Optimization Of Waste Water Utilization In Irrigation

Optimization of Wastewater Utilization in Irrigation: A Case Study in Lebanon

Jennifer Gliem Emily Parker

BE 431 Resource Optimization

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Table of Contents Introduction………………………………………………………………………………..3 Objective…………………………………………………………………………………..4 Methods……………………………………………………………………………………5 Results……………………………………………………………………………………..7 Discussion…………………………………………………………………………………9 Conclusions………………………………………………………………………………10 Appendix…………………………………………………………………………………11 Bibliography……………………………………………………………………………..20

List of Figures & Tables Figure 1 (Location of study area)……………………………………………………….…4 Table 1 ( Net revenue and nitrogen and water requirements for specified crops)………………………………………………………………………………………6 Table 2 (Net return, secondary treated wastewater and nitrogen consumption, and optimal cropping pattern for the feasible scenarios)……………………………………………….7 Figure 2 (Total net revenue of scenarios 1 through 4)…………………………………….8 Figure 3 (Water consumption of scenarios 1 through 4b)………………………………...8 Figure 4 (Nitrogen consumption of scenarios 1 through 4b)……………………………...9

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Introduction Wastewater is defined as the used water and solids from the community that flow to a treatment plant (1). Wastewater is also made up of water that collects in street drains during storms, as well as groundwater that enters through cracks in sewers. On average, each person in the U.S. contributes 50 to 100 gallons of wastewater everyday (2). Typically wastewater is subjected to three levels of treatment; primary, secondary, and tertiary treatment. Primary treatment removes approximately 60% of the suspended solids from wastewater, and reduces the BOD content by 35% (3). Secondary treatment removes 90% of the rest of the suspended solids and BOD (1). The final treatment process, tertiary treatment, removes 95% or more of the remaining contaminants (4). Tertiary treatment of wastewater in the municipal system can be avoided by using secondary treated wastewater for irrigation. The reuse of treated wastewater in irrigation provides a cost effective and environmentally friendly alternative to wastewater disposal. When wastewater is adequately treated it can be safely applied to crops, and poses no greater risk to the consumer than conventional sources (5). Currently both Myrtle Beach, South Carolina and Santa Rosa, California successfully use secondary treated wastewater for supplemental irrigation (6). Currently in Lebanon, there are no regulations on the levels of wastewater treatment requirements. In many coastal areas wastewater is disposed directly into the Mediterranean Sea without any preliminary treatment. These practices can lead to human health hazards and eutrophication (7). The use of treated wastewater for irrigation in Lebanon may be a practical alternative for wastewater disposal. There are several benefits to using secondary treated wastewater for irrigation in Lebanon. • Nutrients in wastewater can supply much of the phosphorus and nitrogen required by plants previously supplied by fertilizers. • Application of wastewater to crops prevents the contamination of Lebanon’s surface and groundwater supplies. • Provides a solution to future water scarcity. • Reduces the risk to human health caused by dumping untreated wastewater into surface water. • Provides an economically sound alternative to municipal wastewater treatment.

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Objective In this study, we will apply secondary treated wastewater for irrigation in Lebanon to determine which cropping patterns will provide the best alternative for Lebanon by looking at 1. Net revenue 2. Nitrogen removal 3. Water consumption We will accomplish these objectives by utilizing linear programming for optimization of wastewater reuse in irrigation. We have optimized five scenarios for wastewater irrigation in Lebanon. These scenarios examine the net revenue produced by the optimal cropping pattern, as well as nitrogen removal and water consumption.

Methods We conducted a study on the use of waste water for irrigation in Lebanon, based on a previous study by Darwish et al.(7). The study area selected is in the Tyre region of south Lebanon (Figure1). The land available in this region for agriculture use is 1500 ha. The study area currently cultivates 930 ha of citrus, 330 ha of bananas, 36 ha of tomato greenhouses, 7 ha of cucumber greenhouses. The remaining 197 ha are divided between melons, peppers, field tomatoes, and field cucumbers. The wastewater produced by the study area is primarily domestic (7). Further characteristics of the study area can be found in Darwish et al. (7).

