Order picking in a parallel-aisle warehouse with turn penalties

In many real-life routing problems, incorporating the negative effects of turns is an important, but often overlooked aspect. This is especially true for order picking in warehouses, where making the turns not only decreases the picking efficiency by reducing the speed of the vehicle, but it also results in other unquantifiable effects such as vehicle tipovers, increased congestion and increased risk of collision with pedestrians or other vehicles. In this paper, we consider the order picking problem in a parallel-aisle warehouse by taking into account the number and effect of the turns. In particular, we show that the problem of minimising the number of turns, minimising travel time under turn penalties, the biobjective problem that involves turn and travel time minimisation as separate objectives, and the triobjective problem with U-turn minimisation as a third objective can all be solved in polynomial time. Our computational results show that the algorithms we develop can generate the corresponding Pareto front very quickly, and significantly outperform heuristic approaches used in practice.


Introduction
The efficiency of most real-life vehicle routing applications is measured by the travel time for the vehicle(s), where the main factor determining the total travel time of a route is generally the time required to cover the distance. In many cases, such as those involving the routing of large vehicles or those arising in urban traffic environments, the effects of making the left, right and U-turns on the total travel time cannot be neglected. This also applies to routing applications in warehouses, where the speed of the vehicle decreases by as much as 80% while making the turns. In addition, the existence of narrow aisles and the intensity of other ongoing activities on the floor result in other unquantifiable effects while making the turns, such as vehicle tipovers, increased congestion and risks of collision with a pedestrian or another vehicle (Occupational Safety and Health Administration 2012). Consequently, the design of an efficient route in a warehouse requires the consideration of the number of turns incurred during travel, in addition to travel distance or time.
In this paper, we take into account the effect of turns on the performance of warehouse order picking activities, where the aim is to collect the items in a list corresponding to an order. Order picking is generally regarded as the most important activity in a warehouse, as it accounts for more than 55% of all warehouse-related costs (Bartholdi and Hackman 2014). Motivated by the fact that around half of the order picking time is spent for travelling between locations of items in the list (Tompkins et al. 2010), the order picking problem (OPP) aims to find a pick sequence for items in an order so that the travel time required to complete the picking process is minimised.
The literature on the OPP assumes that travel time solely depends on the distance travelled. This paper extends the definition of the OPP by accounting for the effects of turns by means of time penalties, and develops polynomial-time exact solution approaches for this extended problem under various objectives. In doing so, the main motivation is that planning of the picker's route by considering the contributions of both travel distance and number of turns can not only result in a more time-efficient route compared to considering the travel distance alone, but also reduces the risks and challenges associated with the turns. To the best of our knowledge, this is the first study that considers the effects of turns on order picking efficiency and/or the effects of travel time and number of turns together.
We consider the OPP in a parallel-aisle warehouse, an example of which is given in Figure 1. Items are located on pick aisles that are parallel to one another and expand from the front cross aisle to the back cross aisle. The warehouse may also consist of middle aisles, which intersect pick aisles perpendicularly, and the existence of which divides the warehouse into multiple blocks. In the OPP, given a pick list, a feasible route of a picker starts from the pickup-and-deposit (P&D) point, picks all the items given in the list and returns to the P&D point. We refer to the OPP with m −1 blocks as m-OPP. Throughout the remainder of the paper, we deal with the case where no middle aisles exist, that is, the case of 2-OPP.  A 2-OPP instance can be modelled on a graph G in the following manner: Each pick item i is represented by a node v i , with node v 0 representing the P&D point. The intersection of pick aisle j with back and front cross aisles are represented by nodes a j and b j , respectively. The path providing direct access from a node to another without visiting a third node is represented by an edge. Associated with edge (i, j) is the corresponding travel distance/time c i j . Graph representation of the 2-OPP instance in Figure 1 is given in Figure 2.
The cases of 2-OPP and 3-OPP have been shown to be polynomially solvable by Ratliff and Rosenthal (1983) and Roodbergen and De Koster (2001b), respectively. In this paper, we show that the travel time minimisation problem on 2-OPP graphs with turn penalties can be solved in polynomial time (i) by modifying the Ratliff and Rosenthal algorithm on the original 2-OPP graph, or (ii) by modifying the 2-OPP graph and applying the Ratliff and Rosenthal algorithm directly. Since this approach only takes into account the effect of turns on the travel time and ignores the adverse unquantifiable effects, we also treat travel distance and number of turns as separate performance measures. In particular, we show that the biobjective problem of minimising turns and travel time, and the triobjective problem that considers U-turn minimisation as a separate objective are both polynomially solvable on 2-OPP graphs. The approaches we develop contribute not only to the practice of picker routing in warehouses, but also to the literature on multiobjective combinatorial optimisation (MOCO), where there is lack of connection between theory and practice as well as efficient solution results on three objective problems (Ehrgott and Gandibleux 2000).
The remainder of this paper is organised as follows: In the next section, we review the literature on distance (or time) minimisation problems in warehouses as well as turn minimisation problems in graphs. In Section 3, we give polynomial time algorithms solving the turn minimisation problem and the time minimisation problem with turn penalties on 2-OPP graphs. Section 4 discusses the polynomial time algorithms that solve the biobjective problem with turn and distance minimisation and the triobjective problem with U-turns. Our computational experiments are provided in Sections 5 and 6 presents our conclusions as well as further research directions.

