Problem $1(25$ points$)$: Distance Measures

Calculate the Euclidean and Rectilinear distance measures for the distance between the following two points: $(2\mathrm{.}3,43\mathrm{.}5)$ and $(15\mathrm{.}3,6\mathrm{.}34).$

Problem $2(75$ points$)$: Facilities Layout Problem

The following questions are associated with the facilities layout problem shown below. The problem contains three plants supplying four customers or warehouses.

Customer i $1234$

Demand d$\_$i $10025075300$

Facility j c$\_($i$,$j$)$ f$\_$j M$\_$j

$110512201000500$

$212724152000300$

$32397181000500$

Recall that c$\_($i$,$j$)$ refers to the cost associated with servicing customer i from facility j$.$ A solution $\{$x$\_($i$,$j$)$:i$=1,2,3,4,$j$=1,2,3\}$ refers to the amount of product that flows to customer i from facility j$.$ You are given the following solution: x$\_1,1=100,$ x$\_2,1=250,$ x$\_3,3=75,$ x$\_4,3=300,$ and all other x$\_($i$,$j$)=0.$ Please answer the following questions.

Objective Function: Use the solution and the information found in c$\_($i$,$j$)$ to find the value of the objective function.

Capacity Constraint: Use the information contained in the Facility Layout Problem and solution to verify that the capacity constraints are not violated.

Demand Constraint: Use the information contained in the Facility Layout Problem and solution to verify that the customer demand constraints are satisfied.