## 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:

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:

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.

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.

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.

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.

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

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.

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.

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).

Your Mobile number and Email id will not be published. Required fields are marked *

Request OTP on Voice Call

Post My Comment

• Share Share

## Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

• DSA for Beginners
• DSA Tutorial
• Data Structures
• Dynamic Programming
• Binary Tree
• Binary Search Tree
• Divide & Conquer
• Mathematical
• Backtracking
• Branch and Bound
• Pattern Searching

• Explore Our Geeks Community
• Transportation Problem | Set 4 (Vogel's Approximation Method)
• Transportation Problem | Set 7 ( Degeneracy in Transportation Problem )
• Transportation Problem | Set 5 ( Unbalanced )
• Transportation Problem | Set 6 (MODI Method - UV Method)
• Multiply N complex numbers given as strings
• Check if any large number is divisible by 19 or not
• Check if any large number is divisible by 17 or not
• Transportation Problem | Set 2 (NorthWest Corner Method)
• Transportation Problem | Set 3 (Least Cost Cell Method)
• Reservoir Sampling
• Game Theory (Normal-form Game) | Set 7 (Graphical Method [M X 2] Game)
• Find the type of triangle from the given sides
• Game Theory (Normal-form Game) | Set 6 (Graphical Method [2 X N] Game)
• Determine whether the given integer N is a Peculiar Number or not
• Count number of pairs with positive sum in an array
• Game Theory (Normal-form Game) | Set 4 (Dominance Property-Pure Strategy)
• Game Theory (Normal-form Game) | Set 5 (Dominance Property-Mixed Strategy)
• What is the result of ∞ - ∞?
• Aronson's Sequence

## Transportation Problem | Set 1 (Introduction)

Transportation problem is a special kind of Linear Programming Problem (LPP) in which goods are transported from a set of sources to a set of destinations subject to the supply and demand of the sources and destination respectively such that the total cost of transportation is minimized. It is also sometimes called as Hitchcock problem.

Types of Transportation problems: Balanced: When both supplies and demands are equal then the problem is said to be a balanced transportation problem.

Unbalanced: When the supply and demand are not equal then it is said to be an unbalanced transportation problem. In this type of problem, either a dummy row or a dummy column is added according to the requirement to make it a balanced problem. Then it can be solved similar to the balanced problem.

Methods to Solve: To find the initial basic feasible solution there are three methods:

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

Solve DSA problems on GfG Practice.

• DSA in Java
• DSA in Python
• DSA in JavaScript

• sumitlovanshi

Please write us at contrib[email protected] to report any issue with the above content

## Quantitative Techniques: Theory and Problems by P. C. Tulsian, Vishal Pandey

Get full access to Quantitative Techniques: Theory and Problems and 60K+ other titles, with a free 10-day trial of O'Reilly.

There are also live events, courses curated by job role, and more.

## WHAT IS TRANSPORTATION PROBLEM

The transportation problem is a special type of linear programming problem where the objective is to minimise the cost of distributing a product from a number of sources or origins to a number of destinations. Because of its special structure the usual simplex method is not suitable for solving transportation problems. These problems require a special method of solution. The origin of a transportation problem is the location from which shipments are despatched. The destination of a transportation problem is the location to which shipments are transported. The unit transportation cost is the cost of transporting one unit of the consignment from an origin to a destination.

In the most general form, a transportation ...

Get Quantitative Techniques: Theory and Problems now with the O’Reilly learning platform.

O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.

## Don’t leave empty-handed

Get Mark Richards’s Software Architecture Patterns ebook to better understand how to design components—and how they should interact.

## Check it out now on O’Reilly

Dive in for free with a 10-day trial of the O’Reilly learning platform—then explore all the other resources our members count on to build skills and solve problems every day.

