Research presented at an OPTIMA seminar by Elsa Silva shows the fixes cost far less capacity than most operators assume, and that they will not happen by accident.
Picture a Tuesday at a furniture distributor. The rigid truck is loaded the way it always is: heaviest items over the rear axle where the forklift can reach them, everything squared off, wrap on, about 15 tonnes on board. The driver checks the plate, the total is under the limit, and the truck leaves.
The first customer takes the two heavy lounge suites from the back. The truck is now carrying 10 tonnes. Nobody thinks twice, because a lighter truck is a safer truck.
Except that it is not.
With the rear weight gone, the remaining load sits well forward, and at 10 tonnes with that much weight over the cab, the front axle is now carrying more than the manufacturer rates it for. The steering and braking behaviour has changed. If the truck is weighed at the next site, or is involved in an incident, the business is the one explaining why a compliant departure became a non-compliant vehicle at the first stop.
Same truck, one drop later
Departure · 15 t Inside safe zone
After drop 1 · 10 t Front axle over rating
Illustrative. The two heavy lounge suites at the rear (highlighted) come off first. The lighter load now sits further forward.
That example is Silva’s own, from her OPTIMA seminar Beyond Geometry: Optimising Cutting and Packing in Practice. [1] Silva is an operations researcher at the University of Minho in Portugal who has spent years building loading and cutting models with real companies: a Port wine exporter, a textile plant, a fuel distributor.
Her message to industry is simple. The academic field has spent decades on how to fit the most into the least space. The things that actually cost businesses money and put people at risk are pallets that collapse, trucks that breach axle limits, and cutting plans that look efficient on paper and cause chaos on the floor. None of that shows up in a plan unless someone deliberately puts it there.
This article sets out what her work means for a business that moves goods, in plain terms first. The mathematics and the published papers are at the end for anyone who wants to go deeper.
Why this matters to your business
Getting a load wrong costs money in four places, and most businesses only track one of them.
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Damaged stock
A pallet that shifts in transit means write-offs, credits, re-deliveries and a customer who remembers. The standard failure: column-stacked cartons, wrapped, leaning and then down. [1]
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Injured people
The same collapse lands on whoever opens the trailer door. Interlocking is used in practice precisely to reduce product damage and the risk of injury to people handling the pallet. [4]
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Compliance exposure
Axle mass limits are law, and being under the gross limit is not the same as being compliant. Under the Heavy Vehicle National Law, chain of responsibility duties extend to the parties who pack, load and consign.
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Wasted space
The one everyone measures, and the reason the other three get neglected. Teams resist stability and balance rules because they assume the rules cost cube.
The most useful thing in Silva’s work is evidence that they mostly do not. Across her cases, the pattern repeats. When a safety or handling rule is left out of the plan, the plan breaks it a large share of the time. When the rule is built into the plan from the start, the capacity given up is small, and in the pallet case almost nothing. [1,2,4]
In other words, the trade-off most businesses believe they are making does not exist at the size they think it does. The rest of this article works through the five examples behind that claim, each with what it means on the floor.
Example 1 · Pallets
The stable pallet costs one case in fourteen plans
The pallet work started with a Port wine exporter. Cases of wine, all the same size for a given product, had to be stacked onto pallets, and the company wanted as many cases per pallet as possible. [1] Every manufacturer shipping a single SKU on a pallet, from cartons of tiles to flat-pack furniture, has the same problem.
The layout that fits the most cases is usually a column stack: every layer identical, every case directly on top of the one below. It is also the layout that falls over. Anyone who has run a warehouse knows the fix. You interlock: rotate or mirror each layer so the cases overlap the joints below, the way a bricklayer staggers a wall. Silva is blunt that stretch-wrap on a column stack does not reliably hold it in a moving truck. [1]
What the warehouse does not know is what interlocking costs in cases per pallet, and that is what the research measured. Araújo, Ramos, Silva and Moura built a planning model that requires the case orientation at each corner of the pallet to change between one layer and the next, then ran it across 532 different pallet and case combinations. [1,4]
- 494 combinations: interlocking cost nothing
- 38 : cost one case in a layer
532 pallet and case combinations · never more than one case lost [1]
On the standard measure of how many cases are boxed in by their neighbours, the interlocked plans scored far higher than the plans that chased occupancy alone. [1]
Example 2 · Vehicle balance
Your truck has a safe zone, and total weight does not tell you where it is
Ask most loading supervisors whether a truck is safe and they will tell you the total mass and compare it to the gross limit. That answers one question. It does not answer whether the front axle is overloaded, whether the rear axle is overloaded, or whether there is enough weight on the steering axle for the driver to keep the truck pointed where they want it.
- Weight distribution How the mass is spread across the floor. Matters for the floor.
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Load balance
Where the load’s centre of gravity sits along the vehicle. Matters for steering, braking and axle loads, and most loading tools ignore it. [1,2]
Every vehicle has a safe zone for its centre of gravity, and it is not a single number. It depends on how much is on board. The manufacturer’s data (tare weight on each axle, wheelbase, maximum front and rear axle loads, gross limit, and the minimum weight that has to stay on the steering and drive axles) can be turned into a diagram. On one axis is where the centre of gravity of the load sits, on the other is how much load there is, and the diagram shows the region where every legal and technical limit is met at once. [1,2] Silva calls it the load distribution diagram.
