Optimization: Placement

Purpose

This calculator assigns entities to locations under per-location capacity while balancing score and cost.

It answers a question that comes up whenever options differ in both quality and price: which entity should go where so you get the best overall outcome, without pushing any location past its capacity?

Background

Problem Domain

Placement optimization is a binary assignment with either score-first or cost-first orientation, using a blended score-cost objective. This mirrors a common real-world dilemma — the best-performing option is rarely the cheapest one — and lets you dial in exactly how much extra performance is worth through score_weight.

Real-World Uses

• VM/container placement: Each workload can run on multiple hosts with different performance scores and operating costs. Use the model to assign workloads within host capacity while optimizing score-cost tradeoff.
• Warehouse slotting: Products can be placed in alternative locations with different handling efficiency and cost. Use the model to choose the best location per entity under slot capacity limits.
• Team/site assignment: Teams can be deployed across sites with varying benefit and expense. Use the model to enforce one-site-per-team placement under the selected objective mode.
• Asset placement decisions: High-performing locations can also be more expensive. Use the model as a transparent score-vs-cost framework when assigning assets to limited-capacity locations.

Inputs

• placement_object: Input table with columns Entity, Location, Score, and Cost.
• objective_mode: Optimization orientation. Use maximize_score to maximize blended utility or minimize_cost to minimize the cost-oriented variant of the same blended expression.
• location_capacity: Maximum entities allowed per location.
• score_weight: Weight applied to Score in the blended objective.
• show_zero: Controls output display. Turn on to include non-selected entity-location rows; turn off to show only selected placements.

Results

• Summary: Overall placement quality with objective mode, location capacity, total selected score, and total cost. This shows the headline tradeoff the optimizer achieved.
• Decision Table: Entity-location assignment output (Selected, Score, Cost) so you can directly see chosen placements and rejected alternatives.
• Constraint Slack: Slack for one-location-per-entity and location-capacity constraints, indicating where capacity limits are binding.

Understanding the Calculation

Binary variable $x_{e,l}$ indicates whether entity $e$ is placed at location $l$.

Blended expression per pair is $(w \cdot \text{Score}_{e,l} - \text{Cost}_{e,l})$.

• In maximize_score, this expression is maximized.
• In minimize_cost, its negation is minimized.

Constraints:

• Each entity assigned exactly once.
• Per-location selected entities do not exceed location_capacity.

Example

Default run (maximize_score, capacity 2) returns Optimal objective 11, with assignments like:

• E1 -> L1
• E2 -> L2
• E3 -> L2

Interpretation: this arrangement gives best blended score-cost outcome under location caps. Try raising score_weight and re-running — entities may shift toward higher-score locations even at higher cost, revealing how sensitive the plan is to how much performance is worth to you.

Important Assumptions and Limitations

• Score and cost are linearly combined.
• Same location capacity applies to every location.
• No affinity/anti-affinity or compatibility constraints beyond table rows.