In this research three methods have been used to find an initial basic feasible solution for the balanced transportation model. We have used a new method of Minimum Transportation Cost Method (MTCM) to find the initial basic feasible solution for the solved problem by Hakim [2]. Hakim used Proposed Approximation Method (PAM) to find initial basic feasible solution for balanced transportation model and then compared the results with Vogel’s Approximation Method (VAM) [2]. The results of both methods were noted to be the same but here we have taken the same transportation model and used MTCM to find its initial basic feasible solution and compared the result with PAM and VAM. It is noted that the MTCM process provides not only the minimum transportation cost but also an optimal solution.   Keywords: Transportation problem, Vogel’s Approximation Method (VAM), Maximum  Penalty of largest numbers of each Ro

Bhayo, Ghulam Murtaa

Soomro, Abdul Sattar

Tularam, Gurudeo Anand

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.1, 2014  11  A comparative study of initial basic feasible solution methods for transportation problems  Abdul Sattar Soomro1  Gurudeo Anand Tularam2 Ghulam Murtaa Bhayo3 dr_sattarsoomro@yahoo.co.in, a.tularam@griffith.edu.au, gmsindhi@yahoo.com 1 Professor of Mathematics, Institute of Mathematics and Computer Science, University of Sindh, Jamshoro,  Sindh, Pakistan 2Senior Lecturer, Mathematics and Statistics, Science Environment Engineering and Technology [ENV], Griffith University, Brisbane Australia 3Lecturer, Govt. Degree College, Pano Akil, Sukkur, Sindh, Pakistan Abstract  In this research three methods have been used to find an initial basic feasible solution for the balanced transportation model. We have used a new method of Minimum Transportation Cost Method (MTCM) to find the initial basic feasible solution for the solved problem by Hakim [2]. Hakim used Proposed Approximation Method (PAM) to find initial basic feasible solution for balanced transportation model and then compared the results with Vogel’s Approximation Method (VAM) [2]. The results of both methods were noted to be the same but here we have taken the same transportation model and used MTCM to find its initial basic feasible solution and compared the result with PAM and VAM. It is noted that the MTCM process provides not only the minimum transportation cost but also an optimal solution.     Keywords: Transportation problem, Vogel’s Approximation Method (VAM), Maximum                                                     Penalty of largest numbers of each Row  1. Introduction  The transportation problem is a special linear programming problem which arises in many practical applications in other areas of operation, including, among others, inventory control, employment scheduling, and personnel assignment [1].  In this problem we determine optimal shipping patterns between origins or sources and destinations. The transportation problem deals with the distribution of goods from the various points of supply, such as factories, often known as sources, to a number of points of demand, such as warehouses, often known as destinations. Each source is able to supply a fixed number of units of the product, usually called the capacity or availability and each destination has a fixed demand, usually called the requirements. The objective is to schedule shipments from sources to destinations so that the total transportation cost is a minimum.   There are various types of transportation models and the simplest of them was first presented by Hitchcock (1941). It was further developed by developed by Koopmans (1949) and Dantzig (1951). Several extensions of transportation model and methods have been subsequently developed. In general, the Vogel’s approximation method yields the best starting solution and the north-west corner method yields the worst. However, the latter is easier, quick and involves the least computations to get the initial solution [5]. Goyal (1984) improved VAM for the unbalanced transportation problem, while Ramakrishnan (1988) discussed some improvement to Goyal’s Modified Vogel’s Approximation method for unbalanced transportation problem [6].  Adlakha and Kowalski (2009) suggested a systematic analysis for allocating loads to obtain an alternate optimal solution [7]. However, the study on alternate optimal solutions is clearly limited in the literature of transportation Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.1, 2014  12  with the exception of Sudhakar VJ, Arunnsankar N, Karpagam T (2012) who suggested a  new approach for finding  an optimal solution for transportation problems [8].  2. Transportation problem and General Computational Procedures  The transportation model of LP can be modeled as follows:  ;,0)()()cos(111 1jandiallforxnsdestinatiofromDemandbxsourcesfromSupplyxtoSubjectttiontransportaTotalxCZMinimizeijmijijnjiijminjijij a  where   Z   :  Total transportation cost to be minimized.  