A Game Theoretic Competitive Supply Chain Network Model with Green Investments and Labour

MENDEL — Soft Computing Journal, Volume 29, No.g1, June 2023, Brno, Czech RepublicX ISSN: 1803-3814 (Printed), 2571-3701 (Online) https://doi.org/10.13164/mendel.2023.1.025 A Game Theoretic Competitive Supply Chain Network Model with Green Investments and Labour Kurt Pace Debono  , Maria Kontorinaki, Monique Sciortino Department of Statistics and Operations Research, University of Malta, Malta kurt.pace-deb[email protected]  , maria.konto[email protected], monique.scio[email protected] Abstract In light of the recent severe Supply Chain (SC) disruptions that have occurred across multiple industries around the globe, three essential and linked themes have emerged in SC management: the well-being of employees, SC sustainability, and competition between SCs for limited resources. In this paper, we create a game-theoretic SC network model that incorporates together non-cooperative SC competition, employee productivity and engagement, and green investing. Each competing firm within the network seeks to maximise its profit by determining an optimal flow of products and allocation of green investments across the SC according to a predetermined budget. A carbon tax on emissions and consumer sustainability preferences are also included in the model. The model is solved using a Variational Inequality reformulation. The illustrative numerical examples presented in this paper have been inspired by the Maltese dairy industry and demonstrate the applicability of the model to real-world problems. The results highlight the significance of the employee engagement factor in enabling firms to adopt and realise more sustainable SC practices. Keywords: Green Supply Chain Management, Supply Chain Network Modelling, Optimisation, Game Theory, Employee Engagement, Labour, Sustainability Received: 27 April 2023 Accepted: 01 June 2023 Online: 06 June 2023 Published: 30 June 2023 1 Introduction The interconnectedness within and between Supply Chains (SCs), both on a local and global scale, has never been more apparent; SC disruptions caused by the global pandemic have occurred in multiple industries worldwide, while competition between SCs for limited natural and human resources has intensified. With heightened awareness around climate change and its direct effects being felt across the globe, governments and international institutions are pushing to legislate towards greener initiatives. For example, the European Union launched the European Green Deal in 2019, which aims for the bloc to be climate neutral by 2050, with significant investments being planned to decarbonise significant polluters [6]. What is more, employers and researchers are realising that even employees are expecting more when it comes to the Environmental, Social and Governance (ESG) credentials of the company they work with. However, in Supply Chain Management (SCM) research, the human element is often overlooked. This was highlighted in [29], where over a hundred SCM researchers were asked which research themes they felt had been under-researched. Indeed, it was found that the most common answer was the people dimension of SCM, noting that only a few studies researched the “dynamics of consumers, managers, or other individual actors within a supply chain system”. Nonetheless, we can still look at business management research as well as current trends that explore the link between employees and the environment. For example, in a 2020 global survey on employee expectations carried out amongst 14 million respondents, a 52% increase in environmental concern was registered over previous years [23]. This increase was significantly higher amongst the youngest generation of workers, with Generation Z respondents (born 1997-2007) registering a 128% increase, indicating that this issue is only going to become more important. This sentiment was amplified in a 2022 survey, which found that employees of purpose-driven employers are three times more likely to continue working with them, while 75% of respondents stated that they would be more likely to buy from a business that incorporates ESG credentials [24]. Various studies have theorised and tested the impact that going green has on employee productivity and employee engagement, the latter of which referring to the “level of commitment and involvement an employee has towards their organisation and its values” [3]. Employee engagement literature has honed in on the idea that employees who feel that the values of a firm align with their personal values, and that their contributions are meaningful, are more engaged in their work and, as a result, more productive [28]. This idea was incorporated into a theoretical model in [11] that links the impact of sustainability with employee engagement, with the main bridge between the two being the increased sense of meaningfulness that 25 MENDEL — Soft Computing Journal, Volume 29, No.g1, June 2023, Brno, Czech RepublicX sustainable practices promote. Also, in [4], the authors sought to test the hypothesis that the adoption of sustainable practices is associated with increased labour productivity. Through an employee survey from 5220 firms, they found that firms that implemented environmental standards had one standard deviation of labour productivity higher than those that did not. In addition, the authors of [22], who studied the relationship between green human resource management and Green SCM (GSCM), found evidence to support the