Transportation Method
Initial Basic Feasible Solution-IBFS
North West Corner Method--NWCM ,
Least Cost Method--LCM and
Vogel’s Approximation Method--VAM
Optimality Test using Modified Distribution Method-MODI method.
Variation in transportation
Unbalance Supply and Demand
Degeneracy and its resolution
Maximization Problem
The transportation problem is a special type of linear programming problem where the objective is to minimize the cost of distributing a product from a number of sources or origins to a number of destinations.
Because of its special structure, the usual simplex method is not suitable for solving transportation problems. These problems require a special method of solution.
The transportation problem is a special type of linear programming problem where the objective is to minimize the cost of distributing a product from a number of sources or origins to a number of destinations.
Because of its special structure, the usual simplex method is not suitable for solving transportation problems. These problems require a special method of solution.
The Modified Distribution Method or MODI is an efficient method of checking the optimality of the initial feasible solution. MODI provides a new means of finding the unused route with the largest negative improvement index. Once the largest index is identified, we are required to trace only one closed path. This path helps determine the maximum number of units that can be shipped via the best unused route.
The Least Cost Method is another method used to obtain the initial feasible solution for the transportation problem. Here, the allocation begins with the cell which has the minimum cost. The lower cost cells are chosen over the higher-cost cell with the objective to have the least cost of transportation.
Steps to solve Transportation models by North west corner method are given the presentation. North west corner method is one of the well known methods used to solve the transportation models.
The Modified Distribution Method or MODI is an efficient method of checking the optimality of the initial feasible solution. MODI provides a new means of finding the unused route with the largest negative improvement index. Once the largest index is identified, we are required to trace only one closed path. This path helps determine the maximum number of units that can be shipped via the best unused route.
The Least Cost Method is another method used to obtain the initial feasible solution for the transportation problem. Here, the allocation begins with the cell which has the minimum cost. The lower cost cells are chosen over the higher-cost cell with the objective to have the least cost of transportation.
Steps to solve Transportation models by North west corner method are given the presentation. North west corner method is one of the well known methods used to solve the transportation models.
In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources.A transportation matrix is a way of understanding the maximum possibilities the shipment can be done. It is also known as decision variables because these are the variables of interest that we will change to achieve the objective, that is, minimizing the cost function.
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The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
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Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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Transportation Method Operation Research
1. Operations Research
Chapter : Transportation
Find Initial Basic Feasible Solution-IBFS Using
North West Corner Method-- NWCM ,
Least Cost Method-- LCM and
Vogel’s Approximation Method-- VAM
Optimality Test using Modified Distribution Method-
MODI method.
Variation in transportation
Unbalance Supply and Demand
Degeneracy and its resolution
Maximization Problem
2. Introduction
• Transportation Models are used to find out optimum cost of
transportation of goods.
• Company ABC has three plants at Ahmedabad, Surat and Rajkot and
four demand stations at Visnagar, Baroda and Vapi and Bhuj. We
can formulate transportation matrix as below
Visnagar Baroda Vapi Bhuj Capacity
Ahmedabad 10 20 30 40 7
Surat 20 30 10 60 8
Rajkot 20 15 40 20 5
Demand 2 7 6 5 20
3. Here,
In table cell value 10,20,30,40 Rs. shows cost of transportation.
Last column value shows daily maximum supply.
Last row value shows daily maximum demand of each city.
4. Terms:
1) Balanced Model
If, Total Supply=Total Demand
2) Non Degenerate Solution
If, m+n-1=number of allocated cell
Where,
m=total number of rows
n=total number of columns
3) Feasible solution
All supply and demand constraints are satisfied.
5. Methods to find out Initial Basic Feasible Solution (IBFS):
1. NWCM-- North West Corner Method
2. LCM-- Least Cost Method
3. VAM-- Vogel’s Approximation Method
6. Example 1 : (Problem Type: Balanced Problem)
A Company has 3 production facilities S1, S2 and S3 with
production capacity of 7, 9 and 18 units per week of a product,
respectively.
