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                Mathematics and Statistics 10(4): 741-746, 2022                                                              http://www.hrpub.org 
                DOI: 10.13189/ms.2022.100404 
                         Analysis of IBFS for Transportation Problem by   
                                                      Using Various Methods 
                                                               S. K. Sharma, Keshav Goel* 
                                   Department of Mathematics, Chandigarh University, Gharuan, Mohali, 140413, Punjab, India 
                                        Received November 17, 2021; Revised February 15, 2022; Accepted March 16, 2022 
                Cite This Paper in the following Citation Styles 
                (a):  [1]  S. K. Sharma, Keshav Goel, "Analysis of IBFS for Transportation Problem by Using Various Methods," 
                Mathematics and Statistics, Vol. 10, No. 4, pp. 741 - 746, 2022. DOI: 10.13189/ms.2022.100404. 
                (b): S. K. Sharma, Keshav Goel (2022). Analysis of IBFS for Transportation Problem by Using Various Methods. 
                Mathematics and Statistics, 10(4), 741 - 746. DOI: 10.13189/ms.2022.100404. 
                Copyright©2022 by authors, all rights reserved. Authors agree that this article remains permanently open access under the 
                terms of the Creative Commons Attribution License 4.0 International License 
                Abstract  The supply, demand and transportation cost               1. Introduction
                in  transportation problem cannot be obtained by all 
                existing methods directly. In the existing literature,               TP is the important type of LPP for solving routing 
                various methods have been proposed for calculating  problems. It gives the supply of any object from the various 
                transportation cost.  In this paper, we are comparing  supply sources to the diverse sink of mandate in a manner 
                various methods for measuring the optimal cost. The  that the entire transportation charge would be minimum. In 
                objective of this paper is obtaining IBFS of real-life             operation research, transportation problem is most 
                problems by various methods. In this paper, we include  essential application in the field of LPP. There are lots of 
                various methods such as AMM (Arithmetic Mean Method),  development in different areas of transportation such as 
                ASM (Assigning Shortest Minimax Method) etc. The  shipping, networking etc. The transport problem is to 
                Initial Basic Feasible solution is one of the most important       transport a single homogeneous good, which is mainly 
                parts for analyzing the optimal cost of  transportation            stored in different places of origin to different destinations, 
                Problem. For many applications of transportation problem           so the total transport costs are minimal. There is a 
                such as image registration and wrapping, reflector design          challenge to introduce a new method for IBFS. Due to 
                seismic tomography and reflection seismology etc, we  traffic and hike of fuel prices in day-to-day life is very 
                analyze the transportation cost. TP is used to find the best       challenging to all humans.  TP was firstly expressed as 
                solution in such a way in which product  produced at  supply of a product from various sources to numerous 
                several sources (origins) are supply  to the various  destinations by Hitchcock and Koopmans [1], [2], found 
                destinations. To fulfil all requirement of destination at  Optimum consumption of the transportation classification. 
                lowest cost possible is the main objective of a  For the development of various methods, these papers are 
                transportation problem. All transport companies are  the milestones for solving transportation problem. Simplex 
                looking  forward  to  adopting  a new approach for  method is  given by G.B. Danzing in 1995 to solve the 
                minimizing the cost. Along these lines, it is essential just as    transportation problem for LPP then it takes a big number 
                an adequate condition for the transportation problem to  of variables, constraints, and take some time for solving the 
                have an attainable arrangement. A numerical example is  problem. Some researchers developed different methods 
                solved by different approaches for obtaining IBFS.                 for finding an IBFS which takes costs into account. There 
                Keywords  TP, LPP, IBFS, LCM, Optimization  are some methods namely, (LCM) Least cost Method, 
                Problem                                                            (VAM) Vogel’s Approximation Method, NW Corner 
                                                                                   Method, Row Minima Method, Column Minima Method 
                                                                                   for obtaining the IBFS of a transportation problem. There 
                                                                                   are many applications of transportation problem such as 
                                                                                   image registration and wrapping, reflector design seismic 
                                 742                                                             Analysis of IBFS for Transportation Problem by Using Various Methods                                                                                                                                                  
                                                                                                                                                                             
                                 tomography and reflection seismology etc. Advancements                                                                                        j (j=1,  2,  3,  ……,  n)  be  Bj.  Transporting  the units 
                                 in data and correspondence innovations also, expanding  available from sources to destination has  some cost 
                                 rivalry, especially in the assembling area, have prompted                                                                                     known as  transportation cost represented by cij. The 
                                 the requirement for viable and modest conveyance of crude                                                                                     intention of transported the number of items from source i 
                                 materials, work in progress, completed items or related  to destination j so that the total cost of transportation 
                                 data from starting place to end of utilization. This need can                                                                                 should be minimum. In accumulation, the limits of supply 
                                 be met specifically with the assistance of ideas for  at the origin and the demand at the destination required 
                                 everything identified with coordination. Now, coordination                                                                                    must be fulfill exactly. 