hypothesis that employees’ empowerment in sustainable progression positively influences the implementation of GSCM practices. GSCM concepts have entered the Supply Chain Network (SCN) modelling literature, with researchers adding environmental aspects to their models and studying their interactions with other parts of the model. For example, in [30], the authors modelled different carbon tax policies and found that such taxes can encourage firms to reduce emissions. The relationship between consumer environmental awareness and green SCs was studied in a model by [14], which found that the more aware consumers are, the more profitable sustainable firms will be. In [31], the authors developed a multi-period model that incorporated the relation between green investments and consumer purchasing behaviour, which highlighted that consumers have the power to encourage firms to go green. For an overview of the different components studied within green SC models, one can refer to [1]. The inclusion of labour in SCN models is a more recent area in SCN modelling literature, and to our knowledge is one that has not been studied in conjunction with green SCN models. It was first studied in [18], where the product flows in a competitive SCN were modelled as a function of labour, with firms also competing on the availability of human resources in the labour market. Following this initial paper which highlighted the importance of safeguarding employee health, the impact that investments in labour productivity can have on the SC profitability of a single firm was studied in [17,19]. In [19], a single-firm model was created with the aim of optimising the firm’s product flows and investments in labour productivity enhancements such as physical workplace improvements, training or health and safety, with labour availability being dependent on the wage offered by the firm. This model was extended to a multi-period model in [17], with labour productivity investments being incorporated into the demand-price function to model consumer sensitivity to the working conditions of the firm’s workers. These papers both concluded that investments in labour productivity increased profits. In light of this discussion, we can see that there is an interesting rationale behind modelling the interplay between green investments, employee engagement and labour productivity in our SCM model, which would be an original contribution to SCM literature. In this paper, we have developed a game-theoretic SC competition model, with a particular focus on the aspects of employee productivity and engagement, investments in green initiatives, and the link between the two. The model consists of a number of firms competing in an oligopolistic industry, whereby each firm seeks to maximise its profit by determining product flows and green investment allocations, within a predetermined budget, throughout the SCN. The element of labour is incorporated into the model by linking product flows with the amount of labour hours available to each firm and the employees’ productivity. In turn, employee productivity is partially dependent on the employee engagement with the green investments that the firm makes. Furthermore, a carbon tax is included in the model, where each firm is taxed based on the amount of CO2emissions it produces. Since the firms compete within the same demand markets, the production and investment decisions made by each firm impact the profitability of all the firms. Assuming that each firm makes the decisions once, and at the same time as all the competing firms, the proposed model is created within a Game Theory (GT) framework as a static, non-cooperative game. Therefore, under this framework, we seek to find a Nash Equilibrium (NE) solution that ensures that no firm will be able to individually improve its profit, given the decisions of the other firms. In order to find such a NE, we use Variational Inequality (VI) theory to reformulate and solve the model. To this end, in Section 2, we construct the SCN competition model with the inclusion of green investments and labour. This will be followed by a VI reformulation of the model and related VI theoretical results concerning the NE solution and its existence. In Section 3, we apply the model using scenarios inspired by the Maltese dairy industry and discuss the resultant managerial insights. Sensitivity analysis is also carried out to study the interplay between green investments and employee engagement introduced for the first time in our model, and its effect on SC profitability, demands and prices. Finally, in Section 4, we present a summary of the results and discuss the conclusions of this paper, as well as the direction for future research. 2 The SCN Competition Model with Green Investments and Labour We consider an industry/network in which there are I firms seeking to maximise their profits by determining optimal product path flows and green investments. In this network, we assume that the firms compete noncooperatively in the delivery of a substitutable product to customers in Rdemand markets. By delivery, we mean the entire set of processes carried out in order to convert raw materials into a product to be sold to customers, while note that a demand market could refer to an individual consumer, a business, an organisation, a retailer or a specific segment of customers. 