How to cite: P. T. Bantining N, B. Surarso, and Sutimin, “Comparison Between Zero Point and Zero Suffix 

Methods in Fuzzy Transportation Problems”, JMM, vol. 6, no. 1, pp. 38-46, May 2020. 

Comparison Between Zero Point and Zero Suffix Methods in 

Fuzzy Transportation Problems 
 

 

Pukky T. B. Ngastiti1, Bayu Surarso2, Sutimin3 

1Department of Mathematics, Universitas Billfath, tetralian@billfath.ac.id 

2Department of Mathematics, Universitas Diponegoro, bayusurarso@yahoo.com 

3Department of Mathematics, Universitas Diponegoro, sutimin@undip.ac.id 

 
doi: https://doi.org/10.15642/mantik.2020.6.1.38-46 

 

 
Abstrak: Persoalan transportasi membahas tentang masalah pendistribusian suatu barang dari 

sejumlah sumber kepada sejumlah tujuan dengan tujuan meminumkan biaya pengangkutan. Masalah 

transportasi fuzzy merupakan biaya transportasi, persediaan dan permintaan dengan kuantitas 

jumlah fuzzy. Tujuan dari penelitian ini adalah untuk mempelajari komparasi teoritis dan numerik 

antara metode zero-point dan metode zero suffix dalam menentukan solusi optimal pada biaya 

pengangkutan barang. Berdasarkan hasil komparasi didapatkan iterasi pada metode zero-point lebih 

besar dalam mencapai nilai optimal dibandingkan dengan metode zero suffix. Berdasarkan hasil 

perbandingan teoritis dapat disimpulkan bahwa proses menggunakan metode zero-suffix lebih 

pendek dalam menentukan solusi optimal yaitu 6 langkah daripada metode zero-point yaitu 11 

langkah. Untuk mencapai nilai optimal menunjukkan bahwa, untuk metode zero suffix terjadi iterasi 

pada langkah keenam, namun untuk metode zero-point iterasi terjadi pada tahap kesembilan. Hasil 

perbandingan numerik, kami menyimpulkan biaya distribusi dengan menggunakan dua metode 

adalah sama, berdasarkan permintaan dan penawaran diperoleh 7 kali iterasi dan 7 item alokasi 

untuk metode zero point, sedangkan 6 kali iterasi dan 7 item alokasi untuk metode zero suffix. 

Kata Kunci: Metode zero-point; Metode zero-suffix; Masalah transportasi fuzzy 

Abstract: Transportation is discussing the problems of distribution items from a source to a 

destination with an aim to minimize transportation costs. The problem of fuzzy transport is the cost 

of transportation, supply, and demand with a quantity of fuzzy. The purpose of the research is a 

study of a comparison of theories from the zero-point method and the zero-suffix method in 

determining the optimal solution on cost transportation. Based on the result of the theoretical 

comparison, it can be concluded that the process of using the zero-suffix method is shorter in 

determining an optimal solution in 6 steps than that of a zero-point method in 11 steps. For achieving 

the optimal value shows that for zero-suffix the method of occurrence iteration in the sixth step, but 

for the zero-point method the iteration occurs in the ninth step. The results in the numerical 

comparison we conclude the distribution cost using two methods is the same, based on the demand 

and supply obtained 7 times iteration and 7 items allocation for zero point method, while 6 times 

iteration and 7 items allocation for zero suffix method. 

 
Keywords: Zero-point method; Zero-suffix Method; Fuzzy transportation problems 

 

  

Jurnal Matematika MANTIK 

Vol. 6, No. 1, May 2020, pp. 38-46 

 

ISSN: 2527-3159 (print) 2527-3167 (online) 

http://u.lipi.go.id/1458103791


P. T. Bantining N, B. Surarso, and Sutimin 
Comparison Between Zero Point and Zero Suffix Methods in Fuzzy Transportation Problems 

 

39 

1. Introduction  

Many applications have in solving problems in everyday life. The parameters of the 

transportation problem consist of the amount of costs, supply, and demand [1]. Conditions 

that occur in the field of transportation problems of the value of an item often occur 

uncertainty and tend to change from time to time. This happens because, lack of 

information data related to the value of these problems [2]. Based on Zadeh introduces 

theory fuzzy and problems fuzzy that have been studied in relation to daily life [3]-[4]. The 

problem of transportation fuzzy is the amount of costs, supply, and demand of value fuzzy 

[5]-[6]. One method used for transportation problems fuzzy is the zero-point method and 

the method zero suffix to find the optimal solution for transportation costs.  

