# American Institute of Mathematical Sciences

April  2013, 9(2): 437-454. doi: 10.3934/jimo.2013.9.437

## Joint pricing and ordering policies for deteriorating item with retail price-dependent demand in response to announced supply price increase

 1 Department of Industrial Management, Chien Hsin University of Science and Technology, Jung-Li, Taoyuan 320, Taiwan, Taiwan 2 Department of Management Sciences, Tamkang University, Tamsui, Taipei 251, Taiwan 3 Department of Accounting, Tamkang University, Tamsui, Taipei 251, Taiwan

Received  August 2011 Revised  January 2013 Published  February 2013

Recently, due to rapid economic development in emerging nations, the world's raw material prices have been rising. In today's unrestricted information environment, suppliers typically announce impending supply price increases at specific times. This allows retailers to replenish their stock at the present price, before the price increase takes effect. The supplier, however, will generally offer only limited quantities prior to the price increase, so as to avoid excessive orders. The retail price will usually reflect any supply price increases, as market demand is dependent on retail price. This paper considers deteriorating items and investigates (1) the possible effects of a supply price increase on retail pricing, and (2) ordering policies under the conditions that special order quantities are limited and demand is dependent on retail price. The purpose of this paper is to determine the optimal special order quantity and retail price to maximize profit. Our theoretical analysis examines the necessary and sufficient conditions for an optimal solution, and an algorithm is established to obtain the optimal solution. Furthermore, several numerical examples are given to illustrate the developed model and the solution procedure. Finally, a sensitivity analysis is conducted on the optimal solutions with respect to major parameters.
Citation: Chih-Te Yang, Liang-Yuh Ouyang, Hsiu-Feng Yen, Kuo-Liang Lee. Joint pricing and ordering policies for deteriorating item with retail price-dependent demand in response to announced supply price increase. Journal of Industrial & Management Optimization, 2013, 9 (2) : 437-454. doi: 10.3934/jimo.2013.9.437
##### References:
 [1] R. P. Covert and G. C. Philip, An EOQ model for items with Weibull distribution deterioration,, AIIE Transactions, 5 (1973), 323.   Google Scholar [2] P. S. Deng, R. H. Lin and P. A. Chu, A note on the inventory models for deteriorating items with ramp type demand rate,, European Journal of Operations Research, 178 (2007), 112.  doi: 10.1016/j.ejor.2006.01.028.  Google Scholar [3] C. Y. Dye, Joint pricing and ordering policy for a deteriorating inventory with partial backlogging,, Omega, 35 (2007), 184.   Google Scholar [4] P. M. Ghare and G. H. Schrader, A model for exponentially decaying inventory system,, Journal of Industrial Engineering, 163 (1963), 238.   Google Scholar [5] A. K. Ghosh, On some inventory models involving shortages under an announced price increase,, International Journal of Systems Science, 34 (2003), 129.  doi: 10.1080/0020772031000152956.  Google Scholar [6] S. K. Goyal, A note on the paper: An inventory model with finite horizon and price changes,, Journal of the Operational Research Society, 30 (1979), 839.   Google Scholar [7] S. K. Goyal and S. K. Bhatt, A generalized lot size ordering policy for price increases,, Opsearch, 25 (1988), 272.   Google Scholar [8] S. K. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory,, European Journal of Operations Research, 134 (2001), 1.  doi: 10.1016/S0377-2217(00)00248-4.  Google Scholar [9] Y. He, S. Y. Wang and K. K. Lai, An optimal production-inventory model for deteriorating items with multiple-market demand,, European Journal of Operations Research, 203 (2010), 593.   Google Scholar [10] W. Huang and V. G. Kulkarni, Optimal EOQ for announced price increases in infinite horizon,, Operations Research, 51 (2003), 336.  doi: 10.1287/opre.51.2.336.12785.  Google Scholar [11] P. C. Jordan, Purchasing decisions considering future price increases: An empirical approach,, Journal of Purchasing and Materials Management, 23 (1987), 25.   Google Scholar [12] B. Lev and A. L. Soyster, An inventory model with finite horizon and price changes,, Journal of the Operational Research Society, 30 (1979), 43.   Google Scholar [13] B. Lev B. and H. J. Weiss, Inventory models with cost changes,, Operations Research, 38 (1990), 53.  doi: 10.1287/opre.38.1.53.  Google Scholar [14] E. P. Markowski, Criteria for evaluating purchase quantity decisions in response to future price increases,, European Journal of Operations Research, 47 (1990), 364.   Google Scholar [15] I. Moon, B. C. Giri and B. Ko, Economic order quantity models for ameliorating/deteriorating items under inflation and time discounting,, European Journal of Operations Research, 162 (2005), 773.  doi: 10.1016/j.ejor.2003.09.025.  Google Scholar [16] E. Naddor, "Inventory Systems,", John Wiley $&$ Sons, (1966).   Google Scholar [17] G. C. Philip, A generalized EOQ model for items with Weibull distribution,, AIIE Transactions, 6 (1974), 159.   Google Scholar [18] N. H. Shah, An order-level lot size inventory model for deteriorating items,, AIIE Transactions, 9 (1977), 108.   Google Scholar [19] N. H. Shah, A discrete-time probabilistic inventory model for deteriorating items under a known price increase,, International Journal of Systems Science, 29 (1998), 823.   Google Scholar [20] S. G. Taylor and C. E. Bradley, Optimal ordering strategies for announced price increases,, Operations Research, 33 (1985), 312.   Google Scholar [21] R. J. Tersine and E. T. Grasso, Forward buying in response to announced price increases,, Journal of Purchasing and Materials Management, 14 (1978), 20.   Google Scholar [22] K. S. Wu, L. Y. Ouyang and C. T. Yang, Coordinating replenishment and pricing policies for non-instantaneous deteriorating items with price sensitive demand,, International Journal of Systems Science, 40 (2009), 1273.  doi: 10.1080/00207720903038093.  Google Scholar [23] H. H. Yanasse, EOQ systems: The case of an increase in purchase cost,, Journal of the Operational Research Society, 41 (1990), 633.   Google Scholar [24] C. T. Yang, L. Y. Ouyang, and H. H. Wu, Retailer's optimal pricing and ordering policies for non-instantaneous deteriorating items with price-dependent demand and partial backlogging,, Mathematical Problems in Engineering, (2009).  doi: 10.1155/2009/198305.  Google Scholar [25] M. J. Yao and Y. C. Wang, Theoretical analysis and a search procedure for the joint replenishment problem with deteriorating products,, Journal of Industrial and Management Optimization, 1 (2005), 359.  doi: 10.3934/jimo.2005.1.359.  Google Scholar [26] J. C. P. Yu, H. M. Wee and K. J. Wang, Supply chain partnership for Three-Echelon deteriorating inventory model,, Journal of Industrial and Management Optimization, 4 (2008), 827.  doi: 10.3934/jimo.2008.4.827.  Google Scholar