26 MENDEL — Soft Computing Journal, Volume 29, No.g1, June 2023, Brno, Czech RepublicX Debono-g2igHX,gA Game Theoretic Competitive Supply Chain Network Model with Green Investments and Labour ˆ F1 M1 1M1 n1 M D1 1,1D1 n1 D,1 D1 1,2D1 n1 D,2 ˆ FI MI 1MI nI M DI 1,1DI nI D,1 DI 1,2DI nI D,2 Q1QR . . . . . . . . . . . . . . . ... ... ... ... ............ . . . . . . . . . . . . . . . . . . ... ... Firm 1 Firm I Production Transportation Storage Sales Manufacturing Distribution Demand Centres Markets Facilities Links Links Links Links ...... ... ... ... ... ... ... ............ Figure 1: The SCN competition model topology. 2.1 Variables, Parameters and Functions We denote the set of all firms by I={1,2, . . . , I}and the set of all demand markets by R={1,2, . . . , R}. We can represent each firm i∈ I competing in this industry as a SCN of its economic activities consisting of five tiers: the firm node with label ˆ Fi, i ∈ I; the firm’s ni Mmanufacturing facilities {Mi 1, Mi 2, . . . , Mi ni M }; the first level of the firm’s ni Ddistribution centres {Di 1,1, Di 2,1, . . . , Di ni D,1}, representing the receiving of products from the manufacturing facilities; the second level of the same ni Ddistribution centres {Di 1,2, Di 2,2, . . . , Di ni D,2}, representing the storage facilities; and, the demand market nodes with labels Qr, r ∈ R. Each link between a pair of nodes in different tiers represents a SC process. The links between: the firm node and the manufacturing facilities represent the production processes of each firm; the manufacturing facilities and the first level of the distribution centres represent the transportation of the finished products; the first and second levels within the same distribution centres represent the storage of the products; the second level of the distribution centres and the demand markets represent the sales of the products. It is possible to have the same pair of nodes be connected by more than one link, adding the flexibility to allow for different options for each process, such as different production methods or modes of transport. Links can be grouped together to form a path (having one link of each type), which is a series of links that starts from a firm node and ends at a market node. Paths can be grouped into three sets: Pi r, the set of all paths that join firm i∈ I with demand market r∈ R;Pi, the set of all paths that join firm i∈ I with all the Rdemand markets; and, P, the set of all paths in the SCN. We depict the network of all the firms’ nodes and links in the graph G= (N, L) in Fig. 1, where Nis the set of all nodes and Lis the set of all links. Note that the SCs of the individual firms share no links with one another, thus we can group all the links representing the SC processes of firm i∈ I into the set Li. Each firm i∈ I seeks to maximise its profit by optimising two strategic vectors of decision variables: the vector of product path flows xi={xp}p∈Piand the vector of green investments vi={vl}l∈Li. A product path flow, which we denote by xp, refers to the flow of products along the path p∈Pi. On the other hand, vlrepresents the amount invested in green initiatives on link l∈L. For example, on production links, these could represent the investment in solar panels or the introduction of environmentally friendly materials. On the transportation links, green investments could represent new electric vehicles, while on the storage links these could include the purchasing of energy efficient refrigeration units or the upgrading of climate control systems. Central to this model, inspired by [18], is that the product path flows will be determined by the availability of labour hours hl, l ∈Li, the firm i∈ I has at its disposal. Two variables related to xpwill aid us in the formulation and interpretation of the model: the link flows fl,l∈Land demands di r,i∈ I,r∈ R. The link flow flrepresents the amount of flow along link l∈L in the SCN, while the demand di ris the total amount of products delivered by firm i∈ I to demand market r∈ R. Demands can be grouped into two vectors: di={di r}r∈R, the vector of demands of firm i∈ I at all demand markets, and dr={di r}i∈I, the vector of demands of all firms at demand market r∈ R. Having defined the decision variables, we now define the functions that will make up our model’s objective function. Firstly, we define the function that will determine the price of the competing products being sold. To this end, we define the demand price function ρi r(dr,v) which calculates the unit price of a product 27 MENDEL — Soft Computing Journal, Volume 29, No.g1, June 2023, Brno, Czech RepublicX of firm i∈ I at demand market r∈ R as: ρi r(dr,v) = πi−X j∈I σj rdj r+ηi rX l∈Li vl−X j∈I j=i ηj rX l∈Lj vl, ∀i∈ I,∀r∈ R. (1) This function is an adaptation of the inverse demand function used in economics to express price as a function of the quantity demanded. The parameter πirepresents all the factors affecting the price of the product of firm i∈ I other than the total demand and green investment. The parameters σj rrepresent the effect that the demand of each of the firms’ product at demand market r∈ R has on the price of the product of firm i∈ I at that same demand market. The parameters ηj rrepresent the effect of the total green investment of each firm on the price of the product of firm i∈ I at demand market r∈ R. This signifies the idea that consumers may be willing to pay more to purchase products that are environmentally friendly, a concept which was explored in [14,27,31]. Since the firms’ products are substitutable, we note that the demands and investments of one firm also impacts the prices of other competing firms’ products. Next, we define the