Novel Russell's approximation method for transportation problem under hesitant bifuzzy environment

Ismat Beg; Soniya Gupta; Arpa Ghosh; Dheeraj Kumar Joshi

doi:10.55976/dma.4202614771-9

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Novel Russell's approximation method for transportation problem under hesitant bifuzzy environment

Ismat Beg ¹ ^* , Soniya Gupta ² , Arpa Ghosh ³ , Dheeraj Kumar Joshi ⁴

1. Lahore School of Economics, Lahore 53200, Pakistan
2. School of Physical Sciences, DIT University Dehradun, Uttarakhand, 248009, India
3. School of Physical Sciences, DIT University Dehradun, Uttarakhand, 248009, India
4. School of Physical Sciences, DIT University Dehradun, Uttarakhand, 248009, India

Correspondence: ibeg@lahoreschool.edu.pk

DOI: https://doi.org/10.55976/dma.4202614771-9

Received

03 November 2025
Revised

10 December 2025
Accepted

16 December 2025
Published

30 December 2025

RAM Hesitant bifuzzy set Fuzzy set Transportation problem

Abstract

Transportation problem is one of the significant linear programming problems that emerges in several conditions and gains attention of many researchers. Reducing a commodity's transportation expense to meet demand at destination is the goal of the transportation problem. However, uncertainty plays a vital role in real life, making it challenging for decision-makers to provide precise values for the coefficients related to the transportation problem. Hesitant bifuzzy sets are a notable advancement of fuzzy set theory, as they allow decision-makers to deal with every hesitancy without any restriction. This study aims to introduce a formulation of transportation problem under a hesitant bifuzzy environment. In this study, we have introduced a novel Russel approximation method (RAM) for the hesitant bifuzzy transportation problem, where every parameter is represented by hesitant bifuzzy elements. To show the applicability of the proposed method, a real-life illustration has been taken in this article. Additionally, to demonstrate the superiority of the proposed method, a comparative study has also been conducted with other existing methods, which results in the minimum transportation cost.

