how to solve unbalanced transportation problem of maximization type

Maximization Transportation Problem

There are certain types of transportation problems where the objective function is to be maximized instead of being minimized.

These problems can be solved by converting the maximization problem into a minimization problem.

"Profit maximization is the single universal objective for most commercial organizations." -Vinay Chhabra & Manish Dewan

Example: Maximization Problem in Transportation

Surya Roshni Ltd. has three factories - X, Y, and Z. It supplies goods to four dealers spread all over the country. The production capacities of these factories are 200, 500 and 300 per month respectively.

Determine a suitable allocation to maximize the total net return.

Maximization transportation problem can be converted into minimization transportation problem by subtracting each transportation cost from maximum transportation cost.

Here, the maximum transportation cost is 25. So subtract each value from 25. The revised transportation problem is shown below.

An initial basic feasible solution is obtained by matrix minimum method and is shown in the final table.

Final table

Use Horizontal Scrollbar to View Full Table Calculation.

The maximum net return is

25 X 200 + 8 X 80 + 7 X 320 + 10 X 100 + 14 X 100 + 20 X 200 = 14280.

Share this article with your friends

Operations Research Simplified Back Next

Goal programming Linear programming Simplex Method Assignment Problem

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).

Leave a Comment Cancel reply

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.

Transportation Problem, Data Analysis and Modeling || Bcis Notes

April 5, 2021 Ritik Bhujel DAM 0

Transportations Problems

1 Introduction: The transportation problem is concerned to determine the amount that should be shipped from each source to each destination to make the total transportation cost minimum and at the same time satisfying the supply limits and the demand requirements. The transportation problem is concerned with selecting routes to distribute the goods in the different destinations in order to minimize the total transportation cost or to maximize the total revenue of the problem by satisfying the requirements of the different destinations and supply quantities of different sources.

2  Formulation of transportation problems as a linear programming problem: The transportation problem is in the tabular form (matrix form) as shown

Where, there are ‘m’ sources S 1 , S 2 , ………, S m having a i (i=1,2,…..,m ) units of supplies or capacity respectively to be transported among ‘n’ destinations D 1 , D 2 , ………, D n  with b j ( j= 1,2,……..,n ) units of requirements respectively.

C ij be the cost of shipping one unit of the commodity from sources ‘i’ to destination ‘j’ and X ij denotes the number of units shipped per route from source ‘i’ to destination ‘j’.

Now, this transportation problem is formulated to minimize the total transportation cost as

Minimize Z = C 11 X 11 + C 12 X 12 +………+ C m n X m n

Z = C ij X ij

Subjected to constraints

Supply constraints

X 11 + X 12 +………+   X 1 n = a 1

X 21 + X 22 +………+   X 2 n = a 2

………………………………

X m1 + X m2 +………+   X m n = a m

Demand constraints

X 11 + X 21 +………+   X m1 = b 1

X 12 + X 22 +………+   X   m2 = b 2

X 1 n + X 2 n +………+   X m n = b n

X i j  0 , for all i and j .

3  Types of transportations problems: There are two types of transportation problems depending upon supply and demand.

• Balanced transportation problem: If total supply equals total demand. Then problem is called a balanced transportation problem.
• Unbalanced transportation problem: If total supply does not equal to total demands. Then problem is called an unbalanced transportation problem.

  Note: It should be noted that the transportation problem will be solved when we convert unbalanced transportation problem into balanced transportation problem.

4 Methods of converting unbalanced transportation problem into balanced transportation problem:

• When demand exceeds supply. When total supply (a i ) is less than total demand (b j ), at that time, introduce a dummy source (dummy row) in the transportation table to make total supply equals to total demand. Here, in this dummy row, the unit cost of transporting from this source to any destination is set to zero.
• When supply exceeds demand. When total supply (a i ) is greater than total demand ( bj ), at that time, introduce a dummy destination (dummy column) in the transportation table to make total demand equals to total supply. Here, in this dummy column, the unit cost of transporting from different sources to this dummy destination is set to zero.

