Product allocation problem for an AS/RS with multiple in-the-aisle pick positions

An automated storage/retrieval system with multiple in-the-aisle pick positions is a semi-automated case-level order fulfillment technology that is widely used in distribution centers. We study the impact of product to pick position assignments on the expected throughput for different operating policies, demand profiles, and shape factors. We develop efficient algorithms of complexity O(nlog(n)) that provide the assignment that minimizes the expected travel time. Also, for different operating policies, shape configurations, and demand curves, we explore the structure of the optimal assignment of products to pick positions and quantify the difference between using a simple, practical assignment policy versus the optimal assignment. Finally, we derive closed-form analytical travel time models by approximating the optimal assignment's expected travel time using continuous demand curves and assuming an infinite number of pick positions in the aisle. We illustrate that these continuous models work well in estimating the travel time of a discrete rack and use them to find optimal design configurations.


Introduction
According to the U.S. Roadmap for Material Handling & Logistics (2014), currently 350 000 distribution centers operate in the United States. A distribution center's most important, as well as most costly and labor-intensive operation, is to fulfill customer order requests. Thus, careful design and control of order fulfillment operations can result in considerable improvements in the performance of a distribution center. In this work, we are interested in case-level order fulfillment when non-identical demand is experienced for different products (i.e., item demand varies among the set of items managed). A typical phenomenon exists where a small percentage of Stock-Keeping Units (SKUs) will make up a large percentage of the total demand. This skewed demand is captured in a well-known method called ABC analysis, which ranks the SKUs based on their contribution to the total demand. The "A" class represents the fast-moving SKUs, the "B" class represents the medium-moving SKUs, and the "C" class represents the slow-moving SKUs. * Corresponding author Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/uiie. An Automated Storage/Retrieval System (AS/RS) with Multiple In-The-Aisle Pick Positions (MIAPP-AS/RS) is an example of a case-level order fulfillment technology that is frequently used in distribution centers (see Fig. 1). These systems are common in temperature-controlled distribution centers due to their ability to increase the space utilization and energy efficiency and to reduce the number of employees that have to work in the harsh environment.
The MIAPP-AS/RS consists of crane aisles and picking aisles. The crane aisles are dedicated to the movement of the cranes and the picking aisles to the travel of the order pickers. Distribution centers that utilize MIAPP-AS/RS receive full pallets from an upstream member of their supply chain and are tasked with supplying a variety of products fulfilled at the case level to a downstream member of their supply chain. In these systems, each opening can store one pallet (in single-deep systems) or two pallets (in double-deep systems). As shown in Fig. 1, the openings in the first row of the rack are for pick positions and the other openings in the rack are for storage locations. Pallets are put away and stored in full pallet quantities in the storage openings. When all of the cases of a particular SKU are picked from a pick position, the crane replenishes another full pallet of that SKU to the pick position. Generally, for the convenience of the picking process, each pick position in the aisle is dedicated to a particular type of 1380 Ramtin and Pazour SKU. Therefore, an MIAPP-AS/RS is a semi-automated order fulfillment strategy because the full-pallet put-awayto-storage and replenishment-to-picking position movements are completed with cranes. On the other hand, the picking of cases from pick positions in the aisle is done via manual order pickers.
In this research, we are interested in the assignment of the SKUs to pick positions for MIAPP-AS/RS. We call this problem the SKU Assignment Problem (SAP). There are two key research questions regarding the SAP.
1. How should SKUs be assigned to pick positions such that the maximum expected crane throughput occurs? 2. What is the impact of different SKU to pick position assignments on throughput performance under different operating policies, demand curves, and system configurations?
The goal of our study is to understand the impact of different SKU assignments on the MIAPP-AS/RS throughput, as well as system design trade-offs that occur when an MIAPP-AS/RS is operated using different policies and demand profiles. To do so, we begin by reviewing the relevant AS/RS and SKU assignment problem literature in Section 2. We describe the MIAPP-AS/RS physical configuration, demand curve models, operating policies, and key assumptions in Section 3. Next, in Section 4, we develop mathematical optimization models to find the optimal assignment that minimizes the expected travel time and provide an optimal assignment algorithm of complexityO(n log(n)). We also provide structural properties of the optimal assignment of SKUs to pick positions in an MIAPP-AS/RS for different operating policies, demand profiles, and shape factors. We quantify the performance difference between optimal assignments and a common assignment used in practice that assigns the most demanded SKUs to the pick positions that are closest to the input point. In Section 5, we derive and validate analytical closed-form models to approximate the expected travel time for different assignments; also, we use these models to obtain the optimal shape factors for different policies. These closed-form analytical travel time models are derived by using continuous demand curves and assuming an infinite number of pick positions in the aisle to approximate the optimal assignment's expected travel time. In Section 6, we illustrate that these models work well in estimating the travel time of a discrete rack and use them to derive optimal design configurations and provide managerial insights. We also provide numerical results and managerial insights by exploring the tradeoffs that occur while implementing different assignments. Finally, in Section 7, we review our main contributions and then provide insights from our study and directions for future research.

