Optimal resource allocation in hierarchical organizations under uncertainty: An interval-valued n-person cooperative game approach

2.1 Operations on intervals

Operations on intervals refer to mathematical techniques utilized for performing operations between intervals. These operations encompass addition, subtraction, multiplication, division, and other mathematical procedures executed on intervals to generate interval outcomes that inherently possess uncertainty. In this study, the utilization of operations on intervals plays a pivotal role in effectively addressing uncertainty in resource allocation problems arising from insufficient information or data.

Let R be the set of real numbers. An interval may be expressed with a ˆ = [ a ¯ , a ¯ ] = { a | a ¯ ⩽ a ⩽ a ¯ , a ¯ ∈ R , a ¯ ∈ R } , in which y ¯ and a ¯ are called the lower and upper bounds of the interval a ˆ , respectively. If a ¯ = a ¯ , then a ˆ = [ a ¯ , a ¯ ] is reduced to a real number a, where a = a ¯ = a ¯ .

If y ¯ ⩾ 0, then a ˆ is called a non-negative interval, denoted by a ˆ ⩾ 0 ; if a ¯ ⩽ 0 , then a ˆ is called a non-positive interval, denoted by a ˆ ⩽ 0 . If y ¯ > 0, then a ˆ is called a positive interval, denoted by a ˆ > 0 .

Alternatively, an interval a ˆ can be expressed in mean-width or center-radius, such as a ˆ = 〈 m ( a ˆ ) , w ( a ˆ ) 〉 ; in which m ( a ˆ ) = ( a ¯ + a ¯ ) / 2 is the mid-point of a ˆ and w ( a ˆ ) = ( a ¯ - a ¯ ) / 2 is the half-width of a ˆ , respectively. The set of intervals in the real number set R is denoted by R ˆ .

Based on the definition of operations on intervals in Ref. [23], the definition is improved by introducing equality and scalar multiplication. This improvement makes the definition more rigorous and reflects the essence of interval operations more accurately, thus laying a solid foundation for subsequent research and application. The operations on intervals of equality, addition, and scalar multiplication are defined as follows:

Definition 1. Let a ˆ = [ a ¯ , a ¯ ] and b ˆ = [ b ¯ , b ¯ ] be two intervals in the set R ˆ . The interval arithmetic operations are stipulated as follows:

Equality of two intervals: a ˆ = b ˆ if and only if y ¯ =  b ¯ and a ¯ = b ¯ ;

γ a ˆ = [ γ a ¯ , γ a ¯ ] = { [ γ a ¯ , γ a ¯ ] if γ ⩾ 0 [ γ a ¯ , γ a ¯ ] if γ < 0 ;

Where γ ∈ R is a real number.

Using the mean-width or center-radius form mentioned above, we can rewrite the operations of a ˆ = b ˆ as follows:

Equality of two intervals: a ˆ = b ˆ if and only if m ( a ˆ ) = m ( b ˆ ) and w ( a ˆ ) = w ( b ˆ ) ;

γ a ˆ = γ 〈 m ( a ˆ ) , w ( a ˆ ) 〉 = 〈 γ m ( a ˆ ) , | γ | w ( a ˆ ) 〉 { 〈 γ m ( a ˆ ) , γ w ( a ˆ ) 〉 if γ ⩾ 0 〈 γ m ( a ˆ ) , - γ w ( a ˆ ) 〉 if γ < 0 .

Obviously, the arithmetic operations above are the generalization of real numbers.

2.2 Satisfactory degrees of comparing intervals

Satisfactory degrees of comparing intervals are used to measure the degree of comparison between two intervals. This concept provides a systematic method for measuring the relative position and degree of overlap between intervals. In this study, the satisfactory degree is utilized to analyze the correlation and relative advantages of intervals to aid in identifying opportunities for cooperation and develop strategies to maximize the benefits of resource allocation.

