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Scan for outdated or missing drivers - takes under a minuteDriver Scan →Clear out junk files and repair common Windows errorsFree Scan →To model a placement decision as a linear assignment problem, define a binary variable for each possible item–position pairing, minimize the sum of selected pairing costs, and require every item and every position to appear exactly once. This standard model fits only when placements are one-to-one and each pairing’s cost is independent of the other placements.
Define the items, positions, and costs
Let I be the set of items to place and J the set of available positions. For every feasible pair of item i and position j, define cij as the cost of assigning item i to position j. Use one consistent, meaningful measure—for example distance, time, or a penalty—and ensure that lower values really represent better outcomes if the model is minimizing.
Define the decision variable xij as 1 if item i is placed in position j, and 0 otherwise. The standard one-to-one formulation is:
Minimize ∑i∈I ∑j∈J cijxij
Subject to
- ∑j∈J xij = 1 for every item i ∈ I.
- ∑i∈I xij = 1 for every position j ∈ J.
- xij ∈ {0, 1} for every item–position pair.
The first constraint assigns each item to exactly one position. The second ensures each position is used by exactly one item. The binary domain makes each pairing a yes-or-no choice. This is the standard linear assignment formulation described in the scholarly treatment of the linear assignment problem.
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Check whether the placement decision fits
The basic model is appropriate when every item must be placed once, every position must be occupied once, and the total objective is the sum of independent item–position costs. A cost matrix captures the cost of each pairing; it does not capture how one assignment changes the value or feasibility of another.
- Pair costs depend on other placements: If the cost of placing one item changes depending on where another item is placed, an additive cost matrix is insufficient. A quadratic assignment formulation may be needed.
- Positions have capacity for multiple items: The one-item-per-position equality is too restrictive. Add capacity constraints and reassess the model class.
- Items consume a limited resource: Resource limits require additional constraints; the generalized assignment problem, for example, assigns each job once while limiting resource consumption on each agent.
- The goal is a score to maximize: Formulate a consistent maximization objective, or convert scores to costs only when the conversion preserves the intended ranking. Kuhn’s foundational paper describes the assignment problem as maximizing the sum of selected person–job scores (Kuhn, 1955).
Handle unequal set sizes and unavailable pairings
The displayed equalities require the item and position sets to have the same size: summing the item constraints requires one assignment per item, while summing the position constraints requires one assignment per position. If their sizes differ, first decide which side is allowed to remain unmatched and what that means in the real process.
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A rectangular assignment solver can be useful, but verify that its unmatched-item or unmatched-position behavior matches the requirement. SciPy documents its linear_sum_assignment interface for linear sum assignment. If both sides must be fully matched, dummy rows or columns can balance the formulation only when an unmatched choice has a deliberate interpretation and defensible penalty; otherwise, a dummy assignment can conceal an infeasible real problem.
For a pairing that is impossible, exclude it from the feasible choices or use the solver’s documented forbidden-pair mechanism. Then check that the remaining feasible pairings still permit a full assignment. Avoid arbitrary large penalties: unless their scale is chosen carefully, they can distort the optimum rather than reliably represent a forbidden choice.
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Build and validate the model
- List both sets. Specify every item and position, and define what one placement means in the actual process.
- Populate the pairing costs. Set cij for each allowed pair using the decision criterion the model is meant to optimize. Mark or exclude impossible pairs rather than treating them as ordinary choices.
- Create binary variables. Include xij only for feasible pairings, or apply the solver’s supported method for forbidden pairs.
- Add one constraint per item. Require the sum of that item’s assignment variables to equal 1.
- Add one constraint per position. Require the sum of variables assigned to that position to equal 1.
- Set the objective and domain. Minimize the sum of cijxij and enforce binary variables. The classic assignment formulation is commonly handled with specialized assignment algorithms.
- Verify the result independently. Confirm every item and every position occurs exactly once, and recompute the objective by summing the costs of the selected pairings.
Choose a solver for the assignment model
The Hungarian method is a classical method for the assignment problem. Kuhn’s 1955 paper introduced the score-maximization framing, seeking an assignment whose selected scores have the largest sum. A scholarly 2016 paper reports an O(n³) running-time bound for the classical Hungarian algorithm; this is an algorithmic complexity statement, not a runtime guarantee for a particular machine, implementation, or input.
For a Python implementation, SciPy provides scipy.optimize.linear_sum_assignment. Check the documentation for the installed SciPy version and confirm its input and output conventions before relying on it in production. The solver can choose an assignment from the supplied cost data, but it cannot repair a cost matrix or model that fails to represent the actual placement rules.
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