Chapter 4, Problem 22RP

Explanation of Solution

Optimal solution:

• Consider the preemptive programming problem of the Monroe Country of building a fire station.
• This fire station covers four major towns that are mentioned in the given diagram:

• The below given table specifies the average number of fire incidents in each of the towns:

Town	Location	Fire
1	(10,20)	20
2	(60,20)	30
3	(40,30)	40
4	(80,60)	25

• Assume that (x,y) is the point where the fire station is build. Suppose a town with fire is located at point (a,b).
• Depending upon the location of the fire station in town A, the value $x-a$ will be positive or negative. That is if (x,y) is in the east, then $x-a$ is positive, and therefore,
  • $x-a=e-w$
• e and w are defined as non-negative. Hence, when $x-a$ is positive, $x-a=e$ and w=0. When $x-a$ is negative, $x-a=-w$ and e=0.
• Thus, the total distance between fire station and town is given by
  • $d=e+w+n+s$
• Hence, the optimization equation is
• Minimize z=$20d1+30d2+40d3+25d4$
• The constraints for each town are
  • $x-10=e1-w1$
  • $y-20=s1-n1$
  • $x-60=e2-w2$
  • $y-20=s2-n2$
  • $x-40=e3-w3$
  • $y-30=s3-n3$
  • $x-80=e4-w4$
  • $y-60=s4-n4$
• Thus the following LP formulation:
• Minimize z=$20d1+30d2+40d3+25d4$
• Subject to
  • $x-10=e1-w1$
  • $y-20=s1-n1$
  • $x-60=e2-w2$
  • $y-20=s2-n2$
  • $x-40=e3-w3$
  • $y-30=s3-n3$
  • $x-80=e4-w4$
  • $y-60=s4-n4$
  • $d1=e1+w1+n1+s1$
  • $d2=e2+w2+n2+s2$
  • $d3=e3+w3+n3+s3$
  • $d4=e4+w4+n4+s4$
• The LP problem is solved using Lindo software