Most traditional ranking methods [24, 25] are stricter, as shown in Fig. 1, which only considers strict intersection, greater than, and other relations without considering inclusion and/or overlapping relations between intervals. In fact, for fuzzy sets, the statement “the interval a ˆ is not greater than the interval b ˆ ” can be regarded as a fuzzy relation between a ˆ and b ˆ , denoted by a ˆ ⩽ b ˆ . Therefore, inspired by Li et al. [21], this paper defines the fuzzy partial order relation of intervals, which fully considers the inclusion and/or overlapping relationship between intervals, as shown in Fig. 2, with novelty and mathematical rigour.

OPEN IN VIEWER

Fig. 1

Relations of a ˆ and b ˆ by using the traditional ranking methods.

OPEN IN VIEWER

Fig. 2

Relations of a ˆ and b ˆ by using a ˆ ⩽ I b ˆ with inclusion and overlap relations.

Definition 2. The premise “ λ ( a ˆ ⩽ I b ˆ ) ” is regarded as a fuzzy set, whose membership function is defined as follows:

λ ( a ˆ ⩽ I b ˆ ) = { 1 if a ¯ < b ¯ 1 - if a ¯ < b ¯ < a ¯ < b ¯ ( b ¯ - a ¯ ) 2 ( w ( b ˆ ) - w ( a ˆ ) ) if b ¯ ⩽ a ¯ ⩽ a ¯ ⩽ b ¯ and w ( b ˆ ) > w ( a ˆ ) 0 . 5 if a ¯ = b ¯ and w ( a ˆ ) = w ( b ˆ )

Definition 3. The premise “ λ ( a ˆ ⩾ I b ˆ ) ” is regarded as a fuzzy set, whose membership function is defined as λ ( a ˆ ⩾ I b ˆ ) = 1 - λ ( a ˆ ⩽ I b ˆ ) , i.e.,

λ ( a ˆ ⩾ I b ˆ ) = { 0 if a ¯ < b ¯ 0 + if a ¯ < b ¯ < a ¯ < b ¯ ( a ¯ - b ¯ ) 2 ( w ( b ˆ ) - w ( a ˆ ) ) if b ¯ ⩽ a ¯ ⩽ a ¯ ⩽ b ¯ and w ( b ˆ ) > w ( a ˆ ) 0 . 5 if a ¯ = b ¯ and w ( a ˆ ) = w ( b ˆ )

In the sequent, it is easy to prove that the satisfactory degree λ is continuous except a single special case, i.e., y ¯ =  b ¯ and w ( a ˆ ) = w ( b ˆ ) . Moreover, for any intervals a ˆ and b ˆ , we can easily prove that the following properties are valid:

0 ⩽ λ ( a ˆ ⩽ I b ˆ ) ⩽ 1

Thus, “ ⩽ _I ” and “⩾ _I ” are well established as fuzzy partial orders of intervals. Definitions 2 and 3 may provide quantitative ways to determine the exact degree of satisfaction in ranking two intervals. Then, the satisfaction equivalent form of the interval-valued inequality relation is defined using the satisfactory degree λ.

3.1 Modeling optimal resource allocation in hierarchical organizations under uncertainty

In practical resource allocation scenarios, problems such as incomplete data, unclear data, and complex behavioural patterns are often encountered. As a result, many model parameters are defined as interval values rather than exact values in such situations. Imagine a scenario where a superior department is responsible for allocating a limited number of resources among its n (n ⩾ 2) subordinate departments.

