The optimum output quantity of a duopoly market under a fuzzy decision environment

Transcription

1 Computers and Mathematics with pplications wwwelseviercom/locate/camwa The optimum output quantity of a duopoly market under a fuzzy decision environment Gin-Shuh Liang a,, Ling-Yuan Lin b, Chin-Feng Liu c a Department of Shipping and Transportation Management, National Taiwan Ocean University, 2 Pei Ning Road, Keelung, Taiwan, ROC b Department of Business dministration, Ching Yun University, 229 Chien Hsin Road, Jung-Li, Taiwan, ROC c Department of Finance and Taxation, Dahan Institute of Technology, Shjen Street, Sincheng, Hualien, Taiwan, ROC Received 8 pril 2005; received in revised form 9 July 2007; accepted February 2008 bstract The main purpose of this paper is to develop a new optimum output quantity decision analysis of a duopoly market under a fuzzy decision environment To efficiently handle the fuzziness of the decision variables, the linguistic values, subjectively represented by the trapezoidal fuzzy numbers, are used to act as the evaluation tool of decision variables such as fixed cost and unit variable cost This paper will apply fuzzy set theory to construct an optimum output quantity decision model based on aiming for the maximum profit of a duopoly market By using this decision model, the decision-makers fuzzy assessments with various variables can be considered in the decision process to assure more convincing and accurate decision-making c 2008 Elsevier Ltd ll rights reserved Keywords: Trapezoidal fuzzy number; Cournot model; Stackelberg model; Optimum output quantity; Duopoly market Introduction To maximize profit and minimize overall cost, the optimum output quantity decision has been an important issue for industrial organization In a duopoly market, two models can be used to optimize the output quantity One of them is the Cournot model [ 3], and the other is the Stackelberg model [4] The Cournot and Stackelberg models are similar because, in both, competition occurs in terms of quantity The Cournot model forms a situation in which each firm chooses its output independently The Stackelberg model is a two-stage leadership in which the leader chooses its output quantity before the follower does The follower then notes the leader s choice of output quantity and chooses its own output quantity [5] Both Cournot and Stackelberg model play a vital role in such fields as economics, behavioral sciences, management, and politics [ 8] lepuz and Urbano [9] analyzed how learning behavior can change the outcome of competition in a duopoly industry facing demand uncertainty They found that each firm will increase its first period quantity for the myopic choice to make price a more informative signal Wang [0] studied and compared fee and royalty licensing in a differentiated Cournot duopoly The study results showed licensing by a royalty may be better than a fixed-fee from the Corresponding author Tel: ; fax: address: gsliang@mailntouedutw G-S Liang /$ - see front matter c 2008 Elsevier Ltd ll rights reserved doi:00/jcamwa

2 G-S Liang et al / Computers and Mathematics with pplications viewpoint of the patent-holding firm However, the consumer preferred the fixed-fee licensing Barr and Saraceno [] examined the effects of both environmental and organizational factors on the outcome of repeated Cournot games Eiselt [] examined a facility planner s advantage resulting from knowledge of his competitor s opinion in the location Stackelberg games Some valuable findings are obtained For example, given perfect information, the leader always has an advantage over the follower Lavigne et al [3] presented a general method to study the electricity market of a country or region, under various pricing mechanisms The study results showed monopolistic pricing chosen by the producer to minimize its costs while knowing the optimal consumers reaction to the proposed price of electricity leads to a Stackelberg-type equilibrium Nie [] explored discrete time dynamic Stackelberg games with an open loop complete state The study pointed out that both feedback and closed loop dynamic Stackelberg games with complete information are valuable in explaining some social and economic phenomena Yang and Zhou [8] analyzed the effects of the duopolistic retailers different competitive behaviors - Cournot Collusion and Stackelberg - on the optimal decisions of the manufacturer and the duopolistic retailers themselves The results showed that the total profit of the duopolistic retailers who act as the followers will exceed the more powerful manufacturer s profit as long as the dissimilarity between the duopolistic retailers market demands is sufficient In a duopoly market, four patterns to market structure can be formed They are: both companies and B are followers; 2 company is a leader and company B is a follower; 3 company B is a leader and company is a follower; 4 both companies and B are leaders In pattern 2 and 3, the leader makes the optimum output quantity decision by considering the response function of the follower, then, the follower decides his optimum output quantity based on the leader s decision In pattern, the optimum output quantities of companies and B can be solved by the simultaneous-equation models constructed by their response function respectively In pattern 4, the optimum output quantities of companies and B can be achieved when they are recognized as leaders In conventional precision-based models of duopoly, decision variables are expressed in crisp values [4,5] However, because of the insufficiency and uncertainty of information in the decision environment, it is difficult to find the exact economic assessment data, such as the prices of products, volume of activity, per unit variable costs, and total fixed costs Therefore, the precision-based decision may be ineffective In fact, when decision-makers decide, they assess based on their professional knowledge, experience, and subjective judgment Linguistic values, such as about 2000 dollars, about 40%, are usually used to suggest their estimations Fuzzy set theory can play a significant role in this decision-making environment Fuzzy set theory was introduced by Zadeh [] to solve problems in which a source of vagueness existed Linguistic values can be expressed fittingly by the approximate reasoning of fuzzy set theory [7] To deal with the ambiguities involved in the process of linguistic estimations effectively, the trapezoidal fuzzy numbers are used to characterize fuzzy measure of linguistic values [8] t the same time, combining the Cournot and Stackelberg models, a fuzzy duopoly model which considers the factors of market demand, business cost and market position is developed to answer the optimum output quantity of duopoly market under a fuzzy decision environment This paper is organized as follows Section 2 introduces fuzzy set theory The fuzzy model of duopoly is developed in Section 3 numerical example to explain the computational process of a fuzzy model of duopoly will be presented in Section 4 Finally, conclusions are given in Section 5 2 Fuzzy set theory In this section, some concepts of fuzzy set theory used in this paper are briefly introduced 2 Trapezoidal fuzzy numbers fuzzy number [9] is described as a special