<h5>1. Introduction</h5>
Linear programming problems have a significant impact on our daily lives. Transportation problems (TP) are among the most important linear programming problems. They play a significant role in developing strong client relationships in a skillful manner.  Transportation problems encounter the challenges of delivering reliable, secure, and efficient transportation while reducing the negative effects on the environment and local populations. The primary goal of TP is to reduce transportation costs from sources to multiple destinations.  TP was primarily developed in 1941 by Hitchcock <a href="#reference_list_0">[1]</a>. Thereafter, Dantzig <a href="#reference_list_1">[2]</a> conducted research and provided various methods to solve TP. Several methods have been developed to obtain an Initial Basic Feasible Solution (IBFS) namely, North-West corner, least cost method, Raw-minima method, Column-minima method, and Vogel’s Approximation Method (VAM). The modified distribution (MODI) method and Stepping stone method are used to determine the most efficient transportation cost. In everyday life, optimization tasks are often imprecise and vague in nature. To deal with vagueness and uncertainty in our daily lives, Zadeh <a href="#reference_list_2">[3]</a> introduced the concept of fuzzy sets. Since then, many extensions have been made by Atanassov <a href="#reference_list_3">[4,5]</a>, Chaube et al. <a href="#reference_list_5">[6]</a>, Torra <a href="#reference_list_6">[7]</a>, and Zhu et al. <a href="#reference_list_7">[8]</a>.
Fuzzy transportation problems (FTP) are problems in which the parameters of transportation are given by fuzzy numbers. It has been predicted that fuzzy linear programming would be more consistent, as it can deal with uncertainty and vagueness in an efficient manner. Zimmermann <a href="#reference_list_8">[9]</a> developed the fuzzy optimisation method to solve TP. Afterwards, Oheigeartaigh <a href="#reference_list_9">[10]</a> suggested a method for addressing TP where supply and demand were given as fuzzy numbers. Later in 1984, Chanas et al. <a href="#reference_list_10">[11]</a> invented a fuzzy linear programming model with crisp co-efficient and fuzzy supply and demand values. Then Chanas and Kuchta <a href="#reference_list_11">[12]</a> proposed a technique to solve TP in which the terms of fuzzy coefficient were expressed as fuzzy numbers. And later Kaur & Kumar<a href="#reference_list_12">[13]</a> also suggested an algorithm to address FTP. Afterwards, the fuzzy transportation problem has gained the attention of many researchers, and several methods to generalize fuzzy numbers and to solve transportation problems in various methods have been developed (Gupta et al. <a href="#reference_list_13">[14,15]</a>; Liu & Kao <a href="#reference_list_15">[16]</a>; Pandian & Natarajan <a href="#reference_list_16">[17]</a>; Singh <a href="#reference_list_17">[18]</a>. Several methods were proposed to solve transportation problems under different fuzzy environments (Pandian & Natarajan <a href="#reference_list_16">[17]</a>; Singh <a href="#reference_list_17">[18]</a>). In 2010, Pandian and Natarajan <a href="#reference_list_16">[17]</a> introduced the zero-point method to solve FTP. The Russell’s Approximation Method (RAM) was introduced by Hasibuan <a href="#reference_list_18">[19]</a> for solving basic TP. Since, the RAM for the hesitant bifuzzy transportation problem (HBFTP) has not yet been developed in the literature, there is a need to develop a novel Russel’s Approximation Method (RAM) for getting an initial basic feasible solution (IBFS) for hesitant bifuzzy TP.
In this work, we develop methods for HBFTP by taking into account HBFNs. The main contributions of this study are as follows:
<ul>
<li>A novel RAM-based approach has been developed to improve the initial basic feasible solution (IBFS) of transportation problems.    </li>
<li>An algorithm for the Generalized Hesitant Bifuzzy Modified Distribution Method has been formulated.</li>
<li>A practical real-life case study has been conducted to demonstrate the applicability, feasibility, and effectiveness of the proposed RAM and Generalized Hesitant Bifuzzy MODI techniques in solving transportation-related problems.</li>
<li>A comparative analysis with existing methods confirms the superior robustness and performance of the developed approach, establishing its potential for broader use in optimization and transportation research.</li>
</ul>
The remainder of the article is structured as follows: primary definitions of HBFSs and the bifuzzy score function are given in section 2. In section 3, the problem formulation is defined. Section 4 presents the rules and steps of RAM for HBFTP. In section 5, a practical problem has been taken to demonstrate the relevance of the suggested method in a real-life situation. In section 6, a comparative study was conducted to show the superiority of the suggested method over other existing methods. Concluding remarks and future scope are given in section 7.   
<h5>2. Preliminaries</h5>
The primary definition of hesitant bifuzzy set and the hesitant bifuzzy score function are given in this section.
2.1 The definition of fuzzy set by Zadeh <a href="#reference_list_2">[3]</a>
If X  is a collection of objects denoted by x , then a fuzzy set Ã in X is a set of ordered pair
<img src="https://file.luminescience.cn/69523f748d25f1766997876" alt="" width="180" height="38" />