All Courses

• Interview Questions
• Free Courses
• Career Guide
• PGP in Data Science and Business Analytics
• PG Program in Data Science and Business Analytics Classroom
• PGP in Data Science and Engineering (Data Science Specialization)
• PGP in Data Science and Engineering (Bootcamp)
• PGP in Data Science & Engineering (Data Engineering Specialization)
• NUS Decision Making Data Science Course Online
• Master of Data Science (Global) – Deakin University
• MIT Data Science and Machine Learning Course Online
• Master’s (MS) in Data Science Online Degree Programme
• MTech in Data Science & Machine Learning by PES University
• Data Analytics Essentials by UT Austin
• Data Science & Business Analytics Program by McCombs School of Business
• MTech In Big Data Analytics by SRM
• M.Tech in Data Engineering Specialization by SRM University
• M.Tech in Big Data Analytics by SRM University
• PG in AI & Machine Learning Course
• Weekend Classroom PG Program For AI & ML
• AI for Leaders & Managers (PG Certificate Course)
• Artificial Intelligence Course for School Students
• IIIT Delhi: PG Diploma in Artificial Intelligence
• Machine Learning PG Program
• MIT No-Code AI and Machine Learning Course
• Study Abroad: Masters Programs
• MS in Information Science: Machine Learning From University of Arizon
• SRM M Tech in AI and ML for Working Professionals Program
• UT Austin Artificial Intelligence (AI) for Leaders & Managers
• UT Austin Artificial Intelligence and Machine Learning Program Online
• MS in Machine Learning
• IIT Roorkee Full Stack Developer Course
• IIT Madras Blockchain Course (Online Software Engineering)
• IIIT Hyderabad Software Engg for Data Science Course (Comprehensive)
• IIIT Hyderabad Software Engg for Data Science Course (Accelerated)
• IIT Bombay UX Design Course – Online PG Certificate Program
• Online MCA Degree Course by JAIN (Deemed-to-be University)
• Cybersecurity PG Course
• Online Post Graduate Executive Management Program
• Product Management Course Online in India
• NUS Future Leadership Program for Business Managers and Leaders
• PES Executive MBA Degree Program for Working Professionals
• Online BBA Degree Course by JAIN (Deemed-to-be University)
• MBA in Digital Marketing or Data Science by JAIN (Deemed-to-be University)
• Post Graduate Diploma in Management (Online) by Great Lakes
• Online MBA Program by Shiv Nadar University
• Cloud Computing PG Program by Great Lakes
• University Programs
• Stanford Design Thinking Course Online
• Design Thinking : From Insights to Viability
• PGP In Strategic Digital Marketing
• Post Graduate Diploma in Management
• Master of Business Administration Degree Program
• MS Artificial Intelligence and Machine Learning
• MS in Data Analytics
• Study MBA in USA
• Study MS in USA
• Data Analytics Course with Job Placement Guarantee
• Software Development Course with Placement Guarantee
• MIT Data Science Program
• AI For Leaders Course
• Data Science and Business Analytics Course
• Cyber Security Course
• Pg Program Online Artificial Intelligence Machine Learning
• Pg Program Online Cloud Computing Course
• Data Analytics Essentials Online Course
• MIT Programa Ciencia De Dados Machine Learning
• MIT Programa Ciencia De Datos Aprendizaje Automatico
• Program PG Ciencia Datos Analitica Empresarial Curso Online
• Mit Programa Ciencia De Datos Aprendizaje Automatico
• Program Pg Ciencia Datos Analitica Empresarial Curso Online
• Online Data Science Business Analytics Course
• Online Ai Machine Learning Course
• Online Full Stack Software Development Course
• Online Cloud Computing Course
• Cybersecurity Course Online
• Online Data Analytics Essentials Course
• Mit Data Science Program
• No Code Artificial Intelligence Machine Learning Program
• Ms Information Science Machine Learning University Arizona
• Wharton Online Advanced Digital Marketing Program
• Data Science
• Introduction to Data Science
• Data Scientist Skills
• Get Into Data Science From Non IT Background
• Data Scientist Salary
• Data Science Job Roles
• Data Science Resume
• Data Scientist Interview Questions
• Data Science Solving Real Business Problems
• Business Analyst Vs Data Scientis
• Data Science Applications
• Must Watch Data Science Movies
• Data Science Projects
• Free Datasets for Analytics
• Data Analytics Project Ideas
• Mean Square Error Explained
• Hypothesis Testing in R
• Understanding Distributions in Statistics
• Bernoulli Distribution
• Inferential Statistics
• Analysis of Variance (ANOVA)
• Sampling Techniques
• Outlier Analysis Explained
• Outlier Detection
• Data Science with K-Means Clustering
• Support Vector Regression
• Multivariate Analysis
• What is Regression?
• An Introduction to R – Square
• Why is Time Complexity essential?
• Gaussian Mixture Model
• Genetic Algorithm
• What is Business Analytics?
• Business Analytics Career
• Major Misconceptions About a Career in Business Analytics
• Business Analytics and Business Intelligence Possible Career Paths for Analytics Professionals
• Business Analytics Companies
• Business Analytics Tools
• Business Analytics Jobs
• Business Analytics Course
• Difference Between Business Intelligence and Business Analytics
• Python Tutorial for Beginners
• Python Cheat Sheet
• Career in Python
• Python Developer Salary
• Python Interview Questions
• Python Project for Beginners
• Python Books
• Python Real World Examples
• Python 2 Vs. Python 3
• Free Online Courses for Python
• Flask Vs. Django
• Python Stack
• Python Switch Case
• Python Main
• Data Types in Python
• Mutable & Immutable in Python
• Python Dictionary
• Python Queue
• Iterator in Python
• Regular Expression in Python
• Eval in Python
• Classes & Objects in Python
• OOPs Concepts in Python
• Inheritance in Python
• Abstraction in Python
• Polymorphism in Python
• Fibonacci Series in Python
• Factorial Program in Python
• Armstrong Number in Python
• Reverse a String in Python
• Prime Numbers in Python
• Pattern Program in Python
• Palindrome in Python
• Convert List to String in Python
• Append Function in Python
• REST API in Python
• Python Web Scraping using BeautifulSoup
• Scrapy Tutorial
• Web Scraping using Python
• Jupyter Notebook
• Spyder Python IDE
• Free Data Science Course
• Free Data Science Courses
• Data Visualization Courses

## 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.

## Mastering Pivot Tables in Excel: A Comprehensive Guide

Save my name, email, and website in this browser for the next time I comment.

Learn data analytics or software development & get guaranteed* placement opportunities.

• 7 guaranteed* placement opportunities
• 3-6 Lakh Per Annum salary range.
• Suited for freshers & recent graduates
• Choose between classroom learning or live online classes
• 4-month full-time program
• Placement opportunities with top companies
• Open access
• Published: 20 July 2023

## Solving a multi-objective solid transportation problem: a comparative study of alternative methods for decision-making

• Mohamed H. Abdelati   ORCID: orcid.org/0000-0002-5034-7323 1 ,
• Ali M. Abd-El-Tawwab 1 ,
• Elsayed Elsayed M. Ellimony 2 &
• M Rabie 1

Journal of Engineering and Applied Science volume  70 , Article number:  82 ( 2023 ) Cite this article

721 Accesses

Metrics details

The transportation problem in operations research aims to minimize costs by optimizing the allocation of goods from multiple sources to destinations, considering supply, demand, and transportation constraints. This paper applies the multi-dimensional solid transportation problem approach to a private sector company in Egypt, aiming to determine the ideal allocation of their truck fleet.

In order to provide decision-makers with a comprehensive set of options to reduce fuel consumption costs during transportation or minimize total transportation time, a multi-objective approach is employed. The study explores the best compromise solution by leveraging three multi-objective approaches: the Zimmermann Programming Technique, Global Criteria Method, and Minimum Distance Method. Optimal solutions are derived for time and fuel consumption objectives, offering decision-makers a broad range to make informed decisions for the company and the flexibility to adapt them as needed.

Lingo codes are developed to facilitate the identification of the best compromise solution using different methods. Furthermore, non-dominated extreme points are established based on the weights assigned to the different objectives. This approach expands the potential ranges for enhancing the transfer problem, yielding more comprehensive solutions.