Literature review
During the past three decades, AS/RS systems have received considerable attention from different perspectives that can be categorized into system configuration, travel time estimation, storage assignment, dwell-point, and batching/sequencing (Roodbergen and Vis, 2009). In addition, some AS/RS research has been addressed in warehouse design papers; interested readers may refer to review papers such as De Koster et al. (2007) and Gu et al. (2007).
In fact, the literature on AS/RS systems is the exclusive focus in the following review papers: Sarker and Babu (1995), Vis (2009), Gagliardi et al. (2012), and Vasili et al. (2012). Among these review papers, Sarker and Babu (1995) is specifically focused on travel time models. According to the AS/RS review papers, AS/RS with multiple Input/Output (I/O) points is a subject that deserves further study. Topics similar to the SAP include the Storage Location Assignment Problem (SLAP) and the Product Allocation Problem (PAP), which have been widely studied in traditional order-picking and warehouse management literature. The SLAP represents a notable portion of the AS/RS literature. A storage assignment policy identifies how items are assigned to storage locations. Three main storage assignment policies have received the most attention in the literature: randomized storage, turnoverbased storage, and class-based storage. Generally, randomized storage has the best space utilization; turnover-based storage has the shortest expected travel time; and classbased storage is a compromise between these two policies (Pohl et al., 2011). The turnover-based and class-based storage policies incorporate that different SKUs have different demand profiles when determining the assignment of SKUs to storage locations. For the special cases of two or three classes in class-based storage, explicit travel time analytical expressions have been developed by Hausman et al. (1976), Eynan and Rosenblatt (1994), and Kouvelis and Papanicolaou (1995). Rosenblatt and Eynan (1989) and Eynan and Rosenblatt (1994) provided a procedure to obtain optimal boundaries for n-classes under a specific layout. Van den Berg (1996) developed a dynamic programming algorithm that optimally assigns products and locations into a certain number of classes. He considered travel time minimization under single command cycles. Goetschalckx and Ratliff (1990) considered a duration-of-stay shared policy for unit-load AS/RS and showed the optimality of this policy under an assumption of perfectly balanced I/O flows. All of the above studies focused on the assignment of the SKUs to storage locations in traditional AS/RS that have a single I/O point and thus no SKU to pick positions assignment problem exists. In contrast, we focus on the assignment of the non-identical demanded SKUs to multiple in-the-aisle pick positions.
Also, the PAP has been widely studied in the literature on traditional order-picking systems where a manual order picker traverses aisles to pick a set of items (Francis et al., 1992;De Koster et al., 2007). The PAP determines the SKUs that should be assigned to picking locations in order to minimize a given performance measure, such as travel time. For example, Francis et al. (1992) analyzed various assignment models in a traditional manual picking warehouse subject to different space and throughput constraints. Jarvis and McDowell (1991) considered the PAP in different types of order picking warehouses. They presented a stochastic model to obtain the optimal results. Petersen and Aase (2004) built a simulation model to examine the effect of picking, storage, and routing on order picking travel time. They showed that order batching of products results in the highest savings, particularly when the order sizes are small. Heragu et al. (2005) developed a mathematical model and a heuristic procedure that simultaneously determined the allocation of the products, as well as the size of the functional areas. Pazour and Meller (2011) analyzed the SKU assignment and allocation problem in an A-frame dispenser system, which is a fully automated order picking technology. Guerriero et al. (2013) developed a nonlinear integer mathematical model to minimize the handling cost considering compatibility constraints among classes. They used an iterated local search-based algorithm to solve large-scale instances. Our work is similar to this area of research, in that we are also interested in assigning SKUs to pick position locations. However, the focus of our study is on the assignment of SKUs to pick positions in a MIAPP-AS/RS, which have different travel time dynamics than manual order picking operations, including the need to consider simultaneous vertical and horizontal travel movements and retrieving, as well as storage operations.
Only a few studies exist that study an AS/RS with either multiple I/O points or non-end-of-aisle I/O points. Bozer and White (1984) derived and analyzed the expected travel time for different alternative I/O point locations; however, all of the alternatives considered a single input and single output point. Randhawa et al. (1991) and Randhawa and Shroff (1995) focused on simulation analysis to identify the impact of changing the location and number of I/O points on travel time. They considered different layouts and performance measures to evaluate their simulation studies. Vasili et al. (2008) developed a continuous travel time model for the split-platform AS/RS, where an I/O point is located at the center of the rack. Ramtin and Pazour (2014) were the first to study the AS/RS with multiple in-the-aisle pick positions and developed expected travel time models for different operating policies and physical configurations. For their closed-form travel time models and numerical analysis, they assumed that the demand for SKUs was identical and thus did not consider the assignment of SKUs to pick positions. To our knowledge, no study exists that addresses the assignment of SKUs to multiple pick positions in an AS/RS.

