A new modeling approach for the unrestricted block relocation problem

We consider the block relocation problem (BRP), a combinatorial optimization problem that may arise in storage systems where items are organized in stacks. The objective is to retrieve all items in a predefined order with a minimal number of relocations. It can be distinguished between a restricted and an unrestricted version of the BRP. While in the restricted BRP (R-BRP) only relocations of items located above the item to be retrieved next are permitted, in the unrestricted BRP (U-BRP) all possible relocations are allowed. Existing exact methods concerning the BRP are frequently search-based methods which appear to be very effective. Nevertheless, recent literature concerning the R-BRP has shown that model-based methods can be competitive and therefore should also be taken into consideration. In this paper, we propose a new model-based approach for the U-BRP. It eliminates the fact that the number of variables is increasing with the number of necessary relocations; a disadvantage most mathematical models for the U-BRP have in common.


Introduction and problem description
In a storage system, the space available for storing items is scarce.Due to that fact, items are often stored in stacks, i.e., on top of each other.For example, this is a common approach for storing containers in a container terminal.Due to technical conditions, items are only accessible from above, e.g., via rail mounted gantry cranes.Consequently, an item can be blocked by other items stored above it.If an item to be retrieved is blocked, repositioning moves are necessary.Since repositioning moves may be very time consuming, minimizing the total number of these moves can have crucial impact on the efficiency of such a storing system.
An optimization problem that may arise in a storing system like this is the wellknown block relocation problem (BRP) which can be stated as follows: There is a given number N of items with labels 1 to N piled up in W last-in-first-out stacks with limited height H.The N items have to be retrieved in a predefined order 1, … , N until all stacks are cleared.The item to be retrieved next is called target item and the stack in which it is located is called target stack.If a target item is not the topmost of its stack, all items located above it must be relocated to other stacks first.So there is a distinction between two types of moves: relocations and retrievals.A relocation moves an item from its current position to another position within the stacking area.The new position is always on top of another (possibly empty) stack.A retrieval removes an item from the stacking area.The objective is to retrieve all items in the predefined order with a minimal number of relocations.Note that the number of retrievals is a constant since it always equals the number N of items and therefore does not need to be taken into consideration within the objective.
In the top left corner of Fig. 1 (at c = 0 ), a stacking area with an initial configu- ration consisting of W = 3 stacks (stack 3 is an empty stack) with a limited height H = 3 is depicted.A position within the two-dimensional stacking area is denoted as a slot (i, j) which is a 2-tuple consisting of a stack i ∈ {1, … , W} and a tier j ∈ {1, … , H} , e.g., item 1 is located in slot (1, 2).A configuration (1) each slot is occupied by at most one item and (2) each item is located on top of another item or on the floor, i.e., there are no floating items.
A solution of the BRP is a sequence of moves (relocations and retrievals).Each move of such a sequence leads to a transition from a configuration c − 1 to a con- figuration c.The transition which results in configuration c is called the c-th transition.A solution is feasible if (1) the resulting configurations after each single move within the sequence are feasible, (2) the retrieval order is correct and (3) the stacking area is cleared in the end.
Figure 1 presents an optimal solution of the unrestricted BRP (U-BRP) for an exemplary initial configuration.Let the occurring configurations be numbered by A new modeling approach for the unrestricted block relocation… c = 0, … , C .After relocating item 5 from slot (2, 3) to slot (3, 1) within the first transition, item 2 is only blocked by one item, namely item 6.Within the second transition, item 4 can be relocated onto item 5 without creating a further blockage.Since item 1 is the topmost of its stack in configuration c = 2 it can be retrieved within the third transition.After two more relocations the configuration c = 5 is free of blockages and hence all items can be retrieved without further relocations.Therefore, an optimal solution of the U-BRP contains four relocations to clear the stacking area.It can be observed that the first relocation does not originate in the target stack.This opportunity of presorting items characterizes the U-BRP and can be advantageous with respect to the objective of minimizing the total number of relocations.
A second version of the BRP is the restricted BRP (R-BRP).It is based on an assumption proposed by Kim and Hong (2006) which prevents relocations of items not located in the target stack.This reduces the complexity of the problem since the decision on which item to relocate next is fixed.The only decision to make is determining the stack to which a relocation moves a blocking item.An optimal solution of the R-BRP for the identical example of Fig. 1 requires five relocations.
The main contribution of our paper is a new modeling approach for the U-BRP with a reduced number C of transitions (or configurations) that are considered.To the best of our knowledge, all mathematical model formulations for the U-BRP in the literature allow at most one relocation per transition.Thus, the size of these model formulations increases with the number of necessary moves.In our approach, the number C of transitions (or configurations) which need to be considered equals the number N of items, achieved by allowing multiple moves within each transition.We show that modifying a model formulation in such a manner may significantly increase its performance.
This paper is organized as follows: In Sect.2, we give an overview of relevant literature.A mixed integer linear model formulation whose size is independent of the number of relocations required is presented in Sect.3. In Sect.4, a row generation framework is proposed to construct an optimal solution for the original problem U-BRP by means of the model formulation.Acceleration methods to speed up the row generation procedure are proposed in Sect. 5.In Sect.6, the results of a computational study are presented which examines if a modeling approach may benefit from the reduction of the number of considered configurations.Finally, a short conclusion is given in Sect.7.