function that will capture the production/transportation/storage costs associated with the link flow fl, as well as additional costs associated with green investments. Note that labour costs will not be considered in this function and will be inserted into the objective function separately by multiplying the number of hours worked hlwith the hourly wage ωl. We define the operational cost function associated with link l∈Las: ˆcl(fl, vl) = γlf2 l+µlflvl,∀l∈L. (2) The parameter γlrepresents the cost per unit squared of fl. The term flis squared to model the economic concept of marginal cost; as the flow along a link lnears its maximum capacity, which is dictated by the upper bound on labour available, denoted by ¯ hl, the cost per unit increases [27]. The parameter µlrepresents the additional marginal cost per unit of flow that may arise out of the investment, such as increased maintenance requirements. Since one of the main features of our model is the inclusion of labour, we construct a function that relates the amount of labour hours hlworked with the product output on each link l∈L. A novel feature of our model is the relation of the amount vlinvested in green initiatives with productivity. To this end, we define the labour productivity function associated with link l∈L as: ˆgl(hl, vl)=(αl+βlvl)hl,∀l∈L. (3) The parameter αlrepresents the factor directly relating the labour input to production output, such that one labour hour on link l∈Lproduces αlunits of flow. On the other hand, the parameter βlrepresents the impact that green investments have on the productivity of employees. This can be interpreted as a metric of employee engagement, whereby the more engaged employees are with the firm’s investments, the more productive they are at their jobs. We also include a function that tracks the amount of carbon emissions being generated throughout the SCN. The emissions function associated with link l∈Lcan be defined as: ˆel(fl, vl) = ξlfl−φlflvl,∀l∈L. (4) In this equation, similar to that defined in [31], the CO2 emissions in tonnes are calculated as a function of the product flows and green investments. A relationship is modelled between the product flows and the emissions on a link l∈L, with every unit flow creating ξlunits of CO2. However, for every e1 invested in green initiatives, the emissions generated by one unit of flow are reduced by φl. Finally, we model the introduction of a carbon tax on emissions. The carbon tax function associated with link l∈Lcan be defined as: ˆ tl(ˆel(fl, vl)) = τˆel(fl, vl),∀l∈L, (5) where τis the flat tax rate per tonne of CO2emitted. For example, τcould be equal to e50 per tonne emitted. 2.2 Objective Function and Constraints Recall that each firm i∈ I seeks to maximise its profit by deciding its strategic product flows and green investments. Thus, we define the objective function of firm i∈ I as the profit function: Ui= R X r=1 ρi r(dr,v)di r−X l∈Li ˆcl(fl, vl)−X l∈Li ωlhl −X l∈Li ˆ tl(ˆel(fl, vl)) −X l∈Li vl. (6) The first term PR r=1 ρi r(dr,v)di rof (6) is the total revenue of firm i∈ I across the Rdemand markets, calculated by multiplying the price of the product of firm i∈ I at demand market r∈ R by the demand of that product in that market. To arrive at the profit figure for firm i∈ I, we then subtract from the total revenue term the total operational costs Pl∈Liˆcl(fl, vl), wages Pl∈Liωlhl, carbon taxes Pl∈Liˆ tl(ˆel(fl, vl)) and investments in green initiatives Pl∈Livlacross the nLi SC process links of the firm. The optimisation of (6) is subject to a number of constraints. First, we require that the flow flalong a link l∈Liequals the sum of that product’s flow along all the paths xpthat contain that link, such that: fl=X p∈Pi xpδl,p,∀l∈Li,∀i∈ I,(7) where δl,p is a parameter that indicates whether link l is contained in path por not; δl,p = 1 if link l∈Lis 28 MENDEL — Soft Computing Journal, Volume 29, No.g1, June 2023, Brno, Czech RepublicX Debono-g2igHX,gA Game Theoretic Competitive Supply Chain Network Model with Green Investments and Labour contained in path p∈Pand δl,p = 0 otherwise. Also, the path flows must be non-negative: xp≥0,∀p∈Pi,∀i∈ I.(8) Additionally, we ensure that the demand for the product of firm i∈ I at market r∈ R is satisfied by the sum of product flows along all the paths p∈Pi r starting from said firm and ending at said demand market: X p∈Pi r xp=di r,∀i∈ I,∀r∈ R.(9) To relate flows with labour, we equate the product flow flon link l∈Li, i ∈ I, to the labour productivity function (3): fl= (αl+βlvl)hl,∀l∈Li,∀i∈ I,(10) which is capped by the upper bound ¯ hlthat we set on the total labour hours hlavailable on link l∈Li, i ∈ I: hl≤¯ hl,∀l∈Li,∀i∈ I.(11) The last two sets of constraints which we consider relate to the green investments made by the firms: First, we set bounds for the green investment vl, i.e.,: vmin lfl≤vl≤vmax lfl,∀l∈Li,∀i∈ I,(12) where vmin land vmax ldenote the minimum and maximum amounts of green investment per unit flow through link l∈Li, respectively. For most links, vmin l can be defined as 0, meaning no minimum investment would be required. However, there could exist capitalintensive links with corresponding positive vmin l, which cater for scenarios where firms deem a minimum investment amount per unit of flow necessary to set up such a link. For example, a transportation link could represent the option a firm has to invest in an electric vehicle; for such an investment to be feasible, the minimum investment required per unit of flow to use this link would be the cost of one vehicle divided by the amount of units projected to flow through the link. By multiplying the parameters vmin land vmax lby the flow variable flin (12), we ensure that if a firm decides not to make use of a link, then no investment will be made, i.e., fl= 0 =⇒vl= 0. Second, the total budget constraint for firm i∈ I, where the total sum of green investments over the firm’s entire set of links Li cannot exceed the firm’s budget Θi, i.e.,: X l∈Li vl≤Θi,∀i∈ I.