Column generation algorithms - Revision history

Youfengqi at 23:07, 1 February 2026

2026-02-01T23:07:21Z

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	then <math>y^*</math> is an optimal solution of the full continuous relaxed problem (that is, including all patterns in ''<math display="inline">S</math>'')		then <math>y^*</math> is an optimal solution of the full continuous relaxed problem (that is, including all patterns in ''<math display="inline">S</math>'')

	If this condition is not true, we go ahead and update the master problem by including in ''<math display="inline">S^'</math>'' the pattern <math>\lambda</math> defined by <math>N_{s,\lambda}</math> (in practical terms this means that the column '''<math>y^*</math>''' needs to be included in the constraint matrix)		If this condition is not true, we go ahead and update the master problem by including in ''<math display="inline">S^{'}</math>'' the pattern <math>\lambda</math> defined by <math>N_{s,\lambda}</math> (in practical terms this means that the column '''<math>y^*</math>''' needs to be included in the constraint matrix)

	For this example we find that the optimality test is met as <math>\sum_{k=1}^{K}z_{k}^{}u_{k}^{}=0.4 \leq 1</math> so we have have found an optimal solution of the relaxed continuous problem (if this was not the case we would have had to go back to reformulating and solving the master problem, as discussed in the methodology section of this page)		For this example we find that the optimality test is met as <math>\sum_{k=1}^{K}z_{k}^{}u_{k}^{}=0.4 \leq 1</math> so we have have found an optimal solution of the relaxed continuous problem (if this was not the case we would have had to go back to reformulating and solving the master problem, as discussed in the methodology section of this page)

Youfengqi at 23:06, 1 February 2026

2026-02-01T23:06:07Z

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	s.t.&\\		s.t.&\\
	& \sum_sN_{ks}y_s\geq R_k & \forall k\in K \\		& \sum_sN_{ks}y_s\geq R_k & \forall k\in K \\
	& y_s\in ~~\Zeta_~~+ & \forall s\in S \\		& y_s\in Z_+ & \forall s\in S \\
	\end{align}		\end{align}

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	</math> to be a possible column of the constraint matrix, the following condition must be satisfied:		</math> to be a possible column of the constraint matrix, the following condition must be satisfied:

	<math display="inline">\begin{matrix}z_k\in ~~\Zeta_~~+\forall k\in K \\ \ \sum_kL_kz_k \leq M \end{matrix}		<math display="inline">\begin{matrix}z_k\in Z_+\forall k\in K \\ \ \sum_kL_kz_k \leq M \end{matrix}

	</math>		</math>
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	And by so doing, it enables the conversion of the problem of finding a variable with negative reduced cost into the integer linear programming problem below:		And by so doing, it enables the conversion of the problem of finding a variable with negative reduced cost into the integer linear programming problem below:

	<math>\begin{matrix}\min\ \bar{c} = 1 - sum_{k=1}^K u_k \times z_k \\ \ s.t. \sum_kL_kz_k \leq M \\ z_k\in ~~\Zeta_~~+\forall k\in K \end{matrix}		<math>\begin{matrix}\min\ \bar{c} = 1 - sum_{k=1}^K u_k \times z_k \\ \ s.t. \sum_kL_kz_k \leq M \\ z_k\in Z_+\forall k\in K \end{matrix}

	</math>		</math>
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	which, in turn, would be equivalent to the below formulation (we just write the objective in maximization form and ignore the additive constant <math>1</math>):		which, in turn, would be equivalent to the below formulation (we just write the objective in maximization form and ignore the additive constant <math>1</math>):

	<math>\begin{matrix} \max\sum_{k=1}^K u_k \times z_k \\ \ s.t. \sum_kL_kz_k \leq M \\ z_k\in ~~\Zeta_~~+\forall k\in K \end{matrix}</math>		<math>\begin{matrix} \max\sum_{k=1}^K u_k \times z_k \\ \ s.t. \sum_kL_kz_k \leq M \\ z_k\in Z_+\forall k\in K \end{matrix}</math>

Youfengqi at 23:05, 1 February 2026

2026-02-01T23:05:24Z

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	''Optimality check for LPM:'' It is required to check whether <math>\left ( \pi ,\mu \right )</math> is dual feasible for ''LPM''. This means checking for each column, that is for each <math>k</math>, and for each <math>x\; \epsilon \; X^{k}</math> if the reduced price <math>c^{k}x-\pi A^{k}x-\mu _{k}\leq 0</math>. Rather than examining each point separately, we treat all points in <math>X^{k}</math> implicitly, by solving an optimization subproblem:		''Optimality check for LPM:'' It is required to check whether <math>\left ( \pi ,\mu \right )</math> is dual feasible for ''LPM''. This means checking for each column, that is for each <math>k</math>, and for each <math>x\; \epsilon \; X^{k}</math> if the reduced price <math>c^{k}x-\pi A^{k}x-\mu _{k}\leq 0</math>. Rather than examining each point separately, we treat all points in <math>X^{k}</math> implicitly, by solving an optimization subproblem:

	<math>~~\zeta _~~{k}=max\left \{ \left (c^{k}-\pi A^{k} \right )x-\mu _{k}\; :\; x\; \epsilon \; X^{k}\right \}.</math>		<math>Z_{k}=max\left \{ \left (c^{k}-\pi A^{k} \right )x-\mu _{k}\; :\; x\; \epsilon \; X^{k}\right \}.</math>


	''Stopping criteria:'' If <math>~~\zeta _~~{k}> 0</math> for <math>k=1,...,K</math> the solution <math>\left ( \pi ,\mu \right )</math> is dual feasible for ''LPM'', and hence <math>z^{LPM}\leq \sum_{i=1}^{m}\pi _{i}b_{i}+\sum_{k=1}^{K}\mu _{k}</math>. As the value of the primal feasible solution <math>\tilde{\lambda }</math> equals that of this upper bound, <math>\tilde{\lambda }</math> is optimal for ''LPM''.		''Stopping criteria:'' If <math>Z_{k}> 0</math> for <math>k=1,...,K</math> the solution <math>\left ( \pi ,\mu \right )</math> is dual feasible for ''LPM'', and hence <math>z^{LPM}\leq \sum_{i=1}^{m}\pi _{i}b_{i}+\sum_{k=1}^{K}\mu _{k}</math>. As the value of the primal feasible solution <math>\tilde{\lambda }</math> equals that of this upper bound, <math>\tilde{\lambda }</math> is optimal for ''LPM''.


	''Generating a new column:'' If <math>~~\zeta _~~{k}> 0</math> for some <math>k</math>, the column corresponding to the optimal solution <math>\tilde{x}^{k}</math> of the subproblem has a positive reduced price. Introducing the column <math>\begin{pmatrix}		''Generating a new column:'' If <math>Z_{k}> 0</math> for some <math>k</math>, the column corresponding to the optimal solution <math>\tilde{x}^{k}</math> of the subproblem has a positive reduced price. Introducing the column <math>\begin{pmatrix}
	c^{k}x\\		c^{k}x\\
	A^{k}x\\		A^{k}x\\

Midahoru at 18:55, 16 January 2025

2025-01-16T18:55:59Z

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	It is important to note that CG can also be applied to solve integer or mixed-integer problems if used to solve the linear relaxation and is implemented within a Branch-and-Bound or Branch-Cut-and-Price algorithm.		It is important to note that CG can also be applied to solve integer or mixed-integer problems if used to solve the linear relaxation and is implemented within a Branch-and-Bound or Branch-Cut-and-Price algorithm.

	== Theory~~, methodology and algorithmic discussions ==~~		== Theory ==

	~~=== '''''Theory''''' =~~==
	Column Generation can be described as a variation of the Simplex Method. Instead of evaluating the reduced costs of all the variables and selecting the one with the minimum value (for minimization problems, maximum for maximization problems), it dynamically generates a new variable by solving an auxiliary optimization problem called the Subproblem or the Pricing Problem.		Column Generation can be described as a variation of the Simplex Method. Instead of evaluating the reduced costs of all the variables and selecting the one with the minimum value (for minimization problems, maximum for maximization problems), it dynamically generates a new variable by solving an auxiliary optimization problem called the Subproblem or the Pricing Problem.

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	In summary, when the master problem is solved, we are able to obtain dual prices for each of the constraints in the master problem. This information is then utilized in the objective function of the subproblem. The subproblem is solved. If the objective value of the subproblem is negative, a variable with negative reduced cost has been identified. This variable is then added to the master problem, and the master problem is re-solved. Re-solving the master problem will generate a new set of dual values, and the process is repeated until no negative reduced cost variables are identified. The subproblem returns a solution with non-negative reduced cost, we can conclude that the solution to the master problem is optimal.<ref>Wikipedia, the free encyclopeda. Column Generation. URL: https://en.wikipedia.org/wiki/Column_generation</ref>		In summary, when the master problem is solved, we are able to obtain dual prices for each of the constraints in the master problem. This information is then utilized in the objective function of the subproblem. The subproblem is solved. If the objective value of the subproblem is negative, a variable with negative reduced cost has been identified. This variable is then added to the master problem, and the master problem is re-solved. Re-solving the master problem will generate a new set of dual values, and the process is repeated until no negative reduced cost variables are identified. The subproblem returns a solution with non-negative reduced cost, we can conclude that the solution to the master problem is optimal.<ref>Wikipedia, the free encyclopeda. Column Generation. URL: https://en.wikipedia.org/wiki/Column_generation</ref>

