Introduction

Recent studies reveal that the human consumption of earth’s resources outstrips the planet’s capacity to regenerate by 30% (WWF 2008). The surge of resource use intensity is rooted in the increased population growth especially in the industrialised and developing countries (IEA 2008). The OECD Environmental Outlook 2008 projected that by 2030, 3.9 billion people will live in areas under severe water stress, which is an additional of 1 billion people over those of today (OECD 2008). All these challenges urged for efficient use of resources by adopting sustainable resource management strategies. As concluded in the UNEP Year Book, efficient use of resources should be implemented (UNEP 2009). In relation to the sustainable resource management strategies, industrial symbiosis (IS) has been identified as an innovative waste management approach (UNEP 2009).

IS engages traditionally individual industries in a collective physical exchange of materials, energy, water and by-products to gain both environmental and economic benefits (Chertow 2007). As a result, industrial owners gained benefits from IS through fresh resource and waste treatment costs savings as the consumption of resources in the processes are optimised (Frosch and Gallopoulos 1989). Despite the benefits from the established inter-plant resource conservation networks (IPRCNs), the different interests exist among the plant owners often becomes the barrier for such collaboration work. Therefore, the IPRCNs need to be carefully designed to achieve a mutually beneficial cooperation among the participating plants (Chertow 2007; Schwarz and Steininger 1997; Fichtner et al. 2005).

Until recently, various process integration tools such as insight-based pinch analysis (Olesen and Polley 1996; Spriggs et al. 2004; Chew et al. 2007; Foo 2008; Bandyopadhyay et al. 2009; Chew et al. 2009a) and mathematical optimisation (Keckler and Allen 1998; Nobel and Allen 2000; Liao et al. 2007; Lovelady et al. 2007; Chew et al. 2008; Chen et al. 2009; Chew and Foo 2009; Lovelady and El-Halwagi 2009; Lovelady et al. 2009) were used to synthesise optimum IPRCNs. Although these techniques are useful in synthesising the IPRCN with overall minimum flowrates or cost, it, however, does not guarantee the maximum benefits for each participating network (in an individual basis). Nor do these methods fully reflect that these individual networks act in self-interest that may conflict with the overall goals of the IPRCN. A recent work by Chew et al. (2009b) adopted game theory approach to assist the selection of optimum inter-plant water integration (IPWI) scheme which guaranteed the maximum benefits for each participating water networks. In their work, alternative IPWI schemes with direct integration schemes (participating water networks are integrated directly through cross-plant pipelines) were analysed through non-cooperative and cooperative game approaches. In game theory perspective, a game is categorised as non-cooperative if the participants do not make binding commitments to coordinate their strategies. In contrast, a cooperative game is defined as a game in which participants make binding agreements to coordinate their strategies (Nash 1950; Nash 1951; McCain 2004; Osborne 2004). Optimum integration schemes for these approaches, i.e. Nash equilibrium (for non-cooperative approach) and Pareto optimal solutions (cooperative approach), are selected based on rational self-interest of each participant. Nash equilibrium in any non-cooperative game is a solution from which no player (i.e. individual water network) can unilaterally deviate to improve his/her payoff (Nash 1950; Nash 1951; McCain 2004; Osborne 2004). On the other hand, Pareto optimal is the best solution (refers to IPWI scheme in this case) that one can make in a cooperative game. A solution is Pareto optimal if no one can be made better off without making someone worse off (McCain 2004).

On the other hand, Aviso et al. utilised fuzzy mathematical programming in synthesising the optimum IPRCNs (Aviso et al. 2010a, b). In the fuzzy optimisation model, the upper and lower limits of the fuzzy goal for each water network are specified, and the model is solved to maximise the level of satisfaction achieved by each participating networks. The optimum solution obtained in the fuzzy optimisation is the Pareto optimal solution through cooperative game approach (Aviso et al. 2010a). Further work integrates the role of an external agent (e.g. government) in inducing cooperation among companies using economic instruments such as fees and subsidies (Aviso et al. 2010b). Other works using game theory approach in the analysis of industrial ecology are such as those in carbon emission reduction (Jørgensen and Zaccour 2001; Breton et al. 2008), emergy analysis (Lou et al. 2004), drinking water-supply system improvement (Peldschus and Zavadskas 2005), industrial ecology optimisation (Singh and Lou 2006), analysis of post-consumer beverage packaging waste (Grimes-Casey et al. 2007) and the synthesis of wastewater neutralisation network (Kim and Lee 2007).

