Optimal replenishment, pricing and preservation technology investment policies for non-instantaneous deteriorating items under two-level trade credit policy

Chandan Mahato; Gour Chandra Mahata

doi:10.3934/jimo.2021123

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2022, Volume 18, Issue 5: 3499-3537. 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

Chandan Mahato and
Gour Chandra Mahata^,

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

^* Corresponding author: Gour Chandra Mahata
^* Corresponding author: Gour Chandra Mahata

Received: June 30, 2020

Revised: February 28, 2021

Early access: July 2021

Published: November 2022

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 90B05; Secondary: 90C26.

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Figure 1. Inventory Level

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Figure 2. Inventory system for Case 1 when $ T \leq t_d $

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Figure 3. Inventory system for Case 1 when $ T \leq t_d $

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Figure 4. Inventory system for Case 2 when $ T \leq M-N $

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Figure 5. Inventory system for Case 2 when $ M-N \leq T \leq t_d $

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Figure 6. Inventory system for Case 2 when $ T \geq t_d $

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Figure 7. Inventory system for Case 3 when $ T \leq t_d $

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Figure 8. Inventory system for Case 3 when $ t_d \leq T \leq M-N $

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Figure 9. Inventory system for Case 3 when $ T \leq M-N $

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Table 1. 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

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Table 2. 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 $

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Table 3. 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 $

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