Silvopastoral and agroforestry systems: An integer linear programming model for investment decisions

Rodrigo Conde; Franco Robledo; Agustín López de Lacalle

doi:10.3934/jdg.2023013

Article Contents

2023, Volume 10, Issue 4: 304-329. Doi: 10.3934/jdg.2023013

This issue Previous Article Optimal economic policy and growth in an open economy Next Article Producer behavior with rules of origin

Silvopastoral and agroforestry systems: An integer linear programming model for investment decisions

Facultad de Ingeniería, Universidad de la República, Uruguay

^*Corresponding author: Rodrigo Conde
^*Corresponding author: Rodrigo Conde

Received: August 2023

Early access: September 19, 2023

Published online: September 19, 2023

Abstract / Introduction Full Text(HTML) Figure(11) / Table(22) Related Papers Cited by

Abstract

Since the countryside is one of the main drivers of the global economy, it is vital to support its technological development. When a potential investor contemplates agribusiness investments, several alternatives are presented to him, including forestry, livestock raising, and silvopastoralism, which consists of integrating both activities into a single agroforestry system. In this paper, we address the problem of designing a portfolio of agroforestry investments that maximizes economic results. In order to do so, we developed an integer linear programming model, which selects the optimal combination of the three aforementioned activities for a zoned field. This model was created from a national perspective, considering elements specific to the Uruguayan reality. Additionally, we produced a model implementation utilizing cutting-edge software such as AMPL and CPLEX. This solution was empirically validated using a battery of 56 test cases, based on actual data gathered from multiple sources.

Keywords:

Mathematics Subject Classification: Primary: 90-10, 91G10; Secondary: 90C11.

Citation:

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Figure 1. Satelital image (left) and zoning plan (right) of property 4 (extracted from Google Maps and [10])

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Figure 2. Proportion of activities by soil

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Figure 3. NPV evolution as a function of the initial budget

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Figure 4. Proportion of activities by initial budget

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Figure 5. Proportion of activities by operational budget

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Figure 6. NPV for different activity restrictions

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Figure 7. Evolution of the NPV as a function of the surface

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Figure 8. Total execution time as a function of the problem's size

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Figure 9. Total execution time as a function of the problem's size

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Figure 10. Solve and load time evolution as a function of the number of forestry and livestock species ($ \mid E_f \mid $ and $ \mid E_g \mid $ respectively)

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Figure 11. Solve and load time evolution as a function of the problem's size (The notation used is the same as in the model's specification)

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Table 1. Assumed costs for the main initial activities of a forestry system

Activity	Assumed cost (US$ \$/ha)
Site preparation	$ 302 $
Plowing	$ 74 $
Plantation	$ 0.21 \times d(m) $

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Table 2. Estimated volume per tree ($ m^3 $) by year 10 as a function of soil and tree species

Soil region	Eucalyptus grandis	Eucalyptus dunni
Basaltic	$ 0.178 $	$ 0.16 $
Central-south	$ 0.20 $	$ 0.18 $
Eastern hills	$ 0.224 $	$ 0.202 $
Western basin	$ 0.188 $	$ 0.169 $
Northwestern basin	$ 0.21 $	$ 0.189 $
Eastern hill ranges	$ 0.214 $	$ 0.193 $
Plains systems	$ 0.198 $	$ 0.178 $

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Table 3. Assumed costs for the main operational activities of a forestry system

Activity	Estimated cost (US$ \$/ha)
Insects control	$ 43 $
Weed control	$ 52 $
Harvest, extraction and load	$ 14 \times V $
Transport	$ 15 \times V $

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Table 4. Wood selling price per $ m^3 $ according to species

Species	Price (US$ \$/$ {\bf m^3} $)
Eucalyptus grandis	$ 60 $
Eucalyptus dunni	$ 55 $

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Table 5. Annual dry matter production and cattle units (CU) supported per hectare by region

