# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2018050

## Dynamic optimal decision making for manufacturers with limited attention based on sparse dynamic programming

 1 School of Accountancy, Hunan University of Finance and Economics, Changsha 410205, China 2 Business School, Central South University, Changsha 410083, China 3 Department of Mathematics and Statistics, Curtin University, Australia

* Corresponding author: beenjoy@126.com (Wenjie Bi)

Received  June 2017 Revised  January 2018 Published  April 2018

Fund Project: This work is supported by National Natural Science Foundation of China, NO.91646115, 71371191, 71790615 71631008, and Natural Science Foundation of Hunan Province, NO.2018JJ3012

In a fully competitive industry, the market demand is changing rapidly. Thus, it is important for manufacturers to manage their inventory effectively as well as to determine the best order quantity and optimal production strategy. In this paper, our concern is how shall a manufacturer with limited attention determine his optimal order quantity and optimal production strategy in an environment when many factors are volatile, such as the price of raw materials (respectively, finished goods) and attrition rate of inventory of raw materials (respectively, finished product). Under this environment, it is observed, according to various empirical studies, that decision makers tend to focus their attention on factors with major changes. Taking all these into account, our problem is formulated as a discrete-time stochastic dynamic programming. We propose a general approach based on the sparse dynamic programming method to solve this multidimensional dynamic programming problem. From the numerical examples solved using the proposed method, it is interesting to observe that decision makers with limited attention do not adjust their final decision when the volatility is small.