Study Area

The amount of wastewater produced annually is 8,891,035 m3. The total nitrogen provided from this wastewater is 350 thousand tons per year. The cropping patterns currently in use generate a total net revenue of $7,406,437 per year.

Figure 1 – Location of the Study Area

We examined five scenarios to maximize total net revenue grossed in the study area. Scenarios 2 through 5 contain sub-scenarios, a and b. In sub-scenarios a only the crops currently in cultivation in the study are considered when determining the cropping pattern which maximizes total net revenue. In sub-scenarios b new crops are introduced. These crops include alfalfa, roses in greenhouse, carnation in greenhouse, and gerbera in greenhouse. The new crops were chosen primarily for their high nitrogen uptake and the

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amount of revenue they produce. Other reasons for including these crops in this study can be found in Darwish et al. (7). Each scenario was subject to constraints land use. The total amount of land used by the crops could not exceed the available 1500 ha. The amount of land used for citrus and bananas could not be less than the amount currently in use for these crops (930 ha for citrus and 330 for bananas). We did not deem it economically sound to remove citrus and banana trees currently planted. To maintain the economic stability if the region, we placed constraints which ensured that more than one type of crop would be produced. Therefore, there had to be at least 100 ha of vegetable crop. Vegetable crops were defined as tomatoes, cucumbers, melons, and peppers for scenarios a. Scenarios b included alfalfa as a vegetable crop. Constraints were also placed on the amount of land available for greenhouse crops. Since there are currently 36 ha allotted to greenhouse production, we did not find it economical to remove or build new greenhouses. Therefore, the amount of land used for greenhouse crops must be equal to 36 ha. Scenario 1 Scenario 1 represents status quo. We used scenario 1 as a point of comparison for each additional scenario. Scenario 2a and 2b Scenarios 2a and 2b maximize total net revenue in the study area with no constraints placed on available wastewater or nitrogen. The only constraints are in reference to land use. Scenario 3a and 3b Scenarios 3a and 3b maximize total net revenue in the study area while placing constraints on the amount of wastewater available for irrigation. There are no constraints placed on available nitrogen. The water used for irrigation had to be at least 8,891,035 m3, the amount of available wastewater produced by the study area. Therefore, in Scenarios 3a and 3b, the entire amount of wastewater produced by the study area will be used for irrigation. Scenario 4a and 4b Scenarios 4a and 4b maximize total net revenue in the study area while placing constraints on the amount of nitrogen available in the wastewater. There are no constraints placed on available wastewater. Scenarios 4a and 4b ensure that the crops will utilize all of the nitrogen present in the wastewater. This prevents nitrogen contamination of groundwater supplies. Scenario 5a and 5b Scenario 5a and 5b maximize total net revenue in the study area while placing constraints on both the amount of water and the amount of nitrogen available. These scenarios ensure that not only will all of the wastewater be disposed of through irrigation, but also that all of the nitrogen will be removed from the wastewater by the crops.

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Table 1. Net revenue, nitrogen requirements, and water requirements of selected crops.

Citrus

5,000

Nitrogen Requirements (kg/ha) 200

Banana

6,100

400

588,800,000

Alfalfa

870

1

1,600,000

Tomato field

2,990

134.4

600,00

Cucumber field

2,210

89.6

1,168,400

Melon

5,970

112

600,000

Pepper Winter tomato in greenhouse Summer tomato in greenhouse Winter cucumber in greenhouse Summer cucumber in greenhouse Roses in greenhouse Carnation in greenhouse Gerbera in greenhouse

3,900

112

900,000

18,340

134.4

600,000

18,250

134.4

600,000

11,300

89.6

1,168,400

11,260

89.6

1,168,400

185,320

951

1,981,200

183,650

951

1,981,200

112,500

951

1,981,200

Crop

Net Return ($/ha)

Water Requirements (L/ha) 1,200,000

The net revenue, nitrogen, and water requirements for each crop considered in this study can be found in Table 1. For each scenario we maximized profit using a system of linear equations and the ConstrainedMax function in Mathematica. The results of this analyzation can be found in the appendix. The net revenue, wastewater, and nitrogen consumption for each scenario were graphed to illustrate the relationships between each scenario.