Literature review
The BRP may arise in any storage system in which items are organized in last-infirst-out stacks, e.g., containers in a container terminal, pallets or steel plates in a warehouse, freight cars in a shunting yard etc. Due to the large and still increasing amount of cargo being shipped in containers via sea transportation, the field of container terminal logistics is apparently one of the most relevant areas.According to that, the BRP is also known as the container relocation problem (CRP).For further insights into container terminal logistics and operations research applications see, e.g., Steenken et al. (2004), Stahlbock and Voß (2008), Caserta et al. (2011) and Covic (2019).
The BRP is an NP-hard problem (see Caserta et al. (2012)) which belongs to the research field of storage problems.Storage problems can be classified into storage loading, unloading and premarshalling problems.Storage loading problems deal with incoming items which have to be assigned to locations within a storage area.Within storage unloading problems, in turn, items are retrieved and therefore leave the storage area.The BRP is a typical variant of a storage unloading problem where the storage area is organized in last-in-first-out stacks.Lastly, storage premarshalling problems deal with presorting operations without any items entering or leaving the storage area.Additionally, there exist combined problems, e.g., simultaneous planning of loading and unloading operations.For a detailed classification scheme of storage problems and a categorisation of existing literature see Lehnfeld and Knust (2014).
The BRP has been introduced by Kim and Hong (2006) in the restricted version.The authors propose an assumption called Assumption A1 which permits only relocations above the target item.They develop a heuristic based on a calculation of the "expected number of additional relocations" caused by executed moves as well as an exact branch-and-bound procedure.The Assumption A1 is widely used in the literature and therefore a large part of subsequent studies addresses the R-BRP.For the R-BRP, there exist several heuristics (see Bacci et al. 2019;Caserta et al. 2009Caserta et al. , 2011Caserta et al. , 2012;;Jin et al. 2015;Jovanovic and Voß 2014;Jovanovic et al. 2019;Tang et al. 2015;Ting and Wu 2017;Ünlüyurt and Aydın 2012;Wan et al. 2009;Zhang et al. 2020) and exact methods (see Bacci et al. 2020;Expósito-Izquierdo et al. 2014, 2015;Ku and Arthanari 2016;Quispe et al. 2018;Tanaka and Takii 2016;Tanaka and Mizuno 2018;Tanaka and Voß 2022;Ünlüyurt and Aydın 2012;Zhang et al. 2010Zhang et al. , 2020;;Zhu et al. 2012).Some of the most effective exact methods for the R-BRP are search-based algorithms, like the branch-and-bound algorithms proposed by Tanaka and Mizuno (2018) or Zhang et al. (2020) as well as an iterative deepening A* (IDA*) algorithm proposed by Quispe et al. (2018).Nevertheless, Bacci et al. (2020) present a branch-and-cut algorithm and Tanaka and Voß (2022) propose an iterative algorithm based on binary linear models which both have proven to be very competitive model-based approaches for the R-BRP.In particular, Tanaka and Voß (2022) compared their model-based approach with the model-based approach of Bacci et al. (2020) and with three variants of one of the best-performing search-based approaches of Tanaka and Mizuno (2018) on 840 instances [test (CV) instances proposed by Caserta et al. (2011) and Caserta et al. (2012)].The results show that the model-based approach of Tanaka and Voß (2022) outperforms all other investigated methods on these instances and that it is also the first algorithm which can optimally solve all instances of the dataset.
Furthermore, mathematical model formulations for the BRP have been developed.Kim and Hong (2006) and Caserta et al. (2011) present dynamic programming formulations for the R-BRP.Wan et al. (2009) develop a binary linear model called MRIP (Minimization of Reshuffles IP) for the R-BRP which is corrected and further improved by Tang et al. (2015).Caserta et al. (2012) propose another binary linear model for the R-BRP called BRP-II.For specific instances, the BRP-II falsely has no feasible solution which is corrected by Expósito-Izquierdo et al. (2015) and Zehendner et al. (2015).Furthermore, Zehendner et al. (2015) present a binary linear model called BRP-II-A which is an improved version of the BRP-II with a reduced number of binary variables.Zehendner and Feillet (2014) develop a reformulation of the BRP-II which is used within a branch-and-price framework.A further binary linear model for the R-BRP called CRP-I with an even lower number of binary variables than the BRP-II-A is proposed by Galle et al. (2018).The authors make use of a binary encoding of a configuration as proposed by Caserta et al. (2009).Bacci et al. (2020) develop a compact binary model for the R-BRP which contains logical constraints.Due to the complexity of these constraints, they are relaxed and the remaining model is embedded within a branch-and-cut framework.Most recently, Tanaka and Voß (2022) propose a binary linear model for the R-BRP which depends on sets of all possible relocation sequences for each individual item.Due to the exponential number of these sequences, the authors present a relaxed model which is embedded within an iterative algorithm.
For the U-BRP, Lee and Hsu (2007) formulate an integer model as a multicommodity flow problem.The authors actually address the premarshalling problem, but they propose model modifications which also make it applicable for the U-BRP.A binary linear model for the U-BRP called BRP-I is proposed by Caserta et al. (2012).Petering and Hussein (2013) develop a mixed binary linear model for the U-BRP called BRP-III which has a decreased number of binary variables in comparison with the BRP-I.A unified binary linear model which is applicable for the premarshalling problem, the R-BRP and the U-BRP, is proposed by de Melo da Silva et al. (2018).They present two variants of model formulations called BRP m1 and BRP m2 , whereas BRP m2 outperforms all existing model formulations for the U-BRP so far.Finally, Lu et al. (2020) develop a further unified binary linear model called BRP m3 which is, to the best of our knowledge, the state-of-the-art model formulation for the U-BRP.
A recent overview of existing literature concerning the BRP can be found in Lu et al. (2020).The authors propose a classification scheme which consists of 16 variants of the BRP.Furthermore, they give a recap of existing lower bounds on the number of relocations and their development in the literature as well as a proposal for a new and stronger lower bound.

Mathematical model formulation
In this section, we propose a mixed integer linear model called U-BRP master .The notation and the structure of the model formulation is similar to the BRP-I proposed by Caserta et al. (2012).The BRP-I as well as the other existing mathematical model formulations for the U-BRP has in common that they permit at most one relocation per transition.This property makes it easy to prevent infeasible relocations.
As a major difference to these model formulations, we are allowing multiple relocations per transition.This provides the possibility to decrease the size of the model since the number of configurations which must be considered can be reduced.Even though there exist more efficient model formulations for the U-BRP, as presented in Sect.2, we make use of the BRP-I as a basis since we can exploit some of its structural properties.Furthermore, it is very suitable for a reduction of the number of occurring configurations and therefore for an investigation if such a modification may involve benefits.
Note that within model formulations which ensure the configurations to be feasible after each individual move, the number C of occurring configurations needs to be an upper bound on the total number of moves (relocations and retrievals).In contrast to that, within the proposed U-BRP master the configurations are not ensured to be feasible after every move, but only after each retrieval.Therefore, the number C of configurations which must be considered is determined by the number of retrievals or items (C = N ), respectively.
The transition resulting in configuration c contains the retrieval of item n = c , i.e., within the c-th transition the item c is the target item.In addition, several relocations may take place in a transition as well.Remember the optimal solution of the U-BRP for our example presented in Fig. 1. Figure 2 presents the same solution adapted to the framework of the U-BRP master .Within the U-BRP master , e.g., the first transition contains the retrieval of item 1 and additionally two relocations which results in the feasible configuration c = 1.
A quick recap of the parameters: W is the number of stacks, H is the maximal height of the stacks, N is the number of items and items are numbered 1, … , N which indi- cates the sequence in which items are to be retrieved.Additionally, there exist binary A new modeling approach for the unrestricted block relocation… parameters vnc which take value 1 if item n has already been retrieved and therefore is not anymore located within the stacking area of configuration c (0 otherwise).Note that we know in advance within which (c-th) transition a specific item n is retrieved and therefore vnc are not decision variables but parameters defined as follows: Furthermore, there are four types of decision variables in the U-BRP master .The variables bijnc take value 1 if item n is located in slot (i, j) of configuration c (0 other- wise).Note that bijn0 are parameters and define the initial configuration c = 0 .The value of the variables xijklnc represent the number of how many times an item n is relocated from slot (i, j) of configuration c − 1 to slot (k, l) of configuration c (0 otherwise), i.e., within the c-th transition.Note that usually xijklnc takes values {0, 1} only since it is very unlikely that an optimal solution contains an identical relocation of the same item more than once within a transition.Nevertheless, we do not exclude this opportunity in order not to falsely cut off such an optimal solution (see the Appendix 1 for a feasible solution with an x-variable having the value 2).The binary variables ȳijnc take value 1 if item n is retrieved from slot (i, j) within the c- th transition (0 otherwise) and therefore item n is not anymore located within the stacking area of configuration c.The variables B ij take value 1 if slot (i, j) contains an item within configuration c = N − W which is blocking any of the items located below it (0 otherwise).Finally, the mathematical program can be stated as follows: In configuration c = N − W , the number of remaining items equals the number W of stacks.Thus, each blocking item can be relocated to an individual stack and the problem becomes trivial for all succeeding transitions.For these transitions N − W + 1, … , N , the number of relocations required equals the number of block- ing items and therefore it is not necessary to calculate the moves of these transitions.As a result, most decision variables and constraints are not defined for c > N − W .This insight is based on a model formulation for the R-BRP called CRP-I proposed by Galle et al. (2018).For illustration purposes, most figures depict all transitions 1, … , N and its contained moves, even though the moves of the transitions N − W + 1, … , N are not explicitly calculated within the U-BRP master .
According to that, the objective (2) minimizes the sum of the number of relocations within the transitions 1, … , N − W and the number of blocking items within configuration c = N − W .Note that the number min(H, W) is the maximal height a stack can have within the configuration c = N − W so that blocking items cannot be located on a tier j > min(H, W) .Constraints (3) state that in a configuration an item can either be located inside or outside of the stacking area, not both.Constraints (4) make sure that a slot (i, j) cannot be occupied by more than one item within a configuration.Constraints (5) prevent an empty slot (i, j) within a configuration if an item is located above it in slot (i, j + 1) , i.e., there must not be floating items.Con- straints (6) are "flow balancing constraints" which transfer the correct positions of 1 3 A new modeling approach for the unrestricted block relocation… all items from a configuration c − 1 to its consecutive configuration c considering all possibly performed relocations and retrievals.Note that bijn0 are binary parameters and define the initial configuration of the stacking area.Constraints (7) ensure that an item which is not anymore located within the stacking area of configuration c ( vnc = 1 ) must be retrieved in a preceding transition c � = 1, … , c ( ȳijnc � = 1 ).Addi- tionally, constraints (8) are applied to the U-BRP master to prohibit relocations within one stack since such relocations are never feasible with respect to the original problem.Actually, the variables xijklnc with i = k can be omitted from the U-BRP master , but for a better readability and to go in line with the remainder of this paper, we just state this by (8).Constraints (9) are introduced for counting the number of blocking items by means of the variables B ij .If there is an item n located in slot (i, j) of con- figuration c = N − W and there is an item n ′ < n located in any of the slots below slot (i, j), then the corresponding variable B ij is enforced to take the value 1.The domains of the decision variables are declared by ( 10)-( 13).Note that even though the variables bijnc and B ij are defined continuous, the model enforces these variables to take values {0, 1} only ( bijnc enforced by ( 4) + ( 6), B ij enforced by (2) + ( 9)).
A drawback of the U-BRPmaster is that it may deliver optimal solutions which are infeasible with respect to the original problem U-BRP.Remember that the U-BRP master does not contain a configuration after each move, i.e., there are omitted configurations and therefore not every individual move is checked for feasibility.This means that even though optimally solving the U-BRP master delivers a solution with feasible configurations only, there might be omitted configurations which would be infeasible.
Note that Fig. 2 shows a feasible but not an optimal solution of the U-BRP master .An optimal solution of the U-BRP master for the same instance is presented in Fig. 3.The first and second transitions in Fig. 3 include two and three moves, respectively.The first transition, for example, includes one relocation ( x * 231251 = 1 ) and one retrieval ( ȳ * 1211 = 1 ) leading to the feasible configuration c = 1 , but the U-BRP master does not deliver information about the order of multiple moves within a transition.It is obvious that a solution must lead to a feasible configuration after each single move to be feasible for the original problem U-BRP.In our small example it is easy to see that the moves of the first transition cannot be executed in any order.If the Fig. 3 An optimal solution of the U-BRP master with objective function value 3 relocation of item 5 is performed first, the result would be a configuration with slot (1, 2) being occupied by the two items 1 and 5 which is infeasible.On the other hand, retrieving item 1 first would lead to a configuration including floating items which is also infeasible.For the second transition, there does not exist any feasible order with respect to the original problem U-BRP, either.
To guarantee feasibility (and optimality) of the solutions, the U-BRP master is embedded within a row generation framework.Within this procedure, for each transition including multiple moves that cannot be executed in any order, a constraint is generated and added to the U-BRP master to prohibit a specific combination of moves.This procedure is presented in the next section.

