## 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 |

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