Literature review
The work in this paper is relevant to a number of streams in the literature, which can be classified under three groups: (i) exact and heuristic solution approaches for the OPP, (ii) studies involving turn minimisation in routing problems and (iii) MOCO problems. For a detailed review of warehouse order picking operations, the reader is referred to De Koster, Le-Duc, and Roodbergen (2007).
The OPP is a special case of the steiner travelling salesman problem (STSP), which aims to find the shortest tour visiting a required subset of vertices on a graph at least once. While the STSP is NP-hard, as it includes the travelling salesman problem as a special case, certain cases of the OPP have been shown to be polynomially solvable. The tractability of the 2-OPP is shown by Ratliff and Rosenthal (1983) using a dynamic programming based algorithm that runs in linear time in terms of the number of pick aisles. The algorithm takes pick aisles as stages and uses the concept of a partial tour subgraph (PTS) to proceed. In order to avoid an extensive search over all PTSs, the PTSs are grouped into equivalence classes, which constitute the states of the algorithm.
The algorithm finds the shortest distance solution for each equivalence classes in each aisle. This requires enumeration over a number of PTSs under the same class, which are formed using the intra-aisle connection types (1)-(6) in Figure 3 and the inter-aisle connection types (i)-(v) in Figure 4 (the two item nodes in Figure 3 are the consecutive items which have the maximum pairwise distance among all consecutive item pairs in the aisle). As our approach generalises the distance minimisation algorithm for the 2-OPP, details of the latter are provided in Section 4. Roodbergen and De Koster (2001b) consider the 3-OPP, in which the complexity is still linear in terms of the number of pick aisles. Çelik and Süral (2014) provide a linear algorithm for fishbone and V-shaped warehouse layouts. However, in both cases, the enumerative nature of optimal solution approaches poses a limitation to their applicability in practice. This has resulted in a number of heuristic approaches being proposed for the OPP. For the 2-OPP, Hall (1993) describes a number of simple-to-apply heuristics. Among these, the S-shape (traversal) heuristic starts from the left-most nonempty pick aisle, traverses all nonempty pick aisles completely, and once all items are picked, returns to the P&D point. Largest gap heuristic    aims to avoid the longest gap between any pair of items or between an item and corner node. Since we relate our results with these two heuristics, the solution to the OPP instance in Figure 1 is given in Figures 5 and 6. Other heuristic approaches for the OPP include Roodbergen and De Koster (2001a), Makris and Giakoumakis (2003), Theys et al. (2010), Çelik and Süral (2010) and Roodbergen, Vis, and Taylor (2015). The sole focus of these studies is travel distance, and the effect of turns is not incorporated.
Turn penalties/restrictions in routing have a wide variety of applications such as garbage collection, postal deliveries, snow plowing and maritime routing. Thus, the problem has been handled in a number of different ways in the literature. These include network modification, incorporating turn minimisation into graph algorithms, minimum link-tour path problems, and bicriteria distance and turn minimisation.
To the best of our knowledge, the most commonly used approaches to handle restrictions and penalties focus on transforming the original network. The simplest way to achieve this is to convert the arcs into nodes and each turn into an arc in the transformed graph. Each transformed arc includes the cost of the starting arc on the original graph as well as the cost of the turn. In more recent applications, Añez, Barra, and Pérez (1996) apply dual graph methods, Winter (2002) uses a pseudo-dual graph representation, Gutiérrez and Medaglia (2008) use labelling approaches and Vanhove and Fack (2012) make use of extensive computational experiments to compare various graph transformation methods. For our case, applying such network transformations results in loss of the parallel-aisle properties and may prevent polynomial time solution of the problem.
A second approach is to transform the turn constrained problem into an equivalent problem for which exact or heuristic solution approaches exist. Recent studies applying this approach include Bautista, Fernández, and Pereira (2008), Soler, Martínez, and Micó (2008), Bräysy et al. (2011), and. Using a different approach, Perrier, Langevin, and Amaya (2008) and Kim, Jun, and Kim (2011) heuristically solve turn constrained routing problems by adjusting the heuristic approach to incorporate turn penalties without making such a transformation. These studies involve problems that are NP-hard even without turn constraints. In this study, the distance (or time) minimisation problem we consider is polynomially solvable and we find a polynomial time algorithm for the turn constrained version as well.
While the single objective turn minimisation problem has not been widely studied, a more general version called the minimum link path problem, in which the objective is to minimise the number of arcs used while going from a source to a sink node, has received significant attention. Recent studies on this problem include Fitch, Butler, and Rus (2001), Wagner (2006), Bereg et al. (2009) and Cook and Wenk (2011). Our study builds upon this stream by showing that a special case is polynomially solvable.
Very few studies consider turn and distance minimisation as separate objectives and focus on complexity results. Arkin, Mitchell, and Piatko (1991) show that the bicriteria shortest path problem is NP-complete on general graphs. Mitchell (1998) gives a number of special cases whose complexity is open, and Arkin et al. (2005) give polynomial time approximation schemes for various special cases. To the best of our knowledge, our study is the first that gives a polynomial time algorithm for the biobjective distance (or time) and turn minimisation problem. Ehrgott and Gandibleux (2000) report that while a considerable amount of literature exists on MOCO, very few studies characterise the number and structure of efficient solutions. Furthermore, they observe a lack of consideration of practical applications of the problems and most studies are limited to two objectives only. In this paper, we consider a practical problem in warehouse logistics and develop approaches that can be easily applied. We characterise the maximum number of efficient solutions and extend our polynomial algorithms to the case of three objectives.

