Cooperative/Non-cooperative Supply Chain Models for Imperfect Quality Items with Trade Credit Financing

Abstract

This paper studies the cooperative and non-cooperative models between the two partners of the supply chain system, seller and buyer. In this paper, supply chain models are formulated for imperfect quality items in which end demand of the product depends upon the retail price. The fixed credit period is offered by the seller to the buyer to stimulate his sales. The inspection process is also applied to each supplied lot at buyer’s end, and all the inspected items are separated into perfect quality items and imperfect quality items. Once the inspection process completed, perfect quality items are sold at selling price and imperfect quality items are sold at rebated/discounted price. The selling price and credit period proposed by the seller are considered as decision variables. The lot size and retailer price are decision variables of the buyer. In the proposed model, optimal policies of the seller’s and buyer’s are obtained under cooperative and non-cooperative analogue which will enhance the supply chain profit. Cooperative relationship is derived by a Pareto-efficient solution method, and non-cooperative is obtained by Seller-Stackelberg approach. Finally, numerical illustrations with sensibility analysis are stated to exemplify the theory of the paper.

Appendix 1

Expected profit function for the buyer is given by

$E\left[ {TP^{c}_{b} \left( {p,Q} \right)} \right] = Kp_{b}^{ - e} \left[ {p_{b} - A_{1} Q - A_{2} - \frac{{A_{3} }}{Q}} \right] - c_{b} A_{4} Q$, where $A_{1} ,A_{2} ,A_{3}$ and $A_{4}$ are defined by Eq. (4).

First and second derivative with respect to $p_{b} ,Q$ are given below

$$ \begin{aligned} \frac{{\partial E\left[ {TP^{c}_{b} \left( {p,Q} \right)} \right]}}{{\partial p_{b} }} & = kp_{b}^{ - e} - kep_{b}^{ - e - 1} \left( {p_{b} - A_{1} Q - A_{2} - \frac{{A_{3} }}{Q}} \right) \\ & = \left( {1 - e} \right)kp_{b}^{ - e} + kep_{b}^{ - e - 1} (A_{1} Q + A_{2} + \frac{{A_{3} }}{Q}) \\ \end{aligned} $$

$$ \frac{{\partial^{2} E\left[ {TP^{c}_{b} \left( {p,Q} \right)} \right]}}{{\partial p_{b}^{2} }} = e\left( {e - 1} \right)kp_{b}^{ - e - 1} - ke\left( {e + 1} \right)p_{b}^{ - e - 2} \left( {A_{1} Q + A_{2} + \frac{{A_{3} }}{Q}} \right) $$

$$ \frac{{\partial E\left[ {TP^{c}_{b} \left( {p,Q} \right)} \right]}}{\partial Q} = kp_{b}^{ - e} \left( { - A_{1} + \frac{{A_{3} }}{{Q^{2} }}} \right) - c_{b} A_{4} $$

$$ \frac{{\partial^{2} E\left[ {TP^{c}_{b} \left( {p,Q} \right)} \right]}}{{\partial Q^{2} }} = kp_{b}^{ - e} \left( { - \frac{{2A_{3} }}{{Q^{3} }}} \right) $$

$$ \frac{{\partial^{2} E\left[ {TP^{c}_{b} \left( {p,Q} \right)} \right]}}{{\partial p_{b} \partial Q}} = ekp_{b}^{ - e - 1} \left( {A_{1} - \frac{{A_{3} }}{{Q^{2} }}} \right) $$

where $A_{1} ,A_{2} , A_{3}$ and $A_{4}$ are defined by Eq. 4.