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Optimal Ordering Decisions and Revenue Sharing in a Single Period Split Order Supply Chain

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Technology Operation Management

Abstract

This paper investigates the splitting of single period order into two orderings in a two-echelon supply chain in which the retailer revises the second order quantity after observing the demand in the first ordering instance. This paper also proposes a revenue sharing mechanism in which the retailer shares the revenue with the manufacturer in order to share the risk of manufacturer for holding inventory (half of the optimal order quantity) in the first ordering instance and for the units, which are not ordered by the retailer in the second ordering instance. The objective of this paper is to develop a mathematical model with the help of dynamic programming approach, which incorporates the revenue sharing factor in the model, which maximizes the expected profit of the whole supply chain as well as all the supply chain entities. The performance indicators like expected profit of the supply chain and the number of units ordered in the second ordering instance are evaluated. The proposed model is explained by a numerical illustration using Mathematica 7. The results of the proposed model show better performance than a single period model.

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Notes

  1. All costs and prices are in Rs. or monetary units (MU) per unit.

  2. See Appendix 1 for derivation.

  3. See Appendix 2 for derivation.

  4. See Appendix 3 for derivation.

  5. See Appendix 3 for derivation.

  6. See Appendix 4 for derivation.

  7. See Appendix 5 for derivation.

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Authors and Affiliations

  1. Department of Management Studies, Indian Institute of Technology, Madras, Chennai, 36, India

    P. Kalpana & Arshinder Kaur

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  1. P. Kalpana
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  2. Arshinder Kaur
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Correspondence to Arshinder Kaur.

Appendices

Appendices

Appendix 1: Single Ordering Model for Decentralized Supply Chain

$$ \frac{{d\pi_{1dr} }}{{dQ_{d} }} = 0 $$
$$ \begin{aligned} & \Rightarrow \int\limits_{0}^{{Q_{d} }} {(S - H)f(x)dx} + \int\limits_{{Q_{d} }}^{\infty } {(P + G)f(x)dx} - W = 0 \\ & \Rightarrow (S - H)F(Q_{d} ) + (P + G)[1 - F(Q_{d} )] - W = 0 \\ & \Rightarrow F(Q_{d} ) = \left( {\frac{P + G - W}{P + G + H - S}} \right) \\ \end{aligned} $$

Appendix 2: Single Ordering Model for Centralized Supply Chain

$$ \frac{{d\pi_{1c} }}{{dQ_{c} }} = 0 $$
$$ \begin{aligned} & \Rightarrow (S - H)F(Q_{c} ) + (P + G)[1 - F(Q_{c} )] - C = 0 \\ & \Rightarrow F(Q_{c} )(S - H - P - G) + (P + G - C) = 0 \\ & \Rightarrow F(Q_{c} ) = \left( {\frac{P + G - C}{P + G + H - S}} \right) \\ \end{aligned} $$

Appendix 3: Split Ordering Model for Decentralized Supply Chain

Integrate Eq. (14) then we will get

$$ \begin{aligned} EP_{2d} ((Q_{1d} + Q_{2d} - x_{1} )) = & (S - H)[Q_{1d} + Q_{2d} ]F(Q_{1d} + Q_{2d} - x_{1} ) - (S - H)(x_{1} )F(Q_{1d} + Q_{2d} - x_{1} ) \\ & + (P + G)(Q_{1d} + Q_{2d} )[1 - F(Q_{1d} + Q_{2d} - x_{1} ) - (Px_{1} )F(Q_{1d} + Q_{2d} - x_{1} ) \\ & - Gx_{1} [1 - F(Q_{1d} + Q_{2d} - x_{1} )] - WQ_{2d} + (P - S - H)\int\limits_{0}^{{Q_{1d} + Q_{2d} - x_{1} }} {x_{2} f(x_{2} )dx_{2} )} \\ & - G\int\limits_{{Q_{1d} + Q_{2d} - x_{1} }}^{\infty } {x_{2} f(x_{2} )dx_{2} } \\ \end{aligned} $$
$$ \frac{{dEP_{2d} ((Q_{1d} + Q_{2d} - x_{1} ))}}{{dQ_{2d} }} = 0 $$

Apply Leibniz integral rule for taking first order differentiation

$$ \begin{aligned} & \Rightarrow [( - H + S)F(Q_{1d} + Q_{2d} - x_{1} ) + (P + G)(1 - F(Q_{1d} + Q_{2d} - x_{1} ) - W] = 0 \\ & \Rightarrow F(Q_{1d} + Q_{2d} - x_{1} ) = \left( {\frac{P + G - W}{P + G + H - S}} \right) \\ & \Rightarrow Q_{1d} + Q_{2d} - x_{1} = F^{ - 1} \left( {\frac{P + G - W}{P + G + H - S}} \right) \\ \end{aligned} $$
$$ Q_{2d}^{*} = F^{ - 1} \left( {\frac{P + G - W}{P + G + H - S}} \right) - Q_{1d} + x_{1} $$