This research contributes to the field by addressing the transportation problem practically and applying a multi-objective approach to support decision-making. The findings provide valuable insights for optimizing the distribution of the truck fleet, reducing fuel consumption costs, and improving overall transportation efficiency.

## Introduction

The field of operations research has identified the transportation problem as an optimization issue of significant interest [ 1 , 2 ]. This problem concerns determining the optimal approach to allocate a given set of goods that come from particular sources to the designated destinations to minimize the overall transportation costs [ 3 ]. The transportation problem finds applications in various areas, including logistics planning, distribution network design, and supply chain management. Solving this problem relies on the assumption that the supply and demand of goods are known, as well as the transportation cost for each source–destination pairing [ 4 , 5 ].

Solving the transportation problem means finding the right quantities of goods to be transported from the sources to the destinations, given the supply and demand restrictions. The ultimate goal is to minimize the total transportation cost, which is the sum of the cost for each shipment [ 6 ]. Various optimization algorithms have been developed for this problem, such as the North-West Corner Method, the Least Cost Method, and Vogel’s Approximation Method [ 7 ].

A solid transportation problem (STP) is a related transportation problem that centers around a single commodity, which can be stored at interim points [ 8 ]. These interim points, known as transshipment points, act as origins and destinations. The STP involves determining the most efficient means of transporting the commodity from the sources to the destinations, while minimizing transportation costs by going through the transshipment points. The STP has real-world applications in container shipping, air cargo transportation, and oil and gas pipeline transportation [ 9 , 10 ].

Multi-dimensional solid transportation problem (MDSTP) represents a variation on the STP, incorporating multiple commodities that vary in properties such as volume, weight, and hazard level [ 11 ]. The MDSTP aims to identify the best way to transport each commodity from the sources to the destinations, taking into account the capacity restrictions of transshipment points and hazardous commodity regulations [ 12 ]. The MDSTP is more complex than the STP and requires specific algorithms and models for its resolution.

Solving the STP and MDSTP requires identifying the most effective routing of commodities and considering the storage capacity of transshipment points. The goal is to minimize total transportation costs while satisfying supply and demand constraints and hazardous material regulations. Solutions to these problems include the Network Simplex Method, Branch and Bound Method, and Genetic Algorithm [ 13 ]. Solving the STP and MDSTP contributes valuable insights into the design and operation of transportation systems and supports improved sustainability and efficiency.

In the field of operations research, two critical research areas are the multi-objective transportation problem (MOTP) and the multi-objective solid transportation problem (MOSTP) [ 14 ]. The MOTP aims to optimize the transportation of goods from multiple sources to various destinations by considering multiple objectives, including minimizing cost, transportation time, and environmental impacts. The MOSTP, on the other hand, focuses on the transportation of solid materials, such as minerals or ores, and involves dealing with multiple competing objectives, such as cost, time, and quality of service. These problems are essential in logistics and supply chain management, where decision-makers must make optimal transportation plans by considering multiple objectives. Researchers and practitioners often employ optimization techniques, such as mathematical programming, heuristics, and meta-heuristics, to address these challenges efficiently [ 15 ].

Efficient transportation planning is essential for moving goods from their source to the destination. This process involves booking different types of vehicles and minimizing the total transportation time and cost is a crucial factor to consider. Various challenges can affect the optimal transportation policy, such as the weight and volume of products, the availability of specific vehicles, and other uncertain parameters. In this regard, several studies have proposed different approaches to solve the problem of multi-objective solid transportation under uncertainty. One such study by Kar et al. [ 16 ] used fuzzy parameters to account for uncertain transportation costs and time, and two methods were employed to solve the problem, namely the Zimmermann Method and the Global Criteria Method.

Similarly, Mirmohseni et al. [ 17 ] proposed a fuzzy interactive probabilistic programming approach, while Kakran et al. [ 18 ] addressed a multi-objective capacitated solid transportation problem with uncertain zigzag variables. Additionally, Chen et al. [ 19 ] investigated an uncertain bicriteria solid transportation problem by using uncertainty theory properties to transform the models into deterministic equivalents, proposing two models, namely the expected value goal programming and chance-constrained goal programming models [ 20 ]. These studies have contributed to developing different approaches using fuzzy programming, uncertainty theory, and related concepts to solve multi-objective solid transportation problems with uncertain parameters.

This paper presents a case study carried out on a private sector company in Egypt intending to ascertain the minimum number of trucks required to fulfill the decision-makers’ objectives of transporting the company’s fleet of trucks from multiple sources to various destinations. This objective is complicated by the diversity of truck types and transported products, as well as the decision-makers’ multiple priorities, specifically the cost of fuel consumption and the timeliness of truck arrival.

In contrast to previous research on the transportation problem, this paper introduces a novel approach that combines the multi-dimensional solid transportation problem framework with a multi-objective optimization technique. Building upon previous studies, which often focused on single-objective solutions and overlooked specific constraints, our research critically analyzes the limitations of these approaches. We identify the need for comprehensive solutions that account for the complexities of diverse truck fleets and transported products, as well as the decision-makers’ multiple priorities. By explicitly addressing these shortcomings, our primary goal is to determine the minimum number of trucks required to fulfill the decision-makers’ objectives, while simultaneously optimizing fuel consumption and transportation timeliness. Through this novel approach, we contribute significantly to the field by advancing the understanding of the transportation problem and providing potential applications in various domains. Our research not only offers practical solutions for real-world scenarios but also demonstrates the potential for improving transportation efficiency and cost-effectiveness in other industries or contexts. The following sections will present a comparative analysis of the proposed work, highlighting the advancements and novelty introduced by our approach.

## Methods/experimental

This study uses a case study from Egypt to find the optimal distribution of a private sector company’s truck fleet under various optimization and multi-objective conditions. Specifically, the study aims to optimize the distribution of a private sector company’s truck fleet by solving a multi-objective solid transportation problem (MOSTP) and comparing three different methods for decision-making.

## Design and setting

This study uses a case study design in a private sector company in Egypt. The study focuses on distributing the company’s truck fleet to transport products from factories to distribution centers.