Citation: Haiying Liu, Wenjie Bi, Kok Lay Teo, Naxing Liu. Dynamic optimal decision making for manufacturers with limited attention based on sparse dynamic programming. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018050
##### References:
 [1] A. B. Abel, J. C. Eberly and S. Panageas, Optimal inattention to the stock market with information costs and transactions costs, Econometrica, 81 (2013), 1455-1481. doi: 10.3982/ECTA7624. [2] R. Akella, V. F. Araman and J. Kleinknecht, B2B Markets: Procuremen and Supplier Risk Management in E-Business, in Supply chain management: models, applications, and research directions, Springer, (2005), 33-66. [3] P. Berling and V. Martínez-de-Albéniz, Optimal inventory policies when purchase price and demand are stochastic, Operations Research, 59 (2011), 109-124. doi: 10.1287/opre.1100.0862. [4] P. Berling and K. Rosling, The effects of financial risks on inventory policy, Management Science, 51 (2002), 1804-1815. doi: 10.1287/mnsc.1050.0435. [5] W. Bi, G. Li and M. Liu, Dynamic pricing with stochastic reference effects based on a finite memory window, International Journal of Production Research, 55 (2017), 3331-3348. doi: 10.1080/00207543.2016.1221160. [6] W. Bi, L. Tian, H. LIu and X. Chen, A stochastic dynamic programming approach based on bounded rationality and application to dynamic portfolio choice, Discrete Dynamics in Nature and Society, 2014 (2014), Article ID 840725, 11pages. [7] A. Bouras and L. Tadj, Production planning in a three-stock reverse-logistics system with deteriorating items under a continuous review policy, Journal of Industrial and Management Optimization, 11 (2015), 1041-1058. doi: 10.3934/jimo.2015.11.1041. [8] J.-M. Chen and C.-S. Lin, An optimal replenishment model for inventory items with normally distributed deterioration, Production Planning and Control, 13 (2002), 470-480. [9] S. K. Devalkar, R. Anupindi and A. Sinha, Integrated optimization of procurement, processing, and trade of commodities, Operations Research, 59 (2011), 1369-1381. doi: 10.1287/opre.1110.0959. [10] D. Duffie and T. Sun, Transactions costs and portfolio choice in a discrete-continuous-time setting, Journal of Economic Dynamics and Control, 14 (1990), 35-51. doi: 10.1016/0165-1889(90)90004-Z. [11] Q. Fu, C. Y. Lee and C. P. Teo, Procurement management using option contracts: Random spot price and the portfolio effect, IIE Transactions, 42 (2010), 793-811. [12] X. Gabaix, A sparsity-based model of bounded rationality, Quarterly Journal of Economics, 129 (2014), 1661-1710. doi: 10.3386/w16911. [13] X. Gabaix, Sparse Dynamic Programming and Aggregate Fluctuations, Working Paper, New York University, 2016. [14] V. Gaur and S. Seshadri, Hedging inventory risk through market instruments, Manufacturing and Service Operations Management, 7 (2005), 103-120. doi: 10.1287/msom.1040.0061. [15] S. Gavirneni, Periodic review inventory control with fluctuating purchasing costs, Operations Research Letters, 32 (2004), 374-379. doi: 10.1016/j.orl.2003.11.003. [16] S. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory, European Journal of Operational Research, 134 (2001), 1-16. doi: 10.1016/S0377-2217(00)00248-4. [17] J. Jenkinson, Procurement in action, the efficio grassroots procurement survey 2011, Efficio Consulting, 2011. [18] D. Kahneman, Attention and Effort, Englewood Cliffs, N7T Prentice-Hall, 1973. [19] B. A. Kalymon, Stochastic prices in a single-item inventory purchasing model, Operations Research, 19 (1971), 1434-1458. doi: 10.1287/opre.19.6.1434. [20] M. Lashgari, A. A. Taleizadeh and S. S. Sana, An inventory control problem for deteriorating items with back-ordering and financial considerations under two levels of trade credit linked to order quantity, Journal of Industrial and Management Optimization, 12 (2016), 1091-1119. [21] H. Liu, X. Luo, W. Bi, Y. Man and K. L. Teo, Dynamic pricing of network goods in duopoly markets with boundedly rational consumers, J. Ind. Manag. Optim, 13 (2017), 427-445. [22] B. Mackowiak and M Wiederholt, Information processing and limited liability, The American Economic Review, 102 (2012), 30-34. doi: 10.1257/aer.102.3.30. [23] B. Mackowiak and M. Wiederholt, Inattention to Rare Events, 2015. Available at SSRN 2477548: https://ssrn.com/abstract=2650452. [24] F. Matejka and C. A. Sims, Discrete actions in information -constrained tracking problems, Princeton University Manuscript, (2011). doi: 10.2139/ssrn.1886640. [25] S. Nahmias and W. S. Demmy, Operating characteristics of an inventory system with rationing, Management Science, 27 (1981), 1236-1245. [26] F. Raafat, Survey of literature on continuously deteriorating inventory models, Journal of the Operational Research Society, 42 (1991), 27-37. [27] R. Reis, Inattentive consumers, Journal of Monetary Economics, 53 (2006), 1761-1800. doi: 10.3386/w10883. [28] J. Schwartzstein, Selective attention and learning, Journal of the European Economic Association, 12 (2014), 1423-1452. doi: 10.1111/jeea.12104. [29] K. Sebastian, A. Maessen and S. Strasmann, Mastering the Uniqueness of Commodity Pricing: How to Guide, Set and Control Prices, Simon-Kucher-Whitepaper, 2010. [30] N. H. Shah and Y. Shah, Literature survey on inventory models for deteriorating items, Ekonomski Anali, 44 (2000), 221-237. [31] C. A. Sims, Implications of rational inattention, Journal of Monetary Economics, 50 (2003), 665-690. doi: 10.1016/S0304-3932(03)00029-1. [32] B. Sivakumar, A perishable inventory system with retrial demands and a finite population, Journal of Computational and Applied Mathematics, 224 (2009), 29-38. doi: 10.1016/j.cam.2008.03.041. [33] T. M. Whitin, Inventory control and price theory, Management Science, 2 (1955), 61-68. doi: 10.1287/mnsc.2.1.61. [34] O. Q. Wu and H. Chen, Optimal control and equilibrium behavior of production-inventory systems, Management Science, 56 (2010), 1362-1379. doi: 10.1287/mnsc.1100.1186. [35] J. X. Zhang, Z. Y. Bai and W. S. Tang, Optimal pricing policy for deteriorating items with preservation technology investment, Journal of Industrial and Management Optimization, 10 (2014), 1261-1277. doi: 10.3934/jimo.2014.10.1261. [36] Y.-S. Zheng, Optimal control policy for stochastic inventory systems with Markovian discount opportunities, Operations Research, 42 (1994), 721-738. doi: 10.1287/opre.42.4.721. [37] P. Zipkin, Critical number policies for inventory models with periodic data, Management Science, 35 (1989), 71-80. doi: 10.1287/mnsc.35.1.71.

