A gametheoretic optimisation approach to fair customer allocation in oligopolies
 121 Downloads
Abstract
Under the everincreasing capital intensive environment that contemporary process industries face, oligopolies begin to form in mature markets where a small number of companies regulate and serve the customer base. Strategic and operational decisions are highly dependent on the firms’ customer portfolio and conventional modelling approaches neglect the rational behaviour of the decision makers, with regards to the problem of customer allocation, by assuming either static competition or a leaderfollower structure. In this article, we address the fair customer allocation within oligopolies by employing the Nash bargaining approach. The overall problem is formulated as mixed integer program with linear constraints and a nonlinear objective function which is further linearised following a separable programming approach. Case studies from the industrial liquid market highlight the importance and benefits of the proposed game theoretic approach.
Keywords
Game theory Supply chain optimisation Oligopoly Nash equilibrium Customer allocationList of symbols
Sets
 b
Outsourcing tiers
 c
Customers
 \(cf_{(c,f)}\)
Set of existing firm’s customers
 \(cti_{(c,t,i)}\)
Set of customer’s tanks for product i
 f
Oligopoly firms
 i
Liquid products
 k
Grid points
 t
Customer tanks
Parameters
 \(\alpha _{f}\)
Negotiation power of firm f
 \(\delta ^{f}_{i}\)
Shortcut model parameters dependent on the design of the ASU plant (–)
 \(\eta _{ff^{\prime}}\)
Interfirm swaps premium
 \(\gamma _{b}^{L},\gamma _{b}^{U}\)
Lower and upper bounds of the tiers b for outsourcing product demand (m^{3})
 \(\overline{V}^{f}_{LOX}\)
Upper limit on the volumetric rate flow of liquid oxygen in the ASU of firm f (m^{3}/h)
 \(\pi ^{sq}_{f}\)
Status quo profit of firm f prior to the fair allocation of the customers ($)
 \(\tilde{\pi }_{fk}\)
Profit of firm f at grid point k ($)
 \(\underline{V}^{f}_{air},\overline{V}^{f}_{air}\)
Lower and upper limits on the volumetric rate flow of air in the ASU of firm f (m^{3}/h)
 \(\underline{V}^{f}_{GNI^{Liq}},\overline{V}^{f}_{GNI^{Liq}}\)
Lower and upper limits on the volumetric rate flow of gaseous nitrogen in the liquefier of ASU of firm f (m^{3}/h)
 \(\underline{V}^{f}_{GNI^{Pip}},\overline{V}^{f}_{GNI^{Pip}}\)
Lower and upper limits on the volumetric rate flow of gaseous nitrogen send to product pipeline by firm f (m^{3}/h)
 \(D_{ict}\)
Product demand of customer c for tank t (m^{3})
 \(DC_{ictf}\)
Delivery cost of demand of product i for customer c and tank t served by firm f ($)
 \(E_{cf}\)
1, if customer c is initially contracted to firm f; 0, otherwise
 EPC
Electricity price ($/MWh)
 \(FDC_{cf}\)
Fixed cost of firm f for dropping customer c ($)
 \(FNC_{cf}\)
Fixed cost of firm f for acquiring new customer c ($)
 MT
Average ASU plant uptime (h)
 \(OC_{ifb}\)
Piecewise constant outsourcing premium cost of tier b ($/m^{3})
 \(P_{ictf}\)
Price of product i for customer c and tank t served by firm f ($)
 \(SWC_{icf^{\prime}f}\)
Swap premium cost ($/m^{3})
 \(U_{i}\)
Upper limit on product demand swaps (m^{3})
 \(USC_{ictf}\)
Unit service cost of demand of product i for customer c and tank t served by firm f ($/m^{3})
 \(VDC_{ctf}\)
Variable cost of firm f for dropping customer c ($/m^{3})
 \(VNC_{ctf}\)
Variable cost of firm f for acquiring new customer c ($/m^{3})
Binary variables
 \(X_{cf}\)
1, if customer c is assigned to firm f; 0, otherwise
 \(Y_{ifb}\)
1, if firm i has outsourced production of product i in tier b; 0, otherwise
Continuous variables
 \(\hat{O}_{ictfb}\)
Disaggregated level of outsourcing for demand of product i for tank t of customer c (m^{3})
 \(\lambda _{fk}\)
SOS2 variables associated with the piecewise linear approximation of the profit of firm f over k grid points (–)
 \(\pi _{f}\)
Profit of firm f
 \(Cap_{if}\)
Production capacity for product i of firm f (m^{3})
 \(EC_{f}\)
Total electricity cost of firm f ($)
 \(NC_{f}\)
Total cost of firm f for acquiring new customers ($)
 \(O_{ictf}\)
Firm’s outsourcing level for demand of product i for tank t of customer c (m^{3})
 \(PW_{f}\)
Electricity power consumed by firm f (KW)
 \(Q_{ictf}\)
Firm’s inhouse production for demand of product i for tank t of customer c (m^{3})
 \(RC_{f}\)
Total cost of firm f for dropping customers ($)
 \(SC_{f}\)
Total customer service cost of firm f ($)
 \(SF_{ictff^{\prime}}\)
Swap amount of product for demand of product i for tank t of customer c between firms f and f′ (m^{3})
 \(V^{f}_{air}\)
Volumetric rate flow of air in the ASU of firm f (m^{3}/h)
 \(V^{f}_{GNI^{Liq}}\)
Volumetric rate flow of gaseous nitrogen in the liquefier of ASU of firm f (m^{3}/h)
 \(V^{f}_{GNI^{Pip}}\)
Volumetric rate flow of gaseous nitrogen send to product pipeline by firm f (m^{3}/h)
 \(V^{f}_{GNI^{Vent}}\)
Volumetric rate flow of gaseous nitrogen ventilated by firm f (m^{3}/h)
 \(V^{f}_{LNI^{ASU}}\)
Volumetric rate flow of liquid nitrogen in the ASU of firm f (m^{3}/h)
 \(V^{f}_{LOX}\)