Based on research conducted by Annie Christi MS and Kumari Shoba. K gives the 

conclusion that using the robust ranking method with the zero-suffix method gets the 

optimal solution of the fuzzy problem accurately and effectively [7]. In the following year 

research Chandrasekaran, S, et al concluded that the transportation problem fuzzy using a 

heptagon fuzzy number with the zero-suffix method obtained optimal solutions in solving 

problems fuzzy [8]. Fegade, et al about the method, zero suffix it was concluded that the 

optimal solution by changing the problem  fuzzy to crisp using the method robust ranking 

so that the total fuzzy cost becomes optimal and more effective based on the example of 

the problem  fuzzy [9]. This research was supported by Nirmala. G and Anju R in their 

research concluded that the zero-suffix method produces an optimal solution with few 

iterations in transportation problems [10]. 

Based on research on the zero point method by Ismail Mohideen, S and Senthil Kumar 

P conclude that using the method zero point in multiplication operations is better than the 

Vogel's Approximation method and the method modified distribution in the distribution 

problem fuzzy [11]. P. Pandian and G. Natarajan, it is concluded that the zero-point method 

provides the optimal value of the objective function with the number fuzzy trapezoid for 

transportation fuzzy [12]. Subsequent research carried out by Pukky Tetralian, et al 

concluded that the results of research on CV. Bintang Elektrik Grace associated with 

transportation problems fuzzy obtained optimal solutions that the zero point method and 

zero suffix method are the same but in terms of the number of iterations the method zero 

point is greater in achieving the optimal solution [13]. This is in line with Samuel's research, 

Edward concluded that the method was improved zero point is an efficient and better 

method than the VAM, SVAM, GVAM, RVAM, BVAM methods in achieving optimal 

solutions to the transportation problem fuzzy [14]. Sharma Gaurav et al in his research 

concluded that the method zero point is a symmetrical procedure for transportation 

problems that is easily applied and utilized for all types of transportation problems with 

objective functions in the form of maximum or minimum values, to make decisions when 

there are various types of logistical problems and provide optimal solutions to 

transportation problems [15]. L. Sujatha, P. Vinothini and R. Jothilakshmi conclude that 

the procedure developed in this paper provides the optimal fuzzy solution and the optimal 

fuzzy objective value which are non-negative fuzzy numbers, hence the method developed, 

serve as an important tool for the decision maker while handling the transportation problem 

under fuzzy environment [16]. P. K Senthil conclude that obtained an optimal solution to 

the type-2 IFTP without using the basic feasible solution and the method of testing 

optimality. The main advantage of this method is that the obtained solution is always 

optimal, and it is not required to have (m + n – 1) allotted entries. In feature, the proposed 

method may be modified to find intuitionistic fuzzy optimal solution of solid intuitionistic 

fuzzy transportation problems and solid assignment problems with IFNs [17]-[18]. 

The purpose of this research is to study the theoretical and numerical comparisons of 

transportation problems fuzzy with the zero-point method and zero suffix method. 



Jurnal Matematika MANTIK 

Volume 6, Issue. 1, May 2020, pp. 38-46 

40 

2. Methods 

Following are different theoretical and numerical comparisons obtained from the zero-

point method and zero suffix method. 

2.1. Methods north west corner  

The methods North West Corner used to calculate solution a feasible of the 

transportation problem. Following are the steps to get a solution feasible [19]. 

Step 1:  

Allocate and select cells in the upper left corner on transportation issues. This cell must be 

allocated as many units as possible. the unit must be the same as the minimum requirements 

between demand and available inventory. 

Step 2:  

The next step is to adjust the number of requests and supplies in the allocated row and 

column. The first cell in the second row will only continue, if the first row's supply has run 

out. Likewise, the next cell in the second column will also be continued with the condition 

that the request for the first cell has been fulfilled. 

Step 3:  

For allocated cells, each supply of cells is equal to demand, the next allocation can be made 

in the next row or column. This procedure is repeated until the full allocation of the number 

of successful units according to the cells in need.  

 

2.2. Methods zero point [20] 

Step 1: 

Establish a transport table fuzzy transport problem fuzzy is then converted into transport 

table fuzzy balanced if it is not balanced.  

Step 2: 

Reduce each row in the transportation table fuzzy from the minimum row. 

Step 3: 

Reduce each column in the transportation table results fuzzy in step 2 of the minimum 

column. 

