Pivot-and-reduce cuts: An approach for improving Gomory mixed-integer cuts

European Journal of Operational Research 214 (2011) 15–26

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Discrete Optimization

Pivot-and-reduce cuts: An approach for improving Gomory mixed-integer cuts Franz Wesselmann ⇑, Achim Koberstein, Uwe H. Suhl Decision Support & Operations Research Lab, Universität Paderborn, Warburger Straße 100, 33098 Paderborn, Germany

a r t i c l e

i n f o

Article history: Received 17 December 2008 Accepted 13 April 2011 Available online 22 April 2011 Keyword: Integer programming Cutting planes Gomory mixed-integer cuts

a b s t r a c t Gomory mixed-integer cuts are of crucial importance in solving large-scale mixed-integer linear programs. Recently, there has been considerable research particularly on the strengthening of these cuts. We present a new heuristic algorithm which aims at improving Gomory mixed-integer cuts. Our approach is related to the reduce-and-split cuts. These cuts are based on an algorithm which tries to reduce the coefficients of the continuous variables by forming integer linear combinations of simplex tableau rows. Our algorithm is designed to achieve the same result by performing pivots on the simplex tableau. We give a detailed description of the algorithm and its implementation. Finally, we report on computational results with our approach and analyze its performance. The results indicate that our algorithm can enhance the performance of the Gomory mixed-integer cuts. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction In recent years the performance of software packages for solving mixed-integer programs (MIPs) has increased considerably. While instances with ten thousands of integer variables were impossible to solve a few years ago, today many of them can rapidly be solved using standard software. The improvement of the state-of-the-art becomes especially clear by inspecting the development of the MIPLIB library. Many instances from MIPLIB 2.0 or MIPLIB 3.0 can today be solved to proven optimality within only a few seconds. There are multiple reasons for the gain in performance which the last twenty years brought about. Besides improvements in hardware technology (e.g. multi core technology) and linear programming codes, enhanced cutting plane techniques enable MIP solvers such as CPLEX (ILOG Inc., 2010) or MOPS (Suhl, 1994) to solve large-scale real-world MIPs. For an overview of recent improvements in LP technology see Koberstein (2008), Koberstein and Suhl (2007). The importance of cutting planes in solving MIPs is for instance discussed by Bixby and Rothberg (2007). MIP models from practical applications keep growing in size and complexity along with the performance of MIP solvers. This trend creates a constant need for further theoretical and implementational advances. Gomory (1960) proposed the Gomory mixed-integer (GMI) cuts in the early 1960s. For more than thirty years these cutting planes were considered useless in practice for several reasons. The development of the lift-and-project method (Balas et al., 1993, 1996a) ⇑ Corresponding author. Tel.: +49 5251 60 2385; fax: +49 5251 603542. E-mail addresses: [email protected] (F. Wesselmann), [email protected] (A. Koberstein), [email protected] (U.H. Suhl). 0377-2217/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.04.009

led to a revival of the GMI cuts (Balas et al., 1996b) and they became an important factor in state-of-the-art MIP solvers. An overview of the history of Gomory cuts and the development that led to their rediscovery is given by Cornuéjols (2007). Furthermore, other families of cutting planes such as cover, flow cover or mixedinteger rounding cuts were integrated into MIP solvers and led to further computational progress. In this paper we present a new heuristic algorithm for obtaining improved Gomory mixed-integer cuts. This algorithm is based on the work of Andersen et al. (2005) and tries to reduce the size of the coefficients of the continuous variables in a simplex tableau row by performing a sequence of pivots. We give a detailed description of the implementation incorporated in the MIP solver MOPS. Finally, we present computational results reflecting the effectiveness of our approach. The MOPS MIP solver (Suhl, 1994) is a high performance system for solving large-scale linear and mixed-integer programming problems. The current version of MOPS features state-of-the-art primal simplex, dual simplex and interior point algorithms to solve linear programs. Moreover, the system uses sophisticated heuristics, branch-and-bound (branch-and-cut) algorithms and cutting plane techniques to solve MIPs. We consider mixed-integer programs of the form

min cT x s:t: Ax ¼ b

ð1Þ

xP0 xj 2 Z;

j 2 NI ;

where c; x 2 Rn ; A 2 Rmn ; b 2 Rm and NI # N = {1, . . . ,n}. The linear programming (LP) relaxation of (1) is obtained by dropping the

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F. Wesselmann et al. / European Journal of Operational Research 214 (2011) 15–26

integrality requirements on the integer variables xj for all j 2 NI. A basis B of the LP relaxation is a set of m linearly independent columns of A. We denote by x⁄ the associated basic solution. Moreover, let J = NnB index the non-basic variables, i.e. the remaining columns of A. The rows of the simplex tableau associated with a basis B read

X

xi ¼ xi 

ij xj ; a

i 2 B:

ð2Þ

j2J

xi þ dxk ¼ xi þ dxk 

X ij þ da kj Þxj : ða

ð7Þ

j2J

The procedure aims at reducing the coefficients of the non-basic continuous variables j 2 JnNI. The integer d is thus chosen such that the function

f ðdÞ ¼

X

ij þ da kj Þ2 ða

ð8Þ

j2JnN I

2. Gomory mixed-integer cuts Gomory mixed-integer cuts are generated from rows of the simplex tableau associated with basic integer variables which take fractional values in the basic solution to the LP relaxation. The GMI cut generated from a row of the simplex tableau (2) associated with variable xi where i 2 B \ NI and xi R Z is

X

X

fij xj þ

j2J\N I :fij 6fi0

j2J\NI :fij >fi0

X

þ

ij <0 j2JnNI :a

X fi0 ð1  fij Þ ij xj a xj þ 1  fi0  P0 j2JnN :a I

ij Þ fi0 ða xj P fi0 ; 1  fi0

3.1. Implementation of the reduction algorithm

ij

ð3Þ

ij  ba ij c and fi0 ¼ xi  bxi c > 0. This cut can also be dewhere fij ¼ a rived as an intersection cut (see Balas, 1971) from the basis B and a split disjunction

ðpT x 6 p0 Þ _ ðpT x P p0 þ 1Þ;

ð4Þ

nþ1

where ðp; p0 Þ 2 Z and pj = 0 for all j 2 NnNI (see Cook et al., 1990). The aim of this paper is to present a procedure to strengthen GMI cuts, i.e. to improve their quality. One way to measure the quality of a GMI cut is to compute the Euclidean distance between x⁄ and its orthogonal projection on the cut hyperplane. This value is also referred to as the distance cut off. The distance cut off by a GMI cut aTx P b is given by

b  aT x b ¼ ; kak kak

ð5Þ

where aTx⁄ = 0 as we assume x⁄ to be a basic solution. To enhance the distance cut off, we can either try to increase the fractional part of the right-hand side of a tableau row (numerator) or decrease the size of the coefficients in the GMI cut (denominator). Note also that b represents the violation of the GMI cut. The coefficients of the integer-constrained variables in a GMI cut are in the interval [0, 1] while coefficients on continuous variables are not bounded and depend on the size of the coefficients in the corresponding row of the simplex tableau. 3. Reduce-and-split cuts Andersen et al. (2005) developed an approach to strengthen GMI cuts which is based on the observation that the size of the coefficients of the non-basic continuous variables in a GMI cut has a crucial impact on the quality of the cut (see equation (5)). Their idea is to reduce the size of these coefficients by forming integer linear combinations of the rows of the simplex tableau. Consider two such rows

xi ¼ xi 

X

ij xj ; a

xk ¼ xk 

Our implementation of the R&S reduction algorithm is as follows. First a matrix which consists of the rows of the simplex tableau associated with the basic integer-constrained variables is constructed. Each of these rows is then iteratively replaced by a new row which has smaller coefficients on the continuous variables by forming integer linear combinations. We implemented a slight variation of the R&S cuts. Specifically, we do not perform a strengthening of the split disjunction as proposed by Andersen et al. Instead, we directly generate GMI cuts from each of the reduced tableau rows of the constructed matrix. Finally, note that by restricting d to be integer we ensure that any R&S cut aTx P b with b R Z is violated. Specifically, the combined tableau rows generated by the R&S reduction algorithm have integral coefficients on the basic integer variables, and consequently these variables have zero coefficients in the R&S cuts (i.e. aTx⁄ = 0). If non-integral multipliers d are used the R&S cuts may have non-zero coefficients on the basic integer variables and may thus not be violated even though b R Z. 4. Pivot-and-reduce cuts The Balas–Perregaard procedure (Balas and Perregaard, 2003) for generating lift-and-project (L&P) cuts guides the search for an LP basis where the tableau row associated with a specific basic variable gives a stronger GMI cut than the corresponding row of the optimal LP tableau. In contrast, the R&S reduction algorithm (Andersen et al., 2005) tries to improve the disjunction on which a GMI cut is based by forming integer linear combinations of LP tableau rows while keeping the basis fixed. A natural question is whether it is possible to integrate both approaches in order simultaneously to improve the basis and the disjunction. We approach this question by generating GMI cuts from alternative bases of the LP relaxation. These bases need be neither optimal nor feasible. Consider a row of the simplex tableau (2) associated with a basic integer variable xi with i 2 B \ NI. At each iteration of our algorithm we perform a pivot in row k – i, which produces a linear combination

xi þ cxk ¼ xi þ cxk 

ð6aÞ

j2J

X

is minimized. Andersen et al. (2005) show that f is a quadratic convex function in d and that its minimum can be found efficiently. After a tableau row has been reduced in the above manner, Andersen et al. perform a strengthening of the underlying split disjunction and derive the intersection cut associated with the given basis and the strengthened split disjunction. Andersen et al. call the resulting cuts reduce-and-split (R&S) cuts.

