MMM Optimizer

Overview

MMM Optimizer: Overview

What is the MMM Optimizer?

The MMM Optimizer answers one question:

"Given what the model has learned about each media vehicle, where should the next dollar go?"

It takes the response curves of an approved Naukr AI MMM and a business request, such as keep the budget flat, raise it by 10%, hold one vehicle fixed, or move at most 15% per vehicle. It returns a deterministic, validated allocation: the best split of budget across vehicles (and, for portfolios, across brands) that respects every guardrail.

Budget optimization is a later-stage use of approved model outputs. It never runs on a model that has not passed the quality gates described in the MMM documentation.

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What we offer

Capability	What it gives you
Single brand	Reallocate one brand's budget across its own vehicles, with vehicle-level pins, Traditional / Digital group limits, locks, and a weekly plan
Cross brand	Reallocate one combined budget across several independently fitted brands in one joint solve, with brand-level and vehicle-level limits
Headless	Run either of the above on a saved scenario against a pinned, approved fit, with no interactive session: start it from a scenario, an agent or the API, and collect the outputs afterwards

All three share one solver and one set of outputs, so a result means the same thing whichever way it was started.

Objectives

Objective	Maximizes
Volume	Total incremental response (units, conversions, sales volume)
Revenue	Total incremental revenue (price per unit times response)
Profit	Total incremental profit net of media spend (margin per unit times response, minus spend)

Business controls

• Total budget: keep it flat, raise it, or cut it by a percentage. The change applies to the movable (unlocked) budget.
• Vehicle limits: a one-sided limit pins a vehicle to exactly that change; a two-sided limit sets a range.
• Group limits: Traditional and Digital totals, with a built-in movement cap.
• Brand limits (cross brand): brand totals, with a built-in movement cap.
• Locks: vehicles that must stay at current spend, such as non-paid media.
• Macro assumptions: media-cost inflation and price inflation, applied to the optimized side of the reporting only.

Outputs

Output	Content
Detailed media analysis	A three-level table (portfolio, group or brand, vehicle) with current vs optimized spend, volume, revenue, profit, ROI and cost per unit
Weekly flighting plan	Each vehicle's annual optimum phased across 52 weeks by its own history, tilted toward historically efficient weeks
Optimization summary	Solver status, validation gates, constraint adjustments, and the assumptions used
Report (PDF)	A business-readable summary with weekly budget charts (one per brand for portfolios)

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Why it matters for business

The optimizer uses the marginal return of each vehicle (what the next dollar buys), not the average return. Average ROI rewards vehicles that are already saturated; marginal ROI finds where extra spend still works. At the optimum, every vehicle that is free to move earns the same marginal return, and vehicles held back by a limit earn more or less than that, which tells you which guardrail is costing you.

It helps teams decide:

• how to reshuffle a flat budget,
• where an extra 10% should go,
• which vehicles to cut first when the budget shrinks,
• whether budget should move between brands.

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Where it fits in the product experience

1. Fit and approve an MMM (see the MMM documentation).
2. The approved fit publishes one response curve per vehicle.
3. Create a scenario: objective, budget change, limits, locks.
4. Run the optimizer (single brand, cross brand, or headless).
5. Review the three-level table, the weekly plan, and the validation gates.

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Quick reading guide

Question	Where to look
How are the curves defined and where do they come from?	Methodology
How do I reallocate within one brand?	Single Brand
How do I move budget across brands?	Cross Brand
How do I run an optimization without the UI?	Headless

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What the optimizer does not do

• It optimizes annual spend per vehicle. Carryover is already embedded in each curve; it is not re-simulated week by week.
• It optimizes on each vehicle's fitted mean curve. Credible bands are published for display but do not enter the solve.
• For S-shaped curves the problem is not concave, so the solver returns a local optimum and reports whether it converged and passed validation.
• It does not check that response units match across brands. Cross-brand results are only meaningful when every brand's response and unit value are on the same basis.

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Related documentation

• Methodology — curves, objectives, constraints, solver, diagnostics, weekly phasing (mmm-optimizer-methodology)
• Single Brand — one brand's budget across its vehicles (mmm-optimizer-single-brand)
• Cross Brand — one combined budget across brands (mmm-optimizer-cross-brand)
• Headless — saved scenarios run without an interactive session (mmm-optimizer-headless)

Methodology

MMM Optimizer: Methodology

Problem statement

Let $v$ index a media vehicle (the unit the optimizer allocates to), $x_v$ its annual spend, $x_v^{0}$ its current annual spend, and $B$ the total budget. Each vehicle has a response curve $f_v(x)$ in annual units and a unit value $q_v$ (profit per unit of response).

