A food producer has two processing plants, P1 and P2, in New South Wales. The producer operates 5 days a week. The cost of running plant P1 is $3000 per day. The cost of running plant P2 is $2000 per day. After processing, food is graded into three grades: Grade I, Grade II and Grade III. The producer has contracted to provide each week the following quantities to the bulk buyer.

Grade I: 200kg

Grade II: 200kg

Grade III: 300kg

Each day, plant P1 processes 100kg of Grade 1, 160kg of Grade II and 100kg of Grade III food. The corresponding quantities for plant P2 are 100kg, 50kg and 300kg, respectively. The objective of the producer is to fulfil the contract with a minimum cost.

Use the information given above and set up the linear programming (LP) problem.

Using the linear programming problem, you have set up, answer the following questions:

Calculate and plot all the constraints and clearly shade the feasible region. Clearly mark and label all the constraints and corner point solutions on the graph. Show all the workings and calculations.

Calculate the cost minimizing number of days for which each plant should be operated per week to fulfil the contract? (fractional days allowed).

What is the minimum cost the food producer incurs?