PRODUCTION INVENTORY MODELS WITH INTEGRATED STOCK AND PRICE DEPENDENT DEMANDS FOR DETERIORATIVE ITEMS

θ	T	Q	T₁	Q₁	Setup cost	Holding cost	DC	Total cost	Total Profit
0.01	0.5558	250.12	0.5002	25.01	233.88	162.58	16.25	58912.72	8587.27
0.02	0.5342	240.40	0.4808	24.04	243.34	156.26	31.25	58930.85	8569.14
0.03	0.5150	231.75	0.4635	23.17	252.41	150.64	45.19	58948.25	8551.74
0.04	0.4977	224.00	0.4480	22.40	261.14	145.60	58.24	58964.99	8535.00
0.05	0.4822	217.00	0.4340	21.70	269.57	141.05	70.52	58981.15	8518.84
0.06	0.4680	210.64	0.4212	21.06	277.72	136.91	82.15	58996.71	8503.20
0.07	0.4551	204.81	0.4096	20.48	285.62	133.13	93.19	59011.94	8488.05
0.08	0.4432	199.45	0.3989	19.95	293.29	129.64	103.71	59026.66	8473.33
0.09	0.4322	194.51	0.3890	19.45	300.75	126.43	113.78	59040.97	8459.02
0.10	0.4220	189.91	0.3798	18.99	308.02	123.44	123.44	59054.92	8445.07

x	T	Q	T ₁	Q ₁	Setup cost	Holding cost	DC	Total cost	Total Profit
21	0.3540	111.54	0.2230	41.26	367.13	268.25	26.82	41612.21	5637.78
22	0.3609	119.12	0.2382	40.50	360.11	263.27	26.32	43549.71	5950.28
23	0.3697	127.54	0.2550	39.54	351.62	257.01	25.70	45484.34	6265.65
24	0.3805	137.01	0.2740	38.36	341.56	249.36	24.93	47415.87	6584.12
25	0.3942	147.82	0.2956	36.95	329.77	240.21	24.02	49344.01	6905.98
26	0.4113	160.41	0.3208	35.29	316.06	229.38	22.93	51268.38	7231.61
27	0.4331	175.42	0.3508	33.32	300.13	216.64	21.66	53188.44	7561.55
28	0.4616	193.89	0.3877	31.02	281.60	201.64	20.16	55103.41	7896.58
29	0.5002	217.62	0.4352	28.29	259.84	183.89	18.38	57012.13	8237.86
30	0.5558	250.12	0.5002	25.01	233.88	162.58	16.25	58912.72	8587.27

a	T	Q	T ₁	Q ₁	Setup cost	Holding cost	DC	Total cost	Total Profit
21	0.3745	67.42	0.1348	43.15	347.06	280.47	28.04	24055.59	2944.40
22	0.3625	76.13	0.1522	44.16	358.55	287.04	28.70	27974.30	3525.69
23	0.3565	85.56	0.1711	44.49	364.65	289.19	28.91	31882.76	4117.23
24	0.3556	96.02	0.1020	44.16	365.54	287.09	28.70	35781.35	4718.64
25	0.3599	107.97	0.2159	43.18	361.19	280.73	28.07	39670.00	5329.99
26	0.3700	122.11	0.2442	41.51	351.31	269.86	26.98	43548.17	5951.82
27	0.3876	139.56	0.2791	39.07	335.32	254.00	25.40	47414.77	6585.26
28	0.4164	162.41	0.3248	35.73	312.16	232.25	23.22	51267.64	7232.35
29	0.4646	195.14	0.3902	31.22	279.79	202.94	20.29	55103.03	7896.96
30	0.5558	250.12	0.5002	25.01	233.88	162.58	16.25	58912.72	8587.27

