Formation of Student Groups with the Help of Optimization

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Department of Mathematics

Formation of Student Groups with the Help

of Optimization

Kaj Holmberg

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Department of Mathematics Link¨oping University S-581 83 Link¨oping, Sweden.

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Formation of student groups with the help of optimization

Kaj Holmberg

Department of Mathematics Linköping Institute of Technology

SE-581 83 Linköping, Sweden

September 18, 2014

Abstract: We study the problem of forming groups of students so that the groups are as even as possible with respect to certain aspects, and formulate it as a mixed integer programming problem. We find that standard software cannot solve real life sized instances, so we develop several heuristics and metaheuristics for the problem. Computational tests are made on randomly generated instances as well as real life instances. Some of the heuristics give good solutions in short time, and tests on real life problems indicate that satisfactory solutions can be found within 60 seconds.

1 Introduction

In the IT-program at Linköping University, the students are each term divided into base groups, as a consequence of the problem based learning profile used. Each base group contains around 7 students, depending on the total number of students, which is between 20 and 40. This is done in five terms in sequence. Goals of the base group work are that each student should be active, participate in the discussions, and be able to work together with many different persons.

When forming a base group for a term, some aspects are taken into account. First of all, the number of students should be the same, if that is possible, or otherwise as similar as possible.

One wish is that the students should as much as possible have different comrades in the base group from term to term. The students should learn to work together in groups, and this is best done if the groups are as different as possible.

Another aspect is that after some time, the students show different progress in their studies. This is quantified by the number of examination points each student has obtained. While this measure does not exactly show how well each student will do in

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the base group work, it is still preferred that all the best students should not be in the same group and all the worst students should not be in the same group.

There are more aspects that may be interesting, for example gender. One might want to have similar mixes between boys and girls in the groups, especially if one gender is much less represented than the other. There are also other aspects which one might like to divide evenly among the groups, such as various computer skills etc. We will include a list of aspects that may include whatever one wishes, and the goal will be to spread the measures of the aspects as evenly as possible between the groups.

The formation of base groups has so far been done by hand, based on the information of each student’s points and all the previous groups the student has been in, and to some extent the gender. Sometimes additional information has been taken into account, such as two students not working well together, or two students working too well together (in which case they might dominate the group and give less room for the other students in the group).

There is not much research published in this area, and even less using optimization models and methods. We have not found any model in the literature that resembles ours and takes the aspects we think are important into account. For other models, we have found the following. In Graf and Bekele (2006) ant colony optimization was used, while Yeoh and Nor (2011) propose an ad-hoc heuristic. Spoelstra, van Rosmalen, van de Vrie, Obreza, and Sloep (2013) discuss several different measures, while Dascalu, Bodea, Lytras, de Pablos, and Burlacu (2014) focus on an interactive system, containing particle swarm optimization. Other publications dealing with the issue in various ways are Ounnas, Davis, and Millard (2009), Cocea and Magoulas (2010), Tourtoglou and Virvou (2010), Abnar, Orooji, and Taghiyareh (2012) and Cocea and Magoulas (2012). In this paper we describe a new optimization model for the formation of good groups, and some experiences in using it. In our tests, we first take the three factors number (number of students in each group), points (total number of points of the students in a group) and repetition (a measure of how much two students have been in the same group in previous time periods) into account, but open the model for more aspects, such as gender etc, as discussed in Graf and Bekele (2006).

2 The mathematical model

More specifically, we consider the following problem. We have n persons which shall be divided into m groups. There are q previous time periods, in which the persons have been allocated to different groups. We let g_jt be the number of the group that person j was allocated to in time period t. Apart from the number of persons in each group, we have p different aspects that we wish to spread as evenly as possible between the groups, the first one being the points each person has obtained, and the following anything that is of interest (for example gender etc). We let a_lj be the measure of aspect l for person j. For example, a1j is the number of points person j has obtained. and a2j may be 1 if person j is female and 0 if person j is male.

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Also needed are weights reflecting the importance we put on the different parts of the objective. We let wN be the weight on the wish that the number of persons should be the same in each group, and wT the weight on the wish that the persons should have different group partners from term to term. Furthermore, we let w_lP be the weight on the measure of aspect l, so wP₁ will be the weight on the wish that each group should have a good mixture of high and low points. It is only the relation between these weights that is interesting. However, they must reflect the scaling of the aspects, since the number of persons in our real life examples will be below 10, while the number of points will be up to 120. Unlike the previous indata, which is given with certainty, these weights may need to be adjusted so that they accurately model our wishes.

Furthermore, the disadvantage of putting two persons, that previously were in the same group, in the same group again, may decrease by time. It may not be that serious if they were in the same group several time periods ago. We call this discounting. Therefore we let wA_t be the (relative) weight of time period t, i.e. reflect how much we take previous groups into account. We let wA_t be one for the most recent time period (the last time period in the model), and decrease the weights as we go back in time. One possibility (which is used in our computational tests) is to set wA_t = 1/(1 + q − t). If we don’t use discounting, we set wA_t = 1 for all t.

The basic variable in the model is xij which is equal to 1 if person j is put in group j, and 0 if not. We also need the variable y_jk which is equal to 1 if persons j and k are put in the same group.

Furthermore, in order to control the properties of each group, we let vi be the number of persons in group i and z_li the sum of the measure of aspect l in group i (so z_1i is the total number of points in group i, and z2i could be the number of girls in group i, etc). We will also use vLand vU as the lowest and highest number of persons in a group, and z_lL and z_lU as the lowest and highest sum of measure of aspect l in a group.

The first and most obvious constraint in the model is that each person should be allo-cated to one group.

X i

xij = 1 for all j.