	==~~= '''''~~Methodology~~'''''~~<ref>L.A. Wolsey, Integer programming. Wiley,Column Generation Algorithms p185-p189,1998</ref> ===		== Methodology<ref>L.A. Wolsey, Integer programming. Wiley,Column Generation Algorithms p185-p189,1998</ref> ==
	[[File:Column Generation.png\|thumb\|468x468px\|Column generation schematics<ref name=":4">GERARD. (2005). Personnel and Vehicle scheduling, Column Generation, slide 12. URL: https://slideplayer.com/slide/6574/</ref>]]		[[File:Column Generation.png\|thumb\|468x468px\|Column generation schematics<ref name=":4">GERARD. (2005). Personnel and Vehicle scheduling, Column Generation, slide 12. URL: https://slideplayer.com/slide/6574/</ref>]]
	Consider the problem in the form:		Consider the problem in the form:
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	== Numerical example: The one-dimensional Cutting Stock problem<ref>L.A. Wolsey, Integer programming. Wiley,Column Generation Algorithms p185-p189,1998The Cutting Stock problem</ref> ==		== Numerical example: The one-dimensional Cutting Stock problem<ref>L.A. Wolsey, Integer programming. Wiley,Column Generation Algorithms p185-p189,1998The Cutting Stock problem</ref> ==

	=== ~~''<u>~~Problem Overview~~</u>''~~ ===		=== Problem Overview ===
	A company produces steel bars with a diameter of 45 millimeters and a length of 33 meters. The company also handles cutting the bars to meet the specific length requirements of its customers. Currently, the following demand forecast is expected and must be satisfied:		A company produces steel bars with a diameter of 45 millimeters and a length of 33 meters. The company also handles cutting the bars to meet the specific length requirements of its customers. Currently, the following demand forecast is expected and must be satisfied:
	{\| class="wikitable"		{\| class="wikitable"
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	</math>		</math>

	=== ~~''<u>~~Solving the problem~~</u>''~~ ===		=== Solving the problem ===

	To directly solve the model, it is necessary to know all the parameters, in particular <math display="inline">N_{ks}</math>. This means we need to be able to enumerate all the elements in set <math>S		To directly solve the model, it is necessary to know all the parameters, in particular <math display="inline">N_{ks}</math>. This means we need to be able to enumerate all the elements in set <math>S

Midahoru: Intro

2025-01-16T18:49:10Z

Intro

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	* Miguel Hoyos		* Miguel Hoyos

	== Introduction ==		== Introduction <ref>J. Desrosiers, M. L¨ubbecke, G. Desaulniers, J. B. Gauthier (June 2024). Branch-and-Price, Les Cahiers du GERAD G–2024–36, HEC Montr´eal, Canada. Revised version: October 2024</ref> ==
	Column Generation is an iterative decomposition method for solving linear problems ~~with~~ a ~~very~~ large number of variables. The main idea is to ~~reformulate~~ the original problem ~~such~~ that the variables ~~(or columns)~~ are ~~described implicitly as~~ the ~~incidence vectors of certain subsets of~~ the ~~set~~ of ~~alternatives (e.g., tours, client subsets)~~<ref ~~name=":0"~~>~~Wolsey~~, ~~L. A~~. (~~2021~~). ~~''Integer Programming'' (2nd ed~~.). ~~Wiley. Section 11.4, Solving the Master Problem: Branch~~-~~and~~-~~Price, pp. 219~~-~~222~~.</ref>. The problem is then decomposed into a Master Problem and at least one Subproblem, also called the Pricing problem. At each iteration, the SP provides a new variable to be included in the Master Problem, which then has to be resolved. This process is repeated until a stopping criterion is met.		Column Generation (CG) is an iterative decomposition method for solving linear problems that typically have a prohibitively large number of variables. The main idea of the algorithm is to remove all variables from the original model and then iteratively reintroduce a limited number of them until it is possible to prove that the smaller problem contains at least one optimal solution to the original problem. The rationale behind this algorithm is that, in an optimal solution, nearly all the variables take the value zero, which means those variables are essentially not part of the model at all. This method simplifies the formulation and resolution of large problems since not all variables need to be explicitly listed to find an optimal solution<ref>Desrosiers, Jacques & Lübbecke, Marco. (2006). A Primer in Column Generation.p7-p14 10.1007/0-387-25486-2_1. </ref>.