show all references

##### References:
 [1] A. B. Abel, J. C. Eberly and S. Panageas, Optimal inattention to the stock market with information costs and transactions costs, Econometrica, 81 (2013), 1455-1481. doi: 10.3982/ECTA7624. [2] R. Akella, V. F. Araman and J. Kleinknecht, B2B Markets: Procuremen and Supplier Risk Management in E-Business, in Supply chain management: models, applications, and research directions, Springer, (2005), 33-66. [3] P. Berling and V. Martínez-de-Albéniz, Optimal inventory policies when purchase price and demand are stochastic, Operations Research, 59 (2011), 109-124. doi: 10.1287/opre.1100.0862. [4] P. Berling and K. Rosling, The effects of financial risks on inventory policy, Management Science, 51 (2002), 1804-1815. doi: 10.1287/mnsc.1050.0435. [5] W. Bi, G. Li and M. Liu, Dynamic pricing with stochastic reference effects based on a finite memory window, International Journal of Production Research, 55 (2017), 3331-3348. doi: 10.1080/00207543.2016.1221160. [6] W. Bi, L. Tian, H. LIu and X. Chen, A stochastic dynamic programming approach based on bounded rationality and application to dynamic portfolio choice, Discrete Dynamics in Nature and Society, 2014 (2014), Article ID 840725, 11pages. [7] A. Bouras and L. Tadj, Production planning in a three-stock reverse-logistics system with deteriorating items under a continuous review policy, Journal of Industrial and Management Optimization, 11 (2015), 1041-1058. doi: 10.3934/jimo.2015.11.1041. [8] J.-M. Chen and C.-S. Lin, An optimal replenishment model for inventory items with normally distributed deterioration, Production Planning and Control, 13 (2002), 470-480. [9] S. K. Devalkar, R. Anupindi and A. Sinha, Integrated optimization of procurement, processing, and trade of commodities, Operations Research, 59 (2011), 1369-1381. doi: 10.1287/opre.1110.0959. [10] D. Duffie and T. Sun, Transactions costs and portfolio choice in a discrete-continuous-time setting, Journal of Economic Dynamics and Control, 14 (1990), 35-51. doi: 10.1016/0165-1889(90)90004-Z. [11] Q. Fu, C. Y. Lee and C. P. Teo, Procurement management using option contracts: Random spot price and the portfolio effect, IIE Transactions, 42 (2010), 793-811. [12] X. Gabaix, A sparsity-based model of bounded rationality, Quarterly Journal of Economics, 129 (2014), 1661-1710. doi: 10.3386/w16911. [13] X. Gabaix, Sparse Dynamic Programming and Aggregate Fluctuations, Working Paper, New York University, 2016. [14] V. Gaur and S. Seshadri, Hedging inventory risk through market instruments, Manufacturing and Service Operations Management, 7 (2005), 103-120. doi: 10.1287/msom.1040.0061. [15] S. Gavirneni, Periodic review inventory control with fluctuating purchasing costs, Operations Research Letters, 32 (2004), 374-379. doi: 10.1016/j.orl.2003.11.003. [16] S. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory, European Journal of Operational Research, 134 (2001), 1-16. doi: 10.1016/S0377-2217(00)00248-4. [17] J. Jenkinson, Procurement in action, the efficio grassroots procurement survey 2011, Efficio Consulting, 2011. [18] D. Kahneman, Attention and Effort, Englewood Cliffs, N7T Prentice-Hall, 1973. [19] B. A. Kalymon, Stochastic prices in a single-item inventory purchasing model, Operations Research, 19 (1971), 1434-1458. doi: 10.1287/opre.19.6.1434. [20] M. Lashgari, A. A. Taleizadeh and S. S. Sana, An inventory control