Volumetric rate flow of liquid oxygen in the ASU of firm f (m^{3}/h)
 \(V^{f}_{NI}\)
Volumetric rate flow of nitrogen in the ASU of firm f (m^{3}/h)
1 Introduction
1.1 Motivation
Current socioeconomic trends such as market globalisation, interconnectedness of firms and the everincreasing capital intensive environment begin to lead to a paradigm shift on the market structure of process industries. Financial sustainability, competition elimination and market share growth are a few reasons that lead businesses in mature markets to form coalitions leading to oligopolistic market structures (Ziss 2007; Nagarajan and Sošić 2008). Oligopolies are formed when a limited number of companies rule in a particular market offering similar goods and typical examples from the process industry include the steel, food, pharmaceutical and specialty gases sector. Many of the strategic, tactical and operational decisions of industrial companies are directly related to the firms’ customer portfolio ranging from future investment planning to satisfying daily demands.
In this paper we study the problem of fair customer allocation in oligopolies for the case that new customers appear and provide market share growth opportunity for the firms from a decentralised gametheoretic viewpoint. The firms that comprise the oligopoly agree to allocate the new customers in a fair manner while at the same time they can reassign existing ones so as to maximise their respective profits. The goal of this paper is: (i) to introduce a new approach for the problem of customer allocation in oligopolies and (ii) apply gametheoretic concepts to ensure the fair optimisation of the market under study.
The remaining of the paper is organised as follows: in Sect. 1.2 a literature review on the game theoretic optimisation of supply chain systems is conducted. Section 2 introduces the problem statement that is addressed in the present work whereas the key mathematical developments are detailed in Sect. 3. Next, in Sect. 4 two case studies are examined from the industrial liquids market and the key findings are discussed while conclusions and future research directions are outlined in Sect. 5.
1.2 Literature review
The progressive competition within supply chain systems along with the individual objectives and constraints of the supply chain participants have led many researchers to study the impact of such decentralisation in the optimal decision making (Papageorgiou 2009; BarbosaPóvoa and Pinto 2018). In contrast to centralised modelling approaches, the interdependence of individual decisions, constraints and potentially conflicting objectives of the supply chain participants are explicitly taken into account (Shah 2005; Sahay and Ierapetritou 2013). Due to the nature of such systems, the formalisation of the optimisation problem needs to reflect the hierarchical structure of the market under study and to this end different methods have been employed such as bilevel programming, tailormade iterative frameworks as well as Nash equilibrium methods to name a few.
Sherali and Leleno (1988) studied the existence, characterisation and computation of a twostage oligopolistic network and they provided theoretical results that should hold for a market equilibrium to exist. The multiperiod resource allocation problem was treated by Klein et al. (1992) with a lexicographic minimax algorithm that the authors proposed. Within the lexicographic minimax methods the concept of fairness is closely related to that of Pareto optimality, i.e. the solution returned from these methods is Pareto optimal in the sense that no preference by the decision maker was taken into account and also the optimal decision vector is nondominated. Lexicographic minimax optimisation has also been applied successfully to location problems (Ogryczak 1997) in order to compute fair nondominated solutions.
Later on, Nash strategies among multienterprise supply chains for the fair optimisation of transfer prices were proposed by Gjerdrum et al. (2001, 2002). In a series of papers, the authors considered the problem of transfer price and inventory optimisation and two different solution approaches were presented. A separable programming approach with exact linearisation of bilinear terms and a spatial and binary branch and bound procedure that solved the original MINLP problem as a series of MILPs (Gjerdrum et al. 2001).
