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04q2sol

# 04q2sol - 1.203J/6.281J/15.073J/16.76J/ESD.216J Fall 2004...

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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 state-transition 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 dial-a-ride 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 2-exhange heuristic to the DARP problem is not straightforward. First, it is obvious that a 2-exchange may lead to a violation of the

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