determination of initial basic feasible solution of a transportation problem

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de Universitatea Tehnică „Gheorghe Asachi” din Iaşi Tomul LXI (LXV), Fasc. 1, 2015 SecŃia AUTOMATICĂ şi CALCULATOARE

DETERMINATION OF INITIAL BASIC FEASIBLE SOLUTION OF A TRANSPORTATION PROBLEM: A TOCM-SUM APPROACH BY

AMINUR RAHMAN KHAN1,2∗∗, ADRIAN VILCU2, NAHID SULTANA3 and SYED SABBIR AHMED1 1

Jahangirnagar University, Dhaka-1342, Bangladesh, Department of Mathematics 2 “Gheorghe Asachi” Technical University of Iaşi, România, Department of Management Engineering 3 Bangladesh University, Dhaka-1207, Bangladesh, Department of Mathematics

Received: January 27, 2015 Accepted for publication: March 31, 2015

Abstract. A new heuristic for obtaining an initial basic feasible solution of a transportation problem (TP) is introduced in this paper. The proposed method is illustrated with a number of numerical examples. Comparison of findings obtained by the new heuristic and the existing heuristics show that the method presented herein gives a better result. Key words: VAM; MMM; TOCM; HCDM; Pointer cost; Optimum solution. 2010 Mathematics Subject Classification: 90B50, 90C08.

∗

Corresponding author; e-mail: [email protected]

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1. Introduction The transportation problem is one of the oldest applications of linear programming problems. The basic transportation problem was originally developed by Hitchcock (1941). Efficient methods of solution derived from the simplex algorithm were developed, primarily by Dantzig (1963) and then by Charnes et al. (1953). The problem of minimizing transportation cost has been studied since long and is well known by Abdur Rashid et al. (2012), Aminur Rahman Khan (2011; 2012), Hamdy (2007), Kasana and Kumar (2005), Sharif Uddin et al. (2011), Md. Amirul Islam et al. (2012), Md. Main Uddin et al. (2013a; 2013b), Md. Ashraful Babu et al. (2014a; 2014b), Mollah Mesbahuddin Ahmed et al. (2014), Pandian & Natarajan (2010), Sayedul Anam et al. (2012), Shenoy et al. (1991), Utpal Kanti Das et al. (2014a; 2014b). Several researchers have developed alternative methods for determining an initial basic feasible solution which takes costs into account. Well-known heuristics methods are North West Corner Method (NWCM), Matrix Minima Method (MMM), Vogel’s Approximation Method (VAM), Highest Cost Difference Method (HCDM), Extremum Difference Method (EDM), TOCM-MMM Approach, TOCM-VAM Approach, TOCM-EDM Approach, TOCM-HCDM Approach etc. Reinfeld and Vogel (1958) introduced 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 (2005) proposed EDM where they define the penalty as the difference of highest and lowest unit transportation cost in each row and column and allocate as like as the VAM procedure. Aminur Rahman Khan (2012) presented HCDM by defining 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 cost. Sayedul Anam et al. (2012) determine the impact of transportation cost on potato distribution in Bangladesh. Kirca and Satir (1990) first transform the cost matrix to create what they call the total opportunity cost matrix (TOCM). The TOCM is formed by adding the row opportunity cost matrix (ROCM) and the column opportunity cost matrix (COCM) where, for each row in the initial transportation cost matrix, the ROCM is generated by subtracting the lowest cost in the row from the other cost elements in that row and, for each column in the initial transportation cost matrix, the COCM is generated by subtracting the lowest cost in the column from the other cost elements in that column. Kirca and Satir then essentially use the MMM with some tie-breaking rules on the TOCM to generate a feasible solution to the transportation problem.

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Mathirajan and Meenakshi (2004) applied VAM on the TOCM whereas Md. Amirul Islam et al. applied EDM on TOCM (2012) and allocate to the minimum cost cell corresponding to the highest distribution indicator and again HCDM on TOCM (2012) and allocate to the minimum cost cell corresponding to the highest two distribution indicator. Here, in this paper, we calculate the pointer cost for each row and column of the TOCM by taking sum of all entries in the respective row or column and make maximum possible allocation to the lowest cost cell corresponding to the highest pointer cost. Comparative study shows that the proposed method gives better result in comparison to the other existing heuristics available in the literature. We also coded the proposed heuristic by using MATLAB 7.7.0 and the code is tested via many randomly generated problems of different size to compare the solution obtained manually and using MATLAB 7.7.0 code in order to prove the correctness of the code. Based on the results we show that both the result has the same value when solving the transportation problem. 2. Mathematical Representation of TP The following notation is used for the mathematical representation of the transportation problem. For each source point i (i = 1, 2, … …, m) and destination point j (j = 1, 2, … …, n): m = number of source; n = number of destination; ai = amount of supply at source i; bj = amount of demand at destination j; cij = unit transportation cost between source i and destination j; xij= amount of homogeneous product transported from source i and destination j. Using the above notations, the transportation problem can be expressed in mathematical term as finding a set of xij’s, i = 1, 2, … …, m; j = 1, 2, … …, n to m