## Participants or materials

The participants in this study are the transportation planners and managers of the private sector company in Egypt. The materials used in this study include data on the truck fleet, sources, destinations, and products.

## Processes and methodologies

The study employs the multi-objective multi-dimensional solid transportation problem (MOMDSTP) to determine the optimal solution for the company’s truck fleet distribution, considering two competing objectives: fuel consumption cost and total shipping time. The MOMDSTP considers the number and types of trucks, sources, destinations, and products and considers the supply and demand constraints.

To solve the MOMDSTP, three decision-making methods are employed: Zimmermann Programming Technique, Global Criteria Method, and Minimum Distance Method. The first two methods directly yield the best compromise solution (BCS), whereas the last method generates non-dominated extreme points by assigning different weights to each objective. Lingo software is used to obtain the optimal solutions for fuel consumption cost and time and the BCS and solutions with different weights for both objectives.

## Ethics approval and consent

This study does not involve human participants, data, or tissue, nor does it involve animals. Therefore, ethics approval and consent are not applicable.

## Statistical analysis

Statistical analysis is not conducted in this study. However, the MOMDSTP model and three well-established decision-making methods are employed to derive the optimal distribution of the company’s truck fleet under various optimization and multi-objective conditions.

In summary, this study uses a case study design to find the optimal distribution of a private sector company’s truck fleet under various optimization and multi-objective conditions. The study employs the MOMDSTP and three methods for decision-making, and data on the truck fleet, sources, destinations, and products are used as materials. Ethics approval and consent are not applicable, and statistical analysis is not performed.

## Multi-objective transportation problem

The multi-objective optimization problem is a complex issue that demands diverse approaches to determine the most satisfactory solution. Prevalent techniques employed in this domain include the Weighted Sum Method, Minimum Distance Method, Zimmermann Programming Technique, and Global Criteria Method. Each method offers its own benefits and limitations, and the selection of a specific method depends on the nature of the problem and the preferences of the decision-makers [ 21 ].

This section discusses various methodologies employed to identify the most optimal solution(s) for the multi-objective multi-dimensional solid transportation problem (MOMDSTP), which is utilized as the basis for the case study. These methodologies encompass the Minimum Distance Method (MDM), the Zimmermann Programming Technique, and the Global Criteria Method [ 22 ].

Zimmermann Programming Technique

The Zimmermann Programming Technique (ZPT) is a multi-objective optimization approach that was developed by Professor Hans-Joachim Zimmermann in the late 1970s. This technique addresses complex problems with multiple competing objectives that cannot be optimized simultaneously. Additionally, it incorporates the concept of an “aspiration level,” representing the minimum acceptable level for each objective. The aspiration level ensures that the solution obtained is satisfactory for each objective. If the solution does not meet the aspiration level for any objective, the weights are adjusted, and the optimization process is iterated until a satisfactory solution is obtained.

A key advantage of ZPT is its ability to incorporate decision-makers’ preferences and judgments into the decision-making process. The weights assigned to each objective are based on the decision-maker’s preferences, and the aspiration levels reflect their judgments about what constitutes an acceptable level for each objective [ 23 ].

The Zimmermann Programming Technique empowers decision-makers to incorporate multiple objectives and achieve a balanced solution. By assigning weights to objectives, a trade-off can be made to find a compromise that meets various criteria. For example, this technique can optimize cost, delivery time, and customer satisfaction in supply chain management [ 24 ]. However, the interpretation of results may require careful consideration, and computational intensity can increase with larger-scale and complex problems.

In order to obtain the solution, each objective is considered at a time to get the lower and upper bounds for that objective. Let for objective, and are the lower (min) and upper (max) bounds. The membership functions of the first and second objective functions can be generated based on the following formula [ 25 ]:

Next, the fuzzy linear programming problem is formulated using the max–min operator as follows:

Maximize min $${\mu }_{k}\left({F}_{k}\left(x\right)\right)$$

Subject to $${g}_{i}\left(x\right) \left\{ \le ,= , \ge \right\}{b}_{i}\mathrm{ where }\;i = 1, 2, 3, ..., m.$$

Moreover, x ≥ 0.

Global Criteria Method

The Global Criteria Method is a multi-objective optimization method that aims to identify the set of ideal solutions based on predetermined criteria. This method involves defining a set of decision rules that assess the feasibility and optimality of the solutions based on the objectives and constraints [ 26 ]. By applying decision rules, solutions that fail to meet the predetermined criteria are eliminated, and the remaining solutions are ranked [ 27 ].

The Global Criteria Method assesses overall system performance, aiding decision-makers in selecting solutions that excel in all objectives. However, it may face challenges when dealing with conflicting objectives [ 28 ]. Furthermore, it has the potential to overlook specific details, and the choice of aggregation function or criteria can impact the results by favoring specific solutions or objectives.

Let us consider the following ideal solutions:

f 1* represents the ideal solution for the first objective function,

f 2* represents the ideal solution for the second objective function, and

n 1* represents the ideal solution for the nth objective function.

Objective function formula:

Minimize the objective function F  =  $$\sum_{k=1}^{n}{(\frac{{f}_{k}\left({x}^{*}\right)-{f}_{k}(x)}{{f}_{k}({x}^{*})})}^{p}$$

Subject to the constraints: g i ( x ) $$\le$$ 0, i  = 1, 2,.., m

The function fk( x ) can depend on variables x 1 , x 2 , …, x n .

Minimum Distance Method

The Minimum Distance Method (MDM) is a novel distance-based model that utilizes the goal programming weighted method. The model aims to minimize the distances between the ideal objectives and the feasible objective space, leading to an optimal compromise solution for the multi-objective linear programming problem (MOLPP) [ 29 ]. To solve MOLPP, the proposed model breaks it down into a series of single objective subproblems, with the objectives transformed into constraints. To further enhance the compromise solution, priorities can be defined using weights, and a criterion is provided to determine the best compromise solution. A significant advantage of this approach is its ability to obtain a compromise solution without any specific preference or for various preferences.

