BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de Universitatea Tehnică Gheorghe Asachi din Iaşi Tomul LIX (LXIII), Fasc. 1, 016 Secţia AUTOMATICĂ şi CALCULATOARE A COMPETENT ALGORITHM TO FIND THE INITIAL BASIC FEASIBLE SOLUTION OF COST MINIMIZATION TRANSPORTATION PROBLEM BY AMINUR RAHMAN KHAN 1,*, ADRIAN VILCU, MD. SHARIF UDDIN 1 and FLORINA UNGUREANU 3 1 Jahangirnagar University, Dhaka-134, Bangladesh. Department of Mathematics Gheorghe Asachi Technical University of Iasi, Romania, Department of Management Engineering 3 Gheorghe Asachi Technical University of Iasi, Romania, Department of Automatic Control and Computer Engineering Received: 6 July 015 Accepted for publication: 4 July 015 Abstract. In this paper, we propose a new algorithm along with MATLAB 7.7.0 code for determining the initial basic feasible solution of Cost Minimization Transportation Problem (CMTP). Comparative study is carried out between the proposed algorithm and the other existing algorithm by means of sample examples which shows that the proposed algorithm provides better result. Key words: Cost minimization transportation problem; MATLAB; Distribution indicator; Optimum solution. 010 Mathematics Subject Classification: 90B50, 90C08. * Corresponding author; e-mail: aminurju@yahoo.com
Aminur Rahman Khan et al. 1. Introduction The branch of Linear Programming Problem (LPP) in which a single uniform commodity is shifted from several sources to different localities in such a way to minimize the total transportation cost while fulfilling all supply and demand limitations are CMTP. The basic CMTP was originally developed by Hitchcock (1941). Competent methods of solution derived from the simplex algorithm were developed, mainly by Dantzig (1947) and then by Charnes et al. (1953). The problem of CMTP has been studied since long and is well known by Abdur Rashid (013), Aminur Rahman Khan (011; 01; 015a; 015b), Hamdy (007), Juman & Hoque (015), Kasana & Kumar (005), M. Sharif Uddin et al. (011; 01), Md. Amirul Islam et al. (01a, 01b), Md. Ashraful Babu et al. (013; 014a; 014b), Md. Main Uddin et al. (013; 015), Mollah Mesbahuddin Ahmed et al. (014), Sayedul Anam et al. (01) and Utpal Kanti Das et al. (014a; 014b). For determining the initial basic feasible solution of TP, Reinfeld and Vogel (1958) introduced Vogel s Approximation Method (VAM) by defining penalty as the difference of lowest and next to lowest cost in each row and column of a transportation table and allocate to the minimum cost cell corresponding to the highest penalty; Kasana and Kumar (005) presented Extremum Difference Method (EDM) by calculating the penalty as the difference between the highest and lowest unit transportation cost in each row and column and allocate as like as VAM; Aminur Rahman Khan (011) proposed Highest Cost Difference Method (HCDM) by introducing pointer cost as the difference of highest and next to highest cost in each row and column of a transportation table and allocate to the minimum cost cell corresponding to the highest three pointer costs. Kirca and Satir (1990) first define the Total Opportunity Cost Matrix (TOCM) as the sum of Row Opportunity Cost Matrix (ROCM) and Column Opportunity Cost Matrix (COCM). Where, the ROCM is generated by subtracting the lowest cost of each row from the other cost elements in that row and, the COCM is generated by subtracting the lowest cost of each column from the other cost elements in that column. Kirca and Satir then essentially apply the Least Cost Method with some tie-breaking policies on the TOCM to determine the feasible solution of the transportation problem. Mathirajan and Meenakshi (004) applied VAM on the TOCM, Md. Amirul Islam et al. applied EDM (01a) and HCDM (01b) on TOCM whereas Aminur Rahman Khan (015a) calculate the pointer cost as the sum of all entries in the respective row or column of the TOCM to find the feasible solution of the transportation problem. Here, in this paper, we determine the distribution indicator for each cell of the TOCM by subtracting corresponding row and column highest element of every cell from the respective element. We then make maximum possible
Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 1, 016 3 allocation to the cell having the smallest distribution indicator. The proposed method is also illustrated with several numerical examples. Comparative study shows that the proposed algorithm gives better result in comparison to the other existing heuristics available in the text. We also coded the presented algorithm by using MATLAB 7.7.0 and it is utilized via many randomly generated problems of different order in order to prove the exactness of the code. Based on the results we conclude that the coded algorithm for solving the transportation problem is accurate.. Formulation of Cost Minimization Transportation Problem A general cost minimization transportation problem is represented by the network in the following Fig. 1. S 1 Source 1 C : X 11 11 Locality 1 D 1 Units of Supply S D Units of Demand S m m C mn : X mn n D n Fig. 1 Network representation of Cost Minimization Transportation Problem There are m sources and n destinations, each represented by a node. The arrows joining the sources and the localities represent the route through which the commodity is shifted. Suppose S i denotes the amount of supply at source i (i = 1,,, m), D j represents the amount of demand at destination j (j=1,,, n), C denotes the unit transportation cost from sources i to destination j, X represents the amount transported from sources i to destination j. Then the LPP model of the balanced cost minimization transportation problem is m n Min. Z = i = j = C X 1 1 n s/t, =1 X = i ; i=1,,,m j S
Aminur Rahman Khan et al. m i =1 X = D j ; j=1,,,n X 0 for all i, j. (1) 3. Algorithm of Proposed Method The proposed algorithm for determining the initial basic feasible solution consists of the following steps: Step 1 Step Step 3 : Subtract the smallest entry of every row from each of the element of the subsequent row of the transportation table and place them on the right-top of the corresponding elements. C C ik C, where C ( C, C,, C ) ik = min i1 i in, i = 1,,......, m : Apply the same operation on each of the column and place them on the left-bottom of the corresponding elements. C C kj C, where C kj = min ( C 1 j, C j,, C mj ), j = 1,,......,n : Form the TOCM whose entries are the summation of righttop and left-bottom elements of Steps 1 and. C = C C + C C ( ) ( ) Step 4 : For each cell (i, j), calculate the distribution indicator, Δ = c - ū i - ē j, where, ū i =largest unit time in the ith row and ē j = largest unit time in the jth column. Step 5 : Make maximum possible allocation to the cell having the smallest value of Δ. If tie occurs in the distribution indicator, select any one of them arbitrarily. Step 6 Step 7 Step 8 : No further deliberation is required for the row or column which is satisfied. If both the row and column are satisfied at a time, delete both of them assigning an extra zero supply (or demand) to any one cell of the satisfied row or column. : Calculate fresh distribution indicators for the remaining submatrix as in Step 4 and allocate following the procedure of Steps 5 and 6. : Continue the process until all rows and columns are satisfied. ik kj
Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 1, 016 5 Step 9 : Finally compute the total transportation cost as the sum of the product of original transportation cost and corresponding allocation obtained in step 5. 4. The Novelty of our algorithm Although we have used TOCM of Kirca and Satir in our proposed algorithm; we calculate the distribution indicator (in step 4) for each cell of the TOCM by subtracting corresponding row and column highest element of every cell from the respective element. Whereas Mathirajan and Meenakshi calculate the penalty as the difference of lowest and next to lowest entries of the TOCM; Md. Amirul Islam et al. calculate distribution indicator as the difference of highest and lowest entries of the TOCM and Aminur Rahman Khan et al. calculate pointer cost as the sum of all entries in the respective row or column of the TOCM. 5. Material and Methods Table 1, 10 and 11 shows three sample cost minimizing transportation problem, selected at random to solve by using proposed algorithm and the existing algorithms. Example 1: Table 1 Cost Matrix for the Numerical example Destination 1 3 6 8 4 0 6 1 5 8 3 7 8 3 9 17 Demand 15 19 13 18 Factory Step 1: 3 is the minimum element of the first row, so we subtract 3 from each element of the first row. Similarly, we subtract 1 and 3 from each element of the nd and 3 rd row respectively and place all the differences on the right-top of the corresponding elements in Table. Step : In a similar fashion, we subtract 3, 1, and 4 from each element of the 1 st, nd, 3 rd and 4 th column respectively and place the result on the leftbottom of the corresponding elements in Table.
