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The general MIP model, discussed in Chap. 2, is reconsidered hereinafter, investigating some possible reformulations, from different points of view (Sect. 3.1). The objective of enucleating implicit implications and introducing valid inequalities, to tighten the model, is examined next (Sect. 3.2).

3.1 Alternative Models

The issue discussed in this section focuses mainly on the case occurring when the packing problem is expressed in terms of feasibility, i.e. when all the given items have to be placed and no objective function is stated a priori. This situation can arise, for instance, when the items are the elements of a device and, as such, they all have to be installed inside an appropriate container, as essential parts of the same equipment. The thus defined feasibility subproblem is also of interest, as it represents one of the basic concepts of the heuristic procedures put forward in Chap. 4. As far as this specific subproblem is concerned, since no objective function is specified a priori, an arbitrary one can be introduced, in order to simplify the task of finding an integer-feasible solution.

The general model of Sect. 2.1 (including the additional conditions of Sect. 2.3) is reconsidered hereinafter in terms of feasibility, providing three different reformulations of it (Sects. 3.1.1, 3.1.2 and 3.1.4). In all of them, it is understood that either all the given items can be loaded or the instance to solve is infeasible. In each of these reformulations, an ad hoc objective function is defined, with the scope of minimizing (even if indirectly) the overall overlap of items. In the first (Sect. 3.1.1) and second (Sect. 3.1.2, except the variation outlined at the end), no sooner does the solver obtain the first integer-feasible solution than the optimization is stopped (even if just a suboptimal solution of the ad hoc objective function has been found). In all reformulations, both the orthogonality and domain condition s are maintained, as defined in Sect. 2.1. (i.e. consisting of constraints (2.1), (2.2), (2.3) and (2.4)). The second reformulation (Sect. 3.1.2) is subject to straightforward variations. One in particular (Sect. 3.1.3) is an actual alternative to the general MIP model, no longer restricted to the feasibility subproblem. It could also be utilized (at least partially) in the heuristics of Chap. 4. This aspect would definitely represent an interesting objective for future research.

3.1.1 General MIP Model First Linear Reformulation

The rationale of the general MIP model reformulation presented hereinafter stresses the introduction of an ad hoc objective function. This aims at reducing the solution search region , as much as possible, in order to obtain any integer-feasible solution .

The approach adopted draws on the work achieved by Suhl (1984), dealing with (large-scale) fixed-charge models . Suhl’s work provides an efficient preprocessing technique aimed at reducing the big-M terms, associated to the fixed-charge constraints, i.e. at ‘minimizing’ (a priori) the related region, in the LP relax ation .

As far as the model reformulation in question is concerned, an approach, intended to ‘minimize’ the search region R S , relative to the non-intersection (big-M) constraints (2.5a) and (2.5b), is investigated, to tackle efficiently the relative feasibility subproblem . These constraints are then reformulated in an LP-relax ed form and an ad hoc objective function , substituting (2.7), is introduced. The reformulated model is described as follows.

All variables χ are set to one, as all the given items must be inside the domain and the non-intersection constraint s (2.5a) and (2.5b) are rewritten as

$$ \begin{array}{l}\forall \beta \in B,\forall i,j\in I/i<j,\forall h\in {C}_i,\forall k\in {C}_j\hfill \\ {}{w}_{\beta 0 hi}-{w}_{\beta 0 kj}\ge \frac{1}{2}{\displaystyle \sum_{\omega \in \varOmega}\left({L}_{\omega \beta hi}{\vartheta}_{\omega i}+{L}_{\omega \beta kj}{\vartheta}_{\omega j}\right)}+{d}_{\beta hkij}^{+}-{D}_{\beta },\hfill \end{array} $$
(3.1a)
$$ \begin{array}{l}\forall \beta \in B,\forall i,j\in I/i<j,\forall h\in {C}_i,\forall k\in {C}_j\hfill \\ {}{w}_{\beta 0 kj}-{w}_{\beta 0 hi}\ge \frac{1}{2}{\displaystyle \sum_{\omega \in \varOmega}\left({L}_{\omega \beta hi}{\vartheta}_{\omega i}+{L}_{\omega \beta kj}{\vartheta}_{\omega j}\right)}+{d}_{\beta hkij}^{-}-{D}_{\beta }.\hfill \end{array} $$
(3.1b)

