American Institute of Mathematical Sciences

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June  2020, 13(6): 1653-1682. doi: 10.3934/dcdss.2020097

A multi-stage method for joint pricing and inventory model with promotion constrains

Li Deng 1,2,3, , Wenjie Bi 1, , Haiying Liu 4,, and  Kok Lay Teo 5,6,

1. 

Business School, Central South University, Changsha 410083, China
2. 

Department of Mathematics and Statistics, Curtin University, Perth, 6102, Australia
3. 

School of Economics and Management, Hunan University of Science and Engineering, Yongzhou 425199, China
4. 

School of Accountancy, Hunan University of Finance and Economics, Changsha 410205, China
5. 

Department of Mathematics and Statistics, Curtin University, Perth, 6102, Australia
6. 

Coordinated Innovation Center for Computable Modeling in Management Science, University of Finance and Economics, Tianjin 300222, China

* Corresponding author: Haiying Liu

Received  February 2018 Revised  August 2018 Published  September 2019

In this paper, we consider a joint pricing and inventory problem with promotion constrains over a finite planning horizon for a single fast-moving consumer good under monopolistic environment. The decision on the inventory is realized through the decision on inventory replenishment, i.e., decision on the quantity to be ordered. The demand function takes into account all reference price mechanisms. The main difficulty in solving this problem is how to deal with the binary logical decision variables. It is shown that the problem is equivalent to a quadratic programming problem involving binary decision variables. This quadratic programming problem with binary decision variables can be expressed as a series of conventional quadratic programming problems, each of which can be easily solved. The global optimal solution can then be obtained readily from the global solutions of the conventional quadratic programming problems. This method works well when the planning horizon is short. For longer planning horizon, we propose a multi-stage method for finding a near-optimal solution. In numerical simulation, the accuracy and efficiency of this multi-stage method is compared with a genetic algorithm. The results obtained validate the applicability of the constructed model and the effectiveness of the approach proposed. They also provide several interesting and useful managerial insights.

Keywords: Pricing and Inventory, promotion optimization, promotion constrains, binary decision variables, quadratic programming, multi-stage method.
Mathematics Subject Classification: Primary: 90B50, 90B05.
Citation: Li Deng, Wenjie Bi, Haiying Liu, Kok Lay Teo. A multi-stage method for joint pricing and inventory model with promotion constrains. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1653-1682. doi: 10.3934/dcdss.2020097
References:
[1]

H.-s. Ahn, M. Gümüş and P. Kaminsky, Pricing and manufacturing decisions when demand is a function of prices in multiple periods, Operations Research, 55 (2007), 1039-1057.  doi: 10.1287/opre.1070.0411.  Google Scholar

[2]

H. Arslan and S. Kachani, Dynamic Pricing under Consumer Reference-Price Effects, Wiley Encyclopedia of Operations Research and Management Science, 2010. doi: 10.1002/9780470400531.eorms0273.  Google Scholar

[3]

M. Baucells, M. Weber and F. Welfens, Reference-point formation and updating, Management Science, 57 (2011), 506-519.  doi: 10.1287/mnsc.1100.1286.  Google Scholar

[4]

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

[5]

G. R. Bitran and S. V. Mondschein, Periodic pricing of seasonal products in retailing, Management Science, 43 (1997), 64-79.  doi: 10.1287/mnsc.43.1.64.  Google Scholar

[6]

M. Chen and Z.-L. Chen, Recent developments in dynamic pricing research: Multiple products, competition, and limited demand information, Production and Operations Management, 24 (2015), 704-731.  doi: 10.1111/poms.12295.  Google Scholar

[7]

M. Chen, Z.-L. Chen, G. Pundoor, S. Acharya and J. Yi, Markdown optimization at multiple stores, IIE Transactions, 47 (2015), 84-108.  doi: 10.1080/0740817X.2014.916459.  Google Scholar

[8]

X. Chen, P. Hu and Z. Hu, Efficient algorithms for the dynamic pricing problem with reference price effect, Management Science, 63 (2016), 4389-4408.  doi: 10.1287/mnsc.2016.2554.  Google Scholar

[9]

X. Chen, P. Hu, S. Shum and Y. Zhang, Dynamic stochastic inventory management with reference price effects, Operations Research, 64 (2016), 1529-1536.  doi: 10.1287/opre.2016.1524.  Google Scholar

[10]

X. Chen, S. W. Shum, P. Hu and Y. Zhang, Stochastic inventory model with reference price effects, Operations Research, (2013). Google Scholar

[11]

X. Chen and D. Simchi-Levi, Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: The finite horizon case, Operations Research, 52 (2004), 887-896.  doi: 10.1287/opre.1040.0127.  Google Scholar

[12]

X. Chen and D. Simchi-Levi, Pricing and inventory management, The Oxford Handbook of Pricing Management, (2012), 784Ƀ824. doi: 10.1093/oxfordhb/9780199543175.013.0030.  Google Scholar

[13]

