Takustr. 7 Zuse Institute Berlin 14195 Berlin Germany FELIX PRAUSE1,KAI HOPPMANN-BAUM2,BORIS DEFOURNY3,THORSTEN KOCH4 The Maximum Diversity Assortment Selection Problem 1 0000-0001-9401-3707 2 0000-0001-9184-8215 3 0000-0003-0405-5538 4 0000-0002-1967-0077 ZIB Report 20-34 (Dezember 2020) Zuse Institute Berlin Takustr. 7 14195 Berlin Germany Telephone: +49 30-84185-0 Telefax: +49 30-84185-125 E-mail: [email protected] URL: http://www.zib.de ZIB-Report (Print) ISSN 1438-0064 ZIB-Report (Internet) ISSN 2192-7782 The Maximum Diversity Assortment Selection Problem Felix Prause1[0000−0001−9401−3707], Kai Hoppmann-Baum1;2[0000−0001−9184−8215], Boris Defourny3[0000−0003−0405−5538], and Thorsten Koch1;2[0000−0002−1967−0077] 1 Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany {prause,hoppmann-baum,koch}@zib.de 2 TU Berlin, Chair of Software and Algorithms for Discrete Optimization, Str. des 17. Juni 135, 10623 Berlin, Germany 3 Lehigh University, Department of Industrial and Systems Engineering, 200 W Packer Ave, Bethlehem, PA, 18015, USA [email protected] Abstract. In this paper, we introduce the Maximum Diversity Assort- ment Selection Problem (MADASS), which is a generalization of the 2-dimensional Cutting Stock Problem (2CSP). Given a set of rectan- gles and a rectangular container, the goal of 2CSP is to determine a subset of rectangles that can be placed in the container without overlap- ping, i.e., a feasible assortment, such that a maximum area is covered. In MADASS, we need to determine a set of feasible assortments, each of them covering a certain minimum threshold of the container, such that the diversity among them is maximized. Thereby, diversity is defined as minimum or average normalized Hamming-Distance of all assortment pairs. The MADASS Problem was used in the 11th AIMMS-MOPTA Competition in 2019. The methods we describe in this article and the computational results won the contest. In the following, we give a def- inition of the problem, introduce a mathematical model and solution approaches, determine upper bounds on the diversity, and conclude with computational experiments conducted on test instances derived from the 2CSP literature. 1 Introduction The problem of packing rectangles into rectangular containers or to cut them from rectangular stock sheets arises in a variety of industrial operations. Thereby, one typically aims at determining a feasible solution where the wasted material or space is minimized. Consider for example the paper, glass, wood, textile, or metal industry. Here, rectangular pieces are needed for the production of certain goods, which are typically cut from given stock pieces [43]. Another common application arising in logistics is to load pallets or containers [49]. Furthermore, similar problems appear in the production and operation of microchips, namely 2 Prause et al. in the layout of processor chips [77] and the dynamic allocation of memory as well as in multiprocessor scheduling [43]. Finally, the editing and lay-outing of newspapers [76] or the arrangement of products on supermarket shelves [2] belong to this class of applications, too. These problems are typically modeled as some special variant of the 2-dim Cutting Stock Problem (2CSP), which we introduce in Section 2 and discuss in Section 3. However, for practitioners it is often useful when they are presented not only one optimal but a set of diverse “near-optimal” solutions from which she or he can choose. This holds especially for those tasks where it is hard to formalize or model important side constraints. Consider, for example, the last two exam- ples from above. If a supermarket wants to investigate the buying behavior of its customers, an arrangement of the products minimizing the empty space is certainly desirable. Nevertheless, for this particular task the result is not very meaningful. Instead, the company needs to conduct tests with a variety of ar- rangements to assess whether they increase the purchasing rates [52]. Similarly, when it comes to the layout of texts, pictures, or ads on a newspaper page, the result does not necessarily have to be minimal w.r.t. the resulting empty space, but has to come in some aesthetic appeal. Thus, presenting the user with a se- lection of assortments that cover some minimum threshold of the available area can be advantageous in all areas where experiences and subjective perceptions of humans have an impact on the chosen