These units are to be shipped to 4 warehouses D1, D2, D3 and
D4 with requirement of 5,6,7 and 14 units per week,
respectively.
The transportation costs (in rupees) per unit between factories
to warehouses are given in the table below.
7. D1 D2 D3 D4 Capacity
S1 19 30 50 10 7
S2 70 30 40 60 9
S3 40 8 70 20 18
Demand 5 8 7 14 34
Find initial basic feasible solution for given problem by using
North-West corner method
if the object is to minimize the total transportation cost.
8. Demand Destination
D1 D2 D3 D4 Supply
Supply
Station
S1 19 (5) 30 50 10 7
S2 70 30 40 60 9
S3 40 8 70 20 18
Demand 5 8 7 14 34/34
Solution:
North-West corner method---NWCM
Step 1 : Identify North west corner from all cost cell.
Here S1D1 is North west corner.
Do allocation in this cell.
Demand of D1 = 5 which is less than possible supply from S1= 7. Here
5<7...So allocate (5) in S1D1 Cell.
9. Demand Destination
D1 D2 D3 D4 Supply
Supply
Station
S1 19 (5) 30 (2) 50 10 7
S2 70 30 40 60 9
S3 40 8 70 20 18
Demand 5 8 7 14 34/34
•Step 2 : Now no need to consider D1 column as its demand 5 is
completed.
• Identify North west corner from all remaining cost cell.
•Here S1D2 is North west corner.
• Do allocation in this cell.
•Demand of D1 = 8. Available supply from S1=7-5=2 Here, 2<8...So
allocate (2) in S1D2 Cell.
10. Demand Destination
D1 D2 D3 D4 Supply
Supply
Station
S1 19 (5) 30 (2) 50 10 7
S2 70 30 (6) 40 60 9
S3 40 8 70 20 18
Demand 5 8 7 14 34/34
•Step 3 : Now no need to consider D1 column and S1 row.As its
constraints 5 and 7 are satisfied.
• Identify North west corner from all remaining cost cell...Here S2D2 is
North west corner.
• Do allocation in this cell.
•Remaining Demand of D2 = 8-2 =6. Possible supply from S2=9. Here,
6<9..So allocate (6) in S2D2 Cell.
11. Demand Destination
D1 D2 D3 D4 Supply
Supply
Station
S1 19 (5) 30 (2) 50 10 7
S2 70 30 (6) 40 (3) 60 9
S3 40 8 70 20 18
Demand 5 8 7 14 34/34
•Step 4 : Now no need to consider D1-D2 column and S1 row.
• Identify North west corner from all remaining cost cell.
•Here S2D3 is North west corner.
• Do allocation in this cell.
•Demand of D3 = 7. Available supply from S2=9-6=3. Here, 3<7..So
allocate (3) in S2D3 Cell.
12. Demand Destination
D1 D2 D3 D4 Supply
Supply
Station
S1 19 (5) 30 (2) 50 10 7
S2 70 30 (6) 40 (3) 60 9
S3 40 8 70 (4) 20 18
Demand 5 8 7 14 34/34
•Step 5 : Now no need to consider D1-D2 column and S1-S2 row.
• Identify North west corner from all remaining cost cell.
•Here S3D3 is North west corner.
• Do allocation in this cell.
•Remaining Demand of D3 = 7-3=4 . Possible supply from S2=18. So
allocate (4) in S3D3 Cell.
13. Demand Destination
D1 D2 D3 D4 Supply
Supply
Station
S1 19 (5) 30 (2) 50 10 7
S2 70 30 (6) 40 (3) 60 9
S3 40 8 70 (4) 20 (14) 18
Demand 5 8 7 14 34/34
•Step 6 : Here remaining cell is S3D4.
• Do allocation in this cell.
•Demand of D3 = 14 .
• Possible supply from S3=18-4=14.
•So allocate (14) in S3D4 Cell.
14. From previous table,
The minimum total transportation cost =
(19×5)+(30×2)+(30×6)+(40×3)+(70×4)+(20×14)
=1015 Rs.--------Answer.
