Imperfect Reworking Process Consideration in Integrated Inventory Model under Permissible Delay in Payments

Table 1

Optimal solutions for various values of permissible delay period 𝑚(𝜆=1.50).

𝑚	𝑛^∗	𝑝0	𝑝^∗	𝐿^∗ (day)	𝐷(𝑝^∗)	𝑄^∗	𝑛^∗𝑄^∗	Profit ($/yr)
(day)								Vendor	Buyer	Joint
0	10	—	p2 = 8.6191	L2 = 65.9521	3951.9107	714.0734	7140.7344	6542.7743	15639.3831	22182.1574
5	10	0.2313	p2 = 8.6097	L2 = 65.8780	3958.3844	714.4393	7144.3930	6546.5177	15648.1525	22194.6702
10	10	0.5903	p2 = 8.6007	L2 = 65.7660	3964.5993	714.3443	7143.4434	6549.8114	15658.0354	22207.8468
15	10	1.0290	p2 = 8.5920	L2 = 65.6157	3970.6225	713.7949	7137.9488	6552.7706	15668.9163	22221.6869
20	10	1.5361	p2 = 8.5837	L2 = 65.4278	3976.3830	712.7838	7127.8381	6555.2744	15680.9170	22236.1914
25	10	2.1074	p2 = 8.5758	L2 = 65.2024	3981.8788	711.3094	7113.0936	6557.3194	15694.0429	22251.3623
30	10	2.7421	p2 = 8.5683	L2 = 64.9392	3987.1080	709.3688	7093.6881	6558.9020	15708.3011	22267.2031
40	10	4.2066	p2 = 8.5545	L2 = 64.2988	3996.7599	704.0736	7040.7364	6560.6634	15740.2516	22300.9150
41	10	4.3680	p2 = 8.5532	L2 = 64.2263	3997.6711	703.4398	7034.3977	6560.7450	15743.6916	22304.4366
42	10	4.5321	p2 = 8.5519	L2 = 64.1521	3998.5827	702.7877	7027.8768	6560.8266	15747.1593	22307.9859
43	10	4.6991	p2 = 8.5506	L2 = 64.0763	3999.4946	702.1173	7021.1732	6560.9080	15750.6547	22311.5627
44	10	4.8689	p2 = 8.5494	L2 = 63.9994	4000.3367	701.4221	7014.2213	6560.8706^(b)	15754.2967	22315.1673
45	10	5.0417	p2 = 8.5482	L2 = 63.9208	4001.1791	700.7085	7007.0854	6560.8330	15757.9668	22318.7998
50	10	5.9502	p2 = 8.5424	L2 = 63.5041	4005.2548	696.8497	6968.4967	6560.4028	15776.9794	22337.3822
60	11	8.4755	p2 = 8.5341	L2 = 60.2724	4011.0993	662.3522	7285.8737	6556.3448	15820.4483	22376.7931
70	11	11.0359	p1 = 8.5309	L1 = 60.2343	4013.3564	662.3060^(a)	7285.3655	6545.3169	15872.9112	22418.2281
80	11	14.0380	p1 = 8.5280	L1 = 60.2221	4015.4037	662.5092	7287.6008	6533.9086	15925.7575	22459.6661
90	11	17.5552	p1 = 8.5250	L1 = 60.2094	4017.5234	662.7195	7289.9147	6522.6064	15978.4997	22501.1061

^(a) When 0 < m ≤ 70, the optimal order quantity 𝑄^∗ is negatively correlated to the length of the permissible delay period m.
^(b) When m < 44, the vendor’s expected total annual profit is positively correlated to the length of the permissible delay period m but as m ≥ 44, it is reverse.