Soil region	Dry matter (Kg/ha)	Ideal capacity (CU/ha)	Secure capacity (CU/ha)
Basaltic	$ 4,073.9 $	$ 1.47 $	$ 0.74 $
Central-south	$ 4,599.1 $	$ 1.66 $	$ 0.83 $
Eastern hills	$ 5,185.7 $	$ 1.87 $	$ 0.94 $
Western basin	$ 4,325.0 $	$ 1.56 $	$ 0.78 $
Northwestern basin	$ 4,855.7 $	$ 1.75 $	$ 0.88 $
Eastern hill ranges	$ 4,923.0 $	$ 1.77 $	$ 0.89 $
Plains systems	$ 4,553.9 $	$ 1.64 $	$ 0.82 $

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Table 6. Pasture planting costs by species

Species	Price (US$ \$/ha)
Natural field	0
25% enhanced pasture	200
50% enhanced pasture	375

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Table 7. Livestock prices by category

Category	Price (US$ \$/Kg)	Initial weight (Kg)	Price(US$ \$/CU)
Steers (1 year old)	$ 2.75 $	$ 240 $	$ 660 $
Steers (2+ years old)	$ 2.69 $	$ 330 $	$ 887.7 $
Calf	$ 3.06 $	$ 70 $	$ 214.2 $
Wintering cattle	$ 2.22 $	$ 380 $	$ 843.6 $

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Table 8. Livestock selling prices by initial category

Category	Final weight (Kg)	Price(US$ \$/CU)
Steers (1 year old)	$ 360 $	$ 968.4 $
Steers (2+ years old)	$ 450 $	$ 1,170 $
Calf	$ 190 $	$ 522.5 $
Wintering cattle	$ 500 $	$ 1,110 $

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Table 9. Real properties considered in this paper

Property	Location	$ {\bf \mid Z \mid} $	Soil region	Surface (ha)	Coordinates (S, W)
1	Florida	2	Central-south	33.96	33$ ^\circ$48'10.8" 56$ ^\circ$04'08.4"
2	Rocha	8	Eastern hills	25.47	34$ ^\circ$22'33.6" 54$ ^\circ$19'58.8"
3	Florida	23	Central-south	180.81	33$ ^\circ$39'57.6" 55$ ^\circ$36'50.4"
4	Maldonado	42	Eastern hill ranges	337.17	34$ ^\circ$41'24.0" 55$ ^\circ$08'20.4"

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Table 10. Realistic test cases

Case	Property	$ {\bf P_0} $ (US$ \$)	$ {\bf P_y} $ (US$ \$)	Accepts forestry?	Accepts grazing?
1	1	28,000	3,500	Every zone	Every zone
2	1	28,000	1,750	Every zone	Every zone
3	1	28,000	0	Every zone	Every zone
4	1	14,000	1,750	Every zone	Every zone
5	1	14,000	0	Every zone	Every zone
6	2	20,500	2,500	1 zone doesn't	1 zone doesn't
7	2	20,500	1,250	1 zone doesn't	1 zone doesn't
8	2	20,500	0	1 zone doesn't	1 zone doesn't
9	2	10,200	1,250	1 zone doesn't	1 zone doesn't
10	2	10,200	0	1 zone doesn't	1 zone doesn't
11	3	150,000	18,000	3 zones don't	3 zones don't
12	3	150,000	9,000	3 zones don't	3 zones don't
13	3	150,000	0	3 zones don't	3 zones don't
14	3	72,000	9,000	3 zones don't	3 zones don't
15	3	72,000	0	3 zones don't	3 zones don't
16	4	270,000	33,000	5 zones don't	5 zones don't
17	4	270,000	16,500	5 zones don't	5 zones don't
18	4	270,000	0	5 zones don't	5 zones don't
19	4	135,000	16,500	5 zones don't	5 zones don't
20	4	135,000	0	5 zones don't	5 zones don't

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Table 11. Solutions found for realistic cases

