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Transportation Problem Explained and how to solve it?

  • Introduction
  • Transportation Problem
  • Balanced Problem
  • Unbalanced Problem

Contributed by: Patrick

Operations Research (OR) is a state of art approach used for problem-solving and decision making. OR helps any organization to achieve their best performance under the given constraints or circumstances. The prominent OR techniques are,

  • Linear programming
  • Goal programming
  • Integer programming
  • Dynamic programming
  • Network programming

One of the problems the organizations face is the transportation problem. It originally means the problem of transporting/shipping the commodities from the industry to the destinations with the least possible cost while satisfying the supply and demand limits.  It is a special class of linear programming technique that was designed for models with linear objective and constraint functions. Their application can be extended to other areas of operation, including

  • Scheduling and Time management
  • Network optimization
  • Inventory management
  • Enterprise resource planning
  • Process planning
  • Routing optimization  

The notations of the representation are:

m sources and n destinations

(i , j) joining source (i) and destination (j)

c ij 🡪  transportation cost per unit

x ij 🡪  amount shipped

a i   🡪 the amount of supply at source (i)

b j   🡪 the amount of demand at destination (j)

Transportation problem works in a way of minimizing the cost function. Here, the cost function is the amount of money spent to the logistics provider for transporting the commodities from production or supplier place to the demand place. Many factors decide the cost of transport. It includes the distance between the two locations, the path followed, mode of transport, the number of units that are transported, the speed of transport, etc. So, the focus here is to transport the commodities with minimum transportation cost without any compromise in supply and demand. The transportation problem is an extension of linear programming technique because the transportation costs are formulated as a linear function to the supply capacity and demand. Check out the course on transportation analytics .

Transportation problem exists in two forms. 

  • Balanced 

It is the case where the total supply equals the total demand.

It is the case where either the demand is greater than the supply, or vice versa.

In most cases, the problems take a balanced form. It is because usually, the production units work, taking the inventory and the demand into consideration. Overproduction increases the inventory cost whereas under production is challenged by the demand. Hence the trade-off should be carefully examined. Whereas, the unbalanced form exists in a situation where there is an unprecedented increase or decrease in demand.

Let us understand this in a much simpler way with the help of a basic example. 

Let us assume that there is a leading global automotive supplier company named JIM. JIM has it’s production plants in many countries and supplies products to all the top automotive makers in the world. For instance, let’s consider that there are three plants in India at places M, N, and O. The capacity of the plants is 700, 300, 550 per day. The plant supplies four customers A, B, C, and D, whose demand is 650, 200, 450, 250 per day. The cost of transport per unit per km in INR and the distance between each source and destination in Kms are given in the tables below.

Here, the objective is to determine the unknown while satisfying all the supply and demand restrictions. The cost of shipping from a source to a destination is directly proportional to the number of units shipped.

Many sophisticated programming languages have evolved to solve OR problems in a much simpler and easier way. But the significance of Microsoft Excel cannot be compromised and devalued at any time. It also provides us with a greater understanding of the problem than others. Hence we will use Excel to solve the problem.

It is always better to formulate the working procedure in steps that it helps in better understanding and prevents from committing any error.

Steps to be followed to solve the problem:

  • Create a transportation matrix (define decision variables)
  • Define the objective function
  • Formulate the constraints
  • Solve using LP method 

Creating a transportation matrix:

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.

Define the objective function: 

An objective function is our target variable. It is the cost function, that is, the total cost incurred for transporting. It is known as an objective function because our interest here is to minimize the cost of transporting while satisfying all the supply and demand restrictions.

The objective function is the total cost. It is obtained by the sum product of the cost per unit per km and the decision variables (highlighted in red), as the total cost is directly proportional to the sum product of the number of units shipped and cost of transport per unit per Km.

The column “Total shipped” is the sum of the columns A, B, C, and D for respective rows and the row “Total Demand” is the sum of rows M, N, and O for the respective columns. These two columns are introduced to satisfy the constraints of the amount of supply and demand while solving the cost function. 

Formulate the constraints:

The constraints are formulated concerning the demand and supply for respective rows and columns. The importance of these constraints is to ensure they satisfy all the supply and demand restrictions.

For example, the fourth constraint, x ma + x na + x oa = 650 is used to ensure that the number of units coming from plants M, N, and O to customer A should not go below or above the demand that A has. Similarly the first constraint x ma + x mb + x mc + x md  = 700 will ensure that the capacity of the plant M will not go below or above the given capacity hence, the plant can be utilized to its fullest potential without compromising the inventory. 

Solve using LP method:

The simplest and most effective method to solve is using solver. The input parameters are fed as stated below and proceed to solve. 

This is the best-optimized cost function, and there is no possibility to achieve lesser cost than this having the same constraints.

From the solved solution, it is seen that plant M ships 100 units to customer A, 350 units to C and 250 units to D. But why nothing to customer B? And a similar trend can be seen for other plants as well. 

What could be the reason for this? Yes, you guessed it right! It is because some other plants ship at a profitable rate to a customer than others and as a result, you can find few plants supplying zero units to certain customers. 

So, when will these zero unit suppliers get profitable and can supply to those customers? Wait! Don’t panic. Excel has got away for it too. After proceeding to solve, there appears a dialogue box in which select the sensitivity report and click OK. You will get a wonderful sensitivity report which gives details of the opportunity cost or worthiness of the resource.

Basic explanation for the report variables,

Cell: The cell ID in the excel

Name: The supplier customer pairing

Final value: Number of units shipped (after solving)

Reduced cost: How much should the transportation cost per unit per km should be reduced to make the zero supplying plant profitable and start supplying

Objective coefficient: Current transportation cost per unit per Km for each supplier customer pair

Allowable Increase: It tells us the maximum cost of the current transportation cost per unit per Km can be increased which doesn’t make any changes to the solution

Allowable Decrease: It tells how much maximum the current transportation cost per unit per Km can be lowered which doesn’t make any changes to the solution

Here, look into the first row of the sensitivity report. Plant M supplies to customer A. Here, the transportation cost per unit per Km is ₹14 and 100 units are shipped to customer A. In this case, the transportation cost can increase a maximum of ₹6, and can lower to a maximum of ₹1. For any value within this range, there will not be any change in the final solution. 

Now, something interesting. Look at the second row. Between MB, there is not a single unit supplied to customer B from plant M. The current shipping cost is ₹22 and to make this pair profitable and start a business, the cost should come down by ₹6 per unit per Km. Whereas, there is no possibility of increasing the cost by even a rupee. If the shipping cost for this pair comes down to ₹16, we can expect a business to begin between them, and the final solution changes accordingly.

The above example is a balanced type problem where the supply equals the demand. In case of an unbalanced type, a dummy variable is added with either a supplier or a customer based on how the imbalance occurs.

Thus, the transportation problem in Excel not only solves the problem but also helps us to understand how the model works and what can be changed, and to what extent to modify the solution which in turn helps to determine the cost and an optimal supplier. 

If you found this helpful, and wish to learn more such concepts, head over to Great Learning Academy and enroll in the free online courses today.

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Table of contents

Geography Notes

8 helpful steps for solving the problems of urban transport.

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There is no readymade universally acceptable solution to the urban transport problem. Planners, engineers, economists and transport technologists each have their own views, which when combined, invariably produced a workable strategy. Whatever policy evolved should be considered firstly, in the light of time it takes to implement them and secondly, all policies need to be appraised in terms of their cost.

The following common steps may be helpful in solving the problems of urban transport:

1. Development of Additional Road Capacity:

One of the most commonly adopted methods of combatting road congestion in medium and small towns or in districts of larger centres is the construction of bypasses to divert through-traffic. This practice has been followed throughout the world including India. Mid-twentieth century planners saw the construction of additional road capacity in the form of new or improved highways as the acceptable solution to congestion within major towns and cities.

Since the pioneer transportation studies of the 1950s and 1960s were carried out in the US metropolitan areas, where the needs of an auto-dominated society were seen to be paramount, the provision of additional road capacity was accepted for several decades as the most effective solution to congestion, and urban freeways were built in large cities such as Chicago, San Francisco and Los Angeles.