fuzzy subset of real numbers whose membership function f satisfies five conditions f is a continuous mapping from R real line to a closed interval [0, ], 2 f x 0, for all x, c] [d,, 3 f x is strictly increasing in the interval [c, a], 4 f x, for all x [a, b], 5 f x is strictly decreasing in the interval [b, d] Where c, a, b, d are real numbers, and < c a b d < For convenience, f L is named as the left membership function of fuzzy number, defining f L x f x, for all x [c, a]; and f R is named as the right membership function of fuzzy number, defining f R x f x, for x [b, d] The fuzzy number in R is a trapezoidal fuzzy number if its membership function f : R [0, ] is

3 78 G-S Liang et al / Computers and Mathematics with pplications x c / a c, c x a, a x b f x x d / b d, b x d 0, otherwise where < c a b d < The trapezoidal fuzzy number, as given by Eq, can be represented by c, a, b, d The interval [a, b] of trapezoidal fuzzy number gives the maximal grade of f x, that is, f x, x [a, b]; it is the most probable value of the evaluation data In addition, c and d are the lower and upper bounds of the available area of the evaluation data They are used to reflect the fuzziness of the evaluation data The narrower the interval [c, d] is, the lower the fuzziness of the evaluation data The trapezoidal fuzzy numbers are easily used and interpreted For example, approximately between 800 and 80 can be represented by 795, 800, 80, 85; and the more blurry representation can be represented as 790, 800, 80, 820 In addition, an exact number, a can be represented by a, a, a, a For example, 40 can be represented by 40, 40, 40, 40 By the extension principle [], the extended algebraic operations of two trapezoidal fuzzy numbers, c, a, b, d and 2 c 2, a 2, b 2, d 2, can be expressed as: 2 c + c 2, a + a 2, b + b 2, d + d 2, k kc, ka, kb, kd, k R, k 0 22 The ranking of fuzzy numbers In the fuzzy model of duopoly, ranking the fuzzy profits and fuzzy total profits being considered is important and essential Many methods of ranking fuzzy numbers have been proposed [20 27] The graded mean integration representation method [28] not only improves on some drawbacks of existing ranking methods, but also has the advantage of easy implementation and power in problem solving So, it will be used to characterize the presentation value of the trapezoidal fuzzy number and rank the fuzzy profits and fuzzy total profits Let i c i, a i, b i, d i, i, 2,, n, be n trapezoidal fuzzy numbers The graded mean integration representation P i of i is P i c i + 2a i + 2b i + d i Let P i and P j be the graded mean integration representations of i and j, respectively Define that i > j if and only if P i > P j, i < j if and only if P i < P j, i j if and only if P i P j 2 3 Fuzzy model of duopoly Notation q i : The output quantity of company i Q: The total output quantity of duopoly market PQ: The demand price under Q FTC i : The fuzzy cost function of company i f i : The fuzzy fixed cost of company i c i : The fuzzy unit variable cost of company i FT π i : The fuzzy profit function of company i q Fik : The optimum output quantity of company follower i in pattern k ik : The optimum output quantity of company leader i in pattern k P k : The market equilibrium price in pattern k