In this study, the recent developed game theory approach for direct integration (Chew et al. 2009b) is extended for the analysis of indirect integration. In this scheme, water networks are interconnected via a centralised utility hub (CUH). The main advantage of such arrangement is that, it is more practical in handling large number of water networks in the IPWI scheme (Chew et al. 2008; Chen et al. 2009). Note that the CUH is commonly used in eco-industrial parks (EIPs), wherein the CUH provides shared services to multiple plants which include source interception, wastewater treatment and fresh resource supply services (Spriggs et al. 2004; Lovelady et al. 2007; Lovelady and El-Halwagi 2009; Lovelady et al. 2009). In the following section, a set of indirect integration schemes are first generated (using pinch analysis techniques). Next, the integration schemes are analysed through non-cooperative and cooperative game approaches, from which an optimum integration scheme (Nash equilibrium or Pareto optimality) is to be determined.

Problem statement

Given a set of water networks k of the fixed flowrate type problem, with process sinks j and sources i that may be considered for water reuse/recycle, where each sink and source has a limiting water flowrate and concentration. External fresh water source(s) is used to supplement additional water requirement of the process sink that is not satisfied by the process sources. All water networks share a common CUH where water regeneration and wastewater treatment facilities are made available for service. The alternative IPWI schemes are analysed using game theory approach based on two different payoffs, based on internal and external costs. The former reflects the combined capital and operating costs. In contrast, the external cost is associated with environmental damage for which conventional approaches fail to account. However, such costs may be internalised through the application of incentives (or disincentives) imposed by an authority to induce companies to act in an environmentally responsible manner. Game theory provides a rigorous framework for the analysis of such mechanisms, which in practice may be in the form of taxes, pollution fees, subsidies, etc. (Salman and Cruz 1981; Ho et al. 1982; Cruz and Simaan 1999). In this study, the intervention of the EIP authority in the cooperative game approach by providing green incentive to the participating networks is analysed.

Model for payoff based on internal costs

Figure 1 shows the generic structure of an indirect integration scheme for a set of water network k. As shown, the CUH provides services to regenerate water and to treat wastewater for all networks (with service costs incurred). The effluent from each network is sent to the centralised regeneration and waste treatment units (with flowrate \( f_{i, \, s}^{\text{R}} \) and \( f_{i, \, s}^{\text{T}} \), respectively). The regenerated water (with flowrate of \( f_{{{\text{REG}}k, \, s}} \)) is then imported by network k as a secondary water source besides fresh water. The unused water from the network is then sent for treatment to an allowable quality (C D) prior to environmental discharge. One may also view this as a total water network of multiple plants, conceptually similar to the work presented earlier (Ng et al. 2007a, b; Chew and Foo 2009).

