Imperfection with inspection policy and variable demand under trade-credit: A deteriorating inventory model

Sankar Kumar Roy; Magfura Pervin; Gerhard Wilhelm Weber

doi:10.3934/naco.2019032

Article Contents

2020, Volume 10, Issue 1: 45-74. Doi: 10.3934/naco.2019032

This issue Previous Article Initial guess sensitivity in computational optimal control problems Next Article Existence and iterative approximation method for solving mixed equilibrium problem under generalized monotonicity in Banach spaces

Imperfection with inspection policy and variable demand under trade-credit: A deteriorating inventory model

1.
Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721102, West Bengal, India
2.
Faculty of Engineering Management, Chair of Marketing and Economic Engineering, Poznan University of Technology, ul. Strzelecka 11, 60-965 Poznan, Poland

^* Corresponding Author: sankroy2006@gmail.com
^* Corresponding Author: sankroy2006@gmail.com

The author, Magfura Pervin is very thankful to University Grants Commission (UGC) of India for providing financial support to continue this research work under [MANF(UGC)] scheme: Sanctioned letter number [F1-17.1/2012-13/MANF-2012-13-MUS-WES-19170 /(SA-Ⅲ/Website)] dated 28/02/2013

Received: July 31, 2018

Revised: March 31, 2019

Published: February 2020

The research of Gerhard-Wilhelm Weber (Institute of Applied Mathematics, METU, 06800, Ankara, Turkey) is partially supported by the Portuguese Foundation for Science and Technology ("FCT-Fundação para a Ciência e a Tecnologia"), through the CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/ 04106/2013.

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Abstract

A deteriorating inventory model with imperfect product and variable demand is formulated in this paper. A time-dependent deterioration factor is considered because the rate of deterioration is highly hinging on time. We introduce imperfect quality of production which leads to imperfect items in our proposed model. The retailer adopts inspection policy to pick over the perfect items from imperfect. Type Ⅰ and Type Ⅱ, both type of errors are included and the retailer invest some capital to improve the production process quality of the supplier. There is also a penalty cost for the retailer if they deliver some defective items by mistake. Sometime, there is a high amount of demand and, consequently, we assume shortages and partial backorder in our formulated model. The retailer adopts the trade-credit policy for his customers in order to promote market competition. The main objective of the paper is to show that the total cost is globally minimized and we have aimed at reducing the total cycle length, defectiveness of the system and the optimal order size by maximizing the total profit of the system. Then, we present three theorems and prove them to find an easy solution procedure to reduce the total cost of a system. The results are discussed with the help of numerical examples to approve the proposed model. A sensitivity analysis of the optimal solutions for the parameters is also provided. The paper ends with the conclusions and an outlook to possible future studies.

Keywords:

Mathematics Subject Classification: Primary: 90B05, 91A10; Secondary: 90C26.

Citation:

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Figure 1. Graphical representation of our proposed inventory control model

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Figure 2. Graphical description to show the convexity of the total cost. Included are $ T $, $ \alpha $ and the total cost $ R_2(T) $, along the x-axis, the y-axis and the z-axis, respectively

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Figure 3. Graphical presentation to show the convexity of the total cost. The Portrayed are $ T $, $ q $ and the total cost $ R_1(T) $, along the x-axis, the y-axis and the z-axis, respectively

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Figure 4. Graphical representation to show the convexity of the total cost. Represented are $ \alpha $, $ q $ and the total cost $ R_3(T) $, along the x-axis, the y-axis and the z-axis, respectively

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Table 1. Contributions of many authors related to inventory model