(13) Considering the above, the optimisation problem faced by each firm is therefore to maximise its profit (6) subject to constraints (7) - (13). 2.3 Variational Inequality Reformulation To aid the reformulation of the objective function into a VI problem, we rewrite the optimisation problem of each firm i∈ I in terms of the path flow variables x= {xi}i∈I and green investment variables v={vi}i∈I . By constraint (7), we can replace flwith Pp∈Pxpδl,p wherever it appears in the objective function (6), as well as in the green investment constraint (12), giving us the following constraint: vmin lX p∈P xpδl,p ≤vl≤vmax lX p∈P xpδl,p.(14) A similar replacement can be done for the demand di r using the relation in constraint (9). Constraints (7) and (10) can be equated to each other, and then can be further combined with constraint (11) to give us the following: X p∈P xpδl,p ≤(αl+βlvl)¯ hl,∀l∈L. (15) Thus, our model aims to maximise the profit of each firm i∈ I ˜ Ui(x,v) = R X r=1 ˜ρi r(x,v)X p∈Pi r xp−X l∈Li ˜cl(x, vl) −X l∈Li ωl αl+βlvlX p∈P xpδl,p −X l∈Li ˆ tl(˜el(x, vl)) −X l∈Li vl, (16) subject to constraints (8), (13), (14) and (15), where ˜ρi r(x,v) = ρi r(dr,v), ˜cl(x, vl) = ˆcl(fl, vl) and ˜el(x, vl) = ˆel(fl, vl) for every i∈I,r∈Rand l∈Li. We can define the feasible set of this problem for each firm i∈ I as: Ki≡ {(xi,vi)|(8),(13),(14)&(15) hold}.(17) However, looking at the objective function (16), we notice that the profit ˜ Ui(x,v) of firm i∈ I is determined not only by the firm’s optimal choice of (xi,vi), but also by its competitors’ decisions for their own product flows and green investments. Thus, we can use Game Theory (GT) to solve the SCN optimisation problem using the framework of the non-cooperative game ⟨I,(x,v),U⟩,where Iis the set of firms; (x,v) is the tuple of strategies consisting of the product flow and green investment vectors xiand viof each firm i∈ I; and, U={˜ Ui(x,v)}i∈I is the set of objective functions of all firms. We define the feasible set of this oligopolistic competition problem as the set: K= I Y i=1 Ki=K1×K2× · · · × KI.(18) The optimal solution in such a GT framework would be what is known as a Nash Equilibrium (NE) solution [21]. To define the form of a NE solution, let us first define the decision vectors of the competitors of firm i∈ I, relating to the product flows and the green investments, i.e.,: x−i= (x1,x2, . . . , xi−1,xi+1, . . . , xI),∀i∈ I, 29 MENDEL — Soft Computing Journal, Volume 29, No.g1, June 2023, Brno, Czech RepublicX v−i= (v1,v2, . . . , vi−1,vi+1, . . . , vI),∀i∈ I, respectively. For our model, a NE (x∗,v∗) is established if a firm i∈ I cannot individually improve its profit by changing its decisions (xi∗,vi∗), given the other firms’ decisions (x−i∗,v−i∗). Definition (Model Nash Equilibrium).A tuple of path flows and green investments (x∗,v∗)is said to be a NE for the competitive SCN model if for each firm i∈ I: ˜ Ui(xi∗,vi∗,x−i∗,v−i∗)≥˜ Ui(xi,vi,x−i∗,v−i∗), ∀(xi,vi)∈Ki.(19) Under this definition, an optimal solution to our model is not focused on maximising one individual firm’s profit, but instead optimising all the firms’ decision vectors concurrently in such a way that the SCN is in equilibrium. 2.4 Related VI Theoretical Results VI theory provides the tools necessary to find the equilibrium of mathematical problems, a solution concept that is central to GT. Thus, by reformulating our GT competitive SCN model into a VI problem, we can proceed with finding a NE solution. A background on related theorems and algorithms can be found in [9,16,20]. For our model, we will apply the following theorem, the proof of which can be found in [10]. Theorem (NE solution).Assume that for each firm i∈ I, the profit function ˜ Ui(x,v)is continuously differentiable and concave in xand v. Also, assume that the feasible set Kis convex. Then, (x∗,v∗)∈Kis said to be a NE for our competitive SCN model if and only if it satisfies the VI: − I X i=1 ∇xi˜ Ui(x∗,v∗),xi−xi∗ − I X i=1 ∇vi˜ Ui(x∗,v∗),vi−vi∗≥0,∀(x,v)∈K, (20) where ⟨·,·⟩ is the inner product in the n-dimensional Euclidean space. The theorem above links the solution of a VI to a NE solution. We will now proceed with explaining how the assumptions of the theorem above hold for our SCN model. The feasible set Kin (18) is convex as it is the Cartesian product of convex sets (each Ki,i∈ I, is constructed by considering simple and linear bounds for the decision variables). Also, from the form of (16), it can easily be observed that the utility functions are continuously differentiable in Ki, for each i∈ I. The concavity of the utility functions is typically assumed throughout the SCN modelling literature [18,30,31, 27]. For the sake of completeness, we also provide an existence result that is relevant to our model [26]. Theorem (Existence).The existence of a NE for our competitive SCN model is guaranteed under the compactness of the feasible set Kin (18) and continuous differentiability of each ˜ Ui, i ∈ I. Since K⊂RI, compactness of Kfollows from the compactness of Ki’s. Each Kiis compact. i.e., closed