Then we need to get correct values of the additional variables. yjk should be equal to 1 if both person j and person k belong to some group i, i.e. if xij = 1 and xik = 1. We get the following constraints.

yjk ≥ xij+ xik− 1 for all j, k and i, j 6= k The values of v and z are simply obtained by summing up.

vi =X j

xij for all i. zli=

X j

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Then we get the variable upper and lower bounds on these variables as follows. vL≤ vi ≤ vU for all i.

z_lL≤ zli ≤ zlU for all i and l.

Finally we have the domain of the variables: x_ij ∈ {0, 1} for all i and j, y_jk ∈ {0, 1} for all j and k, zli≥ 0 for all i and l, vi ≥ 0 for all i, zLl , zUl ≥ 0 for all l, vL, vU ≥ 0. (v will automatically become integer, and the same holds for z if all measures a are integral. However, in our real life example, half points are possible.)

Now we need to formulate the objective function. The slightly difficult part is to mini-mize the negative effect of putting two persons in the same group. For this, it is more convenient to have the coefficients djkt = 1 if person j and person k were in the same group in time period t. These coefficients are easily calculated by initiating d to zero, and then setting d_jkt = 1 if gjt= gkt.

The total disadvantage of putting person j in the same group as person k is then cjk =

X t

wTwA_tdjkt for all j and k, j 6= k.

We multiply this by y_jk, sum this over all persons j and k and put it in the objective function. (If we are using discounting, all of c will not be integer, so non-integral objective function values will be possible.)

Next we need to address the number of persons and points in each group. Since we know that vL_{≤ v}U_{, we simply wish to minimize v}U _{− v}L_{, with weight w}N_{. The same} reasoning applies to the number of points and other aspects.

Thus we obtain the following objective function. min X j X k cjkyjk+ X l wP_l (zU_l − zL_l ) + wN(vU− vL) (We let c_jj = 0, and can remove yjj from the model.)

One may note that we only take pairwise effects into account for previous groups. It is conceivable that certain subgroups of three or more persons may have other effects that what is measured by this. These factors were not possible to take into account when this was done by hand. What is possible in this model is to introduce a new aspect with the single purpose of keeping three or more persons apart, by letting a_lj = 1 for those persons and a_lj = 0 for the others, similarly to what was done for gender. In general, this optimization model allows taking more aspects into account then what is possible by hand.

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min X j X k cjkyjk+ X l wP_l (z_lU− zL_l ) + wN(vU − vL) s.t. X i xij = 1 ∀j (1) yjk− xij− xik ≥ −1 ∀j, k, i (2) vi−X j xij = 0 ∀i (3) zli− X j aljxij = 0 ∀i, l (4) vL≤ vi≤ vU ∀i (5) zL_l ≤ z_li ≤ z_lU ∀i, l (6) xij ∈ {0, 1} ∀i, j (7) yjk ∈ {0, 1} ∀j, k (8) zli≥ 0 ∀i, l (9) vi≥ 0 ∀i (10) zL_l , zU_l ≥ 0 ∀l (11) vL, vU ≥ 0 (12)

The size of the model is as follows. There are mn x-variables, n2 _{y-variables, pm} z-variables, m v-z-variables, plus the 2p + 2 bounds, i.e. a total of n2+ mn binary variables and (p + 1)m + 2p + 2 continuous variables. As for the constraints, there are n in the first group and n2q in the second. The following constraints are (2p + 2)m, of which half are variable upper and lower bounds.

3 Constructive heuristics

Preliminary tests (see section 5) reveal that the MIP-model is too hard to be solved by standard MIP-software. (We have tried GLPK and CPLEX.) Therefore we have developed a few constructive heuristics for finding feasible, hopefully good, solutions. The heuristics sequentially build up a feasible solution, by allocating one person at a time to a group. Only information about the previously allocated persons are taken into account.

We start by sorting the persons in order based on the points. Then we allocate the best person first, then the second best, and so on. In each step, when considering a certain person, we calculate penalties for the existing groups based on c, i.e. add a penalty if the previously allocated persons in a group has been in the same group as the person in question in earlier time periods. This is heuristic A.

In heuristics B, C and D, we also add penalties based on the number of persons in each group, and the total number of points in each group etc, as described below.

Let J ⊆ 1, . . . , n be the persons for which the group allocation has been done, and let αj be the group person j is allocated to. Also let Ji ⊆ J be the set of persons that are allocated to group i, i.e. J_i= {j : αj = i}.

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The measures to keep into account are the following. ¯

vi = |Ji| for all i (the number of persons in each group) ¯

zli= X j∈Ji

alj for all i and l (the measure of aspect l in each group) ¯

ri= X j∈Ji

X k∈Ji,k6=j

cjk for all i (the total repetition in each group) With these, we sum up a total cost for allocating person j to group i as follows.

¯

ti = ¯ri+ ¯si for all i

where ¯si is given below, in different ways for the different heuristics. Then we simply choose the group that yields the minimal total cost.

ˆi = argmin_i_ti¯

After this we set J = J ∪ j, Jˆ_i = Jˆ_i∪ j, αj = ˆi, and repeat for the next person. In heuristic A, we set ¯si = 0.

In heuristic B, we use the following, inspired by the objective function. ¯

si = wNvi¯ +X l

w_lPz¯li for all i

In heuristic C, we do a more elaborate evaluation of including person j in group i for the numbers and other aspects. If the group has less than the maximal number of persons, i.e. if ¯vi < ¯vU, then adding a person to the group does not increase the maximal number of students in a group. Then the penalty is set to zero. If ¯vi = ¯vU, then adding a person to the group will increase the maximal number of students in a group, and the penalty will be wN. Furthermore, if the number of persons in the group is equal to the minimal number, and all other groups have more persons, i.e. if ¯vi = ¯vL and ¯vk > ¯vL for all k 6= i, then the minimal number of persons in a group will increase, which is good. Then we set the penalty to −wN.

