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

• * Corresponding author: Kar Hung Wong
• 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.

Mathematics Subject Classification: Primary: 49M15, 65M60; Secondary: 35Q92.

 Citation: • • Figure 1.  Optimal Production Rates of the Manufacturers in Example 7.1

Figure 2.  Optimal Production Rate of the Supplier in Example 7.1

Figure 3.  Optimal Inventory Levels of the Manufacturers in Example 7.1

Figure 4.  Optimal Inventory Level of the Supplier in Example 7.1

Figure 5.  Optimal Production Rates of the Manufacturers in Example 7.2

Figure 6.  Optimal Production Rate of the Supplier in Example 7.2

Figure 7.  Optimal Inventory Levels of the Manufacturers in Example 7.2

Figure 8.  Optimal Inventory Level of the Supplier in Example 7.2

Figure 9.  Optimal Production Rates of the Manufacturers in Example 7.3

Figure 10.  Optimal Production Rate of the Supplier in Example 7.3

Figure 11.  Optimal Inventory Levels of the Manufacturers in Example 7.3

Figure 12.  Optimal Inventory Level of the Supplier in Example 7.3

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