Promotional Planning

Lift modeling · Budget allocation

A category manager must choose which SKUs to promote, at what discount depth, in which week, under a fixed promotional budget — while accounting for cross-product cannibalisation and category halo. The exact problem is a 0-1 knapsack on (SKU × week × depth) triples with a budget constraint and concurrent-promo limits. Foundational papers: Blattberg, Briesch & Fox (1995); Cooper et al. (1999) PromoCast; van Heerde, Leeflang & Wittink (2004); Macé & Neslin (2004).

Why it matters

Scale of promotion in retail · documented optimization lift

~30%

Of US grocery sales occur on promotion — promo planning is a structural revenue mechanism, not an exception.

Source: Blattberg, Briesch & Fox (1995), Marketing Science.

+5–12%

Documented promo-margin lift from optimization vs. rule-of-thumb planning at a major US retailer.

Source: Cooper et al. (1999) PromoCast, Marketing Science 18(3).

20–50%

Of promo lift is cannibalisation (other-SKU substitution); ~10–15% is post-promo dip (forward-buying); only ~30–70% is incremental.

van Heerde, Leeflang & Wittink (2004); Macé & Neslin (2004).

5–15%

Promo budgets as a share of category revenue — top-of-mind lever for category managers.

Phillips (2005), Pricing & Revenue Optimization.

Where the decision sits

Trade-promotion calendar · weekly ad · loyalty offer planning

Promotional planning sits between forecasting and execution. Forecasts give base demand; lift functions quantify how that demand responds to discount depth; the planner decides the SKU/week/depth grid; downstream execution sets prices in POS systems and triggers replenishment. The hard part is not depth alone but the portfolio: too many concurrent promotions cannibalise each other, blow the budget, and shift demand forward without growing the category. End-of-life pricing on promoted SKUs handed off to markdown downstream.

Forecast basedemand without promo

Estimate liftdepth-response curves

Choose SKU/week/depthbudget-constrained

Execute & measureread incremental lift

Problem & formulation

Multi-period, multi-SKU 0-1 knapsack with budget and page constraints

OR family

0-1 Integer Programming

Complexity

NP-hard (multi-knapsack)

Solver realism

★★ Greedy + budget

Reference

Cooper et al. (1999)

Sets and indices

Symbol	Meaning	Domain
$i \in N$	SKU in the category	discrete
$t \in T$	Week in the planning horizon	discrete
$d \in D$	Discount depth (e.g., 0%, 10%, 20%, 30%)	discrete, ordered

Parameters

Symbol	Meaning	Unit
$\text{base}_{i,t}$	Baseline weekly demand for SKU $i$ in week $t$	units / week
$p_i$	List price of SKU $i$	$ / unit
$m_i$	Unit margin of SKU $i$ at full price	$ / unit
$\alpha_i, \beta_i$	Lift coefficients for SKU $i$ (linear, quadratic in depth)	—
$\gamma_{i,j}$	Cannibalisation coefficient: substitution from $j$ when $i$ is promoted	$\in [0,1]$
$B$	Total promotional budget for the horizon	$
$M_t$	Maximum number of SKUs promoted in week $t$ (page constraint)	integer

Decision variable

Symbol	Meaning	Domain
$x_{i,t,d}$	1 if SKU $i$ is promoted at depth $d$ in week $t$, else 0	$\in \{0, 1\}$

Lift function (own-SKU response)

A standard parametric form is concave in depth above a threshold — doubling the depth less than doubles the unit lift:

$$\text{lift}_{i,t}(d) \;=\; \text{base}_{i,t} \cdot \bigl(\alpha_i \, d \,+\, \beta_i \, d^{2}\bigr)$$

With $\alpha_i > 0$ and $\beta_i < 0$ the curve is concave. Calibrated from historic promo-vs-non-promo weeks.

