Transportation ppt

By : Vikrant V.Shaga

TRANSPORTATION MODEL

G.S Mandal’sMarathwada Institute of Technology,

Aurangabad

Department of MCA

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Introduction • Transportation model is a typical OR technique

intended to establish the “Least Cost Route” of Transportation of goods from the company’s plant to its warehouse

• In other words, Transport various quantities of a single homogeneous commodity to different destinations in such a way that total transportation cost is minimum

By Vikrant V. Shaga

By Vikrant V. Shaga 3

Terminology of Transportation Problem

• Balance TP : If total supply from all the sources (origins) is equal to the total demand (requirement) in all destinations

• Unbalanced TP : If total supply from all the sources (origins) is not equal to the total demand (requirement) in all destinations

• Feasible Solution : Non-negative values of xij where i=1,2…..m and j=1,2….n which satisfy the constraints of supply and demand


Terminology of Transportation Problem

• Basic feasible solution : If the number of positive allocations are (m+n-1)

• Optimal Solution : A feasible solution is said to be optimal solution if it minimizes the total TP cost

• Degenerate basic feasible solution: if the number of allocation in basic feasible solutions is less than (m+n-1)


The Transportation Representation

1

2

m

1

2

n

Sources Destinations

… …

Supply s1

Supply s2

Supply sm

Demand d1

Demand d2

Demand dn

xij

Costs cij


The Transportation Simplex Table

DestinationSupply ui

Source 1 2 … n

1c11 c12

…c1n s1

2c21 c22

…c2n s2

… … … … … …

mcm1 cm2

…cmn sm

Demand d1 d2 … dn

Z = vj


Optimal Solution of Transportation Problem

Initial Feasible Solution

Obtain an optimal solution by making

successive improvements in IBFS until no further

decrease in transportation cost is

possible

STEP1 STEP2


Finding an Initial BFS

North West Corner Rule

Least Cost Entry Method

Vogel’s Approximation


Step1: Select the upper left (north-west) cell of the transportation matrix and allocate the maximum possible

value to X11 which is equal to min(s1,d1).Step2: • If allocation made is equal to the supply available at the

first source (s1 in first row), then move vertically down to the cell (2,1).

• If allocation made is equal to demand of the first destination (d1 in first column), then move horizontally to the cell (1,2).

• If s1=d1 , then allocate X11= s1 or d1 and move to cell (2,2).Step3: Continue the process until an allocation is made in

the south-east corner cell of the transportation t

The Northwest Corner Rule

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DestinationSupply

Source N S E W

A16 13 22 17

200

B14 13 19 15

350

C9 20 23 10

150

Demand 100 140 300 250

By Vikrant V. Shaga

Example 1 : Find I.B.F.S of the following Transportation Problem:



DestinationSupply

Source N S E W

A16 13 22 17

200

B14 13 19 15

350

C9 20 23 10

150

Dummy0 0 0 0

90

Demand 100 140 300 250

Solution) The given Transportation Problem is Unbalance Transportation Problem.Make it Balance by adding DUMMY ROW with Supply of 90.


The Northwest Corner RuleDestination

SupplySource N S E W

A16 13 22 17

100100

B14 13 19 15

350

C9 20 23 10

150

Dummy0 0 0 0

90

Demand - 140 300 250

100

13



A16 13 22 17

-100

B14 13 19 15

350

C9 20 23 10

150

Dummy0 0 0 0

90

Demand - 40 300 250

100 100

By Vikrant V. Shaga




A16 13 22 17

-100

B14 13 19 15

310

C9 20 23 10

150

Dummy0 0 0 0

90

Demand - - 300 250

100 100

40




A16 13 22 17

-100

B14 13 19 15

10

C9 20 23 10

150

Dummy0 0 0 0

90

Demand - - - 250

100 100

40 300




A16 13 22 17

-100

B14 13 19 15

-

C9 20 23 10

150

Dummy0 0 0 0

90

Demand - - - 240

100 100

40 300 10




A16 13 22 17

-100

B14 13 19 15

10

C9 20 23 10

-

Dummy0 0 0 0

90

Demand - - - 90

100 100

40 300 10

150




A16 13 22 17

-100

B14 13 19 15

-

C9 20 23 10

-

Dummy0 0 0 0

-

Demand - - - -

100 100

40 300 10

150

90

Z=(16x100)+(13x100)+(13x40)+(19x300)+(15x10)+(10x150)+(0x90)=Rs.10770


Step1: Select the cell having lowest unit cost in the entire table and allocate the minimum of supply or demand values in that cell.