The Minimum Distance Method prioritizes solutions that closely resemble the ideal or utopian solution, assisting decision-makers in ranking and identifying high-performing solutions. It relies on a known and achievable ideal solution, and its sensitivity to the chosen reference point can influence results. However, it does not provide a comprehensive trade-off solution, focusing solely on proximity to the ideal point [ 30 ].

The mathematical formulation for MDM for MOLP is as follows:

The formulation for multi-objective linear programming (MOLP) based on the minimum distance method is referred to[ 31 ]. It is possible to derive the multi-objective transportation problem with two objective functions using this method and its corresponding formula.

Subject to the following constraints:

f * 1 , f * 2 : the obtained ideal objective values by solving single objective STP.

w 1 , w 2 : weights for objective1 and objective2 respectively.

f 1, f 2: the objective values for another efficient solution.

d : general deviational variable for all objectives.

$${{c}_{ij}^{1}, c}_{ij}^{2}$$ : the unit cost for objectives 1 and 2 from source i to destination j .

$${{x}_{ij}^{1}, x}_{ij}^{2}$$ : the amount to be shipped when optimizing for objectives 1 and 2 from source i to destination j .

## Mathematical model for STP

The transportation problem (TP) involves finding the best method to ship a specific product from a defined set of sources to a designated set of destinations, while adhering to specific constraints. In this case, the objective function and constraint sets take into account three-dimensional characteristics instead of solely focusing on the source and destination [ 32 ]. Specifically, the TP considers various modes of transportation, such as ships, freight trains, cargo aircraft, and trucks, which can be used to represent the problem in three dimensions When considering a single mode of transportation, the TP transforms into a solid transportation problem (STP), which can be mathematically formulated as follows:

The mathematical form of the solid transportation problem is given by [ 33 ]:

Subject to:

Z = the objective function to be minimized

m = the number of sources in the STP

n = the number of destinations in the STP

p = the number of different modes of transportation in the STP

x ijk represents the quantity of product transported from source i to destination j using conveyance k

c ijk = the unit transportation cost for each mode of transportation in the STP

a i = the amount of products available at source i

b j = the demand for the product at destination j

e k = the maximum amount of product that can be transported using conveyance k

The determination of the size of the fleet for each type of truck that is dispatched daily from each factory to all destinations for the transportation of various products is expressed formally as follows:

z ik denotes the number of trucks of type k that are dispatched daily from factory i .

C k represents the capacity of truck k in terms of the number of pallets it can transport.

x ijk denotes a binary decision variable that is set to one if truck k is dispatched from factory i to destination j to transport product p , and zero otherwise. The summation is performed over all destinations j and all products p .

This case study focuses on an Egyptian manufacturing company that produces over 70,000 pallets of various water and carbonated products daily. The company has 25 main distribution centers and eight factories located in different industrial cities in Egypt. The company’s transportation fleet consists of hundreds of trucks with varying capacities that are used to transport products from factories to distribution centers. The trucks have been classified into three types (type A, type B, and type C) based on their capacities. The company produces three different types of products that are packaged in pallets. It was observed that the sizes and weights of the pallets are consistent across all product types The main objective of this case study is to determine the minimum number of each truck type required in the manufacturer’s garage to minimize fuel consumption costs and reduce product delivery time.

The problem was addressed by analyzing the benefits of diversifying trucks and implementing the solid transport method. Subsequently, the problem was resolved while considering the capacities of the sources and the requirements of the destinations. The scenario involved shipping products using a single type of truck, and the fuel consumption costs were calculated accordingly. The first objective was to reduce the cost of fuel consumption on the one-way journey from the factories to the distribution centers. The second objective was to reduce the time of arrival of the products to the destinations. The time was calculated based on the average speed of the trucks in the company’s records, which varies depending on the weight and size of the transported goods.

To address the multiple objectives and the uncertainty in supply and demand, an approach was adopted to determine the minimum number of trucks required at each factory. This approach involved determining the maximum number of trucks of each type that should be present in each factory under all previous conditions. The study emphasizes the significance of achieving a balance between reducing transportation costs and time while ensuring trucks are capable of accommodating quantities of any size, thus avoiding underutilization.

Figure  1 presents the mean daily output, measured in pallets, for each factory across three distinct product types. Additionally, Fig.  2 displays the average daily demand, measured in pallets, for the distribution centers of the same three product types.

No. of pallets in each source

No. of pallets in each destination

## Results and discussion

As a result of the case study, the single objective problems of time and fuel consumption cost have been solved. The next step is to prepare a model for the multi-objective multi-dimensional solid transportation problem. Prior to commencing, it is necessary to determine the upper and lower bounds for each objective.

Assuming the first objective is fuel consumption cost and the second objective is time, we calculate the upper and lower bounds as follows:

The lower bound for the first objective, “cost,” is generated from the optimal solution for its single-objective model, denoted as Z 1 ( x 1 ), and equals 70,165.50 L.E.

The lower bound for the second objective, “time,” is generated from the optimal solution for its single-objective model, denoted as Z 2 ( x 2 ), and equals 87,280 min.

The upper bound for the first objective is obtained by multiplying c ijkp for the second objective by x ijkp for the first objective. The resulting value is denoted as Z 1 ( x 2 ) and equals 73,027.50 L.E.

The upper bound for the second objective is obtained by multiplying t ijkp for the first objective by x ijkp for the second objective. The resulting value is denoted as Z 2 ( x 1 ) and equals 88,286.50 min.

As such, the aspiration levels for each objective are defined from the above values by evaluating the maximum and minimum value of each objective.

The aspiration level for the first objective, denoted as F 1, ranges between 70,165.50 and 73,027.50, i.e., 70,165.50 <  =  F 1 <  = 730,27.50.

The aspiration level for the second objective, denoted as F 2, ranges between 87,280 and 88,286.50, i.e., 87,280 <  =  F 2 <  = 88,286.50.