M. C. Cohen, N.-H. Z. Leung, K. Panchamgam, G. Perakis and A. Smith, The impact of linear optimization on promotion planning, Operations Research, 65 (2017), 446-468.  doi: 10.1287/opre.2016.1573.  Google Scholar

[14]

P. R. Dickson and A. G. Sawyer, The price knowledge and search of supermarket shoppers, The Journal of Marketing, (1990), 42-53. Google Scholar

[15]

W. Elmaghraby and P. Keskinocak, Dynamic pricing in the presence of inventory considerations: Research overview, current practices, and future directions, Management Science, 49 (2003), 1287-1309.   Google Scholar

[16]

G. Fibich, A. Gavious and O. Lowengart, Explicit solutions of optimization models and differential games with nonsmooth (asymmetric) reference-price effects, Operations Research, 51 (2003), 721-734.  doi: 10.1287/opre.51.5.721.16758.  Google Scholar

[17]

K. Gedenk, S. A. Neslin and K. L. Ailawadi, Sales promotion, Retailing in the 21st Century, (2006), 345-359. Google Scholar

[18]

L. Gimpl-Heersink, Joint Pricing and Inventory Control under Reference Price Effects, PhD thesis, WU Vienna University of Economics and Business, 2008. doi: 10.3726/b13901.  Google Scholar

[19]

E. A. Greenleaf, The impact of reference price effects on the profitability of price promotions, Marketing Science, 14 (1995), 82-104.  doi: 10.1287/mksc.14.1.82.  Google Scholar

[20]

M. G. Güler, The value of modeling with reference effects in stochastic inventory and pricing problems, Expert Systems with Applications, 40 (2013), 6593-6600.   Google Scholar

[21]

M. G. Güler, T. Bilgiç and R. Güllü, Joint inventory and pricing decisions with reference effects, IIE Transactions, 46 (2014), 330-343.   Google Scholar

[22]

M. G. Güler, T. Bilgiç and R. Güllü, Joint pricing and inventory control for additive demand models with reference effects, Annals of Operations Research, 226 (2015), 255-276.  doi: 10.1007/s10479-014-1706-3.  Google Scholar

[23]

R. L. Hall and C. J. Hitch, Price theory and business behaviour, Oxford Economic Papers, 2 (1939), 12-45.  doi: 10.1093/oxepap/os-2.1.12.  Google Scholar

[24]

P. K. Kopalle, A. G. Rao and J. L. Assuncao, Asymmetric reference price effects and dynamic pricing policies, Marketing Science, 15 (1996), 60-85.  doi: 10.1287/mksc.15.1.60.  Google Scholar

[25]

D. Levy, M. Bergen, S. Dutta and R. Venable, The magnitude of menu costs: direct evidence from large US supermarket chains, The Quarterly Journal of Economics, 112 (1997), 791-824.  doi: 10.1162/003355397555352.  Google Scholar

[26]

D. Levy, S. Dutta, M. Bergen and R. Venable, Price adjustment at multiproduct retailers, Managerial and Decision Economics, 19 (1998), 81-120.  doi: 10.1002/(SICI)1099-1468(199803)19:2<81::AID-MDE867>3.0.CO;2-W.  Google Scholar

[27]

T. Mazumdar, S. Raj and I. Sinha, Reference price research: Review and propositions, Journal of Marketing, 69 (2005), 84-102.  doi: 10.1509/jmkg.2005.69.4.84.  Google Scholar

[28]

D. C. Montgomery, M. Bazaraa and A. K. Keswani, Inventory models with a mixture of backorders and lost sales, Naval Research Logistics (NRL), 20 (1973), 255-263.  doi: 10.1002/nav.3800200205.  Google Scholar

[29]

J. Nasiry and I. Popescu, Dynamic pricing with loss-averse consumers and peak-end anchoring, Operations Research, 59 (2011), 1361-1368.  doi: 10.1287/opre.1110.0952.  Google Scholar

[30]

S. Netessine, Dynamic pricing of inventory/capacity with infrequent price changes, European Journal of Operational Research, 174 (2006), 553-580.  doi: 10.1016/j.ejor.2004.12.015.  Google Scholar

[31]

I. Popescu and Y. Wu, Dynamic pricing strategies with reference effects, Operations Research, 55 (2007), 413-429.  doi: 10.1287/opre.1070.0393.  Google Scholar

[32]

S. A. Smith, N. Agrawal and S. H. McIntyre, A discrete optimization model for seasonal merchandise planning, Journal of Retailing, 74 (1998), 193-221.  doi: 10.1016/S0022-4359(99)80093-1.  Google Scholar

[33]

Y. Song, S. Ray and T. Boyaci, Optimal dynamic joint inventory-pricing control for multiplicative demand with fixed order costs and lost sales, Operations Research, 57 (2009), 245-250.  doi: 10.1287/opre.1080.0530.  Google Scholar

[34]

A. Taudes and C. Rudloff, Integrating inventory control and a price change in the presence of reference price effects: A two-period model, Mathematical Methods of Operations Research, 75 (2012), 29-65.  doi: 10.1007/s00186-011-0374-1.  Google Scholar