                                 as an answer for assembling organizations turns out to be                                                                                           If r (r ≥0) is the numeral of shipping items from basis i 
                                 more significant. However, control of administrations and                                                                                                  ij     ij
                                                                                                                                                                               to the end point j, the corresponding LPP is 
                                 activities, organization additionally offers a solid and                                                                                            Calculate r  (i=1,2, 3, ………, m; j=1, 2, 3, ……., n) in 
                                                                                                                                                                                                              ij
                                 prudent vehicle limit. Organization things might differ by                                                                                    sequence to   
                                 time and industry. Separation in necessities and innovation                                                                                         Minimize z =∑∑c r , 
                                                                                                                                                                                                                                  ij ij
                                 has prompted the way that organization related parts have                                                                                           subject to  ∑r  = P,  i=1,2,3,…………,m, 
                                 changed over the long run. However, transportation costs                                                                                                                             ij         i
                                                                                                                                                                                     and ∑r  = Q, j=1,2,3,………….,n, 
                                 have consistently been for most logistic companies. Linear                                                                                                          ij          j
                                                                                                                                                                                     where x ≥0. 
                                 programming is a method to complete the best result (such                                                                                                             ij
                                                                                                                                                                                     The two sets of restrictions will be reliable  i.e. the 
                                 as  supreme  profit  or  minimum  price).  A transportation  structure will be in steadiness if 
                                 problem is concerned with calculating the lowest cost of                                                                                                                                         ∑ Pi =  ∑Qj 
                                 transporting of a single product from a given quantity of 
                                 initial point to a specified quantity of destinations.                                                                                              In the event that this condition is satisfied, then we have 
                                       A feasible solution is called to be optimum solution if it                                                                              an achievable arrangement of the given transportation 
                                 reduces the transport price. The feasible solution is  problem. At that point we state that a transportation 
                                 supposed to be elementary if the number of allocations  problem will have a possible arrangement if and just if 
                                 like to m+n-1; that is one less than the number of rows                                                                                       ∑P= ∑Q will be fulfilled. The issue which fulfils this 
                                 and columns in a transportation problem. When the                                                                                                     i              j
                                                                                                                                                                               condition is called balanced  transportation problem. 
                                 number of belongings obtainable for shipment to the  Furthermore, the issue which do not fulfill this condition is 
                                 origins equals the request for belongings to the terminuses,  called unbalanced transportation problem. We can't locate 
                                 the transportation problem is titled as the balanced  the feasible solution of transportation problem if 
                                 transportation problem otherwise                                                                             unbalanced                                                                               ∑P ≠ ∑Q
                                 transportation problem. The constraint is a condition of an                                                                                                                                                  i              j 
                                 optimization problem that the solution must satisfy.                                                                                                Note that a transportation problem will have a doable 
                                                                                                                                                                               arrangement just if the above limitation is fulfilled. Along 
                                 2. Formulation of Transportation                                                                                                              these lines, it is essential just as an adequate condition for 
                                                                                                                                                                               the transportation problem to have an attainable 
                                         Problem                                                                                                                               arrangement. 
                                                                                                                                                                                     Problem that fulfills this condition are called adjusted 
                                       Let us consider the number of sources and destinations                                                                                  transportation problem. Where D  is the shipping cost, R
                                 are m and n respectively. Let the number of items for                                                                                                                                                                      ij                                                      ij 
                                                                                                                                                                               is the shipping quantity, P  is the supply available, and Q  
                                 supply existing at source i (i=1,2, 3, ……., m) be Ai and                                                                                                                                                   ij                                                                      ij
                                                                                                                                                                               is the destination demand. 