A row generation framework
In the following, we propose a row generation framework to construct an optimal solution for the original problem U-BRP by means of the model formulation U-BRP master .The idea is to generate and add constraints to the U-BRP master until the optimal solution of the U-BRP master is feasible for the original problem U-BRP.Note that every optimal solution of the U-BRP is feasible for the U-BRP master , but not vice versa.If a specific combination of moves cannot be part of a feasible solution of the U-BRP, then this combination of moves is prohibited within the U-BRP master by an additional constraint.Thus, an optimal solution of the original problem U-BRP is never cut off by a generated constraint.
Given an optimal solution of the U-BRP master , for each transition c = 1, … , N − W a subproblem is to be solved.In each subproblem, it must be examined if there is any order for the corresponding moves leading to feasible configurations after each single move.Thus, if a transition contains only one move, there is no subproblem to be solved.Remember that within the c-th transition, the item c is the target item and is therefore retrieved.So if there is only one move within a transition, this move is a retrieval.Remember that the transitions N − W + 1, … , N are not contained within the U-BRP master and therefore do not need to be considered.
In the next Sect.4.1, the general outline of the row generation procedure is presented and illustrated by means of a small example and due to the small size of the example the subproblems are solved by just looking closely.Actually, within the implementation of the whole procedure, the subproblems are solved with the help of a model formulation.This model formulation for the subproblems (separation problems) to identify infeasible combinations of moves is given in Sect.4.2.

General outline
Recall the optimal solution of the U-BRP master of our example presented in Fig. 3.The first subproblem corresponds to the first transition, i.e., the transition from configuration c = 0 to c = 1 .As already mentioned in Sect.3, there exists no order for the moves of the first transition ( x * 231251 = 1 and ȳ * 1211 = 1 ) which leads to a feasible configuration after each single move.Thus, this combination of moves 1 3 A new modeling approach for the unrestricted block relocation… cannot be part of a feasible solution of the original problem U-BRP.To prohibit this combination, the model U-BRP master is extended by an additional constraint.
Let c be the number of moves within the c-th transition of an optimal solution of the U-BRP master , formally defined as 5-tuples (i, j, k, l, n) corresponding to all relocations that may or may not take place within a transition and let 3-tuples (i, j, n)  corresponding to all retrievals that may or may not take place within a transition. For 1} be the set of the 5-tuples (i, j, k, l, n) corresponding to the relocations contained in the c-th transition of the current optimal solution of the U-BRP master .Analogously, for all be the set including the 3-tuple (i, j, n) corresponding to the retrieval within the cth transition of the optimal solution and let be the set of the 3-tuples (i, j, n) corresponding to the locations of the items within configuration c of the optimal solution.
For each transition c = 1, … , N − W containing a combination of moves which cannot be part of a feasible solution of the original problem U-BRP, a constraint of the form is added to the U-BRP master .By applying a constraint of the form (15), the current combination of moves (left-hand side) within a transition is cut off from the solution space, unless (i) the combination of moves is extended by at least one further relocation ( x-variables on the right-hand side), or (ii) the item locations within the related configurations c − 1 and c are not identical to the corresponding item locations of the current optimal solution ( b-variables on the right-hand side).In such cases, at least one variable on the right-hand side of constraint (15) takes a value greater or equal to 1, so that the respective constraint is disabled.This ensures that a combination of moves is not falsely identified to be infeasible and therefore cut off.For example, a specific combination of moves which is infeasible within a c-th transition with the two related configurations c − 1 and c might be feasible within the same c-th transition if the item locations within the two related configurations c − 1 and c are changed.( 14) In our example (Fig. 3), there are two moves within the first transition and therefore 1 takes value 2. To prohibit the combination of moves within the first transition the additional constraint is added to the U-BRP master .Thus, the current combination of moves ( x * 231251 = 1 and ȳ * 1211 = 1 ) can only remain if at least one further relocation (different from x231251 ) is added to the first transition or if the item locations within the configuration c − 1 (or c) differ.Note that bijn0 are parameters and the corresponding term always has the value 0. Thus, it could actually be omitted from constraint (15) for c = 1.
There also exists no feasible order for the moves of the second transition and therefore additionally the constraint is added to the U-BRP master .The third transition does not need to be considered since it includes only one move.By generating and adding these constraints to the U-BRP master , the first iteration is finished.Iteration 2 begins with optimally solving the extended U-BRP master .The optimal solution of the U-BRP master calculated in iteration 2 is presented in the upper half of Fig. 4. The first and third transitions again contain multiple moves and therefore lead to further subproblems which must be solved.
For clarity reasons, let the following three points recap and define one iteration of the procedure: (1) Optimally solve the U-BRP master .(2) For each transition of the optimal solution of the U-BRP master that contains multiple moves, solve a subproblem.(3) For each subproblem that detects infeasibility, generate and add a constraint to the U-BRP master .
A new modeling approach for the unrestricted block relocation… The procedure is repeated until the optimal solution provided by the U-BRP master is also feasible for the original problem U-BRP, i.e., no subproblem detects infeasibility.This basic version of the procedure needs 26 iterations (when implemented in AMPL/Gurobi) to find the solution which is presented in the bottom half of Fig. 4. Note again that this Sect.4.1 shall explain the general outline of the procedure.Any existing infeasibility within the transitions of the example is identified by just looking closely, so that the constraints are generated without the help of a model formulation.Indeed, the detection of infeasibility and the corresponding separation of constraints is done by means of a model formulation which is presented in the next Sect.4.2.