                                                                                                               (1)
μÃ (x) is called the membership function (generalized characteristic function) which maps X to the membership space and μÃ (x) ∈ <a href="#reference_list_-1">[0,1]</a>.
2.2 The definition of hesitant fuzzy set by Torra <a href="#reference_list_6">[7]</a>
On the universal set X, defined hesitant fuzzy set as:
<img src="https://file.luminescience.cn/69524081230881766998145" alt="" width="180" height="28" />
                                                                                                                  (2)
where the sets g(x) have value in <a href="#reference_list_-1">[0,1]</a>, which signifies the possible membership value for a set and for degrees of x ∈ X .
2.3 The definition of hesitant bifuzzy set by Chaube et al. <a href="#reference_list_5">[6]</a>
Definition 1. Let X be universe of discourse, a hesitant bifuzzy set defined on X is given by
  <img src="https://file.luminescience.cn/6952410bcbb591766998283" alt="" width="250" height="33" />                                                
                                                                                                                     (3)
where hH (x)  and gH (x) ∈<a href="#reference_list_-1">[0, 1]</a>, represents a set of possible membership degree (MD) and a possible non-membership degree (NMD) of x ∈ X to the set H , respectively, with the conditions: 0 ≤ α, β ≤ 1, 0 ≤ α+ + β+ < 2,  where α ϵ hH(x), β ϵ gH(x) , <img src="https://file.luminescience.cn/6952436471a9f1766998884" alt="" width="210" height="24" />  and  <img src="https://file.luminescience.cn/6952459e763041766999454" alt="" width="210" height="25" /> for all x ∈ X. Here, bH(x) = (hH(x), gH(x)) represents an HBFE denoted by bH = (hH, gH) ,  with the condition α ϵ hH , β ϵ gH , <img src="https://file.luminescience.cn/6952436471a9f1766998884" alt="" width="210" height="24" />, and  <img src="https://file.luminescience.cn/6952459e763041766999454" alt="" width="210" height="25" />,  0 ≤ α, β ≤ 1 and 0 ≤ α+ + β+ < 2.
2.4 The definition of bifuzzy score function by Chaube et al. <a href="#reference_list_5">[6]</a>
Definition 2. Let B = {〈x, hH(x), gH(x)〉: x ϵ X } be a hesitant bifuzzy set. The bifuzzy score function is given by
<img src="https://file.luminescience.cn/6952477daa6f61766999933" alt="" width="400" height="73" />
                                                                                                                        (4)
Where, #h and #g are the total number of components in the MD and NMD set respectively. With the help of bifuzzy score function, we converted hesitant bifuzzy element (HBFE) into a conflicting bifuzzy element (CBFE). After aggregating the HBFE and converting into CBFE with the help of score function, we defuzzified the elements with S-operator <a href="#reference_list_8">[9]</a>.
The S-operator is given by
<img src="https://file.luminescience.cn/69524803c71701767000067" alt="" width="130" height="37" />
                                                                                                                 (5)
Where HBFN is in the form 〈 x,μx,ϑx 〉 of a CBFN.
<h5>3. Model formulation</h5>
Uncertainty plays a vital role in our daily life. Therefore, the exact amount of the transportation cost cij may not be known. Studying OR using HBFS is important in several ways. In reality, there are numerous fields to be studied under the HBF environment, such as inventory and sustainable supplier selection (Gupta & Joshi <a href="#reference_list_19">[20]</a>). In this study, taking into account the incomplete data on transportation factors in TP, we have used HBFS to tackle the real-life situation. Since the customer bears the cost of transportation, it is necessary to keep the expense of transportation to a minimum. Hence, the hesitant bifuzzy cost (cij ) is presented in order to create the mathematical model of the transportation problems.
The HBFTP is given as follows:
<img src="https://file.luminescience.cn/6952493a6b2bb1767000378" alt="" width="220" height="29" />
                                                                                                                       (6)
<img src="https://file.luminescience.cn/695249827793e1767000450" alt="" width="300" height="32" />
                                                                                                                        (7)
<img src="https://file.luminescience.cn/695249f812e851767000568" alt="" width="220" height="33" />
                                                                                                                        (8)
<img src="https://file.luminescience.cn/69524a397629b1767000633" alt="" width="100" height="29" />
                                                                                                                        (9)
Where, z denotes the objective function in Eq. (6) which was designed based on hesitant bifuzzy cost Cij (i =1,2,…,m; j =1,2,…,n). Cij is transportation cost for each unit of an item from ith origin to jth destination; ai (i =1,2,…m) and bj (j =1,2,…n) are interpreted as supply and demand respectively, and
<img src="https://file.luminescience.cn/69524b3049b1f1767000880" alt="" width="140" height="41" />
                                                                                                                  (10)
  