Author(s)	Imperfec tness	Variable demand	Deterio rations	Trade-credit policy	Short ages
Sana et al. (2004)	$ \surd $		$ \surd $		$ \surd $
Khan et al. (2014)	$ \surd $
Lin and Hou (2015)	$ \surd $
Teng et al. (2002)		$ \surd $	$ \surd $		$ \surd $
Ghosh and Chaudhuri (2004)		$ \surd $	$ \surd $		$ \surd $
Kumar et al. (2012)	$ \surd $	$ \surd $
Pervin et al. (2016, a)		$ \surd $	$ \surd $	$ \surd $
Mahata (2012)			$ \surd $	$ \surd $
Goswami and Chaudhuri (1991)		$ \surd $	$ \surd $		$ \surd $
Aggarwal and Jaggi (1995)			$ \surd $	$ \surd $
Ting (2015)			$ \surd $	$ \surd $
Ouyang et al. (2005)			$ \surd $	$ \surd $	$ \surd $
Tripathi (2015)		$ \surd $		$ \surd $	$ \surd $
Annadurai and Uthayakumar (2015)		$ \surd $		$ \surd $	$ \surd $
Balkhi (2004)	$ \surd $		$ \surd $
Pervin et al. (2016, b)		$ \surd $	$ \surd $		$ \surd $
Our paper	$ \surd $	$ \surd $	$ \surd $	$ \surd $	$ \surd $

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Table 2. Computational result. Bold style represents the optimal solution

$ q $	$ \alpha^* $	$ T_1^* $	$ T_2^* $	$ T_3^* $	$ R_1(T,q,\alpha) $	$ R_2(T,q,\alpha) $	$ R_3(T,q,\alpha) $
$ 1 $	0.1077	1.235	1.302	1.187	3478.12	3599.45	3517.30
$ 2 $	0.1125	1.301	1.213	1.171	3560.82	3011.67	3069.57
$ 3 $	0.1092	1.420	1.387	1.432	3673.19	3095.07	3112.70
$ {\bf{4}} $	0.1089	1.324	1.112	1.108	3488.22	2973.16	2844.93
$ 5 $	0.1107	1.403	1.274	1.392	3579.08	3077.45	3205.71
$ 6 $	0.1073	1.449	1.171	1.321	3678.99	3360.21	3119.54
$ 7 $	0.128	1.610	1.534	1.558	3879.04	3232.70	3374.56
$ 8 $	0.116	1.791	1.645	1.325	3927.34	3306.18	3317.84
$ 9 $	0.125	1.570	1.349	1.102	4012.57	3812.05	3518.38
$ 10 $	0.144	1.397	1.747	1.560	3939.55	3670.10	3575.68

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Table 3. Sensitivity analysis for various parameters involved in Example 1

