A new energy vehicle supply chain member game based on triangular fuzzy numbers

2.1 The definition of TFN

Definition 1. [24] A real fuzzy number A is more precisely described as any fuzzy subset of the R (real line), whose membership function is a continuous mapping from R to the closed interval [0,1].

Definition 2. [26] Assume the triangular fuzzy number (TFN) A is defined as a fuzzy set on R (real line), whose membership function is: μ A = { ( x - a 1 ) / - ( a 2 - a 1 ) , a 1 < x ⩽ a 2 ( a 3 - x ) / - ( a 3 - a 2 ) , a 2 < x ⩽ a 3 0 , otherwise

With a₁ ⩽ a₂ ⩽ a₃, the TFN is defined as A = (a₁, a₂, a₃).

Definition 3. [26] Let A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃) be two triangular fuzzy numbers (TFNs), then the operations on the TFNs are defined as:

Addition: A ⊕ B = (a₁ + b₁a₂ + b₂a₃ + b₃).

Definition 4. [38] Let A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃) be two TFNs, possible degrees of a A > B is: V ( A > B ) = { 0 , a 3 ⩽ b 1 ( a 3 - b 1 ) / - [ ( a 3 - b 1 ) + ( b 2 - a 2 ) ] , a 3 > b 1 and a 2 < b 2 1 , a 2 ⩾ b 2

Definition 5. [39] For a TFN, the optimistic and pessimistic values of A = (a₁, a₂, a₃) are expressed as: { ɛ α L = a 2 α + a 1 ( 1 - α ) ɛ α U = a 2 α + a 3 ( 1 - α )

The finite expected value of A is expressed as: E ( ɛ ) = 1 2 ∫ 0 1 ɛ α L + ɛ α U d α = a 1 + 2 a 2 + a 3 4

2.2 TFN-based equilibrium strategy solution for the game

The TFN-based game model is denoted as [40]: G = { N 1 , N 2 ; S 1 , S 2 ; P ij }

Where N₁, N₂ are the two players involved in the game, S₁ = { α₁, α₂,  . . . , α _m  } , S₂ =  { β₁, β₂,  . . . , β _n  } are the set of strategies of N₁, N₂, and P _ij is the triangular fuzzy benefit function of the strategy j taken by the game player i. The Nash equilibrium existence theorem states that every finite game has at least one Nash equilibrium. If an optimal solution does not exist for pure strategy, a mixed strategy optimal solution becomes necessary. For a two-by-two game, the mixed strategy of the player N₁ is σ₁ = (x, 1 - x), the player N₁ chooses strategy α₁ with probability (x) and strategy α₂ with probability (1 - x). The mixed strategy of the player N₂ is σ₂ = (y, 1 - y), the player N₂ chooses strategy β₁ with probability (y) and strategy β₂ with probability (1 - y). The game payment matrix based on TFN is shown in Table 1.

There is a unique optimal mixed strategy solution to the game of players N₁, N₂, as follows from Nash theorem. { x * = B 22 - B 21 B 11 - B 12 - B 21 + B 22 y * = A 22 - A 12 A 11 - A 12 - A 21 + A 22

Definition 6. Let E be a fuzzy set over a real number field R and the affiliation function be denoted E (x), x ∈ R. If E (x) satisfies the following properties:

E (0) =1.

Definition 7. [41] For a given fuzzy triangular structured element and any fuzzy number A, there always exists a function f (x) such that A = f (E), where f (x) is a monotonically bounded function on the interval [–1,1], and the affiliation function of the fuzzy number A is E [f^-1 (x)], f^-1 (x) is a rotationally symmetric function of f (x) with respect to the variable x, y. For a triangular fuzzy number A = (a₁, a₂, a₃), the corresponding monotone bounded function is: f ( x ) = { ( a 2 - a 1 ) x + a 2 ( a 3 - a 2 ) x + a 2

Definition 8. [35] For a fuzzy monotone matrix A m × n = ( a ij ) m × n , if A m × n - = ( a ij - ) m × n , a ij - = 1 2 ∫ - 1 1 f ij ( x ) ( 1 - x ) dx , then A m × n - is said to be the projection matrix of A_m×n.