is the feasibility condition.
The cost Cij is usually considered to be essentially deterministic. According to real life situation, we designed the problem based on hesitant bifuzzy sets.  We took the hesitant bifuzzy cost as (hBij, gBij) for a corresponding route. In order to reduce the cost of transportation, DMs need to increase the MD hBij and decrease the NMD gBij.
<h5>4. Solution procedure of HBFTP by Russell's approximation method</h5>
In this section, we discussed the solution procedure of HBFTP with the help of Russell's Approximation Method (RAM).
4.1 Algorithm for Russell’s approximation method 
Step 1: Construct a hesitant bifuzzy cost function Cij by taking into account the routes' priority and the possibility of delivering goods along these routes. Convert the hesitant bifuzzy cost functioninto a crisp cost using the bifuzzy score function. 
Step 2: Check whether the problem is balanced, i.e., whether demand is equal to supply. If they are equal, proceed to the next step. If not, then we have to introduce a dummy row or column to balance the problem.
Step 3: Find the highest cost  (ui) of each row.
Step 4:  Find the highest cost (vj) of  the column.
Step 5:  For each variable in transportation problem obtain Δij = Cij - (ui + vj ).
Step 6:  Choose the variable that has the highest negative value of Δ and allocate as much as possible according to demand and supply. When either demand or supply becomes zero after allocation, cancel that row or column.
Step 7: Repeat steps 3 to 6 until all demand and supply goes to zero.
Step 8:  Then the initial hesitant bifuzzy basic feasible solution (IHBBFS) Cij  and basic hesitant bifuzzy cost is ⅀Cij. mij.
The algorithm for Russell’s approximation method is presented in Figure 1.
<img src="https://file.luminescience.cn/695344dd2dc011767064797" alt="" width="450" height="766" />
                                                                       Figure 1. Algorithm for Russell's approximation method
4.2 Generalized the hesitant bifuzzy modified distribution method
The hesitant bifuzzy modified distribution method is introduced in this section to obtain the optimal outcome of hesitant bifuzzy TP from IHBBFS by using the RAM.
Step 1:  Obtain IHBBFS of HBTP by using the RAM.
Step 2: Then write the HBF variables ui and vj in accordance with each row and column. At the head of each row, write ui ,  and at the bottom of each column, write vj , respectively.
Step 3: Set any ui or vj  to be zero where there is the  maximum number of allocations.
Step 4: Then calculate the Cij  = ui + vj (for every i and j) for each basic cell. From this, we can determine the values of ui and vj.
Step 5:  For each non-basic cell, calculate dij  = Cij - (ui + vj)(for every i and j , respectively). Then the following cases may arise:
Step 5.1:  If every dij  ≥ 0 ∀ i, j, then the obtained solution is optimal.
Step 5.2:  If any dij < 0 , then the resultant answer is not the optimal one. In this case, proceed to step 6.
Step 6: In the IHBBFS, choose dij which has the most negative score value. Let it be at cell (r,s) , where 1 ≤ r ≤ m and 1 ≤ s ≤ n .
Step 7:  After that, choose a quantity θ in the (r,s) cell with the most negative dij,  then make a loop:
Rule to define the loop: Starting with cell (r,s) , allocate the amount θ horizontally or vertically and move to the next basic cell, with the restriction that the entire loop cannot be contained in a non-basic cell (except for the θ cell). In this way, we reach the θ cells and complete the loop.
Step 8:  In the (r,s) cell, add the positive amount θ at the start of the loop, with alternating positive and negative signs at all endpoints to satisfy supply and demand constraints. Then, take the minimum value of θ  that will make non-negative values in the revised solution for each basic variable.
Step 9: Insert the new value of θ and get the new allocation for the transportation table. When the value of θ is inserted, a cell assumes a value of 0. The cell with a value of 0 will become non-basic. This improves the HBFBFS. Now again, set one row or column to be zero and calculate the value of Cij  = ui + vj .
Step 10:  Now repeat step 5. Calculate the value of dij  = Cij - (ui + vj) for new transportation table until all dij  ≥ 0 ∀ i, j.
Step 11: The obtained hesitant bifuzzy optimal solution is xij , and the hesitant bifuzzy optimal transportation cost is ⅀Cij. xij ∀ i,j.
<h5>5. Practical application of the hesitant bifuzzy transportation problem</h5>
A well-known business has three plants of electric appliances namely S1(Bulb), S2(Tube), and S3(Fan) and the electrical appliances are distributed to three markets D1(Dehradun), D2(Delhi), and D3(Haridwar), respectively. There are different routes to transport the appliances from the factories to the markets. Different transportation costs are assigned to each route. It is notified that customers prefer to minimize transportation costs. So, it is impossible for supplier to deliver materials to all customers due to vehicle capacity constraints. Consequently, hesitant bifuzzy sets are used to select the route. The company's primary goal is to reduce shipping costs through a TP under a hesitant bifuzzy environment.
Table 1 denotes the availability ai (i =1,2,3) of the electrical appliances at the three sources Si (i=1,2,3) and the demands bj (j =1,2,3) at the three destinations Dj (j = 1,2,3).. Now we determine the HBFTP and want to find out the optimal solution. According to the stepped algorithm, we first convert the hesitant bifuzzy number to crisp number by using the bifuzzy score function. Then, we use the RAM method to find the IBFS of the HBFTP.
<img src="https://file.luminescience.cn/6953696c36bf51767074156" alt="" width="500" height="265" />
Step 1: Convert the HBFTP into crisp values by using bifuzzy score function given in Eq. (4) and Eq. (5). Table 2 contains the HBFTP converted into crisp values.
Let B = {〈x,hH(x),gH(x): x ϵ X} be a hesitant bifuzzy set. The bifuzzy score function is given by  
<img src="https://file.luminescience.cn/69536a1c652411767074332" alt="" width="350" height="71" />
                                                                                                                 (11)
Here, #h and #g are the total number of components in the MD and NMD sets, respectively.
The HBFTP values in Table 1 are converted to fuzzy cost ranking values as follows:
<img src="https://file.luminescience.cn/69536ae58210a1767074533" alt="" width="350" height="200" />
 
