Heuristic approaches for the DRFLP

There are three reasons for constructing double-row layouts based on single-row layouts. At first, good or optimal single-row layouts can be obtained very fast, see, e. g., [34, 58, 91, 93]. At second, our computational results, see Section 3.5.4, indicate that by going from single-row layouts to double-row layouts the objective value is approximately halved. And the third reason is that these double-row layouts can be calculated very fast, in particular, for heuristically determined single-row layouts.

We present a heuristic based on a single-row layout π and we assume, w. l. o. g., π= (1, . . . , n).

The idea of our heuristic is that the ordering of departments in the same row is given viaπ, so it remains to determine the row assignment of the departments as well as their exact positions. We are given nmin, nmax∈N, nmin≤nmax, and in each step we add a set S of departments which contains, if possible, at leastn_min departments and at mostn_max departments.

Let the set of departments [h],0≤h≤n, h∈N0, be already added to the double-row layout (we start with h= 0 and we stop ifh=n). Ifh+nmax≥n,all remaining departments are added and we set S = {h+ 1, . . . , n}. Otherwise, we interpret the departments H :={h+ 1, . . . , n} as nodes in a complete graph with weights w_ij +w_ji, i, j ∈H, i < j. Our goal is to determine somek⁰ and an associated setS:={h+ 1, h+ 2, . . . , k⁰}such that the sum of the total transport weights between S and [n]\[k⁰] is small. So we detect which departments should be considered together in the next step. We set

k⁰ :=





arg minh+nmin≤k≤h+nmax

k∈N

Pi=h+1,...,k

j=k+1,...,nw_ij +w_ji, h+n_max < n,

n, h+n_max ≥n.

So |S| ≤ n_max, and, if h+n_min ≤n, then |S| ≥n_min. The calculation of k⁰ is related to the calculation of a constrained minimum cut in the graph described above.

Then, we add the dummy departmentn+1 (n+2) to row 1 (row 2) with length`<sub>n+1</sub> =`_n+2 = 0 and weights w_i(n+1) = w_(n+1)i = w_i(n+2) =w_(n+2)i = ¹₂^Pi=h+1,...,k⁰

j=k⁰+1,...,nw_ij +w_ji such that n+ 1 (n+ 2) is the rightmost department in row 1 (row 2). Knowing k⁰ and so S, our goal is to determine a row assignment of the departments S such that departments in the same row are sorted according to π and such that sum ^P_i,j∈[k⁰]∪{n+1,n+2}

i<j

(wij +wji)dij is minimized, i. e., we have to solve a (small) double-row instance where the order of the departments in the same row is known. For solving this problem we apply the approach of [42] and enumerate over all distinguishable assignments of the departments S to the rows. Knowing the order of the departments in the rows, each subproblem reduces to some LP with k⁰+ 2 departments. We choose one of the row assignments for S where the layout has minimal objective value. In the last step, when n ∈ S, the solution of the LP corresponds to a double-row layout including possible free-spaces. The algorithm stops after returning this layout. We denote this heuristic by mc(nmin, n_max). To further improve this layout, we set up anMILPmodel for the 1-opt algorithm.

Then, we apply the 1-opt (2-opt) approach until the solution cannot be improved by a 1-opt (2-opt) step.

3.5.4. Computational results

We apply our heuristics based on best known single-row layouts, see, e. g., [70, 71, 73], and we use a heuristic for the SRFLPwith a short running time and which is easy to implement, i. e., we start with a random single-row layout and apply a 1-opt algorithm and a 2-opt algorithm until the single-row layout cannot be improved by a 1-opt or 2-opt step, respectively.

We start our computational study with the equidistant case. In Table 3.5.1 we display in column two (column three) the objective value of a best known (heuristically determined) single-row layout denoted by “Best known” (“Heuristic”). The objective value of the start layout and of the final layout after applying our exchange algorithm is denoted byH_Best (HHeur) and is based on a best known single-row layout (heuristically determined single-row layout). The current best upper bounds for these instances are derived by the SDPapproach of [15] and the time limit is set to 3 hours.