Minimize z =

n

∑ ∑ ci j xi j i =1

j =1

n

subject to

∑ xi j = ai ;

i = 1, 2 , ... ... , m

j =1

m

∑ xi j = b j ;

j =1, 2 , ... ... , n

i =1

xi j ≥ 0.

for all i and j .

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3. Algorithm of Proposed Method We term the proposed heuristic as ‘TOCM-SUM Approach’ which consists of the following steps: Step 1

: 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 ij − C ik

C ij

(

)

, where C ik = min C i1 , C i 2 ,⋯ ⋯ , C in ,

i = 1, 2 , ... ... , m Step 2

: Apply the same operation on each of the column and place them on the left-bottom of the corresponding elements. C ij − C kj C ij , where C kj = min C 1 j , C 2 j ,⋯ ⋯ , C mj ,

(

)

j =1, 2 , ... ... , n Step 3

: Form the TOCM whose entries are the summation of righttop and left-bottom elements of Steps 1 and 2.

C ij = (C ij − C ik )+ (C ij − C kj ) Step 4

Step 5

Step 6

Step 7

Step 8

: Determine the pointer cost for each row of the TOCM by taking sum of all entries in the respective row and write them in front of the row on the right. Do the same for each column and place them in the bottom of the cost matrix below the corresponding column. : Choose the highest pointer cost and observe the row or column to which this corresponds. Then make maximum possible allocation to the lowest cost cell corresponding to selected row or column. If tie occurs, choose the pointer cost arbitrarily. : No further consideration is required for the row or column which is satisfied. If both the row and column are satisfied at a time, assigned a zero supply (or demand) to any one cell of the row or column and delete both of them. : Calculate fresh pointer costs for the remaining sub-matrix as in Step 4 and allocate following the procedure of Steps 5 and 6. Continue the process until all rows and columns are satisfied. : Compute the total transportation cost using the original transportation cost matrix and allocations obtained in step 5.

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4. Novelty of our Algorithm Although we have used TOCM of Kirca and Satir in our proposed algorithm, we calculate the pointer cost (in Step 4) as the sum of all entries in the respective row or column of the TOCM whereas Mathirajan and Meenakshi calculate the penalty as the difference of lowest and next to lowest entries of the TOCM and Md. Amirul Islam et al. calculate distribution indicator as the difference of highest and lowest entries of the TOCM. 5. Material and Methods Three sample cost minimizing transportation problem of different order were selected at random to solve by using proposed TOCM-SUM Approach and the existing heuristics. The Tables 1, 5 and 7 shows the costs cij, supplies ai, demands bj of the sample transportation problems. Example 1: Table 1 Cost Matrix for the Numerical Example Destination 2

3

4

Supply

1 2 3

3 6 7

6 1 8

8 2 3

4 5 9

20 28 17

Demand

15

19

13

18

Factory

1

Iteration 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 2nd and 3rd row respectively and place all the differences on the right-top of the corresponding elements in Table 2. Iteration 2: In the same manner, we subtract 3, 1, 2 and 4 from each element of the 1st, 2nd, 3rd and 4th column respectively and place the result on the left-bottom of the corresponding elements in Table 2. Table 2 Formation of Total Opportunity Cost Matrix Destination

Factory

1 1 2 3

Demand

2

03

0

36

5

47

4

15

3

56

3

01

0

78

5

19

4

68

5

02

1

13

0

13

Supply

04

1

20

15

4

28

59

6

17

18

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Aminur Rahman Khan et al.