Constraints (2.6) are substituted with the following:

$$ \forall \beta \in B,\forall i,j\in I/i<j,\forall h\in {C}_i,\forall k\in {C}_j\kern1em {d}_{\beta hkij}^{+}\ge {\sigma}_{\beta hkij}^{+}{D}_{\beta }, $$
(3.2a)
$$ \forall \beta \in B,\forall i,j\in I/i<j,\forall h\in {C}_i,\forall k\in {C}_j\kern1em {d}_{\beta hkij}^{-}\ge {\sigma}_{\beta hkij}^{-}{D}_{\beta }, $$
(3.2b)
$$ \begin{array}{l}\forall i,j\in I/i<j,\forall h\in {C}_i,\forall k\in {C}_j\hfill \\ {}{\displaystyle \sum_{\beta \in B}\Big({\sigma}_{\beta hkij}^{+}+{\sigma}_{\beta hkij}^{-}}\Big)=1,\hfill \end{array} $$
(3.3)

where d + βhkij , d − βhkij  ∈ [0, D β ].

The adopted ad hoc objective function is

$$ \max {\displaystyle \sum_{\begin{array}{l}\beta \in B,\\ {}i,j\in I/i<j,\\ {}h\in {C}_i,k\in {C}_j\end{array}}\kern-1em \frac{d_{\beta hkij}^{+}+{d}_{\beta hkij}^{-}}{D_{\beta }}}. $$
(3.4)

Any optimal solution of the reformulated model identifies a minimal subset of the feasibility region , relative to the general MIP model (Sect. 2.1).

Proposition 3.1

For any given set of items, the feasibility region s, associated to the general MIP model and its first linear reformulation respectively (neglecting the subspace associated to the variables d + and d − ), are coincident.

Proof

Dealing with the feasibility subproblem, all variables χ are set to one. Constraints (2.1), (2.2), (2.3) and (2.4) are obviously coincident in both models, and it is thus sufficient to demonstrate that constraints (2.5a), (2.5b) and (2.6) of the general MIP model are equivalent to constraints (3.1a), (3.1b), (3.2a), (3.2b) and (3.3) of the reformulated one. It is immediately seen that given that all variables χ are set to one, constraints (2.6) can be substituted with (3.3). To show that constraints (2.5a) and (2.5b) are equivalent to (3.1a), (3.1b), (3.2a) and (3.2b), we shall distinguish the cases where the variables σ are zero from those where they are equal to one.

Consider, for instance, σ + βhkij  = 0. This implies that constraints (2.5a) become \( {w}_{\beta 0 hi}-{w}_{\beta 0 kj}\ge \frac{1}{2}{\displaystyle \sum_{\omega \in \varOmega}\left({L}_{\omega \beta hi}{\vartheta}_{\omega i}+{L}_{\omega \beta kj}{\vartheta}_{\omega j}\right)}-{D}_{\beta } \).

These are equivalent to constraints (3.1a), with d + βhkij  = 0, in compliance with constraints (3.2a). Considering, instead, σ + βhkij  = 1, this implies that constraints (2.5a) become \( {w}_{\beta 0 hi}-{w}_{\beta 0 kj}\ge \frac{1}{2}{\displaystyle \sum_{\omega \in \varOmega}\left({L}_{\omega \beta hi}{\vartheta}_{\omega i}+{L}_{\omega \beta kj}{\vartheta}_{\omega j}\right)} \).

These are equivalent to constraints (3.1a), with d + βhkij  = D β , in compliance with constraints (3.2a). As the same reasoning can be carried out, taking into account the cases relative to the variables σ − βhkij , the two models are equivalent. □

