The equilibrium quantity and production strategy in a fuzzy random decision environment: Game approach and case study in glass substrates industries

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The equilibrium quantity and production strategy in a fuzzy random

decision environment: Game approach and case study in glass

substrates industries

Jr-Fong Dang

a

, I-Hsuan Hong

b,n,1

a_{Department of Industrial Engineering & Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan} b_{Institute of Industrial Engineering, National Taiwan University, 1 Section 4 Roosevelt Road, Taipei 106, Taiwan}

a r t i c l e i n f o

Article history: Received 13 June 2012 Accepted 16 May 2013 Available online 11 June 2013 Keywords:

Fuzzy random variable Game theory Equilibrium quantity Entropy

Case study

a b s t r a c t

This paper develops a two-stage Cournot production game that integrates strategic and operational planning under the fuzzy random environment, which to our best knowledge has not appeared in the literature. At the strategic level, two competing decision-makers determine the upper bound of a production quantity under a high-production strategy and the lower bound of the production quantity under a low-production strategy. Then at the operational level, the two competitors determine the range-type production quantity that is assumed to be a triangular fuzzy number represented by the apex and the entropies rather than a crisp value. The apex of a fuzzy equilibrium quantity can be obtained by the conventional Cournot game as the membership value is equal to one. A fuzzy random decision can be represented by entropies derived from the fuzzy random profit function of each firm in a specific production strategy. A case study of two leadingfirms in the glass substrates industry demonstrates the applicability of the proposed model. Thefinding that both firms would tend to adopt the common strategy coincides with observed real-world behavior. We conclude that our proposed method can provide decision-makers with a simple mathematical foundation for determining production quantity under a production strategy in a fuzzy random environment.

1. Introduction

Decision-making in complicated and competitive

environ-ments can be a difﬁcult task because of uncertainty in the form

of ambiguity or randomness. After the notion of fuzzy sets theory

was introduced byZadeh (1965)to manage ambiguity, the theory

underwent extensive development and today is routinely applied

to solve a variety of real-world problems, (Kunsch and Fortemps,

2002;Tanaka, 1987;Wong and Lai, 2011). Problems of randomness

can be properly modeled by probability theory; applications to

real problems appear in (Pastor et al., 1999; Valadares Tavares

et al., 1998; Zhang et al., 2004). However, decision-makers often

work in a hybrid (uncertain) environment where ambiguity and randomness exist simultaneously. Given such environments, a

fuzzy random variable as introduced by Kwakernaak (1978)is a

useful tool for solving these two aspects of uncertainty. Other

studies (Colubi et al., 2001;Krätschmer, 2001) also extend several

theories (Wang et al., 2007;Wang et al., 2008) to environments

with these two aspects of uncertainty.

The typical Cournot game (see Cournot, 1838) models a

duopoly in which two competingﬁrms choose their production

quantity. In the equilibrium quantity, noﬁrm can be better off by a

unilateral change in its solution. The exact values of parameters are required information when the Cournot game is applied to decision-making models, but exact values are often unobtainable

in a business environment.Yao and Wu (1999)probably initiated a

non-cooperative game involving fuzzy data by applying the rank-ing method to defuzzify the fuzzy demand and fuzzy supply functions into crisp values such that both consumer surplus and producer surplus can be calculated in a conventional manner. Their method of transforming fuzzy numbers to crisp values is also

utilized to construct the monopoly model in Chang and Yao

(2000).Liang et al. (2008)propose a duopoly model with fuzzy

costs to obtain the optimal quantity of eachﬁrm. RecentlyDang

and Hong (2010)propose a fuzzy Cournot game with rigorous

deﬁnitions ensuring a positive equilibrium quantity and with a

ﬂexible controlling mechanism that adjusts the parameters of associated objective functions. As mentioned, the resulting crisp values derived by previous studies are counter-intuitive outcomes of the problem in the fuzzy sense. A crisp decision is too precise

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Int. J. Production Economics

n_{Corresponding author. Fax:}_{+886 2 2362 5856.} 1

The corresponding author holds a joint appointment with the Department of Mechanical Engineering, National Taiwan University.

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to be believed in a business environment with a fuzzy sense. To bridge this gap, we solve for a range-type solution rather than a crisp-value one. In addition, decision-makers may prefer a decision with more information in a range they themselves can adjust. In this case study, we empirically demonstrate our proposed approach to investigate the behaviors of the equilibrium quantity and production strategy in the glass substrates industry, which is highly volatile and ambiguous in the market demand and production costs.

In reality, the parameters of the Cournot game may behave with the characteristic of randomness in nature. In a business environment, decision-makers may predict demand behavior in a market, i.e. a functional form describes the relationship between price and quantity. The parameters of the functional form of market demand typically can be estimated by the technique of econometrics. Regression is one of the most popular statistical approaches employed and leads to game-theoretical models with

random parameters.Hosomatsu (1969)indicates that the Cournot

solution to an oligopolistic market is based upon the implicit

assumptions given the estimated market demand function.Sato

and Nagatani (1967) propose a model to relax the Cournot

assumption and substituteﬁrms' subjective evaluation of a market

with a more general form with randomness.

Unpredictable events may drive the price _{ﬂuctuation over a}

short period. It is appropriate to apply the fuzzy regression method

(Chen and Dang, 2008;González-Rodríguez et al., 2009) to manage

the data with ambiguity sense. The resulting estimators derived by econometrics involve in ambiguity and randomness, which occur in

manyﬁelds (Guo and Lu, 2009;Xu and Zhao, 2010). In fact,Guo and

Lu (2009)state that the coexistence of ambiguity and randomness

becomes an intrinsic characteristic in the real world. Thus, there is a strong motivation to develop the Cournot production game with parameters of ambiguity and randomness described by fuzzy random parameters. To our best knowledge, none of or very little research involves in such a Cournot production game.