The total amount of these finite resources is represented by an interval b ˆ = [ b ¯ , b ¯ ] , where b ¯ and b ¯ respectively denote the lower and upper bounds of b ˆ . The resources obtained by each subordinate department may be described as an interval α ^ i = [ α i , α ¯ i ] ∈ R ^ + m 1 , i ∈ N = { 1 , 2 , … , n } , α ^ = ( α ^ 1 , α ^ 2 , … , α ^ n ) ∈ R ^ + n m 1 , w h e r e , R ^ + m 1 a n d R ^ + n m 1 signify the non-negative sector of the Euclidean space, and N represents the collect of subordinate departments. Considering factors such as production capacities, resource constraints, and technological requirements, a mathematical model may be built to address this resource allocation problem:

max { F ˆ ( x ˜ ( α ˆ ) , α ˆ ) } s . t . { ∑ i = 1 n α ˆ i ⩽ b ˆ G ˆ ( α ˆ ) ⩽ 0(1)

Where x ˜ i ( i ∈ N ) is the solution to the following programming:

f ˆ i ( x i ) = max { f ˆ i ( x i ) } s . t . { g ˆ i ( x i ) ⩽ α ˆ i h ˆ i ( x i ) ⩽ 0(2)

In which F ˆ ∈ R ˆ is the objective function of the superior departments; x i ∈ R m 2 and f ˆ i ∈ R ˆ are the production activity variable and objective function of the ith subordinate department, which can be expressed as production and profit. x = ( x 1 , x 2 , … , x n ) ∈ R n m 2 , G ˆ ( α ˆ ) ∈ R ˆ m 3 , h ˆ i ( x i ) ∈ R ˆ m 4 ( i ∈ N ) .

This study focuses on a specific situation: when the superior department allocates resources according to the model, it does not consider the cooperation between subordinate departments. However, after the resource allocation scheme is determined, some subordinate departments will redistribute the resources they receive according to their actual production conditions, capacity and efficiency in order to achieve greater benefit among departments. We refer to this collaborative approach as “sharing of results”. By applying the methodology outlined in Ref. [26], we can find out the optimal amount of resources α ˆ i * , allocated by the superior departments to the subordinate departments, and the optimal production activity decisions x i * , objective function values f ˆ i * and objective functions F ˆ * of the superior departments.

Any nonempty subset S in N is called the cooperative coalition of subordinate departments. Then, there are n cooperative coalitions. The purpose of forming cooperative coalition S is to maximize the objective function of S. Since we assume that there is no adversarial competition between subordinate departments, so the mathematical model of cooperative coalition S can be built as follows:

max { ∑ i ∈ S f ˆ i ( x i ) } s . t . { ∑ i ∈ S g ˆ i ( x i ) ⩽ ∑ i ∈ S α ˆ i * h ˆ i ( x i ) ⩽ 0 , i ∈ S(3)

The objective function here is equal to the sum of the objective functions of each department in the cooperative alliance S. Obviously, other forms of the target function are also available. At this point, model (1) can be considered as the resource allocation problem of n - s + 1 subordinate departments without any cooperation, where s is the number of departments in cooperative coalition S. The optimal solution x i * * ( i ∈ S ) and the objective function value ∑ i ∈ S f ˆ i ( x i * * ) of Equation (3) can be obtained by using the optimization method. Obviously, the objective function is dependent on coalition S; in other words, it is a function of S, so it can be written as υ ˆ ( S ) = ∑ i ∈ S f ˆ i ( x i * * ) . If S = {i} (i ∈ N), then the optimal solution and the objective function value of Equation (3) are x i * and f ˆ i ( x i * ) , that is υ ˆ ( { i } ) = f ˆ i ( x i * ) , υ ˆ ( { i } ) is reworded as υ ˆ ( i ) . Here, we set υ (∅) =0, in which ∅ is an empty set. υ ˆ ( S ) is a quantitative index that measures the benefit (or profit) and production capacity of cooperative coalition S. Thus, the following conclusion can be obtained:

Theorem 1. For cooperative coalition S, T ⊆ N, if S∪ T ≠ ∅, then υ ^ ( S ∪ T ) ≥ υ ^ ( S ) + υ ^ ( T )

The property of the value υ ˆ of the objective function is called super-additivity.