Cij :  Unit transportation cost of the commodity from each            source i to destination j.  xij   :  Number of units of commodity sent from source i to destination j.   ia  :  Level of supply at each source i.  bj    :  Level of demand at each destination j.   .bDemandSupplym1iim1ii a       NOTE:  Transportation model is balanced if .bDemandSupplym1iim1ii a                Otherwise unbalanced if .b11miimii DemandSupply a  The total number of variables is mn. The total number of constraints is m+n, while the total number of allocations (m+n–1) should be in feasible solution. Here the letter m denotes the number of rows and n denotes the number of columns.  Solving Transportation Problems The basic steps for solving transportation model are:   Step 1 - Determine a starting basic feasible solution. In this paper we use any one method NWCM, LCM, or VAM, to find initial basic feasible solution.  Step 2 - Optimality condition - If solution is optimal then stop the iterations otherwise go to step 3.  Step 3 - Improve the solution. We use either optimal method: MODI or Stepping Stone method.    Table 1: Transportation array   Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.1, 2014  13  DESTINATIONS Supply a i D1              D2  …….. ………..  Dn   S1   S2 . . . . Sm    S o u r c e s   a 1  x1n C1n  ……..  x12 C12  x11 C11     a 2  x2n C2n ……..  x22 C22  x21 C21    …….. …….. …….. …….. ……..  a m  xmn Cmn ……..  xm2 Cm2  xm1 Cm1    Balanced model    m1in1jji ba  bn …….. b2 b1 Demand bj   2. Methodology  The following methods are always used to find initial basic feasible solution for the transportation problems and are available in almost all text books on Operations Research [5]. The Initial Basic Feasible Solutions Methods are: (i) Column Minimum Method (CMM)  (ii) Row Minimum Method (RMM)  (iii) North West-Corner Method (NWCM)  (iv) Least Cost Method (LCM)  (v) Vogel’s Approximation Method (VAM)  The Optimal Methods used are:  (i) Modified Distribution (MODI) Method or u-v Method (ii) Vogel’s Approximation Method (VAM)  3.       Initial Basic Feasible Solution Methods and Optimal Methods  There are several initial basic feasible solution methods and optimal methods for solving transportation problems satisfying supplying and demand.  Initial Basic Feasible Solution Methods  We have used following three methods to find initial basic feasible solution of the balanced transportation problem:   Vogel’s Approximation Method (VAM)  Proposed Approximation method (PAM)  Minimum Transportation Cost Method (MTCM)  For optimal methods we have used the Modified Distribution (MODI) Method and the Stepping Stone Method  Vogel’s Approximation Method (VAM)  Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.1, 2014  14  This method provides a better starting solution than the North West Corner rule and Least Cost Method. VAM generally yields an optimum or close to optimum solution.   Algorithm  Step 1.   Compute penalty of each row and a column. The penalty will be equal to the difference between the two smallest shipping costs in the row or column.   Step 2.   Identify the row or column with the largest penalty and assign highest possible value to the variable having smallest shipping cost in that row or column.   Step 3.  Cross out the satisfied row or column.  Step 4.  Compute new penalties with same procedure until one row or column is left out.  Note:    Penalty means the difference between two smallest numbers in a row or a column.    Proposed Approximation Method (PAM)   This method provides a better starting solution than the North West Corner rule and Least Cost Method. The PAM generally yields optimum solution or close to optimum solution.  Algorithm   Step 1. Identify the boxes having maximum and minimum transportation cost in             each row and write the difference (penalty) along the side of the table against               the corresponding row.  Step 2.  Identify the boxes having maximum and minimum transportation cost in              each column and write the difference (penalty) against the corresponding              column.  Step 3. Identify the maximum penalty. If it is along the side of the table, make              maximum allotment to the box having minimum cost of transportation in that             row. If it is below the table, make maximum allotment to the box having             minimum cost of transportation in that column.  Step 4. If the penalties corresponding to two or more rows or columns are equal,              select the box where allocation is maximum. Step 5. No further consideration is required for the row or column which is satisfied.             If both the row and column are satisfied at a time, delete the two or column is             assigned zero supply (or demand).  Step 6. Calculate fresh penalty cost for the remaining sub-matrix as in step 1 and              allocate following the procedure of previous step. Continue the process until              all rows and columns are satisfied.  