4 G-S Liang et al / Computers and Mathematics with pplications FT π Fik : The fuzzy maximum profit of company follower i in pattern k FT π Lik : The fuzzy maximum profit of company leader i in pattern k Let q and q B represent the output quantities of companies and B, respectively Let Q be the total output quantity of duopoly market That is, Q q + q B Suppose the demand price PQ is PQ α + β Q α + β q + q B In addition, we define the fuzzy cost functions of companies and B, represented by FTC and FTC B, as follows FTC f c q FTC B f B c B q B where, f and f B stand for the fuzzy fixed costs, c and c B represent the fuzzy unit variable costs, and f f, f 2, f 3, f 4, c c, c2, c3, c4, f B f B, f 2 B, f 3 B, f 4 B, c B c B, c2 B, c3 B, Then the fuzzy profit functions of companies and B, represented by FT π and FT π B, can be calculated by Eqs and 7, respectively FT π T R ΘFTC and pqq ΘFTC αq + βq 2 + βq q B Θ f + c q, f 2 + c2 q, f 3 + c3 q, f 4 + c4 q α c 4 q + βq q B + βq 2 f 4, α c3 q + βq q B + βq 2 f 3, α c 2 q + βq q B + βq 2 f 2, α c q + βq q B + βq 2 f FT π PQq B ΘFTC B α + β q + q B q B Θ f B + c B q B, f 2 B + c2 B q B, f 3 B + c3 B q B, f 4 B + q B α c 4 B q B + βq q B + βq 2 B f 4 B, α c3 B q B + βq q B + βq 2 B f 3 B, α c 2 B q B + βq q B + βqb 2 f B 2, α c B q B + βq q B + βqb 2 f B 7 By Eq 2, the graded mean integration representation P FT π of FT π is [ α c 4 P FT π q + βq q B + βq 2 f 4 ] [ + 2 α c 3 q + βq q B + βq 2 f ] [ α c 2 q + βq q B + βq 2 f 2 ] [ + α c q + βq q B + βq 2 f ] [ ] α c + 2c2 + 2c3 + c4 Similarly, the graded mean integration representation of FT π B is [ ] P FT π B α c B + 2c2 B + 2c3 B + q + βq q B + βq 2 f + 2 f f 3 + f 4 q B + βq q B + βqb 2 f B + 2 f B f B 3 + f B 4 Pattern : Both companies and B are followers Step : Find the response functions of companies and B By considering the maximum profit of company, we can calculate the first-order partial of P FT π versus q and suppose the derivative result equals zero Then the response function of company can be found That is, let P FT π q α + 2βq + βq B c + 2c2 + 2c3 + c4 0 8

5 80 G-S Liang et al / Computers and Mathematics with pplications Then, by Eq 8, we can find the response function of Company q c + 2c2 + 2c3 + c4 α βq B 9 β Similarly, we can find the response function of company B q B c B + 2c2 B + 2c3 B + α βq 0 β Step 2: By Eqs 9 and 0, the optimum output quantities of companies and B, represented by q F and q FB, can be found That is q F 2c + 4c2 + 4c3 + 2c4 c B 2c2 B 2c3 B α 8β q FB 2c B + 4c2 B + 4c3 B + 2 c 2c2 2c3 c4 α 8β Step 3: By taking Eqs and into Eq 3, we can find the market equilibrium price That is, P c + 2c2 + 2c3 + c4 + c B + 2c2 B + 2c3 B + + α 3 8β Step 4: By taking q F and q FB into Eqs and 7, the fuzzy maximum profits FT π F and FT π FB of companies and B can be found FT π F α c 4 q F + βq F q FB + βq 2 F f 4, α c3 q F + βq F q FB + βq 2 F f 3, α c 2 q F + βq F q FB + βq 2 F f 2, α c q F + βq F q FB + βq 2 F f 4 FT π FB α c 4 B q F B + βq F q FB + βq 2 F B f 4 B, α c3 B q F B + βq F q FB + βq 2 F B f 3 B, α c 2 B q F B + βq F q FB + βqf 2 B f B 2, α c B q F B + βq F q FB + βqf 2 B f B 5 Step 5: By Eqs 4 and 5, the fuzzy total profit FT π in pattern is as below FT π FT π F FT π FB Pattern 2: Company is a leader and company B is a follower Step : By taking the response function of company B, represented by Eq 0, into Eq, the fuzzy profits function FT π is as below: α c 4 FT π + c B + 2c2 B + 2c3 B + q + β 2 q2 f 4, α c 3 + c B + 2c2 B + 2c3 B + q + β 2 q2 f 3, α c 2 + c B + 2c2 B + 2c3 B + q + β 2 q2 f 2, α c + c B + 2c2 B + 2c3 B + q + β 2 q2 f 7 Then the graded mean integration representation P FT π of FT π is P FT π α + c B + 2c 2 B + 2c3 B + 2c 4c2 4c3 2c4 q