3.1 Model construction and basic assumptions

3.1.1 Problem description

With the new energy vehicle market expanding quickly, product development cycles have become shorter, and competition has intensified. This has prompted supply chain members to seek cooperative research and development opportunities to reduce costs, shorten cycle times, and enhance competitiveness. Research on the new energy vehicle supply chain mainly focuses on pricing, profitability, and recovery strategies. However, there is a lack of research on collaboration strategies among supply chain members. The new energy industry also faces funding challenges, and government subsidies primarily incentivize sales and buyers, potentially affecting the R&D efforts of supply chain members. To summarize, this paper presents a game model for cooperative R&D within the new energy vehicle supply chain. The model involves upstream battery manufacturers and downstream vehicle manufacturers as participants, and considers their decision-making process in a fuzzy environment. Furthermore, a separate analysis is conducted on how government incentives affect the decision-making process of each party, with the corresponding model illustrated in Fig. 1.

OPEN IN VIEWER

Fig. 1

Supply chain members game model.

When taking into account non-incentives, the advantages for both players consist of regular profit, extra incremental rates, and input costs, as per the aforementioned assumptions. When both players choose to cooperate, they can both benefit from increased revenue and reduced input costs. In the event that both players do not cooperate, their profits will remain regular. However, under certain circumstances, free-riding and penalties may be imposed. The game payment matrix for both players is shown in Table 4.

According to Table 4, the expected utility functions for the battery manufacturer (N₁) and the vehicle producer (N₂) can be obtained as follows. E 1 ( x , y ) = xy [ π 1 ( 1 + λ 1 ) - C 1 ] + x · ( 1 - y ) · ( π 1 - C 1 + P ) + y · ( 1 - x ) · [ π 1 ( 1 + I 1 ) - P ] + ( 1 - x ) · ( 1 - y ) · π 1 E 2 ( x , y ) = xy [ π 2 ( 1 + λ 2 ) - C 2 ] + y · ( 1 - x ) · ( π 2 - C 2 + P ) + x · ( 1 - y ) · [ π 2 ( 1 + I 2 ) - P ] + ( 1 - x ) · ( 1 - y ) · π 2

The probability of the optimal mixed strategy is obtained by applying the calculus to the above expected utility functions and solving for the extrema, respectively: ∂ E 1 ( x , y ) ∂ x = y · π 1 · ( λ 1 - I 1 ) + P - C 1 ∂ E 2 ( x , y ) ∂ y = x · π 2 · ( λ 2 - I 2 ) + P - C 2 { x * = C 2 - P ( λ 2 - I 2 ) π 2 y * = C 1 - P ( λ 1 - I 1 ) π 1

According to the conditions for mixed Nash equilibrium, the above result should satisfy the following conditions. The probability of the battery manufacturer (N₁) and the vehicle producer (N₂) choose to cooperate is influenced by the other player at the same time, λ₁ > I₁, λ₂ > I₂ hold at the same time if C₁, C₂ > P, it means that when the cost C₁, C₂ of N₁, N₂ choosing to cooperate is greater than the penalty P for free-riding, then the additional return to N₁, N₂ is greater than the corresponding probability of free-riding. Similarly, if C₁, C₂ < P, the above applies accordingly. { λ 1 ≠ I 1 λ 2 ≠ I 2 ( λ 1 - I 1 ) ( C 1 - P ) > 0 ( λ 2 - I 2 ) ( C 2 - P ) > 0

The factors influencing the choice of cooperation between the two players are analyzed separately below.

(1) Analysis of factors affecting x^*.

x^* is an increasing function concerning C₂ and I₂. If N₂ chooses to cooperate with an increase in input costs, N₁ is more likely to choose to cooperate. In other words, if N₁ knows N₂’s input cost (C₂), the size of C₂ determines N₁’s strategy: a larger C₂ indicates that it is more likely to engage in resource sharing, and therefore N₁ is also more likely to choose to cooperate. If the probability of N₂ selecting to free-ride (I₂) is higher, N₁ is more inclined to choose cooperation to achieve win-win cooperation.

x^* is a decreasing function concerning λ₂ and P. N₂’s revenue growth rate λ₂ influences N₁’s decision, and if N₁ knows that λ₂ is larger, N₁’s probability of choosing cooperation becomes smaller. Looking at this factor alone, when both players choose the cooperation strategy, the probability of choosing cooperation becomes smaller when N₁ learns that N₂’s larger revenue growth will result in a smaller payoff for itself and a disadvantage in the two-party game. P is another influencing factor that constrains N₁’s choice of cooperation, and the probability of choosing cooperation becomes smaller when P is larger. If N₁ learns that the penalty P of free-riding is greater, the probability of choosing not to cooperate is greater.

