accelerating column generation for variable sized bin-packing problems

European Journal of Operational Research 183 (2007) 1333–1352

www.elsevier.com/locate/ejor

Accelerating column generation for variable sizedbin-packing problems

Claudio Alves *, J.M. Valerio de Carvalho

Departamento de Producao e Sistemas, Escola de Engenharia, Universidade do Minho, 4710-057 Braga, Portugal

Received 30 July 2004; accepted 26 July 2005Available online 12 June 2006

Abstract

In this paper, we study different strategies to stabilize and accelerate the column generation method, when it is appliedspecifically to the variable sized bin-packing problem, or to its cutting stock counterpart, the multiple length cutting stockproblem. Many of the algorithms for these problems discussed in the literature rely on column generation, processes thatare known to converge slowly due to primal degeneracy and the excessive oscillations of the dual variables. In the sequel,we introduce new dual-optimal inequalities, and explore the principle of model aggregation as an alternative way of con-trolling the progress of the dual variables. Two algorithms based on aggregation are proposed. The first one relies on a rowaggregated LP, while the second one solves iteratively sequences of doubly aggregated models. Working with these approx-imations, in the various stages of an iterative solution process, has proven to be an effective way of achieving fasterconvergence.

The computational experiments were conducted on a broad range of instances, many of them published in the litera-ture. They show a significant reduction of the number of column generation iterations and computing time.� 2006 Elsevier B.V. All rights reserved.

Keywords: Integer programming; Column generation; Variable sized bin-packing; Multiple length cutting stock; Convergence

1. Introduction

Packing and cutting problems share the same objective: find the best way of assigning a set of objects toother ones. Many variations are allowed. Dyckhoff [11] suggests a typology in which these problems are clas-sified mainly according to their dimensionality, the shape of the objects, the types of assortments and the avail-ability of large and small objects. This latter point distinguishes bin-packing from cutting stock problems. In abin-packing problem, we have to deal with a limited number of small objects, whereas the cutting stock prob-lems involve quantities that are far greater than the unit.

In this paper, we consider the one-dimensional variable sized bin-packing problem, which is characterizedby bins with different capacities, and a cost function that depends on the total capacity required. We address in

0377-2217/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2005.07.033

* Corresponding author. Tel.: +351 253 604751; fax: +351 253 604741.E-mail address: [email protected] (C. Alves).

mailto:[email protected]

1334 C. Alves, J.M. Valerio de Carvalho / European Journal of Operational Research 183 (2007) 1333–1352

particular the case in which the number of bins is bounded. To clarify the presentation, we will refer only tothe variable sized bin-packing problem, but note that the approaches we devised are identically applicable tothe multiple length cutting stock problem, its cutting stock counterpart. Indeed, no special attention was givento the quantities that are ordered.

The problem is a generalization of the standard bin-packing problem, which deserved in the past muchmore attention from the research community. It is NP-hard [13]; no absolute approximation scheme can bedevised that solves it in fully polynomial time. Therefore, a lot of research has been initially devoted to thedevelopment of approximation algorithms for which only worst-case performance could be guaranteed. Frie-sen and Langston [12], Murgolo [24], Chu and La [6] worked on such algorithms.

Meanwhile, solutions to the multiple length cutting stock problem were proposed. Gilmore and Gomory[15] extend their column generation scheme for the cutting stock problem to what they call the machine bal-ance problem. Roodman [26] describes how to find near optimal solutions starting with the linear program-ming (LP) relaxation, and generating columns heuristically. Recently, Holthaus [17] reports extensivecomputational experiments on combining column generation with different methods to solve the residualproblem arising from the rounding of the continuous solution.

Only recently, attempts have been made to construct efficient algorithms that solve the variable sized bin-packing problem to optimality. Monaci [23] devises an exact branch-and-bound algorithm based on a branch-ing rule from Martello and Toth [22]. Lower bounds are computed along the tree, one of which through acolumn generation procedure. Belov and Scheithauer [3] use Chvatal-Gomory cutting planes embedded ina column generation algorithm to solve the multiple length cutting stock problem. Their pricing subproblembecomes a general integer program without any special structure that has to be solved by branch-and-bound.Alves and Valerio de Carvalho [1] propose a branch-and-price algorithm where branching constraints areimposed on the arc-flow variables of an original compact formulation. The subproblem remains a knapsackproblem, solvable in pseudo-polynomial time.

In fact, column generation is at the heart of most of the solution approaches. The model it solves, that resultsfrom the Dantzig–Wolfe decomposition of a compact formulation [32,30], is stronger and leads to improved lowerbounds. However, it is well known that column generation procedures suffer from slow convergence induced byundesirable behaviors such as primal degeneracy [8], or the excessive oscillations of the dual variables [10].

In the last years, many efforts have been devoted to the topic of stabilized column generation, with the pur-pose of accelerating these processes. One way of accelerating column generation was proposed by Degraeveand Peeters [7], who use a simplex method/subgradient-optimization procedure to solve the LP relaxationof the cutting stock problem. To obtain the optimal solution, the following procedure is repeated: for a specificnumber of iterations, subgradient optimization is used to update the dual prices, and new columns are pricedand generated; then, the simplex method is used to reoptimize the master problem with the new columnsadded. Column generation and Lagrangean relaxation are equivalent [25], and the subgradient method hasbeen extensively used to solve the Lagrangean problem. Column generation is known to be more robust,but it has the burden of reoptimizing the master problem to update the dual variables at every iteration.The subgradient method provides a fast way of updating the dual solution, but may have some convergencedifficulties. The hybrid procedure of Degraeve and Peeters combines the robustness of the column generationmethod with the fast update of the dual prices of the subgradient method, producing an overall acceleration ofthe solution process. It will become clear that the procedures presented in this paper, which are based on adifferent idea, can be combined with the one used by Degraeve and Peeters.

Stabilization is a different way of accelerating the column generation process. Usually, stabilization may beachieved by restricting once the admissible dual solution space or, instead, by guiding the progress of the dualvariables. The boxstep method of Marsten et al. [21] follows this latter strategy by drawing fixed-size boxesaround the solutions of the dual restricted master problem. The trust region method [19] uses a similar conceptbut relies on box constraints whose sizes may be dynamically updated. From the primal standpoint, thesemethods solve successively a restricted master problem in which slack and surplus variables are penalized.In [10], du Merle et al. extend this approach by imposing additional constraints to these variables. They sug-gest strategies to set the box sizes, and report promising results on air transportation and location problems.Other methods, such as bundle [16] and analytic center cutting plane methods [9], have been used to preventthe excessive variations of the dual variables.

C. Alves, J.M. Valerio de Carvalho / European Journal of Operational Research 183 (2007) 1333–1352 1335

Recent publications have shown how to reduce the instability of column generation, when this method isapplied to bin-packing or cutting stock problems. Valerio de Carvalho [31] proposes adding a polynomialnumber of columns (i.e., dual inequalities) prior to the solution of the first restricted master problem. Hederives a family of dual inequalities, and proves that they are weak dual-optimal inequalities in the sense intro-duced by Ben Amor et al. [5] that they do not exclude any dual-optimal solution. Since primal feasibility maybe lost, Ben Amor et al. [5] suggest perturbing the right hand side of the dual inequalities by small amounts,forcing those columns to have null values in any optimal solution. Besides, they show that the aggregation intosingle constraints of items with the same size leads to a substantial acceleration of the overall solution process.In their models for packing and cutting problems, researchers have since long considered this aggregation.This was already a way of implicitly restricting the dual space by enforcing equality between some dual vari-ables. In fact, we know that there always exists an optimal solution in which items of identical sizes have cor-responding dual variables with the same value. In practice, this equality extends frequently to other items withnearly the same size. This phenomenon was early pointed out by Gilmore and Gomory [15], who explore itonly to reduce the size of the knapsack subproblems.