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Results There was no feasible solution for scenario 5a or 5b. Therefore, scenarios 5a and 5b have been omitted from the rest of this study.

Table 2. Net Return above all costs, secondary treated wastewater and nitrogen consumption, and optimal cropping pattern for the feasible scenarios and sub-scenarios Net Return ($ US) Total secondary treated wastewater consumption (L/ha) Used Unused Supplementary Total Nitrogen Uptake (kg/ha) Used Unused Additional Crop area (ha) Citrus Banana Alfalfa Tomato field Cucumber field Melon Pepper Winter tomato in greenhouse Summer tomato in greenhouse Winter cucumber in greenhouse Summer cucumber in greenhouse Roses in greenhouse Carnation in greenhouse Gerbera in greenhouse

1 7406437

2a 8554640

Scenarios and Sub-scenarios 2b 3a 3b 14565920 8554640 14565920

4a 8.55*106

4b 8.55*106

1.85*1011 0 1.76*1011

2.44*1011 0 2.35*1011

2.44*1011 0 2.35*1011

2.44*1011 0 2.35*1011

2.44*1011 0 2.35*1011

2.44*1011 0 2.35*1011

2.44*1011 0 2.35*1011

345529.6 0 28014.9

375638.4 0 58123.7

405036 0 87521.3

375638.4 0 58123.7

405036 0 87521.3

375638.4 0 58123.7

375638.4 0 58123.7

930 330 NA * * * * 36 0 7 0 NA NA NA

930 434 NA 0 0 100 0 36 0 0 0 NA NA NA

930 434 0 0 0 100 0 0 0 0 0 36 0 0

930 434 NA 0 0 100 0 36 0 0 0 NA NA NA

930 434 0 0 0 100 0 0 0 0 0 36 0 0

930 434 NA 0 0 100 0 36 0 0 0 NA NA NA

930 434 0 0 0 100 0 36 0 0 0 0 0 0

The results for each scenario and sub scenario can be found in Table 2. The * indicates that these crops covered a total of 197 ha.

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Total Net Revenue ($/yr)

16000000 14000000 12000000 10000000 8000000 6000000 4000000 2000000 0 1

2a

2b

3a

3b

4a

4b

Scenario

Figure 2 – Total Net Revenue of Scenarios 1 through 4b The total net revenue produced by each scenario can be found in Figure 2. Scenarios 2b and 3b produced the same cropping pattern, and therefore have the same net revenue. Scenarios 2a, 3a, 4a, and 4b also produced the same cropping pattern and net revenue. Scenarios 2b and 3b produced the highest total net revenue. All scenarios produce a greater total net revenue than the cropping pattern currently in practice which is represented by the dotted line in Figure 2.

Water Consumption (L/ha)

3.00E+11 2.50E+11 2.00E+11 1.50E+11 1.00E+11 5.00E+10 0.00E+00 1

2a

2b

3a

3b

4a

4b

Scenario

Figure 3 – Water Consumption of Scenarios 1 through 4b The water requirements for each scenario can be found in Figure 3. Although two cropping patterns were recommended (one for scenarios 2b and 3b, and another for scenarios 2a, 3a, 4a, and 4b) the water requirements for all six scenarios are the same. The two new cropping patterns consume more water than the cropping pattern currently in place. All scenarios use more water than can be provided by the wastewater, which is represented by the dotted line in Figure 3.

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Nitrogen Consumption (kg/ha)

410000 400000 390000 380000 370000 360000 350000 340000 330000 320000 310000 1

2a

2b

3a

3b

4a

4b

Scenario

Figure 4 – Nitrogen Consumption of Scenarios 1 through 4b The nitrogen requirements of each scenario can be found in Figure 4. The cropping pattern prescribed for scenario 2b and 3b uses more nitrogen than the cropping pattern prescribed by scenario 2a, 3a, 4a, 4b, and the current cropping pattern (scenario 1). In addition, scenario 2b and 3b are the only scenarios which consume all of the nitrogen available in the wastewater.