(2) Analysis of factors affecting y^*.

In the same way as x^*, y^* also influenced by C₁, I₁, λ₁ and P. y^* is an increasing function concerning C₁ and I₁. If N₁ chooses to cooperate with an increase in input costs, N₂ is more likely to choose to cooperate. Similarly, when the probability I₁ of N₁ wanting to free-ride is higher, N₂ is more inclined to choose cooperation.

y^* is a decreasing function concerning λ₁ and P. The revenue growth (λ₁) of N₁ will also affect the decision of N₂ and will have the same outcome as N₁. The constraint effect of P on N₁, N₂ is the same, the larger P is, the more likely N₂ will make a conservative decision.

In summary, the value of x^* and y^* is influenced by a combination of facilitating and constraining factors. Both are influenced by each other’s decision parameters, so the values of optimal decision probabilities are mutually constrained.

When government incentives are considered, the government will consider additional incentives r₁, r₂ for N₁, N₂ respectively when both players choose the cooperation strategy, which will eventually affect the probability of both parties choosing to cooperate, and the game payment matrix is shown in Table 5.

According to Table 4, the expected utility functions for the battery manufacturer (N₁) and the vehicle producer (N₂) can be obtained as follows: E 1 , ( x , y ) = xy [ π 1 ( 1 + λ 1 ) - C 1 + r 1 ] + x · ( 1 - y ) · ( π 1 - C 1 + P ) + y · ( 1 - x ) [ π 1 ( 1 + I 1 ) - P ] + ( 1 - x ) · ( 1 - y ) · π 1 E 2 , ( x , y ) = xy [ π 2 ( 1 + λ 2 ) - C 2 + r 2 ] + y · ( 1 - x ) · ( π 2 - C 2 + P ) + x · ( 1 - y ) · [ π 2 ( 1 + I 2 ) - P ] + ( 1 - x ) · ( 1 - y ) · π 2

The probability of the mixed optimal strategy is obtained by applying the calculus to the above expected utility functions and solving for the extrema, respectively: ∂ E 1 , ( x , y ) ∂ x = y [ ( λ 1 - I 1 ) π 1 + r 1 ] + P - C 1 ∂ E 2 , ( x , y ) ∂ y = x [ ( λ 2 - I 2 ) π 2 + r 2 ] + P - C 2 { x , * = C 2 - P ( λ 2 - I 2 ) π 2 + r 2 y , * = C 1 - P ( λ 1 - I 1 ) π 1 + r 1

Compared to the anarchic incentive game model, the results of the optimal solution of the mixed Nash equilibrium are also affected by government incentives. The above result should satisfy the following conditions. { π 1 ( λ 1 - I 1 ) + r 1 ≠ 0 π 2 ( λ 2 - I 2 ) + r 2 ≠ 0 ( C 1 - P ) · [ π 1 ( λ 1 - I 1 ) + r 1 ] > 0 ( C 2 - P ) · [ π 2 ( λ 2 - I 2 ) + r 2 ] > 0

The probability of N₁ and N₂ choosing to cooperate when government incentives are present is influenced by another player and government incentives. Then we analyze x^,* and y^,*.

(1) Analysis of factors affecting x^,*.

As with the analysis of x^* in 3.2, x^,* is an increasing function of C₂ and I₂. The probability that N₁ chooses to cooperate is greater if the input cost (C₂) of N₂ and the probability that N₂ chooses to free-ride I₂ are greater. The probability that N₁ ultimately decides to cooperate is determined by the parameters of N₂.

In contrast to 3.2, x^,* is a decreasing function concerning λ₂, r₂ and P. If the government incentive for N₂ is large, the probability that N₁ will choose to cooperate is smaller. It means that N₁ may perceive itself as disadvantaged when it learns that the government incentive for both players to cooperate is large, therefore the smaller the strategy of choosing to cooperate.