Load distribution diagram Illustrative two-axle rigid truck
The opening example on an illustrative vehicle. At departure the load sits inside the zone. After the rear drop the point moves forward and crosses the front-axle limit. Drawn from the equations in the appendix, not taken from the source papers.
Two trucks with 9-tonne and 19-tonne payloads have differently shaped diagrams, because they are different trucks. [1] This is what makes the opening example work. Take the rear pallets off and the point moves: less weight, but sitting further forward, and for that vehicle the new point is outside the zone. Less cargo, less safe.
Does enforcing it cost capacity?
Ramos, Silva and Oliveira built the diagram into a container loading algorithm as a hard rule, so the algorithm never proposes a load that falls outside the zone. [2] They then did two things worth knowing about.
First, they took packing solutions from the academic literature, which optimise space and ignore balance, and tested them against a real vehicle’s diagram. Almost 32 per cent failed. [1] Those are the solutions commercial load-planning tools are broadly modelled on.
Second, they measured what enforcing the rule cost in trailer fill. The impact on volume utilisation was small. The published paper’s conclusion is that stable, balanced loads can be produced without compromising utilisation. [2]
Example 3 · Multi-drop routes
The load plan has to survive the whole route
A multi-drop run is the normal working day for furniture, appliance, parcel and wholesale distribution. The truck is loaded in reverse order so the first drop is nearest the door. Every stop removes weight, and every stop moves the centre of gravity of what is left.
Silva’s team asked a question most businesses never ask: given a load that is already packed and a delivery sequence that is already fixed, does the truck stay inside its safe zone at every stop, and if not, what is the smallest change to the load that fixes it? [1] Silva, Ramos and Oliveira call this load balance recovery. [5]
Their model looks at each box, works out what is in front of it, beside it and above it, and finds the minimum number of boxes that need to move so that the truck is compliant at departure and after every drop. If nothing can be moved to fix it, the model will say so and drop a box from the load rather than send a non-compliant vehicle. [1,5] Tested with the technical data of a Volvo truck, the approach worked well when the trailer was not completely full, because free space is what gives the model room to rearrange.
Example 4 · Dense product
With dense product, intuition fails four times in five
The last loading case is a fuel tanker, and it is worth a furniture or warehouse reader’s attention because dense product in fixed positions behaves the same way whether it is diesel in compartments or drums, IBCs and stone benchtops in racked slots.
A tanker has several compartments, seven in Silva’s example, and each can hold one product. The planning decision is which product and how much goes into each compartment for a route of service stations. [1] Paixão, Soares, Ramos and Silva built a model for this that keeps the truck inside its safe zone at every delivery. [6]
Same fuel, same truck, opposite result
Intuitive: fill the first four to the top Outside safe zone
Spread across all seven, part-filled Compliant
Illustrative, after Silva’s seminar example. [1] Fill levels are schematic.
The headline number comes from the paper. When the model was run without the safety rule, its plans failed safety standards in 78 per cent of tests. [1,6] Optimising for cost and delivered volume did not produce safe loads on the way through; safety had to be a rule.
The paper also weighed three things a fuel distributor cares about against each other: fewer compartments per customer (less handling at the drop), more fuel delivered (more revenue), and a forward centre of gravity (safer over the route). It found limited conflict between them, and each plan solved in under three seconds on average. [6] Safety was neither expensive nor slow.
Example 5 · Cutting
The plan that wastes the least material can cost the most
Silva’s cutting cases are about manufacturing rather than transport, but they carry the lesson that ties the whole seminar together: a planning tool optimises what it is told to, and if what it is told to optimise is not what the business pays for, the answer will be wrong in a way that looks right.
At a home textile plant, fabric rolls are spread in layers on a cutting table and one pattern is cut through the whole stack. The team built planning models to use the least fabric, and they worked. [1,7] They also produced plans with many different cutting patterns, and every new pattern is a setup: the table cleared, the layers re-spread, the machine reprogrammed. The material saving was real and the labour cost was invisible, because the model had never been told setups cost anything.
- Fewer patterns Faster production A little more fabric
- More patterns Less fabric Slower production
The fix was to give the model both objectives and let it show the trade-off. Silva’s point is that the choice between them belongs to the production manager, and the model’s job is to make the cost of each option visible. [1,7]
Her other cutting case makes a related point about equipment. A panel saw cuts edge to edge; a CNC router or laser can stop mid-sheet. Those used to need different planning models. The Floating Cuts model her group published handles both in one, so a shop with mixed equipment plans once and tells the model which machine the job is going to. [1,8]
Nine things to do this quarter
Silva’s closing line is that geometry is the starting point, and that stability, safety and compliance only appear in a plan if someone puts them in. [1] For a business that moves goods, that becomes a short list.
