Medical Supply Transportation Problem
• A Medical Supply company produces catheters in packs at three productions facilities.
• The company ships the packs from the production facilities to four warehouses.
• The packs are distributed directly to hospitals from the warehouses.
• The table on the next slide shows the costs per pack to ship to the four warehouses.
Medical Supply
Seattle New York Phoenix MiamiFROMPLANT
Juarez $19 $ 7 $ 3 $21Seoul 15 21 18 6Tel Aviv 11 14 15 22
TO WAREHOUSE
CapacityJuarez 100Seoul 300Tel Aviv 200
DemandSeattle 150New York 100Phoenix 200Miami 150
Source: Adapted from Lapin, 1994
J Xjs Xjn Xjp Xjm 100
S N P M
Xss Xsn Xsp Xsm
Xts Xtn Xtp Xtm
150 100 200 150
S 300
T 200
WarehouseDemand 600
TO WAREHOUSEPlant
CapacityFrom Plant
Number of constraints = number of rows + number of columns
Total plant capacity must equal total warehouse demand.Although this may seem unrealistic in real world application, it is possible to construct any transportation problem using this model.
Source: Adapted from Lapin, 1994
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
Northwest Corner Method
Begin with a blank shipment schedule. Note the shipping costs in the upper right hand corner of each cell.
Source: Adapted from Lapin, 1994
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
Northwest Corner Method
100
Start in the upper left-hand corner, “northwest corner” of the schedule and place the largest amount of capacity and demand available in that cell. Seattle demands 150 and Jaurez has a capacity of 100.
Source: Adapted from Lapin, 1994
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
Northwest Corner Method
100
Since Juarez capacity is depleted move down to repeat the process for the Seoul to Seattle cell. Seoul has sufficient capacity but Seattle can only take another 50 packs of demand.
50
Source: Adapted from Lapin, 1994
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
Northwest Corner Method
100
50
Now move to the next cells to the right and assign capacity for Seoul to warehouse demand until depleted. Then move down to the Tel Aviv row and repeat the process.
100 150
50 150
Source: Adapted from Lapin, 1994
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
Northwest Corner Method
The previous slides show the process of satisfying all constraints and allows us to begin with a starting feasible solution. Multiply the quantity in each cell by the cost.
1900
2100750 2700
750 3300
1900
750
2100
2700
750
3300
C =11,500
Source: Adapted from Lapin, 1994
rj = 0
ks = 19
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
100
50 100 150
50 150
a
b
For non empty cells: cij = ri+ kj
Assign zero as the row number for the first row.
19 = (0) + ks
Source: Adapted from Lapin, 1994
rj = 0
rs = -4
ks = 19
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
100
50 100 150
50 150
c
a
b
For non empty cells: cij = ri+ kj
Assign zero as the row number for the first row.
15 = rs + 19rs = -15 + 19 = -4 Note: Always use the newest r value to compute the next k.
Source: Adapted from Lapin, 1994
rj = 0
rs = -4
ks = 19 kp = 22
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
100
50 100 150
50 150
*c
a
b d
For non empty cells: cij = ri+ kj
Assign zero as the row number for the first row.
18 =-4 + kp
18 + 4 = kp
= 22
Skip cell SN, mark it * for later and move on to cell SP .
Source: Adapted from Lapin, 1994
rj = 0
rs = -4
rt = -7
ks = 19 kp = 22
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
100
50 100 150
50 150
*c
a
e
b d
For non empty cells: ctp = rt+ kp
Assign zero as the row number for the first row thenuse the newest r value to compute the next k.
15 = rt + 2215 - 22 = rt
= -7
Source: Adapted from Lapin, 1994
rj = 0
rs = -4
rt = -7
ks = 19 kp = 22 km = 29
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
100
50 100 150
50 150
*c
a
e
b d f
For non empty cells: cij = ri+ kj
Assign zero as the row number for the first row thenuse the newest r value to compute the next k.
22 = -7 + km
22 + 7 = km
= 29
Source: Adapted from Lapin, 1994
rj = 0
rs = -4
rt = -7
ks = 19 kn = 25 kp = 22 km = 29
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
100
50 100 150
50 150
*c
a
e
b g d f
For non empty cells: cij = ri+ kj
Assign zero as the row number for the first row thenuse the newest r value to compute the next k.