Fig. 1
figure 1

Indirect integration scheme

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In the following section, it is assumed that a regeneration unit of the fixed outlet quality type and a wastewater treatment unit of the fixed removal ratio (RR) type are used. The RR is defined as the ratio of the contaminant load removed from the inlet load (Wang and Smith 1994). The internal payoff \( \left( {{\text{PY}}_{k, \, s}^{\text{INT}} } \right) \) is shown in Eq. 1, which considers the annual cost savings (ACSk,s), water regeneration cost (R k,s) and the annualised cross-plant piping cost \( \left( {P_{k, \, s}^{\text{IND}} } \right). \) The ACSk,s is a result of the reduction of fresh water \( \left( {F_{k, \, s}^{\text{save}} } \right) \) flowrate and contaminant mass load in the wastewater treatment \( \left( {E_{k, \, s}^{\text{save}} } \right), \) calculated using Eqs. 2–6. In Eq. 2, AWH corresponds to the annual working hours of the water networks (assuming the same for all networks), while W COST and \( E_{\text{COST}}^{\text{L}} \) are unit cost of fresh water and contaminant load removal in the wastewater treatment, respectively. Equations 3 and 4 determine the reduction of fresh water flowrate and contaminant load upon the implementation of IPWI as compared to the base case (in-plant reuse/recycle without IPWI). Variables \( f_{{{\text{FW}}k}} \) and \( f_{{{\text{FW}}k, \, s}}^{\text{IPWI}} \) in Eq. 3 denoted fresh water flowrate required by network k in scheme s before and after IPWI, respectively. Equation 5 denotes the total contaminant mass load removal from network k (ΔL k ) prior to environment discharge for the base case, with f Wi and C SRi denote the wastewater flowrate from source i and its corresponding concentration. Equation 6 states the total contaminant mass load need to be removed from network k in scheme s \( \left( {\Updelta L_{k, \, s}^{\text{IPWI}} } \right) \) by the centralised treatment unit. Variable \( L_{k, \, s}^{\text{IPWI}} \) denotes the wastewater contaminant mass load from network k in scheme s, while variable \( f_{i, \, s}^{\text{T}} \) denotes the wastewater flowrate from sources i sent to the centralised treatment unit in each integration scheme.

$$ {\text{PY}}_{k, \, s}^{\text{INT}} = \text{ACS}_{k, \, s} - R_{k, \, s} - P_{k, \, s}^{\text{IND}} \quad \forall k \in K,\forall s \in S $$
(1)
$$ {\text{ACS}}_{k, \, s} = \left( {F_{k, \, s}^{\text{save}} W_{\text{COST}} + E_{k, \, s}^{\text{save}} E_{\text{COST}}^{\text{L}} } \right){\text{AWH}}\quad \forall k \in K,\forall s \in S $$
(2)
$$ F_{k, \, s}^{\text{save}} = f_{{{\text{FW}}k}} - f_{{{\text{FW}}k, \, s}}^{\text{IPWI}} \quad \forall k \in K,\forall s \in S $$
(3)
$$ E_{k, \, s}^{\text{save}} = \Updelta L{}_{k} - \Updelta L_{k, \, s}^{\text{IPWI}} \quad \forall k \in K,\forall s \in S $$
(4)
$$ \Updelta L{}_{k} = \sum\limits_{i \in I} {f_{Wi} \left( {C_{{{\text{SR}}i}} - C_{\rm D} } \right)} \quad \forall k \in K $$
(5)
$$ \Updelta L_{k, \, s}^{\text{IPWI}} = L_{k, \, s}^{\text{IPWI}} - \sum\limits_{i \in I} {f_{i, \, s}^{\text{T}} } C_{\rm D} \quad \forall k \in K,\forall s \in S $$
(6)

The water regeneration cost (R k,s in Eq. 1) is determined by Eq. 7. Note that for integration scheme s, each participating network is charged by the total contaminant load sent to the centralised regeneration unit \( \left( {L_{k, \, s}^{\text{R}} } \right), \) as well as the minimum regenerated flowrate \( \left( {f_{{{\text{REG}}k, \, s}} } \right) \) imported by the participating network. Parameters \( R_{\text{COST}}^{\text{m}} \) and \( R_{\text{COST}}^{\text{f}} \) each denote the regeneration costs, expressed per unit of contaminant load removed and per unit of regenerated water imported, respectively. Due to the unequal regeneration flowrate sent from each network for regeneration (usually the ‘cleanest’ wastewater streams are selected for regeneration), the total regeneration cost \( R_{\text{COST}}^{\text{m}} \) is allocated to the different plants based on the unit of mass load removed (instead of unit flowrate). Detailed procedure on the selection of wastewater streams for regeneration is referred to Ng et al. (2007c, 2008). Equation 8 states the total contaminant load for the combined sources being sent to the centralised regeneration unit in each scheme, with \( f_{i, \, s}^{\text{R}} \) and C SRi denoting the water flowrate and concentration of source i, respectively.