5 Definition of some basic terms:

• Source: It is the place of origin from which the goods or services are supplied to the different places of destinations.
• Destination: It is the place of demand where the goods are required in certain finite quantities.
• Occupied (basic)cells and non-occupied (non-basic) cells: Squares (cells) in the transportation table having positive allocation are called occupied cells, otherwise, called empty or non-occupied cells.
• The feasible solution : A set of non-negative values X ij that satisfied the constraints of given T.P. is called a feasible solution to the transportation problem.
• The basic feasible solution: A feasible solution that contains no more than (R+C-1) non-negative allocations is called a basic feasible solution, where ‘R’ is no. of rows and ‘C’ is the no. of columns in the transportation table.
• Degeneracy: When the number of occupied cells (basic cells) of general T.P. is less than (R+C-1), Then it is called degeneracy in transportation problem, and the solution is called degeneracy basic feasible solution.
• Non – degeneracy: when the number of occupied cells (basic cells) of general T.P. is exactly equal to (R+C-1), then it is called non-degeneracy in the transportation problem, and the solution is called non-degeneracy basic feasible solution.

6  Methods of obtaining the initial basic feasible solution of transportation problem:

• Northwest corner method.
• Least cost method Or Greedy Method.
• Vogel’s approximation method or penalty method.

i)   Northwest corner method (NWCM):

In this method, the following systematic steps are used to obtain the initial basic feasible solutions.

• Step 1 : Allocation starts with the cell (1,1) at the upper left ( northwest ) corner of the transportation table and allocates as much as a possible value equal to the minimum of first row values and first column values i.e. min ( a 1, b 1 )
• (a) If the allocation made in step 1 is equal to the capacity of the first row (a 1 ), then move vertically down to the cell (2, 1) to fulfill the remaining demand of the first column. Here allocation is made according to step 1.
• (b) If the allocation made in step 1 is equal to the demand of the first column (b 1 ), then move horizontally to the cell (1, 2) to finish the remaining capacity (sources) of the first row (source). Here allocation is made according to step 1.
• (c) If allocation made in step 1 is equal to the capacity of the first source (a 1 ) and demand of first destination (b 1 ). Then move diagonally to the cell (2, 2). Here allocation is made according to step 1.
• Step 3: continue the procedure step by step till an allocation is made in the southeast corner cell of the transportation table i.e. ( continue the above steps until all the capacity of all the sources are finishes and the demands of all the destinations are full filled).
• Step 4: Calculate the total transportation cost, which is obtained as, at first multiply the allocated values with the corresponding unit cost for all occupied cells separately and then add all the multiple values which you obtained.

ii)  Least cost method (LCM) Or Greedy Method:

In this method, the following systematic steps are used to obtain the initial basic feasible solution.

• Step 1: At first, select the cell having the smallest unit cost in the entire transportation table and allocate as much as possible value to this cell and then eliminate that row if the capacity of that row is finished or eliminate that column if the demand of that column (destination) is fulfilled.

Note 1: If both row (source) and column (destination) are satisfied simultaneously, then only one eliminates (crossed out)

Note 2: In case, the smallest unit cost is not unique, then select the cell where maximum allocation can be made.

• Step 2: After the first step, again select the cell having the smallest unit cost out of uncrossed (non -eliminated) rows and columns. Then allocation is made to this cell according to step 1. Then eliminate that row and column in which either supply or demand is exhausted. This process should repeat until the entire available supply at various sources and demand at various destinations is satisfied.

The solution so obtained need not be non-degeneracy.

• Step 3: Calculate the total transportation cost, which is obtained as, at first multiply the allocated values with the corresponding unit cost for all occupied cells separately and then add all the multiple values which you obtained.

iii)  Vogel’s approximation method:

In this method, the following systematic steps are used to obtain the initial basic feasible solution to the transportation problem.

• Step 1: At first, calculate the difference between the smallest and next to smallest unit cost for each row (source) and column (destination). These differences are known as row and column opportunities (penalties) cost
• Step 2: Select the row or column with the largest difference (opportunities cost). Then allocate as many units as possible to this row or column in the cell having the least unit cost.

Note: if there is a tie in the values of the opportunities cost. At that time, we select that row or column with minimum unit cost. Again if there is a tie in the unit cost, we select that cell of that row or column where maximum allocation can be made.

• Step 3: After step 2, if the capacity of that row is finished, we eliminate that row. If the demand of that column is fulfilled, we eliminate that column. If the capacity of that row and demand of that column are satisfied simultaneously then only one of them is eliminated and in the remaining one, we assign zero supply or zero demand. Any row or column with zero supply or zero demand should not be used in computing future penalties (opportunities cost)
• Step 4: Again, calculate the opportunity cost for the remaining rows and columns which are not eliminated.
• Step 5: Repeat steps 2 to step 4 until the entire available supply at various sources (rows) and demand at various destinations (columns) are satisfied.
• Step 6 : Calculate the total transportation cost, which is obtained as, at first multiply the allocated values with the corresponding unit cost for all occupied cells separately and then add all the multiple values which you obtained.