Problem definition and assumptions
In this section, we provide a description of the environment that we model. To do so, we describe the MIAPP-AS/RS's physical configuration, demand models, crane travels, and operational policies, as well as the key modeling assumptions.

Physical configuration
We assume that the system has a single picking level and each crane aisle has its own crane that is working independently from other cranes, as illustrated in Fig. 1. Each rack has one input point located at the lower-left corner and the pick positions are located on the bottom line of the rack. We develop our models considering a single rack.
Most of the existing AS/RS travel time model studies have used a continuous approximation of the rack to derive expected value models for different configurations (Graves et al., 1977;Bozer and White, 1984;Foley and Frazelle, 1991;Chang et al., 1995;Kouvelis and Papanicolaou, 1995;Sarker and Babu, 1995;Hu et al., 2005). We also use a continuous approximation of the rack for the travel time models in Sections 4 and 5. The storage locations are assumed to be continuously and uniformly distributed over the rack. We follow the methodology in Bozer and White (1984) to "normalize" the rack as follows. We utilize the following notation: t h : time required to travel horizontally from the input point to the most distant point; t v : time required to travel vertically from the input point to the most distant point; By assuming t h ≥ t v , the rack can be normalized as a rectangle with length of 1.0 and height of b, where 0 < b ≤ 1.

Demand curve representation
We denote I and K as the sets of all pick positions and SKUs to be assigned to those pick positions, respectively, such that I = {1, 2, . . . , n} and K = {1, 2, . . . , n}. Let i ∈ I and k ∈ K be the index of pick positions and SKUs, respectively. Let Pr i denote the probability that pick position i is visited such that n i =1 Pr i = 1. Let Pr k denote the probability that a storage/retrieval request is for SKU k among a total of n SKUs, where n k=1 Pr k = 1. We use the continuous demand model in Hausman et al. (1976) to represent the ABC analysis that ranks the items in monotonically decreasing order based on their contribution to the total demand. In this demand model, x is the percentage of inventoried SKUs, x ∈ [0, 1], s represents the skewness of the demand curve, and sx s−1 is the demand of the 100x percentage SKU. We consider 20/20, 20/40, 20/60, 20/80, and 20/90 demand curves, which correspond to s = 1, s = 0.5693, s = 0.3174, s = 0.1386, s = 0.0655, respectively. For example, a 20/60 curve represents the case that 20% of inventoried SKUs contribute 80% of the total demand. SKUs with identical demand can be represented by 20/20 curves. To calculate Pr k , the continuous demand curve model can be discretized by the approach used in Pohl et al. (2011) as shown in Equation (1):

Crane travels and operational policies structure
In an MIAPP-AS/RS, the crane can carry a single unit load and travel between the input point, storage locations, and pick positions. In a Single Command (SC) storage operation, full pallets of different SKUs originate from an input point and are transferred to a storage location in the rack by the crane. An SC retrieval operation is performed by the crane when it moves a SKU stored in a rack location to a pick position located in the aisle. A Dual Command (DC) operation involves the crane conducting a storage operation followed by a retrieval operation. All of the trav- els performed by an MIAPP-AS/RS can be broken into four elements denoted as trips (see Fig. 2). All of the notations for the trips and travel, as well as their expressions, are as follows (for detailed derivation of Equations (2) and (3) please refer to Bozer and white (1984), and for Equations (4) to (7) refer to Ramtin and Pazour (2014)): E(T B): expected travel time between any two randomly selected points in the rack, where E(W m i ): expected travel time between pick position i and any randomly selected point in the rack, where We consider operating policies for the MIAPP-AS/RS that are common for peak and non-peak hours. The first operating policy is a Consecutive Retrievals (CR) policy, where the AS/RS performs only SC retrievals. This policy is used during the peak hours when the priority is to fullfil the customer orders. In the case of a CR policy, the crane only travels between storage locations and pick positions. Thus, all trips are E(W m i ) trips, and the overall expected travel time can be calculated as 2T × E(W) using Equation (5).
If a CR policy is used during peak hours, then the AS/RS will need to perform Consecutive Storages (CS) during the non-peak hours. We call this policy the CR-then-CS policy, and the overall expected travel time for this policy can be calculated as T × (E (W) + E (V)), which is the average of the CR and CS policies. Another policy we consider during non-peak hours is a Mixed policy. In a Mixed policy, the crane performs both SC (storage/retrieval) and DC. In this policy the proportion of the number of SC to DC operations is defined with the parameter α. Let α denote the percentage of storage (or retrieval) operations that are performed under the SC basis. For this policy the combination of all E(V), E(T B), E(W m i ), and m i trips may occur, and the overall expected travel time can be obtained as T × E(M), where E(M) is defined in Equation (7).