Para meter	$ \% $ change	value	*$ T_1^ $**	*$ T_2^ $**	*$ T_3^ $**	$ R_1(T) $	$ R_2(T) $	$ R_3(T) $	Total Cost	Profit
$ A $	+50	1350	1.832	1.572	1.624	3122.05	2705.91	2812.83	$ \downarrow $	$ \uparrow $
	+30	1170	1.510	1.480	1.613	2988.71	2633.61	2694.74	$ \downarrow $	$ \uparrow $
	+10	990	1.257	1.437	1.589	2734.00	2482.09	2503.15	$ \downarrow $	$ \uparrow $
	-10	810	0.941	1.428	1.541	2613.47	2290.41	2387.40	$ \downarrow $	$ \uparrow $
	-30	630	0.703	1.410	1.516	2497.38	2100.16	2206.85	$ \downarrow $	$ \uparrow $
	-50	450	0.627	1.377	1.487	2206.16	2000.00	2183.57	$ \downarrow $	$ \uparrow $
$ s $	+50	600	0.904	0.867	0.880	3385.10	2834.61	2791.38	$ \downarrow $	$ \uparrow $
	+30	520	0.760	0.843	0.861	3174.51	2690.42	2614.50	$ \downarrow $	$ \uparrow $
	+10	440	0.601	0.829	0.847	2893.67	2517.83	2576.31	$ \downarrow $	$ \uparrow $
	-10	360	0.472	0.820	0.831	2635.85	2385.76	2344.10	$ \downarrow $	$ \uparrow $
	-30	280	0.275	0.816	0.822	2483.77	2189.02	2208.47	$ \downarrow $	$ \uparrow $
	-50	200	0.064	0.803	0.815	2305.38	1985.47	2083.17	$ \downarrow $	$ \uparrow $
$ h_1 $	+50	150	1.966	1.973	1.989	3192.70	2911.56	2810.55	$ \downarrow $	$ \uparrow $
	+30	130	1.728	1.854	1.910	2948.16	2738.64	2655.18	$ \downarrow $	$ \uparrow $
	+10	110	1.601	1.779	1.878	2749.03	2500.53	2451.38	$ \downarrow $	$ \uparrow $
	-10	90	1.486	1.611	1.763	2605.74	2374.95	2180.76	$ \downarrow $	$ \uparrow $
	-30	70	1.310	1.572	1.683	2477.19	2285.31	2090.59	$ \downarrow $	$ \uparrow $
	-50	50	1.173	1.489	1.557	2204.60	2069.73	1877.92	$ \downarrow $	$ \uparrow $
$ h_2 $	+50	300	2.765	2.581	2.593	3592.00	2746.31	2522.20	$ \downarrow $	$ \uparrow $
	+30	260	2.303	2.505	2.512	3386.17	2506.57	2257.46	$ \downarrow $	$ \uparrow $
	+10	220	2.007	2.488	2.502	3152.04	2344.10	2037.18	$ \downarrow $	$ \uparrow $
	-10	180	1.841	2.394	2.581	2819.11	2183.51	1734.62	$ \downarrow $	$ \uparrow $
	-30	140	1.573	2.353	2.473	2540.83	1982.86	1504.27	$ \downarrow $	$ \uparrow $
	-50	100	1.250	2.279	2.371	2274.62	1710.27	1374.02	$ \downarrow $	$ \uparrow $
$ a $	+50	22.5	2.791	2.880	2.898	2916.47	3721.40	2883.45	$ \downarrow $	$ \uparrow $
	+30	19.5	2.675	2.741	2.805	2803.15	3569.00	2761.74	$ \downarrow $	$ \uparrow $
	+10	16.5	2.533	2.689	2.775	2785.32	3477.54	2653.14	$ \downarrow $	$ \uparrow $
	-10	13.5	2.412	2.537	2.643	2511.63	3307.11	2578.39	$ \downarrow $	$ \uparrow $
	-30	10.5	2.370	2.481	2.550	2479.59	3258.46	2386.55	$ \downarrow $	$ \uparrow $
	-50	7.5	2.264	2.372	2.445	2333.16	3117.35	2290.28	$ \downarrow $	$ \uparrow $
$ b $	+50	30	2.672	2.503	2.475	3599.89	2800.00	2795.64	$ \downarrow $	$ \uparrow $
	+30	26	2.538	2.484	2.449	3512.04	2715.35	2683.05	$ \downarrow $	$ \uparrow $
	+10	22	2.460	2.336	2.429	3475.68	2610.97	2500.34	$ \downarrow $	$ \uparrow $
	-10	18	2.397	2.260	2.405	3381.62	2573.61	2435.65	$ \downarrow $	$ \uparrow $
	-30	14	2.200	2.175	2.391	3260.91	2483.29	2362.41	$ \downarrow $	$ \uparrow $
	-50	10	2.822	2.124	2.364	3129.56	2321.79	2155.20	$ \downarrow $	$ \uparrow $
$ \gamma $	+50	0.60	1.754	2.134	2.431	3529.87	3138.56	2490.18	$ \downarrow $	$ \uparrow $
	+30	0.52	1.694	2.089	2.379	3348.72	2763.79	2317.93	$ \downarrow $	$ \uparrow $
	+10	0.44	1.504	1.905	2.248	3095.23	2480.58	2188.37	$ \downarrow $	$ \uparrow $
	-10	0.36	1.438	1.881	2.179	2860.93	2201.49	2020.99	$ \downarrow $	$ \uparrow $
	-30	0.28	1.215	1.753	2.098	2547.65	1973.10	1845.73	$ \downarrow $	$ \uparrow $
	-50	0.20	1.093	1.668	1.979	2196.37	1748.53	1549.27	$ \downarrow $	$ \uparrow $
$ r_1 $	+50	75	2.402	1.402	1.657	3891.57	3124.51	3054.36	$ \downarrow $	$ \uparrow $
	+30	65	2.763	1.435	1.661	3522.56	2910.28	2839.17	$ \downarrow $	$ \uparrow $
	+10	55	2.911	1.492	1.704	3205.84	2764.33	2780.10	$ \downarrow $	$ \uparrow $
	-10	45	3.250	1.555	1.783	2935.27	2459.37	2638.61	$ \downarrow $	$ \uparrow $
	-30	35	3.619	1.784	2.044	2789.42	2218.06	2529.50	$ \downarrow $	$ \uparrow $
	-50	25	3.873	1.976	2.423	2642.17	2049.62	2485.73	$ \downarrow $	$ \uparrow $