Step 4: 

Check if each column is fuzzy demand (𝑏𝑗 ) less than the total fuzzy inventory (𝑎𝑖 ) resulting 

from cost reduction in the zero-value column. Also check if each line is fuzzy in inventory 

(𝑎𝑖 ) less than the number of columns fuzzy request (𝑏𝑗 ) where row reduction is zero(𝑐𝑖𝑗 =

0). If fulfilled, continue to step 7. If it is not met, then go to step 5.  
Step 5: 

Closing the minimum value with horizontal lines and vertical lines that are zero from the 

reduction results of the transportation table fuzzy so that some rows or columns that do not 

meet the requirements in step 4 are not closed.   

Step 6: 

Form a transportation improvement table fuzzy follows: 

a. Find the smallest value from the reduction of the cost matrix fuzzy that is not 

covered by a line. 

b. Subtract the smallest reduced cost  to all costs  that are not covered and add 

costs  covered by two intersection lines, then return to step 4. 

Step 7: 

Selecting the reduction cell in the transportation table fuzzy that has the largest reduced 

cost, is called . If there are more than one cell, then choose just one. 

ij
w

ij
w

ij
c

ij
c

),( 



P. T. Bantining N, B. Surarso, and Sutimin 
Comparison Between Zero Point and Zero Suffix Methods in Fuzzy Transportation Problems 

 

41 

Step 8: 

Select cells in the row - or column - in the reduced transportation table fuzzy- in 

which the cost-reduction cell is zero cij = 0 and the maximum allocation to the cell. If the 

cell does not appear at the maximum value, find another maximum value so that the other 

maximum values will be met. If there is no value in the cell that appears, in the 

transportation table fuzzy reduced the reduction cost is zero.   

Step 9: 

Reform the transportation table fuzzy after it is deleted in the supply row and demand 

column. 

Step 10: 

Repeat steps 7 through 9 until inventory fuzzy and demand fuzzy are fulfilled. 

Step 11: 

Allocation produce solutions fuzzy on transportation fuzzy matters. 

 

2.3. Method zero suffix [21]-[22] 

Step 1:  

Build a transportation table. 

Step 2:  

Subtract each row entry from the transportation table from the corresponding minimum 

row after that subtract each column entry from the transportation table to the appropriate 

minimum column. 

Step 3:  

In the cost matrix there will be at least one zero in each row and column, then look for 

suffix values that are denoted by S. S = Add cost from the nearest side zero which is greater 

than zero/ additional cost 

Step 4:  

Select the maximum value of S, if it has one maximum value. if you have two or more of 

the same value then choose one, then the cost becomes the allocation of goods with due 

regard to demand and supply. 

Step 5:  

After step 4, select the minimum {ai, bj} then allocate it to the transportation table. The 

resulting table must have at least one cost worth 0 in each row and column, otherwise repeat 

step 2. 

Step 6:  

Repeat step 3 through step 5 until the optimal cost is obtained. optimal cost is obtained if 

the column or row is saturated (suffix values = 0). 

3. Results and Discussions  

3.1.  Theoretical comparison of zero-point method and zero suffix method  

Step 1 of the two methods is in the first step of the two methods forming the 

transportation table into a transportation fuzzy balanced, so zero suffix method and zero-

point method have the same steps in step 1. 

Step 2 of the two methods is to reduce rows and columns by reducing the minimum 

value in zero suffix method while zero point method in step 2 only reduces rows, so in step 

2 method zero suffix has occurred in column and row reduction. 

Next step 3 of the two methods is the zero suffix method there will be at least one zero 

in rows and columns to get the suffix value (S) while zero point method reduces the column, 

so zero suffix method in step 3 looks for the value suffix and zero point only reduces 

column. 

 



Jurnal Matematika MANTIK 

Volume 6, Issue. 1, May 2020, pp. 38-46 

42 

Step 4 of the two methods is zero suffix method with the maximum S value being the 

allocation of goods by taking into account the amount of supply and demand and zero point 

method checks each column with the condition that demand is less than inventory and 

checks the row with the condition that inventory is less than demand on the row and column 

zero value, so zero suffix method selects the maximum S value and zero point method 

checks the supply and demand requirements. 

While step 5 of the two methods is zero suffix method the minimum inventory or 

demand is selected and then allocated to the transportation table. While zero point method 

closes the minimum horizontal and vertical rows or columns that are not met in step 4 are 

not covered, so zero suffix method in step 5 will go to the iteration process and zero point 

method meets the requirements of step 4. 

Step 6 of the two methods is zero suffix method is iterated to get the optimal value 

where all column or row values are zero. while zero point method still performs a 

transportation improvement table with several iterations until each column with the 

condition that demand is less than inventory and rows with the condition that inventory is 

less than demand on rows and columns has zero value, so that step 6 zero suffix method  

has occurred iterating and zero point method is still form a repair table. 