Single-objective turn minimisation and time minimisation problems
In this section, we consider the single objective turn minimisation problem and travel time minimisation problem with turn penalties in a parallel-aisle warehouse, and present the polynomial-time algorithms to solve the two problems.
Throughout the remainder of this paper, a nonempty pick aisle refers to an aisle containing items and/or the P&D point, and the nonempty aisles are indexed as 1, . . . , n, starting from the left-most one. We assume that aisles are narrow enough so that no time is incurred when moving horizontally on a pick aisle. A left or right turn incurs a single turn, whereas a U-turn incurs two turns. These assumptions are in line with those in the literature.
There are three different cases for where the P&D point might be located in the warehouse: (i) on the corner of a pick aisle and a cross aisle, (ii) on a cross aisle, not on the corner of a pick aisle and a cross aisle, and (iii) on a pick aisle. These will be referred to as the corner-depot, cross-depot and pick-depot cases, respectively. Throughout the remainder of the paper, we state our results for the corner-depot case and provide the results for the cross-depot and pick-depot cases in the online supplemental material.

Single-objective turn minimisation problem
Our results depend on the relationship between the connection types in Figure 3 and the number of turns resulting from them, as turns are only incurred when entering/exiting a pick aisle or making a U-turn within an aisle. The following proposition states that we only need to focus on a subset of these connection types. Proofs of all propositions and theorems throughout the remainder of the paper can be found in the online supplemental material. Figure 3 is not needed for solving the distance (or time) minimisation problem. This does not hold for turn minimisation Proposition 3.1 implies that for turn minimisation, we need to take into account connection type (5) only for the aisle incident to the P&D point. By ignoring the empty pick aisles and the corresponding corner nodes, we can disregard connection type (6) in Figure 3. Since connection type (1) does not create any additional turns to those needed for entering and exiting a pick aisle, we need to consider three connection types, for which p denotes the total number of connection types (2) and (3) and q the number of connection types (4) in a given tour.

Proposition 3.1 On corner-depot graphs, connection type (5) in
Proposition 3.2 Given p, q and n ≥ 2 on a corner-depot graph, the total number of turns in the tour is given by 2n + 2 p + 6q + r − 1, where r = 1 if the connection type in the pick aisle incident to the P&D point is (5), and r = 0 otherwise. Here, p + q + r has to be odd for odd n and even for even n.
Proposition 3.2 allows us to calculate the number of turns for a given set of n, p, q and r . Since the restrictions on these are also given, the minimum number of turns for each setting can now be found.
Proposition 3.3 For the corner-depot case, the minimum number of turns is 2n − 1 for even n and 2n for odd n.
Knowing the minimum number of turns allows us to develop Algorithm 1, which finds the minimum number of turns for any given pick list.
Theorem 3.4 Algorithm 1 finds the turn minimising tour on corner-depot graphs in O(n) time.
Algorithm 1. Turn minimisation algorithm for corner-depot graphs if n is even then Apply connection type (1) for every nonempty aisle else Apply connection type (5) on the aisle incident to v 0 ; and type (1) on the remaining aisles end if Starting with the left-most one, join neighbouring connection type (1) pairs using connection type (i) Join subtours using connection types (ii) and (iii) arbitrarily Once the optimal tour graph is found, we use the tour construction algorithm in Ratliff and Rosenthal (1983) to find the optimal tour. Starting from v 0 , the algorithm checks if there is any unused parallel pair of edges incident to the node. If this is the case, it proceeds by using one of those. Otherwise, it selects any one of the incident edges. The algorithm terminates when there are no longer any unused edges in the tour graph. This algorithm can be used to construct the optimal tours found by the algorithms in the next two sections as well.
The following connects the turn minimisation problem to the NP-hard rural postman problem (RPP).
Proposition 3.5 When edges connecting the same two pick aisles are of equal length, the problem of minimising the number of turns on a corner-depot graph is equivalent to solving the RPP with required edges as the within-aisle connection types determined by Algorithm 1.
In most 2-OPP configurations, the turn minimising tour has a very simple structure.
Proposition 3.6 (Complete traversal property) There exists a turn minimising tour on corner-depot graphs that traverses each nonempty pick aisle completely.
The following corollary is a direct result of complete traversal of nonempty pick aisles under certain cases and provides another way of polynomially solving the problem for these cases when n is even.
Corollary 3.7 For even n, S-shape heuristic finds the turn minimising tour on 2-OPP corner-depot graphs.
Using the number of turns for each move type in Proposition 3.2, a modified version of the Ratliff and Rosenthal algorithm can find the turn minimising tour. Here, inter-aisle connection types incur no costs. The costs associated with each intra-aisle connection type are two turns for connection type (1), four turns for connection types (2), (3) and (5) (two turns if any of these have v 0 as their end node), eight turns for type (4) (six turns if it has v 0 as an end node) and no turns for connection type (6).