                                 let the demand of number of units necessary at destination 
                                                                                                                                                Table 1.    Formulation of TP 
                                                                                                                                                                                 DESTINATION 
                                                                                                                                      1                                  2                          3                      ……j….                                 n                     SUPPLY 
                                                                                                                                    D                                  D                          D                             D                              D
                                                                                                 1                                     11                                 12                         13                             1j                             1n                        P
                                                                                                                                     r                                  r                          r                             r                              r                               1 
                                                                                                                                      11                                 12                         13                             1j                             1n
                                                                                                                                    D                                  D                          D                             D                              D
                                                                                                 2                                     21                                 22                         23                             2j                             2n                        P
                                                                                                                                     r                                  r                          r                             r                              r                               2 
                                                                                                                                      21                                 22                         23                             2j                             2n
                                                                                                                                    D                                  D                          D                             D                              D
                                                    SOURCES                                      3                                     31                                 32                         33                             3j                             3n                        P
                                                                                                                                     r                                  r                          r                             r                              r                               3 
                                                                                                                                      31                                 32                         33                             3j                             3n
                                                                                                …i                                   …                                  …                          …                             …                                                            P 
                                                                                                                                                                                                                                                                 …                              i
                                                                                                                                   D                                   D                         D                              D                              D
                                                                                                 M                                     m1                                 m2                         m3                            mj                             mn                         P
                                                                                                                                    r                                  r                          r                              r                              r                               m 
                                                                                                                                      m1                                 m2                         m3                            mj                             mn
                                                                                           Demand                                    Q                                  Q                          Q                             Q                              Q                    ∑P=  ∑Q 
                                                                                                                                        1                                  2                          3                             j                              n                       i             j
                                        
                                                      Mathematics and Statistics 10(4): 741-746, 2022                                  743 
                                                                              
                  TP is used to find the best solution in such a way in                     Table 2.  Specific data of Column/Row 
               which product produced at several sources (origins) are                          P       P      P       P         Farm 
               supply to the various destinations. To fulfil all requirement                      1      2       3      4      Capability 
               of destination at lowest cost possible is the main objective           R1        19      30     50      10         7 
               of a transportation problem. For determining an optimum                R         70      30     40      60         9 
               result of a transport problem firstly find the IBFS and the             2
                                                                                     R3         40      8      70      20         18 
               IBFS can us find out by any of the methods like as NWCR,                  
               LCM – Method and VAM-Method etc. Quddoos [3] gave a                  Plant        5      8       7      14         34 
               new algorithm for solving transportation problems named            Constraint 
               the method as ASM-Method. A numerical is taken and the            Solution: Since we see that the transportation problem is 
               result efficiency of this method is also verified. The  balanced, we can find solutions to this transportation 
               importance of this method is that it is very easy to  problem by various methods to minimize costs. 
               recognize and also have less number of iterations. Das et       ASM-Method:  
               al.[4] talked about the drawbacks of VAM and introduced 
               a new method for explaining the transportation problem.                         Table 3.  IBFS by ASM Method 
               When biggest cost looks in two or more than two columns 
               or rows  then the VAM Method doesn’t give a logical                               P      P       P        P        Farm 
               solution. Then he gives a new method name as Logical                               1       2      3        4      Capacity 
                                                                                      F          19     30      50       10       7/2/0 
               Development of VAM- Method for the solution of highest                  1          (5)                   (2) 
               cost appearing in rows and columns. Kumar [5] provided                 F          70     30      40       60       9/2/0 
               the relative analysis of ASM Method and North West                      2                (2)     (7) 
               Corner Method for solving transport problem and then                   F          40      8      70       20      18/12/0 
               checked  the efficiency for lowest shipping cost. An                    3                (6)              (12) 
               innovative process for solving the transportation problem             Plant      5/0    8/6/0    7/0   14/12/0      34 
               has  discussed in [6]. The intention of this method is  to        Requirement 
               minimize the shipping cost. This method is solved by using             Z=19*5+10*2+30*2+40*7+8*6+20*12=743 
               the statistical tool called arithmetic Mean. The main  Row-Minima Method:  
               advantage of this method it is very easy to use but finding 
               the solution by this method takes some time. Kumar et al.                    Table 4.    IBFS by Row Minima Method 
               [7]  suggested the transportation problem have  many 
               objectives such as minimize the transportation cost, with                         P      P       P        P        Farm 
               respect to time  we minimize the distance, find the past                           1      2       3        4     Capacity 
               having lowest cost etc. Sharma [8]  also suggested an                  F          19     30      50       10        7/0 
                                                                                       1                                (7) 
               analysis for solving the different types of problems in real.                            30      40 
               Kizolli   [9]   suggested the transportation demand                    F2         70      (8)     (1)     60       9/1/0 
               management through physical improvement as application                 F          40     8       70       20     18/11/6/0 
               in real life. There is a standard way to solve any problems.            3        (5)            (6)      (7) 
               For this we can find the IBFS of the given problem firstly            Plant      5/0    8/0    7/6/0    14/7/0      34 
               by any of the method such as NWCR-Method, least cost              Requirement 
               method, VAM-Method etc and also there are some other                     Z=10*7+30*8+40*1+70*6+20*7=1110. 
