Optimal Block Stacking

FeaturedOptimal Block Stacking

I am pleased to announce that Shahab Derhami, Jeff Smith and I have just had a manuscript accepted in International Journal of Production Research entitled, “Optimizing Space Utilization in Block Stacking Warehouses.” Congratulations especially to Shahab—this is the first paper from his dissertation research!

The origin of this paper was a consulting engagement Jeff and I had with a beverage manufacturer, in which pallets roll off a production line and then are stored for a time before shipping. The finished goods warehouse uses deep lane block stacking, as is common in many large lot, unit load handling environments. The company wanted to know how deep lanes should be to make the best use of space.

Space lost due to honeycombing

Block stacking warehouses waste space in two ways: aisles to allow lift truck travel and partially filled lanes, which another sku cannot occupy because it would block access to the sku already stored there. As explained in Bartholdi and Hackman’s textbook, each lane makes a contribution to both types of waste. If lanes are too shallow, then too much storage space is devoted to aisles; if too deep, then too much storage lane space is wasted. An optimal design balances this tradeoff.

The academic literature has well-known results for optimal lane depth in the case of instantaneous resupply, which is appropriate for warehouses storing product received from suppliers. Pallets for a particular sku arrive at the same time, so the production rate is “infinite.” A second assumption behind existing results is continuous demand, rather than discrete demand. Our paper relaxes both of these assumptions: production rate can be finite—even less than the rate of demand, and demand is discrete. (The production rate can be less than the rate of demand if demand is only intermittent. Imagine all demand for a particular sku happening only on Friday, for example.) The paper develops approximations for the optimal lane depths for block stacking areas with one or multiple skus.

The main insight we develop in the paper is that using the infinite production rate model in a finite production rate environment produces lane depths about twice as deep as they should be, but that the resulting loss of space is modest (less than about 2 percent). Said another way, the space utilization curve, as a function of lane depth, is quite flat.

However, shorter, optimal lanes means more aisles and therefore more flexibility with respect to travel for lift truck drivers or autonomous vehicles. The effects of lane depth on travel costs have not been considered in our work or in the literature.

We welcome your comments.


New Paper: Approximating Sojourn Times

Yet another announcement today!  (I am atoning for months of ignoring my blog.)  My former student Hyun Ho Kim and I are pleased to announce the publication of “An Approximation Model for Sojourn Time Distributions in Acyclic Multi-Server Queueing Networks” in Computers & Operations Research.

Queueing networks have been an attractive method of modeling complex manufacturing and other systems for many decades. Much of this work is limited in that it develops means for performance measures of interest rather than distributions, or it assumes exponential arrival or service processes. Our paper describes a mathematical model that produces a distribution for the sojourn time (total time in the system, including delays and processing), in the presence of general distributions for service processes.

Results of the model
Results of the model. On the left is the probability density function; on the right, the cumulative distribution function. From the plot on the right, an order has approximately 80% chance of being processed in less than 10 time units.

Here’s an example of how this research might be used in practice: Suppose you are interested in determining a cutoff time for accepting orders in an order fulfillment system, before which you guarantee the order will make it onto the last truck leaving in the evening. You have collected data on all the relevant order processing times—picking, transport to packing, packing, transport to shipping, and shipping. These data, of course, are variable, and you would like to assess the risk of setting a particular cutoff time in light of all this variability.

The model we develop in this paper takes the means and variances of all the relevant processing times and combines them to produce a distribution for the sojourn time of an order, so that you can make statements such as, “orders can be fulfilled within 2 hours with 98 percent probability,” or “orders can be fulfilled within 1 hour with 90 percent probability.” These values can then be used to determine an “optimal” cutoff time, such that the expected benefit of getting the order on the truck (revenue from premium shipping) just exceeds the cost of missing the truck (perhaps offering free shipping because the promise wasn’t kept). The tradeoff is “newsvendor like.”

Technical AbstractWe develop an approximation model for the sojourn time distribution of customers or jobs arriving to an acyclic multi-server queueing network. The model accepts general interarrival times and general service times, and is based on the characteristics of phase-type distributions. The model produces excellent results for multi-server networks with a small to medium number of workstations, but is less accurate when the number of workstations is large.

If you would like a copy of the paper, please email me.

GridStore implemented

FeaturedGridStore implemented

One of the most satisfying parts of this job is seeing others take your work and go further. In this case, Benedikt Fuß, who spent time with me at Auburn and who is now a research associate at the Institute for Material Handling and Logistics at Karlsruhe Institute of Technology, has improved the GridStore algorithm by adding asynchronous control.

As originally conceived, GridStore operates in stepwise fashion, with each conveyor module deciding its action and taking it according to a shared clock. An admitted weaknesses of that work was our failure to explain exactly how synchronization would be implemented in modules without a centralized controller. Enter Benedikt’s asynchronous modification, in which modules operate independently and without a common clock, yet seem to synchronize on their own. Notice the slight variations of timing among the modules in the video below:

Do I even need to say how satisfying it is to see this work implemented in a real system? Here’s hoping these ideas someday are applied in industry. Then, I retire.

Human-Centric Design is on the Way

Human-Centric Design is on the Way

Today I met with a company contemplating a greenfield facility design on a large plot of land. The facility will host both production and a warehouse to support raw materials, work in process, and some finished goods storage. During the discussion, the project manager showed me two designs—one a single, large facility and one consisting of several smaller facilities in a “campus” arrangement. When asked what motivated the campus arrangement, which seemed to me much less efficient, the manager stated that a major design goal for this facility was talent retention. Well, imagine that!

As I argued in a conference paper in 2010, there is a major need for industrial engineers to work with architects, interior designers, and industrial psychologists (at least) to design facilities that are not just “efficient,” but also appealing to workers. The talent crisis continues unabated, and major corporations designing entire facilities to attract quality workers is just the latest evidence that the problem is recognized and serious.

Dear colleagues, let us rise to the challenge!