21= -4 + kn
21 + 4 = kn
= 25
Source: Adapted from Lapin, 1994
Source: Adapted from Lapin, 1994
rj = 0
rs = -4
rt = -7
ks = 19 kn = 25 kp = 22 km = 29
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
100
50 100 150
50 150
c
a
e
b g d f
Next calculate empty cells using: cij - ri - kj
JN = 7 – 0 – 25 = -18
-18
Improvement Difference >>
Source: Adapted from Lapin, 1994
rj = 0
rs = -4
rt = -7
ks = 19 kn = 25 kp = 22 km = 29
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
100
50 100 150
50 150
c
a
e
b g d f
Next calculate empty cells using: cij - ri - kj
JP = 3 – 0 – 22 = -19
-18 -19
Improvement Difference >>
Source: Adapted from Lapin, 1994
rj = 0
rs = -4
rt = -7
ks = 19 kn = 25 kp = 22 km = 29
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
100
50 100 150
50 150
c
a
e
b g d f
Next calculate empty cells using: cij - ri - kj
JM = 21 – 0 – 29 = -8
-18 -19 -8
Improvement Difference >>
Source: Adapted from Lapin, 1994
rj = 0
rs = -4
rt = -7
ks = 19 kn = 25 kp = 22 km = 29
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
100
50 100 150
50 150
c
a
e
b g d f
Next calculate empty cells using: cij - ri - kj
SM = 6 – (-4) – 29 = -19
-18 -19 -8
-19
Improvement Difference >>
Source: Adapted from Lapin, 1994
rj = 0
rs = -4
rt = -7
ks = 19 kn = 25 kp = 22 km = 29
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
100
50 100 150
50 150
c
a
e
b g d f
Next calculate empty cells using: cij - ri - kj
TS = 11 – (-7) – 19 = -1
-18
-19
-8
-1
Improvement Difference >>
-19
Source: Adapted from Lapin, 1994
rj = 0
rs = -4
rt = -7
ks = 19 kn = 25 kp = 22 km = 29
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
100
50 100 150
50 150
c
a
e
b g d f
Next calculate empty cells using: cij - ri - kj
TN = 14 – (-7) – 25 = -4
-18
-19
-8
-1 -4
Improvement Difference >>
-19
Source: Adapted from Lapin, 1994
rj = 0
rs = -4
rt = -7
ks = 19 kn = 25 kp = 22 km = 29
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
100
50 100 150
50 150
c
a
e
b g d f
Next calculate empty cells using: cij - ri - kj
-18
-19
-8
-1 -4
Improvement Difference >>
-19
Source: Adapted from Lapin, 1994
rj = 0
rs = -4
rt = -7
ks = 19 kn = 25 kp = 22 km = 29
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
100
50 100 150
50 150
Next calculate the entering cell by finding the empty cell with the greatest absolute negative improvement difference.
-18 -19 -8
-1 -4
-19
Cells JP and SM are tied for the greatest improvement at $19 per pack. Break the tie and arbitrarily choose JP. JP becomes the entering cell. Place a + sign in cell JP
(+)(-)
(+) (-)
Note: Except for the entering cell all changes must involve nonempty cells.
100
Source: Adapted from Lapin, 1994
rj = 0
rs = -4
rt = -7
ks = 19 kn = 25 kp = 22 km = 29
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
50 100
50 150
-18 -19 -8
-1 -4
-19
Continue around the closed loop until all tradeoffs are completed.
(+)(-)
(+) (-)
Note: Except for the entering cell all changes must involve nonempty cells.
100
150 50
Previous cost was $11,500 and the new is:
300 22502100
900750
3300C = $9,600
Source: Adapted from Lapin, 1994
rj = 0
rs = 15
rt = 12
ks = 0 kn = 6 kp = 3 km = 10
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
50 100
50 150
119 11
-1 -4
-19(+)
(-)(+)
(-)
Note: The r and k values and the improvement difference values have changed.
100
150 50
Begin another iteration choosing the empty cell with the greatest absolute negative improvement difference.>>>>>SM
Source: Adapted from Lapin, 1994
rj = 0
rs = 15
rt = 12
ks = 0 kn = 6 kp = 3 km = 10
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
50 100
50 150
119 11
-1 -4
-19(+)
(-)(+)
(-)
Note: The r and k values and the improvement difference values have changed.
100
150 50
Previous cost was $9,600, now the new is:
300 22502100
30015002200
C = $8,650
Begin another iteration choosing the empty cell with the greatest absolute negative improvement difference.SM
50
100100
Source: Adapted from Lapin, 1994
rj = 0
rs = -4
rt = 12
ks = 19 kn = 25 kp = 3 km = 10
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
50 100
-180 11
-20 -23
-19(+)
(-)(+)
(-)
Note: The r and k values and the improvement difference values have changed.
100
150
Previous cost was $8,650, now the new is:
300 2250
90014001500
C = $6,350
Begin another iteration choosing the empty cell with the greatest absolute negative improvement difference.SM
50
100100100
150
0
kn = 2
Source: Adapted from Lapin, 1994
rj = 0
rs = 16
rt = 12
ks = -1 kp = 3 km = 10
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
520 31
20
3 -1(+)
(-)(+)
(-)
Note: The r and k values and the improvement difference values have changed.
100
150
$6,350
300 2250
90014001500
C = $6,350
Begin another iteration choosing the empty cell with the greatest absolute negative improvement difference.SM
100100
150
0
kn = 2
Source: Adapted from Lapin, 1994
rj = 0
rs = 15
rt = 11
ks = 0 kp = 3 km = -9
S N P M
150 100 200 150
J 100
S 300
T 200
Demand 600
CapacityFrom
To
19 7 3 21
6
22
18
15
21
14
15
11
419 30
20
3
1
(+)
(-)(+)
(-)
Note: The r and k values and the improvement difference values have changed.
100
50
$6,250
300 750
1800900
11001400
C = $6,250
Optimal Solution
100
100
150
100
kn = 3
In five iterations the shipping cost has moved from $11,500 to $6,250. There are no remaining empty cells with a negative value.
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