[35]

A. Tversky and D. Kahneman, Loss aversion in riskless choice: A reference-dependent model, The Quarterly Journal of Economics, 106 (1991), 1039-1061.   Google Scholar

[36]

T. L. Urban, Coordinating pricing and inventory decisions under reference price effects, International Journal of Manufacturing Technology and Management, 13 (2008), 78-94.  doi: 10.1504/IJMTM.2008.015975.  Google Scholar

[37]

S. Wu, Q. Liu and R. Q. Zhang, The reference effects on a retailer's eynamic pricing and inventory strategies with strategic consumers, Operations Research, 63 (2015), 1320-1335.  doi: 10.1287/opre.2015.1440.  Google Scholar

[38]

C. A. Yano and S. M. Gilbert, Coordinated pricing and production/procurement decisions: a review, Managing Business Interfaces, 16 (2005), 65-103.  doi: 10.1007/0-387-25002-6_3.  Google Scholar

[39]

M. J. Zbaracki, M. Ritson, D. Levy, S. Dutta and M. Bergen, Managerial and customer costs of price adjustment: Direct evidence from industrial markets, The Review of Economics and Statistics, 86 (2004), 514-533.   Google Scholar

[40]

Y. Zhang, Essays on Robust Optimization, Integrated Inventory and Pricing, and Reference Price Effect, PhD thesis, University of Illinois at Urbana-Champaign, 2010. Google Scholar

show all references

References:
[1]

H.-s. Ahn, M. Gümüş and P. Kaminsky, Pricing and manufacturing decisions when demand is a function of prices in multiple periods, Operations Research, 55 (2007), 1039-1057.  doi: 10.1287/opre.1070.0411.  Google Scholar

[2]

H. Arslan and S. Kachani, Dynamic Pricing under Consumer Reference-Price Effects, Wiley Encyclopedia of Operations Research and Management Science, 2010. doi: 10.1002/9780470400531.eorms0273.  Google Scholar

[3]

M. Baucells, M. Weber and F. Welfens, Reference-point formation and updating, Management Science, 57 (2011), 506-519.  doi: 10.1287/mnsc.1100.1286.  Google Scholar

[4]

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

[5]

G. R. Bitran and S. V. Mondschein, Periodic pricing of seasonal products in retailing, Management Science, 43 (1997), 64-79.  doi: 10.1287/mnsc.43.1.64.  Google Scholar

[6]

M. Chen and Z.-L. Chen, Recent developments in dynamic pricing research: Multiple products, competition, and limited demand information, Production and Operations Management, 24 (2015), 704-731.  doi: 10.1111/poms.12295.  Google Scholar

[7]

M. Chen, Z.-L. Chen, G. Pundoor, S. Acharya and J. Yi, Markdown optimization at multiple stores, IIE Transactions, 47 (2015), 84-108.  doi: 10.1080/0740817X.2014.916459.  Google Scholar

[8]

X. Chen, P. Hu and Z. Hu, Efficient algorithms for the dynamic pricing problem with reference price effect, Management Science, 63 (2016), 4389-4408.  doi: 10.1287/mnsc.2016.2554.  Google Scholar

[9]

X. Chen, P. Hu, S. Shum and Y. Zhang, Dynamic stochastic inventory management with reference price effects, Operations Research, 64 (2016), 1529-1536.  doi: 10.1287/opre.2016.1524.  Google Scholar

[10]

X. Chen, S. W. Shum, P. Hu and Y. Zhang, Stochastic inventory model with reference price effects, Operations Research, (2013). Google Scholar

[11]

X. Chen and D. Simchi-Levi, Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: The finite horizon case, Operations Research, 52 (2004), 887-896.  doi: 10.1287/opre.1040.0127.  Google Scholar

[12]

X. Chen and D. Simchi-Levi, Pricing and inventory management, The Oxford Handbook of Pricing Management, (2012), 784Ƀ824. doi: 10.1093/oxfordhb/9780199543175.013.0030.  Google Scholar

[13]

M. C. Cohen, N.-H. Z. Leung, K. Panchamgam, G. Perakis and A. Smith, The impact of linear optimization on promotion planning, Operations Research, 65 (2017), 446-468.  doi: 10.1287/opre.2016.1573.  Google Scholar

[14]

P. R. Dickson and A. G. Sawyer, The price knowledge and search of supermarket shoppers, The Journal of Marketing, (1990), 42-53. Google Scholar

[15]

W. Elmaghraby and P. Keskinocak, Dynamic pricing in the presence of inventory considerations: Research overview, current practices, and future directions, Management Science, 49 (2003), 1287-1309.   Google Scholar

[16]

G. Fibich, A. Gavious and O. Lowengart, Explicit solutions of optimization models and differential games with nonsmooth (asymmetric) reference-price effects, Operations Research, 51 (2003), 721-734.  doi: 10.1287/opre.51.5.721.16758.  Google Scholar

[17]