Formula

The optimizer chooses annual spend for every vehicle to maximize total value, subject to the budget, vehicle bounds, and group shares.

$$\max_{x}\ \sum_{v} U_v(x_v) \quad \text{subject to} \quad \sum_{v} x_v = B,\qquad \ell_v \le x_v \le u_v,\qquad \underline{\omega}_g B \le \sum_{v \in g} x_v \le \overline{\omega}_g B .$$

Terms: $U_v$ is the vehicle's contribution to the chosen objective; $B$ is the total budget; $\ell_v$ and $u_v$ are the vehicle's lower and upper spend bounds; $g$ indexes a group (Traditional / Digital, or a brand); $\underline{\omega}_g$ and $\overline{\omega}_g$ are the group's minimum and maximum share of the budget.

A scenario changes only the constraints and the total budget. The same solver runs unchanged for every scenario.

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Response curves

Every vehicle's response is a saturating function of annual spend, evaluated directly on spend. Carryover (adstock) is already embedded in the curve, so the optimizer does no time shifting.

Weibull-CDF curve (Naukr AI default)

Formula

Response rises from zero toward a ceiling; the scale sets where the curve bends and the shape sets how sharply.

$$f_v(x) = \alpha_v\left[1 - \exp\!\left(-\left(\frac{x}{\kappa_v}\right)^{s_v}\right)\right],$$

Terms: $f_v(x)$ is annual incremental response at spend $x$; $\alpha_v$ is the response ceiling; $\kappa_v$ is the scale in spend units (response reaches about 63% of the ceiling at $x = \kappa_v$); $s_v > 0$ is the shape.

The spend at which the curve reaches half of its ceiling is

Formula

Half-saturation spend follows from the scale and shape.

$$h_v = \kappa_v\,(\ln 2)^{1/s_v}, \qquad f_v(h_v) = \tfrac{1}{2}\alpha_v .$$

Terms: $h_v$ is the spend giving 50% of the ceiling; $\kappa_v$ and $s_v$ are the scale and shape; $\ln 2$ is the natural logarithm of 2.

This is the same curve family as the model's own saturation. With $h$ and $k$ as in the MMM Models tab, $1 - \exp[-\ln 2\,(a/h)^{k}]$ equals the form above with $\kappa = h/(\ln 2)^{1/k}$ and $s = k$. The model fits it on normalized exposure; the optimizer needs it on annual spend, which is what the next section builds.

Hill curve (compatible engines)

Curves fitted by other supported engines use the Hill family, and the optimizer accepts both:

Formula

Hill saturation on spend.

$$f_v(x) = \alpha_v\,\frac{x^{s_v}}{x^{s_v} + k_v^{\,s_v}},$$

Terms: $\alpha_v$ is the ceiling; $k_v$ is the half-saturation spend, $f_v(k_v) = \tfrac{1}{2}\alpha_v$; $s_v > 0$ is the shape. A shape above 1 gives an S-curve; a shape of 1 or below is concave from the origin.

Marginal response: the optimization signal

The optimizer climbs the marginal response, the exact analytic derivative of the curve. There is no finite differencing and no sampling.

Formula

Marginal response of the next unit of spend, for each family.

$$f_v'(x) = \alpha_v\,\frac{s_v}{\kappa_v}\left(\frac{x}{\kappa_v}\right)^{s_v-1}\exp\!\left(-\left(\frac{x}{\kappa_v}\right)^{s_v}\right) \quad\text{(Weibull-CDF)}, \qquad f_v'(x) = \alpha_v\,\frac{s_v\,k_v^{\,s_v}\,x^{s_v-1}}{\left(x^{s_v}+k_v^{\,s_v}\right)^{2}} \quad\text{(Hill)}.$$

Terms: $f_v'(x)$ is the change in annual response per extra unit of spend; the other symbols are the curve parameters above. A tiny spend floor keeps the derivative finite at zero spend for shapes below 1.

Average ROI is reported to humans but is never the objective:

Formula

Derived business metrics for a vehicle at spend $x$.

$$\mathrm{ROI}^{avg}_v(x) = \frac{q_v\, f_v(x)}{x}, \qquad \mathrm{ROI}^{mar}_v(x) = q_v\, f_v'(x), \qquad \sigma_v(x) = \frac{f_v(x)}{\alpha_v}.$$

Terms: $\mathrm{ROI}^{avg}_v$ is average return per dollar; $\mathrm{ROI}^{mar}_v$ is the return of the next dollar; $q_v$ is profit per unit of response; $\sigma_v$ is the saturation ratio, the share of the ceiling already reached.

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From the Bayesian model to optimizer curves

The MMM fits response on normalized exposure (a multiple of each media node's typical execution level). The optimizer needs annual spend per vehicle. A bridge rebuilds each vehicle's spend-space curve from the approved model and registers it as a Weibull-CDF curve.

