solutions04 - 1 15.093J/2.098J Optimization Methods...

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1 15.093J/2.098J Optimization Methods Assignment 4 Solutions Exercise 4.1 BT, Exercise 7.1. Construct the network as follows. For each day i , create two nodes: node d i for dirty tablecloths and c i for clean tablecloths. There is a supply of r i for node d i and similarly, there is a demand of r i for node c i . The node c i can receive new tablecolths from purchasing; create a node n for this new tablecoth source. For each node c i , there is an incoming arc from node n with arc cost of p (and unlimited capacity). There is also an incoming arc from node d i n j with arc cost of f if i > n + j and an incoming arc from node d i m j with arc cost of g if i > m + j (launder tablecloths can be distributed for many days, not one). Not all the laundered tablecloths will be reused , especially at the very end of the period. These tablecloth are wasted. We could create a waste node for this purpose; however, the network flow needs to be balanced and therefore, we could put all wasted tablecolths back to the new tablecloth source. All these arcs will have zero cost and again, unlimited capacity. The network is then balanced with the equal supply and demand of N i =1 r i . Exercise 4.2 BT, Exercise 7.2. Construct the network as follows. For each wood unit i , create a node w i . For each year j , create two nodes: node h j to collect all harvested wood and node y j for wood demand.
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