	~~This~~ method ~~simplifies~~ the ~~formulation and resolution~~ of ~~large problems since not~~ all variables ~~need to be explicitly listed to find an optimal solution<ref>Desrosiers~~, ~~Jacques & Lübbecke~~, ~~Marco.~~ (~~2006~~)~~. A Primer in Column Generation.p7-p14 10.1007/0-387-25486-2_1. </ref>~~.		CG can be described as the '''primal simplex method''' with a variation on the pricing step. Instead of evaluating the reduced costs of all the variables and selecting the one with the minimum non-positive value (for minimization problems, or the maximum non-negative value for maximization problems), it dynamically generates a new variable that meets that condition (or proves that none exists) by solving an auxiliary optimization problem.

	~~Column Generation~~ can also be applied to solve integer or mixed-integer problems if implemented within a Branch-and-Bound or Branch-Cut-and-Price algorithm.		To implement this algorithm the original problem has to be reformulated in a way that the variables (or columns) are described implicitly as the incidence vectors of certain subsets of the set of alternatives (e.g., tours, client subsets)<ref name=":0">Wolsey, L. A. (2021). ''Integer Programming'' (2nd ed.). Wiley. Section 11.4, Solving the Master Problem: Branch-and-Price, pp. 219-222.</ref>, and then decompose it into a Master Problem and at least one Subproblem, also called the Pricing problem. At each iteration, the Subproblem provides a new variable to be included in the Master Problem, which then has to be resolved. This process is repeated until optimality can be proved.

			CG is often mentioned in the context of Dantzig-Wolfe decomposition, and some even confuse the two, but they are distinct: the former is an independent method from the latter.

			It is important to note that CG can also be applied to solve integer or mixed-integer problems if used to solve the linear relaxation and is implemented within a Branch-and-Bound or Branch-Cut-and-Price algorithm.

	== Theory, methodology and algorithmic discussions ==		== Theory, methodology and algorithmic discussions ==

Midahoru: Include better explanations and references.

2025-01-16T02:07:41Z

Include better explanations and references.

Show changes

Midahoru: In the numerical example. 1 : Different names used for the same parameter. / 2: Clarification of some explanations / 3: Add a citation

2025-01-15T20:19:39Z

In the numerical example. 1 : Different names used for the same parameter. / 2: Clarification of some explanations / 3: Add a citation

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	== Numerical example: The Cutting Stock problem<ref>L.A. Wolsey, Integer programming. Wiley,Column Generation Algorithms p185-p189,1998The Cutting Stock problem</ref> ==		== Numerical example: The Cutting Stock problem<ref>L.A. Wolsey, Integer programming. Wiley,Column Generation Algorithms p185-p189,1998The Cutting Stock problem</ref> ==

	~~Suppose we want to solve~~ a numerical example of the ~~cutting stock problem, specifically a~~ one-dimensional cutting stock problem.		Consider solving a numerical example of the one-dimensional cutting stock problem.

	''<u>Problem Overview</u>''		=== ''<u>Problem Overview</u>'' ===
			A company produces steel bars with a diameter of 45 millimeters and a length of 33 meters. The company also handles cutting the bars to meet the specific length requirements of its customers. Currently, the following demand forecast is expected and must be satisfied:
	A company produces steel bars with diameter ~~<math>~~45~~</math>~~ millimeters and length ~~<math>~~33~~</math>~~ meters. The company also ~~takes care of~~ cutting the bars ~~for their different~~ customers~~, who each require different lengths~~. ~~At the moment~~, the following demand forecast is expected and must be satisfied:
	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+
	\|Pieces needed		\|'''Pieces needed'''
	\|Piece length(m)		\|'''Piece length(m)'''
	\|Type of item		\|'''Type of item'''
	\|-		\|-
	\|144		\|144
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	\|5		\|5
	\|}		\|}
	The objective is to ~~establish what is~~ the minimum number of steel bars ~~that should be used~~ to satisfy the total demand.		The objective is to determine the minimum number of steel bars required to satisfy the total demand.

	A possible model for ~~the~~ problem, proposed by Gilmore and Gomory in the ~~1960ies~~ is ~~the one below~~:		A possible model for this problem, proposed by Gilmore and Gomory in the 1960s, is as follows:

	'''Sets'''		'''Sets'''
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	<math>K=\left \{ 1,2,3,4,5 \right \}</math>: set of item types;		<math>K=\left \{ 1,2,3,4,5 \right \}</math>: set of item types;

	''<math display="inline">S</math>:'' set of patterns (i.e., possible ways) that can be ~~adopted~~ to cut a given bar into portions of the ~~need~~ lengths.		''<math display="inline">S</math>:'' set of patterns (i.e., possible ways) that can be used to cut a given bar into portions of the required lengths. It is important to note that the cardinality of this set is unknown a priori and can be extremely large.

