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Summary

This paper is about solving transportation problem (shortest path model) using Operation

Research (OR) approach in analysis. Transportation model is a special type of networks

problems for shipping a commodity from source (e.g. branch office) to destinations (e.g.,

main hub). Transportation model deal to get the minimum-cost, time or distance plan to

transport a commodity from a number of sources. Using linear programming method to solve

transportation problem, we determine the value of objective function which minimize the

distance for transporting.

Background

About GATI

Gati Limited is a pioneer and leader in the Express Distribution and Supply Chain Solutions

in India. It was the revolutionary approach adopted by Gati that helped launch many path-breaking initiatives in the logistics segment and many were the firsts for the Indian market. In

a span of 20 years, Gati has consistently explored various ways to bring premium value to thecustomer, always setting benchmarks in quality of service and customer satisfaction.

Having started as a cargo management company in 1989, Gati has grown into an organization

with more than 3500 employees and a turnover of Rs 576 Crore covering 603 out of 611

districts in India. Gati has over 4000 vehicles on road, fleet of refrigerated trucks, container

vessels and world class mechantronic warehousing facilities across India. Be it flexible point-

to-point distribution solutions or complex end-to-end integrated logistics solutions or supply

chain management, Gati does it all with great effectiveness and reliability, and enjoys the

trust of a large customer base.

The Gati advantage of seamless connectivity across air, road, ocean and rail has resulted in a

plethora of offerings to the customer unmatched in the industry. Besides having a strong

network in India, Gati has a strong market presence in the Asia Pacific region and SAARC

countries. Today, Gati has offices in China, Singapore, Bhutan, Dubai, Hong Kong, Thailand,

Nepal and Sri Lanka and has plans to foray into other markets.

Gati's business model is well aligned with the customers need, which is why the core

businesses have grown to meet the evolving needs of the customer, and this has resulted in

consolidation of services and in the development of core and critical infrastructure, thus

propelling Gati to the forefront in the logistics segment.

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Problem definition

Shortest path model

Cargo delivery is the transportation of packages or parcels from a source point to a

destination point. Generally key success factors in cargo delivery are speed and reliability.

During the journey from branch office to the main hub the vehicles followed different routes

after it reaches to the main hub they send the cargo to their main destination

Branch office Main hub

Customer either brings the cargo to the office or it is picked up by the company vehicles.

Before being transported to the office cargos wait for other cargo which has different origin

but same destination.

The branch office picks up the cargos and delivers cargos to the main hub.

Shortest path model is to look the map and try to plan best route to the destination. The

models main objective is to minimize the distance, time, or cost from the starting point the

start node to the destination node terminal node.

GATI (south Bangalore branch) wants to determine the route to minimum distance from

Singsandra office to Peenya main hub

METHODS FOR SOLVING TRANSPORTATION PROBLEM

There are five methods to determine the solution for balanced transportation problem:

1. Northwest Corner method

2. Vogels approximation method(VAM)

3. Stepping stone method

4. Hungarian method

5. Linear programming.

The methods are as follows1. North-West corner method (NWCM)

The North West corner rule is a method for computing a basic feasible solution of a

transportation problem where the basic variables are selected from the North Westcorner (i.e., top left corner).

Steps

1. Select the North West (upper left-hand) corner cell of the transportation table and

allocate as many units as possible equal to the minimum between available supplies

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and demand requirements, i.e., min (s1, d1).

2. Adjust the supply and demand numbers in the respective rows and columns

allocation.

3. If the supply for the first row is exhausted then move down to the first cell in the

second row.

4. If the demand for the first cell is satisfied then move horizontally to the next cell

in the second column.

5. If for any cell supply equals demand then the next allocation can be made in cell

either in the next row or column.6. Continue the procedure until the total available quantity is fully allocated to the

cells as required.

Vogels Approximation Method (VAM)

The Vogel approximation method is an iterative procedure for computing a basic feasiblesolution of the transportation problem.

Steps

1. Identify the boxes having minimum and next to minimum transportation cost ineach row and write the difference (penalty) along the side of the table against the

corresponding row.

2 Identify the boxes having minimum and next to minimum transportation cost in

each column and write the difference (penalty) against the corresponding column

3. Identify the maximum penalty. If it is along the side of the table, make maximum

allotment to the box having minimum cost of transportation in that row. If it is

below the table, make maximum allotment to the box having minimum cost oftransportation in that column.

4. If the penalties corresponding to two or more rows or columns are equal, select

the top most row and the extreme left column.

Linear Program Structure

Linear programming models consist of an objective function and the constraints on thatfunction.

The problem is formulated from the problem statement as follows:

1.Identify the objective of the problem; that is, which quantity is to be optimized.

2.Identify the decision variables and the constraints on them.

3. Write the objective function and constraints in terms of the decision variables, using information

from the problem statement to determine the proper coefficient for each term.

4. Add any implicit constraints, such as non-negative restrictions.

5. Arrange the system of equations in a consistent form suitable for solving by computer.

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Stepping stone method

TheStepping stone method is an iterative technique for moving from an initial feasible solution

to an optimal solution.

It is used to evaluate the cost effectiveness of shipping goods via transportation routes not

currently in the solution.

Rules of stepping stone method

1. Select an empty cell to evaluate.

2. Beginning at this cell, trace a closed path back to the cell, turning corners only on filled cells. Only

horizontal & vertical moves are allowed. You may step over any cell.

3. Place alternating +s and -s on this path, beginning with a + on the empty cell being evaluated.

4. Calculate the change index by summing the unit costs, in the tagged cells, accounting for the + & -

signs.

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DATA ANALY (Mathemati al Approach)

Deci i Variable: Xij

Objecti e functi n:

Mini i e t e di tance:

13.3X12

+11.7X13

+16.3X14

+12.6X16

+10.2X25

+5.5X26

+11.3X38

+3.6X47

+8.8X59

+8.5X65

+14.8X

68+16.6X69+11.2X78+7.1X89

Cons

nts:

X12+X13+X14+X16=1

X25+X26-X12=0

X38-X13=O

X47-X14=0

X59-X25-X65=0

X65+X68+X69-X16-X26=0

X78-X47=0

X89-X38=0

-X59-X69-X89=-1

XIJ=0 or1

V

ous outs

om Singasandra branc office to Peenya main hub

Figures are in kilometers

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Dec var x12 x13 x14 x16 x25 x26 x38 x47 x59 x65 x68 x69 x78 x89

2 1 -3 1 1 1 1 -3 2 1 3 -2 -3 1

13.3 11.7 16.3 12.6 10.2 5.5 11.3 3.6 8.8 8.5 14.8 16.6 11.2 7.1

constrain

1st 1 1 1 1

2nd -1 1 1

3rd -1 1

4th -1 1

5th -1 1 -1

6th -1 -1 1 1 1

7th -1 1

8th -1 1

9th -1 -1 -1

constrain

1st 1 1

2nd -1.1E-13 0

3rd 0 0

4th 0 0

5th 1.1E-13 0

6th -1.1E-13 0

7th 0 0

8th 0 0

9th -1 -1

obj.fn. 29

Shortest path: 1-3-8-9