The Cournot production game proposed in this paper incorpo-rates a production strategy which refers to the pattern of produc-tion quantities chosen by a decision-maker. Facing uncertainty with ambiguity and randomness, the choice of production strategy has important consequences for the selection, deployment and management of production resources. In general, there are two major stages of decision sequence: strategic and operational levels

(seeBallou (1992)). Many studies focus only on the strategic level

or operational level and ignore the importance of the interaction between them. In this paper, we propose a model considering both strategic and operational levels. Our model is similar to other

two-stage games (see (Bae et al., 2011;Dhaene and Bouckaert, 2010))

where a player's decision in the ﬁrst stage affects the action taken

in the second stage. We consider two speciﬁc types of production

strategies– high and low – by which the decision-maker's proﬁt

function depends on the highest production quantity or the lowest production quantity, respectively. In the long term, a high- or low-production strategy may be employed because the decision-maker desires to gain market share or to enhance the quality of products

(Stout, 1969; Walters, 1991). Furthermore, Yang and Wee (2010)

indicate that the production strategy is needed to respond to the market demand because of the rapid technology change (see

(Droge et al., 2012; Kenne et al., 2012;Xu et al., 2012) for other

models adopting strategy perspectives). The aim of this paper is to solve for the fuzzy equilibrium quantity of each decision-maker and to provide an appropriate production strategy under the fuzzy random business environment.

The remainder of this paper is organized as follows.Section 2

presents the preliminary knowledge of the fuzzy sets theory,

entropy, and fuzzy random variable. Section 3 addresses the

Cournot game in the fuzzy random environment and solves for

the fuzzy equilibrium quantity of eachﬁrm.Section 4investigates

the insights of the proposed method including the extension and

discussion. Section 5illustrates the applicability of the proposed

model in the real-world situation.Section 6discusses our

conclu-sions and gives suggestions for future research.

2. Deﬁnitions and concepts

This section introduces the fuzzy sets theory, entropy and expected operator which are integral to this paper.

2.1. Fuzzy sets theory

The fuzzy sets theory initiated by Zadeh (1965) attempts to

analyze and solve problems with a source of ambiguity called

fuzziness. In the following, we introduce the deﬁnitions and

notations of triangular fuzzy numbers,α-level cut and the

exten-sion principle.

2.1.1. Triangular fuzzy number

For practical purposes, one of the most commonly used fuzzy numbers is the triangular type because it is easy to handle

arithmetically and has an intuitive interpretation (Dağdeviren

and Yüksel, 2008;Şen et al., 2010).Giannoccaro et al. (2003)and

Petrovic et al. (1999)show that triangular fuzzy numbers are the

most suitable for modeling market demand in the fuzzy sense (see

(Ayağ and Özdemir, 2012;Vijay et al., 2005) for other applications

of triangular fuzzy numbers). The membership functionμ~AðxÞ of a

triangular fuzzy number ~A can be deﬁned by

μ~AðxÞ ¼ x−mAþlA lA ; mA−lA≤x ≤mA mAþrA−x rA ; mA≤x ≤mAþ rA 0; otherwise; 8 > > < > > : ð1Þ

where ~A is represented as a triplet ðmA−lA; mA; mAþ rAÞ and mA, lA

and rAare the apex, left and right spreads of the fuzzy number ~A,

respectively. Furthermore, a triangular fuzzy number ~A can be

shown inFig. 1.

2.1.2. α-Level cut

One of the most important concepts of fuzzy sets is theαlevel

cut given by ~Bα: ¼ fx∈ℝj ~BðxÞ≥αg

whereα∈½0; 1, which means for a fuzzy number ~B, those elements

whose membership values are greater than or equal toα.

2.1.3. Extension principle

Let“⊙” be any binary operation ⊕ and ⊗ between two fuzzy

numbers ~A and ~B. Based on the extension principle, the

member-ship function of ~A⊙ ~B is deﬁned by

μ~A⊙ ~BðzÞ ¼ sup_x_∘yminfμ~AðxÞ; μ~BðyÞg

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where“⊙¼⊕ or ⊗” corresponds to the operation “∘¼+ or ”. This result helps us to derive the membership function of a fuzzy number.

2.2. Entropy

The fuzziness of a fuzzy number can be described by a

membership function applied in many aspects (Petrovic et al.,

1999;Shu and Wu, 2010). However,Kao and Lin (2005)mention a

preference for using a simple index to show the fuzziness rather than using a membership function. They apply entropy, a simple index method, to interpret the fuzziness as a crisp value and consider the randomness of the fuzzy number.

Denote hðμ~AðxÞÞ, h : ½0; 1-½0; 1, as the entropy function that is

monotonically increasing in ½0; 1=2 and monotonically decreasing

in ½1=2; 1. The most well-known entropy function is the Shannon

function (Zimmermann, 1996) as described in(2).

hðuÞ ¼−uln u−ð1−uÞlnð1−uÞ ð2Þ

where u is the membership function of the fuzzy number.

Integrating the entropy over all elements x∈X leads to a global

entropy measure Hð ~AÞ:

Hð ~AÞ ¼ Z

x∈X

hðμ~AðxÞÞpðxÞdx ð3Þ

where p(x) denotes the probability density function of the

avail-able data set deﬁned over X. It is common to assume a uniform

distribution p(x)¼k following Kao and Lin (2005)) andPedrycz

(1994). According to (3), the entropy of the triangular fuzzy

number ~A is calculated by decomposing it into HLð ~AÞ and HRð ~AÞ:

Hð ~AÞ ¼ HLð ~AÞ þ HRð ~AÞ ¼

Z mA mA−lA hð_μ_~AðxÞÞpðxÞdx þZ mAþrA mA hðμ~AðxÞÞpðxÞdx ð4Þ

The three resulting entropies are the left entropy

HLð ~AÞ ¼ k lA=ln 4, the right entropy HRð ~AÞ ¼ k rA=ln 4, and the

entropy Hð ~AÞ ¼ k ðlAþ rAÞ=ln 4 (Kao and Lin, 2005). Furthermore,

Kao and Lin (2005) show that by ignoring the constant k a

triangular fuzzy number can be determined by the unique apex, left and right entropies without complicated membership func-tions. In other words, the left (right) entropy can be regarded as the left (right) spread of a triangular fuzzy number. The concept of entropy can be extended to other types of fuzzy numbers such as

trapezoid, exponential, etc (Kao and Lin, 2005).

