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A transportation problem is a specific type of linear programming problem where the objective is to determine the most cost-effective way of transporting goods from several suppliers to multiple consumers. This concept typically applies to scenarios like supply chain logistics, where there are distribution centers (factories) and destination points (stores). The main goal is to minimize the transportation cost while meeting supply and demand requirements.

In the exercise, the companies have two factories (F1 and F2) and three stores (S1, S2, and S3). Each factory has a certain production capacity, and each store has a specific demand for products. Additionally, there is a cost associated with shipping goods from each factory to each store.

The essence of solving a transportation problem is to allocate the shipment from factories to stores in such a way that all demands are satisfied without exceeding the production capacities, and all of these are done in the most cost-efficient manner.

The objective function in a linear programming problem represents the goal of either minimizing or maximizing a certain value. For transportation problems, this usually refers to minimizing the total cost associated with moving goods or services from one point to another.

In the given step-by-step solution, the objective function is described mathematically. It is the weighted sum of all possible shipping routes between factories and stores. Each term in the function represents the cost for each route multiplied by the number of units transported via that route.

The mathematical representation is given by the equation:
\[Z = c_{11}x_{11} + c_{12}x_{12} + c_{13}x_{13} + c_{21}x_{21} + c_{22}x_{22} + c_{23}x_{23}\]
Here, \(c_{ij}\) denotes the cost per unit for transporting goods from factory \(i\) to store \(j\), while \(x_{ij}\) denotes the quantity transported on each respective route.

Constraints are essential in linear programming as they define the boundaries within which a solution must be found. For the transportation problem, constraints ensure both that supply does not exceed production capacity and that demand is fully met.

In our exercise, we encounter two major types of constraints:

• Supply Constraints: Ensure that the total units shipped from a factory do not surpass its production capacity. For example:
  \[ x_{11} + x_{12} + x_{13} \leq P_1 \]
• Demand Constraints: Ensure that each store receives the required number of products. For example:
  \[ x_{11} + x_{21} \geq D_1 \]

Additionally, there are non-negativity constraints stating that the quantities shipped cannot be negative, \(x_{ij} \geq 0\). These constraints shape the feasible region from which the optimal solution is selected.

In any linear programming problem, decision variables are the unknowns we aim to solve for. They represent the choices available to accomplish the objective.

In our transportation problem, decision variables signify the number of units to be shipped from each factory to each store. They take the form \(x_{ij}\) where \(i\) refers to the factory and \(j\) refers to the store. For each route from a factory to a store, there is an associated decision variable.

For the example given, decision variables include the following:

• \(x_{11}\): Units from Factory 1 to Store 1
• \(x_{12}\): Units from Factory 1 to Store 2
• ... and so forth for each factory-store combination.

The goal is to determine the optimal values for these decision variables to achieve minimum transportation cost, fulfilling the objective function while adhering to all constraints.