In this paper, we present different strategies to stabilize and accelerate column generation for the variablesized bin-packing problem. We concentrate on the LP relaxation. Accelerating the resolution of the LP relax-ation is an important matter, since this a way of getting good lower bounds in less time. However, being able toapply these stabilization techniques to all the nodes of a branching tree is even more interesting. Indeed, in abranch-and-price framework the tailing off effect does not occur only at the root node. In a forthcoming paper,we will show how to use one of the stabilization techniques discussed below, namely the dual inequalities, to thewhole branch-and-bound tree [1]. In the sequel, new dual-optimal inequalities will be analyzed. We discuss theirrelative strength, and proceed by exploring the principle of model aggregation as an alternative way of control-ling the progress of the dual variables. Two algorithms based on the iterative resolution of aggregated modelsare proposed. The first is a simple two-steps procedure that starts by solving a row aggregated LP resulting fromthe juxtaposition of the original items. The second is an iterative algorithm that solves a sequence of smaller sizeapproximations obtained through a double aggregation of variables and constraints. This aggregation schemeamounts to imposing equality constraints between some dual variables. Note that these aggregation strategiesrely essentially on items, which allows us to use them for the single sized bin-packing problem.

This paper is organized as follows. The mathematical formulation for the variable sized bin-packing prob-lem is presented in Section 2. In Section 3, we briefly recall the dual inequalities introduced in [31], which arestill valid even when bins with different capacities are available, and we describe the set of new dual inequal-ities. Section 4 is devoted to our aggregation based procedures. Extensive computational results are presentedin Section 5.

2. Variable sized bin-packing problems: A mathematical formulation

In variable sized bin-packing problems, we are given K classes of bins consisting in Bk bins of integer capac-ity Wk, such that W k 6¼ W k0 , for all k 5 k 0, and m different sets of items, each one with its own integer size wi,i = 1, . . . ,m. The number of items to pack, or demand, in each set i is denoted by bi. Throughout this paper, wewill assume that the items and bins are sorted in order of decreasing sizes and capacities, respectively. Theobjective is to find what packing patterns to use, and in what extent, in order to satisfy at minimum costthe demand using nothing more than the available bins. The cost of a solution is expressed as the total capacityrequired to pack the items. So, finding a minimum cost solution amounts to minimizing the total unfulfilledspace. Note that, due to the availability constraints, the problem may be infeasible.

Using the new typology developed by Wascher et al. [34], we can characterize this problem as a multiple binsize bin packing problem, with a single and fixed dimension, a weakly heterogeneous assortment of large objects,a strongly heterogeneous assortment of small items, and an objective that consists on input minimization.

The problem is formulated through the following integer linear programming (ILP) model:

minXK

k¼1

Xpk

r¼1

W kxkr; ð1Þ


subject toXK

k¼1

Xpk

r¼1

aikrxkr P bi; i ¼ 1; . . . ;m; ð2Þ

Xpk

r¼1

xkr 6 Bk; k ¼ 1; . . . ;K; ð3Þ

xkr P 0; and integer; k ¼ 1; . . . ;K; r ¼ 1; . . . ; pk: ð4Þ

Formulation (1)–(4) has an exponential number of columns, denoted by p. Each column r, r = 1, . . . ,p, con-sists in a vector (a1kr,a2kr, . . . ,amkr; . . . , 1,. . .)T, with integer and non-negative coefficients aikr, identifying a va-lid packing pattern assigned to a bin with capacity Wk, i.e., a combination of aikr items of size wi, i = 1, . . . ,m,that fits in this bin

Pmi¼1aikrwi 6 W k

� �. There are pk admissible patterns per bin class k and, so,

PKk¼1pk ¼ p.

Variables xkr are general integer variables. They represent the number of times a pattern r associated to abin with capacity Wk is used.

Inequalities (2) model the demand constraints, whereas (3) ensure that the number of available bins is notexceeded. Inequalities (2) might be replaced by equalities. Gilmore and Gomory [14] showed that an optimalsolution always exists with null slack variables in (2). Nevertheless, we will keep these inequalities since theylead to a more restricted dual solution space.

In this paper, we consider the resolution by column generation of the LP relaxation of (1)–(4), which will bedenoted by P.

Let ui, i = 1, . . . ,m, and vk, k = 1, . . . ,K, be the dual variables associated to constraints (2) and (3), respec-tively. For ease of presentation, we will frequently refer to the ui and vk variables, respectively, as the items’and bins’ dual variables. Based on the complementary slackness conditions, we can give an interesting inter-pretation to the items’ dual variables. If the availability of the bins was unbounded, the ui variables wouldrepresent the exact ideal sizes the items should have in order to fulfill the bins selected in the primal optimalsolution. From the dual standpoint, a pattern that is in the optimal primal basis with a positive value corre-sponds to a constraint with a slack variable that is equal to 0. Instead of their original sizes, if the items hadsizes equal to the corresponding items’ dual variables, the patterns that are in this optimal primal basis wouldbecome patterns without any unused space. In the bounded case, when the availability constraints are effec-tive, the vk variables relax the dual knapsack constraints to some point. The dual formulation D of P follows:

ðDÞ maxXm

i¼1

uibi þXK

k¼1

vkBk; ð5Þ

subject toXm

i¼1

aikrui þ vk 6 W k; k ¼ 1; . . . ;K; r ¼ 1; . . . ; pk; ð6Þ

ui P 0; i ¼ 1; . . . ;m; ð7Þvk 6 0; k ¼ 1; . . . ;K: ð8Þ

The LP relaxation of (1)–(4) can be obtained through an adequate Dantzig–Wolfe decomposition of two otherLP models: the extended Kantorovich model [20], and the arc-flow model described in [29,30]. These models areusually called compact or original models [33]. The former relies on assignment variables, and is known for itspoor LP bound. It exhibits a high degree of symmetry, which is characterized by many alternative solutions cor-responding in fact to the same packing. The LP lower bound of the arc-flow model is as strong as the one providedby the LP relaxation of (1)–(4), but the high number of constraints of this model may penalize its usability. For-mulation (1)–(4) is in fact the most popular formulation for the class of problem we are studying in this paper.Gilmore and Gomory [15] used it to model their machine balance problem, but it can also be used to model themultiple length cutting stock problem, and the variable sized bin-packing problem as well.

3. Dual-optimal inequalities

In [31], Valerio de Carvalho introduced a certain concept of dual cuts, columns in the primal that do notaffect the optimal value as long as a solution to the original problem can be recovered at no cost with those


columns at the zero level. Later on, Ben Amor et al. [5] used the term dual-optimal inequalities, and dis-tinguished between weak and deep inequalities. In the former, no optimal solution of the original formu-lation is excluded, whereas the deep inequalities may cause the rejection of some alternative optimalsolutions.

The cuts presented in [31] apply to the variable sized bin-packing problem. We recall them briefly in Section3.1. In Section 3.2, we introduce new types of weak and deep dual-optimal inequalities specially devised for thevariable sized bin-packing problem.

3.1. Inequalities on items’ dual variables

If we observe the values of an optimal dual solution to (5)–(8), the ordering of the items’ dual variables willbe quite evident. After all, the items’ dual variables are subject to constraints whose coefficients are directlyrelated to the set of item sizes. This conduces to an ordering that is completely dependent on the size ofthe items. Let S be the set of items of size ws, such that

Ps2Sws 6 wi, for some i = 1, . . . ,m. The relation

can be expressed as follows:

�ui þXs2S

us 6 0; i ¼ 1; . . . ;m; 8S: ð9Þ

Depending on the cardinality of S, different types of dual inequalities may be defined. The result is an expo-nential number of valid constraints. Adding these inequalities to D leads to a so-called extended formulation.In the primal, the respective columns allow an item in a pattern to be exchanged by a combination of otheritems with smaller or equal total size. Valerio de Carvalho showed that any optimal solution to the non-ex-tended formulation verify inequalities (9), which make them weak dual-optimal inequalities. These results re-main even in the case of the variable sized bin-packing problem. Extensions to the proofs are straightforwardand, so, we omit them. During the successive resolutions of the restricted master problems in a column gen-eration process, since the columns are not completely enumerated, these inequalities are frequently violated.The inclusion of a bounded number of such cuts is opportune. Furthermore, when we add inequalities (9) tothe dual, it may happen that some basic dual solutions associated to a given degenerate extreme point in theprimal are cut. As a consequence, adding the corresponding extra columns to the primal may also reduce theprimal degeneracy [31].