Separation problem
A separation of constraints of the form (15) is performed by means of a model formulation called U-BRP sub(c) .The master problem U-BRP master and the subproblem U-BRP sub(c) both include configurations and transitions.For clarity reasons and from now on, a configuration (transition) of the U-BRP master is called m-configuration (m-transition) and a configuration (transition) of the U-BRP sub(c) is called Fig. 4 Optimal solutions provided by the U-BRP master embedded within the row generation framework in iteration 2 and 26, respectively.The bottom solution is optimal with respect to the original problem s-configuration (s-transition).In contrast to the U-BRP master and its m-configurations, the U-BRP sub(c) contains an s-configuration after each single move to easily verify the feasibility of all moves.The U-BRP sub(c) is independently solved for each m-transition 1, … , N − W of the U-BRP master whereby the U-BRP sub(c) corresponds to the c-th m-transition of the U-BRP master .Note that the notation c only appears in connection with the U-BRP sub(c) and allows to distinguish between the s-transitions of the U-BRP sub(c) and the corresponding m-transition c of the U-BRP master .
Given an optimal solution of the U-BRP master with N − W m-transitions and its contained moves.The aim of the subproblem U-BRP sub(c) , c = 1, … , N − W is to find a sequence of the moves contained in the c-th m-transition which leads to a feasible s-configuration after each single move.An obvious approach might be to add the following constraints ( 16) to the U-BRP sub(c) which allow at most one move per s-transition: Nevertheless, this straightforward formulation ( 16) would be insufficient.The available moves for the U-BRP sub(c) are limited to the moves which are used within the corresponding c-th m-transition of the U-BRP master and therefore the constraints ( 16) might cause subproblems which have no feasible solution.To guarantee feasibility of each subproblem U-BRP sub(c) 16', the constraints ( 16) are reformulated as soft constraints () by introducing further variables  c ≥ 0, c = 1, … ,  c .Within an opti- mal solution of the U-BRP sub(c) , the variables c indicate if the moves within the c -th m-transition are feasible with respect to the original problem U-BRP (all c = 0 ) or not (at least one  c > 0).
The number of s-configurations which must be considered within the U-BRP sub(c) (additionally to the initial s-configuration) is given by the number of moves  c within the c-th m-transition, calculated as presented in (14).Furthermore, the initial s-configuration c = 0 and the final s-configuration c =  c of the U-BRP sub(c) are deter- mined by the m-configurations c − 1 and c of the U-BRP master , respectively.
The notation used within the U-BRP sub(c) is like in the U-BRP master .For clarity reasons, there is a small adjustment to clearly mark notations related to the U-BRP sub(c) or the U-BRP master , e.g., x ijklnc is used within the U-BRP sub(c) and xijklnc within the U-BRP master .Finally, the subproblem U-BRP sub(c) , c = 1, … , N − W can be stated as follows: A new modeling approach for the unrestricted block relocation… The constraints ( 18)-( 22) and ( 24) are fundamental restrictions of the U-BRP which are already known from the U-BRP master , constraints (3)-( 8).Each subproblem U-BRP sub(c) corresponds to the c-th m-transition of an optimal solution of the U-BRP master which may contain several relocations and always contains exactly one retrieval, namely the retrieval of item n = c .The number of moves  c within the c-th m-transition determines the number of s-transitions or, respectively, the number of s-configurations which are considered within the U-BRP sub(c) (additionally to the initial s-configuration c = 0 ).Thus, most decision variables and constraints are defined for c = 1, … ,  c.
The objective ( 17) maximizes the number of s-configurations in which the target item c is no longer located within the stacking area.This expression is equiva- lent to a minimization of the number of relocations.As already mentioned, only the retrieval of item n = c takes place within the c-th m-transition and therefore within the U-BRP sub(c) , but it is unknown in which s-transition the retrieval takes place.Thus, v nc , n = 1, … , N, c = 1, … ,  c is a variable iff n = c and so, only v cc occurs in the objective (17).All items n < c are already retrieved in preceding m-transitions and all items n > c will be retrieved in later m-transitions whereby all v nc with n ≠ c are parameters defined as follows: The second part of the objective (17) is a penalization term which is associated with the constraints (16').If the c-th s-transition of the U-BRP sub(c) contains more than one move, then the corresponding variable c takes a value greater than 0 which, in turn, is penalized in the objective (17).Note that the variables c are always integer since their values are determined by binary variables only.The weights within the penalty term ( c c ) are set to the number of moves  c .These are the minimum values required to ensure that an optimal solution of the U-BRP sub(c) violates one or more constraints of (16) only if there exists no feasible solution which does not violate (16), i.e., if there is no feasible solution with c = 0 for all c = 1, … ,  c (see the Appendix 2 for a proof).
Furthermore, the parameters b ijn0 and b ijn c which determine the initial and final s-configuration of the U-BRP sub(c) are defined as follows: Thus, the origin m-configuration of the c-th m-transition (m-configuration c − 1 ) defines the initial s-configuration c = 0 of the subproblem U-BRP sub(c) (by ( 32)) and analogously the resulting m-configuration of the c-th m-transition (m-configuration c ) defines the final s-configuration c =  c of the subproblem U-BRP sub(c) (by ( 33)).( 30) A new modeling approach for the unrestricted block relocation… Finally, the constraints ( 24) and ( 25) enforce that only those moves can be used within the U-BRP sub(c) which are also used within the corresponding m-transition.Thereby, the majority of the variables x ijklnc and y ijnc is fixed to the value 0. Addition- ally, the constraints (23) ensure that each relocation of the c-th m-transition takes place within the U-BRP sub(c) .In most cases the constraints (23) are superfluous since the available relocations (those which are not prohibited by ( 24)) have to be used anyway to reach the given final s-configuration, but in a few cases the final s-configuration within a subproblem can be obtained by using only a subset of these relocations.In this scenario, a relaxation of constraints ( 23) may result in a generation of identical constraints within consecutive iterations and thus the procedure does not terminate.In turn, integrating the constraints (23) into the U-BRP sub(c) guarantees that the procedure can find an optimal solution within a finite number of iterations (see the Appendix 3 for an example in which the constraints ( 23) are mandatory as well as for a clarification that the procedure never causes deadlocks if the constraints (23) are enabled).Similar constraints for the retrievals are not needed.Within each subproblem the given final s-configuration always includes one item less than the given initial s-configuration and there is always exactly one retrieval permitted.Thus, the final s-configuration cannot be obtained if the respective retrieval is not a part of the solution.
The example in Fig. 5 gives an overview of the connection between the master problem and the subproblems.The upper row shows an optimal solution of the Fig. 5 First two m-transitions of an optimal solution of the U-BRP master and optimal solutions of the two corresponding subproblems U-BRP sub(1) and U-BRP sub(2) U-BRP master which occurs within the procedure.Only the first two m-transitions contain relocations and are therefore presented.Furthermore, the corresponding subproblems U-BRP sub(1) and U-BRP sub(2) are presented.Since the first m-transition contains 1 = 4 moves, the corresponding subproblem U-BRP sub(1) consid- ers the s-configurations 0, … , 4 and therefore four s-transitions.Analogously, the U-BRP sub(2) considers the s-configurations 0, … , 2 and therefore two s-transitions.The optimal solution of the first subproblem contains variables  c > 0 so there is no feasible solution which leads to a feasible s-configuration after each single move.Thus, the combination of moves cannot be part of a feasible solution of the original problem U-BRP and therefore is cut off by adding the constraint to the U-BRP master .On the other hand, the optimal solution of the second subproblem contains c = 0 for all c = 1, … , 2 .Thus, the combination of moves of the sec- ond m-transition is identified to be feasible with respect to the original problem and therefore no constraint is generated.

Acceleration methods
The row generation framework presented in Sect. 4 is a basic version.In this section, we propose possible modifications which may accelerate the procedure.In Sect.5.1, we present possibilities to fix a part of the variables within the U-BRP master to the value 0 in a preprocessing step.In the Sect.5.2, we present valid inequalities for the U-BRP master .In the Sects.5.3 and 5.4, variable fixings and symmetry breaking constraints for the U-BRP sub(c) are presented.Finally, in Sect.5.5, we propose three different types of constraints which can be generated within the procedure.