Note: Among the three methods, the total transportation cost obtained by Vogel’s approximation is the least so Vogel’s approximation is a more effective method for obtaining the initial basic feasible solution than other methods.

7  Maximization of transportation problems: Transportation problems may be of the maximization type, maximization may be profit or revenue or productions. To solve such type of maximization transportation problem, at first convert this maximization T.P. into minimization T.P. by subtracting each unit cost from the highest unit cost of maximization transportation problem. Then we will get the minimization transportation problem. Then we proceed with the same process, preceded in case of minimization T.P.

Note: during the calculation of the maximum value, we will consider the original table. In which we multiply allocated goods with corresponding unit cost separately and then add all the multiple values to obtain the maximum value.

You may also like this:  Method of Construction of Index Number 

• Definition of some basic terms
• Formulation of transportation problems as a linear programming problem
• Methods of converting unbalanced transportation problem into balanced transportation problem
• Methods of obtaining the initial basic feasible solution of transportation problem
• Types of transportations problems

Be the first to comment

Leave a reply cancel reply.

Your email address will not be published.

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

Copyright © 2024 | WordPress Theme by MH Themes

• Practice Mathematical Algorithm
• Mathematical Algorithms
• Pythagorean Triplet
• Fibonacci Number
• Euclidean Algorithm
• LCM of Array
• GCD of Array
• Binomial Coefficient
• Catalan Numbers
• Sieve of Eratosthenes
• Euler Totient Function
• Modular Exponentiation
• Modular Multiplicative Inverse
• Stein's Algorithm
• Juggler Sequence
• Chinese Remainder Theorem
• Quiz on Fibonacci Numbers
• Transportation Problem | Set 6 (MODI Method - UV Method)
• Transportation Problem | Set 2 (NorthWest Corner Method)
• Transportation Problem | Set 4 (Vogel's Approximation Method)
• Transportation Problem | Set 1 (Introduction)
• Transportation Problem | Set 3 (Least Cost Cell Method)
• Transportation Problem Set 8 | Transshipment Model-1
• Transportation Problem | Set 5 ( Unbalanced )
• Transportation Problem | Set 7 ( Degeneracy in Transportation Problem )
• Travelling Salesman Problem using Hungarian method
• Travelling Salesman Problem (TSP) using Reduced Matrix Method
• Traveling Salesman Problem (TSP) Implementation
• Approximate solution for Travelling Salesman Problem using MST
• Travelling Salesman Problem implementation using BackTracking
• Bitonic Travelling Salesman Problem
• Bellman Ford Algorithm (Simple Implementation)
• Find the minimum cost to reach destination using a train
• Problem on Trains, Boat and streams
• Java Program to Solve Travelling Salesman Problem Using Incremental Insertion Method
• QA - Placement Quizzes | Mixtures and Alligation | Question 6
• QA - Placement Quizzes | Permutation and Combination | Question 8

Transportation Problem | Set 6 (MODI Method – UV Method)

There are two phases to solve the transportation problem. In the first phase, the initial basic feasible solution has to be found and the second phase involves optimization of the initial basic feasible solution that was obtained in the first phase. There are three methods for finding an initial basic feasible solution,

NorthWest Corner Method

• Least Cost Cell Method
• Vogel’s Approximation Method

This article will discuss how to optimize the initial basic feasible solution through an explained example. Consider the below transportation problem.

Check whether the problem is balanced or not. If the total sum of all the supply from sources

is equal to the total sum of all the demands for destinations

then the transportation problem is a balanced transportation problem.

