity planning has been addressed to a lesser extent compared to production planning.
Table 1. Production and capacity planning works.
  Capacity Production Planning MAS Non-Cooperative Coalition Gale–Shapley Shapley Value Nash
Argoneto and Renna [11] X   X     X    
Krenczyk and Olender [12]   X X X        
Yin et al. [13]   X   X        
Olender and Krenczyk [14]   X   X        
Hafezalkotob et al. [15]   X     X   X  
Bigdeli et al. [16]   X   X        
Renna [17] X         X    
Nishizaki et al. [18]   X            

3. Scheduling

In today’s competitive environment, scheduling models need to be highly responsive to real-time events, leveraging the vast amount of data available from Industry 4.0 technologies. However, the abundance of information also brings about increased computational complexity, necessitating more efficient scheduling algorithms to capitalize on this opportunity. Game theory emerges as a promising model to address scheduling challenges with greater efficiency.
Sun et al. [19] studied the flexible job-shop scheduling problem subject to machine breakdown, considering the objectives of robustness and stability. To optimize these two objectives, they modeled the problem as a non-cooperative game and the Nash equilibrium was derived to optimize the two objectives.
Chandrasekaran et al. [20] studied the n-job, m-machine job-shop scheduling problem using a game theory model to find the optimal makespan, mean flow time, and mean tardiness values. The approach proposed is a simplified heuristic derived from game theory tested in a reduced scheduling problem.
Han et al. [21] studied the flow-shop scheduling problem with component altering times, which is a particular problem for sequence-dependent setup times. They developed six rules for the machine assignment of jobs and proposed a Nash equilibrium model to manage these rules. The numerical results show how the game theory model performs better than a model using a genetic algorithm.
Renna [22] proposed a reconfigurable machine scheduling method based on a Gale–Shapley model. The Gale–Shapley model forms a coupled of overloaded and underloaded machines to allocate the modules for the reconfigurable machines. The numerical results of the simulation model show how the game theory model improves all performance measures with a restricted number of machine reconfigurations.
Wang et al. [23] proposed a multi-agent architecture to support real-time scheduling in flexible job-shop systems. A bargaining game model based on the Nash equilibrium was developed to support coordination among the agents of the architecture.
Nie et al. [24] modeled the flexible job-shop scheduling problem as a game theory model where the manufacturer wants to minimize the makespan of all jobs, and the job wants to minimize its tardiness. The game is solved by searching the Nash equilibrium, supported by a genetic algorithm.
Renna et al. [25] studied the dual resource scheduling problem in job-shop manufacturing systems. They proposed a Gale–Shapley model to support worker assignment for dual resource-constrained job-shop problems. The simulation experiments highlight how the Gale–Shapley model leads to better results, particularly when the workers have different efficiency levels.
Atay et al. [26] studied open-shop scheduling problems to minimize the total completion times. They proposed a cooperative TU game and allocate the affected jobs for each alliance to minimize the makespan.
Han et al. [27] modeled the flow-shop scheduling problem with multiple batches as a game model. The method proposed is based on multi-player cooperation and a static game with complete information. The proposed method allows the waiting time to be reduced and improves other performance measures of the flow line.
Wei et al. [28] addressed the multi-objective dynamic flexible job-shop scheduling problem when unforeseen events such as machine breakdown occur. They developed a model that approximates the Nash equilibrium solution to balance Pareto optimality and fairness between the two objectives of production efficiency and stability. The numerical results of several problem sizes are compared to three meta-heuristics proposed in the literature.
Table 2 summarizes the key findings from recent research on scheduling problems in manufacturing systems. The table highlights various characteristics, including the type of manufacturing system studied (flow line, job-shop, and reconfigurable), the game theory approaches employed (cooperative, Nash equilibrium, and Gale–Shapley), the use of multi-agent system (MAS) architecture, and the integration with genetic algorithms.
Table 2. Studies on scheduling.
  Flow Line Job Shop RMS Open
Shop		 Cooperative Model Nash
Eq.		 Gale–Shapley MAS Genetic Algorithm
Sun et al. [19]   X       X      
Chandrasekaran et al. [20]   X     X        
Han et al. [21] X         X      
Renna [22]     X       X    
Wang et al. [23]   X       X   X  
Nie et al. [24]   X       X     X
Renna et al. [25]   X         X    
Atay et al. [26]       X X        
Han et al. [27] X       X        
Wei et al. [28]   X     X      
Notably, recent research reveals that only one study has explored reconfigurable manufacturing systems and open-shop scenarios. Additionally, the integration of game theory with other optimization techniques, such as genetic algorithms, has only been proposed in a single article.