Next step 7 to step 11 zero-point method is iterated to get the optimal solution value. 

3.2. Numerical Comparison of the zero-point method and zero suffix method  

Following is a numerical example of fuzzy transportation problem: 
 

Table 1. Transportation Fuzzy Full 
 

Destination Inventory 

 

 

 

Source 

 1 2 3 4 5  

1 (5,7,8,11) (1,6,7,12)122 (2,4,5,7) (2,5,7,9) (7,9,10,12) (20,35,45,60) 

2 (5,8,9,12) (2,5,7,9) (1,6,7,12) (5,7,8,11) (5,8,9,12) (15,25,35,45) 

3 (1,6,7,12) (5,8,9,12) (7,9,10,12) (1,6,7,12) (2,5,7,9) (10,15,25,30) 

4 (2,5,7,9) (5,7,8,11) (5,7,8,11) (5,8,9,12) (1,6,7,12) (5,8,12,15) 

Demand (15,25,25,45) (15,25,35,45) (8,14,16,22) (10,15,25,30) (2,4,6,8)  

From transportation problems fuzzy, then changed to the transportation problem crisp using 

robust ranking method [23]. 

𝑅(�̃�) = ∫ (0,5)(𝛼𝛼
𝐿 ,

1

0

𝛼𝛼
𝑈 )𝑑𝛼 

Where: 

𝛼𝛼
𝐿 , 𝛼𝛼

𝑈  = {(𝑎2 − 𝑎1)𝛼 + 𝑎1, 𝑎4 − (𝑎4 − 𝑎3)𝛼} 
R(�̃�11)  = R (5, 7, 8, 11) 

               = ∫ (0,5)(2
1

0

𝛼 + 5 + 11 − 3𝛼)𝑑𝛼 

               = ∫ (−0,5𝛼
1

0

+ 8)𝑑𝛼 

             = 7,75 
 

So that:  

𝑅(�̃�12) = 6,5, 𝑅(�̃�13) =4,5, 
𝑅(�̃�14) = 5,75𝑅(�̃�15) =9,5 
𝑅(�̃�21) = 8,5, 𝑅(�̃�22) =5,75,  
𝑅(�̃�23) = 6,5, 𝑅(�̃�24) =7,75,𝑅(�̃�25) =8,5 

 To 
From 



P. T. Bantining N, B. Surarso, and Sutimin 
Comparison Between Zero Point and Zero Suffix Methods in Fuzzy Transportation Problems 

 

43 

𝑅(�̃�31) = 6,5, 𝑅(�̃�32) =8,5, 𝑅(�̃�33) =9,5 
𝑅(�̃�34) = 6,5𝑅(�̃�35) =5,75 
𝑅(�̃�41) = 5,75, R(c̃42) =7,75, 𝑅(�̃�43) =7,75 
,𝑅(�̃�44) = 8,5𝑅(�̃�45) =5,75 
 

Inventory column: 

R(ã11) = 40R(ã12) =30,  
𝑅(�̃�13) = 20, R(ã14) =10 
 

Request line: 

R(b̃11) = 30R(b̃12) =30, R(b̃13) =15,  

R(b̃14) = 20,R(b̃15) =5 
Next, enter table transportation crisp. 

 

Table 2. Transportation Crisp 
 

Destination Inventory 

Source 

 1 2 3 4 5  

1 7,75 6,5 4,5 5,75 9,5 40 

2 8,5 5,75 6,5 7,75 8,5 30 

3 6,5 8,5 9,5 6,5 5,75 20 

4 5,75 7,75 7,75 8,5 6,5 10 

Demand 30 30 15 20 5  

From table 2, we get the total inventory and demand: 

 ∑ �̃�𝑚𝑖 = 100  
∑ �̃�𝑛𝑗 = 100  

Can be concluded that  ∑ �̃�𝑚𝑖 = ∑ �̃�
𝑛
𝑗 , So the transportation problem is balanced. 

Next calculate the solution feasible using north west corner method  

 

Table 3. Solution feasible method north west corner 
 

Destination Inventory 

Source 

 1 2 3 4 5  

1 7,75 

(30) 

6,5 

(10) 

4,5 

 

5,75 

 

9,5 - 

2 8,5 

 

5,75 

(20) 

6,5 

(10) 

7,75 8,5 - 

3 6,5 

 

8,5 9,5 

(5) 

6,5 

(15) 

5,75 

 

- 

4 5,75 

 

7,75 7,75 8,5 

(5) 

6,5 

(5) 

- 

 - - - - -  

Thus, transportation total costs  

= 7,75(30) + 6,5(10) + 5,75(20) + 6,5(10) + 9,5(5) + 6,5(15) + 8,5(5) + 6,5(5) 

= 697,5. 