(2) Analysis of factors affecting y^,*.

Similar to the analysis of x^,*, y^,* is an increasing function of C₁ and I₁, and a decreasing function of λ₁, r₁ and P. If the probability that N₁ chooses the cost of cooperation and free-riding is greater, the probability that N₂ chooses to cooperate is greater. The probability that N₂ chooses to cooperate becomes smaller when N₁ chooses a higher return to cooperation, government incentives and penalties for free-riding.

In summary, the probabilities that affect N₁, N₂’s decision to cooperate are all influenced by the parameters of another player’s decision. In addition to the influence of their factors, the degree of punishment and incentive must be captured if the players ultimately cooperate.

Our desired result is that both players achieve cooperation. Based on the analysis of x^* and y^* in 3.2 and 3.3, we find that the outcome of the decisions made by both players will be influenced by another player. In the non-incentive model, the desire for both players to adopt a cooperative strategy requires a reduction in the penalty for free-riding. If the difference between the revenue growth rate (λ _i ) and the probability of free-riding (I _i ) is not large, the fact that the benefits of both players choosing to cooperate are greater than non-cooperating will encourage players to tend to cooperate. In the government incentive model, in addition to the above influences, there are government incentives, and the probability of one player cooperating is also influenced by another player’s reward, so the government should reward both players equally. The results of the payment matrix showed that when the two players choose to cooperate, the benefits after the government incentive are greater than the benefits without the government incentive.

In the solution process, we use triangular fuzzy numbers (TFNs) instead of exact numbers to account for the comprehensiveness of the information involved. The payment matrix represented by the TFNs is obtained based on the operation. In the next step, we use the triangular structured element method to solve and transform the original information into crisp numbers. Finally, we can obtain the strategy for both players.

Then we perform a numerical simulation and analysis to obtain the probability of both players choosing the cooperation strategy under different parameters.

4.1 The non-incentive calculation

Based on the above analysis, the fuzzy values for each parameter are assumed to be as shown in Table 6.

Step 1: Calculate the non-incentive payment matrix. According to the rules of triangular fuzzy number operation, the payment matrix of the game between the battery manufacturer (N₁) and the vehicle producer (N₂) in the absence of government incentives is obtained, as shown in Table 7.

We can translate the above matrix into the following equation. A 2 × 2 = [ ( 13 5 , 13 2 , 203 20 ) ( 3 , 7 , 9 ) ( 49 10 , 143 20 , 93 10 ) ( 8 , 9 , 19 2 ) ] B 2 × 2 = [ ( 3 2 , 22 5 , 93 10 ) ( 29 10 , 27 5 , 163 20 ) ( 2 , 5 , 17 2 ) ( 7 , 8 , 9 ) ]

Step 2: Generate linear triangular structured elements. According to the definition of triangular fuzzy elements, the elements in the fuzzy matrix can be linearly composed of triangular structured elements.

A_2×2 = (f _ij  (E)) _2×2, B_2×2 = (g _ij  (E)) _2×2, and f _ij , g _ij is a homogeneous monotonic function, we obtain f _ij  (x) , g _ij  (y) as: f 11 ( x ) = { 39 10 x + 13 2 , x ∈ [ - 1 , 0 ) 73 20 x + 13 2 , x ∈ [ 0 , 1 ] f 12 ( x ) = { 4 x + 7 , x ∈ [ - 1 , 0 ) 2 x + 7 , x ∈ [ 0 , 1 ] f 21 ( x ) = { 9 4 x + 143 20 , x ∈ [ - 1 , 0 ) 43 20 x + 143 20 , x ∈ [ 0 , 1 ] f 22 ( x ) = { x + 9 , x ∈ [ - 1 , 0 ) 1 2 x + 9 , x ∈ [ 0 , 1 ] g 11 ( y ) = { 29 10 y + 22 5 , y ∈ [ - 1 , 0 ) 49 10 y + 22 5 , y ∈ [ 0 , 1 ] g 12 ( y ) = { 5 2 y + 27 5 , y ∈ [ - 1 , 0 ) 11 4 y + 27 5 , y ∈ [ 0 , 1 ] g 21 ( y ) = { 3 y + 5 , y ∈ [ - 1 , 0 ) 7 2 y + 5 , y ∈ [ 0 , 1 ] g 22 ( y ) = { y + 8 , y ∈ [ - 1 , 0 ) y + 8 , y ∈ [ 0 , 1 ]

