An evolutionary approach for the offsetting inventory cycle problem

Figures & data

Table 1. Data of the example

Item	Storage space per unit time (m^3)	Daily demand	Time between orders - TBO (days)	Economic order quantity - EOQ
#1	1	3	3	9
#2	1	2	10	20
#3	1	5	6	30

Figure 1. Space requirement without offsetting.

Figure 2. Space requirement with offsetting.

Figure 3. Flowchart of the genetic algorithm.

Figure 4. Pseudo-code of the genetic algorithm.

Table

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Table 2. Data of Murthy et al.’s example

Item j	1	2	3	4	5	6	7	8	9
sjQj	100	200	81	144	150	160	90	60	50
TBOj	4	5	9	12	15	8	6	12	2

Table 3. Application of the two procedures to Murthy et al.’s example

	Without offsetting		Murthy et al. model		MIP model		Genetic algorithm
Item	gj	Ij,0·sj	gj	Ij,0·sj	gj	Ij,0 · sj	gj	Ij,0 · sj
#1	0	100	0	100	3	100	0	100
#2	0	200	4	160	1	80	1	40
#3	0	81	0	81	5	54	1	9
#4	0	144	0	144	1	24	6	72
#5	0	150	0	150	3	40	8	80
#6	0	160	6	120	5	120	2	40
#7	0	90	5	75	4	75	3	45
#8	0	60	4	20	7	40	0	60
#9	0	50	1	25	0	25	1	25
Max peak (m^3)	1,035		875		760		760
Peak day	0		0		38		42
% Reduction	–		15.46%		26.57%		26.57%

Notes: The values in bold are representative of the results obtained through the application of MIP model and genetic algorithm on the Murthy et al.’s example.

Figure 5. Murthy et al. Case. Comparison between the space requirement with and without offsetting.

Table 4. Characteristics of the items

Characteristic	Measure Unit	Range
Daily demand (dj)	Units/day	1–106
Time between order (TBOj)	Days	1–220
Economic order quantity (Qj)	Units	74–3,748

Table 5. MIP model results

	MIP model
Case study	Upper bound (m^3)	Lower bound (m^3)	Run time (sec)	Stop motivation	GAP (%)
9 items	760.00	760.00	3.55	Optimal	0.00
10 items	2,854.00	2,854.00	229.95	Optimal	0.00
20 items	7,121.00	6,902.06	10,810.01	AbortTimeLim	3.07
30 items	12,352.00	12,210.00	10,810.03	AbortTimeLim	1.14
40 items	13,795.00	13,585.30	10,810.03	AbortTimeLim	1.52
50 items	17,602.00	17,131.80	10,810.04	AbortTimeLim	2.67
80 items	27,689.00	27,024.60	10,810.08	AbortTimeLim	2.40
200 items	73,309.00	72,477.70	10,810.18	AbortTimeLim	1.13

Table 6. Comparison between the results of the two procedures

	MIP model	Genetic Algorithm
Case study	Max Volume Peak (m^3)	Max Volume Peak (m^3)	Run Time (s)	Gap (%)
9 items	760	760	87	0.000
10 items	2,854.00	2,881.00	88	0.946
20 items	7,121.00	7,229.00	110	1.517
30 items	12,352.00	12,660.00	137	2.494
40 items	13,795.00	14,147.00	177	2.552
50 items	17,602.00	18,019.00	206	2.369
80 items	27,689.00	28,489.00	294	2.889
200 items	73,309.00	75,682.00	733	3.237

Table 7. Ranges of values for instances

Characteristic	Measure unit	Range 1,000 items	Range 2,000 items
Daily demand (dj)	Units/day	1–108	1–110
Time between order (TBOj)	Days	1–202	1–373
Specific volume (sj)	m^3/unit	0.002–1	0.002–1
Unit purchasing cost (pj)	€/unit	0.1–7	0.1–7
Unit ordering cost (Clj)	€/order	50–100	50–100
Unit holding cost (kj)	€/(unit·day)	0.001–0.05	0.001–0.05

Figure 6. 1,000 item case. Comparison between the space requirement with and without offsetting.

Figure 7. 2,000 item case. Comparison between the space requirement with and without offsetting.

Table 8. Comparing the costs

	1,000 item instance		2,000 item instance
Characteristic	Wilson model	Genetic algorithm	Wilson model	Genetic Algorithm
Purchasing cost (€)	38,375,050	38,375,050	75,632,639	75,632,639
Ordering cost (€)	1,373,354	1,336,834	2,766,932	2,693,515
Holding cost (€)	1,371,553	1,335,040	2,769,247	2,695,727
Extra ordering cost (€)	–	71,325	–	145,386
Extra holding cost (€)	–	22,865	–	46,464
Total cost (€)	41,119,957	41,141,114	81,168,818	81,213,731
Max volume (m^3)	318,488	175,813	645,813.24	355,386.67

Table

#Item	Q10
1–10	617	677	81	193	238	341	692	219	524	603

Table

#Item	TBO10
1–10	6	20	81	32	30	23	115	17	8	10

Table

#Item	Q20
1–10	113	823	741	423	562	442	274	1,043	443	803
11–20	612	1,343	594	565	406	426	412	508	550	596