Case	Soil region	$ {\bf P_0} $ (US$ \$)	$ {\bf P_y} $ (US$ \$)	Forestry	Grazing	Silvopasture
1	Central-south	28,000	3,500	2	0	0
2	Central-south	28,000	1,750	1	0	1
3	Central-south	28,000	0	1	1	0
4	Central-south	14,000	1,750	1	0	0
5	Central-south	14,000	0	1	1	0
6	Eastern hills	20,500	2,500	7	1	0
7	Eastern hills	20,500	1,250	7	1	0
8	Eastern hills	20,500	0	4	3	1
9	Eastern hills	10,200	1,250	3	4	0
10	Eastern hills	10,200	0	3	4	0
11	Central-south	150,000	18,000	20	3	0
12	Central-south	150,000	9,000	17	6	0
13	Central-south	150,000	0	14	9	0
14	Central-south	72,000	9,000	9	14	0
15	Central-south	72,000	0	9	14	0
16	Eastern hill ranges	270,000	33,000	37	5	0
17	Eastern hill ranges	270,000	16,500	33	9	0
18	Eastern hill ranges	270,000	0	33	9	0
19	Eastern hill ranges	135,000	16,500	19	23	0
20	Eastern hill ranges	135,000	0	19	23	0

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Table 12. Economic results for realistic cases

Case	Soil region	$ {\bf P_0} $ (US$ \$)	$ {\bf P_y} $ (US$ \$)	Initial investment	NPV
1	Central-south	28,000	3,500	27,032.16	218,600.35
2	Central-south	28,000	1,750	27,348.82	184,065.92
3	Central-south	28,000	0	17,381.84	148,737.45
4	Central-south	14,000	1,750	12,425.56	100,481.49
5	Central-south	14,000	0	13,229.36	92,192.64
6	Eastern hills	20,500	2,500	19,012.53	177,833.85
7	Eastern hills	20,500	1,250	19,012.53	177,833.85
8	Eastern hills	20,500	0	17,256.54	161,488.70
9	Eastern hills	10,200	1,250	10,199.60	101,129.33
10	Eastern hills	10,200	0	10,199.60	101,129.33
11	Central-south	150,000	18,000	138,904.86	1,127,531.83
12	Central-south	150,000	9,000	129,501.83	1,059,459.14
13	Central-south	150,000	0	113,650.82	944,706.75
14	Central-south	72,000	9,000	71,995.56	643,146.07
15	Central-south	72,000	0	71,995.56	643,146.07
16	Eastern hill ranges	270,000	33,000	244,808.47	2,168,246.75
17	Eastern hill ranges	270,000	16,500	242,925.79	2,153,030.58
18	Eastern hill ranges	270,000	0	215,884.27	1,934,475.54
19	Eastern hill ranges	135,000	16,500	134,995.80	1,280,718.55
20	Eastern hill ranges	135,000	0	134,995.80	1,280,718.55

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Table 13. Sensitivity test cases

Case	$ {\bf \mid Z \mid} $	Soil region	$ {\bf P_0} $ (US$ \$)	$ {\bf P_y} $ (US$ \$)	Activity limitations
1	42	Max yield	270,000	16,500	Some for both
2	42	Medium yield	270,000	16,500	Some for both
3	42	Min yield	270,000	16,500	Some for both
4	42	50% max 50% min	270,000	16,500	Some for both
5	42	Equal thirds	270,000	16,500	Some for both
6	42	Mixed (good average)	270,000	16,500	Some for both
7	42	Mixed (bad average)	270,000	16,500	Some for both
2	42	Medium yield	270,000	16,500	Some for both
8	42	Medium yield	216,000	16,500	Some for both
9	42	Medium yield	162,000	16,500	Some for both
10	42	Medium yield	108,000	16,500	Some for both
11	42	Medium yield	54,000	16,500	Some for both
12	42	Medium yield	270,000	33,000	Some for both
2	42	Medium yield	270,000	16,500	Some for both
13	42	Medium yield	270,000	0	Some for both
14	42	Medium yield	270,000	33,000 - 3,667 $ \times (y - 1) $	Some for both
15	42	Medium yield	270,000	1320 $ \times (y-5)^2 $	Some for both
16	42	Medium yield	270,000	16,500	None
17	42	Medium yield	270,000	16,500	Forestry 100% banned
18	42	Medium yield	270,000	16,500	Grazing 100% banned
2	42	Medium yield	270,000	16,500	Some for both
19	2	Medium yield	28,000	1,750	Some for both
20	8	Medium yield	20,500	1,250	Some for both
21	23	Medium yield	150,000	9	Some for both
2	42	Medium yield	270,000	16,500	Some for both

