SIT718 Real world Analytics Assignment-1 Total Marks = 100, Weighting - 15% Due date: 20 April 2017 by 11.30 PM Assignment (pdf or MS word doc) must be submitted via CloudDeakin's Assignment Dropbox. You can submit a (reasonably good resolution) scanned or an electronic version of your assignment (photos of written document are not accepted). No hard copy or email submissions are accepted. This assignment assesses : ULO1: Transform a real-life problem into a mathematical model. ULO2: Apply linear programming concepts to make optimal decisions. ULO3: Obtain optimal solutions for quantities that are either continuous or discrete. 1. Adam is making a special perfume, called Dream, for his wife Helen, from two different existing products Spellbound and Bewitched. The composition of Spellbound and Bewitched are given as follows. Amount (ml) in /100 ml of Spellbound and Bewitched Deer Musk Grapefruit Mercaptan Raspberry Ketone Cost ($/ml) Spellbound 3 7 3 10 Bewitched 10 2 6 8 Helen uses at least 40ml of perfume a year, and requires that there must be at least 3ml Grapefruit Mercaptan and at least 4 ml of Raspberry Ketone in per 100ml of Dream Perfume respectively, but no more than 7.5 ml of Deer Musk in per 100 ml of Dream. Formulate a Linear Programming (LP) model for Adam that minimizes the total cost of producing Dream, while satisfying all constraints. Use the graphical method to find the optimal solution. Show the feasible region and the optimal solution on the well-annotated graph. What is the minimal cost? [20 marks] 1SIT718 Assignments 2017 { T1 2 of 3 2. Haus Alloys are blending three types of alloys: Alloy A, Alloy B, and Alloy C from steel, brass, and phosphor bronze. The following table provides details on the sales price and purchase cost per ton of alloys and metals respectively. Sales price per ton Production cost per ton Purchase price per ton Alloy A $60 $4 Steel $50 Alloy B $70 $2.8 Brass $40 Alloy C $45 $2.8 Bronze $30 The maximum demand (in tons) for each alloy, the minimum steel and brass proportion in each alloy is detailed in the following table. max demand min steel proportion min brass proportion Alloy A 4000 0.6 0.3 Alloy B 3800 0.65 0.25 Alloy C 3500 0.3 0.4 Let xij ≥ 0 be a decision variable that denotes the number of tons of Alloy j for j 2 fA; B; Cg to be produced from Metals i 2 fSteel, Brass, Bronzeg. Formulate an LP model for the Haus that maximizes the profit, while satisfying the demand as well as the steel and brass proportion constraints. [Hints: 1. Let xij be a decision variable that indicates the number of tons of Metal i be blended into Alloy j. 2. The proportion of a particular type of Metal in a particular type of Alloy can be calculated as below: E.g., the proportion of Steel in Alloy A is given by: xSteel;A xSteel;A + xBrass;A + xBronze;A .] [20 marks] 3. Vicky and David play a game. David holds 3 chips and Vicky holds 4. They allocate the chips into two piles, then compare the numbers in each pile and between the piles. If the number of chips in a pile from one player is more than that of the opponents, this player will score 1 point from this pile. If the players have the same number of chips in one pile, then nobody will score any points from this pile. If the number of points of one player from one pile is more than the number of points of the opponent from the other pile, then this player will score 1 point from the round (the opponent will loose one point). If one player has a number of points from one pile equal to the number of points of the opponent from the other pile, then both players will score zero points. (a) Formulate the payoff matrix for the game. Does the game have a saddle point? (b) Solve the game for David using linear programming. [Hint: To record the number of chips in each pile for a player you may use the notation (i; j), where i is the number of chips in pile 1 and j is the number of chips in pile 2, e.g. (2,1) means two chips in pile 1 and one chip in pile 2.] [20 marks]SIT718 Assignments 2017 { T1 3 of 3 4. (a) Consider the following network. Find the shortest path from Vertex 1 to Vertex 7. Show the workings. 8 7 6 5 2 4 3 1 7 4 1 1 1 3 2 2 2 2 3 5 5 6 6 8 (b) Consider the following network. Find the Early Event Time and Late Event Time for each vertex, and the Total Float for each arc. Hence identify the critical path from Vertex 1 to Vertex 7. 7 8 3 4 2 5 1 10 7 6 6 9 3 8 4 1 3 6 4 8 5 [20 marks] 5. Co-op store supplies milk to Midsomer Mallow. The demands for the next 4 days are 450, 250, 380, and 420 litres respectively, and the supply capacities for the next 4 days are 580, 520, 200, and 300 litres. The purchase price per litre of milk varies from day to day, and is estimated at $1.1, $1.4, $1.2, and $1.5 respectively. As milk is perishable, today's supply expires on the 4th day, (e.g., Monday's supply can no longer be consumed on Thursday). The storage cost per litre of milk is 3 cents a day. Model the problem for Co-op as a transportation problem. Write the cost matrix for the problem. Formulate an LP model and explain the variables. Solve the LP model using an LP solver. [20 marks]