Proof. Because υ ˆ ( S ) and υ ˆ ( T ) are the optimal objective function values of the following two programs respectively: max x ˆ i , i ∈ S { ∑ i ∈ S f ˆ i ( x i ) } , s . t . { ∑ i ∈ S g ˆ i ( x i ) ⩽ ∑ i ∈ S α ˆ i * h ˆ i ( x i ) ⩽ 0 , i ∈ S max x ˆ i , i ∈ T { ∑ i ∈ T f ˆ i ( x i ) } , s . t . { ∑ i ∈ T g ˆ i ( x i ) ⩽ ∑ i ∈ T α ˆ i * h ˆ i ( x i ) ⩽ 0 , i ∈ T

Obviously, the above two programs are equivalent to the following programs:

max x ˆ i , i ∈ S ∪ T { ∑ i ∈ S ∪ T f ˆ i ( x i ) } , s . t . { ∑ i ∈ S g ˆ i ( x i ) ⩽ ∑ i ∈ S α ˆ i * ∑ i ∈ T g ˆ i ( x i ) ⩽ ∑ i ∈ T α ˆ i * h ˆ i ( x i ) ⩽ 0 , i ∈ S ∪ T(4)

So, υ ˆ ( S ∪ T ) ⩾ υ ˆ ( S ) + υ ˆ ( T ) .

Theorem 1 states that two coalitions that do not contain any of the same departments are likely to produce more benefits, or at least not getting worse due to the cooperation.

In the case of only two subordinate departments, the only kind of cooperative coalition is N = {1, 2}. At this point, the corollary becomes υ ˆ ( { 1 , 2 } ) ⩾ υ ˆ ( 1 ) + υ ˆ ( 2 ) . For essential cooperation, its allocation vector set is E ( υ ˆ ) = { y ˆ 1 , υ ˆ ( 1 , 2 ) - y ˆ 1 | υ ˆ ( 1 ) ⩽ y ˆ 1 ⩽ υ ˆ ( 1 , 2 ) - υ ˆ ( 2 ) }

Because for any two allocation vectors y ˆ and y ˆ ′ in E ( υ ˆ ) , if y ˆ 1 > y ˆ 1 ′ , then y ˆ 2 < y ˆ 2 ′ , we can’t decide which is better. And then we’ll discuss how to determine the benefit distribution.

5.1 Application and computation

Suppose a superior department with three subordinate factoriesT₁, T₂, and T₃, which share the same resource. Each factory produces two products using that resource. Due to lack of information or imprecision of the available information, we assume the following mathematical model: max { F ( x ˜ ( α ˆ ) , α ˆ ) } = [ 4 , 5 ] ( x 11 + x 12 ) + [ 6 , 8 ] ( x 21 + x 22 ) + [ 9 , 10 ] ( x 31 + x 32 ) s . t { ∑ i = 1 3 α ˆ i ⩽ [ 15 , 20 ] α ˆ i ⩾ 0 i ∈ N = { 1 , 2 , 3 } where x ˜ 1 , x ˜ 2 and x ˜ 3 are respectively the solutions of the following programs: max x 1 { f 1 ( x 1 ) = [ 12 , 15 ] x 11 + [ 8 , 10 ] x 12 } s . t { ∑ j = 1 2 x 1 j ⩽ α ˆ 1 x 1 j ⩾ 2 , j = 1 , 2 max x 2 { f 2 ( x 2 ) = [ 8 , 10 ] x 21 + [ 6 , 8 ] x 22 } , s . t { ∑ j = 1 2 x 2 j ⩽ α ˆ 2 x 2 j ⩾ 2 , j = 1 , 2 max x 3 { f 1 ( x 3 ) = [ 4 , 8 ] x 31 + [ 3 , 5 ] x 32 } , s . t { ∑ j = 1 2 x 3 j ⩽ α ˆ 3 x 3 j ⩾ 2 , j = 1 , 2