If the problem is not unbalanced then the concept of a dummy row or a dummy column to transform the unbalanced problem to balanced can be followed as discussed in

Finding the initial basic feasible solution. Any of the three aforementioned methods can be used to find the initial basic feasible solution. Here,

will be used. And according to the NorthWest Corner Method this is the final initial basic feasible solution:

Now, the total cost of transportation will be

(200 * 3) + (50 * 1) + (250 * 6) + (100 * 5) + (250 * 3) + (150 * 2) = 3700

U-V method to optimize the initial basic feasible solution. The following is the initial basic feasible solution:

– For U-V method the values

have to be found for the rows and the columns respectively. As there are three rows so three

values have to be found i.e.

for the first row,

for the second row and

for the third row. Similarly, for four columns four

. Check the image below:

There is a separate formula to find

is the cost value only for the allocated cell. Read more about it

. Before applying the above formula we need to check whether

m + n – 1 is equal to the total number of allocated cells

or not where

is the total number of rows and

is the total number of columns. In this case m = 3, n = 4 and total number of allocated cells is 6 so m + n – 1 = 6. The case when m + n – 1 is not equal to the total number of allocated cells will be discussed in the later posts. Now to find the value for u and v we assign any of the three u or any of the four v as 0. Let we assign

in this case. Then using the above formula we will get

). Similarly, we have got the value for

so we get the value for

which implies

. From the value of

. See the image below:

Now, compute penalties using the formula

only for unallocated cells. We have two unallocated cells in the first row, two in the second row and two in the third row. Lets compute this one by one.

• For C 13 , P 13 = 0 + 0 – 7 = -7 (here C 13 = 7 , u 1 = 0 and v 3 = 0 )
• For C 14 , P 14 = 0 + (-1) -4 = -5
• For C 21 , P 21 = 5 + 3 – 2 = 6
• For C 24 , P 24 = 5 + (-1) – 9 = -5
• For C 31 , P 31 = 3 + 3 – 8 = -2
• For C 32 , P 32 = 3 + 1 – 3 = 1

If we get all the penalties value as zero or negative values that mean the optimality is reached and this answer is the final answer. But if we get any positive value means we need to proceed with the sum in the next step. Now find the maximum positive penalty. Here the maximum value is 6 which corresponds to

cell. Now this cell is new basic cell. This cell will also be included in the solution.

The rule for drawing closed-path or loop.

Starting from the new basic cell draw a closed-path in such a way that the right angle turn is done only at the allocated cell or at the new basic cell. See the below images:

Assign alternate plus-minus sign to all the cells with right angle turn (or the corner) in the loop with plus sign assigned at the new basic cell.

Consider the cells with a negative sign. Compare the allocated value (i.e. 200 and 250 in this case) and select the minimum (i.e. select 200 in this case). Now subtract 200 from the cells with a minus sign and add 200 to the cells with a plus sign. And draw a new iteration. The work of the loop is over and the new solution looks as shown below.

Check the total number of allocated cells is equal to (m + n – 1). Again find u values and v values using the formula

is the cost value only for allocated cell. Assign

then we get

. Similarly, we will get following values for

Find the penalties for all the unallocated cells using the formula

• For C 11 , P 11 = 0 + (-3) – 3 = -6
• For C 13 , P 13 = 0 + 0 – 7 = -7
• For C 14 , P 14 = 0 + (-1) – 4 = -5
• For C 31 , P 31 = 0 + (-3) – 8 = -11

There is one positive value i.e. 1 for

. Now this cell becomes new basic cell.

Now draw a loop starting from the new basic cell. Assign alternate plus and minus sign with new basic cell assigned as a plus sign.

Select the minimum value from allocated values to the cell with a minus sign. Subtract this value from the cell with a minus sign and add to the cell with a plus sign. Now the solution looks as shown in the image below:

Check if the total number of allocated cells is equal to (m + n – 1). Find u and v values as above.

Now again find the penalties for the unallocated cells as above.

• For P 11 = 0 + (-2) – 3 = -5
• For P 13 = 0 + 1 – 7 = -6
• For P 14 = 0 + 0 – 4 = -4
• For P 22 = 4 + 1 – 6 = -1
• For P 24 = 4 + 0 – 9 = -5
• For P 31 = 2 + (-2) – 8 = -8

All the penalty values are negative values. So the optimality is reached. Now, find the total cost i.e.

(250 * 1) + (200 * 2) + (150 * 5) + (50 * 3) + (200 * 3) + (150 * 2) = 2450

Please Login to comment...

Similar reads.

• Mathematical

Improve your Coding Skills with Practice

What kind of Experience do you want to share?