E(W)
: expected one-way travel time between any pick position and any randomly selected point in the rack, where E(R): expected one-way travel time between any pick position and input point, where E(M): expected "per operation" travel time for the mixed policy, where The assumptions used in this article are as follows: 1. Each normalized rack has one input point located at the lower-left corner and the pick positions are located on the bottom line of a normalized rack. 2. Demand for each item type (and thus each pick position location) is known and independent. 3. For assigning SKUs to storage locations, a randomized storage assignment is used; i.e., any point within the rack is equally likely to be selected as a storage location. The assignment of SKUs to pick positions is discussed in Section 4. 4. Each pick position is dedicated to particular type of SKU. 5. We consider a single-deep rack, where each opening can contain a single pallet. 6. The storage and retrieval rate is assumed to be equal in steady state, and operation requests are processed on a first-come-first-serve basis. 7. Only the travel time model of the crane is considered.
The travel time of the human order pickers and the pick/deposit times to handle the loads are ignored. 8. The crane carries unit loads under either SC (storage/retrieval) or DC travels. 9. The crane's travel time follows the Chebyshev metric and moves at constant horizontal and vertical speeds; i.e., acceleration and deceleration are ignored.
10. A stay dwell-point strategy is considered; i.e., after completing any storage or retrieval operation, the crane stays where it is and waits for the next operation request.

The SAP
In this section, we analyze the SAP for the MIAPP-AS/RS, which determines the assignment of SKUs to pick positions to minimize the expected travel time of the crane. When the demands for SKUs are identical, changing the assignment of SKUs to pick positions does not impact the expected travel time for any policy (see Equations (5) and (7)). However, in the case of non-identical demand, as the expected travel time to each pick position is different (see Equation (4)), assigning higher-demanded SKUs to pick positions with expected lower travel time will result in a lower overall expected travel time.

Optimal assignment problem
Our objective is to minimize the expected travel time for different operating policies by changing the assignment of SKUs to pick positions. Referring to Equations (5) and (7), the expected travel time models for CR and mixed policies are linear functions of the demands for the SKUs and the locations of the pick positions. Therefore, we formulate our problems as a Linear Assignment Problem (LAP) for each policy. Let x ik be a binary variable where one denotes that the pick position i is assigned to SKU k and is zero otherwise. We formulate the assignment problem for the CR policy and denote it as Model 1. The objective function is to minimize the expected travel time for a CR policy, which is expressed in Equation (8). Equation (9) provides the constraints that each SKU must be assigned to one pick position, and Equation (10) provides the constraints that each pick position must be assigned to one SKU.

Model 1 (for a CR policy)
Model 2 is the SAP for the Mixed policy. In this case, the assignment problem depends only on the elements of the expected travel time of the Mixed policy, as shown in Equation (7), which are a function of m i (the location of the pick positions). Consequently, E(V) and E(TB) are not impacted. Therefore, the objective function of the assignment problem for the Mixed policy is formulated as Equation (12) subject to constraints (9), (10), and (11).

Model 2 (for a Mixed policy)
Because of the structures of Models 1 and 2, we are able to provide an algorithm for the optimal solution that is computationally tractable. In order to simplify Models 1 and 2, we define f i = E(W m i ) and Therefore, the objective functions of Models 1 and 2 be-