6 G-S Liang et al / Computers and Mathematics with pplications βq2 2 f 4 f 2 4 f 3 2 f 4 Step 2: Let 2 be the optimum quantity of company It can be found by taking first-order partial of P FT π versus q and assuming the derivative result equals zero That is, let P FT π q α + c B + 2c 2 B + 2c3 B + 2c 4c2 4c3 2c4 + βq By solving the equation above, we can find 2 as below: c + 4c2 + 4c3 + 2c4 c B 2c2 B 2c3 B α 9 β Step 3: By taking Eq 9 into Eq 7, we can find the fuzzy maximum profit FT π L 2 of company α c 4 FT π L 2 + c B + 2c2 B + 2c3 B β 2 q2 L 2 f 4, α c 3 + c B + 2c2 B + 2c3 B β 2 q2 L 2 f 3, α c 2 + c B + 2c2 B + 2c3 B β 2 q2 L 2 f 2, α c + c B + 2c2 B + 2c3 B β 2 q2 L 2 f 20 Step 4: By taking Eq 9 into Eq 0, we can find the optimum output quantity q FB2 of company B q FB2 3c B + c2 B + c3 B + 3 2c 4c2 4c3 c4 α 2 24β Step 5: By taking Eqs 9 and 2 into Eq 3, the market equilibrium price P 2 in pattern 2 can be found P 2 2c + 4c2 + 4c3 + 2c4 + c B + 2c2 B + 2c3 B + + α Step : By taking 2 and q FB2 into Eq 7, the fuzzy maximum profit FT π L B2 of company B in pattern 2 is as below: FT π FB2 α c 4 B q F B2 + β 2 q FB2 + βq 2 F B2 f 4 B, α c3 B q F B2 + β 2 q FB2 + βq 2 F B2 f 3 B, α c 2 B q F B2 + β 2 q FB2 + βqf 2 B2 f B 2, α c B q F B2 + β 2 q FB2 + βqf 2 B2 f B 23 Step 7: Then the fuzzy total profit FT π 2 in pattern 2 is FT π 2 FT π L 2 FT π FB2 24 Pattern 3: Company B is a leader and company is a follower By copying the analysis of pattern 2, the following results can be found: The optimum output quantity B3 of company B is B3 2c B + 4c2 B + 4c3 B + 2 c 2c2 2c3 c4 α 25 β

7 82 G-S Liang et al / Computers and Mathematics with pplications The fuzzy maximum profit FT π L B3 of company B is α c 4 FT π L B3 B + c + 2c2 + 2c3 + c4 B3 + β 2 q2 L B3 f B 4, α c 3 B + c + 2c2 + 2c3 + c4 B3 + β 2 q2 L B3 f B 3, α c 2 B + c + 2c2 + 2c3 + c4 B3 + β 2 q2 L B3 f B 2, α c B + c + 2c2 + 2c3 + c4 B3 + β 2 q2 L B3 f B 2 Step : By taking Eq 25 into Eq 9, we can find the optimum output quantity q F3 expressed as of company, it can be q F3 3c + c2 + c3 + 3c4 2c B 4c2 B 4c3 B 2 α 27 24β Step 2: By taking Eqs 25 and 27 into Eq 3, we can find the market equilibrium price P 3 in pattern 3, that is P 3 2c B + 4c2 B + 4c3 B c + 2c2 + 2c3 + c4 + α 24 Step 3: By taking B3 and q F3 into Eq, the fuzzy maximum profit FT π F3 is FT π F3 α c 4 q F 3 + βq F3 B3 + βq 2 F 3 f 4, α c3 q F 3 + βq F3 B3 + βq 2 F 3 f 3, α c 2 q F 3 + βq F3 B3 + βqf 2 3 f 2, α c q F 3 + βq F3 B3 + βqf 2 3 f 28 Step 4: Then the fuzzy total profit FT π 3 of companies and B is FT π 3 FT π F3 FT π L B3 29 Pattern 4: Both companies and B are leaders Step : When company is a leader and company B is a follower, the optimum output quantities of companies and B are as below, respectively 2 2c + 4c2 + 4c3 + 2c4 c B 2c2 B 2c3 B α β q FB2 3c B + c2 B + c3 B + 3 2c 4c2 4c3 c4 α 24β nd the fuzzy maximum profits FT π L 2 and FT π FB2 are expressed as Eqs 20 and 23 Step 2: When company B is a leader and company is a follower, the optimum output quantities of the two companies are as below: B3 2c B + 4c2 B + 4c3 B + 2 c 2c2 2c3 c4 α β q F3 3c + c2 + c3 + 3c4 2c B 4c2 B 4c3 B 2 α 24β nd the fuzzy maximum profits FT π L 2 and FT π FB2 are expressed as Eqs 2 and 28 Step 3: When both companies and B are leaders, the optimum output quantities of the two companies are as below:

8 G-S Liang et al / Computers and Mathematics with pplications Table The optimum output quantities, fuzzy maximum profits, fuzzy total profits and equilibrium market prices of the four different market structures Company B Leader Follower Company Leader FT π L 4 4 FT π L 2 2 FT π L B4 FT π FB2 B2 FT π 4 P 4 FT π 2 P 2 Follower FT π F3 3 FT π F q F FT π L B3 B3 FT π FB q FB FT π 3 P 3 FT π P The optimum output quantity of company is 4 2 2c + 4c2 + 4c3 + 2c4 c B 2c2 B 2c3 B α 30 β The optimum output quantity of company B is B3 2c B + 4c2 B + 4c3 B + 2 c 2c2 2c3 c4 α 3 