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## Optimizing Daily Production Schedule for the Garment Factory

1. {20 Marks} Sam owns a large Garment factory and produces shirts and trousers for Coles supermarket stores in Victoria. Coles will accept all the production supplied by Sam. The production process includes cutting, sewing and packaging. Sam employs 22 workers in the cutting department, 50 workers in the sewing department and 12 workers in the packaging department. The garment factory works 8 hours a day (these are productive hours). There is a daily demand for at least 100 shirts. The table below gives the time requirements (in minutes) and profit per unit for the two garments. minutes per unit Cutting Sewing Packaging Unit profit (\$) Shirts 20 20 10 6 Trousers 10 50 10 10 a) Formulate a Linear Programming model to help Sam determine the optimal daily production schedule. b) Use the graphical method to find the optimum solution. Show the feasible region and the optimal solution on the graph. What is the optimum profit? c) Find a range for the profit (\$) of a shirt that can be changed without affecting the optimum solution obtained above. 1 SIT718 Assignment-1 (2016-T3) 2 of 3 2.

{20 Marks} A food factory makes three types of cereals A, B, and C from a mix of several ingredients Oates, raisins, coconuts and almonds. 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 \$2.50 \$4.00 Cereal B \$2.00 \$2.80 Cereal 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 Raisins \$80 5 Coconut \$120 2 Almonds \$200 2 The minimum daily demand (in boxes) for each cereal and the proportion of the Oates, raisins, coconut and almonds in each cereal is detailed in the following table. proportion of Minimum demand (boxes) Oates Raisins Coconut Almonds Cereal A 1000 0.8 0.1 0.05 0.05 Cereal B 700 0.65 0.2 0.05 0.1 Cereal C 750 0.5 0.1 0.1 0.3 a) Let xij ≥ 0 be a decision variable that denotes the kg of ingredient i ∈ {Oates, Raisins, Coconut, Almonds} used to produce the Cereal j ∈ {A, B, C} (in boxes).

Formulate a linear programming (LP) model to determine the optimal production mix of cereals and the associated amounts of ingredients that maximizes the profit, while satisfying the constraints. b) Find the optimal solution using the IBM CPLEX software or any other LP solver. 3. {20 Marks} Consider the following payoff table for a 2 player zero sum game having Player 1 and Player 2. Player 2 Strategy 1 2 3 4 5 Player 1 1 1 -3 2 -2 -1 2 2 2 -1 1 2 3 0 1 2 -3 -2 4 -3 0 -2 4 1 (a) Show that this game does not have a saddle point. State the range of the value of the game. SIT718 Assignment-1 (2016-T3) 3 of 3 (b) Formulate the linear programs and solve the game for Player 1 and Player 2. 4. {20 Marks}

