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Purpose:
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 zero sum 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.

A factory produces dresses and coats for a chain of departmental  stores in Victoria. The stores will accept all the production supplied  by the factory. The production process includes Cutting, Sewing and  Packaging, in this order. You can assume that each worker participates  in one operation only (Cutting, Sewing, or Packaging). The factory  employs 25 workers in the cutting department, 52 workers in the sewing  department and 14 work-ers in the packaging department. The factory  works 8 hours a day (these are productive hours). There is a daily  demand for at least 120 dresses, and no specific demand for the coats.  The table below gives the time requirements (in minutes) and profit per  unit for the two garments to be produced.

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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
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minutes per unit

Cutting

Sewing

Packaging

Unit profit (\$)

Dresses

25

25

15

8

Coats

12

55

15

15

Explain why a linear programming model would be suitable for this case study.

Formulate a Linear Programming model to help the management of the  factory de-termine the optimal daily production schedule, that is, find  the number of dresses and coats to be produced that would maximize the  profit.

Use the graphical method to find the optimum solution. Show the  feasible region and the optimal solution on the graph. Annotate your  graph. What is the optimum

Find a range for the profit (\$) of a dress that can be changed  without a?ecting the optimum solution obtained above.A food producer  makes three types of cereals A, B, and C from a mix of several  ingredients Oates, Raisins, Apricots and Hazelnuts. The cereals are  produced in 2kg boxes. The following table provides details of the sales  price per box of cereals and the production cost per ton (1000 kg) of  cereals respectively.
A food producer makes three types of cereals A, B, and C from a mix  of several ingredients Oates, Raisins, Apricots and Hazelnuts. The  cereals are produced in 2kg boxes. The following table provides details  of the sales price per box of cereals and the production cost per ton  (1000 kg) of cereals respectively.

Sales price per box

Production cost per ton

Cereal A

\$2.60

\$4.20

Cereal B

\$2.30

\$2.60

Cereal C

\$3.20

\$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

Raisins

\$90

5

Apricots

\$110

2

Hazelnuts

\$200

2

The minimum daily demand (in boxes) for  each cereal and the proportion of the Oates, Raisins, Apricots and  Hazelnuts in each cereal is detailed in the following table.

proportion of

Minimum demand (boxes)

Oates

Raisins

Apricots

Hazelnuts

Cereal A

1000

0.80

0.10

0.05

0.05

Cereal B

800

0.60

0.25

0.05

0.10

Cereal C

750

0.45

0.15

0.10

0.30

Choose appropriate decision variables. Formulate a linear  programming (LP) mod-el to determine the optimal production mix of  cereals that maximises the profit, while satisfying the constraints.  Then compute the associated amounts of ingredients for each cereal.
b) Find the optimal solution using R/R studio.
John and Alice are playing a game by putting chips in two piles  (each player has two piles P1 and P2), respectively. Alice has 4 chips  and John has 5 chips. Each player place 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 Johns P1 vs Alices P1, Johns P1 vs Alices P2, Johns P2 vs  Alices P1, and Johns P2 vs Alices P2. For each comparison, the  player with more chips in the pile will score 1 point (the opponent will  loose 1 point). If the number of chips are 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  Alice puts 4 and 0 chips in her P1 and P2, John puts 1 and 4 chips in  his P1 and P2, respectively. Then Alice will get 1 (4 vs 1) + 0 (4 vs 4)  – 1 (0 vs 1) – 1 (0 vs 4) = -1 as her final score, and John will get  his final score of 1.

(a) Give reasons why/how this game can be described as two-players-zero-sum game.
(b) Formulate the payo? 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 this game;
(e) Produce an appropriate code to solve the linear programming model in part (c).
(f) Solve the game for Alice using the linear programming model you constructed in part (c). Interpret your solution.

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