Storage Location Assignment Problem

A place for everything, and everything in its place Benjamin Franklin

Learning Goals

• Define SLAP and explain dominant cost drivers in picking.
• Distinguish storage policies, ranking rules, and routing impacts.
• Understand DSLAP, including re-warehousing versus healing and how to evaluate net gain.

What is SLAP?

• The Storage Location Assignment Problem (SLAP) assigns stock-keeping units (SKUs) to physical locations in a warehouse.
• The goal is to determine an optimal assignment that minimizes an operational criterion, typically expected picker travel or order cycle time.
• Synonyms include storage assignment, product allocation, slotting, and storage location problem (Medrano-Zarazúa et al. 2024; Reyes et al. 2019).

The storage location assignment problem (SLAP) is an important step when we are designing a warehouse layout, and this step influences in the arranging, picking process, sorting, routing, and sequencing of requests [19]. The problem is to decide the corresponding areas in which the products should be allocated. It can be done using different techniques to establish a specific position for the products. Academic research generally seek to optimize the space used, distances, and cycle times of order picking (OP).

— Storage Location Assignment Problem in a Warehouse: A Literature Review (Medrano-Zarazúa et al. 2024, 16).

The SLAP concerns the allocation of products into a storage space and optimization of the material handling costs or storage space utilization. The problem depends on parameters such as storage area design, storage space availability, warehouse storage capacity, physical characteristics of the products, arrival times, and demand behavior. The main optimization approaches concern warehouse space utilization and the cycle time for order preparation and picking operations, considering restrictions such as available storage capacity, order-picking resource capacities, and dispatching policies (Gu et al., 2007). In terms of complexity, Frazelle (1989) classifies the SLAP as NP-Hard, due to variations caused by the number of products and warehouse storage characteristics. When the number of products is equal to the number of storage spaces, the problem is defined as a quadratic allocation problem (QAP) (Kofler, 2015). Moreover, if the number of products is greater than the number of storage spaces and each storage space can store several products, it becomes a knapsack problem (KP) (Gu et al., 2010).

Reference of the problem: This category captured the terminology used to identify the SLAP. The following terms were considered: Storage location assignment, product allocation/location, storage location/space, storage/reserve/space allocation, and slotting or warehouse layout problem.

— The Storage Location Assignment Problem: A Literature Review (Reyes et al. 2019, 200, 203).

Why SLAP Matters

• Picking is a large share of warehouse labor.
• Travel time often dominates pick time in picker-to-parts systems.
• Proximity to I/O and golden-zone placement (i.e., waist-to-shoulder height) reduces reach and travel (Petersen et al. 2005).
• Assigning most picked items to the most convenient locations reduces overall travel.

The impact that storage has on the cost and efficiency of companies is a critical element for the supply chain. Different studies carried out on inventory management show that storage operations in a company represent approximately 23% of logistics costs in the case of countries like the United States and up to 39% in the case of Europe [35].

The warehouse is responsible for receiving, storing, picking, packing, and shipping products between suppliers and customers. The order picking operation is among the activities that generate the most costs within warehouses, with approximately 55% responsibility of operational costs [66].

The storage location assignment problem (SLAP) is an important step when we are designing a warehouse layout, and this step influences in the arranging, picking process, sorting, routing, and sequencing of requests [19]. The problem is to decide the corresponding areas in which the products should be allocated. It can be done using different techniques to establish a specific position for the products. Academic research generally seek to optimize the space used, distances, and cycle times of order picking (OP).

SLAP is considered an NP-hard due to the impact in operations which depends on the order selection method, design and dimensions of the storage system, material handling, type of products, demand behavior, turn over, and spaces within the warehouse.

An adequate allocation in the warehouses must adapt to the flexibility of the demand, which causes a good flow of the materials to optimize the time in the collection of orders and at the same time that it ensures a cost reduction [57].

— Storage Location Assignment Problem in a Warehouse: A Literature Review (Medrano-Zarazúa et al. 2024, 16).