Western European transport planners incorporated many of their American counterparts’ concepts into their own programmes and the urban motorway featured in many of the larger schemes (Muller, 1995). However, it soon became evident that the generated traffic on these new roads rapidly reduced the initial advantages.

The construction of an urban motorway network with its access junctions requires large areas of land and the inevitable demolition of tracts of housing and commercial properties. By the 1970s planners and policy­makers came to accept that investment in new highways dedicated to the rapid movement of motor traffic was not necessarily the most effective solution to urban transport problems.

2. Traffic Management Measures:

Temporary and partial relief from road traffic congestion may be gained from the introduction of traffic management schemes, involving he reorganisation of traffic flows and direc­tions without any major structural alterations to the existing street pattern. Among the most widely used devices are the extension of one-way systems, the phasing of traffic-light controls to take account of traffic variation, and restrictions on parking and vehicle loading on major roads.

On multi-lane highways that carry heavy volumes of commuter traffic, certain lanes can be allocated to incoming vehicles in the morning and to outgoing traffic in the afternoon, producing a tidal-flow effect. Recent experiments using information technology have been based upon intelligent vehicle highway systems (IVHS), with the computerised control of traffic lights and entrances to freeways, advice to drivers of alternative routes to avoid congestion, and information on weather and general road conditions. The IVHS can be linked up with advanced vehicle control systems, making use of in-car computer to eliminate driver error and control automatic braking and steering when accidents are imminent.

Traffic management has been extensively applied within urban residential areas, where excessive numbers of vehicles produce noise, vibration, pollution and, above all, accident risks, especially to the young. ‘Traffic calming’ has been intro­duced to many European cities and aims at the creation of an environment in which cars are permitted but where the pedes­trian has priority of movement. Carefully planned street-width variations, parking restrictions and speed-control devices such as ramps are combined to secure a safe and acceptable balance between car and pedestrian.

3. Effective Use of Bus Service :

Many transportation planning proposals are aimed specifically at increasing the speed and schedule reliability of bus services, and many European cities have introduced bus priority plans in an attempt to increase the attractions of public transport. Bus-only lanes, with or against the direction of traffic flow, are designated in heavily congested roads to achieve time savings, although such savings may later be dissipated when buses enter inner-city areas where priority lanes at intersections and certain streets may be restricted to buses only, particularly in pedestrianised shopping zones.

Where entirely new towns are planned, there is an oppor­tunity to incorporate separate bus networks within the urban road system, enabling buses to operate free from congestion. In the UK, Runcorn New Town, built as an overspill centre for the Merseyside conurbation, was provided with a double- looped busway linking shopping centre, industrial estates and housing areas.

About 90 per cent of the town’s population was within five minutes’ walk of the busway and operating costs were 33 per cent less than those of buses on the conventional roads. Although the system is not used to the extent originally envisaged, it successfully illustrates how public transport can be integrated with urban development. Bus-only roads can also be adapted to vehicle guidance systems, whereby the bus is not steered but controlled by lateral wheels, with the resumption of conventional control when the public road network is re-entered.

Such systems have been adopted in Adelaide and experiments have been made in many other cities (Adelaide Transit Authority, 1988). The bus can also be given further advantages in city centres where major retailing and transport complexes are being redeveloped. The construction of covered shopping malls and precincts can incorporate bus facilities for shoppers, and reconstruction of rail stations can also allow bus services to be integrated more closely with rail facilities.

The ‘park-and-ride’ system, now adopted by many European cities, enables the number of cars entering city centres to be reduced, particularly at weekend shopping peak periods. Large car-parks, either temporary or permanent according to need, on the urban fringe are connected by bus with city centres, with charges generally lower than central area parking costs.

The advantages of the bus over the car as an efficient carrier are secured, and the costs of providing the fringe car-parks are much less than in inner-city zones. Rail commuters can also be catered for in a similar manner with the provision of large-capacity car-parks adjacent to suburban stations.

Many towns and cities have’ attempted to attract passengers back to bus transport by increasing its flexibility and level of response to market demand. In suburban areas the dial-a-ride system has met with partial success, with prospective passengers booking seats by telephone within a defined area of operation.

Such vehicles typically serve the housing areas around a district shopping centre and capacity is limited, so they are best suited to operations in conditions of low demand or in off-peak periods. Fares are higher than on conventional buses since the vehicle control and booking facilities require financing.

Experiments have also been made with small- capacity buses that can be stopped and boarded in the same way as a taxi and which can negotiate the complex street patterns of housing estates more easily than larger buses. However, with the widespread introduction of scheduled minibus is the problem of overloading has been reduced.

4. Parking Restrictions :

As we have seen, it is not possible to provide sufficient space for all who might like to drive and park in the central areas of large towns. Parking thus must be restricted and this is usually done by banning all-day parking by commuters or making it prohibi­tively expensive. Restrictions are less severe – off-peak, so that shoppers and other short-term visitors who benefit the economy of the centre are not deterred. Separate arrangements must be made for local residents, perhaps through permits or reserved parking.

City authorities can thus control public car-parking places, but many other spaces are privately owned by businesses and reserved for particular employees. The effect of this is to perpetuate commuting to work by car. The future provision of such space can be limited through planning permission for new developments, as is done in London, but controlling the use of existing private spaces raises problematical issues of rights and freedoms that many countries are reluctant to confront.

Overall, parking restrictions have the advantage of being simple to administer, flexible in application and easily under­stood by the public. Their Achilles’ heel is enforcement, for motorists are adept at parking where and when they should not and evading fines once caught.

Fines in many cities are so low that being caught once or twice a week works out cheaper than paying the parking charge. Indeed, in London in 1982, a survey showed that illegal parkers outnumbered legal ones and only 60 per cent of the fines were ever paid. Parking controls have to be stringent and be enforced if they are to make any significant contribution to reducing congestion in the city.

5. Promoting the Bicycle :

The benefits of cycling have long been recognised. The bicycle is cheap to buy and run and is in urban areas often the quickest door-to-door mode (Figure 5.3). It is a benign form of transport, being noiseless, non-polluting, energy-and space-efficient and non-threatening to most other road users. A pro-cycling city would promote fitness among cyclists and health among non-cyclists. Cycling is thus a way of providing mobility, which is cheap for the individual and for society.

Advocates of Environmental Traffic Management (ETM) frequently cast envious glances at the Netherlands, where cycle planning is set in the context of national planning for sustainability. The Master Plan Bicycle, which aims to increase bicycle-kilometers by at least 30 per cent between 1986 and 2010, not only tackles the traditional concerns of cycle infra­structure and road safety, but also addresses issues of mobility and modal choice; how to encourage businesses to improve the role of the bicycle in commuting; reducing bicycle theft and increasing parking quantity and quality; improving the combi­nation of cycling and public transport; and promoting consideration of the bicycle amongst influential decision makers. These ‘pull’ measures are part of a national transport strategy of discouraging car use, which ‘pushes’ motorists towards use of the bicycle.

6. Encouraging Walking :

Walking is the most important mode of transport in cities, yet frequently data on it are not collected and many planners do not think of it as a form of transport. As a result of this neglect, facilities provided specifically for walking are often either absent or badly maintained and pedestrians form the largest single category of road user deaths. There are social, medical, environmental and economic reasons for promoting walking, for it is an equitable, healthy, non-polluting and inexpensive form of transport. Moreover, ‘foot cities’ tend to be pleasurable places in which to live, with access to facilities within walking distance frequently cited as a key indicator of neighbourhood quality of life.

7. Promoting Public Transport :

If ETM aims to shift trips away from cars, then attractive alter­natives are required. Cycling and walking may be appropriate for the shorter distances, but transferring longer trips requires that a good quality public transport system is in place to ensure that the city can function efficiently.

This means that:

1. Fares need to be low enough for poor people to be able to afford them;

2. There must be sufficient vehicles for a frequent service to be run throughout the day;

3. Routes must reflect the dominant desire lines of the travelling public and there should be extensive spatial coverage of the city so that no one is very far from a public transport stop;

4. Speeds of buses need to be raised relative to cars by freeing them from congestion;

5. It is not enough to provide public transport: it also has to be coordinated. Multi-modal tickets may be one essential ingredient of a functional urban transport system, but the key item is the integration of services by the provision of connections between modes.

