MMM Optimizer: Cross Brand

When to use it

Use cross-brand optimization when several brands are fitted as separate models but share one combined media budget, and you want to know whether budget should move between brands as well as between vehicles.

Typical questions:

• "Should some of Brand B's budget move to Brand A?"
• "Keep the combined budget flat. What is the best split across both brands?"
• "Brand A must grow by 5%. What is the best way to fund it?"

A single-brand run can only move money between a brand's own vehicles. A cross-brand run moves it across brand boundaries.

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How it works

Each brand is fitted independently, so each brand's model stays exactly as fitted. For the optimization, the brands' vehicles are merged into one flat list and solved in one joint solve.

• One pinned fit per brand. The request names an exact approved fit for each brand. It is never resolved by brand label, which prevents a run silently picking up a different fit.
• Brand-qualified vehicles. Each vehicle is identified as brand and vehicle, so two brands can both have a TV vehicle without colliding.
• Brands may differ. Brands can have different vehicle sets and very different scales.
• No hierarchy in the solve. The solver sees a flat list of vehicles. Brands exist as groups for share constraints and as labels in the reports.

Formula

The cross-brand problem is the single-brand problem over the union of every brand's vehicles.

$$\max_{x}\ \sum_{b}\sum_{v \in b} U_{bv}(x_{bv}) \quad \text{subject to} \quad \sum_{b}\sum_{v \in b} x_{bv} = B, \qquad \ell_{bv} \le x_{bv} \le u_{bv}, \qquad \underline{\omega}_b B \le \sum_{v \in b} x_{bv} \le \overline{\omega}_b B .$$

Terms: $b$ indexes a brand and $v$ a vehicle within it; $U_{bv}$ is the vehicle's contribution to the objective (see Methodology); $B$ is the combined budget; $\underline{\omega}_b$ and $\overline{\omega}_b$ are the brand's minimum and maximum share of the combined budget.

At the optimum, every vehicle free to move earns the same marginal return across brands, not just within a brand. That is what makes a cross-brand reallocation meaningful.

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Controls

Control	Meaning
Combined budget change	Flat, increase or decrease by a percentage, applied to the movable spend of the whole portfolio
Brand limit	An increase or decrease for a brand's total (one-sided is a target, two-sided is a range), applied to that brand's movable spend
Vehicle limit within a brand	A limit on one brand's vehicle, resolved by brand and vehicle name
Default vehicle guardrail	±15% for every vehicle without an explicit limit, widened only if the combined budget moves further
Brand movement cap	Every brand's total is held within ±10% of its movable spend by default
Locks	Non-paid media and vehicles that failed fit-quality checks stay at current spend
Macro assumptions	Media-cost and price inflation, applied to the optimized side of the reporting

The brand band

Every brand always gets a band, whether or not you asked for a brand limit. The band is the intersection of the brand cap and what is actually reachable.

Formula

Reachable range for brand $b$, given the other brands' own bounds.

$$F_b^{lo} = \max\!\left(\Lambda_b,\ B - U_{-b}\right), \qquad F_b^{hi} = \min\!\left(U_b,\ B - \Lambda_{-b}\right),$$

Terms: $\Lambda_b$ and $U_b$ are the smallest and largest totals brand $b$'s vehicles can reach under their own bounds; $\Lambda_{-b}$ and $U_{-b}$ are the same sums over all other brands. Because the combined budget is fixed, a brand's increase must be funded by other brands' decreases, so its total can only lie in this range.

Formula

Brand band with no explicit request: the brand cap, intersected with the reachable range.

$$\max\!\left(F_b^{lo},\ X_b - m_b\,c^{-}\right) \;\le\; \sum_{v \in b} x_{bv} \;\le\; \min\!\left(F_b^{hi},\ X_b + m_b\,c^{+}\right), \qquad c^{-} = \max(0.10,\ 1-\rho), \quad c^{+} = \max(0.10,\ \rho-1),$$

Terms: $X_b$ is the brand's current spend; $m_b$ is its movable (unlocked) spend; $c^{-}$ and $c^{+}$ are the brand movement cap (10%, widened only when the combined budget moves further); $\rho$ is the ratio of target to current movable budget across the portfolio.

The brand cap is a hard ceiling. Unlike the vehicle guardrail, which fills in only when you gave no limit, an explicit brand request for more than the cap is clamped down and reported.

Explicit brand requests

A percentage on a brand applies to that brand's movable spend, the same basis as the combined-budget change and as the reports (locked spend is excluded). A request of +5% on a brand with 6 million movable and 13 million locked means +0.3 million, not +1 million.

