IQ Smart drives daily to work. Having just completed a course on network analysi

Question

IQ Smart drives daily to work. Having just completed a course on network analysis, Smart is able to determine the shortest route to work. Unfortunately, the selected route is heavily patrolled by police, and with all the fines paid for speeding, the shortest route may not be the best choice. Smart has thus decided to choose a route that maximizes the probability of not being stopped by police. The network on the following slide shows the possible routes between home and work and the associated probabilities of not being stopped on each segment. The probability of not being stopped on the way to work is the product of the probabilities associated with the successive segments of the selected route. Determine his daily route. 0.8 0.35 0.2 0.5 0.6 0.1 0.4 0.9 0.25 0.3

Explanation / Answer

The problem can be formulated as a shortest-route model by using a logarithms transformation that converts the product probability into the sum of the logarithms of probabilities that is, if p1k = p1 * p2 * ....... * pk is the probability of not being stopped, then log p1k = log p1 + log p2 + ....... + log pk .

Mathematically, the maximization of log p1k is equivalent to the min of -log p1k . Using this transformation , the individual probabilities pj are replaced with -log pj .

0.1054

Let -log p17 possible values are as

1) -log p17 = -log p12 - log p24 -log p46 -log p67 = 1.6094 + 0.2231 + 1.0498 + 0.6931 = 3.5755

2) -log p17 = -log p12 - log p24 -log p45 - log p57 = 4.1352

3) -log p17 = -log p12 - log p23 -log p35 - log p57 = 4.7105

4) -log p17 = -log p12 - log p23 -log p34 - log p46 - log p67 = 6.1658

5) -log p17 = -log p12 - log p23 -log p34 - log p45 - log p57 = 6.7254

6) -log p17 = -log p13  -log p35 - log p57 = 2.6956

7) -log p17 = -log p13  -log p34 - log p45 - log p57 = 4.7105

8) -log p17 = -log p13 - log p34 - log p46 - log p67 = 4.1509

From above 8 calculations we see that the nodes 1 ,3,5,7 with a shortest route with a corresponding length is 2.6956 . Thus the maximu probability of not being stopped is p17 = 0.9 * 0.3 * 0.25 = 0.0675 only.

Nodes pj -log pj 1 - 2 0.2 1.6094 1 - 9 0.9

0.1054

2 - 4 0.8 0.2231 2 - 3 0.6 0.5108 3 - 4 0.1 2.3026 3 - 5 0.3 1.204 4 - 5 0.4 0.9163 4 - 6 0.35 1.0498 5 - 7 0.25 1.3863 6 - 7 0.5 0.6931

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