Production, Manufacturing, Transportation and Logistics
Product packing and stacking under uncertainty: A robust approach

https://doi.org/10.1016/j.ejor.2019.03.040Get rights and content

Highlights

  • We present an integrated packing and stacking problem based on a case study.

  • To address highly variable demand, a two-stage robust optimization model is developed.

  • We propose two approximate mixed-integer optimization models for the robust problem.

  • We validate the benefits of the integrated model.

  • We investigate the effects of varying variability and sizes of containers.

Abstract

China, the world’s largest logistics market, is experiencing insatiable demand for logistics services. The boom in China’s e-commerce has led to massive growth in the need for product shipping and warehouse storage. Some of China’s largest retailers have invested tremendous amounts of money in warehouse development. Two important functions of a warehouse are packing and stacking: packing small products into medium-sized containers and stacking the containers on pallets. Often, the packing problem is solved independently from that of stacking or is ignored. Generally, the stacking problem is solved while treating the packing decisions as given. However, considering these two problems separately may result in suboptimal, i.e., less efficient operations. Therefore, we propose a mixed integer linear programming formulation that addresses these two problems as a whole and two relaxations for actual-size examples. Due to the inherent demand uncertainty, we present a two-stage robust model that employs interval-based and scenario-based uncertainty sets. We numerically verify that the integrated model achieves considerable economic benefits and efficient space utilization while exploiting spatial flexibility. The additional computational experiments suggest that two relaxations yield tiny optimality gaps of 0.3% on average (4.6% at maximum), and considerable computation time improvements of 207.9% on average (347.4% at maximum). Our numerical results also illustrate that the scenario-based robust approach outperforms the deterministic approach and interval-based robust approach. We verify the model with varying variability and sizes of containers. The computational results indicate that compared with single type of containers, two or three types would bring significant economic benefits.

Introduction

China, the world’s largest logistics market, is experiencing insatiable demand for logistics services, which are viewed as an important value-adding activity and play a strategic role in the creation of business assets. Therefore, China’s largest retailers are striking out on their own with in-house logistics arms (Zito, 2014; Barreto, 2017). Benefiting from this growing demand for logistics facilities, Global Logistic Properties, which is Asia’s largest warehouse operator, boasts a $41 billion portfolio of assets spread across China (Daga & Barreto, 2017).

Although Chinese companies’ investments in in-house logistics facilities are staggering, poor warehouse management, an impediment to promoting storage efficiency and cost savings, has become an issue for them. For instance, Fig. 1 shows disordered products stacked in an overloaded warehouse owned by STO. Express, a Chinese private domestic enterprise, which had revenues of $14.54 billion in 2016. Only one Chinese company was included in the Gartner Supply Chain Top 25 for 2017 (Anonymous, 2017a). Although Chinese labor costs are much lower than those in the U.S., China’s logistics costs are over twice the costs in the U.S. (Aldred & Jim, 2014). Long afflicted by storage inefficiency and excessive costs, the managers began to recognize the need to improve the level of warehouse management.

To increase storage efficiency and decrease operation costs, the People’s Liberation Army Navy (PLA Navy) of China partnered with researchers from Tsinghua University (in Beijing, China) to apply innovative operation research to develop and implement new software with the ultimate goal of helping the PLA Navy improve its warehouse management. The PLA Navy’s warehouses are categorized into four levels according to their function and size. Our target warehouse (hereafter, warehouse N) is one of the three top-level warehouses. In warehouse N, products are first packed into containers from inbound transport, and then, the containers are packed on stacking pallets to await retrieval.

Warehouse N faces two major challenges. First, its required storage capacity has grown each year since 2013 because the Chinese authorities have poured enormous amounts of direct investment into high-tech weapons and equipment, leading to rapid increases in inventory (Lei, 2017; Anonymous, 2014). Thus, a major difficulty lies in the contradiction between the mushrooming demand for storage and the limited supply of land. Second, a typical feature of military equipment is its tremendous variation in size and value. A submarine or destroyer consists of millions of parts (Anonymous, 2017b). This huge product heterogeneity significantly complicates the product packing process.

The current storage policy in warehouse N focuses on how to assign products to containers, namely, the packing process. When a new stock of goods arrives, warehouse workers identify the type of products and gather the items of the same type into as few containers as possible (see, for example, Fig. 2(a)). However, an improper packing scheme may result in a substantial waste of space in the stacking process. Compared with that in Fig. 2(a), the scheme in Fig. 2(b) employs seven (10−3=7) more containers but occupies one fewer stack. One problem with traditional packing schemes is the preference for large containers and the resulting failure to exploit the reduced spatial flexibility that small containers offer. There is growing consensus among the managers of warehouse N that integrating the packing and stacking processes would offer considerable benefits. However, such an approach has received scant attention in the literature. The managers of warehouse N are interested in an integrated packing and stacking scheme that could help achieve the following three performance goals:

  • (i)

    procuring appropriate containers in terms of both number and type;

  • (ii)

    reducing losses when the warehouse is overloaded; and

  • (iii)

    boosting storage capacity and system utilization through container deployment.