Cost Parameters		Optimum Solutions
Cost Parameters		T	Q	T ₁	Q ₁	Setup cost	Holding cost	DC	Total cost	Total Profit
C ₀	110	0.5156	232.06	0.4641	23.20	213.30	150.84	15.08	58879.22	8620.77
	120	0.5362	241.31	0.4826	24.13	223.77	156.85	15.68	58896.31	8603.68
	130	0.5558	250.12	0.5002	25.01	233.88	162.58	16.25	58912.72	8587.27
	140	0.5745	258.54	0.5171	25.85	243.66	168.05	16.80	58928.53	8571.46
	150	0.5924	266.62	0.5332	26.66	253.16	173.30	17.33	58943.80	8556.19
C _h	11	0.5944	267.51	0.5350	26.75	218.67	147.13	17.38	58883.20	8616.80
	12	0.5741	258.36	0.5167	25.83	226.42	155.01	16.79	58898.23	8601.72
	13	0.5558	250.12	0.5002	25.01	233.88	162.58	16.25	58912.72	8587.27
	14	0.5392	242.66	0.4853	24.26	241.06	169.86	15.77	58926.71	8573.28
	15	0.5241	235.86	0.4717	23.58	248.01	176.90	15.33	58940.25	8559.74
C _d	110	0.5593	251.70	0.5034	25.17	232.41	163.61	13.84	58909.86	8590.13
	120	0.5575	250.91	0.5018	25.09	233.14	163.09	15.05	58911.29	8588.70
	130	0.5558	250.12	0.5002	25.01	233.88	162.58	16.25	58912.72	8587.27
	140	0.5541	249.34	0.4986	24.93	234.61	162.07	17.45	58914.14	8585.85
	150	0.5523	248.57	0.4971	24.85	235.33	161.57	18.64	58915.55	8584.44
y	0.06	0.5816	261.76	0.5235	26.17	223.48	170.14	17.01	58910.64	8589.35
	0.07	0.5746	258.61	0.5172	25.86	226.20	168.09	16.80	58911.11	8588.88
	0.08	0.5680	255.63	0.5112	25.58	228.84	166.16	16.61	58911.62	8588.37
	0.09	0.5618	252.81	0.5056	25.28	231.40	164.32	16.43	58912.15	8587.84
	0.10	0.5558	250.12	0.5002	25.01	233.88	162.58	16.25	58912.72	8587.27

θ	T	Q	T₁	Q₁	Setup cost	Holding cost	DC	Total cost	Profit
0.01	0.6277	282.48	0.5649	28.24	207.09	183.61	18.36	58909.06	8590.93
0.02	0.6006	270.31	0.5406	27.03	216.41	175.70	35.14	58927.25	8572.74
0.03	0.5768	259.57	0.5191	25.95	225.37	168.72	50.61	58944.70	8555.29
0.04	0.5555	250.00	0.5000	25.00	233.99	162.50	65.00	58961.49	8538.50
0.05	0.5364	241.41	0.4828	24.14	242.32	156.91	78.45	58977.70	8522.29
0.06	0.5191	233.63	0.4672	23.35	250.38	151.81	91.11	58993.37	8506.62
0.07	0.5034	226.56	0.4531	22.65	258.20	147.26	103.08	59008.55	8491.44
0.08	0.4890	220.08	0.4401	22.00	265.80	143.05	114.44	59023.30	8476.69
0.09	0.4758	214.12	0.4282	21.41	273.20	139.17	125.26	59037.64	8462.35
0.10	0.4635	208.61	0.4172	20.86	280.42	135.60	135.60	59051.62	8448.37