Step2: Then eliminate the row or column in which supply or demand is exhausted. If both the supply and demand values are same, either of the row or column can be eliminated.

In case, the smallest unit cost is not unique, then select the cell where maximum allocation can be made.

Step3: Repeat the process with next lowest unit cost and continue until the entire available supply at various sources and demand at various destinations is satisfied.




DestinationSupply

Source N S E W

A16 13 22 17

200

B14 13 19 15

350

C9 20 23 10

150

Demand 100 140 300 250

Example 1 : Find I.B.F.S of the following Transportation Problem:



DestinationSupply

Source N S E W

A16 13 22 17

200

B14 13 19 15

350

C9 20 23 10

150

Dummy0 0 0 0

90

Demand 100 140 300 250

Solution) The given Transportation Problem is Unbalance Transportation Problem.Make it Balance by adding DUMMY ROW with Supply of 90.


Least Cost Entry MethodDestination


A16 13 22 17

200

B14 13 19 15

350

C9 20 23 10

150

Dummy0 0 0 0

-

Demand 100 140 300 160

90




A16 13 22 17

200

B14 13 19 15

350

C9 20 23 10

50

Dummy0 0 0 0

-

Demand - 140 300 160

90

100




A16 13 22 17

200

B14 13 19 15

350

C9 20 23 10

-

Dummy0 0 0 0

-

Demand - 140 300 110

90

100 50




A16 13 22 17

60

B14 13 19 15

350

C9 20 23 10

-

Dummy0 0 0 0

-

Demand - - 300 110

90

100 50

140




A16 13 22 17

60

B14 13 19 15

240

C9 20 23 10

-

Dummy0 0 0 0

-

Demand - - 300 -

90

100 50

140

110




A16 13 22 17

60

B14 13 19 15

-

C9 20 23 10

-

Dummy0 0 0 0

-

Demand - - 60 -

90

100 50

140

110240




A16 13 22 17

-

B14 13 19 15

-

C9 20 23 10

-

Dummy0 0 0 0

-

Demand - - - -

90

100 50

140

110240

60

Z=(13x140)+(22x60)+(19x240)+(15x110)+(9x100)+(10x50)+(0x90)=Rs.10750


Vogel’s Approximation MethodStep1: Calculate penalty for each row and column by taking the difference between the two smallest unit costs. This penalty or extra cost has to be paid if one fails to allocate the minimum unit transportation cost.Step2: Select the row or column with the highest penalty and select the minimum unit cost of that row or column. Then, allocate the minimum of supply or demand values in that cell. If there is a tie, then select the cell where maximum allocation could be made.Step3: Adjust the supply and demand and eliminate the satisfied row or column. If a row and column are satisfied simultaneously, only of them is eliminated and the other one is assigned a zero value.Any row or column having zero supply or demand, can not be used in calculating future penalties.Step4: Repeat the process until all the supply sources and demand destinations are satisfied.