Fixing the variables of the master problem
Some moves can be excluded in advance and therefore the corresponding variables of the U-BRP master can be fixed to the value 0 in a preprocessing step.These A new modeling approach for the unrestricted block relocation… variables can actually be omitted from the model, but for a better readability we leave the U-BRP master unchanged and state this by the definitions (34)-( 36).
The first fixings (34) concern the variables bijnc .
Remember that item c is the target item within the c-th m-transition.Thus, all items n ≤ c are not anymore located within m-configuration c.Second, an item n cannot be located on a tier j in an m-configuration c if there are not enough items left within that m-configuration to reach that tier.Therefore, in an m-configuration c there is never an item n located on a tier j higher than the number N − c of remaining items in that m-configuration.
The variables xijklnt are fixed as denoted by ( 35).There are four cases in which a variable is fixed.
The first condition i = k of (35) prohibits a relocation within one stack since such a relocation is never feasible.Note that by applying these fixings, the constraints (8) become redundant and can therefore be relaxed.Second, within the c-th m-transition the target item c is never relocated but only retrieved since a relocation of this item is superfluous.Additionally, all items n < c are already retrieved and therefore can- not be relocated anymore.Thus, all relocations with n ≤ c can be fixed to 0. Third, a relocation of an item from tier j = 1 of one stack to tier l = 1 of another stack is superfluous since it is indifferent in which stack an item is located if it is on the bottom.Fourth, a relocation is excluded from the solution space if the sum j + l of the corresponding tiers of this relocation cannot be reached by the remaining items.Remember that the variables xijklnc represent relocations within the c-th m-transition originating in m-configuration c − 1 .Thus, the number of the remaining items within the concerned m-configuration is N − c + 1 .The sum j + l connected to a relocation can have a maximal value N − c + 2 (number of the remaining items plus 1) which occurs in an m-configuration with all items located in at most two stacks.Figure 6 gives two examples to illustrate the fourth condition.The items are not numbered since it is irrelevant for illustration purposes.Both examples include N = 3 items and concern relocations within the first m-transition ( c = 1 ).Thus, j + l can have a (34 Fig. 6 The relocation on the left is not affected by a variable fixing.The relocation on the right is prohibited by fixing the corresponding variable to the value 0 maximal value of N − c + 2 = 4 .The example on the left shows a relocation which is not affected by a fixing.On the right side an example for a relocation is presented which is prohibited by fixing the corresponding variable to the value 0. The variables ȳijnc are fixed as stated by (36).
Within the c-th m-transition only item n = c is retrieved and thus the variables ȳijnc are fixed to the value 0 if n ≠ c .Since the variables ȳijnc remain unfixed only if n = c , they can be reduced to triple indexed variables ȳijc , but for clarity reasons we leave the variables unchanged for the remainder of this paper.Second, an item cannot be retrieved from a location on a tier j within the c-th m-transition if the number N − c + 1 of items left within m-configuration c − 1 is too low to reach that tier.

Valid inequalities for the master problem
Furthermore, we formulate valid inequalities for strengthening the relaxed model formulation U-BRP master .The intention is to cut off solutions of the solution space which are infeasible with respect to the original problem U-BRP.Since these solutions may be prohibited within the row generation procedure otherwise, it should reduce the number of iterations required within the procedure.
Property 5.1 If there is a relocation of an item n in an m-transition c ending in slot (i, j), then slot (i, j) must not be occupied by another item n ′ ≠ n or there must be a further relocation of an item n ′ ≠ n in that m-transition c starting in slot (i, j) or there must be a retrieval of item c from slot (i, j).
With respect to the U-BRP master it is feasible to relocate an item n to a slot (i, j) which is actually occupied by another item n ′ ≠ n if item n leaves the slot again by a further relocation within the same m-transition.For the original problem U-BRP, this would cause an infeasible solution since it violates the condition that a slot must be occupied by at most one item.Therefore, we are applying the valid inequalities (37) to implement the insight of Property 5.1.
Note that the variables bijn(c−1) are excluded from the constraints (37) to not pro- hibit some sort of "back and forth"-or cyclic relocations within an m-transition. (36) A new modeling approach for the unrestricted block relocation… Relocations are called cyclic relocations if an item is relocated at least twice within an m-transition c whereby it is relocated to the slot in which it was initially located in the originating m-configuration c − 1 .For specific instances, cyclic relocations are mandatory to obtain an optimal solution and therefore shall not be cut off (see the Appendix 4 for an example).
Property 5.2 If there is a relocation in an m-transition c starting or ending in slot (i, j), then the slot (i, j + 1) above must either be a free slot or there must be another relocation in that m-transition c starting in slot (i, j + 1) or there must be a retrieval of item c from slot (i, j + 1) .Analogously, if item c is retrieved from slot (i, j) in m-transition c, then the slot (i, j + 1) above must either be a free slot or there must be another relocation in that m-transition c starting in slot (i, j + 1).
On the basis of Property 5.2 we are applying the valid inequalities ( 38) and ( 39).
Note that a Big-M is not needed in the constraints (39) since only one retrieval takes place within each m-transition and therefore the left-hand sides have a maximal value 1.
Property 5.3 For all c = 1, … , N − W it can be enforced that the retrieval of the target item c is the last move within the c-th m-transition, i.e., if the target item c is located in slot (i, j) at the beginning of the c-th m-transition, then this slot (i, j) and all slots below can be disabled for any relocation in this m-transition.Such relocations can take place in the following m-transition and therefore no feasible solution of the original problem gets lost.
Based on the Property 5.3 we are applying the valid inequalities (40) to enforce that in each m-transition c = 1, … , N − W there are neither relocations starting or ending in the slot in which the target item c is located nor in a slot below. (38 The valid inequalities (40) have two effects.A first (obvious) effect is that some solutions are cut off from the solution space which are infeasible with respect to the original problem U-BRP.An example for such a solution is presented in Fig. 7.
A second effect is that feasible but symmetrical solutions may be prevented.A minimal example for symmetrical solutions is depicted in Fig. 8.Both solutions are feasible with respect to the U-BRP but the upper solution is prohibited by (40).
(40 A new modeling approach for the unrestricted block relocation… For clarity reasons, this reduction is explained in three steps.( 1) According to Property 5.3, within an m-transition, relocations shall neither start or end in the slot of the target item c nor in a slot below, i.e., a retrieval from a slot (i, j) shall neither enable this slot (i, j) nor a slot below for any relocation.Therefore, the ȳ-variables can be omitted from the constraints ( 37) and ( 38). ( 2) Furthermore, the valid inequalities ( 40) are reduced by just disabling the slot, in which the target item is located, for any relocation.All slots below are implicitly disabled by the constraints (38').( 3) Finally, the valid inequalities ( 38) and ( 40) may restrict any relocation which starts or ends in a specific slot (i, j).These constraints can be reduced by only prohibiting any relocation which starts in that specific slot (i, j) (and not those which end there) and therefore the variables xklijnc and xklij ′ nc are omitted from the left-hand sides of constraints ( 38) and ( 40), respectively.A relocation ending in an occupied slot (i, j) automatically leads to a further relocation starting in that slot (i, j) due to constraints (37').Therefore, if a relocation starting in a slot (i, j) is prohibited by (38') or (40'), then all relocations ending in this slot (i, j) are implicitly prohibited by the constraints (37').Note that the reduced constraints (37'), (38') and (40') are mutually dependent and therefore should be applied together and not isolated.
The constraints (41) affect the retrieval moves.Within the c-th m-transition, as soon as the target item c is accessible it can be retrieved and therefore a relocation of it is always superfluous within that m-transition.Therefore, the constraints (41) enforce that item c is retrieved from the slot (i, j) where it is located at the beginning of the c-th m-transition.
Note that a target item is never relocated within an optimal solution of the original problem U-BRP or the U-BRP master (in its initial version) since such relocations are superfluous.Within the row generation procedure such relocations might occur.The reason is that a superfluous relocation may (re-)allow a combination of moves which is actually prohibited by a generated constraint (see the relocations of item 3 in the Appendix 3 for an illustration of a similar effect).Additionally, these superfluous relocations are implicitly prohibited by the valid inequalities (40) or (40') which actually make the constraints (41) redundant.Nevertheless, applying (41) might reduce solution time.

