$(12\mathrm{.}5$ points$)$ In p$-$hub median problem, $p$ hubs are opened, origin$-$destination $($o$-$d$)$ flows

travel via hubs and interhub transportation is discounted by a factor $$ that is between $0$ and $1$

to reflect transportation economies of scale.

Problem has the following setup:

$(1)n$ demand locations $($origin$/$destinations$)$ are given,

$(2)$ the flow for the $n^2$ location $($o$-$d$)$ pairs is given,

$(3)$ the per unit transportation cost between all location $($o$-$d$)$ pairs and a hub$-$to$-$hub

transportation discount factor $$ is given, and

$(4)$ the total amount of flow between hub $k$ and $m$ cannot exceed capacity $C_{km}.$

The objective is to locate $p$ hubs to minimize the total transportation cost of the flows and the

fixed cost of setting up the hubs.

Write down a mixed integer mathematical optimization model of p$-$hub median problem

and use the following decision variable in your model:

$x_{\text{i}\text{j}\text{k}\text{m}}$: the amount $of$ flow $(not$ the percentage$)$ from customer $itoj$ via hubs $k$ and

Notice that $x$ is not between $0$ and $1,$ it denotes total flow in the given route between $0$ and

$h_{ij},$ where

$h_{ij}$: denotes the total amount $of$ demand $of$ customer $j$ from customer $i$

Define other variables and parameters clearly $($if necessary$)($problem must be linear$)$

B$)$ Write down the mathematical model of $p-$median problem. P$-$median problem has the

following setup: $($No partial points will be given!$)$

$(1)$ n potential locations for facilities,

$(2)$ m customers to serve,

$(3)$ the per unit transportation cost between facilities and customers,

$(4)$ each customer's demand $(D_i)$ must be satisfied by one facility,

$(5)$ each facility can serve at most $C_j$ amount of customers' demand.

The objective is to locate p medians to minimize the total transportation cost of the flows and

the fixed cost of setting up the medians.

Write down a mathematical optimization model of $p$ median problem.