A mixed-integer linear programming model for optimal vessel scheduling in offshore oil and gas operations

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October 2017, 13(4): 1601-1623. doi: 10.3934/jimo.2017009

A mixed-integer linear programming model for optimal vessel scheduling in offshore oil and gas operations

Elham Mardaneh ^1, , Ryan Loxton ^1, , Qun Lin ^1, and Phil Schmidli ^2,

1.	Curtin University, Perth, 6102, Australia
2.	Woodside Energy Ltd, Perth, 6000, Australia

Received November 2015 Revised May 2016 Published December 2016

This paper introduces a non-standard vehicle routing problem (VRP) arising in the oil and gas industry. The problem involves multiple offshore production facilities, each of which requires regular servicing by support vessels to replenish essential commodities such as food, water, fuel, and chemicals. The support vessels are also required to assist with oil off-takes, in which oil stored at a production facility is transported via hose to a waiting tanker. The problem is to schedule a series of round trips for the support vessels so that all servicing and off-take requirements are fulfilled, and total cost is minimized. Other constraints that must be considered include vessel suitability constraints (not every vessel is suitable for every facility), depot opening constraints (base servicing can only occur during specified opening periods), and off-take equipment constraints (the equipment needed for off-take support can only be deployed after certain commodities have been offloaded). Because of these additional constraints, the scheduling problem under consideration is far more difficult than the standard VRP. We formulate a mixed-integer linear programming (MILP) model for determining the optimal vessel schedule. We then verify the model theoretically and show how to compute the vessel utilization ratios for any feasible schedule. Finally, simulation results are reported for a real case study commissioned by Woodside Energy Ltd, Australia's largest dedicated oil and gas company.

Keywords: Optimal scheduling, vehicle routing problem, mixed-integer linear programming, oil and gas industry.

Mathematics Subject Classification: Primary: 90B06; Secondary: 90B10, 90C11, 90C27.

Citation: Elham Mardaneh, Ryan Loxton, Qun Lin, Phil Schmidli. A mixed-integer linear programming model for optimal vessel scheduling in offshore oil and gas operations. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1601-1623. doi: 10.3934/jimo.2017009

References:

[1]	AIMMS Modelling Platform, AIMMS B. V., Haarlem, Netherlands, accessed 17 April 2016, <http://www.aimms.com>Google Scholar
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[3]	N. Azi, M. Gendreau and J. Y. Potvin, An exact algorithm for a single-vehicle routing problem with time windows and multiple routes, European Journal of Operational Research, 178 (2007), 755-766. doi: 10.1016/j.ejor.2006.02.019. Google Scholar
[4]	N. Azi, M. Gendreau and J. Y. Potvin, An exact algorithm for a vehicle routing problem with time windows and multiple use of vehicles, European Journal of Operational Research, 202 (2010), 756-763. Google Scholar
[5]	M. Battarra, M. Monaci and D. Vigo, An adaptive guidance approach for the heuristic solution of a minimum multiple trip vehicle routing problem, Computers and Operations Research, 36 (2009), 3041-3050. Google Scholar
[6]	J. Bisschop, AIMMS Optimization Modelling, Haarlem: AIMMS B. V., 2016.Google Scholar
[7]	J. C. S. Brandão and A. Mercer, The multi-trip vehicle routing problem, Journal of the Operational Research Society, 49 (1998), 799-805. Google Scholar
[8]	D. Feillet, P. Dejax, M. Gendreau and C. Gueguen, An exact algorithm for the elementary shortest path problem with resource constraints: Application to some vehicle routing problems, Networks, 44 (2004), 216-229. doi: 10.1002/net.20033. Google Scholar
[9]	B. Fleischmann, The vehicle routing problem with multiple use of vehicles (technical report), Hamburg: Universität Hamburg, 1990.Google Scholar
[10]	Gurobi, Optimizer Reference Manual, Houston: Gurobi Optimization Inc., 2016.Google Scholar
[11]	F. Hernandez, D. Feillet, R. Giroudeau and O. Naud, A new exact algorithm to solve the multi-trip vehicle routing problem with time windows and limited duration, 4OR, 12 (2014), 235-259. doi: 10.1007/s10288-013-0238-z. Google Scholar
[12]	IBM ILOG CPLEX Optimizer, IBM Corporation, New York, USA, accessed 17 April 2016,<http://www.ibm.com/software/commerce/optimization/cplex-optimizer>Google Scholar
[13]	IBM ILOG CPLEX Optimization Studio CPLEX User's Manual, New York, IBM Corporation, 2014.Google Scholar
[14]	R. Macedo, C. Alves, J. M. Valério de Carvalho, F. Clautiaux and S. Hanafi, Solving the vehicle routing problem with time windows and multiple routes exactly using a pseudo-polynomial model, European Journal of Operational Research, 214 (2011), 536-545. doi: 10.1016/j.ejor.2011.04.037. Google Scholar
[15]	A. Olivera and O. Viera, Adaptive memory programming for the vehicle routing problem with multiple trips, Computers and Operations Research, 34 (2007), 28-47. doi: 10.1016/j.cor.2005.02.044. Google Scholar
[16]	R. J. Petch and S. Salhi, A multi-phase constructive heuristic for the vehicle routing problem with multiple trips, Discrete Applied Mathematics, 133 (2003), 69-92. doi: 10.1016/S0166-218X(03)00434-7. Google Scholar
[17]	E. D. Taillard, G. Laporte and G. Gendreau, Vehicle routeing with multiple use of vehicles, Journal of the Operational Research Society, 47 (1996), 1065-1070. Google Scholar
[18]	P. Toth and D. Vigo, editors, The Vehicle Routing Problem, Philadelphia: SIAM, 2002. doi: 10.1137/1.9780898718515. Google Scholar
[19]	S. Salhi and R. J. Petch, A GA based heuristic for the vehicle routing problem with multiple trips, Journal of Mathematical Modelling and Algorithms, 6 (2007), 591-613. doi: 10.1007/s10852-007-9069-2. Google Scholar
[20]	A. Şen and K. Bülbül, A survey on multi trip vehicle routing problem, In: Proceedings of the International Logistics and Supply Chain Congress 2008, Istanbul, Turkey.Google Scholar

show all references

References:

[1]	AIMMS Modelling Platform, AIMMS B. V., Haarlem, Netherlands, accessed 17 April 2016, <http://www.aimms.com>Google Scholar
[2]	F. Alonso, M. J. Alvarez and J. E. Beasley, A tabu search algorithm for the periodic vehicle routing problem with multiple vehicle trips and accessibility restrictions, Journal of the Operational Research Society, 59 (2008), 963-976. Google Scholar
[3]	N. Azi, M. Gendreau and J. Y. Potvin, An exact algorithm for a single-vehicle routing problem with time windows and multiple routes, European Journal of Operational Research, 178 (2007), 755-766. doi: 10.1016/j.ejor.2006.02.019. Google Scholar
[4]	N. Azi, M. Gendreau and J. Y. Potvin, An exact algorithm for a vehicle routing problem with time windows and multiple use of vehicles, European Journal of Operational Research, 202 (2010), 756-763. Google Scholar
[5]	M. Battarra, M. Monaci and D. Vigo, An adaptive guidance approach for the heuristic solution of a minimum multiple trip vehicle routing problem, Computers and Operations Research, 36 (2009), 3041-3050. Google Scholar
[6]	J. Bisschop, AIMMS Optimization Modelling, Haarlem: AIMMS B. V., 2016.Google Scholar
[7]	J. C. S. Brandão and A. Mercer, The multi-trip vehicle routing problem, Journal of the Operational Research Society, 49 (1998), 799-805. Google Scholar
[8]	D. Feillet, P. Dejax, M. Gendreau and C. Gueguen, An exact algorithm for the elementary shortest path problem with resource constraints: Application to some vehicle routing problems, Networks, 44 (2004), 216-229. doi: 10.1002/net.20033. Google Scholar
[9]	B. Fleischmann, The vehicle routing problem with multiple use of vehicles (technical report), Hamburg: Universität Hamburg, 1990.Google Scholar
[10]	Gurobi, Optimizer Reference Manual, Houston: Gurobi Optimization Inc., 2016.Google Scholar
[11]	F. Hernandez, D. Feillet, R. Giroudeau and O. Naud, A new exact algorithm to solve the multi-trip vehicle routing problem with time windows and limited duration, 4OR, 12 (2014), 235-259. doi: 10.1007/s10288-013-0238-z. Google Scholar
[12]	IBM ILOG CPLEX Optimizer, IBM Corporation, New York, USA, accessed 17 April 2016,<http://www.ibm.com/software/commerce/optimization/cplex-optimizer>Google Scholar
[13]	IBM ILOG CPLEX Optimization Studio CPLEX User's Manual, New York, IBM Corporation, 2014.Google Scholar
[14]	R. Macedo, C. Alves, J. M. Valério de Carvalho, F. Clautiaux and S. Hanafi, Solving the vehicle routing problem with time windows and multiple routes exactly using a pseudo-polynomial model, European Journal of Operational Research, 214 (2011), 536-545. doi: 10.1016/j.ejor.2011.04.037. Google Scholar
[15]	A. Olivera and O. Viera, Adaptive memory programming for the vehicle routing problem with multiple trips, Computers and Operations Research, 34 (2007), 28-47. doi: 10.1016/j.cor.2005.02.044. Google Scholar
[16]	R. J. Petch and S. Salhi, A multi-phase constructive heuristic for the vehicle routing problem with multiple trips, Discrete Applied Mathematics, 133 (2003), 69-92. doi: 10.1016/S0166-218X(03)00434-7. Google Scholar
[17]	E. D. Taillard, G. Laporte and G. Gendreau, Vehicle routeing with multiple use of vehicles, Journal of the Operational Research Society, 47 (1996), 1065-1070. Google Scholar
[18]	P. Toth and D. Vigo, editors, The Vehicle Routing Problem, Philadelphia: SIAM, 2002. doi: 10.1137/1.9780898718515. Google Scholar
[19]	S. Salhi and R. J. Petch, A GA based heuristic for the vehicle routing problem with multiple trips, Journal of Mathematical Modelling and Algorithms, 6 (2007), 591-613. doi: 10.1007/s10852-007-9069-2. Google Scholar
[20]	A. Şen and K. Bülbül, A survey on multi trip vehicle routing problem, In: Proceedings of the International Logistics and Supply Chain Congress 2008, Istanbul, Turkey.Google Scholar

Figure 1. An example of our arrival/departure time convention: if $d_i^k=3$, then a vessel arriving during period 2 and can depart any time from period 6 onwards

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Figure 2. An example of base servicing with $d_{\text{base}}^k=2$. Closed periods are shaded in grey. After arriving at the base, the vessel must stay for at least $\delta^k(t)=5$ periods to complete the service. For option (i), $\omega_{\text{base}}^k=2$ since the 3 closed periods during service are not considered active periods. For option (ii), $\omega_{\text{base}}^k=5$ since the 3 closed periods are considered active periods

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Figure 3. Woodside's offshore facilities in the North West Shelf region and Carnarvon Basin

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Figure 4. Historical and optimized vessel schedules for Scenario 1

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Figure 5. Historical and optimized vessel schedules for Scenario 2

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Figure 6. Historical and optimized vessel schedules for Scenario 3

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Figure 7. Historical and optimized vessel schedules for Scenario 4