Step 7. Compute total transportation cost for the feasible cost for the feasible              allocations using the original balanced transportation cost matrix.  Minimum Transportation Cost Method (MTCM)  This method provides a better starting solution than the North West Corner rule and Least Cost Method. The MTCM generally yields optimum solution or close to optimum solution.  Algorithm Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.1, 2014  15  Step 1.   Make the table balanced. Compute penalty of each row. The penalty will be equal to the difference between the two largest shipping costs in the row.  Step 2. Identify the row or column with the maximum penalty and assign possible value to the variable having smallest shipping cost in that row. If two or more rows corresponding equal penalty then select the cell with minimum cost of that maximum penalty row.  Step 3.  Cross out the satisfied row or column.  Step 4.  Write the reduced table and compute new penalties with same procedure until one row or column is left out. Determine the total minimum cost of occupied cells satisfying m+n-1allocations. Note:   Penalty means the difference between two largest numbers in a row. Optimal Method  Modified Distribution (MODI) Method  This method always gives the total minimum transportation cost to transport the goods from sources to the destinations.  Algorithm  1. If the problem is unbalanced, balance it. Setup the transportation tableau 2. Find a basic feasible solution. 3. Set 01 u and determine  sui ' and  sv j ' such that ijji cvu  for all basic variables.                4. If the reduced cost 0 jiij vuc  for all non-basic variables (minimization problem), then the current BFS is optimal. Stop! Else, enter variable with most negative reduced cost and find leaving variable by looping. 5. Using the new BFS, repeat steps 3 and 4.  4. The Numerical Problem   We have used three methods to find an initial basic feasible solution for the balanced transportation problem [4]. The problem was developed by Hakim [2]. Consider the transportation problem presented in Table 2 - where there are 4 sources, 6 destinations; the cost is given in the cells, and the supply and demand given in bottom and right hand end row and column respectively in Table 2.                                  Table 2: Example problem                                                                      Destinations                Sources 1 2 3 4 5 6 Supply 1 1 2 1 4 5 2 30 2 3 3 2 1 4 3 50 3 4 2 5 9 6 2 75 4 3 1 7 3 4 6 20 Demand 20 40 30 10 50 25 175  Solution Three methods have been used here to find initial basic feasible solution of the above problem and these are presented in turn.  Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.1, 2014  16   Vogel’s Approximation Method (VAM) Using VAM the final solution is presented in the Table 3.                                                       Table 3: Solution using VAM                 Sources 1 2 3 4 5 6 Supply 1 20  10    30 2   20 10 20  50 3  20   30 25 75 4  20     20 Demand 20 40 30 10 50 25 175  Therefore the total transportation cost determined by the Vogel’s Approximation Method is:  Minimize Z = (20) (1) + (10) (1) + (20) (2) + (10) (1) + (20) (4) +(20) (2) + (30) (6)                        + (25) (2) + (20) (1)                              = 20 + 10 + 40 + 10 + 80 + 40 + 180 + 50 + 20         = 450  Solution Proposed Approximation Method (PAM) The final solution completed using PAM is presented in the Table 4.                                                          Table 4: Solution by PAM                 Sources 1 2 3 4 5 6 Supply 1   30    30 2    10 40  50 3 20 20   10 25 75 4  20     20 Demand 20 40 30 10 50 25 175  The total transportation cost by Proposed Approximation Method can be given as:  Minimize Z = (30) (1) + (10) (1) + (40) (4) + (20) (4) + (20) (2) +(10) (6) + (25) (2)                        + (20) (1)                              = 30 + 10 + 160 + 80 + 40 + 60 + 50 + 20         = 450 Minimum Transportation Cost Method (MTCM) The total transportation cost by Minimum Transportation Cost Method is given in Table 5.  Table 5: Solution by MTCM                 Sources 1 2  3 4 5 6 Supply 1 20  10    30 2   20  30  50 3  40   10 25 75 4    10 10  20 Demand 20 40 30 10 50 25 175  Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.1, 2014  17  The total transportation Cost by MCTM is given as:  Minimize Z = (20) (1) + (10) (1) + (20) (2) + (30) (4) + (40) (2) +(10) (6) + (25) (2)                        + (10) (3) + (10) (4)                              = 20 + 10 + 40 + 120 + 80 + 60 + 50 + 30 +40         = 450  Modified Distribution (MODI) Method We have found total minimum transportation cost using MODI method by taking initial basic feasible solution obtained by Minimum Transportation Cost Method (MTCM). The final solution is shown in Table 6.      