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Table 4. Sensitivity analysis for various parameters involved in Example 1

Para meter	$ \% $ change	value	*$ T_1^ $**	*$ T_2^ $**	*$ T_3^ $**	$ R_1(T) $	$ R_2(T) $	$ R_3(T) $	Total Cost	Profit
$ r_2 $	+50	60	2.402	2.265	1.875	3891.57	3124.51	3054.36	$ \downarrow $	$ \uparrow $
	+30	52	2.257	2.203	1.713	3522.56	2910.28	2839.17	$ \downarrow $	$ \uparrow $
	+10	44	2.105	2.165	1.681	3205.84	2764.33	2780.10	$ \downarrow $	$ \uparrow $
	-10	36	1.943	2.081	1.578	2935.27	2459.37	2638.61	$ \downarrow $	$ \uparrow $
	-30	28	1.870	1.863	1.475	2789.42	2218.06	2529.50	$ \downarrow $	$ \uparrow $
	-50	20	1.655	1.756	1.409	2642.17	2049.62	2485.73	$ \downarrow $	$ \uparrow $
$ r_3 $	+50	15	2.795	2.531	2.320	3891.57	3124.51	3054.36	$ \downarrow $	$ \uparrow $
	+30	13	2.661	2.457	2.259	3522.56	2910.28	2839.17	$ \downarrow $	$ \uparrow $
	+10	11	2.573	2.347	2.188	3205.84	2764.33	2780.10	$ \downarrow $	$ \uparrow $
	-10	9	2.412	2.201	2.096	2935.27	2459.37	2638.61	$ \downarrow $	$ \uparrow $
	-30	7	2.345	2.179	2.016	2789.42	2218.06	2529.50	$ \downarrow $	$ \uparrow $
	-50	5	2.235	2.153	1.875	2642.17	2049.62	2485.73	$ \downarrow $	$ \uparrow $
$ r_4 $	+50	60	1.987	1.853	1.760	3891.57	3124.51	3054.36	$ \downarrow $	$ \uparrow $
	+30	52	1.664	1.571	1.483	3522.56	2910.28	2839.17	$ \downarrow $	$ \uparrow $
	+10	44	1.370	1.356	1.338	3205.84	2764.33	2780.10	$ \downarrow $	$ \uparrow $
	-10	36	1.297	1.283	1.275	2935.27	2459.37	2638.61	$ \downarrow $	$ \uparrow $
	-30	28	1.264	1.257	1.241	2789.42	2218.06	2529.50	$ \downarrow $	$ \uparrow $
	-50	20	1.239	1.221	1.195	2642.17	2049.62	2485.73	$ \downarrow $	$ \uparrow $
$ r_5 $	+50	45	2.354	2.279	2.165	3567.10	3049.31	3098.21	$ \downarrow $	$ \uparrow $
	+30	39	2.170	2.086	1.952	3417.05	2994.22	2947.18	$ \downarrow $	$ \uparrow $
	+10	33	1.941	1.876	1.740	3307.53	2885.31	2856.37	$ \downarrow $	$ \uparrow $
	-10	27	1.769	1.635	1.509	3268.37	2765.42	2664.00	$ \downarrow $	$ \uparrow $
	-30	21	1.526	1.470	1.358	3191.30	2670.51	2509.11	$ \downarrow $	$ \uparrow $