 Engine: GRG Nonlinear Solution Time: 0.203 Seconds. Iterations: 11 Subproblems: 0 Max Time Unlimited,  Iterations Unlimited, Precision 0.000001, Use Automatic Scaling Convergence 0.0001, Population Size 100, Random Seed 0, Derivatives Forward, Require Bounds Max Subproblems Unlimited, Max Integer Sols Unlimited, Integer Tolerance 1%, Assume NonNegative Cell Name Original Value Final Value \$H\$13 Total profit Profit 0 29101.28 Cell Name Original Value Final Value Integer \$D\$4 Cereal A Oates 0 4.476115 Contin \$E\$4 Cereal A Raisins 0 16930.89 Contin \$F\$4 Cereal A Coconut 0 6064.997 Contin \$G\$4 Cereal A Almonds 0 1.6 Contin \$D\$5 Cereal B Oates 0 1.401601 Contin \$E\$5 Cereal B Raisins 0 6704.362 Contin \$F\$5 Cereal B Coconut 0 1161.854 Contin \$G\$5 Cereal B Almonds 0 1.239517 Contin \$D\$6 Cereal C Oates 0 19991.02 Contin \$E\$6 Cereal C Raisins 0 19660.38 Contin \$F\$6 Cereal C Coconut 0 16386.57 Contin \$G\$6 Cereal C Almonds 0 6665.987 Contin Cell Name Cell Value Formula Status Slack \$K\$10 Cereal A Total production 2000 \$K\$10>=\$M\$10 Binding 0 \$K\$11 Cereal B Total production 1400 \$K\$11>=\$M\$11 Binding 0 \$K\$12 Cereal C Total production 15600 \$K\$12>=\$M\$12 Not Binding 14100 \$K\$15 Oates Total production 10000 \$K\$15<=\$M\$15 Binding 0 \$K\$16 Raisins Total production 5000 \$K\$16<=\$M\$16 Binding 0 \$K\$17 Coconut Total production 2000 \$K\$17<=\$M\$17 Binding 0 \$K\$18 Almonds Total production 2000 \$K\$18<=\$M\$18 Binding 0 \$D\$4 Cereal A Oates 4.476115 \$D\$4>=0 Not Binding 4.476115 \$E\$4 Cereal A Raisins 16930.89 \$E\$4>=0 Not Binding 16930.89 \$F\$4 Cereal A Coconut 6064.997 \$F\$4>=0 Not Binding 6064.997 \$G\$4 Cereal A Almonds 1.6 \$G\$4>=0 Not Binding 1.6 \$D\$5 Cereal B Oates 1.401601 \$D\$5>=0 Not Binding 1.401601 \$E\$5 Cereal B Raisins 6704.362 \$E\$5>=0 Not Binding 6704.362 \$F\$5 Cereal B Coconut 1161.854 \$F\$5>=0 Not Binding 1161.854 \$G\$5 Cereal B Almonds 1.239517 \$G\$5>=0 Not Binding 1.239517 \$D\$6 Cereal C Oates 19991.02 \$D\$6>=0 Not Binding 19991.02 \$E\$6 Cereal C Raisins 19660.38 \$E\$6>=0 Not Binding 19660.38 \$F\$6 Cereal C Coconut 16386.57 \$F\$6>=0 Not Binding 16386.57 \$G\$6 Cereal C Almonds 6665.987 \$G\$6>=0 Not Binding 6665.987 Ingredients Purchased needs to be done [KG] Proportion of Ingredients Oates Raisins Coconut Almonds Oates Raisins Coconut Almonds Cereal A 4.476115449 16930.89 6064.997 1.6 Cereal A 0.8 0.1 0.05 0.05 Cereal B 1.401600877 6704.362 1161.854 1.239517 Cereal B 0.65 0.2 0.05 0.1 Cereal C 19991.01614 19660.38 16386.57 6665.987 Cereal C 0.5 0.1 0.1 0.3 Production Constraints Sales Price Production Cost Material Cost Profit Total production Cereal A 2500 8 172.2112 2319.789 Cereal A 2000 ≥ 2000 Cereal B 1400 3.92 114.3568 1281.723 Cereal B 1400 ≥ 1400 Cereal C 27300 46.8 1753.432 25499.77 Cereal C 15600 ≥ 1500 Total profit 29101.28 Material constraints Oates 10000 ≤ 10000 Raisins 5000 ≤ 5000 Coconut 2000 ≤ 2000 