β By taking 4 and into Eqs and 7, the fuzzy maximum profits and fuzzy total profit, FT π L 4, FT π L B4 and FT π 4, are as below: FT π L 4 α c β 4 + βq 2 L 4 f 4, α c3 4 + β 4 + βq 2 L 4 f 3, α c β 4 + βq 2 L 4 f 2, α c 4 + β 4 + βq 2 L 4 f FT π L B4 α c 4 B + β 4 + βq 2 L B4 f 4 B, α c3 B + β 4 + βq 2 L B4 f 3 B, α c 2 B + β 4 + βq 2 L B4 f 2 B, α c B + β 4 + βq 2 L B4 f B 32 FT π 4 FT π L 4 FT π L B4 nd by taking Eqs 30 and 3 into Eq 3, we can find the market equilibrium price P 4 in pattern 4, that is, P 4 c + 2c2 + 2c3 + c4 + c B + 2c2 B + 2c3 B + By understanding the analysis of four cases described above, we can obtain the optimum output quantities, fuzzy maximum profits, fuzzy total profits and market equilibrium prices of the four different market structures These results are shown as Table 4 Numerical example In this section, a hypothetical optimum output quantity decision problem of duopoly market is designed to explain the computational process of the fuzzy Stackelberg model proposed here ssume that companies and B need to make decision of the optimum output quantities in a duopoly market The estimated fuzzy fixed costs, fuzzy variable costs are given as follows: For company, the fuzzy fixed cost is about 5,000 dollars, and the fuzzy variable cost a unit is about 200 dollars That is, f 0000, 5000, 5000, 8000 and c 50, 200, 200, 230 For company B, the fuzzy fixed cost is about 0,000 dollars, and the fuzzy variable cost a unit is about 250 dollars That is, f B 55000, 0000, 0000, 3000, and c 200, 250, 250, 280 Then, the fuzzy total costs of companies and B are FTC q, q, q, q FTC B q B, q B, q B, q B In addition, let demand price P be P q + q B

9 84 G-S Liang et al / Computers and Mathematics with pplications Pattern : Both companies and B are followers In this pattern, the optimum output quantities, fuzzy maximum profits, fuzzy total profit and market equilibrium price of the two companies are as below: q F 304, q FB 308, P 848 FT π F 888, 4033,4033, 052; FT π FB 0458,32708, 32708, 535; FT π 2484, 27302, 27302, 334 Pattern 2: Company is a leader and company B is a follower In this pattern, the optimum output quantities, fuzzy maximum profits, fuzzy total profit and market equilibrium price of the two companies are as below: 2 488, q FB2 29, P 2 85 FT π L ,7842,7842,2058; FT π FB2 2573,35338,35338,59; FT π ,20779,20779, Pattern 3: Company B is a leader and company is a follower In this pattern, the optimum output quantities, fuzzy maximum profits, fuzzy total profit and market equilibrium price of the two companies are as below: B3 45, q F3 250, P 3 98 FT π L B3 5223, 4748, 4748, 9290; FT π F , 59582, 59582, 7703; FT π , 20330, 20330, Pattern 4: Both companies and B are leaders When both companies and B are leaders, they decide the optimum output quantity is based on profit itself Company is a leader and company B is a follower: 2 488, q FB2 29 FT π L , 7842, 7842, 2058; FT π FB2 2573, 35338, 35338, 59; FT π , 20779, 20779, Company B is a leader and company is a follower: B3 45, q F3 250 FT π L B3 5223, 4748, 4748, 9290 ; FT π F , 59582, 59582, 7703; FT π 74293, 20330, 20330, When both companies and B are leaders, they want their fuzzy profit to be maximal To pursue the maximum individual fuzzy profit, they want to be leaders The company will produce the output quantity nd company B will produce the output quantity 45 FT π L , 5424, 5424, 2484; FT π L B4 9304, 734, 734, 4424; FT π , 7928, 7928, 98; P 4 222