Once slotting has been completed, a warehouse manager must still decide on which warehouse locations to assign to the slotted SKUs. The literature on storage assignment strategies is fairly extensive. Gibson and Sharp (1992) conducted an analysis of batching algorithms in conjunction with random and popularity-based storage and found that the most popular SKUs should be assigned to storage locations in a diagonal pattern from the pickup/drop-off (p/d) point. Jarvis and McDowell (1991) determined that within-aisle (W) storage was the optimal storage assignment policy for a warehouse, assuming traversal routing and one-way picking aisles using picker travel distance. Petersen (1999) and Petersen and Schmenner (1999) evaluated Gibson and Sharp’s diagonal and Jarvis and McDowell’s W storage assignment strategies and found that their performance was dependent on the routing of the order picker and the size of the order to be picked; however, W storage was the best overall storage assignment strategy. Other studies confirming the superiority of W storage for picker-to-part systems include Petersen and Aase (2004) and Petersen et al. (2004). Other research on storage assignment strategies has primarily occurred in the area of automated storage/retrieval systems utilizing class-based storage (Hausman et al., 1976; Eynan and Rosenblatt, 1994). Class-based storage divides SKUs into storage classes based on a slotting measure and then randomly locates the SKUs within their storage class.

In all the aforementioned papers on storage assignment strategies, the warehouse appears in only two dimensions (length and width). Vertical dimension (height) is not taken into consideration. Saccomano (1996) and Jones and Battieste (2004) suggests that higher demand SKUs should be stored in the “golden zone”, the area between a picker’s waist and shoulders. The time to retrieve a SKU from a storage location is dependent on the height of the storage location, in addition to size and weight of the SKU. Storage locations above the picker’s shoulder or below the picker’s waist require more time to retrieve. It has been suggested that pick activity is maximized when performed at or near the golden zone, in order to improve picking efficiency and to reduce picker fatigue (Saccomano, 1996; Jones and Battieste, 2004). Frazelle (2002) suggests that golden zone picking used in conjunction with pick density slotting can be used with SKUs with a high correlation to reduce picker travel. Studies have looked into the vertical dimension of picking but only in pallet applications (Malmborg, 1996) and man-aboard systems (Malmborg and Altassan, 1998). However, since this paper deals with a manual bin-shelving picking environment, studies related to pallet applications and man-aboard systems have limited relevance.

— Improving Order Picking Performance Utilizing Slotting and Golden Zone Storage (Petersen et al. 2005, 999).

The goal of slotting—also called storage assignment optimization, inventory slotting, or inventory profiling—is to determine the best place to store each stock keeping unit (SKU) in a warehouse according to a variety of factors. Two of the most common incentives for companies to slot their warehouse include the need to squeeze more SKUs into an already overflowing warehouse and the desire to reduce overall handling costs and effort [1]. In many order picking environments the travel time to retrieve an order has been found to be the largest component of labour, amounting to 50% or more of total order picking time [2]. Therefore this study will focus on the reduction of travel distances as an approximation of travel time. It should be noted that other factors, such as load balancing across warehouse zones, work ergonomics (e.g. reduce bending and reaching activity [3]) or pre-consolidation (reduce downstream sorting) can also be of importance for a particular scenario.

— Re-Warehousing vs. Healing: Strategies for Warehouse Storage Location Assignment (Kofler et al. 2011, 77).

SLAP Baseline Formulation

Let \(I\) be items and \(L\) locations. Decision \(x_{i\ell}\in\{0,1\}\) equals 1 if item \(i\) is at location \(\ell\) with cost \(c_{i\ell}\).

• Objective:

  \[\min \sum_{i\in I}\sum_{\ell\in L} c_{i\ell}\,x_{i\ell}\]

• Constraints:

  • \(\sum_{\ell\in L} x_{i\ell}=1\ \forall i\) (assign each item)
  • \(\sum_{i\in I} x_{i\ell}\le 1\ \forall \ell\) (assign at most one item per location)
  • \(x_{i\ell}\in\{0,1\}\)

Where:

• \(c_{i\ell}\) is a travel or time proxy derived from a routing policy.