Remark 3.1

To better understand the meaning of the general MIP model first linear reformulation , we shall make some intuitive considerations. Let us define, for each β, for every pair of components h and k of item i and j, respectively, the squared subspace S β  = [0, D β ] × [0, D β ] ⊂ R 2, associated to variables d − βhkij and d + βhkij . The bound \( {d}_{\beta hkij}^{+}+\kern0.5em {d}_{\beta hkij}^{-}\le 2{D}_{\beta }-{\displaystyle \sum_{\omega \in \varOmega}\Big({L}_{\omega \beta hi}{\vartheta}_{\omega i}+{L}_{\omega \beta kj}{\vartheta}_{\omega j}}\Big) \) is implicitly determined by inequalities (3.1a) and (3.1b). The objective function induces the solution projection on S β to stay along the straight line \( {d}_{\beta hkij}^{+}+{d}_{\beta hkij}^{-}=2{D}_{\beta }-{\displaystyle \sum_{\omega \in \varOmega}\left({L}_{\omega \beta hi}{\vartheta}_{\omega i}+{L}_{\omega \beta kj}{\vartheta}_{\omega j}\right)} \). If \( {D}_{\beta }-{\displaystyle \sum_{\omega \in \varOmega}\left({L}_{\omega \beta hi}{\vartheta}_{\omega i}+{L}_{\omega \beta kj}{\vartheta}_{\omega j}\right)}\ge 0 \), this intersects S β in the points \( \left({D}_{\beta },{D}_{\beta }-{\displaystyle \sum_{\omega \in \varOmega}\left({L}_{\omega \beta hi}{\vartheta}_{\omega i}+{L}_{\omega \beta kj}{\vartheta}_{\omega j}\right)}\right) \) and \( \left({D}_{\beta }-{\displaystyle \sum_{\omega \in \varOmega}\left({L}_{\omega \beta hi}{\vartheta}_{\omega i}+{L}_{\omega \beta kj}{\vartheta}_{\omega j}\right)},{D}_{\beta}\right) \), respectively, determining an internal segment. In this occurrence, if the linear solver (utilized by the MIP optimizer) looks for vertex solution s (as in the case of a simplex -based one), the above extreme points are more likely to be selected than the ones internal to the segment (although this expectation is not based on rigorous reasoning). One has to bear in mind, moreover, that either d + βhkij  = D β or d − βhkij  = D β (for any β) guarantees that no intersection occurs between the two corresponding items.

As a partially alternative version of this model reformulation, the constraints ∀ β ∈ B, ∀ i, j ∈ I/i < j, ∀ h ∈ C i , ∀ k ∈ C j , d + βhkij  + d − βhkij  ≤ D β could also be added to tighten the feasibility region (creating in the subspace S β the two extreme points (D β , 0) and (0, D β ), without excluding any solution. These inequalities are obviously tighter than the bounds \( {d}_{\beta hkij}^{+}+{d}_{\beta hkij}^{-}\le 2{D}_{\beta }-{\displaystyle \sum_{\omega \in \varOmega}\Big({L}_{\omega \beta hi}{\vartheta}_{\omega i}+{L}_{\omega \beta kj}{\vartheta}_{\omega j}}\Big) \), when \( {D}_{\beta }-{\displaystyle \sum_{\omega \in \varOmega}\left({L}_{\omega \beta hi}{\vartheta}_{\omega i}+{L}_{\omega \beta kj}{\vartheta}_{\omega j}\right)}\ge 0 \). The conditions d − βhkij , d + βhkij  ∈ [0, D β ], moreover, if explicitly introduced in the model, can be of computational advantage, when the linear solver adopted treats the variable bounds independently (as in the case of simplex-based ones).

3.1.2 General MIP Model Second Linear Reformulation

To discuss this alternative model, we shall consider, for each item component, the set of all concentric parallelepipeds containing it. The reformulation examined hereinafter is also based on an ad hoc objective function. It is aimed at finding, for each component, the enclosing parallelepiped (included in D) of maximum volume that does not intersect any other enclosing parallelepipeds, associated to components of different items.