	'''Parameters'''		'''Parameters'''
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	'''Decision variables'''		'''Decision variables'''

	<math display="inline">~~Y_s~~</math> : number of bars that should be portioned using pattern ''<math display="inline">s</math>'' ''<math display="inline">\in</math> <math display="inline">S</math>''.		<math display="inline">y_s</math> : number of bars that should be portioned using pattern ''<math display="inline">s</math>'' ''<math display="inline">\in</math> <math display="inline">S</math>''.

	'''Model'''		'''Model'''

	<math>\begin{matrix}\min(y)\sum_{s=1}^Sy_s \\ \ s.t. \~~sum_kN_~~{ks}y_s\geq ~~J_k~~ \forall k\in K \\ y_s\in \Zeta_+\forall s\in S \end{matrix}		<math>\begin{matrix}\min(y)\sum_{s=1}^Sy_s \\ \ s.t. \sum_sN_{ks}y_s\geq R_k \quad \forall k\in K \\ y_s\in \Zeta_+\forall s\in S \end{matrix}

	</math>		</math>

	''<u>Solving the problem</u>''		=== ''<u>Solving the problem</u>'' ===

	~~The~~ model ~~assumes the availability of the set ''<math display="inline">K</math>'' and~~ the parameters <math display="inline">N_{k,s}</math> . To generate this data, you would have to list all possible cutting patterns. However, the number of possible cutting patterns is a big number. This is why a direct implementation of the model above is not practical in real-world problems. In this case is when it makes sense to solve the continuous relaxation of the above model. This ~~is because, in reality, the demand figures are so high that the number of bars~~ to ~~cut is also a large number, and therefore a good solution can~~ be ~~determined by rounding up~~ to the ~~next integer each variable~~ <math>~~y_s~~		To directly solve the model, it is necessary to know all the parameters, in particular <math display="inline">N_{k,s}</math>. This means we need to be able to enumerate all the elements in set <math>S

	</math>~~found by solving~~ the ~~continuous relaxation~~. ~~In addition to~~ that, the solution of the ~~relaxed problem~~ will ~~become~~ the ~~starting point for~~ the ~~application~~ of ~~an exact~~ solution ~~method (~~for ~~instance, the Branch~~-and ~~Bound)~~.<blockquote><u>''Key take-away: In the next steps of this example we will analyze how to solve the continuous relaxation of the model.''</u></blockquote>As a starting point, we need any feasible solution. Such a solution can be constructed as follows:		</math> and determine hoy many pieces of each item are present. The problem lies in the fact that the cardinality of <math>S

			</math> (i.e. the number of elements in the set) is extremely large, making the direct solve an impractical task.

			Another issue with the formulation presented above is that the <math display="inline">y_{s}</math> variables are integer. As stated in <ref>Gilmore, P. C., & Gomory, R. E. (1961). A Linear Programming Approach to the Cutting-Stock Problem. ''Operations Research, 9''(6), 849-859. INFORMS.</ref>, we can relax this constraint, which will result in a non-integer solution most of the time. We can then either round up, resulting in a higher cost, or round down, leaving some demand unmet, which would require the use of ad hoc methods. The authors also note that if the non-integer values are high, any rounding will produce only a marginal change in the cost (here, the cost is equivalent to the number of bars). The common practice is to use the relaxed solution as a starting point for a branch-and-bound algorithm.<blockquote><u>''Key take-away: In the next steps of this example we will analyze how to solve the continuous relaxation of the model.''</u></blockquote>As a starting point, we need any feasible solution. Such a solution can be constructed as follows:

	# We consider any single-item cutting patterns, i.e., <math>\\|K\\|		# We consider any single-item cutting patterns, i.e., <math>\\|K\\|

Wc593 at 11:37, 21 December 2020

2020-12-21T11:37:13Z

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	Author: Lorena Garcia Fernandez (lgf572)		Author: Lorena Garcia Fernandez (lgf572) (SysEn 5800 Fall 2020)

	== Introduction ==		== Introduction ==

Wc593 at 10:46, 21 December 2020

2020-12-21T10:46:30Z

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	''<u>Solving the problem</u>''		''<u>Solving the problem</u>''

	The model assumes the availability of the set ''<math display="inline">K</math>'' and the parameters <math display="inline">N_{k,s}</math> . To generate this data, you would have to list all possible cutting patterns. However, the number of possible cutting patterns is a big number. This is why a direct implementation of the model above is not ~~partical~~ in real-world problems. In this case is when it makes sense to solve the continuous relaxation of the above model. This is because, in reality, the demand figures are so high that the number of bars to cut is also a large number, and therefore a good solution can be determined by rounding up to the next integer each variable <math>y_s		The model assumes the availability of the set ''<math display="inline">K</math>'' and the parameters <math display="inline">N_{k,s}</math> . To generate this data, you would have to list all possible cutting patterns. However, the number of possible cutting patterns is a big number. This is why a direct implementation of the model above is not practical in real-world problems. In this case is when it makes sense to solve the continuous relaxation of the above model. This is because, in reality, the demand figures are so high that the number of bars to cut is also a large number, and therefore a good solution can be determined by rounding up to the next integer each variable <math>y_s