This baseline is a course re-indexing of Heragu’s item–location assignment model. Set \(S_i=1\) so that each item occupies one location. If there are more locations than items, the slide’s \(\leq 1\) location constraint allows unused locations; Heragu’s equivalent construction adds a dummy item to fill surplus locations. The slide’s \(c_{i\ell}\) is an aggregate placement-cost proxy. In the source, this cost is built from item trip frequency, movement cost, and I/O-to-location distance, so its interpretation depends on the travel assumptions used to estimate those terms.

“Consider the following warehouse storage problem. A warehouse has \(p\) I/O points through which \(m\) items enter and leave the warehouse. The items are stored in one of \(n\) storage spaces or locations. Each location requires the same storage space. It is known that item \(i\) requires \(S_i\) storage spaces. Ideally, we would like \(\sum_{i=1}^{m}S_i=n\). However, if the left-hand side is less than that of the right-hand side, we can always add a dummy product \((m+1)\) to take up the remaining \(n-\sum_{i=1}^{m}S_i\) spaces. We will assume throughout our discussion that the aforementioned equality holds. (Of course, if the right-hand side is less than the left-hand side, there is no feasible solution to the problem as there are not enough storage spaces to accommodate all the items.)

There are \(f_{ik}\) trips of item \(i\) through I/O point \(k\). The cost of moving a unit load of item \(i\) to and from I/O point \(k\) is \(c_{ik}\) and the distance of storage space \(j\) from I/O point \(k\) is \(d_{kj}\). Given the aforementioned and a binary decision variable \(x_{ij}\) that specifies whether or not item \(i\) is assigned to storage space \(j\), we are ready to formulate a model to assign the items to storage spaces in such a way that the cost of moving the items in and out of the I/O points is minimized. This is shown next in model 2. Note that the objective spreads the cost of moving item \(i\) in and out of storage evenly to all the \(S_i\) spaces occupied by the item. The cost for each item–space pair is calculated as a product of three factors: (1) frequency of trips made to each I/O point, (2) travel distance to each I/O point, and (3) cost per unit distance. It assumes that the cost per unit distance may vary for each I/O point–storage space combination.”

— Facilities Design (Heragu 2022, sec. 8.6.7.1 and Model 2, pp. 295–296).

Travel-time Models and Cost Construction

• The choice of \(c_{i\ell}\) is critical: how should we estimate the cost of placing item \(i\) at location \(\ell\)?
• Closed-form expressions for route length under standard routing policies (e.g., Return, S-shape, Largest-gap) can inform \(c_{i\ell}\) (see, e.g., Dijkstra and Roodbergen 2017).

Before customer orders can be retrieved, items must first have been stored in the available locations. The problem of choosing appropriate storage locations for the items is called the storage location assignment problem (Hausman et al., 1976; Gu et al., 2007). This problem shows similarities with the Assignment Problem, a fundamental problem from the field of Operations Research. Both problems require the matching of objects from two mutually exclusive sets, in our case a matching of items to locations. However, the two problems differ strongly in their objective functions. The objective function of the standard Assignment Problem has a simple structure, which makes the problem solvable in polynomial time. The storage location assignment problem, on the other hand, has the objective to minimize the average route length traveled by the workers (order pickers) while retrieving items from locations in the warehouse. This results in a complex function that depends on the layout of the warehouse, the routing method employed, the demand frequencies of all items, and the item-to-location assignment itself.

The routing method prescribes how an order picker should navigate the warehouse to collect the items in an order. A good routing method yields short route lengths, which decreases the time to pick an order. The physical structure of the warehouse limits the movements of the order picker, who has to navigate between the racks that store the items. Essentially, the routing problem in warehouses classifies as a special case of the Steiner Traveling Salesman Problem (see De Koster et al., 2007), that in some layouts is solvable to optimality in polynomial time (Ratliff and Rosenthal, 1983). However, as De Koster et al. (2007) note, “in practice, the problem of routing order pickers in a warehouse is mainly solved by using heuristics”. For this purpose, a number of heuristic routing methods are available from literature. These heuristics can be of a constructive (greedy) nature, employing properties of the layout to find solutions (Hall, 1993) or contain a procedure to search the solution space (Theys et al., 2010).

