Reference no: EM132379415
SIT718 Real World Analytics Assessment Task - Problem Solving, Deakin University, Australia
Learning Outcomes - This assessment assesses the following Unit Learning Outcomes (ULO) and related Graduate Learning Outcomes (GLO):
Unit Learning Outcome (ULO) -
- Assessed through student ability to apply game theory, and linear programming skills and models, to make optimal decisions.
- Assessed through student ability to develop software codes to solve computational problems for real world analytics.
Graduate Learning Outcome (GLO) -
- Discipline knowledge and capabilities
- Critical thinking
- Problem solving
Purpose - Assignment assesses your abilities to build linear programming models and solving them. You will demonstrate your skills in using linear programming to model real life case studies. You will consider cases with two variables and solve them using the graphical method. For problems with more than two variables, you will solve the linear programming models that you built using linear programming solvers in appropriate software, such as R. You will consider game theory - two players game - to build appropriate models that describe different game scenarios. You will demonstrate your knowledge in investigating the existence of equilibrium (stable solution). You will use mixed models to find appropriate solutions and solve the models you constructed with appropriate software such as R.
Instructions - The assessment consists of FOUR question sets. Students must attempt all tasks and provide an individual written report in appropriate word processor.
Your final submission must include the following two files:
1. "name-report.pdf"A report, in pdf format (created in any word processor), covering all of the items in above (where "name" is replaced with your name -you can use your surname or first name). With plots and tables, it should be up to 8 pages.
2. "name-code.R"The R code file (that you have written to produce your results) named(where "name" is replaced with your name - you can use your surname or first name). Your assignment will not be assessed if we cannot reproduce your results with your R code.
Question 1 - A food factory is making a beverage for a customer from mixing two different existing products A and B. The compositions of A and B and prices ($/L) are given as follows,
|
Amount (L) in / 100 L of A and B
|
Cost ($/L)
|
|
Lime
|
Orange
|
Mango
|
|
A
|
3
|
6
|
4
|
5
|
|
B
|
8
|
4
|
6
|
7
|
The customer requires that there must be at least 4.5 Litres (L) Orange and at least 5 Litres of Mango concentrate per 100 Litres of the beverage respectively, but no more than 6 Litres of Lime concentrate per 100 Litres of beverage. The customer needs at least 80 Litres of the beverage per week.
a) Explain why a linear programming model would be suitable for this case study.
b) Formulate a Linear Programming (LP) model for the factory that minimises the total cost of producing the beverage while satisfying all constraints.
c) 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?
Note: you can use graphical solvers available online but make sure that your graph is clear, all variables involved are clearly represented and annotated, and each line is clearly marked and related to the corresponding equation.
d) Is there a range for the cost ($) of A that can be changed without affecting the optimum solution obtained above?
Question 2 - A factory makes three products called Spring, Autumn, and Winter, from three materials containing Cotton, Wool and Silk. The following table provides details on the sales price, production cost and purchase cost per ton of products and materials respectively.
|
Sales price
|
Production cost
|
|
Purchase price
|
|
Spring
|
$60
|
$5
|
Cotton
|
$30
|
|
Autumn
|
$55
|
$4
|
Wool
|
$45
|
|
Winter
|
$60
|
$5
|
Silk
|
$50
|
The maximal demand (in tons) for each product, the minimum cotton and wool proportion in each product is as follows:
|
Demand
|
min Cotton proportion
|
min Wool proportion
|
|
Spring
|
4500
|
50%
|
30%
|
|
Autumn
|
3000
|
50%
|
40%
|
|
Winter
|
3500
|
40%
|
50%
|
a) Formulate an LP model for the factory that maximises the profit, while satisfying the demand and the cotton and wool proportion constraints.
b) Solve the model using R/R Studio. Find the optimal profit and optimal values of the decision variables.
Hints: 1. Let xij ≥ 0 be a decision variable that denotes the number of tons of products j for j ∈ {1 = Spring; 2 = Autumn; 3 = Winter} to be produced from Materials i ∈ {C=Cotton, W=Wool, S=Silk}.
2. The proportion of a particular type of Material in a particular type of Product can be calculated as: e.g., the proportion of Cotton in product Spring is given by: xC1/(xC1 + xW1 + xS1).
Question 3 - Helen and David are playing a game by putting chips in two piles (each player has two piles P1 and P2), respectively. Helen has 6 chips and David has 5 chips. Each player places his/her chips in his/her two piles, then compare the number of chips in his/her two piles with that of the other player's two piles. Note that once a chip is placed in one pile it cannot be moved to another pile. There are four comparisons including Helen's P1 vs David's P1, Helen's P1 vs David's P2, Helen's P2 vs David's P1, and Helen's P2 vs David's P2. For each comparison, the player with more chips in the pile will score 5 point (the opponent will lose 5 point). If the number of chips is the same in the two piles, then nobody will score any points from this comparison. The final score of the game is the sum score over the four comparisons. For example, if Helen puts 5 and 1 chips in her P1 and P2, David puts 4 and 1 chips in his P1 and P2, respectively. Then Helen will get 5 (5 vs 4) + 5 (5 vs 1) - 5 (1 vs 4) + 0 (1 vs 1) = 5 as her final score, and David will get his final score of -5.
(a) Give reasons why/how this game can be described as a two-players-zero-sum game.
(b) Formulate the payoff matrix for the game.
(c) Explain what is a saddle point. Verify: does the game have a saddle point?
(d) Construct a linear programming model for each player in this game;
(e) Produce an appropriate code to solve the linear programming model in part (c).
(f) Solve the game for David using the linear programming model you constructed in part (c). Interpret your solution.
[Hint: To record the number of chips in each pile for each player you may use the notation (I, j), where i is the number of chips in P1 and j is the number of chips in P2, for example (2, 4) means two chips in P1 and four chips in P2.]
Question 4 - If there are two companies making the same model of cellphones. Assuming the demand for the cellphones produced by Company 1 is Q1, and the demand for the cellphones produced by Compnay 2 is Q2, are described by the following functions:
Q1 = 200 - P1 - (P1 - P-) (1)
Q2 = 200 - P2 - (P2 - P-) (2)
where P- is the average price over the prices of the two companies. Each company has the cost of C = 15 for producing one cellphone. Suppose each company can only choose one of the three prices {70, 80, 90} for sale.
(a) Compute the profits of each company under all sale price combinations and make the payoff matrix for the two companies. [Hint: the total profits = the demand for the cellphones x the profit of one cellphone after sale.]
(b) Find the Nash equilibrium of this game. What are the profits at this equilibrium? Explain your reason clearly.
(C) If the cost C = 40, would your the Nash equilibrium change? Why?