Factory

Iteration 3: We add the right-top and left-bottom entry of each element of the transportation table obtained in Iteration 1 and Iteration 2 and formed the TOCM as in Table 3. Table 3 Total Opportunity Cost Matrix (TOCM) Destination 1 2 3 4 1 0 8 11 1 2 8 0 1 5 3 8 12 1 11 Demand 15 19 13 18

Supply 20 28 17

Iteration 4: We determine the pointer cost for each row of the TOCM by taking the sum of all entries in the respective row and write them in front of the row on the right (e.g. 0+8+11+1=20, 8+0+1+5=14 and 8+12+1+11=32). Do the same for each column and place them in the bottom of the cost matrix below the corresponding columns (e.g. 0+8+8=16, 8+0+12=20, 11+1+1=13 and 1+5+11=17). Table 4 Initial Basic Feasible Solution Using TOCM-SUM Approach 1 2 3 4 Supply Row Pointer 11

1

9 0

8

11

19

2

9

8

0

4

3

1

1

5

1

11

13 12

15 16

19 20

13 13

18 17

16

20

--

17

8

8

--

6

8

--

--

6

Pointer

Demand Column

8

20

20

9

9

1

28

14

13

13

13

17

32

31

--

--

Iteration 5: Here, maximum pointer cost is 32 and minimum transportation cost corresponding to this is 1 in the cell (3, 3). So we allocate 13 units (minimum of 17 and 13) to the cell (3, 3).

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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 pointer cost again for the resulting reduced transportation table. Iteration 6: In this stage, maximum pointer cost is 31 and minimum transportation cost corresponding to this is 8 in the cell (3, 1). So we allocate 4 units (minimum of 4 and 15) to the cell (3, 1). We adjust the supply and demand requirements corresponding to the cell (3, 1) and since the supply for the cell (3, 1) is depleted, we delete the third row and calculate the pointer cost again for the resulting reduced transportation table. Iteration 7: In this case, maximum pointer cost is 13 and minimum transportation cost corresponding to this is 0 in the cell (2, 2). So we allocate 19 units (minimum of 28 and 19) to the cell (2, 2). We adjust the supply and demand requirements corresponding to the cell (2, 2) and since the demand for the cell (2, 2) is satisfied, we delete the second column and calculate the pointer cost again for the resulting reduced transportation table. Iteration 8: In this stage, maximum pointer cost is 13 and minimum transportation cost corresponding to this is 5 in the cell (2, 4). So we allocate 9 units (minimum of 9 and 18) to the cell (2, 4). We adjust the supply and demand requirements corresponding to the cell (2, 4) and since the supply for the cell (2, 4) is depleted, we delete the second row and calculate the pointer cost again for the resulting reduced transportation table. Iteration 9: Since only the first row is remaining with two unallocated cell in this case, we allocate 11 units (minimum of 20 and 11) to the cell (1, 1) and 9 units (minimum of 9 and 9) to the cell (1, 4). We adjust the supply and demand requirements again and we see that all supply and demand values are exhausted.

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Iteration 10: Since all the rim conditions are satisfied and total number of allocation is 6. Therefore, the solution for the given problem is

x11 = 11 , x14 = 9 , x22 = 19 , x24 = 9 , x31 = 4 and x33 = 13 . for a flow of 65 units with the total transportation cost

z = 11× 3 + 9 × 4 + 19 × 1 + 9 × 5 + 4 × 7 + 13 × 3 = 200 Example 2:

Source

Table 5 Cost Matrix for the Numerical Example Destination 1 2 3 Supply 1 6 10 14 50 2 12 19 21 50 3 15 14 17 50 Demand 30 40 55

Example 3:

Factory

1 2 3 4 5 6 Demand

1 12 9 4 9 7 16 75

Table 6 Cost Matrix for the Numerical Example Warehouse 2 3 4 5 4 13 18 9 16 10 7 15 9 10 8 9 3 12 6 4 11 5 18 2 8 4 5 1 85 140 40 95

6 2 11 7 5 7 10 65

Supply 120 80 50 90 100 60

6. Result Table 7 shows a comparison among the solutions obtained by our proposed TOCM-SUM Approach and the 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.

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Table 7 A Comparative Study of Different Solutions Total Transportation Cost Solution obtained by Ex. 2 Ex. 1 Ex. 3 Size of the matrix 3×4 3×3 6×6 273 1815 4285 North West Corner Method Matrix Minima Method 231 1885 2455 Vogel’s Approximation Method 204 1745 2220 Extremum Difference Method 218 1695 2580 Highest Cost Difference Method 231 1885 2455 TOCM-MMM Approach 231 1795 2470 TOCM-VAM Approach 204 1695 2170 TOCM-EDM Approach 204 1695 2470 TOCM-HCDM Approach 255 1755 2470 TOCM-SUM Approach (Proposed) 200 1690 2170 Optimum Solution 200 1650 2170