Here,
the number of allocated cells = 6
is equal to
m + n - 1 = 3 + 4 - 1 = 6
So,
∴ This solution is non-degenerate----Answer.
Where, m= total rows = 3.
n= total columns = 4.
15. Example 2:
(Similar as example 1 NWCM METHOD)
Find Solution using North-West Corner method
D1 D2 D3 D4 Supply
S1 11 13 17 14 250
S2 16 18 14 10 300
S3 21 24 13 10 400
Demand 200 225 275 250 950/950
Type-Balanced supply and demand
22. The minimum total transportation cost =
(11×200) +(13×50)+(18×175)+(14×125)+(13×150)+(10×250)
=12200
Here,
the number of allocated cells = 6
is equal to
m + n - 1 = 3 + 4 - 1 = 6
∴ This solution is non-degenerate
23. Example 3 (Problem Type-Unbalanced supply and demand
example)
Find Solution using North-West Corner method
24. D1 D2 D3 Supply
S1 4 8 8 76
S2 16 24 16 82
S3 8 16 24 77
Demand 72 102 41
Here Total Demand = 215 is less than Total Supply = 235
25. D1 D2 D3 Ddummy Supply
S1 4 8 8 0 76
S2 16 24 16 0 82
S3 8 16 24 0 77
Demand 72 102 41 20
So
add a dummy demand column
with 0 unit cost with allocation 20.
Now, the modified table is
32. The minimum total transportation cost
=(4×72)+(8×4)+(24×82)+(16×16)+(24×41)+(0×20)=3528
Here, the number of allocated cells = 6
is equal to
m + n - 1 = 3 + 4 - 1 = 6
∴ This solution is non-degenerate
34. Step-1. Identify Least Cost value Out of all Cost values…
Here 8 Rs. is least Cost
Step2. Do First allocation in this S3-D2 cell
D2 demand 8 is less than S3 supply 18, so allocate 8
Now no need to consider D2 column value further.
D1 D2 D3 D4 Supply
S1 19 30 50 10 7
S2 70 30 40 60 9
S3 40 8 (8) 70 20 18
Demand 5 8 7 14
35. Step 3: Identify next to smallest cost value that is 10 Rs.
Step 4: Do allocate in S1-D4 Cell.
S1 Supply 7 is less than D4 Demand 14 so allocate 7
Now no need to consider S1 row value further.
D1 D2 D3 D4 Supply
S1 19 30 50 10 (7) 7
S2 70 30 40 60 9
S3 40 8 (8) 70 20 18
Demand 5 8 7 14
36. Follow similar steps.
Next smallest cost value is 20 (As we have no need to
consider S1 row values).
Possible Allocation in S3-D4 cell is 7
D1 D2 D3 D4 Supply
S1 19 30 50 10 (7) 7
S2 70 30 40 60 9
S3 40 8 (8) 70 20 (7) 18
Demand 5 8 7 14
37. Here, next two cost 40 Rs. are same in two cell .
It is Situation of tie.
Do allocate where maximum allocation possible that is in cell S2-D3.
Allocate 7 , as demand in D3 is 7 which is less than supply in S2, 9.
D1 D2 D3 D4 Supply
S1 19 30 50 10 (7) 7
S2 70 30 40 (7) 60 9
S3 40 8 (8) 70 20 (7) 18
Demand 5 8 7 14
38. Next allocation is in S3-D1 that is 18-8-7=3
D1 D2 D3 D4 Supply
S1 19 30 50 10 (7) 7
S2 70 30 40 (7) 60 9
S3 40 (3) 8 (8) 70 20 (7) 18
Demand 5 8 7 14
39. Next allocation is in S2-D1 that is 9-7= 5-3 =2
D1 D2 D3 D4 Supply
S1 19 30 50 10 (7) 7
S2 70 (2) 30 40 (7) 60 9
S3 40 (3) 8 (8) 70 20 (7) 18
Demand 5 8 7 14
40. The minimum total transportation cost =
(10×7)+(70×2)+(40×7)+(40×3)+(8×8)+(20×7) = 814 Rs.
