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.
minutes per unit
Unit profit ($)
Explain why a linear programming model would be suitable for this case study.
Formulate a Linear Programming model to help the management of the factory de-termine the optimal daily production schedule, that is, find the number of dresses and coats to be produced that would maximize the profit.
Use the graphical method to find the optimum solution. Show the feasible region and the optimal solution on the graph. Annotate your graph. What is the optimum
Find a range for the profit ($) 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
The following table provides the purchase price per ton of ingredients and the maximum availability of the ingredients in tons respectively.
Purchase price per ton
Maximum availability in tons
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.
Minimum demand (boxes)
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 players two piles. Note that once a chip is placed in one pile it cannot be moved to another pile. There are four comparisons including Johns P1 vs Alices P1, Johns P1 vs Alices P2, Johns P2 vs Alices P1, and Johns P2 vs Alices 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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