and bounded. From (8), we have that xp≥0,∀p∈ Pi,∀i∈ I. Thus, each path flow fl=Pp∈Pxpδl,p is bounded below. Moreover, from (15), we know that each path flow flis bounded above by (αl+βlvl)¯ hl. From (14), we have that vmin lfl≤vl≤vmax lfl,∀l∈ Li,∀i∈ I. Thus, vlis bounded below and above. Inequality (13) is another constraint on the vl’s, such that Pl∈Livl≤Θi,∀i∈ I, imposing an upper bound on the sum of all green investments for each firm. Since all the inequalities that make up each Kiare not strict, we have that each Kiis closed as well. To solve our VI problem, we will be making use of the Extragradient Algorithm [13]. Assuming that the function Fis monotone and Lipschitz continuous, this algorithm is guaranteed to converge to a solution with a polynomial rate of convergence (see Theorem 12.6.4 in [9]). Algorithm 1 Extragradient Algorithm Step 0: Initialisation Set initial solution z0= (x0,v0)∈K. Let the iteration counter t= 1. Let ζbe a scalar such that 0 < ζ ≤1 L, where Lis the Lipschitz continuity constant. Set tolerance ε > 0. Step 1: Computation Compute ¯ zt−1by solving the VI subproblem: ⟨¯ zt−1+ζF (zt−1)−zt−1,z−¯ zt−1⟩ ≥ 0,∀z∈K. Step 2: Adaptation Compute ztby solving the VI subproblem: ⟨zt+ζF (¯ zt−1)−zt−1,z−zt⟩ ≥ 0,∀z∈K. Step 3: Convergence Verification If |zt−zt−1| ≤ ε, then stop; else, set t:= t+ 1 and go to Step 1. 3 Numerical Application In this section, we will be constructing scenarios inspired by the Maltese dairy industry to illustrate the properties of the proposed model, while sensitivity analysis on the most important parameters will also be carried out. 3.1 Data and Scenarios According to [7], the average cost of raw milk in Malta in 2021 was 57.44 cents per kilogram (kg). Competition in this industry exists between farms, as well as with alternative milk products such as dairy free and long-life 30 MENDEL — Soft Computing Journal, Volume 29, No.g1, June 2023, Brno, Czech RepublicX Debono-g2igHX,gA Game Theoretic Competitive Supply Chain Network Model with Green Investments and Labour milk. A breakdown of milk production costs was studied in [8]; on average, labour costs represented around 23% of these costs in 2019, with feed, machinery and equipment costs representing a further 39%, and general operating costs representing the remaining 38%. Interestingly, the authors of the latter report highlight that better recognition of labour costs is required, with the amount of labour hours, experience and knowledge not corresponding with the average labour costs computed. The median annual salary for persons working with livestock in 2018 was e12,500 per annum [12], equating to a wage of approximately e6 per hour. Estimating productivity, in 2019 the average yield was 6,843kg of milk per cow, with the average amount of cows per farm being 67.9 [5], meaning that each Maltese farm had an average output of 464,639kg per annum. With an average labour input of 6,115 hours per farm per annum [5], the milk production per hour of labour can therefore be estimated as approximately 76kg/hour. With regards to emissions, in a study of twelve Maltese dairy farms [25], it was found that the amount of CO2equivalent emissions per kilogram of milk ranged between 1.14kg and 3.00kg. In an analysis of the entire local dairy production process, the author of [2] commented that almost half of the energy consumption occurs during the refrigeration (storage) stage. Based on this data, let us consider a scenario where two competing dairy farms, Farm 1 and Farm 2, would like to optimise the flow of milk and green investments within their SCs, depicted in the SCN topolgy in Fig. 2. Farm 1 has two production facilities M1 1and M1 2 and a distribution centre D1 1, while Farm 2 has one production facility M2 1and one distribution centre D2 1. Both farms serve two demand markets, Q1and Q2. l1 l3 l2 l4 l5 l8 l9 l10 l6l7l11 l12 M1 1 ˆ F1 M1 2 D1 1,1 D1 1,2 Q1Q2 ˆ F2 M2 1 D2 1,1 D2 1,2 Figure 2: SCN topology for two competing farms. We define the operational cost (in cents), labour productivity (in kgs of milk produced per labour hour) and carbon emissions (in kgs of CO2per kg of flow of milk) functions in Table 1. Looking at the parameters in this table, we note that the cost and emissions parameters for Farm 1’s second production facility M1 2are lower since it is equipped with more modern machinery. Similarly, Farm 2 has recently invested heavily in the latest technologies across its SC, thus we notice lower parameters in the emissions functions for Farm 2 compared to those of Farm 1, as well as some slightly higher costs due to using more sustainable materials. With regards to the cost of labour, Farm 1 pays its workers e6 per hour, while Farm 2 opts to pay a higher wage of e7 per hour. Thus, ωl= 600,∀l∈L1and ωl= 700,∀l∈L2. Assuming a 40 hour week and that Farm 1 employs 3 full-time workers while Farm 2 employs 6 full-time workers and one part timer, the upper bounds on labour ¯ hl= 120,∀l∈L1and ¯ hl= 250,∀l∈L2. In the demand price functions of Farm 1 (in cents), we set the baseline price at π1= 100, which falls within the range of current market prices. When setting the parameters η, we specify higher values at Market 2 to simulate customers at this market being more environmentally conscious than those at Market 1. Thus, we set σ1 1=σ1 2= 0.001, σ2 1=σ2 2= 0.0003, η1 1= 0.001, η1 2= 0.0015, η2 1= 0.0008 and η2 2= 0.001. For Farm 2’s functions, we set π2= 130, σ1 1=σ1 2= 0.0002, σ2 1=σ2 2= 0.001, η1 1= 0.0008, η1 2= 0.001, η2 1= 0.001 and η2 2= 0.0015. We note