1. Group curves. For each pooling group $p$, take the published posterior-median half-response point $h_p$ and shape $k_p$, and the group's response share

Formula

The model's own response share at normalized exposure $z$.

$$S_p(z) = 1 - \exp\!\left[-\ln 2\left(\frac{z}{h_p}\right)^{k_p}\right].$$

Terms: $S_p(z)$ is the response share between zero and one; $z$ is exposure as a multiple of the node's typical level; $h_p$ is the 50% response point; $k_p$ is the shape.

The published parameters are used directly when they reproduce the published mean curve to within 2% of the ceiling; otherwise they are refitted from the published curve points.

2. Vehicle curve. A vehicle is a set of media nodes and weeks. Scaling the vehicle's spend by a multiplier $m$ scales every node's exposure by $m$, so

Formula

Vehicle response at spend multiplier $m$, anchored to the reference-year incremental volume.

$$R_v(m) = V_v\,\frac{\sum_{i \in I_v} S_{p(i)}\!\left(m\,u_i\right)}{\sum_{i \in I_v} S_{p(i)}\!\left(u_i\right)}, \qquad u_i = \frac{\text{exposure}_i}{\text{typical exposure of node}(i)} ,$$

Terms: $R_v(m)$ is the vehicle's annual response when spend is $m$ times the reference level; $V_v$ is its reported incremental volume in the reference year; $I_v$ is the set of node-week cells with positive exposure in that year; $p(i)$ is the pooling group of cell $i$; $u_i$ is the cell's exposure relative to its node's typical level (the median positive exposure over the history).

The curve is evaluated on a grid of 61 multipliers from 0 to 3, at spend $x = m\,s_v$ where $s_v$ is the vehicle's reference-year spend.

3. Single Weibull in spend space. One Weibull-CDF is fitted to those points, with the ceiling at least the largest response on the grid, the scale between 2% and 50 times the reference spend, and the shape between 0.5 and 6.

4. Unit value. The profit basis comes from the same reference year:

Formula

Profit per unit of response.

$$q_v = \frac{\text{Profit}_v}{V_v}.$$

Terms: $q_v$ is the unit value used by the profit objective; $\text{Profit}_v$ and $V_v$ are the vehicle's incremental profit and incremental volume in the reference year.

Reference year. The latest complete year: the latest year whose total spend is at least 80% of the median year's total spend.

Quality gates. A vehicle is registered only if all of these hold:

Gate	Threshold
Each group's published curve follows the model's own form	Maximum gap at most 2% of the ceiling
The vehicle curve is well described by one Weibull	Maximum gap at most 3% of the vehicle's yearly response
The registered curve reproduces the reported outcome at current spend	Difference at most 3%
Shape stays in a stable range	Between 0.5 and 6
The vehicle had positive incremental volume and profit and execution in the reference year	Required

A run is optimizable only when every vehicle that spent in the reference year converts cleanly, and only when it is a production run. A smoke (non-production) run publishes curves for display but is never used to move budget.

Assumptions recorded with every conversion: spend and exposure move together; saturation is evaluated on the period's exposure, with carryover not re-simulated; a node's typical level is the median of its positive exposure over the whole history.

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Objective functions

The objective is a sum over every vehicle of every brand in the solve.

Formula

Per-vehicle contribution for each objective.

$$U_v^{volume}(x) = f_v(x), \qquad U_v^{revenue}(x) = p_v\, f_v(x), \qquad U_v^{profit}(x) = q_v\, f_v(x) - x .$$

Terms: $p_v$ is revenue per unit of response (it defaults to $q_v$ when no price is set); $q_v$ is profit per unit of response; $x$ is the vehicle's spend.

The gradient of each contribution is the marginal response scaled the same way ($f_v'$, $p_v f_v'$, or $q_v f_v' - 1$), so the objective and its gradient always share one basis.

Because the budget constraint fixes $\sum_v x_v = B$, the profit objective equals $\sum_v q_v f_v(x_v) - B$. For a given budget, the profit and margin-weighted revenue objectives therefore choose the same allocation; the profit objective value is simply reported net of spend.

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What an optimum looks like

Attach a multiplier $\lambda$ to the budget constraint. At an optimum, the first-order conditions say that every vehicle that is free to move earns the same marginal value, and every vehicle held at a bound is on the correct side of it.

Formula

First-order (equal-marginal-return) conditions at the optimum, for the profit objective.

$$q_v\, f_v'(x_v^{*}) = \lambda \quad \text{if } \ell_v < x_v^{*} < u_v, \qquad q_v\, f_v'(x_v^{*}) \le \lambda \quad \text{if } x_v^{*} = \ell_v, \qquad q_v\, f_v'(x_v^{*}) \ge \lambda \quad \text{if } x_v^{*} = u_v .$$

Terms: $x_v^{*}$ is the optimal spend; $\lambda$ is the common marginal return implied by the budget; $\ell_v$ and $u_v$ are the bounds. With a binding group-share constraint, the vehicles in that group share a value shifted by that constraint's multiplier. This is a standard consequence of the Lagrangian, not an extra rule in the solver.