Proposition 1a. The optimal solution to Model 1 is obtained by the assignment permutation that has the pick positions sorted in ascending order based on their f i values and the SKUs sorted in descending order based on their Pr k values.
Proof. f i is independent of the demand for a particular SKU. Hence, the cost of assigning SKU k to pick position i is obtained by multiplying two independent parameters ( f i and Pr k ). Therefore, by introducing permutation functions W and P, Model 1 can be represented as its combinatorial formulation, Min W,P∈Z n i =1 f W(i ) × Pr P(i ) . Let Z be the set of all permutations on W : I → I and P : K → K. According to the rearrangement inequality, the minimum value to this model is obtained by the optimal permutations (W * andP * ) such that f W * (1) ≤ f W * (2) ≤ . . . ≤ f W * (n) and Pr P * (1) ≥ Pr P * (2) ≥ . . . ≥ Pr P * (n) (For an extended proof, please refer to Hardy et al. (1988)). Therefore, the optimal assignment for Model 1 can be obtained by bijective mapping of permutation W * and P * , which is represented as Equation (13): This completes the proof.
Proposition 1b. The optimal solution to Model 2 is obtained by the assignment permutation that has the pick positions sorted in ascending order based on their g i values and the SKUs sorted in descending order based on their Pr k values.
Proof. Applying the arguments used in the proof of Proposition 1a, let Z be the set of all permutations on M : The minimum value to this model is obtained by the permutations (M * andP * ) such that g M * (1) ≤ g M * (2) ≤ . . . ≤ g M * (n) and Pr P * (1) ≥ Pr P * (2) ≥ . . . ≥ Pr P * (n) . Therefore, the optimal assignment for Model 2 can be obtained by bijective mapping of permutation M * and P * , which is represented as Equation (14): This completes the proof.
Corollary 1. The minimum expected travel time "per operation" with respect to the optimal assignment of SKUs to pick positions is equal to Equations (15) and (16) for a CR and a Mixed policy, respectively.
According to Propositions 1a and 1b, the optimal algorithm for the SKU assignment problems is of complexityO(nlog(n)), as it sorts two sets of cardinality n. The optimal assignment can be obtained using the following procedure: Step 1: Calculate f i and g i for ∀i ∈ I, as well as Pr k for ∀k ∈ K.
Step 2: Find permutations W * and M * by sorting f i and g i in monotonically increasing order, respectively. Find permutation P * by sorting Pr k in monotonically decreasing order.
Step 3: Provide the optimal assignment for Model 1 and 2 by Equations (13) and (14), respectively.  positions for a CR and a Mixed policy are obtained by bijective mapping W * and M * , respectively, on P * . Based on Equations (13) and (14), the optimal assignment for the CR policy has x 31 = x 22 = x 43 = x 14 = x 55 = 1 and for the Mixed policy it has x 11 = x 22 = x 33 = x 44 = x 55 = 1. Also, according to Equations (15) and (16) Proof. Appendix A.
For the sake of convenience, we call the assignment of SKUs ordered based on monotonically decreasing demand to pick positions ordered from nearest to farthest from the input point the Monotonically Decreasing Demand (MDD) assignment and is shown in Fig. 4(c). The insight behind Proposition 2 as well as the other assignment characteristics will be discussed in Section 4.2. Based on the definition of the MDD assignment, the most demanded SKU is assigned to the pick position closest to the input point, and the next-highest-demanded SKU to the next pick position closest to input point, and so on. The expected travel time for CR and mixed policies under the MDD assignment can be calculated by substituting any monotonically decreasing demand for Pr i into either Equation (5) or Equation (7).

Structure of the optimal assignment
In this section, we explore the structure of the optimal assignment of SKUs to pick positions introduced in Section 4.1. Recall that the set of f i is used to find the optimal assignment for a CR policy and the set of g i for a Mixed policy. If we denote m as the location of a pick position from the input point (0 ≤ m ≤ 1) these sets can be defined as functions of m, such that f = E(W m ) and To observe the behavior of the expected travel time, Fig Let m * denote the location of a pick position to which the expected travel time is minimum for each f and g function. Function f has m * = 0.5 for any shape factor (a proof is provided in the online supplement A). The intuitive reason for this result is that a CR policy performs only SC retrievals; thus, the crane never travels to the input point. For a randomized storage policy, storage location decisions are made independent of pick positions. Therefore, the pick position located in the middle of the rack has the shortest expected travel time to conduct retrievals. Comparing Fig. 3(a) with Fig. 3(b), another observation for function f is that the difference between the expected travel times for the largest and smallest pick position locations increases by decreasing the b value. The optimal assignment of the SKUs to the pick positions for a CR policy are illustrated in Fig. 4(a), where the most demanded SKU is assigned to the pick position in the middle of the rack.
In contrast, a Mixed policy performs both SC and DC operations. Therefore, to perform a storage operation after a retrieval operation, the crane is required to travel from a pick position to the input point. Consequently, m * shifts to the left and will be located between the input point location and the middle of the rack (0 ≤ m * < 0.5). For this case, we show the optimal assignment of SKUs in Fig. 4(b) (e.g., when α = 0.4 and b = 0.48). When the function g is monotonically increasing, then m * is equal to zero and the MDD assignment is optimal for the Mixed policy. We illustrate such a case (e.g., when α = 0.4 and b = 1) in Fig 4(c). The MDD assignment is optimal for several combination of α and b; however, in Proposition 2, we prove this is always true for α = 0 regardless of the shape factor. This occurs because when all of the operations are performed by DC (α = 0), the crane has to travel back from a pick position to the input point at the end of each cycle. However, as α increases, the number of these returns to the input point decreases because the chance of consecutive SC storages or consecutive SC retrievals is higher with higher α values. As an example for the case of α = 0 or 0.4 and b = 0.48, as shown in Fig. 3(b), m * shifts to the right (from the location of the input point to a point towards the middle of the rack).