Discussion Given the model assumptions and specifications stated earlier the following conclusions can be drawn. Water consumption did not play an active role in deciding which scenario benefited the study area the most. The water required by the crops in all scenarios were equivalent and therefore water did not recommend one scenario over another. Scenarios 2b and 3b require more nitrogen than scenarios 2a, 3a, 4a and 4b. Since all of the scenarios consume more than the amount of nitrogen present in the wastewater all of the scenarios satisfy our requirements. Therefore, the greatest impact nitrogen will have on the solution will be the added cost of additional nitrogen. The greatest profit for the study area would be achieved by implementing the cropping pattern recommended by scenarios 2b and 3b. However, these figures are not entirely accurate. The excessive nitrogen consumption of scenarios 2b and 3b would detract from the profit gained from producing these crops. We estimate that although there will be the added cost of nitrogen fertilizer, scenarios 2b and 3b will still provide higher profits than scenarios 2a, 3a, 4a and 4b. We recognize the possibility of local resistance to the addition of new crops which were examined in scenarios 2b, 3b and 4b. However, the addition of the new crops will

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provide higher net profits and serve to diversify the agricultural commodities on which the study area is dependent.

Conclusions In conclusion, we recommend the implementation of the cropping pattern suggested by scenarios 2b and 3b. This cropping pattern will provide the study area with increased net revenue, a diversified agricultural commodities market and an economically and environmentally sound method of wastewater disposal.

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Appendix Mathematical calculations for Scenario 2a

f = 5000 x1 + 6100 x2 + 2990 x4 + 2210 x5 + 5970 x6 + 3900 x7 + 18340 x8 + 18250 x9 + 11300 x10 + 11260 x11 5000 x1 + 11300 x10 + 11260 x11 + 6100 x2 + 2990 x4 + 2210 x5 + 5970 x6 + 3900 x7 + 18340 x8 + 18250 x9

c1 = x1 + x2 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 £ 1500 c2 = x1 ³ 930 c5 = x2 ³ 330 c3 = x4 + x5 + x6 + x7 ³ 100 c4 = x8 + x9 + x10 + x11 Š 36 x1 + x10 + x11 + x2 + x4 + x5 + x6 + x7 + x8 + x9 £ 1500 x1 ³ 930 x2 ³ 330 x4 + x5 + x6 + x7 ³ 100 x10 + x11 + x8 + x9 == 36

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ConstrainedMax f, c1, c2, c3, c4, c5 , x1, x2, x4, x5, x6, x7, x8, x9, x10, x11

8554640, x1 ® 930, x2 ® 434, x4 ® 0, x5 ® 0, x6 ® 100, x7 ® 0, x8 ® 36, x9 ® 0, x10 ® 0, x11 ® 0

Mathematica calculations for Scenario 2b

f = 5000 x1 + 6100 x2 + 870 x3 + 2990 x4 + 2210 x5 + 5970 x6 + 3900 x7 + 18340 x8 + 18250 x9 + 11300 x10 + 11260 x11 + 185320 x12 + 183650 x13 + 112500 x14 5000 x1 + 11300 x10 + 11260 x11 + 185320 x12 + 183650 x13 + 112500 x14 + 6100 x2 + 870 x3 + 2990 x4 + 2210 x5 + 5970 x6 + 3900 x7 + 18340 x8 + 18250 x9

c1 = x1 + x2 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x3 + x12 + x13 + x14 £ 1500 c2 = x1 ³ 930 c5 = x2 ³ 330 c3 = x4 + x5 + x6 + x7 + x3 ³ 100 c4 = x8 + x9 + x10 + x11 + x12 + x13 + x14 Š 36 x1 + x10 + x11 + x12 + x13 + x14 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 £ 1500 x1 ³ 930 x2 ³ 330 x3 + x4 + x5 + x6 + x7 ³ 100 x10 + x11 + x12 + x13 + x14 + x8 + x9 == 36