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Table

Algorithm 1 Returns the value of $\delta^k(t)$
Set $t\rightarrow t'$ ▷Initialization step; $t'$ is the period counter
Set $0\rightarrow d$ ▷Initialization step; $d$ is the working period counter
while $d < d_{\text{base}}^k$ do ▷Iterate for $d_{\text{base}}^k$ working periods
Set $t'+1\rightarrow t'$ ▷Increment period counter
if $t'>T$ then
Set $+\infty\rightarrow\delta^k(t)$ ▷Insufficient time to conduct base service
return $\delta^k(t)$
else if $t'\in\mathcal{O}_{\text{base}}$ then
Set $d+1\rightarrow d$ ▷Increment working period counter if base is open
end if
end while
Set $t'-t\rightarrow\delta^k(t)$ ▷Calculate $\delta^k(t)$
return $\delta^k(t)$

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Algorithm 1 Returns the value of $\delta^k(t)$
Set $t\rightarrow t'$ ▷Initialization step; $t'$ is the period counter
Set $0\rightarrow d$ ▷Initialization step; $d$ is the working period counter
while $d < d_{\text{base}}^k$ do ▷Iterate for $d_{\text{base}}^k$ working periods
Set $t'+1\rightarrow t'$ ▷Increment period counter
if $t'>T$ then
Set $+\infty\rightarrow\delta^k(t)$ ▷Insufficient time to conduct base service
return $\delta^k(t)$
else if $t'\in\mathcal{O}_{\text{base}}$ then
Set $d+1\rightarrow d$ ▷Increment working period counter if base is open
end if
end while
Set $t'-t\rightarrow\delta^k(t)$ ▷Calculate $\delta^k(t)$
return $\delta^k(t)$

Table 1. Distances (in nautical miles) between facilities

	Karratha	Angel	Goodwyn	Nganhurra	Ngujima-Yin	North Rankin	Okha	Pluto
Karratha	-	68.4	78.4	180.0	175.0	75.0	65.0	95.9
Angel	68.4	-	38.4	188.4	181.7	27.5	10.0	75.0
Goodwyn	78.4	38.4	-	155.0	155.9	12.5	30.0	38.4
Nganhurra	180.0	188.4	155.0	-	5.0	165.0	170.0	117.5
Ngujima-Yin	175.0	181.7	155.9	5.0	-	160.0	165.0	112.5
North Rankin	75.0	27.5	12.5	165.0	160.0	-	18.4	50.0
Okha	65.0	10.0	30.0	170.0	165.0	18.4	-	65.0
Pluto	95.9	75.0	38.4	117.5	112.5	50.0	65.0	-

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	Karratha	Angel	Goodwyn	Nganhurra	Ngujima-Yin	North Rankin	Okha	Pluto
Karratha	-	68.4	78.4	180.0	175.0	75.0	65.0	95.9
Angel	68.4	-	38.4	188.4	181.7	27.5	10.0	75.0
Goodwyn	78.4	38.4	-	155.0	155.9	12.5	30.0	38.4
Nganhurra	180.0	188.4	155.0	-	5.0	165.0	170.0	117.5
Ngujima-Yin	175.0	181.7	155.9	5.0	-	160.0	165.0	112.5
North Rankin	75.0	27.5	12.5	165.0	160.0	-	18.4	50.0
Okha	65.0	10.0	30.0	170.0	165.0	18.4	-	65.0
Pluto	95.9	75.0	38.4	117.5	112.5	50.0	65.0	-

Table 2. Service requirements for Scenario 1

Facility	Day	Demand	Time Window	Duration	Suitable Vessels	Off-take?
Ngujima-Yin	1	300 m²	0:00-24:00	30 hours	OSV	Yes
Goodwyn	2	100 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	2	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	3	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	6	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	6	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	8	300 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	9	100 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	9	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	9	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	10	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	16	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Nganhurra	16	300 m²	0:00-24:00	30 hours	OSV	Yes
North Rankin	16	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Pluto	17	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	19	200 m²	0:00-24:00	30 hours	OSV	Yes
Goodwyn	20	100 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	20	200 m²	0:00-24:00	3 hours	PSV, OSV	No