that Farm 2 has a higher baseline price of 130 cents, owing to the significant investments it has made to set up sustainable operations. To combat a newly introduced carbon tax, management at the farms would like to invest in green initiatives in order to reduce their CO2emissions. To this end, they allocate a budget of e5,000 and set vmin l= 0, vmax l= 0.5∀l∈L, such that they would not like to spend more than 0.5c per kg of flow of milk on any specific link. We can define the paths in this model as p1=(l1, l3, l5, l6), p2=(l2, l4, l5, l6), p3=(l1, l3, l5, l7), p4 = (l2, l4, l5, l7), p5= (l8, l9, l10, l11), and p6= (l8, l9, l10, l12). The set of links L={l1, l2, . . . , l12}can be split into those in Farm 1’s SC as L1={l1, l2, . . . , l7} and those in Farm 2’s as L2={l8, l9, . . . , l12}. Scenario 1 (Base Case).We solve the base scenario in MATLAB1using the Extragradient Algorithm with a step size of ζ= 50, tolerance of ε= 0.01 and all path flows xpinitialised at 3000 and green investments vl initialised at 0. The parameter ζhas been chosen by performing a grid search and by assuming a sufficiently large Lipschitz constant L. We obtain the following NE solution: x∗ p1= 4599, x∗ p2= 5780, x∗ p3= 4900, x∗ p4= 6080, x∗ p5= 16543, x∗ p6= 17207, with a profit of e11,804 for Farm 1 and e24,603 for Farm 2, equilibrium demands d1∗ 1= 10379 and d1∗ 2= 10981 for Farm 1 with corresponding prices ρ1 1(d∗ 1,v∗) = 86 and ρ1 2(d∗ 2,v∗) = 86 c/kg, and equilibrium demands d2∗ 1= 16543 and d2∗ 2= 17207 for 1The code for the numerical scenarios can be accessed at: https://github.com/kurtpacedebono/A-Game-Theoretic-Com petitive-Supply-Chain-Network-Model-with-Green-Investm ents-and-Labour.git 31 MENDEL — Soft Computing Journal, Volume 29, No.g1, June 2023, Brno, Czech RepublicX Table 1: Operational cost, labour productivity and carbon emissions functions for Farms 1 and 2. Link ˆcl(fl, vl) ˆgl(hl, vl) ˆel(fl, vl) Farm 1: l10.00042f2 l1+ 0.0005fl1vl1(76 + 0.01vl1)hl10.45fl1−0.000005fl1vl1 l20.00032f2 l2+ 0.0004fl2vl2(80 + 0.01vl2)hl20.34fl2−0.000004fl2vl2 l30.00008f2 l3+ 0.0001fl3vl3(300 + 0.01vl3)hl30.05fl3−0.000001fl3vl3 l40.00008f2 l4+ 0.0001fl4vl4(300 + 0.01vl4)hl40.05fl4−0.000001fl4vl4 l50.0001f2 l5+ 0.0005fl5vl5(150 + 0.01vl5)hl50.45fl5−0.000005fl5vl5 l60.00008f2 l6+ 0.0001fl6vl6(300 + 0.01vl6)hl60.05fl6−0.000001fl6vl6 l70.00008f2 l7+ 0.0001fl7vl7(300 + 0.01vl7)hl70.05fl7−0.000001fl7vl7 Farm 2: l80.00052f2 l8+ 0.0001fl8vl8(85 + 0.01vl8)hl80.25fl8−0.000003fl8vl8 l90.0001f2 l9+ 0.0001fl9vl9(450 + 0.01vl9)hl90.01fl9−0.000001fl9vl9 l10 0.00006f2 l10 + 0.0001fl10 vl10 (250 + 0.01vl10 )hl10 0.3fl10 −0.000004fl10 vl10 l11 0.00004f2 l11 + 0.0001fl11 vl11 (450 + 0.01vl11 )hl11 0 l12 0.00004f2 l12 + 0.0001fl12 vl12 (450 + 0.01vl12 )hl12 0 Farm 2 with corresponding prices ρ2 1(d∗ 1,v∗) = 112 and ρ2 2(d∗ 2,v∗) = 113 c/kg. The equilibrium link flows, green investments, labour requirements and emissions can be seen in Table 2. Table 2: Results for Scenario 1. Link f∗ lv∗ lh∗ lˆel(f∗ l, v∗ l) l19500 316 120 4260 l211860 1884 120 3943 l39500 0 32 475 l411860 0 40 593 l521360 2800 120 9313 l610379 0 35 519 l710981 0 37 549 l833750 5000 250 7931 l933750 0 75 337 l10 33750 0 135 10125 l11 16543 0 37 0 l12 17207 0 38 0 From these results, we can see how Farm 2 performs strongly in both markets, managing to attract higher demands whilst still maintaining higher prices that reflect the higher sustainability of the farm’s practices. As a testament to this better environmental track record, Farm 2 produces 6.4% less total emissions than Farm 1 whilst having a 58% higher total flow of milk throughout its SC. Should Farm 1 wish to improve its position in the market, it should consider investing more into sustainable operations. These observations highlight the impact that competition can have in the market, such that if one firm in a SCN opts to go green and this is well received by the consumers, then this may have a domino effect and convince other competitors to become more sustainable themselves. Scenario 2 (No Employee Engagement with Sustainability).In this scenario, we explore what happens when we remove the increase in labour productivity experienced due to employee engagement with green investments. Thus, we set the parameter βl= 0 for all links l∈L, affecting the productivity functions ˆgl(hl, vl), as well as the product flow constraints (15). Solving this scenario, we obtain the following NE solution: x∗ p1= 4048, x∗ p2= 4648, x∗ p3= 4352, x∗ p4= 4952, x∗ p5= 10294, x∗ p6= 10956, with a profit of e10,825 for Farm 1 and e18,462 for Farm 2, equilibrium demands d1∗ 1= 8695 and d1∗ 2= 9305 for Farm 1 with corresponding prices ρ1 1(d∗ 1,v∗) = 89 and ρ1 2(d∗ 2,v∗) = 90 c/kg, and equilibrium demands d2∗ 1= 10294 and d2∗ 2= 10956 for Farm 2 with corresponding prices ρ2 1(d∗ 1,v∗) = 119 and ρ2 2(d∗ 2,v∗) = 120 c/kg. We can note that the profit declined for both farms, declining by 8.3% for Farm 1 when compared to the Base Case, and by 25% for Farm 2. The dramatic decline for Farm 2 can be attributed to the significantly lower output from its manufacturing link; with productivity falling from 33,750kg to 21,250kg. We also note that the decrease in output, especially that experienced by Farm 2, has driven up the prices, with the two farms’ prices increasing between 3.5% and 6.3% at the two demand markets. In Table 3, we note that due to the removal of the relationship between productivity and investments, we can see a shift in the way the e5,000 is invested by both firms. The results of