## 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?

### Assumptions and notations

#### Assumptions

The following assumptions are considered to develop the model:

(i) Models are developed for single item product.

(ii) Lead time is negligible.

(iii) Joint effect of supplier, manufacturer, retailer is consider in a supply chain management.

(iv) Supplier produced the item with constant rate unit per unit time, which is a decision variable.

(v) Total production rate of manufacturer is equal to the demand rate of manufacturer.

(vi) The manufacturer give the opportunity to the retailer conditionally

permissible delay in payment.

(vii) Idle cost of suppliers, manufacturer and retailer are also assumed.

#### Notations

The following notations are considered to develop the model:

• $p_s$ = production rate for the suppliers, which is a decision variable.
• $p_m$ = demand rate or production rate for the manufacturer.
• $D_r$ = constant demand rate for the retailer.
• $D_c$ = constant demand rate of customer.
• $C_s$ = purchase cost of unit item for suppliers.
• $C_m$ = selling price of unit item for suppliers which is also the purchase cost for manufacturer.
• $C_r$ = selling price of unit item for manufacturer which is also the purchase cost for retailers.
• $C_{r1}$ = selling price for retailers.
• $t_s$ = production time for supplers.
• $T_s$ = cycle length for the suppliers.
• $T_r$ = time duration where order is supplied by the manufacturer, by retailer's cycle length.
• $T′$ = last cycle length of the retailer.
• $T$ = total time for the integrated model.
• $h_s$ = holding cost per unit per unit time for suppliers.
• $h_m$ = holding cost per unit per unit time for manufacturer.
• $h_r$ = holding cost per unit per unit time for retailers.
• $A_s$ = ordering cost for suppliers.
• $A_m$ = ordering cost for manufacturer.
• $h_r$ = ordering cost for retailers.
• $id_s , \widetilde {id}_s$ = idle cost per unit time for suppliers in crisp and uncertain environments, respectively.
• $id_m , \widetilde {id} _m$ = idle cost per unit time for manufacturer in crisp and uncertain environments, respectively.
• $id_r ,\widetilde {id}_r$ = idle cost per unit time for retailers in crisp and uncertain environments, respectively.
• $n$ = number of cycle for retailers.
• $r$ = number of cycles where manufacturer stops production.
• $M$ = retailer's trade credit period offered by the manufacturer to the retailers in years, which is the fraction of the years.
• $I_p$ = interest payable to the manufacturer by the retailers.
• $I_e, \widetilde {I}_e$ = interest earned by the retailers in crisp and uncertain environments, respectively.
• $ATP$ = average total profit for the integrated models.
• $p^{*}_{m}$ = optimum value of P m for integrated models.
• $ATP^∗$ = optimum value of average total profit for the integrated models.