4. Sustainable Production Systems

In recent years, the growing relevance of climate change, coupled with rising energy costs, has prompted manufacturing systems managers to prioritize energy efficiency and the utilization of renewable energy sources. These factors underscore the increasing significance of sustainable production systems.
Zhang et al. [29] proposed a dynamic game model based on the Nash equilibrium to improve production efficiency further and reduce processing costs, including total energy consumption for flexible job-shop problems. The numerical test highlighted reducing makespan, the total workload of machines, and the total energy consumption compared to genetic algorithm solutions.
Renna [30] developed a model to allocate the power to machines using the Gale–Shapley algorithm. The model exchanges the power from the underloaded to overloaded machines. The simulation results show how the model can improve the performance of a manufacturing system under a constraint power limit.
Wang et al. [31] studied the real-time scheduling problem in a job shop with the application of Internet of Things technology to improve production efficiency and reduce energy consumption. An infinitely repeated game optimization approach is developed, and the numerical results show that game theory can improve results compared to other dynamic scheduling methods.
Schwung et al. [32] presented a multi-agent architecture for a decentralized control design of modular production units. The interactions among the agents are supported by a game theory approach. The numerical tests show promising results for improvements in production efficiency in terms of energy consumption as well as throughput times.
Wang et al. [33] proposed a scheduling model for a flexible job shop in real-time. An evolutionary game-based solver method was proposed to support the scheduling model improving energy efficiency.
Sun et al. [34] proposed a digital twin framework to support process planning and scheduling in job-shop systems. Then, a dynamic game theory was adopted to improve production efficiency and reduce energy consumption. The model considered two sub-games, the process planning sub-game and scheduling sub-game, integrated with the Nash equilibrium solution.
Zhao et al. [35] proposed an optimization method for shared energy storage in microgrids using negotiation game theory. This establishes a cooperative interaction mechanism between Microgrid Cluster Operator (MGCO) and Shared Energy Storage Operator (SESO), leading to an optimization framework for microgrid clusters. The dynamic leasing of shared energy storage is considered, resulting in a negotiation game-based capacity configuration model for MGCO and SESO, demonstrating a cost reduction for MGCO and revenue increase for SESO.
Table 3 summarizes the key findings from recent research on applying game theory to sustainable production systems. While the majority of studies focus on job-shop systems, only one addresses peak power constraints. Significantly, no studies explored using game theory to optimize the adoption and integration of renewable energy sources within manufacturing systems.
Table 3. Studies on sustainable production systems.
  Job Shop Nash
Equilibrium		 Gale–Shapley MAS Cooperative Model Energy Peak
Power
Zhang et al. [29] X X       X  
Renna [30] X   X X     X
Wang et al. [31] X       X X  
Schwung et al. [32] X     X X X  
Wang et al. [33] X       X X  
Sun et al. [34] X X       X  
Zhao et al. [35]         X X

