BSc: Introduction to Optimization.F22 - Revision history

Mo.bahrami: /* Course activities and grading breakdown */

2022-09-01T14:30:42Z

Course activities and grading breakdown

← Older revision		Revision as of 14:30, 1 September 2022
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	! Activity Type !! Percentage of the overall course grade		! Activity Type !! Percentage of the overall course grade
	\|-		\|-
−	\| Midterm \|\| 25	+	\| Midterm \|\| 30
	\|-		\|-
	\| 2 Intermediate tests \|\| 30 (15 for each)		\| 2 Intermediate tests \|\| 30 (15 for each)
	\|-		\|-
−	\| Final exam \|\| 30	+	\| Final exam \|\| 40
		+
−	\|-
−	\| Final presentation \|\| 15
	\|}		\|}

I.konyukhov: /* Final assessment */

2022-06-28T08:17:48Z

Final assessment

← Older revision		Revision as of 08:17, 28 June 2022
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	=== Final assessment ===		=== Final assessment ===
−	~~'''~~Section 1~~'''~~	+	==== Section 1 ====
	# Solve the following linear programming problems.		# Solve the following linear programming problems.
	#: maximize <math>x_1 + 3x_2</math>		#: maximize <math>x_1 + 3x_2</math>
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	#: subject to <math>x_1 + x_2 + x_3 + x_4 = 1</math>		#: subject to <math>x_1 + x_2 + x_3 + x_4 = 1</math>
	#:: <math>x_1, x_2, x_3, x_4 \geq 0</math>		#:: <math>x_1, x_2, x_3, x_4 \geq 0</math>
		⚫	==== Section 2 ====
−
⚫	~~'''~~Section 2~~'''~~
	# Perform two iterations of the steepest descent method to minimize the function given from the stated starting point <math>(-1.2, 1)</math>		# Perform two iterations of the steepest descent method to minimize the function given from the stated starting point <math>(-1.2, 1)</math>
	#:: <math>f(x_1, x_2) = 100(x_2 - x_1^2 )^2 + (1 - x_1)^2</math>		#:: <math>f(x_1, x_2) = 100(x_2 - x_1^2 )^2 + (1 - x_1)^2</math>
	# Minimize <math>f = 2x_1^2 + x_2^2</math> by using the steepest descent method with the starting point <math>(1, 2)</math> (two iterations only).		# Minimize <math>f = 2x_1^2 + x_2^2</math> by using the steepest descent method with the starting point <math>(1, 2)</math> (two iterations only).
	# Minimize <math>f = x_1^2 + 3x_2^2 + 6x_3^2</math> by Newton's method using the starting point as <math>(2,-1, 1)</math> .		# Minimize <math>f = x_1^2 + 3x_2^2 + 6x_3^2</math> by Newton's method using the starting point as <math>(2,-1, 1)</math> .
		⚫	==== Section 3 ====
−
⚫	~~'''~~Section 3~~'''~~
	* It is proposed to build thermal stations at three different sites. The total budget available is 3 units (1 unit = $10 million) and the feasible levels of investment on any thermal station are 0, 1, 2, or 3 units. The electric power obtainable (return function) for different investments is given below. Find the investment policy for maximizing the total electric power generated.		* It is proposed to build thermal stations at three different sites. The total budget available is 3 units (1 unit = $10 million) and the feasible levels of investment on any thermal station are 0, 1, 2, or 3 units. The electric power obtainable (return function) for different investments is given below. Find the investment policy for maximizing the total electric power generated.
	{\| class="wikitable" style="height:14em;"		{\| class="wikitable" style="height:14em;"

I.konyukhov: /* Final assessment */

2022-06-23T11:22:18Z

Final assessment

← Older revision		Revision as of 11:22, 23 June 2022
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	*: [[Image:ItO_p2.png]]		*: [[Image:ItO_p2.png]]
	*: As usual, bold arcs represent arcs on the spanning tree, numbers next to the bold arcs are primal flows, numbers next to non-bold arcs are dual slacks, and numbers next to nodes are dual variables.		*: As usual, bold arcs represent arcs on the spanning tree, numbers next to the bold arcs are primal flows, numbers next to non-bold arcs are dual slacks, and numbers next to nodes are dual variables.
−	** For what values of is this tree solution optimal?	+	*:: For what values of is this tree solution optimal?
−	** What are the entering and leaving arcs?	+	*:: What are the entering and leaving arcs?
−	** After one pivot, what is the new tree solution?	+	*:: After one pivot, what is the new tree solution?
−	** For what values of is the new tree solution optimal?	+	*:: For what values of is the new tree solution optimal?

