We are evaluating a project that costs $650, 000, has a five-year life, and has no salvage value. Assume that depreciation is straight-lint to zero over the life of the project. Sales are projected at 47,000 units per year. Price per unit is $56, variable cost per unit is $26, and fixed costs are $860, 000 per year. The tax rate is 35 percent, and we require a return of 10 percent on this project. Suppose the projections given for price, quantity, variable costs, and fixed costs are all accurate to within plus or minus 10 percent. Calculate the best-case and worst-case NPV figures (Negative amounts should be indicated by a minus sign. Do not round intermediate calculations and round your final answers to 2 decimal places, e.g., 32.16.) Solution.

1. We are given an undirected graph G = (N, E), where N is the set of nodes and E is the set of
edges. Suppose that we can color a node in the graph G so that the number of edges connected to
two nodes with different colors is maximized. Formulate this problem as an integer linear
program.
Solution
Let node i have 3 indicator variables ri, yi, bi
ri = 1 if the node is red, 0 otherwise.
yi = 1 if the node is yellow, 0 otherwise.
bi= 1 if the node is blue, 0 otherwise.
Then, let edge e ij be the edge from i to j where there is an edge in the above diagram.
Thus, we have e12, e13, e15, e24, e23, e27, e35, e34, e36, e45, e46, e47, e 56, e57, and e67.
We will let this have the value 1 if nodes i and j have different color and otherwise have value 0.
max e12 + e13 + e15 + e24 + e23+ e27 + e35 + e34 + e36 + e45 + e46 + e47 + e56 + e57 + e67
subject to
ri + rj + eij <= 2.
yi + yj + eij <= 2.
yi + yj + eij <= 2.
for ij {12, 13, 15, 24, 23, 27, 35, 34, 36, 45, 46, 47, 56, 57, 67}
Note that these inequalities work. If the colors are the same, for that color, they add to 2, so the
edge has value 0, and they sum to 2 is less than or equal to 2. For the other colors, they add to 0,
+ 0 for the edge is 0 is less than or equal to 2.
If the colors are different, then the sum over each of the 3 colors will be 1 in 2 cases and 0 in the
other, so the edge will have value 1, and the sum will be 1 in 1 case and 2 in the other 2.
(While the edge could have a smaller value, this would not be the solution at the maximum, so
the formulation works.)
At each node, we must have one of ri, yi, and b i equal to 1 and the others equal to 0.
Have them sum to 1 so there is a color at each node ri + yi + bi = 1 i = 1, 2, 3, 4, 5, 6, 7
All variables are greater than or equal to 0.
All variables in this problem are integers.