Discussion Question: $$Linear programming, a mathematical technique, has become an indispensable tool for optimizing decision$-$making processes in industries worldwide. It offers significant benefits and improves efficiency across diverse sectors$.$ Let's explore some real$-$world applications and discuss the advantages that linear programming brings to different industries. Choose only one industry for Module $6$ and a different one for Module $7($you may choose a different industry of interest to you if you prefer$).$

In the transportation industry, linear programming models can optimize routes and schedules to minimize costs and maximize efficiency. How can variables like distance, fuel consumption, and delivery schedules be incorporated into a linear programming model to find the most cost$-$effective transportation plan?

Financial institutions utilize linear programming for portfolio optimization. How can this technique help investment managers maximize returns while managing risk? Discuss the use of variables such as expected returns, risk measures, and constraints to develop a linear programming model that achieves the best portfolio composition.

In manufacturing, linear programming assists in optimizing production and resource allocation. How can variables like demand, production capacities, and resource constraints be incorporated into a linear programming model to maximize output and minimize costs?