QUANTI
Experience is not what happens to a man, it is what
a man does with what happens to him
-Aldous Huxley
TP
TRANSPORTATION
PROBLEM
VOGEL METHOD
Engr. Edward P. Lacson, MBM, CLSSGB
Transportation
The transportation problem arises frequently in planning
for the distribution of goods and services from several
supply locations to several demand locations.
Typically, the quantity of goods available at each supply
location (origin) is limited, and the quantity of goods
needed at each of several demand locations
(destination) is known.
The usual objective in a transportation problem is to
minimize the cost of shipping goods from the origin to
destinations.
ORIGIN
1
2
3
TOTAL
DESTINATION
A
B
C
D
TOTAL
ORIGIN
1
2
3
SAMPLE PROBLEM
PLANT
CLEVELAND
BEDFORD
YORK
DISTRIBUTION
CENTER
BOSTON
CHICAGO
ST. LOUIS
LEXINGTON
A
3
7
2
CAPACIY
5000
6000
2500
13500
FORECAST DAMAND
6000
4000
2000
1500
13500
DISTRIBUTION COST PER UNIT
DESTINATIONS
B
2
5
5
C
7
2
4
D
6
3
5
STEP 1 Transform the problem into table by specifying the cost
relative to the destination and origin (transportation table)
Origin/Destination
A
B
3
1
C
2
D
7
CAPACITY
6
5000
7
2
5
2
3
6000
2
3
5
4
5
2500
DEMAND
6000
4000
2000
1500
13500/13500
STEP 2 Get the opportunity cost per row and per column, by
subtracting the lowest value to the second lower value. Then allocate the
highest opportunity cost until all destination will be satisfied.
Origin/Destination
A
B
3
1
2
7
2
D
7
5
2
3
C
R1
3
2
1
5000
R2
3
2
1
4
2
2
6000
R3
COLUM
N
CA
3
2
1
CB
5
2
3
CC
4
2
2
CD
5
3
2
6
2
5
ROW
OPPORTUNITY
COST
CAPACITY
3
4
5
2500
DEMAND
6000
4000
2000
1500
13500/13500
STEP 2 Get the opportunity cost per row and per column, by
subtracting the lowest value to the second lower value. Then allocate the
highest opportunity cost until all destination will be satisfied.
Origin/Destination
A
B
C
3 4000 2
1
D
7
CAPACITY
6
5000
7
2
5
2
3
6000
2
3
5
4
5
2500
DEMAND
6000
4000
2000
1500
13500/13500
ROW
OPPORTUNITY
COST
R1
3
2
1
R2
3
2
1
R3
4
2
2
CA
3
2
1
CB
5
2
3
CC
4
2
2
CD
5
3
2
COLUMN
STEP 2 Get the opportunity cost per row and per column, by
subtracting the lowest value to the second lower value. Then allocate the
highest opportunity cost until all destination will be satisfied.
Origin/Destination
A
B
C
3 4000 2
1
D
7
CAPACITY
6
5000
7
2
5
2
3
6000
2
3
5
4
5
ROW
OPPORTUNITY
COST
R1
6
3
3
R2
3
2
1
R3
4
2
2
CA
3
2
1
CC
4
2
2
CD
5
3
2
COLUMN
2500
DEMAND
6000
4000
2000
1500
13500/13500
STEP 2 Get the opportunity cost per row and per column, by
subtracting the lowest value to the second lower value. Then allocate the
highest opportunity cost until all destination will be satisfied.
Origin/Destination
1
A
B
C
1000 3 4000 2
D
7
CAPACITY
6
5000
7
2
5
2
3
6000
2
3
5
4
5
ROW
OPPORTUNITY
COST
R1
6
3
3
R2
3
2
1
R3
4
2
2
CA
3
2
1
CC
4
2
2
CD
5
3
2
COLUMN
2500
DEMAND
6000
4000
2000
1500
13500/13500
STEP 2 Get the opportunity cost per row and per column, by
subtracting the lowest value to the second lower value. Then allocate the
highest opportunity cost until all destination will be satisfied.
Origin/Destination
1
A
B
C
1000 3 4000 2
D
7
CAPACITY
7
5
2
3
6000
2
3
5
R2
3
2
1
R3
4
2
2
CA
7
2
5
CC
4
2
2
CD
5
3
2
6
5000
2
ROW
OPPORTUNITY
COST
4
5
COLUMN
2500
DEMAND
6000
4000
2000
1500
13500/13500
STEP 2 Get the opportunity cost per row and per column, by
subtracting the lowest value to the second lower value. Then allocate the
highest opportunity cost until all destination will be satisfied.
Origin/Destination
1
A
B
C
1000 3 4000 2
D
7
CAPACITY
7
5
2
3
6000
3
2500 2
5
R2
3
2
1
R3
4
2
2
CA
7
2
5
CC
4
2
2
CD
5
3
2
6
5000
2
ROW
OPPORTUNITY
COST
4
5
COLUMN
2500
DEMAND
6000
4000
2000
1500
13500/13500
STEP 2 Get the opportunity cost per row and per column, by
subtracting the lowest value to the second lower value. Then allocate the
highest opportunity cost until all destination will be satisfied.
