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166 Biomimetics: Biologically Inspired Technologies

Table 5.2 Total Enumeration of Branch Locations Combinations
# x 1 x 2 x 3 x 4 x 5 x 6 x 7 f(x)
1 0 0 0 0 1 1 1 36.04
2 0 0 0 1 0 1 1 39.92
3 0 0 0 1 1 0 1 40.41
4 0 0 0 1 1 1 0 46.07
5 0 0 1 0 0 1 1 35.98
6 0 0 1 0 1 0 1 36.61
7 0 0 1 0 1 1 0 55.88
8 0 0 1 1 0 0 1 36.32
9 0 0 1 1 0 1 0 43.79
10 0 0 1 1 1 0 0 44.42
11 0 1 0 0 0 1 1 35.80
12 0 1 0 0 1 0 1 39.57
13 0 1 0 0 1 1 0 47.41
14 0 1 0 1 0 0 1 39.11
15 0 1 0 1 0 1 0 41.14
16 0 1 0 1 1 0 0 44.90
17 0 1 1 0 0 0 1 36.61
18 0 1 1 0 0 1 0 46.83
19 0 1 1 0 1 0 0 48.81
20 0 1 1 1 0 0 0 41.95
21 1 0 0 0 0 1 1 34.61
22 1 0 0 0 1 0 1 33.79
23 1 0 0 0 1 1 0 36.16
24 1 0 0 1 0 0 1 41.60
25 1 0 0 1 0 1 0 39.39
26 1 0 0 1 1 0 0 35.61
27 1 0 1 0 0 0 1 31.01
28 1 0 1 0 0 1 0 35.53
29 1 0 1 0 1 0 0 36.17
30 1 0 1 1 0 0 0 32.59
31 1 1 0 0 0 0 1 33.40
32 1 1 0 0 0 1 0 32.73
33 1 1 0 0 1 0 0 36.09
34 1 1 0 1 0 0 0 33.35
35 1 1 1 0 0 0 0 33.54

The p-median problem is well researched in the literature. Various optimization techniques
have been developed for its solution (see Daskin, 1995; Current et al., 2002). Typically, it is
formulated as integer programming (Daskin, 1995; Current et al., 2002). We present here a
different, intuitive formulation. Suppose that x ¼ {x 1 , x 2 ,.. ., x 7 } are seven binary variables.
A variable is assigned a value of ‘‘1,’’ if the corresponding community is selected for a branch and
‘‘0,’’ if it is not. The set I is defined to include all variables with a value of ‘‘1.’’ For a feasible
(acceptable) solution, the set I must be of cardinality 3. The service distance for community j is

defined as d j ¼ min d ij , where d ij is the distance between communities i and j. By these
i2I
definitions, the formulation is:
( )
X
20
Minimize f(x) ¼ d j (5:1)
j¼1
X
7
Subject to: x i ¼ 3 (5:2)
i¼1
x i 2f0; 1g (5:3)