Q1 A company currently produces 3 products (A, B, C) with overall profit as follows:

The requirements are that:

There should be at least 100 A, 20 B, and 50 C. A total of 350 items should be made. How would be the optimal quantities to maximize profit? Describe your findings.

Q2 A company wishes to introduce a packaged trail mix as a new product Ingredients: seeds, raisins, flakes, pecans, walnuts, each of which contains some amount of vitamins, minerals, protein, and calories and costs some amount per pound. The marketing department has specified that a minimum nutritional profile is met for each bag. How can this be achieved at minimum production cost? Based on the shadow price, which constraint has the greatest impact on cost? How does it affect the objective?

Q3 In World War I, a soldier named Hans has been dropped behind enemy lines. He is posing as a farmer who makes and sells butter by the ton, but he is really helping the resistance by secretly building and selling them guns. In order to not be suspected for helping the resistance he must produce at least some butter to keep up his disguise as a farmer. That means he cannot spend all of his resources on only guns.

In addition to hiding his covert actions, he has two key limitations. First, he has a limit on his storage space of 21 units. He can only store a limited number of guns and tons of butter. He also has a limited budget of $1,800 to buy supplies. It costs $150 to produce each gun and $100 for each ton of butter. Currently, the price for gun is $195 and each ton of butter is $150. He clearly wants to maximize his revenue to continue the mission. Given his limits on storage space and money, what is the optimal number of guns and tons of butter that Hans should produce?

Complete the initial pass and determine the optimal number of guns or tons of butter, in whole number, to produce to maximize Hans' revenue.

As the war goes on, there have been some changes in his circumstances. His budget has increased from $1,800 to $2,400. And he found a second warehouse that allows him to double his available warehouse space from 21 to 42. But the revenue for a ton of butter has dropped to $125. However, he must continue to produce some tons of butter, or the enemy will know that he is up to something. He needs to produce at least 5 tons of butter. How many guns and tons of butter, in whole number, can he now produce? Create a copy and run the analysis with the new circumstances.

Q4 A brand manager for ColPal Products must determine how much time to allocate between radio and television advertising during the next month. Market research has provided estimates of the audience exposure for each minute of advertising in each medium, which it would like to maximize. Costs per minute of advertising are also known, and the manager has a limited budget of $25,000. The manager has decided that because television ads have been found to be much more effective than radio ads, at least 70% of the time should be allocated to television. Suppose we have the following data:

Formulate a linear optimization model and identify the optimal solution set.

Q5 A company has three employees (A, B, C) and four machines (x1, x2, x3, x4). It wishes to assign the employees to machines to minimize total costs. The cost matrix showing the cost incurred by each employee on each machine is given below.

Determine the optimal assignment and calculate the total minimum cost.