NEW POST MAT540

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week_8_solutions.xlsx
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P1.a&b

Graftann's Metal work company
Variables u: % of metal allocated to Ore 1 v: % of metal allocated to Ore 2
w: % of metal allocated to Ore 3 x: % of metal allocated to Ore 4
y: % of metal allocated to Ore 5 z: % of metal allocated to Ore 6
minimize: 27 u+25v+32w+22x+20y+24z
subject to 0.19u+0.43v+0.17w+0.2x+0y+0.12z >= 0.21 Metal A
0.15u+0.1v+0w+0.12x+0.24y+0.18z <= 0.12 Metal B
0.12u+0.25v+0w+0x+0.1y+0.16z <= 0.07 Metal C
0.14u+0.07v+0.53w+0.18x+0.31y+0.25z >= 0.3 Metal D1
0.14u+0.07v+0.53w+0.18x+0.31y+0.25z <= 0.65 Metal D2
0.6u+0.85v+0.7w+0.5x+0.65y+0.71z =1 Alloy
Ore Metal (%) Constraints
Cost/Ton A B C D1 D2 Alloy Impurities
1 0
Lenovo User: variables; changing cells 		27 0.19 0.15 0.12 0.14 0.14 0.6 0.4
2 0.1153468463 25 0.43 0.1 0.25 0.07 0.07 0.85 0.15
3 0.8486969659 32 0.17 0 0 0.53 0.53 0.7 0.3
4 0.0806118595 22 0.2 0.12 0 0.18 0.18 0.5 0.5
5 0.4116328843 20 0 0.24 0.1 0.31 0.31 0.65 0.35
6 0 24 0.12 0.18 0.16 0.25 0.25 0.71 0.29
Usage 0.21 0.12 0.07 0.6 0.6 1
>= <= <= => <= =
Available 0.21 0.12 0.07 0.3 0.65 1
Cost 40.0480926602
Lenovo User: objective

P2.a&b

congressman’s district
Variables J: $ allocated to Job P: $ allocated to Parks
S: $ allocated to Sanitation L: $ allocated to Library
maximize 0.02J + 0.09P + 0.06S + 0.04L
subject to J <= 1600000 Const 1
P <= 1600000 Const 2
S <= 1600000 Const 3
L <= 1600000 Const 4
P - S - L <= 0 Const 5
J - S => 0 Const 6
J + P + S + L = 4000000 Budget
Votes/$ Const 1 Const 2 Const 3 Const 4 Const 5 Const 6 Budget
Job 800000
Lenovo User: variables; changing cells 		0.02 1 1 1
Parks 1600000 0.09 1 1 1
Sanitation 800000 0.06 1 -1 -1 1
Library 800000 0.04 1 -1 1
Usage 800000 1600000 800000 800000 -0.0000000003 0 4000000
<= <= <= <= <= >= =
Available 1600000 1600000 1600000 1600000 0 0 4000000
Votes 240000
Lenovo User: objective

P3.a&b

Anna Broderick Lunch Menu
Variables t: contibution of chicken in menu
u: contibution of fish in menu v: contibution of ground beef in menu
w: contibution of dried beans in menu x: contibution of Lettuce in menu
y: contibution of potatoes in menu z: contibution ofmilk in menu
minimize: 0.8t+3.7u+2.3v+0.9w+0.75x+0.4y+0.83 z
subject to 520t+500u+860v+600w+50x+460y+240z>=1500 Calories 1
520t+500u+860v+600w+50x+460y+240z<=2000 Calories 2
4.4t+3.3u+0.3v+3.4w+0.5x+2.2y+0.2z >=5 Iron
17t+85u+82v+10w+6x+10y+16z >=30 Protein
0t+0u+0v+30w+0x+70y+22z>=40 Carbohydrates
30t+5u+75v+3w+0x+0y+10z >=20 Fat1
30t+5u+75v+3w+0x+0y+10z <=60 Fat2
180t+90u+350v+0w+0x+0y+20z<=30 Cholestrol
Cost Calories 1 Calories 2 Iron Protein Carbo-hydrates Fat 1 (g/lb.) Fat 2 (g/lb.) Chol-esterol
$/lb. (per lb.) (per lb.) (mg/lb.) (g/lb.) (g/lb.) (mg/lb.)
Chicken 0.8 520 520 4.4 17 0 30 30 180
Fish 3.7 500 500 3.3 85 0 5 5 90
Ground beef 2.3 860 860 0.3 82 0 75 75 350
Dried beans 0.9 600 600 3.4 10 30 3 3 0
Lettuce 0.75 50 50 0.5 6 0 0 0 0
Potatoes 0.4 460 460 2.2 10 70 0 0 0
Milk (2%) 0.83 240 240 0.2 16 22 10 10 20
1500.0000000017 1500.0000000017 6.6362318841 43.7101449276 104.304347827 20 20 30
Constraints >= <= >= >= >= >= <= <=
1500 2000 5 30 40 20 60 30
Decision Variables Chicken 0.00
Lenovo User: changing cells; variables 		<= 0.5
Fish 0.00 <= 0.5
Ground beef 0.00 <= 0.5
Dried beans 1.67 <= 0.5
Lettuce 0.00 <= 0.5
Potatoes 0.30 <= 0.5
Milk (2%) 1.50
Objective Minimize Cost ($) 2.87
Lenovo User: objective
c.
If a serving of each of the food items (other than milk) was limited to no more than a half pound:
Answer:
Decision Variables Chicken 0.00
Lenovo User: changing cells; variables
Lenovo User: changing cells; variables 		<= 0.5
Fish 0.00 <= 0.5
Ground beef 0.00 <= 0.5
Dried beans 0.50 <= 0.5
Lettuce 0.50 <= 0.5
Potatoes 0.50 <= 0.5
Milk (2%) 1.50
Objective Minimize Cost ($) 2.27
Lenovo User: objective

Lenovo User: objective		 The minimum cost will go down to $2.27 / pound.