Using <img src="https://file.luminescience.cn/69536b34622951767074612" alt="" width="120" height="43" />
<img src="https://file.luminescience.cn/69536b778d0b61767074679" alt="" width="180" height="77" />
<img src="https://file.luminescience.cn/69536bc6b61921767074758" alt="" width="500" height="218" />
Step 2: The given HBFTP is balanced in nature. Step 3: Find the highest cost ui of each row. Step 4: Find the highest cost vj of each column. The determined values of the row's highest cost (ui) and the column's highest cost (vj), as shown in Tables 3 and 4, are calculated as follows: ui = max (0.535, 0.512, 0.550) =0.550 vj = max (0.535, 0.565, 0.520) =0.565
<img src="https://file.luminescience.cn/69536dce387591767075278" alt="" width="500" height="163" />
<img src="https://file.luminescience.cn/69536e4a33f4e1767075402" alt="" width="420" height="187" />
Step 5: Compute the cost of Δij , where Δij = Cij − (ui+ vj), Δ11= C11 − (u11 + v11) = 0.535− (0.550+0.565) = −0.580 Δ12= C12 − (u12 + v12) = 0.512 − (0.550+0.585) = −0.623 Δ13= C13 − (u13+ v13) = 0.550 − (0.550+0.565) = −0.565 Δ21= C21 − (u21 + v21) = 0.565 − (0.585+0.565) = −0.585 Δ22= C22 − (u22+ v22) = 0.585 − (0.585+0.585) = −0.585 Δ23= C23 − (u23+ v23) = 0.530 − (0.585+0.550) = −0.605 Δ31= C31 − (u31+ v31) = 0.520 − (0.565+0.520) = −0.565 Δ32= C32 − (u32 + v32) = 0.515 − (0.585+0.520) = −0.590
The values of reduced cost to each Δij are given in Table 5.
<img src="https://file.luminescience.cn/695371164272c1767076118" alt="" width="600" height="190" />
Step 6: Select the variable with the most negative Δij  and select the minimum demand and supply. Allocate to the cell with the lowest unit cost.
The most negative Δij is (-0.623) in cell S2D2.
The most allocation to the cell is min (22,38) = 22.
This leaves the supply unchanged and completely meets the demand of 16 (38-22). The first allocation is given in Table 6.
<img src="https://file.luminescience.cn/695371a86b4891767076264" alt="" width="430" height="169" />
Step 7: Eliminate row S1 as demand becomes 0. Return to step 3 and repeat the procedure as many times as you can, ensuring the required cell is removed each time. The final allocations by RAM are given in Table 7.
<img src="https://file.luminescience.cn/695372bb468191767076539" alt="" width="430" height="172" />
d11=C11−(u1+v1)=0.535−(0.520−0.003) =0.018 d13=C21−(u1+v3)=0.550−(0.485−0.003) =0.068 d22=C22−(u2+v2)=0.585−(0.515+0.045) =0.025 d33=C33−(u3+v3)=0.512−(0.485+0) =0.027 Since all dij ≥0 ∀ i,j. Therefore, the obtained solution is optimal. The final allocations are x12= 22, x21= 8, x23= 22, x31= 28, and x32= 16. The minimum transportation cost is calculated as: Min =(22∗23+16∗28.66+22∗27.66+8∗25.66+28∗36) =2786.36 The minimum allocated cells are = m + n – 1 = 3+3-1 = 5, therefore, it is a non-de-generate solution.
<h5>6. Comparison to the existing methods</h5>
The transportation problem is a common decision-making problem in daily life. The aim of TP is to minimize the transportation costs while meeting the constraints on supply and demand. It is very important to develop a model that can minimize the transportation costs. In literature, many TP have been studied, and various methods have been introduced to solve them. In order to confirm the superiority of our proposed method, a comparative analysis is carried out with other existing methods. We have summarized all transportation costs by different existing methods in Table 8.