Step 3: Generate the projection matrix. According to Definition 8, we can obtain the projection matrix A - 2 × 2 , B - 2 × 2 of A_2×2, B_2×2. A - 2 × 2 = [ 1243 240 65 12 767 120 69 8 ] B - 2 × 2 [ 18 5 367 80 97 24 23 3 ]

Step 4: Obtain the expected utility functions of N₁ and N₂. E 1 ( x , y ) = ( x , 1 - x ) [ 1243 240 65 12 767 120 69 8 ] ( y 1 - y ) = 479 240 xy - 37 24 x - 67 30 y + 69 8 E 2 ( x , y ) = ( y , 1 - y ) [ 18 5 367 80 97 24 23 3 ] ( x 1 - x ) = 211 80 xy - 87 24 x - 539 240 y + 23 3

Step 5: Calculate the optimal mixing strategy. A first order derivative of E₁ (x, y) and E₂ (x, y) respectively gives: ∂ E 1 ( xy ) ∂ x = ∂ ( 479 240 xy - 37 24 x - 67 30 y + 69 8 ) ∂ x = 479 240 y - 37 24 = 0 ∂ E 2 ( xy ) ∂ y = ∂ ( 211 80 xy - 87 24 x - 539 240 y + 23 3 ) ∂ y = 211 80 x - 539 240 = 0

Further solving for this results in the following: x * = 539 633 ≈ 0.85 , y * = 370 479 ≈ 0 , 77

Therefore, the optimal strategy for N₁ and N₂ is: { ( x , 1 - x ) = ( 0 . 85 , 0 . 15 ) ( y , 1 - y ) = ( 0 . 77 , 0 . 23 )

Step 6: Calculate the fuzzy payoffs. Based on the optimal strategy solution, the fuzzy payoffs for N₁ and N₂ are found to be: V 1 = ( 0.85 , 0.15 ) · [ ( 13 5 , 13 2 , 203 20 ) ( 3 , 7 , 9 ) ( 49 10 , 143 20 , 93 10 ) ( 8 , 9 , 19 2 ) ] · ( 0.77 0.23 ) = ( 3 , 6.7 , 9.7 ) V 2 = ( 0.77 , 0.23 ) · [ ( 3 2 , 22 5 , 93 10 ) ( 29 10 , 27 5 , 163 20 ) ( 2 , 5 , 17 2 ) ( 7 , 8 , 9 ) ] · ( 0.85 0.15 ) = ( 1.9 , 4.8 , 9 )

Through the above series of solutions, we find that the probability of N₁ and N₂ choosing to cooperate in the non-incentive case is x^* = 0.85, y^* = 0, 77 and the final payoff for each side is V₁ = (3, 6.7, 9.7) and V₂ = (1.9, 4.8, 9). The value of the payoff for N₁ is greater than that for N₂. Next, we discuss the model in the incentive case, assuming in 4.2 that the incentive for N₂ is greater than that for N₁.

When both players choose the cooperative strategy, the government will consider an additional incentive r₁, r₂ for N₁, N₂ separately, assuming r₂ > r₁, the fuzzy values for each parameter as shown in Table 8:

Step 1: Calculate the incentive payment matrix. According to the triangular fuzzy number operation rules, the game payment matrix of N₁ and N₂ at the time of government incentive is obtained as shown in Table 9.