P4.a&b

Cabin Creek Coal (CCC) Note: No quality constraints
Plant
Mine 1 2 3 4
1 $7 $9 $10 $12
2 $9 $7 $8 $12
3 $11 $14 $5 $7
DV Plant
Mine 1 2 3 4 Shipped from Mine Mine Capacity (tons) Cost (Processing & Mining)
1 110.00
Lenovo User: variables i.e. changing cells 		80.00 0.00 30.00 220.00 <= 220 62
2 0.00 80.00 90.00 0.00 170.00 <= 170 67
3 0.00 0.00 0.00 150.00 150.00 <= 280 75
Received by Plant 110.00 160.00 90.00 180.00
= = = =
Demand (tons) 110 160 90 180
Objective Function Minimize Cost $40,460.00
Lenovo User: objective

P5.a&b

Joe Henderson Assignments
1, 2 Operator Drill Press (min) Lathe (min) Grinder (min)
1 23 18 35
2 41 30 28
3 25 36 18
DV Operator Drill Press Lathe Grinder Operator Constraints Assignment
1 0
Lenovo User: Varibles i.e. changing cells 		1 0 1 = 1
2 0 0 1 1 = 1
3 1 0 0 1 = 1
Machine Constraints 1 1 1
= = =
1 1 1
Objective Minimize Time 71
Lenovo User: objective

Lenovo User: Varibles i.e. changing cells 		c.
Allocation at the optimal solution
Operators Machine Time required
Operator 1 Lathe 18
Operator 2 Grinder 28
Operator 3 Drill press 25
total 71
Operator Drill Press (min) Lathe (min) Grinder (min) Kelly (min)
1 23 18 35 20
2 41 30 28 20
3 25 36 18 20
Allocation if Joe hire kelly
Operators Machine Time required
Operator 1 Lathe 18
Operator 2 Nil 0
Operator 3 Grinder 18
Operator 4(kelly) Drill press 20
total 56
The minimum time will decrease to 56 minutes, therefore joe must hire kelly.

Kelly, who is a machine operator. Kelly can perform each of the three required machine operations in 20 minutes. Therfore kelly is an efficient worker and can be assigned at the place of operator 2 who is taking 28 minute. Now, operator 3 will be assigned to grinder as it will take the minimum of time and kelly would be assigned Drill press to minimize the total time.

P6.a&b

Cash and Carry Building Supply Company
The company determines how to cut the 25-foot boards to meet the order requirements and minimize the number of standard-length boards used
Note: There are 6 possible patterns you can cut the board (shown in #1 through #6)
Example #1 pattern cut the 25-foot board into 3 7-foot board (waste 25-3*7 = 4 feet)
Example #4 pattern cut 25-foot board into 1 7-foot, 2 9-foot boards (waste = 25-(1*7+2*9) = 0)
See table for the rest of the patterns
Variables: a : number of 25 ft boards cut using pattern #1 b: number of 25 ft boards cut using pattern #2
c: number of 25 ft boards cut using pattern #3 d: number of 25 ft boards cut using pattern #4
e : number of 25 ft boards cut using pattern #5 f: number of 25 ft boards cut using pattern #6
minimize a + b + c + d + e + f
subject to 3a + 2b + 2c + 1d = 700
1c + 2d + 1e = 1200
2b + 5e + 2f = 300
Objective Coeffcients 7 ft boards 9 ft boards 10 ft boards Wastage
Pattern #1 (a) 0
Lenovo User: changing cells 		1 3 0 0 4
Pattern #2 (b) 50.0000000011 1 2 0 1 1
Pattern #3 ( c) 0 1 2 1 0 2
Pattern #4 (d) 600 1 1 2 0 0
Pattern #5 (e) 0 1 0 1 1 6
Pattern #6 (f) 125 1 0 0 2 5
Usage 700.0000000023 1200 300.0000000011 675.0000000011
= = =
Available 700 1200 300
Total Standard (25 ft) boards 775.0000000011
Lenovo User: objective
c.
object: To minimize the wastage
Objective Coeffcients 7 ft boards 9 ft boards 10 ft boards Wastage
Pattern #1 (a) 0
Lenovo User: changing cells
Lenovo User: changing cells 		1 3 0 0 4
Pattern #2 (b) 50 1 2 0 1 1
Pattern #3 ( c) 0 1 2 1 0 2
Pattern #4 (d) 600 1 1 2 0 0
Pattern #5 (e) 0 1 0 1 1 6
Pattern #6 (f) 125 1 0 0 2 5
Usage 700 1200 300 675
SERVER: SERVER: objective function
= = =
Available 700 1200 300
Total Standard (25 ft) boards 775

There would be no effect on the solution and the solution will remain same.