The other aspects are handled similarly, but is slightly more complicated, since the coefficient a_lj may be other than zero or one. If ¯zli+ alj ≤ ¯zUl (and ¯zli> ¯zlL), then the penalty will be zero, while if ¯zli+ alj > ¯zlU, we use the penalty wlPalj. If ¯zli = ¯zlL, and ¯

zlk > ¯z_lL for all k 6= i, we use the penalty −wP_l alj.

Therefore, in heuristic C, we let LU_i = {l : ¯zli + alj > ¯zlU} and LLi = {l : ¯zli = ¯

z_lL and ¯zlk> ¯z_lL for all k 6= i}, and calculate ¯ sP_i = X l∈LU i wP_l alj− X l∈LL i

wP_l alj for all i.

Furthermore, for each i, we let ¯sN_i = wN if ¯vi= ¯vU, ¯s_iN = −wN if ¯vi = ¯vLand ¯vk> ¯vL for all k 6= i, and ¯sN_i = 0 otherwise. We then use

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¯

si = ¯sNi + ¯sPi for all i

In heuristic D, we take this further and calculate the exact increase/decrease of the bounds for the aspects. For the numbers, there is no difference, but for the other aspects, we find the following. If ¯zli+ alj > ¯z_lU, then the upper bound will not increase by a_lj, but only by ¯zli+ alj− ¯zlU, so the penalty will be wlP(¯zli+ alj− ¯zlU).

Furthermore, the increase of the lower bound will be ¯zli+ alj− ¯z_lL, so the penalty will be −w_lP(¯zli+ alj− ¯zlL). So in heuristic D, we use ¯ sP_i = X l∈LU i wP_l (¯zli+ alj− ¯zU_l ) − X l∈LL i

w_lP(¯zli+ alj− ¯z_lL) for all i. Otherwise we use the same values as in heuristic C.

A partly manual version of the heuristics has been constructed. The difference is only that the allocation is made by the user. The method loops over the persons, highest points first, presents the current situation (of the partly fixed solutions, ¯vi, ¯zli, ¯ri and ¯ti for each i) and asks the user to decide in which group to put the person. This is possible as long as the number of groups is not to large. In our real life case m is around 5, and we believe that the user will be able to quickly scan five values. It is also possible for the user to deviate from the given weights by taking certain aspects more into account, and to take other (non-numerical) aspects into consideration.

4 Local search and metaheuristics

It is also possible to use metaheuristics on the problem, and we have chosen tabu search, Glover (1989), Glover (1990), and simulated annealing, Kirkpatrick, Gelatt, and Vecchi (1983), Cerny (1985). Considering neighborhoods for local search, we can use move, i.e. moving a person from one group to another, and swap, i.e. swapping groups between two persons. We have implemented the following methods with these two neighborhoods. Method 20: Heuristic A.

Method 21: Start by heuristic A. Then check all move-neighbors once, directly do all changes that yield improvement.

Method 22: Start by heuristic A. Then check all swap-neighbors once, directly do all changes that yield improvement.

Method 23: Start by heuristic A. Then check all move-neighbors repeatedly, doing all changes that yield improvement, until no more improvement is obtained.

Method 24: Start by heuristic A. Then check all swap-neighbors repeatedly, doing all changes that yield improvement, until no more improvement is obtained.

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Method 25: Start by heuristic A. Then use methods 23 and 24 mixed, i.e. iterate between move and swap moves, doing all changes that yield improvement, until no more improvement is obtained.

Method 26: Start by heuristic A. Then run tabu search with moves. In each step the best move is made.

Method 27: Start by heuristic A. Then run tabu search with swaps. In each step the best swap is made.

Method 28: Start by heuristic A. Then run simulated annealing with moves. Method 29: Start by heuristic A. Then run simulated annealing with swaps. Methods 30, . . . , 39 are the same as 20, . . . , 29, but initiated with heuristic B. Methods 40, . . . , 49 are the same as 20, . . . , 29, but initiated with heuristic C. Methods 50, . . . , 59 are the same as 20, . . . , 29, but initiated with heuristic D. Methods 60, . . . , 69 are the same as 20, . . . , 29, but initiated with a random solution.

5 Computational tests

The model is implemented in GMPL and solved by glpsol. Calculation of the coefficients d and c is done by a code in Python, which also reads indata in a simple format and writes an indata file in GMPL/AMPL-format. The heuristics are also implemented in Python. Some tests are also done with CPLEX, just in order to check if a better code easily solves all problems of interest.

The main tests are run on computer 1, which is an Acer Aspire X3 X3995, Intel Core i7-3770, 3.4GHz, running Linux, Fedora Core 19, and the CPLEX-tests are run on computer 2, which is a HP Compaq 8100 Elite, Intel Core i5-650 3,2 GHz, running Linux, CentOS 6. Only one processor was used during the runs with the Python code and GLPK, while CPLEX uses more (up to 4). The CPU-time was limited to 60 seconds.

The tabu search was made with a tabu list length of 7, and 100 iterations. For simulated annealing, the starting temperature was 100, the cooling factor 0.7, and 50 outer itera-tions was made, each with 20 inner iteraitera-tions with the same temperature. We did not attempt to optimize the values of these parameters, so it is likely that the results can be improved somewhat by tuning of the methods. (Therefore we believe that a comparison to GLPK is more relevant than to CPLEX or another highly optimized code.)