In this paper, we develop methods for determining storage location assignments in warehouses where order pickers span multiple aisles in each route, and where each order may contain any number of items to be picked (multi-aisle multi-item picking). We solve a number of multi-aisle multi-item storage location assignment problems and can guarantee optimality for the first time. As a core component for achieving this, we present formulas for four routing methods that give the value for expected route length in multi-aisle multi-item picking warehouses. In contrast to existing research work, these formulas give the exact expected route length, not an approximation, and they hold for any storage location assignment. Furthermore, structural properties of optimal solutions are derived, which aid in narrowing down the solution space. Due to the low computational requirements for our formulas, in combination with the structural properties, a number of instances can be solved to proven optimality by means of complete enumeration. Using the formulas at its core, we present a dynamic programming approach for determining storage location assignments. The dynamic program provides optimal solutions for one routing method, and near-optimal solutions for another routing method. For the remaining two routing methods, we have no optimal benchmark, but the dynamic program is shown to consistently outperform common storage location assignment rules from literature. Results from our experiments demonstrate patterns for storage location assignment that have not been described before in literature, and that challenge current assumptions about predetermined storage location assignment patterns. A detailed comparison of our work to existing literature is given in Sections 3 and 4.

— Exact Route-Length Formulas and a Storage Location Assignment Heuristic for Picker-to-Parts Warehouses (Dijkstra and Roodbergen 2017, 38–39).

Storage Policies, Ranking Rules, and KPIs

• Policies. Dedicated, Random/Open-location, Class-based, Golden-zone, Similarity or family grouping, Forward–reserve two-tier, and shared storage with duration-of-stay (see, e.g., Petersen et al. 2005; Manzini et al. 2015; Walter et al. 2013).
  
• Ranking rules. Popularity or frequency, Turnover, Volume, Pick-density, COI, Velocity-based, Recency-based (see, e.g., Heskett 1963; Chou et al. 2012; Reyes et al. 2019).
  
• KPIs. Distance and time, aisle changes, relocations per day, labor-hours, energy and ergonomics when relevant.

Variants and Modeling Extensions

SLAP variants reflect constraints and objectives found in practice, such as:

1. Class-based. Decide class boundaries and map classes to areas (Manzini et al. 2015; Muppani and Adil 2008).
   
2. Shared storage with duration-of-stay. Minimize peak load and balance space when items share locations over time (Chen et al. 2010).
   
3. Forward–reserve sizing and allocation. Jointly size forward and reserve areas and assign SKUs (Walter et al. 2013; Heragu et al. 2005).
4. Correlated demand and affinity. Co-locate items that are often co-picked or linked in BOMs (Bill of Materials) (Mantel et al. 2007; Zhang et al. 2019; Xiao and Zheng 2010).
   
5. Ergonomics and human factors. Integrate golden-zone reach and energy expenditure (Petersen et al. 2005; Battini et al. 2016).
   
6. Automation contexts. AS/RS, shuttles, robotic systems, and RFID-enabled storage decisions (Chen et al. 2010; Ramtin and Pazour 2015; Yang et al. 2015; Choy et al. 2017).
   
7. Multi-criteria decision making. ELECTRE TRI, TOPSIS, efficient frontiers, and hybrid MCDM–optimization (Micale et al. 2019; Fontana and Cavalcante 2014; Fontana and Nepomuceno 2017).

Reassignment Upon Demand Changes

• Static SLAP assumes constant demand and a one-time assignment.
• Dynamic/multi-period SLAP reassigns SKUs over time as demand changes (see, e.g., Kofler et al. 2012).
• Goal: Balance reassignment costs with operational savings in picking efficiency.
• Strategies:
  • Re-warehousing performs large periodic redesigns (many moves, high disruption).
  • Healing performs frequent small swaps (fewer moves, low disruption) (see, e.g., Kofler et al. 2011, 2012).

