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July  2018, 14(3): 877-894. doi: 10.3934/jimo.2017080

## Optimal production schedule in a single-supplier multi-manufacturer supply chain involving time delays in both levels

 1 School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa, 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

* Corresponding author: Kar Hung Wong

Received  December 2015 Revised  August 2017 Published  July 2018 Early access  September 2017

This paper considers an optimal production scheduling problem in a single-supplier-multi-manufacturer supply chain involving production and delivery time-delays, where the time-delays for the supplier and the manufacturers can have different values. The objective of both levels is to find an optimal production schedule so that their production rates and their inventory levels are close to the ideal values as much as possible in the whole planning horizon. Each manufacturer's problem, which involves one time-delayed argument, can be solved analytically by using the necessary condition of optimality. To tackle the supplier's problem involving $n+1$ different time-delayed arguments (where $n$ is the number of manufacturers) by the above approach, we need to introduce a model transformation technique which converts the original system of combined algebraic/differential equations with $n+1$ time-delayed arguments into a sum of $n$ sub-systems, each of which consists of only two time-delayed arguments. Thus, the supplier's problem can also be solved analytically. Numerical examples consisting of a single supplier and four manufacturers are solved to provide insight of the optimal strategies of both levels.

Citation: Kar Hung Wong, Yu Chung Eugene Lee, Heung Wing Joseph Lee, Chi Kin Chan. Optimal production schedule in a single-supplier multi-manufacturer supply chain involving time delays in both levels. Journal of Industrial & Management Optimization, 2018, 14 (3) : 877-894. doi: 10.3934/jimo.2017080
##### References:

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##### References:
Optimal Production Rates of the Manufacturers in Example 7.1
Optimal Production Rate of the Supplier in Example 7.1
Optimal Inventory Levels of the Manufacturers in Example 7.1
Optimal Inventory Level of the Supplier in Example 7.1
Optimal Production Rates of the Manufacturers in Example 7.2
Optimal Production Rate of the Supplier in Example 7.2
Optimal Inventory Levels of the Manufacturers in Example 7.2
Optimal Inventory Level of the Supplier in Example 7.2
Optimal Production Rates of the Manufacturers in Example 7.3
Optimal Production Rate of the Supplier in Example 7.3
Optimal Inventory Levels of the Manufacturers in Example 7.3
Optimal Inventory Level of the Supplier in Example 7.3
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