Where the Budget Goes, Explained with Examples

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Where the Budget Goes

Quantitative Analysis · Chapter 11: Linear Programming ApplicationsSection 11.1 · lesson 48 of 5512 min · free with a free account

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In this lesson

Applying LP to audience targeting and marketing research allocation.

  • Formulate a media-selection LP with reach, budget and demographic constraints
  • Build a marketing-research LP that allocates sample size across segments
  • Set up a customer-segment targeting model with expected revenue as the objective
  • Interpret the optimal solution and shadow prices as managerial guidance

Transcript

Everything said in the video, section by section (1,706 words). Read along, or search it.

Linear Programming · Section 11.1

Where the Budget Goes

Welcome to Chapter eleven point one of Quantitative Analysis. Where the Budget Goes. This section is about the oldest question in marketing. You have a fixed pot of money, and far more places to spend it than you can afford. Linear programming turns that complaint into a question with an actual answer. By the end you will be able to formulate a media selection model with reach and budget constraints. You will build a research model that allocates sample size across market segments. You will set up a customer targeting model with expected revenue as the objective. And you will read shadow prices as managerial advice, not as algebra.

Common Template

One Structure, Three Applications

Media choice, research design and audience targeting look like three different problems. Structurally, they are the same problem. A finite pool of money or attention is split across competing alternatives. Each alternative carries its own cost, and its own payoff. Every one of these models has exactly three parts. Decision variables, which say how much of each option you buy. An objective function, which scores a whole plan with a single number. And constraints, which describe what is actually allowed. The three classic applications differ only in what those parts stand for. Underneath all of them sits one question. Where does the next dollar do the most good?

Media Selection

Insertions, Reach and a Quality Weight

Start with media selection. The question is how many insertions to buy in each available channel. Television spots, print advertisements, social media campaigns, radio commercials. We want total audience reach as large as possible, without exceeding the budget. So the decision variable for each channel counts the insertions placed there. It is non-negative and, in strict practice, it ought to be a whole number. Next, the reach coefficient. The audience delivered by a single insertion in that channel. And then a quality weight, which is where the model gets interesting. The weight says how closely a channel's audience matches the people you actually want. The objective maximises the sum of quality, times reach, times insertions. Set every weight to one and you are simply maximising raw reach. Give digital one point two and radio zero point seven, and you reward the right eyeballs.

Four Constraints

Budget, Floors, Ceilings, Coverage

Four kinds of constraint appear in almost every media selection model. First, the budget. Cost per insertion times insertions, summed over channels, cannot exceed the budget. Second, minimum insertions. Some channels carry contractual or strategic floors. We must run at least five prime time television spots, for instance. Third, maximum insertions. These ceilings encode diminishing returns. Above some threshold, extra insertions in one channel keep reaching the same people. Fourth, demographic coverage. Impressions delivered to a segment, summed over channels, must clear a required minimum. Two assumptions are quietly doing all the work here. Every term is a constant multiplied by a decision variable. That is linearity. And every variable is non-negative. Drop either one, and this stops being a linear program.

Reach Vs Frequency

Floors Buy Recall, Ceilings Prevent Waste

Now a warning about what the objective does not capture. The linear program maximises total reach. Seasoned media planners want a balance between reach and frequency. Reach is how many distinct people see the advertisement. Frequency is how many times each of them sees it. A high reach, low frequency plan introduces the product to many people once. A low reach, high frequency plan engages a much smaller group deeply. The solver has no opinion about that trade off, so planners impose it through the bounds. A lower bound keeps a channel in the plan often enough to build recall. Without it, the model would happily drop that channel altogether. The ceilings do the opposite job, preventing over frequency in one place. Reach saturates as insertions rise. Each new one buys less fresh audience than the last.

Research Budgets

How Many Respondents, From Which Segment

The second model moves from buying attention to buying information. A research team designing a survey has to decide who to recruit. How many respondents should come from each market segment? Larger samples give more reliable estimates, and they cost more to field. The decision variable is the sample size from each segment, or the number of interviews under each methodology. In person interviews, online panels, telephone surveys, focus groups. Two formulations are common, and they are equivalent. Minimise total cost, subject to statistical power requirements. Or fix the budget, and maximise a weighted sum of sample sizes. Three constraints then do the real work. The budget, of course. A minimum sample per segment, often thirty, for normal approximation inference. And interview duration capacity, because fieldwork hours are finite too.

Segment Targeting

Buying Expected Revenue, Not Impressions

The third model targets audiences rather than channels. Marketing spend is split across customer segments: loyal high value customers, lapsed customers, and prospects. The goal is to maximise total expected revenue. The decision variable is the number of dollars directed at each segment. Each dollar earns the segment's response rate, times the revenue per responder. That product is then scaled by an efficiency factor. The factor captures diminishing returns on incremental spend. Constraints include the total budget, and channel capacity. Call centre capacity and email send volume limits are real ceilings. And brand image floors keep loyal customers in the plan regardless of marginal economics. That protects retention, and it protects the brand.

Three Models

Same Template, Different Fillings

It is worth seeing the three models side by side. Media selection. The variable is insertions per channel, and the objective is weighted reach. Its constraints are the budget, per channel floors and ceilings, and demographics. Marketing research. The variable is sample size per segment. Its objective is minimum cost, or maximum statistical power. Its constraints are the budget, minimum sample, and interview hours. Customer targeting. The variable is spend per segment, maximising expected revenue. Its constraints are the budget, channel capacity, and coverage floors. A budget constraint appears in every single one of them. What changes is the currency of the objective. Impressions, statistical power, or dollars.