Fixing the variables of the subproblems
As in the master problem U-BRP master , there are some variables within the subproblems U-BRP sub(c) which can be fixed in a preprocessing step.Note that fixing the variables of the U-BRP master as presented in Sect.5.1 implicitly fixes the corresponding variables of the U-BRP sub(c) within the row generation procedure due to the constraints ( 24) and ( 25).Nevertheless, further variables can be fixed within the U-BRP sub(c) which may not yet be affected.Due to the Property 5.3, the retrieval of the target item can be assumed to be the last move within the respective m-transition.Thus, for each U-BRP sub(c) , c = 1, … , N − W it can be enforced by the fixings (42) that the retrieval of the tar- get item c takes place in the last s-transition c =  c and from the slot (i, j) where the target item is initially located.In turn, all retrievals which are related to the target item c and to its initial location but not to the last s-transition c =  c can be prohibited by the fixings (43).All retrievals which are not related to the target item c or not to the initial location of the target item are already excluded within the row generation procedure by the constraints (25) and are therefore not part of this subsection.
Taking together the constraints (25) and the fixings ( 42) and ( 43), all variables y ijnc are fixed and therefore become parameters.
According to that, it can be assumed that the target item remains in its initial location within all s-configurations c = 1, … ,  c − 1 except in the last one c =  c since it is retrieved in the last s-transition.Therefore, further variables can be fixed according to (44) and ( 45).
Note that one could apply additional fixings to forbid any relocation of the target item, but since it is implicitly done within the row generation procedure by the constraints (24), no further x-variables are fixed within the subproblems.
Enforcing the retrieval of the target item c within the subproblem U-BRP sub(c) to take place in the last s-transition c =  c , v cc are not anymore variables but parameters defined as follows: (42) A new modeling approach for the unrestricted block relocation… Therefore, by applying the variable fixings ( 42)-( 45), v cc can be omitted from the objective ( 17) of the U-BRP sub(c) such that the objective reduces to max ).Furthermore, the constraints ( 18) can be relaxed since v nc are not anymore variables but parameters and the correct values for the variables b ijnc are implicitly ensured by the constraints ( 21).The constraints ( 22) and ( 25) contain parameters only and can therefore also be relaxed.Note that the valid inequalities ( 40) or (40') within the U-BRP master should be activated when fixing the variables within the U-BRP sub(c) as denoted by ( 42)-( 45) to ensure that the Property 5.3 holds true.

Symmetry breaking constraints for the subproblems
The symmetry breaking constraints (47) within the U-BRP sub(c) can be stated as follows: If an optimal solution of the U-BRP sub(c) contains multiple moves within an s-transition c = 1, … ,  c , then there must be at least one s-transition c ′ ≠ c without any move.If one or more s-transitions without any move exist, then the constraints (47) enforce that these s-transitions must take place at the beginning.The last s-transition c =  c is omitted from the domain of the constraints since there is always a move, namely the retrieval of item c , in this s-transition (enforced by ( 42)).Therefore, not to perform a relocation in s-transition  c must not prohibit a relocation in the preced- ing s-transition  c − 1 .Note that the variable fixings ( 42)-( 45) should be applied when adding the constraints (47) to the U-BRP sub(c) to ensure the retrieval to take place within the last s-transition  c.
The sum on the right-hand sides of constraints ( 47) is multiplied by the factor (c − 1) .If a relocation takes place in an s-transition c, then c − 1 is the maxi- mal number of relocations that can take place in the preceding s-transition c − 1 .Therefore, the factor (c − 1) is large enough to not falsely cut off an optimal solution.
Note that the computational effort for optimally solving the U-BRP sub(c) is negligible in comparison with the U-BRP master due to fixing the most variables.Nevertheless, applying the constraints (47) might reduce solution time, especially caused by a reduced number of iterations required.

Alternative types of generated constraints
In this subsection, we propose three types of constraints that can be generated within the row generation procedure.Constraints of Type 1 are the generated constraints presented in Sect.4.1.Alternatively, a constraint generation of Type 2 or 3 can be applied.For illustration purposes, Fig. 9 introduces a new example with N = 6 items, W = 2 stacks and a limited height H = 5 .The upper row shows the first m-transition of an optimal solution for the U-BRP master which occurs within the row generation procedure.The bottom row shows an optimal solution of the corresponding subproblem U-BRP sub(1) consisting of five s-transitions.Additionally, it is illustrated which combinations of moves are prohibited by the different types of constraints.Note that the example of Fig. 9 contains the U-BRP master and the U-BRP sub(c) in its basic versions.This example cannot arise while applying the acceleration methods presented in the Sect.5.1 -5.4 since the solutions violate many of the valid inequalities, symmetry breaking constraints and variable fixings mentioned so far (see the Appendix 5 for further information on the fact that Type 2 is tighter than Type 1, and Type 3 is tighter than Type 2.).Type 1 The first type of generated constraints is like the explanations and the example within Sect.4.1.If a subproblem U-BRP sub(c) has an optimal solution with at least one s-transition c = 1, … ,  c containing more than one move (determined by  c > 0 ), then a constraint is added to the U-BRP master .This constraint prohibits all moves, which are contained in the U-BRP sub(c) , in this combination in the c-th m-transition.In our example of Fig. 9, the additional constraint Fig. 9 First m-transition of an optimal solution of the U-BRP master and an optimal solution of the corresponding subproblem U-BRP sub(1) 1 3 A new modeling approach for the unrestricted block relocation… is added to the U-BRP master .
Type 2 The second of generated constraints is similar to Type 1, but only such moves are considered which are contained in an s-transition c = 1, … ,  c with  c > 0 .In this case, the additional constraint is added to the U-BRP master of our example.Actually, the relocation x132331 does not cause infeasibility since it can take place as the only move within an s-transition (see the third s-transition of the U-BRP sub(1) in Fig. 9).Thus, the decision variable x132331 is not part of the left-hand side of the generated constraint, so the value of x132331 is not affected by this constraint.Note that constraints of Type 2 are tighter than constraints of Type 1 since they always cut off the same solutions as Type 1 constraints but prohibit further solutions if Type 1 and 2 constraints are not identical.In our example, the partial solution x132151 = 1, x211331 = 1, x231161 = 1, ȳ1111 = 1 (and all other xijkln1 and ȳijn1 with value 0) is still feasible with the additional constraint of Type 1, but not anymore with the constraint of Type 2.

Tightening the generated constraints
Remember that a generated constraint (15) does not prohibit a specific combination of moves within an m-transition if it is extended by at least one further relocation or if the adjacent m-configurations differ (see Sect. 4.1).This is ensured by the variables on the right-hand side of (15).In turn, it is very rare that any of these variables on the right-hand side take a value unequal to 0, i.e., that an optimal solution would be falsely cut off if the variables on the right-hand side of (15) were omitted.To make use of this insight, we partition the procedure in two phases and tighten the generated constraints within the first phase.Note that this modification is applicable for all three types of generated constraints presented in Sect.5.5.
Phase 1 The procedure starts with a generation of tighter constraints denoted by (15') instead of (15).
With the exception of the tighter cuts (15'), the procedure is performed as presented in Sect. 4. After a termination of the procedure, the second phase starts.
Phase 2 The result of Phase 1 is the U-BRP master extended by all generated constraints in all iterations of Phase 1 and the corresponding optimal solution of the extended U-BRP master .Since constraints of the form (15') are generated in Phase 1 A new modeling approach for the unrestricted block relocation… instead of constraints of the form (15), the final solution of the U-BRP master obtained in Phase 1 is not proven to be In order not to lose optimality guarantee, all generated constraints of the form (15') within Phase 1 are once again replaced by ( 15) and the procedure is continued by usage of (15).More specifically, if the result of Phase 1 is the U-BRP master extended by x generated constraints of the form (15'), then the input to Phase 2 is the U-BRP master extended by x corresponding constraints of the form (15). Thus, Phase 2 is an optimality check for the solution found in Phase 1.It is worth mentioning that in none of our instances, the optimal solution found at the end of Phase 2 was better than the feasible solution found at the end of Phase 1, and therefore, in all these cases Phase 1 already delivered an optimal solution of the original problem.

