A transportation problem involving multiple objectives, multiple products, and three constraints, namely, source, destination, and conveyance, is called the multiobjective multiitem solid transportation problem (MOMISTP). Recently, Kundu et al. (2013) proposed a method to solve an unbalanced MOMISTP. In this paper, we suggest a method, which first converts an unbalanced problem to a balanced one. In one case of an example, while the method proposed by Kundu et al. concludes infeasibility, our method gives a feasible solution.
In today’s business environment, competition is increasing day by day and each organization aims to find better ways to deliver values to the customers in a cost effective manner within the specified time and fulfill their demands. For this they think of different ideas. One of the ideas can be using different type of transportation modes to save time and money.
Taking into account this factor, the conventional transportation problem proposed by Hitchcock in 1941 [
In many realworld problems, there are situations where several objectives are to be considered and optimized at the same time. Such problems are called multiobjective problems. In multiobjective transportation problems, instead of optimal solution, optimal compromise solution or efficient solution (feasible solution for which no improvement in any objective function is possible without sacrificing at least one of the objective functions) is considered. Zimmermann [
The geometric programming approach for multiobjective transportation problems was considered by Islam and Roy [
The multiobjective transportation problems in which heterogeneous items are to be transported from different production points to different consumer points using different modes of conveyance are called multiobjective multiitem solid transportation problems (MOMISTPs).
Due to shortage of information, insufficient data, lack of evidence, and so forth, the data for a transportation system such as availabilities, demands, and conveyance capacities are not always exact but can be fuzzy or arbitrary or both. Bector and Chandra [
Genetic algorithm was used by Li et al. [
In this paper a new method is proposed to solve the fuzzy MOMISTP. The application of proposed method is shown by solving two numerical examples in which objective function coefficients, availabilities, and demand parameters are represented by trapezoidal fuzzy numbers. In one case of an example, while the method proposed in [
The present paper is organized as follows: Section
In this section, some basic definitions, arithmetic operations, and ranking approach of trapezoidal fuzzy numbers are presented.
A fuzzy number
A trapezoidal fuzzy number
A trapezoidal fuzzy number
The support of a fuzzy number
The core of a fuzzy number
If, for a trapezoidal fuzzy number
Two trapezoidal fuzzy numbers
Let
where
In this paper, we shall use the ranking approach suggested by Liou and Wang [
Let
Let
A multiobjective multiitem solid transportation problem with
For the above problem to be balanced, it should satisfy the following conditions.
Total availability of an item at all sources should be equal to its demand at all the destinations.
Overall availability of all the items at all the sources, overall demand of all the items at all the destinations, and total conveyance capacity should be equal.
Mathematically, it means
Recently, Kundu et al. [
In this paper, we propose a method for MOMISTPs. The method first converts an unbalanced problem to a balanced one. Therefore the expected value model gives a feasible solution. This is then used to obtain its optimal compromise solution. Our method gives better value of the objective function than that obtained in [
In this section a new method has been proposed to find the optimal compromise solution of fuzzy MOMISTP in which all the parameters except the decision variables are represented by trapezoidal fuzzy numbers. The method consists of the following steps.
Check whether the problem under consideration is balanced or not (according to the definition given in Section
For this, find
Now proceed to Step
Using Step
Assume the unit fuzzy cost of transportation from the newly introduced source to all the destinations via any of the conveyance as zero trapezoidal fuzzy number.
The problem is balanced now.
The next step is to convert the balanced fuzzy MOMISTP problem, obtained by using Steps
In the above multiobjective problem, we wish to
maximize the possibility of getting values less than
minimize the possibility of getting values more than
This multiobjective problem is then changed to the following single objective optimization problem:
Solve the crisp multiobjective linear programming problem, obtained in Step
There are several defuzzification methods (e.g., [
A flowchart of the proposed method is shown in Figure
Flow chart for the proposed method.
Consider the fuzzy MOMISTP solved by Kundu et al. [
Unit penalties of transportation for item 1 in the first objective.
Conveyance 


Sources  Destinations  


 

(5, 8, 9, 11)  (4, 6, 9, 11)  (10, 12, 14, 16) 