3.2. Inequalities on bins’ dual variables

Let De (Pe) correspond to D (P) extended with additional inequalities on the vk dual variables (primal col-umns, respectively)

ðP eÞ min cxþ fy; ðDeÞ max ubþ vB;

Ax P b; uAþ vE 6 c;

Exþ Fy 6 B; vF 6 f ;

x; y P 0: u P 0; v 6 0:

A family of dual inequalities can be stated using the following argument: a packing pattern associated tosome bin can always be reassigned to another bigger bin at a cost that equals the difference between the twocapacities. In the primal, this operation is allowed by including columns with a +1 in row m + k, a �1 in rowm + k 0, and a cost of W k � W k0 , k = 1, . . . ,K � 1, k 0 = 2, . . . ,K, and W k > W k0 . In the dual, we will be insertingcuts with the form:

vk � vk0 6 W k � W k0 ; k ¼ 1; . . . ;K � 1; k0 ¼ 2; . . . ;K; W k > W k0 : ð10Þ

Let ð~x; ~yÞ be a valid solution for Pe. In the case where F ~y 6¼ 0, ~x may be infeasible for P, as illustrated in Exam-ple 3.1. The next proposition, and the corollary that follows, show that the LP lower bounds of P and Pe re-main the same.


Proposition 3.1. Given a feasible solution ð~x; ~yÞ to Pe, we can always recover a feasible solution to P with an equal

or lower cost.

Proof. Let AE and NF be the subset of columns in Pe associated to the patterns and to the dual inequalities(10), respectively. Starting with ð~x; ~yÞ, the following step-by-step procedure shows how to build a solutionexpressed only as a non-negative combination of feasible packing patterns. To clarify the presentation, thek index was dropped from the columns (a1kr,a2kr, . . . ,amkr;e1kr, . . . ,eKkr)

T of Pe.

for k :¼ K, . . ., 2
Let AE be the subset of columns (a1r,a2r, . . . ,amr;e1r, . . . ,eKr)
T of AE with ~xr > 0, and an unit coefficientin row m + k (ekr = 1).Let NF be the subset of columns NFj = (0, . . . , 0; . . . , 1, . . . ,�1, . . .)T from NF with ~yj > 0, a cost of fj

units, the �1 occurring at position m + k and the +1 at position sj.for all j : NFj 2 NF

while ~yj > 0 andP

r:AEr2AE~xr > BkT
Select a column ða1r; a2r; . . . ; amr; . . . ; esjr; . . . ; ekr; . . . Þ from AE with ~xr > 0
AEnew ¼ ða1r; a2r; . . . ; amr; . . . ; esjr þ 1; . . . ; ekr � 1; . . . ÞT. This column has a cost of Wk + fj; xnew isthe associated primal variable.if ð~yj > ~xrÞ

~yj :¼ ~yj � ~xr, xnew :¼ ~xr and ~xr :¼ 0else

~xr :¼ ~xr � ~yj, xnew :¼ ~yj and ~yj :¼ 0end if

Add AEnew to Pe and update the solution with xnew

end while

if ~yj > 0, ~yj :¼ 0end for

end for

The final solution is valid for Pe and have all the columns referring to the dual inequalities (10) at the zerolevel. Therefore, we get a feasible solution for P. If condition ~yj > 0 at the end of the algorithm is true at leastonce, the resulting solution will have a cost that is lower than the one associated with the original solutionð~x; ~yÞ. Otherwise, the cost remains the same. h

Corollary 3.1. Since Pe is a relaxation of P, the optimal solutions of P and Pe have the same cost.

We prove next that inequalities (10) are in fact weak dual-optimal inequalities.

Proposition 3.2. Any optimal dual solution (u*, v*) to D satisfies inequalities (10).

Proof. Consider an optimal primal-dual solution pair x* and (u*,v*) to P and D, respectively. Using argu-ments based on the complementary slackness conditions, we will show that this dual optimal solution obeysto inequalities (10). For this purpose, we consider each bin type separately, and start with a bin type k. Inthe optimal primal solution x*, two cases may occur: (1) at least one pattern associated to bin k is in theoptimal basis ðx�kr > 0, for some r), or (2) no pattern associated to bin k is in the optimal basisPpk

r¼1x�kr ¼ 0� �

.For the case (1), let Akr = (a1kr,a2kr, . . . ,amkr)

T, and Ekr = (. . . , 1 , . . .)T, the +1 occurring at position k, bethe elements of a column in P with a positive value in x*. This column has a null reduced cost, i.e.,Wk � u* ÆAkr � v* ÆEkr = 0. If we assign the items of the corresponding pattern to a bigger bin, say k 0, we getanother valid pattern; let Ak0r0 and Ek0r0 be the elements of the corresponding column. That new pattern may bein the optimal basis, or not, but its reduced cost will always be non-negative, i.e., W k0 � u� � Ak0r0 � v� � Ek0r0 P0. Subtracting the reduced costs, we get W k0 � W k � u� � ðAk0r0 � AkrÞ � v� � ðEk0r0 � EkrÞ ¼ W k0 � W k�v�k0 þ v�k P 0. Thus, v�k0 � v�k 6 W k0 � W k. This inequality holds no matter what bin k 0 is considered, as long


as W k0 > W k. This means that, if there is in x* a pattern associated to bin type k with a positive value, all theinequalities (10) relating k with other bigger bins will be satisfied.

For the case (2), sincePpk

r¼1x�kr ¼ 0, the slack associated to the availability constraint of bin k will bepositive, and, by the complementary slackness conditions, v�k is equal to 0. Since the v variables in D are non-positive, for all the k 0 such that W k0 > W k, inequality v�k0 � v�k 6 W k0 � W k still holds.

The point here is that, whatever the structure of x* concerning the inclusion or not of patterns associated toa particular bin type k, we are always able to show that inequalities (10) relating k with other bigger bins aresatisfied. The same reasoning applies for any bin type k, and hence, all the inequalities (10) are satisfied by(u*,v*). h

Example 3.1. Consider an instance with a set W of bins, W = (7,7,5,4,4,4), and a set w of items,w = (3,3,3,2,2,2,1,1). A possible restricted master problem comprising the set of linearly independent dualinequalities associated to (10) ð~y1; ~y2Þ is illustrated in Fig. 1. A solution ð~x; ~yÞ with ~x1, ~x3, ~x4 and ~y1 equal toone, and the other variables equal to zero, is feasible for the extended model. However, ~x is infeasible forthe non-extended model since the limit on bins with capacity 5 is exceeded.

Additionally, two valid dual inequalities were derived that strengthen D even more. In the sequel, e is asmall positive value. The first dual inequality is as follows:

vK P W K � W 1 � e: ð11Þ

Proposition 3.3. Let ð~x�; y�Þ be an optimal solution to P e0 , the extended version of P comprising the additional

primal decision variable y* associated to the dual inequality (11). If P is feasible, then y* = 0, and ~x� is optimal forP.

Proof. First, note that, in the primal, the column associated to (11) has a positive cost equal to W1 �WK + e.Our proof will be made by contradiction. Assume that y* is positive. Hence, we must have

PpKr¼1~x

�Kr > BK , and

y� ¼PpK

r¼1~x�Kr � BK . If this was not the case, clearly, a solution would exist with a lower cost than ð~x�; y�Þ, con-

tradicting the optimality of ð~x�; y�Þ. Furthermore, for ð~x�; y�Þ, the sum of the slack variables of the availabilityconstraints must be greater than or equal to y*. Again, if this was not the case, this would imply that ð~x�; y�Þhas a cost greater than

PkW kBk contradicting the feasibility of P.

In practice, having y* > 0 means that we are using extra units of the bin type K, which are not available.According to the complementary slackness conditions, we have �v�K ¼ W 1 � W K þ e.

As mentioned, there will be a bin, larger than the smallest, with positive slack. Without loss of generality,assume that this positive slack occurs in the biggest bin (the results holds for any other bin larger than thesmallest). This implies that v�1 ¼ 0. Let Ptr = (a1Kr,a2Kr, . . . ,amKr; 0,0, . . . , 1) be one of the patterns associatedto the bin type K with a positive value in ~x�, i.e., such that

Pmi¼1u�i aiKr þ v�K ¼ W K . Such a pattern must exist.

The equalityPm

i¼1u�i aiKr ¼ W 1 þ e follows. A pattern, say r 0, associated to the biggest bin, and with the sameset of items as Ptr, is a valid pattern. This pattern has the form Ptr0 ¼ ða1Kr; a2Kr; . . . ; amKr; 1; 0; . . . ; 0Þ. Itsreduced cost, which is given by W 1 �

Pmi¼1u�i aiKr þ v�1, is equal to �e. This contradicts the optimality of

ð~x�; y�Þ. h

Fig. 1. Restricted master problem (Example 3.1).