10 G-S Liang et al / Computers and Mathematics with pplications Table 2 The optimum output quantities, fuzzy maximum profits, fuzzy total profits and equilibrium market prices of four different market structures Company B Leader Follower Company Leader FT π L , 5424, 5424, 2484 FT π L , 7842, 7842, 2058 FT π L B4 9304, 734, 734, 4424 FT π FB2 2573, 35338, 35338, 59 FT π , 7928, 7928, 98 FT π , 20779, 20779, ; 45; P ; q FB2 29; P 2 85 Follower FT π L B3 5223, 4748, 4748, 9290 FT π F 888, 4033, 4033, 052 FT π F , 59582, 59582, 7703 FT π FB 0458, 32708, 32708, 535 FT π , 20330, 20330, FT π 2484, 27302, 27302, 334 B3 45; q F3 250; P 3 98 q F 304; q FB 308; P 848 By understanding the analysis results of four cases described above, we can find the optimum output quantities, fuzzy maximum profits, fuzzy total profits and market equilibrium prices of four different market structures They are expressed as Table 2 For company, the fuzzy profits of four different market structures are as below: FT π F 888, 403, 403, 052; FT π L , 7842, 7842, 2058; FT π F , 59582, 59582, 7703; FT π L , 5424, 5424, 2484 Based on the graded mean integration representation method, P FT π F 8, , ,33 + 0, 52 P FT π L 2 54, , , , 258 P FT π F3 49, , , , 03 0,750 4,0 73,803 P FT π L 4 7, , , ,84 Since P FT π L 2 > P FT πf > P FT πf3 > P FT πl 4, So FT π L 2 > FT π F > FT π F3 > FT π L 4 For company B, the fuzzy profits of four different market structures are as below: FT π FB 0458, 32708, 32708, 535; FT π FB2 2573, 35338, 35338, 59; FT π L B3 5223, 4748, 4748, 9290; FT π L B4 9304, 734, 734, 4424 Based on the graded mean integration representation method, P FT π FB 0, , , , 5 P FT π FB2 25, , , , 29 P FT π L B3 5, , , , 290 3,402 34,09 43,584 P FT π L B4 9, , ,4 + 44,24 52,304 7,704

11 8 G-S Liang et al / Computers and Mathematics with pplications Since PFT π L B3 > PFT π FB > PFT π FB2 > PFT π L B4, So FT π L B3 > FT π FB > FT π FB2 > FT π L B4 The fuzzy total profits of companies and B are as below: FT π 2484, 27302, 27302, 334; FT π , 20779, 20779, ; FT π , 20330, 20330, 24393; FT π , 7928, 7928, 98 Based on the graded mean integration representation method, 248, , , , 4 P FT π 275,729 79, , , , 554 P FT π 2 20,204 74, , , , 393 P FT π 3 204,334 3, , , ,8 P FT π 4 4,008 Since PFT π > PFT π 2 > PFT π 3 > PFT π 4 So FT π > FT π 2 > FT π 3 > FT π 4 Based on the analysis results stated above, we find: When both companies and B are followers The equilibrium market price and the total fuzzy profit are maximal That is, 848 > 98 > 85 > 222 FT π > FT π 2 > FT π 3 > FT π 4 nd, the company individual fuzzy profit is almost maximal Since, for company : FT π L 2 > FT π F > FT π F3 > FT π L 4 nd for company B: FT π L B3 > FT π L B > FT π FB2 > FT π L B4 2 When both companies and B are leaders and both companies produce many products, the equilibrium market price and total fuzzy profit are minimal That is, 222 < 85 < 98 < 848 FT π < FT π 2 < FT π 3 < FT π 4 nd, the company individual fuzzy profit will be the lowest Since, for company : FT π L 4 > FT π F3 > FT π F > FT π L 2 nd, for company B: FT π L B4 > FT π L B2 > FT π FB > FT π L B3 Therefore, the best decision is both companies and B are followers 5 Conclusions In business decision analysis, obviously, the difficulty in finding precise assessment data such as fixed cost, unit variable cost of product caused the vagueness and uncertainty of the relevant information in deciding So, the conventional precision-based decision analysis method is less effective in suggesting available information in such an imprecise and fuzzy decision environment This paper improves the estimating methods in which we can send the estimation of decision variables by the linguistic values characterized by trapezoidal fuzzy numbers Further, in this paper we examined an analytical method of effectively carrying out the optimum output quantity decision of duopoly market under a fuzzy environment The proposed fuzzy model of duopoly not only considered the factors of market demand, business cost and market position but also could be used to analyze the interaction tactic behavior between enterprises Besides, the proposed model can