8. Other Measures :

Some of the other measures useful for urban transport planning are:

1. Restrictions on road capacity and traffic speeds,

2. Regulating traffic access to a link or area,

3. Charging for the use of roads on a link, or area basis,

4. Vehicle restraint schemes,

5. Rail rapid transit,

6. Transport coordination, and

7. Public transport improvement, etc.

The urban transport planning is a continuous process and it should be done through a process, as Figure 5.4 shows, are the pre-analysis, technical analysis and the post analysis phases.

Once the goals are established, data need to be collected in order to prepare land use, transport and travel inventories of the study area. The availability of good quality, extensive and up-to-date data is an essential precondition for the preparation of an urban transport plan. Accordingly, there will need to be an inventory of the existing transport system and the present distribution of land uses; a description of current travel patterns; and data on such matters as population growth, economic activity, employment, income levels, car ownership, housing and preferred travel modes.

In brief, urban transport process has four principal charac­teristics – quantification, comprehensiveness, systems thinking and a scientific approach. The environmental traffic management system should be adopted both in developed and developing countries in order to check the increasing problems of the urban transport.

Related Articles:

  • 7 Problems of Urban Transport (Explained With Diagram)
  • Essay on the Problems of Urban Growth | World | Human Geography
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  • Modes of Transport Suitable For Urban Areas

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Solving Transportation Problem using Linear Programming in Python

Learn how to use Python PuLP to solve transportation problems using Linear Programming.

In this tutorial, we will broaden the horizon of linear programming problems. We will discuss the Transportation problem. It offers various applications involving the optimal transportation of goods. The transportation model is basically a minimization model.

The transportation problem is a type of Linear Programming problem. In this type of problem, the main objective is to transport goods from source warehouses to various destination locations at minimum cost. In order to solve such problems, we should have demand quantities, supply quantities, and the cost of shipping from source and destination. There are m sources or origin and n destinations, each represented by a node. The edges represent the routes linking the sources and the destinations.

how to solve problem of transportation

In this tutorial, we are going to cover the following topics:

Transportation Problem

The transportation models deal with a special type of linear programming problem in which the objective is to minimize the cost. Here, we have a homogeneous commodity that needs to be transferred from various origins or factories to different destinations or warehouses.

Types of Transportation problems

  • Balanced Transportation Problem :  In such type of problem, total supplies and demands are equal.
  • Unbalanced Transportation Problem : In such type of problem, total supplies and demands are not equal.

Methods for Solving Transportation Problem:

  • NorthWest Corner Method
  • Least Cost Method
  • Vogel’s Approximation Method (VAM)

Let’s see one example below. A company contacted the three warehouses to provide the raw material for their 3 projects.

how to solve problem of transportation

This constitutes the information needed to solve the problem. The next step is to organize the information into a solvable transportation problem.

Formulate Problem

Let’s first formulate the problem. first, we define the warehouse and its supplies, the project and its demands, and the cost matrix.

Initialize LP Model

In this step, we will import all the classes and functions of pulp module and create a Minimization LP problem using LpProblem class.

Define Decision Variable

In this step, we will define the decision variables. In our problem, we have various Route variables. Let’s create them using  LpVariable.dicts()  class.  LpVariable.dicts()  used with Python’s list comprehension.  LpVariable.dicts()  will take the following four values:

  • First, prefix name of what this variable represents.
  • Second is the list of all the variables.
  • Third is the lower bound on this variable.
  • Fourth variable is the upper bound.
  • Fourth is essentially the type of data (discrete or continuous). The options for the fourth parameter are  LpContinuous  or  LpInteger .

Let’s first create a list route for the route between warehouse and project site and create the decision variables using LpVariable.dicts() the method.

Define Objective Function

In this step, we will define the minimum objective function by adding it to the LpProblem  object. lpSum(vector)is used here to define multiple linear expressions. It also used list comprehension to add multiple variables.

In this code, we have summed up the two variables(full-time and part-time) list values in an additive fashion.

Define the Constraints

Here, we are adding two types of constraints: supply maximum constraints and demand minimum constraints. We have added the 4 constraints defined in the problem by adding them to the LpProblem  object.

Solve Model

In this step, we will solve the LP problem by calling solve() method. We can print the final value by using the following for loop.

From the above results, we can infer that Warehouse-A supplies the 300 units to Project -2. Warehouse-B supplies 150, 150, and 300 to respective project sites. And finally, Warehouse-C supplies 600 units to Project-3.

In this article, we have learned about Transportation problems, Problem Formulation, and implementation using the python PuLp library. We have solved the transportation problem using a Linear programming problem in Python. Of course, this is just a simple case study, we can add more constraints to it and make it more complicated. In upcoming articles, we will write more on different optimization problems such as transshipment problem, assignment problem, balanced diet problem. You can revise the basics of mathematical concepts in  this article  and learn about Linear Programming  in this article .

  • Solving Cargo Loading Problem using Integer Programming in Python
  • Solving Blending Problem in Python using Gurobi

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Innovative Uses of Data Science to Solve Transportation Problems

State DOTs, metropolitan planning organizations (MPO), and regional transportation planning agencies rely on complex data to solve the nation’s congestion, infrastructure, and equity issues. U.S. DOT Volpe Center data experts are using tools from the data science and machine learning toolbox to identify roadway safety issues and address old transportation problems in new ways. Using big data and computer vision solutions to provide state DOTs with new sources of roadway infrastructure data and natural language processing (NLP), the U.S. DOT Volpe Center is creating new methods of ingesting and processing data from multiple sources. 

Understanding Travel Patterns and Roadway Safety in Public Lands

Heatmap of hard braking events around Lincoln Memorial Circle at National Mall, Washington, D.C. (NPS image)

Efforts to reduce traffic congestion, identify safety risks, and improve visitor experience are often hampered by a lack of relevant data. U.S. DOT Volpe Center researchers are evaluating and analyzing emerging data sources to understand visitor movement and improve planning, management, and operations at public lands. A selection of these data include Strava Metro, Wejo, StreetLight Data, AllTrails, and U.S. DOT’s nationwide archive of Waze traffic alerts and traffic jams.  

Planning efforts at Adirondack High Peaks Region used Waze’s traffic alerts, along with trail information on AllTrails, to identify key time periods when shuttle service might alleviate traffic congestion while still providing trail access, which are detailed in the New York State Adirondack High Peaks Region Shuttle Feasibility Study . Other planning efforts at Mount Rainier National Park included location based services (LBS) data, Waze traffic alerts, and biking and hiking data from Strava Metro to provide visitor home regions and a holistic view of visitor travel within and en route to the Park.  

Rural Roadway Safety 

The U.S. DOT Volpe Center is also collaborating with the University of North Carolina Renaissance Computing Institute (RENCI) and the North Carolina Department of Transportation (NCDOT) to develop an artificial intelligence (AI) computer vision tool for automated analysis, extraction, and annotation of roadside features on rural roads from existing video data. This information can be used to by NCDOT to more efficiently assess the safety of the roadside. The tool will help to facilitate safety initiatives more efficiently and effectively on North Carolina’s rural roadway system. AI components will also provide a blueprint for other jurisdictions and states. This project has a public GitHub repository for the code used in the tool. For more information about this project, see NCDOT’s final report or the IEEE published research paper for a more technical overview. 

Waze Traffic Trends

Waze Traffic Trends Graphic of U.S.

The U.S. DOT Volpe Center developed a COVID-19 Waze Traffic Alert Dashboard in March 2020 to track relative changes in weekly traffic jam alerts for all U.S. metropolitan areas. The team has continued to provide weekly updates through 2022. The Waze dashboard provides a rapid indicator of traffic jams covering all U.S. metropolitan areas, increasing accessibility to state, metropolitan, and county-level time trends.  

Text Data Toolbox

Text data is a large untapped resource for the U.S. DOT. U.S. DOT Volpe Center data scientists worked OST-R to build a toolbox of NLP approaches to make tasks like summarization, comparison, and exploration of text data easier for U.S. DOT offices. For example, OST-R asked the U.S. DOT Volpe Center to develop tools to identify points of similarity for all departmental-funded research projects, which was not possible to do manually. U.S. DOT Volpe Center experts used open-source similarity techniques, tuned for the kind of text used across U.S. DOT, and created a dashboard for OST-R staff to quickly find previously unknown complementarity for research projects across the Department. U.S. DOT Volpe Center data scientists continue to build on this concept as the project develops.