$$ R_{k, \, s} = \left( {L_{k, \, s}^{\text{R}} R_{\text{COST}}^{\text{m}} + f_{{{\text{REG}}k, \, s}} R_{\text{COST}}^{\text{f}} } \right){\text{AWH}}\quad \forall k \in K,\forall s \in S $$
(7)
$$ L_{k, \, s}^{\text{R}} = \sum\limits_{i \in I} {f_{i, \, s}^{\text{R}} C_{{{\text{SR}}i}} } \quad \forall k \in K,\forall s \in S $$
(8)

The capital cost for the cross-plant pipeline (\( P_{k, \, s}^{\text{IND}} \) in Eq. 1) is adapted from Kim and Smith (2004). Equation 9 shows the total annualised cross-plant piping cost, \( P_{k, \, s}^{\text{IND}} \) for each network in scheme s, which consists of export \( \left( {P_{k, \, s}^{\text{EXP}} } \right) \) and import \( \left( {P_{k, \, s}^{\text{IMP}} } \right) \) piping costs. The former is calculated in Eq. 10, considering two export cross-plant pipings, i.e. flowrates sent from sources i to the centralised regeneration \( \left( {f_{i, \, s}^{\text{R}} } \right) \) and treatment \( \left( {f_{i, \, s}^{\text{T}} } \right) \) units. On the other hand, one import cross-plant pipeline is required to transfer the regenerated flowrate \( \left( {f_{{{\text{REG}}k, \, s}} } \right) \) from the centralised regeneration unit to each network, with the piping cost shown in Eq. 11. Piping capital cost within the individual network as well as that in the CUH is assumed negligible. Binary variables \( x_{k, \, s}^{\text{IND}} \) and \( y_{k, \, s}^{\text{IND}} \) in Eq. 10 indicate the presence of the export cross-plant flowrates to regeneration and treatment units, respectively; while binary variable \( z_{k, \, s}^{\text{IND}} \) in Eq. 11 indicates the presence of import cross-plant flowrate. Parameter D is the distance between the CUH and water networks k and cpI and cpII are the cross-plant pipeline cost parameters.

$$ P_{k, \, s}^{\text{IND}} = P_{k, \, s}^{\text{EXP}} + P_{k, \, s}^{\text{IMP}} \quad \forall k \in K,\forall s \in S $$
(9)
$$ P_{k, \, s}^{\text{EXP}} = D\left[ {\left( {{\text{cp}}_{\text{I}} \sum\limits_{i \in I} {\left( {{\frac{{f_{i, \, s}^{\text{R}} }}{3600\rho \nu }} + {\text{cp}}_{\text{II}} x_{k, \, s}^{\text{IND}} } \right)} } \right) + \left( {{\text{cp}}_{\text{I}} \sum\limits_{i \in I} {\left( {{\frac{{f_{i, \, s}^{\text{T}} }}{3600\rho \nu }} + {\text{cp}}_{\text{II}} y_{k, \, s}^{\text{IND}} } \right)} } \right)} \right]{\text{AF}}\quad \forall k \in K,\forall s \in S $$
(10)
$$ P_{k, \, s}^{\text{IMP}} = D\left( {{\text{cp}}_{\text{I}} {\frac{{f_{{{\text{REG}}k, \, s}} }}{3600\rho \nu }} + {\text{cp}}_{\text{II}} z_{k, \, s}^{\text{IND}} } \right){\text{AF}}\quad \forall k \in K,\forall s \in S $$
(11)

An annualising factor (AF) is used to annualise the piping capital cost in Eqs. 10 and 11, defined as (Smith 2005):

$$ {\text{AF }} = \, {\frac{{a\left( {1 + a} \right)^{y} }}{{\left( {1 + a} \right)^{y} - 1}}} $$
(12)

where a is the fractional interest rate per year and y is the number of years.