Table 6: Solution using MODI                 Sources 1 2  3 4 5 6 Supply 1 20  10    30 2   20 10 20  50 3  40   10 25 75 4     20  20 Demand 20 40 30 10 50 25 175  The total transportation cost by Modified Distribution Method is given as:  Minimize Z = (20) (1) + (10) (1) + (20) (2) + (10) (1) + (20) (4) + (40) (2) + (10) (6)                        + (25) (2) + (20) (4)                              = 20 + 10 + 40 + 10 + 80 + 80 + 60 + 50 +80         = 430  Table 5: A comparison of the methods - VAM, PAM and MTCM and MODI  Initial Basic Feasible methods Value of the objective function   Minimum Value (cost) VAM 450 Same Result PAM 450 Same Result MTCM 450 Same Result MODI Method 430 Optimal or minimum cost   The cost of transportation shows that the: (i) Minimum Transportation Cost Method (MTCM), Vogel’s approximation method (VAM), and Proposed Approximation Method (PAM) provide the same result, not optimal but close to optimal;              (ii) In MTCM, we have used penalty of maximum numbers of each row yet not of each              column; (iii) In VAM and PAM, the penalty of smallest numbers of each row and column are applied;    (iv) In MTCM, the penalty of each row makes the problem simple, easy and takes a short time in calculation; and (v) In VAM and PAM, the penalty of each row and column makes the problem lengthy and the calculation time is longer.      Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.1, 2014  18  5. Conclusion  As transportation problem is a special linear programming problem having many practical applications in other areas of operations, including, among others, inventory control, employment scheduling, and personnel assignment as mentioned earlier.  Here in our research work we have used three methods, The Minimum Transportation Cost Method (MTCM), Vogel’s Approximation Method (VAM) and Proposed Approximation Method (PAM). These were used to find an initial basic feasible solution for the transportation balanced model. The results are noted to be the same. It is important to note that we have used only penalty of each row of maximum numbers that is a simpler option and thus takes much less time in the calculation. In contrast, other methods using maximum penalty of smallest numbers of each row and column makes the problem lengthy and the calculation takes longer. Moreover, the method presented here is simpler in comparison of other presented methods earlier and can be easily applied to find the initial basic feasible solution for the balanced and unbalanced transportation problems.    REFERENCES [1]    Hamdy A Taha, Prentice Hall 2002. Operations Research: An introduction 7th Edition,   p.165  [2]    M.A. Hakim, An Alternative Method to Find Initial Basic Feasible Solution of a Transportation         Problem, Annals of Pure and Applied Mathematics, Vol. 1, No. 2, 2012, 203-209. [3]    S. K Goyal, Improving VAM for unbalanced transportation problems, Journal of Operational                  Research Society, 35(12) (1984) 1113-1114. [4]    P. K. Gupta and Man Mohan. (1993). Linear Programming and Theory of Games, 7th edition,          Sultan Chand & Sons, New Delhi (1988) 285-318. [5]    Operations Research  by Prem Kumar Gupta and D.S. Hira, Page 228-235. [6]    Goyal (1984) improving VAM for the Unbalanced Transportation Problem, Ramakrishnan         (1988) discussed some improvement to Goyal’s Modified Vogel’s Approximation method              for Unbalanced Transportation Problem.  [7]    Veena Adlakha, Krzysztof Kowalski (2009), Alternate Solutions Analysis For              Transportation problems, Journal of Business & Economics Research – November,Vol 7.        [8]    Sudhakar VJ, Arunnsankar N, Karpagam T (2012). A new approach for find an Optimal             Solution for Trasportation Problems, European Journal of Scientific Research 68 254-257. 

A comparative study of initial basic feasible solution methods for transportation problems

In this research three methods have been used to find an initial basic feasible solution for the balanced transportation model. We have used a new method of Minimum Transportation Cost Method (MTCM) to find the initial basic feasible solution for the solved problem by Hakim [2]. Hakim used Proposed Approximation Method (PAM) to find initial basic feasible solution for balanced transportation model and then compared the results with Vogel's Approximation Method (VAM) [2]. The results of both methods were noted to be the same but here we have taken the same transportation model and used MTCM to find its initial basic feasible solution and compared the result with PAM and VAM. It is noted that the MTCM process provides not only the minimum transportation cost but also an optimal solution.Full Tex

Tularam, Gurudeo

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https://www.iiste.org/Journals/index.php/MTM/article/download/10500/10693

A comparative study of initial basic feasible solution methods for transportation problems

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