Lecture 3.4 Recap
3.4: Transshipment · ISE 754, Fall 2026 · in class 8 October 2026
A recap answers what students found most difficult in their pre-class review of the lecture, along with anything the instructor adds. Blue Lecture panels quote the lecture; orange Another view panels look at the same idea from another side.
What changed in the lecture
Added by the instructor Lecture 3.4, Sec. 2.3, revised 7 and 8 OctoberLecture 3.4 was revised after the review, in Sec. 2.3, where the three models are stated. The changes are to how the models are stated, not to any result: every number in the lecture is as it was. Anyone who read the lecture before 8 October should read Table 1 below.
Lecture New: Table 1, and the terms it uses
“A cross-docking set is a group of products whose suppliers all ship on one interval, the set’s own, so that its products arrive at the DC together and leave it the same day. A product in no set is stocked: held at the DC, and shipped in on its own interval. A delivery is everything a customer receives from the DC on one day. Each set makes a delivery to each customer on its interval, and a stocked product goes out either in one set’s deliveries or in a delivery that carries only stocked products. A shipment is uncoordinated when its size is its own optimal size, as in Lecture 3.3, and coordinated when its size is its annual flow times its set’s interval. Every shipment is therefore one of the four kinds in Table 1.”
| Uncoordinated | Coordinated | |
|---|---|---|
| In from a supplier | stocked product’s shipment (\(\alpha_S + \frac{1}{2}\)) | cross-docked product’s shipment (\(\alpha_S\)) |
| Out to a customer | delivery that carries only stocked products (\(\alpha_C\)) | set’s delivery, with any stocked products it carries (\(\alpha_C\)) |
“A coordinated shipment can be larger. One in from a supplier is still held to one truck, which limits how long its set’s interval can be. A set’s delivery that is larger than one truck goes as two or more shipments on the same day, since its products arrived together and leave together.”
Also changed in Sec. 2.3
- Solve for, in all three models. Part (b) now reads “shipment interval of each set, and shipment size of each uncoordinated shipment,” and a new (c) asks, “for each stocked product, the set, if any, whose deliveries to the customers also carry it.”
- Assumptions, in all three models. Rewritten as four: the DC location is given; each supplier's and each customer's inventory fraction, \(\alpha_S\) and \(\alpha_C\), is given; every shipment is an independent shipment; and stocked products are held at the DC, while the products of each set arrive together and leave the same day.
- Model 1's formulation. It now adds up the cost of every shipment and delivery, with row (a), payload, for the shipments in from the suppliers, row (b), interval, for each set, and one row with no letter that puts each set's shipments on its interval. The three unlettered rows the review's Q3 asked about are gone.
- Model 3's longest interval. Its payload row is now written for the shipments in from the suppliers only, and the longest common interval is an equation of its own, \(t_{\max} = \min_{i \in I_S} q_{\max,i}/f_i\) (Eq. 8), which Code blocks 15 and 23 compute over the inbound shipments.
- Numbering. Table 1 is new, so the shipment-by-shipment table of Example 1(g) is now Table 2. Eq. 8 is new, so every equation after it is one number higher. Figures and code blocks keep their numbers.
Cross-docking, from the top
Asked in the review Lecture 3.4, Sec. 2The questionThe cross-docking example, and the big picture behind it.
Lecture The whole idea in three sentences
In transshipment, inventory is stored time: the gap between when freight arrives and when it is wanted. Cross-docking closes that gap by timing the arrivals, trading the cost of storage for the cost of coordination. That trade is made product by product.
Lecture Direct, or through a DC
“Direct shipment means P2P shipments from suppliers to customers… Transshipment is the use of a DC to consolidate outbound shipments, as on the right, and it runs in one of two ways.”
“In this distribution problem each customer wants both products A and B, which are not combined into a different product as they would be in manufacturing. Shipped direct, every supplier ships to every customer; with transshipment, the DC lets each outbound shipment carry a mix of the two products.”
Lecture The two ways to run the DC
“Uncoordinated transshipment determines separately each optimal inbound and outbound shipment, and so it must hold inventory at the DC. A (perfect) cross-dock uses a single shipment interval for all inbound and outbound shipments, so there is no inventory at the DC; a non-perfect cross-dock would only cross-dock a selected subset of shipments.”
“At one, termed uncoordinated, the shipment from each supplier is determined to be optimal for that supplier and the shipment to each customer optimal for that customer, with no coordination between when products arrive at the DC and when they are demanded from it. At the other, perfect cross-docking, all inbound shipments have the same shipment interval: everything arrives at the DC, say, once a week, is offloaded onto the floor rather than put into storage racks, and is immediately switched around and shipped out to the customers, much as at an LTL terminal. The advantage is that no inventory is held at all. The difficulty is that every shipment must arrive on time, which would have taken good luck thirty years ago; tracking has made it better, but things still happen.”
