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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 and Management Optimization, 2018, 14 (3) : 877-894. doi: 10.3934/jimo.2017080
##### References:
 [1] S. Axsater, Control theory concepts in production and inventory control, International Journal of Systems Science, 16 (1985), 161-169.  doi: 10.1080/00207728508926662. [2] A. Bradshaw and D. Daintith, Synthesis of control policies for cascaded production inventory systems, International Journal of Systems Science, 7 (1976), 1053-1070. [3] C. K. Chan, H. W. J. Lee and K. H. Wong, Optimal feedback production for a two-Level supply chain, International Journal of Production Economics, 113 (2008), 619-625.  doi: 10.1016/j.ijpe.2007.12.012. [4] Y. H. Dai and K. Schittkowski, A sequential quadratic programming algorithm with non-monotone line search, Pacific Journal of Optimization, 4 (2008), 335-351. [5] J. Dejonckheere, S. M. Disney, M. R. Lambrecht and D. R. Towill, Measuring and avoiding the bullwhip effect: A control theoretic approach, European Journal of Operational Research, 147 (2003), 567-590.  doi: 10.1016/S0377-2217(02)00369-7. [6] S. M. Disney and D. R. Towill, On the bullwhip and inventory variance produced by an ordering policy, Omega, 31 (2003), 157-167.  doi: 10.1016/S0305-0483(03)00028-8. [7] J. W. Forrester, Industrial Dynamics, Cambridge MA: MIT Press, 1961. [8] K. Kaji and K. H. Wong, Nonlinearly constrained time-delayed optimal control problems, Journal of Optimization Theory and Applications, 82 (1994), 295-313.  doi: 10.1007/BF02191855. [9] K. Kogan, Y. Y. Leu and J. R. Perkins, Parallel-machine, multiple-product-type, continuous-time Scheduling: Decomposable cases, IEE Transactions, 34 (2002), 11-22.  doi: 10.1080/07408170208928846. [10] M. Miranbeigi, A. Jalali and A. Miranbeigi, A constrained inventory level optimal control on supply chain management System, International Journal of Innovation, 1 (2010), 69-74. [11] M. Miranbeigi, B. Moshiri, A. Rahimi-Kian and J. Razmi, Demand satisfaction in supply chain management System using a full online optimal control method, International Journal of Advanced Manufacturing Technology, 77 (2015), 1401-1417.  doi: 10.1007/s00170-014-6513-0. [12] B. Porter and A. Bradshaw, Modal control of production inventory systems using piecewise constant control policies, International Journal of Systems Science, 7 (1976), 1053-1070. [13] B. Porter and F. Taylor, Modal control of production inventory systems, International Journal of Systems Science, 3 (1972), 325-331. [14] C. E. Riddalls and S. Bennett, Modelling the dynamics of supply chains, International Journal of Systems Science, 31 (2000), 969-976.  doi: 10.1080/002077200412122. [15] C. E. Riddalls and S. Bennett, The stability of supply chains, International Journal of Production Research, 40 (2000), 459-475.  doi: 10.1080/00207540110085629. [16] K. Schittkowski, A robust implementation of a sequential quadratic programming algorithm with successive error restoration, Optimization Letters, 5 (2011), 283-296.  doi: 10.1007/s11590-010-0207-9. [17] H. A. Simon, On the application of servomechanism theory in the study of production control, Econometrica, 20 (1952), 247-268.  doi: 10.2307/1907849. [18] K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, England Longman Scientific and Technical, 1991. [19] K. L. Teo, K. H. Wong and D. J. Clements, Optimal control computation for linear time-lag systems with linear terminal constraints, Journal of Optimization Theory and Applications, 44 (1984), 509-526.  doi: 10.1007/BF00935465. [20] K. H. Wong, C. K. Chan and H. W. J. Lee, Optimal feedback production for a single-echelon supply chain, Journal of Discrete and Continuous Dynamical Systems, Series B, 6 (2006), 1431-1444.  doi: 10.3934/dcdsb.2006.6.1431. [21] K. H. Wong, D. J. Clements and K. L. Teo, Optimal control computation for nonlinear systems, Journal of Optimization Theory and Applications, 47 (1985), 91-107.  doi: 10.1007/BF00941318. [22] K. H. Wong, L. S. Jennings and F. Benyah, Control parametrization method for free planning time optimal control problems with time-delayed arguments, Journal of Nonlinear Analysis, Series A, 47 (2001), 5679-5689.  doi: 10.1016/S0362-546X(01)00669-1. [23] F. Yang, K. L. Teo, R. Loxton, V. Rehbock, L. Bin, Y. Changjun and L. S. Jennings, Visual Miser: an efficient user-friendly visual problem for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781. [24] C. Yu, Q. Lin, R. Loxton, K. L. Teo and G. Wang, A hybrid time-scaling transformation for time-delay optimal control problems, Journal of Optimization Theory and Applications, 169 (2016), 876-901.  doi: 10.1007/s10957-015-0783-z. [25] H. Zaher and T. T. Zaki, Optimal control theory to solve production inventory system in supply chain management, Journal of Mathematics Research, 6 (2014), 109-117.  doi: 10.5539/jmr.v6n4p109.