kj xj ; a

ð6bÞ

where i, k 2 B \ NI. To improve the GMI cut generated from tableau row (6a), the latter is combined with tableau row (6b) by adding d 2 Z times (6b) to (6a). We obtain

 ij þ ca kj xj ; a

ð9Þ

j2J  a

j2J

X

kp – 0 such that the squared where c ¼  a ip for some p 2 J with a kp Euclidean norm of the coefficients of the continuous variables decreases. We want to obtain a basis where the representation of the simplex tableau row corresponding to xi is ‘‘better’’ with respect to size of the coefficients of the continuous variables. We thereby also modify the underlying split disjunction (see Andersen et al., 2005). Suppose that the pivot simulated in

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equation (9) makes the integer-constrained variable xk non-basic at its lower bound and the integer-constrained variable xp basic. ij þ We have B:¼(B [ {p})n{k} and J:¼(J [ {k})n{p}. Define fij ¼ a cakj  baij þ cakj c for j 2 J and fi0 ¼ xi þ cxk  bxi þ cxk c. We obtain a new split disjunction pTx 6 p0 or pTx P p0 + 1 where p is given by

pj

 8 ij þ ca kj a > >   > < kj ij þ ca a ¼ >1 > > : 0

if j 2 J \ N I and f ij 6 fi0 ; if j 2 J \ N I and f ij > fi0 ; if j ¼ i;

ð10Þ

otherwise

and p0 ¼ bxi þ cxk c. In particular this means that pp = 0 and that pk either has the value bcc or dce. In the following we discuss our pivoting algorithm in detail. As mentioned above, the objective of our algorithm is to reduce the squared Euclidean norm on the continuous variables in the selected reference row i of the simplex tableau. To state this objective more formally, we introduce some additional notation. Given k 2 B, Ck : define the vector a

8 kj > : 0

if j 2 J n N I ; if j ¼ k and j 2 B n NI ;

ð15Þ

lRJ[fig

The second important step of the procedure is choosing a variable xp which enters the basis. Again the critical question is how to select this variable. Remember that, given a pair (i, k) of rows of the simplex tableau, we want to minimize equation (12). Therefore the effect that pivoting the non-basic variable xp into the basis has on the size of the coefficients of the continuous variables can be measured  a kj – 0g be the by computing f(cp) where cp ¼  a ip . Let J0 ¼ fj 2 J : a kp set of admissible pivots in tableau row k. From all of these pivots we select the one which brings about the largest possible reduction

  ij a : p ¼ arg min f ð c Þ : c ¼  j j kj a j2J 0

ð16Þ

After the leaving and entering variable have been selected the pivot is performed. The method is iterated until a pivot limit is reached or no improving reduction is found. We call the resulting cuts pivotand-reduce (P&R) cuts. 4.1. Implementation of the pivoting procedure

We want to select a basic variable xk and a non-basic variable xp kp – 0 minimizes such that pivoting on the element a

  C T  C  Ck a  i þ ca Ck ¼ ka Ck ;  i þ ca Ck k2 þ 2c ða Ci ÞT a Ci k2 þ c2 ka f ðcÞ ¼ a ð12Þ  a  a ip . kp

where c ¼ It follows that a pivot reduces the squared Euclidean norm if the inequality

ð13Þ

holds. Observe that the right-hand side of inequality (13) is nonnegative. Thus there is no improving pivot in row k with respect to the of the coefficients of the continuous variables in row i  size Ck P 0. Ci ÞT a if c ða Ci of coefficients of the continuous variables in Given the vector a Ck and the scalar c which minimize the reference row, the vector a (12) can be identified by computing the zeros of the first derivaCk and c. However, our algorithm pertives of (12) with respect to a forms pivots on the simplex tableau and can, therefore, only Ck and scalars choose from a limited number of possible vectors a c. Moreover, the choices of a pivot row and pivot column are not independent because every row of the simplex tableau only contains a limited number of possible pivots. Since evaluating every possible pivot row and pivot column is computationally expensive, we develop a heuristic strategy to select them. The first main step of the algorithm is to identify a variable xk which leaves the basis, i.e. a pivot row k R J [ {i} which we combine our reference row i with. The heuristic we use to select this row measures the similarity between the reference row and possible pivot rows in terms of the coefficients of the continuous variables. Specifically, we compute the absolute value of the cosine of the anCi and a Ck gle between the vectors a



C T C

 C C  ða i Þ ak

i ; a k ¼   h a a Ci ka Ck k

C C i ; a l Þ : k ¼ arg max hða

ð11Þ

otherwise:

 Ck > c2 ka Ci ÞT a Ck k2 2c ða

uous variables (see inequality (13)). Therefore the pivot row k – i is chosen such that

ð14Þ

 C C Ci ; a Ck Þ 6 1. The larger h a i ; a k is, for each k R J [ {i}. Note that 0 6 hða the more similar are the vectors with respect to the coefficients of the continuous variables. On the other hand, if we have  C C  C T C i ; a k ¼ 0), every possible pivot will not be k ¼ 0 (i.e. a i a h a improving with respect to the size of the coefficients of the contin-

In this section we discuss our implementation of the previously discussed pivoting procedure. There are a few aspects one has to consider to implement this procedure efficiently. We first perform an initialization of the data structures. We then enter the main loop of the procedure and check whether the termination criteria are satisfied. If the pivot limit is reached, we exit the main loop and generate a P&R cut. Otherwise the algorithm computes the reference row of the simplex tableau. We stop the search for an improving pivot if the coefficients of the continuous variables in the reference row i are relatively small, i.e. if the Ci k2 6  with  = 105. norm satisfies ka Given a basic integer variable xi, we have to identify a pivot row k R J [ {i} which is likely to contain pivots which reduce the size of the coefficients on the continuous variables in row i. We want to choose the index k that maximizes the value of equation (14). It turns out that the numerator of equation (14) can be computed very efficiently. We get

Ck ¼ Ci ÞT a kk ¼ ða

X j2JnNI

¼

kj a ij ¼ a

X

A1 B AJ

j2JnNI

kj

ij a

X  1 X ij ; ij ¼ A1 ABðkÞ ðAJ Þj a ðAJ Þj a BðkÞ j2JnN I

ð17Þ

j2JnN I

where B(k) is the position of xk in the basis B, A1 BðkÞ is the row of the basis inverse associated with xk and (AJ)j is the jth non-basic column of A. To obtain the vector k, i.e. the inner product for all candidate rows k R J [ {i}, we have to solve the system

AB k ¼

X

ij ðAJ Þj a

ð18Þ

j2JnNI

for k. Fortunately, solving this system is a standard operation in state-of-the-art simplex engines. It involves one forward transformation (or ftran) operation which can be carried out in a highly efficient manner. The denominator of equation (14) cannot be calculated so efficiently. The Euclidean norm of the tableau row associated with xi is of course a constant in all calculations. On the other hand, the Euclidean norms of the remaining rows of the simplex tableau may change after each pivot. Therefore there are two ways to compute the norms for all rows k R J [ {i}. Firstly, one can construct each tableau row in order to calculate its Euclidean norm. This is a computationally very expensive operation. Secondly, one can

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F. Wesselmann et al. / European Journal of Operational Research 214 (2011) 15–26

initially compute the squared Euclidean norm for all rows of the simplex tableau and then update them after each pivot using equation (12). This is also quite costly from a computational point of view. Accordingly, we only compute the numerator for all candidates and select k such that

o n

 C T C

k ¼ arg max ða i Þ al : lRJ[fig

ð19Þ

Once a leaving variable xk with k – i has been selected, we compute the associated row of the simplex tableau and the Euclidean norm Ck . Suppose that we have computed the index k of the vector a according to the above formula. The algorithm is stopped if the inner product with the largest absolute value is sufficiently small. More precisely, the algorithm is stopped if



C T C

ðai Þ ak 6 ;

ð20Þ

where  = 5104 in our code. The second main step of the procedure is selecting the variable which will enter the basis. It can be implemented in a straightforward way. As elaborated above, we select the entering variable xp from row k such that the resulting pivot will produce the largest possible reduction in the coefficients on the continuous variables in row i. In our current implementation the algorithm does not, however, check whether the P&R cut derived from the updated row i obtained after the pivot is violated by the optimal LP solution kj which are smaller than the pix⁄. We reject pivots on elements a vot tolerance 105. The overall computational effort necessary to select the entering and leaving variable is comparable to the work carried out by the Balas–Perregaard method for generating L&P cuts. But the L&P algorithm has to maintain a lot of data, e.g. the value of the objective function of the cut generating linear program and the partition of the non-basic variables (see Balas and Bonami, 2009). An advantage of the P&R heuristic is that it only has to retain the Ci and a Ck for each values of the inner products between the vectors a possible pivot row k R J [ {i} (see equation (19)). Moreover, these inner products are the direct result of an ftran operation and can therefore be scanned and handled very efficiently, e.g. by sorting according to descending absolute values. After a pivot has been performed no additional work is necessary for updates of existing data. 5. Computational experiments In this section we report on computational experiments we performed with the previously discussed approaches to strengthen GMI cuts. The computations were carried out on 107 MIP instances taken from the Mittelmann MIP collection (Mittelmann, 2010), MIPLIB 3.0 (Bixby et al., 1998), MIPLIB 2003 (Achterberg et al., 2006) and the CORAL library (Linderoth and Ralphs, 2005). All tests runs were conducted on an Intel Core 2 Duo PC with 3.16 GHz and 8 GB of main memory. We only discuss cumulative results in this section while the detailed results can be found in Tables 4–6. Given a series of observed values we compute the arithmetic mean, the geometric mean and the shifted geometric mean (see Achterberg, 2009). The shifted geometric mean is a variant of the geometric mean which is parameterized by the shifting parameter s in order to decrease the influence of very small values on the geometric mean. Given the mean values mA and mB obtained with two configurations A and B we calculate the ratio mB/mA  1 in order to compare the performance of these configurations. Note that when mA and mB are arithmetic means the ratio between them is just the ratio between the sums of the observed values.

Table 1 Properties of the matrices generated by the reduction algorithms. The values represent percentage changes in the mean values compared with the original unmodified matrices. R&S algorithm

arithm. mean geom. mean shifted geom. mean

P&R algorithm

cont.

int.

cont.

int.