problem for deteriorating items with back-ordering and financial considerations under two levels of trade credit linked to order quantity, Journal of Industrial and Management Optimization, 12 (2016), 1091-1119. [21] H. Liu, X. Luo, W. Bi, Y. Man and K. L. Teo, Dynamic pricing of network goods in duopoly markets with boundedly rational consumers, J. Ind. Manag. Optim, 13 (2017), 427-445. [22] B. Mackowiak and M Wiederholt, Information processing and limited liability, The American Economic Review, 102 (2012), 30-34. doi: 10.1257/aer.102.3.30. [23] B. Mackowiak and M. Wiederholt, Inattention to Rare Events, 2015. Available at SSRN 2477548: https://ssrn.com/abstract=2650452. [24] F. Matejka and C. A. Sims, Discrete actions in information -constrained tracking problems, Princeton University Manuscript, (2011). doi: 10.2139/ssrn.1886640. [25] S. Nahmias and W. S. Demmy, Operating characteristics of an inventory system with rationing, Management Science, 27 (1981), 1236-1245. [26] F. Raafat, Survey of literature on continuously deteriorating inventory models, Journal of the Operational Research Society, 42 (1991), 27-37. [27] R. Reis, Inattentive consumers, Journal of Monetary Economics, 53 (2006), 1761-1800. doi: 10.3386/w10883. [28] J. Schwartzstein, Selective attention and learning, Journal of the European Economic Association, 12 (2014), 1423-1452. doi: 10.1111/jeea.12104. [29] K. Sebastian, A. Maessen and S. Strasmann, Mastering the Uniqueness of Commodity Pricing: How to Guide, Set and Control Prices, Simon-Kucher-Whitepaper, 2010. [30] N. H. Shah and Y. Shah, Literature survey on inventory models for deteriorating items, Ekonomski Anali, 44 (2000), 221-237. [31] C. A. Sims, Implications of rational inattention, Journal of Monetary Economics, 50 (2003), 665-690. doi: 10.1016/S0304-3932(03)00029-1. [32] B. Sivakumar, A perishable inventory system with retrial demands and a finite population, Journal of Computational and Applied Mathematics, 224 (2009), 29-38. doi: 10.1016/j.cam.2008.03.041. [33] T. M. Whitin, Inventory control and price theory, Management Science, 2 (1955), 61-68. doi: 10.1287/mnsc.2.1.61. [34] O. Q. Wu and H. Chen, Optimal control and equilibrium behavior of production-inventory systems, Management Science, 56 (2010), 1362-1379. doi: 10.1287/mnsc.1100.1186. [35] J. X. Zhang, Z. Y. Bai and W. S. Tang, Optimal pricing policy for deteriorating items with preservation technology investment, Journal of Industrial and Management Optimization, 10 (2014), 1261-1277. doi: 10.3934/jimo.2014.10.1261. [36] Y.-S. Zheng, Optimal control policy for stochastic inventory systems with Markovian discount opportunities, Operations Research, 42 (1994), 721-738. doi: 10.1287/opre.42.4.721. [37] P. Zipkin, Critical number policies for inventory models with periodic data, Management Science, 35 (1989), 71-80. doi: 10.1287/mnsc.35.1.71.
Production inventory system
Attention function
Truncation function
The influence of ${\hat p_1}$ on $\lambda$
The influence of ${\hat p_2}$ on $\lambda$
The influence of ${\hat \theta _1}$ on $\lambda$
The influence of ${\hat p_1}$ on $q$
The influence of ${\hat p_2}$ on $q$
The influence of ${\hat \theta _1}$ on $q$
The influence of ${\hat \theta _2}$ on $q$
The influence of ${\hat p_1}$ on $s$
The influence of ${\hat p_2}$ on $s$
The influence of ${\hat \theta _2}$ on $s$
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