Where α ˆ = ( α ˆ 1 , α ˆ 2 , α ˆ 3 ) , x = (x₁, x₂, x₃), x _i  = (x_i1, x_i2) (i ∈ N); x_i1 and x_i2 are the production of the first and second products of the factory T _i , the objective function f ˆ i ( x i ) of the subordinate factory is product profit, the objective function F ˆ ( x , α ˆ ) of the superior department is the profit paid by the subordinate factories. For the situation there is no cooperation between the three factories, According to Theorem 1 and Theorem 2 and given λ₀ = 0.8 (determined by all the factories’ intentions), we can calculate the optimal solution is α ˆ * = ( [ 3 , 4 ] , [ 3 , 4 ] , [ 9 , 12 ] ) , f ˆ 1 * = [ 37.5 , 50 ] , f ˆ 2 * = [ 27 , 36 ] , f ˆ 3 * = [ 67.5 , 90 ] , then υ ˆ 1 = [ 37.5 , 50 ] , υ ˆ 2 = [ 27 , 36 ] , υ ˆ 3 = [ 67.5 , 90 ] .

Similarly, if the factory T₂ and factory T₃ cooperate, the model after cooperation is as follows: max { F ( x ˜ ( α ˆ ) , α ˆ ) } = [ 4 , 5 ] ( x 11 + x 12 ) + [ 6 , 8 ] ( x 21 + x 22 ) + [ 9 , 10 ] ( x 31 + x 32 ) s . t { ∑ i = 1 3 α ˆ i ⩽ [ 15 , 20 ] α ˆ i ⩾ 0 i ∈ N = { 1 , 2 , 3 } max x 1 { f 1 ( x 1 ) = 15 x 11 + 10 x 12 } s . t { ∑ j = 1 2 x 1 j ⩽ α ˆ 1 x ˆ 1 j ⩾ 2 , j = 1 , 2 max x 2 , x 3 { f 2 ( x ˆ 2 ) = 10 x 21 + 8 x 22 + 8 x 31 + 5 x 32 } s . t { ∑ j = 1 2 x 2 j + ∑ j = 1 2 x 3 j ⩽ α ˆ 2 + α ˆ 3 x 2 j ⩾ 2 , j = 1 , 2 x 3 j ⩾ 2 , j = 1 , 2 .

we can obtain the interval value of coalition S = {2, 3} is υ ˆ ( { 2 , 3 } ) = [ 128 , 142 ] .Similarly, we can also obtain υ ˆ ( { 1 , 3 } ) = [ 127 , 185 ] , υ ˆ ( { 1 , 2 } ) = [ 84.5 , 106 ] , υ ˆ ( { 1 , 2 , 3 } ) = [ 174 , 232 ] . Obviously, the interval-valued core allocation vector set has an infinite number of allocation schemes, and each scheme is acceptable to all players.

According to Equation (8), the nonlinear programming model can be constructed as follows:

max { δ } s . t . { y ¯ 1 ⩽ 50 y ¯ 2 ⩽ 36 y ¯ 3 ⩽ 90 y ¯ 1 + y ¯ 2 ⩽ 106 y ¯ 2 + y ¯ 3 ⩽ 142 y ¯ 1 + y ¯ 3 ⩽ 185 y ¯ 1 + y ¯ 1 ⩾ 87.5 y ¯ 1 + y ¯ 1 - 87.5 y ¯ 1 - y ¯ 1 + 12.5 ⩾ δ y ¯ 2 + y ¯ 2 ⩾ 63 y ¯ 2 + y ¯ 2 - 63 y ¯ 2 - y ¯ 2 + 9 ⩾ δ y ¯ 3 + y ¯ 3 ⩾ 157.5 y ¯ 3 + y ¯ 3 - 157.5 y ¯ 3 - y ¯ 3 + 22.5 ⩾ δ y ¯ 1 + y ¯ 1 + y ¯ 2 + y ¯ 2 ⩾ 190.5 y ¯ 1 + y ¯ 1 + y ¯ 2 + y ¯ 2 - 190.5 y ¯ 1 - y ¯ 1 + y ¯ 2 - y ¯ 2 + 21.5 ⩾ δ y ¯ 2 + y ¯ 2 + y ¯ 3 + y ¯ 3 ⩾ 270 y ¯ 2 + y ¯ 2 + y ¯ 3 + y ¯ 3 - 270 y ¯ 2 - y ¯ 2 + y ¯ 3 - y ¯ 3 + 14 ⩾ δ y ¯ 1 + y ¯ 1 + y ¯ 3 + y ¯ 3 ⩾ 312 y ¯ 1 + y ¯ 1 + y ¯ 3 + y ¯ 3 - 312 y ¯ 1 - y ¯ 1 + y ¯ 3 - y ¯ 3 + 58 ⩾ δ y ¯ 1 + y ¯ 2 + y ¯ 3 = 232 y ¯ 1 + y ¯ 2 + y ¯ 3 = 174 y ¯ i ⩾ y ¯ i ( i = 1 , 2 , 3 ) y ¯ i ⩾ 0 ( i = 1 , 2 , 3 ) 0 ⩽ δ ⩽ 1(11)