Multi-dimensional transportation problems in multiple environments: a simulation based heuristic approach

• Soft computing in decision making and in modeling in economics
• Published: 14 May 2023
• Volume 27 , pages 11603–11628, ( 2023 )

Cite this article

• Sova Pal 1 ,
• Prasenjit Pramanik   ORCID: orcid.org/0000-0002-0957-8875 2 ,
• Ajoy Kumar Maiti 3 &
• Manas Kumar Maiti 4  

146 Accesses

Explore all metrics

Here, a general methodology is proposed to formulate and solve any multidimensional balanced/unbalanced, constrained/unconstrained transportation problems(TP) in different environments(crisp/fuzzy/rough). To understand the general model easily, here, at first, a multi-item 5-dimensional fixed charge profit maximization TP under budget and time constraint is presented. A potential solution of the problem is coded as a permutation of the different cells of the allocation matrix. A general decoding rule is proposed to determine the actual allocation from this coded solution. A heuristic approach is applied on a set of randomly generated coded solution of the target problem to determine the marketing decision. Applying swap operations on the coded solutions, the perturbation rules of the heuristic Particle Swarm Optimization(PSO) are modified to solve the problem. In a particular case, the problem is analysed as a bi-criteria decision making problem with the maximization of the total profit as well as the minimization of the total shipment time under a budget constraint. The bi-criteria TP is formulated as a single objective optimisation problem using a proposed rule and the same heuristic is run for a finite number of times to determine the pareto optimal front. To formulate the problem in the fuzzy(rough) environment an approach is proposed using credibility(trust) measure of fuzzy(rough) events. Proper fuzzy(rough) simulation algorithms are also proposed to solve the problem for any type of fuzzy(rough) estimation. Using this approach no crisp equivalent of any imprecise parameters is used for the marketing decision. Due the unavailability of the test data in the literature, different hypothetical data sets are used for the illustration of the models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price includes VAT (Russian Federation)

Instant access to the full article PDF.

Rent this article via DeepDyve

Institutional subscriptions

Similar content being viewed by others

Multi-objective capacitated transportation: a problem of parameters estimation, goodness of fit and optimization.

Some Transportation Problems Under Uncertain Environments

A direct solution approach based on constrained fuzzy arithmetic and metaheuristic for fuzzy transportation problems, data availability.

Enquiries about data availability should be directed to the authors.

Akhand MAH, Akter S, Rashid MA (2013) Velocity tentative particle swarm optimiza- tion to solve TSP, International Conference on Electrical Information and Communication Technology (EICT)

Bakhayt A-GK (2016) Solving bi-objective 4-dimensional transportation problem by using PSO. Sci Int Lahore 28:2403–2410

Google Scholar  

Bit AK, Biswal MP, Alam SS (1993) Fuzzy programming approach to multi-objective solid transportation problem. Fuzzy Sets Syst 57:183–194

Article   MATH   Google Scholar  

Dubois D, Prade H (1983) Ranking fuzzy numbers in the setting of possibility theory, Information. Science 30:183–224

MATH   Google Scholar  

Eberhart RC, Kennedy J (1995) A new optimizer using Particle swarm theory, In: Proceedings of the Sixth International Symposium on micro machine and human science, pp. 39-43

Engelbrecht AP (2005) Fundamentals of Computational Swarm Intelligence, John Wiley and Sons, Ltd

Esmin A, Aoki A, Lambert-Torres RG (2002) Particle swarm optimization for fuzzy mem- bership functions optimization. IEEE Int Conf Syst Man Cybernet 3:6–9

Article   Google Scholar  

Feng HM (2005) Particle swarm optimization learning fuzzy systems design, In: Proceedings of the ICITA 3rd International Conference on Information Technology and Applications, 1: 363-366

Gottlieb J, Paulmann L (1998) Genetic algorithms for the fixed charge transportation problems, In: Proceedings of the IEEE Conference on Evolutionary Computation, ICEC, 330-335

Giri PK, Maiti MK, Maiti M (2013) Entropy based solid transportation problems with discounted unit costs under fuzzy random environment. Opsearch. https://doi.org/10.1007/s12597-013-0155-0

Giri PK, Maiti MK, Maiti M (2014) Fuzzy stochastic solid transportation problem using fuzzy goal programming approach. Comput Ind Eng 72:160–168

Giri PK, Maiti MK, Maiti M (2015) Fully fuzzy fixed charge multi-item solid transportation problem. Appl Soft Comput 27:77–91