Model for payoff based on external costs

The payoff based on external costs is defined as the overall reduction of fresh water and wastewater flowrates upon the implementation of IPWI. These flowrates are assumed to provide an indication of environmental impacts through resource depletion and pollution, respectively. As shown in Eq. 13, the external payoff \( \left( {{\text{PY}}_{s}^{\text{EXT}} } \right) \) used here is the summation of the overall fresh water \( \left( {F_{k, \, s}^{\text{save}} } \right) \) and wastewater \( \left( {Q_{k, \, s}^{\text{save}} } \right) \) flowrates reduction, given as in Eqs. 3 and 14, respectively. An IPWI scheme with a higher overall flowrates reduction indicates higher external payoff, as it is more environmental friendly.

$$ {\text{PY}}_{s}^{\text{EXT}} = \sum\limits_{k \in K} {F_{k, \, s}^{\text{save}} } + \sum\limits_{k \in K} {Q_{k, \, s}^{\text{save}} } $$
(13)
$$ Q_{k, \, s}^{\text{save}} = f_{{{\text{WW}}k}} - f_{{{\text{WW}}k, \, s}}^{\text{IPWI}} $$
(14)

Note that Eq. 13 assumes that fresh water and wastewater reductions are of equal importance. If necessary, the payoff function may be modified to account for weights based on the concept of water footprint (Hoekstra 2009). Alternatively, weights may be used to directly convert the flowrates into monetary equivalents.

Illustrative example

This example illustrates an indirect integration scheme that involves four water networks, taken from Wang and Smith (1994), Polley and Polley (2000), Sorin and Bédard (1999) and Wang and Smith (1995). The example is used to elucidate game theory approach in the analysis of indirect integration scheme, with its limiting water data tabulated in Table 1.

Table 1 Limiting water data for illustrative example
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Table 2 shows the minimum fresh water (f FWk ) and wastewater (f WWk ) flowrates for each network in the base case. These flowrates can be determined using any developed pinch targeting tools such as material recovery pinch diagram (El-Halwagi et al. 2003; Prakash and Shenoy 2005), water cascade analysis (Manan et al. 2004; Foo et al. 2006), material surplus composite curve (Saw et al. 2009), etc. When no IPWI is performed, de-centralised wastewater treatment unit (RR = 0.9) is used, operated by network k. The fourth column in Table 2 summarises the individual wastewater streams identified through the waste stream identification technique of Ng et al. (2007a, b). Equation 5 is used to calculate the contaminant load removed in the decentralised wastewater treatment unit (ΔL k ), with results shown in the last column of Table 2. Note that in the base case, water regeneration is not considered.

Table 2 Minimum flowrate targets and contaminant load removal in the base case (in-plant reuse/recycle without IPWI)
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In this example, game theoretic analysis is carried out for five indirect integration schemes, assuming that each network is operated by different business entities. Each participating water network will seek for maximum benefits based on the internal and external payoffs. The following assumptions are made in the calculation of internal and external payoffs:

  1. 1.

    Equal distance of D = 200 m between all water networks and the CUH.

  2. 2.

    The capital cost for the cross-plant pipelines was annualised to a period of 5 years (y) with a fixed interest rate (a) of 5%. This leads to an AF (Eq. 12) of 0.231.

  3. 3.

    The piping cost (Eqs. 10–11) considers the use of carbon steel pipes, with the cost parameters of cpI = 7,200 and cpII = 250 (given in USD; CE plant index = 318.3). The stream flowrate velocity (ν) is assumed as 1 m s−1 and water density (ρ) as 1,000 kg m−3.

  4. 4.

    AWH is assumed to be 7,920.

  5. 5.

    Unit cost for fresh water (W COST) is assumed at $1/t.

  6. 6.

    Regeneration unit of fixed outlet concentration (C R = 10 ppm) is used.

  7. 7.

    Discharge concentration (C D) is assumed at 30 ppm.

  8. 8.

    Regeneration service costs are assumed as \( R_{\text{COST}}^{\text{m}} = \$ 2/{\text{kg}} \) load removed (for water that is sent to the centralised regeneration unit) and \( R_{\text{COST}}^{\text{f}} = \$ 0. 5/{\text{t}} \) treatment flowrate (imported from the CUH).

  9. 9.

    Wastewater treatment unit with RR = 0.9 is used, with its treatment cost \( \left( {E_{\text{COST}}^{\text{L}} } \right) \) of $2.36/kg load removed (El-Halwagi 2006).