	</math>found by solving the continuous relaxation. In addition to that, the solution of the relaxed problem will become the starting point for the application of an exact solution method (for instance, the Branch-and Bound).<blockquote><u>''Key take-away: In the next steps of this example we will analyze how to solve the continuous relaxation of the model.''</u></blockquote>As a starting point, we need any feasible solution. Such a solution can be constructed as follows:		</math>found by solving the continuous relaxation. In addition to that, the solution of the relaxed problem will become the starting point for the application of an exact solution method (for instance, the Branch-and Bound).<blockquote><u>''Key take-away: In the next steps of this example we will analyze how to solve the continuous relaxation of the model.''</u></blockquote>As a starting point, we need any feasible solution. Such a solution can be constructed as follows:
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	The column generation subproblem is the critical part of the method is generating the new columns. It is not reasonable to compute the reduced costs of all variables <math>y_s		The column generation subproblem is the critical part of the method is generating the new columns. It is not reasonable to compute the reduced costs of all variables <math>y_s

	</math> for <math>s=1,...,S</math>, otherwise this procedure would reduce to the simplex method. In fact, n<math>n</math> can be very large (as in the cutting-stock problem) or, for some reason, it might not be possible or convenient to enumerate all decision variables. This is when it would be necessary to study a specific column generation algorithm for each problem; ''only if such an algorithm exists (and is ~~partical~~)'', the method can be fully applied. In the one-dimensional cutting stock problem, we transformed the column generation subproblem into an easily solvable integer linear programming problem. In other cases, the computational effort required to solve the subproblem is too high, such that appying this full procedure becomes unefficient.		</math> for <math>s=1,...,S</math>, otherwise this procedure would reduce to the simplex method. In fact, n<math>n</math> can be very large (as in the cutting-stock problem) or, for some reason, it might not be possible or convenient to enumerate all decision variables. This is when it would be necessary to study a specific column generation algorithm for each problem; ''only if such an algorithm exists (and is practical)'', the method can be fully applied. In the one-dimensional cutting stock problem, we transformed the column generation subproblem into an easily solvable integer linear programming problem. In other cases, the computational effort required to solve the subproblem is too high, such that appying this full procedure becomes unefficient.

	== Applications ==		== Applications ==

Wc593 at 10:45, 21 December 2020

2020-12-21T10:45:10Z

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	== Introduction ==		== Introduction ==
	Column Generation techniques have the scope of solving large linear optimization problems by generating only the variables that will have an influence on the objective function. This is important for big problems with many variables where the formulation with these techniques would simplify the problem formulation, since not all the possibilities need to be explicitly listed<ref>Desrosiers, Jacques & Lübbecke, Marco. (2006). A Primer in Column Generation.p7-p14 10.1007/0-387-25486-2_1. </ref>.		Column Generation techniques have the scope of solving large linear optimization problems by generating only the variables that will have an influence on the objective function. This is important for big problems with many variables where the formulation with these techniques would simplify the problem formulation, since not all the possibilities need to be explicitly listed.<ref>Desrosiers, Jacques & Lübbecke, Marco. (2006). A Primer in Column Generation.p7-p14 10.1007/0-387-25486-2_1. </ref>

	== Theory, methodology and algorithmic discussions ==		== Theory, methodology and algorithmic discussions ==
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	The way this method work is as follows; first, the original problem that is being solved needs to be split into two problems: the master problem and the sub-problem.		The way this method work is as follows; first, the original problem that is being solved needs to be split into two problems: the master problem and the sub-problem.

	* The master problem is the original column-wise (i.e: one column at a time) formulation of the problem with only a subset of variables being considered<ref>		* The master problem is the original column-wise (i.e: one column at a time) formulation of the problem with only a subset of variables being considered.<ref>
	AlainChabrier, Column Generation techniques, 2019 URL: https://medium.com/@AlainChabrier/column-generation-techniques-6a414d723a64		AlainChabrier, Column Generation techniques, 2019 URL: https://medium.com/@AlainChabrier/column-generation-techniques-6a414d723a64
	</ref>.		</ref>