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5.1 Test problems

Some artificial test problems of different sizes were generated and solved. There is only one aspect in the first set of instances, namely the points, which were randomly generated between 10 and 100. The group allocation in the first of the previous time periods was to put person 1 in group 1, person 2 in group 2, . . . person m in group m, person m + 1 in group 1, person m + 2 in group 2, etc. The subsequent allocations were random permutations of the first. The parameter values are wP = 10, wN = 10 and wT = 100, and we use discounting. The maximal size of these instances is n = 35, m = 5 and q = 4, which is the approximate size of the real life instance we wish to solve.

The second set of test problems contains more instances of different sizes. They are similar to the first, but with other weights, namely wP = 1, wN = 10 and wT = 100. The third set of test problems has two aspects (points and gender), but the same parameter setting as the second set. In order to better investigate the limits of our solutions methods, more instances with sizes at the higher end are used. The maximal size of these instances is n = 40, m = 6 and q = 5.

Finally a fourth set of problems contains three real life instances, which however are rather similar.

The two first set of problems have been used for preliminary tests, mainly to see how far exact optimization with the MIP-model can go. The third and the fourth sets of problems are considered to be the main tests.

5.2 Computational results

In table 1 we give the results by the first tests with GLPK on computer 1. The table gives problem size, in the form of n, m and q, as well as the number of rows, columns and binary variables. The number of nodes in the branch-and-bound tree are given, as well as the solution time and the objective function value (rounded). If the CPU-time of 60 seconds was not enough to finish the solution procedure, we report the number of nodes at termination, the remaining nodes + those investigated, as a lower bound on the total number of nodes, and the best integer solution found, as an upper bound on the optimal value (the usage of < does not necessarily mean that the optimal value is strictly lower).

Table 2 give the results of the initial constructive heuristics, on computer 1. Here we see that several methods yield the optimal solution to inst1-1, and a couple for inst1-2. Some other heuristics, however, yield solutions that are not very good. For inst-13, inst1-4 and inst1-5, GLPK yields the best solutions, which are optimal for inst1-3 and inst1-4. For inst1-6, method 32 gives the best solution, while for inst1-7, method 22 gives the best solution. Preliminary conclusions are that GLPK cannot solve the larger problems to optimality, and that heuristics using swap seem to be stronger than heuristics using move. The simple methods 20 and 30 may give quite bad solutions, see for example

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Instance n m q Rows Cols Bin Nodes Time Obj inst1-1 5 2 2 68 43 35 31 0.0 710 inst1-2 10 3 2 329 140 130 99 0.1 420 inst1-3 12 3 3 463 190 180 2651 2.2 1160 inst1-4 15 4 3 940 297 285 15497 28.5 420 inst1-5 20 5 3 2051 514 500 >5394+1350 >60 <317 inst1-6 30 5 4 4561 1064 1050 >2271+67 >60 <3560 inst1-7 35 5 4 6191 1414 1400 >1240+147 >60 <5323

Table 1: Results by GLPK on the first problem set. Instance n m q GLPK M20 M21 M22 M30 M31 M32 inst1-1 5 2 2 710 1150 730 710 710 710 710 inst1-2 10 3 2 420 700 700 420 810 810 420 inst1-3 12 3 3 1160 2340 1530 1230 1960 1780 1450 inst1-4 15 4 3 420 1743 1146 673 1310 1310 693 inst1-5 20 5 3 <317 2093 1053 623 983 983 476 inst1-6 30 5 4 <3560 3573 3020 3080 3390 3313 2836 inst1-7 35 5 4 <5323 6193 4890 4606 5296 5183 4620 Table 2: Results by the heuristics on the first problem set. inst1-3 and inst1-4.

Table 3 give the results for CPLEX (on computer 2). Here we see that CPLEX is better than GLPK, but not good enough to solve the larger problems to optimality in 60 seconds. (There is one unit difference in the optimal objective function value for some instances, which is due to different rounding in the output.)

A conclusion of the firsts tests is that problems of realistic size cannot be solved by GLPK or CPLEX. Tests allowing more than 60 seconds reveal that the time is increasing quickly, and solving for example inst1-5 would take several hours. Another conclusion is that the initial heuristics sometimes yield good solutions, but sometimes quite bad. The second set of tests have wP = 1, wN = 10 and wT = 100, and discounting. The weight on the points is more realistic than in the first set, so we rely a bit more on these tests. Here we only made tests with the MIP-model, which again were unable to solve problems of interesting sizes. The results can be found in tables 7 and 8 in the

Instance n m q Nodes Time Obj inst1-1 5 2 2 0 0.0 710 inst1-2 10 3 2 31 0.07 420 inst1-3 12 3 3 1512 0.14 1159 inst1-4 15 4 3 802 0.47 419 inst1-5 20 5 3 32970 35.82 233 inst1-6 30 5 4 >58006 >60 <2799 inst1-7 35 5 4 >35396 >60 <4799