show all references

##### References:
 [1] R. P. Covert and G. C. Philip, An EOQ model for items with Weibull distribution deterioration,, AIIE Transactions, 5 (1973), 323.   Google Scholar [2] P. S. Deng, R. H. Lin and P. A. Chu, A note on the inventory models for deteriorating items with ramp type demand rate,, European Journal of Operations Research, 178 (2007), 112.  doi: 10.1016/j.ejor.2006.01.028.  Google Scholar [3] C. Y. Dye, Joint pricing and ordering policy for a deteriorating inventory with partial backlogging,, Omega, 35 (2007), 184.   Google Scholar [4] P. M. Ghare and G. H. Schrader, A model for exponentially decaying inventory system,, Journal of Industrial Engineering, 163 (1963), 238.   Google Scholar [5] A. K. Ghosh, On some inventory models involving shortages under an announced price increase,, International Journal of Systems Science, 34 (2003), 129.  doi: 10.1080/0020772031000152956.  Google Scholar [6] S. K. Goyal, A note on the paper: An inventory model with finite horizon and price changes,, Journal of the Operational Research Society, 30 (1979), 839.   Google Scholar [7] S. K. Goyal and S. K. Bhatt, A generalized lot size ordering policy for price increases,, Opsearch, 25 (1988), 272.   Google Scholar [8] S. K. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory,, European Journal of Operations Research, 134 (2001), 1.  doi: 10.1016/S0377-2217(00)00248-4.  Google Scholar [9] Y. He, S. Y. Wang and K. K. Lai, An optimal production-inventory model for deteriorating items with multiple-market demand,, European Journal of Operations Research, 203 (2010), 593.   Google Scholar [10] W. Huang and V. G. Kulkarni, Optimal EOQ for announced price increases in infinite horizon,, Operations Research, 51 (2003), 336.  doi: 10.1287/opre.51.2.336.12785.  Google Scholar [11] P. C. Jordan, Purchasing decisions considering future price increases: An empirical approach,, Journal of Purchasing and Materials Management, 23 (1987), 25.   Google Scholar [12] B. Lev and A. L. Soyster, An inventory model with finite horizon and price changes,, Journal of the Operational Research Society, 30 (1979), 43.   Google Scholar [13] B. Lev B. and H. J. Weiss, Inventory models with cost changes,, Operations Research, 38 (1990), 53.  doi: 10.1287/opre.38.1.53.  Google Scholar [14] E. P. Markowski, Criteria for evaluating purchase quantity decisions in response to future price increases,, European Journal of Operations Research, 47 (1990), 364.   Google Scholar [15] I. Moon, B. C. Giri and B. Ko, Economic order quantity models for ameliorating/deteriorating items under inflation and time discounting,, European Journal of Operations Research, 162 (2005), 773.  doi: 10.1016/j.ejor.2003.09.025.  Google Scholar [16] E. Naddor, "Inventory Systems,", John Wiley $&$ Sons, (1966).   Google Scholar [17] G. C. Philip, A generalized EOQ model for items with Weibull distribution,, AIIE Transactions, 6 (1974), 159.   Google Scholar [18] N. H. Shah, An order-level lot size inventory model for deteriorating items,, AIIE Transactions, 9 (1977), 108.   Google Scholar [19] N. H. Shah, A discrete-time probabilistic inventory model for deteriorating items under a known price increase,, International Journal of Systems Science, 29 (1998), 823.   Google Scholar [20] S. G. Taylor and C. E. Bradley, Optimal ordering strategies for announced price increases,, Operations Research, 33 (1985), 312.   Google Scholar [21] R. J. Tersine and E. T. Grasso, Forward buying in response to announced price increases,, Journal of Purchasing and Materials Management, 14 (1978), 20.   Google Scholar [22] K. S. Wu, L. Y. Ouyang and C. T. Yang, Coordinating replenishment and pricing policies for non-instantaneous deteriorating items with price sensitive demand,, International Journal of Systems Science, 40 (2009), 1273.  doi: 10.1080/00207720903038093.  Google Scholar [23] H. H. Yanasse, EOQ systems: The case of an increase in purchase cost,, Journal of the Operational Research Society, 41 (1990), 633.   Google Scholar [24] C. T. Yang, L. Y. Ouyang, and H. H. Wu, Retailer's optimal pricing and ordering policies for non-instantaneous deteriorating items with price-dependent demand and partial backlogging,, Mathematical Problems in Engineering, (2009).  doi: 10.1155/2009/198305.  Google Scholar [25] M. J. Yao and Y. C. Wang, Theoretical analysis and a search procedure for the joint replenishment problem with deteriorating products,, Journal of Industrial and Management Optimization, 1 (2005), 359.  doi: 10.3934/jimo.2005.1.359.  Google Scholar [26] J. C. P. Yu, H. M. Wee and K. J. Wang, Supply chain partnership for Three-Echelon deteriorating inventory model,, Journal of Industrial and Management Optimization, 4 (2008), 827.  doi: 10.3934/jimo.2008.4.827.  Google Scholar
 [1] Gaurav Nagpal, Udayan Chanda, Nitant Upasani. Inventory replenishment policies for two successive generations price-sensitive technology products. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021036 [2] Frank Sottile. The special Schubert calculus is real. Electronic Research Announcements, 1999, 5: 35-39. [3] Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341 [4] Chaudry Masood Khalique, Muhammad Usman, Maria Luz Gandarais. Special issue dedicated to Professor David Paul Mason. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : iii-iv. doi: 10.3934/dcdss.2020416 [5] Eduardo Casas, Christian Clason, Arnd Rösch. Preface special issue on system modeling and optimization. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021008 [6] Michel Chipot, Mingmin Zhang. On some model problem for the propagation of interacting species in a special environment. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020401 [7] Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133 [8] Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597 [9] Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021017 [10] Yunfei Lv, Rong Yuan, Yuan He. Wavefronts of a stage structured model with state--dependent delay. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4931-4954. doi: 10.3934/dcds.2015.35.4931 [11] Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454 [12] Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020136 [13] Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028 [14] Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269 [15] Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065 [16] Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 [17] Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 [18] A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044 [19] Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 [20] Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

2019 Impact Factor: 1.366