The objective function for the multi-objective multidimensional solid transport problem was determined using the Zimmermann Programming Technique, Global Criteria Method, and Minimum Distance Method. The first two methods directly provided the best compromise solution (BCS), while the last method generated non-dominated extreme points by assigning different weights to each objective and finding the BCS from them. The best compromise solution was obtained using the Lingo software [ 34 ]. Table 1 and Fig.  3 present the objective values for the optimal solutions of fuel consumption cost and time, the best compromise solution, and solutions with different weights for both objectives. Figure  4 illustrates the minimum required number of each type of truck for daily transportation of various products from sources to destinations.

Objective value in different cases

Ideal distribution of the company’s truck fleet

The primary objective of the case study is to determine the minimum number of trucks of each type required daily at each garage for transporting products from factories to distribution centers. The minimum number of trucks needs to be flexible, allowing decision-makers to make various choices, such as minimizing fuel consumption cost, delivery time, or achieving the best compromise between different objectives. To determine the minimum number of required trucks, we compare all the previously studied cases and select the largest number that satisfies the condition: min Zik (should be set) = max Zik (from different cases). Due to the discrepancy between the truck capacity and the quantity of products to be transported, the required number of trucks may have decimal places. In such cases, the fraction is rounded to the nearest whole number. For example, if the quantity of items from a location requires one and a half trucks, two trucks of the specified type are transported on the first day, one and a half trucks are distributed, and half a truck remains in stock at the distribution center. On the next day, only one truck is transferred to the same distribution center, along with the semi-truck left over from the previous day, and so on. This solution may be preferable to transporting trucks that are not at full capacity. Table 2 and Fig.  5 depict the ideal distribution of the company’s truck fleet under various optimization and multi-objective conditions.

Min. No. of trucks should be set for different cases

## Conclusions

In conclusion, this research paper addresses the critical issue of optimizing transportation within the context of logistics and supply chain management, specifically focusing on the methods known as the solid transportation problem (STP) and the multi-dimensional solid transportation problem (MDSTP). The study presents a case study conducted on a private sector company in Egypt to determine the optimal distribution of its truck fleet under different optimization and multi-objective conditions.

The research utilizes the multi-objective multi-dimensional solid transportation problem (MOMDSTP) to identify the best compromise solution, taking into account fuel consumption costs and total shipping time. Three decision-making methods, namely the Zimmermann Programming Technique, the Global Criteria Method, and the Minimum Distance Method, are employed to derive optimal solutions for the objectives.

The findings of this study make a significant contribution to the development of approaches for solving multi-objective solid transportation problems with uncertain parameters. The research addresses the complexities of diverse truck fleets and transported products by incorporating fuzzy programming, uncertainty theory, and related concepts. It critically examines the limitations of previous approaches that often focused solely on single-objective solutions and overlooked specific constraints.

The primary objective of this research is to determine the minimum number of trucks required to fulfill decision-makers objectives while optimizing fuel consumption and transportation timeliness. The proposed approach combines the framework of the multi-dimensional solid transportation problem with a multi-objective optimization technique, offering comprehensive solutions for decision-makers with multiple priorities.

This study provides practical solutions for real-world transportation scenarios and demonstrates the potential for enhancing transportation efficiency and cost-effectiveness in various industries or contexts. The comparative analysis of the proposed work highlights the advancements and novelty introduced by the approach, emphasizing its significant contributions to the field of transportation problem research.

Future research should explore additional dimensions of the multi-objective solid transportation problem and incorporate other decision-making methods or optimization techniques. Additionally, incorporating uncertainty analysis and sensitivity analysis can enhance the robustness and reliability of the proposed solutions. Investigating the applicability of the approach in diverse geographical contexts or industries would yield further insights and broaden the potential applications of the research findings.

## Availability of data and materials

The data that support the findings of this study are available from the company but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available. Data are however available from the authors upon reasonable request. Please note that some data has been mentioned in the form of charts as agreed with the company.

## Abbreviations

Solid transportation problem

Multi-objective solid transportation problems

Multi-dimensional solid transportation problem

Multi-objective multi-dimensional solid transportation problem

Best compromise solution

Taha HA (2013) Operations research: an introduction. Pearson Education India

Li T et al (2020) Federated optimization in heterogeneous networks. Proc Mach Learn Syst 2:429–450

Babu MA et al (2020) A brief overview of the classical transportation problem.

Winston WL (2022) Operations research: applications and algorithms. Cengage Learning

Pratihar J et al (2020) Transportation problem in neutrosophic environment, in Neutrosophic graph theory and algorithms. IGI Global, p 180–212

Guo G, Obłój J (2019) Computational methods for martingale optimal transport problems. Ann Appl Probab 29(6):3311–3347

Article   MathSciNet   MATH   Google Scholar

Marwan M (2022) Optimasi biaya distribusi material Dengan Metode NWC (North West Corner) DAN Metode VAM (Vogel Approximation Method) PADA PT XYZ. IESM J (Indust Eng Syst Manage J) 2(2):137–146

Qiuping N et al (2023) A parametric neutrosophic model for the solid transportation problem. Manag Decis 61(2):421–442

Singh S, Tuli R, Sarode D (2017) A review on fuzzy and stochastic extensions of the multi index transportation problem. Yugoslav J Oper Res 27(1):3–29

Baidya A, Bera UK (2019) Solid transportation problem under fully fuzzy environment. Int J Math Oper Res 15(4):498–539

Berbatov K et al (2022) Diffusion in multi-dimensional solids using Forman’s combinatorial differential forms. Appl Math Model 110:172–192

Carlier G (2003) On a class of multidimensional optimal transportation problems. J Convex Anal 10(2):517–530

MathSciNet   MATH   Google Scholar

Zaki SA et al (2012) Efficient multiobjective genetic algorithm for solving transportation, assignment, and transshipment problems. Appl Math 03(01):92–99

Article   MathSciNet   Google Scholar

Latpate R, Kurade SS (2022) Multi-objective multi-index transportation model for crude oil using fuzzy NSGA-II. IEEE Trans Intell Transp Syst 23(2):1347–1356