	* The sub-problem is a new problem created to identify a new promising variable. The objective function of the sub-problem is the reduced cost of the new variable with respect to the current dual variables, and the constraints require that the variable obeys the naturally occurring constraints. The subproblem is also referred to as the RMP or “restricted master problem”. From this we can infer that this method will be a good fit for problems whose constraint set admit a natural breakdown (i.e: decomposition) into sub-systems representing a well understood combinatorial structure<ref>		* The sub-problem is a new problem created to identify a new promising variable. The objective function of the sub-problem is the reduced cost of the new variable with respect to the current dual variables, and the constraints require that the variable obeys the naturally occurring constraints. The subproblem is also referred to as the RMP or “restricted master problem”. From this we can infer that this method will be a good fit for problems whose constraint set admit a natural breakdown (i.e: decomposition) into sub-systems representing a well understood combinatorial structure.<ref>
	AlainChabrier, Column Generation techniques, 2019 URL: https://medium.com/@AlainChabrier/column-generation-techniques-6a414d723a64		AlainChabrier, Column Generation techniques, 2019 URL: https://medium.com/@AlainChabrier/column-generation-techniques-6a414d723a64
	</ref>.		</ref>

	To execute that decomposition from the original problem into Master and subproblems there are different techniques. The theory behind this method relies on the Dantzig-Wolfe decomposition<ref>Dantzig-Wolfe decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dantzig-Wolfe_decomposition&oldid=50750</ref>.		To execute that decomposition from the original problem into Master and subproblems there are different techniques. The theory behind this method relies on the Dantzig-Wolfe decomposition.<ref>Dantzig-Wolfe decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dantzig-Wolfe_decomposition&oldid=50750</ref>

	In summary, when the master problem is solved, we are able to obtain dual prices for each of the constraints in the master problem. This information is then utilized in the objective function of the subproblem. The subproblem is solved. If the objective value of the subproblem is negative, a variable with negative reduced cost has been identified. This variable is then added to the master problem, and the master problem is re-solved. Re-solving the master problem will generate a new set of dual values, and the process is repeated until no negative reduced cost variables are identified. The subproblem returns a solution with non-negative reduced cost, we can conclude that the solution to the master problem is optimal<ref>Wikipedia, the free encyclopeda. Column Generation. URL: https://en.wikipedia.org/wiki/Column_generation</ref>.		In summary, when the master problem is solved, we are able to obtain dual prices for each of the constraints in the master problem. This information is then utilized in the objective function of the subproblem. The subproblem is solved. If the objective value of the subproblem is negative, a variable with negative reduced cost has been identified. This variable is then added to the master problem, and the master problem is re-solved. Re-solving the master problem will generate a new set of dual values, and the process is repeated until no negative reduced cost variables are identified. The subproblem returns a solution with non-negative reduced cost, we can conclude that the solution to the master problem is optimal.<ref>Wikipedia, the free encyclopeda. Column Generation. URL: https://en.wikipedia.org/wiki/Column_generation</ref>

	'''''Methodology'''''<ref>L.A. Wolsey, Integer programming. Wiley,Column Generation Algorithms p185-p189,1998</ref>		'''''Methodology'''''<ref>L.A. Wolsey, Integer programming. Wiley,Column Generation Algorithms p185-p189,1998</ref>
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	* Bandwith packing		* Bandwith packing
	* Bus driver scheduling		* Bus driver scheduling
	* Generally, column generation algorithms are used for large delivery networks, often in combination with other methods, helping to implement real-time solutions for on-demand logistics.We discuss a supply chain scheduling application below.		* Generally, column generation algorithms are used for large delivery networks, often in combination with other methods, helping to implement real-time solutions for on-demand logistics. We discuss a supply chain scheduling application below.

	'''''Bandwidth packing'''''		'''''Bandwidth packing'''''
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	'''''Bus driver scheduling'''''		'''''Bus driver scheduling'''''

	Bus driver scheduling aims to find the minimum number of bus drivers to cover a published timetable of a bus company. When scheduling bus drivers, contractual working rules must be enforced, thus complicating the problem. A column generation algorithm can decompose this complicated problem into a master problem and a series of pricing subproblems. The master problem would select optimal duties from a set of known feasible duties, and the pricing subproblem would augment the feasible duty set to improve the solution obtained in the master problem. <ref name=":2">Dung‐Ying Lin, Ching‐Lan Hsu. Journal of Advanced Transportation. Volume50, Issue8, December 2016, Pages 1598-1615. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/atr.1417</ref>.		Bus driver scheduling aims to find the minimum number of bus drivers to cover a published timetable of a bus company. When scheduling bus drivers, contractual working rules must be enforced, thus complicating the problem. A column generation algorithm can decompose this complicated problem into a master problem and a series of pricing subproblems. The master problem would select optimal duties from a set of known feasible duties, and the pricing subproblem would augment the feasible duty set to improve the solution obtained in the master problem.<ref name=":2">Dung‐Ying Lin, Ching‐Lan Hsu. Journal of Advanced Transportation. Volume50, Issue8, December 2016, Pages 1598-1615. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/atr.1417</ref>

