Ben and Jerry's manufacturing system starts at the local Vermont farms which sell their raw milk to the St. Albans Cooperative Creamery, where the milk is then separated into heavy cream and shipped by tanker truck to the St. Albans and Waterbury Vermont factories. 

Ben and Jerry's are planning a fundraising event where they'll be selling their top 3 products which are the Chocolate Fudge Brownie, the Strawberry Cheesecake and the Mint Chocolate cookie. A batch is equivalent to 8 ice cream cups. 

A batch of Product 1 which is the Chocolate Fudge Brownie takes 8 hours to make at the St. Albans factory and 10 hours to make at the Waterbury Vermont factory. 

A batch of Product 2 which is the Strawberry Cheesecake takes 7 hours to make at the St. Albans factory and cannot be made at the Waterbury factory as they do not have the swirl machine. 

A batch of Product 3 which is the Mint Chocolate Cookie takes 9 hours to make at the St. Albans factory and 8 hours to make at the Waterbury factory. 

Considering that the St Albans factory operates for 96 hours a week and that the Waterbury Factory operates for 72 hours a week and that Ben and Jerry's will be preparing for the fundraiser for one month, how many batches of each product should they produce to maximize profits ?

 

Prod time (hours) per batch Prod time (hours) per batch Prod time (hours) Available per Prod time (hours) per batch Product 1 Product 2 Product 3 week Plant 1 8 7 96 Plant 2 10 8 72 Profit / 300 $ 400 $ 500 $ batch a) Explain the objective function and constraints and find the solution by the software (CPLEX) including postoptimality analysis (shadow prices and sensitivity analysis for LP problems only)