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Facility	Day	Demand	Time Window	Duration	Suitable Vessels	Off-take?
Ngujima-Yin	1	300 m²	0:00-24:00	30 hours	OSV	Yes
Goodwyn	2	100 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	2	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	3	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	6	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	6	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	8	300 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	9	100 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	9	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	9	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	10	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	16	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Nganhurra	16	300 m²	0:00-24:00	30 hours	OSV	Yes
North Rankin	16	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Pluto	17	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	19	200 m²	0:00-24:00	30 hours	OSV	Yes
Goodwyn	20	100 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	20	200 m²	0:00-24:00	3 hours	PSV, OSV	No

Table 3. Service requirements for Scenario 2

Facility	Day	Demand	Time Window	Duration	Suitable Vessels	Off-take?
Goodwyn	1	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	2	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	3	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	4	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	5	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	5	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	5	100 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	6	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Nganhurra	7	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	7	100 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	8	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	9	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	9	200 m²	0:00-24:00	30 hours	OSV	Yes
Okha	11	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	12	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	12	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	13	200 m²	0:00-24:00	30 hours	OSV	Yes
Goodwyn	14	250 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	15	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	18	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	19	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	19	150 m²	0:00-24:00	3 hours	PSV, OSV	No

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Facility	Day	Demand	Time Window	Duration	Suitable Vessels	Off-take?
Goodwyn	1	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	2	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	3	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	4	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	5	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	5	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	5	100 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	6	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Nganhurra	7	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	7	100 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	8	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	9	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	9	200 m²	0:00-24:00	30 hours	OSV	Yes
Okha	11	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	12	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	12	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	13	200 m²	0:00-24:00	30 hours	OSV	Yes
Goodwyn	14	250 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	15	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	18	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	19	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	19	150 m²	0:00-24:00	3 hours	PSV, OSV	No

Table 4. Service requirements for Scenario 3

Facility	Day	Demand	Time Window	Duration	Suitable Vessels	Off-take?
Angel	1	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	1	250 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	1	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Nganhurra	3	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	3	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Angel	4	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	4	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	4	100 m²	0:00-24:00	30 hours	OSV	Yes
North Rankin	5	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	7	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	8	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	8	200 m²	0:00-24:00	30 hours	OSV	Yes
North Rankin	8	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	10	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	11	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	12	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	14	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	14	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Pluto	14	300 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	15	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Nganhurra	17	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	17	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	18	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	19	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	21	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	21	100 m²	0:00-24:00	3 hours	PSV, OSV	No

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Facility	Day	Demand	Time Window	Duration	Suitable Vessels	Off-take?
Angel	1	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	1	250 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	1	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Nganhurra	3	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	3	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Angel	4	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	4	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	4	100 m²	0:00-24:00	30 hours	OSV	Yes
North Rankin	5	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	7	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	8	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	8	200 m²	0:00-24:00	30 hours	OSV	Yes
North Rankin	8	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	10	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	11	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	12	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	14	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	14	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Pluto	14	300 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	15	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Nganhurra	17	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	17	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	18	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	19	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	21	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	21	100 m²	0:00-24:00	3 hours	PSV, OSV	No

Table 5. Service requirements for Scenario 4

Facility	Day	Demand	Time Window	Duration	Suitable Vessels	Off-take?
Goodwyn	1	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Nganhurra	1	250 m²	0:00-24:00	30 hours	OSV	Yes
North Rankin	1	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	3	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	3	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	3	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Nganhurra	4	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	5	150 m²	0:00-24:00	30 hours	OSV	Yes
Goodwyn	7	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	7	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	8	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Pluto	9	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	10	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	11	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Nganhurra	11	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	11	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Pluto	12	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	13	100 m²	0:00-24:00	30 hours	OSV	Yes
Goodwyn	14	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	14	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	15	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	16	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	17	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	18	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Nganhurra	18	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	18	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Angel	21	300 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	21	100 m²	0:00-24:00	3 hours	PSV, OSV	No