K. Gedenk, S. A. Neslin and K. L. Ailawadi, Sales promotion, Retailing in the 21st Century, (2006), 345-359. Google Scholar

[18]

L. Gimpl-Heersink, Joint Pricing and Inventory Control under Reference Price Effects, PhD thesis, WU Vienna University of Economics and Business, 2008. doi: 10.3726/b13901.  Google Scholar

[19]

E. A. Greenleaf, The impact of reference price effects on the profitability of price promotions, Marketing Science, 14 (1995), 82-104.  doi: 10.1287/mksc.14.1.82.  Google Scholar

[20]

M. G. Güler, The value of modeling with reference effects in stochastic inventory and pricing problems, Expert Systems with Applications, 40 (2013), 6593-6600.   Google Scholar

[21]

M. G. Güler, T. Bilgiç and R. Güllü, Joint inventory and pricing decisions with reference effects, IIE Transactions, 46 (2014), 330-343.   Google Scholar

[22]

M. G. Güler, T. Bilgiç and R. Güllü, Joint pricing and inventory control for additive demand models with reference effects, Annals of Operations Research, 226 (2015), 255-276.  doi: 10.1007/s10479-014-1706-3.  Google Scholar

[23]

R. L. Hall and C. J. Hitch, Price theory and business behaviour, Oxford Economic Papers, 2 (1939), 12-45.  doi: 10.1093/oxepap/os-2.1.12.  Google Scholar

[24]

P. K. Kopalle, A. G. Rao and J. L. Assuncao, Asymmetric reference price effects and dynamic pricing policies, Marketing Science, 15 (1996), 60-85.  doi: 10.1287/mksc.15.1.60.  Google Scholar

[25]

D. Levy, M. Bergen, S. Dutta and R. Venable, The magnitude of menu costs: direct evidence from large US supermarket chains, The Quarterly Journal of Economics, 112 (1997), 791-824.  doi: 10.1162/003355397555352.  Google Scholar

[26]

D. Levy, S. Dutta, M. Bergen and R. Venable, Price adjustment at multiproduct retailers, Managerial and Decision Economics, 19 (1998), 81-120.  doi: 10.1002/(SICI)1099-1468(199803)19:2<81::AID-MDE867>3.0.CO;2-W.  Google Scholar

[27]

T. Mazumdar, S. Raj and I. Sinha, Reference price research: Review and propositions, Journal of Marketing, 69 (2005), 84-102.  doi: 10.1509/jmkg.2005.69.4.84.  Google Scholar

[28]

D. C. Montgomery, M. Bazaraa and A. K. Keswani, Inventory models with a mixture of backorders and lost sales, Naval Research Logistics (NRL), 20 (1973), 255-263.  doi: 10.1002/nav.3800200205.  Google Scholar

[29]

J. Nasiry and I. Popescu, Dynamic pricing with loss-averse consumers and peak-end anchoring, Operations Research, 59 (2011), 1361-1368.  doi: 10.1287/opre.1110.0952.  Google Scholar

[30]

S. Netessine, Dynamic pricing of inventory/capacity with infrequent price changes, European Journal of Operational Research, 174 (2006), 553-580.  doi: 10.1016/j.ejor.2004.12.015.  Google Scholar

[31]

I. Popescu and Y. Wu, Dynamic pricing strategies with reference effects, Operations Research, 55 (2007), 413-429.  doi: 10.1287/opre.1070.0393.  Google Scholar

[32]

S. A. Smith, N. Agrawal and S. H. McIntyre, A discrete optimization model for seasonal merchandise planning, Journal of Retailing, 74 (1998), 193-221.  doi: 10.1016/S0022-4359(99)80093-1.  Google Scholar

[33]

Y. Song, S. Ray and T. Boyaci, Optimal dynamic joint inventory-pricing control for multiplicative demand with fixed order costs and lost sales, Operations Research, 57 (2009), 245-250.  doi: 10.1287/opre.1080.0530.  Google Scholar

[34]

A. Taudes and C. Rudloff, Integrating inventory control and a price change in the presence of reference price effects: A two-period model, Mathematical Methods of Operations Research, 75 (2012), 29-65.  doi: 10.1007/s00186-011-0374-1.  Google Scholar

[35]

A. Tversky and D. Kahneman, Loss aversion in riskless choice: A reference-dependent model, The Quarterly Journal of Economics, 106 (1991), 1039-1061.   Google Scholar

[36]

T. L. Urban, Coordinating pricing and inventory decisions under reference price effects, International Journal of Manufacturing Technology and Management, 13 (2008), 78-94.  doi: 10.1504/IJMTM.2008.015975.  Google Scholar

[37]

S. Wu, Q. Liu and R. Q. Zhang, The reference effects on a retailer's eynamic pricing and inventory strategies with strategic consumers, Operations Research, 63 (2015), 1320-1335.  doi: 10.1287/opre.2015.1440.  Google Scholar

[38]

C. A. Yano and S. M. Gilbert, Coordinated pricing and production/procurement decisions: a review, Managing Business Interfaces, 16 (2005), 65-103.  doi: 10.1007/0-387-25002-6_3.  Google Scholar

