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# Formulation of linear programming model

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A manufacturing firm located in Chicago ships its product by railroad to Detroit. Several different routes are available, as shown in the attached diagram, referred to as a network.

Each circle in the network represents a railroad junction. Each arrow is a railroad branch between two junctions. The number above each arrow is the cost (\$1,000s) necessary to ship 1 ton of product from junction to junction. The firm wants to ship 5 tons of its product from Chicago to Detroit at the minimum cost.

Formulate a linear programming model for this problem.

So, I believe there are 7 decision variables in this problem. I know that I want to minimize Z = x13 + x14 + x12=5 tons . I am lost at this point. Not sure how to correctly label and use double subscripts on my decision variables.

https://brainmass.com/math/linear-programming/341140

#### Solution Preview

Actually, there is no need to use DOUBLE subscripts.

Here is the basic setup of the problem.

Some of the product will get to Chicago to Detroit by (first) going through junction 3, some will get there by (first) going through junction 4, and the rest will get there by (first) going through junction 2. For the sake of convenience, let x_3, x_4, and x_2 denote the amounts (in tons) that will get to Detroit by (first) going through junctions 3, 4, and 2, respectively.

The quantity to be MINIMIZED is the total COST of TRANSPORTING the ...

#### Solution Summary

The choice and labeling of the decision variables is developed, and a detailed explanation is provided.

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