Since the 1960s researcher have developed various storage assignment strategies for different warehouse scenarios. Much of this research has been devoted to re-warehousing, which involves extensive re-arrangements akin to filling a warehouse from scratch. However, due to seasonal fluctuations in demand or product mix, operations managers need to periodically review item placement in practise and re-organize the warehouse to keep it operating efficiently. In this paper we demonstrate for a sample warehouse how applying a small number of cleanup tasks (=healing) every day will lead to a good total warehouse assignment over the course of many months and show that the results are competitive to re-warehousing.

Most of the existing scientific studies present results obtained by completely re-slotting a warehouse according to some assignment strategy, a process commonly referred to as re-warehousing. Migrating from a randomly slotted warehouse to class-based storage, changing the number of classes or the class characteristics in a live operating warehouse is more problematic. Warehouse managers might wish to implement a target assignment gradually during normal operations, rather than interrupt the normal workflow to move hundreds of items. In addition, an optimal storage assignment created by re-warehousing might become outdated before long due to fluctuations in product demand caused by seasonal variations and product life cycle characteristics. In this study, we therefore also distinguish between construction and improvement approaches. Construction algorithms build a feasible slotting from scratch, assuming that the warehouse is initially empty. On the other hand, improvement algorithms try to enhance an existing feasible solution. We will compare the results of both construction and various improvement algorithms, distinguishing further between greedy and heuristic search methods. We briefly introduce the approaches in subsection I-A to I-C.

— Re-Warehousing vs. Healing: Strategies for Warehouse Storage Location Assignment (Kofler et al. 2011, 77).

Overview of Approaches for Product Reassignment

Table 1: Overview of approaches for product reassignment by Chabot et al. (2024).

Strategy name	Full rearrangement	Healing	Healing
Impact on daily operation	High	Medium	Low
Problem name	Storage location assignment problem	Rearrange-while-working problem	Warehouse reassignment problem
Customer demand	Static	Dynamic	Dynamic
Typical objective	Minimize average order-picking time	Minimize total time of picking + relocations	Minimize time to perform relocations
Typical constraints	None	Picking jobs + desired final storage location assignment	Desired final storage location assignment

Finally, Carlo and Giraldo (2012) study a rearrange-while-working, in which unit-load warehouses can reassign a pallet to another location upon retrieval. A summary of the approaches mentioned for product reassignment in warehouses is presented in Table 1.

Strategy name	Full rearrangement	Healing	Healing
Impact on daily operation	High	Medium	Low
Problem name	Storage location assignment problem	Rearrange-while-working problem	Warehouse reassignment problem
Customer demand	Static	Dynamic	Dynamic
Typical objective	Minimize average order-picking time	Minimize total time of picking + relocations	Minimize time to perform relocations
Typical constraints	None	Picking jobs + desired final storage location assignment	Desired final storage location assignment
Literature	Hausman et al. (1976), Silva et al. (2020)	Carlo and Giraldo (2012)	Christofides and Colloff (1973), Pazour and Carlo (2015); this chapter

— Mathematical Models for the Warehouse Reassignment Problem (Chabot et al. 2024, 141).

How to Transition from One Setup to Another

• Transitioning from one assignment to another incurs material handling costs.
• How can several product switches achieve an optimal transition?

Example of warehouse reassignment problem. Locations are colored by picking frequency: black = most frequent, white = least frequent. Left: Initial assignment. Right: New best assignment based on demand fluctuations (Chabot et al. 2024).

In order to understand our context easily, it is important to properly define the term product switch. Let product A be at an occupied location, and product B already on the vehicle to be dropped at the position currently occupied A. The product switch is defined as dropping B on the floor (near the new location), removing A and also setting it aside on the floor, picking up and placing B inside the now empty location, and picking up A to move it towards its destination. By performing a series of product switches, it is possible to reposition a large number of products and effectively obtain a new storage location warehouse-wide. Figure 1 shows an example of reassignment of products inside a warehouse. In the left part, we see the initial assignment A. The locations are colored according to the picking frequency of their products. From white, the less frequent, to black, the more frequently picked. Suppose that according to the fluctuation of demand and products availability, the new best assignment should be setup B, on the right side. We have to compute the material handling effort required to move from setup A to B. This can be achieved by making several product switches.