Origin/Destination
1
A
B
C
1000 3 4000 2
D
7
CAPACITY
7
5
2
3
6000
3
2500 2
5
R2
3
2
1
R3
4
2
2
CA
7
2
5
CC
4
2
2
CD
5
3
2
6
5000
2
ROW
OPPORTUNITY
COST
4
5
COLUMN
2500
DEMAND
6000
4000
2000
1500
13500/13500
STEP 2 Get the opportunity cost per row and per column, by
subtracting the lowest value to the second lower value. Then allocate the
highest opportunity cost until all destination will be satisfied.
Origin/Destination
1
A
B
C
1000 3 4000 2
7
2
D
7
5
2500 2
2
5
5000
R2
3
2
1
6000
COLUMN
CC
4
2
2
CD
5
3
2
6
3
2000
3
ROW
OPPORTUNITY
COST
CAPACITY
4
5
2500
DEMAND
6000
4000
2000
1500
13500/13500
STEP 2 Get the opportunity cost per row and per column, by
subtracting the lowest value to the second lower value. Then allocate the
highest opportunity cost until all destination will be satisfied.
Origin/Destination
1
A
B
C
1000 3 4000 2
7
2
3
7
5
2500
D
2500 2
5
5000
R2
3
2
1
6000
COLUMN
CC
4
2
2
CD
5
3
2
6
2
2000
ROW
OPPORTUNITY
COST
CAPACITY
3
1500
4
5
2500
DEMAND
6000
4000
2000
1500
13500/13500
STEP 2 Get the opportunity cost per row and per column, by
subtracting the lowest value to the second lower value. Then allocate the
highest opportunity cost until all destination will be satisfied.
Origin/Destination
1
A
B
C
1000 3 4000 2
D
7
CAPACITY
6
5000
7
2
5
2500
3
2
2000
2500 2
5
3
1500
4
6000
5
2500
CELLS ALLOCATION COST
1A
1000 3
3000
1B
4000 2
8000
2A
2500 7
17500
2C
2000 2
4000
2D
1500 3
4500
3A
2500 2
5000
TOTAL
DEMAND
6000
4000
2000
1500
13500/13500
42000
STEP 3 Test for unallocated cells by moving vertical and horizontal by
assigning plus (+) starting from the unallocated cell and minus (-) to the next
allocated cell alternately and get their difference.
Origin/Destination
1
A
-
B
3
1000
2
+
C
2
7
7
6
-
2
2000
5
3
1500
4
6000
4000
2000
1500
2B
3B
2500
DEMAND
1C
1D
5
2500
Test for unallocated cells
Cell
5000
5
2
6000
CAPACITY
4000
2500
3
+
D
13500/
13500
3C
3D
7-3+7-2
STEP 3 Test for unallocated cells by moving vertical and horizontal by
assigning plus (+) starting from the unallocated cell and minus (-) to the next
allocated cell alternately and get their difference.
Origin/Destination
1
A
-
B
3
1000
2
+
C
2
7
+
7
6
2
2000
5
-
3
1500
4
6000
5
2500
4000
2000
1500
1C
7-3+7-2
1D
6-3+7-3
2B
3B
2500
DEMAND
Test for unallocated cells
Cell
5000
5
2
6000
CAPACITY
4000
2500
3
D
13500/
13500
3C
3D
STEP 3 Test for unallocated cells by moving vertical and horizontal by
assigning plus (+) starting from the unallocated cell and minus (-) to the next
allocated cell alternately and get their difference.
Origin/Destination
1
A
+
B
3
1000
2
-
-
C
2
7
7
+
6
5
2
5
3
1500
4
6000
5
2500
4000
2000
1500
1C
7-3+7-2
1D
6-3+7-3
2B
5-2+3-7
3B
2500
DEMAND
Test for unallocated cells
Cell
5000
2000
2
6000
CAPACITY
4000
2500
3
D
13500/
13500
3C
3D
STEP 3 Test for unallocated cells by moving vertical and horizontal by
assigning plus (+) starting from the unallocated cell and minus (-) to the next
allocated cell alternately and get their difference.
Origin/Destination
1
A
+
B
3
1000
-
2
-
7
6
5
2
2000
2
+
5
1500
4
6000
5
2500
DEMAND
4000
2000
1500
Test for unallocated cells
Cell
3
2500
6000
CAPACITY
5000
2500
3
D
4000
7
2
C
13500/
13500
1C
7-3+7-2
1D
6-3+7-3
2B
5-2+3-7
3B
5-2+3-2
3C
3D
STEP 3 Test for unallocated cells by moving vertical and horizontal by
assigning plus (+) starting from the unallocated cell and minus (-) to the next
allocated cell alternately and get their difference.
A
Origin/Destination
B
3
1
1000
2
+
C
2
-
7
6
7
5
-
2
2000
2
5
+
3
1500
4
6000
5
2500
DEMAND
4000
2000
1500
Test for unallocated cells
Cell
5000
2500
6000
CAPACITY
4000
2500
3
D
13500/
13500
1C
7-3+7-2
1D
6-3+7-3
2B
5-2+3-7
3B
5-2+3-2
3C
4-2+7-2
3D
STEP 3 Test for unallocated cells by moving vertical and horizontal by
assigning plus (+) starting from the unallocated cell and minus (-) to the next
allocated cell alternately and get their difference.