The following calculations are then performed to obtain the payoffs for the various IPWI schemes.

Minimum fresh water and regeneration flowrates

Table 3 shows five proposed indirect integration schemes that represent participation of different water networks. Scheme 1 involves the integration of four networks, while Schemes 2–5 involve the integration of three networks. Following the algebraic regeneration targeting techniques of Ng et al. (2007c, 2008), the minimum regenerated \( \left( {f_{{{\text{REG}}k, \, s}} } \right) \) and fresh water \( \left( {f_{{{\text{FW}}k, \, s}}^{\text{IPWI}} } \right) \) flowrates required by each network in scheme s are determined and summarised in Table 3. The total regenerated water flowrate for each scheme can then be determined, as given in the last column of Table 3. Note that no regeneration flowrate is received in the network that does not participate in the integration scheme; hence its fresh water flowrate is referred to that in the base case (as shown in Table 2). For instance, Network D has a fresh water flowrate of 92 t/h since it does not participate in Scheme 2 (see Tables 2, 3).

Table 3 Minimum fresh and regenerated water flowrates
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In order to achieve the total regeneration flowrate as targeted in Table 3b, water sources are to be extracted from the individual network. These water sources are summarised in Table 4a. Based on the concentration of these sources, the total contaminant load sent to the regeneration unit \( \left( {L_{k, \, s}^{\text{R}} } \right) \) from each network can be calculated using Eq. 8, with the results summarised in Table 4b. Note that the regenerated flowrate needed by an individual network does not need to be the same as its source flowrate that is sent for regeneration. For instance, Network B in Scheme 1 required minimum regenerated flowrate of 75 t/h (see Table 3), however, only 55 t/h of water flowrate is available for regeneration (see Table 4a). On the other hand, Network C sends 128.89 t/h of water flowrate to the regeneration unit (see Table 4a), even though it only requires 88.89 t/h of regenerated water (see Table 3). This means that Network C will supply additional wastewater flowrates to supplement regeneration flowrate requirement in other networks.

Table 4 Flowrate and contaminant load sent to the centralised regeneration unit
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Wastewater treatment flowrate

Table 5a shows the individual source streams \( \sum\limits_{i \in I} {f_{i, \, s}^{\text{T}} } \) sent to the wastewater treatment unit. The total wastewater contaminant load removed in the wastewater treatment unit \( \left( {\Updelta L_{k, \, s}^{\text{IPWI}} } \right) \) can be calculated using Eq. 6, with the results summarised in Table 5b. As in the case of regeneration, a network that does not participate in the integration scheme will keep its wastewater treatment flowrate and contaminant load the same as in the base case (Table 2).

Table 5 Flowrate and contaminant load sent to the wastewater treatment unit
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With the determination of the above flowrates, the IPRCN for all schemes are synthesised and are shown in Fig. 2. Note that the design within the individual network is not shown for simplicity. In the next section, all integration schemes are analysed through two different approaches, i.e. non-cooperative and cooperative games. Finally, sensitivity analysis is carried out to check the robustness of the Nash equilibrium and Pareto optimal solutions obtained through the non-cooperative and cooperative games, respectively.

Fig. 2
figure 2figure 2figure 2

Inter-plant water network design for illustrative example: a Scheme 1, b Scheme 2, c Scheme 3, d Scheme 4 and e Scheme 5

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Non-cooperative game approach