12 G-S Liang et al / Computers and Mathematics with pplications be computerized to enable the decision-makers to achieve the best decision automatically either by conducting fuzzy or non-fuzzy assessment References [] D John, C Schultz, Renegotiation in a repeated Cournot duopoly, Economics Letters [2] C James, M Walker, Learning to play Cournot duopoly strategies, Journal of Economic Behavior and Organization [3] Morton, cake for Cournot, Economics Letters [4] HV Stackelberg, Marktform und Gleichgewicht, Juliu Springer, Vienna, 934 [5] MR Caputo, dual vista of the Stackelberg duopoly reveals it fundamental qualitative structure, International Journal of Industrial organization [] PY Nie, Dynamic Stackelberg games under open-loop complete information, Journal of Franklin Institute [7] JF Tsai, CP Chu, Economic analysis of collecting parking fees by a private firm, Transportation Research Part [8] SN Yang, YW Zhou, Two-echelon supply chain models: Considering duopolistic retailer s different competitive behaviors, International Journal of Production Economics [9] MD lepuz, Urbano, Duopoly experimentation: Cournot competition, Mathenatical Social Sciences [0] XH Wang, Fee versus royalty licensing in a differentiated Cournot duopoly, Journal of Economics and Business [] J Barr, F Saraceno, Cournot competition, organization and learning, Journal of Economic & Control [] H Eiselt, Perception and information in a competitive location model, European Journal of Operational Research [3] D Lavigne, R Loulou, G Savard, Pure competition, regulated and Stackelberg equilibria: pplication to the energy system of Quebec, European Journal of Operational Research [4] T u, TP u, Engineering Economics for Capital Investment nalysis, llyn Bacon Inc, Boston, 983 [5] FE Brigmann, LC Gapensiki, Financial Management: Theory and Practice, 7th ed, The Dryden Press, 994 [] L Zadeh, Fuzzy sets, Information and Control [7] L Zadeh, The concept of a linguistic variable and its application to approximate reasoning, part, 2 and 3, Information Sciences ; 9, [8] GS Liang, MJJ Wang, fuzzy multi-criteria decision-making method for facility site selection, International Journal of Production of Production Research [9] D Dubois, H Prade, Operations on fuzzy numbers, The International Journal of Systems Sciences [20] RR Yager, procedure for ordering fuzzy subsets of the unit interval, Information Sciences [2] G Bortolan, R Degani, review of some methods for ranking fuzzy subsets, Fuzzy Sets and Systems [22] SH Chen, Ranking fuzzy numbers with maximizing and minimizing set, Fuzzy Sets and Systems [23] JJ Buckley, fuzzy ranking of fuzzy numbers, Fuzzy Sets and Systems [24] F Choobineh, H Li, new index for ranking fuzzy numbers, in: Proceeding of International Symposium on Uncertainty Modeling and nalysis, IEEE Computer Press, 990, pp [25] K Kim, KS Park, Ranking fuzzy numbers with index of optimism, Fuzzy Sets and Systems [2] TS Liou, MJJ Wang, Ranking fuzzy numbers with integral value, Fuzzy Sets and Systems [27] OP Dias, Ranking alternatives using fuzzy numbers a computational approach, Fuzzy Sets and Systems [28] SH Chen, CH Hsieh, Representation, ranking, distance, and similarity of L-R type fuzzy number and pplication, ustralian Journal of Intelligent Processing System