<img src="https://file.luminescience.cn/69537447e9c431767076935" alt="" width="650" height="171" />
From Table 8, we can conclude that the transportation cost obtained for HBFTP by RAM is the minimum. Table 8 validates the superiority of the developed method over other existing methods, as it gives the minimum optimal transportation cost. Additionally, the HBFIBFS obtained by RAM is also the minimum, therefore it can be used to obtain IBFS under a hesitant bifuzzy environment. The comparison to the existing methods is shown in Figure 2.
<img src="https://file.luminescience.cn/6953747ada8b61767076986" alt="" width="455" height="261" />
                                                                                      Figure 2. Comparison to the existing methods
<h5>7. Conclusion</h5>
Uncertainty has a significant impact on our daily lives; therefore, it is necessary to develop uncertain transportation problems that can address real-life decision-making problems. One such uncertain transportation problem is the fuzzy TP. Contrary to classical TPs under fuzziness, we have developed a new class of uncertain TPs, namely HBFTPs. In this study, we defined TP under a hesitant bifuzzy environment and formulated its mathematical model. To obtain the IBFS for HBFTP, an algorithm for RAM has also been designed. An algorithm to obtain the optimal solution for HBFTP has also been developed in this article. The effectiveness of the proposed method and the underlying model has been demonstrated using a numerical example. To illustrate the advantages of the developed method, a comparative study with other existing methods has also been conducted.  In the future, we may create the transportation models with fixed charges under a hesitant bifuzzy environment and solve them by using any other suitable approach. Our method may also be used to solve problems in supply chain management, inventory optimisation, and other operations research and economics-related fields.
<h5>Acknowledgement</h5>
The authors would like to thank the reviewers for their valuable suggestions to the improvement of this article.
<h5>Authors' contribution</h5>
Ismat Beg: Conceptualization, methodology, validation and editing. Soniya Gupta: Conceptualization, methodology and writing – original draft. Arpa Ghosh: Conceptualization, methodology and writing – original draft.  Dheeraj Kumar Joshi: Conceptualization, methodology, validation, supervision and writing – review & editing.
<h5>Conflict of interests</h5>
The authors declare that there is no conflict of interest.
<h5>Funding</h5>
No funding was received for this work
<h5>Availability of data and materials</h5>
The data sets generated and/or analysed during the current study are included in this article. And if more information is needed, it is available from the corresponding author on reasonable request.

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Beg, I., Gupta, S., Ghosh, A., & Kumar Joshi, D. (2025). Novel Russell’s approximation method for transportation problem under hesitant bifuzzy environment. Decision Making and Analysis, 4(1), 1–9. https://doi.org/10.55976/dma.4202614771-9

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Novel Russell's approximation method for transportation problem under hesitant bifuzzy environment

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Novel Russell's approximation method for transportation problem under hesitant bifuzzy environment

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