This is why the result is explainable: a vehicle sitting at its upper bound with a marginal ROI above the shared value is being held back by a guardrail, and loosening that guardrail would raise the objective.

For S-shaped curves ($s > 1$) the problem is not concave, so these conditions identify a local optimum. The solver safeguards below, and the validation gates, exist to catch the failure modes this creates.

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Constraints and bounds

Vehicle bounds

For each vehicle, in this order of precedence:

1. Locked vehicles are pinned at current spend: $\ell_v = u_v = x_v^{0}$.
2. Otherwise start from the hard guardrails $m_v^{\min}$ (default 0) and $m_v^{\max}$ (default unbounded).
3. Narrow by the scenario's change limits.
4. If the result is inverted, collapse to the upper bound.

Formula

Effective bounds for an unlocked vehicle.

$$\ell_v = \max\!\left(m_v^{\min},\ x_v^{0}\,(1-\delta^{-})\right), \qquad u_v = \min\!\left(m_v^{\max},\ x_v^{0}\,(1+\delta^{+})\right).$$

Terms: $x_v^{0}$ is current spend; $\delta^{-}$ and $\delta^{+}$ are the allowed decrease and increase as fractions of current spend; $m_v^{\min}$ and $m_v^{\max}$ are hard dollar limits.

Budget and feasibility

The budget is an equality constraint, $\sum_v x_v = B$. Before solving, the optimizer checks that the problem is feasible:

Formula

Pre-solve feasibility check.

$$\sum_v \ell_v \;\le\; B \;\le\; \sum_v u_v .$$

Terms: $\sum_v \ell_v$ is the smallest total the bounds allow; $\sum_v u_v$ is the largest. If the budget falls outside, the run stops with a plain-language error that says to reduce a pinned vehicle's change, unlock a vehicle, or change the budget.

Group shares

A group $g$ (Traditional, Digital, or a brand) is constrained as a share of the budget: $\underline{\omega}_g B \le \sum_{v\in g} x_v \le \overline{\omega}_g B$. Both limits are enforced natively by the solver.

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From a business request to constraints

The same mapping is used for every surface (see the Single Brand and Cross Brand tabs for the group and brand layers).

Total budget

The requested change applies to the movable slice of the budget; locked spend is added back unchanged.

Formula

Target total budget for a requested change $p$.

$$B = L + R\,(1 + p),$$

Terms: $L$ is the current spend of locked vehicles; $R$ is the current spend of unlocked (reallocatable) vehicles; $p$ is the requested change as a fraction (positive for an increase, negative for a decrease, zero for no change).

Applying $p$ to the full current total instead would demand that the movable slice absorb a disproportionately large change whenever much of the budget is locked.

Default vehicle guardrail

Vehicles without an explicit limit get a hard default of $\pm g$ with $g = 15\%$. The guardrail widens only if the budget itself moves further than $g$, otherwise the target would be unreachable.

Formula

Default guardrail, widened only when the budget moves more than the default.

$$\rho = \frac{B - L}{R}, \qquad \delta^{+} = \max\!\left(g,\ \rho - 1\right), \qquad \delta^{-} = \max\!\left(g,\ 1 - \rho\right).$$

Terms: $\rho$ is the ratio of the target movable budget to the current movable budget; $g$ is the default guardrail (15%); $\delta^{+}$ and $\delta^{-}$ are the effective increase and decrease limits.

Explicit vehicle limits

Request	Meaning
Increase 0, decrease 0	No explicit limit; the default guardrail applies
Only an increase or only a decrease	A directional target: the vehicle is pinned to exactly $x_v^{0}(1 \pm p)$
Both an increase and a decrease	A range: $\left[x_v^{0}(1-d),\ x_v^{0}(1+i)\right]$

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The solver

The problem is solved with sequential least squares programming (SLSQP) using the exact analytic gradient.

Normalization. Raw spend is around ten million, which flattens the objective in the saturated regime and makes SLSQP stall. The solve runs in budget-normalized space $y = x / S$ with $S = \max(B, 1)$, so variables are of order 0.1 and the budget equality is of order 1. The objective, gradient and constraints are converted in kind (the chain-rule factor cancels exactly in the gradient).

Feasible start. The starting point clamps current spend into the bounds, then redistributes the remaining budget into vehicles with headroom (or out of vehicles with room), for up to 64 passes.

Settings. At most 300 iterations, with a function tolerance of $10^{-9}$.