2.3. Expected value of the fuzzy random variable

The mathematical notations of fuzzy random variables are

given in this section. Let ðΩ; Σ; PÞ be a probability space where Σ

is as− ﬁeld and P is a probability measure.

Deﬁnition 1. (Liu and Liu, 2003) Let ðΩ; Σ; PÞ be a probability

space. A fuzzy random variable is mappingξ : Ω-FvðℜÞ such that

for any closed C ofℜ, and the function

ξn_{ðCÞðωÞ ¼ sup}

x∈CμξðωÞðxÞ

is a measurable function of ω, where μξðωÞ is the possibility

distribution function of a fuzzy variable ξðωÞ and FvðℜÞ is a

collection of fuzzy variables deﬁned on a possibility space.

A triangular fuzzy random variable can be described by a triplet composed of the apex and the two crisp values of the left and right spreads. A detailed representation of a triangular fuzzy random

variableξ is deﬁned as

ξ ¼ ðX−l; X; X þ rÞ

where X is the apex distributed with the normal distribution,

Nðμ; s2_{Þ, having mean μ and variance s}2_.

Deﬁnition 2. (Liu and Liu, 2003) Let ξ be a triangular fuzzy

random variable. The expected value EðξÞ of ξ is calculated as:

EðξÞ ¼ μ þ1₄ðr−lÞ

Next, we introduce the linearity of the expected operator of fuzzy random variables.

Theorem 1. (Liu and Liu, 2003) Assume that ξ and η are fuzzy

random variables, then for any real numbers a and b, we have

Eðaξ þ bηÞ ¼ aEðξÞ þ bEðηÞ ð5Þ

3. Model and methodology

The Cournot game is a situation where eachﬁrm independently

chooses its production quantity in order to maximize the

respec-tive proﬁt function. In the real world, ambiguity and randomness

may appear simultaneously. If the parameters are fuzzy random

variables, the proﬁt function with these parameters is possibly a

fuzzy random variable (Liu and Liu, 2003). This section proposes

the theoretical model as a coherent, rigorous and novel philosophy position that not only substantiates the case study of glass substrates industries, but also provides helpful implications for

both theoretical development and real-world applications.

Furthermore, the proposed method solves for the closed-form equilibrium quantity with entropy spreads under a given

produc-tion strategy of each_ﬁrm.

3.1. The Cournot production game under the fuzzy random environment

At the strategic level of the game, it is essential to consider a high- or low-production strategy in response to the market because of the rapid change resulting in increasing or decreasing

market demand. Under a high-production,ﬁrm i, i¼1, 2, supplies a

highest quantity (the highest in the range-type solution) to the market, but a lowest quantity (the lowest in the range-type solution) under a production strategy. The high- or low-production strategy assists us to derive a range-type low-production

quantity of each ﬁrm. Furthermore, this leads to four (2 2)

strategic scenarios in a duopoly where eachﬁrm adopts a

high-or low-production strategy, respectively, in the strategic level.

At the operational level, each ﬁrm determines its production

quantity so as to maximize its Fuzzy Random Proﬁt Function (FRPF).

Fig. 2represents the decision sequence of our model. Assuming it

behaves rationally, ﬁrms anticipate their best response in the

operational level to the chosen production strategy in the strategic level. This allows us to solve this two-stage sequential game by moving from the operational level to the strategic level based on the chosen production strategy.

We assume the production quantity of each ﬁrm to be a

triangular fuzzy number represented by the apex and the entro-pies. We develop the Cournot production game including the parameters with triangular fuzzy random variables, a linear

TIME HORIZON

Operational Level Strategic Level

Fig. 2. Decision timeline of the Cournot production game under the fuzzy random environment.

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inverse demand and cost functions as shown below. Consider the fuzzy random inverse demand function as

pðξÞ ¼ ξa−ξb~Q ð6Þ

where ξ ¼ ðξa; ξbÞ is a fuzzy random vector representing the

parameters of market demand. The total market quantity is that

~Q ¼ ~q1þ~q2; ξa and ξb are triangular fuzzy random variables

deﬁned as

ξa¼ ðXa−la; Xa; Xaþ raÞ;

ξb¼ ðXb−lb; Xb; Xbþ rbÞ: ð7Þ

In(7),_ξais with left spread la, right spread raand apex Xawhere

Xafollows the normal distribution with meanμaand variances2a.

Parameterξbcan be explained in a similar manner. In addition, the

total cost function ofﬁrm i with fuzzy random parameters denoted

by TCiðξÞ is

TCiðξÞ ¼ ξdi~qi ð8Þ

whereξdi is a triangular fuzzy random variable deﬁned by

ξdi¼ ðXdi−ldi; Xdi; Xdiþ rdiÞ ð9Þ

In(9), the fuzzy random variable cost ofﬁrm i, ξdi, is with left

spread ldi, right spread rdi and apex Xdi where Xdi follows the

normal distribution with meanμdiand variances

2

di. Thus, the FRPF

ofﬁrm i is

πi¼ ðξa−ξb~Q Þ~qi−ξdi~qi ð10Þ

In (10), we can recognize that the problem involves in the

ambiguous uncertainty according to Liu and Liu (2003). For

simplicity, we take the expected value of(10)as:

Eð_πiÞ ¼ E½ðξa−ξb~Q Þ~qi−E½ξdi~qi ð11Þ

Similar approaches in different applications with fuzzy random

parameters appear in (Dutta et al., 2005; Kwakernaak, 1978).

