Assignment 4

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Sheet1

BUS404 Assignment #4
Maximise 45 X + 30 Y This table is only used in the creation of the graph
Subject to
Constraint #1 3 X + 1.5 Y ≤ 400 X C1 C2 C3 C4 C5
Constraint #2 4 X + 3 Y ≤ 600 C1 0 266.6666666667
Constraint #3 1 X ≥ 20 C1 133.3333333333 0
Constraint #4 1 Y ≥ 25 C2 0 200
Constraint #5 3 X + 4 Y ≤ 720 C2 150 0
C3 20 0
Constraint #1 3 X + 1.5 Y = 400 C3 20 200
1.5 Y = -3 X + 400 C4 0 25
Y = -2 X + 266.6666666667 C4 240 25
C5 0 180
Constraint #2 4 X + 3 Y ≤ 600 C5 240 0
3 Y = -4 X + 600
Y = -1.3333333333 X + 200
Constraint #3 X = 20
Constraint #4 Y = 25
Constraint #5 3 X + 4 Y ≤ 720
4 Y = -3 X + 720
Y = -0.75 X + 180
Objective function slope -1.5
This is between the slopes of constraints 1 and 2, so these are the binding constraints.
Equate the two constraint lines to get the optimal decision.
Y = -2 X + 266.6666666667 = -1.33 X + 200
∴ 66.6666666667 = 0.67 X
∴ X = 100
Y = -1.33 X + 200
∴ Y = 66.6666666667
Profit $6,500.00
b. Constraints 1 and 2 are binding, ∴ slack is zero.
Constraint #3 value= 100
∴ slack is 80
Constraint #4 value 66.67
∴ slack is 41.67
Constraint #5 value 566.67
∴ slack is 153.33
c. Objective function slope = -C₁/30, currently -45/30
C₁ can increase until the slope matches that of constraint #1, i.e. = -2
At that stage C₁ = 60
C₁ can decrease until the slope matches the slope of constraint #2.
At that stage C₁ = 40
Allowable decrease 5
d. 3 X + 1.5 Y ≤ 401
∴ 1.5 Y = -3 X + 401
∴ Y = -2 X + 267.3333333333
Y = -2 X + 267.3333333333 = -1.3333333333 X + 200
∴ 0.6666666667 X = 67.3333333333
∴ X = 101
Y = -1.3333333333 X + 200
∴ Y = -1.3333333333 * 101 + 200
∴ Y = 65.3333333333
Profit $6,505.00
Shadow price $5.00
Allowable increase - until junction of constraints 2 and 4
Y = -1.3333333333 X + 200
Y = 25
∴ 1.3333333333 X = 200 - Y
∴ 1.3333333333 X = 175
∴ X = 131.25
Constraint #1 RHS 431.25
Allowable increase 31.25
Allowable decrease - until junction of constraints 2 and 5
Y = -1.3333333333 X + 200 = -0.75 X + 180
∴ 0.5833333333 X = 20
∴ X = 34.2857142857
∴ Y = 154.2857142857
Constraint #1 RHS 334.29
Allowable decrease 65.71
C1 0 133.33333333333334 0 150 20 20 0 240 0 240 266.66666666666669 0 C2 0 133.33333333333334 0 150 20 20 0 240 0 240 200 0 C3 0 133.33333333333334 0 150 20 20 0 240 0 240 0 200 C4 0 133.33333333333334 0 150 20 20 0 240 0 240 25 25 C5 0 133.33333333333334 0 150 20 20 0 240 0 240 180 0