Stall recovery. SLSQP can report success after a single iteration while having moved essentially nowhere. A stall is detected when the solve took at most one iteration, moved less than $10^{-6}$ from its start, and some vehicle actually had freedom to move. The solver then retries from the start nudged along its own gradient:

Formula

Restart point after a detected stall.

$$x^{retry} = x^{0} + 0.10\,\frac{B}{N}\,\hat{d}, \qquad \hat{d} = \frac{\nabla U(x^{0})}{\max_v \left|\nabla U_v(x^{0})\right|},$$

Terms: $x^{0}$ is the original feasible start; $N$ is the number of vehicles; $\hat{d}$ is the ascent direction scaled to unit maximum. The retry is adopted only if it moved, converged, met the budget equality, and reached an objective at least as good as the stalled solution, so recovery can never make a result worse.

Each iteration's objective and spend are recorded, so a run shows the objective climbing toward the optimum.

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Diagnostics and validation

Saturation zones

Each vehicle is labelled by how much of its ceiling it has reached. The band edges are an internal rule, not universal constants.

Saturation ratio $\sigma_v$	Zone
Below 0.10	Threshold
Below 0.45	Growth
Below 0.70	Optimal
Below 0.90	Diminishing
0.90 and above	Saturation

Per-vehicle rows

For every vehicle the output reports current and optimized spend, the change in dollars and percent, current and optimized response, average and marginal ROI, the saturation ratio, the zone, and whether the optimum sits on its lower or upper bound.

Validation gates

Five checks run on every solution and are published in the summary:

Gate	Passes when
Budget equality	Total optimized spend equals the budget within a relative tolerance of $10^{-4}$
Bounds respected	Every vehicle lies inside its own bounds
Locked unchanged	Every locked vehicle still equals its current spend
Marginal declines	Marginal response does not rise as spend grows, checked only past 50% saturation (below that, an S-curve legitimately rises)
Solver success	The solver reported convergence

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Weekly flighting

The solver produces one annual number per vehicle. The flighting engine phases it into 52 weeks.

Historical-share phasing

Formula

Each vehicle's annual optimum is phased by its own historical weekly share.

$$w_{v,t} = \frac{h_{v,t}}{\sum_{t'} h_{v,t'}}\; x_v^{*},$$

Terms: $w_{v,t}$ is the planned spend for vehicle $v$ in week $t$; $h_{v,t}$ is that vehicle's historical spend in week $t$; $x_v^{*}$ is its optimized annual spend. A vehicle with no history is phased flat at $x_v^{*}/52$. Without a historical file, every vehicle is phased flat.

Efficiency tilt

When the model provides per-week attribution, spend is tilted toward weeks that were historically more efficient, then rescaled so the annual total is unchanged.

Formula

Weekly efficiency, tilt, and per-week cap.

$$e_{v,t} = \frac{\mathrm{MaCo}_{v,t}}{\mathrm{Spend}_{v,t}}, \qquad \rho_{v,t} = \frac{e_{v,t}}{\bar{e}_v}, \qquad \tilde{w}_{v,t} = w_{v,t}\,\rho_{v,t}^{\,0.3},$$$$\gamma_{v,t} = 1 + 0.5\left(1 - e^{-0.5\,\left|\ln \rho_{v,t}\right|}\right), \qquad \left|\hat{w}_{v,t} - w_{v,t}\right| \le 0.20\,\gamma_{v,t}\, w_{v,t}, \qquad \sum_t \hat{w}_{v,t} = x_v^{*} .$$

Terms: $\mathrm{MaCo}_{v,t}$ is the model's own attributed contribution in week $t$; $\bar{e}_v$ is the vehicle's average efficiency (ratio of sums); $\rho_{v,t}$ is the week's efficiency relative to average; the exponent 0.3 keeps the tilt gentle (a ratio of 2 scales a week by about 1.23); $\gamma_{v,t}$ widens a week's cap from 1 to at most 1.5 times the base cap when its signal is strong; $\hat{w}_{v,t}$ is the final weekly spend.

A vehicle needs at least 8 weeks with positive attribution and spend before it is tilted at all; otherwise it keeps its historical-share phasing unchanged.

Aggregate weekly cap

A second, cross-vehicle cap bounds each week's combined spend against that week's historical combined spend:

Formula

Aggregate weekly band.

$$(1-0.15)\, H_t \;\le\; \sum_v \hat{w}_{v,t} \;\le\; (1+0.15)\, H_t, \qquad H_t = \sum_v h_{v,t}.$$

Terms: $H_t$ is the historical combined spend in week $t$; $\hat{w}_{v,t}$ is each vehicle's planned spend. Violating weeks are scaled to the nearest edge, and each vehicle's other weeks are rescaled to restore its annual total, for up to 5 rounds.

A feasibility guard runs first. If the optimized annual total already falls outside the sum of every week's band (for example a large annual increase phased into 52 weeks each capped at 15%), no weekly plan can satisfy both, so the cap is skipped rather than silently dropping budget. The annual decision made by the solver always wins.