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Instance n m q p Rows Cols Iter Nodes Obj1 Obj2 Time inst3-1 10 3 2 2 339 145 6033 453 377 0.0 0.2 inst3-2 10 3 2 2 339 145 1648 99 339 0.0 0.1 inst3-3 12 3 3 2 473 195 30366 1709 1126 0.0 1.5 inst3-4 12 3 4 2 473 195 28729 1311 1269 0.0 1.2 inst3-5 15 3 4 2 719 285 244979 11157 2534 0.0 17.7 inst3-6 17 3 4 2 913 355 624661 5264 3466 2870.5 60.0 inst3-7 17 4 4 2 1211 375 309396 9629 609 0.0 26.0 inst3-8 20 4 4 2 1658 498 498447 1118 1728 388.0 60.0 inst3-9 21 4 4 2 1823 543 434746 874 2251 303.9 60.0 inst3-10 23 4 4 2 2177 639 341924 409 3049 356.5 60.1 inst3-11 23 5 4 2 2715 665 325924 885 539 52.0 60.1 inst3-12 25 5 4 2 3197 771 265773 323 839 65.0 60.2 inst3-13 27 5 4 2 3719 885 225309 446 1781 108.5 60.1 inst3-14 29 5 4 2 4281 1007 186192 155 2372 105.0 60.2 inst3-15 30 5 4 2 4577 1071 167016 41 2666 22.6 60.2 inst3-16 32 5 4 2 5199 1205 145616 28 3539 31.2 60.3 inst3-17 35 5 4 2 6207 1421 127429 141 4712 113.2 60.4 inst3-18 36 6 4 2 7868 1536 112455 96 1433 0.0 60.5 inst3-19 38 6 5 2 8758 1696 93825 164 2558 10.0 60.6 inst3-20 40 6 5 2 9696 1864 91725 11 3779 0.0 60.8

Table 4: Results by GLPK on the third problem set.

Appendix. We see that CPLEX is better than GLPK, but not efficient enough. Both codes run into difficulties when the number of persons is around 20 or more. Another observation is that inst2-7 is easier to solve than inst2-6, although inst2-7 has one group more, and thus is a larger problem. The same holds for inst2-8 and inst2-9.

The third set of instances has two aspects, the points and the gender (or any other binary measure). In table 4, we give the results from GLPK (on computer 1). We give n, m, q, p, followed by the number of rows and columns of the MIP-model, the total number of simplex iterations and the number of branch-and-bound nodes that were investigated. We give two objective function values, the first, obj1, is the optimal value, if optimality was verified, which can be seen by a solution time less than 60, or the best upper bound that was found in 60 seconds. The second value is only given if optimality is not verified, and is the best lower bound that was obtained in 60 seconds.

Some reflections of table 4 are that increasing q, as done between inst3-3 and inst3-4 does not seem to increase the difficulty of the problem. Increasing n, as between inst3-5 and inst3-6, definitely makes the problem harder to solve. And again we find that increasing m, as between inst3-6 and inst3-7, may make the problem easier to solve. In tables 9 - 13 in the Appendix, we give the results for all the heuristics, methods 20 - 69. We have also added the results from GLPK for comparison. The tables give the instance name on top, the name of the method, followed by the objective function value and the solution time for each instance. All these test are made on computer 1. Here we find that for small problems, some heuristics give the optimal solution in a

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fraction of the time needed by GLPK. For larger problems, some of the heuristics yield better solutions in much shorter time. In no case does GLPK yield a better solution than all heuristics (in 60 seconds).

In more detail, we find the following. For the smallest problem, inst3-1, many methods yield the optimal solution (377). The fastest of those is method 22 (as well as 32, 42, 52 and 62), which only needs 0.02 seconds. For inst3-2 even more methods yield the optimal value (339), and here methods 41 and 50 are the fastest. Similar results hold for inst3-3 and inst3-4, where the best method 62 (and 32) yield the optimal solution in 0.03 seconds.

For problems up to this, GLPK is a viable option. However, for inst3-5 and upwards, GLPK needs more time than the alternatives. For inst3-5 only four methods give the optimal value (2534), with method 49 as the fastest, 0.57 seconds.

For inst3-6, GLPK failed to find the optimum within 60 seconds. The best solution was found by five methods, of which method 64 is the fastest, 0.17 seconds.

Inst3-7 is slightly easier than inst3-6, and here GLPK and six other methods give the optimal solution. Method 62 does this in 0.12 seconds, while GLPK needs 26 seconds. From inst3-8 and upwards, GLPK fails to find and/or verify the optimum within 60 seconds, so for these problems, we don’t know the optimal solution. For inst3-8, method 57 gives the best solution in 17.63 seconds. One interesting question is how fast a method can yield a solution that is better than what GLPK gave in 60 seconds. Here method 64 does this in 0.36 seconds.

For inst3-9, method 37 gives the best solution in 21.33 seconds. Here 29 methods give better solutions than GLPK, often in much shorter time, the fastest being 0.21 seconds for methods 32 and 42.

For inst3-10, method 47 gives the best solution, in 29.29 seconds. Also here many (32) methods give better solutions than GLPK, and the fastest is 0.06 seconds for method 61.

For inst3-11, method 57 gives the best solution, in 30.18 seconds. Here the GLPK-solution is rather good, and only 6 methods find better GLPK-solutions.

For inst3-12, method 67 gives the best solution, in 40.61 seconds, while method 32 finds a better solution than GLPK in 0.40 seconds.

For inst3-13, method 67 gives the best solution, in 53.10 seconds, while methods 22 and 32 give a better solution than GLPK in 0.52 seconds. For inst3-14, method 37 gives the best solution, in 60 seconds. Here the method is terminated because of the time limit of 60 seconds. Even so, it found a much better solution than GLPK. A slightly better solution than GLPK was found in 0.02 seconds by method 50. For these more difficult problems, we find that GLPK does not find a good solution at all within 60 seconds, so it is easy for a heuristic to find a better one.

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67 gives the best solution, in 60 seconds. For both these instances, a simple method as 20 finds a better solution than GLPK in 0.02 seconds.

For inst3-17, method 64 gives the best solution, in 5.23 seconds. For inst3-18, method 47 gives the best solution, in 60 seconds. For inst3-19, method 47 gives the best solution, in 60 seconds. For inst3-20, method 67 gives the best solution, in 60 seconds. For these harder problem, the stronger heuristics are terminated because of the maximal time allowed.