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Appendix

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Mathematica calculations for Scenario 2b continued


ConstrainedMax f, c1, c2, c3, c4, c5 , x1, x2, x4, x5, x6, x7, x8, x9, x10, x11, x3, x12, x13, x14

14565920, x1 ® 930, x2 ® 434, x4 ® 0, x5 ® 0, x6 ® 100, x7 ® 0, x8 ® 0, x9 ® 0, x10 ® 0, x11 ® 0, x3 ® 0, x12 ® 36, x13 ® 0, x14 ® 0

Mathematica calculations for Scenario 3a

f = 5000 x1 + 6100 x2 + 2990 x4 + 2210 x5 + 5970 x6 + 3900 x7 + 18340 x8 + 18250 x9 + 11300 x10 + 11260 x11 5000 x1 + 11300 x10 + 11260 x11 + 6100 x2 + 2990 x4 + 2210 x5 + 5970 x6 + 3900 x7 + 18340 x8 + 18250 x9

c1 = x1 + x2 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 £ 1500 c2 = x1 ³ 930 c5 = x2 ³ 330 c3 = x4 + x5 + x6 + x7 ³ 100 c4 = x8 + x9 + x10 + x11 Š 36 c6 = 1200000 x1 + 1168400 x10 + 1168400 x11 + 558800000 x2 + 600000 x4 + 1168400 x5 + 600000 x6 + 900000 x7 + 600000 x8 + 600000 x9 >= 8890000000 x1 + x10 + x11 + x2 + x4 + x5 + x6 + x7 + x8 + x9 £ 1500 x1 ³ 930 x2 ³ 330 x4 + x5 + x6 + x7 ³ 100 x10 + x11 + x8 + x9 == 36 1200000 x1 + 1168400 x10 + 1168400 x11 + 558800000 x2 + 600000 x4 + 1168400 x5 + 600000 x6 + 900000 x7 + 600000 x8 + 600000 x9 ³ 8890000000

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ConstrainedMax f, c1, c2, c3, c4, c5, c6 , x1, x2, x4, x5, x6, x7, x8, x9, x10, x11

<

8554640, x1 ® 930, x2 ® 434, x4 ® 0, x5 ® 0, x6 ® 100, x7 ® 0, x8 ® 36, x9 ® 0, x10 ® 0, x11 ® 0

Mathematica calculations for Scenario 3b

f = 5000 x1 + 6100 x2 + 870 x3 + 2990 x4 + 2210 x5 + 5970 x6 + 3900 x7 + 18340 x8 + 18250 x9 + 11300 x10 + 11260 x11 + 185320 x12 + 183650 x13 + 112500 x14

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Appendix Mathematica calculations for Scenario 3b continued 5000 x1 + 11300 x10 + 11260 x11 + 185320 x12 + 183650 x13 + 112500 x14 + 6100 x2 + 870 x3 + 2990 x4 + 2210 x5 + 5970 x6 + 3900 x7 + 18340 x8 + 18250 x9

c1 = x1 + x2 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x3 + x12 + x13 + x14 £ 1500 c2 = x1 ³ 930 c5 = x2 ³ 330 c3 = x4 + x5 + x6 + x7 + x3 ³ 100 c4 = x8 + x9 + x10 + x11 + x12 + x13 + x14 Š 36 c6 = 1200000 x1 + 1168400 x10 + 1168400 x11 + 558800000 x2 + 600000 x4 + 1168400 x5 + 600000 x6 + 900000 x7 + 600000 x8 + 600000 x9 + 1981200 x12 + 1981200 x13 + 1981200 x14 + 1600000 x3 >= 8890000000 x1 + x10 + x11 + x12 + x13 + x14 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 £ 1500 x1 ³ 930 x2 ³ 330 x3 + x4 + x5 + x6 + x7 ³ 100 x10 + x11 + x12 + x13 + x14 + x8 + x9 == 36 1200000 x1 + 1168400 x10 + 1168400 x11 + 1981200 x12 + 1981200 x13 + 1981200 x14 + 558800000 x2 + 1600000 x3 + 600000 x4 + 1168400 x5 + 600000 x6 + 900000 x7 + 600000 x8 + 600000 x9 ³ 8890000000