The second dual inequality restricts the set of possible values for the v1 dual variable, i.e., the one relatedwith the biggest bin. This inequality is valid for instances with items smaller than the smallest bin, and is statedas follows:

v1 P W 1 �W 1

wm

� �W 1 � e: ð12Þ

Proposition 3.4. Let ð~x�; y�Þ be an optimal solution to P e0 , the extended version of P comprising the additional

primal decision variable y* associated to the dual inequality (12). If P is feasible, then y* = 0, and ~x� is optimal for

P.

Proof. This proof is made by contradiction. Assume that y* > 0. This implies thatPp1

r¼1~x�1r > B1 and

y� ¼Pp1

r¼1~x�1r � B1. By the complementary slackness conditions, we have also �v�1 ¼ �W 1 þ W 1

wm

l mW 1 þ e,

andPm

i¼1ai1ru�i þ v�1 ¼ W 1, for at least one pattern r associated to the bin of capacity W1. Combining both

the equations we getPm

i¼1ai1ru�i ¼ W 1

wm

l mW 1 þ e.

At least one constraint on the availability of the bins must have a corresponding slack variable that ispositive. Indeed, since P is feasible and P e0 is a relaxation of P, the following holds: zP e0 6

PKk¼1W kBk. The

value of zP e0 is given by W 1B1 þ y� W 1

wm

l mW 1 þ y�eþ

PKk¼2

Ppkr¼1x�krW k. The inequality

PKk¼2W kBk�PK

k¼2

Ppkr¼1x�krW k P y� W 1

wm

l mW 1 þ y�e follows, indicating that there is a positive slack in at least one availability

constraint.Since all the items fit in the smallest bin, all the patterns that consist in a single item i assigned to a bin k are

feasible, whatever the bin k. The corresponding dual constraints u�i þ v�k 6 W k, i = 1, . . . ,m,k = 1, . . . ,K, arealso valid for the dual problem. Since there is a positive slack variable in at least one availability constraint,say k 0, by the complementary slackness conditions, we have v�k0 ¼ 0, and u�i 6 W k0 6 W 1, for i = 1, . . . ,m.Furthermore, given that wm is the size of the smallest item, whatever the pattern (a1kr,a2kr, . . . ,amkr; . . .), the

following inequality holds:Pm

i¼1aikr 6W 1

wm

l m. Hence, for any pattern r associated to a bin with capacity W1, we

havePm

i¼1ai1ru�i 6W 1

wm

l mW 1, which contradicts the conclusion stated above that

Pmi¼1ai1ru�i ¼ W 1

wm

l mW 1 þ e. h

With the inequalities (11) and (12), the dual variables are confined to a fixed and non-empty box, which canbe defined before starting column generation. This box contains at least an optimal dual solution. In the lit-erature, there are many evidences pointing to the efficiency of column generation when this kind of boxes areused [4]. The following example shows that inequalities (11) and (12) are deep dual-optimal inequalities.

Example 3.2. Consider an instance similar to the one presented in Example 3.1, except for the demand vectorb = (5, 6,4). In the optimal solution, all the bins are used, because

Pmi¼1wibi ¼ 5� 3þ 6� 2þ 4 ¼ 31. An

optimal primal solution is x1 = 2, x4 = 1, x5 = 1 and x6 = 2.An optimal dual solution is u1 = 24, u2 = 16, u3 = 8, v1 = �49, v2 = �35 and v3 = �28, showing that

v3 P �3 � e and v1 P �42 � e, inequalities (11) and (12), respectively, for this instance, are deep dual-optimalinequalities. The alternative optimal dual solution u1 = 3, u2 = 2, u3 = 1, v1 = v2 = v3 = 0 is not cut by the deepdual-optimal inequalities.

Moving all the items from a bin to another empty bin leads to an increase of the unused space equal to thedifference between the bin capacities, which, if it is big enough, may be fulfilled with other items. However, thisoperation does not translate into valid dual inequalities. In fact, if we consider adding items’ dual variables to(10), and if we add the corresponding columns to P, the result will be a true relaxation of P with a poorerlower bound. This is illustrated by the following trivial example.

Example 3.3. Assume that two items of size 5 and 2, respectively, have to be packed into bins with capacities 5and 3, and suppose that 2 bins of each type are available. The feasible packing patterns can be denoted asfollows: (1,0;1,0)T, (0,1;1,0)T and (0,1;0,1)T. For example, the second pattern means that an item with a sizeequal to 2 is packed in a bin of capacity 5. The LP optimal solution of this problem has a cost equal to 8, andconsists in using an unit of the first pattern, and another unit of the last pattern. By adding the extra column


(0,1;1, � 1)T to the primal with a cost equal to 2 (the difference between the two capacities), the LP optimalsolution cost reduces to 7 (an unit of the first pattern, plus an unit of the extra column). In the primal, thisextra column means that a pattern related to a bin with capacity 3 can be transferred to a bin with capacity 5at a cost equal to 2. With this transfer, the unused space in the pattern is at least equal to 2 units, which isenough to accommodate the smallest item of the instance. However, translating this operation into a singleprimal column, and adding it to the LP master leads to a relaxation of the original LP.

4. Alternative aggregation schemes

In the past, aggregation and disaggregation techniques were essentially used to deal with problems comingfrom memory space restrictions. These difficulties were becoming even more constraining as researchers weretrying to solve problems with a growing level of precision. By concentrating on a restricted but sufficientlyrepresentative set of data, one will also expect to see the computational burden reduced. Rogers et al. [27] gavea comprehensive survey on the contributions made in the field, and synthesized the different elements of anaggregation/disaggregation process.

In this section, we describe two alternative aggregation schemes. The goal is to accelerate the search for anoptimal solution through column generation by solving sequences of easier approximations, i.e., aggregatedproblems. Here, aggregation is seen as an implicit form of controlling the dual variables.

The methods rely exclusively on the aggregation of items and, therefore, they can be applied to the singlesized bin-packing problem. In Section 4.1, we present a simple two-phase column generation algorithm thatstarts by solving a row aggregated model, which corresponds to a problem with larger items. In Section 4.2, aniterative algorithm is presented that takes advantage of the phenomenon of identical prices, soon pointed outby Gilmore and Gomory [15], using smaller LPs where both variables and constraints are aggregated. Theresulting doubly aggregated models give a good approximation to the disaggregated problems.

4.1. A simple row aggregation scheme

In a variable sized bin-packing problem, when the number of items that can fit in the bins is high, the LPmatrix has an high density, and this favors the occurrence of primal degeneracy. If we were able to anticipatehow some of the items are combined in the optimal solution, larger items could be defined, and density couldbe reduced. Unfortunately, it is difficult to find the right combinations and, so, we will approach this problemheuristically.

We solve the variable sized bin-packing problem by column generation in two steps. In the first, the itemsof the original problem are combined pairwise, leading to an approximation Pra of P. Since we are restrict-ing the original solution space, we get an upper bound on the optimal solution of P; its quality will be asgood as will our guess for the combination of the items. Note that the quality of the approximation willcertainly tend to decrease if greater combinations are tried. In the second step, we guarantee the convergenceto the optimal solution of P by solving P starting with the columns of the last restricted master problemrelative to Pra, properly disaggregated. We use the simple scheme RA to aggregate the items of the originalproblem.

Aggregation scheme RA

Let wra and bra be, respectively, the set of item sizes and demands of the aggregated problem. Initially,wra = ; and bra = ;.

Let b, indexed by i, represent the set of demands of the original problem, and q be the number of positivevalues in b.

i :¼ 1while i 6 m and q P 2

while bi > 0 and q P 2Let kj be the index of the smallest bin where the item i and another item, say j, fit together, i.e.,wi þ wj 6 W kj .


Let j be the index of an item of the original problem between i + 1 and m, with size wj, and bj > 0, suchthat the difference between wi + wj and W kj is minimized.if (bj P bi)

bj :¼ bj � bi, branew :¼ bi and bi :¼ 0

else

bi :¼ bi � bj, branew :¼ bj and bj :¼ 0

end if

Add size wi + wj to wra with a demand of branew units if it does not already exist or increment by bra

new itsdemand, otherwise.i :¼ i + 1

end while

end whileif bi > 0, add wi to wra with a demand of bi units if it does not already exist or increment by bi its demand,otherwise.