this scenario therefore highlight the importance that employee engagement has on the effectiveness of green investments and the overall profitability of a SCN. Scenario 3 (Labour Shortages).Inspired by the COVID-19 pandemic, we construct a scenario where an outbreak occurs at Farm 1’s manufacturing facility M1 2with corresponding link l2, leaving only 10 labour hours available out of the usual 120. Solving such a scenario yields the following NE solution: x∗ p1= 7409, x∗ p2= 249, x∗ p3= 7710, x∗ p4= 551, x∗ p5= 16543, x∗ p6= 17207, 32 MENDEL — Soft Computing Journal, Volume 29, No.g1, June 2023, Brno, Czech RepublicX Debono-g2igHX,gA Game Theoretic Competitive Supply Chain Network Model with Green Investments and Labour Table 3: Results for Scenario 2. Link f∗ lv∗ lh∗ lˆel(f∗ l, v∗ l) l18400 0 111 3780 l29600 0 120 3264 l38400 348 28 417 l49600 0 32 480 l518000 0 120 8100 l68695 0 29 435 l79305 4652 31 422 l821250 0 250 5312 l921250 0 47 212 l10 21250 5000 85 5950 l11 10294 0 23 0 l12 10956 0 24 0 with a profit of e9,014 for Farm 1 and e25,378 for Farm 2, equilibrium demands d1∗ 1= 7659 and d1∗ 2= 8261 for Farm 1 with corresponding prices ρ1 1(d∗ 1,v∗) = 88 and ρ1 2(d∗ 2,v∗) = 90 c/kg, and equilibrium demands d2∗ 1= 16543 and d2∗ 2= 17207 for Farm 2 with corresponding prices ρ2 1(d∗ 1,v∗) = 113 and ρ2 2(d∗ 2,v∗) = 114 c/kg. We can see that these results represent a 23.6% decrease in profit for Farm 1 when compared to the base case, owing to a 25.5% decrease in output capacity. From the equilibrium results in Table 4, we can see that even though production is ramped up at the first manufacturing facility M1 1to make up for the severely restricted capacity at M1 2, this is not enough to make up for the lost output. We can also note how the e5,000 budget is fully allocated to the first manufacturing facility to increase the flow capacity as much as possible. This highlights the importance of safeguarding employee health, a theme that has emerged and been strongly prioritised throughout the pandemic. Table 4: Results for Scenario 3. Link f∗ lv∗ lh∗ lˆel(f∗ l, v∗ l) l115120 5000 120 6426 l2800 0 10 272 l315120 0 50 756 l4800 0 3 40 l515920 0 106 7164 l67659 0 26 383 l78261 0 28 413 l833750 5000 250 7931 l933750 0 75 337 l10 33750 0 135 10125 l11 16543 0 37 0 l12 17207 0 38 0 3.2 Sensitivity Analysis Since the scenarios highlighted an important link between employees, green investments and profitability, using the same functions and topology, we will be conducting sensitivity analysis on the employee engagement with green investments β, and the green investment budget Θ. To study the impact that employee engagement with green investments has on the SC of a firm, we consider productivity factors βbetween 0 and 1 kg/einvested, increasing in increments of 0.05. We note that we vary the productivity factor for Farm 1, while those of Farm 2 remain fixed at 0.1kg/e. In Fig. 3, we can note that Farm 2’s profits are not impacted by the variations in the employee engagement at Farm 1, which is to be expected. For Farm 1, up to a productivity factor of 0.25kg/einvested, its profit increases at an average rate of 3.3% per 0.05 increment. However, at the 0.3kg/epoint we note that this profit takes a hit, initially decreasing by 14.4% and then climbing at an average rate of 0.5% thereafter. This drop can be attributed to the employees’ perceptions and expectations regarding green investments. Keeping in mind that the budget Θ is fixed at e5,000, initially the ratio between the productivity factor and the budget results in an increase in profits due to the increased employee engagement. However, beyond the 0.25kg/e point, productivity diminishes due to the gap between the employees’ environmental expectations and what is actually being carried out by the firm. This is in line with findings that employees at purpose-driven firms are more likely to remain working with them [24], with employee engagement being one of the linking factors. This also highlights that the more environmentally aware employees are, the more they will demand from their employer. To confirm this thinking, we carried out the same sensitivity analysis again with a higher budget of e7,000 and could note that the inflection point occurred at a higher productivity factor of 0.35kg/e. Finally, we carry out sensitivity analysis on the budget parameter Θ to study the impact the amount that a firm invests in green initiatives has on its SC. To this end, we consider budgets between e0 and e100,000 in increments of e5,000 for Farm 1, while keeping Farm 2’s budget fixed at e10,000. From Fig. 4, we can see that an increase in budget leads to an increase in productivity, demands and ultimately profits for Farm 1. This in turn impacts Farm 2’s performance, which loses market share and as a result sees a decline in profit. On average, profits for Farm 1 increase by e2,598 for every additional e5,000 invested, representing an average return on investment of 52%, while profits for Farm 2 decrease by e1,420 with every increment. Farm 1 overtakes Farm 2 in terms of profits at an investment level of e30,000, at which point Farm 1 invests 3 times that of Farm 2. At a budget of e90,000, Farm 1’s profit reaches a maximum of e60,223, which cannot be improved beyond this point, regardless of the budget increase. This is because at this point, the ratio of green investments to link flows is at 0.5 for all links, which we recall is the upper bound per kg of flow. We can also see how Farm 1 prioritises Market 2 over Market 33