Extreme travel time models for non-identical demand curves
In Section 4.1, we derived the expected travel time for discrete pick position locations for CR and mixed policies under the optimal and MDD assignment (which we denote as base models). In this section, we derive closed-form travel time models for the CR and mixed policies (which we denote as extreme models). We derive extreme models by assuming that (i) there are an infinite number of pick positions in the aisle and (ii) demand curves are continuous.
The motivation for deriving extreme models is twofold. First, the closed-form solutions provide a computationally efficient way to evaluate the expected travel time in an MIAPP-AS/RS with non-identical demand. Second, we use the models to derive optimal design parameters of an MIAPP-AS/RS with non-identical demand.
In the previous section, we showed that m * = 0.5 is optimal for the CR policy regardless of the shape factor. Therefore, to approximate the optimal assignment for the CR policy shown in Fig. 4(a), we introduce a new continuous demand curve defined in Equation (17) and denoted as D OPT (m, s). Let m and s denote the location of the pick position and demand curve skewness factor, respectively. As illustrated in Fig. 5(a), this curve assigns the highest demanded SKUs to locations in the middle of the rack. We use this curve to represent the optimal assignment for the CR policy: This demand curve is obtained by truncating and shifting the demand curve expressed in Hausman et al. (1976). Therefore, to represent the different demand curves (e.g., 20/20, 20/40, 20/60, 20/80, and 20/90), the same s values provided in Section 3 can be used. D OPT (m, s) is a valid Probability Distribution Function (PDF) for 0 ≤ m ≤ 1 and 0 ≤ s ≤ 1 (a proof is provided in the online supplement B). We use the continuous monotonically decreasing demand curve defined in Equation (18) to represent the MDD assignment. As illustrated in Fig. 5(b), this curve is the continuous representation of the MDD assignment shown in Fig. 4(c). This curve will be used for representing the MDD assignment and we derive extreme models for both CR and Mixed policies with an MDD assignment: In Sections 5.1 and 5.2, we will develop extreme travel time models for CR policy under optimal and MDD curves mentioned above. For the Mixed policy, m * varies based on the value of α and b. In Section 4.2, we showed that for several combinations of α and b where m * = 0, and the MDD assignment for a Mixed policy is the optimal assignment. Also, later in Section 6, we will show that for several cases where m * = 0 (i.e., applying the MDD assignment is not optimal), we do not lose much efficiency compared with the optimal assignment due to the small percentage differences (see Section 6, see Table 7). Therefore, in Section 5.3, we will only develop extreme travel time model for the Mixed policy under the MDD curve.

Extreme travel time models for the CR policy under optimal assignment
We denote the extreme value of E (W) under the optimal assignment asE ∞ OPT (W), which is calculated as in Equation (19a). This extreme value is based on letting n go to infinity in Equation (5), as well as using Equation (17), which is an optimal assignment. Each of the integrals on the right-hand side of Equation (19a)

separate into two integrals as the value of E(W m ) changes for different m and b values based on Equation (4). The final closed-form result is given in Equation (19b). See Appendix B for the complete derivation of E ∞
OPT (W) for the CR policy:
Note that E ∞ OPT (W) derived as Equation (19b) varies from E ∞ MDD (W) in Equation (20b) and are not replaceable because E ∞ OPT (W) is derived based on the continuous demand curve used for approximating optimal assignment for CR policy (Equation (17)) and E ∞ MDD (W) is derived based on the MDD assignment continuous demand curve (Equation (18)).

Extreme travel time models for the Mixed policy under MDD assignment
To derive the extreme travel time model for the mixed policy under an MDD assignment, note that E (W) and E (R) are the only components of E (M) impacted by varying the location of the pick positions and the demand for their assigned SKU (see Equation (7)). Therefore, the extreme value of E (M) under an MDD assignment can be ob-  (21). Finally, by substituting E ∞ MDD (W) and E ∞ MDD (R) for E (W) and E (R), respectively, into Equation (7), the extreme value of E (M) under an MDD assignment is obtained as Equation (22):

Validation of the extreme travel time models
In this section the results of the extreme models developed in Sections 5.1 to 5.3 are compared to results from a discrete-event simulation. The purpose is to understand the impact of the extreme models' assumptions (i.e., a continuous rack, an infinite number of pick positions, and continuous demand curves) on the expected travel time. We used MATLAB 2013a to code and run the simulation.
In the discrete-event simulation, we assumed that the number of pick positions and the number of SKUs are equal to the number of columns. The openings in the first row of the rack are for pick positions and the other openings in the rack are for storage locations. We calculated the distance between any two openings' centroids using the Chebychev distance metric. We considered six different configurations as shown in Table 1. To make an equal comparison, we set the number of storage locations in each configuration to be approximately equal to 950. To simulate the expected travel time, we ran five replications of a sequence of 100 000 storage and retrieval operations for each of six configurations, five demand curves, and two operating policies. In the simulation, we modeled discrete pick positions and SKU demand. The demand for each SKU was obtained using Equation (1). To simulate the CR policy under optimal assignment, the optimal assignment of a discrete number of SKUs to pick positions was obtained using Proposition 1a for each shape configuration. Also, we applied an MDD assignment to simulate the CR and mixed policies under the MDD assignment. These results were used as inputs to the simulation model. To compare the accuracy of the results obtained from the continuous extreme models versus the simulation, we calculated the "% deviation" as shown in Equation (23) We report the results obtained for the CR policy under an optimal assignment, the CR policy under an MDD assignment, and the Mixed policy under the MDD assignment in Table 2, Table 3, and Table 4, respectively. For the simulation results, we report the average and variance of the travel time "per operation" for each configuration (as "Sim. Mean" and "Sim. variance" in Tables 2 to 4). The un-normalized expected travel time for the extreme models were calculated as 2T × E ∞ OPT (W) using Equations (19b) for the CR policy under an optimal assignment; as 2T × E ∞ MDD (W) using Equation (20b) for the CR policy under an MDD assignment; and as T × E ∞ MDD (M) using Equation (22) for the Mixed policy under the MDD assignment.
From Tables 2, 3, and 4, the "% deviation" measures between the simulation results and extreme models are less than 1.3% in all cases. Therefore, the extreme models can be used to represent a discrete rack with a finite number of pick positions without substantial loss of accuracy. We also calculated the percentage differences between the extreme and discrete models for both the CR and Mixed policies. The differences are extremely small and less than 0.1% in all cases; therefore, the continuous approximations of the number of the pick positions and demand profiles do not have a   (19b) and (20b) can be substituted for the discrete models to estimate the expected travel time with an optimal and MDD assignment, respectively. Also, for the mixed policy, the extreme model developed as Equation (22) can be used to estimate the expected travel time of the MDD assignment.