Single-objective travel time minimisation problem with turn penalties
In this section, we consider the problem of minimising total travel time by taking into account the time lost due to both travel distance and making the turns. Without loss of generality, we assume that each right and left turn incurs k units of time, and it takes 2k time units to perform a U-turn.
It can be inferred from Proposition 3.2 that the number of turns in a given tour depends only on the intra-aisle connection types in Figure 3. Using this inference, we propose two approaches, both of which run in O(n) time, to solve the time minimisation problem optimally.
The first approach involves modification of the Ratliff and Rosenthal algorithm, which is originally used for distance (or time) minimisation on the original corner-depot graph. Using the results of Proposition 3.2, when calculating the time for each equivalence class, intra aisle connection types (1) through (4) incur additional turn times of 2k, 4k, 4k and 8k, respectively. The only exception is on the aisle incident to the P&D node, where the additional turn times are reduced to k, 3k, 3k and 7k, respectively. An example for the application of this algorithm is given in the online supplement.
The second approach uses an extended version of the Ratliff and Rosenthal algorithm on a transformed version of the corner-depot graph, which can be formulated in the following manner: • Each node v i corresponding to the items i closest to the front and back cross aisles are duplicated as v i and v i , which are connected to each other with a travel time of 2k units. • If i is the only node in the aisle, v i and v i are connected to the corner nodes of the cross aisle with the original travel times between node i and the corner nodes unchanged. • For the aisles not incident to the P&D point and with multiple item nodes, the largest travel time between any two consecutive items within the aisle is decremented by 2k time units. • The P&D node is moved into the incident aisle. The travel time between (i) v 0 and the corresponding corner node is set as −0.5k units, (ii) the opposite corner node and the item nearest to it is decremented by 0.5k units, (iii) v 0 and the closest item node is incremented by 0.5k time units and (iv) any two consecutive items with the largest time between is decremented by 2.5k time units.
It is easy to check that the original Ratliff and Rosenthal algorithm correctly runs on this transformed graph, and that the transformation and algorithm also apply to more general cases where turn times can be different on each pick aisle. It should be noted here that although the transformation may result in edges of negative travel time, infinite loops on these edges are avoided by the fact that the Ratliff and Rosenthal algorithm prohibits traversal of any edge by more than twice.
To illustrate this procedure, Figure 7 provides an example with eight aisles and 15 items in the list, along with the corresponding transformed graph.

Biobjective travel time and turn minimisation
In this section, we consider total travel time and turn minimisation as two separate objectives and develop an algorithm that finds the set of all efficient solutions. As the first step, we show that the travel time minimisation problem under a fixed number of turns is polynomially solvable.