Table

#Item	TBO20
1–10	23	39	9	11	6	8	14	33	44	38
11–20	10	14	14	14	9	7	9	9	7	7

Table

#Item	Q30
1–10	544	326	370	726	547	249	160	1,470	1,316	607
11–20	618	544	583	972	1,145	485	986	200	335	520
21–30	2,961	420	178	756	686	410	1,086	441	505	189

Table

#Item	TBO30
1–10	7	8	10	10	13	83	40	22	14	9
11–20	7	8	10	13	19	6	10	25	19	10
21–30	51	16	15	10	8	7	33	5	12	38

Table

#Item	Q40
1–10	261	521	643	408	1,169	192	592	696	396	143
11–20	535	514	706	303	692	501	1,510	614	330	459
21–30	331	74	108	354	263	422	654	327	548	1,378
31–40	862	792	962	197	304	1,424	442	636	2,023	345

Table

#Item	TBO40
1–10	11	5	10	19	32	19	13	10	11	20
11–20	7	9	9	14	10	8	21	11	13	9
21–30	19	74	36	13	10	11	9	8	91	15
31–40	10	24	13	25	10	24	25	6	38	10

Table

#Item	Q50
1–10	502	108	652	844	558	428	399	332	520	585
11–20	1,647	465	857	865	325	339	741	478	356	891
21–30	1,021	150	2,424	989	332	501	641	1,076	486	273
31–40	127	735	1,298	385	762	189	1,183	570	382	293
41–50	130	297	850	679	354	162	944	563	438	560

Table

#Item	TBO50
1–10	63	27	7	11	8	9	10	7	8	39
11–20	36	13	15	12	11	12	9	9	18	14
21–30	13	21	25	19	14	12	15	11	6	39
31–40	42	7	16	9	9	27	15	6	10	8
41–50	33	15	12	10	44	23	21	24	219	22

Table

#Item	Q80
1–10	576	1,269	644	560	334	363	376	381	501	298
11–20	525	503	1,115	475	612	674	1,781	474	775	737
21–30	192	482	524	581	363	97	307	624	407	753
31–40	435	539	329	331	179	469	1,200	789	708	263
41–50	466	485	1,010	453	484	433	544	956	551	679
51–60	583	914	755	339	648	964	422	461	233	883
61–70	354	778	683	174	429	413	2,551	227	1,188	1,237
71–80	1,533	484	566	1,604	458	244	465	374	200	786

Table

#Item	TBO80
1–10	9	14	8	11	8	16	7	22	20	27
11–20	12	42	31	8	8	7	21	5	30	8
21–30	19	14	7	13	14	49	16	7	10	58
31–40	12	6	10	9	45	6	15	9	13	88
41–50	33	5	11	7	8	6	21	10	6	9
51–60	8	10	31	10	7	37	8	8	16	16
61–70	14	9	11	58	15	14	43	17	33	39
71–80	26	6	13	25	7	22	9	8	20	24

Table

#Item	Q200
1–10	518	769	808	450	322	730	725	739	415	603
11–20	508	353	894	266	3,140	1,348	552	138	415	324
21–30	105	543	388	1,068	710	289	579	288	1,030	758
31–40	473	238	622	646	732	820	529	449	2,923	1,106
41–50	461	491	1,017	517	599	569	192	371	692	422
51–60	299	1,013	1,033	1,532	1,538	501	1,519	1,704	507	810
61–70	1,015	660	326	1,143	427	610	559	690	364	295
71–80	191	1,095	424	782	315	606	537	520	1,511	628
81–90	150	622	644	546	573	775	101	484	414	184
91–100	320	494	361	342	363	558	157	631	634	1,120
101–110	2,258	466	174	1,387	416	496	444	519	395	525
111–120	1,375	392	3,748	1,061	456	549	160	289	519	91
121–130	251	597	236	1,895	558	983	544	416	2,265	709
131–140	369	746	554	770	315	474	681	729	247	646
141–150	551	598	387	806	701	1,472	1,287	506	581	827
151–160	831	190	743	151	932	400	515	369	345	351
161–170	405	468	449	417	124	595	707	251	472	1,044
171–180	566	661	582	1,063	360	840	777	519	718	386
181–190	681	306	646	391	618	679	589	1,471	85	562
191–200	534	588	561	1,982	647	475	1,524	594	355	915

Table

#Item	TBO200
1–10	10	17	135	10	7	8	9	18	9	10
11–20	6	10	10	10	34	17	15	23	13	20
21–30	35	9	15	33	15	13	6	17	13	10
31–40	9	22	11	9	8	12	9	9	29	17
41–50	6	5	20	7	12	11	21	13	9	10
51–60	13	18	13	26	17	10	17	34	9	8
61–70	11	9	82	52	5	7	7	33	11	10
71–80	27	17	6	17	8	17	23	20	16	37
81–90	25	16	19	12	15	12	51	9	10	20
91–100	11	7	9	14	11	8	26	10	14	14
101–110	24	11	22	14	35	7	9	7	10	6
111–120	15	12	44	14	16	55	18	11	7	46
121–130	63	10	12	20	7	13	11	13	33	10
131–140	7	9	9	9	45	53	15	9	12	9
141–150	8	6	10	10	9	49	16	8	7	13
151–160	17	38	8	38	18	11	17	17	11	13
161–170	6	7	220	7	41	10	10	13	9	20
171–180	8	8	8	15	8	11	18	10	8	10
181–190	8	7	7	14	13	7	6	16	43	8
191–200	31	7	6	20	9	6	54	13	6	2