Where δ ∈ [0, 1] and y ¯ _i and y ¯ i (i = 1, 2, 3)are decision variables. Solving Equation (11) by using the MATLAB tool, the global optimal solution ( δ * , y ˆ * ) of Equation (11) at a given precision can be estimated, where δ^* = 0.797, y ˆ 1 * = [ 50 , 71 ] , y ˆ 2 * = [ 35 , 58 ] , and y ˆ 3 * = [ 90 , 115 ] . Thus, we obtain an element y ˆ * of the interval-valued core C ( υ ˆ ) of the interval-valued 3-person cooperative games with the maximum satisfactory degree δ^* = 0.797. In other words, if the satisfactory degrees of ∑ i ∈ S y ˆ i ⩾ υ ˆ ( S ) for these three factoriesT₁, T₂, and T₃ are not greater than 0.797, the interval-valued core of the interval-valued 3-person cooperative game exists and hereby these three factories may choose product cooperative production.

According to Equation (9), the system of linear inequalities can be constructed as follows:

s . t . { y ¯ 1 + y ¯ 1 ⩽ 87.5 y ¯ 2 + y ¯ 2 ⩽ 63 y ¯ 3 + y ¯ 3 ⩽ 177.5 y ¯ 1 + y ¯ 1 + y ¯ 2 + y ¯ 2 ⩽ 150.5 y ¯ 2 + y ¯ 2 + y ¯ 3 + y ¯ 3 ⩽ 258 y ¯ 1 + y ¯ 1 + y ¯ 3 + y ¯ 3 ⩽ 312 y ¯ 1 + y ¯ 2 + y ¯ 3 = 232 y ¯ 1 + y ¯ 2 + y ¯ 3 = 174 y ¯ i ⩾ y ¯ i ( i = 1 , 2 , 3 ) y ¯ i ⩾ 0 ( i = 1 , 2 , 3 )(12)

Where y ¯ _i and y ¯ i (i = 1, 2, 3)are decision variables which need to be determined.

Solving Equation (12) by using the LINGO tool, we find there is no solution that simultaneously satisfies all the constraints. In other words, there is no feasible solution of Equation (12) and hereby these three factories may have not any cooperative desire for this situation.

Analogously, according to Equation (10) for other situation, the systems of linear inequalities can be constructed as follows:

s . t . { y ¯ 1 > 50 y ¯ 2 > 36 y ¯ 3 > 90 y ¯ 1 + y ¯ 2 > 86 y ¯ 2 + y ¯ 3 > 142 y ¯ 1 + y ¯ 3 > 185 y ¯ 1 + y ¯ 1 ⩾ 87.5 y ¯ 2 + y ¯ 2 ⩾ 63 y ¯ 3 + y ¯ 3 ⩾ 177.5 y ¯ 1 + y ¯ 1 + y ¯ 2 + y ¯ 2 ⩾ 150.5 y ¯ 2 + y ¯ 2 + y ¯ 3 + y ¯ 3 ⩾ 258 y ¯ 1 + y ¯ 1 + y ¯ 3 + y ¯ 3 ⩾ 312 y ¯ 1 + y ¯ 2 + y ¯ 3 = 232 y ¯ 1 + y ¯ 2 + y ¯ 3 = 174 y ¯ i ⩾ y ¯ i ( i = 1 , 2 , 3 ) y ¯ i ⩾ 0 ( i = 1 , 2 , 3 )(13)