Clearly, the total number of items ðPjwraj

i¼1 brai Þ decreases, while the number of different types of items in the

aggregated problem will only be smaller compared to the original one if equality bj = bi holds, or if bi is equalto 0 in the last step. Since the combination of items with the same size is not allowed, the number of differenttypes of items will never increase.

An item is combined with a smaller one if the difference between the sum of their sizes and the capacity ofthe smaller bin where this combination fits is the minimum over all the possible combinations that include thefirst item. With such a criterion, if an item i goes with an item j, item i + 1 will frequently be combined with theitem j � 1, particularly if wi + wj is almost equal to wi+1 + wj�1. This type of association is frequent in optimalsolutions, in which some of the patterns differ only in a small number of items. The unused space within apattern is used to cope with the small difference between pairs of items. These pairs finally appear in very sim-ilar patterns. Another interesting point is the fact that the dual inequalities of Section 3.1 are particularly effi-cient in terms of stabilization for instances with groups of almost identical items.

The rows of P are subject to an equivalent aggregation process. The following example illustrates the stepsfollowed by the aggregation scheme.

Example 4.1. Consider an instance with a set of item sizes w = (90, 59,58,57,25,20) and demandsb = (5, 5,7,3,7,1). Ten bins with capacities 160, 120 and 100 are available. Fig. 2 illustrates a possiblerestricted master problem. The aggregation proceeds as follows. We start with wra = ; and bra = ;. Five unitsof w1 = 90 are initially aggregated to 5 other units of w5=25, leading to wra = (115) and bra = (5). The 5 unitsleft in W2 = 120, the smaller bin where an item of size 115 fits, are the smaller unused space over all the othercombinations that include w1. Subsequently, 5 items of size 59 are combined with 5 items of size 58(wra = (117, 115), bra = (5, 5)), 2 items of size 58 with 2 items of size 57 (wra = (127, 115), bra = (5,7)), 1 item ofsize 57 with 1 item of size 25 (wra = (117, 115,82), bra = (5, 7,1)), and finally 1 item of size 25 with 1 item of size20 (wra = (117, 115,82,45), bra = (5, 7,1,1)). The aggregated problem has four different types of items for atotal demand of 14 units, corresponding to half the 28 items of the original problem.

Fig. 2. Restricted master problem (Example 4.1).

Fig. 3. Row aggregated LP (Example 4.1).


The aggregation of the items leads to a similar row aggregation in P. Fig. 3 shows the result for the restrictedmaster problem of Fig. 2. A column in P will have an associated column in the aggregated model only if all theitems of the corresponding pattern can be combined among them according to the item aggregation defined byRA. Thus, the first column in Fig. 2 will have a counterpart in the aggregated LP, while the second will not. Forthe latter, the unique item of size 90 combines with the item of size 25 and the remaining item of size 25 goes withthe item of size 20. The disaggregation process is straightforward. Note that a column of the aggregatedformulation may generate one or more columns for the disaggregated problem.

Besides the sparser density of the aggregated problem, which reduces its degeneracy, this problem is alsoeasier to solve for other obvious reasons. The set of valid packing patterns is smaller compared to the originalproblem. Fewer items are needed to fulfill the bins. Their demands are also lower. As a consequence, the col-umn generation subproblems are solved faster. Moreover, the disaggregation cost is not high. The efficacy ofthe algorithm depends on how accurate is the combination of items. We are looking for a good set of initialcolumns to start the resolution of the original problem by column generation, and we expect that the effort tofind them will be essentially concentrated in the first phase where a simpler and less degenerated problem issolved. The computational experiments presented in Section 5 show that for some instances this algorithmbrings some appreciable improvements.

4.2. Implicit dual constraints: A double aggregation scheme

The optimal solutions of bin-packing problems have an interesting characteristic: items that are almostidentical have dual variables that are frequently equal. Gilmore and Gomory [15] already experienced this phe-nomenon of identical prices in various stages of the column generation algorithm, for low and high wasteproblems. For the cutting stock problem, they give an explanation based on the fact that the trim loss in somecutting patterns belonging to the basis allows some of their items to be interchanged.

In formulations in which a demand constraint is defined for each single item, there exists an optimal solu-tion with equal dual variables for the items of the same size. This result is well known. Ben Amor et al. [5]report computational results where it is shown that convergence is improved when a unique constraint peritem size is considered. What happens is that the dual variables of identical items are forced to be equal. Inthis section, we propose an algorithm that solves successive approximations obtained by imposing these equal-ity constraints to items with different sizes. From the problem definition standpoint, items with nearly thesame size are replaced by a single item with a size equal to the average size of the original items. This operationamounts to aggregating columns followed by a constraint aggregation. A column of the aggregated model willcorrespond to a set of columns with more rows in the disaggregated model. The following example illustratesthis correspondence.

Example 4.2. Remember the problem of Example 4.1. The patterns associated to x9, x10 and x11 (respectively(0,1,1,0,0,0;0,1,0)T, (0,1,0,1,0,0;0,1,0)T and (0,0,1,1,0,0;0,1,0)T) differ in a single item. There are otherpatterns that differ from the previous in a single item such as (0,2,0,0,0,0;0,1,0)T, (0, 0,2,0,0,0;0,1,0)T and(0,0,0,2,0,0;0,1,0)T. If we replace the items of size 59, 58 and 57 by an item with size 58 (the average of 59, 58and 57), and a demand of 15 units (the sum of the corresponding original demands), the pattern(0,2,0,0;0,1,0)T in the aggregated model will now stand for all the aforementioned patterns.


The aggregation scheme is now described more formally. We define an equivalence group Gl as the set ofsuccessive items i, i + 1, . . . , i + jGlj such that wi � wi+1 6 D. Let mda be the total number of equivalence groupsin the original problem for some positive value D. We have mda

6 m. Two patterns AE1 and AE2 are said to beadjacent if they are associated to the same bin type, and if AE1 is obtained from AE2, and vice versa, by replac-ing some items of AE2 each one with an item belonging to the same equivalence group. The columns of P, theoriginal formulation, are divided in groups of adjacent patterns, say AEr, r ¼ 1; . . . ; �p. The aggregated modelPda results from the aggregation of columns and rows within each set AEr defined by the following linearoperations:

T � AEr � Sr;

where T is a mda · m matrix with coefficients equal to 1 and a block diagonal structure, and Sr is a j AEr j �1column vector, r ¼ 1; . . . ; �p. There is a unit coefficient in row l and column i of T iff i 2 Gl. All the elements ofSr are equal to 1=jAErj.

The question now is which size should have the mda items of the aggregated problem, one per each originalequivalence group, such that all the columns in Pda (and no others) are feasible patterns. According to theaggregation scheme, these item sizes depend on each column of Pda and, in particular, on the original setof adjacent columns AEr in P that originate them. A way to have a unique size for each of the mda items isto make the equivalence groups coincide with the set of dual variables that are equal in an optimal dual solu-tion, and to set the sizes equal to the optimal dual values accordingly. However, an optimal dual solution israrely available beforehand. Searching for a set of sizes such that all the columns in Pda are feasible startingwith a set of predefined equivalence groups is not guaranteed to succeed. In fact, such sizes may not exist. Thealternative consists in using approximate sizes. Setting the size of the ith item, i = 1, . . . ,mda, equal to the sizeof the smallest item in Gi leads to an aggregated problem that is a relaxation of P. On the other hand, if wechoose the size of the largest item in Gi, Pda will be a restriction of P. We chose to use the first integer valuelower than the average size of the items in Gi, i = 1, . . . ,mda. The consequence is that some columns of P maynot have a representation in the resulting Pda.

Example 4.3. Consider an instance with a set of items w = (80, 70,60,55,43,38,33,20,16) to be packed in binsof capacity 100. Fig. 4 shows the complete set of feasible patterns for this instance. With D = 5, fiveequivalence groups can be defined: G1 = (80), G2 = (70), G3 = (60,55), G4 = (43,38,33) and G5 = (20,16).They lead to the seven different sets of adjacent columns identified in Fig. 4.