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Table 14. Solutions found for the sensitivity test cases

Case	Description	Forestry	Grazing	Silvopasture
1	Max yield soil	35	7	0
2	Medium yield soil	33	8	1
3	Min yield soil	33	9	0
4	Soils 50% max yield, 50% min yield	31	10	1
5	Soils distributed in thirds	35	7	0
6	Soils mostly good	32	10	0
7	Soils mostly bad	34	8	0
2	High initial budget	33	8	1
8	80% initial budget	30	12	0
9	60% initial budget	22	20	0
10	40% initial budget	12	30	0
11	20% initial budget	0	31	0
12	High operational budget	37	5	0
2	Medium operational budget	33	8	1
13	Low operational budget	31	11	0
14	Linearly decreasing operational budget	34	8	0
15	Parabolic operational budget	31	11	0
16	No activity restricted	35	7	0
17	Forestry totally banned	0	42	0
18	Grazing totally banned	26	0	0
2	Some bans for both activities	33	8	1
19	2 zones	1	1	0
20	8 zones	5	2	1
21	23 zones	17	6	0
2	42 zones	33	8	1

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Table 15. Economic results for sensitivity cases

Case	Description	Initial investment	NPV
1	Max yield soil	244,381.22	2,284,562.33
2	Medium yield soil	241,045.83	1,972,096.52
3	Min yield soil	237,736.70	1,691,039.62
4	Soils 50% max yield, 50% min yield	242,947.50	2,044,119.08
5	Soils distributed in thirds	242,015.14	2,034,251.20
6	Soils mostly good	241,155.92	2,097,111.14
7	Soils mostly bad	240,758, 75	1,872,426.03
2	High initial budget	241,045.83	1,972,096.52
8	80% initial budget	215,993.68	1,791,061.11
9	60% initial budget	161,992.69	1,400,124.25
10	40% initial budget	107,997.88	1,009,232.14
11	20% initial budget	53,998.94	595,758.15
12	High operational budget	244,307.88	1,996,040.03
2	Medium operational budget	241,045.83	1,972,096.52
13	Low operational budget	211,950.56	1,761,791.20
14	Linearly decreasing operational budget	218,386.17	1,808,381.38
15	Parabolic operational budget	214,362.42	1,766,503.98
16	No activity restricted	241,018.99	1,972,230.25
17	Forestry totally banned	59,944.11	661,349.81
18	Grazing totally banned	138,241.32	1,117,912.93
2	Some bans for both activities	241,045.83	1,972,096.52
19	2 zones	17,381.84	148,737.45
20	8 zones	18,904.19	153,601.66
21	23 zones	129,501.83	1,059,459.14
2	42 zones	241,045.83	1,972,096.52

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Table 16. Reference values for devising performance test cases

Parameter	Description	Reference value
$ T $	Zone's surface	10
$ I_f $	Initial forestry investment	638.5
$ C_f[y< H] $	Forestry operational cost	95
$ C_f[H] $	Forestry operational cost (harvest year)	6,018.77
$ R_f $	Forestry returns	12,256.09
$ I_p $	Initial grazing investment	845
$ C_p $	Grazing operational cost	845
$ R_p $	Grazing returns	1,111.91

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Table 17. Performance test plan

Case	$ {\bf \mid Z \mid} $	$ {\bf \mid E_f \mid} $	$ {\bf \mid E_g \mid} $	$ {\bf \mid E_p \mid} $	$ {\bf \mid M \mid} $	$ {\bf \mid S \mid} $	$ {\bf H} $
1	20	20	20	20	20	20	20
2	100	20	20	20	20	20	20
3	200	20	20	20	20	20	20
4	20	100	20	20	20	20	20
5	20	200	20	20	20	20	20
6	20	20	100	20	20	20	20
7	20	20	200	20	20	20	20
8	20	20	20	100	20	20	20
9	20	20	20	200	20	20	20
10	20	20	20	20	100	20	20
11	20	20	20	20	200	20	20
12	20	20	20	20	20	100	20
13	20	20	20	20	20	200	20
14	20	20	20	20	20	20	100
15	20	20	20	20	20	20	200