Travel time minimisation under fixed number of turns
Since the algorithm that finds the minimum time solution under a fixed number of turns makes use of the Ratliff and Rosenthal algorithm, we give the specifics of this algorithm before proceeding further.
To define a PTS, let L ⊆ G and G − L be the subgraph of G induced by the nodes not in L. A subgraph T ⊆ L is a PTS if there exists a completion C ⊆ G − L such that T ∪ C forms a feasible OPP tour. Of these PTSs, the DP approach only needs to focus on two specific groups of subgraphs of G. The first group, denoted by L − j , is the subgraph induced by a j , b j and all the nodes and edges of G to the left of these two nodes. The second group, denoted by L + j , is the subgraph induced by aisles 1, 2, . . . , j. These two groups constitute the stages of the algorithm.
The PTSs can be grouped into equivalence classes based on a set of attributes: (i) the degree parity of a j , (ii) that of b j and (iii) the number of connected components in the PTS. Two PTSs for which these three attributes are identical have the same set of possible completions, and hence can be used as states in the algorithm. In a class, the first two components can be zero (0) Figure 4 to each L + j equivalence class, dashed lines indicating suboptimality or infeasibility.
Inter-aisle connection types in Figure 4 classes  class, determined from Table 1; and finds the optimal solution for each class on L + j by adding to the optimal L − j solutions the connection types in Figure 3 and finding the minimum time solution for each class, determined by Table 2.
Assume that for a biobjective 2-OPP on a corner-depot graph, we are given the set of p, q and r values. To solve the travel time minimisation problem constrained by these values, the equivalence class definition consists of the degree parities of a j and b j as well as the number of connected components in the PTS. Furthermore, it also includes (i) the remaining number of connection type (2) or (3) moves, denoted byp, (ii) the remaining number of connection type (4) moves, denoted byq, and (iii) the remaining number of connection type (5) moves, denoted byr . These parameters are initalised at p, q and r , and whenever any of these connection types is used, the corresponding parameter is decremented by one. For brevity, we leave out the parameters that initially have zero value from the equivalence class definition. Therefore, each class can have up to six components, depending on the total number of each connection type allowed in the solution.
The algorithm is similar to that by Ratliff and Rosenthal (1983). First, to obtain a L + j PTS from a L − j PTS, a connection type (2) or (3) in Figure 3 can only be used forp > 0 in the class definition. Similarly, connection type (4) can only be used for classes withq > 0. If r = 1, only connection type (5) can be used in the aisle incident to the P&D point.
Under the given feasibility conditions, an L + j equivalence class can be obtained from an L − j one using Table 2 with the sole difference that the resulting class hasp,q orr , decremented by one, depending on the connection type used. Obtaining the L − j+1 classes is also carried out similarly, using the connection types in Figure 4 and finding the minimum time solution for each class, determined by Table 1. In this case, the remaining numbers of connection types in the equivalence class Table 2. Resulting L + j equivalence classes after adding the connection types in Figure 3 to each L − j equivalence class, dashed lines indicating suboptimality.