Bélanger V, Ruiz A, Soriano P (2019) Recent optimization models and trends in location, relocation, and dispatching of emergency medical vehicles. Eur J Oper Res 272(1):1–23

Kar MB et al (2018) A multi-objective multi-item solid transportation problem with vehicle cost, volume and weight capacity under fuzzy environment. J Intell Fuzzy Syst 35(2):1991–1999

Mirmohseni SM, Nasseri SH, Zabihi A (2017) An interactive possibilistic programming for fuzzy multi objective solid transportation problem. Appl Math Sci 11:2209–2217

Kakran VY, Dhodiya JM (2021) Multi-objective capacitated solid transportation problem with uncertain variables. Int J Math, Eng Manage Sci 6(5):1406–1422

Chen L, Peng J, Zhang B (2017) Uncertain goal programming models for bicriteria solid transportation problem. Appl Soft Comput 51:49–59

Khalifa HAE-W, Kumar P, Alharbi MG (2021) On characterizing solution for multi-objective fractional two-stage solid transportation problem under fuzzy environment. J Intell Syst 30(1):620–635

El-Shorbagy MA et al (2020) Evolutionary algorithm for multi-objective multi-index transportation problem under fuzziness. J Appl Res Ind Eng 7(1):36–56

Uddin MS et al (2021) Goal programming tactic for uncertain multi-objective transportation problem using fuzzy linear membership function. Alex Eng J 60(2):2525–2533

Hosseinzadeh E (2023) A solution procedure to solve multi-objective linear fractional programming problem in neutrosophic fuzzy environment . J Mahani Math Res. 111–126.  https://jmmrc.uk.ac.ir/article_3728_bc0be59dc0f595cc32faae1991cd12f9.pdf

Jagtap K, and Kawale S (2017) Multi-Dimensional-Multi-Objective-Transportation-Problem-by-Goal-Programming . Int J Sci Eng Res 8(6):568–573

Paratne P, and Bit A (2019) Fuzzy programming technique with new exponential membership function for the solution of multiobjective transportation problem with mixed constraints. J Emerg Technol Innov Res.  https://www.researchgate.net/profile/Mohammed-Rabie-3/publication/363480949_A_case_study_on_the_optimization_of_multi-objective_functions_transportation_model_for_public_transport_authority_Egypt/links/631f0549071ea12e362a9214/A-case-study-on-the-optimization-of-multi-objective-functions-transportation-model-for-public-transport-authority-Egypt.pdf

Annamalaınatarajan R, and Swaminathan M (2021) Uncertain multi–objective multi–item four dimensional fractional transportation model . Ann Rom Soc Cell Biol. 231–247.  https://www.annalsofrscb.ro/index.php/journal/article/download/2457/2063

Mohammed A (2020) Towards a sustainable assessment of suppliers: an integrated fuzzy TOPSIS-possibilistic multi-objective approach. Ann Oper Res 293:639–668

Umarusman N (2019) Using global criterion method to define priorities in Lexicographic goal programming and an application for optimal system design . MANAS Sosyal Araştırmalar Dergisi. 8(1):326–341

Kamal M et al (2018) A distance based method for solving multi-objective optimization problems . J Mod Appl Stat Methods 17(1).  https://digitalcommons.wayne.edu/jmasm/vol17/iss1/21

Kaur L, Rakshit M, Singh S (2018) A new approach to solve multi-objective transportation problem. Appl Appl Math: Int J (AAM) 13(1):10

Kamal M et al (2018) A distance based method for solving multi-objective optimization problems. J Mod Appl Stat Methods 17(1):21

Yang L, Feng Y (2007) A bicriteria solid transportation problem with fixed charge under stochastic environment. Appl Math Model 31(12):2668–2683

Article   MATH   Google Scholar

Munot DA, Ghadle KP (2022) A GM method for solving solid transportation problem. J Algebraic Stat 13(3):4841–4846

Gupta N, and Ali I (2021) Optimization with LINGO-18 problems and applications. CRC Press

Not applicable.

## Author information

Authors and affiliations.

Automotive and Tractor Engineering Department, Minia University, Minia, Egypt

Mohamed H. Abdelati, Ali M. Abd-El-Tawwab & M Rabie

Automotive and Tractor Engineering Department, Helwan University, Mataria, Egypt

Elsayed Elsayed M. Ellimony

You can also search for this author in PubMed   Google Scholar

## Contributions

MHA designed the research study, conducted data collection, analyzed the data, contributed to the writing of the paper, and reviewed and edited the final manuscript. AMA contributed to the design of the research study, conducted a literature review, analyzed the data, contributed to the writing of the paper, and reviewed and edited the final manuscript. EEME contributed to the design of the research study, conducted data collection, analyzed the data, contributed to the writing of the paper, and reviewed and edited the final manuscript.MR contributed to the design of the research study, conducted programming using Lingo software and others, analyzed the data, contributed to the writing of the paper, and reviewed and edited the final manuscript. All authors have read and approved the manuscript.

## Corresponding author

Correspondence to Mohamed H. Abdelati .

## Ethics declarations

Competing interests.

The authors declare that they have no competing interests.

## Rights and permissions

Reprints and Permissions

Abdelati, M.H., Abd-El-Tawwab, A.M., Ellimony, E.E.M. et al. Solving a multi-objective solid transportation problem: a comparative study of alternative methods for decision-making. J. Eng. Appl. Sci. 70 , 82 (2023). https://doi.org/10.1186/s44147-023-00247-z

Received : 19 April 2023

Accepted : 27 June 2023

Published : 20 July 2023

DOI : https://doi.org/10.1186/s44147-023-00247-z

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Decision-making
• Multi-objective
• Solid transpiration

#### IMAGES

1. PPT

2. Transportation problem

3. PPT

4. PPT

5. PPT

6. PPT

#### VIDEO

1. Means of transport

2. Transportation problems/Operation Research/Lec.-2/Vam Method/B Com-6th sem/P U Chd

3. Transportation problems/MODI Method/Operation Research/B Com-6th sem/P U Chd

4. Transportation problems/Stepping Stone Method with Epsilon/Operation Research/B Com-6th sem/P U Chd

5. Transportation problems/Operation Research/Lec.-1/B Com-6th sem/P U Chd

6. Smart Transportation Methods #satisfying #shot

1. PDF Solving Transportation Problem by Various Methods and Their Comaprison

Introduction The first main purpose is solving transportation problem using three methods of transportation model by linear programming (LP).The three methods for solving Transportation problem are: North West Corner Method Minimum Cost Method Vogel's approximation Method Trannsportation Model

2. Solving the Transportation Problem

Solving the Transportation Problem Algorithms & Functions Minimizing the cost of transporting products from production and storage locations to demand centers is an essential part of maintaining profitability for companies who deal with product distribution.

