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doi: 10.3934/jimo.2021123
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## Optimal replenishment, pricing and preservation technology investment policies for non-instantaneous deteriorating items under two-level trade credit policy

 Department of Mathematics, Sidho-Kanho-Birsha University, P.O. – Purulia Sainik School, Dist. – Purulia, PIN – 723104, W.B., India

* Corresponding author: Gour Chandra Mahata

Received  July 2020 Revised  March 2021 Early access July 2021

In the business world, both the supplier and the retailer accept the credit to make their business position strong, because the credit not only strengthens their business relationships but also increases the scale of their profits. In this paper, we consider an inventory model for non-instantaneous deteriorating items with price sensitive demand, time varying deterioration rate under two-level trade credit policy. Besides, to reduce deterioration rate, retailers invest some cost to prevent product degradation/decay, known as preservation technology, is also inserted. Consumption of such items within shelf life prevents to deterioration, which can be achieved by bulk sale. In order to stimulate the selling, trade-credit policy is also considered here. In the sequel, not only the supplier would offer fixed credit period to the retailer, but retailer also adopt the trade credit policy to the customers in order to promote the market competition. The retailer can accumulate revenue and interest after the customer pays for the amount of purchasing cost to the retailer until the end of the trade credit period offered by the supplier. The main objective is to determine the optimal replenishment, pricing and preservation technology investment strategies including whether or not invest in preservation technology and how much to invest in order to maximize the average profit of the system. It is proved that the optimal replenishment policy not only exists but is unique for any given selling price and preservation technology cost. An algorithm is presented to derive the optimal solutions of the model. Numerous theorems and lemmas have been inserted to obtain the optimal solution. Finally, numerical examples and managerial implications are incorporated to validate the proposed model.

Citation: Chandan Mahato, Gour Chandra Mahata. Optimal replenishment, pricing and preservation technology investment policies for non-instantaneous deteriorating items under two-level trade credit policy. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021123
##### References:

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##### References:
Inventory Level
Inventory system for Case 1 when $T \leq t_d$
Inventory system for Case 1 when $T \leq t_d$
Inventory system for Case 2 when $T \leq M-N$
Inventory system for Case 2 when $M-N \leq T \leq t_d$
Inventory system for Case 2 when $T \geq t_d$
Inventory system for Case 3 when $T \leq t_d$
Inventory system for Case 3 when $t_d \leq T \leq M-N$
Inventory system for Case 3 when $T \leq M-N$
Summary of the related research
 Authors Demand factors Demand patterns Deterio-ration Deterio-ration Pattern Non-instan-taneous Trade Credit Level of trade credit Preservation technology Wu et al. [48] Inventory level Linear Yes Constant Yes No - No Chang et al. [3] Constant - Yes Constant Yes Yes One No Hsu et al. [12] Constant - Yes Constant No No - Yes Shastri et al. [40] Selling price Power form Yes Constant No Yes Two Yes Dye & Hsieh [7] Constant - Yes Constant No No - Yes Mahata et al. [28] Selling price Iso-elastic Yes Constant No Yes One No Maihami and Karimi [29] Selling price General Yes Constant Yes No - No Mukherjee & Mahata et al. [32] Time & Credit period General Yes General type No Yes Two No Jaggi et al. [17] Selling price Power form Yes Constant Yes Yes One No Soni [41] Selling price & stock level General Yes Constant No Yes One No Shah et al. [37] Advert-isement of an item & selling price power form Yes General type Yes No - Yes Mishra et al. [31] Selling price Linear Yes Constant No Yes One Yes Yang et al. [49] Time & Credit period General Yes General type No Yes One Yes Present paper Selling price General form Yes General type Yes Yes Two Yes
 Authors Demand factors Demand patterns Deterio-ration Deterio-ration Pattern Non-instan-taneous Trade Credit Level of trade credit Preservation technology Wu et al. [48] Inventory level Linear Yes Constant Yes No - No Chang et al. [3] Constant - Yes Constant Yes Yes One No Hsu et al. [12] Constant - Yes Constant No No - Yes Shastri et al. [40] Selling price Power form Yes Constant No Yes Two Yes Dye & Hsieh [7] Constant - Yes Constant No No - Yes Mahata et al. [28] Selling price Iso-elastic Yes Constant No Yes One No Maihami and Karimi [29] Selling price General Yes Constant Yes No - No Mukherjee & Mahata et al. [32] Time & Credit period General Yes General type No Yes Two No Jaggi et al. [17] Selling price Power form Yes Constant Yes Yes One No Soni [41] Selling price & stock level General Yes Constant No Yes One No Shah et al. [37] Advert-isement of an item & selling price power form Yes General type Yes No - Yes Mishra et al. [31] Selling price Linear Yes Constant No Yes One Yes Yang et al. [49] Time & Credit period General Yes General type No Yes One Yes Present paper Selling price General form Yes General type Yes Yes Two Yes
Sensitivity analysis with respect to different parameters
 Parameter Values $P^{*}$ $\xi^{*}$ $T^{*}$ $Q^{*}$ $TP^{*}$ 80 60.0939 4.7922 0.2097 83.1619 $TP^{*}_{23}=15543.44$ 100 60.1823 4.9543 0.2514 99.7122 $TP^{*}_{23}=15456.70$ A 120 60.2498 5.1039 0.2868 113.7114 $TP^{*}_{23}=15382.39$ 140 60.3056 5.2469 0.3180 126.0302 $TP^{*}_{23}=15316.27$ 160 60.3540 5.3864 0.3463 137.1341 $TP^{*}_{23}=15256.08$ 10 55.2849 5.3094 0.3289 146.8334 $TP^{*}_{33}=19629.55$ 15 57.7692 5.1910 0.3053 128.6614 $TP^{*}_{23}=17441.33$ c 20 60.2498 5.1039 0.2868 113.7114 $TP^{*}_{23}=15382.39$ 25 62.7284 5.0393 0.2725 101.2497 $TP^{*}_{23}=13451.34$ 30 65.2071 4.9925 0.2618 90.7764 $TP^{*}_{23}=11647.08$ 1 60.2001 5.4634 0.3623 144.0668 $TP^{*}_{33}=15509.14$ 2 60.2272 5.2450 0.3185 126.4466 $TP^{*}_{23}=15441.97$ h 3 60.2498 5.1039 0.2868 113.7114 $TP^{*}_{23}=15382.39$ 4 60.2691 5.0048 0.2626 103.9802 $TP^{*}_{23}=15328.48$ 5 60.2859 4.9310 0.2433 96.2400 $TP^{*}_{23}=15279.01$ 0.02 60.2890 5.5741 0.3011 119.4436 $TP^{*}_{33}=15371.08$ 0.05 60.2768 5.3873 0.2964 117.5873 $TP^{*}_{23}=15374.53$ $t_{0}$ 0.10 60.2498 5.1039 0.2868 113.7114 $TP^{*}_{23}=15382.39$ 0.15 60.2115 4.8468 0.2742 108.6560 $TP^{*}_{23}=15393.68$ 0.20 60.1570 4.6081 0.2580 102.1061 $TP^{*}_{23}=15409.61$