Because the production quantity of eachﬁrm is assumed to be a

triangular fuzzy number, this allows us to characterize the pro-duction quantity by the apex and the entropies. When the

membership value,α, is equal to 1, the values of the production

quantity and the fuzzy random parameters are apexes based on

the extension principle (seeZadeh, 1965). Thus, the case involving

an ambiguous uncertainty can be treated in a crisp manner. Furthermore, the right and left entropies are decision variables to characterize the highest and lowest production levels. It follows

that the expected FRPF of ﬁrm i is with the highest entropies

under a high-production strategy, and vice versa. Therefore, we can decompose the original problem into (i) the center problem that solves for the apex of the fuzzy equilibrium quantity by the conventional Cournot game given that each parameter has a membership value equal to 1, and (ii) the spreads problem that

maximizes each_{ﬁrm's expected FRPF over its entropies under a}

production strategy. The solutions of these two problems give a triangular fuzzy equilibrium quantity, which provides decision-makers with a range-type solution instead of a crisp-value

solu-tion. In the following section, weﬁrst solve for the apex of the

fuzzy equilibrium quantity of ﬁrm i followed by the spreads

problem.

3.2. The operational-level decision: The center problem

As mentioned, when the membership value is equal to 1, the

expected FRPF of ﬁrm i can be represented by the apex of each

parameter as follows:

EðπiÞ ¼ E½ðξa−ξbQ Þqi−EðξdiÞqi ð12Þ

We derive the apex of the fuzzy equilibrium quantity qi

according to the conventional Cournot game (see Rasmusen,

2001). The best response function is obtained by maximizingﬁrm

i's expected FRPF over the apex of the production quantity, qi. This

generates theﬁrst-order condition returning ﬁrm i's best response

function toﬁrm j's production quantity, qj, as:

qi¼

μa−qj−μdi

2 ð13Þ

Similarly, we deriveﬁrm j's best response function to ﬁrm i's

production quantity as qj¼

μa−qi−μdj

2 ð14Þ

The apex of the fuzzy equilibrium quantity ofﬁrm i, shown in

(15), is obtained by simultaneously solving(13)and(14).

qi¼ μaþ μdj−2μdi 3μb ð15Þ Similarly, qj¼ μaþ μdi−2μdj 3μb ð16Þ

It is clear that qiis the solution of the center problem given that

the membership value is 1. Utilizing the resulting outcome we can now solve the spreads problem.

3.3. The operational-level decision: The spreads problem

We note that the upper bound of the total production quantity can be achieved since the lower bound of the market price occurs because of the law of demand. Substituting the highest production

quantity of each ﬁrm into the market demand, we deﬁne the

expected FRPF ofﬁrm i under a high-production strategy as:

EðπH iÞ ¼ Eðξa−ξbQHÞqHi−EðξdiÞq H i ð17Þ where QH¼ qiþ eRi þ qjþ eRj

In(17), we note that the upper bound of market demand, QH_{, is}

the production quantity determined by each ﬁrm plus the right

entropy. The right entropy of the fuzzy equilibrium quantity can be

interpreted as the increasing quantity of oneﬁrm.

Similarly, the expected FRPF ofﬁrm i under a low-production

strategy is EðπL iÞ ¼ Eðξa−ξbQLÞqLi−EðξdiÞq L i ð18Þ where QL¼ qi−eLiþ qj−eLj

The lower bound of the market demand, QL_{, is the production}

quantity determined by eachﬁrm minus the left entropy of each

ﬁrm. In the real word, ﬁrms may adjust their capacities to produce products in peak and off-peak seasons so that the relation between the designed capacities in peak and off-peak seasons

practically behaves in a ﬁxed ratio manner. This allows us to

assume that the ratio of the right entropy of ﬁrm i to its left

entropy is a given parameter,λi; that is, eRi ¼ λieLi whereλi40. This

assumption assists in obtaining the qualitative managerial insights

with less analytical complexity. Substituting eR

i ¼ λieLi into(17), the

expected FRPF under a high-production strategy is concave in eL

i

since ð∂2_Eð_πH iÞ=∂ðeLiÞ

2_{Þ ¼ −2λ}2

iEðξbÞo0. Similarly, in(18), the

expec-ted FRPF under a low-production strategy is concave in eL

i. In the

following, we derive the resulting entropies of each ﬁrm for our

four strategic scenarios where eachﬁrm maximizes its expected

FRPF. Furthermore, we adopt the concept of the production

strategy to construct the fuzzy equilibrium quantity of eachﬁrm

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Strategic Scenario 1. Bothﬁrms i and j adopt the high-production strategy.

Firm i maximizes its expected FRPF over eL

i. The ﬁrst-order

condition returnsﬁrm i's best response function to ﬁrm j's decision

variable, eL j; that is

−λiEðξbÞðqiþ λieLiÞ

þ λi½EðξaÞ−EðξbÞðqiþ λieLiþ qjþ λjeLjÞ−λiEðξdiÞ ¼ 0 ð19Þ

Similarly, the ﬁrst-order condition returning ﬁrm j's best

response function toﬁrm i's decision variable, eL

i, is

−λjEðξbÞðqjþ λjeLjÞ

þ λj½EðξaÞ−EðξbÞðqiþ λieLiþ qjþ λjeLjÞ−λjEðξdjÞ ¼ 0: ð20Þ

Let eL

i1 be the left equilibrium entropy of ﬁrm i in Strategic

Scenario 1 derived by simultaneously solving(19)and(20).

eL i1¼

EðξaÞ−3EðξbÞqi−2EðξdiÞ þ EðξdjÞ

3λiEðξbÞ ; i; j ¼ 1; 2; i≠j:

ð21Þ

To ensure a non-negative left equilibrium entropy ofﬁrm i, we

impose the condition such that eL

i1≥0, i¼1, 2.Assumption 1follows

from the condition, where qiis derived in(15).

Assumption 1. EðξaÞ−3EðξbÞqi−2EðξdiÞ þ EðξdjÞ≥0.

Combining(15)and(21), the fuzzy equilibrium quantity ofﬁrm

i in Strategic Scenario 1 becomes ðqi−eLi1; qi; qiþ λieLi1Þ; i ¼ 1; 2

Strategic Scenario 2. Firm i adopts the low-production strategy

andﬁrm j adopts the high-production strategy.