A one-sided request becomes a target:

Formula

Target for a directional brand request, then clamped in this order.

$$T_b = X_b + m_b\,p_b \;\;\longrightarrow\;\; \text{clamp to the brand's own vehicle bounds} \;\;\longrightarrow\;\; \text{clamp to } \left[F_b^{lo},\,F_b^{hi}\right] \;\;\longrightarrow\;\; \text{clamp to the brand cap}.$$

Terms: $p_b$ is the requested change as a fraction of movable spend; $T_b$ is the requested brand total. When a target sits exactly on a limit, a 0.1% band is kept around it so the solver retains a sliver of room. Interior targets stay exact.

Requests are processed from the smallest dollar change to the largest, and each clamped target becomes a fixed bound for the next, so the full set is jointly feasible, the smaller request is met exactly, and the larger one is adjusted. The order you submit them in does not matter.

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What is reported when a request cannot be met

Nothing is silently dropped. Every adjustment is reported with the requested and applied values and a reason:

Reason	Meaning
Brand cap exceeded	The request was larger than the brand movement cap and was clamped to it
Channel limits	The brand's own vehicle limits cannot deliver that change
Other brands' capacity	With the combined budget fixed, the other brands cannot give up or absorb that much
All locked / no spend	The brand has nothing movable, so its limit was ignored
Brand cap infeasible	The cap could not be met together with the other brands' limits, so the wider reachable range was used
Unknown brand	The brand is not in this run, so its limit was ignored

If every brand asked for a fixed change but the combined budget cannot fund them all, the output also states the combined-budget change that would make the requests consistent.

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Worked example

This is an illustrative example on synthetic numbers, computed with the production solver. It is not client data.

Setup. Two brands, Weibull-CDF curves, a combined budget of 9.2 million. Brand A has a profit value of 4.0 per unit and Brand B 3.4 per unit, on the same basis.

Vehicle	Ceiling $\alpha$	Scale $\kappa$	Shape $s$	Current spend
A: TV	1,000,000	1,600,000	1.3	2,000,000
A: OOH	450,000	1,300,000	1.2	800,000
A: Digital Video	700,000	1,100,000	1.1	600,000
B: TV	2,100,000	3,800,000	1.3	4,000,000
B: Radio	300,000	500,000	1.5	600,000
B: Search	600,000	900,000	1.0	1,200,000

Reshuffle across both brands

Vehicle	Current	Optimized	Change	Marginal ROI	At bound
A: TV	2,000,000	2,086,126	+4.3%	0.858	No
A: OOH	800,000	809,776	+1.2%	0.858	No
A: Digital Video	600,000	690,000	+15.0%	1.469	Upper
B: TV	4,000,000	3,974,571	−0.6%	0.858	No
B: Radio	600,000	619,527	+3.3%	0.858	No
B: Search	1,200,000	1,020,000	−15.0%	0.730	Lower

Four vehicles across the two brands end at the same marginal ROI (0.858): that is the cross-brand equal-return condition. Brand A's total rises 5.5% and Brand B's falls 3.2%, with the combined budget unchanged. All validation gates pass.

Adding a brand-level request

Request	Brand A total	Profit objective	Adjustments reported
None (reshuffle)	3,585,902 (+5.5%)	2,678,186	None
Brand A +5%	3,570,000 (+5.0%)	2,678,124	None; met exactly
Brand A +25%	3,740,000 (+10.0%)	2,672,342	Clamped: first by the vehicle limits to +15%, then by the brand cap to +10%

The +5% target is met exactly and costs almost nothing in profit. The +25% request is clamped to the 10% brand cap, and the output says so. The profit objective falls only slightly with each added constraint, which is how you read the price of a constraint. Lift in raw response units is not the right comparison here, because the two brands carry different unit values and the objective is profit.

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Outputs

Output	Content
Detailed media analysis	Level 1 is the whole portfolio, Level 2 one row per brand, Level 3 one row per vehicle labelled by brand and vehicle
Weekly flighting plan	Built once per brand from that brand's own history and attribution, then combined; the media type column carries the brand
Optimization summary	Solver status, validation gates, adjustments, assumptions, and any brand that was skipped
Report (PDF)	One weekly budget chart per brand, because a combined chart is misleading when brands differ greatly in scale

If one brand's pinned fit is missing or no longer matches the request, that brand is skipped and reported, and the run continues with the rest. The run fails only if no brand can be resolved. The phasing baseline is the earliest of the brands' most recent complete years, so every brand is phased from a full year of history. A partly refreshed trailing year is never used as the baseline.

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What you need to ensure

Requirement	Why
Same response unit and value basis across brands	The solve equalizes marginal value across all vehicles, which is only meaningful if every brand's response and unit value are on one basis (for example incremental profit per unit). The optimizer does not check units or currency.
Annual spend in one currency and year basis	Curves are fitted on annual spend
One fit per brand	A brand appears once, with an exact approved fit
Independent source data	Two entries built from the same underlying data would count the same budget twice

For the formulas behind each step, see mmm-optimizer-methodology.