Homework SourseTo properly stock his restaurant each week John needs at lea
To properly stock his restaurant each week, John needs at least 10lbs of carrots, 12lbs of onions, and 20lbs of potatoes. One of his suppliers offers three packages containing combinations of these vegetables. Package 1 contains 4lbs of carrots and 3lbs of onions. Package 2 contains 1lb of carrots, 2lbs of onions, and 4lbs of potatoes. Package 3 contains 10 pounds of carrots, 1lb of onions, and 5lbs of potatoes. Package 1 costs $40, Package 2 costs $50, and package 3 costs $10. Determine how many of each package John should purchase each week to minimize his costs.
Solution
Lets assume he purchases x pkg of pkg1
Lets assume he purchases y pkg of pkg2
Lets assume he purchases z pkg of pkg3
Carrots : 4x +y +10z >= 10 ----(1)
Onions : 3x +2y + z>= 12 ----(2)
Potatoes : 4y +5z >= 20 -----(3)
And , x>=0 ; y>=0 ; z>=0
We have to minimise , C = 40x +50y +10z
Use Simplex method to solve :
Tableau #1
x y z s1 s2 s3 s4 s5 s6 -p
4 1 10 -1 0 0 0 0 0 0 10
3 2 1 0 -1 0 0 0 0 0 12
0 4 5 0 0 -1 0 0 0 0 20
1 0 0 0 0 0 -1 0 0 0 0
0 1 0 0 0 0 0 -1 0 0 0
0 0 1 0 0 0 0 0 -1 0 0
40 50 10 0 0 0 0 0 0 1 0
Tableau #2
x y z s1 s2 s3 s4 s5 s6 -p
4 1 0 -1 0 0 0 0 10 0 10
3 2 0 0 -1 0 0 0 1 0 12
0 4 0 0 0 -1 0 0 5 0 20
1 0 0 0 0 0 -1 0 0 0 0
0 1 0 0 0 0 0 -1 0 0 0
0 0 1 0 0 0 0 0 -1 0 0
40 50 0 0 0 0 0 0 10 1 0
Tableau #3
x y z s1 s2 s3 s4 s5 s6 -p
0.4 0.1 0 -0.1 0 0 0 0 1 0 1
2.6 1.9 0 0.1 -1 0 0 0 0 0 11
-2 3.5 0 0.5 0 -1 0 0 0 0 15
1 0 0 0 0 0 -1 0 0 0 0
0 1 0 0 0 0 0 -1 0 0 0
0.4 0.1 1 -0.1 0 0 0 0 0 0 1
36 49 0 1 0 0 0 0 0 1 -10
Tableau #4
x y z s1 s2 s3 s4 s5 s6 -p
0 0.1 0 -0.1 0 0 0.4 0 1 0 1
0 1.9 0 0.1 -1 0 2.6 0 0 0 11
0 3.5 0 0.5 0 -1 -2 0 0 0 15
1 0 0 0 0 0 -1 0 0 0 0
0 1 0 0 0 0 0 -1 0 0 0
0 0.1 1 -0.1 0 0 0.4 0 0 0 1
0 49 0 1 0 0 36 0 0 1 -10
Tableau #5
x y z s1 s2 s3 s4 s5 s6 -p
0 0.25 0 -0.25 0 0 1 0 2.5 0 2.5
0 1.25 0 0.75 -1 0 0 0 -6.5 0 4.5
0 4 0 0 0 -1 0 0 5 0 20
1 0.25 0 -0.25 0 0 0 0 2.5 0 2.5
0 1 0 0 0 0 0 -1 0 0 0
0 0 1 0 0 0 0 0 -1 0 0
0 40 0 10 0 0 0 0 -90 1 -100
Tableau #6
x y z s1 s2 s3 s4 s5 s6 -p
0 0 0 -0.25 0 0 1 0.25 2.5 0 2.5
0 0 0 0.75 -1 0 0 1.25 -6.5 0 4.5
0 0 0 0 0 -1 0 4 5 0 20
1 0 0 -0.25 0 0 0 0.25 2.5 0 2.5
0 1 0 0 0 0 0 -1 0 0 0
0 0 1 0 0 0 0 0 -1 0 0
0 0 0 10 0 0 0 40 -90 1 -100
Tableau #7
x y z s1 s2 s3 s4 s5 s6 -p
0 0 0 -0.4 0.2 0 1 0 3.8 0 1.6
0 0 0 0.6 -0.8 0 0 1 -5.2 0 3.6
0 0 0 -2.4 3.2 -1 0 0 25.8 0 5.6
1 0 0 -0.4 0.2 0 0 0 3.8 0 1.6
0 1 0 0.6 -0.8 0 0 0 -5.2 0 3.6
0 0 1 0 0 0 0 0 -1 0 0
0 0 0 -14 32 0 0 0 118 1 -244
Tableau #8
x y z s1 s2 s3 s4 s5 s6 -p
0 0 0 -0.0465116 -0.271318 0.147287 1 0 0 0 0.775194
0 0 0 0.116279 -0.155039 -0.20155 0 1 0 0 4.72868
0 0 0 -0.0930233 0.124031 -0.0387597 0 0 1 0 0.217054
1 0 0 -0.0465116 -0.271318 0.147287 0 0 0 0 0.775194
0 1 0 0.116279 -0.155039 -0.20155 0 0 0 0 4.72868
0 0 1 -0.0930233 0.124031 -0.0387597 0 0 0 0 0.217054
0 0 0 -3.02326 17.3643 4.57364 0 0 0 1 -269.612
Tableau #9
x y z s1 s2 s3 s4 s5 s6 -p
0 0 0 0 -0.333333 0.0666667 1 0.4 0 0 2.66667
0 0 0 1 -1.33333 -1.73333 0 8.6 0 0 40.6667
0 0 0 0 0 -0.2 0 0.8 1 0 4
1 0 0 0 -0.333333 0.0666667 0 0.4 0 0 2.66667
0 1 0 0 0 0 0 -1 0 0 0
0 0 1 0 0 -0.2 0 0.8 0 0 4
0 0 0 0 13.3333 -0.666667 0 26 0 1 -146.667
Tableau #10
x y z s1 s2 s3 s4 s5 s6 -p
0 0 0 0 -5 1 15 6 0 0 40
0 0 0 1 -10 0 26 19 0 0 110
0 0 0 0 -1 0 3 2 1 0 12
1 0 0 0 0 0 -1 0 0 0 0
0 1 0 0 0 0 0 -1 0 0 0
0 0 1 0 -1 0 3 2 0 0 12
0 0 0 0 10 0 10 30 0 1 -120
WE get minimium at C = 120; x = 0, y = 0, z = 12
So, x = 0 pkg ; y = 0 pkg ; z = 12 pkg