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Table 18. Execution times for performance tests

Case	$ {\bf \mid Z \mid} $	$ {\bf \mid E_f \mid} $	$ {\bf \mid E_g \mid} $	$ {\bf \mid E_p \mid} $	$ {\bf \mid M \mid} $	$ {\bf \mid S \mid} $	$ {\bf H} $	$ {\bf T_{solve}} $	$ {\bf T_{load}} $	$ {\bf T_{total}} $
1	20	20	20	20	20	20	20	0:04:31	0:01:26	0:05:57
2	100	20	20	20	20	20	20	0:31:07	0:08:02	0:39:09
3	200	20	20	20	20	20	20	1:12:21	0:19:08	1:31:29
4	20	100	20	20	20	20	20	0:47:29	0:12:07	0:59:36
5	20	200	20	20	20	20	20	1:25:45	0:28:08	1:53:53
6	20	20	100	20	20	20	20	0:31:36	0:19:58	0:51:35
7	20	20	200	20	20	20	20	1:26:57	0:40:12	2:07:10
8	20	20	20	100	20	20	20	0:30:57	0:21:33	0:52:30
9	20	20	20	200	20	20	20	1:14:54	0:45:40	2:00:34
10	20	20	20	20	100	20	20	0:31:19	0:28:22	0:59:41
11	20	20	20	20	200	20	20	1:14:17	0:54:20	2:08:36
12	20	20	20	20	20	100	20	0:04:37	0:09:09	0:13:46
13	20	20	20	20	20	200	20	0:04:42	0:06:49	0:11:30
14	20	20	20	20	20	20	100	0:26:32	0:23:30	0:50:02
15	20	20	20	20	20	20	200	0:45:27	0:46:38	1:32:05

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Table 19. Realistic tests' execution time

Case	Soil region	$ {\bf P_0} $ (US$ \$)	$ {\bf P_y} $ (US$ \$)	$ {\bf T_{solve}(s)} $	$ {\bf T_{load}(s)} $
1	Central-south	28,000	3,500	0.153	0.546
2	Central-south	28,000	1,750	0.103	0.624
3	Central-south	28,000	0	0.094	0.662
4	Central-south	14,000	1,750	0.068	0.613
5	Central-south	14,000	0	0.102	0.888
6	Eastern hills	20,500	2,500	0.081	0.521
7	Eastern hills	20,500	1,250	0.077	0.41
8	Eastern hills	20,500	0	6.633	0.462
9	Eastern hills	10,200	1,250	21.152	0.812
10	Eastern hills	10,200	0	6.120	0.564
11	Central-south	150,000	18,000	0.115	0.384
12	Central-south	150,000	9,000	46.138	0.706
13	Central-south	150,000	0	48.650	0.69
14	Central-south	72,000	9,000	34.377	0.611
15	Central-south	72,000	0	0.151	0.591
16	Eastern hill ranges	270,000	33,000	0.154	0.566
17	Eastern hill ranges	270,000	16,500	12.338	0.95
18	Eastern hill ranges	270,000	0	28.670	0.742
19	Eastern hill ranges	135,000	16,500	17.095	0.547
20	Eastern hill ranges	135,000	0	0.183	0.637

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Table 20. Annual balances for the solutions of realistic cases