To account for all of these performance goals and to formulate corresponding strategies, we present a two-stage robust optimization model. In the first stage (the packing process), we calculate the number of each type of container. The key trade-off is between assigning each product to containers of appropriate sizes and adopting fewer types of containers. The aim of the former approach is to improve space utilization, but it employs more types of containers. The latter approach facilitates the rapid retrieval of unit loads and high throughput of stacks. In the second stage (the stacking process), we optimize the stacking scheme for all the loaded containers. Moreover, prioritizing products according to their value makes it possible to alleviate economic losses. In this study, we find that integrating packing decision-making with the stacking process could offer advantageous inputs for the stacking scheme, which treats container quality as an endogenous variable. Thus, because we also consider the packing process, our final solution is concerned with not only how to arrange the loaded containers but also which containers need to be stacked.

Our numerical results in Section 5.2 indicate that the integrated packing and stacking scheme outperforms the decentralized scheme for warehouse N. The reformed policy makes use of container combination patterns to simultaneously avoid wasting space on each stack and to minimize operating costs. Variations in product demand, which can be observed from historical data, are formulated in our robust model for the worst-case analysis. This paper contributes to the literature in the following respects:

  • 1.

    We formulate an integrated packing and stacking problem as a two-stage robust optimization model. To the best of our knowledge, this paper is the first attempt to jointly determine the product packing process and container stacking process. Furthermore, our robust model considers two types of uncertainty sets in order to address the highly variable demand.

  • 2.

    We propose equivalent mixed-integer formulations for the two-stage robust model. For the second stage of the model, two relaxations are presented. One of the relaxations is proven to be equivalent to the original model in certain real-life scenarios. Furthermore, the quantitative relationships and monotonicity are provided.

  • 3.

    We use a set of computational experiments to verify that our integrated packing and stacking policy is superior to the decentralized policy. In addition, we evaluate the relaxations in terms of computation time and optimality. The robustness of the storage system is also examined. The robust model appears to exhibit better performance with the volatile inputs. Finally, we also verify the model with varying variability and sizes of containers.

The remainder of this paper is organized as follows. We review relevant literature in Section 2. Section 3 formulates the problem as a two-stage robust mixed-integer programming model. In Section 4, we propose several solution approaches, including relaxations and a lower bound based on dynamic programming. Section 5 examines the effect of our model by using numerical experiments. Section 6 offers our conclusions and remarks on directions for further research.

Section snippets

Literature review

Stacking, a widely used storage method, has attracted considerable attention in the academic literature. The existing models tend to develop and evaluate policies for assigning containers to stacks and strategies for rehandling containers when orders arrive (Bruns, Knust, Shakhlevich, 2016, Dekker, Voogd, van Asperen, 2006, Junqueira, Morabito, Yamashita, 2012, Le, Knust, 2017, Rei, Pedroso, 2013). Those policies and strategies share the assumption that the size of containers is identical and

Model

The fundamental model consists of two major processes: packing process and stacking process. Fig. 3 graphically illustrates those two processes. First, warehouse N receives P types of products shipped from upper-level suppliers. Then, those products are packed into K types of containers after regular maintenance. Finally, those containers are stacked on pallets waiting for retrieval orders. We first describe those two processes sequentially, and then describe the system performance evaluation.

Solution approaches

In Section 4.1, we first propose two relaxations for Problem (4) and prove that one relaxation does not comprise under certain conditions, which are usually satisfied in our real application. In Section 4.2, we propose two equivalent mixed-integer optimization models for robust Problems (5) and (6).

Numerical studies

First, we illustrate the setup of numerical studies in Section 5.1. Second, we numerically verify the benefits of the integrated packing and stacking processes in Section 5.2. Third, we test the quality of relaxation Problems (7) and (8) in Section 5.3. Forth, we discuss how to optimize the integrated packing and stacking processes with uncertain product demand in Section 5.4. Finally, we investigate the effect of variability and sizes of containers, and propose some findings.

We use programming

Conclusions

In this study, we investigate China’s military equipment logistics with a special focus on a top-level warehouse, which has been long afflicted by storage inefficiency and excessive costs. Two important functions of our target warehouse N are to pack and stack: pack small products into medium-sized containers and stack the containers on pallets. Often, the problem of stacking is solved independently from that of packing or is ignored. However, considering these two problems separately may

Acknowledgments

The authors thank the editor and two anonymous referees for their constructive suggestions that have significantly improved this paper. This work was partially supported by the National Natural Science Foundation of China (Grant 71822105) and Tsinghua University Initiative Scientific Research Program.

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