Figs. 5 and 6 illustrate the result of the successive column and row aggregation, respectively. The new set ofitems for the aggregated problem is wda = (80, 70,57,38,18). For the pattern (0, 0,1,1,0)T, which replaces thecolumns of AE1, the size of the item that stands for those of G4 should be 37 ((1/5 · 43 + 2/5 · 38 + 2/5 · 33)/(1/5 + 2/5 + 2/5)), while for the pattern (0,0,0,2,1)T this size should be 36.6 ((3/9 · 43 + 7/9 · 38 + 8/9 · 33)/2). By using the averages for each equivalence group, pattern (0, 0,0,0,0,0,3,0,0)T can no longer berepresented in the aggregated model (3 · 38 > 100) and the aggregation of (0,0,0,0,2,0,0,0,0)T results in apattern (0, 0,0,2,0)T that has enough waste to cope with another item. Note that it is convenient for the itemsof an equivalence group to appear in quite similar patterns. This situation is likely to appear when the itemsizes are relatively near. Suppose, for example, that we include the items of size 80 and 70 in the sameequivalence group. This can be done with D = 10. In this case, there are three equivalence groups: G1

Fig. 4. Set of feasible patterns (Example 4.3).

Fig. 5. Set of patterns after column aggregation (Example 4.3).

Fig. 6. Final set of patterns (Example 4.3).


comprises the items with sizes 80, 70, 60 and 55, G2 the items with sizes 43, 38 and 33, and G3 the items withsizes 20 and 16. The items of these equivalence groups are replaced by items with sizes 66, 38 and 18 (theaverage of the original sizes). Items of size 66 cannot combine with items of size 38 and hence, this choice willincrease the number of patterns that cannot be represented in the aggregated model.

This aggregation can be done repeatedly. A problem P can be aggregated using D = D1, leading to an aggre-gated problem P da

1 that can in turn be aggregated with D = D2, and so on. Disaggregation is applied to P dai to

recover the set of original adjacent columns of P dai�1, or P if i = 1.

To solve P, we use a n-phase algorithm that solves the sequence P dan�1, P da

n�2; . . . ; P by column generation. Alimit is imposed in the number of items in each equivalence group. Ideally, this control should be madethrough the parameters Di. The columns in the final restricted master problem of P da

i are partially disaggre-gated. For each column that is not in the basis, only one column of the original set of adjacent columns isgenerated. For those that are in the basis, we generate a subset of original columns keeping the coefficientsof the aggregated column. Therefore, if pattern (0,0,0,2,1)T in Example 4.3 is in the optimal solution, only(0,0,0,0,0,2,0,1,0)T, (0,0,0,0,0,0,2,1, 0)T, (0, 0,0,0,0,2,0,0,1)T and (0, 0,0,0,0,0,2,0,1)T will be generated.The resolution of P da

i�1 starts with this initial set of disaggregated columns. An artificial column with a highcost guarantees the feasibility of the disaggregated master problem. The nth iteration consists in solving theoriginal problem P starting with the disaggregated columns of P da

1 .

5. Computational experiments

We report the results of three groups of tests performed on randomly generated instances and other wellknown instances from the literature. We compare the effectiveness of the different approaches based on thenumber of column generation iterations and the total computing time.

A starting set of columns was computed through a FFD heuristic, where bins were filled in order of increas-ing capacities. An artificial column was added to the restricted master problem in case the heuristic does notprovide a valid solution due to the availability constraints on the bins. The knapsack subproblems were solvedusing the mt1r procedure of Martello and Toth [22]. At most one column per bin class was generated in eachiteration.

Our computational experiments were conducted on a 700 MHz Pentium III with 128 Mb of RAM underWindows ME operating system. The algorithms were coded using C++ and CPLEX 6.5 [18].

Table 1 illustrates the relative performance of the dual inequalities introduced in Section 3. Comparison ismade between the standard column generation algorithm, column generation with the additional dual

Table 1Performance of the inequalities on items’ and bins’ dual variables

m k Strategy splp % red tlp % red

168.0 5.0 Standard column generation 162.6 16.9Inequalities on items’ dual variables 132.1 18.8 14.7 13.0Inequalities on items’ and bins’ dual variables 80.0 50.8 8.7 48.5

102.6 11.0 Standard column generation 50.8 3.2Inequalities on items’ dual variables 49.5 2.6 3.5 �9.4Inequalities on items’ and bins’ dual variables 35.4 30.3 2.4 25.0

128.0 11.0 Standard column generation 72.3 7.6Inequalities on items’ dual variables 64.2 11.2 7.4 2.6Inequalities on items’ and bins’ dual variables 45.5 37.1 4.8 36.8





inequalities (9) and column generation considering inequalities (9)–(12). Dual inequalities were added prior tothe first resolution of the LP relaxation with a small perturbation of their right hand side for (9) and (10) sothat the respective columns are at the zero level in the final solution. To limit the number of columns inserted,the maximum cardinality of S has been set to jSj = 2. Only one column per item was considered when jSj = 2.The tests were carried out on a set of randomly generated problems inspired in the triplets instances from theOR-library [2]. For these instances, the solution consists in bins receiving exactly three items. The only differ-ence between the triplets instances of the OR-library and our test-problems is that a limited number of binswith various capacities are now considered. Six groups of 50 instances each were generated with m rangingfrom 100 to 180 and K from 5 to 15. Items have integer sizes between 100 and 360 while the capacities ofthe bins vary between 400 and 720.

Table 1 shows the average computing time in seconds (tlp) and the average number of column generationiterations (splp) obtained with the three alternative strategies. A percentage of reduction (% red) in time anditerations is indicated comparing the standard column generation algorithm to the other two methods.

The inequalities on items’ dual variables seem to have only a limited impact on those instances. The set ofinequalities imposed on the bins’ dual variables appears as a good complement to reach an interesting level ofstabilization. For the first group of instances, which is also the harder with an average number of iterations of162.6, the combined application of these dual inequalities yields a reduction of approximately 50% in the com-puting time and number of subproblems solved. Note that the instances were generated such that the con-straints on bins’ availability were truly effective. Indeed, dual inequalities (10)–(12) will be as strong as willthose restrictions. In practice, we will surely found many problems with this characteristic.

In Table 2, we report the average computational results obtained for a subset of the instances used in arecent publication of Belov and Scheithauer [3]. These results illustrate the impact of the two-phase algorithmbased on the aggregation scheme RA of Section 4.1 (procedure RA, for short) compared to the standard col-umn generation algorithm with and without the valid dual inequalities discussed in this paper. We chose 15groups of 50 instances each, with m = 100 and m = 150 and a number of bin types varying between 2 and 10.

With procedure RA, the reduction in the number of column generation iterations achieved with dualinequalities is almost always duplicated. For one group of instances (m = 99.40 and K = 3.80), the improve-ment reaches 36.2%. The computing time does not generally decrease in the same proportion. However, there

Table 2Performance of aggregation scheme RA


99.10 3.70 Standard column generation 135.3 11.7Dual inequalities 110.9 18.0 5.4 53.8Procedure RA + dual inequalities 86.1 36.4 5.1 56.4



99.08 3.73 Standard column generation 121.3 5.1Dual inequalities 99.0 18.4 5.3 �3.9Procedure RA + dual inequalities 78.9 35.0 5.6 �9.8



99.10 3.74 Standard column generation 128.1 2.4Dual inequalities 101.4 20.8 2.7 �12.5Procedure RA + dual inequalities 81.7 36.2 2.2 8.3


99.08 3.78 Standard column generation 123.1 4.3Dual inequalities 100.7 18.2 5.5 �27.9Procedure RA + dual inequalities 79.8 35.2 4.5 �4.7








Table 3Computational results for the best 10 instances of each group in Table 2



















exists an appreciable number of instances for which the RA procedure gave very good results. In Table 3, wecompiled the results obtained with the best 10 instances within each group. This represents 20% of the initialinstances. Surprisingly, we were able to cut between 43.3% and 83.5% of the column generation iterations and,even with the disaggregation overhead, the total computing time was reduced by up to 71.4%. In some runs ofthe column generation algorithm with dual inequalities, we observed an increase of the computing time. Thissituation was due to a limited number of instances for which the mt1r algorithm found some difficulties.

In all the experiments conducted, the n-phase algorithm of Section 4.2 (we will also refer to it as procedureDA) yields far more regular results than those achieved with procedure RA. To illustrate this conclusion, wepresent the results obtained with a set of well known bin-packing instances for which procedure RA gave onlymarginal improvements. Three problems sets are considered. The Hard28 problems consists in 28 instancesoriginally generated by Schoenfield [28]. As we can see in Table 4, the standard column generation algorithmtakes on average 532.4 iterations to solve them, which is considerable. The two remaining problem sets are thet501 and t249 triplet instances from the already referred OR-library [2].