Concerning the comparison to GLPK, we find that for easy problems, is is not that hard to find the optimal solution. For hard problems, GLPK gives a quite bad solution in 60 seconds, so then it is easy to find a better solution. In intermediate cases, GLPK finds a rather good solution, so the heuristics need to work more to find a better solution, and some of them fail to find a better solution. However, we can note that in all cases, some heuristic finds a better or equal solution in much shorter time.

Summing up the number of times a method gives the best solution in shortest time (of the ones found), we get the following result. One best solution was found by the methods 22, 41, 42, 49, 50, 52 and 65. Many of these were best only for the smallest problems. Two best solutions were found by methods 32, 57 and 64. Three best solutions were found by method 47. Four best solutions were found by methods 62 and 67. Five best solutions were found by method 37, and this is the maximum, so by this measure, method 37 would be a winner. However, methods 47 and 67 are also rather strong. All in all we find that methods ending with 7, i.e. use tabu search with swaps, are quite strong. They take more time than many other heuristics, but yield better solutions. A comparison of the simple and very fast one-shot heuristics 20, 30, 40, 50 and 60 is made in table 5. The time is in principle negligible for all of them, so the question is simply which one gives the best solutions. Obviously the random solution (60) is worse by far (which however does not stop it from giving good final results of the metaheuristics). Apart from this, the conclusions are not very evident. Summing up, method 20 gives the best solution in 4 cases, method 30 in 4 cases, method 40 in 10 cases and method 50 in 5 cases. Thus each method has its merits, but method 40 seems to be slightly better than the others. It is interesting to see that the simple method 20 is almost as good as the more advanced method 50.

Finally tests were made on real life data. The first problem has n = 31, m = 5, q = 4 and p = 1, which yields 4868 rows, 1130 columns, 15284 non-zeros and 1116 binary variables in the MIP-model. The second problem has n = 30, which gives 4562 rows, 1064 columns, 14327 non-zeros and 1050 binary variables. The third problem has p = 2 (points and gender), but is otherwise similar to the first problem. Table 6 gives the results of these tests for GLPK, and table 14 in the Appendix gives the results for the heuristics.

For all of these three problems, method 20 gives a better solution in 0.01 seconds than GLPK did in 60 seconds. These three problems also have in common that method 37 yields the best solution (using all the 60 seconds).

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Method Obj Time Obj Time Obj Time Obj Time inst3-1 inst3-2 inst3-3 inst3-4 20 396.00 0.00 449.00 0.00 1217.00 0.00 1413.67 0.00 30 387.00 0.01 339.00 0.01 1207.67 0.00 1441.00 0.01 40 378.00 0.00 359.00 0.00 1345.00 0.00 1356.67 0.01 50 387.00 0.00 339.00 0.00 1401.67 0.01 1441.00 0.00 60 835.00 0.00 968.00 0.00 1362.00 0.00 2414.00 0.00

inst3-5 inst3-6 inst3-7 inst3-8 20 2760.67 0.00 3821.00 0.00 1154.33 0.01 2069.33 0.01 30 2682.67 0.01 3809.00 0.01 1080.67 0.00 2391.00 0.01 40 2644.67 0.01 3809.00 0.00 1290.33 0.00 2095.67 0.02 50 2682.67 0.00 3933.00 0.00 1080.67 0.01 2095.67 0.01 60 2940.00 0.00 5191.00 0.00 2676.67 0.00 3071.00 0.01

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Instance n m q p Rows Cols Iter Nodes Obj1 Obj2 Time instre1 31 5 4 2 4868 1130 163134 357 2592 228.7 60.3 instre2 30 5 4 1 4562 1064 173501 128 2404 194.4 60.2 instre3 31 5 4 2 4883 1137 162922 272 2986 223.7 60.3

Table 6: Results by GLPK, computer 1, on the fourth problem set.

For the first problem, the objective function value of GLPK was 2592 after 60 seconds, while heuristic 20 yields objective function value 2530 after 0.01 second. Local search with move-neighborhood (method 21) decreased it slightly to 2526. Local search with swap-neighborhood (method 22) decreased it to 2493. Continued local search (methods 23 and 24) does not improve these values, but using both neighborhoods (method 25) improves it to 2323. Tabu search with move-neighborhood (method 26) gave the value 2396, while tabu search with swap-neighborhood (method 27) decreased it to 2129. Using simulated annealing (instead of tabu search) (methods 28 and 29) yields 2478 and 2530, so here tabu search with swaps is the best method.

Starting with heuristic B (methods 30-39) gives similar results, most solutions somewhat worse, but slightly better for method 37. Starting with heuristic C (methods 40-49) gives in general worse solutions, and the same applies for heuristic D (methods 50-59) as well as starting with a random solution (methods 60-69).

Similar results were obtained for the second and third problems.

Methods 30 and 20 are very similar to the procedure used when this is done by hand, and it is interesting to see that both these give a better solution than GLPK in a much shorter time. For these problems, the simple heuristics give better solutions than GLPK very quickly, and tabu search improves these solutions further. However, in a fair amount of cases, a random starting solution (which may be very bad) enables the metaheuristics to find even better solutions.

5.3 Summary of the computational tests

We find that for small instances, a MIP-solver can be used, but when the size grows, it is better to use heuristics. For large problems, even the simplest heuristics produce better solutions than the MIP-solver. Further improvement can be obtained by many of the methods, but the best method seems to be tabu search with swap neighborhood. However, the performance of simulated annealing can probably be improved by tuning the parameters.

Considering the starting solution, it is interesting to see that although method 20 some-times gives lower objective function value than method 30, the final result is often worse than starting by 30. Also random starting points can yield good results. Heuristics C and D can be seen as more intricate extensions of A and B, so it is slightly surprising to see that they are not always better.