1200000 x1 + 1168400 x10 + 1168400 x11 + 1981200 x12 + 1981200 x13 + 1981200 x14 + 558800000 x2 + 1600000 x3 + 600000 x4 + 1168400 x5 + 600000 x6 + 900000 x7 + 600000 x8 + 600000 x98890000000 ConstrainedMax f, c1, c2, c3, c4, c5, c6 , x1, x2, x4, x5, x6, x7, x8, x9, x10, x11, x3, x12, x13, x14

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14565920, x1 ® 930, x2 ® 434, x4 ® 0, x5 ® 0, x6 ® 100, x7 ® 0, x8 ® 0, x9 ® 0, x10 ® 0, x11 ® 0, x3 ® 0, x12 ® 36, x13 ® 0, x14 ® 0

Mathematica calculations for Scenario 4a

f = 5000 x1 + 6100 x2 + 2990 x4 + 2210 x5 + 5970 x6 + 3900 x7 + 18340 x8 + 18250 x9 + 11300 x10 + 11260 x11 5000 x1 + 11300 x10 + 11260 x11 + 6100 x2 + 2990 x4 + 2210 x5 + 5970 x6 + 3900 x7 + 18340 x8 + 18250 x9

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Appendix Mathematica calculations for Scenario 4a continued

c1 = x1 + x2 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 £ 1500 c2 = x1 ³ 930 c5 = x2 ³ 330 c3 = x4 + x5 + x6 + x7 ³ 100 c4 = x8 + x9 + x10 + x11 Š 36 c6 = 200 x1 + 400 x2 + 134.4 x4 + 89.6 x5 + 112 x6 + 112 x7 + 134.4 x8 + 134.4 x9 + 89.6 x10 + 89.6 x11 ³ 317514.7 x1 + x10 + x11 + x2 + x4 + x5 + x6 + x7 + x8 + x9 £ 1500 x1 ³ 930 x2 ³ 330 x4 + x5 + x6 + x7 ³ 100 x10 + x11 + x8 + x9 == 36 200 x1 + 89.6 x10 + 89.6 x11 + 400 x2 + 134.4 x4 + 89.6 x5 + 112 x6 + 112 x7 + 134.4 x8 + 134.4 x9 ³ 317515.

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ConstrainedMax f, c1, c2, c3, c4, c5, c6 , x1, x2, x4, x5, x6, x7, x8, x9, x10, x11

<

8.55464 ´ 106, x1 ® 930., x2 ® 434., x4 ® 0, x5 ® 0, x6 ® 100., x7 ® 0, x8 ® 36., x9 ® 0, x10 ® 0, x11 ® 0

Mathematica calculations for Scenario 4b

f = 5000 x1 + 6100 x2 + 870 x3 + 2990 x4 + 2210 x5 + 5970 x6 + 3900 x7 + 18340 x8 + 18250 x9 + 11300 x10 + 11260 x11 + 185320 x12 + 183650 x13 + 112500 x14 c1 = x1 + x2 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x3 + x12 + x13 + x14 £ 1500 c2 = x1 ³ 930 c5 = x2 ³ 330 c3 = x4 + x5 + x6 + x7 + x3 ³ 100 c4 = x8 + x9 + x10 + x11 + x12 + x13 + x14 Š 36 c6 = 200 x1 + 400 x2 + 134.4 x4 + 89.6 x5 + 112 x6 + 112 x7 + 134.4 x8 + 134.4 x9 + 89.6 x10 + 89.6 x11 + x3 + 951 x12 + 951 x13 + 951 x14 ³ 317514.7 x1 + x10 + x11 + x12 + x13 + x14 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 £ 1500 x1 ³ 930

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Appendix Mathematica calculations for Scenario 4b continued x2 ³ 330 x3 + x4 + x5 + x6 + x7 ³ 100 x10 + x11 + x12 + x13 + x14 + x8 + x9 == 36 200 x1 + 89.6 x10 + 89.6 x11 + 951 x12 + 951 x13 + 951 x14 + 400 x2 + x3 + 134.4 x4 + 89.6 x5 + 112 x6 + 112 x7 + 134.4 x8 + 134.4 x9 ³ 317515.