Computational study
In this section, the results of a computational study are presented.All model formulations and the row generation method are implemented in AMPL.The commercial solver Gurobi (version 9.1.1)is used to solve the models.The computational experiments are conducted on an Intel(R) Core(TM) i7-4770 CPU @ 3.4 GHz desktop computer with 16.0 GB RAM and Windows 10 Professional (64-bit).
The computational experiments are performed on the test (CV) instances proposed by Caserta et al. (2011) and Caserta et al. (2012).The authors randomly generated instances for different sizes of stacking areas, determined by the number of stacks W ∈ {3, 4, … , 10} and the maximal stack height H ∈ {5, 6, 7, 8, 12} .Each size of a stacking area defines an instance class (H-W) containing 40 instances.Any stack of the stacking area contains H − 2 items, i.e., the two upper slots of each In total, the CV instances contain 21 instance classes and therefore a total of 840 test instances.
Within preliminary computational experiments, we compared the three proposed types of constraints (see Sect. 5.5).The results show no significant differences in performance for most test instances.Only for the largest test instances (5-6) of the preliminary study, Type 3 slightly outperforms Type 1 and 2 and therefore we decided to generate constraints of Type 3 within the computational study.Additionally, we make use of the tighter cuts (15') in Phase 1 of the procedure and (15) in Phase 2 (see Sect. 5.6).All proposed opportunities for fixing the variables (see Sect. 5.1 and 5.3), the valid inequalities (37'), (38'), ( 39), (40') and ( 41) and the symmetry breaking constraints (47) are applied.
Table 1 compares the performances of the new modeling approach (RG), its underlying basic model formulation BRP-I proposed by Caserta et al. (2012) and the state-of-the-art model formulation BRP m3 proposed by Lu et al. (2020).There is a need to calculate an upper and a lower bound on the total number of relocations, as there are parameters within the model formulations that require these values.Thus, we calculate an upper bound by means of a heuristic proposed by Caserta et al. (2012) and a lower bound called LB 3 proposed by Tricoire et al. (2018).The first column of Table 1 gives the instance classes.The columns two, three and four give the number of instances which could be optimally solved by the RG, the BRP-I and the BRP m3 , respectively.The columns five, six and seven give the average runtimes in seconds for the RG, the BRP-I and the BRP m3 , respectively, calculated over all instances of each class.Furthermore, the columns eight, nine and ten depict the respective average runtimes in seconds, calculated over the instances which could A new modeling approach for the unrestricted block relocation… be optimally solved by the RG.A time limit of 3600 s for each instance is implemented.Instances larger than = (7-5) are not presented since none could be optimally solved within the time limit.
The results given in Table 1 confirm that the model BRP m3 outperforms the model BRP-I.Furthermore, the results show that the RG outperforms the model formulation BRP-I.However, the difference in performance between the two model formulations BRP-I and BRP m3 is quite large, so that although the RG framework greatly increases the efficiency of the underlying model BRP-I, it cannot yet compete with the model BRP m3 .
Nevertheless, it is worth noting that 14 of the instances are solved faster by the RG as by the BRP m3 and all these instances have an optimal solution value (number of relocations) that is greater than or equal to the average of the corresponding instance class.Details on this are presented in Table 2.The first and second columns give the CV instance number and the corresponding instance class, respectively.The third column presents the optimal numbers of relocations for the specific instances.The column four indicates the average optimal numbers of relocations for the respective instance classes.Finally, the columns five and six give the runtimes in seconds used by the RG and the BRP m3 , respectively, for solving the corresponding CV instance.For example, the CV instance # 359 (instance class 6-6) requires 13 relocations in an optimal solution and can be optimally solved by the RG in 1410.2 s, while the BRP m3 hits the time limit of 3600 s.Thus, although the BRP m3 performs better than the RG on average, the RG may have advantages on specific instances, especially when an optimal solution requires a relatively large number of relocations.This may indicate that a restriction to allow at most one relocation per transition within a mathematical model (like in the BRP-I and the BRP m3 ) is a Fig. 10 Average runtimes with valid inequalities enabled/disabled.The numbers within the brackets denote the numbers of instances running into the time limit of 3600 s, respectively limiting factor.Thus, a relaxation of this restriction has high potential to speed up the solution time.
The valid inequalities (see Sect. 5.2) improve the solution quality of the U-BRP master and therefore reduces the number of iterations required, but most of the valid inequalities exploit a Big-M, the time for solving the U-BRP master tends to increase.Therefore, the impact of the valid inequalities are investigated by further computational experiments.Figure 10 depicts the results of the comparison of the procedure with the valid inequalities enabled and disabled.The numbers within the brackets indicate the respective numbers of instances that ran into the time limit of 3600 s.The settings are like in the main study, with the exception that only 1 is investigated.Additionally, we generated 40 further instances with (H-W) = (4-3) and N = 6 randomly allocated items.It can be observed that the valid inequalities significantly reduce the runtimes.Without applying the valid inequalities, the number of instances which cannot be solved increases very strongly by extending the instance size.For (H-W) = (5-6) , e.g., the major part (33 of 40) could no longer be solved.None of the instances greater than (5-6) are solved within the time limit unless the valid inequalities are applied.
Finally, Table 3 provides more detailed results of the new modeling approach.The first column of Table 3 again gives the instance classes.The second and third columns present the number of instances for which the first phase and both phases could be finished, respectively.Remember that a solution provided by Phase 1 is feasible for the U-BRP but only proven to be optimal after finishing Phase 2. In this context, it is worth mentioning that for the instances which are optimally solved (proven by finishing Phase 2), all solutions obtained after Phase 1 were already optimal.Thus, in these cases, the initial solution in Phase 2 is already optimal with respect to the original problem, even though it is not yet proven at this point.Already having an optimal solution for the original problem is obviously a great initial situation for the optimization procedure within Phase 2 and therefore Phase 2 has shorter A new modeling approach for the unrestricted block relocation… average runtimes compared to Phase 1 in many cases.This can be observed in the columns four, five and six which give the average runtimes for the completion of Phase 1, Phase 2 and the whole procedure, respectively.The last states the median of the runtimes for the RG which are lower than the average runtimes in all cases, in most cases even significantly lower.The measures presented in Table 3 are computed over the instances which could be optimally solved by the RG.
Even though the new modeling approach cannot yet compete with the model formulation BRP m3 from Lu et al. (2020) or the most effective search-based methods, e.g., the branch-and-bound algorithms from Jin and Tanaka (2023), Tanaka and Mizuno (2018) and Tricoire et al. (2018), it gives rise to spend more effort on model-based approaches with a focus on the reduction of occurring configurations.

Conclusion
This paper adresses the unresticted block relocation problem (U-BRP).A mathematical model formulation is developed to enable multiple moves within each transition, which is prohibited in most models from the literature.This significantly decreases the size of such model formulations and may speed up the solution time.The proposed model formulation is embedded within a row generation framework to construct an optimal solution for the U-BRP.The results of the computational study show that this modeling approach outperforms the underlying basic model formulation.
Although there are more effective procedures for solving the U-BRP optimally, e.g., the branch-and-bound algorithms developed by Jin and Tanaka (2023), Tanaka and Mizuno (2018) and Tricoire et al. (2018), or the model formulation proposed by Lu et al. (2020), the results of the computational study show that the proposed modifications can significantly speed up model-based approaches.Like in the literature concerning the R-BRP, search-based methods have been considered to be superior to model-based approaches.Nevertheless, a recent publication provides a very competitive model-based branch-and-cut approach for the R-BRP.Future research on model-based approaches for the U-BRP could focus on reducing the number of variables by means of a reduction of the number of configurations that are considered, i.e., enable multiple moves within each transition.With respect to the proposed row generation framework, it could be worth to spend more effort on developing techniques for a further reduction of the number of iterations required.A further and more promising approach might be to make use of an adapted modification of a better performing model formulation, e.g., the BRP m2 or the BRP m3 , and embed it within the row generation framework.