	=== The retake exam ===		=== The retake exam ===

I.konyukhov: /* Final assessment */

2022-06-23T11:21:20Z

Final assessment

← Older revision		Revision as of 11:21, 23 June 2022
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	'''Section 3'''		'''Section 3'''
−	# It is proposed to build thermal stations at three different sites. The total budget available is 3 units (1 unit = $10 million) and the feasible levels of investment on any thermal station are 0, 1, 2, or 3 units. The electric power obtainable (return function) for different investments is given below. Find the investment policy for maximizing the total electric power generated.	+	* It is proposed to build thermal stations at three different sites. The total budget available is 3 units (1 unit = $10 million) and the feasible levels of investment on any thermal station are 0, 1, 2, or 3 units. The electric power obtainable (return function) for different investments is given below. Find the investment policy for maximizing the total electric power generated.
	{\| class="wikitable" style="height:14em;"		{\| class="wikitable" style="height:14em;"
	\|-		\|-
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	\| R3(x) \|\| 6 \|\| 6 \|\| 6		\| R3(x) \|\| 6 \|\| 6 \|\| 6
	\|}		\|}
−	# A fertilizer company needs to supply 50 tons of fertilizer at the end of the first month, 70 tons at the end of the second month, and 90 tons at the end of the third month. The cost of producing x tons of fertilizer in any month is given by $<math>(4500x + 20x^2)</math>. It can produce more fertilizer in any month and supply it in the next month. However, there is an inventory carrying cost of $400 per ton per month. Find the optimal level of production in each of the three periods and the total cost involved by solving it as an initial value problem.	+	* A fertilizer company needs to supply 50 tons of fertilizer at the end of the first month, 70 tons at the end of the second month, and 90 tons at the end of the third month. The cost of producing x tons of fertilizer in any month is given by $<math>(4500x + 20x^2)</math>. It can produce more fertilizer in any month and supply it in the next month. However, there is an inventory carrying cost of $400 per ton per month. Find the optimal level of production in each of the three periods and the total cost involved by solving it as an initial value problem.
−	# Consider the following tree solution for a minimum cost network flow problem:	+	* Consider the following tree solution for a minimum cost network flow problem:
−	#: [[Image:ItO_p2.png]]	+	*: [[Image:ItO_p2.png]]
−	#: As usual, bold arcs represent arcs on the spanning tree, numbers next to the bold arcs are primal flows, numbers next to non-bold arcs are dual slacks, and numbers next to nodes are dual variables.	+	*: As usual, bold arcs represent arcs on the spanning tree, numbers next to the bold arcs are primal flows, numbers next to non-bold arcs are dual slacks, and numbers next to nodes are dual variables.
−	#* For what values of is this tree solution optimal?	+	** For what values of is this tree solution optimal?
−	#* What are the entering and leaving arcs?	+	** What are the entering and leaving arcs?
−	#* After one pivot, what is the new tree solution?	+	** After one pivot, what is the new tree solution?
−	#* For what values of is the new tree solution optimal?	+	** For what values of is the new tree solution optimal?

	=== The retake exam ===		=== The retake exam ===

I.konyukhov: /* Final assessment */

2022-06-23T11:18:55Z

Final assessment

← Older revision		Revision as of 11:18, 23 June 2022
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	'''Section 3'''		'''Section 3'''
−	# It is proposed to build thermal stations at three different sites. The total budget available is 3 units (1 unit = $10 million) and the feasible levels of investment on any thermal station are 0, 1, 2, or 3 units. The electric power obtainable (return function) for different investments is given below:	+	# It is proposed to build thermal stations at three different sites. The total budget available is 3 units (1 unit = $10 million) and the feasible levels of investment on any thermal station are 0, 1, 2, or 3 units. The electric power obtainable (return function) for different investments is given below. Find the investment policy for maximizing the total electric power generated.
	{\| class="wikitable" style="height:14em;"		{\| class="wikitable" style="height:14em;"
	\|-		\|-
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	\| R3(x) \|\| 6 \|\| 6 \|\| 6		\| R3(x) \|\| 6 \|\| 6 \|\| 6
	\|}		\|}
−	#: Find the investment policy for maximizing the total electric power generated.
	# A fertilizer company needs to supply 50 tons of fertilizer at the end of the first month, 70 tons at the end of the second month, and 90 tons at the end of the third month. The cost of producing x tons of fertilizer in any month is given by $<math>(4500x + 20x^2)</math>. It can produce more fertilizer in any month and supply it in the next month. However, there is an inventory carrying cost of $400 per ton per month. Find the optimal level of production in each of the three periods and the total cost involved by solving it as an initial value problem.		# A fertilizer company needs to supply 50 tons of fertilizer at the end of the first month, 70 tons at the end of the second month, and 90 tons at the end of the third month. The cost of producing x tons of fertilizer in any month is given by $<math>(4500x + 20x^2)</math>. It can produce more fertilizer in any month and supply it in the next month. However, there is an inventory carrying cost of $400 per ton per month. Find the optimal level of production in each of the three periods and the total cost involved by solving it as an initial value problem.
	# Consider the following tree solution for a minimum cost network flow problem:		# Consider the following tree solution for a minimum cost network flow problem:

I.konyukhov: /* Final assessment */

2022-06-23T11:18:22Z

Final assessment

← Older revision		Revision as of 11:18, 23 June 2022
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	\| R3(x) \|\| 6 \|\| 6 \|\| 6		\| R3(x) \|\| 6 \|\| 6 \|\| 6
	\|}		\|}
−	#:Find the investment policy for maximizing the total electric power generated.	+	#: Find the investment policy for maximizing the total electric power generated.
	# A fertilizer company needs to supply 50 tons of fertilizer at the end of the first month, 70 tons at the end of the second month, and 90 tons at the end of the third month. The cost of producing x tons of fertilizer in any month is given by $<math>(4500x + 20x^2)</math>. It can produce more fertilizer in any month and supply it in the next month. However, there is an inventory carrying cost of $400 per ton per month. Find the optimal level of production in each of the three periods and the total cost involved by solving it as an initial value problem.		# A fertilizer company needs to supply 50 tons of fertilizer at the end of the first month, 70 tons at the end of the second month, and 90 tons at the end of the third month. The cost of producing x tons of fertilizer in any month is given by $<math>(4500x + 20x^2)</math>. It can produce more fertilizer in any month and supply it in the next month. However, there is an inventory carrying cost of $400 per ton per month. Find the optimal level of production in each of the three periods and the total cost involved by solving it as an initial value problem.
	# Consider the following tree solution for a minimum cost network flow problem:		# Consider the following tree solution for a minimum cost network flow problem:

I.konyukhov: /* Final assessment */

2022-06-23T11:17:50Z

Final assessment

← Older revision		Revision as of 11:17, 23 June 2022
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	'''Section 3'''		'''Section 3'''
	# It is proposed to build thermal stations at three different sites. The total budget available is 3 units (1 unit = $10 million) and the feasible levels of investment on any thermal station are 0, 1, 2, or 3 units. The electric power obtainable (return function) for different investments is given below:		# It is proposed to build thermal stations at three different sites. The total budget available is 3 units (1 unit = $10 million) and the feasible levels of investment on any thermal station are 0, 1, 2, or 3 units. The electric power obtainable (return function) for different investments is given below:
−	[{\| class="wikitable" style="height:14em;"	+	{\| class="wikitable" style="height:14em;"
	\|-		\|-
	! Return function Ri(x) !! i = 1 !! i = 2 !! i = 3		! Return function Ri(x) !! i = 1 !! i = 2 !! i = 3
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	\|-		\|-
	\| R3(x) \|\| 6 \|\| 6 \|\| 6		\| R3(x) \|\| 6 \|\| 6 \|\| 6
−	\|}]	+	\|}
	#:Find the investment policy for maximizing the total electric power generated.		#:Find the investment policy for maximizing the total electric power generated.
	# A fertilizer company needs to supply 50 tons of fertilizer at the end of the first month, 70 tons at the end of the second month, and 90 tons at the end of the third month. The cost of producing x tons of fertilizer in any month is given by $<math>(4500x + 20x^2)</math>. It can produce more fertilizer in any month and supply it in the next month. However, there is an inventory carrying cost of $400 per ton per month. Find the optimal level of production in each of the three periods and the total cost involved by solving it as an initial value problem.		# A fertilizer company needs to supply 50 tons of fertilizer at the end of the first month, 70 tons at the end of the second month, and 90 tons at the end of the third month. The cost of producing x tons of fertilizer in any month is given by $<math>(4500x + 20x^2)</math>. It can produce more fertilizer in any month and supply it in the next month. However, there is an inventory carrying cost of $400 per ton per month. Find the optimal level of production in each of the three periods and the total cost involved by solving it as an initial value problem.