100 72 62

+1 +99 +86

97 88 68

0 25 28

5.1. Performance of the reduction algorithms We first analyze the properties of the rows generated by the R&S and the P&R algorithm. As described above, the R&S algorithm produces a matrix whose rows are integer linear combinations of the rows of the optimal simplex tableau which are associated with basic integer variables. To compare the P&R algorithm with the R&S algorithm, we also construct a matrix which consists of the rows of the simplex tableau obtained by applying the P&R pivoting procedure. Observe that each of the tableau rows generated by the latter algorithm may be associated with a different basis. To obtain comparable results we exclude those 10 instances from our analysis where the R&S algorithm is unable to save the rows of the simplex tableau due to its size.1 To measure the performance of the reduction algorithms, we calculate the size of the reduced matrices by summing up the squared norms of the rows on the continuous and integer variables respectively. We then compute the mean size of the matrices generated by the R&S and P&R reduction algorithms respectively and the mean size of the matrices containing the original rows of the respective optimal simplex tableaus (i.e. the matrices before applying the algorithms). We set the shifting parameter s = 100 to compute the shifted geometric means. Table 1 shows the percentage changes in the mean values calculated for the R&S and P&R algorithms relative to the mean values obtained if no reduction algorithm is applied. The detailed results can be found in Table 4. The columns of Table 1 headed ‘‘cont.’’ show the changes in the size of the matrices in terms of the coefficients of the continuous variables. When the R&S algorithm is applied the geometric mean of the sums of the squared norms of the rows on the continuous variables decreases by 72% while the shifted geometric mean decreases by 62%. Concerning the P&R algorithm a decrease in the geometric mean of 88% can be observed whereas the shifted geometric mean decreases by 68%. The arithmetic mean of the sizes of the matrices decreases by about 100% for both reduction algorithms. The P&R algorithm is thus competitive with the R&S algorithm with respect to the reduction in the coefficients of the continuous variables in the tableau rows. Information on the size of the coefficients on the integer variables is not considered in both algorithms. An interesting question is thus how the reduction algorithms affect the coefficients of the integer variables. The percentage changes in the size of the matrices in terms of the coefficients of the integer variables can be found in the columns ‘‘int.’’ of Table 1. The geometric mean of the sums of the squared norms of the rows on the integer variables increases by 99% and the shifted geometric mean by 86% when the R&S algorithm is applied. Concerning the P&R algorithm the geometric mean decreases by 25% and the shifted geometric mean by 28%. The arithmetic mean does not change significantly for both algorithms.

1 neos-1367061, neos-503737, neos-824695, neos-827175, neos-933638, neos-933966, neos-935769, neos-936660, neos-937446, neos-937511

19

F. Wesselmann et al. / European Journal of Operational Research 214 (2011) 15–26 Table 2 Properties of the cutting planes generated by the reduction algorithms. The values represent percentage changes in the mean values compared with the GMI cuts. R&S cuts

arithm. mean geom. mean shifted geom. mean

P&R cuts

#cuts

#nz

cont. coef.

int. coef.

violation

distance

#cuts

#nz

cont. coef.

int. coef.

violation

distance

43 45 41

5 18 15

96 41 34

41 2 3

0 +3 +2

+2 +17 +13

31 32 31

18 33 28

47 41 41

36 8 7

+14 +9 +9

+5 +46 +36

Table 3 Performance of R&S and P&R cuts in a cut-and-branch scheme. The values represent percentage changes in the mean values compared with a reference setting in which only GMI cuts are applied. R&S cuts

P&R cuts

GMI + R&S cuts

GMI + P&R cuts

time

arithm. mean geom. mean shifted geom. mean

+3 1 +1

+9 +4 +1

+3 0 1

21 8 11

nodes

arithm. mean geom. mean shifted geom. mean

+7 12 7

+35 10 7

10 18 13

22 31 26

gap cl.

arithm. mean geom. mean shifted geom. mean

2 10 7

1 10 7

+4 +6 +6

+11 +14 +13

We next analyze the properties of the cutting planes generated by the R&S and P&R reduction algorithms. We examine the size of the coefficients of integer and continuous variables in the R&S and P&R cuts. We also investigate the quality of R&S and P&R cuts in terms of violation and distance cut off. For each instance in our test set we record the number of GMI cuts generated from the initial optimal basis of the LP relaxation. We compute the average number of non-zero coefficients and the average coefficient of an integer and a continuous variable in a GMI cut. We also compute the average violation and the average distance cut off by a GMI cut. We perform analogous calculations for the R&S and P&R cuts. We then use these averages to compute mean values over all instances. To calculate the shifted geometric means we set s = 10 for the number of cuts and the average number of non-zeros, s = 0.1 for the average coefficient of an integer and continuous variable and for the average violation and s = 0.01 for the average distance cut off. To increase comparability, we exclude instances for which our code could not generate GMI, R&S or P&R cuts from the computation of the mean values.2 In these cases cuts were rejected due to density or numerical issues. We also do not use (combined) tableau rows for cut generation whose right-hand sides are almost integral. Recall that the fractional part of the right-hand side gives the violation of the resulting GMI cut. Table 2 shows the percentage changes in the mean values calculated for the R&S and P&R cuts relative to the mean values obtained for the GMI cuts. The detailed results can be found in Table 5. Table 2 shows that our code generates fewer R&S and P&R cuts than GMI cuts (see columns ‘‘#cuts’’). The geometric mean of the number of generated cutting planes reduces by 45% for the R&S cuts and by 32% for the P&R cuts. One possible explanation for the reduction in the number of cutting planes lies in the fact that the R&S and P&R reduction algorithms do not control the violation of the generated cuts. If the right-hand side of a row generated by one of the reduction algorithms is only slightly fractional, we do not use this row for cut generation. Table 2 also shows that the R&S and P&R cuts contain fewer non-zero coefficients than the GMI cuts (see columns ‘‘#nz’’). The geometric mean of the average 2 blend2, neos-1211578, neos-1330635, neos-1367061, neos-1396125, neos-1489999, neos-555694, neos-799711, neos-799716, neos-827175, neos-863472, neos-880324, pk1, rentacar

number of non-zero elements in the R&S cuts is 18% smaller than in the GMI cuts. The shifted geometric mean reports on a reduction in the average number of non-zero elements of 15%. Concerning the P&R cuts, the geometric mean of the average number of nonzero elements reduces by 33% and the shifted geometric mean by 28% respectively. Moreover, Table 2 reveals that the R&S and P&R reduction algorithms are successful in producing cutting planes which have smaller coefficients on the continuous variables (see columns ‘‘cont. coef.’’). The geometric mean of the average sizes of a non-zero coefficient of a continuous variable in a R&S cut is by 41% smaller than in a GMI cut while the shifted geometric mean reduces by 34%. Concerning the P&R cuts both the geometric and the shifted geometric mean indicate a reduction of the average non-zero coefficient of a continuous variable of 41%. With respect to the reductions of the coefficients of continuous variables both algorithms thus produce similar results in terms of the geometric and shifted geometric means. On the other hand, both reduction algorithms do not control the size of the coefficients of integer variables. The coefficients of integer variables in a GMI cut of the form (3) are bounded in the interval [0, 1]. The coefficients of integer variables thus have a smaller range than the coefficients of continuous variables. However, a GMI cut written in the space of the structural variables (i.e. after projecting away slack or surplus variables) may have arbitrary coefficients on the integer variables. The columns of Table 2 headed ‘‘int. coef.’’ indicate slight reductions in the average coefficients of the integer variables in the R&S and P&R cuts. The shifted geometric mean of the average sizes of the coefficient of an integer variable reduces by 3% for the R&S cuts and by 7% for the P&R cuts. Finally, we examine the quality of the cutting planes generated by the R&S and P&R reduction algorithms in terms of violation and distance cut off (see columns ‘‘violation’’ and ‘‘distance’’). While both algorithms do not control the violation of the generated cuts, the shifted geometric mean of the average violation of a R&S and P&R cut is 2% and 9% respectively larger than the violation of the GMI cuts. On the other hand, the distance cut off by R&S and P&R cuts is clearly larger than the distance cut off by GMI cuts. The shifted geometric mean of the average distance cut off by a R&S cut is 13% larger. For the P&R cuts the shifted geometric mean shows an increase of 36% in the distance cut off.

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Table 4 Sizes of the matrices generated by the reduction algorithms (i.e. the sums of the squared norms of the rows on the continuous and integral variables). original

R&S algorithm

P&R algorithm

instance

cont.

int.

cont.

int.

cont.

int.