— Mathematical Models for the Warehouse Reassignment Problem (Chabot et al. 2024, 142).

Evaluation by Discrete-Event Simulation

• Frequently, closed-form cost proxies \(c_{i\ell}\) omit key operational factors.
• Discrete-event simulation (DES) can capture batching, routing, congestion, calendars, and stochasticity (see, e.g., León et al. 2023).

In the context of increasing complexity in manufacturing and logistic systems, the combination of optimization and simulation can be considered a versatile tool for supporting managerial decision-making. An informed storage location assignment policy is key for improving warehouse operations, which play a vital role in the efficiency of supply chains. Traditional approaches in the literature to solve the storage location assignment problem present some limitations, such as excluding the stochastic variability of processes or the interaction among different warehouse activities. This work addresses those limitations by proposing a discrete-event simheuristic framework that ensures robust solutions in the face of real-life warehouse conditions. The approach followed embraces the complexity of the problem by integrating the order sequence and picking route in the solution construction and uses commercial simulation software to reduce the impact of stochastic events on the quality of the solution. The implementation of this type of novel methodology within a warehouse management system can enhance warehouse efficiency without requiring an increase in automation level. The method developed is tested under a number of computational experiments that show its convenience and point toward future lines of research.

— A Discrete-Event Simheuristic for Solving a Realistic Storage Location Assignment Problem (León et al. 2023, 1).

Very often, people in charge of managing the daily operations of a warehouse face challenging decisions such as: “What would be the best way to organize the products in the warehouse?”, “what products should I place differently to increase overall warehouse productivity?”, “do I have enough resources to fulfill tomorrow’s orders?”. These questions are difficult to answer due to (i) the vagueness of some terms, such as “best way”; (ii) the non-mentioned aspects, which are, in fact, relevant, such as considering the variability in the x and y parameters; and (iii) the interconnection between all the activities that take place in the warehouse, such as assigning a product to a place due to its own characteristics (weight, fragility, etc.), and the frequency with which this place is visited by workers. Nonetheless, science and technology can illuminate and guide managerial decisions of this sort if the questions are properly framed, and the appropriate tools are employed. As pointed out by Zhang et al. [1], warehouse daily operations can be improved by different approaches, including: (i) by assigning items to appropriate storage locations (storage location assignment); (ii) by determining the optimal retrieval route for the items in an order (retrieval operation); and (iii) by picking orders in batches (batching). These points are always present for warehouse management as a single problem (i.e., improving warehouse efficiency), but very often, they are treated as separate problems in the scientific literature. Separating a complex problem into smaller problems is a natural consequence of applying an analytical approach. For instance, focusing on the question of where to best place items in the available storage locations gave rise to the storage location assignment problem (SLAP) definition. However, isolating this product assignment decision has impacts on other operational areas of the warehouse, such as the product picking routes. Since some operational areas are removed from the scope of the SLAP, the particular solution found to the isolated problem can potentially result in an overall drop in warehouse efficiency. This shows a sharp conflict between the current scientific methods and the warehouse’s real-world needs. Having that in mind, the main objective of this paper is to develop a solving methodology that involves the combined use of heuristics and simulation (simheuristics) to improve the storage location assignment of products in a warehouse, integrating (rather than isolating) different aspects of the warehouse operations, such as routing planning and order fulfillment. It is worth noting that the use of simulation in order to address an optimization problem can have a number of advantages: (i) it allows the creation and evaluation of complex scenarios with little effort, where many interconnected systems interact with each other; (ii) it generates additional data that can be employed for a stochastic evaluation, or even as a guide, of the optimization component; and (iii) it allows integration of different optimization problems in a single framework. It also has some disadvantages: (i) it significantly increases the total computing time, since simulation can be a resource-hungry activity; and (ii) it can be difficult to replicate by the scientific community since most high-quality simulation software is proprietary and require licensing to be employed.

— A Discrete-Event Simheuristic for Solving a Realistic Storage Location Assignment Problem (León et al. 2023, 1–2).

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