Algorithmic Maximisation#

Week 10 — The ABV-to-Dollar Index#

Where we left off#

• Week 9: a scraped, normalised dataset of 100+ SKUs
• name, volume, ABV, and price, cleaned of the failure modes covered
• a dataset is not yet an answer — it is only rankable once it is scored
• this week: the formula that does the scoring, and how to maximise it

Learning objectives#

By the end of this lecture you should be able to:

• define the ABV-to-dollar index from a normalised SKU record
• rank a dataset by index with a naive greedy approach
• explain why a budget cap breaks the greedy approach’s optimality
• apply a dynamic-programming formulation under budget and variety constraints

Defining the index#

For a single SKU:

standard_drinks = (volume_ml × abv_percent × 0.789) / 1000
index = standard_drinks / price_dollars

Every ratio this course has discussed qualitatively since week 2 is this one number, computed per SKU, from fields week 9’s scrape already extracted.

The naive approach: greedy ranking#

• sort every SKU in the dataset by index, descending
• for an unconstrained comparison — “which single product is best value” — this is not just adequate, it is the correct answer
• most of weeks 3 through 8’s case studies are, in effect, one-row greedy comparisons done by hand

Where greedy fails#

Add a fixed budget and a minimum-variety requirement — the shape of both the scraping exercise’s ranking task and the closing event portfolio — and greedy selection stops being optimal:

Greedy overspends on the single highest-index SKU and has nothing left to satisfy the variety constraint efficiently.

The dynamic-programming formulation#

• state: (SKUs considered so far, budget remaining, distinct types so far)
• transition: include the next SKU, or skip it
• objective: maximise total standard drinks across included SKUs
• constraints: total price ≤ budget; distinct types ≥ minimum variety

This is a bounded knapsack problem with an added distinctness constraint, not a novel algorithm — the reading’s contribution is showing that the retail case genuinely needs it.

Common errors#

• ranking by index without checking whether a budget constraint applies
• forgetting that standard_drinks must use week 2’s conversion exactly, or every downstream ranking is silently wrong
• treating “minimum variety” as a soft preference rather than a hard constraint in the state space
• optimising for total standard drinks while ignoring week 11’s caution that the objective function itself has a cost the model doesn’t price

Where this connects#

• the data scraping exercise’s ranking step uses the unconstrained greedy version directly against week 9’s dataset
• the event portfolio, due at the semester’s close, is the fully constrained version — fixed budget, minimum variety, and a guest count that turns this week’s per-shopper formulation into next week’s harder, event-scale one

Next week#

Ethical implications, and why an index that is mathematically optimal is not, on its own, a recommendation.