I.konyukhov: /* Final assessment */

2022-06-23T11:17:36Z

Final assessment

← Older revision		Revision as of 11:17, 23 June 2022
Line 278:		Line 278:

	'''Section 3'''		'''Section 3'''
−	# It is proposed to build thermal stations at three different sites. The total budget available is 3 units (1 unit = $10 million) and the feasible levels of investment on any thermal station are 0, 1, 2, or 3 units. The electric power obtainable (return function) for different investments is given below: ~~{\| class="wikitable" style="height:14em;"~~	+	# It is proposed to build thermal stations at three different sites. The total budget available is 3 units (1 unit = $10 million) and the feasible levels of investment on any thermal station are 0, 1, 2, or 3 units. The electric power obtainable (return function) for different investments is given below:
		+	[{\| class="wikitable" style="height:14em;"
	\|-		\|-
	! Return function Ri(x) !! i = 1 !! i = 2 !! i = 3		! Return function Ri(x) !! i = 1 !! i = 2 !! i = 3
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	\|-		\|-
	\| R3(x) \|\| 6 \|\| 6 \|\| 6		\| R3(x) \|\| 6 \|\| 6 \|\| 6
−	\|}	+	\|}]
	#:Find the investment policy for maximizing the total electric power generated.		#:Find the investment policy for maximizing the total electric power generated.
	# A fertilizer company needs to supply 50 tons of fertilizer at the end of the first month, 70 tons at the end of the second month, and 90 tons at the end of the third month. The cost of producing x tons of fertilizer in any month is given by $<math>(4500x + 20x^2)</math>. It can produce more fertilizer in any month and supply it in the next month. However, there is an inventory carrying cost of $400 per ton per month. Find the optimal level of production in each of the three periods and the total cost involved by solving it as an initial value problem.		# A fertilizer company needs to supply 50 tons of fertilizer at the end of the first month, 70 tons at the end of the second month, and 90 tons at the end of the third month. The cost of producing x tons of fertilizer in any month is given by $<math>(4500x + 20x^2)</math>. It can produce more fertilizer in any month and supply it in the next month. However, there is an inventory carrying cost of $400 per ton per month. Find the optimal level of production in each of the three periods and the total cost involved by solving it as an initial value problem.

I.konyukhov: /* Final assessment */

2022-06-23T11:17:12Z

Final assessment

← Older revision		Revision as of 11:17, 23 June 2022
Line 278:		Line 278:

	'''Section 3'''		'''Section 3'''
−	# It is proposed to build thermal stations at three different sites. The total budget available is 3 units (1 unit = $10 million) and the feasible levels of investment on any thermal station are 0, 1, 2, or 3 units. The electric power obtainable (return function) for different investments is given below:	+	# It is proposed to build thermal stations at three different sites. The total budget available is 3 units (1 unit = $10 million) and the feasible levels of investment on any thermal station are 0, 1, 2, or 3 units. The electric power obtainable (return function) for different investments is given below: {\| class="wikitable" style="height:14em;"
−	#{\| class="wikitable" style="height:14em;"
	\|-		\|-
	! Return function Ri(x) !! i = 1 !! i = 2 !! i = 3		! Return function Ri(x) !! i = 1 !! i = 2 !! i = 3

I.konyukhov: /* Final assessment */

2022-06-23T11:16:27Z

Final assessment

← Older revision		Revision as of 11:16, 23 June 2022
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	'''Section 3'''		'''Section 3'''
	# It is proposed to build thermal stations at three different sites. The total budget available is 3 units (1 unit = $10 million) and the feasible levels of investment on any thermal station are 0, 1, 2, or 3 units. The electric power obtainable (return function) for different investments is given below:		# It is proposed to build thermal stations at three different sites. The total budget available is 3 units (1 unit = $10 million) and the feasible levels of investment on any thermal station are 0, 1, 2, or 3 units. The electric power obtainable (return function) for different investments is given below:
−	#:{\| class="wikitable" style="height:14em;"	+	#{\| class="wikitable" style="height:14em;"
	\|-		\|-
	! Return function Ri(x) !! i = 1 !! i = 2 !! i = 3		! Return function Ri(x) !! i = 1 !! i = 2 !! i = 3