show all references

##### References:
 [1] S. Axsater, Control theory concepts in production and inventory control, International Journal of Systems Science, 16 (1985), 161-169.  doi: 10.1080/00207728508926662. [2] A. Bradshaw and D. Daintith, Synthesis of control policies for cascaded production inventory systems, International Journal of Systems Science, 7 (1976), 1053-1070. [3] C. K. Chan, H. W. J. Lee and K. H. Wong, Optimal feedback production for a two-Level supply chain, International Journal of Production Economics, 113 (2008), 619-625.  doi: 10.1016/j.ijpe.2007.12.012. [4] Y. H. Dai and K. Schittkowski, A sequential quadratic programming algorithm with non-monotone line search, Pacific Journal of Optimization, 4 (2008), 335-351. [5] J. Dejonckheere, S. M. Disney, M. R. Lambrecht and D. R. Towill, Measuring and avoiding the bullwhip effect: A control theoretic approach, European Journal of Operational Research, 147 (2003), 567-590.  doi: 10.1016/S0377-2217(02)00369-7. [6] S. M. Disney and D. R. Towill, On the bullwhip and inventory variance produced by an ordering policy, Omega, 31 (2003), 157-167.  doi: 10.1016/S0305-0483(03)00028-8. [7] J. W. Forrester, Industrial Dynamics, Cambridge MA: MIT Press, 1961. [8] K. Kaji and K. H. Wong, Nonlinearly constrained time-delayed optimal control problems, Journal of Optimization Theory and Applications, 82 (1994), 295-313.  doi: 10.1007/BF02191855. [9] K. Kogan, Y. Y. Leu and J. R. Perkins, Parallel-machine, multiple-product-type, continuous-time Scheduling: Decomposable cases, IEE Transactions, 34 (2002), 11-22.  doi: 10.1080/07408170208928846. [10] M. Miranbeigi, A. Jalali and A. Miranbeigi, A constrained inventory level optimal control on supply chain management System, International Journal of Innovation, 1 (2010), 69-74. [11] M. Miranbeigi, B. Moshiri, A. Rahimi-Kian and J. Razmi, Demand satisfaction in supply chain management System using a full online optimal control method, International Journal of Advanced Manufacturing Technology, 77 (2015), 1401-1417.  doi: 10.1007/s00170-014-6513-0. [12] B. Porter and A. Bradshaw, Modal control of production inventory systems using piecewise constant control policies, International Journal of Systems Science, 7 (1976), 1053-1070. [13] B. Porter and F. Taylor, Modal control of production inventory systems, International Journal of Systems Science, 3 (1972), 325-331. [14] C. E. Riddalls and S. Bennett, Modelling the dynamics of supply chains, International Journal of Systems Science, 31 (2000), 969-976.  doi: 10.1080/002077200412122. [15] C. E. Riddalls and S. Bennett, The stability of supply chains, International Journal of Production Research, 40 (2000), 459-475.  doi: 10.1080/00207540110085629. [16] K. Schittkowski, A robust implementation of a sequential quadratic programming algorithm with successive error restoration, Optimization Letters, 5 (2011), 283-296.  doi: 10.1007/s11590-010-0207-9. [17] H. A. Simon, On the application of servomechanism theory in the study of production control, Econometrica, 20 (1952), 247-268.  doi: 10.2307/1907849. [18] K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, England Longman Scientific and Technical, 1991. [19] K. L. Teo, K. H. Wong and D. J. Clements, Optimal control computation for linear time-lag systems with linear terminal constraints, Journal of Optimization Theory and Applications, 44 (1984), 509-526.  doi: 10.1007/BF00935465. [20] K. H. Wong, C. K. Chan and H. W. J. Lee, Optimal feedback production for a single-echelon supply chain, Journal of Discrete and Continuous Dynamical Systems, Series B, 6 (2006), 1431-1444.  doi: 10.3934/dcdsb.2006.6.1431. [21] K. H. Wong, D. J. Clements and K. L. Teo, Optimal control computation for nonlinear systems, Journal of Optimization Theory and Applications, 47 (1985), 91-107.  doi: 10.1007/BF00941318. [22] K. H. Wong, L. S. Jennings and F. Benyah, Control parametrization method for free planning time optimal control problems with time-delayed arguments, Journal of Nonlinear Analysis, Series A, 47 (2001), 5679-5689.  doi: 10.1016/S0362-546X(01)00669-1. [23] F. Yang, K. L. Teo, R. Loxton, V. Rehbock, L. Bin, Y. Changjun and L. S. Jennings, Visual Miser: an efficient user-friendly visual problem for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781. [24] C. Yu, Q. Lin, R. Loxton, K. L. Teo and G. Wang, A hybrid time-scaling transformation for time-delay optimal control problems, Journal of Optimization Theory and Applications, 169 (2016), 876-901.  doi: 10.1007/s10957-015-0783-z. [25] H. Zaher and T. T. Zaki, Optimal control theory to solve production inventory system in supply chain management, Journal of Mathematics Research, 6 (2014), 109-117.  doi: 10.5539/jmr.v6n4p109.
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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