With the instances of the Hard28 set, we were able to control the number of items in each equivalence groupusing only the parameters Di. In fact, considering a 4 iterations process, equivalence groups with more than 5items were rare. This is an a priori guarantee that the time needed to perform the disaggregation will never behigh. With the triplet instances, the situation is different. Item sizes differ only in small amounts and, for exam-ple, an aggregation with D = 1 leads to a high number of items going to the same equivalence group. Therefore,the maximum number of items per equivalence group was restricted to 5 elements. Since the number of columngeneration iterations achieved with dual inequalities is reasonably low, we tested a 2 iterations procedure.

In Tables 4–6, we present the results obtained with standard column generation, column generation usingdual inequalities on the items’ dual variables and procedure DA. In the latter, dual inequalities were also

Table 4Solution data for the Hard28 instances

Problem m Standard CG Dual inequalities Procedure DA + dual inequalities

splp tlp splp % red tlp % red splp % red tlp % red

bpp14 136 472 6.1 290 38.6 4.6 24.6 142 69.9 3.2 47.5bpp832 139 497 6.9 317 36.2 5.4 21.7 172 65.4 5.6 18.8bpp40 144 594 9.2 369 37.9 10.9 �18.5 199 66.5 8.2 10.9bpp360 148 419 5.3 207 50.6 3.4 35.8 53 87.4 1.7 67.9bpp645 141 561 8.4 344 38.7 6.2 26.2 174 69.0 4.3 48.8bpp742 148 394 4.7 229 41.9 3.9 17.0 87 77.9 3.5 25.5bpp766 143 521 7.3 363 30.3 6.3 13.7 187 64.1 5.4 26.0bpp60 144 518 7.1 292 43.6 4.8 32.4 161 68.9 4.5 36.6bpp13 161 682 12.6 413 39.4 9.5 24.6 260 61.9 9.9 21.4bpp195 161 596 10.9 446 25.2 13.7 �25.7 282 52.7 9.6 11.9bpp709 160 550 9.3 379 31.1 10.5 �12.9 182 66.9 6.9 25.8bpp785 163 679 13.2 459 32.4 10.9 17.4 251 63.0 8.9 32.6bpp47 158 476 7 220 53.8 3.9 44.3 104 78.2 3.4 51.4bpp181 157 546 8.4 363 33.5 7.1 15.5 176 67.8 5.9 29.8bpp359 164 417 5.9 252 39.6 4.9 16.9 142 65.9 4.2 28.8bpp485 163 552 9 329 40.4 6.8 24.4 166 69.9 5.7 36.7bpp640 165 396 5.3 211 46.7 4.1 22.6 53 86.6 2.4 54.7bpp716 158 383 4.8 206 46.2 3.8 20.8 86 77.5 2.9 39.6bpp119 173 629 12.5 371 41.0 9.2 26.4 274 56.4 10.9 12.8bpp144 173 548 10.4 354 35.4 9.2 11.5 185 66.2 7.3 29.8bpp561 177 598 12.4 414 30.8 11.5 7.3 273 54.3 9.9 20.2bpp781 174 606 12.5 374 38.3 10.5 16.0 167 72.4 6.5 48.0bpp900 173 632 12.6 403 36.2 10.5 16.7 168 73.4 6.3 50.0bpp175 185 528 9.3 264 50.0 6.2 33.3 154 70.8 6.6 29.0bpp178 178 608 11.7 357 41.3 9.1 22.2 248 59.2 10.4 11.1bpp419 189 673 14.5 390 42.1 11.2 22.8 201 70.1 7.9 45.5bpp531 175 443 6.7 220 50.3 4.7 29.9 55 87.6 2.4 64.2bpp814 179 388 5.7 208 46.4 4.3 24.6 68 82.5 3.0 47.4

Average 161.8 532.4 8.9 323.0 39.3 7.4 16.9 166.8 68.7 6.0 32.6


added to the aggregated master problems. In the next comments, we will denote these procedures by SCG,CG–DI and DA–DI, respectively.

Table 6Solution data for the t249 instances



t249_00 134 146 4.2 82 43.8 2.5 40.5 48 67.1 2.1 50.0t249_01 140 151 4.5 87 42.4 3.8 15.6 49 67.5 2.1 53.3t249_02 139 152 4.5 89 41.5 2.9 35.6 46 69.7 2.2 51.1t249_03 142 150 4.5 75 50.0 2.5 44.4 48 68.0 2.3 48.9t249_04 134 136 3.5 76 44.1 2.3 34.3 42 69.1 2.0 42.9t249_05 145 158 5.0 80 49.4 2.8 44.0 41 74.1 3.0 40.0t249_06 138 148 4.1 84 43.2 2.7 34.1 47 68.2 1.6 61.0t249_07 137 136 3.8 80 41.2 2.6 31.6 42 69.1 2.9 23.7t249_08 139 153 4.5 93 39.2 4.8 �6.7 43 71.9 2.0 55.6t249_09 141 155 5.0 92 40.7 3.2 36.0 41 73.5 1.8 64.0t249_10 140 146 4.0 83 43.2 2.7 32.5 49 66.4 2.4 40.0t249_11 141 149 4.3 83 44.3 2.8 34.9 38 74.5 2.2 48.8t249_12 141 146 4.2 83 43.2 2.9 31.0 42 71.2 2.0 52.4t249_13 141 153 4.6 84 45.1 2.8 39.1 42 72.5 2.0 56.5t249_14 145 157 5.1 103 34.4 3.8 25.5 54 65.6 2.3 54.9t249_15 142 145 4.3 80 44.8 2.6 39.5 49 66.2 1.9 55.8t249_16 144 151 4.6 81 46.4 2.8 39.1 43 71.5 2.2 52.2t249_17 145 158 4.9 92 41.8 3.2 34.7 43 72.8 2.3 53.1t249_18 138 144 4.1 82 43.1 2.7 34.1 48 66.7 1.6 61.0t249_19 136 137 3.6 77 43.8 2.6 27.8 41 70.1 1.9 47.2

Average 140.1 148.6 4.4 84.3 43.3 3.0 31.8 44.8 69.9 2.1 52.3

Table 5Solution data for the t501 instances



t501_00 190 173 11.3 104 39.9 7.5 33.6 44 74.6 5.3 53.1t501_01 192 152 9.6 87 42.8 6.6 31.3 38 75.0 4.3 55.2t501_02 190 174 11.7 100 42.5 8.7 25.6 43 75.3 3.6 69.2t501_03 199 169 15.7 103 39.1 14.1 10.2 55 67.5 5.7 63.7t501_04 195 168 11.4 96 42.9 12.4 �8.8 44 73.8 6.0 47.4t501_05 195 182 12.3 96 47.3 7.4 39.8 54 70.3 4.5 63.4t501_06 196 180 12.4 96 46.7 7.4 40.3 44 75.6 6.1 50.8t501_07 192 168 10.8 91 45.8 6.7 38.0 38 77.4 4.4 59.3t501_08 196 170 11.4 103 39.4 8.0 29.8 47 72.4 4.4 61.4t501_09 189 155 9.8 96 38.1 7.2 26.5 42 72.9 3.8 61.2t501_10 190 165 10.6 92 44.2 6.8 35.8 39 76.4 4.7 55.7t501_11 195 178 12.0 102 42.7 7.9 34.2 44 75.3 6.1 49.2t501_12 189 172 10.9 91 47.1 6.5 40.4 48 72.1 5.0 54.1t501_13 198 187 13.1 104 44.4 8.3 36.6 47 74.9 6.3 51.9t501_14 203 190 14.4 95 50.0 7.7 46.5 45 76.3 4.8 66.7t501_15 197 181 13.0 102 43.6 8.1 37.7 39 78.5 6.1 53.1t501_16 198 176 11.9 95 46.0 7.2 39.5 48 72.7 6.0 49.6t501_17 196 171 12.0 92 46.2 7.7 35.8 43 74.9 3.7 69.2t501_18 193 189 12.4 99 47.6 7.4 40.3 52 72.5 4.1 66.9t501_19 192 182 11.6 95 47.8 7.0 39.7 50 72.5 5.7 50.9

Average 194.3 174.1 11.9 97.0 44.3 8.0 32.8 45.2 74.0 5.0 58.0


With procedure DA–DI, the number of column generation iterations needed to reach the optimal solutiondecreases significantly. Compared to CG–DI, the percentage of reduction is approximately 50% for the threesets of instances. The number of columns in the final restricted master problem increases, but these extra col-umns are generated faster via the disaggregation processes. Indeed, the computational results show that it ismore efficient to perform even a restricted disaggregation rather than generating columns by solving a knap-sack problem, whatever the algorithm we choose. For the three sets of instances, the average computing timesdecrease with procedure DA–DI. For the Hard28 set, standard column generation needs 532.4 iterations onaverage and runs in 8.9 seconds. With CG–DI, we get the optimal solution 16.9% faster. Procedure DA–DIalmost doubles this value with 32.6% of time reduction. For the t501 set, CG–DI yields a reduction of 32.8%,while procedure DA–DI improves this value in 25.2%. The maximum reduction in the computing time isachieved for the instance t501_02 with 69.2%. For this instance, procedure CG–DI yields a reduction of25.6%. With problems growing in size and complexity, this tendency of time reduction will surely become evenmore significant.