Here,
m=total rows=3
n=total columns=4
the number of allocated cells = 6
is equal to
m + n - 1 = 3 + 4 - 1 = 6
∴ This solution is non-degenerate
47. The minimum total transportation cost =
11*200 + 13*50 + 18*175 + 14*125 + 13*150 + 10*250
= 12200 Rs.
Here,
the number of allocated cells = 6
is equal to
m + n - 1 = 3 + 4 - 1 = 6
∴ This solution is non-degenerate
56. The minimum total transportation cost =
19×5 + 10×2 + 40×7 + 60×2 + 8×8 + 20×10 = 779 Rs.
Here, the number of allocated cells = 6 is equal to
m + n - 1 = 3 + 4 - 1 = 6
∴ This solution is non-degenerate
65. The minimum total transportation cost
=11×200+13×50+18×175+10×125+13×275+10×125
=12075
Here, the number of allocated cells = 6 is equal to
m + n - 1 = 3 + 4 - 1 = 6
∴ This solution is non-degenerate
67. Step-1:Find an initial basic feasible solution--IBFS--using any
one of the three methods NWCM, LCM or VAM.
Step-2:Find ui and vj for rows and columns.
To start
a. assign 0 to ui or vj where maximum number of allocation in
a row or column respectively.
b. Calculate other ui's and vj's using cij=ui+vj, for all occupied
cells.
Step-3:
For all unoccupied cells (i , j),
calculate dij=cij-(ui+vj)
68. Step-4:Check the sign of dij
a. If dij>0, then current basic feasible solution is optimal and stop
this procedure.
b. If dij=0 then alternative solution exists, with different set
allocation and same transportation cost.
C. If dij<0, then the given solution is not an optimal solution and
further improvement in the solution is possible.
Step-5:Select the unoccupied cell with the largest negative value of
dij
69. Step-6:Draw a closed path (or loop) from the unoccupied cell
(selected in the previous step).
Mark (+) and (-) sign alternatively at each corner, starting from
the original unoccupied cell.
Step-7: Repeat Step-2 to step-7 until optimal solution is
obtained.
This procedure stops when all dij≥0 for unoccupied cells
72. Step-2:Find ui and vj for rows and columns.
To start
a. assign 0 to ui or vj where maximum number of allocation in a
row or column respectively.
b. Calculate other ui's and vj's using cij=ui+vj, for all occupied cells.
74. Iteration-1 of optimality test
Find ui and vj for all occupied cells(i , j), where cij=ui+vj= cost from
i row to j column
1. Substituting, v4=0, we get
2.c14=u1+v4⇒u1=c14-v4⇒u1=10-0⇒u1=10
3.c11=u1+v1⇒v1=c11-u1⇒v1=19-10⇒v1=9
80. Step-4:Check the sign of dij
a. If dij>0, then current basic feasible solution is optimal and
stop this procedure.
b. If dij=0 then alternative solution exists, with different set
allocation and same transportation cost.
C. If dij<0, then the given solution is not an optimal solution and
further improvement in the solution is possible.
Here d22= -18 which is less than zero,
so it is non optimal solution
81. Step-5:Select the unoccupied cell with the largest negative
value of dij
Step-6:Draw a closed path (or loop) from the unoccupied cell
(selected in the previous step).
Mark (+) and (-) sign alternatively at each corner, starting from
the original unoccupied cell.
Now choose the minimum negative value from all dij
(opportunity cost) = d22 = [-18]
and draw a closed path from S2D2.
Closed path is S2D2→S2D4→S3D4→S3D2
82. Closed path loop and plus/minus sign allocation…
1. Select the minimum value from cells marked with (-) sign of the
closed path.
2. Assign this value to selected unoccupied cell (So unoccupied cell
becomes occupied
3. Add this value to the other occupied cells marked with (+) sign.
4. Subtract this value to the other occupied cells marked with (-)
sign.