Under a low-production strategy, ﬁrm i solves the spreads

problem by maximizing its expected FRPF over eL

i. Eq. (18) is

maximized when theorder condition holds. Using the

first-order condition to derivefirm i's best response function to firm j's

decision variable, eL

j, gives

EðξbÞðqi−eLiÞ−½EðξaÞ−EðξbÞðqi−eLi þ qj−eLjÞ þ EðξdiÞ ¼ 0 ð22Þ

Firm j maximizes its expected FRPF, as shown in(17), under a

high-production strategy. As mentioned, EðπH

jÞ is concave in eLj so

theﬁrst-order condition of(17)gives

−λjEðξbÞðqjþ λjeLjÞ

þ λj½EðξaÞ−EðξbÞðqiþ λieLiþ qjþ λjeLjÞ−λjEðξdjÞ ¼ 0 ð23Þ

Let eL

Scenario 2 derived by solving(22)and(23). Theﬁnal results of eL

i2

and eL

j2 are

eL i2¼

−EðξaÞð1 þ 2λjÞ þ EðξbÞðqið1 þ 4λjÞ þ 2qjð1 þ λjÞÞ þ 2λjEðξdiÞ þ EðξdjÞ

Eð_ξbÞð4λj−λiÞ

ð24Þ and

eL j2¼

EðξaÞð2 þ λiÞ−EðξbÞ2qið1 þ λiÞ−EðξbÞqjð4 þ λiÞ−λiEðξdiÞ−2EðξdjÞ

EðξbÞð4λj−λiÞ :

ð25Þ

To ensure non-negative left equilibrium entropies ofﬁrms i and j,

we impose the condition such that eL

i2≥0 and eLj2≥0.Assumption 2

follows the condition, where qiand qjare derived in(15)and(16).

Assumption 2.

ð4λj−λiÞ½−EðξaÞð1 þ 2λjÞ þ EðξbÞðqið1 þ 4λjÞ þ 2qjð1 þ λjÞÞ þ 2λjEðξdiÞ

þ EðξdjÞ≥0

and

ð4λj−λiÞ½EðξaÞð2 þ λiÞ−EðξbÞ2qið1 þ λiÞ−EðξbÞqjð4

þ λiÞ−λiEðξdiÞ−2EðξdjÞ≥0:

The fuzzy equilibrium quantity ofﬁrm i can be constructed by

(15)and(24)as:

ðqi−eLi2; qi; qiþ λieLi2Þ; i ¼ 1; 2

Strategic Scenario 3. Firm i adopts the high-production strategy

and_{ﬁrm j adopts the low-production strategy.}

Here, the solution procedure to derive the entropies of each

ﬁrm is similar to Strategic Scenario 2. Let eL

i3be the left equilibrium

entropy of_{ﬁrm i in Strategic Scenario 3. The resulting outcomes of}

eL

i3and eLj3 can be obtained as

eLi3¼

EðξaÞð2 þ λjÞ−EðξbÞqið4 þ λjÞ−EðξbÞ2qjð1 þ λjÞ−2EðξdiÞ−λjEðξdjÞ

EðξbÞð4λi−λjÞ

ð26Þ and

eL j3¼

−EðξaÞð1 þ 2λiÞ þ EðξbÞð2qið1 þ λiÞ þ qjð1 þ 4λiÞÞ þ EðξdiÞ þ 2λiEðξdjÞ

EðξbÞð4λi−λjÞ

ð27Þ Similarly, to ensure non-negative left equilibrium entropies of

ﬁrms i and j, we imposeAssumption 3, where qiand qjare derived

in(15)and(16).

Assumption 3.

ð4λi−λjÞ½EðξaÞð2 þ λjÞ−EðξbÞqið4 þ λjÞ−EðξbÞ2qjð1 þ λjÞ−2EðξdiÞ−λjEðξdjÞ≥0

and

ð4λi−λjÞ½−EðξaÞð1 þ 2λiÞ þ EðξbÞð2qið1 þ λiÞ þ qjð1 þ 4λiÞÞ þ EðξdiÞ

þ 2λiEðξdjÞ≥0:

The fuzzy equilibrium quantity ofﬁrm i can be constructed by

(15)and(26)as

Strategic Scenario 4. Bothﬁrms i and j adopt the low-production

strategy.

As mentioned, the expected FRPF under a low-production

strategy is concave in eL

i, so (18) is maximized when the

ﬁrst-order condition holds. From theﬁrst-order condition, we have

EðξbÞðqi−eLiÞ−½EðξaÞ−EðξbÞðqi−eLiþ qj−eLjÞ þ EðξdiÞ ¼ 0 ð28Þ

Similarly, we can obtain theﬁrst-order condition of ﬁrm j as

EðξbÞðqj−eLjÞ−½EðξaÞ−EðξbÞðqi−eLiþ qj−eLjÞ þ EðξdjÞ ¼ 0 ð29Þ

Let eL

Scenario 4 derived by solving(28)and(29):

eL i4¼

−EðξaÞ þ 3EðξbÞqiþ 2EðξdiÞ−EðξdjÞ

3EðξbÞ ; i; j ¼ 1; 2; i≠j:

ð30Þ

To ensure a non-negative left equilibrium entropy ofﬁrm i, we

impose the condition such that eL

i4≥0, i¼1, 2.Assumption 4follows

from this condition, where qiis derived in(15).

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Due to(15)and(30), the fuzzy equilibrium quantity ofﬁrm i in Strategic Scenario 4 is

4. Analysis at the strategic level

In this section, we derive the conditions such that one of the four strategy combinations is the Nash equilibrium outcome in the strategic level. The Nash equilibrium, where no player has an incentive to deviate from its strategy given that the other players do not change their strategies, allows us to analyze the

relation-ship between eachﬁrm's production strategies.

Proposition 1. The expected FRPF ofﬁrm i, i¼1, 2 under a

high-production strategy is equal to the FRPF ofﬁrm i, i¼1, 2 under a

low-production strategy if bothﬁrms adopt the common strategy;

that is, EðπH

i1Þ ¼ EðπLi4Þ, where EðπHi1Þ and EðπLi4Þ are the expected FRPF

ofﬁrm i under a high-production strategy in Strategic Scenario 1

and a low-production strategy in Strategic Scenario 4, respectively.