solution. Problems of this kind are the motivation for the Maximum Diversity As- sortment Selection Problem (MADASS), which we introduce in Section 2. We review the literature on two inherent subproblems in Section 3: the above men- tioned 2CSP and the Maximum Diversity Problem (MDP), where a predefined number of elements has to be selected from a given set such that the diversity among them is maximized. Before discussing an MIQP model for MADASS in Section 5, we introduce MIQP formulations to determine upper bounds on the maximum diversity in Section 4. Next, we present a generic two-stage heuris- tic in Section 6. We present extensive computational experiments in Section 7. Finally, we conclude and give an outlook on future research in Section 8. 2 Definitions and Problem Setup In MADASS, we are given a rectangle C with width w 2 Z≥0 and height h 2 Z≥0, which we call container. Furthermore, we are given a set of rectangles R := fR1; :::; Rng and each of them is associated with its width wi 2 Z≥0 and its height hi 2 Z≥0. Next, an assortment is a subset A ⊆ R of rectangles, i.e., A 2 P(R) where P(R) denotes the powerset of R. We call an assortment A feasible if it can be placed in the container C without overlapping, i.e., if we can 2 assign a bottom-left corner coordinate (xi; yi) 2 R to each rectangle Ri 2 A such that [xi; xi + wi) × [yi; yi + hi) ⊆ [0; w) × [0; h) and for all Ri;Rj 2 A with i 6= j we have [xi; xi + wi)×[yi; yi + hi)\[xj; xj + wj)×[yj; yj + hj) = ;. In the sequel, we denote the set of all feasible assortments by F. Next, each assortment A has an associated value v(A) = P w h , which is the sum of the areas of Ri2A i i The Maximum Diversity Assortment Selection Problem 3 the rectangles it contains. We call an assortment A∗ optimal if it is feasible and if v(A∗) ≥ v(A) holds for every A 2 F. Determining an optimal assortment is called the 2-dimensional Cutting Stock Problem (2CSP). Note that we otherwise allow an arbitrary placement of the rectangles inside the container, i.e., we do not impose any further conditions, and we do not allow the rotation of rectangles. An example can be found in Figure 1. R2 h2 R3 h3 R1 h1 w2 w3 h w1 R5 h5 R4 h4 w4 w5 C w R R4 R4 R4 R3 R5 R5 R1 R1 R1 R1 R2 R2 R2 C C C C A1 A2 A3 A4 Fig. 1: Container C with rectangle set R. While assortment A1 is feasible, A2 is even optimal. On the other hand, assortments A3 and A4 are not feasible. In MADASS we are furthermore given a threshold value v 2 [0; v(C)], with v(C) := wh denoting the area of the container, as well as a natural number m 2 N with m ≥ 2. We call an assortment A v-good if it is feasible and if v(A) ≥ v and we denote the set of v-good assortments by Fv ⊆ F. Furthermore, a selection is a multi-subset S ⊆ P(R) of assortments of cardinality jSj = m, i.e., we allow that an assortment is contained more than once in S. We call S feasible if all of its assortments are v-good, i.e., if S ⊆ Fv. Additionally, we are given a diversity function δ for selections and call δ(S) the diversity of selection S. The two diversity functions that we consider in this paper are based on the Hamming-Distances between assortment pairs contained in S and are discussed in Subsection 2.1. Finally, a selection S∗ is called optimal if it is feasible and if δ(S∗) ≥ δ(S) holds for all feasible selections S. In the following, we denote a MADASS instance I as quintuple I := (C; R; m; v; δ). Lemma 1. MADASS is NP-hard. 4 Prause et al. Proof. We prove this by reduction from 2CSP, which is NP-hard, see Fekete and Schepers [29] and Garey and Johnson [32]. Given an instance of 2CSP with container C and set of rectangles R, for arbitrary m and δ, there exists a feasible assortment of value at least v if and only if there exists a feasible selection for MADASS instance I := (C; R; m; v; δ). 2.1 Diversity of Selections A common distance measure applied to the subsets of a common superset is the Hamming-Distance [41]. Given an assortment A, we associate the vector n i vA 2 f0; 1g with it, where the i-th entry vA is equal to 1 if rectangle Ri is contained in A and 0 otherwise. For two assortments A and A0, the (normalized) Hamming-Distance is n P i i jvA − vA0 j d (A; A0) := i=1 : H jAj + jA0j 0 0 Furthermore, we set dH (A; A ) = 0 in the case A = A = ;. Note that we consider the normalized Hamming-Distance, i.e., we divide by jAj + jA0j, to ensure that the number of rectangles forming the assortments has no impact on the distance.