Cross-product cannibalisation

A fraction of SKU $i$’s lift is stolen from substitute SKU $j$:

$$\text{cannibal}_{i,j,t}(d) \;=\; -\,\gamma_{i,j} \cdot \text{lift}_{i,t}(d) \quad \text{for substitutes } j \neq i$$

Net incremental margin

Margin earned from incremental lift on SKU $i$ minus margin lost to substitutes:

$$\text{incremental}_{i,t,d} \;=\; (m_i - d \cdot p_i) \cdot \text{lift}_{i,t}(d) \;-\; \sum_{j \neq i} \gamma_{i,j} \cdot m_j \cdot \text{lift}_{i,t}(d)$$

Objective

$$\max \;\; \sum_{i \in N} \sum_{t \in T} \sum_{d \in D} \text{incremental}_{i,t,d} \cdot x_{i,t,d}$$

Constraints

One depth per SKU per week; total promo cost within budget; at most $M_t$ SKUs promoted per week (circular page / display capacity):

$$\sum_{d \in D} x_{i,t,d} \;\leq\; 1, \quad \sum_{i,t,d} d \cdot p_i \cdot \text{lift}_{i,t}(d) \cdot x_{i,t,d} \;\leq\; B, \quad \sum_{i,\, d > 0} x_{i,t,d} \;\leq\; M_t$$

Real-world → OR mapping

How a category manager’s vocabulary translates to the IP

On the planning desk	In the IP
Category SKU range	$N$
Promotional calendar (weeks ahead)	$T$
Available discount tiers (15% off, 30% off, BOGO)	$D$
Baseline forecast at full price	$\text{base}_{i,t}$
Depth-response curve from past promos	$\alpha_i, \beta_i$
Substitution between SKUs in same need-state	$\gamma_{i,j}$
Trade-promotion budget	$B$
Slots on weekly circular / endcap count	$M_t$

Interactive solver

Greedy by margin-per-budget-dollar with budget & page checks; re-solves in browser

Promo Planner

6 SKUs · 4 weeks · 4 depth levels · budget + concurrent caps

★★ Greedy + budget

Budget $B$ ($)

Total promo dollars

Max SKUs / week $M_t$

Page / endcap capacity

Cannibalisation $\gamma$

Avg. substitute steal rate

Margin / price (avg.)

Used to derive $m_i = r \cdot p_i$

SKU	Price ($)	Base demand (units/wk)	Lift α

—

Incremental margin ($)

—

Lift vs uniform baseline

—

SKU-weeks promoted

—

Budget used ($)

—

Average depth chosen

—

Cannibalisation share

15% depth 30% depth 50% depth No promo

Under the hood

We enumerate all (SKU, week, depth>0) triples and compute their net incremental margin and promo-dollar cost. Triples are sorted by margin per dollar and inserted greedily, skipping any that would violate the SKU-per-week constraint, the per-week SKU cap $M_t$, or push the cumulative cost above $B$. Cannibalisation is approximated as a flat $\gamma$ of the chosen SKU’s own lift, multiplied by the average margin of substitutes. The baseline is a uniform-rotation policy that promotes $M_t$ SKUs per week at the cheapest depth that fits the budget. Runtime is $\mathcal{O}(|N| \cdot |T| \cdot |D| \log(\cdot))$ — sub-millisecond at this scale.

Reading the solution

What a category manager actually does with the optimal plan

Three patterns to watch for

High-base SKUs at shallow depth. Heavy sellers maximise margin-per-dollar at 15-20% off — the curve is steep enough that you don’t need to discount deeper. Deep depths are reserved for SKUs that genuinely need the depth to move.
Concentration on low-cannibalisation SKUs. When $\gamma$ is high, the planner avoids promoting close substitutes in the same week — promoting one already steals from the others. This shows up as a sparser plan when $\gamma$ rises.
Budget binds before page capacity. In small-budget runs the plan stops well before $M_t$ slots are full each week. Raising $B$ by 50% typically buys 70-80% more incremental margin (roughly linear) until page constraints bind.

Sensitivity questions the model answers instantly

Add a 50% depth tier? — rarely chosen unless lift coefficients are unusually steep; flags overly aggressive depth menus.
Cut budget by 30%? — concentrates the plan on the most efficient (high-base, low-cannibalisation) SKUs first.
Raise $\gamma$ from 0.20 to 0.40? — incremental margin drops sharply; planner spreads promos across weeks rather than across SKUs.

Model extensions

From single-category baseline to retail-OR variants that matter

Stochastic baseline demand

Replace deterministic $\text{base}_{i,t}$ with a forecast distribution; objective becomes expected incremental margin under chance constraints on stockouts.

Dynamic budget rebalancing

Re-optimise the remaining budget mid-season after observing promo response; rolling-horizon implementation common in retail planning systems.

Joint markdown + promo

End-of-life promo decisions interact with seasonal markdown; jointly decide both to avoid double-discounting and inventory mismatch.