θ	T	Q	T₁	Q₁	Setup cost	Holding cost	DC	Total cost	Profit
0.01	0.6349	285.70	0.5714	28.57	204.75	185.70	18.57	58909.03	8590.96
0.02	0.6072	273.25	0.5465	27.32	214.08	177.61	35.52	58927.22	8572.77
0.03	0.5828	262.27	0.5245	26.22	223.04	170.48	51.14	58944.67	8555.32
0.04	0.5611	252.50	0.5050	25.25	231.67	164.13	65.65	58961.45	8538.54
0.05	0.5416	243.74	0.4874	24.37	240.00	158.43	79.21	58977.65	8522.34
0.06	0.5240	235.81	0.4716	23.58	248.07	153.28	91.96	58993.32	8506.67
0.07	0.5080	228.60	0.4572	22.86	255.90	148.59	104.01	59008.50	8491.49
0.08	0.4933	222.00	0.4440	22.20	263.50	144.30	115.44	59023.25	8476.74
0.09	0.4798	215.93	0.4318	21.59	270.90	140.36	126.32	59037.58	8462.40
0.10	0.4674	210.33	0.4206	21.03	278.12	136.71	136.71	59051.56	8448.43

S.No.	Inventory Parameters	Stock Dependent Demand	Price Dependent Demand	Stock and Price Dependent Demand
1.	Optimum cycle time	0.6277	0.6349	0.5558
2.	Optimum quantity	282.48	285.70	250.12
3.	Production time	0.5649	0.5714	0.5002
4.	Maximum inventory	28.24	28.57	25.01
5.	Demand	450	450	450
6.	Production cost	58500.00	58500.00	58500.00
7.	Setup cost	207.09	204.75	233.88
8.	Holding cost	183.61	185.71	162.58
9.	Deteriorative cost	18.36	18.57	16.25
10.	Total Cost	58909.06	58909.03	58912.72
11.	Total Sales	67500.00	67500.00	67500.00
12.	Total profit	8590.93	8590.96	8587.27

\frac{(P_{c} - x (a - b P))}{θ + y (a - b P)} (1 - e^{- (θ + y (a - b P)) T_{1}}) = \frac{x (a - b P)}{θ + y (a - b P)} (e^{- (θ + y (a - b P)) (T - T_{1})} - 1) .

\frac{C_{h}}{T} [\int_{0}^{T_{1}} \frac{(P_{c} - x (a - b P))}{θ + y (a - b P)} (1 - e^{- (θ + y (a - b P) t)}) d t + \int_{T_{1}}^{T} \frac{x (a - b P)}{θ + y (a - b P)} (e^{(θ + y (a - b P)) (T - t)} - 1)] = \frac{C_{h}}{T} [\frac{(P_{c} - x (a - b P))}{{(θ + y (a - b P))}^{2}} [(θ + y (a - b P)) T_{1} + e^{- (θ + y (a - b P)) T_{1}} - 1] - \frac{x (a - b P)}{{(θ + y (a - b P))}^{2}} (1 + (θ + y (a - b P)) T - e^{(θ + y (a - b P)) (T - T_{1})} - (θ + y (a - b P)) T_{1})]

\frac{θ C_{d}}{T} [\frac{(P_{c} - x (a - b P))}{{(θ + y (a - b P))}^{2}} [(θ + y (a - b P)) T_{1} + e^{- (θ + y (a - b P)) T_{1}} - 1] - \frac{x (a - b P)}{{(θ + y (a - b P))}^{2}} (1 + (θ + y (a - b P)) T - e^{(θ + y (a - b P)) (T - T_{1})} - (θ + y (a - b P)) T_{1})]

T C (T) = \frac{C_{0}}{T} + \frac{C_{h} + θ C_{d}}{T} [\frac{(P_{c} - x (a - b P))}{{(θ + y (a - b P))}^{2}} [(θ + y (a - b P)) T_{1} + e^{- (θ + y (a - b P)) T_{1}} - 1] - \frac{x (a - b P)}{{(θ + y (a - b P))}^{2}} (1 + (θ + y (a - b P)) T - e^{(θ + y (a - b P)) (T - T_{1})} - (θ + y (a - b P)) T_{1})]

