1. In some cases, the waste generated by the production of material at a facility must be disposed of at special waste disposal locations. We need to identify which waste facilities to open, given a capacity for each waste unit and per-unit transportation costs for shipping waste from a plant facility to a waste facility. We can model this problem by extending our facility location models. Suppose we have the network shown on the next slide and let K={4, 9, 10} be possible locations for our waste disposal facilities. At each plant j  J, the amount of waste material produced is proportional (with proportionality constant 0<  j <1) to the amount of goods produced; for example, if plant 5 produces 500 units, then it also produces 500  units of waste. These proportionality constants are The one-time fixed cost associated with opening a waste disposal facility at k  K is and their capacities are The cost of shipping one unit of waste from plant j  J to waste disposal unit k  K is $10d jk, where d jk is the shortest distance from j to k, while the cost of shipping one unit of demand from plant j to customer i is $20d ij. Assuming that a customer can receive its demand from multiple plants and that the waste from a plant can be shipped to multiple disposal locations, determine how to satisfy customer demands and waste removal requirements at minimum cost. Where are the plants and waste disposal units?

3. Fixed-Charge Location Problem

5. Solution Optimal cost is $42,813.80 Optimal solution opens plants in all possible locations. Plant 3 services customer 3, plant 5 serves customers 2 and 5, plant 6 serves customers 1, 6, and 9, plant 7 serves customer 7, and plant 8 serves customers 4, 8, and 10. Optimal solution opens waste disposal facilities at locations 4 and 10, with location 4 handling waste from plants 3, 5, and 7, and location 10 handling waste from plants 6 and 8.