Aminur Rahman Khan et al. Table Formation of Total Opportunity Cost Matrix Destination 1 03 0 56 3 68 5 04 1 0 36 5 01 0 0 1 15 4 8 3 47 4 78 5 13 0 59 6 17 Demand 15 19 13 18 Factory Step 3: We add the right-top and left-bottom entry of each element of the transportation table obtained in Iteration 1 and Iteration and formed the TOCM as in Table 3. Table 3 Total Opportunity Cost Matrix (TOCM) Destination 1 0 8 11 1 0 8 0 1 5 8 3 8 1 1 11 17 Demand 15 19 13 18 Factory Step 4: We determine the distribution indicator for each cell of the TOCM by subtracting corresponding row and column highest element of every cell from the respective element. Here, c 11 =0, highest entry in the first row is 11 and in the first column is 8, so distribution indicator, Δ 11 =0-8-11=-19. Do the same for each entry and place them in the right-top of every cell of the cost matrix. Table 4 Determination of distribution indicator after Step 4 1-19 -15-11 -1 0 8 11 1 0-8 -0-18 -14 8 0 1 5 8 3-1 -1 - -1 8 1 1 11 17 Demand 15 19 13 18
Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 1, 016 7 Step 5: Here, smallest value of Δ = - corresponding to the cell (3, 3). So we allocate 13 units (minimum of 17 and 13) to the cell (3, 3). We adjust the supply and demand requirements corresponding to the cell (3, 3) and since the demand for the cell (3, 3) is satisfied, we delete the third column and calculate the distribution indicator again for the resulting reduced transportation table. Table 5 Determination of distribution indicator after Step 5 1-16 -1-18 0 8 11 1 0-8 -0-14 8 0 1 5 8 3-1 -1 13-1 8 1 1 11 4 Demand 15 19 0 18 Step 6: Here, smallest value of Δ = -0 corresponding to the cell (, ). So we allocate 19 units (minimum of 8 and 19) to the cell (, ). We adjust the supply and demand requirements corresponding to the cell (, ) and since the demand for the cell (, ) is satisfied, we delete the third column and calculate the distribution indicator again for the resulting reduced transportation table. Table 6 Determination of distribution indicator after Step 6 1-9 -1-11 0 8 11 1 0-8 19-0 -14 8 0 1 5 9 3-11 -1 13-11 8 1 1 11 4 Demand 15 0 0 18
Aminur Rahman Khan et al. Step 7: Here, smallest value of Δ = -14 corresponding to the cell (, 4). So we allocate 9 units (minimum of 9 and 18) to the cell (, 4). We adjust the supply and demand requirements corresponding to the cell (, 4) and since the supply for the cell (, 4) is satisfied, we delete the second row and calculate the distribution indicator again for the resulting reduced transportation table. Table 7 Determination of distribution indicator after Step 7 1-9 -1-11 0 0 8 11 1-8 19-0 9-14 0 8 0 1 5 3-11 -1 13-11 4 8 1 1 11 Demand 15 0 0 9 Step 8: Here, smallest value of Δ = -11 corresponding to the cell (1, 4), (3, 1) and (3, 4). So we allocate 9 units (minimum of 0 and 9) to the cell (1, 4). We adjust the supply and demand requirements corresponding to the cell (1, 4) and since the demand for the cell (1, 4) is satisfied, we delete the fourth column and calculate the distribution indicator again for the resulting reduced transportation table. Table 8 Determination of distribution indicator after Step 8 1-9 -1 9-11 0 8 11 1 11-8 19-0 9-14 8 0 1 5 0 3-11 -1 13-11 4 8 1 1 11 Demand 15 0 0 0 Step 9: Since only the first column is remaining with two unallocated cell in this case, we allocate 11 units (minimum of 11 and 15) to the cell (1, 1) and 4 units (minimum of 4 and 4) to the cell (3, 1).
Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 1, 016 9 We adjust the supply and demand requirements again and we see that all supply and demand values are exhausted. Table 9 Determination of distribution indicator after Step 9 1 11-9 -1 9-11 11 0 8 11 1-8 19-0 9-14 0 8 0 1 5 3 4-11 -1 13-11 4 8 1 1 11 Demand 15 0 0 0 Step 10: Since all the rim conditions are satisfied and total number of allocation is 6. Therefore, the solution for the given problem is x =11, 9 11 x 14=, x = 19, x 9 4 =, x = 31 4 and x = 33 13. for a flow of 65 units with the total transportation cost Example : z = 11 3+ 9 4 + 19 1+ 9 5 + 4 7 + 13 3 = 00 Table 10 Cost Matrix for the Numerical example Destination 1 3 Supply 1 7 4 5 3 3 1 8 3 5 4 7 7 4 1 6 14 Demand 7 9 18 Factory Example 3: Fac tory Table 11 Cost Matrix for the Numerical example Destination 1 6 1 9 3 70 11 5 8 55