90. 1. Unbalance Supply and Demand
Solution : Add dummy row or column
Supply less Add Dummy Row
Demand less Add Dummy Column
91. D1 D2 D3 Supply
S1 5 4 7 120
S2 10 15 12 80
S3 3 5 12 60
Demand 150 80 50 280/260
Unbalance Supply and Demand
Here, Total supply = 260
Total Demand = 280
Add dummy row
Solution : Supply 260 is less than demand 280,
92. D1 D2 D3 Supply
S1 5 4 7 120
S2 10 15 12 80
S3 3 5 12 60
Sdummy 0 0 0 20
Demand 150 80 50 280/280
Now, Total supply = Total Demand = 280,
So it is balanced Problem,
Now we can find IBFS using NWCM, LCM Or VAM Method.
Supply less Add Dummy Row
93. D1 D2 D3 Supply
S1 5 4 7 100
S2 10 15 12 100
S3 3 5 12 50
Demand 80 80 50 210/250
Unbalance Supply and Demand
Here, Total supply = 250
Total Demand = 210
Solution : Demand 210 is less than supply 250,
Add dummy column
94. D1 D2 D3 Ddummy Supply
S1 5 4 7 0 100
S2 10 15 12 0 100
S3 3 5 12 0 50
Demand 80 80 50 40 250/250
Now, Total supply = Total Demand = 250,
So it is balanced Problem,
Now we can find IBFS using NWCM, LCM Or VAM Method.
Demand less Add Dummy Column
95. 2. Degeneracy and its resolution
(Convert Degenerate solution into non degenerate solution)
Solution :
• To resolve degeneracy, we make use of an artificial quantity
delta (Δ )
• The quantity Δ is assigned to that unoccupied cell, which
has the minimum transportation cost.
96. D1 D2 D3 Supply
S1 8(70) 5 6(50) 120
S2 15 10(80) 12 80
S3 3(80) 9 10 80
Demand 150 80 50 280/280
Initial feasible solution by VAM method is
Here, the number of allocated cells = 4,
which is one less than m + n - 1 = 3 + 3 - 1 = 5
∴ This solution is degenerate
97. Here, the number of allocated cells = 4,
which is one less than m + n - 1 = 3 + 3 - 1 = 5
∴ This solution is degenerate
To Make it Non degenerate,
The quantity Δ is assigned to S1D2, which has the
minimum transportation cost = 5.
98. D1 D2 D3 Supply
S1 8(70) 5 (Δ) 6(50) 120
S2 15 10(80) 12 80
S3 3(80) 9 10 80
Demand 150 80 50 280/280
Here, Now Number of allocated cell = 5 And
m+n-1=5,
Now, Solution is non degenerate.
Now we can do optimality Test
Minimum transportation cost = 5 Rs. Here Assign (Δ) in this (S1, D2) cell
99. 3. Maximization Problem
Solution: Convert into Minimization Problem
How ?? :: By subtracting all cost values from maximum
cost value.
----Than apply LCM, NWCM Or VAM Method
100. In data it is mentioned-Find maximum profit .
D1 D2 D3 D4 Supply
S1 10 20 30 20 7
S2 30 10 40 60 9
S3 40 50 70 20 18
Demand 5 8 7 14
10, 20, 30, 20…etc are profit values.
Convert into Minimization problem subtracting all values from
maximum value (70 Rs).
101. Convert into Minimization problem subtracting all values from
maximum value (70 Rs).
Updated New Table is below,
D1 D2 D3 D4 Supply
S1 60 50 40 50 7
S2 40 60 30 10 9
S3 30 20 0 50 18
Demand 5 8 7 14
Now find IBFS using NWCM, LCM or VAM Method.
102. Summery Of Transportation Chapter
1. Find Initial Basic Feasible Solution-IBFS Using
a) North West Corner Method--NWCM ,
b) Least Cost Method--LCM and
c) Vogel’s Approximation Method--VAM
2. Optimality Test using Modified Distribution Method-
MODI method.
3. Variation in transportation
a) Unbalance Supply and Demand
b) Degeneracy and its resolution
c) Maximization Problem