22433 23588 aflow30a aligninq bell3a bell5 bienst1 bienst2 blend2 dcmulti dsbmip egout fiber fixnet6 flugpl gen gesa2 gesa2_o gesa3 gesa3_o khb05250 markshare_4_0 mas76 misc06 mod011 modglob neos-1062641 neos11 neos-1122047 neos-1171448 neos-1171692 neos-1200887 neos-1211578 neos-1330635 neos-1396125 neos-1440447 neos-1480121 neos-1489999 neos2 neos20 neos22 neos-430149 neos-480878 neos-501453 neos-501474 neos-504674 neos-504815 neos-506422 neos-512201 neos-525149 neos-530627 neos-555694 neos-555771 neos-584851 neos-598183 neos6 neos-603073 neos-611838 neos-612125 neos-612143 neos-612162 neos-717614 neos-799711 neos-799716 neos-803219 neos-806323 neos-807639 neos-807705 neos-826250 neos-826812 neos-830439 neos-839859 neos-863472

9012.10 805.14 1.27 0.00 0.84 67.29 37.35 36.44 0.00 0.03 0.77 0.00 0.35 0.00 2.59 749.68 1344.75 4.13 4141.97 3.09 0.00 0.00 0.00 1486.22 0.00 0.00 0.00 15195.48 0.00 873.39 678.42 2264.31 191.04 0.00 16169.79 39.20 0.00 348.99 3928.77 0.00 921.00 10.00 2893.99 40.30 286.94 394.23 321.24 0.66 361.23 1782.45 0.00 4091.66 675.04 2755.30 145.80 2697.62 17035.51 0.01 0.00 0.01 0.01 410217748.36 0.00 0.00 0.08 0.11 0.10 0.37 134286.75 162617.67 0.02 0.00 0.32

2638.95 767.23 2480.33 24522.76 4071.43 533.23 43.00 56.00 7280.06 133.71 237.65 9.00 19187.77 58.00 9.00 288.33 186.77 3662.28 173.23 4972.79 24.00 46.40 785.58 29.68 71.50 78.67 31.83 15487.23 150.60 38706.37 25012.11 4311.00 881.71 4605.21 5685.54 306.11 4.00 2699.28 731.34 18147325685254.90 454.00 25.00 183.49 136.33 1252.57 352.55 301.06 99.50 330.22 99985.75 23.00 68696.48 21698.49 523.71 521.21 827501.39 526.05 41.00 39.00 39.00 40.00 844788733.99 838.33 70.14 30.38 53.85 31.66 32.53 3134164383.50 828645712.50 66.00 148422.33 5161.00

1111.42 291.95 0.53 0.00 0.84 0.13 18.29 17.33 0.00 0.00 0.01 0.00 0.06 0.00 0.20 73.93 54.49 1.45 142.81 1.05 0.00 0.00 0.00 991.62 0.00 0.00 0.00 1601.67 0.00 195.66 65.87 132.77 23.00 0.00 164.50 22.81 0.00 0.72 1663.50 0.00 477.00 10.00 587.26 40.30 51.92 178.71 128.53 0.14 183.34 160.00 0.00 96.03 110.61 885.11 52.96 961.75 7811.20 0.00 0.00 0.00 0.00 358876.35 0.00 0.00 0.03 0.03 0.09 0.09 10830.38 155396.33 0.02 0.00 0.08

1268.50 632.16 1386.14 24522.76 4071.43 2051.50 47.00 76.00 819.87 2581.06 5823.93 9.00 5471.51 35.00 67.00 225921437014.21 3029.06 897.86 1278.84 2812.70 24.00 31.42 785.58 41.81 71.50 78.67 31.83 10739.63 150.60 8635.22 2588.45 1412.14 392.01 4605.21 1193161.58 244.01 5.00 3781.05 1821.61 18147325685254.90 618.00 25.00 298.40 136.33 1447.23 396.29 559.62 285.00 402.68 294380.73 28.00 43546.94 36708.94 808.55 1408110.42 5545270.73 1918243.13 81.00 62.00 90.00 94.00 3387065114.91 838.33 70.14 70.49 130.29 37.92 108.56 2972410205.05 774141471.33 72.00 148422.33 6007.00

275.21 126.53 0.48 0.00 0.04 0.05 16.75 11.37 0.00 0.00 0.00 0.00 0.04 0.00 1.47 45.86 87.14 1.23 309.00 0.63 0.00 0.00 0.00 1058.19 0.00 0.00 0.00 1430.53 0.00 46.87 41.46 23.74 9.00 0.00 118.58 8.14 0.00 0.00 1021.31 0.00 507.00 0.00 282.93 40.30 48.80 84.42 58.92 0.08 87.76 157.26 0.00 104.09 76.32 647.97 47.24 96.48 15830.21 0.01 0.00 0.01 0.01 12474543.60 0.00 0.00 0.01 0.02 0.02 0.01 26275.65 48108.92 0.00 0.00 0.00

350.68 320.75 1282.89 24522.76 4421.00 2448.05 58.75 77.96 3123.41 148.89 1239.13 9.00 3753.60 32.00 14.34 223.13 663.84 680.93 467.72 1218.26 24.00 28.19 785.58 44.78 71.50 78.67 31.83 2492.33 150.60 3029.89 1701.70 238.25 195.02 4605.21 471.70 181.14 4.00 6210.00 1259.25 18147325685254.90 618.00 30.00 907.33 136.33 476.25 295.26 254.63 145.00 311.29 20018.83 23.00 6095.68 9620.64 597.14 3119.68 146571.11 454.54 41.12 39.00 63.74 49.14 901701159.69 838.15 70.16 84.73 160.11 88.05 100.93 664764378.53 305087945.91 22.25 148422.34 5636.00

21

F. Wesselmann et al. / European Journal of Operational Research 214 (2011) 15–26 Table 4 (continued) original instance neos-880324 neos-885086 neos-906865 neos-941698 pg5_34 pk1 pp08a pp08acuts prod1 prod2 qiu ran10x26 ran12x21 ran13x13 rentacar rgn rout roy set1ch seymour1 swath1 swath2 vpm1 vpm2

cont.

R&S algorithm int.

0.65 1842.40 1145.08 601.07 27.29 0.07 0.01 0.02 1.00 0.46 1438.23 14.82 23.01 42.48 0.00 13.47 217.51 97.84 0.05 6351.32 12360.43 12810.05 212.35 285.54

cont. 262.52 29114.13 2314.02 50184.97 94.92 155.59 83.12 65.95 113.58 163.76 47.95 477.14 1182.13 1308.39 5.00 644.00 7707.05 53.00 201.92 3102.84 6264.69 7107.00 130.46 130.49

0.16 426.48 248.72 19.79 27.29 0.04 0.01 0.01 1.00 0.46 320.34 11.63 14.17 17.95 0.00 3.26 27.18 41.08 0.03 643.27 4700.66 4858.47 127.90 163.11

5.2. Computational usefulness in a cut-and-branch framework In the preceding part of this section we discussed properties of the rows and the cutting planes generated by the P&R algorithm. To assess the computational usefulness of P&R cuts we now adopt a cut-and-branch approach where P&R cuts are used to strengthen the LP relaxation of MIP problems before running the branch-andbound code of MOPS. We add twenty rounds of cuts at the root node of the branchand-bound tree. In each round cutting planes are added to a cut pool. If R&S or P&R cuts are applied in conjunction with GMI cuts duplicate cuts may be added to the cut pool when the reduction algorithms do not change a row of the optimal simplex tableau. We simply remove these duplicates. We then choose the ‘‘best’’ cutting planes in terms of the distance cut off. These cuts are then added to the MIP formulation. The cut pool assures that roughly the same number of cuts are added in each of the considered variants. We impose a maximum computation time of two hours per instance. We compare P&R cuts to R&S and GMI cuts. We also present results obtained with two mixed approaches which use GMI cuts in combination with R&S cuts or P&R cuts respectively. In order to compare the performance of different configurations, we compute mean values of the solution times, numbers of branching nodes and amounts of integrality gap closed at the root node. To compute the shifted geometric means, we set the shifting parameter s = 100 for node counts, s = 10 for solution times and s = 0.05 (i.e. 5%) for integrality gaps. If an instance is not solved within the time limit, we assume a computation time of two hours and treat the current number of computed branching nodes as the final node count in our evaluations. For instances which do not have an integrality gap we assume a value of 1 (i.e. 100% gap closed) in the calculation of the mean values. Table 3 shows the percentage changes in the mean values calculated for the R&S cuts, P&R cuts and the mixed approaches relative to the mean values obtained if only GMI cuts are applied. The detailed results can be found in Table 6. Table 3 shows that, concerning the times needed to solve the instances in the test set to optimality, the configuration which applies GMI cuts in conjunction with P&R cuts yields the best

P&R algorithm int.

cont. 261.63 8390.12 671.35 437399430.95 94.92 160.18 118.73 127.20 113.58 163.76 1144.80 532.63 759.49 786.46 5.00 442.00 8297.66 63.00 182.03 3131.76 4334.94 4921.38 93.35 94.11

int. 0.00 123.80 195.80 24.71 27.29 0.02 0.01 0.02 1.00 0.46 387.25 13.37 17.47 26.66 0.00 2.98 1.02 42.50 0.03 88.39 1858.00 1651.75 172.14 141.05

201.00 2915.04 533.38 18234.75 94.92 100.77 89.85 67.58 113.58 163.76 514.04 316.90 606.32 517.53 5.00 268.50 2856.70 56.00 178.14 2462.36 2990.47 3661.30 108.44 91.91

results (see column ‘‘GMI + P&R cuts’’). In comparison with the reference setting, the geometric and shifted geometric means of the running times decrease by 8% and 11% respectively. By contrast, the configuration which uses GMI cuts together with R&S cuts cannot significantly improve upon the results obtained with GMI cuts alone (see column ‘‘GMI + R&S cuts’’). Table 3 also indicates that the two configurations which use R&S or P&R cuts as a replacement for GMI cuts are not effective with respect to solution times. As shown in the columns headed ‘‘R&S cuts’’ and ‘‘P&R cuts’’ the shifted geometric means of the solution times increase by 1% if only R&S or P&R cuts are separated. Concerning the enumeration in the branch-and-bound algorithm, the geometric and shifted geometric means of the number of branching nodes are smaller than in the reference setting in any configuration presented in Table 3. As in our results on solution times, the combination of GMI and P&R cuts results in the largest reductions of the mean values. The geometric and shifted geometric means decrease by 31% and 26% respectively. Applying GMI together with R&S cuts results in a reduction of 13% in the shifted geometric mean of the number of branching nodes computed. The smallest reductions in the number of branching nodes are obtained when applying R&S or P&R cuts alone. For both configurations the shifted geometric mean of the number of branching nodes decreases by 7%. These two are also the only configurations in which the total number of branching nodes increases by 7% and 35% respectively. Moreover, Table 3 presents results concerning the amounts of integrality gap closed at the root node. Applying GMI cuts in conjunction with P&R cuts again results in the largest improvement in the mean values compared with the GMI cuts. Specifically, the geometric and shifted geometric means of the amounts of gap closed increase by 14% and 13% respectively. The mixed approach which separates GMI and R&S cuts also closes larger amounts of the integrality gap than GMI cuts alone. The geometric and the shifted geometric mean increase by 6%. By contrast, the configurations which do not apply GMI cuts deteriorate the performance in terms of integrality gap closed. For these configurations the geometric means of the amounts of integrality gap closed reduce by 10% while the shifted geometric means deteriorate by 7%.

