Mixed-integer linear fractional programming (MILFP) - Revision history

Wc593 at 11:38, 21 December 2020

2020-12-21T11:38:35Z

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	Author: Xiang Zhao (SysEn 6800 Fall 2020)		Author: Xiang Zhao (SysEn 6800 Fall 2020)

	~~Steward: Allen Yang, Fengqi You~~

	==Introduction==		==Introduction==

Xz643 at 14:09, 12 December 2020

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	===Computational performance===		===Computational performance===
	The computational performances of branch-and-refine algorithm and BARON are shown in the table below, where we find that the former algorithm has advantage over the latter one.		The computational performances of branch-and-refine algorithm and BARON are shown in the table below, where we find that the former algorithm has advantage over the latter one. The optimal solutions for both algorithm are the same, which illustrates the global optimality of the solution from branch-and-refine algorithm.
	{\| class="wikitable"		{\| class="wikitable"
	\|+ Computational Performance for maximizing unit NPV		\|+ Computational Performance for maximizing unit NPV
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	\|-		\|-
	\| 0.125 \|\| 98.2		\| 0.125 \|\| 98.2
			\|}

			{\| class="wikitable"
			\|+ Performance for BARON algorithm
			\|-
			! Production Amount (ton/yr) !! Unit NPV ($/ton) !! Unit GHG Emissions (ton CO<sub>2</sub>-eq/ton products)
			\|-
			\| 768,000 \|\| 187.75 \|\| 10.18
	\|}		\|}

Xz643 at 14:06, 12 December 2020

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@@ Line 377: / Line 377: @@
 |-
 | 768,000 || 186.23 || 9.68
 |}

Xz643: /* References */

2020-12-12T13:54:10Z

References

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	# Quesada, I., & Grossmann, I. E. (1995). A global optimization algorithm for linear fractional and bilinear programs. Journal of Global Optimization, 6(1), 39-76.		# Quesada, I., & Grossmann, I. E. (1995). A global optimization algorithm for linear fractional and bilinear programs. Journal of Global Optimization, 6(1), 39-76.
	# Zhong, Z., & You, F. (2014, June). Parametric algorithms for global optimization of mixed-integer fractional programming problems in process engineering. In 2014 American Control Conference (pp. 3609-3614). IEEE.		# Zhong, Z., & You, F. (2014, June). Parametric algorithms for global optimization of mixed-integer fractional programming problems in process engineering. In 2014 American Control Conference (pp. 3609-3614). IEEE.
	~~# https://glossary.informs.org/ver2/mpgwiki/index.php/Glovers_linearization~~
	# Charnes, A., & Cooper, W. W. (1973). An explicit general solution in linear fractional programming. Naval Research Logistics Quarterly, 20(3), 449-467.		# Charnes, A., & Cooper, W. W. (1973). An explicit general solution in linear fractional programming. Naval Research Logistics Quarterly, 20(3), 449-467.
	# Gao, J., & You, F. (2015). Optimal design and operations of supply chain networks for water management in shale gas production: MILFP model and algorithms for the water‐energy nexus. AIChE Journal, 61(4), 1184-1208.		# Gao, J., & You, F. (2015). Optimal design and operations of supply chain networks for water management in shale gas production: MILFP model and algorithms for the water‐energy nexus. AIChE Journal, 61(4), 1184-1208.

Xz643: /* Reformulation-Linearization Method */

2020-12-12T13:53:33Z

Reformulation-Linearization Method

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	==Reformulation-Linearization Method==		==Reformulation-Linearization Method==
	The reformulation-linearization method, which incorporates the Glover’s linearization~~<sup>[https://glossary.informs.org/ver2/mpgwiki/index.php/Glovers_linearization]</sup>~~ into the Charnes-Cooper transformation,<sup>[https://onlinelibrary.wiley.com/doi/abs/10.1002/nav.3800200308?casa_token=BsaOkI0dilIAAAAA:wPELPH83o1FuB9xHW8rRDhwInT3xsGjqqqk6LWID7WYpLexkAhgiymU-4-ew7c0nEoC3wM49-oFHa1m5]</sup> introduce auxiliary variables to reformulate the MILFP into equivalent MINLP. The resulting MINLP is subsequently transformed into MILP, which can be efficiently solved by typical MILP solvers like [[wikipedia:CPLEX\|CPLEX]], via using Glover’s linearization.The reformulation approach is shown as follows:		The reformulation-linearization method, which incorporates the Glover’s linearization into the Charnes-Cooper transformation,<sup>[https://onlinelibrary.wiley.com/doi/abs/10.1002/nav.3800200308?casa_token=BsaOkI0dilIAAAAA:wPELPH83o1FuB9xHW8rRDhwInT3xsGjqqqk6LWID7WYpLexkAhgiymU-4-ew7c0nEoC3wM49-oFHa1m5]</sup> introduce auxiliary variables to reformulate the MILFP into equivalent MINLP. The resulting MINLP is subsequently transformed into MILP, which can be efficiently solved by typical MILP solvers like [[wikipedia:CPLEX\|CPLEX]], via using Glover’s linearization.The reformulation approach is shown as follows:

	The original form of the optimization model is:		The original form of the optimization model is:

Xz643: /* Economic Evaluation Constraints */

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Economic Evaluation Constraints

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	The net present value is calculated in the constraint below (<math>fix</math>), where we account for the total discounted cash flow and <math>SP</math> represents for the lifespan of the this project.		The net present value is calculated in the constraint below (<math>fix</math>), where we account for the total discounted cash flow and <math>SP</math> represents for the lifespan of the this project.

	<math>npv={DR(1+DR)^{SP} \over (1+DR)^{SP}-1}{(sale-(vai+oper+fec))-fix}</math>		<math>npv={DR(1+DR)^{SP} \over (1+DR)^{SP}-1}(sale-(vai+oper+fec))-fix</math>

	===== Environmental Evaluation Constraint =====		===== Environmental Evaluation Constraint =====

Xz643: /* Economic Evaluation Constraints */

2020-12-12T13:50:48Z

Economic Evaluation Constraints

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	The net present value is calculated in the constraint below (<math>fix</math>), where we account for the total discounted cash flow and <math>SP</math> represents for the lifespan of the this project.		The net present value is calculated in the constraint below (<math>fix</math>), where we account for the total discounted cash flow and <math>SP</math> represents for the lifespan of the this project.

	<math>npv={DR(1+DR)^{SP} \over (1+DR)^{SP}-1}(sale-(vai+oper+fec))-fix</math>		<math>npv={DR(1+DR)^{SP} \over (1+DR)^{SP}-1}{(sale-(vai+oper+fec))-fix}</math>

	===== Environmental Evaluation Constraint =====		===== Environmental Evaluation Constraint =====

Xz643: /* Economic Evaluation Constraints */

2020-12-12T13:49:07Z

Economic Evaluation Constraints

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	The net present value is calculated in the constraint below (<math>fix</math>), where we account for the total discounted cash flow and <math>SP</math> represents for the lifespan of the this project.		The net present value is calculated in the constraint below (<math>fix</math>), where we account for the total discounted cash flow and <math>SP</math> represents for the lifespan of the this project.

	<math>npv={DR(1+DR)^{SP} \over (1+DR)^{SP}-1}(sale-vai-oper-fec)-fix</math>		<math>npv={DR(1+DR)^{SP} \over (1+DR)^{SP}-1}(sale-(vai+oper+fec))-fix</math>

	===== Environmental Evaluation Constraint =====		===== Environmental Evaluation Constraint =====

Xz643: /* Economic Evaluation Constraints */

2020-12-12T13:48:31Z

Economic Evaluation Constraints

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	<math>sale=PRI*\sum_{j}mo_{3,j}</math>		<math>sale=PRI*\sum_{j}mo_{3,j}</math>

	The net present value is calculated in the constraint below (<math>fix</math>), where we account for the total discounted cash flow ~~(<math>{DR(1+DR)^{SP} \over (1+DR)^{SP}-1}(sale-vai-oper-fec)</math>)~~ and <math>SP</math> represents for the lifespan of the this project.		The net present value is calculated in the constraint below (<math>fix</math>), where we account for the total discounted cash flow and <math>SP</math> represents for the lifespan of the this project.

	<math>npv={DR(1+DR)^{SP} \over (1+DR)^{SP}-1}(sale-vai-oper-fec)-fix</math>		<math>npv={DR(1+DR)^{SP} \over (1+DR)^{SP}-1}(sale-vai-oper-fec)-fix</math>

Xz643: /* Mass Balance Constraints */

2020-12-12T13:46:29Z

Mass Balance Constraints

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	This aforementioned constraint illustrates the conversion of the inlet flow and the outlet flow.		This aforementioned constraint illustrates the conversion of the inlet flow and the outlet flow.

	<math>\sum_{j}mo_{i-1,j} = \sum_{j}mi_{i,j},\quad \forall i \geq 2,\forall j \in J</math>		<math>\sum_{j}mo_{(i-1),j} = \sum_{j}mi_{i,j},\quad \forall i \geq 2,\forall j \in J</math>

	This aforementioned constraint denotes that the summation of mass flow rates of the outlet flows from the previous processing level equals to those of the inlet flows in the next processing level.		This aforementioned constraint denotes that the summation of mass flow rates of the outlet flows from the previous processing level equals to those of the inlet flows in the next processing level.