Aminur Rahman Khan et al. 3 10 1 4 7 90 Demand 85 35 50 45 6. Result Table 1 shows a comparison among the solutions obtained by our Proposed Approach and the other existing methods and also with the optimal solution by means of the above three sample examples and it is seen that our proposed method gives better results. Table 1 A comparative study of different solutions Solution obtained by Total Transportation Cost Ex. 1 Ex. Ex. 3 North West Corner Method 73 10 165 Row Minimum Method 31 80 1165 Column Minimum Method 31 111 10 Least Cost Method 31 83 1165 Vogel s Approximation Method 04 80 10 Extremum Difference Method 18 83 1165 Highest Cost Difference Method 31 83 1165 TOCM-MMM Approach 31 83 1165 TOCM-VAM Approach 04 76 1165 TOCM-EDM Approach 04 83 1165 TOCM-HCDM Approach 55 81 1165 TOCM-SUM Approach 00 76 180 Average Cost Method 18 80 180 Proposed Approach 00 76 1165 Optimum Solution 00 76 1160 We also solve randomly selected transportation problem of order 3 3, 3 4, 3 5, 4 3, 4 4, 4 5, 4 6, 6 6 and see that the MATLAB code presented by us gives identical result as the manual solution which proves the correctness of our code. 7. Conclusion A new algorithm along with MATLAB 7.7.0 code for finding an initial basic feasible solution of cost minimization transportation problem is introduced. We also illustrate this algorithm numerically to test the efficiency of
Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 1, 016 11 the proposed method. Comparative study among the solution obtained by proposed method and the other existing methods by means of sample examples show that our proposed method gives better result. Acknowledgement The first author acknowledges the financial support provided by the EU Erasmus Mundus Project-cLINK, Grant Agreement No: 1-645/001-001- EM, Action. REFERENCES Abdur Rashid, Syed Sabbir Ahmed, Md. Sharif Uddin, Development of a New Heuristic for Improvement of Initial Basic feasible solution of a Balanced Transportation Problem, Jahangirnagar University Journal of Mathematics and Mathematical Sciences, 8, 105-11, 013. Aminur Rahman Khan, A Re-solution of the Transportation Problem: An Algorithmic Approach, Jahangirnagar University Journal of Science, 34,, 49-6, 011. Aminur Rahman Khan, Analysis and Resolution of the Transportation Problem: An Algorithmic Approach, M. Phil. Thesis, Dept. of Mathematics, Jahangirnagar University, 01. Aminur Rahman Khan, Adrian Vilcu, Nahid Sultana, Syed Sabbir Ahmed, Determination of Initial Basic Feasible Solution of a Transportation Problem: A TOCM-SUM Approach, Buletinul Institutului Politehnic Din Iasi, Romania, Secţia Automatica si Calculatoare, LXI (LXV), 1, 39-49, 015a. Aminur Rahman Khan, Avishek Banerjee, Nahid Sultana, M Nazrul Islam, Solution Analysis of a Transportation Problem: A Comparative Study of Different Algorithms accepted for publication in the Bulletin of the Polytechnic Institute of Iasi, Romania, Section Textile. Leathership, in issue of 015b. Charnes A., Cooper W.W, Henderson A., An Introduction to Linear Programming, John Wiley & Sons, New York, 1953. Dantzig G.B., Linear Programming and Extentions, Princeton University Press, Princeton, N J, 1963. Hamdy A.T., Operations Research: An Introduction, 8 th Edition, Pearson Prentice Hall, Upper Saddle River, New Jersey 07458, 007. Hitchcock F.L., The distribution of a Product from Several Sources to Numerous Localities, Journal of Mathematics and Physics, 0, 4-30, 1941. Juman Z.A.M.S., Hoque M.A., An Efficient heuristic to obtain a Better Initial Feasible Solution to the Transportation Problem, Applied Soft Computing, 34, 813-86, 015. Kasana H.S., Kumar K.D., Introductory Operations Research: Theory and Applications, Springer International Edition, New Delhi, 005. Koopmans T.C., Optimum Utilization of the Transportation System, Econometrica, 17, 3-4, 1947.