To this purpose, the non-intersection condition s of Sect. 2.1 are properly changed. Whilst (2.6) is kept, inequalities (2.5a) and (2.5b) are substituted with the constraints below. For each component h of i, the non-negative variable s l βhi are introduced, assuming that all variables χ are set to one:

$$ \begin{array}{l}\forall \beta \in B,\forall i,j\in I/i<j,\forall h\in {C}_i,\forall k\in {C}_j\hfill \\ {}{w}_{\beta 0 hi}-{w}_{\beta 0 kj}\ge \frac{1}{2}\left({l}_{\beta hi}+{l}_{\beta kj}\right)-{D}_{\beta}\left(1-{\sigma}_{\beta hkij}^{+}\right),\hfill \end{array} $$
(3.5a)
$$ \begin{array}{l}\forall \beta \in B,\forall i,j\in I/i<j,\forall h\in {C}_i,\forall k\in {C}_j\hfill \\ {}{w}_{\beta 0 kj}-{w}_{\beta 0 hi}\ge \frac{1}{2}\left({l}_{\beta hi}+{l}_{\beta kj}\right)-{D}_{\beta}\left(1-{\sigma}_{\beta hkij}^{-}\right),\hfill \end{array} $$
(3.5b)
$$ \begin{array}{l}\forall \omega \in \varOmega, \forall \beta \in B,\forall i\in I,\forall h\in {C}_i\hfill \\ {}{l}_{\beta hi}\ge {L}_{\omega \beta hi}{\vartheta}_{\omega i}.\hfill \end{array} $$
(3.6)

The following (surrogate ) objective function is defined:

$$ \max {\displaystyle \sum_{\begin{array}{l}\beta \in B,\\ {}i\in I,h\in {C}_i\end{array}}{l}_{\beta hi}}. $$
(3.7)

For each component h of each item i, the terms l βhi represent (for the orientation ω assumed by i) the projections, on the axes w β , of an enclosing parallelepiped, containing component h and centred with it. Inequalities (3.5a), (3.5b) and (3.6) (together with (2.6)) guarantee that the enclosing parallelepipeds, belonging to different items, do not intersect.

Remark 3.2

Rigorously speaking, as the objective function (3.7) refers to the total sum of the component sides, it should be considered as a surrogate expression of \( \max {\displaystyle \sum_{i\in I,h\in {C}_i}{\displaystyle \prod_{\beta \in B}{l}_{\beta hi}}} \).

As previously mentioned, possible variations of the approach discussed above could be considered. One is obtained simply by inverting inequalities (3.6) as follows and keeping all remaining constraints, as well as the objective function, unaltered:

$$ \begin{array}{l}\forall \omega \in \varOmega, \forall \beta \in B,\forall i\in I,\forall h\in {C}_i\hfill \\ {}{l}_{\beta hi}\le {L}_{\omega \beta hi}{\vartheta}_{\omega i}.\hfill \end{array} $$
(3.8)

In this case, an integer-optimal solution (and not just any integer-feasible one) has necessarily to be found, in order to guarantee that no intersections occur among the given items. It should be noticed that, at each step, the optimization process is induced to minimize the overall overlap , without assigning items a volume that exceeds their own. Moreover, since, in this case, the value of the global optima l solution is known a priori, it can be advantageously utilized as cutoff parameter (to get rid of suboptimal solution s).

3.1.3 A Non-restrictive Reformulation of the General MIP Model

A possible reformulation of the general MIP model, without renouncing its original objective of maximizing either the overall loaded volume or mass, is also quite straightforward. The problem is no longer expressed in terms of feasibility (i.e. without the possibility of rejecting items), so that all variables χ are set free again, as in Sect. 2.1.

As a first step, inequalities (3.6) are transformed into the equations:

$$ \begin{array}{l}\forall \omega \in \varOmega, \forall \beta \in B,\forall i\in I,\forall h\in {C}_i\hfill \\ {}{l}_{\beta hi}={L}_{\omega \beta hi}{\vartheta}_{\omega i}.\hfill \end{array} $$
(3.9)

In order to define the new objective function (substituting (2.7)), the terms K hi are introduced (with obvious meaning) for each component h of each item i, where \( \forall i\in I\kern1em {\displaystyle \sum_{h\in {C}_i}{K}_{hi}}={K}_i \), cf. (2.7). The dimensions of component h of i are indicated with L αhi , α ∈ {1, 2, 3} = A, assuming, from now on, that L 1hi  ≤ L 2hi  ≤ L 3hi . The new objective function is then expressed by the following:

$$ \max {\displaystyle \sum_{\begin{array}{l}\beta \in B,\\ {}i\in I,h\in {C}_i\end{array}}\frac{K_{hi}}{{\displaystyle \sum_{\alpha \in A}{L}_{\alpha hi}}}{l}_{\beta hi}}. $$
(3.10)