Optimal shape factors
We use the extreme models developed in Sections 5.1 to 5.3 to derive the optimal shape factor that minimizes the expected travel time for different combinations of operating and assignment policies. We consider three operating policies: CR (with optimal and MDD assignment), CR-then-CS (with optimal and MDD assignment), and mixed (with MDD assignment). The expected travel time for these policies were calculated as , and T × E ∞ MDD (M), respectively. Note that the optimal assignment for CR-then-CS policy is the same as optimal assignment for CR policy because the performance of a CS policy is not impacted by the SKU assignment. To find the optimal shape factor for each policy, we considered a system that has constant area space (denoted as A) and an equality constraint of A = bT 2 . By substituting T as √ A/b in the extreme travel time models, we can find the optimal shape factor, denoted as b * , which achieves the minimum expected travel time for the range of b (0 < b ≤ 1). Because of the complexity of the extreme models, we used the "fminbnd" function of MAT-LAB 2013a, which applies numerical search methods to find the minimum of a function for a given range of a vari-able. The results for b * for different policies and demand curves are shown in Table 5. For the CR policy and the CR-then-CS policy using an optimal assignment, increasing the skewness of the demand curve results in decreasing the optimal shape factor. However, for these policies with an MDD assignment, the optimal shape factor increases when the skewness of the demand curve increases. For the same demand skewness the corresponding optimal shape factor for CR-then-CS policy is greater than a CR policy. This occurs because when we are only performing storages (a CS policy), the optimal shape factor is equal to 1.0 (Bozer and White, 1984); therefore, for a CR-then-CS policy b * is between the optimal shape of a CR policy and unity. Also, we observe that b * = 1 for a Mixed policy under an MDD assignment for any demand curve skewness.