22

Table 5 Properties of the cuts generated by the reduction algorithms. GMI cuts #cuts

#nz

22433 23588 aflow30a aligninq bell3a bell5 bienst1 bienst2 blend2 dcmulti dsbmip egout fiber fixnet6 flugpl gen gesa2 gesa2_o gesa3 gesa3_o khb05250 markshare_4_0 mas76 misc06 mod011 modglob neos-1062641 neos11 neos-1122047 neos-1171448 neos-1171692 neos-1200887 neos-1211578 neos-1330635 neos-1367061 neos-1396125 neos-1440447 neos-1480121 neos-1489999 neos2 neos20 neos22 neos-430149 neos-480878 neos-501453 neos-501474 neos-503737 neos-504674 neos-504815 neos-506422 neos-512201 neos-525149 neos-530627

87 57 25 157 9 20 12 16 0 49 24 7 46 6 9 16 54 67 77 87 19 3 6 4 3 16 20 183 3 143 59 74 14 0 9 97 4 2 484 22 23 454 25 4 2 30 173 113 84 28 113 82 3

242.43 245.58 89.80 1821.67 6.11 8.30 85.92 93.50 0.00 88.08 14.58 2.57 56.98 61.67 7.22 22.19 8.91 17.78 21.01 30.32 58.89 30.00 150.00 55.00 430.67 32.56 8.70 484.20 36.00 1497.36 761.53 143.65 52.00 0.00 6152.67 495.71 43.75 20.00 78.00 121.50 7.52 3.17 2.80 170.75 51.50 49.90 2161.90 4.47 4.55 40.29 4.72 1184.18 2.00

R&S cuts cont. coef. 0.30 0.28 0.08 0.00 0.05 0.05 0.01 0.01 0.00 0.01 0.00 0.01 0.00 0.00 0.01 1.02 0.42 0.16 2.35 0.10 0.00 0.00 0.00 1.68 0.00 0.00 0.00 0.56 0.00 0.23 0.53 1.37 0.92 0.00 0.00 0.45 0.46 0.00 1.23 1.70 0.00 1.21 0.09 0.19 5.08 0.98 0.62 0.76 0.94 0.03 0.76 1.87 0.00

int. coef. 1.23 0.71 2.18 0.47 0.76 1.23 0.29 0.30 0.00 1.05 0.59 0.20 1.36 0.56 1.11 5.66 2.06 0.94 4.61 0.99 1.02 4.58 4.34 1.00 0.24 0.46 0.42 0.35 1.18 0.55 1.12 0.76 1.64 0.00 15.17 0.52 0.85 0.63 0.12 1.16 0.68 1.07 0.09 0.49 6.86 1.38 0.53 0.79 0.91 1.01 0.80 0.60 2.62

P&R cuts

violation

distance

#cuts

#nz

cont. coef.

0.47 0.39 0.48 0.40 0.25 0.34 0.15 0.15 0.00 0.35 0.40 0.11 0.32 0.25 0.46 0.62 0.48 0.46 0.65 0.62 0.35 0.69 0.43 0.26 0.19 0.23 0.36 0.21 0.65 0.27 0.39 0.43 0.50 0.00 0.70 0.17 0.50 0.35 0.49 0.41 0.49 0.50 0.28 0.20 0.55 0.45 0.28 0.45 0.45 0.25 0.46 0.43 0.72

0.02 0.04 0.04 0.02 0.56 0.52 0.69 0.66 0.00 0.86 0.80 0.73 0.04 0.30 0.31 0.08 0.27 0.27 0.14 0.16 0.64 0.03 0.01 0.02 0.37 0.59 0.44 0.02 0.48 0.10 0.03 0.03 0.06 0.00 0.00 0.03 0.13 0.65 0.20 0.05 0.27 0.29 29.26 0.02 0.01 0.08 0.01 0.34 0.31 0.15 0.34 0.04 0.20

57 27 11 157 9 13 12 13 0 18 20 7 19 5 5 7 67 48 46 33 19 3 6 2 3 16 20 37 3 40 5 2 0 0 9 0 3 2 1 15 23 290 25 1 2 11 158 89 53 21 87 20 1

244.14 245.52 105.36 1821.67 6.11 3.85 85.58 79.92 0.00 64.67 12.00 2.57 146.42 33.60 7.00 8.71 11.37 15.19 14.33 21.12 58.89 30.00 150.00 22.50 430.67 32.56 8.70 481.97 36.00 225.30 46.00 142.00 0.00 0.00 6152.67 0.00 33.67 20.00 78.00 109.27 7.52 2.00 2.80 162.00 51.50 42.45 2170.10 2.96 3.32 22.05 2.99 1200.05 2.00

0.41 0.45 0.03 0.00 0.05 0.08 0.03 0.01 0.00 0.00 0.00 0.01 0.00 0.00 0.01 0.02 0.22 0.11 0.18 0.11 0.00 0.00 0.00 1.18 0.00 0.00 0.00 0.72 0.00 0.22 0.18 1.88 0.00 0.00 0.00 0.00 0.42 0.00 1.00 0.87 0.00 1.00 0.09 0.07 5.08 0.61 0.64 0.78 0.75 0.06 0.79 1.32 0.00

int. coef. 1.66 0.88 0.98 0.47 0.76 1.72 0.54 0.31 0.00 0.42 0.60 0.20 0.53 0.62 1.00 2.17 3.35 0.67 2.01 0.64 1.02 2.49 4.34 1.00 0.24 0.46 0.42 0.58 1.18 0.65 0.17 1.07 0.00 0.00 15.17 0.00 0.90 0.63 0.10 0.92 0.68 1.00 0.09 0.43 6.86 0.99 0.53 0.78 0.85 1.36 0.79 0.61 2.62

violation

distance

#cuts

#nz

0.51 0.51 0.29 0.40 0.25 0.43 0.20 0.17 0.00 0.35 0.38 0.11 0.33 0.23 0.47 0.49 0.54 0.51 0.47 0.50 0.35 0.46 0.43 0.17 0.19 0.23 0.36 0.47 0.65 0.37 0.14 0.70 0.00 0.00 0.70 0.00 0.50 0.35 0.40 0.40 0.49 0.50 0.28 0.24 0.55 0.33 0.31 0.43 0.40 0.33 0.44 0.50 0.72

0.03 0.04 0.04 0.02 0.56 0.50 0.63 0.66 0.00 1.28 0.79 0.73 0.08 0.32 0.32 0.15 0.18 0.30 0.18 0.20 0.64 0.03 0.01 0.03 0.37 0.59 0.44 0.03 0.48 0.28 0.12 0.03 0.00 0.00 0.00 0.00 0.15 0.65 0.20 0.07 0.27 0.35 29.26 0.05 0.01 0.10 0.01 0.36 0.33 0.18 0.36 0.04 0.20

34 23 14 157 9 12 14 18 0 34 19 7 8 4 7 18 32 6 26 35 19 3 5 4 3 16 20 94 3 12 16 64 0 0 0 93 3 3 0 13 23 290 25 10 2 3 143 72 41 17 71 7 3

158.00 217.43 33.86 1821.67 2.89 2.83 92.79 98.67 0.00 86.53 10.21 2.57 28.63 31.00 6.14 7.89 6.69 11.50 9.65 15.91 58.89 30.00 150.00 53.75 430.67 32.56 8.70 513.71 36.00 236.67 631.50 58.66 0.00 0.00 0.00 21.05 37.00 25.33 0.00 91.92 7.52 2.00 2.80 22.20 51.50 148.33 1589.99 2.32 2.71 8.65 2.96 720.00 2.00

cont. coef. 0.27 0.33 0.06 0.00 0.05 0.09 0.02 0.02 0.00 0.00 0.00 0.01 0.00 0.00 0.01 0.39 0.25 0.09 0.66 0.12 0.00 0.00 0.00 2.25 0.00 0.00 0.00 0.15 0.00 0.21 0.45 0.09 0.00 0.00 0.00 0.14 0.14 0.00 0.00 0.98 0.00 1.00 0.00 0.14 5.08 10.82 0.09 0.95 1.02 0.07 0.91 1.28 0.00

int. coef. 0.57 0.62 1.51 0.47 0.76 1.86 0.45 0.65 0.00 0.44 0.67 0.20 0.86 0.30 0.79 2.69 3.24 0.50 3.98 1.27 1.02 2.38 5.52 1.10 0.24 0.46 0.42 0.13 1.18 0.68 1.03 0.64 0.00 0.00 0.00 0.20 0.53 0.45 0.00 1.60 0.68 1.00 0.10 1.71 6.86 12.62 0.22 0.97 1.28 2.20 0.84 0.89 2.62

violation

distance

0.51 0.54 0.42 0.40 0.25 0.46 0.31 1.67 0.00 0.38 0.42 0.11 0.12 0.14 0.48 0.51 0.52 0.32 3.39 0.74 0.35 0.58 0.51 0.30 0.19 0.23 0.36 0.26 0.65 0.30 0.49 0.48 0.00 0.00 0.00 0.27 0.39 0.27 0.00 0.48 0.49 0.50 0.28 0.49 0.55 0.81 0.41 0.45 0.44 0.50 0.44 0.50 0.72

0.12 0.10 0.07 0.02 0.63 0.48 0.52 0.59 0.00 1.04 0.76 0.73 0.15 0.39 0.37 0.16 0.19 0.24 0.33 0.19 0.64 0.05 0.01 0.02 0.37 0.59 0.44 0.07 0.48 0.12 0.04 0.22 0.00 0.00 0.00 0.31 0.24 0.73 0.00 0.09 0.27 0.35 29.34 0.26 0.01 0.01 0.06 0.35 0.31 0.23 0.35 0.12 0.20

