Nagpur, a prominent city in India boasting a population exceeding $2\mathrm{.}4$ million, is a crucial mecca for transportation, encompassing an expansive network of roads, railroads, and an airport. The issue of congested traffic during rush hours poses a significant challenge in Nagpur. The city gets high volumes of vehicles on its roads, leading to severe traffic, particularly during peak hours. The Nagpur Improvement Trust $($NIT$)$ is laboriously seeking results to palliate rush hour traffic and enhance the overall ease of exchanging within the city.

Problem Statement:

The Nagpur Improvement Trust $($NIT$)$ is trying to minimize the number of vehicles on the road during rush hour. They have collected data on the percentage of different types of vehicles with the highest values of Passenger Car Units $($PCUs$).$ PCUs are a measure of a vehicle's impact on traffic congestion.

NIT wants to develop a plan to minimize the number of vehicles on the road during rush hour, while still meeting the needs of the community. They need to consider the following factors:

$$ The percentage of different types of vehicles on the road in Nagpur

$$ The PCU values of different types of vehicles

$$ The needs of the community, such as the number of people who need to commute to work or school

The transportation authority is also interested in finding a solution that is cost$-$effective and easy to implement.

Some specific challenges that the Nagpur transportation authority faces include:

$$ The city has a high percentage of two$-$wheelers on the road, which have a relatively high PCU value.

$$ The city has a growing number of cars on the road.

$$ The city has a limited public transportation system.

METHODOLOGY:

The following constraints are taken into considering by NIT for reducing the traffic in the rush hour:

$1.$ Collection of data for different types of vehicles:

$$ The percentage of different types of vehicles on the road during rush hour

$$ Number of vehicles of each type

$2.$ Identifying the problem:

$$ The primary objective of NIT is to alleviate traffic congestion in the vicinity during the peak hours of rush

NIT is trying to minimize the number of vehicles on the road during rush hour. They have collected data on the percentage of different types of vehicles with the highest values of Passenger Car Units $($PCUs$).$ The following table shows the data:

Vehicle Type Urban Rural

Buses $2\mathrm{.}23\mathrm{.}7$

Minibus $1\mathrm{.}51\mathrm{.}5$

Car$/$Jeep$/$Van $11$

Two Wheeler $0\mathrm{.}50\mathrm{.}75$

Auto Rickshaw $1\mathrm{.}22$

Trucks $2\mathrm{.}23\mathrm{.}7$

MAV $45$

LCV $1\mathrm{.}42$

Cycles $0\mathrm{.}40\mathrm{.}5$

Carts $23$

Cycle Rickshaw $23$

Mofussil Bus $2\mathrm{.}23\mathrm{.}7$

Other Bus $2\mathrm{.}23\mathrm{.}7$

The transportation authority wants to develop a linear programming $($LP$)$ model to minimize the number of vehicles on the road, subject to the following constraints:

$$ The percentage of urban vehicles with the highest values of PCUs must be at least $40\%.$

$$ The percentage of rural vehicles with the highest values of PCUs must be at least $30\%.$

Assumptions

$$ Here, we are assuming that the total number of vehicles on the road must be at least $1000.$

$$ Here we are considering only $3$ categories as they are the only mostly moved vehicles.

$$ Heavy vehicles $=$ number of buses $+$ number of minibuses $+$ number of cars$/$jeeps$/$vans $(40\%)$

$$ Light vehicles $=$ number of two wheelers $+$ number of auto rickshaws $(30\%)$

$$ Commercial vehicles $=$ number of trucks $+$ number of MAVs $+$ number of LCVs $(30\%)$