L − j equivalence Intra-aisle connection types in Figure 3 classes definition remain unchanged. The optimal solution can be found by backtracking from the minimum time solution among the optimal (E, 0, 1C), (0, E, 1C), (E, E, 1C) and (0, 0, 1C) L + n PTSs withp =q = 0 andr = 0. Unlike the 2-OPP, the number of possible equivalence classes depends on the number of nonempty pick aisles in turnconstrained 2-OPP. Given p, q and r , the total number of possible classes is 7( p+1)(q +1)(r +1)−1, where the multiplicative term comes from all possible combinations of the parameters under seven different classes from Ratliff and Rosenthal (1983), and the last term is caused by the fact that classes of type (E, E, 2C) are not possible whenp = p andq = q. Since r takes values in {0, 1} and p and q are both bounded by n, the maximum number of equivalence classes is in O(n 2 ).
For the example in Figure 7 with p = q = 1 and r = 0, Table 3 gives the solution procedure, assuming that each left and right turn takes 2 units of time (hence k = 2), whereas a U-turn incurs 4 units of time. Each entry in the table consists of three elements: (i) the total length of the PTS corresponding to the equivalence class at that stage, (ii) the equivalence class from which the PTS was obtained using the connection types from Tables 1 or 2, and (iii) the connection type used to obtain the PTS. The optimal solution, which has 282 units of travel time and 23 turns, can be backtracked from class (E, E, 1C, 0, 0) and is shown in bold in the table. The resulting tour is shown in Figure 8.
Determining the minimum travel time under fixed p, q and r values yields the following theorem.

Biobjective travel time and turn minimisation algorithm
As the turn-constrained travel time minimisation problem can be solved in polynomial time, the approach can be extended to an -constrained solution procedure for the biobjective problem. Algorithm 2 starts by finding the minimum time solution and records the number of turns in this solution. It then proceeds by decreasing the number of allowed turns and solving the travel time minimisation problem introduced in Section 3 subject to the fixed number of turns using the aforementioned algorithm. The procedure terminates after the step in which the number of allowed turns is constrained by the minimum possible number of turns, which can be determined from Proposition 3.3.
In each iteration, the algorithm either terminates or decrements the turn counter. If there is no termination, it determines all possible combinations of p, q and r corresponding to the given number of turns and calls the algorithm that solves the restricted travel time minimisation problem. If any of the new solutions is dominated, it is discarded. If it dominates any of the existing nondominated solutions, the latter is discarded. If the new solution is nondominated, it is added to the list of currently nondominated solutions. For the example given in Figure 7, the set of solutions found by Algorithm 2 is given in Figure 9. The algorithm finds 21 solutions in total, six of which form the Pareto front. Here, nondominated solutions for the problem are not necessarily supported, as the nondominated solution with 23 turns and 274 units of travel time cannot be dominated by a weighted sum of any group of nondominated solutions. This implies that finding nondominated solutions by optimising weighted sums of the two objectives will not give the complete Pareto front for this biobjective problem. Table 3. Solution of the example problem in Figure 7 for p = q = 1, r = 0.