Levis and Papageorgiou (2007) derived analytical formulae for the Nash equilibrium of Bertrand games for the special case of single product duopolies. Apart from the analytical formulae, an iterative algorithm for game theoretic price optimisation of multiproduct competing companies was proposed and allowed the computation of the corresponding Nash equilibrium. Erkut et al. (2008) reported a multiobjective optimisation approach for the solution of the locationallocation problem in solid waste management and applied their methodology in a case study from Greece. The authors seeked for fair solutions in a Paretooptimality fashion and devised an approach for the iterative solution of the resulting lexicographic minimax problem. It is interesting to note that their approach is applicable for either convex or nonconvex decision spaces. Zhao et al. (2010) as well as Cao et al. (2013) studied the cooperative game of decentralised supply chain between manufacturer and retailers under demand disruptions. In the first study a Nash approach was employed whereas in the second the authors followed the Stackelberg approach. In general, the Stackelberg approach is employed in cases of hierarchical games with a certain leader and a number of followers while such assumption in the Nash setting is not required. The construction of Paretooptimal fair solution in a game theoretic context has been proposed by Zamarripa et al. (2012) where the authors computed the payoff matrix of the game through a series of multiobjective MILPs.
Cooperative allocation of cost savings in decentralised supply chains using different methods from cooperative game theory was proposed by Lozano et al. (2013). Berglund and Kwon (2014), studied the facility location problem of a Stackelberg firm in the case of a CournotNash game. The price functions of the different firms are modelled via inverse demand correlation while the cost function is convex and monotonically increasing with the amount manufactured by each firm. The overall problem was formulated as a mixed integer program with variational inequalities that enforce the achievement of Nash equilibrium for the competing firms. While exhaustive enumeration approaches had been proposed in the past, the authors proposed a heuristic approach that has at its core the simulated annealing paradigm. Zhang et al. (2014) employed the minimax approach for the fair cost distribution of homes that belong to a microgrid. Yue and You (2014a) studied a threeechelon supply chain system where decisions about its design and operation were made by solving a monolithic nonconvex MINLP. Their work assumes a single leader, i.e. the manufacturer, and multiple followers, i.e suppliers and customers. The authors model the interaction between manufacturer and suppliers using the Nash bargaining approach while a bilevel formulation is employed to model the arrangements among suppliers and customers. The case of capacity planning in a competitive environment has recently been addressed from Grossmann and coworkers both from a bilevel and a trilevel scope (GarciaHerreros et al. 2016; Florensa et al. 2017). Game theoretic optimisation of decentralised supply chains under uncertain environment was recently studied by Hjaila et al. (2017). The authors formulated the problem as a noncooperative, nonzero sum game where apart from the leader/follower set, their interactions with third parties were also considered. The Stackelberg approach was employed and at the end a set “Pareto solutions” were computed and analysed through the solution of a series of nonconvex MINLPs. Key decisions in this work were the transfer prices and resource amounts exchanged between the players while Monte Carlo simulations were conducted for the evaluation of uncertainty regarding a set of product prices and demand. The strategic planning of petroleum refineries was recently studied through a game theoretic perspective by Tominac and Mahalec (2017). The authors formulated the problem as a potential game where a number of refineries engaged in a Cournot oligopoly game and solved the resulting nonconvex (MI)NLP. Yue and You (2017), proposed a decomposition framework for the solution of a mixed integer bilevel programming (MIBP) problems with application on Stackelberg games for supply chain design and operation. The authors considered the discrete, apart from continuous, decisions of the follower and the leader resulting in computational difficulties for the solution of a single level optimisation problem. To circumvent this issue, initially the discrete decisions of the follower are enumerated thus allowing for the introduction of the KKT conditions of the follower’s problem and then an iterative procedure is employed to avoid complete enumeration. Later on, Liu and Papageorgiou (2018) examined the fair profit allocation of an active ingredient supply chain through the solution of a Nash bargaining game and lexicographic maximin optimisation. Finally, Gao and You (2019) incorporated uncertainty considerations in the game theoretic optimisation of multistakeholder supply chains.
2 Problem statement
We study the problem of fair customer allocation in existing oligopolies. It is assumed that the firms which form the oligopoly are rational and that a firm has estimates of the other firms’ information . The fair customer allocation problem is formulated as a static MINLP following the Nash bargaining approach so as to compute the resulting firm profits in the equilibrium along with the assignment of the customers. The MINLP model is then linearised into an MILP using a separable programming approach. Concisely, the problem statement has as follows:

Customer portfolio including existing and new customers

Average customer demand per demand tank and liquid product

Delivery cost to the customer’s demand tank from firm

Customer acquisition/forfeit variable and fixed costs

Thirdparty production outsourcing costs and tiers

Product demand swap costs between firms

Liquid products production capacities of the firms

Product prices per customer and firm

Cost of electricity

Optimal customer assignment to the firms

Optimal production levels of the firms

Optimal product demand outsourcing levels for the firms

Optimal product demand swap levels for the firms

Maximise fairly the firms’ profit
3 Mathematical developments
In this section, first the model formulation for the customer allocation in oligopolies is presented and next the gametheoretic framework for its fair optimisation is introduced. The nomenclature of the mathematical developments is provided at the end of the article. The key assumptions in the present work are summarised as follows: (i) firms will participate in the game only if they can achieve greater profit than their current one, (ii) deterministic production and service cost, (iii) customer demands are given as their deterministic average values, (iv) firms should serve all customers and (v) decentralised decision making.
3.1 Model formulation
3.1.1 Customer assignment and demand satisfaction
3.1.2 Plant production shortcut model
3.1.3 Spot market product acquisition
3.1.4 Interfirm swap agreements
3.1.5 Customer service cost
3.1.6 Customer acquisition cost
3.1.7 Customer forfeit cost
3.1.8 Power consumption cost
3.1.9 Profit calculation
3.2 Nash gametheoretic solution
4 Case studies
To demonstrate the proposed modelling and solution framework, we present two case studies on the fair customer allocation in industrial liquid oligopolistic markets. Firstly, the case of a duopoly is examined to illustrate the efficiency of the proposed framework and results from the sensitivity analysis with regards to the firms’ negotiation power are presented. Next, the case of an oligopoly comprised by three firms is investigated. All the computations were performed using a Dell workstation with Intel\(\circledR\) Core^{TM} i75600 CPU @ 2.60 GHz and 16.00 GB RAM. The implementation and computations were executed using GAMS 25.1.2 (Rosenthal 2012). CPLEX 12.8 was used for solving the MILP while BARON 18.5.8 (Tawarmalani and Sahinidis 2005) for the global solution of the singlelevel MINLP problem.
For the customers that have monthly demand between 0 and 14,000 m^{3} the variable cost of acquisition is \(VNC_{ctf}\), for customers that have demand levels between 14,000 and 28,000 m^{3} is \(2\times VNC_{ctf}\) and for the customers that demand more than 28,000 m^{3} the variable cost of acquisition is \(2.5\times VNC_{ctf}\) . The fixed acquisition cost is assumed to be \(FNC_{cf}\) while the fixed forfeit cost is \(FDC_{cf}=2\times FNC_{cf}\). The variable dropping cost for each customer (\(VDC_{ctf}\)), depending on the total customer’s demand, is assumed to be 2%, 5% and 10% of their unit service cost.
4.1 Liquid market duopoly
In this case study, the fair customer allocation in a duopoly was considered. Each customer can be served by a number of storage tanks and places orders for LOX and LNI there is no LAR demand. Initially, firm A serves 44 customers and firm B 38 customers while there exist 16 new customers that provide opportunity for market share growth.
First, the model is run with fixed customer assignment decisions and no new customers are allowed to be allocated so as to compute the status quo profits of the firms. The market share of each company is computed as the percentage of the firm’s profit with respect to the overall profit generated by the market. Firm A holds 63.1% of the market share while Firm B 36.9%, based solely on the customers served.
4.1.1 Nash equilibrium results
With the status quo profits computed, the gametheoretic model is subsequently solved both as an MINLP and MILP and the related results are compared. The MINLP model comprises of 2,669 equations, 4,003 continuous variables and 208 binary variables and is solved to global optimality using BARON 18.5.8 within 62.5 CPU (s). The Nash equilibrium results in a 7.3% profit increase for firm A while firm’s B profit is increased by 34.2% compared to their status quo values and a 55/45% market share allocation for the two firms respectively. Acquisition of new customers is one of the two main causes for the profit increase while the other one is the reallocation of existing customers among the firms which leads to reduction in distribution costs.
The importance of the game theoretic solution is further underlined when compared to the naive approach where one would seek maximisation of the overall profit generated by the two firms regardless of the decentralised structure of the market. Such a “naive approach”, results in significant market share reductions for firm A down to 50.6% while firm’s B market share is increased to 49.4%. However such case is not realisable since it benefits greatly the weak player of the game who increases their profit by 72% when firm A achieves merely 3.2% profit increase in this case. A graphical illustration of the aforementioned results is given in Fig. 8.
An interesting insight with regards to the effect of negotiation power and the game theoretic solution on the overall profit generated by the duopoly is given by Fig. 8. The highest overall profit is achieved when the decentralised nature of the market is completely neglected and the naive optimisation is employed. The biggest impact on the overall profit is attributed the negotiation power of the firm with the highest market share before the game, in this case firm A. As indicated by the graph in Fig. 8, when A is assumed to be the follower, i.e. when its negotiation power is assumed to be negligible (\(\alpha _{A}=0\)), the profit approaches the one computed by the naive approach and it is the second highest.
Customer allocation and flows between firms on the Nash equilibrium for the duopoly case study
Acquired by Firm A  Acquired by Firm B  

