Transportation model

Transportation model is a mathematical optimization model used to determine the most efficient method of transporting goods or resources from multiple sources to multiple destinations while minimizing total transportation cost or maximizing overall efficiency. It is a specialized form of Linear programming that allocates shipments between supply locations and demand locations subject to capacity and demand constraints. The transportation model is one of the fundamental optimization techniques in operations research, logistics, supply chain management, and industrial engineering.

The transportation model assumes that a fixed quantity of goods is available at one or more sources, while specified quantities are required at one or more destinations. Each possible shipment route has an associated transportation cost per unit. The objective is to determine the quantity to be transported along each route so that all supply and demand requirements are satisfied at the lowest possible total cost.

Transportation models are widely applied in manufacturing, retail distribution, agriculture, mining, energy, humanitarian logistics, military planning, and public transportation. They also serve as building blocks for more advanced optimization models, including network flow models, transshipment models, and supply chain optimization systems.

Because of their computational efficiency and practical applicability, transportation models remain among the most widely used optimization techniques in operations research and management science.

History

The transportation model emerged during the development of operations research in the mid-twentieth century as organizations sought scientific methods for optimizing logistics and resource allocation.

The model became closely associated with the development of Linear programming, particularly following the introduction of the simplex method by mathematicians studying large-scale optimization problems. Specialized solution procedures, such as the transportation simplex method, were later developed to solve transportation problems more efficiently than general linear programming algorithms.

Today, transportation models are routinely solved using commercial optimization software and are integrated into enterprise resource planning and supply chain management systems.

Objectives

The principal objectives of a transportation model are to:

minimize transportation costs;
satisfy supply constraints;
satisfy demand requirements;
optimize resource allocation;
improve logistical efficiency;
support operational decision-making.

These objectives contribute to more efficient distribution systems.

Model structure

A transportation model generally consists of:

one or more supply locations;
one or more demand locations;
transportation routes;
transportation costs;
supply constraints;
demand constraints.

The model determines the optimal shipment quantities between sources and destinations.

Mathematical formulation

Let:

$S_{i}$ represent the supply available at source $i$ ;
$D_{j}$ represent the demand at destination $j$ ;
$c_{i j}$ represent the transportation cost per unit from source $i$ to destination $j$ ;
$x_{i j}$ represent the quantity transported.

The objective function is:

$min Z = \sum_{i} \sum_{j} c_{i j} x_{i j}$

subject to:

$\sum_{j} x_{i j} = S_{i}$

for each source,

$\sum_{i} x_{i j} = D_{j}$

for each destination, and

$x_{i j} \geq 0$

for all shipment quantities.

Example

Suppose a company operates two factories and supplies two warehouses.

Factory	Supply
Factory A	100 units
Factory B	150 units

Warehouse	Demand
Warehouse X	120 units
Warehouse Y	130 units

The transportation costs are:

From	To	Cost per unit
Factory A	Warehouse X	US$4
Factory A	Warehouse Y	US$6
Factory B	Warehouse X	US$5
Factory B	Warehouse Y	US$3

The transportation model determines the shipment quantities that satisfy all supply and demand requirements while minimizing the total transportation cost.

Balanced and unbalanced models

Transportation problems are commonly classified into two categories.

Balanced transportation model

A balanced transportation model exists when:

$Total Supply = Total Demand$

Every unit supplied can be allocated to satisfy demand.

Unbalanced transportation model

An unbalanced transportation model occurs when total supply differs from total demand.

Such problems are commonly balanced by introducing a dummy source or dummy destination with zero transportation cost.

Solution methods

Several algorithms are commonly used to solve transportation models.

These include:

northwest corner method;
least-cost method;
Vogel's approximation method;
transportation simplex method;
modified distribution method (MODI).

Modern optimization software generally solves transportation models automatically.

Relationship with linear programming

The transportation model is a specialized application of Linear programming.

Because of its unique mathematical structure, it can often be solved more efficiently than a general linear programming problem of comparable size.

Relationship with operations research

Transportation models are among the classical optimization techniques studied in Operations research.

They are frequently combined with:

assignment models;
network flow models;
inventory models;
scheduling models;
supply chain optimization.

Relationship with supply chain management

Transportation models are widely used in supply chain management to:

optimize distribution networks;
reduce logistics costs;
allocate inventory;
improve delivery efficiency;
support production planning.

They assist organizations in designing cost-effective distribution systems.

Applications

Transportation models are widely applied in:

manufacturing;
retail logistics;
agricultural distribution;
mining;
humanitarian relief operations;
military logistics;
energy distribution;
public transportation planning.

They are useful whenever goods or resources must be transported efficiently between multiple locations.

Advantages

Transportation models provide several important benefits.

They minimize transportation costs, improve resource utilization, support efficient logistics planning, simplify complex distribution decisions, and provide mathematically optimal allocation solutions.

They are computationally efficient even for large distribution networks.

Limitations

Despite their usefulness, transportation models rely on simplifying assumptions, including known transportation costs, fixed supplies, fixed demands, and deterministic conditions. Real-world logistics networks may involve uncertain demand, variable transportation costs, traffic disruptions, capacity limitations, and multiple operational objectives that require more advanced optimization techniques.

Large-scale transportation problems may also require specialized optimization software and high-quality data.

Importance

The transportation model is one of the most important optimization models in operations research because it provides an efficient mathematical framework for minimizing transportation costs while satisfying supply and demand constraints. Its widespread application across manufacturing, logistics, supply chain management, and public planning has made it a cornerstone of quantitative decision-making and resource allocation.