end while
An important issue upon the generation of the Pareto front is how to aid the decision-maker in selecting the solution corresponding to the 'optimal' preference. This issue falls into the category of posterior articulation of preferences in the multicriteria decision-making literature. The main idea here is to find a set of solutions that represent the Pareto front well and upon presenting these to the decision-maker, update the set of representative points with the aim to converge to the most preferred solution after a number of iterations. Such approaches include physical programming, normal boundary intersection method, normal constraint method and surface approximation. Interested reader is referred to Marler and Arora (2004) and Karasakal and Köksalan (2009) for an overview of these approaches.

Extension to the triobjective problem with U-turn minimisation
In the previous section, we assumed that a U-turn is equivalent to two turns. However, in reality, a U-turn generally requires more effort than that required by two single turns combined, especially in warehouses with narrow pick aisles. In addition, U-turns also contribute to congestion and accident risk more significantly, since they require significant manoeuvring of the vehicle. Consequently, it might be useful to consider U-turns as a third objective in the OPP. Our results in the previous sections can be extended to generate the Pareto front for this triobjective problem in polynomial time.
The extension of our previous approaches to handle the triobjective problem with U-turn minimisation is provided in the online supplement, where it is shown that the complete Pareto front for this problem can also be generated in O(n 7 ) time.
As stated by Ehrgott and Gandibleux (2000), most of the results in the MOCO literature on the tractability of the solution algorithms are limited to two objectives. Our results on this problem imply a case where finding all efficient solutions of a practical triobjective combinatorial problem takes polynomial time.

Computational experiments
In this section, we test the performance of the exact approach for the biobjective turn and travel time minimisation algorithm on a set of randomly generated instances. The objectives of our experiments are mainly threefold: (1) to measure the time required to generate the Pareto front for instances of various sizes, (2) to compare the optimality gaps of the single objective Ratliff and Rosenthal algorithm as well as S-shape and largest gap heuristics in terms of travel time and number of turns and (3) to quantify the value of the exact approach for the biobjective problem, as opposed to using the approaches for time minimisation.
Our instances are generated in line with those in the literature. We use 12 different combinations of the number of aisles, aisle length and number of items. For the first eight, we follow Roodbergen and De Koster (2001b) and set the possible values as 7 or 15 for the number of aisles, 10 or 30 time units for the aisle length, and 10 or 30 for the number of items. For the last four, instance settings are based on Theys et al. (2010), where number of aisles is set at 60, aisle length is varied as 10 or 30 units and the number of items can be 60 or 240. For each combination, we generate 2000 instances with different locations of items in the pick list and report the averages.
In order to evaluate the travel time and turn results of each approach, we denote by GAP1 and GAP2 the per cent travel time and turn gap from the min-time and min-turn solutions, respectively. More formally, G AP1 =  Table 4, which gives the average number of nondominated solutions for the biobjective problem, as well as the average GAP1 and GAP2 values. The CPU times of the proposed algorithm and the remaining benchmark are within 0.03 and 0.01 s on average, respectively.
It can be inferred from Table 4 that the number of nondominated solutions is mainly driven by the number of aisles. While the average is slightly above 5 for our smallest instances, it increases above 54 when the number of aisles is 60. The table also shows that aisle length and the number of items also play a role in increasing the number of nondominated solutions, although their effect is not as significant as that of the number of aisles. The results indicate that our algorithm generates the full Pareto front very quickly; even for the largest instances, the average CPU time never exceeds 0.03 s.
The value of applying a biobjective method becomes more apparent when the per cent gaps of the heuristics and Ratliff and Rosenthal algorithm are observed. It has been shown in Corollary 3.3 that S-shape heuristic finds the optimal number of turns in a number of cases. The high turn performance of the heuristic is also evident in our experiments, particularly when the number of aisles is even (below 1% average gap for 60 aisles). However, average travel time gap can be as high as 37%, especially when the pick list size per aisle length is small, where complete traversal of the aisles results in detrimental effects on the travel time. Exactly opposite results are observed for the largest gap heuristic, which results in an excessive number of turns in all cases. The average turn gaps range between 90 and 342%. Although the average travel time gaps are slightly better than those of the S-shape heuristic, these values worsen when the pick list size per aisle length is larger. Even when the travel time-optimal Ratliff and Rosenthal algorithm is considered, there is a 56% gap from the min-turn solutions over all instances. The algorithm performs particularly poorly in terms of turns when the number of items per number of aisles ratio is small, which leads to numerous partial aisle traversals and hence excessive turns.
The results in Table 4 show that using the practical heuristics or the time-optimal Ratliff and Rosenthal algorithm results in large gaps in terms of at least one of the objectives. While this already points to a need for a biobjective approach to reconcile between the two conflicting objectives, we also make a comparison between the proposed algorithm and the aforementioned approaches from a biobjective standpoint as well. The comparison is made between the optimal Pareto front generated by  our proposed approach and the approximate front generated by the union of S-shape, largest gap, and Ratliff and Rosenthal algorithm solutions. The basis of comparison between the exact and approximate fronts is the hypervolume ratio, which captures both the accuracy of the approximate front and the diversity of its solutions. The hypervolume of a set is defined as the hyperarea or Lebesgue integral of the region dominated by a front and bounded by a reference vector (Helbig and Engelbrecht 2013). The hypervolume ratio is the ratio of hypervolume of the approximate front to that of the optimal Pareto front. In our case, the reference vector is one that consists of the worst value for each objective over the union of all nondominated solutions of the two fronts. For a good approximation, the hypervolume ratio should be as close to 1 as possible. Table 5 displays the average hypervolume ratios for each of the 12 settings. The results clearly show that the three approaches perform quite poorly from a biobjective standpoint, as the hypervolume ratios do not exceed 0.30 even in the best case. While the ratios are slightly better when the ratio of the number of items to the number of aisles is high, lower values of the ratio results in average hypervolume ratios of as low as 0.12, which underlines the fact that to take into account the effect of travel time and turns together, practical approaches are not sufficient, whereas our proposed approach yields the exact front very quickly.