New customers  5  11 
Forfeited by Firm A  –  13 
Forfeited by Firm B  5  – 
Total customers  41  57 
Comparison of the MILP and MINLP model results
Model  Grid points (k)  Error \(\left( \frac{{\tilde{\varPsi }}\varPsi }{\varPsi }\%\right)\)  CPU (s)  Market share (Firm A/B %)  Customers (Firm A/B)  Hamming Distance 

MILP  5  12.99  0.29  59.8/40.2  37/61  24 
MILP  25  0.656  0.39  54.1/45.9  46/52  24 
MILP  50  0.08  0.43  54.8/45.2  42/56  15 
MILP  100  0.054  0.75  54.8/45.2  40/58  3 
MILP  300  0.021  1.23  54.8/45.2  41/57  – 
B&R^{b}  11  0.015  2.84  54.8/45.2  41/57  – 
MINLP^{a}  –  –  62.3  54.8/45.2  41/57  – 
4.1.2 Negotiation power sensitivity analysis
As mentioned in the Sect. 3.2, the Nash bargaining approach can facilitate the case where the different players have unequal negotiation power by adjusting the parameter \(\alpha _{f}\) so as to indicate the hierarchy of the market. In order to evaluate the effect of the negotiation power on the resulting Nash equilibrium sensitivity analysis was conducted with varying the negotiation power of each player from 0 to 1.
4.2 Liquid market oligopoly
Next, in this case study two variants of the naive approach, where the objective is to maximise the overall profit of the market, were examined. In the naive approach (I) each player neglects their status quo profits while in the second case (naive approach (II)) each player requires their resulting profit to be increased with regards to their status quo values. As indicated by Fig. 14, the naive approach (I) benefits greatly the weaker firm (B) while the strongest firm (C) loses a significant amount of its profit. This instance is unrealistic as no firm would engage in a game where it will lose their profit for the sake of market equality. From an operational point of view, firm B before the game utilises only 15% of its LOX capacity and 50% of its LNI capacity capabilities while firm A is on average at 70% production capacity and firm C 60%. Following the naive approach (I) firm A reaches capacity bound on its LOX production capability while due its advantageous position, firm B manages to reach 55.5% of its LOX and 65% of its LNI production capacity.
Customer allocation and flows between firms on the Nash equilibrium for the oligopoly case study
Acquired by Firm A  Acquired by Firm B  Acquired by Firm C  

