Explain the initial methods to solve
ransportation problem with an example.
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Answers
Step-by-step explanation:
The transportation problem is a type of linear programming problem designed to minimize the cost of distributing a product from \(M\) sources to \(N\) destinations.
The transportation problem can be described using examples from many fields. One application is the problem of efficiently moving troops from bases to battleground locations. Another is the optimal assignment of agents or workers to different jobs or positions. By far the most common application is of moving goods from multiple factories to multiple warehouse locations, or from warehouses to storefronts.
Transportation Problem Example
Consider the following example:
XYZ Inc. has two factories in different locations around the country where they produce widgets. Their sales partner has three central warehouses where they ship these widgets to their various customers. The factories can produce a given number of widgets per week each and the expected demand for each warehouse is also known. There is a shipping cost from each factory to each warehouse. Which factory should produce and ship how many widgets to which warehouses to meet the demand at each location with minimal cost?
This problem statement has all the components of a typical transportation problem. The sources and destinations are generic — they could be logging sites and sawmills, factories and warehouses, warehouses and stores, bases, and battlefields, and so on.
In each case, there is some demand or need \(D\) at each of \(N\) locations, some supply \(S\) at each of \(M\) locations, and a cost, \(c\), associated with transporting (or using) one unit from a particular \(M\) location to a particular \(N\) location. There will be a total of \(M\) x \(N\) such costs.
The cost, \(c\), can be a calculation involving factors such as time, distance, material costs, and so on, but it may be any quantity that is relevant to the problem.
Once supplies, demands, and costs are known, the problem is to determine the number of units, \(x\), that should be produced and sent from each of the \(M\) supply centers to each of the \(N\) demand locations.
The total cost is the sum of all individual costs times the individual units to be produced and shipped from each supply center to each demand center. Framing this as an optimization problem, the goal is to minimize the total cost:
$$ \sum_{i=1}^M \sum_{j=1}^N c_{ij}x_{ij} $$
Simultaneously, there are some rules (constraints) that must be satisfied:
The number of units shipped must be less than or equal to the total supply.
The number shipped must match, or meet, the demand at each location.
The number of units to ship must be greater than or equal to zero (no negative values).
In the case of a balanced transportation problem, for which the total demand is equal to the total supply, the constraints have the following mathematical representation:
$$ \sum_{j=1}^N x_{ij} = S_{i}, i=1, …, M $$
$$ \sum_{i=1}^M x_{ij} = D_{j}, j=1, …, N $$
$$ x_{ij} \geqslant 0, i=1 …,M, j=1 …N $$
Note that there may be cases of excess supply or excess demand leading to an unbalanced problem. This case is addressed below and can be solved similarly using dummy variables and possibly penalties for unmet demand or storage costs for excess supply.
The total cost function along with the three constraints define a well-formed linear programming (LP) optimization problem with linear constraints. It is known specifically as the balanced transportation problem.
Solving the Transportation Problem
Unlike many LP problems, the transportation problem is feasible to solve by hand using a series of tables and well-documented strategies such as the Northwest-Corner Method to find an initial basic feasible solution and then using techniques like the Least-Cost Method or the Stepping Stone Method.
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