Almonds 2000 ≤ 2000 Engine: GRG Nonlinear Solution Time: 0.078 Seconds. Iterations: 3 Subproblems: 0 Max Time Unlimited,  Iterations Unlimited, Precision 0.000001, Use Automatic Scaling Convergence 0.0001, Population Size 100, Random Seed 0, Derivatives Forward, Require Bounds Max Subproblems Unlimited, Max Integer Sols Unlimited, Integer Tolerance 1%, Assume NonNegative Cell Name Original Value Final Value \$N\$9 Max y1 4 7.793103 Cell Name Original Value Final Value Integer \$N\$8 Optimal Sol y1 0 0.448276 Contin \$O\$8 Optimal Sol y2 0 0.758621 Contin \$P\$8 Optimal Sol y3 2 4 Contin \$Q\$8 Optimal Sol y4 1 2.586207 Contin \$R\$8 Optimal Sol y5 1 0 Contin Cell Name Cell Value Formula Status Slack \$S\$12 Player 2 1 \$S\$12<=\$U\$12 Binding 0 \$S\$13 Strategy 1 \$S\$13<=\$U\$13 Binding 0 \$S\$14 p1 1 \$S\$14<=\$U\$14 Binding 0 \$S\$15 p2 1 \$S\$15<=\$U\$15 Binding 0 \$N\$8 Optimal Sol y1 0.448276 \$N\$8>=0 Not Binding 0.448276 \$O\$8 Optimal Sol y2 0.758621 \$O\$8>=0 Not Binding 0.758621 \$P\$8 Optimal Sol y3 4 \$P\$8>=0 Not Binding 4 \$Q\$8 Optimal Sol y4 2.586207 \$Q\$8>=0 Not Binding 2.586207 \$R\$8 Optimal Sol y5 0 \$R\$8>=0 Binding 0 Player 2 Strategy 1 2 3 4 5 Player 1 1 1 -3 2 -2 -1 -3 2 2 2 -1 1 2 -1 3 0 1 2 -3 -2 -3 y1 y2 y3 y4 y5 4 -3 0 -2 4 1 -3 Coefficient 1 1 1 1 1 2 2 2 4 2 Optimal Sol 0.448276 0.758621 4 2.586207 0 Max 7.793103 constraints Player 2 1 -3 2 -2 -1 1 <= Strategy 1 2 3 4 5 2 2 -1 1 2 1 <= Player 1 1 3 -1 4 0 1 p1 0 1 2 -3 -2 1 <= 2 4 4 1 3 4 p2 -3 0 -2 4 1 1 <= 3 2 3 4 -1 0 p3 4 -1 2 0 6 3 p4 q1 q2 q3 q4 q5 q1 0.057522 q2 0.097345 q3 0.513274 q4 0.331858 q5 0 Engine: GRG Nonlinear Solution Time: 0.109 Seconds. Iterations: 3 Subproblems: 2 Max Time Unlimited,  Iterations Unlimited, Precision 0.000001, Use Automatic Scaling Convergence 0.0001, Population Size 100, Random Seed 0, Derivatives Forward, Require Bounds Max Subproblems Unlimited, Max Integer Sols Unlimited, Integer Tolerance 1%, Assume NonNegative Cell Name Original Value Final Value \$E\$20 Min X1 46 46 Cell Name Original Value Final Value Integer \$E\$19 X1 4.850626 5 Integer \$F\$19 X2 14.85063 15 Integer \$G\$19 X3 18.85063 19 Integer \$H\$19 X4 24.85063 25 Integer \$I\$19 X5 26.85063 27 Integer \$J\$19 X6 32.85063 33 Integer \$K\$19 X7 40.85063 41 Integer \$L\$19 X8 50.85063 51 Integer Cell Name Cell Value Formula Status Slack \$F\$22 X2-x1>=10 X2 10 \$F\$22>=\$H\$22 Binding 0 \$F\$23 X3-x1>=12 X2 14 \$F\$23>=\$H\$23 Not Binding 2 \$F\$24 X4-x1>=8 X2 20 \$F\$24>=\$H\$24 Not Binding 12 \$F\$25 X3-x2>=4 X2 4 \$F\$25>=\$H\$25 Binding 0 \$F\$26 X4-x3>=6 X2 6 \$F\$26>=\$H\$26 Binding 0 \$F\$27 X5-x2>=8 X2 12 \$F\$27>=\$H\$27 Not Binding 4 \$F\$28 X6-x4>=8 X2 8 \$F\$28>=\$H\$28 Binding 0 \$F\$29 X6-x5>=6 X2 6 \$F\$29>=\$H\$29 Binding 0 \$F\$30 X7-x6>=8 X2 8 \$F\$30>=\$H\$30 Binding 0 \$F\$31 X8-x5>=12 X2 24 \$F\$31>=\$H\$31 Not Binding 12 \$F\$32 X8-x7>=10 X2 10 \$F\$32>=\$H\$32 Binding 0 \$E\$19:\$L\$19=Integer