Interactive GIS Tool for Tracking Transportation, Demographic Data  

Historically disadvantaged populations continue to suffer from systemic inequities. Gaps exist in available knowledge and tools that hinder the ability of transportation agencies to integrate social equity considerations into transportation programs, policies, projects, and other activities. A U.S. DOT Volpe Center team developed a prototype Transportation for Social Equity (TransportSE) tool to work toward closing those gaps. The tool can improve the ability of transportation agencies at all levels to understand, visualize, and analyze transportation equity indicators (including benefits, such as access to transportation services, and burdens, such as noise and pollution) in relation to social equity variables such as race and income.  

The approaches highlighted here demonstrate the U.S. DOT Volpe Center’s capabilities to use data science innovations for transportation planning, operations, safety analysis, and to improve business processes. Further expansion of these capabilities will help the U.S. DOT Volpe Center meet the requirements of increasing volumes and complexity of data, as well as the need for advanced analytics in all areas of transportation. 

how to solve problem of transportation

  • Logistics , Transportation , Truck Aug 18, 2017
  • Subhasis Das

Solving The Transportation Problem Using Technology

how to solve problem of transportation

The future of modern India is built upon transportation. Mobility not only keeps the economy moving but has also made the life fast progressive. As per World Bank, in India Roads carry more than 60 percent of its freight. The density of India’s highway network is 0.66 km of roads per square kilometer, which is similar to the United States (0.65) and greater than China’s (0.16).

Connectivity improved further by information technology has changed the way mankind thinks and India is no exception. As more and more goods are produced in factories, they need to reach the warehouses and to the consumers to cater to the ever-growing demand. The entire structure of our society is based on trade. And if something disrupts this processor doesn’t produce effective results, it results in transportation problem.

Identifying the transportation problem

architecture, blur, bridge

In simple terms, the transportation problem is a classic operations problem. Where the basic objective of determining the accurate schedule for transporting goods from the source to its destination is not met, it causes problems for the transporter.

However, where the entire transportation process runs smoothly minimizing the shipping cost, meeting demand and supply restraints, there is no problem. This is where the role of information technology is very important as it helps by offering linear programming solutions, tracking, saving time, and eventually an effective connected transportation solution.

The transportation problem has a direct impact on the business. It can cause an increase in total transportation cost , more consumption of limited resources, such as diesel, damage of goods etc have a bad effect on business.

The role of technology

auto, automobile, automotive

Trucks and transporters are a vital part of the Indian economy. New technology is making the transportation job easier by overcoming the inefficiencies and shortcomings. The new technologies are emerging to solve the transportation problems and meet the challenges by bridging the gap by real-time information flow which eventually will enhance better pricing, availability and service delivery. They are making fleet connected and autonomous, alternative fuel solutions, hassle-free fleet management and using traffic analytics to identify shortcomings and fixing them.

Trukky is proud to be able to solve India’s transportation challenges with the simple clicks of technology.

Trukky is India’s fast emerging full truckload (3 Ton to 30 Ton) and part load or parcel service provider on major routes comprising of Delhi-Mumbai , Ahmedabad – Mumbai , Mumbai – Bangalore , and Delhi- Chennai .

Our web and mobile interface enables you to book your truck from anywhere. You can make the booking in three quick steps- enter your details, get the price quote and receive the transport service details on your mobile. Our technology brings to you economy and expresses delivery at the doorstep anywhere in the country.

We offer top three technology-based services in India’s trucking industry

  • GPS-based transportation management system software, i.e. vendor management, comparison of rates, scheduling and tracking of trucks
  • Data drove decision making to drive pricing, service delivery, and fleet mobilization.
  • End to end solution through mobile and web application, i.e. registration, ordering, tracking, Documentation, alert notifications, delivery time estimations, invoice,  & payments.

Connected transportation is the only way forward to stay competitive. For businesses, it is now possible to keep a track of almost anything and stay competitive and meet dynamic consumer demand. Earlier, real-time data was never available and consignors were heavily dependent on unreliable manual data provided by service providers which affected operations and deliveries.

For the fleet owners, the real-time information makes it absolutely easy and simple to manage all the vehicles that they own. They would know where is their truck and therefore there is no scope of delay in delivery or false information or misinterpretation.

auto, autocar, automobile

The live GPS technology not only helps in tracking a fleet of trucks but also helps the staff in the security of their vehicles. For instance, if you have placed a live GPS tracking device in your truck, then the telematics will prevent it from getting stolen. If it gets stolen, you can see exactly where the truck is right now. Not only the location, the device will give you a detailed map that will show the route that the thieves had taken. Moreover, you will have the details like the speed at which they were driving and the places where they went and stopped. This information is very critical because it tells you where exactly your car is and you can inform the police at once.

However, to some of our client’s data can appear quite overwhelming. We, at Trukky, help you in understanding that all the data is relevant and useful to match your operational metrics and measure logistics performance with detailed analytics report. We understand that businesses don’t always have the time to sit and understand operational data and that’s where Trukky is at your service by not only providing you an efficient service but also help you understand your logistics performance through simple and automated metrics dashboard, which again can be customized as per business specification.

New technologies are taking on-road communications to another level. They regularly keep updating their article for free guest post sites about the transport industry.They are changing the way the transporters perceive and manage their business. They are dramatically changing the way vehicles operate and make information accessible and build capabilities for real-time traffic management, provided the necessary network infrastructure is present.

The transportation players with good name and bigger market share are better to leverage technology to improve your transportation experience. Technology sets benchmarks and helps you in making the right choice. 

Is the trucking industry ready to embrace and follow the new technology? Is it ready to be smart enough to bring in the revolution? Indeed, there are drivers and transporters who can’t wait for the trucking industry to enjoy the best technologies.

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  • Transportation

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  • About Trukky

Trukky.com is the India’s first portal that allows a customer to book a truck online. Trukky.com is a Logistics aggregator offering On-demand Transportation solutions to a cater to goods movement in both Full load & Part load segment. We aim to deliver quality service at competitive price and back every shipment with technology & outstanding customer support service. Trukky offers a single stop solution for Pan India deliveries to customers. We at Trukky try to support our customers for their customized requirements which are not feasible for a local transporter. Direct connection with the Drivers and Fleetowners allows eradicating the Brokers / Transporters margin and hence reducing the expenses / cost for the customer. Large network of Drivers / Fleet owners from across the country allows Trukky to act as one stop solution for PAN India needs. With 5,000+ supply attached in the system, Trukky services clients/customers from various industries like Paper & Pulp, FMCG, Cement, Agricultural products, Ecommerce, FMCD,and Chemical.

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Solving a Transportation Problem

  • Watching now: Chapter 1: Description of a Transportation Problem to Be Solved Using MS Excel Start time: 00:00:00 End time: 00:01:52
  • Chapter 2: Tabulating the Data to Solve a Transportation Problem Using MS Excel Start time: 00:01:53 End time: 00:06:03
  • Chapter 3: Using MS Excel Goal-Seek to Solve a Transportation Problem Start time: 00:06:04 End time: 00:11:06

Video Type: Tutorial

(2020). Solving a transportation problem [Video]. Sage Research Methods. https:// doi. org/10.4135/9781529630527

"Solving a Transportation Problem." In Sage Video . : Starttech Educational Services LLP, 2020. Video, 00:11:06. https:// doi. org/10.4135/9781529630527.

, 2020. Solving a Transportation Problem , Sage Video. [Streaming Video] London: Sage Publications Ltd. Available at: <https:// doi. org/10.4135/9781529630527 & gt; [Accessed 18 Feb 2024].

Solving a Transportation Problem . Online video clip. SAGE Video. London: SAGE Publications, Ltd., 14 Dec 2022. doi: https:// doi. org/10.4135/9781529630527. 18 Feb 2024.

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How to solve a transportation problem using the MS Excel Solver add-in for business statistic analytics.