We observe that for all instances in Table 3.5.1 our heuristic based on best known single-row layouts is better than the one based on heuristically determined single-row layouts. Note that the obtained gaps of our single-row heuristic are rather small and the running time is at most one minute, even for n= 100. For all large sko-instances withn≥49 and given some best known single-row layout, we improve the previously known best upper bounds in [15] with a significantly smaller running time. Using our approach based on a heuristically determined single-row layout, we obtain small gaps to the approach of [15], however, these layouts can be calculated in a few minutes, including the corresponding single-row layout. Comparing the best solution values of the SREFLPand theDREFLPone can see that the value of the DREFLPis strictly less than halve the value of the SREFLP, but rather close to this value in our tests.

In Table 3.5.2 we consider theDRFLP and the notation is similar to Table 3.5.1. We compare our results with the heuristic approach of [33]. We focus on sko-instances where good heuristi-cally determined single-row layouts are available athttps://www.philipphungerlaender.com/

benchmark-libraries/layout-lib/row-layout-instances/. Looking at the results for the sko-instances in Table 3.5.2 all solutions derived using the mc heuristic based on best known single-row layouts are better than the results of [33]. If we use themc heuristic in combination with our simple single-row heuristic, we could improve 5 out of 9 upper bounds in comparison to the approach in [33]. The running time of the mc heuristic is slightly reduced by using a best known single-row layout instead of a heuristically determined single-row layout and the heuristic of [33] is a bit faster than the mc heuristic. For themc heuristic based on some known single-row layout, the exchange algorithm only slightly improves the start layout, so the running time could be improved, by neglecting the exchange algorithms. Note that the layouts of [33] are

SRFLP Start layout Exchange Time

Instance Best known Heuristic H_Best H_Heur H_Best H_Heur [15] H_Best H_Heur

sko42-1 25525 25525 12749 12749 12743 12743 12731 2 2

sko49-1 40967 42469 20477 21226 20470 21224 20512 6 6

sko56-1 64024 66083 31975 33011 31972 32932 31988 11 24

sko64-1 96883 98122 48418 49052 48409 49004 48574 23 45

sko72-1 139150 143317 69535 71607 69531 71603 69621 41 42

sko81-1 205106 208554 102549 104263 102549 104067 102793 1:10 4:36

sko100-1 378234 384049 189062 191982 189056 191964 - 3:31 3:56

Table 3.5.1: Heuristically determined upper bounds for equidistant double-row instances from the literature [15]. Our heuristics are based on best known single-row layouts as well as heuristically determined single-row layouts.

significantly improved by the exchange algorithms.

The author’s contribution:

The proofs of Theorem 5, Proposition 9 and Theorem 12 were done by the author in consultation with Anja Fischer. The proof of Lemma 11, which is by far the largest proof in this publication, was mainly done by the author and revised multiple times by Anja Fischer. The author’s ideas concern the mincut heuristic and the MILPmodel for the 1-opt algorithm. He is responsible for a large part of the implementation and fully responsible for the computational experiments.

SRFLPStartlayoutExchangeTime InstanceBestknownHeuristicmc(8,14)Bestmc(8,14)Heur[33]mc(8,14)Bestmc(8,14)Heur[33]mc(8,14)Bestmc(8,14)Heur[33] sko56-3170449.0171884.085267.085927.088326.585185.085905.586960.519:0226:389:44 sko56-4313388.0316222.0156772.0158140.0160847.0156690.0158118.0156946.020:3625:109:50 sko56-5592299.5600080.0296336.0300138.0307706.0296306.0300046.0304137.021:0223:529:02 sko64-3414323.5420810.0207278.0210392.0214524.0207160.0210372.0210634.039:3639:2922:50 sko64-4297129.0299193.0148546.0149559.0154923.0148508.0149500.0149844.039:2836:1724:24 sko72-2711998.0719188.0355990.0359486.0373665.0355975.0359482.0364886.01:03:3354:5739:32 sko72-31054110.51067699.5527030.0533922.0539676.0526956.0533748.0530166.01:06:411:22:4153:26 sko72-4919586.0931922.0459892.0466200.0476218.0459824.0466144.0463502.01:12:321:17:091:03:50 sko72-5428226.5433700.0214087.0216798.0219243.0214080.0216782.0215986.056:281:02:3040:46 Table3.5.2:Resultsofthemcheuristicsfordouble-rowinstanceswithupto72departmentsbasedonbestknownandheuristicallydetermined single-rowlayouts.Allsolutionsderivedbyusingthemcheuristicbasedonbestknownsingle-rowlayoutsarebetterthantheresultsof [33].Therunningtimesaregiveninsec,min:sec,h:min:sec.