Case	1	2	3	4	5	6	7	8	9	10
1	$ -3,226.20 $	$ -3,226.20 $	$ -3,226.20 $	$ -3,226.20 $	$ -3,226.20 $	$ -3,226.20 $	$ -3,226.20 $	$ -3,226.20 $	$ -3,226.20 $	$ 417,877.80 $
2	$ -1,728.29 $	$ -1,728.29 $	$ -1,728.29 $	$ -1,728.29 $	$ -1,728.29 $	$ -1,728.29 $	$ -1,728.29 $	$ -1,728.29 $	$ -1,728.29 $	$ 347,829.93 $
3	$ 2,251.18 $	$ 2,251.18 $	$ 2,251.18 $	$ 2,251.18 $	$ 2,251.18 $	$ 2,251.18 $	$ 2,251.18 $	$ 2,251.18 $	$ 2,251.18 $	$ 232,566.42 $
4	$ -1,482.95 $	$ -1,482.95 $	$ -1,482.95 $	$ -1,482.95 $	$ -1,482.95 $	$ -1,482.95 $	$ -1,482.95 $	$ -1,482.95 $	$ -1,482.95 $	$ 192,081.05 $
5	$ 3,212.61 $	$ 3,212.61 $	$ 3,212.61 $	$ 3,212.61 $	$ 3,212.61 $	$ 3,212.61 $	$ 3,212.61 $	$ 3,212.61 $	$ 3,212.61 $	$ 127,452.49 $
6	$ -1,195.78 $	$ -1,195.78 $	$ -1,195.78 $	$ -1,195.78 $	$ -1,195.78 $	$ -1,195.78 $	$ -1,195.78 $	$ -1,195.78 $	$ -1,195.78 $	$ 319,194.59 $
7	$ -1,195.78 $	$ -1,195.78 $	$ -1,195.78 $	$ -1,195.78 $	$ -1,195.78 $	$ -1,195.78 $	$ -1,195.78 $	$ -1,195.78 $	$ -1,195.78 $	$ 319,194.59 $
8	$ 1.71 $	$ 1.71 $	$ 1.71 $	$ 1.71 $	$ 1.71 $	$ 1.71 $	$ 1.71 $	$ 1.71 $	$ 1.71 $	$ 277,566.55 $
9	$ 4,481.90 $	$ 4,481.90 $	$ 4,481.90 $	$ 4,481.90 $	$ 4,481.90 $	$ 4,481.90 $	$ 4,481.90 $	$ 4,481.90 $	$ 4,481.90 $	$ 122,297.74 $
10	$ 4,481.90 $	$ 4,481.90 $	$ 4,481.90 $	$ 4,481.90 $	$ 4,481.90 $	$ 4,481.90 $	$ 4,481.90 $	$ 4,481.90 $	$ 4,481.90 $	$ 122,297.74 $
11	$ -14,327.73 $	$ -14,327.73 $	$ -14,327.73 $	$ -14,327.73 $	$ -14,327.73 $	$ -14,327.73 $	$ -14,327.73 $	$ -14,327.73 $	$ -14,327.73 $	$ 2,128,471.89 $
12	$ -8,990.71 $	$ -8,990.71 $	$ -8,990.71 $	$ -8,990.71 $	$ -8,990.71 $	$ -8,990.71 $	$ -8,990.71 $	$ -8,990.71 $	$ -8,990.71 $	$ 1,947,909.04 $
13	$ 6.08 $	$ 6.08 $	$ 6.08 $	$ 6.08 $	$ 6.08 $	$ 6.08 $	$ 6.08 $	$ 6.08 $	$ 6.08 $	$ 1,643,528.26 $
14	$ 23,648.99 $	$ 23,648.99 $	$ 23,648.99 $	$ 23,648.99 $	$ 23,648.99 $	$ 23,648.99 $	$ 23,648.99 $	$ 23,648.99 $	$ 23,648.99 $	$ 843,638.39 $
15	$ 23,648.99 $	$ 23,648.99 $	$ 23,648.99 $	$ 23,648.99 $	$ 23,648.99 $	$ 23,648.99 $	$ 23,648.99 $	$ 23,648.99 $	$ 23,648.99 $	$ 843,638.39 $
16	$ -17,643.53 $	$ -17,643.53 $	$ -17,643.53 $	$ -17,643.53 $	$ -17,643.53 $	$ -17,643.53 $	$ -17,643.53 $	$ -17,643.53 $	$ -17,643.53 $	$ 3,946,564.78 $
17	$ -16,494.73 $	$ -16,494.73 $	$ -16,494.73 $	$ -16,494.73 $	$ -16,494.73 $	$ -16,494.73 $	$ -16,494.73 $	$ -16,494.73 $	$ -16,494.73 $	$ 3,907,042.98 $
18	$ 5.78 $	$ 5.78 $	$ 5.78 $	$ 5.78 $	$ 5.78 $	$ 5.78 $	$ 5.78 $	$ 5.78 $	$ 5.78 $	$ 3,339,377.74 $
19	$ 49,363.28 $	$ 49,363.28 $	$ 49,363.28 $	$ 49,363.28 $	$ 49,363.28 $	$ 49,363.28 $	$ 49,363.28 $	$ 49,363.28 $	$ 49,363.28 $	$ 1,641,338.12 $
20	$ 49,363.28 $	$ 49,363.28 $	$ 49,363.28 $	$ 49,363.28 $	$ 49,363.28 $	$ 49,363.28 $	$ 49,363.28 $	$ 49,363.28 $	$ 49,363.28 $	$ 1,641,338.12 $