Proof. Substituting eL

i1in(21)into(17), we have

EðπH i1Þ ¼ ð

EðξaÞ−2EðξdiÞ þ EðξdjÞ

3 Þð

3EðξbÞ Þ Similarly EðπL i4Þ can be obtained EðπL i4Þ ¼ ð

3 Þð

3EðξbÞ

Þ

It is clear that Eð_πH

i1Þ is equal to EðπLi4Þ and this completes

the proof.

We can calculate the expected FRPF of eachﬁrm under a

high-or low-production strategy by substituting the resulting entropies

derived inSection 3.3. As mentioned, four strategic scenarios are

considered in our model. We let theﬁrst and second attributes of

(⋅; ⋅) denote the production strategy adopted by ﬁrm i and ﬁrm j,

respectively. Eachﬁrm chooses the optimal strategy for the long

term to maximize its expected FRPF in the short term. Table 1

represents the expected FRPF of ﬁrms i and j under a speciﬁc

combination of production strategies chosen by the two ﬁrms.

Next, we utilize the results inTable 1to derive the conditions such

that a production strategy combination is the Nash equilibrium outcome.

Proposition 2. The conditions for the four possible Nash

equili-brium outcomes are given inTable 2.

Proof. (i) Based on the deﬁnition of the Nash equilibrium, if the

strategy combination (high-production, high-production) is the

Nash equilibrium outcome, it means that Eð_πH

i1Þ≥EðπLi2Þ and

EðπH

j1Þ≥EðπLj3Þ. First, we have

EðπH

i1Þ−EðπLi2Þ ¼ ½EðξaÞ−EðξbÞðqiþ λieLi1þ qjþ λjeLj1Þ−EðξdiÞðqiþ λie L i1Þ−

½EðξaÞ−EðξbÞðqi−eLi2þ qj−eLj2Þ−EðξdiÞðqi−e L

i2Þ≥0: ð31Þ

For notational simplicity, let Δ1¼ EðξaÞ−EðξbÞðqiþ λieLi1þ qjþ

λjeLj1Þ−EðξdiÞ and Δ2¼ EðξaÞ−EðξbÞðqi−e L

i2þ qj−eLj2Þ−EðξdiÞ. Note that

Δ1andΔ2areﬁrm i's expected unit proﬁts, which are reasonably

assumed non-negative. SubstitutingΔ1andΔ2into(31), we have

EðπH

i1Þ−EðπLi2Þ ¼ ðΔ1−Δ2Þqiþ Δ1λieLi1þ Δ2eLi2≥0 ð32Þ

LetΔn_{¼ minfΔ}

1; Δ2−Δ1g. Since Δ24Δ2−Δ1,Δ1, the terms ofΔ2

andΔ2−Δ1in(32)can be replaced by the smaller termΔn,we have

qi≤eLi2þ λieLi1: ð33Þ

Next, to satisfy the Nash equilibrium requirement, we have EðπH

j1Þ−EðπLj3Þ ¼ ½EðξaÞ−EðξbÞðqiþ λieLi1þ qjþ λjeLj1Þ−EðξdjÞðqjþ λje L j1Þ−

½EðξaÞ−EðξbÞðqi−eLi3þ qj−eLj3Þ−EðξdjÞðqj−e L

j3Þ≥0: ð34Þ

Similarly, letΔ3¼ EðξaÞ−EðξbÞðqiþ λiei1L þ qjþ λjeLj1Þ−EðξdjÞ and Δ4¼

EðξaÞ−EðξbÞðqi−ei3L þ qj−eLj3Þ−EðξdjÞ. Because of non-negative expected

unit proﬁts, Δ3≥0 and Δ4≥0. Now(34)can be rewritten as

EðπH

j1Þ−EðπLj3Þ ¼ ðΔ3−Δ4Þqjþ Δ3λjeLj1þ Δ4eLj3≥0 ð35Þ

LetΔnn_{¼ minfΔ}

3; Δ4−Δ3g. Since Δ4≥Δ4−Δ3, the terms ofΔ3,Δ4

and Δ4−Δ3in (35)can be replaced by the smaller term Δnn, we

have

qj≤eLj3þ λjeLj1 ð36Þ

Combining (33) with (36) results in the strategy combination

(high-production, high-production) being the Nash equilibrium outcome.

Table 1

The expected FRPF under a high- or low-production strategy in four strategic scenarios.

Firm j's production strategy

High-production

Low-production Firm i's production

strategy

High-production

ðEðπH

i1Þ; EðπHj1ÞÞ ðEðπHi3Þ; EðπLj3ÞÞ

Low-production

ðEðπL

i2Þ; EðπHj2ÞÞ ðEðπLi4Þ; EðπLj4ÞÞ

Table 2

Conditions for the four possible Nash equilibrium outcomes.

Firm j's production strategy

High-production Low-production

Firm i's production strategy High-production qi≤eLi2þ λieLi1

qj≤eLj3þ λjeLj1 qi≤eLi4þ λieLi3 qj≥eLj3þ λjeLj1 Low-production qi≥eLi2þ λieLi1 qj≤eLj4þ λjeLj2 qi≥eLi4þ λieLi3 qj≥eLj4þ λjeLj2

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(ii) The strategy combination (low-production, low-production)

satisfying the condition EðπL

i4Þ≥Eðπ H i3Þ and Eðπ L j4Þ≥Eðπ H j2Þ is the Nash

equilibrium outcome. Weﬁrst discuss

Eð_πL i4Þ−Eðπ

H

i3Þ ¼ Δ5ðqi−eLi4Þ−Δ6ðqiþ λieLi3Þ≥0 ð37Þ

where

Δ5¼ EðξaÞ−EðξbÞðqi−ei4L þ qj−eLj4Þ−EðξdiÞ

and

Δ6¼ EðξaÞ−EðξbÞðqiþ λiei3L þ qjþ λjeLj3Þ−EðξdiÞ

The terms, Δ5 and Δ6, are greater than 0 due to non-negative

expected unit proﬁts. Based on the solution procedure of (i),we have

qi≥eLi4þ λieLi3 ð38Þ

Similarly,

qj≥eLj4þ λjeLj2 ð39Þ

(iii) The strategy combination (low-production,

high-produc-tion) is the Nash equilibrium outcome if EðπL

i2Þ≥EðπHi1Þ and

Eð_πH

j2Þ≥EðπLj4Þ. By changing “ ≤” to “≥” in(33)and“≥” to “ ≤” in(39),

we have EðπL

i2Þ≥EðπHi1Þ and EðπHj2Þ≥EðπLj4Þ. Therefore, the strategy

combination (low-production, high-production) is the Nash equi-librium outcome if qi≥eLi2þ λieLi1and qj≤eLj4þ λjeLj2.