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Reporting metrics

The detailed media analysis is built from the optimized spend and the curves.

Formula

Reported metrics for a vehicle or any aggregate.

$$\text{Gross revenue} = p \times V, \qquad \text{Profit} = q \times V, \qquad \mathrm{ROI} = \frac{\text{Profit}}{\text{Spend}}, \qquad \text{Cost per unit} = \frac{\text{Spend}}{V},$$

Terms: $p$ is price (revenue per unit); $q$ is margin (profit per unit); $V$ is response volume at the shown spend. ROI here is profit over spend, not revenue over spend.

Macro assumptions. Media-cost inflation (CPM) and price inflation apply to the optimized, forward-looking side only; the current column is always the untouched historical reference:

Formula

Macro adjustment of the optimized side.

$$V^{opt} \leftarrow \frac{V^{opt}}{1 + c}, \qquad p^{opt} \leftarrow p\,(1 + \pi), \qquad q^{opt} \leftarrow q\,(1 + \pi),$$

Terms: $c$ is the net media-cost inflation (increase minus decrease): more cost means fewer response units per dollar; $\pi$ is the net price inflation, which raises value per unit.

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Assumptions and limits

Assumption	Consequence
Carryover is embedded in each curve	The optimizer does not re-simulate adstock; weekly plans phase by history, not by model
Annual spend maps to annual response	Curves are fitted and compared on an annual basis
One fitted mean curve per vehicle	Credible bands are published for display; they do not enter the solve
S-shaped curves are not concave	The result is a local optimum; the gates report convergence and validity
Guardrails are business inputs	A vehicle at a bound is a decision to review, not a modeling result

Single Brand

MMM Optimizer: Single Brand

When to use it

Use single-brand optimization when one approved model covers one brand and you want to decide how that brand's annual budget should be split across its own vehicles.

Typical questions:

• "Keep the budget flat. Where should we shift spend?"
• "We can add 10%. Which vehicles should get it?"
• "The budget is being cut by 10%. What should go first?"
• "TV must come down by exactly 10%. What is the best split of the rest?"

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What you provide and what the system supplies

You provide	The system supplies
Objective: volume, revenue or profit	One approved response curve per vehicle
Budget change: none, increase, or decrease by a percentage	Current annual spend per vehicle (from the model baseline or your merged file)
Optional per-vehicle limits (directional or range)	Unit value (profit per unit) and price (revenue per unit) per vehicle
Optional Traditional / Digital group limits	Historical weekly spend, for the weekly plan
Optional macro assumptions (media-cost and price inflation)	Per-week attribution, when available, for the efficiency tilt
Target year for the plan	Locks for non-paid media and vehicles that failed fit-quality checks

If no approved model exists, the run stops with a clear message. The optimizer never fabricates curves from raw data.

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How a request becomes an optimization problem

Total budget

The requested change applies to the movable (unlocked) spend; locked spend is added back unchanged:

Formula

Target total budget for a requested change $p$.

$$B = L + R\,(1 + p).$$

Terms: $L$ is the current spend of locked vehicles; $R$ is the current spend of unlocked vehicles; $p$ is the requested change as a fraction.

Vehicle limits

• No explicit limit (increase 0, decrease 0): the vehicle gets the default guardrail of ±15%, widened only if the budget itself moves further.
• One-sided limit: a directional target, pinned to exactly $x_v^{0}(1 \pm p)$.
• Two-sided limit: a range, $\left[x_v^{0}(1-d),\ x_v^{0}(1+i)\right]$.

Campaign-level names (such as a TV campaign) roll up to their parent vehicle. If two campaign keys roll up to the same vehicle with different limits, the request is rejected as conflicting rather than silently merged. A directional limit that cannot be matched to any vehicle is rejected, and the message lists the available vehicles.

Traditional and Digital groups

Every vehicle belongs to Traditional or Digital. Each group's total is held in a band, whether or not you asked for a group limit. The movement cap is 10% by default and widens only if the budget moves further:

Formula

Group share band, with no explicit group request.

$$\frac{\max\!\left(\Lambda_g,\ X_g\,(1-c^{-})\right)}{B} \;\le\; \omega_g \;\le\; \frac{\min\!\left(U_g,\ X_g\,(1+c^{+})\right)}{B}, \qquad c^{-} = \max(0.10,\ 1-\rho), \quad c^{+} = \max(0.10,\ \rho-1),$$

Terms: $\omega_g$ is the group's share of the budget; $X_g$ is the group's current spend; $\Lambda_g$ and $U_g$ are the smallest and largest totals its member vehicles can reach under their own bounds; $\rho$ is the ratio of target to current movable budget.