Another view Example 1's DC at Statesville, run both ways
Uncoordinated, each truck is sized for itself, so product A comes in every 9.43 days and goes out on three other schedules: every 8.56 days to Winston-Salem, 12.11 to Durham, 14.07 to Wilmington. Between an arrival and the deliveries that take it, A waits at the DC. That wait is the stored time.
Cross-docked, every truck runs on one interval, 13.94 days, so what arrives leaves the same day and nothing waits. Over a year the lecture finds uncoordinated inventory at $64,820.57 and cross-docking at $55,539.18.
Lecture Where the stock sits, and how it is counted
“The average inventory at the DC turns out to be one half of each inbound shipment… That is the justification for one half: it matches what actual inventory profiles show, and it works in practice.”
Uncoordinated, the DC's half rides on each inbound shipment:
Cross-docked, the half is gone: “With nothing held at the DC, each shipment’s inventory fraction is \(\alpha_S\) inbound and \(\alpha_C\) outbound.” Here \(\alpha_S\) is the supplier's share, 0 for batch production, and \(\alpha_C\) the customer's, \(\tfrac{1}{2}\) for steady use.
Lecture Why not cross-dock everything?
“In practice it is difficult to put every shipment on a single interval: the DC would sit empty most of the day, and then everything would arrive, be cross-docked and leave within a two-hour window. What happens at most DCs is a mix of stocked and cross-docked products, and the cross-docked products are usually not a single set. Coordination makes sense for certain high-value products, while low-value, more general products are probably best left uncoordinated. Seasonal goods are a classic case. A large retailer’s DC does not receive Halloween products from January on and fill a warehouse with them; they arrive over the month or two before the holiday, and a container or truckload of them, once unloaded, is not put into storage, because the stores need it quickly. It is cross-docked at once, one truckload holding enough to supply perhaps 20 to 25 stores, as whole pallets that go straight onto the store floor and are opened there. The trend is toward ever more cross-docking, since cross-docking needs good coordination and companies have become better at tracking their shipments.”
“For a product to be cross-docked, its shipments into and out of the DC must have the same interval; if they differ, inventory is stored at the DC, and the extreme of that is no coordination at all. In practice most DCs are a mix of cross-docking and no coordination: each shipment is examined to decide whether it should be cross-docked or simply stored with no coordination, and different factors lead to different sets of shipments being cross-docked.”
Three types of warehouse
Asked in the review Lecture 3.4, Sec. 4.2The questionThe differences between the types of warehouse.
Lecture Consolidation, cross-dock, break-bulk
“A warehouse provides two kinds of economy, storage and transport, and on the transport side, consolidating loads into aggregate shipments is one of the main economies a warehouse or DC gives.”
“The design of a logistics network has a large impact on transport costs. Raw material or finished goods warehouses can be used for consolidation, cross-docking, or breaking bulk purposes in order to achieve transport economies.”
- “Consolidation warehouse: A consolidation warehouse is used to combine multiple loads into a single load. Instead of costly LTL or infrequent TL shipments from each supplier directly to the customer, a consolidation warehouse provides less-costly and more frequent TL shipments to the customer. Small delivery vans can be used for short-distance transport between the suppliers and the warehouse, and tractor-trailers for long-distance TL transport to the customer. Consolidation warehouses are typically used by wholesalers.”
- “Cross-dock warehouse: A cross-dock warehouse is used to mix freight so that TL shipments can be used for all transport between suppliers and customers. Receiving and shipping are usually coordinated so that no storage is required at the warehouse.”
- “Break-bulk warehouse: At a break-bulk warehouse, a large long-distance TL shipment from a supplier is broken down into smaller loads that are delivered a short-distance to each customer. Break-bulk warehouses are usually located close to or in each major market served.”
Another view The three side by side
| Type | Shape | What comes in | What goes out |
|---|---|---|---|
| Consolidation | several suppliers, one customer | each supplier's load, by small delivery van over a short distance | one TL to the customer, over a long distance, more often than each supplier could send its own |
| Cross-dock | several suppliers, several customers | a TL of each supplier's own product | a TL to each customer, carrying a mix of every product; receiving and shipping are coordinated, so nothing is stored |
| Break-bulk | one supplier, several customers | one large TL over a long distance | smaller loads, each a short distance to a customer near the warehouse, which sits in or near the market |
The row in blue is the one Lecture 3.4 has been working with. In Example 1, Asheville sends only A and Statesville only B, and every truck to a customer carries both.
Lecture Where this lecture's DC fits
“Each inbound truck carries a full truckload of a single product, A or B, and at the DC the two products are combined onto the same trucks to the customers.”
“…the DC of Secs. 2 and 3 has essentially been a cross-dock warehouse; the other two are the consolidation and the break-bulk warehouse.”
“A cross-dock warehouse works like the LTL terminal of Lecture 3.2: freight comes off one truck and goes straight onto another, with nothing stored in between. A DC is a mix of all three types.”