Aminur Rahman Khan et al. M. Sharif Uddin, Sayedul Anam, Abdur Rashid, Aminur R. Khan, Minimization of Transportation Cost by Developing an Efficient Network Model, Jahangirnagar Journal of Mathematics & Mathematical Sciences, 6, 13-130, 011. M. Sharif Uddin, Transportation Time Minimization: An Algorithmic Approach, Journal of Physical Sciences, Vidyasagar University, 16, 59-64, 01. Mathirajan M., Meenakshi B., Experimental Analysis of Some Variants of Vogel s Approximation Method, Asia-Pacific Journal of Operational Research, 1, 4, 447-46, 004. Md. Amirul Islam, Md. Masudul Haque, Md. Sharif Uddin, Extremum Difference Formula on Total Opportunity Cost: A Transportation Cost Minimization Technique, Prime University Journal of Multidisciplinary Quest, 6, 1, 15-130, 01a. Md. Amirul Islam, Aminur Rahman Khan, M. Sharif Uddin, M. Abdul Malek, Determination of Basic Feasible Solution of Transportation Problem: A New Approach, Jahangirnagar University Journal of Science, 35, 1, 101 108, 01b. Md. Ashraful Babu, Md. Abu Helal, Mohammad Sazzad Hasan, Utpal Kanti Das, Lowest Allocation Method (LAM): A New Approach to Obtain Feasible Solution of Transportation Model, International Journal of Scientific and Engineering Research, 4, 11, 1344-1348, 013. Md. Ashraful Babu, Md. Abu Helal, Mohammad Sazzad Hasan, Utpal Kanti Das, Implied Cost Method (ICM): An Alternative Approach to Find the Feasible Solution of Transportation Problem, Global Journal of Science Frontier Research-F: Mathematics and Decision Sciences, 14, 1, 5-13, 014a. Md. Ashraful Babu, Utpal Kanti Das, Aminur Rahman Khan, Md. Sharif Uddin, A Simple Experimental Analysis on Transportation Problem: A New Approach to Allocate Zero Supply or Demand for All Transportation Algorithm, International Journal of Engineering Research & Applications (IJERA), 4, 1, 418-4, 014b. Md. Main Uddin, Md. Azizur Rahaman, Faruque Ahmed, M. Sharif Uddin, Md. Rashed Kabir, Minimization of Transportation Cost on the basis of Time Allocation : An Algorithmic Approach, Jahangirnagar Journal of Mathematics & Mathematical Sciences, 8, 47-53, 013. Md. Main Uddin, Aminur Rahman Khan, Sushanta Kumer Roy, Md. Sharif Uddin, A New Approach for Solving Unbalanced Transportation Problem due to Additional Supply, accepted for publication in the Bulletin of the Polytechnic Institute of Iasi, Romania, Section Textile. Leathership, in issue of 015. Mollah Mesbahuddin Ahmed, Abu Sadat Muhammad Tanvir, Shirin Sultana, Sultan Mahmud, Md. Sharif Uddin, An Effective Modification to Solve Transportation Problems: A Cost Minimization Approach, Annals of Pure and Applied Mathematics, 6,, 199-06, 014a. Mollah Mesbahuddin Ahmed, Algorithmic Approach to Solve Transportation Problems: Minimization of Cost and Time, M. Phil. Thesis, Dept. of Mathematics, Jahangirnagar University, 014b. Omer Kirca, Ahmet Satir, A heuristic for obtaining an initial solution for the transportation problem, Journal of the Operational Research Society, 41, 865 871, 1990.
Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 1, 016 13 Reinfeld N. V., Vogel W. R., Mathematical Programming, Englewood Cliffs, NJ: Prentice-Hall, 1958. Sayedul Anam, Aminur Rahman Khan, Md. Minarul Haque, Reza Shahbaz Hadi, The Impact of Transportation Cost on Potato Price: A Case Study of Potato Distribution in Bangladesh, The International Journal of Management, 1, 3, 1-1, 01. Utpal Kanti Das, Md. Ashraful Babu, Aminur Rahman Khan, Md. Abu Helal, Md. Sharif Uddin, Logical Development of Vogel s Approximation Method (LD- VAM): An Approach to Find Basic Feasible Solution of Transportation Problem, International Journal of Scientific & Technology Research (IJSTR), 3,, 4-48, 014a. Utpal Kanti Das, Md. Ashraful Babu, Aminur Rahman Khan, Md. Sharif Uddin, Advanced Vogel s Approximation Method (AVAM): A New Approach to Determine Penalty Cost for Better Feasible Solution of Transportation Problem, International Journal of Engineering Research & Technology (IJERT), 3, 1, 18-187, 014b. ALGORITM PENTRU DETERMINAREA SOLUȚIEI INIȚIALE ÎN CAZUL MINIMIZĂRII COSTULUI PROBLEMEI DE TRANSPORT (Rezumat) Lucrarea propune un nou algoritm pentru determinarea unei soluții inițiale pentru o problemă clasică de transport bazată pe cost. Se prezintă problema, este descrisă metoda, se aplică algoritmul pe un exemplu numeric și se compară, pe trei instante, rezultatele algoritmului propus cu cele furnizate de alți algoritmi citați în literatura de specialitate.