American Institute of Mathematical Sciences

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April  2019, 15(2): 445-464. doi: 10.3934/jimo.2018050

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

Haiying Liu 1,2, , Wenjie Bi 2,, , Kok Lay Teo 3, and  Naxing Liu 1,

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.

Keywords: Dynamic optimization, limited attention, inventory management, sparse dynamic programming.
Mathematics Subject Classification: Primary: 1201; Secondary: 120100.
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, 2019, 15 (2) : 445-464. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

X. Gabaix, A sparsity-based model of bounded rationality, Quarterly Journal of Economics, 129 (2014), 1661-1710. doi: 10.3386/w16911. Google Scholar

[13]

X. Gabaix, Sparse Dynamic Programming and Aggregate Fluctuations, Working Paper, New York University, 2016.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

J. Jenkinson, Procurement in action, the efficio grassroots procurement survey 2011, Efficio Consulting, 2011.Google Scholar

[18]

D. Kahneman, Attention and Effort, Englewood Cliffs, N7T Prentice-Hall, 1973.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

B. Mackowiak and M. Wiederholt, Inattention to Rare Events, 2015. Available at SSRN 2477548: https://ssrn.com/abstract=2650452.Google Scholar

[24]

F. Matejka and C. A. Sims, Discrete actions in information -constrained tracking problems, Princeton University Manuscript, (2011). doi: 10.2139/ssrn.1886640. Google Scholar

[25]

S. Nahmias and W. S. Demmy, Operating characteristics of an inventory system with rationing, Management Science, 27 (1981), 1236-1245. Google Scholar

[26]

F. Raafat, Survey of literature on continuously deteriorating inventory models, Journal of the Operational Research Society, 42 (1991), 27-37. Google Scholar

[27]

R. Reis, Inattentive consumers, Journal of Monetary Economics, 53 (2006), 1761-1800. doi: 10.3386/w10883. Google Scholar

[28]

J. Schwartzstein, Selective attention and learning, Journal of the European Economic Association, 12 (2014), 1423-1452. doi: 10.1111/jeea.12104. Google Scholar

[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.Google Scholar

[30]

N. H. Shah and Y. Shah, Literature survey on inventory models for deteriorating items, Ekonomski Anali, 44 (2000), 221-237. Google Scholar

[31]

C. A. Sims, Implications of rational inattention, Journal of Monetary Economics, 50 (2003), 665-690. doi: 10.1016/S0304-3932(03)00029-1. Google Scholar

[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. Google Scholar

[33]

T. M. Whitin, Inventory control and price theory, Management Science, 2 (1955), 61-68. doi: 10.1287/mnsc.2.1.61. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

X. Gabaix, A sparsity-based model of bounded rationality, Quarterly Journal of Economics, 129 (2014), 1661-1710. doi: 10.3386/w16911. Google Scholar

[13]

X. Gabaix, Sparse Dynamic Programming and Aggregate Fluctuations, Working Paper, New York University, 2016.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

J. Jenkinson, Procurement in action, the efficio grassroots procurement survey 2011, Efficio Consulting, 2011.Google Scholar

[18]

D. Kahneman, Attention and Effort, Englewood Cliffs, N7T Prentice-Hall, 1973.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

B. Mackowiak and M. Wiederholt, Inattention to Rare Events, 2015. Available at SSRN 2477548: https://ssrn.com/abstract=2650452.Google Scholar

[24]

F. Matejka and C. A. Sims, Discrete actions in information -constrained tracking problems, Princeton University Manuscript, (2011). doi: 10.2139/ssrn.1886640. Google Scholar

[25]

S. Nahmias and W. S. Demmy, Operating characteristics of an inventory system with rationing, Management Science, 27 (1981), 1236-1245. Google Scholar

[26]

F. Raafat, Survey of literature on continuously deteriorating inventory models, Journal of the Operational Research Society, 42 (1991), 27-37. Google Scholar

[27]

R. Reis, Inattentive consumers, Journal of Monetary Economics, 53 (2006), 1761-1800. doi: 10.3386/w10883. Google Scholar

[28]

J. Schwartzstein, Selective attention and learning, Journal of the European Economic Association, 12 (2014), 1423-1452. doi: 10.1111/jeea.12104. Google Scholar

[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.Google Scholar

[30]

N. H. Shah and Y. Shah, Literature survey on inventory models for deteriorating items, Ekonomski Anali, 44 (2000), 221-237. Google Scholar

[31]

C. A. Sims, Implications of rational inattention, Journal of Monetary Economics, 50 (2003), 665-690. doi: 10.1016/S0304-3932(03)00029-1. Google Scholar

[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. Google Scholar

[33]

T. M. Whitin, Inventory control and price theory, Management Science, 2 (1955), 61-68. doi: 10.1287/mnsc.2.1.61. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

Figure 1.  Production inventory system
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Figure 2.  Attention function
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Figure 3.  Truncation function
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Figure 4.  The influence of ${\hat p_1}$ on $\lambda$
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Figure 5.  The influence of ${\hat p_2}$ on $\lambda$
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Figure 6.  The influence of ${\hat \theta _1}$ on $\lambda$
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Figure 7.  The influence of ${\hat p_1}$ on $q$
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Figure 8.  The influence of ${\hat p_2}$ on $q$
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Figure 9.  The influence of ${\hat \theta _1}$ on $q$
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Figure 10.  The influence of ${\hat \theta _2}$ on $q$
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Figure 11.  The influence of ${\hat p_1}$ on $s$
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Figure 12.  The influence of ${\hat p_2}$ on $s$
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Figure 13.  The influence of ${\hat \theta _2}$ on $s$
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