In order to analyse the five integration schemes with non-cooperative game, the internal and external payoffs for each integration scheme are calculated based on the flowrates and contaminant loads determined in Tables 2, 3, 4, and 5. The results are summarised in Table 6. Note that in the last two columns of Table 6 where the external payoff is shown, the total reduction of the fresh water \( \left( {\sum\limits_{k \in K} {F_{k, \, s}^{\text{save}} } } \right) \) and wastewater \( \left( {\sum\limits_{k \in K} {Q_{k, \, s}^{\text{save}} } } \right) \) flowrates is determined using Eq. 13. For instance, Networks A, B and C participated in Scheme 2. Hence, their total fresh water flowrate \( \left( {\sum\limits_{k \in K} {f_{{{\text{FW}}k}} } } \right) \) before IPWI is determined as 360 t/h (=90 + 70 + 200 t/h, see the second column of Table 2). After IPWI, the total fresh water flowrate \( \left( {\sum\limits_{k \in K} {f_{{{\text{FW }}k, \, s}}^{\text{IPWI}} } } \right) \) for this scheme is reduced to 232 t/h (see Table 3a). Therefore, the total fresh water flowrate reduction \( \left( {\sum\limits_{k \in K} {F_{k, \, s}^{\text{save}} } } \right) \) is calculated as 128 t/h (=360 – 232 t/h). The same steps are followed to obtain the total wastewater flowrate reduction \( \left( {\sum\limits_{k \in K} {Q_{k, \, s}^{\text{save}} } } \right), \) in which the wastewater flowrates before and after IPWI are obtained from the third column of Table 2 and the last column of Table 5a, respectively.

Table 6 Evaluation of internal and external payoffs in non-cooperative game
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Analysing the payoffs in Table 6 reveals that Networks A, B and D achieve the highest individual annual internal payoff by choosing Scheme 5 (\( {\text{PY}}_{A, \, 5}^{\text{INT}} = \$ 236{,}129, \) \( {\text{PY}}_{B, \, 5}^{\text{INT}} = \$ 210{,}883, \) \( {\text{PY}}_{D, \, 5}^{\text{INT}} = \$ 243{,}960 \)), while Scheme 1 gives the highest external payoff \( \left( {{\text{PY}}_{1}^{\text{EXT}} = 292 + 292\;{\text{t/h}}} \right). \) This makes Schemes 1 and 5 the Nash equilibrium solutions for the external and internal payoffs, respectively. However, implementing Scheme 5 indicates that Network C will have to be eliminated from the IPWI scheme. Although Scheme 5 guaranteed the highest individual internal payoff (for Networks A, B and D), however, it does not possess the highest overall internal payoff ($857,720/y) as shown in Scheme 1.

In the following section, the IPWI schemes are analysed through cooperative game approach in which Pareto optimal solution is generated.

Cooperative game approach

Knowing the fact that there is always an imbalance segregation of benefits gained by each participating network, the policy maker (e.g. EIP authority, government, etc.) can play an important role to encourage the participation of the individual plant owners through policy setting (Salman and Cruz 1981; Ho et al. 1982; Cruz and Simaan 1999; Schwarz and Steininger 1997; UNEP 2009; Chertow 2007). In the cooperative game approach, it is assumed that a cooperative partnership between authority and the participating water networks is taking place in the IPWI scheme. In the proposed cooperative game, a market-based green incentive given by the EIP authority is introduced in the payoff evaluation. In this case, a uniform incentive is introduced to improve the environmental performance of the entire EIP (Bansal and Gangopadhyay 2003). The introduction of green incentive in the payoff evaluation will contribute to the overall increase of profits gained by each participating network.

Equation 15 shows the revised payoff adapted from Eq. 1, in which a market-based green incentive (G s ) is added on top of the internal payoff in the non-cooperative game. Fraction, x s , in Eq. 15 represents the level of participation of an individual network in an IPWI scheme. As shown in Eq. 16, x s is defined as the fraction of wastewater generation by network k′ in the total wastewater flowrates when all networks participate in scheme s. Hence, the maximum incentive (x s  = 1) will be given if all four networks are participating in the IPWI. As shown in Eq. 17, the green incentive is directly proportional to the total amount of wastewater reduction and is shared equally among all participating networks \( \left( {N_{s}^{\text{Networks}} } \right) \) in each scheme. The incentive rate (IR) is usually set by the policy maker and is given per unit of wastewater flowrate reduced.