	-50	15	1.391	1.258	1.174	3034.67	2500.00	2481.03	$ \downarrow $	$ \uparrow $
$ \beta $	+50	0.45	1.310	1.465	1.589	2473.11	1730.22	2017.27	$ \uparrow $	$ \downarrow $
	+30	0.39	1.574	1.640	1.736	2705.43	1918.47	2384.53	$ \uparrow $	$ \downarrow $
	+10	0.33	1.812	1.922	1.989	3058.24	2164.38	2673.40	$ \uparrow $	$ \downarrow $
	-10	0.27	2.123	2.070	2.145	3347.80	2351.06	2945.82	$ \uparrow $	$ \downarrow $
	-30	0.21	2.492	2.195	2.287	3692.47	2537.64	3255.70	$ \uparrow $	$ \downarrow $
	-50	0.15	2.760	2.358	2.460	3903.12	2799.00	3509.35	$ \uparrow $	$ \downarrow $
$ \delta $	+50	30	2.385	2.072	2.110	3392.46	2918.36	2840.79	$ \downarrow $	$ \uparrow $
	+30	26	2.200	1.915	2.048	3175.13	2875.42	2711.54	$ \downarrow $	$ \uparrow $
	+10	22	2.071	1.854	1.991	2917.00	2704.60	2657.18	$ \downarrow $	$ \uparrow $
	-10	18	1.848	1.774	1.867	2739.26	2538.11	2302.63	$ \downarrow $	$ \uparrow $
	-30	14	1.639	1.640	1.790	2543.35	2326.83	2199.29	$ \downarrow $	$ \uparrow $
	-50	10	1.325	1.573	1.615	2319.67	2100.00	1978.63	$ \downarrow $	$ \uparrow $
$ M $	+50	1.8	1.367	1.470	1.585	3141.47	2758.10	2890.05	$ \uparrow $	$ \downarrow $
	+30	1.56	1.532	1.609	1.779	3250.26	2893.55	2918.62	$ \uparrow $	$ \downarrow $
	+10	1.32	1.727	1.826	1.963	3319.51	2964.28	3022.37	$ \uparrow $	$ \downarrow $
	-10	1.08	1.904	1.995	2.057	3471.43	3027.83	3126.29	$ \uparrow $	$ \downarrow $
	-30	0.84	2.157	2.257	2.376	3520.79	3119.44	3257.81	$ \uparrow $	$ \downarrow $
	-50	0.6	2.370	2.450	2.538	3752.00	3348.63	3486.90	$ \uparrow $	$ \downarrow $
$ \theta $	+50	0.6	2.752	2.568	2.330	4180.57	3042.78	3506.79	$ \downarrow $	$ \uparrow $
	+30	0.52	2.695	2.418	2.257	3912.36	2965.03	3481.20	$ \downarrow $	$ \uparrow $
	+10	0.44	2.516	2.343	2.194	3764.25	2860.24	3358.18	$ \downarrow $	$ \uparrow $
	-10	0.36	2.404	2.267	2.210	3580.19	2725.57	3293.49	$ \downarrow $	$ \uparrow $
	-30	0.28	2.279	2.001	1.935	3318.38	2530.29	3167.16	$ \downarrow $	$ \uparrow $
	-50	0.20	2.011	1.883	1.520	3127.93	2350.03	2948.57	$ \downarrow $	$ \uparrow $

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