Assignment Problems

Overview

Assignment problems represent a fundamental class of combinatorial optimization where a set of agents (workers, machines, facilities) must be matched to a set of tasks (jobs, locations, resources) in a one-to-one manner that optimizes some measure of total cost or benefit. The mathematical elegance of these problems lies in their formulation as permutation optimization: finding the best way to reorder one set relative to another.

First studied systematically in the 1950s in operations research contexts, assignment problems have become ubiquitous in logistics, scheduling, resource allocation, and matching markets. The classic example involves assigning n workers to n jobs, where each worker has a different cost for performing each job, and the goal is to minimize total cost while ensuring each worker gets exactly one job and each job is performed by exactly one worker.

Linear Assignment Problems

The linear assignment problem (LAP) is defined by a cost matrix C where C_{ij} represents the cost of assigning agent i to task j. The objective is to find a permutation \pi that minimizes:

\min_{\pi} \sum_{i=1}^n C_{i,\pi(i)}

The problem can be formulated as an integer linear program, but its special structure allows polynomial-time solution via the Hungarian algorithm (also called the Kuhn-Munkres algorithm), which runs in O(n^3) time. The algorithm works by iteratively transforming the cost matrix through row and column reductions, identifying augmenting paths in a bipartite graph representation, and updating a partial assignment until optimality is reached.

The LINEAR_ASSIGNMENT function wraps scipy.optimize.linear_sum_assignment, which implements a modern variant of the Hungarian algorithm. Applications include:

  • Task scheduling: Assigning workers to shifts or machines to jobs
  • Transportation logistics: Matching delivery vehicles to routes
  • Cloud computing: Allocating computational tasks to servers
  • Bioinformatics: Aligning protein sequences or matching genetic markers

LAP can handle rectangular matrices (unequal numbers of agents and tasks) by conceptually padding the matrix with dummy elements at zero cost.

Quadratic Assignment Problems

The quadratic assignment problem (QAP) extends LAP by incorporating interaction effects between assignments. Instead of a single cost matrix, QAP uses two matrices: a flow matrix F (describing interactions between facilities) and a distance matrix D (describing separation between locations). The objective becomes:

\min_{\pi} \sum_{i=1}^n \sum_{j=1}^n F_{ij} \cdot D_{\pi(i),\pi(j)}

This formulation captures scenarios where the cost of assigning facility i to location \pi(i) depends on where other facilities are placed. For example, if two factories have high material flow between them, placing them far apart incurs additional transportation cost.

Unlike LAP, QAP is NP-hard, meaning no polynomial-time algorithm is known for large instances. State-of-the-art solvers use branch-and-bound methods with linear assignment relaxations as lower bounds, combined with heuristics like simulated annealing or tabu search for finding good feasible solutions.

The QUAD_ASSIGN function leverages scipy.optimize.quadratic_assignment, which implements the FAQAP (Fast Approximate QAP) algorithm. Classic applications include:

  • Facility layout planning: Minimizing material handling costs in manufacturing plants
  • Circuit board design: Placing electronic components to minimize wire length
  • Campus planning: Positioning academic departments to reduce travel between collaborating groups
  • Keyboard layout optimization: Arranging keys to minimize typing effort (e.g., Dvorak layout)

Problem Variants and Extensions

Beyond the standard forms, several important variants exist:

  • Bottleneck assignment: Minimize the maximum cost of any individual assignment (rather than total cost)
  • Multi-dimensional assignment: Assign multiple types of resources simultaneously (e.g., workers and tools to jobs)
  • Dynamic assignment: Assignments evolve over time as new tasks arrive
  • Stochastic assignment: Costs are random variables rather than deterministic values

Native Excel capabilities

Excel does not have dedicated built-in functions for assignment problems. However, two native tools can address small-scale instances:

  • Solver Add-in: Can solve LAP by setting up binary decision variables x_{ij} \in \{0,1\} (1 if agent i is assigned to task j, 0 otherwise) with row-sum and column-sum constraints equal to 1. For small problems (up to ~50×50), Solver’s Simplex LP engine handles this well. However, the standard Excel Solver is limited to 200 decision variables, which constrains problem size.

    For QAP, Solver can formulate the problem but struggles with even modest sizes (e.g., 10×10) because the number of variables and constraints grows quadratically. The nonlinearity and combinatorial structure make it computationally intensive.

  • Goal Seek: Not applicable; Goal Seek is for single-variable root finding, not combinatorial optimization.

Critical limitations: Excel Solver lacks specialized algorithms like the Hungarian method for LAP, meaning it treats LAP as a generic integer program (much slower). For QAP, Solver’s GRG Nonlinear or Evolutionary engines are not competitive with dedicated QAP solvers that exploit problem structure.

Third-party Excel add-ins

  • OpenSolver: An open-source add-in that interfaces Excel with industrial-strength solvers from COIN-OR (CBC for integer programming). It can handle larger LAP instances (hundreds of agents/tasks) by formulating LAP as an integer program. While CBC doesn’t implement the Hungarian algorithm directly, it’s far more efficient than Excel’s native Solver for large LPs and MILPs. OpenSolver does not offer specialized QAP algorithms.

  • Frontline Solvers: Commercial upgrade to Excel Solver. The “Premium Solver Platform” supports larger models (up to 2,000 decision variables in standard edition, unlimited in Pro edition). It includes both the Simplex LP engine (for LAP formulations) and metaheuristic engines (Evolutionary, OptQuest) that can tackle small QAP instances via genetic algorithms or scatter search. Still not as efficient as specialized QAP implementations.

  • FICO Xpress Insight: Enterprise-grade optimization with an Excel interface. Xpress can solve large-scale assignment problems efficiently, including variants like bottleneck assignment or multi-period extensions. Primarily used in industry for supply chain and operations planning.

For serious QAP applications (e.g., facility layout with 20+ facilities), dedicated software like QAPLIB benchmark solvers or commercial tools (CPLEX, Gurobi with quadratic programming extensions) are typically required, though Python-based solutions via QUAD_ASSIGN offer a practical middle ground.

Tools

Tool Description
LINEAR_ASSIGNMENT Solve the linear assignment problem using scipy.optimize.linear_sum_assignment.
QUAD_ASSIGN Solve a quadratic assignment problem using SciPy’s implementation.