Suppose there are four cities at locations x1 = 2; x2 = 5; x3 = 9, and x4 = 11,

Question

Suppose there are four cities at locations x1 = 2; x2 = 5; x3 = 9, and x4 = 11, requiring

respectively Ni = 6; 2; 3; 7 visits, respectively. Suppose that the cost-per-mile to ship goods

is $4 between x1 and x2, but $8 between x2 and x3, and $10 between x3 and x4. What is

the cost of keeping the PDC at x = 6? What about at x = 10?

1. Suppose there are four cities at locations r1 2, r2 0, r3 9, and r4 11, requiring respectively N 6, 2, 3,7 visits, respectively. Suppose that the cost-per-mile to ship goods is $4 between r1 and ro, but $8 between 2 and ra, and $10 between T2 and ra. What is the cost of keeping the PDC at a 6? What about at T 10?

Explanation / Answer

We want a formula for the cost of servicing city xj
for a year, but that cost depends on the location x of the PDC. Let i be the number of cities to the left (west) of the PDC, so that xi < x xi+1.
Let’s write Cij (x) for the cost of servicing city xj
instead of Cj (x), to emphasize that our
calculations depend on the fact that x lies in the i-th interval. Let ki be the unit cost to ship goods between xi and xi+1.

Show that when j i we have
Cij (x) = Nj{kj (xj+1 xj ) + · · · + ki1(xi xi1) + ki(x xi)},
while if j i + 1 then we have
Cij (x) = Nj{ki(xi+1 x) + ki+1(xi+2 xi+1) + · · · + kj1(xj xj1)}.

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