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January  2017, 13(1): 187-206. doi: 10.3934/jimo.2016011

## Preservation of deteriorating seasonal products with stock-dependent consumption rate and shortages

 1 Department of Industrial & Management Engineering, Hanyang University, Ansan Gyeonggi-do, 426 791, South Korea 2 Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721 102, India

* Corresponding author: ss.sumonsarkar@gmail.com, Office Ph. No. 031-436-8118

1Dr. Biswajit Sarkar is in leave on lien from Vidyasagar University.

Received  March 2015 Revised  December 2015 Published  March 2016

In literature, many inventory studies have been developed by assuming deterioration of items as either a variable or constant. But in real life situation, deterioration of goods can be reduced by adding some extra effective capital investment in preservation technology. In this paper, a deteriorating inventory model with ramp-type demand under stock-dependent consumption rate by assuming preservation technology cost as a decision variable is formulated. Shortages are allowed and the unsatisfied demand is partially backlogged at a negative exponential rate with the waiting time. The objective of this study is to obtain the optimal replenishment and preservation technology investment strategies so that the total profit per unit time is maximum. Further, the necessary and sufficient conditions are considered to prove the existence and uniqueness of the optimal solution. Some numerical examples along with graphical representations are provided to illustrate the proposed model. Sensitivity analysis of the optimal solution with respect to major parameters of the system has been carried out and the implications are discussed.

Citation: Biswajit Sarkar, Buddhadev Mandal, Sumon Sarkar. Preservation of deteriorating seasonal products with stock-dependent consumption rate and shortages. Journal of Industrial & Management Optimization, 2017, 13 (1) : 187-206. doi: 10.3934/jimo.2016011
##### References:

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##### References:
Graphical presentation of the inventory system (Case 1: μ>t1)
Graphical presentation of the inventory system (Case 2: μ>t1)
Graphical presentation of total profit function versus time and preservation technology cost (Example 1)
Graphical presentation of total profit function versus time and preservation technology cost (Example 2)
Comparison between the contributions of different authors
 Author (s) Ramp-type demand Deterio-ration Preser-vation Stock-dependent consumption rate Partial back-logging Ghare and Schrader [17] √ Covert and Philip [9] √ Sana, Goyal, and Chaudhuri [30] √ Dye, Chang, and Teng [13] √ Chung and Wee [7] √ √ Widyadana and Wee [54] √ Chung and Wee [6] √ Wee, Lee, Yu, and Wang [52] √ Sarkar [34] √ Sett, Sarkar, and Goswami [46] √ Sarkar [35] √ Sarkar and Sarkar [36] √ Sarkar and Sarkar [40] √ Sarkar, Sarkar, and Yun [41] √ √ √ Sarkar [42] √ Bouras and Tadj [4] √ Sarkar, Saren, and C$\acute{a}$rdenas-Barr$\acute{o}$n [44] √ Hsu, Wee, and Teng [18] √ √ Dye and Hsieh [14] √ √ √ Dye [15] √ √ √ Zhang, Bai, and Tang [58] √ √ Montgomery, Bazaraa, and Keswani [24] √ Park [28] √ Rosenberg [29] √ Abad [1] √ Chang and Dye [5] √ √ Papachristos and Skouri [27] √ √ Skouri and Papachristos [48] √ √ C$\acute{a}$rdenas-Barr$\acute{o}$n [10] √ Sana [31] √ √ C$\acute{a}$rdenas-Barr$\acute{o}$n [11] √ Widyadana, C$\acute{a}$rdenas-Barr$\acute{o}$n, and Wee [55] √ √ Sarkar and Sarkar [43] √ √ √ Wee, Huang, Wang and Cheng [53] √ Sarkar [37] √ Sarkar, Mandal, and Sarkar [39] √ Mandal and Pal [23] √ √ Wu [56] √ √ √ Wu, Ouyang, and Yang [57] √ √ √ √ Skouri, Konstantaras, Papachristos, and Ganas [49] √ √ √ Sarkar, Sett, Goswami, and Sarkar [45] √ √ √ √ Levin, McLaughlin, Lemone, and Kottas [21] √ Silver and Peterson [47] √ Padmanabhan and Vrat [26] √ √ √ Liao, Tsai, and Su [22] √ √ Dye and Ouyang [16] √ √ Alfares [2] √ Chung and Wee [8] √ √ Sana and Chaudhuri [32] √ Sana [33] √ √ Sarkar [38] √
 Author (s) Ramp-type demand Deterio-ration Preser-vation Stock-dependent consumption rate Partial back-logging Ghare and Schrader [17] √ Covert and Philip [9] √ Sana, Goyal, and Chaudhuri [30] √ Dye, Chang, and Teng [13] √ Chung and Wee [7] √ √ Widyadana and Wee [54] √ Chung and Wee [6] √ Wee, Lee, Yu, and Wang [52] √ Sarkar [34] √ Sett, Sarkar, and Goswami [46] √ Sarkar [35] √ Sarkar and Sarkar [36] √ Sarkar and Sarkar [40] √ Sarkar, Sarkar, and Yun [41] √ √ √ Sarkar [42] √ Bouras and Tadj [4] √ Sarkar, Saren, and C$\acute{a}$rdenas-Barr$\acute{o}$n [44] √ Hsu, Wee, and Teng [18] √ √ Dye and Hsieh [14] √ √ √ Dye [15] √ √ √ Zhang, Bai, and Tang [58] √ √ Montgomery, Bazaraa, and Keswani [24] √ Park [28] √ Rosenberg [29] √ Abad [1] √ Chang and Dye [5] √ √ Papachristos and Skouri [27] √ √ Skouri and Papachristos [48] √ √ C$\acute{a}$rdenas-Barr$\acute{o}$n [10] √ Sana [31] √ √ C$\acute{a}$rdenas-Barr$\acute{o}$n [11] √ Widyadana, C$\acute{a}$rdenas-Barr$\acute{o}$n, and Wee [55] √ √ Sarkar and Sarkar [43] √ √ √ Wee, Huang, Wang and Cheng [53] √ Sarkar [37] √ Sarkar, Mandal, and Sarkar [39] √ Mandal and Pal [23] √ √ Wu [56] √ √ √ Wu, Ouyang, and Yang [57] √ √ √ √ Skouri, Konstantaras, Papachristos, and Ganas [49] √ √ √ Sarkar, Sett, Goswami, and Sarkar [45] √ √ √ √ Levin, McLaughlin, Lemone, and Kottas [21] √ Silver and Peterson [47] √ Padmanabhan and Vrat [26] √ √ √ Liao, Tsai, and Su [22] √ √ Dye and Ouyang [16] √ √ Alfares [2] √ Chung and Wee [8] √ √ Sana and Chaudhuri [32] √ Sana [33] √ √ Sarkar [38] √
Sensitivity analysis of the key parameters.
 Parameters Changes (in %) Model 1 Model 2 -50% +00.27 +00.76 -25% +00.14 +00.38 A +25% -00.14 -00.38 +50% -00.27 -00.76 -50% -95.65 -97.17 -25% -48.39 -49.19 s +25% +49.34 +49.97 +50% +99.48 +100.52 -50% +42.97 +44.83 -25% +21.34 +22.18 p +25% -21.10 -21.84 +50% -41.96 -43.41 -50% +04.81 +03.57 -25% +02.12 +01.63 h +25% -01.69 -01.37 +50% -03.04 -02.53 -50% +00.22 +00.42 -25% +00.11 +00.18 b +25% -00.08 -00.14 +50% -00.15 -00.25 -50% +00.06 +00.11 -25% +00.03 +00.05 l +25% -00.03 -00.05 +50% -00.06 -00.09
 Parameters Changes (in %) Model 1 Model 2 -50% +00.27 +00.76 -25% +00.14 +00.38 A +25% -00.14 -00.38 +50% -00.27 -00.76 -50% -95.65 -97.17 -25% -48.39 -49.19 s +25% +49.34 +49.97 +50% +99.48 +100.52 -50% +42.97 +44.83 -25% +21.34 +22.18 p +25% -21.10 -21.84 +50% -41.96 -43.41 -50% +04.81 +03.57 -25% +02.12 +01.63 h +25% -01.69 -01.37 +50% -03.04 -02.53 -50% +00.22 +00.42 -25% +00.11 +00.18 b +25% -00.08 -00.14 +50% -00.15 -00.25 -50% +00.06 +00.11 -25% +00.03 +00.05 l +25% -00.03 -00.05 +50% -00.06 -00.09
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