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Table 21. Sensitivity tests' execution time

Case	Soil region	$ {\bf P_0} $ (US$ \$)	$ {\bf P_y} $ (US$ \$)	$ {\bf T_{solve}(s)} $	$ {\bf T_{load}(s)} $
1	Eastern hills	270,000	16,500	7.512	3.377
2	Central-south	270,000	16,500	8.440	0.81
3	Basaltic	270,000	16,500	15.510	0.798
4	Basaltic	270,000	16,500	18.584	0.779
5	Basaltic	270,000	16,500	55.411	0.623
6	Eastern hills	270,000	16,500	27.218	0.765
7	Basaltic	270,000	16,500	11.222	0.627
8	Central-south	216,000	16,500	50.483	0.794
9	Central-south	162,000	16,500	21.461	0.685
10	Central-south	108,000	16,500	28.265	2.812
11	Central-south	54,000	16,500	1.900	1.046
12	Central-south	270,000	33,000	0.207	0.736
13	Central-south	270,000	0	26.524	0.661
14	Central-south	270,000	33,000	6.592	0.93
15	Central-south	270,000	21,120	13.022	0.718
16	Central-south	270,000	16,500	0.559	0.682
17	Central-south	270,000	16,500	0.108	1.978
18	Central-south	270,000	16,500	0.203	1.098
19	Central-south	28,000	1,750	0.089	0.811
20	Central-south	20,500	1,250	6.088	0.752
21	Central-south	150,000	9,000	24.213	0.593

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Table 22. Annual balances for the solutions of sensitivity cases

