The Optimum Output Quantity for Different Competitive Behaviors under a Fuzzy Decision Environment

Due to the uncertainty of information and complexity of decision-making environment, the optimum output quantity is studied under a fuzzy decision environment. Firstly, the triangular intuitionistic fuzzy model is proposed. Secondly, the optimum output quantity is discussed for four patterns to market structure. Thirdly, the effect of fuzzy parameter on optimum output quantity and total market demand is discussed. Finally, a numerical example is given to illustrate the concrete application of the proposedmodel.


Introduction
In the fierce market competition, in order to gain competitive advantage and maximize profit, the optimum output quantity decision among different enterprises or coalitions in supply chain has been an important issue.In a duopoly market, two models can be used to optimize the output quantity.One is the Cournot model [1]; the other is the Stackelberg model [2].The Cournot model forms a situation in which each firm chooses its output independently.The classical Stackelberg model is composed of one leader and one follower.The leader makes its decision taking into account the reaction of the follower.The follower, on the other hand, makes its decision assuming the leader will keep its supply quantity fixed.
The Cournot and Stackelberg models have been widely studied in the literature.Much of the literature about the Cournot model has focused on the extension, equilibrium, and application.For example, Dang et al. (2014) [3] developed the Cournot production game with multiple firms in an ambiguous decision environment, where the form of ambiguity is described by a set of fuzzy parameters.Van den Berg et al. (2012) [4] studied a dynamic Cournot duopoly in which suppliers have a limited amount of products available for two consecutive periods.Barr and Saraceno (2005) [5] examined the effects of both environmental and organizational factors on the outcome of repeated Cournot games.Guo (2010) [6] initially proposed a one-shot decision approach to solve for the Cournot equilibrium.Guo et al. (2010) [7] further extended the method proposed in Guo [6] to a duopoly market with asymmetric possibilistic information describing the demand uncertainty only known by one firm.Colombo and Labrecciosa (2013) [8] pointed out that when the asset stock grows sufficiently fast, the dynamic Cournot game corresponds to the static Cournot solution.Fang and Shou (2015) [9] applied the Cournot model in supply chain competition.Hu et al. (2014) [10] developed the entry and competition of a plant factory supply chain in vegetable markets, using a Nash-Cournot model to simulate this competition.Some researchers focused on the Stackelberg model.For instance, DeMiguel and Xu (2009) [11] studied an oligopoly consisting of  leaders and  followers that supply a homogeneous product noncooperatively.De Wolf and Smeers (1997) [12] made an interesting extension of the singleleader Stackelberg-Nash-Cournot model by incorporating demand uncertainty.Nakamura (2015) [13] analyzed oneleader and multiple-follower Stackelberg games with demand uncertainty.In the research of application, Kim (2012) [14] proposed a new Cognitive Radio network spectrum sharing scheme based on the multileader multifollower Stackelberg game model.Alvarez-Vázquez et al. (2015) [15] applied the Stackelberg techniques to the environmental problem related to determining the optimal location and purification profile when building a new treatment plant in a wastewater depuration system.Yang and Zhou (2006) [16] analyzed the effects of the duopolistic retailers in different competitive behaviors: Cournot, Collusion, and Stackelberg on the optimal decisions of the manufacturer and the duopolistic retailers themselves.
In practical application problems, the economic assessment data, such as the fixed costs and the per unit cost, are not exact.It is necessary to consider information of market uncertainty.Liang et al. (2008) [17] developed optimum output quantity decision analysis of a duopoly market under a fuzzy environment, where the form of ambiguity is described by trapezoidal fuzzy numbers.Triangular intuitionistic fuzzy numbers (TIFNs) may express an ill-known quantity with different degree of membership and degree of nonmembership.It is an extension of triangular fuzzy number which can represent the fuzzy essence of uncertain information and depict fuzzy preference information of decision makers.Therefore, TIFNs will be used in this paper to describe the inverse demand functions of market and cost functions, and the optimum output quantity of duopoly market will be discussed.
This paper is organized as follows.In Section 2, the operations and ranking method of TIFNs are reviewed.Section 3 describes the triangular intuitionistic fuzzy model of duopoly; then the optimum output quantity is discussed for four patterns to market structure.Section 4 discusses the effect of fuzzy parameter changing on optimum output quantity and total market demand.Section 5 presents a numerical example to illustrate the concrete application of the proposed model.Conclusions are made in Section 6.

