# Formulate the following network problem

(See the attached file for the diagram)

- In the above network, a given flow Q is transferred through the network to a demand node N. The goal is to route the flows in such a way that total channel loss is minimized, where channel loss coefficients clij per unit length are given for each link or arc of length Lij. Ignore the maximum capacities of the links for this. Specify how the arc parameters [lij, uij, cij] for any link (i,j) in this network should be defined, as well as the demand node parameters (i.e., priority and demand), that will result in minimization of total channel loss.

- Instead of Q being specified in the above network, suppose that now the objective is to find the maximum flow that can be delivered to demand node N, assuming that upper bounds uij are now specified on the flow in each link or arc (i,j) and channel losses are ignored. Specify the arc parameters and demand node parameters that would be required to solve this problem [Hint: you may need to add a supply node such as a dummy reservoir].

- How would you define the arc parameters and demand node parameters in order to find the shortest path through this network, assuming each link is of length Lij? We are not trying to route flows in this network, but simply identify the shortest connected path through the network.

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#### Solution Summary

This solution gives a step-by-step explanation on how to formulate the given problem using the network flow model. The solution discussed some basic techniques in the formulation of network flow problem.

Formulate the problem as the proper network problem and solve

Topless City is a small chain of car dealerships that sells vintage convertibles throughout the Southern United States. It is owned and managed by Brandon and Kyle Winslow. Each month Brandon and Kyle attend two car auctions, at which they purchase convertibles: one in Atlanta, the other in Miami. The cars are then shipped to one of three locations: Jackson Mississippi, Birmingham Alabama, or Orlando Florida. There, the cars are refurbished, repainted, safety inspected, and sold at the Topless City dealership in that city. In August, Brandon found 20 cars at the Atlanta auction, and Kyle found 50 cars at the Miami auction which met the needs of the company. Only 15 cars can be worked on at each city during the month, however. Another auction is coming up in September; thus, only 45 cars are to be purchased in August. Topless City wishes to minimize its costs of transporting the cars to the refurbishing locations. The cost to transport cars between cities is shown in the table below.

Jackson Birmingham Orlando

Atlanta $200 $100 $175

Miami $250 $200 $125

b. Formulate the problem as the proper network problem and solve.

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