Trucks
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Get the safe-zone diagram for every truck you load. It comes from the axle limits, tare weights and wheelbase on the vehicle’s plate and spec sheet. Without it you know the gross mass and nothing else.
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Check the load at every drop, not just at the gate. Remaining mass, and where it sits. A spreadsheet handles a normal route.
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Treat loading order as a safety decision. It sets unloading speed and it sets whether the truck is compliant at stop three.
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Do not front-fill dense product. Spread the mass along the vehicle. Left to intuition, the tanker plans were wrong 78 per cent of the time. [6]
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Leave some free space in the trailer. A 100 per cent load cannot be rearranged when something changes at a stop.
Pallets
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Interlock every pallet and stop arguing about cube. One case lost in 38 of 532 plans is the price. [1,4] Column stacks to hit a count are a damage claim waiting to happen.
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Put a stability number on the pallet plan. Average support and share of boxed-in cases are both easy to compute and easy to track.
Planning tools
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Ask any planning tool what it is optimising. If the answer is yield or fill rate alone, the handling and safety costs are unpriced.
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Put the rules inside the plan, not after it. Checked afterwards, the balance rule rejected one container plan in three. Built in, it costs almost no capacity. [1,2,4,6]
Appendix · for practitioners
Going deeper: the working behind the claims
This section is for readers who want to build the checks above or read the source papers. Each part points to the reference that treats it fully.
Show the working (equations and methods) Hide the working
Centre of gravity of a mixed load
For n boxes, where box i has mass mi and its centre of gravity is at longitudinal position xi measured from the front of the load space, the cargo’s centre of gravity is the mass-weighted average:
\bar{x} = \frac{\sum_{i=1}^{n} m_i \, x_i}{\sum_{i=1}^{n} m_i}Ramos, Silva and Oliveira treat each box as a rigid body with its centre of gravity at its geometric centre, so xi is the box’s front coordinate plus half its depth. [2] For a tanker, replace boxes with compartments: with vc the volume in compartment c, ρc the product density and xc the compartment’s fixed centre, the same expression becomes a sum of ρcvcxc over a sum of ρcvc. [6]
From axle limits to the safe-zone diagram
Ramos, Silva and Oliveira derive the load distribution diagram from static force and moment equilibrium on the vehicle as a rigid body. [2] For a two-axle rigid vehicle, let L be the wheelbase, P the payload mass, d the payload’s centre of gravity measured from the front axle, and TF, TR the unladen front and rear axle loads. Taking moments about each axle:
F(P, d) = T_F + P \cdot \frac{L - d}{L} \qquad R(P, d) = T_R + P \cdot \frac{d}{L}The legal and technical limits are then five inequalities in P and d:
\begin{aligned} F(P, d) &\leq F_{\max} && \text{(front axle limit)} \\ R(P, d) &\leq R_{\max} && \text{(rear axle limit)} \\ T_F + T_R + P &\leq \text{GVW} && \text{(gross vehicle weight)} \\ F(P, d) &\geq F_{\min} && \text{(minimum steering-axle load)} \\ R(P, d) &\geq R_{\min} && \text{(minimum drive-axle load)} \end{aligned}Each is a straight line in the (d, P) plane and their intersection is the diagram: every admissible pair of payload mass and centre-of-gravity position. [1,2] Vehicles with more than two axles, or with tag or lift axles, need a more involved split of load across the axle group; the principle is unchanged and the paper covers the general case.
The route check
If Sk is the set of boxes still on board after the k-th drop, the requirement in the recovery model is that the remaining mass and its centre of gravity lie in the diagram at every stage, with k = 0 the fully loaded departure: [5]
\left( \sum_{i \in S_k} m_i \;,\; \frac{\sum_{i \in S_k} m_i x_i}{\sum_{i \in S_k} m_i} \right) \in \text{LDD} \qquad \forall\, k = 0, 1, \ldots, KThe model’s decision variables record whether each box moves, constraints keep the rearranged load free of overlap, and the objective minimises the number of boxes moved. [1,5]
Pallet stability metrics
For a load of N boxes above the first layer, where box i rests on s(i) boxes in the layer below, the support metric is the average:
M_1 = \frac{1}{N} \sum_{i=1}^{N} s(i)A column stack scores exactly 1; interlocked stacks score above it. The contact metric counts the share of boxes that have three or more of their four vertical faces against another box or the notional wall at the pallet edge, with f(i) that count for box i:
M_2 = \frac{\left|\{\, i : f(i) \geq 3 \,\}\right|}{N}Silva describes both as established metrics from the loading literature and notes that the paper proposes a third. [1,4]
The setup trade-off
The textile models trace the frontier between material used and number of patterns with the epsilon-constraint method: one objective is optimised while the other is capped, and the cap is swept across its range: [7]
\min \; \text{Length}(x) \quad \text{subject to} \quad \text{Patterns}(x) \leq \varepsilon, \; x \in XThe Floating Cuts model divides a sheet into five sub-rectangles (four corners and a centre) recursively to a chosen depth, leaves cut positions floating until an item is assigned, and recovers the guillotine case by forcing the centre rectangle to zero size. [1,8]