Worked Example

Splitting a $500,000 Media Budget

Time to make all of this concrete. A consumer goods company is launching a new product, with a media budget of five hundred thousand dollars. Four channels are on the table. Television, digital, print and radio. Television costs ten thousand dollars an insertion, and reaches zero point two zero million people. Its quality weight is zero point nine, and it must run between five and twenty five times. Digital costs two thousand dollars, and reaches zero point three zero million. Its weight is one point two, with bounds of twenty and two hundred. Print costs five thousand dollars, and reaches zero point zero eight million. Its weight is zero point eight, between four and thirty insertions. Radio costs one thousand five hundred dollars, and reaches zero point one two million. Its weight is zero point seven, between ten and eighty insertions. Television delivers the most audience per insertion, radio the cheapest. Neither fact decides anything on its own.

Formulation

The LP, Written Out

Now write the linear program out, with four variables counting insertions in television, digital, print and radio. Each objective coefficient is the quality weight multiplied by the reach. Television. Zero point nine times zero point two zero gives zero point one eight. Digital. One point two times zero point three zero gives zero point three six. Print. Zero point eight times zero point zero eight gives zero point zero six four. Radio. Zero point seven times zero point one two gives zero point zero eight four. Those coefficients are in millions of weighted impressions per insertion. The budget row multiplies each variable by its cost, and caps the total at five hundred thousand. The bounds then carry the floors and ceilings straight from the table. One detail is worth pausing on before we solve it. Holding every channel at its floor already consumes one hundred and twenty five thousand dollars. So only three hundred and seventy five thousand dollars is genuinely free to allocate.

Optimal Solution

Fill the Best Ratio First

With one budget constraint and simple bounds, you can follow the simplex method by hand. Rank the channels by weighted reach per thousand dollars. Digital is far ahead at zero point one eight. Radio is second, at zero point zero five six. Television is third at zero point zero one eight, and print is last. Start every channel at its floor, then pour the free budget into the best ratio. Digital absorbs money until it reaches its ceiling of two hundred insertions. That costs four hundred thousand dollars, and returns seventy two million weighted impressions. Fifteen thousand dollars are left, and they go to radio, the next best ratio. Radio rises from its floor of ten, to twenty insertions. Television and print stay pinned at their floors. Five insertions, and four. The plan spends exactly five hundred thousand dollars, and delivers seventy four point eight three six million weighted impressions. Now a caution, because the chapter prints a different allocation. Twenty, one hundred, ten and thirty three insertions is feasible, and spends the whole budget. But it returns only forty three million weighted impressions, well short of seventy four point eight. Its printed total of thirty eight point eight is unweighted reach, a different unit entirely.

Shadow Prices

Turning an Optimum Into Management Language

The last step is turning that answer into something a manager can act on, and shadow prices do exactly that. The budget shadow price is the value of one more dollar. Here it is zero point zero five six million weighted impressions per extra thousand dollars. Notice that it is not digital's ratio of zero point one eight. Digital is already capped, so the marginal dollar goes to radio instead. That is the counter intuitive part. A shadow price belongs to the plan, not to the best channel. The chapter quotes zero point two zero, but that beats every channel's ratio, so no dollar can buy it. The floors have shadow prices too, and theirs are negative. Raising the print minimum by one insertion costs zero point two one six million weighted impressions. Raising the television minimum by one costs zero point three eight zero million. That is the price of a contract clause, and it is quotable in next year's negotiation. Reach then grows linearly with the budget, until another bound binds. At five hundred and ninety thousand dollars, radio hits its ceiling and the slope flattens. That is what shadow prices are for. The worth of another dollar, and the cost of every promise you made.

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FAQ

Questions students ask us

About Where the Budget Goes

How do you formulate a media selection linear program?
Define one variable per channel counting insertions, such as TV spots or digital ads. Maximize weighted reach: the sum of each channel's quality weight times its audience reach per insertion times its insertions. Constrain total cost to the budget, add minimum insertions for contractual or recall reasons, maximum insertions to reflect diminishing returns, and demographic coverage targets. Every term is a constant times a variable, and all variables are non-negative.
How is linear programming used in marketing research survey design?
The decision variables are sample sizes for each market segment or interview method, such as online panels, telephone surveys or in-person interviews. One formulation minimizes total cost subject to statistical requirements; an equivalent one fixes the budget and maximizes a weighted sum of sample sizes. Typical constraints include the budget, a minimum sample per segment, often 30 for normal-approximation inference, and limits on available interviewing hours.
How do you allocate a marketing budget across customer segments with linear programming?
Let each variable be the dollars spent on a segment, such as loyal customers, lapsed customers or prospects. Each dollar earns the segment's response rate times revenue per responder, scaled by an efficiency factor for diminishing returns, and the objective maximizes total expected revenue. Constraints cover the total budget, channel capacities such as call center or email volume, and minimum spending on loyal customers to protect retention and the brand.
What does the budget shadow price mean in a media allocation model?
It is the extra weighted reach from one more unit of budget, and it equals the return of the channel the marginal money actually buys, not the best channel. In one $500,000 plan, digital earns 0.18 million weighted impressions per $1,000 but is already at its ceiling, so extra money goes to radio at 0.056. The shadow price is therefore 0.056 million per $1,000 until radio also reaches its cap.

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