Giri PK, Maiti MK, Maiti M (2016) Profit maximization of solid transportation problem under budget constraint using fuzzy measures. Iran J Fuzzy Syst 13(5):35–63

MathSciNet   MATH   Google Scholar  

Haley KB (1962) The solid transportation problem. Oper Res 11:446–448

Haque S, Bhurjee AK, Kumar P (2022) Multi-objective non-linear solid transportation problem with fixed charge, budget constraints under uncertain environments. Syst Sci Control Eng 10(1):899–909. https://doi.org/10.1080/21642583.2022.2137707

Hitchcock FL (1941) The distribution of a product from several sources to numerous localities. J Math Phys 20:224–230

Article   MathSciNet   MATH   Google Scholar  

Hirsch WM, Dantzig GB (1968) The fixed charge transportation problem. Naval Res Logis Q 15:413–424

Jimenez F, Verdegay JL (1999) Solving fuzzy solid transportation problems by an evolutionary algorithm based parametric approach. Eur J Oper Res 117:485–510

Kar MB, Kundu P, Kar S, Pal T (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. https://doi.org/10.3233/JIFS-171717

Kennedy J, Eberhart RC (1995) Particle swarm optimisation, In: Proceedings of the IEEE International Joint Conference on Neural Network, IEEE Press, 4, 1942-1948

Kennington JL, Unger VE (1976) A new branch and bound algorithm for the fixed charge transportation problem. Manage Sci 22:1116–1126

Kundu P, Kar S, Maiti M (2013) Multi-objective multi-item solid transportation problem in fuzzy environment. Appl Math Model 37:2028–2038

Kundu P, Kar S, Maiti M (2014) Multi-objective solid transportation problems with budget constraint in uncertain environment. Int J Syst Sci 45(8):1668–1682. https://doi.org/10.1080/00207721.2012.748944

Kundu P, Kar MB, Kar S, Pal T, Maiti M (2017) A solid transportation model with product blending and parameters as rough variables. Soft Comput 21:2297–2306. https://doi.org/10.1007/s00500-015-1941-9

Kundu P, Kar S, Maiti M (2015) Multi-item solid transportation problem with type-2 fuzzy parameters. Appl Soft Comput 31:61–80

Kocken HG, Sivri M (2016) A simple parametric method to generate all optimal solutions of Fuzzy Solid Transportation Problem. Appl Math Model 40(7–8):4612–4624

Li Y, Ida K, Gen M (1997) Improved genetic algorithm for solving multi objective solid transportation problem with fuzzy numbers. Comput Ind Eng 33:589–592

Liang JJ, Qin AK, Suganthan PN, Baskar S (2006) Comprehensive learning particle swarm optimizer for global optimization of multimodal functions. IEEE Trans Evol Comput 10:281–295

Liu B (2004) Uncertain Programming- An Introduction to its Axiomatic Foundations. Physica-Verlag, Heidelberg

Liu Y, Liu B (2003) A class of fuzzy random optimization: expected value models, Information. Science 155:89–102

Liu B, Iwamura K (1998) A note on chance constrained programming with fuzzy coefficients. Fuzzy Sets Syst 100:229–233

Liu P, Yang L, L-W, Li S, (2014) A solid transportation problem with type-2 fuzzy variables. Appl Soft Comput 24:543–558

Majumder S, Kundu P, Kar S, Pal T (2018) Uncertain multi-objective multi-item fixed charge solid transportation problem with budget constraint. Soft Comput. https://doi.org/10.1007/s00500-017-2987-7

Maiti MK, Maiti M (2006) Fuzzy inventory model with two warehouses under possibility constraints. Fuzzy Sets Syst 157:52–73

Nagarjan A, Jeyaraman K (2010) Solution of chance constrained programming problem for multi-objective interval solid transportation problem under stochastic environment using fuzzy approach. Int J Comput Appl 10(9):19–29

Niksirat M (2022) A new approach to solve fully fuzzy multi-objective transportation problem. Fuzzy Inform Eng 14(4):456–467

Ojha A, Das B, Mondal S, Maiti M (2010) A stochastic discounted multi-objective solid transportation problem for breakable items using analytical hierarchy process. Appl Math Model 34(2):2256–2271

Ojha A, Das B, Mondal S, Maiti M (2010) A Solid Transportation Problem for an item with fixed charge vechicle cost and price discounted varying charge using Genetic Algorithm. Appl Soft Comput 10:100–110