Origin/Destination
A
B
3
1
1000
2
+
C
2
-
7
6
7
5
2
2000
2
5
-
3
1500
4
+
6000
5
2500
DEMAND
4000
2000
1500
Test for unallocated cells
Cell
5000
2500
6000
CAPACITY
4000
2500
3
D
13500/
13500
1C
7-3+7-2
1D
6-3+7-3
2B
5-2+3-7
3B
5-2+3-2
3C
4-2+7-2
3D
5-2+7-2
STEP 3 Test for unallocated cells by moving vertical and horizontal by
assigning plus (+) starting from the unallocated cell and minus (-) to the next
allocated cell alternately and get their difference.
Origin/Destination
A
B
3
1
1000
2
+
C
2
-
7
6
7
5
2
2000
2
5
-
3
1500
4
+
6000
5
2500
DEMAND
4000
2000
1500
Test for unallocated cells
Cell
5000
2500
6000
CAPACITY
4000
2500
3
D
13500/
13500
1C
7-3+7-2
9
1D
6-3+7-3
7
2B
5-2+3-7
-1
3B
5-2+3-2
4
3C
4-2+7-2
7
3D
5-3+7-2
7
STEP 4 In the second and succeeding iteration start the allocation on
the cell that has the lower negative value and follow the + and – sign, then
repeat step no 3 until optimal value is achieve (if there is no negative in the
test for unallocated cells).
Origin/Destination
A
B
3
1
1000
2
7
5
6
2
2000
2
5
1500
4
6000
5
2500
DEMAND
4000
2000
1500
Test for unallocated cells
Cell
3
2500
6000
CAPACITY
5000
2500
3
D
4000
7
2
C
13500/
13500
1C
7-3+7-2
9
1D
6-3+7-3
7
2B
5-2+3-7
-1
3B
5-2+3-2
4
3C
4-2+7-2
7
3D
5-3+7-2
7
STEP 4 In the second and succeeding iteration start the allocation on
the cell that has the lower negative value and follow the + and – sign, then
repeat step no 3 until optimal value is achieve (if there is no negative in the
test for unallocated cells).
Origin/Destination
1
A
+
B
3
1000
2
-
-
C
2
7
7
+
6
5
2
5
3
1500
4
6000
5
2500
2500
DEMAND
4000
2000
1500
Test for unallocated cells
Cell
5000
2000
2
6000
CAPACITY
4000
2500
3
D
13500/
13500
1C
7-3+7-2
9
1D
6-3+7-3
7
2B
5-2+3-7
-1
3B
5-2+3-2
4
3C
4-2+7-2
7
3D
5-3+7-2
7
STEP 4 In the second and succeeding iteration start the allocation on
the cell that has the lower negative value and follow the + and – sign, then
repeat step no 3 until optimal value is achieve (if there is no negative in the
test for unallocated cells).
Origin/Destination
1
A
B
+ 2500 3 - 2500 2
1000
2
C
7
6
2
2000
2
5
3
1500
4
6000
5
2500
2500
DEMAND
4000
2000
1500
Test for unallocated cells
Cell
5000
2500
6000
CAPACITY
4000
-2500 7 +2500 5
3
D
13500/
13500
1C
7-3+7-2
9
1D
6-3+7-3
7
2B
5-2+3-7
-1
3B
5-2+3-2
4
3C
4-2+7-2
7
3D
5-3+7-2
7
STEP 4 In the second and succeeding iteration start the allocation on
the cell that has the lower negative value and follow the + and – sign, then
repeat step no 3 until optimal value is achieve (if there is no negative in the
test for unallocated cells).
Origin/Destination
A
B
3
1
3500
2
7
5
2
6
2
2000
5
1500
4
6000
5
2500
DEMAND
4000
2000
1500
Test for unallocated cells
Cell
3
2500
6000
CAPACITY
5000
2500
3
D
1500
7
2
C
13500/
13500
1C
7-2+5-2
8
1D
6-2+5-3
6
2A
7-5-2+3
3
3B
5-2+3-2
4
3C 4-2+5-2+3-2
6
3D 5-3+5-2+3-2
6
STEP 4 In the second and succeeding iteration start the allocation on
the cell that has the lower negative value and follow the + and – sign, then
repeat step no 3 until optimal value is achieve (if there is no negative in the
test for unallocated cells).
Origin/Destination
A
B
3
1
3500
2
7
5
2
6
2
2000
5
1500
4
6000
5
2500
DEMAND
4000
2000
1500
Allocation Cost
Cell
3
2500
6000
CAPACITY
5000
2500
3
D
1500
7
2
C
13500/
13500
1A
3500*3
10500
1B
1500*2
3000
2B
2500*5
12500
2C
2000*2
4000
2D
1500*3
4500
3A
2500*2
5000
TOTAL
39500