6. Conclusions

We analyzed different strategies to stabilize and accelerate column generation in the context of variablesized bin-packing problems. Some of the prescribed methods are directly applicable to the standard bin-pack-ing problem, the well known special case in which only one bin type is available.

New weak and deep dual-optimal inequalities were introduced. We showed that they are a good comple-ment to the dual inequalities imposed on items’ dual variables. Their number is limited and, so, we can nor-mally enumerate them all before the resolution of the first master problem.

Additionally, we explored the idea of aggregation to control the progress of the dual variables and thusaccelerate the resolution of the master problems. Procedure RA, despite its extreme simplicity, gave very inter-esting results but was outperformed by the more sophisticated n-phase algorithm of Section 4.2. This latterforces sets of dual variables to be equal during its various stages. The respective approximations are solvedby column generation with a reduced tailing off, which leads to an overall process with better convergenceproperties.

Other approaches have been proposed with the purpose of accelerating the column generation algorithm,such as the procedure proposed by Degraeve and Peeters [7], which is based on a simplex method/subgradient-optimization algorithm to solve the LP relaxation of the cutting stock problem. Our approaches can be com-bined with their algorithm. In fact, we believe that the fastest procedure should be the one that combines them.

To evaluate the performance of the approaches, extensive computational experiments were carried out onvarious sets of randomly generated instances and other instances found in the literature. The appreciableimprovements we achieved allow us to claim the effectiveness of these strategies.

Acknowledgements

We thank two anonymous referees for their constructive comments, which led to a clearer presentation ofthe material. We also want to thank Gleb Belov and Guntran Scheithauer for providing us the Hard28 set.

This work was partially supported by the portuguese Science and Technology Foundation (Projecto POSI/1999/ SRI/35568) and by the Algoritmi Research Center of the University of Minho, and was developed in theIndustrial and Systems Engineering Group.

References

[1] C. Alves, J.M. Valerio de Carvalho, A stabilized branch-and-price-and-cut algorithm for the multiple length cutting stock problem,Working Paper, Universidade do Minho, Portugal, 2006.

[2] J.E. Beasley, OR-library: Distributing test problems by electronic email, Journal of the Operational Research Society 41 (1990) 1069–1072.

[3] G. Belov, G. Scheithauer, A cutting plane algorithm for the one-dimensional cutting stock problem with multiple stock lengths,European Journal of Operational Research 141 (2002) 274–294.

[4] H. Ben Amor, Stabilisation de l’algorithme de generation de colonnes, PhD Thesis, Ecole Polytechnique de Montreal, Canada, 2002.


[5] H. Ben Amor, J. Desrosiers, J.M. Valerio de Carvalho, Dual-optimal inequalities for stabilized column generation, OperationsResearch, in press.

[6] C. Chu, R. La, Variable-sized bin packing: Tight absolute worst-case performance ratios for four approximation algorithms, SIAMJournal on Computing 30 (6) (2001) 2069–2083.

[7] Z. Degraeve, M. Peeters, Optimal integer solutions to industrial cutting-stock problems: Part 2, Benchmark results, INFORMSJournal on Computing 15 (1) (2003) 58–81.

[8] J. Desrosiers, M. Luebbecke, Selected topics in column generation, Technical Paper, Les Cahiers du GERAD G-2002-64, Canada,2002.

[9] O. du Merle, J. Goffin, J. Vial, On improvements to the analytic center cutting plane method, Computational Optimization andApplications 11 (1998) 37–52.

[10] O. du Merle, D. Villeneuve, J. Desrosiers, P. Hansen, Stabilized column generation, Discrete Mathematics 194 (1999) 229–237.[11] H. Dyckhoff, A typology of cutting and packing problems, European Journal of Operational Research 44 (1990) 145–159.[12] D. Friesen, M. Langston, Variable sized bin packing, SIAM Journal on Computing 15 (1986) 222–230.[13] M. Garey, D.S. Johnson, Computers and Intractability. A Guide to the Theory of NP-Completeness, W.H. Freeman and Company,

1979.[14] P.C. Gilmore, R.E. Gomory, A linear programming approach to the cutting stock problem, Operations Research 9 (1961) 849–859.[15] P.C. Gilmore, R.E. Gomory, A linear programming approach to the cutting stock problem – Part ii, Operations Research 11 (1963)

863–888.[16] J.-B. Hiriart-Urruty, C. Lemarechal, Convex analysis and minimization algorithms II: Advanced theory and bundle methods,

Springer-Verlag, Berlin, 1993.[17] O. Holthaus, Decomposition approaches for solving the integer one-dimensional cutting stock problem with different types of

standard lengths, European Journal of Operational Research 141 (2002) 295–312.[18] ILOG, CPLEX 6.5, User’s Manual, 1999.[19] B. Kallehauge, J. Larsen, O. Madsen, Lagrangean duality applied on vehicle routing with time windows, Technical Paper, Technical

University of Denmark, Informatics and Mathematical Modelling IMM-TR-2001-9, Kgs Lyngby, Denmark, 2001.[20] L. Kantorovich, Mathematical methods of organising and planning production (translated from a report in Russian, dated 1939),

Management Science 6 (1960) 366–422.[21] R. Marsten, W. Hogan, J. Blankenship, The Boxstep method for large-scale optimization, Operations Research 23 (1975) 389–405.[22] S. Martello, P. Toth, Knapsack Problems: Algorithms and Computer Implementations, Wiley, New York, 1990.[23] M. Monaci. Algorithms for packing and scheduling problems, PhD Thesis, Universita di Bologna, 2002.[24] F. Murgolo, An efficient approximation scheme for variable-sized bin packing, SIAM Journal on Computing 16 (1987) 149–161.[25] G.L. Nemhauser, L.A. Wolsey, Integer and Combinatorial Optimization, Wiley, New York, 1988.[26] G. Roodman, Near optimal solutions to one-dimensional cutting stock problem, Computers and Operations Research 13 (1986) 713–

719.[27] D. Rogers, R. Plante, R. Wong, J. Evans, Aggregation and disaggregation techniques and methodology in optimization, Operations

Research 39 (1991) 553–582.[28] J.E. Schoenfield, Fast, exact solution of bin-packing problems without linear programming, Technical Paper, US Army Space and

Missile Defense Command, Alabama, USA, 2002.[29] J.M. Valerio de Carvalho, Exact solution of bin-packing problems using column generation and branch-and-bound, Annals of

Operations Research 86 (1999) 629–659.[30] J.M. Valerio de Carvalho, LP models for bin-packing and cutting stock problems, European Journal of Operational Research 141

(2002) 253–273.[31] J.M. Valerio de Carvalho, Using extra dual cuts to accelerate convergence in column generation, INFORMS Journal on Computing

17 (2) (2005) 175–182.[32] P. Vance, Branch-and-price algorithms for the one-dimensional cutting stock problem, Computational Optimization and

Applications 9 (1998) 211–228.[33] D. Villeneuve, J. Desrosiers, M. Lubbecke, F. Soumis, On compact formulations for integer programs solved by column generation,

Technical Paper, Les Cahiers du GERAD G-2003-6, Canada, 2003.[34] G. Wascher, H. Haussner, H. Schumann, An improved typology of cutting and packing problems, European Journal of Operational

Research, this issue, doi:10.1016/j.ejor.2005.12.047.

http://dx.doi.org/10.1016/j.ejor.2005.12.047

accelerating column generation for variable sized bin-packing problems

Documents