Integrated Production-Inventory Supply Chain Model

To better describe subjective imprecise quantity, Liu in proposed an uncertain measure and further developed an uncertainty theory which is an axiomatic system of normality, monotonicity, self-duality, countable subadditivity and product measure.

Definition 1

Let $Γ$ be a non-empty set and $L$ be a $σ$ algebra over $Γ$. Each element $Λ ∈ L$ is called an event. A set function $M{Λ}$ is called an uncertain measure if it satisfies the following four axioms of Liu:

Axiom 1
(Normality) $M{Λ}=1$

Axiom 2
(Monotonicity) $M{Λ}+M{Λ^C}=1$, for any event $Λ$

Axiom 3
(Countable subadditivity) For every countable sequence of events $Λ_1,Λ_2,…,$ we have $M ⎨ ∑_{i=1} ^{∞} Λ_i⎬ ≤ ∑_{i=1} ^∞ M {Λ_i}$.

Definition 2

The uncertainty distribution $Φ : R → [0,1]$ of an uncertain variable $ξ$ is defined by $Φ(t) = M {ξ≤t}$.

Definition 3

Let $ξ$ be an uncertain variable. Then the expected value of $ξ$ is defined by $E[ξ]=∫^{∞}_{0} M {ξ≥r} dr−∫^{0}_{−∞} M{ξ≤r}dr$, provided that at least one of the two integrals is finite.

Theorem 1
Let $ξ$ be an uncertain variable with uncertainty distribution $Φ$. If the expected value exists, then $E[ξ]=∫^{1}_{0}Φ^{−1}(α)dα$.

Lemma 1
Let \$ξ∼L(a,b,c)$ be a zigzag uncertain variable. Then its inverse uncertainty distribution $Φ^{−1}(α)=\dfrac{1}{2}[(1−α)a+b+αc]$, and it can be expressed as $E[ξ]=∫^{1}_{0} \dfrac{1}{2} [(1−α)a+b+αc]dα=\dfrac{a+2b+c}{4}$.
(1)

Theorem 2
Let $ξ_1,ξ_2,…,ξ_n$  be independent uncertain variables with uncertainty distributions $ϕ_1,ϕ_2,…,ϕ_n$, respectively. If f is a strictly increasing function, then $ξ = f(ξ_1,ξ_2,…,ξ_n)$ is an uncertain variable with inverse uncertainty distribution

$Φ^{−1}(α)=f(Φ^{−1}_1(α),Φ^{−1}_2 (α),…,Φ^{−1}_n(α))$.

Theorem 3
Let $ξ_1$ and $ξ_2$ be independent uncertain variables with finite expected values. Then for any real numbers a 1 and a 2, we have

$E[a_1ξ+a_2η]=a_1E[ξ]+a_2E[η]$.
(2)