New customers  5  4  4 
Forfeited by Firm A  –  –  5 
Forfeited by Firm B  1  –  2 
Forfeited by Firm C  5  1  1 
Total customers  27  19  35 
5 Concluding remarks
Considering explicitly the decentralised nature of contemporary supply chain systems when optimal strategic decisions are sought is of great importance. In this paper the problem of fair customer allocation in oligopolies was addressed with emphasis on the industrial liquid markets. A novel singleperiod mathematical model was introduced for the problem under study and in order to address the fairness considerations the Nash bargaining was proposed. The nonlinear nature of the Nash approach resulted in the model formulation as an MINLP for which globally optimal solutions were computed. A separable programming approach was also discussed for its approximate solution as an MILP with considerable computational savings. As shown by the case studies examined the MILP model solution asymptotically converges to the global solution of the MINLP as the number of points increases. The results indicate the computation of more realistic solutions that account for market power dynamics when the Nash approach is employed in comparison to a centralised naive profit maximisation of the entire system. Future work aims at the extension of the model to the multiperiod case in order to account for capacity expansion considerations and the impact of customer contract design on the fair allocation as well as the exploration of uncertainty considerations through the proposed MILP formulation.
Notes
Acknowledgements
The authors gratefully acknowledge financial support from EPSRC Grants EP/M027856/1 and EP/M028240/1.
References
 BarbosaPóvoa AP, Pinto JM (2018) Challenges and perspectives of process systems engineering in supply chain management. Comput Aided Chem Eng 44:87–96CrossRefGoogle Scholar
 Bergamini ML, Grossmann I, Scenna N, Aguirre P (2008) An improved piecewise outerapproximation algorithm for the global optimization of minlp models involving concave and bilinear terms. Comput Chem Eng 32(3):477–493CrossRefGoogle Scholar
 Berglund PG, Kwon C (2014) Solving a location problem of a Stackelberg firm competing with cournotnash firms. Netw Spat Econ 14(1):117–132MathSciNetCrossRefGoogle Scholar
 Cao E, Wan C, Lai M (2013) Coordination of a supply chain with one manufacturer and multiple competing retailers under simultaneous demand and cost disruptions. Int J Prod Econ 141(1):425–433CrossRefGoogle Scholar
 Erkut E, Karagiannidis A, Perkoulidis G, Tjandra SA (2008) A multicriteria facility location model for municipal solid waste management in north greece. Eur J Oper Res 187(3):1402–1421MathSciNetCrossRefGoogle Scholar
 Florensa C, GarciaHerreros P, Misra P, Arslan E, Mehta S, Grossmann IE (2017) Capacity planning with competitive decisionmakers: trilevel MILP formulation, degeneracy, and solution approaches. Eur J Oper Res 262(2):449–463MathSciNetCrossRefGoogle Scholar
 Gao J, You F (2019) A stochastic game theoretic framework for decentralized optimization of multistakeholder supply chains under uncertainty. Comput Chem Eng 122:31–46CrossRefGoogle Scholar
 GarciaHerreros P, Zhang L, Misra P, Arslan E, Mehta S, Grossmann IE (2016) Mixedinteger bilevel optimization for capacity planning with rational markets. Comput Chem Eng 86:33–47CrossRefGoogle Scholar
 Gjerdrum J, Shah N, Papageorgiou LG (2001) Transfer prices for multienterprise supply chain optimization. Ind Eng Chem Res 40(7):1650–1660CrossRefGoogle Scholar
 Gjerdrum J, Shah N, Papageorgiou LG (2002) Fair transfer price and inventory holding policies in twoenterprise supply chains. Eur J Oper Res 143(3):582–599CrossRefGoogle Scholar
 Harsanyi JC, Selten R (1972) A generalized Nash solution for twoperson bargaining games with incomplete information. Manag Sci 18(5–part–2):80–106MathSciNetCrossRefGoogle Scholar