If you request a group move, it is treated like a vehicle limit (one-sided means a target, two-sided means a range), clamped first into what the vehicles can deliver and then into the movement cap. If the request had to change, the output says so, for example:

digital: channel-level moves are capped at 10% up, so the request was applied as +10.0%.

If the group bands could not jointly sum to the budget, they are widened only as much as needed, starting with groups you did not ask about, and the change is reported.

Locked vehicles

Vehicles that must not move are pinned at current spend and still count toward the total:

• non-paid media such as trade marketing and sponsorship,
• vehicles that failed fit-quality checks at model time,
• vehicles with no current spend (there is nothing to reallocate from).

Locked non-paid media is excluded from the paid-media rollups in the reports.

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Run steps

1. Resolve the approved model for the brand (stops with a clear message if none exists).
2. Load vehicles and current spend.
3. Build the scenario: total budget, vehicle bounds, group bands, locks.
4. Solve and validate: the SLSQP solve, then the five validation gates.
5. Build deliverables: the three-level table, the weekly plan, the summary, and the report.

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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. Five vehicles with Weibull-CDF curves, a profit value of 4.0 per unit of response for every vehicle, and a combined budget of 6.5 million.

Vehicle	Group	Ceiling $\alpha$	Scale $\kappa$	Shape $s$	Current spend
TV	Traditional	1,300,000	2,000,000	1.3	3,000,000
OOH	Traditional	600,000	1,800,000	1.2	1,200,000
Radio	Traditional	450,000	700,000	1.5	500,000
Digital Video	Digital	950,000	1,700,000	1.1	1,000,000
Search	Digital	450,000	500,000	1.0	800,000

Scenario A: reshuffle (budget unchanged)

Vehicle	Current	Optimized	Change	Marginal ROI	Saturation	Zone	At bound
TV	3,000,000	2,803,978	−6.5%	0.793	0.79	Diminishing	No
OOH	1,200,000	1,214,400	+1.2%	0.793	0.46	Optimal	No
Radio	500,000	575,000	+15.0%	1.660	0.53	Optimal	Upper
Digital Video	1,000,000	1,150,000	+15.0%	1.234	0.48	Optimal	Upper
Search	800,000	756,622	−5.4%	0.793	0.78	Diminishing	No

The three vehicles free to move share one marginal ROI (0.793). Radio and Digital Video earn more at the margin but are held by the 15% guardrail, which is exactly the signal that a looser guardrail there would help. Total response rises by 38,514 units (+1.67%), worth about 154 thousand in profit at 4.0 per unit, with no extra spend. All five validation gates pass.

Scenarios B and C

Vehicle	Current	B: budget +10%	Change	C: TV −10% and Digital +12%	Change
TV	3,000,000	3,215,000	+7.2%	2,700,000	−10.0%
OOH	1,200,000	1,380,000	+15.0%	1,245,000	+3.8%
Radio	500,000	575,000	+15.0%	575,000	+15.0%
Digital Video	1,000,000	1,150,000	+15.0%	1,150,000	+15.0%
Search	800,000	830,000	+3.7%	830,000	+3.8%
Total	6,500,000	7,150,000		6,500,000
Response lift		+155,430 (+6.74%)		+36,790 (+1.60%)

• B: a 10% budget increase is spread toward the vehicles with the highest marginal return, with three vehicles reaching their upper limit.
• C: TV is pinned to exactly −10%. The request for Digital +12% exceeds the 10% group cap, so it is applied as +10% and the output reports the adjustment. The cost of the pin is visible: lift is 36,790 against 38,514 for the free reshuffle.

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Reading the result

Item	How to read it
Zone	Where the optimized spend sits on the curve: Threshold, Growth, Optimal, Diminishing or Saturation
At bound	The vehicle is held by a limit; compare its marginal ROI with the shared value to see what the limit costs
Adjustments	Any request that had to be clamped, with the requested and applied values and the reason
Validation gates	All five should pass; a failure is shown in the summary

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What can go wrong

Situation	What happens
The budget falls outside what the bounds allow	The run stops with the totals and what to change: reduce a pinned vehicle's change, unlock a vehicle, or change the budget
Heavy locking and a large budget cut	The cut applies only to the movable slice, so the target stays reachable
A group request exceeds the movement cap	It is clamped and reported in the adjustments
A directional vehicle key matches nothing	The run is rejected and lists the available vehicles
Two campaign keys with different limits roll up to one vehicle	The run is rejected as conflicting
The PDF report cannot be rendered	The CSV and summary are still delivered

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

Cross Brand

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.

Headless

MMM Optimizer: Headless

What is the headless optimizer?

The headless optimizer runs an optimization on a saved scenario against a pinned, approved model fit, with no interactive session. Nobody has to sit in the optimizer screen while it works:

• you start it from a saved scenario, an agent, or the API;
• it returns immediately with a run identifier;
• the solve runs as a background job;
• you collect the same outputs (tables, weekly plan, summary, report) when it completes.