F. Wesselmann et al. / European Journal of Operational Research 214 (2011) 15–26

instance

29 50 269 45 75 30 41 39 39 40 145 0 0 20 40 20 22 34 57 123 0 12 3 22 4 76 7 106 104 79 71 65 65 223 86 0 48 8 49 46 36 12 17 18 0 13 31 11 119 160 12 13 15 25

520.86 697.96 50.46 51.73 5628.57 65.93 240.63 348.03 274.90 286.25 32.91 0.00 0.00 36.30 37.75 28.50 31.14 1518.82 964.86 1190.15 0.00 5.17 79.00 2.18 7.00 882.86 151.14 5591.02 5626.78 5549.33 6273.00 6938.89 6968.35 820.15 29.50 0.00 4.75 10.25 22.71 47.76 213.28 142.42 163.65 112.00 0.00 42.31 336.87 24.18 8.09 335.44 1194.83 1281.85 18.07 17.80

2.79 0.81 0.65 0.69 1.02 6.56 0.00 0.00 0.00 0.00 1001.85 0.00 0.00 0.01 0.01 0.03 0.02 1.09 1.99 1.52 0.00 0.01 0.00 0.19 0.46 0.37 1.45 3.33 2.34 1.42 2.00 1.77 1.96 0.30 0.19 0.00 0.01 0.01 0.00 0.00 0.07 0.11 0.20 0.42 0.00 0.28 0.22 0.95 0.01 0.47 5.62 5.22 1.78 2.20

1.26 0.52 0.23 1.82 1.13 2.03 0.63 0.56 0.74 0.70 354.82 0.00 0.00 0.28 0.32 2.49 1.83 34.94 11.22 10.31 0.00 0.07 0.84 0.66 1.71 0.84 2.04 2.01 1.35 1.22 1.78 1.34 1.95 0.40 1.00 0.00 0.24 0.30 0.10 0.06 0.25 0.82 1.13 1.97 0.00 1.01 0.60 2.03 0.27 0.84 4.74 4.40 1.36 1.53

0.47 0.45 0.09 0.44 0.41 0.43 0.30 0.31 0.31 0.30 0.46 0.00 0.00 0.20 0.21 0.69 0.52 0.50 0.50 0.40 0.00 0.07 0.72 0.45 0.49 0.38 0.59 0.88 0.78 0.85 0.90 0.88 0.95 0.33 0.45 0.00 0.17 0.15 0.31 0.32 0.33 0.43 0.45 0.60 0.00 0.31 0.40 0.60 0.18 0.48 0.58 0.53 0.49 0.47

0.02 0.03 0.03 0.45 0.01 0.36 0.53 0.51 0.49 0.48 0.24 0.00 0.00 0.39 0.35 0.21 0.26 0.00 0.00 0.00 0.00 0.87 0.06 0.67 0.11 0.06 0.02 0.01 0.01 0.01 0.01 0.01 0.00 0.02 0.30 0.00 0.78 0.32 0.46 0.45 0.11 0.08 0.06 0.05 0.00 0.06 0.07 0.10 0.77 0.12 0.00 0.00 0.08 0.08

0 1 24 36 52 21 35 39 34 36 33 0 0 19 26 15 23 11 1 101 0 12 3 5 1 49 7 13 68 56 13 17 48 18 86 0 48 4 49 46 4 16 15 15 0 9 14 4 119 110 10 11 11 13

0.00 902.00 51.21 40.56 6141.35 63.00 142.69 401.21 234.76 247.53 41.79 0.00 0.00 39.26 42.50 23.33 44.61 1050.64 7.00 1189.61 0.00 5.00 79.00 2.00 8.00 304.18 180.71 5640.46 5629.59 5563.45 6239.00 6842.18 7020.04 830.56 29.50 0.00 5.46 12.50 22.71 47.76 276.75 148.38 187.00 132.33 0.00 36.44 396.79 14.25 9.20 149.64 1116.60 1360.09 12.00 13.62

0.00 0.40 1.62 0.09 0.66 0.14 0.00 0.00 0.00 0.00 0.06 0.00 0.00 0.01 0.00 0.02 0.04 0.85 0.00 1.55 0.00 0.03 0.00 0.09 0.17 0.32 0.55 1.20 2.72 1.19 0.89 0.91 2.03 0.44 0.19 0.00 0.01 0.01 0.00 0.00 0.57 0.14 0.15 0.21 0.00 0.64 0.30 0.61 0.00 0.13 4.37 3.32 1.80 1.58

0.00 0.37 0.64 0.78 1.28 11.26 0.77 0.88 0.96 0.96 0.51 0.00 0.00 0.45 0.45 3.14 2.06 43.15 143.71 10.65 0.00 1.26 0.84 0.41 0.83 0.76 1.02 0.93 1.61 1.00 0.83 0.73 2.07 0.77 1.00 0.00 0.36 0.51 0.10 0.06 3.36 0.86 0.93 0.83 0.00 1.45 0.83 2.38 0.21 0.88 3.86 2.81 1.48 1.05

0.00 0.40 0.30 0.29 0.44 0.33 0.31 0.36 0.36 0.35 0.32 0.00 0.00 0.25 0.26 0.75 0.50 0.50 0.50 0.39 0.00 0.14 0.72 0.25 0.21 0.45 0.56 0.66 0.79 0.75 0.71 0.60 0.95 0.59 0.45 0.00 0.20 0.30 0.31 0.32 0.54 0.33 0.44 0.47 0.00 0.46 0.46 0.50 0.17 0.49 0.67 0.53 0.54 0.45

0.00 0.03 0.04 0.52 0.01 0.36 0.54 0.44 0.46 0.49 0.49 0.00 0.00 0.31 0.37 0.17 0.15 0.00 0.00 0.00 0.00 0.77 0.06 0.68 0.10 0.08 0.04 0.01 0.01 0.01 0.01 0.01 0.00 0.02 0.30 0.00 0.76 0.38 0.46 0.45 0.02 0.06 0.09 0.08 0.00 0.07 0.08 0.13 0.78 0.16 0.04 0.02 0.10 0.10

24 27 138 27 80 14 41 39 39 40 51 0 0 11 39 16 18 18 28 16 0 9 3 0 0 19 11 109 134 99 80 74 72 212 86 2 50 10 49 46 17 6 16 14 0 10 6 6 116 132 6 11 14 18

281.63 437.15 98.47 13.74 7056.21 44.29 251.44 348.03 277.08 313.35 68.39 0.00 0.00 34.18 38.64 34.88 26.83 527.06 827.07 120.50 0.00 2.44 79.00 0.00 0.00 620.21 99.45 5450.42 4364.98 4476.02 5574.86 6111.69 6276.60 721.03 29.50 67.00 4.90 9.00 22.71 47.76 267.35 40.50 46.19 38.21 0.00 31.10 136.67 3.50 4.59 60.90 153.83 185.64 12.43 8.89

1.19 0.57 0.26 0.04 0.10 0.02 0.00 0.00 0.00 0.00 522.65 0.00 0.00 0.03 0.01 0.01 0.02 1.02 1.02 0.80 0.00 0.01 0.00 0.00 0.00 0.33 0.42 0.28 0.30 0.31 0.29 0.37 0.29 0.10 0.19 0.11 0.01 0.01 0.00 0.00 0.19 0.07 0.21 0.36 0.00 0.23 0.01 0.23 0.00 0.05 3.52 0.37 1.73 2.98

0.79 0.48 0.10 0.89 0.66 1.37 0.61 0.56 1.35 0.97 114.72 0.00 0.00 0.78 0.78 3.00 2.61 42.39 43.01 28.61 0.00 0.07 0.84 0.00 0.00 0.85 0.55 0.47 0.47 0.39 0.37 0.48 0.37 0.45 1.00 14.73 0.26 0.37 0.10 0.06 1.22 0.47 1.19 2.09 0.00 0.79 0.91 1.95 0.23 0.87 2.35 0.80 1.37 2.38

0.51 0.57 0.12 0.33 0.50 0.44 0.30 0.31 0.34 0.32 0.43 0.00 0.00 0.48 0.42 0.75 0.64 0.50 0.44 0.35 0.00 0.07 0.72 0.00 0.00 0.48 0.39 0.65 0.60 0.69 0.64 0.72 0.71 0.46 0.45 0.94 0.18 0.17 0.31 0.32 0.57 0.37 0.45 0.52 0.00 0.28 0.53 0.44 0.17 0.50 0.78 0.61 0.53 0.52

0.06 0.07 0.05 0.62 0.01 0.50 0.54 0.51 0.49 0.49 0.44 0.00 0.00 0.23 0.33 0.11 0.13 0.00 0.00 0.00 0.00 0.92 0.06 0.00 0.00 0.09 0.08 0.01 0.03 0.05 0.02 0.02 0.02 0.04 0.30 0.01 0.77 0.29 0.46 0.45 0.09 0.15 0.09 0.10 0.00 0.08 0.06 0.16 0.79 0.17 0.08 0.10 0.10 0.09

F. Wesselmann et al. / European Journal of Operational Research 214 (2011) 15–26

neos-555694 neos-555771 neos-584851 neos-598183 neos6 neos-603073 neos-611838 neos-612125 neos-612143 neos-612162 neos-717614 neos-799711 neos-799716 neos-803219 neos-806323 neos-807639 neos-807705 neos-824695 neos-826250 neos-826812 neos-827175 neos-830439 neos-839859 neos-863472 neos-880324 neos-885086 neos-906865 neos-933638 neos-933966 neos-935769 neos-936660 neos-937446 neos-937511 neos-941698 pg5_34 pk1 pp08a pp08acuts prod1 prod2 qiu ran10x26 ran12x21 ran13x13 rentacar rgn rout roy set1ch seymour1 swath1 swath2 vpm1 vpm2

23

GMI cuts instance

R&S cuts time

nodes

gap cl.