 Hjaila K, Puigjaner L, Laínez JM, Espuña A (2017) Integrated gametheory modelling for multi enterprisewide coordination and collaboration under uncertain competitive environment. Comput Chem Eng 98:209–235CrossRefGoogle Scholar
 Klein RS, Luss H, Smith DR (1992) A lexicographic minimax algorithm for multiperiod resource allocation. Math Prog 55(1–3):213–234MathSciNetCrossRefGoogle Scholar
 Levis AA, Papageorgiou LG (2007) Active demand management for substitute products through price optimisation. OR Spectrum 29:551–577CrossRefGoogle Scholar
 Liu S, Papageorgiou LG (2018) Fair profit distribution in multiechelon supply chains via transfer prices. Omega 80:77–94CrossRefGoogle Scholar
 Lozano S, Moreno P, AdensoDíaz B, Algaba E (2013) Cooperative game theory approach to allocating benefits of horizontal cooperation. Eur J Oper Res 229(2):444–452MathSciNetCrossRefGoogle Scholar
 Nagarajan M, Sošić G (2008) Gametheoretic analysis of cooperation among supply chain agents: review and extensions. Eur J Oper Res 187(3):719–745MathSciNetCrossRefGoogle Scholar
 Nash JF (1951) Noncooperative games. Ann Math 54:286–295MathSciNetCrossRefGoogle Scholar
 Norouzi M, Fleet DJ, Salakhutdinov RR (2012) Hamming distance metric learning. In: Pereira F, Burges CJC, Bottou L, Weinberger KQ (eds) Advances in neural information processing systems 25. Curran Associates, Inc., pp 1061–1069. http://papers.nips.cc/paper/4808hammingdistancemetriclearning.pdf
 Ogryczak W (1997) On the lexicographic minimax approach to location problems. Eur J Oper Res 100(3):566–585CrossRefGoogle Scholar
 Papageorgiou LG (2009) Supply chain optimisation for the process industries: advances and opportunities. Comput Chem Eng 33(12):1931–1938CrossRefGoogle Scholar
 Rosenthal RE (2012) GAMS—a user’s guide: GAMS development corporation. Washington, DCGoogle Scholar
 Sahay N, Ierapetritou M (2013) Supply chain management using an optimization driven simulation approach. AIChE J 59(12):4612–4626CrossRefGoogle Scholar
 Shah N (2005) Process industry supply chains: advances and challenges. Comput Chem Eng 29(6):1225–1235CrossRefGoogle Scholar
 Sherali HD, Leleno JM (1988) A mathematical programming approach to a NashCournot equilibrium analysis for a twostage network of oligopolies. Oper Res 36(5):682–702MathSciNetCrossRefGoogle Scholar
 Tawarmalani M, Sahinidis NV (2005) A polyhedral branchandcut approach to global optimization. Math Prog 103:225–249MathSciNetCrossRefGoogle Scholar
 Tominac P, Mahalec V (2017) A game theoretic framework for petroleum refinery strategic production planning. AIChE J 63(7):2751–2763CrossRefGoogle Scholar
 Williams HP (2013) Model building in mathematical programming. Wiley, New YorkzbMATHGoogle Scholar
 Yue D, You F (2014a) Fair profit allocation in supply chain optimization with transfer price and revenue sharing: MINLP model and algorithm for cellulosic biofuel supply chains. AIChE J 60(9):3211–3229CrossRefGoogle Scholar
 Yue D, You F (2014b) Gametheoretic modeling and optimization of multiechelon supply chain design and operation under Stackelberg game and market equilibrium. Comput Chem Eng 71:347–361CrossRefGoogle Scholar
 Yue D, You F (2017) Stackelberggamebased modeling and optimization for supply chain design and operations: a mixed integer bilevel programming framework. Comp Chem Eng 102:81–95CrossRefGoogle Scholar
 Zamarripa MA, Aguirre AM, Méndez CA, Espuña A (2012) Improving supply chain planning in a competitive environment. Comput Chem Eng 42:178–188CrossRefGoogle Scholar
 Zhang D, Liu S, Papageorgiou LG (2014) Fair cost distribution among smart homes with microgrid. Energy Convers Manag 80:498–508CrossRefGoogle Scholar
 Zhao Y, Wang S, Cheng TE, Yang X, Huang Z (2010) Coordination of supply chains by option contracts: a cooperative game theory approach. Eur J Oper Res 207(2):668–675MathSciNetCrossRefGoogle Scholar
 Ziss S (2007) Hierarchies, intrafirm competition and mergers. Int J Ind Organ 25(2):237–260CrossRefGoogle Scholar
Copyright information
Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.