 

Then the optimal solution will be calculated using zero-point method. 
 

  

From  

From  

  
To 

 To 



Jurnal Matematika MANTIK 

Volume 6, Issue. 1, May 2020, pp. 38-46 

44 

Table 4. Optimal solutions for zero-point method 
 

Destination Inventory 

 

 

 

Source 

 1 2 3 4 5  

1 7,75 

(5) 

6,5 4,5 

(15) 

5,75 

(20) 

9,5 40 

2 8,5 5,75 

(30) 

6,5 7,75 8,5 30 

3 6,5 

(15) 

8,5 9,5 6,5 5,75 

(5) 

20 

4 5,75 

(10) 

7,75 7,75 8,5 6,5 10 

Request 30 30 15 20 5  

Allocation from the method is obtained zero-point,  

𝑥11 = 5, 𝑥13 = 15, 𝑥14 = 20 
𝑥22 = 30, 𝑥31 = 15, 𝑥35 = 5 
𝑥41 = 10 
so that the minimum transportation cost is to 

𝑍   = ∑ ∑ 𝑐𝑖𝑗 𝑥𝑖𝑗

𝑛

𝑗

𝑚

𝑖

  

= 𝑐11(𝑥11) + 𝑐13(𝑥13) + 𝑐14(𝑥14) + 𝑐22(𝑥22) + 𝑐31(𝑥31) + 𝑐35(𝑥35) + 𝑐41(𝑥41) 
= 7,75(5) + 4,5(15) + 5,75(20) + 5,75(30) + 6,5(15) + 5,75(5) + 5,75(10) 
= 577,5 

 

Calculate the optimal solution using zero suffix method. 

 
Table 5. Optimal solutions to zero suffix method 

 

Objective Inventory 

 

 

 

Source 

 1 2 3 4 5  

1 7,75 

(5) 

6,5 4,5 

(15) 

5,75 

(20) 

9,5 40 

2 8,5 5,75 

(30) 

6,5 7,75 8,5 30 

3 6,5 

(15) 

8,5 9,5 6,5 5,75 

(5) 

20 

4 5,75 

(10) 

7,75 7,75 8,5 6,5 10 

Requests 30 30 15 20 5  

 

Allocation is obtained from the method zero suffix: 

𝑥11 = 5, 𝑥13 = 15, 𝑥14 = 20 
𝑥22 = 30, 𝑥31 = 15, 𝑥35 = 5 
𝑥41 = 10 
Thus, the minimum transportation cost is:  

𝑍  = ∑ ∑ 𝑐𝑖𝑗 𝑥𝑖𝑗

𝑛

𝑗

𝑚

𝑖

 

= 𝑐11(𝑥11) + 𝑐13(𝑥13) + 𝑐14(𝑥14) + 𝑐22(𝑥22) + 𝑐31(𝑥31) + 𝑐35(𝑥35) + 𝑐41(𝑥41) 
= 7,75(5) + 4,5(15) + 5,75(20) + 5,75(30) + 6,5(15) + 5,75(5) + 5,75(10) 
= 577,5 

 

 

 

From  
To 

To 
From 



P. T. Bantining N, B. Surarso, and Sutimin 
Comparison Between Zero Point and Zero Suffix Methods in Fuzzy Transportation Problems 

 

45 

From the results of the comparison of zero-point methods and zero suffix methods 

then arranged into a table: 
 

Table 6. Comparison Results of Zero-Point Method and Zero Suffix Method 

Method Name 
The number of 

iterations 
Allocation 

Optimum 

Solution 

Zero Point 

method 
7 

𝑥11 = 5, 𝑥13 = 15, 
𝑥14 = 20, 𝑥22 = 30, 
𝑥31 = 15, 𝑥35 = 5, 
𝑥41 = 10 

577,5 

Zero suffix 

method 
6 

𝑥11 = 5, 𝑥13 = 15, 
𝑥14 = 20, 𝑥22 = 30, 
𝑥31 = 15, 𝑥35 = 5, 
𝑥41 = 10 

577,5 

 

4. Conclusions 

The results of the comparison of zero point and zero suffix methods concluded that in 

the sixth step the theoretical comparative zero suffix method, has happened iteration  while 

the method in the ninth step just iterates so that zero point method is greater in achieving 

optimal values than zero suffix method. In numerical comparison the number of allocations 

and optimal solutions in both methods is equal, while the number of iterations of zero-point 

method is greater than the zero-suffix method. 

 

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