(8, 10, 13, 15)  (6, 7, 8, 9)  (11, 13, 15, 17) 
Conveyance 


Sources  Destinations  


 

(9, 11, 13, 15)  (6, 8, 10, 12)  (7, 9, 12, 14) 

(10, 11, 13, 15)  (6, 8, 10, 12)  (14, 16, 18, 20) 
Unit penalties of transportation for item 2 in the first objective.
Conveyance 


Sources  Destinations  


 

(9, 10, 12, 13)  (5, 8, 10, 12)  (10, 11, 12, 13) 

(11, 13, 14, 16)  (7, 9, 12, 14)  (12, 14, 16, 18) 
Conveyance 


Sources  Destinations  


 

(11, 13, 14, 15)  (6, 7, 9, 11)  (8, 10, 11, 13) 

(14, 16, 18, 20)  (9, 11, 13, 14)  (13, 14, 15, 16) 
Unit penalties of transportation for item 1 in the second objective.
Conveyance 


Sources  Destinations  


 

(4, 5, 7, 8)  (3, 5, 6, 8)  (7, 9, 10, 12) 

(6, 8, 9, 11)  (5, 6, 7, 8)  (6, 7, 9, 10) 
Conveyance 


Sources  Destinations  


 

(6, 7, 8, 9)  (4, 6, 7, 9)  (5, 7, 9, 11) 

(4, 6, 8, 10)  (7, 9, 11, 13)  (9, 10, 11, 12) 
Unit penalties of transportation for item 2 in the second objective.
Conveyance 


Sources  Destinations  


 

(5, 7, 9, 10)  (4, 6, 7, 9)  (9, 11, 12, 13) 

(10, 11, 13, 14)  (6, 7, 8, 9)  (7, 9, 11, 12) 
Conveyance 


Sources  Destinations  


 

(7, 8, 9, 10)  (4, 5, 7, 8)  (8, 10, 11, 12) 

(6, 8, 10, 12)  (5, 7, 9, 11)  (9, 10, 12, 14) 
Availability and demand data.
Fuzzy availability  Fuzzy demand  Conveyance capacity 
















From Table
Similarly, for the second item, total availability
After calculating the rank for all these, we find that
The considered problem is now balanced and consists of two objectives, two items, two sources, four destinations, and three different modes of transportation. The number of constraints is
Forming the crisp model by using the ranking technique as explained in Step 3.1, the model is as shown in the following.
Solving this problem using fuzzy programming technique, obtained results are as follows:
If a MOMISTP in which availability of one or more items is less than the corresponding demand and/or total conveyance capacity is less than total demand is solved by the method presented in [
Availability and demand data.
Fuzzy availability  Fuzzy demand  Conveyance capacity 
















We find that
After balancing the problem and forming the expected value model using ranking technique the obtained problem is shown as follows.
Solving this problem using fuzzy programming technique, the obtained results are as follows:
Similarly, other problems, in which availability of all the items is less/more than its demand and/or the total conveyance capacity is less/more than the total availability or total demand, can be solved by the proposed method.
The above models are solved with the help of MAPPLE software. Kundu et al. [
Results using the existing method [
Example  Optimal compromise solution 



Ranking method  
1 



2  Infeasible solution  —  — 


Minimum of fuzzy number  
1 



2  Infeasible solution  —  — 
Results using the proposed method.
Example  Optimal compromise solution 



Ranking method  
1 



2 





Minimum of fuzzy number  
1 



2 



The objective values found using the concept of minimum of fuzzy numbers for Example 1 (Section
Optimal value of
Optimal value of
From the membership function of the objective function
According to the decision maker minimum transportation cost for the transportation will be greater than 731.18 units and will be less than 1202.49 units.
The maximum chances are that the minimum transportation cost will lie in the range 872.37–1053.68 units.
The overall level of satisfaction for other values of the minimum transportation cost (say
The results of Example 2 (Section
Our results show that, unlike [
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are thankful to the reviewers for their valuable comments and suggestions, which improved the presentation of the paper. The first author is also thankful to CSIR, Government of India, for providing financial support.