Conclusions and further research directions
In this paper, we have emphasised the effects of left, right and U-turns on the order picking route and shown that the problems of single objective turn minimisation, single objective travel time minimisation, biobjective travel time and turn minimisation, and the triobjective problem with U-turn minimisation are all polynomially solvable.
The paper fills a number of gaps in the warehouse logistics and MOCO literature in the context of the OPP with turn and U-turn objectives. First, we consider a practical logistics problem addressing the decisions in the costliest stage of warehouse operations and all three objectives considered pose practical issues. Second, we can characterise the number of efficient solutions. Since the number of steps in both algorithms for the biobjective and triobjective cases have number of steps linear in the number of nonempty pick aisles, this is also the bound on the number of efficient solutions. Our computational experiments on randomly generated instances show that the proposed algorithm generates the whole set of efficient solutions very quickly.
An immediate extension of the work in this paper is to consider the turn minimisation problem in warehouses with different layout patterns . Different turn metrics, including minimising the total turn angle, have been used in the literature. These metrics can be especially useful for layout types in which pick aisles and cross aisles do not necessarily intersect each other in right angles.
The work in this paper can also be extended to the case of multiple pickers in parallel-aisle warehouses, where congestions due to turns is a substantial problem. Heuristics using graph-based approaches for the multiple-block case (e.g. Çelik and Süral 2010) can be modified to handle the number of turns in addition to minimising the travel time.
Parallel-aisle warehouse graphs are among the special cases of series-parallel graphs, on which the distance minimisation problem has been found out to be solvable in polynomial time. Extension of the work in this paper to series-parallel graphs might yield interesting results, although this requires modification of the turn cost definitions for applicability on these graphs.

Disclosure statement
No potential conflict of interest was reported by the authors.

Supplemental data
The supplemental material provides proofs for part of the statements and gives extensions of the results to cross-dept and pick-depot graphs. Supplemental data for this article can be accessed here http://dx.doi.org/10.1080/00207543.2016.1154624.