3. Balanced and Unbalanced Transportation Problems

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. Balanced Transportation Problem The following is the basic framework of the balanced transportation problem:

4. How to Solve the Transportation Problem

LpSolve R is an easy-to-use R program for solving linear programming (LP) transportation problems. The transportation challenge concerns the movement of items from manufacturing units to...

5. Transportation Problem

Methods to Solve: To find the initial basic feasible solution there are three methods: NorthWest Corner Cell Method. Least Call Cell Method. Vogel's Approximation Method (VAM). Basic structure of transportation problem:

6. The Transportation Problem: Features, Types, & Solutions

Autoplay 80K views Types Transportation problems can be classified into different groups based on their main objective and origin supply versus destination demand. Transportation problems...

7. What is Transportation Problem

The transportation problem is a special type of linear programming problem where the objective is to minimise the cost of distributing a product from a number of sources or origins to a number of destinations. Because of its special structure the usual simplex method is not suitable for solving transportation problems.

8. [PDF] Solving Transportation Problem by Various Methods and Their

1 A Proposed Method for Finding Initial Solutions to Transportation Problems Joseph Ackora-Prah Valentine Acheson Emmanuel Owusu-Ansah Seth K. Nkrumah Business Pakistan Journal of Statistics and Operation… 2023 The Transportation Model (TM) in the application of Linear Programming (LP) is very useful in optimal distribution of goods.

9. Statistical Methods for Solving Transportation Problems

There are three well-known methods namely, North West Corner Method Least Cost Method, Vogel's Approximation Method to find the initial basic feasible solution of a transportation...

10. A method for solving the transportation problem

To solve the transportation problem we need to find a feasible solution. The feasible solution of the transportation problem can be obtained by using the least cost, Vogel or other methods. After obtaining feasible solutions, we use the existing methods such as multiples method or the method of stepping stones to achieve the optimal solution.

11. PDF STATISTICAL METHODS FOR SOLVING TRANSPORTATION PROBLEMS ...

A new method to solve transportation problem- Harmonic Mean approach, Engineering Technology Journal, Vol.2, Issue 3, (2018). 13. Title: KBO Template Author: AFT Subject: KBO Conference

12. Transportation Problem Explained and how to solve it?

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.

13. Solving Transportation Problem with Four Different Proposed Mean Method

A simple penalty and rapid process is used in this paper in order to obtain the lowest shipping cost for the transportation problems. A new method is proposed to extract the optimal solution of ...

14. On Solving the Transportation Problem

that time, i.e. the minimum-cost network out-of-kilter method adapted to solve the TP,22 the standard simplex method for solving the general linear-programming problem and a dual simplex method for solving a TP. The results of the comparison showed that the Glover et al.2" method was six times faster than the best of the competitive methods.

15. PDF A New Method for Solving Transportation Problems Considering Average

A New Method for Solving Transportation Problems Considering Average Penalty DOI: 10.9790/5728-1301044043 www.iosrjournals.org 42 | Page Factories Showrooms Supply a i 1 D D 2 D 3 D 4 S 1 2 2 2 1 3 S 2 10 8 5 4 7 S 3 7 6 6 8 5 Demand b j 4 3 4 4 15 Table: 2.1 We want to solve the transportation problem by the current algorithm. ...

16. optimize Transportation problem

To solve a transportation problem, the following information must be given: m= The number of sources. n= The number of destinations. The total quantity available at each source. The total quantity required at each destination. The cost of transportation of one unit of the commodity from each source to each destination.

17. Solving Transportation Problem by Various Methods and ...

The details method using Stepping Stone method and the Modified Distribution Method (MODI) are used to solve the problem solved using the management scientist 5.0 software. Shraddha (2017 ...

18. PDF Comparison of Methods of Solving Transportation Problems (TP) and

In an attempt to proffer solutions to transportation problems, several methods have been presented, developed and subsequently extended by various researchers at different time [8, 2, 6, 3]. However, this study compares various methods of solving transportation problems, and also reviews the associated

19. PDF Oregon State University

You are being redirected.

20. Operations Research with R

The transportation problem represents a particular type of linear programming problem used for allocating resources in an optimal way; it is a highly useful tool for managers and supply chain engineers for optimizing costs. The lpSolve R package allows to solve LP transportation problems with just a few lines of code.

21. A New Method of Solving the Transportation Problem

Ulrich A. Wagener - A New Method of Solving the Transportation Problem any source. The method will be illustrated with an example, and the different points explained as they arise. It may be noted that the original allocation is the ideal, if all sources had unlimited availability. In some cases this may be useful additional information.

22. PDF Optimization Techniques for Transportation Problems of Three Variables

II. TRANSPORTATION METHOD When transportation method is employed in solving a transportation problem, the very initial step that has to be undertaken is to obtain a feasible solution satisfying demand and supply requirement (Lu 2010). Several methods will be used in this paper to obtain this initial feasible problem. As mentioned earlier in this

23. Solving a multi-objective solid transportation problem: a comparative

The transportation problem in operations research aims to minimize costs by optimizing the allocation of goods from multiple sources to destinations, considering supply, demand, and transportation constraints. This paper applies the multi-dimensional solid transportation problem approach to a private sector company in Egypt, aiming to determine the ideal allocation of their truck fleet.In ...