Appendix 4: Split Ordering Model for Centralized Supply Chain

After integration, Eq. (23) is rewritten as

$$ \begin{aligned} ECP_{2c} ((Q_{1c} + Q_{2c} - x_{1} )) = & (S - H)[Q_{1d} + Q_{2d} ]F(Q_{1d} + Q_{2d} - x_{1} ) - (S - H)(x_{1} )F(Q_{1d} + Q_{2d} - x_{1} ) \\ & + (P + G)(Q_{1d} + Q_{2d} )[1 - F(Q_{1d} + Q_{2d} - x_{1} ) - (Px_{1} )F(Q_{1d} + Q_{2d} - x_{1} ) \\ & - Gx_{1} [1 - F(Q_{1d} + Q_{2d} - x_{1} ) + (P - S - H)\int\limits_{0}^{{Q_{1d} + Q_{2d} - x_{1} }} {x_{2} f(x_{2} )dx_{2} )} \\ & - G\int\limits_{{Q_{1d} + Q_{2d} - x_{1} }}^{\infty } {x_{2} f(x_{2} )dx_{2} } - WQ_{2c} + WQ_{2c} + W^{\prime}(Q_{c}^{*} - (Q_{1c} + Q_{2c} )) \\ \end{aligned} $$

The first order condition of ECP 2c (Q 1c  + Q 2c  − x 1) is given as follows:

$$ \frac{{dECP_{2c} ((Q_{1c} + Q_{2c} - x_{1} ))}}{{dQ_{2c} }} = 0 $$

Apply Leibniz integral rule for taking first order differentiation

$$ \begin{aligned} & \Rightarrow [( - H + S)F(Q_{1c} + Q_{2c} - x_{1} ) + (P + G)(1 - F(Q_{1c} + Q_{2c} - x_{1} ) - W^{\prime}] = 0 \\ & \Rightarrow F(Q_{1c} + Q_{2c} - x_{1} ) = \left( {\frac{{P + G - W^{\prime}}}{P + G + H - S}} \right) \\ & \Rightarrow Q_{1c} + Q_{2c} - x_{1} = F^{ - 1} \left( {\frac{{P + G - W^{\prime}}}{P + G + H - S}} \right) \\ \end{aligned} $$
$$ Q_{2c}^{*} = F^{ - 1} \left( {\frac{{P + G - W^{\prime}}}{P + G + H - S}} \right) - Q_{1c} + x_{1} $$

The second order condition of the function π 2c is given as follows:

$$ \frac{{d^{2} ECP_{2c} (Q_{1c} + Q_{2d} - x_{1} )}}{{d^{2} Q_{2c} }} = [( - H + S)f(Q_{2c} ) - (P + G)f(Q_{2c} )]( - H + S - P - G)f(Q_{2c} ) < 0 $$

So the objective function is concave function [f(Q 2c ) > 0 and (−H + S − P − G) < 0 since P > S > H&G]

Appendix 5: Revenue Sharing Model of Centralized Supply Chain with Split Orders

After integration, Eq. (31) is rewritten as

$$ \begin{aligned} EP_{2c\alpha } ((Q_{1c} + Q_{2c} - x_{1} )) = & (S - H)[Q_{1d} + Q_{2d} ]F(Q_{1d} + Q_{2d} - x_{1} ) - (S - H)(x_{1} )F(Q_{1d} + Q_{2d} - x_{1} ) \\ & + (P + G)(Q_{1d} + Q_{2d} )[1 - F(Q_{1d} + Q_{2d} - x_{1} ) - (Px_{1} )F(Q_{1d} + Q_{2d} - x_{1} ) \\ & - Gx_{1} [1 - F(Q_{1d} + Q_{2d} - x_{1} ) + (P - S - H)\int\limits_{0}^{{Q_{1d} + Q_{2d} - x_{1} }} {x_{2} f(x_{2} )dx_{2} )} \\ & - G\int\limits_{{Q_{1d} + Q_{2d} - x_{1} }}^{\infty } {x_{2} f(x_{2} )dx_{2} } - WQ_{2c} + WQ_{2c} + W^{\prime}(Q_{c}^{*} - (Q_{1c} + Q_{2c} )) \\ \end{aligned} $$

The revenue sharing factor α gets cancelled in the second period channel profit after revenue sharing is in the above equation.

The first order condition of EP 2cα (Q 1c  + Q 2c  − x 1) is given as follows:

$$ \frac{{dEP_{2c\alpha } ((Q_{1c} + Q_{2c} - x_{1} ))}}{{dQ_{2c} }} = 0 $$

Apply Leibniz integral rule for taking first order differentiation

$$ \begin{aligned} & \Rightarrow [( - H + S)F(Q_{1c} + Q_{2c} - x_{1} ) + (P + G)(1 - F(Q_{1c} + Q_{2c} - x_{1} ) - W^{\prime}] = 0 \\ & \Rightarrow F(Q_{1c} + Q_{2c} - x_{1} ) = \left( {\frac{{P + G - W^{\prime}}}{P + G + H - S}} \right) \\ & \Rightarrow Q_{1c} + Q_{2c} - x_{1} = F^{ - 1} \left( {\frac{{P + G - W^{\prime}}}{P + G + H - S}} \right) \\ \end{aligned} $$
$$ Q_{2c}^{*} = F^{ - 1} \left( {\frac{{P + G - W^{\prime}}}{P + G + H - S}} \right) - Q_{1c} + x_{1} $$

The second order condition of the function π 2c is given as follows:

$$ \frac{{d^{2} EP_{2c\alpha } (Q_{1c} + Q_{2d} - x_{1} )}}{{d^{2} Q_{2c} }} = [( - H + S)f(Q_{2c} ) - (P + G)f(Q_{2c} )]( - H + S - P - G)f(Q_{2c} ) < 0 $$

So the objective function is concave function [f(Q 2c ) > 0 and (−H + S − P − G) < 0 since P > S > H&G].

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Kalpana, P., Kaur, A. Optimal Ordering Decisions and Revenue Sharing in a Single Period Split Order Supply Chain. Technol. Oper. Manag 2, 61–79 (2011). https://doi.org/10.1007/s13727-012-0007-9

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