Assignment Task
A cheese factory is making a new cheese from mixing two products A and B, each made of three different types of milk - sheep, cow, and goat. The compositions of A and B and prices ($/kg) are given as follows:
| Amount (litres) per 1000 kg of A and B | Sheep | Cow | Goat | Cost ($/kg) |
|---|---|---|---|---|
| A | 30 | 60 | 40 | 5 |
| B | 80 | 40 | 70 | 8 |
The recipes for the production of the new cheese require that there must be at least 45 litres of Cow milk and at least 50 litres of Goat milk per 1000 kg of cheese, but no more than 60 litres of Sheep milk per 1000 kg of cheese. The factory needs to produce at least 60 kg of cheese per week.
Explain why a linear programming model would be suitable for this case. Formulate a Linear Programming (LP) model for the factory that minimizes the total cost of producing the cheese while satisfying all constraints. Use the graphical method to find the optimal solution. Show the feasible region and the optimal solution on the graph. Annotate all lines on your graph. What is the minimal cost for the product? Is there a range for the cost ($) of A that can be changed without affecting the optimum point obtained above?
A food factory makes three types of cereals, A, B, and C, from a mix of several ingredients: Oates, Apricots, Coconuts, and Hazelnuts. The cereals are packaged in 1 kg boxes. The following table provides details of the sales price per box of cereals and the production cost per ton (1000 kg) of cereals:
| Cereal | Sales price per box ($) | Production cost per ton |
|---|---|---|
| A | 2.50 | 4.00 |
| B | 2.00 | 2.80 |
| C | 3.50 | 3.00 |
The following table provides the purchase price per ton of ingredients and the maximum availability of the ingredients in tons respectively:
| Ingredients | Purchase price ($) per ton | Maximum availability in tons |
|---|---|---|
| Oates | 100 | 10 |
| Apricots | 120 | 5 |
| Coconuts | 80 | 2 |
| Hazelnuts | 200 | 2 |
The minimum daily demand (in boxes) for each cereal and the proportion of the Oates, Apricots, Coconut, and Hazelnuts in each cereal is detailed in the following table:
| Cereal | Minimum demand (boxes) | Proportion of Oates | Proportion of Apricots | Proportion of Coconuts | Proportion of Hazelnuts |
|---|---|---|---|---|---|
| A | 1000 | 0.8 | 0.1 | 0.05 | 0.05 |
| B | 700 | 0.65 | 0.2 | 0.05 | 0.1 |
| C | 750 | 0.5 | 0.1 | 0.1 | 0.3 |
Let xij≥0x_{ij} geq 0xij≥0 be a decision variable that denotes the number of kg of ingredient iii, where iii could be Oates, Apricots, Coconuts, Hazelnuts, used to produce Cereal jjj, here jjj is one of A, B, C (in boxes). Formulate an LP model to determine the optimal production mix of cereals and the associated amounts of ingredients that maximize the profit while satisfying the constraints. Solve the model in R/R. Find the optimal profit and optimal values of the decision variables.
Two mining companies, Red and Blue, bid for the right to drill. The possible bids are $15 Million, $25 Million, $35 Million, $45 Million, and $50 Million. The winner is the company with the higher bid. The two companies decide that in the case of a tie (equal bids), Red is the winner and will get the field. Company Red has ordered a geological survey and, based on the report from the survey, concludes that getting the field for more than $45 Million is as bad as not getting it (assume loss), except in the case of a tie (assume win).
State reasons why/how this game can be described as a two-player zero-sum game. Considering all possible combinations of bids, formulate the payoff matrix for the game. Explain what is a saddle point. Verify: does the game have a saddle point? Construct a linear programming model for Company Blue in this context. Produce an appropriate code to solve the linear programming model. Solve the game for Blue using the linear programming model and the code you constructed. Interpret your solution.