Case	1	2	3	4	5	6	7	8	9	10
1	$ -16,496.69 $	$ -16,496.69 $	$ -16,496.69 $	$ -16,496.69 $	$ -16,496.69 $	$ -16,496.69 $	$ -16,496.69 $	$ -16,496.69 $	$ -16,496.69 $	$ 4,113,590.13 $
2	$ -16,493.89 $	$ -16,493.89 $	$ -16,493.89 $	$ -16,493.89 $	$ -16,493.89 $	$ -16,493.89 $	$ -16,493.89 $	$ -16,493.89 $	$ -16,493.89 $	$ 3,623,128.91 $
3	$ -16,494.48 $	$ -16,494.48 $	$ -16,494.48 $	$ -16,494.48 $	$ -16,494.48 $	$ -16,494.48 $	$ -16,494.48 $	$ -16,494.48 $	$ -16,494.48 $	$ 3,181,523.82 $
4	$ -16,499.65 $	$ -16,499.65 $	$ -16,499.65 $	$ -16,499.65 $	$ -16,499.65 $	$ -16,499.65 $	$ -16,499.65 $	$ -16,499.65 $	$ -16,499.65 $	$ 3,737,995.96 $
5	$ -16,490.31 $	$ -16,490.31 $	$ -16,490.31 $	$ -16,490.31 $	$ -16,490.31 $	$ -16,490.31 $	$ -16,490.31 $	$ -16,490.31 $	$ -16,490.31 $	$ 3,721,118.14 $
6	$ -16,498.71 $	$ -16,498.71 $	$ -16,498.71 $	$ -16,498.71 $	$ -16,498.71 $	$ -16,498.71 $	$ -16,498.71 $	$ -16,498.71 $	$ -16,498.71 $	$ 3,817,498.17 $
7	$ -16,487.24 $	$ -16,487.24 $	$ -16,487.24 $	$ -16,487.24 $	$ -16,487.24 $	$ -16,487.24 $	$ -16,487.24 $	$ -16,487.24 $	$ -16,487.24 $	$ 3,467,822.72 $
8	$ -2,293.31 $	$ -2,293.31 $	$ -2,293.31 $	$ -2,293.31 $	$ -2,293.31 $	$ -2,293.31 $	$ -2,293.31 $	$ -2,293.31 $	$ -2,293.31 $	$ 3,142,782.06 $
9	$ 28,356.85 $	$ 28,356.85 $	$ 28,356.85 $	$ 28,356.85 $	$ 28,356.85 $	$ 28,356.85 $	$ 28,356.85 $	$ 28,356.85 $	$ 28,356.85 $	$ 2,105,821.82 $
10	$ 59,003.49 $	$ 59,003.49 $	$ 59,003.49 $	$ 59,003.49 $	$ 59,003.49 $	$ 59,003.49 $	$ 59,003.49 $	$ 59,003.49 $	$ 59,003.49 $	$ 1,068,980.29 $
11	$ 77,721.17 $	$ 77,721.17 $	$ 77,721.17 $	$ 77,721.17 $	$ 77,721.17 $	$ 77,721.17 $	$ 77,721.17 $	$ 77,721.17 $	$ 77,721.17 $	$ 131,720.11 $
12	$ -18,364.02 $	$ -18,364.02 $	$ -18,364.02 $	$ -18,364.02 $	$ -18,364.02 $	$ -18,364.02 $	$ -18,364.02 $	$ -18,364.02 $	$ -18,364.02 $	$ 3,686,488.74 $
13	$ 1.51 $	$ 1.51 $	$ 1.51 $	$ 1.51 $	$ 1.51 $	$ 1.51 $	$ 1.51 $	$ 1.51 $	$ 1.51 $	$ 3,065,143.59 $
14	$ -3,651.25 $	$ -3,651.25 $	$ -3,651.25 $	$ -3,651.25 $	$ -3,651.25 $	$ -3,651.25 $	$ -3,651.25 $	$ -3,651.25 $	$ -3,651.25 $	$ 3,188,724.08 $
15	$ 9.75 $	$ 9.75 $	$ -1,310.25 $	$ 9.75 $	$ 9.75 $	$ -1,310.25 $	$ 9.75 $	$ 9.75 $	$ -1,310.25 $	$ 3,080,864.73 $
16	$ -16,497.29 $	$ -16,497.29 $	$ -16,497.29 $	$ -16,497.29 $	$ -16,497.29 $	$ -16,497.29 $	$ -16,497.29 $	$ -16,497.29 $	$ -16,497.29 $	$ 3,623,333.29 $
17	$ 86,278.09 $	$ 86,278.09 $	$ 86,278.09 $	$ 86,278.09 $	$ 86,278.09 $	$ 86,278.09 $	$ 86,278.09 $	$ 86,278.09 $	$ 86,278.09 $	$ 146,222.20 $
18	$ -16,498.65 $	$ -16,498.65 $	$ -16,498.65 $	$ -16,498.65 $	$ -16,498.65 $	$ -16,498.65 $	$ -16,498.65 $	$ -16,498.65 $	$ -16,498.65 $	$ 2,137,009.35 $
19	$ 2,251.18 $	$ 2,251.18 $	$ 2,251.18 $	$ 2,251.18 $	$ 2,251.18 $	$ 2,251.18 $	$ 2,251.18 $	$ 2,251.18 $	$ 2,251.18 $	$ 232,566.42 $
20	$ -1,246.57 $	$ -1,246.57 $	$ -1,246.57 $	$ -1,246.57 $	$ -1,246.57 $	$ -1,246.57 $	$ -1,246.57 $	$ -1,246.57 $	$ -1,246.57 $	$ 281,967.80 $
21	$ -8,990.71 $	$ -8,990.71 $	$ -8,990.71 $	$ -8,990.71 $	$ -8,990.71 $	$ -8,990.71 $	$ -8,990.71 $	$ -8,990.71 $	$ -8,990.71 $	$ 1,947,909.04 $

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