0.2 1.8 225.6 10.2 4.2 76.8 132.6 916.8 0.6 0.6 0.6 0.2 0.6 0.6 0.2 0.6 38.4 274.2 1.2 1.2 0.6 166.2 265.8 0.2 295.8 736.2 0.2 10.2 46.2 7.2 726.0 382.2 5167.8 1174.2 2364.6 651.6 151.8 658.8 2080.8 2809.2 9.6 12.0 102.6 1031.4 99.0 46.2 459.6 280.2 907.2 4.8 618.0 132.6 1170.0

41 2245 111396 3566 28507 284364 45796 394041 2329 121 104 1 477 248 225 418 20330 211307 1115 776 85 2067260 1123613 87 111221 1849555 515 3 268 253 374891 855855 2578354 165 610930 977053 532050 374241 2537897 327887 64789 30041 3259 890308 111714 5488 387647 97001 3402578 1100 215693 184718 295427

0.00 0.00 15.92 0.00 57.50 87.53 14.01 10.25 18.78 81.72 100.00 100.00 88.17 82.77 15.48 36.93 86.54 72.12 52.66 43.05 93.96 0.00 0.00 30.96 7.04 32.09 100.00 100.00 100.00 100.00 0.00 0.00 0.00 100.00 0.98 0.00 0.00 4.76 0.00 64.06 87.94 99.02 100.00 24.87 20.91 100.00 25.77 0.00 46.91 0.00 0.00 0.00 30.84

status

P&R cuts time

nodes

gap cl.

1.2 1.8 205.2 10.2 7.2 350.4 98.4 782.4 1.8 1.2 0.6 0.2 2.4 1.2 0.2 0.2 133.2 1536.0 1.8 1.2 0.6 166.2 265.8 0.2 297.0 1120.8 0.2 10.2 42.0 10.2 727.2 358.8 5172.0 1459.2 1544.4 450.6 151.8 466.8 2082.0 2809.2 0.2 115.8 60.6 489.6 70.2 46.2 370.8 233.4 0.2 5.4 105.6 160.8 2997.6

41 2245 140507 3566 50768 1343432 55889 424571 4034 223 177 13 593 273 196 275 58192 1048661 942 715 159 2067260 1123613 107 120615 2663787 515 3 260 242 374891 835809 2578354 213 462073 660389 532050 350923 2537897 327887 1 253592 1610 437998 88972 5488 344312 82980 1 1677 34173 203488 501179

0.00 0.00 22.78 0.00 57.11 71.94 1.65 2.45 19.02 61.16 100.00 99.07 86.53 82.49 17.29 47.92 90.61 88.16 35.15 51.03 88.43 0.00 0.00 28.17 5.14 27.45 100.00 100.00 100.00 100.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 64.06 100.00 87.65 100.00 23.99 13.15 100.00 19.29 0.00 100.00 0.00 0.00 0.00 36.63

status

t

t

GMI + R&S cuts time

nodes

gap cl.

0.6 1.8 178.8 11.4 6.6 7200.0 193.2 445.2 1.2 3.0 0.6 0.2 1.2 1.2 0.2 0.6 54.0 643.2 1.8 3.6 0.6 177.6 265.8 0.2 297.6 783.0 0.2 10.2 32.4 76.2 508.8 297.0 5171.4 1521.0 1878.0 1107.6 97.8 382.2 1487.4 1321.2 9.6 7200.0 189.0 1082.4 79.8 28.8 799.2 255.0 964.2 5.4 54.6 37.8 526.2

41 2241 122664 3827 43047 14058106 53082 255311 3412 195 151 21 1212 310 177 4 58292 790550 864 4624 233 2123125 1123613 119 121465 1634791 145 1 176 2289 280266 682445 2578354 213 503595 1634365 315955 287516 1784970 130452 86517 6232256 5064 1171823 106239 3221 850613 82138 3388606 1628 17660 49193 140799

0.00 0.00 22.21 0.00 57.43 12.62 3.24 2.61 18.78 69.21 100.00 97.16 68.69 82.06 11.34 86.81 58.22 6.46 57.59 2.06 81.18 0.00 0.00 62.45 5.14 38.05 100.00 100.00 100.00 100.00 0.00 0.00 0.00 100.00 19.88 0.00 0.00 0.00 0.00 64.06 84.99 0.00 100.00 11.14 12.48 100.00 10.39 0.00 45.30 0.00 0.00 0.00 41.91

status

GMI + P&R cuts

time

nodes

gap cl.

1.2 1.8 244.8 10.2 7.8 99.0 165.0 951.0 1.8 2.4 0.6 0.2 1.2 1.8 0.2 0.2 28.8 279.0 4.2 3.0 1.2 166.8 265.8 0.2 295.8 736.2 0.6 10.2 47.4 21.0 726.6 382.8 5171.4 1659.6 2361.0 451.2 151.8 663.6 1975.2 2812.8 0.2 8.4 20.4 1503.0 77.4 46.2 419.4 282.6 0.2 4.8 618.0 160.8 930.6

41 2245 137988 3566 52266 310988 56905 320893 4034 110 104 23 1444 432 113 294 21247 218987 715 1437 111 2067260 1123613 125 111221 1849555 515 3 268 584 374891 855855 2578354 195 610930 660389 532050 374241 2465454 327887 1 20867 412 1340828 83111 5488 360435 97001 1 1100 215693 203488 256363

0.00 0.00 18.21 0.00 57.11 88.13 15.12 19.30 19.02 83.78 100.00 98.80 82.91 82.21 22.70 42.36 91.35 57.15 58.74 49.70 93.32 0.00 0.00 30.96 7.04 32.09 100.00 100.00 100.00 100.00 0.00 0.00 0.00 100.00 0.98 0.00 0.00 4.76 0.00 64.06 100.00 99.02 100.00 24.87 22.06 100.00 25.77 0.00 100.00 0.00 0.00 0.00 75.42

status

time

nodes

gap cl.

0.6 1.8 214.8 12.0 4.8 365.4 153.6 953.4 1.2 3.0 0.6 0.2 3.6 1.8 0.2 0.6 24.0 337.8 4.2 4.8 1.2 178.2 265.8 0.2 294.6 781.2 0.2 10.2 64.8 18.6 508.8 340.8 5170.8 1323.0 1915.8 733.8 97.8 721.8 1515.0 1321.8 3.0 13.2 84.0 281.4 71.4 15.0 235.2 280.8 725.4 5.4 173.4 37.8 2659.8

41 2241 162624 3827 29918 1227253 48483 401942 3412 129 189 3 272 242 244 1 17213 200457 654 702 69 2123125 1123613 77 110247 1590565 322 1 301 556 280266 766279 2578458 177 435170 1051840 315955 399096 1878623 130452 27127 31495 2559 218213 81433 1428 193152 97001 2824980 1628 53781 49193 632475

0.00 0.00 6.20 0.00 57.11 88.50 5.30 4.33 18.78 86.41 100.00 99.99 89.74 85.03 15.82 100.00 92.69 87.22 59.43 62.44 93.86 0.00 0.00 30.96 7.04 36.68 100.00 100.00 100.00 100.00 0.00 0.00 0.00 100.00 18.55 0.00 0.00 4.76 0.00 64.06 96.65 99.02 100.00 37.75 21.77 100.00 30.17 0.00 47.71 0.00 0.00 0.00 37.13

F. Wesselmann et al. / European Journal of Operational Research 214 (2011) 15–26

22433 23588 aflow30a aligninq bell3a bell5 bienst1 bienst2 blend2 dcmulti dsbmip egout fiber fixnet6 flugpl gen gesa2 gesa2_o gesa3 gesa3_o khb05250 markshare_4_0 mas76 misc06 mod011 modglob neos-1062641 neos-1122047 neos-1171448 neos-1171692 neos-1200887 neos-1211578 neos-1330635 neos-1367061 neos-1396125 neos-1440447 neos-1480121 neos-1489999 neos-430149 neos-480878 neos-501453 neos-501474 neos-503737 neos-504674 neos-504815 neos-506422 neos-512201 neos-525149 neos-530627 neos-555694 neos-555771 neos-584851 neos-598183

status

24

Table 6 Detailed cut-and-branch results. Status ‘‘t’’ indicates that an instance was not solved within the time limit of two hours.

t

92.4 54.0 21.0 57.0 85.8 139.2 171.6 6.0 64.8 31.8 60.0 21.6 112.2 895.8 18.6 35.4 0.2 61.8 1199.4 6.0 751.2 324.6 660.0 585.0 373.8 1860.6 178.2 453.6 3.0 1608.6 227.4 302.4 157.8 724.8 1246.8 127.2 95.4 174.6 2757.0 960.0 136.2 246.0 2201.4 346.2 1.2 0.6 1108.2 0.6 7200.0 5803.8 26.4 4734.6 13.8 145.2

22741 8663 2381 8069 12833 85375 7072 698 53783 19880 44835 11307 2644 23898 155 53 233 19300 2129044 13855 3068 154487 592 3583 821 2751 503 1444 866 54166 163016 218766 58595 12633 264333 507599 220497 364873 4827104 1347521 25417 265953 2318511 602806 20 3077 1800578 3143 1808007 173944 2780 525925 63404 254683

4.66 70.06 64.07 66.19 64.90 26.19 0.00 100.00 30.36 27.89 17.92 18.57 100.00 100.00 100.00 100.00 100.00 2.59 0.00 0.00 100.00 7.82 100.00 100.00 100.00 100.00 100.00 100.00 100.00 0.00 11.03 0.00 99.31 100.00 90.71 0.00 73.07 25.94 42.88 38.28 10.25 26.26 8.72 25.94 0.00 24.59 0.00 12.96 64.85 10.40 10.28 11.99 56.36 46.85

t

25.8 126.0 25.2 62.4 144.0 213.0 171.6 6.0 52.2 32.4 39.0 27.0 113.4 151.8 229.8 35.4 0.2 61.8 1164.0 6.0 222.0 367.2 324.6 180.6 319.2 6339.0 366.0 453.6 4.2 2422.2 219.6 301.8 143.4 725.4 1206.6 127.2 1399.8 181.2 2756.4 960.0 82.8 257.4 1464.0 427.8 0.6 0.6 985.2 0.2 7200.0 5804.4 278.4 1122.6 2.4 288.0