[\frac{(P_{c} - x (a - b P))}{{(θ + y (a - b P))}^{2}} (θ + y (a - b P)) - (θ + y (a - b P)) e^{- (θ + y (a - b P)) T_{1}} - \frac{x (a - b P)}{{(θ + y (a - b P))}^{2}} - (θ + y (a - b P)) + (θ + y (a - b P)) e^{(θ + y (a - b P)) {(T - T_{1})}_{1}}] = 0

\frac{\partial}{\partial T} T C (T) = - C_{0} + \frac{(C_{h} + θ C_{d})}{{(θ + y (a - b p))}^{2}} [- (P_{c} - x (a - b p)) (e^{- (θ + y (a - b p)) T_{1}} + (θ + y (a - b p)) T_{1} - 1) + x (a - b p) [T \{(θ + y (a - b p)) e^{(θ + y (a - b p)) (T - T_{1})} - (θ + y (a - b p))\} - e^{(θ + y (a - b p)) (T - T_{1})} - (θ + y (a - b p)) (T - T_{1}) - 1]] = 0

[\frac{- (P_{c} - x (a - b P))}{{(θ + y (a - b P))}^{2}} [(θ + y (a - b P)) T_{1} + e^{- (θ + y (a - b P)) T_{1}} - 1] - \frac{x (a - b P)}{{(θ + y (a - b P))}^{2}} [(θ + y (a - b P)) T - (θ + y (a - b P)) T e^{(θ + y (a - b P)) (T - T_{1})} - 1 - (θ + y (a - b P)) (T - T_{1}) + e^{(θ + y (a - b P)) (T - T_{1})}]] = \frac{C_{0}}{(C_{h} + θ C_{d})}

[\frac{- (P_{c} - x (a - b P)) T_{1}^{2}}{2} + x (a - b P) T (T - T_{1}) - \frac{x (a - b P) (T - T_{1})^{2}}{2} + \frac{(P_{c} - x (a - b P)) (θ + y (a - b P)) T_{1}^{3}}{6} + \frac{x (a - b P) (θ + y (a - b P)) T (T - T_{1})^{2}}{2} - \frac{x (a - b P) (θ + y (a - b P)) (T - T_{1})^{3}}{6} - \frac{(P_{c} - x (a - b P)) (θ + y (a - b P))^{2} T_{1}^{4}}{24} + \frac{x (a - b P) {(θ + y (a - b P))}^{2} (T - T_{1})^{3} T}{6} + \frac{x (a - b P) {(θ + y (a - b P))}^{2} (T - T_{1})^{4}}{24}] = \frac{C_{0}}{(C_{h} + θ C_{d})}

[\frac{- (P_{c} - x (a - b P)) T_{1}^{2}}{2} + x (a - b P) T (T - T_{1}) - \frac{x (a - b P) (T - T_{1})^{2}}{2} + \frac{(P_{c} - x (a - b P)) (θ + y (a - b P)) T_{1}^{3}}{6} + \frac{x (a - b P) (θ + y (a - b P)) T (T - T_{1})^{2}}{2} - \frac{x (a - b P) (θ + y (a - b P)) (T - T_{1})^{3}}{6}] = \frac{C_{0}}{(C_{h} + θ C_{d})}

[\begin{matrix} \land \frac{(P_{c} - x (a - b P)) (θ + y (a - b P)) x (a - b P)}{6 P^{2}} [2 P - x (a - b P)] T^{3} \\ \land + \frac{(P_{c} - x (a - b P)) x (a - b P)}{2 P^{2}} T^{2} \end{matrix}] = \frac{C_{0}}{C_{h} + θ C_{d}}

