## Integrated Production-Inventory Supply Chain Model

Read this article. An integrated production-inventory model is constructed to address supplier, manufacturer, and retailer uncertainties. According to the author, what are the three types of uncertainties in supply chain management?

### Necessary knowledge about uncertain variables

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 be a non-empty set and be a algebra over . Each element is called an event. A set function is called an uncertain measure if it satisfies the following four axioms of Liu:

Axiom 1

(Normality)

Axiom 2

(Monotonicity) , for any event

Axiom 3

(Countable subadditivity) For every countable sequence of events we have .

##### Definition 2

The uncertainty distribution of an uncertain variable is defined by .

#### Definition 3

Let be an uncertain variable. Then the expected value of is defined by , provided that at least one of the two integrals is finite.

Theorem 1

Let be an uncertain variable with uncertainty distribution . If the expected value exists, then .

Lemma 1

Let \ be a zigzag uncertain variable. Then its inverse uncertainty distribution , and it can be expressed as .

(1)

Theorem 2

Let be independent uncertain variables with
uncertainty distributions , respectively. If f is a
strictly increasing function, then is an uncertain
variable with inverse uncertainty distribution

Theorem 3

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