It is the same solver and the same outputs as the interactive experience, so a headless result means exactly what an interactive one does. Headless is a way of running, not a different model.

It never refits the model. It reallocates budget from a fit that has already been approved, which is why it can run repeatedly and quickly.

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How it is started

Surface	How it starts a headless run
Media Workspace	Run the optimizer on a saved project scenario; it uses the project's primary model
Agent Builder	An optimization agent submits the request and collects the result
Chat assistant	The assistant runs the optimization as a long-running step and resumes with the result
API	A request with the datasource, objective and constraints

Single-brand and cross-brand runs both work this way. Cross-brand runs carry one pinned fit per brand.

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Lifecycle

1. Request. A scenario, agent or API call supplies the objective, budget change and constraints.
2. Resolve the fit. The run targets one approved fit. In Media Workspace that is the project's primary fit, and in a cross-brand run it is the pinned fit for each brand, rather than simply the newest one.
3. Run guards. The run is refused if a guard fails (see below).
4. Load vehicle curves and spend. Current spend comes from the model baseline, or from the supplied merged file or override.
5. Build the scenario. The same mapping as the interactive path: total budget, guardrails, group or brand bands, locks.
6. Solve and validate. SLSQP, the stall safeguard, and the five validation gates. The convergence trace, the per-vehicle curve parameters, and the average ROI before and after are logged.
7. Build deliverables. The detailed media analysis, the weekly plan, the summary and the report.
8. Register outputs. The files are stored and linked to the run.
9. Status and results. The run moves from pending to completed or failed, and the outputs can be read.

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Guarantees

Guarantee	What it means
Fails loudly	If no fitted model exists, or the model has no response curves, the run stops with a clear message. The optimizer never invents curves from raw data.
Pinned where a fit is chosen	A Media Workspace scenario runs against the project's primary fit, and a cross-brand run against each brand's pinned fit. A newer fit appearing later never silently replaces it.
Deterministic	There is no sampling. The same fit, request and data produce the same allocation.
Nothing silent	Requests that had to be clamped, locks that were applied, assumptions used, and solver status are all written to the summary.
Best-effort report	If the PDF cannot be rendered, the tables, the weekly plan and the summary are still delivered.

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Run guards

A headless run is refused, with a clear message, when:

Guard	Why
No primary model is set for the project	There is no approved fit to optimize against
The primary model has no channel response data	The optimizer would otherwise fall back to a different model, so it refuses instead. A smoke run, or a model with a vehicle that failed the conversion gates, has no registered response curves, so it is refused here
The scenario does not belong to the project, or you cannot see it	Only the author, or a shared reader, may run a scenario
The scenario is already running	A second run would overwrite the first run's result

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Scenarios and staleness

A scenario stores the objective, budget change and constraints. In a project, it runs against the project's primary fit. If the primary model is later changed, a scenario whose last result came from a different fit is marked stale, so you can see that its numbers no longer reflect the current primary model, and re-run it deliberately.

Repeated runs of an unchanged scenario against an unchanged fit give identical numbers. A newer fit never replaces a pinned one automatically.

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Outputs

The outputs are the same as the interactive path:

Output	Content
Detailed media analysis	Portfolio, group or brand, and vehicle rows with current vs optimized spend, volume, revenue, profit, ROI and cost per unit
Weekly flighting plan	Each vehicle's annual optimum phased by its own history and attribution
Optimization summary	Solver status, validation gates, adjustments, macro assumptions used, model used, target year
Report (PDF)	Business-readable summary and weekly budget charts

Scenario comparisons surface the ROI change of the All Media row, not the profit change: a spend-cutting scenario can show a profit drop while still improving ROI, so per-dollar efficiency is the fairer comparison.

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Typical use cases

Use case	How headless helps
Compare scenarios	Run several saved scenarios against the same primary fit and compare their ROI change
Repeatable planning cycles	Re-run a saved scenario after the fit is re-approved, and see that it is flagged stale until you do
Agent-driven planning	Let an agent run the optimization and write the result into a report
Portfolio planning	Run a cross-brand scenario with one pinned fit per brand

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When a run does not complete

Symptom	Likely cause	What to do
Refused at start	A run guard failed	Read the message; it names the missing model, the permission, or the running scenario
Stops with an infeasible-budget error	Locks and limits leave no room for the budget	Reduce a pinned vehicle's change, unlock a vehicle, or change the budget
Completes but a validation gate fails	Solver or constraint issue in the solution	Read the summary; do not use the allocation until it passes
Completes with adjustments	Some requests were clamped	Read the adjustments for the requested and applied values
No PDF	The report render failed	Use the tables and summary; they are delivered regardless

For the formulas behind the solve, see mmm-optimizer-methodology.