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ConstrainedMax f, c1, c2, c3, c4, c5, c6 , x1, x2, x4, x5, x6, x7, x8, x9, x10, x11, x3, x12, x13, x14

<

8.55464 ´ 106, x1 ® 930., x2 ® 434., x4 ® 0, x5 ® 0, x6 ® 100., x7 ® 0, x8 ® 36., x9 ® 0, x10 ® 0, x11 ® 0, x3 ® 0, x12 ® 0, x13 ® 0, x14 ® 0

Mathematica calculations for Scenario 5a

f = 5000 x1 + 6100 x2 + 2990 x4 + 2210 x5 + 5970 x6 + 3900 x7 + 18340 x8 + 18250 x9 + 11300 x10 + 11260 x11 5000 x1 + 11300 x10 + 11260 x11 + 6100 x2 + 2990 x4 + 2210 x5 + 5970 x6 + 3900 x7 + 18340 x8 + 18250 x9

c1 = x1 + x2 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 £ 1500 c2 = x1 ³ 930 c5 = x2 ³ 330 c3 = x4 + x5 + x6 + x7 ³ 100 c4 = x8 + x9 + x10 + x11 Š 36 c6 = 1200000 x1 + 1168400 x10 + 1168400 x11 + 558800000 x2 + 600000 x4 + 1168400 x5 + 600000 x6 + 900000 x7 + 600000 x8 + 600000 x9 ³ 8890000000 c7 = 200 x1 + 400 x2 + 134.4 x4 + 89.6 x5 + 112 x6 + 112 x7 + 134.4 x8 + 134.4 x9 + 89.6 x10 + 89.6 x11 ³ 317514.7 x1 + x10 + x11 + x2 + x4 + x5 + x6 + x7 + x8 + x9 £ 1500 x1 ³ 930 x2 ³ 330 x4 + x5 + x6 + x7 ³ 100 x10 + x11 + x8 + x9 == 36

15

Appendix Mathematica calculations for Scenario 5a continued 1200000 x1 + 1168400 x10 + 1168400 x11 + 558800000 x2 + 600000 x4 + 1168400 x5 + 600000 x6 + 900000 x7 + 600000 x8 + 600000 x9 ³ 8890000000 200 x1 + 89.6 x10 + 89.6 x11 + 400 x2 + 134.4 x4 + 89.6 x5 + 112 x6 + 112 x7 + 134.4 x8 + 134.4 x9 ³ 317515.

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ConstrainedMax f, c1, c2, c3, c4, c5, c6, c7 , x1, x2, x4, x5, x6, x7, x8, x9, x10, x11 ConstrainedMax::nsatt : The specified constraints cannot be satisfied with tolerance 1.`*^-6.

ConstrainedMax 5000 x1 + 11300 x10 + 11260 x11 + 6100 x2 + 2990 x4 + 2210 x5 + 5970 x6 + 3900 x7 + 18340 x8 + 18250 x9, x1 + x10 + x11 + x2 + x4 + x5 + x6 + x7 + x8 + x9 £ 1500, x1 ³ 930, x4 + x5 + x6 + x7 ³ 100, x10 + x11 + x8 + x9 == 36, x2 ³ 330, 1200000 x1 + 1168400 x10 + 1168400 x11 + 558800000 x2 + 600000 x4 + 1168400 x5 + 600000 x6 + 900000 x7 + 600000 x8 + 600000 x9 ³ 8890000000, 200 x1 + 89.6 x10 + 89.6 x11 + 400 x2 + 134.4 x4 + 89.6 x5 + 112 x6 + 112 x7 + 134.4 x8 + 134.4 x9 ³ 317515. , x1, x2, x4, x5, x6, x7, x8, x9, x10, x11