Step 2: Generate linear triangular structured elements. A 2 × 2 , = ( f ij , ( E ) ) 2 × 2 , B 2 × 2 , = ( g ij , ( E ) ) 2 × 2 , and f ij , , g ij , is a homogeneous monotonic function, we obtain f ij , ( x ) , g ij , ( y ) as: f 11 , ( x ) = { 22 5 x + 19 2 , x ∈ [ - 1 , 0 ) 93 20 x + 19 2 , x ∈ [ 0 , 1 ] f 12 , ( x ) = { 4 x + 7 , x ∈ [ - 1 , 0 ) 2 x + 7 , x ∈ [ 0 , 1 ] f 21 , ( x ) = { 9 4 x + 143 20 , x ∈ [ - 1 , 0 ) 43 20 x + 143 20 , x ∈ [ 0 , 1 ] f 22 , ( x ) = { x + 9 , x ∈ [ - 1 , 0 ) 1 2 x + 9 , x ∈ [ 0 , 1 ] g 11 , ( y ) = { 22 5 y + 79 10 , y ∈ [ - 1 , 0 ) 59 10 y + 79 10 , y ∈ [ 0 , 1 ] g 12 , ( y ) = { 5 2 y + 27 5 , y ∈ [ - 1 , 0 ) 11 4 y + 27 5 , y ∈ [ 0 , 1 ] g 21 , ( y ) = { 3 y + 5 , y ∈ [ - 1 , 0 ) 7 2 y + 5 , y ∈ [ 0 , 1 ] g 22 , ( y ) = { y + 8 , y ∈ [ - 1 , 0 ) y + 8 , y ∈ [ 0 , 1 ]

Step 3: Generate the projection matrix. A 2 × 2 ′ , B 2 × 2 ′ generates the respective projection matrix A 2 × 2 - ′ , B 2 × 2 - ′ as: A 2 × 2 - ′ = [ 1933 240 65 12 767 120 69 8 ] B 2 × 2 - ′ = [ 787 120 367 80 97 24 23 3 ]

Step 4: Obtain the expected utility functions of N₁ and N₂. E 1 ′ ( x , y ) = ( x , 1 - x ) [ 1933 240 65 12 767 120 69 8 ] ( y 1 - y ) = 903 80 xy - 77 8 x - 268 120 y + 69 8 E 2 ′ ( x , y ) = ( y , 1 - y ) [ 787 120 367 80 97 24 23 3 ] ( x 1 - x ) = 1343 240 xy - 87 24 x - 739 240 y + 23 3

Step 5: Calculate the optimal mixing strategy. A first order derivative of E 1 , ( x , y ) and E 2 , ( x , y ) respectively gives: ∂ E 1 ′ ( x , y ) ∂ x = ∂ ( 903 80 xy - 77 8 x - 268 120 y + 69 8 ∂ x ) = 903 80 y - 77 8 = 0 ∂ E 2 ′ ( x , y ) ∂ y = ∂ ( 1343 240 xy - 87 24 x - 739 240 y + 23 3 ) ∂ y = 1343 240 x - 739 240 = 0

Further solving for this results in the following: x , * = 739 1343 ≈ 0.55 , y , * = 770 903 ≈ 0.85

Therefore, the optimal strategy for N₁ and N₂ is: { ( x ′ , 1 - x ′ ) = ( 0 . 55 , 0 . 45 ) ( y , , 1 - y , ) = ( 0 . 85 , 0 . 15 )

Step 6: Calculate the fuzzy payoffs. Based on the optimal strategy solution, the fuzzy payoffs for N₁ and N₂ are found to be: V 1 , = ( 0 . 55 , 0 . 45 ) · [ ( 51 10 , 19 2 , 283 20 ) ( 3 , 7 , 9 ) ( 49 10 , 143 20 , 93 10 ) ( 8 , 9 , 19 2 ) ] · ( 0.85 0.15 ) = ( 5.1 , 8.3 , 11.6 ) V 2 , = ( 0 . 85 , 0 . 15 ) · [ ( 7 2 , 79 10 , 138 10 ) ( 29 10 , 27 5 , 163 20 ) ( 2 , 5 , 17 2 ) ( 7 , 8 , 9 ) ] · ( 0.55 0.45 ) = ( 3 . 4 , 6 . 7 , 12 . 6 )

Through the above series of solutions, we obtain the probability of N₁ and N₂ choosing to cooperate in the incentive case as x^,* = 0.55, y^,* = 0, 85 and the fuzzy utility value as V₁ = (5.1, 8.3, 11.6) and V₂ = (3.4, 6.7, 12.6). We find that the probability of N₁ choosing to cooperate decreases when the reward for N₂ is greater than that for N₁. It is possible that N₁ believes that the choice of cooperation is different from the investment and benefits. As for N₂, the probability of choosing to cooperate is greater than the result in the non-incentive model, which can prove that government incentives positively impact encouraging both players to choose cooperation. The results of the two models are analyzed in more detail below.