## 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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## Appendices

### Appendices

### Appendix 1: Single Ordering Model for Decentralized Supply Chain

### Appendix 2: Single Ordering Model for Centralized Supply Chain

### Appendix 3: Split Ordering Model for Decentralized Supply Chain

Integrate Eq. (14) then we will get

Apply Leibniz integral rule for taking first order differentiation

### Appendix 4: Split Ordering Model for Centralized Supply Chain

After integration, Eq. (23) is rewritten as

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

Apply Leibniz integral rule for taking first order differentiation

The second order condition of the function *π*
_{2c
} is given as follows:

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

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:

Apply Leibniz integral rule for taking first order differentiation

The second order condition of the function *π*
_{2c
} is given as follows:

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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### Cite this article

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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DOI: https://doi.org/10.1007/s13727-012-0007-9

### Keywords

- Dynamic programming
- Revenue sharing
- Single period inventory models
- Supply chain coordination
- Two ordering opportunity