(iv) The strategy combination (high-production,

low-produc-tion) is the Nash equilibrium outcome if EðπH

i3Þ≥EðπLi4Þ and

EðπL

j3Þ≥EðπHj1Þ. By changing “≥” to “ ≤” in (38) and changing“ ≤” to

“≥” in (36), we have EðπH i3Þ≥Eðπ L i4Þ and Eðπ L j3Þ≥Eðπ H j1Þ. Therefore, the

strategy combination (high-production, low-production) is the Nash equilibrium outcome if qi≤eLi4þ λieLi3 and qj≥eLj3þ λjeLj1. This

completes the proof.■

Proposition 2. shows that the strategy combination becoming the Nash equilibrium outcome is based on both the apex and the entropies. In other words, our model provides decision-makers with both the fuzzy equilibrium quantity in the short term as well as the equilibrium production strategy in the long term.

5. Case study

In this section, we utilize the model presented inSection 3as a

planning tool to demonstrate how the two competing ﬁrms

determine the equilibrium quantity against ambiguity in the glass substrates industry.

5.1. Industry background

During the last decade, aggressive marketing strategies coupled

with low-cost thin-ﬁlm transistor liquid crystal display (TFT–LCD)

production have induced increasing numbers of consumers to

favor ﬂat screens over conventional cathode ray tube (CRT)

products. The physical sizes of glass substrates required for various

TFT–LCD products play a key role in the growing demand. As

mentioned earlier, there are two prohibitive barriers to entry into the glass substrates industry: capital outlay and the materials.

Since the market share of the two majorﬁrms in our case study

totals approximately 90% in Taiwan (Hwang and Lin, 2008), we

consider the glass substrates industry a duopoly market.

A recent report by DisplaySearch1_{indicates that the production}

of TFT–LCD glass substrates reached a peak of 14.2 million square

meters in second quarter 2010 and then dropped to 12.2 million square meters in the third quarter, a reduction of 14% from last

season. Thus, despite apparent consumer demand, global _ﬂat

screen manufacturers still need to adjust their production

strate-gies. It implies that because of the TFT–LCD panel prices and weak

demand, the manufactures have to adjust their production strat-egy to meet the market demand. This results in twofold produc-tion strategies: high- and low-producproduc-tion strategy.

Research on the Cournot game applied in the real world includes the world oil, electricity and petroleum products markets

(see (Ruiz et al., 2008; Slade, 1986; Salant, 1976)). The previous

studies have proposed to assist decision makers to determine the

equilibrium quantity or analyze the market ef_ﬁciency.

Acknowl-edging the need for improved decision-making, the model

pro-posed inSection 3depicts the behavior of two competing glass

substrates manufacturers in a hybrid uncertain environment, and

constructs eachﬁrm's fuzzy equilibrium quantity. After

determin-ing their production strategies, we obtain the apex of the fuzzy

equilibrium quantity by the center problem. We then deﬁne each

ﬁrm's proﬁt function considering the production strategy by utilizing the resulting apexes. Due to the special characteristics of the glass substrates industry, we apply the model to demon-strate how to obtain the fuzzy equilibrium quantity and the

production strategy of eachﬁrm.

5.2. Insights from the Cournot production game case study 5.2.1. Case study overview and input data

Our case study is based upon timely representative data for the glass substrates industry. We note that the data will differ for other industry sectors, geographic regions, and/or time epochs.

As mentioned earlier, the demand function of the market is

given by pðξÞ ¼ ξa−ξb~Q . The intercept, ξa, represents the amount of

glass substrates sold by the glass substrates manufacturers where

the price, p, is zero and the slope,_ξb, is the price sensitivity to the

increase in the amount of glass substrates per unit of the price

added. Based on the information published by DisplaySearch2_{, we}

arrange the unit price of the glass substrates as shown inTable 3.

Table 3

Market price of the glass substrates corresponding to the size type. Size type

1 2 3 4 5 6

Price (USD) High 57 55 61 63 76 81

Apex 56 53 60 60 73 78

Low 54 50 58 59 71 76

Table 4

Market demand of the glass substrates in each period of the case study (Shao and Lin, 2009). Size type 1 2 3 4 5 6 Period 1 1800 5000 20,000 7000 20,000 19,000 2 1800 3000 12,600 8600 16,000 14,000 3 1500 3000 16,000 5500 20,000 16,000 4 1680 4700 17,000 6000 22,000 16,000 1 http://www.newso.org/ITNews/Trade/DisplaySearch-LCD-substrate-in to-the-third-quarter-will-be-reduced/dc242f45-3d93-48b1-b410-0c2b114a0da1 2 http://www.displaysearch.com/cps/rde/xchg/displaysearch/hs.xsl/resource s_pricewise.asp

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Table 3shows the six types of glass substrates (denoted by 1 to 6), and the three prices in USD (high, apex and low) for each size type.