= \frac{C_{h}}{T} [\int_{0}^{T_{1}} \frac{P_{c} - x}{y + θ} (1 - e^{- (y + θ) t}) d t + \int_{T_{1}}^{T} \frac{x}{y + θ} (e^{(y + θ) (T - t)} - 1) d t] = \frac{C_{h}}{T} [\frac{P_{c} - x}{(y + θ)^{2}} ((y + θ) T_{1} + e^{- (y + θ) T_{1}} - 1) - \frac{x}{(y + θ)^{2}} (1 + (y + θ) T - e^{(y + θ) (T - T_{1})} - (y + θ) T_{1})] = \frac{C_{h}}{T} [\frac{P_{c} - x}{(y + θ)^{2}} (e^{- (y + θ) T_{1} - 1} + (y + θ) T_{1} - 1) + \frac{x}{(y + θ)^{2}} (e^{(y + θ) (T - T_{1})} - (y + θ) (T - T_{1}) - 1)]

\frac{θ C_{d}}{T} [\frac{P_{c} - x}{(y + θ)^{2}} (e^{- (y + θ) T_{1} - 1} + (y + θ) T_{1} - 1) + \frac{x}{(y + θ)^{2}} (e^{(y + θ) (T - T_{1})} - (y + θ) (T - T_{1}) - 1)]

T C (T) = D C_{P} + \frac{C_{0}}{T} + \frac{(C_{h} + θ C_{d})}{T} [\frac{P_{c} - x}{(y + θ)^{2}} (e^{- (y + θ) T_{1} - 1} + (y + θ) T_{1} - 1) + \frac{x}{(y + θ)^{2}} (e^{(y + θ) (T - T_{1})} - (y + θ) (T - T_{1}) - 1)]

\frac{P_{c} - x}{(y + θ)^{2}} ((y + θ) - (y + θ) e^{- (y + θ) T_{1}}) + \frac{x}{(y + θ)^{2}} (- (y + θ) e^{(y + θ) (T - T_{1})} + (y + θ)) = 0

[\begin{matrix} & - \frac{P_{c} - x}{{(y + θ)}^{2}} [\frac{{(y + θ)}^{2} T_{1}^{2}}{2} - \frac{{(y + θ)}^{3} T_{1}^{3}}{6} + \frac{{(y + θ)}^{4} T_{1}^{4}}{24}] (e^{- (y + θ) T_{1}} + (y + θ) T_{1} - 1) \\ & + \frac{x}{{(y + θ)}^{2}} [(y + θ) T e^{(y + θ) (T - T_{1})} - (y + θ) T - e^{(y + θ) (T - T_{1})} + (y + θ) (T - T_{1}) + 1] \end{matrix}] = \frac{C_{0}}{(C_{h} + θ C_{d})}

[- (P_{c} - x) [\frac{T_{1}^{2}}{2} - \frac{(y + θ) T_{1}^{3}}{6} + \frac{{(y + θ)}^{4} T_{1}^{4}}{24}] + \frac{x}{{(y + θ)}^{2}} [{(y + θ)}^{2} T (T - T_{1}) + \frac{{(y + θ)}^{3} T {(T - T_{1})}^{2}}{2} + \frac{{(y + θ)}^{4} T {(T - T_{1})}^{4}}{6} - \frac{{(y + θ)}^{2} {(T - T_{1})}^{2}}{2} \frac{{(y + θ)}^{3} {(T - T_{1})}^{3}}{6} - \frac{{(y + θ)}^{4} {(T - T_{1})}^{4}}{24}]] = \frac{C_{0}}{C_{h} + θ C_{d}} .

[- \frac{(P_{c} - x) x^{2} T^{2}}{2 {P_{c}}^{2}} + \frac{x (P_{c} - x) T^{2}}{P_{c}} - \frac{x {(P_{c} - x)}^{2} T^{2}}{2 {P_{c}}^{2}} + \frac{(P_{c} - x) (y + θ) x^{3} T^{3}}{6 {P_{c}}^{3}} + \frac{x (y + θ) {(P_{c} - x)}^{2} T^{3}}{2 {P_{c}}^{2}} - \frac{x (y + θ) {(P_{c} - x)}^{3} T^{3}}{6 {P_{c}}^{3}} - \frac{(P_{c} - x) {(y + θ)}^{2} x^{4} T^{4}}{24 {P_{c}}^{4}} + \frac{x {(y + θ)}^{2} {(P_{c} - x)}^{3} T^{4}}{6 {P_{c}}^{3}} - \frac{x {(y + θ)}^{2} (P_{c} - x) T^{3}}{24 {P_{c}}^{4}}] = \frac{C_{0}}{C_{h} + θ C_{d}},