Path Duration Early Event Time EF LS Late Event time Total Float 1-2 A 10 0 10 14 24 14 1-3 B 12 0 12 0 12 0 1-4 C 8 0 8 2 10 2 2-3 D 4 10 14 8 12 -2 2-5 E 8 10 18 24 32 14 3-4 F 6 12 18 12 18 0 4-6 G 8 18 26 18 26 0 5-6 H 6 18 24 20 26 2 5-8 I 12 18 30 32 44 14 6 -7 J 8 26 34 26 34 0 7-8 K 10 34 44 34 44 0 X1 X2 X3 X4 X5 X6 X7 X8 5 15 19 25 27 33 41 51 Min 46 X2-x1>=10 10 >= 10 X3-x1>=12 14 >= 12 X4-x1>=8 20 >= 8 X3-x2>=4 4 >= 4 X4-x3>=6 6 >= 6 X5-x2>=8 12 >= 8 X5-X4>=0 2 >= 0 X6-x4>=8 8 >= 8 X6-x5>=6 6 >= 6 X7-x6>=8 8 >= 8 X8-x5>=12 24 >= 12 X8-x7>=10 10 >= 10 Engine: GRG Nonlinear Solution Time: 0.078 Seconds. Iterations: 0 Subproblems: 0 Max Time Unlimited,  Iterations Unlimited, Precision 0.000001, Use Automatic Scaling Convergence 0.0001, Population Size 100, Random Seed 0, Derivatives Forward, Require Bounds Max Subproblems Unlimited, Max Integer Sols Unlimited, Integer Tolerance 1%, Assume NonNegative Cell Name Original Value Final Value \$J\$13 Total Profit = 29250 29250 Cell Name Original Value Final Value Integer \$J\$5 F1 M1 0 0 Contin \$K\$5 F1 M2 0 0 Contin \$L\$5 F1 M3 1000 1000 Contin \$J\$6 F2 M1 500 500 Contin \$K\$6 F2 M2 1000 1000 Contin \$L\$6 F2 M3 0 0 Contin \$J\$7 F3 M1 1500 1500 Contin \$K\$7 F3 M2 0 0 Contin \$L\$7 F3 M3 0 0 Contin \$J\$8 F4 M1 0 0 Contin \$K\$8 F4 M2 500 500 Contin \$L\$8 F4 M3 1500 1500 Contin Cell Name Cell Value Formula Status Slack \$J\$9 M1 2000 \$J\$9=\$J\$11 Binding 0 \$K\$9 M2 1500 \$K\$9<=\$K\$11 Not Binding 1.12E-06 \$L\$9 M3 2500 \$L\$9<=\$L\$11 Not Binding 1500 \$M\$5 F1 1000 \$M\$5=\$O\$5 Binding 0 \$M\$6 F2 1500 \$M\$6=\$O\$6 Binding 0 \$M\$7 F3 1500 \$M\$7=\$O\$7 Binding 0 \$M\$8 F4 2000 \$M\$8=\$O\$8 Binding 0 M1 M2 M3 M1 M2 M3 F1 0 0 1000 1000 = 1000 F1 \$1.00 \$1.50 \$2.00 F2 500.0000073 1000 0 1500 = 1500 F2 \$2.50 \$4.00 \$1.00 F3 1499.999999 0 0 1500 = 1500 F3 \$3.00 \$4.00 \$2.00 F4 0 500 1500 2000 = 2000 F4 \$2.50 \$5.00 \$3.00 2000.000006 1500 2500 = <= <= 2000 1500 4000 Total Profit \$29,250.00
Cite This Work

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My Assignment Help (2020) Optimizing Linear Programming For Garment & Cereal Factories And 2-Player Zero Sum Game Theory. [Online]. Available from: https://myassignmenthelp.com/free-samples/sit718-real-world-analytics/linear-programming.html
[Accessed 24 May 2024].

My Assignment Help. 'Optimizing Linear Programming For Garment & Cereal Factories And 2-Player Zero Sum Game Theory.' (My Assignment Help, 2020) <https://myassignmenthelp.com/free-samples/sit718-real-world-analytics/linear-programming.html> accessed 24 May 2024.

My Assignment Help. Optimizing Linear Programming For Garment & Cereal Factories And 2-Player Zero Sum Game Theory. [Internet]. My Assignment Help. 2020 [cited 24 May 2024]. Available from: https://myassignmenthelp.com/free-samples/sit718-real-world-analytics/linear-programming.html.

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