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6 Conclusions

We have formulated the problem of forming suitable groups as a MIP-problem. Com-putational tests reveal that standard MIP-codes, such as GLPK and CPLEX, cannot solve problems of realistic size.

We present a couple of constructive heuristics for the problem, and describe how meta-heuristics, such as tabu search and simulated annealing, can be used to improve the solutions. Computational results are presented, showing that tabu search with swap neighborhood is a good method for the problem.

The code, together with an interface in Python and Tkinter that allows manual changes, will be used in the future when forming certain student groups at Linköping University.

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Appendix

Instance n m q Rows Cols Bin Nodes Time Obj inst2-1 5 2 2 69 43 32 5 0.0 317 inst2-2 10 3 2 330 140 130 77 0.0 331 inst2-3 12 3 3 464 190 180 1531 1.3 1110 inst2-4 13 2 3 365 203 195 375 0.3 5512 inst2-5 13 3 5 540 218 201 2435 2.4 1850 inst2-6 15 3 4 710 280 270 197554 30.5 2506 inst2-7 15 4 4 941 297 285 5221 9.6 333 inst2-8 20 4 4 1646 492 475 >7822+2075 >60 <1838 inst2-9 20 5 4 2052 508 500 >5163+1186 >60 <222 inst2-10 24 5 4 2936 710 696 >4095+357 >60 <872 inst2-11 27 5 4 3704 872 864 >2598+401 >60 <1600 inst2-12 30 5 4 4562 1064 1050 >2142+233 >60 <2646 inst2-13 33 5 4 5510 1268 1254 >1627+84 >60 <3849 inst2-14 35 5 4 6192 1414 1400 >1266+106 >60 <4751 Table 7: Results by GLPK, computer 1, on the second problem set. Instance n m q Rows Cols Bin Nodes Time Obj inst2-1 5 2 2 69 43 32 0 0.02 317 inst2-2 10 3 2 330 140 130 0 0.04 331 inst2-3 12 3 3 464 190 180 516 0.21 1109 inst2-4 13 2 3 365 203 195 319 0.16 5511 inst2-5 13 3 5 540 218 201 811 0.41 1849 inst2-6 15 3 4 710 280 270 9743 7.04 2505 inst2-7 15 4 4 941 297 285 332 0.23 332 inst2-8 20 4 4 1646 492 475 >72552 >60 <1713 inst2-9 20 5 4 2052 508 500 2173 2.72 212 inst2-10 24 5 4 2936 710 696 >50325 >60 <550 inst2-11 27 5 4 3704 872 864 >70205 >60 <1194 inst2-12 30 5 4 4562 1064 1050 >60649 >60 <2274 inst2-13 33 5 4 5510 1268 1254 >44907 >60 <3477 inst2-14 35 5 4 6192 1414 1400 >37273 >60 <4385

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inst3-1 inst3-2 inst3-3 inst3-4 Method Obj Time Obj Time Obj Time Obj Time 1 377.00 0.20 339.00 0.10 1126.00 1.50 1269.00 1.20 20 396.00 0.00 449.00 0.00 1217.00 0.00 1413.67 0.00 21 396.00 0.01 423.00 0.01 1147.00 0.01 1305.00 0.01 22 377.00 0.02 356.00 0.02 1199.00 0.03 1405.00 0.03 23 396.00 0.01 423.00 0.01 1147.00 0.02 1305.00 0.02 24 377.00 0.03 356.00 0.04 1199.00 0.07 1405.00 0.06 25 377.00 0.04 339.00 0.07 1126.00 0.07 1269.00 0.08 26 377.00 0.57 342.00 0.55 1126.00 0.78 1290.00 0.81 27 377.00 1.64 339.00 1.73 1196.00 2.79 1405.00 2.98 28 379.00 0.34 339.00 0.35 1126.00 0.42 1290.00 0.42 29 377.00 0.33 356.00 0.35 1199.00 0.43 1389.67 0.42 30 387.00 0.01 339.00 0.01 1207.67 0.00 1441.00 0.01 31 387.00 0.01 339.00 0.01 1207.67 0.02 1441.00 0.02 32 377.00 0.02 339.00 0.02 1126.00 0.03 1291.00 0.03 33 387.00 0.01 339.00 0.01 1207.67 0.01 1441.00 0.01 34 377.00 0.04 339.00 0.02 1126.00 0.07 1269.00 0.10 35 377.00 0.04 339.00 0.03 1126.00 0.08 1269.00 0.13 36 377.00 0.56 339.00 0.54 1133.00 0.82 1290.00 0.82 37 377.00 1.63 339.00 1.66 1126.00 2.96 1269.00 3.06 38 387.00 0.33 339.00 0.34 1207.67 0.42 1417.00 0.43 39 379.00 0.35 339.00 0.32 1153.00 0.42 1269.00 0.41 40 378.00 0.00 359.00 0.00 1345.00 0.00 1356.67 0.01 41 378.00 0.01 339.00 0.00 1345.00 0.01 1356.67 0.02 42 377.00 0.02 356.00 0.02 1126.00 0.04 1269.00 0.04 43 378.00 0.01 339.00 0.02 1345.00 0.01 1356.67 0.01 44 377.00 0.04 356.00 0.03 1126.00 0.07 1269.00 0.06 45 377.00 0.05 339.00 0.05 1126.00 0.08 1269.00 0.08 46 377.00 0.53 339.00 0.57 1157.00 0.84 1269.00 0.79 47 377.00 1.61 339.00 1.68 1126.00 3.40 1269.00 2.98 48 378.00 0.35 339.00 0.34 1233.00 0.42 1356.67 0.44 49 377.00 0.35 356.00 0.33 1126.00 0.42 1290.00 0.43 50 387.00 0.00 339.00 0.00 1401.67 0.01 1441.00 0.00 51 387.00 0.01 339.00 0.01 1248.67 0.02 1441.00 0.01 52 377.00 0.02 339.00 0.02 1126.00 0.04 1291.00 0.04 53 387.00 0.01 339.00 0.01 1248.67 0.02 1441.00 0.01 54 377.00 0.04 339.00 0.02 1126.00 0.07 1269.00 0.10 55 377.00 0.05 339.00 0.03 1171.00 0.08 1269.00 0.12 56 377.00 0.59 339.00 0.59 1126.00 0.81 1290.00 0.82 57 377.00 1.63 339.00 1.60 1126.00 2.95 1269.00 2.98 58 387.00 0.34 339.00 0.35 1168.00 0.41 1421.00 0.45 59 377.00 0.33 339.00 0.34 1157.00 0.41 1269.00 0.42 60 835.00 0.00 968.00 0.00 1362.00 0.00 2414.00 0.00 61 439.00 0.00 549.00 0.00 1169.00 0.01 1587.00 0.01 62 377.00 0.02 339.00 0.02 1126.00 0.03 1269.00 0.03 63 380.00 0.01 549.00 0.02 1246.00 0.01 1438.00 0.02 64 377.00 0.04 356.00 0.04 1126.00 0.13 1269.00 0.06 65 379.00 0.05 339.00 0.07 1171.00 0.08 1269.00 0.10 66 378.00 0.59 342.00 0.54 1133.00 0.84 1290.00 0.84 67 377.00 1.61 339.00 1.66 1126.00 3.02 1269.00 3.01 68 401.00 0.33 557.00 0.33 1166.00 0.41 1492.00 0.47 69 435.00 0.34 339.00 0.31 1126.00 0.41 1302.00 0.41