	'''''Supply Chain scheduling problem'''''		'''''Supply Chain scheduling problem'''''

	A typical application is where we consider the problem of scheduling a set of shipments between different nodes of a supply chain network. Each shipment has a fixed departure time, as well as an origin and a destination node, which, combined, determine the duration of the associated trip. The aim is to schedule as many shipments as possible, while also minimizing the number of vehicles utilized for this purpose. This problem can be formulated by an integer programming model and an associated branch and price solution algorithm. The optimal solution to the LP relaxation of the problem can be obtained through column generation, solving the linear program a huge number of variables, without explicitly considering all of them. In the context of this application, the master problem schedules the maximum possible number of shipments using only a small set of vehicle-routes, and a column generation (colgen) sub-problem would generate cost-effective vehicle-routes to be fed fed into the master problem. After finding the optimal solution to the LP relaxation of the problem, the algorithm would branch on the fractional decision variables (vehicle-routes), in order to reach the optimal integer solution<ref name=":1">Kozanidis, George. (2014). Column generation for scheduling shipments within a supply chain network with the minimum number of vehicles. OPT-i 2014 - 1st International Conference on Engineering and Applied Sciences Optimization, Proceedings. 888-898</ref>.		A typical application is where we consider the problem of scheduling a set of shipments between different nodes of a supply chain network. Each shipment has a fixed departure time, as well as an origin and a destination node, which, combined, determine the duration of the associated trip. The aim is to schedule as many shipments as possible, while also minimizing the number of vehicles utilized for this purpose. This problem can be formulated by an integer programming model and an associated branch and price solution algorithm. The optimal solution to the LP relaxation of the problem can be obtained through column generation, solving the linear program a huge number of variables, without explicitly considering all of them. In the context of this application, the master problem schedules the maximum possible number of shipments using only a small set of vehicle-routes, and a column generation (colgen) sub-problem would generate cost-effective vehicle-routes to be fed fed into the master problem. After finding the optimal solution to the LP relaxation of the problem, the algorithm would branch on the fractional decision variables (vehicle-routes), in order to reach the optimal integer solution.<ref name=":1">Kozanidis, George. (2014). Column generation for scheduling shipments within a supply chain network with the minimum number of vehicles. OPT-i 2014 - 1st International Conference on Engineering and Applied Sciences Optimization, Proceedings. 888-898</ref>

	== Conclusions ==		== Conclusions ==
	Column generation is a way of starting with a small, manageable part of a problem (specifically, with some of the variables), solving that part, analyzing that interim solution to find the next part of the problem (specifically, one or more variables) to add to the model, and then solving the full or extended model. In the column generation method, the algorithm steps are repeated until an optimal solution to the entire problem is achieved<ref> ILOG CPLEX 11.0 User's Manual > Discrete Optimization > Using Column Generation: a Cutting Stock Example > What Is Column Generation? 1997-2007. URL:http://www-eio.upc.es/lceio/manuals/cplex-11/html/usrcplex/usingColumnGen2.html#:~:text=In%20formal%20terms%2C%20column%20generation,method%20of%20solving%20the%20problem.&text=By%201960%2C%20Dantzig%20and%20Wolfe,problems%20with%20a%20decomposable%20structure</ref>.		Column generation is a way of starting with a small, manageable part of a problem (specifically, with some of the variables), solving that part, analyzing that interim solution to find the next part of the problem (specifically, one or more variables) to add to the model, and then solving the full or extended model. In the column generation method, the algorithm steps are repeated until an optimal solution to the entire problem is achieved.<ref> ILOG CPLEX 11.0 User's Manual > Discrete Optimization > Using Column Generation: a Cutting Stock Example > What Is Column Generation? 1997-2007. URL:http://www-eio.upc.es/lceio/manuals/cplex-11/html/usrcplex/usingColumnGen2.html#:~:text=In%20formal%20terms%2C%20column%20generation,method%20of%20solving%20the%20problem.&text=By%201960%2C%20Dantzig%20and%20Wolfe,problems%20with%20a%20decomposable%20structure</ref>

	This algorithm provides a way of solving a linear programming problem adding columns (corresponding to constrained variables) during the pricing phase of the problem solving phase, that would otherwise be very tedious to formulate and compute. Generating a column in the primal formulation of a linear programming problem corresponds to adding a constraint in its dual formulation.		This algorithm provides a way of solving a linear programming problem adding columns (corresponding to constrained variables) during the pricing phase of the problem solving phase, that would otherwise be very tedious to formulate and compute. Generating a column in the primal formulation of a linear programming problem corresponds to adding a constraint in its dual formulation.

	== References ==		== References ==