## 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?

### Mathematical formulation of the model

Formulation of suppliers' individual average profit

Differential equation for the supplier in Figure 2 in [0,T] is given by

(3)

##### Case 2

(19)

where A and B are given in (16) and (17), respectively and

Differential equation for the supplier in Figure 2 in [0,T] is given by

with boundary conditions and . Solving the differential equation with the boundary condition, we have

(3)

**Figure 2**Inventory level of supplier.

(4)

#### Formulation of manufacturer individual average profit

Inventory level of manufacturer in Figure 3 in [0,T] is given by#### Formulation of retailer individual average profit

Inventory level of retailer in Figure 4 in [0,T] is given by**Figure 4**Inventory level of retailer.

(10)

(11)

##### Case 2

(19)

where A and B are given in (16) and (17), respectively and

F=[hmn2+n−2r−22+hr2n+12]TRDR+(n+1)cr1IeDcM22

+(nidm−ids−idr)TR+idsTs−(As+Am+Ar) (20)