# Using the Transportation Simplex Method to Solve Transportation Problems

Instructor: Lucinda Stanley

Lucinda has taught business and information technology and has a PhD in Education.

In this lesson, we will explore how to solve transportation problems using the transportation simplex method. We will investigate the data needed and follow an example from beginning to end.

## What's the Problem?

What's all the concern with transportation? Can't we just ship our product to where it needs to go without all this fuss? Well, if we only had one product and one customer it would be simple. Unfortunately, if we want to make money we probably want to sell (and therefore transport) our product to more than one destination. If we're really on top of our game, we could even have multiple facilities that could supply our products. That could get messy and look something like this:

As you can see, each source could potentially supply each destination, so we are looking at a potential for nine transportation routes. But, in the real world, there are typically many more possibilities. How can a transportation manager figure out which source should supply the product for which destination? There are a number of processes that the transportation manager can use. We're going to concentrate on the transportation simplex method.

## Transportation Simplex Algorithm

The transportation simplex algorithm is a linear program, a mathematical model representing linear relationships, like the transportation between a supplier and a destination. Linear programming allows the user to find the optimal or best outcome. We want to transport our product for the least amount of money so we get the most profit.

First, we need to know our parameters or arguments, essentially the data we need to solve the problem:

• Unit Shipping Costs: How much does it cost to ship each truckload of product?
• Supply: The number of truckloads each facility can provide.
• Demand: The number of truckloads each destination requires.

## What It Looks Like

Let's start filling in a transportation matrix so we have all of our data in one place. First, we find the cost per truckload from each of our suppliers to each of our destinations. It will cost \$464 per truckload to ship from Supplier 1 to Destination 1 and \$513 per truckload to ship from Supplier 1 to Destination 2, and so forth.

Next, we include the number of truckloads that each supplier can supply and that each destination demands. We'll make this easy and have our supply equal to our demand.

### Least Cost Method

We want to find the basic feasible solution (BFS), meaning a potential solution to our transportation problem. We can use the northwest corner rule, the least cost method, or Vogel's approximation method. Since we want to maximize our profits, let's use the minimum cell cost method.

We search our transportation matrix for the cell with the smallest cost (noted here in red). We want to assign as much supply to that transportation route as we can. Destination 2 needs 85 truckloads, Supply 2 has 125 so we are going to fill all of Destination 2's order from Supply 2 (in blue).

Now we find the next smallest cost, Supplier 2 for Destination 1 (again shown in red). We've already assigned 85 of Supply 2's available supply so we only have 40 to assign to Destination 1 from Supply 2.

That wipes out Supply 2 so we get rid of the Supply 2 to Destination 3 and find the next lowest cost, which is Supply 3 to Destination 3. We can fill 100 of the 110 truckloads which wipes out Supply 3.

The next lowest cost is Supply 1 to Destination 1.

That leaves us 10 from Supply 1 to fill Destination 3 at \$654.

### Calculate Costs

Now let's calculate our costs by multiplying the number of truckloads times the cost for the assigned transportation route.

How do we know we have the best solution to our transportation problem? We have to test it!

## Optimality Test

We could do a whole lot of linear programming math to test this, but why do a bunch of math when we have access to a great tool in Excel called Solver? Solver is available as an add-in to your Excel program and makes optimality testing much easier.

First, just like our transportation matrix, we'll set up the data about costs:

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