You are the pyrotechnics manager for DS City's New Year's Eve celebration. You have a limited budget to purchase exactly 12 premium firework display components. The audience satisfaction from each component is partially based on a factor you can't control until the night itself: the crowd's energy level.
- Low Energy (L): The crowd is tired, maybe cold, and needs a quick, high-impact component to wake them up. Favors Mines and Comets. This crowd is easiest to impress if you can distract them from their cold hands and feet.
- Mellow Energy (M): The crowd is relaxed and enjoying the atmosphere. Favors beautiful, slow-moving, artistic components. Favors Peony and Willows. This crowd is hard to excite but can be satisfied.
- High Energy (H): The crowd is loud, excited, and ready to party. Favors big, bold, continuous components. Favors Willows and Chrysanthemum.
The costs and satisfaction points for each batch are given below. The costs and satisfaction points are additive. For example, if you bought 2 Comet batches and one Roman batch, you would pay $13,000 and achieve 29 satisfaction points with a low energy crowd, but only 5 satisfaction points with a high energy crowd.
Some constraints were set by the city:
- You may use at most 2 batches from each type of component.
- You have to use at least three aerial and three ground components.
- You have to use at least 6 different types of components.
- Your total budget is $65,000.
- You have to use exactly 12 batches so that the timing with the music will work out.
From experience you know that a total satisfaction level under 43 will hurt your business going forward. You estimate a penalty of $ -5,000 if you underwhelm the crowd.
On the other hand, for every satisfaction point over 50 you estimate an additional $1,000 in extra business coming your way. No change in business is expected if you achieve satisfaction between 43 and 50.
What batches will you purchase?
Whatever selection you make, you will have three potential outcomes. How will you measure the quality of your selection given that it is best described by three different numbers?
Seeker's proprietary multi-objective optimizer suggests this solution: 2 Comet batches, 1 Peony batch, 2 Chrysanthemum batches, no Willows, 2 Mine batches, 1 Roman batch, 2 Crossette batches and 2 Salute batches. This will cost $65,000 and bring estimated $28,000 (78 satisfaction points for a low energy crowd), $0 (45 SPs mid energy), or $9,000 (59 SPs high energy) in extra future business.
Other alternatives that Seeker deems inferior:
- Optimize for each energy level separately. If we evaluate the future business for each such firework as a vector of three numbers (low, mid, and high future business in $1,000), we get: [36, -5, 2] (optimized for low energy), [27, 0, 6] (optimized for mid energy), or [21, 0, 12] (optimized for high energy). Now we have three solutions that each work for one scenario but not so well for the others. We need to find a compromise that will work reasonably for all scenarios.
- A typical way of getting a solution that fares reasonably against all futures is the max min profit solution. If we build that, we may end up with a selection that results in [24, 0, 10], whereby the $0 in (no) extra future business for the mid-energy crowd determines our proxy min profit objective that we sought to maximize. Obviously, that is a bad idea here, as the mid energy crowd just cannot be pleased.
- Based on the fireworks optimized for each individual scenario, we see that the low energy crowd is the one we can excite the most, with $36,000 of potential new business for us. The mid-energy crowd cannot be pleased, though, but we can and should satisfy them so as not to ruin our future business prospects. We may try a min max regret strategy and aim to minimize the potential we leave on the table for each crowd. If we do so, we can select batches that will result in [30, -5, 6], with a maximum regret of $6,000 (for low and high energy crowds) and a regret of $5,000 for the mid-level crowd.
The table below shows an overview of all potential outcomes for all solutions. We highlight in different colors how each outcome ranks among these solutions for each scenario. We observe that only the Max-Min Profit and the Seeker Multi-Objective solutions hold up under all scenarios. Comparing those two, Seeker trades $1,000 in future business should we get a high-energy crowd for $4,000 more in additional business if we get a low energy crowd. A well-justified trade.
This concludes our puzzle series this year. We hope you enjoyed them!
The team at InsideOpt wishes you all a happy new year. May 2026 keep you healthy and bring you much happiness and success.