Numerical results and managerial insights
In this section, we report the results of a numerical study and provide the numerical results for the CR and mixed policies under optimal and MDD assignments using the models derived in Sections 4 and 5. We also provide the percentage difference between the optimal and MDD assignments expected travel time to identify how much efficiency is lost by adapting the MDD assignment over the optimal assignment. We consider the same six different shape configurations and five different demand curves as in Section 5.4.
In Tables 6 and 7, we list the expected "per operation" travel times with respect to the optimal assignment of SKUs to pick positions versus the MDD assignment for different demand curves and different shape configurations. In Table 6, the expected "per operation" travel time for the CR policy was calculated using the extreme models as 2T × E ∞ OPT (W) and 2T × E ∞ MDD (W) for the optimal and MDD assignments (as "OPT extreme" and "MDD extreme"), respectively. In Table 7, the expected "per operation" travel time for the Mixed policy (for α = 0.4, 0.6, and 1.0) with optimal assignment was calculated using the discrete base model as T × E * (M) using Equation (16) (as "OPT base"), and the Mixed policy with the MDD assignment was calculated using the extreme model as 2T × E ∞ MDD (M) using Equation (22) (as "MDD extreme"). Note that all of the extreme models are based on continuous demand curves that approximate the optimal and MDD assignments. In Table 7, we do not provide the results for α = 0, as we proved in Proposition 2 that results for the optimal and MDD assignment are equal for the Mixed policy with α = 0. For demand profiles that have identical demand (i.e., 20/20 curves), the assignment problem does not impact the expected travel times and thus the difference between the optimal and MDD assignments is zero for both CR and mixed policies.
From Tables 6 and 7 it can be seen that when an optimal assignment is adopted, increasing the skewness of the demand curve improves the expected travel time for all policies and configurations. Intuitively, when the skewness of the demand curve increases, the pick positions with shorter expected travel time are visited more frequently, because the SKUs assigned to these pick positions have higher levels of demand. However, when the MDD assignment is applied for the CR policy, the expected travel degrades by increasing the level of skewness of the demand curve. The reason for this behavior is that m * = 0.5 in the CR policy (as mentioned in Section 4.2). That is, the pick position located in the middle of the rack has the shortest expected travel time. Thus, the MDD assignment is far from optimal because it allocates the highest demanded SKUs to pick positions nearest to the input point (which as shown in Fig. 4(c) have the highest expected travel time in a CR policy).
For all operating and assignment policies, the expected travel time increases as the shape of the rack become more rectangular (i.e., b decreases). This occurs because the difference between the highest and smallest expected travel time increases by decreasing the b value (see Fig. 3(a) and Fig. 3(b)). For these reasons, when a distribution center adopts an MDD assignment instead of the optimal assignment and uses a CR policy, it loses more efficiency by increasing the skewness of the demand and decreasing the shape factor. From Table 7, the same observation can be observed for a Mixed policy. However, the percentage differences between the optimal and MDD assignments are much smaller compared with the CR policy (i.e., all are less than 8%). For the Mixed policy, we have 0 ≤ m * < 0.5. As m * be-comes closer to zero, the percentage difference between the optimal assignment and the MDD assignment becomes smaller as the pick positions with the shortest expected travel time are located closer to the input point where the highest demanded SKUs are assigned in an MDD assignment (see Figs. 4(a) and (b)). We observe that the percentage difference increases by increasing α, increasing the skewness of the demand (s), and decreasing the shape factor (b). Therefore, the high percentage differences occur in extreme cases of system parameters (e.g., for highly skewed demand curves or relatively small shape factor configuration), and for most of the practical cases of α, b, and s, the percentage differences are relatively small (less that 5%). In these cases, a system will not lose considerable throughput by implementing an MDD assignment instead of an optimal assignment for mixed policies.
In Fig. 6, we illustrate the impact of demand curve skewness on the performance of different policies. For both the optimal and MDD assignments, a CR-then-CS policy has a lower performance on a peroperation basis than a Mixed policy for all of the demand curves. Also, for the Mixed policy, decreasing the α value results in performance improvement because decreasing α decreases the probability of empty trips that occur from a pick position back to the input point. However, we observe that increasing the demand curve skewness increases the difference between policies' performance. As an example, for the optimal assignment, the difference between a CR-then-CS policy and a Mixed policy with α = 0 is 18.1% for a 20/40 curve, but the difference is equal to 31.1% for a 20/90 curve. Moreover, the differences between policies are even higher for the MDD assignment; e.g., the difference between the CRthen-CS policy and Mixed policy with α = 0 is 20.7% for a 20/40 curve, but the difference is equal to 42.2% for a 20/90 curve.

Conclusions and future research directions
This article investigated the effect of assigning the mostactive SKUs to the best pick positions in an MIAPP-AS/RS system. We presented mathematical models to find the optimal assignment of SKUs with non-identical demand to pick positions that minimizes the expected travel time for MIAPP-AS/RS under different operating policies. We provided an optimal SKU to pick position procedure of complexity O(nlog(n)). We also derived a continuous demand curve that can be used to model the optimal assignment in a CR policy. Based on the proposed continuous demand curves, we derived closed-form expressions for a CR and a Mixed policy that approximate the expected travel time by assuming there are an infinite number of pick positions, as well as continuous demand curves. We validated these continuous extreme models through a set of discreteevent simulations that enforce the discreteness of the rack and observed that all percentage deviations were less than 1%. This showed that our continuous extreme models can accurately approximate a discrete rack. Also, by comparing the results obtained from the extreme models versus the discrete models under the same assignment, we observed less than 0.1% difference for all cases. Using extreme models, we calculated the optimal shape configurations for each demand curve and operating policy.
We analyzed structural results of the optimal assignment problem. For a CR policy with an optimal assignment, we observed that regardless of the shape factor the most demanded SKU is assigned to the pick position located in the middle of the rack, and the next-most-demanded SKU is assigned to the next closest pick position to the middle of the rack and so on. For the Mixed policy the smallest expected travel time occurs at a pick position located between the input point and the middle of the rack.
We explored the impact of different SKU assignments on the expected throughput rates that can be obtained under different system configurations, demand curves, and operating policies for peak and non-peak hours. We compared the results obtained from the optimal assignment with an easy-to-implement assignment that is often seen in practice-the MDD assignment, which assigns the highest demanded SKUs to the locations closest to the input point. We observed that maximum percentage difference between the optimal and the MDD assignment for a mixed policy was less that 8% and in most practical MIAPP-AS/RS configurations less than 5%.
As future research directions, this work can be extended by considering different storage policies (such as turnover-based storage and class-based storage policy), different dwell-point strategies, and different sequencing rules. Applying these directions can help to increase understanding of the impact of different storage policies and demand skewness on the throughput and space utilization of MIAPP-AS/RS.