5. Cloud Manufacturing

Cloud manufacturing represents an emerging paradigm where distributed resources are encapsulated into cloud services and centrally managed. This network of shared resources enables customers to access on-demand services supporting the entire product lifecycle. The efficiency of cloud manufacturing is heavily dependent on coordination models.
Su et al. [36] studied the problem of manufacturing resource allocation, in which the manufacturing service demander and cloud manufacturing service platform operator are considered gamers. They proposed a non-cooperative game approach to support the problem of resource allocation.
Liu et al. [37] proposed a model of resource and service sharing in cloud manufacturing sharing based on the Gale–Shapley algorithm. The results of the proposed model highlighted that there are always enterprises of the network that perform worse.
Carlucci et al. [38] proposed a coordination model based on a minority game to allocate resources/services among partners of a cloud manufacturing system. The proposed model was tested in a simulation environment compared to a model with complete information among the partners.
Xiaoning et al. [39] investigated three resource-sharing strategies: independently, as an alliance, and by cooperating with a cloud platform operator. The interactions between the operator and suppliers were modeled as a two-stage Stackelberg game that contains a simultaneous sub-game. They found the highest system profit when the suppliers cooperate with the operator.
Xiao et al. [40] proposed a cloud manufacturing multi-task scheduling model based on game theory from a customer perspective. The model is derived from the Nash equilibrium game. The simulation results highlight how the proposed model leads to better results compared to basic biogeography-based optimization algorithms, genetic algorithms, and particle swarm optimization.
Wang et al. [41] studied decentralized decision making in the management of manufacturing service allocation in cloud manufacturing systems. The model is based on an evolutionary game approach able to converge to equilibrium.
Zhang et al. [42] considered a cloud manufacturing system where each manufacturer provides manufacturing resources; when the cloud manufacturing received an order, it coordinated manufacturing resources to satisfy order requirements. To solve the scheduling problem of cloud manufacturing, they proposed a genetic algorithm with the use of the Nash equilibrium for a non-cooperative game model.
Liu et al. [43] studied the application of a cloud manufacturing approach for 3D printing services. They proposed a non-cooperative game model for a 3D printing service scheduling problem. The non-cooperative game is based on Nash equilibrium points supported by a genetic algorithm.
Liu et al. [44] proposed a game-theory-based collaborative scheduling approach for cloud manufacturing (CMfg), addressing dynamics and uncertainties. It optimizes manufacturing and logistic resources efficiently, considering fuzzy uncertain task migration. The model achieves the Nash equilibrium through a decision tree optimization algorithm, enhancing transportation efficiency. Simulation results validate its effectiveness and performance in dynamic CMfg environments.
Koochaksaraei et al. [45] presented a novel approach for cloud service providers (CSPs) to efficiently allocate resources through a barter-based auction market, using evolutionary game theory. CSPs estimate and bid their resources without monetary exchange, fostering cooperation and reducing SLA violations. The simulation results demonstrate improved social welfare and fewer contracts.
Zhang et al. [46] proposed a real-time strategy for a flexible job-shop scheduling problem-based on game theory. The solution and optimization strategy for process tasks using the Nash equilibrium was designed and developed to implement the dynamic optimization model. A case study is presented to demonstrate the efficiency of the proposed strategy and method.
An emerging issue concerns using the circular economy to improve the sustainability of different enterprise sectors, such as manufacturing systems [47], the apparel industry [48], and civil engineering [49].
Tushar [50] provided a recent overview of the literature on cyber–physical systems supported by different game theory models. They argued that multi-agent and game theory are adapted to support cyber–physical systems.
Table 4 summarizes the key findings from recent research on applying game theory to cloud manufacturing systems. Notably, the Nash equilibrium is the dominant approach, with fewer studies exploring alternative methods such as the Gale–Shapley model or minority game models. This suggests potential avenues for future research.
Table 4. Studies on cloud manufacturing systems.
  Resources Service Nash
Eq.		 Gale–Shapley Cooperative Model Non Cooperative Model Minority Game
Su et al. [36] X         X  
Liu et al. [37] X X   X      
Carlucci et al. [38] X X         X
Xiaoning et al. [39] X       X    
Xiao et al. [40] X   X        
Zhang et al. [44] X   X        
Wang et al. [41]   X X        
Zhang et al. [42] X   X     X  
Liu et al. [43] X   X        
Liu et al. [44]   X X   X    
Koochaksaraei et al. [45]   X       X

This entry is adapted from the peer-reviewed paper 10.3390/designs8020026

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