[39]

M. J. Zbaracki, M. Ritson, D. Levy, S. Dutta and M. Bergen, Managerial and customer costs of price adjustment: Direct evidence from industrial markets, The Review of Economics and Statistics, 86 (2004), 514-533.   Google Scholar

[40]

Y. Zhang, Essays on Robust Optimization, Integrated Inventory and Pricing, and Reference Price Effect, PhD thesis, University of Illinois at Urbana-Champaign, 2010. Google Scholar

Figure 1.  The paths of optimal prices
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Figure 2.  The paths of reference prices
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Figure 3.  The optimal pricing paths of models without promotion constraints
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Figure 4.  Comparison of the optimal pricing paths ($ 1\leq M\leq8 $)
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Figure 5.  Comparison of the optimal pricing paths ($ 9\leq M\leq19 $)
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Figure 6.  The paths of optimal replenishment quantities
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Figure 7.  Change of the inventory level
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Figure 8.  Change of the backorder level
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Figure 9.  Comparison of the total profit obtained by different methods
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Figure 10.  Comparison of the computational time used by different methods
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Figure 11.  Relations between L/S and the total profit
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Figure 12.  Relations between L/S and the total profit with L/S cost being taken into consideration
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Figure 13.  Relations between T and the total profit
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Figure 14.  Relations between h/b and the total profit
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Table 1.  Notations
Variable Description
$ T\; (T \in Z^+) $ length of the planning horizon
$ t\; (t \in \{1,\ldots,T\}) $ time period
$ L\; (L \in \{1,\ldots,T\}) $ the maximum total number of promotion times in the planning horizon
$ S\; (S \in \{0,\ldots,T-1\}) $ the minimum separating period (a separating period is a period which spaces out the two successive promotions)
$ M\; (M \in \{0,\ldots,T-1\}) $ length of consumers' memory window
$ \underline{u_1}\; (\underline{u_1}\in (0,\infty)) $ permitted minimum promotion price
$ p_0\; (p_0\in (\underline{u_1},\infty)) $ full/normal price without promotion
$ \overline{u_1}\; (\overline{u_1}\in (\underline{u_1},p_0)) $ permitted maximum promotion price
$ u_1(t)\; (u_1(t) \in [\underline{u_1},\overline{u_1}]\; or\; u_1(t)=p_0) $ price in period $ t $
$ \underline{u_2}\; (\underline{u_2}\in (0,\infty)) $ permitted minimum ordering quantity
$ \overline{u_2}\; (\overline{u_2} \in (\underline{u_2},\infty)) $ permitted maximum ordering quantity
$ u_2(t)\; (u_2(t) \in [\underline{u_2},\overline{u_2}]) $ ordering quantity in period $ t $
$ u_3(t)\; (u_3(t)\in \{0,1\}) $ marking decision variable of promotion in period t, set $ u_3(t)=1 $ when there exists a price discount in period t and $ u_3(t)=0 $ when the item's price in period t equals to $ p_0 $
$ y_{max}\; (y_{max}\in (0,\infty)) $ permitted maximum inventory level in each period
$ y_t\; (y_t\in (-\infty,y_{max}]) $ initial inventory level in period t, where $ y_t <0 $ means the back-order demand is $ |y_t| $ at period $ t $
$ y_1\; (y_1\in [0,\infty)) $ initial inventory level at the beginning of the planning horizon
$ c\; (c\in (0,\infty)) $ per unit ordering cost
$ h\; (h\in (0,\infty)) $ per unit inventory cost
$ b\; (b\in (0,\infty)) $ per unit back-order cost