Chapter 1: Description of a Transportation Problem to Be Solved Using MS Excel

  • Start time: 00:00:00
  • End time: 00:01:52

Chapter 2: Tabulating the Data to Solve a Transportation Problem Using MS Excel

  • Start time: 00:01:53
  • End time: 00:06:03

Chapter 3: Using MS Excel Goal-Seek to Solve a Transportation Problem

  • Start time: 00:06:04
  • End time: 00:11:06
  • Product: Sage Research Methods: Business
  • Type of Content: Tutorial
  • Title: Solving a Transportation Problem
  • Publisher: Starttech Educational Services LLP
  • Series: Statistics for Business Analytics using MS Excel
  • Publication year: 2020
  • Online pub date: December 14, 2022
  • Discipline: Business and Management
  • Methods: Statistical packages , Data analysis skills
  • Duration: 00:11:06
  • DOI: https:// doi. org/10.4135/9781529630527
  • Keywords: business skills , data analysis , Statistical packages Show all Show less
  • Online ISBN: 9781529630527 Copyright: Copyright © 2020 Starttech Educational Services LLP More information Less information

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Geektonight

Transportation Problem: Finding an Optimal Solution

  • Post last modified: 27 July 2022
  • Reading time: 15 mins read
  • Post category: Operations Research

The transportation problem is an important Linear Programming Problem (LPP). This problem depicts the transportation of goods from a group of sources to a group of destinations. The whole process is subject to the availability and demand of the sources as well as destination, respectively in a way, where entire cost of transportation is minimised.

Sometimes, it is known as Hitchcock problem. Generally, transportation costs are involved in such problems but the scope of problems extends well beyond to hide situations that do not have anything to try with these costs.

Table of Content

  • 1.1 Step 1: Obtaining the Initial Feasible Solution
  • 1.2 Step 2: Testing the Optimality
  • 1.3 Step 3: Improving the Solution
  • 2 Stepping Stone Method
  • 3.1 Degeneracy in Transportation Problem
  • 3.2 Unbalanced Transportation Problem
  • 3.3 Alternative Optimal Solutions
  • 3.4 Maximisation Transportation Problem
  • 3.5 Prohibited Routes

The term ‘transportation’ is related to such problems principally because in studying efficient transportation routes, a special procedure was used which has come to be referred to as the transportation method.

A typical transportation problem is a distribution problem where transfers are made from various sources to different destinations, with known unit costs of transfer for all source-destination combinations, in a manner that the total cost of transfers is the minimum. In this chapter, you will discuss how to improve an optimal solution by stepping stone method and describe the special cases in the transportation problems.

Finding Optimal Solution Using the Stepping Stone Method

A typical transportation problem is like this. Let’s consider that a man- ufacturer of refrigerators runs three plants located at different places called A, B and C. Suppose further that his buyers are located in three regions X, Y and Z where he has got to supply them the products.

So, the need within the three regions as well as production in several plants per unit time period are known and equal in aggregate and further that the cost of one transporting refrigerator from each plant to each of the requirement centres is given and constant.

The manufacturer’s problem is to determine as to how he should route his product from his plants to the marketplaces so that the total cost involved in the transportation is minimized. In other words, he needs to decide on how many refrigerators should be supplied from A to X, Y and Z, how many from B to X, Y and Z and how many from C to X, Y and Z to attain it at the least cost.

The places where the products originate from (the plants in our example) are called the sources or the origins and places where they are to be supplied are the destinations. In this terminology, the trouble of the manufacturer is to decide on how many units are transported from different origins to different destinations in order that the overall transportation cost is the minimum.

The transportation method is an efficient alternative to the simplex method for solving transportation problems.

Step 1: Obtaining the Initial Feasible Solution

To use the transportation method is to get a feasible solution, namely, the one that satisfies the rim requirements (i.e., the requirements of demand and supply). The initial feasible solution may be obtained by various methods.

  • Row Minima Method
  • Column Minima Method
  • North-West Corner (NWC) Rule
  • Least Cost Method (LCM)
  • The Vogel’s Approximation Method (VAM)

Step 2: Testing the Optimality

After obtaining the initial basic feasible solution, the successive step is to check whether it is optimal or not. There are two methods of testing the optimality of a basic feasible solution. One of these is named the stepping-stone method within which the optimality test is applied by calculating the opportunity cost of each empty cell.

The other method is named as the Modified Distribution Method (MODI). It is based on the concept of dual variables that are used to evaluate the empty cell. Using these dual variables, the opportunity cost of each of the empty cell is determined. Thus, while both methods involves determining opportunity costs of empty cells, the methodology employed by them differs.

Both the methods can be used only when the solution is a basic feasible solution so that it has m + n – 1 basic variables. If a basic feasible solution contains less than m + n – 1 non-negative allocation, then the transportation problem is said to be a degenerate one. Incidentally, none of the methods used to find initial solution would yield a solution with greater than m + n – 1 number of occupied cells.

Step 3: Improving the Solution

By applying stepping stone method, if the answer is found to be optimal, then the process terminates because the problem is solved. If the answer is not seen to be optimal, then a revised and improved basic feasible solution is obtained. This can be done by exchanging a non-basic variable for a basic variable.

In simple terms, rearrangement is formed by transferring units from an occupied cell to an empty cell that has the largest opportunity cost, then adjusting the units in other related cells in a way that each one of the rim requirements are satisfied. The solution obtained is again tested for optimality and revised, if necessary. You continue this manner until an optimal solution is finally obtained.

Stepping Stone Method

This is a procedure for determining the potential, if any, of improving upon each of the non-basic variables in terms of the objective function.

To determine this potential, each of the non-basic variables (empty cells) is taken into account one by one. For each such cell, you discover what effect on the overall cost would be if one unit is assigned to the present cell. With this information, then, you come to understand whether the solution is optimal or not.

If not, you improve that solution. This method is derived from the analogy of crossing a pond using stepping stones. It concludes that the whole transportation table is assumed to be a pond and the occupied cells are the stones required to build specific movements inside the pond.

Stepping stone method helps in determining the change in net cost by presenting any of the vacant cells into the solution. The main rule of the stepping stone method is that every increase (or decrease) in supply in one occupied cell must be associated with a decrease (or increase) in supply in another cell. The same rule also holds for demand.

The steps involved in the stepping stone method are as follows:

  • Determine Initial Basic Feasible Solution (IBFS). Make sure the number of occupied cells is exactly equal to m + n − 1. 2. Evaluate the cost-effectiveness of shipping goods via transpor- tation routes for the testing of each unoccupied cell. For this, se- lect an unoccupied cell and trace a closed path using the straight route in which at least three occupied cells are used.
  • Assign plus (+) and minus (−) signs alternatively in the corner cells of the closed path (identified in step 2). The unoccupied cell should be assigned with a plus sign.
  • Add the unit transportation costs associated with each of the cell traced in the closed path. This would give the net change in terms of cost.
  • Repeat steps 2 to 4 until all unoccupied cells are evaluated.
  • Check the sign of each of the net change in the unit transportation costs. If all the net changes calculated are more than or equal to zero, an optimal solution has been attained. If not, then it is possible to advance the current solution and minimise the total transportation cost.
  • Select the vacant cell with the highest negative net cost change and calculate the maximum number of units that can be assigned to this cell. The smallest value with a negative position on the closed path indicates the number of units that can be shipped to the entering cell. Add this number to the unoccupied cell and all other cells on the route having a plus sign and subtract it from the cells marked with a minus sign.
  • Repeat the procedure until we get an optimal solution.

Special Cases in the Transportation Problems

Transportation is all about getting a product from one place to another, put the product on a truck or railcar and you are good to go. Well, not exactly. There’s a bit more that goes into it. It becomes particularly complicated when there are multiple places the product is coming from, and multiple places the product is going to.

Transportation managers must do to some calculations to find the optimum path for getting their product to the customer. Let us look at some common problems a transportation manager might encounter. One common transportation issue has to do with supply and demand.

Some variations that often arise while solving the transportation problem could be as follows:

Degeneracy in Transportation Problem

In a standard transportation problem with m sources of supply and n demand destinations, the test of optimality of any feasible solution requires allocations in m + n – 1 independent cells. Degeneracy occurs whenever the number of individual allocations are but m + n – 1, where m and n are the number of rows and columns of the transportation problem, respectively. Degeneracy in transportation problem can develop in two ways.