3
A new modeling approach for the unrestricted block relocation… tion with one s-transition containing two moves, one s-transition containing no move and all other s-transitions containing exactly one move.Furthermore, the term ∑  c c=1 v cc has its maximum value if the retrieval takes place in the first s-transition ( ⇒ ∑  c c=1 v cc =  c ).In total, such solutions cannot have a greater optimal objective function value than  c −  c = 0.
Thus, feasible solution of the U-BRP sub(c) with at least one  c > 0 is never optimal if there exists a feasible solution with c = 0 for all c = 1, … ,  c , i.e., if an optimal solu- tion of the U-BRP sub(c) has at least one  c > 0 , then there is no feasible solution with c = 0 for all c = 1, … ,  c.Appendix 3: Discussion on the constraints (23)

3.1: Example which requires the constraints (23)
Given an instance with W = 3, H = 5, N = 8 and the initial m-configuration given in Fig. 12 at c = 0 .Figure 12 presents an initial optimal solution of the U-BRP master , i.e., no constraints are generated yet.Note that only the first m-transition is of interest and therefore presented.
The corresponding subproblem U-BRP sub(1) has the optimal solution presented in Fig. 13.Since there is no sequence for the two moves which leads to a feasible s-configuration after each single move, this combination of moves is prohibited by adding the constraint to the U-BRP master .From now on, these two moves can only take place together within the first m-transition of the U-BRP master if at least one further relocation is added to this m-transition (or if the adjacent m-configurations differ).In iteration 57 this scenario occurs and the two moves from iteration 1 take place in an extended combination of moves (Fig. 14).
It is obvious that the two relocations of item 3 are superfluous since no item located below item 3 is relocated or retrieved.The only reason for the "back and forth"-relocations of item 3 is that it enables the moves from iteration 1 ( x122161 and ȳ2111 ) to take place once again within the first m-transition.
These four moves of the first m-transition should now take place in the resulting subproblem U-BRP sub(1) , but without constraints (23), the optimal solution of the resulting subproblem U-BRP sub(1) contains only a subset of the moves and not all.A new modeling approach for the unrestricted block relocation… The optimal solution (Fig. 15) is very similar to that from iteration 1.The only difference is that 1 = 4 s-configurations are considered.Thus, the constraint of iteration 1 is generated again which does not affect the solution space.From now on, the optimal solution of the U-BRP master remains the same and thus the procedure does not terminate.

3.2:
The procedure never causes a deadlock if the constraints (23) are Enabled In general, a row generation procedure never causes a deadlock if (i) the set of feasible solutions of the relaxed problem is finite and (ii) at least one feasible solution is cut off from the solution space of the relaxed problem per iteration.
Let x * be a feasible solution of the U-BRP master which is optimal with respect to the original problem U-BRP.Furthermore, let X be the set of all feasible solutions of the U-BRP master .For any solution x ∈ X , let r(x) be the corresponding number of relocations.Furthermore, let Xr ∶= {x ∈ X ∶ r(x) ≤ r(x * )} be the set of all feasible solutions of the U-BRP master with at most r(x * ) relocations.
Condition (i) is fulfilled: Due to the fact that the stacking area (and therefore the number of possible relocations) is limited by the number W of stacks, the height limit H and the number N of items, the set Xr is finite.In case of an unlimited maxi- mal height, without loss of generality, H = N can be assumed since this is the maxi- mal stacking height which can be obtained.
Condition (ii) is fulfilled: The constraints (23) ensure that each relocation of the c-th m-transition takes place within the solution of the U-BRP sub(c) .Thus, if the U-BRP sub(c) detects infeasibility, then all moves contained in the U-BRP sub(c) (i.e., contained in the c-th m-transition) are involved in a newly generated constraint of the form (15).This cuts off at least one feasible solution x ∈ Xr from Xr .Note that condition (ii) is also fulfilled for cuts of Type 2 and Type 3 presented in Sect.5.5.These types do not involve all moves contained in the U-BRP sub(c) in a newly generated constraint, but all moves which cause infeasibility.Thus, there is also at least one feasible solution x ∈ Xr being cut off from Xr .
Fig. 16 Feasible solution of the U-BRP master with cyclic relocations which is optimal with respect to the original problem Since x * ∈ Xr and | Xr | < ∞ , the worst case scenario is a complete enumeration of all feasible solutions of the U-BRP master with at most r(x * ) relocations, i.e., a com- plete calculation of the set Xr for finding x * .

Appendix 4: Instance with mandatory cyclic relocations
Figure 16 depicts an instance which has no optimal solution free of cyclic relocations, i.e., an item must be relocated at least twice within an m-transition c whereby it is relocated to the slot in which it was initially located in the originating m-configuration c − 1 .The solution presented in Fig. 16 is optimal with respect to the origi- nal problem U-BRP and is depicted within the framework of the U-BRP master .The relocations of item 12 within the first m-transition are of interest in this example.The optimal solution contains 8 relocations.If any relocation of item 12 ending in its initial slot (2, 3) is prohibited within the first m-transition, then there does not exist anymore a solution with 8 relocations and therefore cyclic relocations of item 12 are mandatory.For a better readability, the configurations which are actually omitted within the U-BRP master are presented in light gray.
Appendix 5: Further clarification that constraints of type 3 are the tightest constraints The following example is a recap of the same example presented in Sect.5.5.Initially, the first m-transition contains 4 relocations and 1 retrieval defined by the partial solution x132151 = 1, x211331 = 1, x231161 = 1, x132331 = 1, ȳ1111 = 1 .All other x -and ȳ -variables within the first m-transition have the value 0 in this example as well as in all following exemplary partial solutions (all xijkln1 and ȳijn1 have the value 0).Thus, for a better readibility, all decision variables on the right-hand side can be omitted.The three different types of constraints generated by the procedure are as follows: A new modeling approach for the unrestricted block relocation… Given the constraints of the different types, the following Table 4 gives some exemplary partial solutions.Furthermore, it indicates whether the solutions are feasible or infeasible (and thus cut off from the solution space) with respect to the constraints of Type 1, 2 and 3, respectively.⇒ Type 2 constraints cut off all (partial) solutions which are cut off by Type 1 constraints (solution 1 in this example).Additionally, Type 2 constraints cut off further (partial) solutions which are not cut off by Type 1 constraints (solution 2 in this example).Thus, Type 2 constraints are tighter than Type 1 constraints.
⇒ Type 3 constraints cut off all (partial) solutions which are cut off by Type 2 constraints (solutions 1 and 2 in this example).Additionally, Type 3 constraints cut off further (partial) solutions which are not cut off by Type 2 constraints (solutions 3 and 4 in this example).Thus, Type 3 constraints are tighter than Type 2 constraints.
Funding Open Access funding enabled and organized by Projekt DEAL.We acknowledge support by the Open Access Publication Fund of the University of Duisburg-Essen.

Fig. 1
Fig.1Optimal solution of the U-BRP with objective function value 4 (the total number of relocations)

Fig. 2
Fig. 2 A feasible solution of the U-BRP master

Fig. 7
Fig. 7 Infeasible solution with respect to the original problem U-BRP

Fig. 11
Fig. 11Feasible solution of the U-BRP master with two identical relocations of the same item

Fig. 12
Fig.12First m-transition of an optimal solution of the U-BRP master (iteration 1)

Fig. 14
Fig.14First m-transition of an optimal solution of the U-BRP master (iteration 57)

Table 1
Comparison of the RG, the BRP-I and the BRP m3 Thus, a (H-W) instance contains N = W(H − 2) items.The item numbers 1, … , N are randomly allocated to the available slots the stacking area.

Table 2
Instances for which the RG outperforms the BRP m3

Table 3
Results