Vogel’s Approximation MethodDestination

Supply diffSource N S E W

A16 13 22 17

200 3

B14 13 19 15

350 1

C9 20 23 10

150 1

Dummy0 0 0 0

90 0

Demand 100 140 300 250

diff 9 13 19 10




A16 13 22 17

200 3

B14 13 19 15

350 1

C9 20 23 10

150 1

Dummy0 0 0 0

90 0

Demand 100 140 300 250

diff 9 13 19 10

90




A16 13 22 17

200 3

B14 13 19 15

350 1

C9 20 23 10

150 1

Dummy0 0 0 0

--- ---

Demand 100 140 210 250

diff 5 0 3 5

90




A16 13 22 17

200 3

B14 13 19 15

350 1

C9 20 23 10

150 1

Dummy0 0 0 0

--- ---

Demand 100 140 210 250

diff 5 0 3 5

90

100




A16 13 22 17

200 4

B14 13 19 15

350 2

C9 20 23 10

50 10

Dummy0 0 0 0

--- ---

Demand --- 140 210 250

diff --- 0 3 5

90

100




A16 13 22 17

200 4

B14 13 19 15

350 2

C9 20 23 10

50 10

Dummy0 0 0 0

--- ---

Demand --- 140 210 250

diff --- 0 3 5

90

100 50




A16 13 22 17

200 4

B14 13 19 15

350 2

C9 20 23 10

--- ---

Dummy0 0 0 0

--- ---

Demand --- 140 210 200

diff --- 0 3 2

90

100 50




A16 13 22 17

200 4

B14 13 19 15

350 2

C9 20 23 10

--- ---

Dummy0 0 0 0

--- ---

Demand --- 140 210 200

diff --- 0 3 2

90

100 50

140




A16 13 22 17

60 5

B14 13 19 15

350 4

C9 20 23 10

--- ---

Dummy0 0 0 0

--- ---

Demand --- --- 210 200

diff --- --- 3 2

90

100 50

140




A16 13 22 17

60 5

B14 13 19 15

350 4

C9 20 23 10

--- ---

Dummy0 0 0 0

--- ---

Demand --- --- 210 200

diff --- --- 3 2

90

100 50

140 60


Vogel’s Approximation Method Destination


A16 13 22 17

--- ---

B14 13 19 15

350 4

C9 20 23 10

--- ---

Dummy0 0 0 0

--- ---

Demand --- --- 210 140

diff --- ---

90

100 50

140 60




A16 13 22 17

--- ---

B14 13 19 15

---

C9 20 23 10

--- ---

Dummy0 0 0 0

--- ---

Demand --- --- --- ---

diff --- ---

90

100 50

140 60

140210

Z=(13x140)+(17x60)+(19x210)+(15x140)+(9x100)+(10x50)+(0x90)=Rs.10330


ConclusionSrno Method Optimum

Solution(Cost in Rs)

1 Northwest Corner Rule 10770

2 Least Cost Entry Method 10750

3 Vogel’s Approximation 10330


In this method , we calculate the net cost change that can be obtained by introducing any of the unoccupied cells into the Solution.

Step1: Determine an I.B.F.S using any of the three Method (i.e. NWCR,LCM or VAM)

Step2: Make sure that no. of occupied cells is exactly equal to (m+n-1).

Step3: Evaluate the cost Effectiveness of shipping goods via transportation routes not currently in the solution. This testing of each unoccupied cell is conducted by 5 steps:

Stepping Stone Method


a) Select an unoccupied cellb) Beginning at this cell , trace a closed path using

the most direct route through at least 3 occupied cells used in the solution and then back to the original occupied cell and moving with only horizontal and vertical moves. Further since only the cell at the turning points are considered to be on the closed path ; both unoccupied & occupied boxes may be skipped over.

The cells at the turning points are called ‘Stepping Stones’ on the path.



c) Assign (+) & (-) signs alternatively on each corner cell of the closed path; starting with (+) sign at the unoccupied cell to be evaluated.

d) Compute ‘Net Change in the Cost’ along the closed path.

e) Repeat sub steps (a) through (d) . Calculate the ‘Net Change in the Cost’ for all the unoccupied cells.



Step 4) Check the sign of each ‘Net Change’. If all net cost changes >= 0, an optimum solution reached else go to next step.

Step 5) Select the Unoccupied cell with Highest Negative Net Cost Change and determine the maximum no. of units that can be assigned to a cell marked with (-) sign on the closed path corresponding to this cell. Add this no. to the unoccupied cell and to all other cells on the path marked with (+) sign. Subtract this no. from cells on the closed path marked with (-) sign.

Step 6) Go to Step (2) & repeat procedure until we get an optimum solution.



The MODI (modified distribution) method allows us to compute improvement indices quickly for each unused square without drawing all of the closed paths. Because of this, it can often provide considerable time savings over other methods for solving transportation problems.

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

MODI (Modified Distribution)Method

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Step1) Determine I.B.F.S consisting of (m+n-1) allocations using any of the three methods.Step2) Determine a set of numbers for each row and each column. To compute Ui (i=1,2..m) for each row & Vj (j=1,2..n) for each column

Cij=Ui+Vj for each of the (m+n-1) occupied cells. Step3) For Unoccupied cells, Compute the opportunity cost(improvement index) by:

dij=Cij-(Ui+Vj)By Vikrant V. Shaga


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Step4) Check the sign of each opportunity cost. If opportunity cost of all unoccupied cells are either POSITIVE or ZERO, the given solution is optimum else it not optimum and further savings in TP cost are possible.Step5) Select the Unoccupied Cell with largest negative opportunity cost as the cell to be included in the next solution.Step6) Draw a closed path or loop for the unoccupied cells selected in step5.By Vikrant V. Shaga


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Step7) Assign (+) & (-) signs alternatively on each corner cell of the closed path; starting with (+) sign at the unoccupied cell to be evaluated.

Step8) Determine the maximum no. of units that should be shipped to this unoccupied cell.

Step9) Repeat the whole procedure until an optimum solution is attained.

Step10) Finally obtain the total TP cost for new solution.

By Vikrant V. Shaga


Transportation ppt

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