\frac{C_{h}}{T} [\int_{0}^{T_{1}} \frac{P_{C} - (a - b P)}{θ} (1 - e^{- θ r}) d t + \int_{T_{1}}^{T} \frac{(a - b P)}{θ} (e^{θ (T - t)} - 1) d t] = \frac{C_{h}}{T} [\frac{P_{C} - (a - b P)}{θ^{2}} (θ T_{1} + e^{- θ T_{1}} - 1) - \frac{a - b P}{θ^{2}} (1 + θ T - e^{θ (T - T_{1})} - θ T_{1})] = \frac{C_{h}}{θ^{2} T} [(P_{C} - (a - b P)) (θ T_{1} + e^{- θ T_{1}} - 1) - (a - b P) (1 + θ T - e^{θ (T - T_{1})} - θ T_{1})]

= \frac{θ C_{d}}{θ^{2} T} [(P_{C} - (a - b P)) (θ T_{1} + e^{- θ T_{1}} - 1) - (a - b P) (1 + θ (T - T_{1}) - e^{θ (T - T_{1})})]

T C (T) = D C_{p} + \frac{C_{0}}{T} + \frac{C_{h} + C_{d}}{θ^{2} T} [\begin{matrix} \land (P_{C} - (a - b P)) (θ T_{1} + e^{- θ T_{1}} - 1) \\ \land - (a - b P) (1 + θ (T - T_{1}) - e^{θ (T - T_{1})}) \end{matrix}]

[\begin{matrix} \land - (P_{C} - (a - b P)) (e^{- θ T_{1}} + θ T_{1} - 1) \\ \land + T \{(a - b P) (θ - θ e^{θ (T - T_{1})})\} + (a - b P) (1 + θ (T - T_{1}) - e^{θ (T - T_{1})}) \end{matrix}] = \frac{θ^{2} C_{0}}{C_{h} + θ C_{d}}

[\begin{matrix} \land - (P_{C} - (a - b P)) (e^{- θ T_{1}} + θ T_{1} - 1) \\ \land + θ (a - b P) T (e^{θ (T - T_{1})} - 1) + (a - b P) (1 + θ (T - T_{1}) - e^{θ (T - T_{1})}) \end{matrix}] = \frac{θ^{2} C_{0}}{C_{h} + θ C_{d}}

[- \frac{(P_{C} - (a - b P)) T_{1}^{2}}{2} + \frac{θ (P_{C} - (a - b P)) T_{1}^{3}}{6} - \frac{θ^{2} (P_{C} - (a - b P)) T_{1}^{4}}{24} + (a - b P) (T (T - T_{1}) + \frac{θ T (T - T_{1})^{2}}{2} + \frac{θ^{2} T (T - T_{1})^{3}}{6} - \frac{(T - T_{1})^{2}}{2} - \frac{θ (T - T_{1})^{3}}{6} - \frac{θ^{2} (T - T_{1})^{4}}{24})] = \frac{C_{0}}{C_{h} + θ C_{d}}

\frac{(P_{C} - (a - b P)) θ (a - b P) (2 P_{C} - (a - b P)) T^{3}}{6 P_{C}^{2}} + \frac{(P_{C} - (a - b P)) (a - b P) T^{2}}{2 P_{C}} = \frac{C_{0}}{C_{h} + θ C_{d}} .

[1] *Corresponding author

Brasil

Brasil

PRODUCTION INVENTORY MODELS WITH INTEGRATED STOCK AND PRICE DEPENDENT DEMANDS FOR DETERIORATIVE ITEMS

ABSTRACT