It is easily seen that the two objective functions (2.7) and (3.10) are equivalent for any integer-feasible solution (by (3.9)). The expression (3.10), differently from (2.7), provides the significant computational advantage of minimizing the item overall overlap at each step of the optimization process. Just to summarize the reformulation in question, we could point out that it consists of constraints (2.1), (2.2) (orthogonality), (2.3), (2.4) (domain), (2.6), (3.5a), (3.5b) and (3.9) (non-intersection), in addition to objective function (3.10). It is also understood that in all the relevant expressions above, the variables l βhi could be eliminated. They may, indeed, be substituted by their corresponding terms, on the basis of (3.9) (that could also be eliminated).

3.1.4 General MIP Model Nonlinear Reformulation

The general packing problem presented in Sect. 2.1 is notoriously subject to nonlinear (MINLP ) formulations (e.g. Birgin and Lobato 2010; Birgin et al. 2006; Cassioli and Locatelli 2011). We shall introduce, hereinafter, a nonlinear reformulation of the general MIP model non-intersection constraints, assuming, as previously, that all variables χ are set to one (as the feasibility subproblem is in question). It is straightforward to prove that the nonlinear constraint s below are equivalent to (2.5a), (2.5b) and (2.6):

$$ \begin{array}{l}\forall \beta \in B,\forall i,j\in I/i<j,\forall h\in {C}_i,\forall k\in {C}_j\hfill \\ {}{\left({w}_{\beta 0 hi}-{w}_{\beta 0 kj}\right)}^2-{\left[\frac{1}{2}{\displaystyle \sum_{\omega \in \varOmega}\left({L}_{\omega \beta hi}{\vartheta}_{\omega i}+{L}_{\omega \beta kj}{\vartheta}_{\omega j}\right)}\right]}^2={s}_{\beta hkij}-{r}_{\beta hkij},\hfill \end{array} $$
(3.11)
$$ \begin{array}{l}\forall \beta \in B,\forall i,j\in I/i<j,\forall h\in {C}_i,\forall k\in {C}_j\hfill \\ {}{\displaystyle \prod_{\beta \in B}{r}_{\beta hkij}}=0,\hfill \end{array} $$
(3.12)

where s βhkij  ∈ [0, D 2 β ] and r βhkij  ∈ [0, D 2 β ] (actually, smaller upper bounds could be chosen for both sets of variables).

Indeed, for each pair of components h and k, of items i and j, respectively, equations (3.12) guarantee that for at least one β, the corresponding term r βhkij is zero, and equations (3.11) that the non-intersection conditions hold for such a β, i.e. \( \left|{w}_{\beta 0 hi}-{w}_{\beta 0 kj}\right|\ge \frac{1}{2}{\displaystyle \sum_{\omega \in \varOmega}\left({L}_{\omega \beta hi}{\vartheta}_{\omega i}+{L}_{\omega \beta kj}{\vartheta}_{\omega j}\right)} \). More precisely, constraints (2.5a) and (2.5b) correspond to equations (3.11), whilst equations (2.6) correspond to (3.12).

As the non-intersection constraint s (3.11) and (3.12) are most likely hard to tackle, they are therefore considered in terms of fixed penalization in the ad hoc objective function we are going to introduce. All remaining linear (MIP), constraints are kept as such. A formulation aimed at satisfying as much non-intersection conditions as possible is the following:

$$ \begin{array}{l} \min \left\{{\displaystyle \sum_{\begin{array}{l}\beta \in B,\\ {}i,j\in I/i<j,\\ {}h\in {C}_i,k\in {C}_j\end{array}}{\left\{{\left({w}_{\beta 0 hi}-{w}_{\beta 0 kj}\right)}^2-{\left[\frac{1}{2}{\displaystyle \sum_{\omega \in \varOmega}\left({L}_{\omega \beta hi}{\vartheta}_{\omega i}+{L}_{\omega \beta kj}{\vartheta}_{\omega j}\right)}\right]}^2-{s}_{\beta hkij}+{r}_{\beta hkij}\right\}}^2}\right.\\ {}\left.+{K}_P{\displaystyle \sum_{\begin{array}{l}i,j\in I/i<j,\\ {}h\in {C}_i,k\in {C}_j\end{array}}{\displaystyle \prod_{\beta \in B}{r}_{\beta hkij}}}\right\}\end{array} $$
(3.13)

where K P is a positive coefficient (that represents an appropriate ‘weight’ associated to the product terms).