Triangular Intuitionistic Fuzzy
Numbers (TIFNs) 2.1.The Definition and Operations of TIFNs.In this section, TIFNs and their operations are defined as follows.
Let Π ã() = 1 −  ã() −  ã(), which is called the intuitionistic fuzzy index of an element  in ã.It is the degree of indeterminacy membership of the element  to ã.
A TIFN ã = ⟨(, , );  ã,  ã⟩ may express an ill-known quantity "approximate ," which is approximately equal to .Namely, the ill-known quantity "approximate " is expressed using any value between  and  with different degree of membership and degree of nonmembership.The pessimistic value is  with the degree of membership 0 and the degree of nonmembership 1; the optimistic value is  with the degree of membership 0 and the degree of nonmembership 1; other value is any  in the open interval (, ) with the membership degree  ã() and the nonmembership degree  ã().It is easy to see that  ã() +  ã() = 1 for any  ∈ R if  ã = 1 and  ã = 0. Hence, the TIFN ã = ⟨(, , );  ã,  ã⟩ degenerates to ã = ⟨(, , ); 1, 0⟩, which is just about a triangular fuzzy number.Therefore, the concept of the TIFN is a generalization of the triangular fuzzy number [19].

Triangular Intuitionistic Fuzzy Model of Duopoly
In a duopoly market, assume that there are two competitive players, denoted by enterprise  and enterprise , respectively.These two enterprises produce homogeneous products.Each enterprise's objective is to select output quantity to maximize their profit.In general, it is almost impossible to find the exact economic assessment of data for parameters' estimation in the real world.In this section, the TIFNs will be applied to study the equilibrium quantity between two enterprises.We will consider four patterns to market structure as follows: (1) Both of enterprises  and  are followers.
(2) Enterprise  is a leader and enterprise  is a follower.
(3) Enterprise  is a leader and enterprise  is a follower.
(4) Both of enterprises  and  are leaders.
The fuzzy cost functions of enterprises  and , denoted by TIFC  and TIFC  , are defined as follows: where f and f denote the fuzzy fixed costs of enterprises  and , respectively.c and c represent the fuzzy unit variable costs, and Then the fuzzy profit functions of enterprises  and , denoted by TIFΠ  and TIFΠ  , can be calculated, respectively, by The fuzzy profit functions TIFΠ  and TIFΠ  are TIFNs.Their fuzziness results from the fuzzy parameters of the inverse demand function and the cost function.We utilize (2) to defuzzify the fuzzy profit function into a crisp value.If Similarly, where Remark 4. In ( 9),

Pattern 1:
Both of Enterprises  and  Are Followers.
In this pattern, both of enterprises  and  independently respond with output quantities, and their optimum output quantities can be solved by the simultaneous-equation models constructed by their response functions, respectively.By considering the maximum profit of enterprise , the first-order derivative of   (TIFΠ  ) with respect to   is as follows: Solving   (TIFΠ  )/  = 0 will obtain the response function of enterprise : Similarly, we can find the response function of enterprise : By ( 13) and ( 14), the optimum output quantities of enterprises  and , represented by  *  1 and  *  1 , can be found.That is, By taking ( 15) into ( 4), the market equilibrium price p * 1 is solved: By taking  *  1 and  *  1 into ( 7) and ( 8), the fuzzy maximum profits TIFΠ *  1 and TIFΠ *  1 of enterprises  and  can be found: By ( 17) and ( 18), the fuzzy total profit TIFΠ * 1 in pattern 1 is as below: Then the   (TIFΠ  ) of TIFΠ  is By solving   (TIFΠ  )/  = 0, the optimum quantity  *  2 of enterprise  is as follows: By taking ( 22) into (20), we can find the fuzzy maximum profit TIFΠ *  2 of enterprise : By taking  *  2 and  *  2 into (8), the fuzzy maximum profit TIFΠ *  2 of enterprise  in pattern 2 is as below: Then the fuzzy total profit TIFΠ * 2 in pattern 2 is The fuzzy total profit TIFΠ * 3 in pattern 3 is