$ d_t\; (d_t\in [0,\infty)) $ demand at period t
$ \gamma\; (\gamma\in (0,1)) $ discount factor
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Variable Description
$ T\; (T \in Z^+) $ length of the planning horizon
$ t\; (t \in \{1,\ldots,T\}) $ time period
$ L\; (L \in \{1,\ldots,T\}) $ the maximum total number of promotion times in the planning horizon
$ S\; (S \in \{0,\ldots,T-1\}) $ the minimum separating period (a separating period is a period which spaces out the two successive promotions)
$ M\; (M \in \{0,\ldots,T-1\}) $ length of consumers' memory window
$ \underline{u_1}\; (\underline{u_1}\in (0,\infty)) $ permitted minimum promotion price
$ p_0\; (p_0\in (\underline{u_1},\infty)) $ full/normal price without promotion
$ \overline{u_1}\; (\overline{u_1}\in (\underline{u_1},p_0)) $ permitted maximum promotion price
$ u_1(t)\; (u_1(t) \in [\underline{u_1},\overline{u_1}]\; or\; u_1(t)=p_0) $ price in period $ t $
$ \underline{u_2}\; (\underline{u_2}\in (0,\infty)) $ permitted minimum ordering quantity
$ \overline{u_2}\; (\overline{u_2} \in (\underline{u_2},\infty)) $ permitted maximum ordering quantity
$ u_2(t)\; (u_2(t) \in [\underline{u_2},\overline{u_2}]) $ ordering quantity in period $ t $
$ u_3(t)\; (u_3(t)\in \{0,1\}) $ marking decision variable of promotion in period t, set $ u_3(t)=1 $ when there exists a price discount in period t and $ u_3(t)=0 $ when the item's price in period t equals to $ p_0 $
$ y_{max}\; (y_{max}\in (0,\infty)) $ permitted maximum inventory level in each period
$ y_t\; (y_t\in (-\infty,y_{max}]) $ initial inventory level in period t, where $ y_t <0 $ means the back-order demand is $ |y_t| $ at period $ t $
$ y_1\; (y_1\in [0,\infty)) $ initial inventory level at the beginning of the planning horizon
$ c\; (c\in (0,\infty)) $ per unit ordering cost
$ h\; (h\in (0,\infty)) $ per unit inventory cost
$ b\; (b\in (0,\infty)) $ per unit back-order cost
$ d_t\; (d_t\in [0,\infty)) $ demand at period t
$ \gamma\; (\gamma\in (0,1)) $ discount factor
Table 2.  Comparison of the total profit obtained by different methods ($ T\in \{x\mid x = 20+5y,\; y = 1,2,\ldots,7\} $)
Conditions (T, L, S)Enumeration Method (＄) 2-stage Method (＄) 3-stage Stage (＄) 4-stage Method (＄) GA-Roulette Wheel (＄) GA-Tournament (＄) GA-Random (＄)
T=20;S=4;320.75318.93320.75320.75319.86314.74320.75
T=20;S=3;325.35324.92320.75325.16325.16324.34315.19
T=20;S=2;334.74334.74329.34329.03324.10323.36334.74
T=25;S=4;354.58354.05353.16354.42348.62348.11352.96
T=25;S=3;361.58357.05356.39357.59352.84357.74348.27
T=25;S=2;364.49363.36363.31362.25364.33361.73352.20
T=30;S=4;380.76378.88377.86379.27370.11380.08375.14
T=30;S=3;385.20384.27383.96380.66373.42384.22382.70
T=30;S=2;388.12387.42385.57384.21379.88385.67385.22
T=35;S=4;-400.41400.61397.40396.31398.86396.73
T=35;S=3;-403.35401.16400.62391.81398.54391.37
T=35;S=2;-406.24401.68400.97402.85402.59397.94
T=40;S=4;-413.71413.32411.90411.97410.17412.60
T=40;S=3;-416.81414.87412.92413.76409.39414.07
T=40;S=2;-418.16417.45418.16410.04410.81412.92
T=45;S=4;-425.79423.87423.34420.72413.68421.14
T=45;S=3;-428.44425.32423.90418.40422.65423.58
T=45;S=2;-429.99428.48429.41416.05417.76419.69
T=50;S=4;-433.74432.99431.70425.82426.32426.94
T=50;S=3;-436.57434.09433.75429.96421.54422.15
T=50;S=2;-439.00439.75437.98436.14427.98423.56
T=55;S=4;-441.03439.20438.03438.62429.32430.26
T=55;S=3;-443.12442.14443.22442.49436.26433.66
T=55;S=2;-446.04446.39446.14439.20443.98440.99
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Conditions (T, L, S)Enumeration Method (＄) 2-stage Method (＄) 3-stage Stage (＄) 4-stage Method (＄) GA-Roulette Wheel (＄) GA-Tournament (＄) GA-Random (＄)
T=20;S=4;320.75318.93320.75320.75319.86314.74320.75
T=20;S=3;325.35324.92320.75325.16325.16324.34315.19
T=20;S=2;334.74334.74329.34329.03324.10323.36334.74