It is immediately seen that the objective function (3.13) is non-negative. A zero-global -optimal solution exists if and only if the constraints ((2.1), (2.2), (2.3), (2.4), (2.5a), (2.5b) and (2.6) of the general MIP model of Sect. 2.1 (with all variables χ set to one) delimit a feasible region . This objective function thus ‘minimizes’ the intersection between items. Its global optima, moreover, guarantee an ultimate (non-approximate) solution to the feasibility subproblem under discussion. It could be observed that for each set of variables ϑ, (3.13) is a polynomial function (providing, as such, potential algorithmic advantages; on global polynomial optimization, see, for instance, De Loera et al. 2012; Hanzon and Jibetean 2003; Schweighofer 2006).

Alternative fixed penalization can be considered (e.g. Cassioli and Locatelli 2011). We shall introduce here one objective function with fixed penalization correlated to the non-intersection constraint s only:

$$ \begin{array}{l} \min \left\{{\displaystyle \sum_{\begin{array}{l}\beta \in B,\\ {}i,j\in I/i<j,\\ {}h\in {C}_i,k\in {C}_j\end{array}} \max \left\{-{\left({w}_{\beta 0 hi}-{w}_{\beta 0 kj}\right)}^2+{\left[\frac{1}{2}{\displaystyle \sum_{\omega \in \varOmega}\left({L}_{\omega \beta hi}{\vartheta}_{\omega i}+{L}_{\omega \beta kj}{\vartheta}_{\omega j}\right)}\right]}^2,0\right\}}\right.\\ {}\left.+{K}_P{\displaystyle \sum_{\begin{array}{l}i,j\in I/i<j,\\ {}h\in {C}_i,k\in {C}_j,\end{array}}{\displaystyle \prod_{\beta \in B}{r}_{\beta hkij}}}\right\}\end{array} $$
(3.14)

As the previous one, this objective function is also non-negative and each zero-global -optimum corresponds to a solution of the feasibility problem.

Remark 3.3

Both the MINLP formulation s discussed above contain only linear (MIP) constraints. This aspect could be advantageous, when the MINLP solver s utilized treat the model linear sub-structure independently (e.g. The MathWorks 2012). It is moreover worth noticing that all functions involved in both MINLP formulation s are Lipschitz-continuous and, consequently, Lipschitzian solver s can be profitably adopted (e.g. Pintér 1997, 2009). Indeed, all constraints are of the MIP type and (3.13) is smooth. As far as (3.14) is concerned, it is sufficient to observe that the terms \( \max \left\{-{\left({w}_{\beta 0 hi}-{w}_{\beta 0 kj}\right)}^2+{\left[\frac{1}{2}{\displaystyle \sum_{\omega \in \varOmega}\left({L}_{\omega \beta hi}{\vartheta}_{\omega i}+{L}_{\omega \beta kj}{\vartheta}_{\omega j}\right)}\right]}^2,0\right\} \) keep their Lipshitz-continuous characteristic (e.g. Pintér 1996).

3.2 Implications and Valid Inequalities

As is well known, in the MIP context, remarkable research effort has been devoted to looking into general approaches to tighten the model. This means to make its linear relaxation an as precise as possible approximation of the convex hull relative to the mixed-integer solution s (e.g. Andersen et al. 2005; Ceria et al. 1998; De Farias et al. 1998; Jünger et al. 2009; Marchand et al 1999; Nemhauser and Wolsey 1990; Van Roy and Wolsey 1987; Weismantel 1996; Wolsey 1989). Polyhedral analysis (e.g. Atamtürk 2005; Constantino 1998; Dash et al. 2010; Hamacher et al. 2004; Padberg 1995; Pochet and Wolsey 1994; Yaman 2009) is adopted to this purpose, in order to find valid inequalit ies (e.g. Aardal et al. 1995; Cornuéjols 2008; Padberg et al. 1985; Wolsey 1990, 2003). These are aimed at tightening the MIP model under consideration. The introduction of such auxiliary conditions is particularly suitable when a branch-and-cut approach (e.g. Andreello et al. 2007; Balas et al. 1996; Cordier et al. 2001; Padberg 2001; Padberg and Rinaldi 1991) is followed.

