Stochastic Server Location Problem $($SSLP$)$

We consider a set of customer buildings in a metropolitan area for which a service provider

is interested in installing optical fibers and switching equipment in the most profitable man$-$

ner. Due to the uncertainty regarding the customer base for high speed services, telecommu$-$

nication providers often adopt a very conservative approach to capital investment, leading

to potential losses in revenue. Such problems are common in practice and can be formulated

as the SSLP$.$

The essence of the SSLP may be described as follows. Suppose that we place a server

at location jinJ where $J$ is the set of possible server locations. The fixed cost of locating

a server at location jinJ is $c_j.$ A server can provide enough capacity to serve up to $u$

amount of resource to clients. There is also a shortage cost $($penalty$)q_{j0}$ for each unit of

demand that remains unserved among the clients assigned to server $j.$ As far as operational

considerations are concerned, we allow only one server to be installed at each location and

each client can only be served by one server.

The revenue generated by serving client iinI from location jinJ, is denoted $q_{ij}$ where $I$

is the set of clients. If client $i$ is served by a server at location $j,$ it uses $d_{ij}$ units of resource

from the server. However, we know that some clients may not always exist at a given time,

that is$,$ presence of clients is subject to uncertainty. Let $S$ be the set of all possible scenarios

and $p_s$ be the probability associated with scenario $sinS.$ Also let $h_{is}=1$ if client $i$ is present

in scenario $sinS$ and $0,$ otherwise. Note that for a given scenario $s,$ if $h_{is}=1$ and the client

$i$ is assigned to a server located at $j,$ client i has a demand of $d_{ij}$ from the server located at

$j.$ If $h_{is}=0,$ then the client has no demand for scenario $sinS.$

Note that we first decide locations of servers, then observe existing customers and finally

decide on client$-$server assignments.

SSLP is to choose locations of servers and client$-$server assignments that minimize the

total expected operating cost subject to the given constraints.

Construct a linear model to solve SSLP and explain the statements of constraints clearly.

Define your variables indicating that whether they are first or second stage variables.

What to hand in:

In a pdf document:

$($a$)$ List the group members names and IDs.

$($b$)$ Mathematical formulation. Explanation of parameters and decision variables, objec$-$

tive function and constraints.