T=25;S=4;354.58354.05353.16354.42348.62348.11352.96
T=25;S=3;361.58357.05356.39357.59352.84357.74348.27
T=25;S=2;364.49363.36363.31362.25364.33361.73352.20
T=30;S=4;380.76378.88377.86379.27370.11380.08375.14
T=30;S=3;385.20384.27383.96380.66373.42384.22382.70
T=30;S=2;388.12387.42385.57384.21379.88385.67385.22
T=35;S=4;-400.41400.61397.40396.31398.86396.73
T=35;S=3;-403.35401.16400.62391.81398.54391.37
T=35;S=2;-406.24401.68400.97402.85402.59397.94
T=40;S=4;-413.71413.32411.90411.97410.17412.60
T=40;S=3;-416.81414.87412.92413.76409.39414.07
T=40;S=2;-418.16417.45418.16410.04410.81412.92
T=45;S=4;-425.79423.87423.34420.72413.68421.14
T=45;S=3;-428.44425.32423.90418.40422.65423.58
T=45;S=2;-429.99428.48429.41416.05417.76419.69
T=50;S=4;-433.74432.99431.70425.82426.32426.94
T=50;S=3;-436.57434.09433.75429.96421.54422.15
T=50;S=2;-439.00439.75437.98436.14427.98423.56
T=55;S=4;-441.03439.20438.03438.62429.32430.26
T=55;S=3;-443.12442.14443.22442.49436.26433.66
T=55;S=2;-446.04446.39446.14439.20443.98440.99
Table 3.  Comparison of the computational time for using different methods ($ T\in \{x\mid x = 20+5y,\; y = 1,2,\ldots,7\} $)
Conditions (T, L, S)Enumeration Method (s)2-stage Method (s)3-stage Stage (s)4-stage Method (s)GA-Roulette Wheel (s)GA-Tournament (s)GA-Random (s)
T=20;S=4;102.521.261.021.2344.1940.7661.43
T=20;S=3;117.193.941.151.3280.0284.2554.55
T=20;S=2;186.408.661.811.6878.61102.34100.20
T=25;S=4;3119.324.591.930.9882.8378.7699.75
T=25;S=3;3302.937.211.754.2698.68109.26102.26
T=25;S=2;4183.1716.694.182.54121.41108.78118.15
T=30;S=4;98508.3713.993.092.75107.32128.91154.09
T=30;S=3;100972.6426.125.463.03111.41148.17134.50
T=30;S=2;104669.8531.563.423.42111.41137.66138.36
T=35;S=4;-40.806.764.67164.83151.35126.96
T=35;S=3;-64.8011.004.24148.59177.05166.54
T=35;S=2;-63.958.866.16138.41151.68122.52
T=40;S=4;-177.3912.915.52166.76197.82166.82
T=40;S=3;-208.6412.115.30172.36136.09165.40
T=40;S=2;-284.3517.294.02136.64145.17136.92
T=45;S=4;-529.4019.538.33217.47179.08161.77
T=45;S=3;-611.0724.0310.58194.80177.25211.04
T=45;S=2;-946.1931.3510.06180.60182.13184.05
T=50;S=4;-3810.1923.299.68160.34142.00130.60
T=50;S=3;-4411.6639.1814.79149.26128.57147.92
T=50;S=2;-6273.4771.0019.41146.46124.79147.17
T=55;S=4;-14704.6864.8714.13172.13162.80168.13
T=55;S=3;-17253.2084.4130.32163.70164.13163.31
T=55;S=2;-23839.38239.7449.88158.11178.72172.18
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Conditions (T, L, S)Enumeration Method (s)2-stage Method (s)3-stage Stage (s)4-stage Method (s)GA-Roulette Wheel (s)GA-Tournament (s)GA-Random (s)
T=20;S=4;102.521.261.021.2344.1940.7661.43
T=20;S=3;117.193.941.151.3280.0284.2554.55
T=20;S=2;186.408.661.811.6878.61102.34100.20
T=25;S=4;3119.324.591.930.9882.8378.7699.75
T=25;S=3;3302.937.211.754.2698.68109.26102.26
T=25;S=2;4183.1716.694.182.54121.41108.78118.15
T=30;S=4;98508.3713.993.092.75107.32128.91154.09
T=30;S=3;100972.6426.125.463.03111.41148.17134.50
T=30;S=2;104669.8531.563.423.42111.41137.66138.36
T=35;S=4;-40.806.764.67164.83151.35126.96
T=35;S=3;-64.8011.004.24148.59177.05166.54
T=35;S=2;-63.958.866.16138.41151.68122.52
T=40;S=4;-177.3912.915.52166.76197.82166.82
T=40;S=3;-208.6412.115.30172.36136.09165.40
T=40;S=2;-284.3517.294.02136.64145.17136.92
T=45;S=4;-529.4019.538.33217.47179.08161.77
T=45;S=3;-611.0724.0310.58194.80177.25211.04
T=45;S=2;-946.1931.3510.06180.60182.13184.05
T=50;S=4;-3810.1923.299.68160.34142.00130.60
T=50;S=3;-4411.6639.1814.79149.26128.57147.92
T=50;S=2;-6273.4771.0019.41146.46124.79147.17
T=55;S=4;-14704.6864.8714.13172.13162.80168.13
T=55;S=3;-17253.2084.4130.32163.70164.13163.31
T=55;S=2;-23839.38239.7449.88158.11178.72172.18
Table 4.  Comparison of the total profit obtained by different methods ($ T\in \{x\mid x = 50+5y,\; y = 1,2,\ldots,7\} $)