Effect of Fuzzy Parameters
As discussed in Section 3, fuzzy parameters are defuzzified into a crisp value when calculating optimum output quantity.
In this section, we use left and right spread of fuzzy parameters to discuss the effect of fuzzy parameter on optimum output quantity and market demand for the four competitive behaviors.
The TIFNs α, β, c , and c can be rewritten as follows: where  and  represent left and right spread of fuzzy parameters, respectively.Firstly, in pattern 1, by ( 15) and (36), the optimum output quantity of  and  can be rewritten as Then the total market demand is In the following, the sensitivity analysis of the optimum output quantity and total market demand is discussed.
Taking the partial derivative of (37) and (38) with respect to   , which means that  *  1 ,  *  1 , and  * 1 decrease in   .Similarly, we also take the partial derivative of (37) and (38) with respect to other fuzzy parameters, such as   ,   ,   ,   ,   ,   , and   .In addition, we can also analyze the sensitivity of the optimum output quantity and total market demand of enterprises  or  for the other patterns.The results are shown in Table 1.
From the results, we can make three observations.
According to the analysis of Section 3, the optimum output quantities of the four patterns are shown in Table 2.
The fuzzy maximum profits and the fuzzy total profits of the four patterns are shown in Table 3.The optimum price of the four patterns is shown in Table 4.
By (2) and In the same way, the following results can be obtained: Based on the analysis results stated above, some conclusions are be got as follows.
When both of enterprises  and  are followers, the equilibrium market price and the total fuzzy profit are maximal.When both of enterprises  and  are leaders, the equilibrium market price and the total fuzzy profit are minimal.When one is a leader, and the other is a follower; that is, enterprises  and  play Stackelberg game: the leader will obtain more profit than the follower.Therefore, the best decision is that both of enterprises  and  are followers.