  • The basic feasible solution might have been degenerate from the initial stage
  • They may become degenerate at any immediate stage

To resolve degeneracy a little positive number, Δ is assigned to at least one or more unoccupied cell that have lowest transportation costs so on make N = m + n – 1 allocations. (Δ is an infinitesimally small number almost equal to zero.)

Although there is a great deal of flexibility in choosing the unused square for the Δ stone, the general procedure, when using the North West Corner Rule, is to assign it to a square in such how that it maintains an unbroken chain of stone squares. However, where Vogel’s method is used, the Δ allocation is carried during a least cost independent cell.

An independent cell during this context means a cell which cannot cause a closed path on such allocation. After this, the test of optimality is applied and if necessary, the solution is improved within the normal way until optimality is reached.

Unbalanced Transportation Problem

When the total supply of all the sources is not equal to the total demand of all destinations, the problem is an unbalanced transportation problem.

Two situations are possible: 1. If supply < demand, a dummy supply variable is introduced in the equation to make it equal to demand 2. If demand < supply, a dummy demand variable is introduced in the equation to make it equal to supply

Then before solving you must balance the demand and supply. The unit transportation cost for the dummy column and dummy row are assigned zero values, because no shipment is really made just in case of a dummy source and dummy destination.

Alternative Optimal Solutions

An alternate optimal solution is additionally called as an alternate optima, which is when a linear / integer programming problem has more than one optimal solution. Typically, an optimal solution may be a solution to a problem which satisfies the set of constraints of the problem and, therefore, the objective function which is to maximise or minimise. It is possible for a transportation problem to possess multiple optimal solutions.

This happens when one or more of the development indices zero within the optimal solution, which suggests that it’s possible to style routes with an equivalent total shipping cost. The alternate optimal solution is often found by shipping the foremost to the present unused square employing a stepping-stone path. Within the world, alternate optimal solutions provide management with greater flexibility in selecting and using resources.

Maximisation Transportation Problem

The main motive of transportation model is used to minimise transportation cost. However, it also can be used to get a solution with the objective of maximising the overall value or returns. Since the criterion of optimality is maximisation, the converse of the rule for minimisation will be used. The rule is: a solution is optimal if all – opportunity costs dij for the unoccupied cell are zero or negative.

Hence, how does all this help the business overall? If a business’ objective is to maximise profits, then finding the answer to transportation problems allows the companies to use the results from the matrixes to maximise their objective and obtain the foremost profit they will. Profit is often calculated by using this easy formula.

Profit = Selling price – Production cost – Transportation cost

If the objective of a transportation problem is to maximise profit, a minor change is required in the transportation algorithm. Now, the optimal solution is reached when all the development indices are negative or zero. The cell with the most important positive improvement index is chosen to be filled by employing a stepping-stone path. This new solution is evaluated and therefore the process continues until there are not any positive improvement indices.

Prohibited Routes

At times there is transportation problems during which one among the sources is unable to ship to at least one or more of the destinations. When this happens, the problem is claimed to have an unacceptable or prohibited route.

In a minimisation problem, such a prohibited route is assigned a really high cost to prevent this route from ever getting used within the optimal solution. In a maximisation problem, the very high cost utilised in minimisation problems is given a negative sign, turning it into an awfully bad profit.

Sometimes there could also be situations, where it is unacceptable to use certain routes during a transportation problem. For example, road construction, bad road conditions, strike, unexpected floods, local traffic rules, etc.

Such restrictions (or prohibitions) will be handled within the transportation problem by assigning a very high cost (say M or [infinity]) to the prohibited routes to make sure that routes will not be included within the optimal solution then the matter is solved within the usual manner.

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Supply Chain Management Design & Simulation Online

How to Solve Transportation Problems Using Excel Solver

May 17, 2021 By Mohammed BOUALAM

Transportation costs are a significant component of the overall supply chain costs. Therefore, supply chain and logistics managers must take advantage of transportation’s potential in terms of opportunities for cost reduction. In this article, we will be exploring one of the fundamental cases of market and supply allocation; it’s called the Transportation problem .

The goal is to satisfy demand, which means being responsive, while also being efficient, which means having the lowest transportation cost. Allocating supply sources to facilities properly has a considerable impact on your supply chain’s financial and operating performance. By doing it right, you will be managing production and inventory effectively, speeding processes, and improving customer service.

The Transportation Problem

The transportation problem is one of the subclasses of a linear programming problem in which the objective is to transport products stored in a facility (e.g. a warehouse) to different destinations or markets in such a way as to minimize total transportation cost while satisfying all the supply and demand constraints.

Here is a list of information needed to solve a transportation problem:

  • Product demand at each destination facility.
  • Facility locations and distances between each source and destination facility.
  • Cost per kilometer traveled.
  • The maximum storage capacity of each source facility.

Below is the supply chain network we are going to use. It’s a simple network with two stages: there are three Distribution Centers, and the other eight facilities are Stores. Facilities are defined and their icons are put on the map using the SCM Globe supply chain modeling and simulation app. In the screenshot below you can see the facilities and the network of possible roads to use in delivering products from one facility to another.

how to solve problem of transportation

We are glad to provide a  free evaluation account  to instructors, students and supply chain professionals interested in exploring SCM Globe simulations — click here to request an account —  Get Your Free Trial Demo

picture of supply chain network design

Moving to Excel Solver

Now that we have defined the problem and the network model, it’s time to move to Excel to enter the data and make our analysis.

how to solve problem of transportation

The first step in our process is to enter the data we need from the supply chain model in an organized manner. Click on each facility in the SCM Globe supply chain model to see information for product demand, storage capacity, etc. Start by entering each store’s demand; that gives us a total demand of 1190 units in this case. Then enter the unit storage capacity for each distribution center. That gives us a total capacity of 2100 units. We can easily see we have enough capacity to cover market demand.

The second step is to enter the distances between all sources and destinations (between DCs and Stores). To find these distances, create routes between each DC and each of the stores. At each DC define a truck and create routes for it to each store (you don’t need to save these routes, just create them to get distances to the stores). When you do this the route distances shown will be round trip distances, so divide these distances in half . The cost per Km is $ 0.9 which is a default value in the software for large trucks. You can change this default value to more accurately reflect current actual costs per Km based on your research. Later, after the solver shows which are the best routes to use, then you will come back to the supply chain model to create and save those routes.

Next, create the changing variables in your spreadsheet template. Remember, our goal is to find delivery quantities from DCs to stores that minimize transportation costs while still meeting store demand. Therefore, the changing variables in our case are the number of units moving on each route.

how to solve problem of transportation

The decision variables that will be changing are colored in yellow. We can make the value in the yellow matrix zeros, but to make sure that our rows and columns sums are correct, we made them equal to one. The values on the right side of the matrix are the total quantity delivered to each store, and the values below the matrix are the total quantity that each DC will transfer. We also copied the DC’s and capacity matrix, and we pasted its Transpose. We did that so we can add quickly and easily the capacity constraint in Solver.

B efore creating and running our solver model, we should define the total cost formula. Here, we have two options: 

  • The total transportation cost can be equal to the cost per Km multiplied by the distance of each route and then multiplied by the number of deliveries needed since vehicles have a maximum carry volume.
  • If we have the cost per Km per unit, then the total cost will be equal to the cost per km per unit, multiplied by the distance of each route, and then multiplied by the number of units sent on those routes. 

In our case, we only have the cost per Km traveled, and so we will use the first option. To know how many trucks we need to deliver the quantity incurred from one DC to a Store, we have to divide this quantity by the maximum carry volume of our trucks and round that number up ( The ROUNDUP Function in Excel ). In our example, we suppose that one large truck has a maximum carry volume of 110 units. The number of deliveries needed will be then equal to “ ROUNDUP(I15:K22/110;0)” . And the total transportation cost formula is “ SUMPRODUCT(C15:E22;ROUNDUP(I15:K22/110;0))*C24 ”.