Differently from more traditional MIP algorithm s, such as branch-and-bound (where all model constraints have to be set a priori) with a branch-and-cut process, the valid inequalit ies are activated just when needed and dropped when not required.

With reference to the general MIP model (Sect. 2.1), for items consisting of single parallelepipeds to load into a parallelepiped (see Sect. 2.1, special case), some valid inequalit ies, holding under specific assumptions, have been put forward by Padberg (1999). This has been done to tackle the problem by means of a dedicated branch-and-cut approach. Some quite simple conditions, not restricted to the case of single parallelepipeds, are considered hereinafter (limited subsets of them can be advantageously taken into account also when a branch-and-bound approach is adopted). A first group of inequalities is hence introduced:

$$ \begin{array}{l}\forall i,j\in I/i<j,\forall h\in {C}_i,\forall k\in {C}_j\hfill \\ {}{\displaystyle \sum_{\beta \in B}\left({\sigma}_{\beta hkij}^{+}+{\sigma}_{\beta hkij}^{-}\right)}\le {\chi}_i,\hfill \end{array} $$
(3.15a)
$$ \begin{array}{l}\forall i,j\in I/i<j,\forall h\in {C}_i,\forall k\in {C}_j\hfill \\ {}{\displaystyle \sum_{\beta \in B}\left({\sigma}_{\beta hkij}^{+}+{\sigma}_{\beta hkij}^{-}\right)}\le {\chi}_j.\hfill \end{array} $$
(3.15b)

These, together with (2.6), for each pair of components h and k of items i and j, respectively, imply that one, and only one, of the relative variables σ + βhkij and σ − βhkij has to be equal to one if both items are loaded; all of them are equal to zero otherwise. It is immediate to notice that in the general MIP model of Sect. 2.1, in case both items are picked, more than one of the variables σ + βhkij and σ − βhkij could be non-zero. The above extended version is hence tighter than the previous, without any loss of generality, as no integer-feasible solution s are excluded.

Some straightforward examples of necessary conditions, concerning pairs of items, in particular situations, can be considered. Firstly, let us consider the very simple case when item i and j cannot be aligned with respect to the axis w β (because they would exceed the dimension D β , for all possible orientations of both). In such an occurrence, the conditions below can be explicitly posed:

$$ \forall h\in {C}_i,\forall k\in {C}_j\kern1em {\sigma}_{\beta hkij}^{+}={\sigma}_{\beta hkij}^{-}=0. $$

In addition to these, a set of more complicated implications, correlating alignment and orientation, could be introduced. An example, dealing with the special case of Sect. 2.1, relative to single parallelepipeds , is reported here.Footnote 1 Considering items i and j, if L 1i  + L 2j  > D β , they cannot be aligned along the axis w β , with either L 2j or L 3j parallel to it. And analogously, this holds if L 1j  + L 2i  > D β . The following inequalities can hence be set:

$$ \forall \beta, \forall i,j\in I/i<j,{L}_{1i}+{L}_{2j}>{D}_{\beta}\kern1em {\delta}_{2\beta j}+{\delta}_{3\beta j}\le 1-{\sigma}_{\beta ij}^{+}-{\sigma}_{\beta ij}^{-}, $$
$$ \forall \beta, \forall i,j\in I/i<j,{L}_{1j}+{L}_{2i}>{D}_{\beta}\kern1em {\delta}_{2\beta i}+{\delta}_{3\beta i}\le 1-{\sigma}_{\beta i j}^{+}-{\sigma}_{\beta i j}^{-}. $$

These conditions can easily be extended when tetris-like items are involved, i.e. when the general MIP model of Sect. 2.1 is considered. This gives rise to inequalities of the type \( {\displaystyle \sum_{\omega \in {\varOmega}_{\beta hkij}^{\prime }}{\vartheta}_{\omega j}}\le 1-{\sigma}_{\beta hkij}^{+}-{\sigma}_{\beta hkij}^{-} \), where Ω ′ βhkij (i < j) is the set of orientations (of j), incompatible with the alignment conditions of the components h (of i) and k (of j). Similar expressions hold for i, with Ω ′ βhkji (i < j).