Table 4shows the sold quantity in the market for each size type after

Shao and Lin (2009). From these tables, we can derive the

para-metersξaandξb by the fuzzy regression methodChen and Dang

(2008). In the case study,ξaandξbcan be estimated as ξa¼ ðXa−

3_{:2300; X}a; Xaþ 2:8040Þ with Xa Nð52:7089; 4:9773Þ and ξb¼ ðXb

−0:000248; Xb; Xbþ 0:000303Þ with ξb¼ ð0:000987 ; 3⋅10−8Þ. In

Table 4, we observe that the peak-season is period 1 and the

off-season is period 2 because the total sold quantity in period 1, 89,000, is the highest, and in period 2, 56,000, is the lowest. Therefore, we estimate the ratio of production quantity in the peak-season to the

off-season as 1.59 (89,000/56,000), which can be viewed as isλ in

our model. We approximately estimate each ﬁrm's variable cost

based onﬁrm 1's consolidated ﬁnancial report3_{which indicates a net}

income of around 22.6%, i.e. ﬁrm 1's total cost is about 77.4%

(100%−22:6% ¼ 77:4%). We know that the unit net income can be

simply derived by market price minus the variable cost. OurTable 3

shows an average market price of 63.33 USD. Therefore, the ballpark

estimate of the variable cost of_{ﬁrm 1 is 63:33⋅77:4% ¼ 49:02 and}

similarly the variable cost ofﬁrm 2 is 50.03 (63:33⋅79%), as a result

ofﬁrm 2's 21%4_{net income.}

5.2.2. Case study results and sensitivity analysis

According to the proposed method inSection 3, we can derive

the apex of the fuzzy equilibrium ofﬁrm 1 by substituting the

parameters estimated in the case study into (18). Then we have

q1¼1,587. Similarly, the apex of the fuzzy equilibrium quantity of

ﬁrm 2 is q2¼570. It is obvious that the apex of the fuzzy

equilibrium quantity offirm 1 is higher than of firm 2 due to firm

1's low variable cost. Next, to solve for the entropy of each ﬁrm, we

consider four strategic scenarios in the spread problem with

assumptions. With the available data, weﬁnd that the resulting

solutions only satisfyAssumption 4, in other words, the entropy of

eachﬁrm can be obtained in Strategic Scenario 4 where both ﬁrms

adopt low-production strategies. As a result, the left entropy is 57

forﬁrm 1 and 43 for ﬁrm 2. Then we have the fuzzy equilibrium

quantity ofﬁrm 1 ~q1¼(1,530, 1,587, 1,677) and the fuzzy

equili-brium quantity of_{ﬁrm 2 ~q}2¼(527, 570, 638). In addition, we know

that the production quantity ranges from 1530 to 1677 forﬁrm

1 and 527 to 638 forﬁrm 2. The report by DisplaySearch5_indicates

that the glass substrates industry tends to decrease production quantities, which coincides with the behaviors predicted in our model.

Next, we investigate the impacts of market demand,μaandμb,

on eachﬁrm's choice of strategic scenarios. Obviously, two zones

exist where bothﬁrms adopt the high- or low-production

strate-gies shown inFig. 3. Knowing thatμacan be interpreted as the

potential demand in the market and given a speciﬁc value of μb,

we ﬁnd that each ﬁrm adopts the low-production strategy

(Strategic Scenario 4) as an increase in_μaas shown in Fig. 3(a).

Therefore, if the potential demand is high enough, bothﬁrms will

determine the lower production quantities in order to maximize

their pro_{ﬁts and vice versa. Similarly, given a speciﬁc value of μ}a,

an increase inμbresults in a scenario whereby bothﬁrms employ

the low-production strategies (Strategic Scenario 4). In other

words, bothﬁrms adopt the low-production strategy once market

demand becomes sensitive.

Fig. 3(b) shows how market demand affects the choice of

strategic scenarios, given the variable cost ofﬁrm 1 being equal

to ﬁrm 2 and all other parameters remaining the same. We

observe that Fig. 3(a) is similar to Fig. 3(b), i.e. market demand

heavily impacts eachﬁrm's production strategy rather than each

ﬁrm's cost structure. Thus, in our case study both ﬁrms tend to simultaneously adopt low- or high-production strategies.

6. Conclusions

Decision-making in a complicated and competitive

environ-ment is often made more difﬁcult due to uncertainty, e.g. customer

demand, production ﬂuctuations, etc. Furthermore, real-world

problems frequently involve ambiguity and randomness. This paper has described a new version of a two-stage Cournot production game, which embeds an operational-level decision in the short term within a strategic-level decision in the long term. In

our model, two ﬁrms determined a high or low production

strategy at the strategic level, followed by determining their

production quantities at the operational level under the speciﬁc

production strategy. The concept of the production strategy was

utilized to construct eachﬁrm's the range-type production

quan-tity. At the operational level, the production quantity of eachﬁrm

was assumed to be a triangular fuzzy number, which allowed the production quantity to be represented by an apex and entropies.

At the operational level, the game was divided into the center and

spreads problems and the fuzzy equilibrium quantity of eachﬁrm

constructed from the outcomes of the two problems. Unlike previous

studies, the equilibrium fuzzy production quantity gave eachﬁrm a

production interval when obtaining accurate parameters is impos-sible. At the strategic level, the Nash equilibrium concept was applied to derive the conditions such that a strategy combination became the Nash equilibrium outcome. Applying the proposed model to the case

study derived the fuzzy equilibrium quantity of eachﬁrm in the glass

substrates industry. The results showed that both ﬁrms tended to

Fig. 3. Impacts ofμaandμbon twoﬁrms' strategic scenarios. (a): The impacts of μaandμbon the strategic scenarios whenμd1oμd2. (b): The impacts ofμaandμbon the strategic scenarios whenμd1¼ μd2.

3 http://www.agc.com/english/news/2012/0208e_1.pdf 4_{http://www.corning.com/tw/tc/news_center/news_releases/2012/} 2012012501.aspx 5 http://www.honghaiglass.com/en/nshow.aspx?id=31

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adopt the common strategy, aﬁnding which coincides with the real-world situation. In addition, sensitivity analysis revealed that the

potential market demand,μa, plays a key role in determining aﬁrm's

production strategy. We suggest that further research should explore

the issue of spreads with probability distributions by reﬁning our

proposed model. Another interesting extension is to investigate combinations of production strategies, where the market demand depends on the considered combination of strategies. Detailed

technical explanations can be found in (Dang, 2012).

Acknowledgement

This paper has bene_{ﬁted from comments and suggestions on}

earlier drafts from anonymous referees and the editor. This research was supported in part by the National Science Council of Taiwan under Grant NSC99-2221-E-002-151-MY3.

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