Optimization: Assignment

Purpose

This calculator assigns tasks to agents at minimum total cost while honoring agent capacity limits.

It is designed for one-to-one task coverage: each task must be assigned exactly once. In practice, it answers a concrete question: given everyone's cost or effort profile, who should be paired with which task so total cost stays as low as possible while nobody is overloaded?

Background

Problem Domain

Assignment optimization is a binary integer program where each decision is either assign (1) or do not assign (0). The classic one-to-one version of this problem can be solved in polynomial time by the Hungarian algorithm; this model generalizes it with per-agent capacity, so one agent can take on several tasks.

Real-World Uses

• Staff-to-ticket assignment: Workload is uneven and handling cost differs by staff member. Use the model to assign one owner per ticket at minimum total cost.
• Machine-to-work-order assignment: Each machine has limited capacity and different processing efficiency by job. Use the model to find a feasible, low-cost machine-job mapping.
• Field engineer dispatch: Several engineers can handle each call, but travel/time cost differs. Use the model to assign calls at lowest cost while keeping engineer load within capacity.
• Reviewer/proctor allocation: Every task must be covered, but reviewer bandwidth is finite. Use the model to guarantee full coverage at minimum assignment cost.

Inputs

• agents: Input table with columns Agent and Capacity. Capacity is the maximum number of tasks that each agent can take.
• tasks: Input table with column Task, listing all tasks to be assigned.
• assign_cost: Input table with columns Agent, Task, and Cost. Cost is the assignment cost for that agent-task pair. This table must contain the complete Agent-Task pair grid.
• show_zero: Controls output display. Turn on to include Assigned = 0 rows; turn off to show only selected assignments.

Results

• Summary: High-level optimization outcome with Status, minimum Objective, and Decision Count. This is your quick feasibility and performance checkpoint before acting on detailed rows.
• Decision Table: Pair-level assignment output (Agent, Task, Assigned, Cost). This is the actionable allocation plan to execute.
• Resource Utilization: Per-agent workload view (Assigned Tasks, Capacity, Utilization %) showing where agents are underused or near saturation.
• Constraint Slack: Tightness of task-coverage and agent-capacity constraints. Helps identify whether cost is being driven by scarce capacity.

Understanding the Calculation

The model solves:

• Minimize: $\sum_{a,t} c_{at} x_{at}$
• Task coverage: $\sum_a x_{at} = 1$ for each task
• Agent capacity: $\sum_t x_{at} \leq \text{Capacity}_a$
• Binary: $x_{at} \in \{0,1\}$.

Example

With defaults, solution is Optimal with objective 10 and assignments:

• A1 -> T2
• A2 -> T1
• A3 -> T3

Interpretation: all tasks are covered exactly once at minimum total cost. A single changed cost cell can flip an entire assignment pattern, which is why it's worth re-running this model whenever pricing or effort estimates shift.

Important Assumptions and Limitations

• Each task is assigned once, not fractionally.
• Agent capacities are upper bounds only.
• No skill/precedence/time-window constraints in this model.