![]() ![]() The IMSL Library algorithm allows users to solve problems of nearly any size with a simple programming interface. Spreadsheet solutions are possible as well, but often require significant rework whenever the number of supply or demand centers changes. While it can be instructive, solving by hand is not practical or scalable for real-world problems.įor larger problems or to realize any level of scalability, a computer-based method is preferred. Working through a problem by hand simply requires repeatedly drawing tables and manual refinement starting with an initial basic feasible solution. 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. It is known specifically as the balanced transportation problem. The total cost function along with the three constraints define a well-formed linear programming (LP) optimization problem with linear constraints. ![]() 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. Note that there may be cases of excess supply or excess demand leading to an unbalanced problem. Framing this as an optimization problem, the goal is to minimize the total cost: 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. 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 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. There will be a total of \(M\) x \(N\) such costs. 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. 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. This problem statement has all the components of a typical transportation problem. Which factory should produce and ship how many widgets to which warehouses to meet the demand at each location with minimal cost? There is a shipping cost from each factory to each warehouse. The factories can produce a given number of widgets per week each and the expected demand for each warehouse is also known. Their sales partner has three central warehouses where they ship these widgets to their various customers. has two factories in different locations around the country where they produce widgets. ![]() By far the most common application is of moving goods from multiple factories to multiple warehouse locations, or from warehouses to storefronts. Another is the optimal assignment of agents or workers to different jobs or positions. One application is the problem of efficiently moving troops from bases to battleground locations. The transportation problem can be described using examples from many fields. What Is the Transportation Problem? 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. Unbalanced Transportation Problem Example.Balanced Transportation Problem Example.In this blog, we give an overview of the transportation problem and how the transportation algorithm - as implemented in IMSL - can be used to solve different variations of the transportation problem. This essential problem was first formulated as a linear programming problem in the early 1940’s and is popularly known as the transportation problem. Since transportation costs are generally not controllable, minimizing total cost requires making the best product routing decisions. Minimizing the cost of transporting products from production and storage locations to demand centers is an essential part of maintaining profitability for companies who deal with product distribution. ![]()
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