$$ {\text{PY}}_{k, \, s}^{\text{INT}} = {\text{ACS}}_{k, \, s} - R_{k, \, s} - P_{k, \, s}^{\text{IND}} + x_{s} G_{s} \quad \forall k \in K,\forall s \in S $$
(15)
$$ x_{s} = {\frac{{\sum\limits_{{k^{\prime}_{s} \in K}} {f_{{{\text{WW }}k^{\prime}}} } }}{{\sum\limits_{k \in K} {f_{{{\text{WW }}k}} } }}}\quad \forall s \in S $$
(16)
$$ G_{s} = {\frac{{{\text{IR}}\left( {\sum\limits_{{k^{\prime} \in K}} {f_{{{\text{WW }}k^{\prime}}} - \sum\limits_{{k^{\prime} \in K}} {f_{{{\text{WW }}k^{\prime}, \, s}}^{\text{IPWI}} } } } \right){\text{AWH}}}}{{N_{s}^{\text{Networks}} }}}\quad \forall s \in S $$
(17)

where k′ denotes the participating networks in an integration scheme s.

Assuming the basis of green incentive rate (IR) of $ 0.2/t wastewater reduction, the annual green incentive (G s ) calculated for each scheme is summarised in Table 7. As the first three terms in Eq. 15 denote the payoff in the non-cooperative game (see Eq. 1), the new internal payoff for cooperative game is obtained by summing the payoff in Table 6 with the annual green incentive in Table 7. For instance, the internal payoff for Network A in Scheme 2 is calculated as $269,146/y (=116,161(0.76) + 180,837). Table 8 summarises the internal and external payoffs for the cooperative game approach. As shown, all networks gain additional internal payoff in all schemes due to the implementation of the green incentive, which internalizes the external costs. Besides, the external payoff component in each scheme remains identical to that of the non-cooperative game approach (see Tables 6, 8). As shown in Table 8, all four networks receive the highest individual annual internal (\( {\text{PY}}_{A, \, 1}^{\text{INT}} = \$ 317{,}356, \) \( {\text{PY}}_{B, \, 1}^{\text{INT}} = \$ 326{,}516, \) \( {\text{PY}}_{C, \, 1}^{\text{INT}} = \$ 351{,}193, \) \( {\text{PY}}_{D, \, 1}^{\text{INT}} = \$ 325{,}187 \)) and external payoffs \( \left( {{\text{PY}}_{1}^{\text{EXT}} = 292 + 292} \right) \) by choosing Scheme 1. Note also in Scheme 1, Network C which was previously eliminated in the non-cooperative game is now being accepted in the cooperative game, which then generates an extra annual profit of $351,193. It is important to note that, Scheme 1 of the cooperative game (Table 8) possesses higher individual internal payoff as compared to Scheme 5 in the non-cooperative game (Table 6). Scheme 1 is hence the Pareto optimal solution in a cooperative game. The inter-plant water network for this scheme is shown in Fig. 2a.

Table 7 Green incentive for each integration scheme
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Table 8 Evaluation of internal and external payoffs in cooperative game
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Sensitivity analysis

Sensitivity analysis is carried out to analyse the internal payoff for the Nash equilibrium solution in non-cooperative game (Scheme 5) and Pareto optimal solution in cooperative game (Scheme 1). In this example, the effect of the changes in wastewater treatment cost (ranges between $2 and $4 per kg load removed) is analysed. As shown in Figs. 3 and 4, the individual internal payoff for all participating networks increases proportionally with the wastewater treatment cost. This indicates that both Nash equilibrium and Pareto optimal solutions are robust to the fluctuation of wastewater treatment cost.

Fig. 3
figure 3

Sensitivity analysis of wastewater treatment cost on internal payoff for Nash equilibrium solution in non-cooperative game (Scheme 5)

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Fig. 4
figure 4

Sensitivity analysis of wastewater treatment cost on internal payoff for Pareto optimal solution in cooperative game (Scheme 1)

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Conclusion

A game theory-based approach is developed as a decision-making tool to analyse various indirect IPWI schemes. The intervention of the EIP authority in the form of providing green incentive induces the plant owners to cooperate among themselves in the IPWI scheme. Analysing the IPWI schemes through cooperative game will generate Pareto optimal solution, and exceed the individual payoffs that the players would receive under non-cooperative game. Although the proposed game theory model in this article is demonstrated for IPWI, it can readily be extended for other types of IPRCNs, including full-fledged EIP’s involving exchange of multiple material and energy streams.