9050 19021 2045 9425 24289 207697 7072 698 52075 19936 28004 14156 1160 3573 1892 53 261 19300 2029239 13855 428 187757 262 756 1370 9900 671 1444 866 89730 179786 218766 52883 12633 276915 507599 3423595 530579 4827104 1347521 19105 329936 1792069 753906 20 3365 1602592 932 3011242 173944 63701 214516 11292 519193

3.94 68.20 68.31 66.54 59.64 3.29 0.00 100.00 12.68 27.89 16.97 18.32 100.00 100.00 100.00 100.00 100.00 2.59 0.00 0.00 100.00 5.75 100.00 100.00 100.00 100.00 100.00 100.00 100.00 0.00 8.39 0.00 99.31 100.00 90.61 0.00 57.49 21.40 42.88 38.28 0.00 23.91 24.58 27.58 0.00 21.18 0.00 13.73 53.69 10.40 13.94 13.65 56.36 42.36

t

t

152.4 150.6 168.6 100.8 248.4 46.2 109.8 4.2 57.0 42.0 36.6 23.4 156.6 29.4 99.0 36.6 0.2 66.0 1521.6 4.8 542.4 79.8 7200.6 76.8 194.4 768.0 566.4 70.2 27.6 1794.0 297.0 210.0 131.4 415.2 1251.0 127.2 39.6 36.6 291.0 850.2 151.8 231.6 1759.8 212.4 0.6 1.2 800.4 0.2 7200.0 2857.2 417.6 1593.0 0.2 208.8

36036 7333 3761 7605 21545 42141 5159 524 52403 20370 24660 12618 932 916 528 53 276 20342 2471585 10941 1024 23427 15140 248 620 1121 1563 112 7910 67916 235817 135138 51061 7402 261207 507599 98955 86741 456228 1214216 30271 250098 1550858 195424 20 2829 1456939 271 2510720 101201 106872 429415 17 314021

5.09 75.12 66.86 69.23 63.14 10.68 0.00 100.00 15.31 23.50 5.11 8.95 100.00 100.00 100.00 100.00 100.00 2.59 0.00 0.00 100.00 46.97 100.00 100.00 100.00 100.00 100.00 100.00 100.00 0.00 6.81 0.00 99.31 100.00 92.97 0.00 79.20 54.58 64.92 38.28 3.17 23.04 36.58 49.26 0.00 71.79 0.00 64.11 70.70 16.55 9.49 16.95 100.00 57.41

t

83.4 75.0 82.2 84.0 108.6 217.8 171.6 6.0 64.8 32.4 59.4 23.4 112.2 895.8 18.6 35.4 0.2 61.8 1200.0 6.0 755.4 367.2 660.6 82.2 319.2 6333.0 366.6 453.0 4.8 1609.8 421.2 302.4 157.8 725.4 1353.6 127.2 157.2 198.6 2756.4 960.6 90.6 234.0 1197.0 463.2 0.6 1.2 985.2 0.6 7200.0 5800.2 135.0 3377.4 4.2 179.4

16074 7483 4263 7915 11611 90972 7072 698 53783 19880 44835 10946 2644 23898 155 53 237 19300 2129044 13855 3068 187757 592 261 1370 9900 671 1444 866 54166 311674 218766 58595 12633 288837 507599 335129 476937 4827104 1347521 17939 286513 1364070 691711 20 3109 1602592 2113 2061808 173944 29475 420695 17591 276693

5.38 67.55 63.64 67.32 65.48 21.13 0.00 100.00 30.36 27.89 17.92 17.69 100.00 100.00 100.00 100.00 100.00 2.59 0.00 0.00 100.00 5.75 100.00 100.00 100.00 100.00 100.00 100.00 100.00 0.00 10.93 0.00 99.31 100.00 90.57 0.00 76.88 25.78 42.88 38.28 10.40 23.27 24.27 29.91 0.00 44.08 0.00 21.79 66.49 10.40 13.86 23.19 56.36 51.58

t

125.4 173.4 193.8 267.0 249.6 127.8 109.8 4.2 70.2 33.0 54.0 26.4 113.4 29.4 20.4 36.6 0.2 66.0 1228.8 6.0 1377.6 47.4 106.2 463.8 194.4 768.0 564.6 69.6 28.2 1793.4 264.6 210.0 172.2 415.8 1319.4 127.2 12.6 36.6 614.4 850.2 133.8 320.4 1130.4 177.0 0.6 0.6 1023.6 0.2 7200.0 2857.8 366.6 2004.6 0.2 126.0

22404 7373 4257 12765 9619 82439 5159 524 53994 19879 39765 11090 2644 916 155 53 256 20342 2110225 14103 5019 13427 88 2176 620 1121 1563 112 7910 67916 196983 135138 63955 7402 257721 507599 25054 87257 1081335 1214216 21931 214244 1078784 202149 20 1813 1681031 70 1780696 101201 67113 396284 41 163201

5.18 75.41 66.78 66.64 72.43 26.97 0.00 100.00 30.36 27.89 17.92 17.11 100.00 100.00 100.00 100.00 100.00 2.59 0.00 0.00 100.00 43.50 100.00 100.00 100.00 100.00 100.00 100.00 100.00 0.00 10.48 0.00 99.31 100.00 90.68 0.00 85.31 52.70 60.58 38.28 10.51 33.62 39.18 36.91 0.00 86.16 0.00 85.25 72.32 16.55 13.69 22.92 100.00 62.48

F. Wesselmann et al. / European Journal of Operational Research 214 (2011) 15–26

neos-603073 neos-611838 neos-612125 neos-612143 neos-612162 neos-717614 neos-799711 neos-799716 neos-803219 neos-806323 neos-807639 neos-807705 neos-824695 neos-826250 neos-826812 neos-827175 neos-830439 neos-839859 neos-863472 neos-880324 neos-885086 neos-906865 neos-933638 neos-933966 neos-935769 neos-936660 neos-937446 neos-937511 neos-941698 neos11 neos2 neos20 neos22 neos6 pg5_34 pk1 pp08a pp08acuts prod1 prod2 qiu ran10x26 ran12x21 ran13x13 rentacar rgn rout roy set1ch seymour1 swath1 swath2 vpm1 vpm2

25

26

F. Wesselmann et al. / European Journal of Operational Research 214 (2011) 15–26

6. Conclusion In this paper, we presented a new algorithm to obtain improved Gomory mixed-integer cuts which is based on pivoting. We described the implementation of this algorithm. We also analyzed the properties of our algorithm with respect to the rows of the simplex tableau it creates. The results showed that our algorithm can drastically reduce the size of the coefficients of the continuous variables in a simplex tableau row. Moreover, we conducted detailed computational experiments in the cut-and-branch framework of the MIP solver MOPS. These experiments showed that our algorithm can actually be helpful to solve challenging MIP problems faster. References Achterberg, T., 2009. SCIP: solving constraint integer programs. Mathematical Programming Computation 1 (1), 1–41. Achterberg, T., Koch, T., Martin, A., 2006. MIPLIB 2003. Operations Research Letters 34 (4), 1–12 . Andersen, K., Cornuéjols, G., Li, Y., 2005. Reduce-and-split cuts: Improving the performance of mixed-integer Gomory cuts. Management Science 51 (11), 1720–1732. Balas, E., 1971. Intersection cuts - a new type of cutting planes for integer programming. Operations Research 19 (1), 19–39. Balas, E., Bonami, P., 2009. Generating lift-and-project cuts from the LP simplex tableau: open source implementation and testing of new variants. Mathematical Programming Computation 1 (2–3), 165–199. Balas, E., Perregaard, M., 2003. A precise correspondence between lift-and-project cuts, simple disjunctive cuts, and mixed integer Gomory cuts for 0-1 programming. Mathematical Programming 94 (2–3), 221–245.

Balas, E., Ceria, S., Cornuéjols, G., 1993. A lift-and-project cutting plane algorithm for mixed 0-1 programs. Mathematical Programming 58 (1–3), 295–324. Balas, E., Ceria, S., Cornuéjols, G., 1996a. Mixed 0-1 programming by lift-and-project in a branch-and-cut framework. Management Science 42 (9), 1229–1246. Balas, E., Ceria, S., Cornuéjols, G., Natraj, N., 1996b. Gomory cuts revisited. Operations Research Letters 19, 1–9. Bixby, R.E., Rothberg, E., 2007. Progress in computational mixed integer programming - a look back from the other side of the tipping point. Annals of Operations Research 149 (1), 37–41. Bixby, R.E., Ceria, S., McZeal, C.M., Savelsbergh, M.W.P., 1998. An updated mixed integer programming library: MIPLIB 3.0. Optima 58, pp.12–15, . Cook, W.J., Kannan, R., Schrijver, A., 1990. Chvátal closures for mixed integer programming problems. Mathematical Programming 47 (1–3), 155–174. Cornuéjols, G., 2007. Revival of the Gomory cuts in the 1990’s. Annals of Operations Research 149 (1), 63–66. Gomory, R.E., 1960. An algorithm for the mixed integer problem. Technical Report RM-2597, The RAND Cooperation. ILOG Inc., 2010. Cplex. . Koberstein, A., 2008. Progress in the dual simplex method for solving large scale LP problems: techniques for a fast and stable implementation. Computational Optimization and Applications 41 (2), 185–204. Koberstein, A., Suhl, U.H., 2007. Progress in the dual simplex method for large scale LP problems: practical dual phase 1 algorithms. Computational Optimization and Applications 37 (1), 49–65. Linderoth, J.T., Ralphs, T.K., 2005. Noncommercial software for mixed-integer linear programming. In: Karlof, J.K. (Ed.), Integer Programming: Theory and Practice. Operations Research Series. CRC Press, pp. 253–303. Mittelmann, H., 2010. Decision tree for optimization software: Benchmarks for optimization software. . Suhl, U.H., 1994. MOPS - mathematical optimization system. European Journal of Operational Research 72, 312–322.