Comparison
Analysis.This section includes three aspects: firstly, the analysis of results about Section 5.1; secondly, the comparison of the results with Liang et al. (2008) [17]; and finally, the advantages of modeling fuzziness through TIFNs are discussed.
Firstly, based on the above conclusions in Section 5.1, we find the orders of the equilibrium market price and the total fuzzy profit in patterns 2 and 3 will exchange when some parameters are changing.For example, if c = ⟨(5, 6, 7); 0.9, 0.1⟩ changes and the other parameters keep unchanging, the order of the fuzzy total profit is as follows: Because the market structures in patterns 2 and 3 are similar, by analyzing (22) to (31), the orders of the equilibrium market price and the total fuzzy profit in these two patterns are correlated with the position of (  + 4  +   ) and (  + 4  +   ). If If then Secondly, Liang et al. ( 2008) [17] designed a hypothetical optimum output quantity decision problem of duopoly market to explain the computational process.The cost functions and the parameters of market demand in [17] are triangular fuzzy numbers.Because TIFNs are an extension of triangular fuzzy numbers, those triangular fuzzy numbers can be rewritten to TIFNs.For example, the estimated fuzzy fixed costs for enterprise  in [17] are f = (60000, 65000, 65000, 68000), which can be rewritten to f = ⟨(60000, 65000, 68000); 1, 0⟩.Similarly, the estimated fuzzy fixed costs, fuzzy unit variable cost, and the parameters of market demand in [17]   Similar to the analysis in Section 5.1, the orders of fuzzy profits and total fuzzy profit for enterprises  and  are as below: From the analysis results, the conclusions are in accordance with [17].That means the approach using TIFNs in this paper is feasible to solve the optimum output quantity problem in practical application and is an extension method of the approach used in [17].
Thirdly, the advantages of the proposed model in this paper are analyzed.The effect of the degree of membership, the degree of nonmembership, and  on the fuzzy maximum profit of enterprise will be discussed based on the above example at first.And we suppose that the triangular intuitionistic fuzzy variables to be analyzed are the same as the above example used in [17], but the degree of membership and the degree of nonmembership are changing.
Let us take enterprise  in pattern 1 as an example.According to ( 9) and ( 17), the   (TIFΠ *  1 ) of TIFΠ *  1 can be found as follows: Taking the partial derivative of   (TIFΠ *  1 ) with respect to  Ã,  Ã, and , respectively, which means that the fuzzy maximum profit of enterprise  in pattern 1 increases in  Ã but decreases in  Ã and .
Similarly, we can also analyze the effect of these three fuzzy variables on the fuzzy maximum profit of enterprise  for the other three patterns or enterprise  for four patterns in the same way.Therefore, we can find that the degree of membership or nonmembership of TIFNs and the preference information of decision makers () will affect the results of the fuzzy maximum profit.
Based on the above analysis, the approach using trapezoidal fuzzy numbers proposed in [17] and the approach using TIFNs in this paper will be compared, and some differences and advantages of the proposed fuzzy approach in this paper will be found as follows.
Firstly, the difference between this two approaches is the way authors treat the fuzziness of the data.In [17], the author used trapezoidal fuzzy numbers to handle the fuzziness of the decision variables; however, TIFNs are used in this paper to describe the market uncertainty.Intuitionistic fuzzy numbers are an extension of fuzzy numbers, which can depict comprehensively the fuzzy essence of uncertainty information by the degree of membership and nonmembership.Therefore, the approach in this paper is better than in [17].Secondly, the fixed cost and unit variable cost are described by fuzzy numbers without considering the parameters of inverse demand function in [17].However, the inverse demand functions of market and the cost functions are both described by TIFNs in this paper, which can be better to describe uncertain market and improve the realism of the model.Thirdly, according to Section 5.2, we find that the degree of membership, the degree of nonmembership of TIFNs, and the preference information of decision makers () have effect on the results of the fuzzy maximum profit.Therefore, introducing TIFNs and  to construct fuzzy model in this paper can depict fuzzy preference information of decision makers and be convenient for managers or decision makers to make decision for their enterprise.

Conclusions
Due to the influences of some factors, such as paucity of data and ambiguous environment, it is not easy to obtain the exact economic assessment of data in the real world situation.The intuitionistic fuzzy numbers are just the suitable tool to express these ill-known quantities.Thus, in this paper, we apply TIFNs to solve the fuzziness aspect of demand and cost uncertainty for two enterprises.For simplicity, we assume that the inverse demand and cost functions of enterprises are TIFNs behaving in a linear form.The equilibrium quantity of each enterprise in a competitive market can be found.Furthermore, we conduct a sensitivity analysis to discuss the impacts of fuzzy parameter on optimum output quantity and market demand for the four competitive behaviors.Finally, a numerical example is given and the comparison between two approaches is analyzed, concluding three advantages of the proposed model at last.
In order to obtain optimum output quantities of enterprises, it is needed to defuzzify the fuzzy profit function into a crisp value.In this paper, we only use the value-index that proposed in [21] to defuzzify the fuzzy profit function.Some results proposed in this paper showed that the degree of membership or nonmembership of TIFNs and the preference information of decision makers cannot adequately reflect some characteristics of TIFNs, such as the results of output quantity or the sequence of the fuzzy profit in four patterns for one enterprise.Therefore, how to find a good defuzzification method of a fuzzy function is also an important issue in the future.

Step 1 .
Compare   (ã) and   ( b) for a given weight .If they are equal, then go to Step 2. Otherwise, rank ã and b according to the relative positions of   (ã) and   ( b).Namely, if   (ã) >   ( b), then ã is greater than b, denoted by ã > b; if   (ã) <   ( b), then ã is smaller than b, denoted by ã < b.Step 2. Compare   (ã) and   ( b) for the same given .If they are equal, then ã and b are equal.Otherwise, rank ã and b according to the relative positions of −  (ã) and −  ( b).

Table 2 :
The optimum output quantities of the four patterns.

Table 3 :
The fuzzy maximum profits and the fuzzy total profits of the four patterns.

Table 4 :
The optimum price of the four patterns.