Question $1$:

The city management is developing a plan for the collection and disposal of household waste. Households that generate waste are mainly clustered in $n$ districts. Currently, waste is collected from each district and disposed of at the landfill. The management encountered a problem: The landfill was full and could no longer provide service. Suppose the location of the landfill is denoted by node $0.$ Industrial engineers of city management have identified $m$ points where a new landfill can be established. The waste of a district can be collected if there is a landfill at most $$ away from that district. However, due to social opposition, the distance between a landfill and the districts where people live should be at least $.$ The parameter $d_{ij}$ represents the distance between districts $)=(\{a,b,$dots,$n\}$ and alternative landfill locations $\{1,2,$dots,$m\}.$ Furthermore, $p_i$ is the amount of waste from district iinN that needs to be disposed of$,$ and $K_j$ is the capacity of landfill jinM. The management wants each district to be assigned to a single landfill.

a$)$ Landfill site $0$ can be reused, and its capacity increased to $K_0$ after a clean$-$up operation costing $C$ TL$.$ Furthermore, the cost of establishing a new landfill site is determined as $B$ TL$.$ Accordingly, develop a linear mathematical model to minimize total cost while all waste is collected and disposed of$.$

b$)$ The management has concluded that the current landfill is no longer usable, and three new landfills are needed. In addition to the requirements $$ and $$ and other factors, the number of landfills within $2$ distance of each district is required to be at most $2.$ Develop a linear mathematical model that minimizes the maximum distance between districts and the landfills serving them.