The Shortest Path to
Destruction: A Floral Shop
Kathedra Burton
Mari Juno
Casey Haley
December 3, 2004
Overview
•
•
•
•
Grummins Engine
Wheat Stock
Floral Dilemma
Conclusion
Overview
•
•
•
•
•
Look at 3 linear programming problems
Formulation
Optimal Solution
Sensitivity Analysis
Results
Grummins Engine
• Truck Co. needs to determine the number of
trucks it will produce over a 3-yr. period
• Decision variables
– Xij = number of trucks made
– Yij = number of trucks sold
– Zij = number of trucks in inventory and carried over
where i = year made from 1 to 3 and j = type of truck 1
and 2
• Solution found in 10 iterations
Formulation
Max 20000y11 + 20000y21 + 20000y31 + 17000y12 + 17000y22
+ 17000y32 – 15000x11 – 15000x21 – 15000x31 – 14000x12 –
14000x22 – 14000x32 - 2000z11 – 2000z12 - 2000z21 2000z22
s.t.
y12
< 200
y11
< 100
x11-y11-z11
=0
x11+x12
< 320
x12-y12-z12
=0
x21+x22
< 320
z12+y22
< 100
z11+y21
< 200
x21+z11-y21-z21 = 0
x22+z12-y22-z22 = 0
z21+y31
< 300
x31+x32
< 320
x31+z21-y31
=0
z22+y32
< 150
x32+z22-y32
=0
5y11+5y21+5y31-5y12-5y22-5y32 < 0
Optimal Solution
Year
Truck Type
Number
Produced
Number Sold
Number
Inventoried
1
1
100
100
0
2
200
200
0
1
180
180
0
2
100
100
0
1
170
170
0
2
150
150
0
2
3
Sensitivity Analysis
VARIABLE
Y11
Y21
Y31
Y12
Y22
Y32
X11
X21
CURRENT
ALLOWABLE
ALLOWABLE
COEF
INCREASE
DECREASE
20000.000000
INFINITY
0.000000
20000.000000
0.000000
2000.000000
20000.000000
2000.000000
0.000000
17000.000000
INFINITY
8000.000000
17000.000000
INFINITY
8000.000000
17000.000000
INFINITY
8000.000000
-15000.000000
2000.000000
0.000000
-15000.000000
0.000000
2000.000000
Wheat Stock
• Warehouse needs to max profit by selling
and buying wheat
• Decision Variables
– Xi = initial stock of wheat
– Yi = amount of wheat sold
– Zi = amount of wheat bought
• Solution found in 20 iterations
Formulation
Maximize 3y1 + 6y2 + 7y3 + y4 + 4y5 + 5y6 + 5y7 + y8 + 3y9 + 2y10 8z1 - 8z2- 2z3 - 3z4 - 4z5 - 3z6 - 3z7 - 2z8 - 5z9 - 5z10
s.t.
z1+x1-y1 < 20
x1 = 6
y1-x1 < 0
z2+x2-y2 < 20
x2-z1-x1+y1 = 0
y2-x2 < 0
z3+x3-y3 < 20
x3-z2-x2+y2 = 0
y3-x3 < 0
z4+x4-y4 < 20
x4-z3-x3+y3 = 0
y4-x4 < 0
z5+x5-y5 < 20
x5-z4-x4+y4 = 0
y5-x5 < 0
z6+x6-y6 < 20
x6-z5-x5+y5 = 0
y6-x6 < 0
z7+x7-y7 < 20
x7-z6-x6+y6 = 0
y7-x7 < 0
z8+x8-y8 < 20
x8-z7-x7+y7 = 0
y8-x8 < 0
z9+x9-y9 < 20
x9-z8-x8+y8 = 0
y9-x9 < 0
z10+x10-y10 < 20
x10-z9-x9+y9 = 0
y10-x10 < 0
Optimal Solution
Month
Initial Stock
Wheat Sold
Wheat Bought
1
6
0
0
2
6
0
0
3
6
6
20
4
20
0
0
5
20
0
0
6
20
20
20
7
20
20
0
8
0
0
20
9
20
20
0
10
0
0
0
Sensitivity Analysis
VARIABLE
Y1
Y2
Y3
Y4
Y5
Y6
Y7
Y8
CURRENT
COEF
3.000000
6.000000
7.000000
1.000000
4.000000
5.000000
5.000000
1.000000
ALLOWABLE
INCREASE
4.000000
1.000000
1.000000
1.000000
0.000000
INFINITY
INFINITY
1.000000
ALLOWABLE
DECREASE
INFINITY
INFINITY
1.000000
INFINITY
2.000000
1.000000
2.000000
INFINITY
Floral Dilemma
•
•
Florist must decide what month and how
many of each specific color flower to buy
Decision Variables
– xij = number of bundles of flowers
purchased each month
where i = color of flower (from 1 to 5)
and j = month purchased (from 1 to 8 )
•
Solution found in 16 iterations
Formulation
Min z = 15x11 + 14x12 + 5x13 + 2x14 + 4x15 + 2x16 +
9x17 + 14x18 + 14x21 + 9x22 + 4x23 + 5x24 + 1x25
+ 4x26 + 6x27 + 12x28 + 16x31 + 4x32 + 3x33 +
3x34 + 3x35 + 7x36 + 5x37 + 10x38 + 13x41 +
10x42 + 7x43 + 1x44 + 2x45 + 6x46 + 7x47 + 16x48
+ 12x51 + 11x52 + 8x53 + 4x54 + 6x55 + 5x56 +
15x57 + 13x58
Formulation
s. t.
x11 + x21 + x31 + x41 + x51 >= 500
x12 + x22 + x32 + x42 + x52 >= 500
x13 + x23 + x33 + x43 + x53 >= 500
x14 + x24 + x34 + x44 + x54 >= 500
x15 + x25 + x35 + x45 + x55 >= 500
x16 + x26 + x36 + x46 + x56 >= 500
x17 + x27 + x37 + x47 + x57 >= 500
x18 + x28 + x38 + x48 + x58 >= 500
x11 + x12 + x13 + x14 + x15 + x16 + x17 + x18 <= 1000
x21 + x22 + x23 + x24 + x25 + x26 + x27 + x28 <= 1000
x31 + x32 + x33 + x34 + x35 + x36 + x37 + x38 <= 1000
x41 + x42 + x43 + x44 + x45 + x46 + x47 + x48 <= 1000
x51 + x52 + x53 + x54 + x55 + x56 + x57 + x58 <= 1000
Optimal Solution
Color
Feb
Mar
Red
Apr
May
500
500
Pink
500
Yellow
Aug
Sep
500
500
500
500
July
500
White
Purple
June
Sensitivity Analysis
VARIABLE
X11
X12
X13
X14
X15
X16
X17
X18
X21
X22
CURRENT
COEF
15.000000
14.000000
5.000000
2.000000
4.000000
2.000000
9.000000
14.000000
14.000000
9.000000
ALLOWABLE
INCREASE
INFINITY
INFINITY
0.000000
INFINITY
INFINITY
3.000000
INFINITY
INFINITY
INFINITY
INFINITY
ALLOWABLE
DECREASE
3.000000
7.000000
5.000000
1.000000
2.000000
2.000000
2.000000
1.000000
3.000000
3.000000
Conclusion
• Grummins will realize $3,600,000 in profits
• Wheat will have $162 in profits
• Floral Shop will have a minimum cost of
$20,500