               methods to solve such type of problems. In this paper, we       Column-Minima Method:  
               developed a new mathematical method named as 
               DSM-Method for finding an IBFS of the transportation                       Table 5.    IBFS by Column Minima Method 
               problem and the effectiveness by this method is also 
               compared with the other methods.                                                  P      P      P        P         Farm 
                                                                                                  1      2      3        4       Capacity 
                                                                                      F          19     30     50       10         7/2 
               3. Numerical Example                                                    1        (5)                     (2) 
                                                                                      F          70     30     40       60        9/2/0 
                                                                                       2                      (7)       (2) 
                  A farm has three utility companies that manufacture 7,                                8               20 
               9, and 18 vehicles. The farm stores four customers whose               F3         40    (8)     70      (10)      18/10/0 
               company constraints are 5, 8, 7, and 14. Consider the                 Plant      5/0    8/0    7/0    14/12/2/0     34 
               following cost minimization problem for  linear                   Requirement 
                                                                          
               programming with 3 farms and 4 plants given in table 2.                Z=19*5+10*2+40*7+60*2+8*8+20*10=779 
               744                          Analysis of IBFS for Transportation Problem by Using Various Methods                               
                                                                                
               LCM- Method:                                                     Arithmetic-Mean Method:  
                                Table 6.  IBFS by LCM Method                                Table 9.  IBFS by Arithmetic Mean Method 
                                   P       P      P       P        Farm                           P       P       P        P         Farm 
                                    1       2      3        4     Capacity                         1       2       3        4      Capacity 
                       F           19      30     50      10         7/0                F         19      30      50       10        7/2/0 
                        1                                   (7)                          1          (5)                    (2) 
                       F           70      30     40      60        9/2/0               F         70      30      40       60        9/2/0 
                        2          (2)            (7)                                    2                (2)     (7) 
                       F           40      8      70      20      18/10/3/0             F         40       8      70       20       18/6/0 
                        3          (3)      (8)           (7)                            3                (6)             (12) 
                     Plant        5/2/0   8/0     7/0    14/7/0      34               Plant       5/0    8/6/0    7/0    14/12/0      34 
                   Requirement                                                     Requirement 
                       Z=10*7+70*2+40*7+40*3+8*8+20*7=814                              Z=19*5+10*2+30*2+40*7+8*6+20*12=743 
               NWCR-Method:                                                     DSM-Method: 
                                                                                                Table 10.    IBFS by DSM Method 
                               Table 7.    IBFS by NWCR Method                                                                      Farm 
                                                                   Farm                            P1      P2      P3       P4     Capacity 
                                  P1      P2      P3       P4     Capacity                         19              50 
                                                                                        F                  30               10       7/1/0 
                       F         19      30       50       10       7/2/0                1         (1)             (6) 
                        1        (5)     (2)                                                               30      40 
                                                                                        F          70                       60       9/1/0 
                       F         70      30       40       60       9/3/0                2                 (8)     (1) 
                        2                (6)      (3)                                              40                       20 
                                                                                        F                   8      70               18/4/0 
                       F         40       8       70       20      18/14/0               3         (4)                     (14) 
                        3                         (4)     (14)                        Plant 
                     Plant                                                         Requirement    5/4/0    8/0    7/6/0    14/0       34 
                  Requirement    5/0    8/6/0    7/4/0    14/0       34 
                                                                                      Z=19*1+50*6+30*8+40*1+40*4+20*14=1039 
                     Z=19*5+30*2+30*6+40*3+70*4+20*14=1015                                Table 11.    Comparison of IBFS with all Method 
               VAM-Method:                                                                    Methods                        Cost 
                         Table 8.  IBFS by Vogel’s Approximation Method                     ASM Method                        743 
                                                                                         Row-Minima Method                   1110 
                                  P       P      P        P         Farm 
                                   1       2       3       4      Capacity             Column-Minima Method                   779 
                       F          19      30     50       10        7/2/0                   LCM-Method                        814 
                        1         (5)                     (2) 
                                                 40       60                               NWCR Method                       1015 
                       F          70      30                         9/0 
                        2                        (7)      (2)                               VAM Method                        779 
                       F          40      8      70       20       18/10/0             Arithmetic Mean Method                 743 
                        3                (8)             (10) 
                     Plant        5/0    8/0     7/0    14/4/0       34                     DSM Method                       1039 
                  Requirement 
                                                                                   By using above desired obtained values construct a 
                      Z=19*5+10*2+40*7+60*2+8*8+20*10=779                       diagram which shown below. 
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...Mathematics and statistics http www hrpub org doi ms analysis of ibfs for transportation problem by using various methods s k sharma keshav goel department chandigarh university gharuan mohali punjab india received november revised february accepted march cite this paper in the following citation styles a vol no pp b copyright authors all rights reserved agree that article remains permanently open access under terms creative commons attribution license international abstract supply demand cost introduction cannot be obtained existing directly literature tp is important type lpp solving routing have been proposed calculating problems it gives any object from we are comparing sources to diverse sink mandate manner measuring optimal entire charge would minimum objective obtaining real life operation research most include essential application field there lots such as amm arithmetic mean method development different areas asm assigning shortest minimax etc shipping networking transport ini...

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