If you have the cost per Km per unit, then the formula will be basically “ SUMPRODUCT(C15:E22;I15:K22)*C24” . Something to keep in mind here is that, in this simple transportation problem, the cost per Km or the cost per Km per unit will not change the solution, but it only affects the total cost.

how to solve problem of transportation

Creating and Running the Solver Model

Let’s define the objective function and the constraints of our model:

  • The objective function : The objective of our model is to minimize the total cost.
  • Constraints : First, we have demand constraints, therefore to meet demand, the total quantity delivered to each store must be equal to the demand of that store. Second, the quantity delivered from each DC must not exceed its unit capacity. And third, the decisions variable, which are the quantities delivered, must be an integer and greater than or equal to zero.

Now, let’s see how we can apply this to the Excel Solver add-in. First, go to the “Data” tab, and click on the “Solver” button that can be found on the right side. [If you have not installed the Solver yet visit this link to learn how to install it: https://www.solver.com/excel-solver-how-load-or-start-solver .] 

Once the solver tab opens, we have to set the objective. In our case, we select the “I25” cell in green; this is our total cost. The variables that we are changing are grouped in the yellow matrix, and so we select this matrix in the second block. 

Concerning the constraints, we have to click on the “Add” button. The three constraints created are shown in the figures below. After adding all these constraints, we have first to make unconstrained variables non-negative by selecting the checkbox. By respecting these installed constraints we can run our Solver model by selecting the Simplex LP (a solving method since our model is linear).

how to solve problem of transportation

Our model is now ready, so click the “Solve” button. We can see that the decision variables and the total cost are changing. The result is a preliminary optimal solution for our transportation problem.

how to solve problem of transportation

Interpretation of Results

We can verify results by seeing if we meet all demands and if capacity constraints are respected.

In this example, we got a total cost of $ 3,991.20, which is acceptable, but we notice that the only warehouse that will be using its max storage capacity is the DC Nbr 1. DC Nbr 3 is using 60% of its max storage capacity, and DC Nbr 2 is only using 10%. Hence the question: Do we need to open all three distribution centers for the next period?

One can say that by using only two facilities, the costs of operating facilities will decrease, but the transportation costs may increase. To find the optimal solution, we have to compare the total cost of two different scenarios. Which scenario offers the lowest total cost while still meeting store demand? In the first scenario, the company opens and uses all its DCs, paying all its fixed costs. In the second scenario, the company operates only two of its DCs while closing the third one.

Make copies of your original SCM Globe supply chain model and change them to reflect these two potential supply chain designs. Run those models in simulation to see how they work, and generate simulation data to create Profit & Loss Reports and KPIs . Compare the simulations and their performance reports. Get review and input from relevant parties, and pick the supply chain design that reviewers feel best meets the needs of the company in this situation.

Supply chain management is both a science and an art . In addition to numeric calculations, the role of professional judgment is also important as it can find growth opportunities, while numeric calculations find ways to reduce costs. Simulations help people explore different ideas and find new opportunities. Simulations bring together numeric calculations and professional judgment.

Situations change from month to month. This month’s optimal solution can quickly become next month’s big mistake. Companies plot their course each month by combining numeric calculations with professional judgment to navigate successfully through a constantly changing world.

[ See more about using the solver for supply chain route selection and scheduling in our blog post “ Manage Your Transportation by Solving The Vehicle Routing Problem “ ]

Special thanks to Mr. Curtis Frye for the quality of his course titled: “ Excel Supply Chain Analysis: Solving Transportation Problems ” on LinkedIn Learning. We based our Excel demonstration on his way of organizing the spreadsheet template.

Balanced and Unbalanced Transportation Problems

The two categories of transportation problems are balanced and unbalanced transportation problems . As we all know, a transportation problem is a type of Linear Programming Problem (LPP) in which items are carried from a set of sources to a set of destinations based on the supply and demand of the sources and destinations, with the goal of minimizing the total transportation cost. It is also known as the Hitchcock problem.

Introduction to Balanced and Unbalanced Transportation Problems

Balanced transportation problem.

The problem is considered to be a balanced transportation problem when both supplies and demands are equal.

Unbalanced Transportation Problem

Unbalanced transportation problem is defined as a situation in which supply and demand are not equal. A dummy row or a dummy column is added to this type of problem, depending on the necessity, to make it a balanced problem. The problem can then be addressed in the same way as the balanced problem.

Methods of Solving Transportation Problems

There are three ways for determining the initial basic feasible solution. They are

1. NorthWest Corner Cell Method.

2. Vogel’s Approximation Method (VAM).

3. Least Call Cell Method.

The following is the basic framework of the balanced transportation problem:

Basic Structure of Balanced Transportation Problem

The destinations D1, D2, D3, and D4 in the above table are where the products/goods will be transported from various sources O1, O2, O3, and O4. The supply from the source Oi is represented by S i . The demand for the destination Dj is d j . If a product is delivered from source Si to destination Dj, then the cost is called C ij .

Let us now explore the process of solving the balanced transportation problem using one of the ways known as the NorthWest Corner Method in this article.

Solving Balanced Transportation problem by Northwest Corner Method

Consider this scenario:

Balanced Transportation Problem -1

With three sources (O1, O2, and O3) and four destinations (D1, D2, D3, and D4), what is the best way to solve this problem? The supply for the sources O1, O2, and O3 are 300, 400, and 500, respectively. Demands for the destination D1, D2, D3, and D4 are 250, 350, 400, and 200, respectively.

The starting point for the North West Corner technique is (O1, D1), which is the table’s northwest corner. The cost of transportation is calculated for each value in the cell. As indicated in the diagram, compare the demand for column D1 with the supply from source O1 and assign a minimum of two to the cell (O1, D1).

Column D1’s demand has been met, hence the entire column will be canceled. The supply from the source O1 is still 300 – 250 = 50.

Balanced Transportation Problem - 2

Analyze the northwest corner, i.e. (O1, D2), of the remaining table, excluding column D1, and assign the lowest among the supply for the appropriate column and rows. Because the supply from O1 is 50 and the demand for D2 is 350, allocate 50 to the cell (O1, D2).

Now, row O1 is canceled because the supply from row O1 has been completed. Hence, the demand for Column D2 has become 350 – 50 = 50.

Balanced Transportation Problem - 3

The northwest corner cell in the remaining table is (O2, D2). The shortest supply from source O2 (400) and the demand for column D2 (300) is 300, thus putting 300 in the cell (O2, D2). Because the demand for column D2 has been met, the column can be deleted, and the remaining supply from source O2 is 400 – 300 = 100.

Balanced Transportation Problem - 4

Again, find the northwest corner of the table, i.e. (O2, D3), and compare the O2 supply (i.e. 100) to the D2 demand (i.e. 400) and assign the smaller (i.e. 100) to the cell (O2, D2). Row O2 has been canceled because the supply from O2 has been completed. Column D3 has a leftover demand of 400 – 100 = 300.

Balanced Transportation Problem -5

Continuing in the same manner, the final cell values will be:

Balanced Transportation Problem - 6

It should be observed that the demand for the relevant columns and rows is equal in the last remaining cell, which was cell (O3, D4). In this situation, the supply from O3 was 200, and the demand for D4 was 200, therefore this cell was assigned to it. Nothing was left for any row or column at the end.

To achieve the basic solution, multiply the allotted value by the respective cell value (i.e. the cost) and add them all together.

I.e., (250 × 3) + (50 × 1) + (300 × 6) + (100 × 5) + (300 × 3) + (200 × 2) = 4400.

Solving Unbalanced Transportation Problem

An unbalanced transportation problem is provided below. Because the sum of all the supplies, O1, O2, O3, and O4, does not equal the sum of all the demands, D1, D2, D3, D4, and D5, the situation is unbalanced.

Unbalanced Transportation Problem - 1

The idea of a dummy row or dummy column will be applied in this type of scenario. Because the supply is more than the demand in this situation, a fake demand column will be inserted, with a demand of (total supply – total demand), i.e. 117 – 95 = 22, as seen in the image below. A fake supply row would have been introduced if demand was greater than supply.

Unbalanced Transportation Problem - 2

Now this problem has been changed to a balanced transportation problem, and it can be addressed using any of the ways listed below to solve a balanced transportation problem, such as the northwest corner method mentioned earlier.

Frequently Asked Questions on Balanced and Unbalanced Transportation Problems

What is meant by balanced and unbalanced transportation problems.

The problem is referred to as a balanced transportation problem when both supplies and demands are equal. Unbalanced transportation is defined as a situation where supply and demand are not equal.

