Optimization: Transportation

Purpose

This calculator finds the lowest-cost shipping plan from multiple sources to multiple destinations while meeting all destination demand and respecting each source capacity.

It is useful when you must decide how much to ship on each source-destination lane, not just whether a lane is open. In practice, it answers a concrete question before you commit budget: which combination of routes gets every customer their required quantity for the least total freight spend?

Background

Problem Domain

Transportation optimization is one of the oldest and most studied problems in linear programming, with roots in WWII-era logistics planning. It still quietly powers modern distribution software: you choose shipment quantities to minimize total variable cost, and the same math scales from a handful of plants to national supply networks.

Real-World Uses

Plant-to-customer dispatch planning: Multiple plants can serve the same customer at different freight rates. Use the model to set lane shipment quantities so demand is met at minimum cost.
Regional warehouse replenishment: Demand spikes across regions while upstream supply is limited. Use the model to allocate shipments by lane without exceeding source capacities.
Inter-branch stock balancing: Some branches hold surplus stock while others have deficits, and transfer costs differ by route. Use the model to build a least-cost rebalancing plan.
Budget-constrained distribution planning: Freight spend is under pressure and lane economics are uneven. Use the model to test feasible low-cost allocations and identify expensive lanes.

Inputs

supply: Input table with columns Source and Capacity. Source is the origin node name/id, and Capacity is the maximum shippable quantity from that source. Use the same quantity unit basis as demand (for example units, tons, pallets).
demand: Input table with columns Destination and Demand. Demand is the required received quantity at each destination and should use the same unit basis as supply capacity.
ship_cost: Input table with columns Source, Destination, and Cost. Cost is per-unit shipping cost on that lane. This table must contain the full Source-Destination pair grid (every source with every destination). Cost is interpreted per one flow unit in the same quantity basis as Capacity/Demand.
flow_type: Shipment variable type. Use continuous for fractional flow and integer for whole-number flow.
show_zero: Controls output display. Turn on to include zero-flow decisions; turn off to show only active lanes.

Results

Summary: High-level optimization outcome with Model, Status, Objective, and Decision Count. Objective is minimum total shipping cost, and Status = Optimal confirms a feasible minimum-cost plan was found.
Decision Table: Lane-level shipping plan (Source, Destination, Flow, Unit Cost). This tells you exactly how much to move on each route.
Capacity Utilization: Per-source utilization (Used, Capacity, Utilization %) to show where supply is tight or has headroom.
Constraint Slack: Constraint-by-constraint slack values that help explain which limits are binding and where capacity expansions might reduce cost.

Understanding the Calculation

The model solves:

Minimize: $\sum_{s,d} c_{sd} x_{sd}$
Source capacity: $\sum_d x_{sd} \leq \text{Capacity}_s$
Demand balance: $\sum_s x_{sd} = \text{Demand}_d$
$x_{sd} \ge 0$ and optionally integer.

Example

Using defaults, the model returns Status: Optimal, objective 66625, with active flows such as:

P1 -> C1 = 25
P1 -> C3 = 75
P2 -> C2 = 95

Interpretation: these lane quantities satisfy all demand at minimum total cost for the given cost table. Notice that not every source-destination pair is used — the optimizer naturally skips costlier lanes even when they still have spare capacity, which is often the most useful insight for negotiating freight rates.

Important Assumptions and Limitations

Costs are linear per unit (no tier pricing).
No explicit lane fixed-charge or minimum-lot logic.
No transit-time/service-level constraints.
Full pair grid is required in ship_cost.