 Parameter Values $P^{*}$ $\xi^{*}$ $T^{*}$ $Q^{*}$ $TP^{*}$ 80 60.0939 4.7922 0.2097 83.1619 $TP^{*}_{23}=15543.44$ 100 60.1823 4.9543 0.2514 99.7122 $TP^{*}_{23}=15456.70$ A 120 60.2498 5.1039 0.2868 113.7114 $TP^{*}_{23}=15382.39$ 140 60.3056 5.2469 0.3180 126.0302 $TP^{*}_{23}=15316.27$ 160 60.3540 5.3864 0.3463 137.1341 $TP^{*}_{23}=15256.08$ 10 55.2849 5.3094 0.3289 146.8334 $TP^{*}_{33}=19629.55$ 15 57.7692 5.1910 0.3053 128.6614 $TP^{*}_{23}=17441.33$ c 20 60.2498 5.1039 0.2868 113.7114 $TP^{*}_{23}=15382.39$ 25 62.7284 5.0393 0.2725 101.2497 $TP^{*}_{23}=13451.34$ 30 65.2071 4.9925 0.2618 90.7764 $TP^{*}_{23}=11647.08$ 1 60.2001 5.4634 0.3623 144.0668 $TP^{*}_{33}=15509.14$ 2 60.2272 5.2450 0.3185 126.4466 $TP^{*}_{23}=15441.97$ h 3 60.2498 5.1039 0.2868 113.7114 $TP^{*}_{23}=15382.39$ 4 60.2691 5.0048 0.2626 103.9802 $TP^{*}_{23}=15328.48$ 5 60.2859 4.9310 0.2433 96.2400 $TP^{*}_{23}=15279.01$ 0.02 60.2890 5.5741 0.3011 119.4436 $TP^{*}_{33}=15371.08$ 0.05 60.2768 5.3873 0.2964 117.5873 $TP^{*}_{23}=15374.53$ $t_{0}$ 0.10 60.2498 5.1039 0.2868 113.7114 $TP^{*}_{23}=15382.39$ 0.15 60.2115 4.8468 0.2742 108.6560 $TP^{*}_{23}=15393.68$ 0.20 60.1570 4.6081 0.2580 102.1061 $TP^{*}_{23}=15409.61$
The numerical results for different values of M and N
 M N $P^{*}$ $\xi^{*}$ $T^{*}$ $Q^{*}$ $TP^{*}$ 0.0 60.4012 5.1072 0.2875 112.4756 $TP^{*}_{23}=15263.37$ 0.2 60.7039 5.1139 0.2890 112.7563 $TP^{*}_{12}=15026.68$ 0.0 0.4 61.0067 5.1207 0.2906 113.0335 $TP^{*}_{12}=14791.82$ 0.6 61.3095 5.1276 0.2921 113.3073 $TP^{*}_{12}=14558.77$ 0.8 61.6124 5.1345 0.2937 113.5776 $TP^{*}_{12}=14327.53$ 0.0 60.0984 5.1006 0.2860 112.7563 $TP^{*}_{33}=15501.87$ 0.2 60.4212 5.1085 0.2879 113.0335 $TP^{*}_{23}=15287.52$ 0.2 0.4 61.7039 5.1139 0.2890 113.3073 $TP^{*}_{12}=15026.68$ 0.6 61.0067 5.1207 0.2906 113.5776 $TP^{*}_{12}=14791.82$ 0.8 61.3095 5.1276 0.2921 113.8444 $TP^{*}_{12}=14558.77$ 0.0 59.7957 5.0941 0.2845 113.0335 $TP^{*}_{32}=15742.19$ 0.2 60.0984 5.1006 0.2860 113.3073 $TP^{*}_{33}=15501.87$ 0.4 0.4 60.4302 5.1092 0.2881 113.5776 $TP^{*}_{32}=15293.23$ 0.6 60.7039 5.1139 0.2892 113.8444 $TP^{*}_{12}=15026.68$ 0.8 61.0067 5.1207 0.2906 114.1078 $TP^{*}_{12}=14791.82$ 0.0 59.4930 5.0877 0.2831 113.3073 $TP^{*}_{32}=15984.32$ 0.2 59.7957 5.0941 0.2845 113.5776 $TP^{*}_{32}=15742.19$ 0.6 0.4 60.0984 5.1006 0.2860 113.8444 $TP^{*}_{33}=15501.87$ 0.6 60.4612 5.1102 0.2878 114.1078 $TP^{*}_{23}=15301.15$ 0.8 60.7039 5.1139 0.2890 114.3679 $TP^{*}_{12}=15026.68$ 0.0 59.1903 5.0814 0.2816 113.5776 $TP^{*}_{32}=16228.27$ 0.2 59.4930 5.0877 0.2831 113.8444 $TP^{*}_{32}=15984.32$ 0.8 0.4 59.7957 5.0941 0.2845 114.1078 $TP^{*}_{32}=15742.19$ 0.6 60.0984 5.1006 0.2867 114.3679 $TP^{*}_{33}=15501.87$ 0.8 60.4825 5.1172 0.2885 114.6246 $TP^{*}_{23}=15374.12$
 M N $P^{*}$ $\xi^{*}$ $T^{*}$ $Q^{*}$ $TP^{*}$ 0.0 60.4012 5.1072 0.2875 112.4756 $TP^{*}_{23}=15263.37$ 0.2 60.7039 5.1139 0.2890 112.7563 $TP^{*}_{12}=15026.68$ 0.0 0.4 61.0067 5.1207 0.2906 113.0335 $TP^{*}_{12}=14791.82$ 0.6 61.3095 5.1276 0.2921 113.3073 $TP^{*}_{12}=14558.77$ 0.8 61.6124 5.1345 0.2937 113.5776 $TP^{*}_{12}=14327.53$ 0.0 60.0984 5.1006 0.2860 112.7563 $TP^{*}_{33}=15501.87$ 0.2 60.4212 5.1085 0.2879 113.0335 $TP^{*}_{23}=15287.52$ 0.2 0.4 61.7039 5.1139 0.2890 113.3073 $TP^{*}_{12}=15026.68$ 0.6 61.0067 5.1207 0.2906 113.5776 $TP^{*}_{12}=14791.82$ 0.8 61.3095 5.1276 0.2921 113.8444 $TP^{*}_{12}=14558.77$ 0.0 59.7957 5.0941 0.2845 113.0335 $TP^{*}_{32}=15742.19$ 0.2 60.0984 5.1006 0.2860 113.3073 $TP^{*}_{33}=15501.87$ 0.4 0.4 60.4302 5.1092 0.2881 113.5776 $TP^{*}_{32}=15293.23$ 0.6 60.7039 5.1139 0.2892 113.8444 $TP^{*}_{12}=15026.68$ 0.8 61.0067 5.1207 0.2906 114.1078 $TP^{*}_{12}=14791.82$ 0.0 59.4930 5.0877 0.2831 113.3073 $TP^{*}_{32}=15984.32$ 0.2 59.7957 5.0941 0.2845 113.5776 $TP^{*}_{32}=15742.19$ 0.6 0.4 60.0984 5.1006 0.2860 113.8444 $TP^{*}_{33}=15501.87$ 0.6 60.4612 5.1102 0.2878 114.1078 $TP^{*}_{23}=15301.15$ 0.8 60.7039 5.1139 0.2890 114.3679 $TP^{*}_{12}=15026.68$ 0.0 59.1903 5.0814 0.2816 113.5776 $TP^{*}_{32}=16228.27$ 0.2 59.4930 5.0877 0.2831 113.8444 $TP^{*}_{32}=15984.32$ 0.8 0.4 59.7957 5.0941 0.2845 114.1078 $TP^{*}_{32}=15742.19$ 0.6 60.0984 5.1006 0.2867 114.3679 $TP^{*}_{33}=15501.87$ 0.8 60.4825 5.1172 0.2885 114.6246 $TP^{*}_{23}=15374.12$

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