What is called a transportation problem?

The transportation problem is a type of Linear Programming Problem in which commodities are carried from a set of sources to a set of destinations while taking into account the supply and demand of the sources and destinations, respectively, in order to reduce the total cost of transportation.

What are the different methods to solve transportation problems?

The following are three approaches to solve the transportation issue:

  • NorthWest Corner Cell Method.
  • Least Call Cell Method.
  • Vogel’s Approximation Method (VAM).

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Experience has shown that promoting safety requires a strong team effort. President Biden’s visit to East Palestine, OH and the announcement from the National Transportation Safety Board regarding its final report on the February 3, 2023 derailment demonstrates that everyone has an important role when it comes to advancing rail safety .

Connected By Chemistry

Just about every aspect of our daily lives depends on chemistry and every corner of the economy – from farms to factories – depends on chemical shipments. Chemicals, including some that are classified as hazardous materials, are essential for growing food, protecting the safety of our water and food supply, making life-saving medicines and equipment, and producing energy.

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‘A nightmare’: Special counsel’s assessment of Biden’s mental fitness triggers Democratic panic

WASHINGTON — President Joe Biden sidestepped any criminal charges as the investigation into his handling of classified documents concluded, but the political blowback from the special counsel’s report Thursday could prove even more devastating, reinforcing impressions that he is too old and impaired to hold the highest office.

Special counsel Robert Hur’s portrait of a man who couldn’t remember when he served as Barack Obama’s vice president, or the year when his beloved son Beau died, dealt a blow to Biden’s argument that he is still sharp and fit enough to serve another four-year term.

In deciding not to charge Biden with any crimes, the special counsel wrote that in a potential trial, “Mr. Biden would likely present himself to a jury, as he did during our interview with him, as a sympathetic, well-meaning, elderly man with a poor memory.”

It was tough enough for Biden to reassure voters about his health before Hur’s report hit like a thunderclap Thursday afternoon, prompting members of his own party to question whether he could remain the nominee in November.

“It’s a nightmare,” said a Democratic House member who asked to speak anonymously to provide a frank assessment, adding that “it weakens President Biden electorally, and Donald Trump would be a disaster and an authoritarian.”

“For Democrats, we’re in a grim situation.”

Biden wasted little time before attempting to minimize the fallout. He held an unexpected exchange with reporters in the White House on Thursday night, in which he disputed Hur's assessment of his mental acuity.

Biden grew emotional when invoking the part of the report addressing the date of his son's death.

"How in the hell dare you raise that?" Biden said. "Frankly, when I was asked the question I thought to myself, 'It wasn't any of their damn business.' "

‘Beyond devastating’

Polling has long shown that age looms as Biden’s greatest liability in his expected rematch with Trump. A January poll by NBC News found that 76% of voters have major or moderate concerns about Biden’s mental and physical health.

“It’s been a problem since way before this ever happened,” said a longtime Democratic operative who noted that when focus groups are asked to apply one word to Biden, it is often “old.”

Just this week, Biden twice referred to conversations he’s had as president with foreign leaders who’ve long since died. In his remarks Thursday night defending his competency, while talking about the war in Gaza, he referred to Egyptian President Abdel Fattah el-Sissi as being the head of Mexico. White House press aides have downplayed such lapses as the sort of mistake anyone in public life can make.

The Hur report strips away the defenses that Biden’s press operation has used to protect him and raises fresh doubts about whether Biden is up to the rigors of the presidency, Democratic strategists said in interviews.

“This is beyond devastating,” said another Democratic operative, speaking on condition of anonymity to talk candidly about Biden’s shortcomings. “It confirms every doubt and concern that voters have. If the only reason they didn’t charge him is because he’s too old to be charged, then how can he be president of the United States?”

Asked if Hur’s report changes the calculus for Democrats who expect Biden to be the party’s nominee, this person said: “How the f--- does it not?”

Another Biden ally called it “the worst day of his presidency.”

“I think he needs to show us this is a demonstrably false characterization of him and that he has what it takes to win and govern.”

Biden has overwhelmingly won the first primary contests — notching victories in New Hampshire, South Carolina and Nevada. It would be virtually impossible for anyone else to challenge him at this point; the deadline has passed in more than 30 states to get on primary ballots.

Some of the president’s allies were quick to defend him. They pointed to the timing of the interview with the special counsel — days after Hamas’ attack on Israel, which had captured much of the president’s focus. Others said that in their own dealings with Biden, he shows no sign of infirmity.

“He did so well in this discussion with members,” Rep. Susan Wild, D-Pa., told NBC News after seeing the president on Thursday. “He’s very sharp, no memory issues, and his only stumbling is when he trips over words consistent with his lifelong speech impediment.”

‘Prejudicial language’

Though Biden was fortunate to escape indictment, the special counsel report may give Trump additional fodder as he fights charges for allegedly mishandling classified records at his Mar-a-Lago social club. Republicans are already accusing Biden of benefiting from a double standard . Trump will likely brandish the Hur report as proof that Biden has “weaponized” the Justice Department for political advantage.

What’s more, Democrats will now be hard-pressed to capitalize on Trump’s indictment over retaining classified records. Before Hur’s report came out, Democrats argued that the two cases were very different. Whereas Trump failed to turn over classified records even after he was asked to do so, Biden willingly cooperated with authorities and relinquished all the material he had, Biden allies had argued.

“The public understands the essential difference between presidents or vice presidents like Joe Biden who occasionally behaved in sloppy ways with respect to where they were taking documents, and a president like Trump, who deliberately makes off with hundreds of classified government documents and then hides them and refuses to return them,” Rep. Jamie Raskin, D-Md., said on Wednesday, before the report was released. (Trump has denied any wrongdoing.)

Now, the distinctions may be harder for Biden allies to draw, given that Hur wrote that there was evidence Biden “willfully retained and disclosed classified material after his vice presidency when he was a private citizen.”

The report mentions an instance in February 2017, when he was no longer vice president, when Biden read notes containing classified information “nearly verbatim” to a ghostwriter helping him with his book, “Promise Me, Dad.”

Storage of sensitive government secrets was haphazard. The report describes certain classified records involving the war in Afghanistan in Biden’s Delaware garage inside a “badly damaged box surrounded by household detritus.”

Before the report was released, Biden aides had been bracing for a finding that he had simply been careless in his treatment of classified records, a person familiar with the White House’s thinking said.

The political fallout from the report, though, is likely to be “worse,” this person said. What will stick in people’s minds is what Hur said about Biden’s memory, the person added.

Biden’s lawyers disputed the report’s description of Biden’s forgetfulness.

“We do not believe that the report’s treatment of President Biden’s memory is accurate or appropriate,” two of his lawyers wrote in a letter to Hur. “The report uses highly prejudicial language to describe a commonplace occurrence among witnesses: a lack of recall of years-old events.”

In the hours after the report was released, people close to the Biden campaign rolled out a different rebuttal. Jim Messina, who ran Obama’s 2012 re-election campaign, wrote on X, the site formerly known as Twitter, that Hur is a Republican who “knew exactly how his swipes could hurt Biden politically.”

That’s a familiar argument. Trump has also claimed that law enforcement is trying to sway the election, meaning both sides are now claiming victimization at the hands of partisan prosecutors.

“Hur knew exactly what he was doing here,” Stephanie Cutter, a veteran Democratic operative, wrote on X. “To provide political cover for himself for not prosecuting, he gratuitously leveled a personal (not legal) charge against the president that he absolutely knows is a gift to Trump. And, guess what we are all talking about?”

how to solve problem of transportation

Peter Nicholas is a senior national political reporter for NBC News.

IMAGES

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  21. How to Solve Transportation Problems Using Excel Solver

    The first step in our process is to enter the data we need from the supply chain model in an organized manner. Click on each facility in the SCM Globe supply chain model to see information for product demand, storage capacity, etc. Start by entering each store's demand; that gives us a total demand of 1190 units in this case.

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    Experience has shown that promoting safety requires a strong team effort. President Biden's visit to East Palestine, OH and the announcement from the National Transportation Safety Board regarding its final report on the February 3, 2023 derailment demonstrates that everyone has an important role when it comes to advancing rail safety.. Connected By Chemistry

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