Straightforward transitivity conditions (e.g. Padberg 1999; Fasano 2008) can, moreover, be looked upon, when triplets of single parallelepipeds are taken into account. They can easily be extended when actual tetris-like items are involved. Focusing on the triplet of components h, h′, h″ of items i, i′, i″, respectively, if, along the axis w β , h precedes h′ and h′ precedes h″, then h precedes h″, along the same axis. This implication is expressed by

$$ \begin{array}{l}\forall \beta \in B,\forall i,{i}^{\prime },{i}^{{\prime\prime}}\in I/i<{i}^{\prime }<{i}^{{\prime\prime} },\forall h\in {C}_i,\forall {h}^{\prime}\in {C}_{i^{\prime }},\forall {h}^{{\prime\prime}}\in {C}_{i^{{\prime\prime} }}\\ {}{\sigma}_{\beta h{h}^{{\prime\prime} }i{i}^{{\prime\prime}}}^{-}\ge {\sigma}_{\beta h{h}^{\prime }i{i}^{\prime}}^{-}+{\sigma}_{\beta {h}^{\prime }{h}^{{\prime\prime} }{i}^{\prime }{i}^{{\prime\prime}}}^{-}-1.\end{array} $$

Still referring to the same triplet of components, the further implication holds: if \( {L}_{1 hi}+{L}_{1{h}^{\prime }{i}^{\prime }}+{L}_{1{h}^{{\prime\prime} }{i}^{{\prime\prime} }}>{D}_{\beta } \), then the whole triplet cannot be aligned along the axis w β . This is expressed by the following constraints:

$$ \begin{array}{l}\forall \beta \in B,\forall i,{i}^{\prime },{i}^{{\prime\prime}}\in I/i<{i}^{\prime }<{i}^{{\prime\prime} },\forall h\in {C}_i,\forall {h}^{\prime}\in {C}_{i^{\prime }},\forall {h}^{{\prime\prime}}\in {C}_{i^{{\prime\prime} }}/{L}_{1 hi}+{L}_{1{h}^{\prime }{i}^{\prime }}+{L}_{1{h}^{{\prime\prime} }{i}^{{\prime\prime} }}>{D}_{\beta}\hfill \\ {}{\sigma}_{\beta h{h}^{\prime }i{i}^{\prime}}^{+}+{\sigma}_{\beta h{h}^{\prime }i{i}^{\prime}}^{-}+{\sigma}_{\beta {h}^{\prime }{h}^{{\prime\prime} }{i}^{\prime }{i}^{{\prime\prime}}}^{+}+{\sigma}_{\beta {h}^{\prime }{h}^{{\prime\prime} }{i}^{\prime }{i}^{{\prime\prime}}}^{-}+{\sigma}_{\beta h{h}^{{\prime\prime} }i{i}^{{\prime\prime}}}^{+}+{\sigma}_{\beta h{h}^{{\prime\prime} }i{i}^{\prime}}^{-}\le 2.\hfill \end{array} $$

The proof is straightforward. It is sufficient to notice that being the hypothesis stated, at the most, two components may be aligned along the axis w β and that for each pair of components, either the corresponding variable σ + β or σ − β must be zero.

As a further observation, note that the implications correlating alignment and orientation, as presented in this section, would be susceptible to extensions involving chains of more than three components. Their introduction could provide practical advantages in the perspective of a dedicated branch-and-cut approach.

Remark 3.4

When the layer constraints reported in Sect. 2.3.5 are introduced in the model, inequalities (3.15a) and (3.15b) can properly be extended. Moreover, the necessary condition s \( \forall i\in I\kern0.35em {w}_{3i}\ge \underset{i^{\prime}\ne i}{ \min}\left\{{L}_{1{i}^{\prime }}\right\}{\overset{\frown }{\chi}}_i \) can explicitly be added, following the perspective presented in this section.