Markdown →

Vendor-funded promo

Trade allowances reduce effective cost of depth from the retailer’s side; objective shifts toward maximising trade-fund pull-through.

Category cross-effects

Halo (lift in adjacent categories) and pantry-loading effects cross category boundaries; requires cross-category lift coefficients.

Omnichannel promo

Online and in-store have different lift coefficients and budget pools; channel-specific decision variables and budget constraints.

Personalised promo

Customer-specific offers via loyalty programs; per-customer lift functions and budget per segment.

Personalised promo →

ML lift estimation

Uplift modelling with double-machine-learning or causal-forest estimators for $\alpha_i, \beta_i$ and $\gamma_{i,j}$ at scale.

Key references

Foundational promotion-response and planning literature

Blattberg, R. C., Briesch, R. & Fox, E. J. (1995).

How promotions work.

Marketing Science 14(3 supp): G122–G132. doi:10.1287/mksc.14.3.G122

Cooper, L. G., Baron, P., Levy, W., Swisher, M. & Gogos, P. (1999).

PromoCast: A new forecasting method for promotion planning.

Marketing Science 18(3): 301–316. doi:10.1287/mksc.18.3.301

van Heerde, H. J., Leeflang, P. S. H. & Wittink, D. R. (2004).

Decomposing the sales promotion bump with store data.

Marketing Science 23(3): 317–334. doi:10.1287/mksc.1040.0064

Macé, S. & Neslin, S. A. (2004).

The determinants of pre- and postpromotion dips in sales of frequently purchased goods.

Journal of Marketing Research 41(3): 339–350. doi:10.1509/jmkr.41.3.339.35992

Silva-Risso, J. M., Bucklin, R. E. & Morrison, D. G. (1999).

A decision support system for planning manufacturers’ sales promotion calendars.

Marketing Science 18(3): 274–300. doi:10.1287/mksc.18.3.274

Felgate, M., Fearne, A., DiFalco, S. & Garcia Martinez, M. (2012).

Using supermarket loyalty card data to analyse the impact of promotions.

International Journal of Market Research / Kent Business School working paper.

Pauwels, K. (2007).

How retailer and competitor decisions drive the long-term effectiveness of manufacturer promotions.

Journal of Retailing 83(3): 297–308.

Ailawadi, K. L., Gedenk, K., Lutzky, C. & Neslin, S. A. (2007).

Decomposition of the sales impact of promotion-induced stockpiling.

Journal of Marketing Research 44(3): 450–467. doi:10.1509/jmkr.44.3.450

Back to the retail domain

Promotional planning sits in the Promotion × Tactical cell of the 4P decision matrix — the lever that drives ~30% of grocery and ~25% of mass-merchant revenue.

Open Retail Landing

Educational solver · flat cannibalisation & quadratic-lift assumptions · calibrate $\alpha, \beta, \gamma$ on your own promo history before live planning.

Symbol	Meaning	Unit
\(\text{base}_{i,t}\)	Baseline weekly demand for SKU \(i\) in week \(t\)	units / week
\(p_i\)	List price of SKU \(i\)	$ / unit
\(m_i\)	Unit margin of SKU \(i\) at full price	$ / unit
\(\alpha_i, \beta_i\)	Lift coefficients for SKU \(i\) (linear, quadratic in depth)	—
\(\gamma_{i,j}\)	Cannibalisation coefficient: substitution from \(j\) when \(i\) is promoted	\(\in [0,1]\)
\(B\)	Total promotional budget for the horizon	$
\(M_t\)	Maximum number of SKUs promoted in week \(t\) (page constraint)	integer

Symbol	Meaning	Domain
\(i \in N\)	SKU in the category	discrete
\(t \in T\)	Week in the planning horizon	discrete
\(d \in D\)	Discount depth (e.g., 0%, 10%, 20%, 30%)	discrete, ordered

On the planning desk	In the IP
Category SKU range	\(N\)
Promotional calendar (weeks ahead)	\(T\)
Available discount tiers (15% off, 30% off, BOGO)	\(D\)
Baseline forecast at full price	\(\text{base}_{i,t}\)
Depth-response curve from past promos	\(\alpha_i, \beta_i\)
Substitution between SKUs in same need-state	\(\gamma_{i,j}\)
Trade-promotion budget	\(B\)
Slots on weekly circular / endcap count	\(M_t\)