Conditions (T, L, S)6-stage Method (＄) GA-Roulette Wheel (＄) GA-Tournament (＄) GA-Random (＄)
T=50;S=4;430.93425.82426.32426.94
T=50;S=3;432.92429.96421.54422.15
T=50;S=2;437.60436.14427.98423.56
T=55;S=4;436.84438.62429.32430.26
T=55;S=3;442.84442.49436.26433.66
T=55;S=2;445.07439.20443.98440.99
T=60;S=4;442.36438.03441.41444.65
T=60;S=3;447.21440.00436.47444.08
T=60;S=2;452.45449.31449.03448.02
T=65;S=4;446.70446.36446.56440.53
T=65;S=3;451.55449.16451.34450.10
T=65;S=2;456.79450.53449.03447.76
T=70;S=4;453.23447.34443.15451.60
T=70;S=3;453.79448.39450.38453.16
T=70;S=2;459.30448.72456.29450.92
T=75;S=4;454.83450.42450.39455.13
T=75;S=3;456.88447.96451.87443.83
T=75;S=2;461.11461.54453.54450.14
T=80;S=4;455.93446.26443.80453.12
T=80;S=3;461.12456.58447.33448.42
T=80;S=2;464.04453.66460.24449.64
T=85;S=4;456.66449.92448.51455.99
T=85;S=3;462.00459.30459.91451.51
T=85;S=2;465.04460.63452.55454.77
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Conditions (T, L, S)6-stage Method (＄) GA-Roulette Wheel (＄) GA-Tournament (＄) GA-Random (＄)
T=50;S=4;430.93425.82426.32426.94
T=50;S=3;432.92429.96421.54422.15
T=50;S=2;437.60436.14427.98423.56
T=55;S=4;436.84438.62429.32430.26
T=55;S=3;442.84442.49436.26433.66
T=55;S=2;445.07439.20443.98440.99
T=60;S=4;442.36438.03441.41444.65
T=60;S=3;447.21440.00436.47444.08
T=60;S=2;452.45449.31449.03448.02
T=65;S=4;446.70446.36446.56440.53
T=65;S=3;451.55449.16451.34450.10
T=65;S=2;456.79450.53449.03447.76
T=70;S=4;453.23447.34443.15451.60
T=70;S=3;453.79448.39450.38453.16
T=70;S=2;459.30448.72456.29450.92
T=75;S=4;454.83450.42450.39455.13
T=75;S=3;456.88447.96451.87443.83
T=75;S=2;461.11461.54453.54450.14
T=80;S=4;455.93446.26443.80453.12
T=80;S=3;461.12456.58447.33448.42
T=80;S=2;464.04453.66460.24449.64
T=85;S=4;456.66449.92448.51455.99
T=85;S=3;462.00459.30459.91451.51
T=85;S=2;465.04460.63452.55454.77
Table 5.  Comparison of the computational time for using different methods ($T\in \{x\mid x = 50+5y,\; y = 1,2,\ldots,7\} $)
Conditions (T, L, S)6-stage Method (s)GA-Roulette Wheel (s)GA-Tournament (s)GA-Random (s)
T=50;S=4;5.66160.34142.00130.60
T=50;S=3;3.49149.26128.57147.92
T=50;S=2;4.43146.46124.79147.17
T=55;S=4;9.60172.13162.80168.13
T=55;S=3;5.13163.70164.13163.31
T=55;S=2;6.49158.11178.72172.18
T=60;S=4;5.51184.39219.30212.04
T=60;S=3;5.97200.68190.23203.26
T=60;S=2;4.93192.55196.28199.34
T=65;S=4;13.83224.00253.94246.07
T=65;S=3;14.60217.75217.02225.30
T=65;S=2;14.16211.35214.32211.38
T=70;S=4;16.32283.38274.18273.71
T=70;S=3;18.70246.01258.15257.03
T=70;S=2;18.78238.08237.44276.35
T=75;S=4;18.17273.58280.91304.42
T=75;S=3;21.14284.10289.44301.87
T=75;S=2;24.02306.91293.79281.39
T=80;S=4;22.20349.83333.17338.17
T=80;S=3;24.50338.76331.27336.95
T=80;S=2;34.04355.38344.48342.70
T=85;S=4;32.86382.19363.28438.18
T=85;S=3;31.13413.49386.06378.90
T=85;S=2;38.55410.78386.99407.43
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Conditions (T, L, S)6-stage Method (s)GA-Roulette Wheel (s)GA-Tournament (s)GA-Random (s)
T=50;S=4;5.66160.34142.00130.60
T=50;S=3;3.49149.26128.57147.92
T=50;S=2;4.43146.46124.79147.17
T=55;S=4;9.60172.13162.80168.13
T=55;S=3;5.13163.70164.13163.31
T=55;S=2;6.49158.11178.72172.18
T=60;S=4;5.51184.39219.30212.04
T=60;S=3;5.97200.68190.23203.26
T=60;S=2;4.93192.55196.28199.34
T=65;S=4;13.83224.00253.94246.07
T=65;S=3;14.60217.75217.02225.30
T=65;S=2;14.16211.35214.32211.38
T=70;S=4;16.32283.38274.18273.71
T=70;S=3;18.70246.01258.15257.03
T=70;S=2;18.78238.08237.44276.35
T=75;S=4;18.17273.58280.91304.42
T=75;S=3;21.14284.10289.44301.87
T=75;S=2;24.02306.91293.79281.39
T=80;S=4;22.20349.83333.17338.17
T=80;S=3;24.50338.76331.27336.95
T=80;S=2;34.04355.38344.48342.70
T=85;S=4;32.86382.19363.28438.18
T=85;S=3;31.13413.49386.06378.90
T=85;S=2;38.55410.78386.99407.43
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