1.203J/6.281J/15.073J/16.76J/ESD.216J
Fall 2004
Quiz 2: Solutions
Problem 1:
(a) There are several, essentially equivalent ways to define the state of the system. One
possibility is: (k, j, i) where
k = the type of customer (0, 1, or 2) currently in service (note that you cannot
have one server occupied by a Type 1 customer and the other by a Type 2 customer)
j = the number of Type 2 customers (0, 1 or 2) in the system
i = the number of Type 1 customers (0, 1, 2, 3 or 4) in the system
The statetransition diagram for the system is then as shown on Figure 1.
(b) From Figure 1 it can be seen that the event of interest can happen only by having a set
of three consecutive transitions from state (1, 0, 4) to (1, 0, 3) to (1, 0, 2) to (1, 1, 2).
(Note that the probability of the first of these transitions is 1.) It can be seen that:
2
2
µ
(a) Note that T
0
is a traveling salesman tour through the 2n+1 points using the
Christofides heuristic. We know that
⎛
⎜
⎜
⎝
Problem 2:
L(T
0
) < (3/2)L(TSP)
where TSP is the optimal traveling salesman tour through the same 2n+1 points.
Note that TSP does not necessarily observe the precedence constraints and, thus,
L(TSP)
≤
L(DARP)
Where DARP is the optimal solution to the dialaride problem.
Finally and obviously: L(T
1
) < 2L(T
0
).
Putting these together, we have:
L(T
1
) < 2L(T
0
) < 2[(3/2)L(TSP)] = 3L(TSP)
≤
3L(DARP)
(b) The application of the 2exhange heuristic to the DARP problem is not
straightforward. First, it is obvious that a 2exchange may lead to a violation of the
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 Fall '06
 hansman
 Travelling salesman problem, TSP, optimal location, weighted distance matrix

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