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instre1 instre2 instre3 Method Obj Time Obj Time Obj Time 1 2591.67 60.30 2404.33 60.20 2985.67 60.30 20 2530.33 0.01 2243.67 0.01 2552.67 0.01 21 2526.33 0.14 2182.67 0.13 2496.67 0.14 22 2493.33 0.79 2186.67 0.71 2519.67 0.86 23 2526.33 0.25 2182.67 0.25 2496.67 0.29 24 2493.33 1.60 2186.67 1.39 2488.67 2.45 25 2323.00 3.50 2036.33 3.20 2455.67 2.96 26 2396.33 13.03 2076.00 12.30 2482.67 13.91 27 2129.33 60.17 1934.00 60.14 2247.33 60.28 28 2487.67 1.72 2182.67 1.55 2482.67 1.98 29 2530.33 1.71 2243.67 1.65 2552.67 1.91 30 2872.33 0.01 2473.00 0.01 2720.67 0.01 31 2872.33 0.15 2473.00 0.13 2691.00 0.16 32 2509.67 0.77 2007.33 0.68 2501.00 0.85 33 2872.33 0.14 2473.00 0.14 2691.00 0.30 34 2259.00 2.54 2007.33 1.46 2501.00 1.68 35 2251.33 3.49 2007.33 1.70 2344.33 3.11 36 2659.67 12.93 2293.00 11.73 2491.67 14.31 37 2119.33 60.69 1881.33 60.25 2152.33 60.04 38 2746.00 1.65 2234.00 1.55 2720.67 1.91 39 2593.00 1.67 2235.67 1.62 2356.00 1.76 40 3230.67 0.01 2684.33 0.01 2887.33 0.02 41 3114.67 0.14 2527.67 0.13 2633.33 0.17 42 2560.00 0.78 2194.67 0.75 2532.33 0.92 43 3005.67 0.38 2527.67 0.26 2633.33 0.31 44 2238.00 2.29 2194.67 1.55 2462.00 2.92 45 2315.67 2.84 1981.33 2.36 2201.33 4.23 46 2284.67 13.27 2405.67 13.09 2461.00 14.89 47 2418.67 60.10 2185.67 60.48 2354.67 60.04 48 2809.33 1.62 2639.33 1.55 2474.67 1.84 49 2463.67 1.58 2163.33 1.53 2542.00 1.80 50 3516.33 0.01 2994.67 0.01 3327.67 0.01 51 3311.67 0.15 2698.33 0.13 2995.00 0.16 52 2553.00 0.78 2254.00 0.68 2536.67 0.86 53 3118.67 0.42 2485.67 0.34 2737.67 0.57 54 2532.33 2.27 2222.33 2.09 2536.67 1.72 55 2303.67 3.55 1894.33 2.53 2316.33 3.08 56 2637.67 14.52 2312.33 11.86 2602.00 14.76 57 2337.67 60.02 1918.33 60.17 2169.33 60.56 58 2615.00 1.61 2459.33 1.55 2514.67 1.82 59 2408.67 1.71 2239.33 1.55 2693.67 1.83 60 5522.33 0.00 5533.67 0.01 5961.33 0.00 61 3302.00 0.14 2951.33 0.12 2678.67 0.15 62 2415.00 0.75 2088.33 0.70 2493.33 0.90 63 2933.00 0.26 2648.00 0.23 3213.00 0.29 64 2367.00 3.07 2108.00 1.72 2230.33 3.42 65 2298.67 1.83 2169.33 2.52 2346.00 4.03 66 2632.00 13.08 2156.67 12.14 2503.67 14.06 67 2231.00 60.46 1925.33 60.59 2198.33 60.79 68 2959.00 1.75 2467.33 1.51 2709.67 1.90 69 2447.67 1.63 2052.00 1.49 2612.00 1.83