Ojha A, Das B, Mondal S, Maiti M (2011) Transportation policies for single and multi-objective transportation problem using fuzzy logic. Math Comput Model 53:1637–1646

Ojha A, Das B, Mondal SK, Maiti M (2013) A multi-item transportation problem with fuzzy tolerance. Appl Soft Comput 13(8):3703–3712

Pramanik P, Maiti MK, Maiti M (2017) A supply chain with variable demand under three level trade credit policy. Comput Indu Eng 106:205–221

Pramanik P, Maiti MK, Maiti M (2017) Three level partial trade credit with promotional cost sharing. Appl Soft Comput 58:553–575

Pramanik P, Maiti MK (2019) An inventory model for deteriorating items with inflation induced variable demand under two level partial trade credit: a hybrid ABC-GA approach. Eng Appl Artif Intell 85:194–207

Pramanik P, Maiti MK (2020) Trade credit policy of an inventory model with imprecise variable demand: an ABC-GA approach. Soft Comput 24:9857–9874

Schell ED (1955) Distribution of a product by several properties, in: Proceedings of 2nd Symposium in Linear Programming, DCS/comptroller, HQ US Air Force, Washington,DC, 615-642

Sun M, Aronson JE, Mckeown PG, Dennis D (1998) A tabu search heuristic procedure for fixed charge transportation problem. Eur J Oper Res 106:411–456

Tao Z, Xu J (2012) A class of rough multiple objective programming and its application to solid transportation problem. Inf Sci 188:215-235

Wang KP, Huang L, Zhou CG, Pang W (2003) Particle swarm optimization for travelling salesman problem, In: Proc. International Conference on Machine Learning and Cybernetics, pp. 1583-1585

Yan X, Zhang C, Luo W, Li W, Chen W, Liu H (2012) Solve travelling salesman prob- lem using particle swarm optimization algorithm. Int J Comput Sci Issues 9:264–271

Yang L, Liu L (2007) Fuzzy fixed charge solid transportation problem and algorithm. Appl Soft Comput 7:879–889

Yang L, Yuan F (2007) A bi-criteria solid transportation problem with fixed charge under stochastic environment. Appl Math Model 31:2668–2683

Zadeh LA (1965) Fuzzy Sets. Inform Control 8:338–353

Zadeh LA (1978) Fuzzy Set as a basis for a theory of possibility. Fuzzy Sets Syst 1:3–28

Zimmermann H-J (1978) Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst 1:45–55

Download references

There is no funding regarding this research work.

Author information

Authors and affiliations.

Department of Computer Science, Yogoda Satsanga Palpara Mahavidyalaya, Palpara, Purba, Medinipur, West Bengal, 721458, India

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Paschim-Medinipur, Midnapore, West Bengal, 721102, India

Prasenjit Pramanik

Department of Mathematics, Raja Narendra Lal Khan Women’s College, Midnapore, West Bengal, 721102, India

Ajoy Kumar Maiti

Department of Mathematics, Mahishadal Raj College, Purba-Medinipur, Mahishadal, West Bengal, 721628, India

Manas Kumar Maiti

You can also search for this author in PubMed   Google Scholar

Contributions

5-dimensional multi item TP under different constraints in both precise and imprecise environments: Discussion, mathematical formulation, solution methodology, numerical illustration. Multi-dimensional multi item TP: theoretical discussion, mathematical formulation, solution methodology. Fuzzy and Rough Simulation: Algorithm developed and used. Soft Computing technique: Swap sequence based PSO.

Corresponding author

Correspondence to Prasenjit Pramanik .

Ethics declarations

Conflict of interest.

The authors declares that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed Consent

Not applicable.

Additional information

Publisher's note.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Pal, S., Pramanik, P., Maiti, A.K. et al. Multi-dimensional transportation problems in multiple environments: a simulation based heuristic approach. Soft Comput 27 , 11603–11628 (2023). https://doi.org/10.1007/s00500-023-08204-x

Download citation

Accepted : 27 March 2023

Published : 14 May 2023

Issue Date : August 2023

DOI : https://doi.org/10.1007/s00500-023-08204-x

Share this article

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

• Multi-dimensional profit maximization transportation problem
• Imprecise environment
• Swap sequence based PSO
• Fuzzy simulation
• Rough simulation
• Find a journal
• Publish with us
• Track your research