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Facility	Day	Demand	Time Window	Duration	Suitable Vessels	Off-take?
Goodwyn	1	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Nganhurra	1	250 m²	0:00-24:00	30 hours	OSV	Yes
North Rankin	1	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	3	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	3	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	3	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Nganhurra	4	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	5	150 m²	0:00-24:00	30 hours	OSV	Yes
Goodwyn	7	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	7	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	8	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Pluto	9	250 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	10	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	11	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Nganhurra	11	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	11	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Pluto	12	100 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	13	100 m²	0:00-24:00	30 hours	OSV	Yes
Goodwyn	14	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	14	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	15	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	16	200 m²	0:00-24:00	3 hours	PSV, OSV	No
Ngujima-Yin	17	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Goodwyn	18	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Nganhurra	18	150 m²	0:00-24:00	3 hours	PSV, OSV	No
North Rankin	18	150 m²	0:00-24:00	3 hours	PSV, OSV	No
Angel	21	300 m²	0:00-24:00	3 hours	PSV, OSV	No
Okha	21	100 m²	0:00-24:00	3 hours	PSV, OSV	No

Table 6. Model dimensions in terms of binary variables (BVs), continuous-valued variables (CVs), and constraints

	Original Model			Simplified Model
	BVs	CVs	Constraints	BVs	CVs	Constraints
Scenario 1	1,646,568	1,646,569	1,646,870	206,868	20,571	207,167
Scenario 2	2,069,928	2,069,929	2,070,258	292,443	31,582	292,771
Scenario 3	2,541,672	2,541,673	2,542,030	322,105	35,540	322,461
Scenario 4	2,795,688	2,795,689	2,796,060	330,484	39,859	330,853

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	Original Model			Simplified Model
	BVs	CVs	Constraints	BVs	CVs	Constraints
Scenario 1	1,646,568	1,646,569	1,646,870	206,868	20,571	207,167
Scenario 2	2,069,928	2,069,929	2,070,258	292,443	31,582	292,771
Scenario 3	2,541,672	2,541,673	2,542,030	322,105	35,540	322,461
Scenario 4	2,795,688	2,795,689	2,796,060	330,484	39,859	330,853

Table 7. Optimal fuel consumption for Scenarios 1-4

	Total Fuel Use (L)
	Historical	Initial	Optimized	Improvement
Scenario 1	108,620	107,560	97,440	10.29%
Scenario 2	124,460	113,880	96,040	22.83%
Scenario 3	139,500	138,400	125,820	9.81%
Scenario 4	170,680	168,960	148,640	12.91%

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	Total Fuel Use (L)
	Historical	Initial	Optimized	Improvement
Scenario 1	108,620	107,560	97,440	10.29%
Scenario 2	124,460	113,880	96,040	22.83%
Scenario 3	139,500	138,400	125,820	9.81%
Scenario 4	170,680	168,960	148,640	12.91%

Table 8. Optimal vessel utilization for Scenarios 1-4

	Deck-space Utilization			Time Utilization
	PSV	OSV 1	OSV 2	PSV	OSV 1	OSV 2
Scenario 1	100%	100%	100%	30%	37%	31%
Scenario 2	80%	88%	89%	26%	31%	31%
Scenario 3	100%	78%	83%	51%	33%	27%
Scenario 4	92%	96%	100%	45%	41%	43%

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	Deck-space Utilization			Time Utilization
	PSV	OSV 1	OSV 2	PSV	OSV 1	OSV 2
Scenario 1	100%	100%	100%	30%	37%	31%
Scenario 2	80%	88%	89%	26%	31%	31%
Scenario 3	100%	78%	83%	51%	33%	27%
Scenario 4	92%	96%	100%	45%	41%	43%

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2018 Impact Factor: 1.025

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2018 Impact Factor: 1.025

American Institute of Mathematical Sciences