Where y ¯ _i and y ¯ i (i = 1, 2, 3)are decision variables which need to be determined. Similarly, solving Equation (13) by using the LINGO tool, we find there is no feasible solution of Equation (13) and hereby these three factories may have not any cooperative desire for this situation.

In order to show the applicability and superiority of the proposed method, we compare it with the LR method. The linear programming model is constructed according to Equation (10) as follows:

{ y ˆ 1 + y ˆ 2 ⩽ [ 84.5 , 106 ] y ˆ 1 + y ˆ 3 ⩽ [ 127 , 185 ] y ˆ 2 + y ˆ 3 ⩽ [ 128 , 142 ] y ˆ 1 + y ˆ 2 + y ˆ 3 = [ 174 , 232 ] y ¯ i ⩾ y ¯ i ( i = 1 , 2 , 3 ) y ¯ i ⩾ 0 ( i = 1 , 2 , 3 )(14)

where y ¯ i and y ¯ _i (i = 1, 2, 3) are decision variables that need to be determined.

By using the LR method of comparing intervals (i.e., if y ¯ ⩽  b ¯ and a ¯ ⩽ b ¯ , then a ˆ ⩽ b ˆ ), then Equation (14) can be rewritten as the following system of inequalities:

{ y ¯ 1 + y ¯ 2 ⩽ 84.5 y ¯ 1 + y ¯ 2 ⩽ 106 y ¯ 1 + y ¯ 3 ⩽ 185 y ¯ 1 + y ¯ 3 ⩽ 127 y ¯ 2 + y ¯ 3 ⩽ 142 y ¯ 2 + y ¯ 3 ⩽ 128 y ¯ 1 + y ¯ 2 + y ¯ 3 = 232 y ¯ 1 + y ¯ 2 + y ¯ 3 = 174 y ¯ i ⩾ y ¯ i ( i = 1 , 2 , 3 ) y ¯ i ⩾ 0 ( i = 1 , 2 , 3 )(15)

Solving Equation (15) by using the LINGO tool, we find there is not a solution that simultaneously satisfies all the above constraints. In other words, there is no feasible solution of Equation (15) and hereby these three factories may have not any cooperative desire. Obviously, it is shown that there is not any feasible solution with the traditional LR interval ranking method. On the contrary, we can obtain the feasible solution by introducing the satisfactory degrees of comparing intervals, which can give more scientific suggestions for players (or managers).

Furthermore, we find that when δ^* = 0.875, then y ˆ 1 * + y ˆ 2 * = [ 89.5 , 107.5 ] , which is essentially greater than υ ˆ ( { 1 , 2 } ) = [ 84.5 , 106 ] . Namely, y ˆ 1 * + y ˆ 2 * ⩾ I υ ˆ ( { 1 , 2 } ) is conformed to the concept of satisfactory degrees of comparing intervals in Sect. 2

Therefore, it is shown that the traditional LR interval ranking method, which strictly considers relationships of intersection and superiority, may limit cooperative intent and decision-making in scenarios involving interval-valued payoffs. In contrast, our nonlinear mathematical programming model, accounting for inclusion and/or overlap relations between intervals, identifies maximum satisfaction levels and feasible solutions, thereby uncovering potential cooperative opportunities previously overlooked. These findings align with real-world expectations. Moreover, this approach complements, rather than replaces, traditional methods by offering alternative solutions when traditional interval ranking fails to yield feasible outcomes. This flexibility enhances decision-making, especially in complex situations characterized by interval relations of inclusion and/or overlap.