A simulation algorithm with uncertain random variables

Article History: Received 05 April 2018 Accepted 22 April 2018 Available 25 April 2018 In many situations, uncertainty and randomness concurrently occur in a system. Thus this paper presents a new concept for uncertain random variable. Also, a simulation algorithm based on uncertain random variables is presented to approximate the chance distribution using pessimistic value and optimistic value. An example is also given to illustrate how to use the presented simulation algorithm.


Introduction
Liu [1] introduced the uncertain random variable for modeling complex systems.In other words; uncertain random variable is improved to illustrate the phenomenon which mixes uncertainty with randomness.If we receive historical data from the sample, we can estimate the probability distribution.But if we have a new product, we can not achieve the probability distribution of this new product owing to lack of data.In this case, we run across both randomness and human uncertainty.Human uncertainty is investigated by some scholars.As a branch of mathematics based on normality, duality, subadditivity and product axioms, uncertainty theory was introduced by Liu [2] in 2007.Gao [3] presented uncertain bimatrix game.Yang and Gao [4] studied uncertain differential game.Gao and Qin [5] introduced the degree connectivity of uncertain graph.Dalman [6] presented a model for the uncertain multi-item solid transportation problem.Dalman [7] constructed models of uncertain random multi-item solid transportation problem.To model uncertain random event, Liu [8] introduced the chance theory to networks optimization problem.Some scholars derived properties of uncertain random entropy [9,10].By employing chance theory, an uncertain random project scheduling problem is presented by Ke et al. [11].They introduced an uncertain random simulation which randomly produces the sample points.But, this algorithm is ambivalent due to produces different values at different time.A simulation algorithm is presented to solve uncertain random shortest path problem by Sheng and Gao [12].Therefore this paper presents an algorithm which includes the inverse uncertainty distribution and uniformly produces the sample points.It has powerful performances on the reliability and stability than the algorithms in [11,12].The paper is built as follows: some basic knowledge of uncertainty theory and chance theory is presented in section II.Section III shows some formulas for uncertain random variables presents an uncertain random simulation algorithm.A numerical example is presented in Section IV.Finally, this paper closes in Section V.

Axiom 2. (Duality Axiom)(Liu
where Λ k are arbitrarily chosen events from L k for k = 1, 2, • • • , respectively.Definition 1. (see [2]): An uncertain variable is a function ξ from an uncertainty space (Γ, L, M) to the set of real numbers such that {ξ ∈ B} is an event for any Borel set B of real numbers.
Definition 2. (Liu [14]): An uncertainty distribution Φ(x) is said to be regular if it is a continuous and strictly increasing function with respect to x at which 0 < Φ(x) < 1, and Definition 3. (Liu [14]): Let ξ be an uncertain variable with regular uncertainty distribution Φ(x).Then the inverse function Φ −1 (α) is called the inverse uncertainty distribution of ξ.

Chance theory
Definition 4. (Liu [1]): Let (Γ, L, M) be an uncertainty space and let (Ω, A, Pr) be a probability space.Then the product (Γ, L, M) × (Ω, A, Pr) is called a chance space Definition 5. (see [1]): An uncertain random variable is a function ξ from a chance space (Γ, L, M) × (Ω, A, Pr) to the set of real numbers such that {ξ ∈ B} is an event in an event in L×A for any Borel set B of real numbers.
Theorem 2. (see [14]): Let ξ 1 , ξ 2 , • • • , ξ n be uncertain random variables on the chance space (Γ, L, M) × (Ω, A, Pr), and let f be a measurable function.Then where Definition 6. (Liu [1]): Let ξ be an uncertain random variable.Then its chance distribution is defined by Theorem 4. Let ξ be an uncertain random variable.Then its expected value is provided that at least one of the two integrals is finite.
Theorem 5. Let ξ be an uncertain random variable with regular chance distribution Φ.If the expected value exists, then In order to obtain the chance distribution, we prove following theorems and develop a simulation algorithm.

A simulation algorithm with uncertain random variables
In this section, by employing the above definition and theorems, the chance distribution will be obtained.To do this, the following theorems is proved and besides a simulation algorithm is developed.
In fact, Formula (3) is a theoretical formula, which is not easy to use in most cases due to the complexity of chance distribution function.
To overcome the difficulty, an uncertain random simulation is proposed to evaluate the chance distribution.First, we introduce the concepts of α−pessimistic value and α−optimistic value for an uncertain random variable.Then, we approximate the chance distribution, α−pessimistic value and α−optimistic value by using a numerical integration method.
Proof.It follows from Definition 7 immediately.
Theorem 11.Let ξ an ordinary uncertain random variable.Then, we have Proof.Since Φ(x) is strictly increasing and continuous, we get According to Definition (), we have The Theorem is proved.
Theorem 12. Let ξ be an ordinary uncertain random variable.Then, we have Proof.Since Φ(x) is strictly increasing and continuous, we get Thus, for all α ∈ (0, 1), we have The theorem is proved.
Theorem 13.Let ξ be an ordinary uncertain random variable.Then, we have Proof.It follows directly from Theorem 11 and Theorem 12. where Proof.It follows from Theorem 1 that inverse uncertainty distribution of F (x; y 1 , y 2 , • • • , y m ) is determined by According to Theorem 10, we substitute Υ . Thus, Formula (19) holds.The theorem is completed.
where F (x; y 1 , y 2 , • • • , y m ) is determined by its inverse uncertainty distribution Proof.The proof is similar to that of Theorem 14.
According to Theorems 1, 17, 8, 9 and 10, we design the following uniform discretization algorithm to simulate Φ(x), α-pessimistic value and α-optimistic value.The presented algorithm is very flexible.Because it can even simulate the empirical distribution.Algorithm 1 (Uniform Discretization Algorithm) Step 1. Discretize the range of the random variable η i into N i equally spaced points.
The uniform discretization algorithm is illustrated by the following example.
For the sake of simplicity, we set N 1 = 10, N 2 = 10, K = 10.We also can assign a large number to N 1 , N 2 and K.The probability distribution function of η 1 and η 2 are Then, we can have discrete forms of Ψ 1 (y 1 ) and Ψ 1 (y 2 ) in which Applying the above presented algorithm, the chance distribution of ξ is obtained as (Figure 1): ξ inf (0.2)=3.9 and ξ sup (0.2)=7.4.

Conclusion
Here, an uncertain random simulation algorithm is presented to illustrate chance distribution.The presented simulation method can be generally applied to uncertain random optimization by approximating chance constraints.The results obtained show that the presented algorithm is the successful for the uncertain random simulation.