Mathematica calculations for Scenario 5b

f = 5000 x1 + 6100 x2 + 870 x3 + 2990 x4 + 2210 x5 + 5970 x6 + 3900 x7 + 18340 x8 + 18250 x9 + 11300 x10 + 11260 x11 + 185320 x12 + 183650 x13 + 112500 x14 5000 x1 + 11300 x10 + 11260 x11 + 185320 x12 + 183650 x13 + 112500 x14 + 6100 x2 + 870 x3 + 2990 x4 + 2210 x5 + 5970 x6 + 3900 x7 + 18340 x8 + 18250 x9

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Appendix Mathematica calculations for Scenario 5b continued

c1 = x1 + x2 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x3 + x12 + x13 + x14 £ 1500 c2 = x1 ³ 930 c5 = x2 ³ 330 c3 = x4 + x5 + x6 + x7 + x3 ³ 100 c4 = x8 + x9 + x10 + x11 + x12 + x13 + x14 Š 36 c6 = 200 x1 + 400 x2 + 134.4 x4 + 89.6 x5 + 112 x6 + 112 x7 + 134.4 x8 + 134.4 x9 + 89.6 x10 + 89.6 x11 + x3 + 951 x12 + 951 x13 + 951 x14 ³ 317514.7 c7 = 1200000 x1 + 1168400 x10 + 1168400 x11 + 558800000 x2 + 600000 x4 + 1168400 x5 + 600000 x6 + 900000 x7 + 600000 x8 + 600000 x9 + 1981200 x12 + 1981200 x13 + 1981200 x14 + 1600000 x3 >= 8890000000 x1 + x10 + x11 + x12 + x13 + x14 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 £ 1500 x1 ³ 930 x2 ³ 330 x3 + x4 + x5 + x6 + x7 ³ 100 x10 + x11 + x12 + x13 + x14 + x8 + x9 == 36 200 x1 + 89.6 x10 + 89.6 x11 + 951 x12 + 951 x13 + 951 x14 + 400 x2 + x3 + 134.4 x4 + 89.6 x5 + 112 x6 + 112 x7 + 134.4 x8 + 134.4 x9 ³ 317515. 1200000 x1 + 1168400 x10 + 1168400 x11 + 1981200 x12 + 1981200 x13 + 1981200 x14 + 558800000 x2 + 1600000 x3 + 600000 x4 + 1168400 x5 + 600000 x6 + 900000 x7 + 600000 x8 + 600000 x9 ³ 8890000000

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ConstrainedMax f, c1, c2, c3, c4, c5, c6, c7 , x1, x2, x4, x5, x6, x7, x8, x9, x10, x11, x3, x12, x13, x14 ConstrainedMax::nsatt : The specified constraints cannot be satisfied with tolerance 1.`*^-6.

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Appendix

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Mathematica calculations for Scenario 5b continued ConstrainedMax 5000 x1 + 11300 x10 + 11260 x11 + 185320 x12 + 183650 x13 + 112500 x14 + 6100 x2 + 870 x3 + 2990 x4 + 2210 x5 + 5970 x6 + 3900 x7 + 18340 x8 + 18250 x9, x1 + x10 + x11 + x12 + x13 + x14 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 £ 1500, x1 ³ 930, x3 + x4 + x5 + x6 + x7 ³ 100, x10 + x11 + x12 + x13 + x14 + x8 + x9 == 36, x2 ³ 330, 200 x1 + 89.6 x10 + 89.6 x11 + 951 x12 + 951 x13 + 951 x14 + 400 x2 + x3 + 134.4 x4 + 89.6 x5 + 112 x6 + 112 x7 + 134.4 x8 + 134.4 x9 ³ 317515., 1200000 x1 + 1168400 x10 + 1168400 x11 + 1981200 x12 + 1981200 x13 + 1981200 x14 + 558800000 x2 + 1600000 x3 + 600000 x4 + 1168400 x5 + 600000 x6 + 900000 x7 + 600000 x8 + 600000 x9 ³ 8890000000 , x1, x2, x4, x5, x6, x7, x8, x9, x10, x11, x3, x12, x13, x14

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