We also solve randomly selected transportation problem of order 3×3, 3×4, 4×3, 4×4, 4×5, 4×6, 5×5, 6×6, 8×8, 10×10, 15×15 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 In this paper, we developed an efficient algorithm for cost minimization of transportation problem which is very easy to understand and provides better result in comparison to the existing methods available in the literature. We also present MATLAB 7.7.0 code for the developed method and test the correctness of the code through different examples which proves that the code provides the identical result. Acknowledgements. The first author acknowledges the financial support provided by the EU Erasmus Mundus Project-cLINK, Grant Agreement No: 2122645/001-001-EM, Action 2. REFERENCES Abdur Rashid, Md. Sharif Uddin, Faruque Ahmed, Md. Rashed Kabir, An Effective Approach for Profit Maximization in a Transportation Problem. Jahangirnagar University Journal of Science, 35, 2, 37−43, 2012. Aminur Rahman Khan, A Re-Solution of the Transportation Problem: An Algorithmic Approach. Jahangirnagar University Journal of Science, 34, 2, 49−62, 2011.

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Aminur Rahman Khan, Analysis and Resolution of the Transportation Problem: An Algorithmic Approach. M. Phil. Thesis, Dept. of Mathematics, Jahangirnagar University, 2012. Aminur Rahman Khan, Avishek Banerjee, Nahid Sultana, Nazrul Islam M., Solution Analysis of a Transportation Problem: A Comparative Study of Different Algorithms. Accepted for Publication in the Bulletin of the Polytechnic Institute of Iaşi, România, Section Textile. Leathership, In Issue of 2015(1). Amirul I., Haque M., Uddin M.S., Extremum Difference Formula on Total Opportunity Cost: A Transportation Cost Minimization Technique. Prime University Journal of Multidisciplinary Quest, 6, 1, 125−130, 2012. 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. 8th Edition, Pearson Prentice Hall, Upper Saddle River, New Jersey 07458, 2007. Hitchcock F.L., The Distribution of a Product from Several Sources to Numerous Localities. Journal of Mathematics and Physics, 20, 224−230, 1941. Kasana H.S., Kumar K.D., Introductory Operations Research: Theory and Applications. Springer International Edition, New Delhi, 2005. Koopmans T.C., Optimum Utilization of the Transportation System. Econometrica, 17, 3−4, 1947. Mathirajan M., Meenakshi B., Experimental Analysis of Some Variants of Vogel’s Approximation Method. Asia-Pacific Journal of Operational Research, 21, 4, 447−462, 2004. Md. Amirul Islam, Aminur Rahman Khan, Sharif Uddin M., Abdul Malek M., Determination of Basic Feasible Solution of Transportation Problem: A New Approach. Jahangirnagar University Journal of Science, 35, 1, 101–108, 2012. 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, 2013a. 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, 2014a. 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, 418−422, 2014b. Md. Main Uddin, Md. Azizur Rahaman, Faruque Ahmed, Sharif Uddin M., Md. Rashed Kabir, Minimization of Transportation Cost on the basis of Time Allocation : An Algorithmic Approach. Jahangirnagar Journal of Mathematics & Mathematical Sciences, 28, 47−53, 2013b.

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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, 2, 199−206, 2014. Mollah Mesbahuddin Ahmed, Algorithmic Approach to Solve Transportation Problems: Minimization of Cost and Time. M. Phil. Thesis, Dept. of Mathematics, Jahangirnagar University, 2014. Nagraj Balakrishnan, Modified Vogel’s Approximation Method for the Unbalanced Transportation Problem. Applied Mathematics Letters, 3, 2, 9−11, 1990. 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. Pandian P., Natarajan G., A New Approach for Solving Transportation Problems with Mixed Constraints. Journal of Physical Sciences, 14, 53−61, 2010. 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, 1−12, 2012. Sharif Uddin M., Sayedul Anam, Abdur Rashid, Aminur R. Khan, Minimization of Transportation Cost by Developing an Efficient Network Model. Jahangirnagar Journal of Mathematics & Mathematical Sciences, 26, 123−130, 2011. Sharif Uddin M., Transportation Time Minimization: An Algorithmic Approach. Journal of Physical Sciences, Vidyasagar University, 16, 59−64, 2012. Shenoy G.V., Srivastava U.K., Sharma S.C., Operations Research for Management. 2nd Edition, New Age International (P) Limited Publishers, New Delhi, 110002, 1991. 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, 2, 42−48, 2014b. 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, 182−187, 2014a.

METODA EURISTICĂ PENTRU DETERMINAREA UNEI SOLUłII FEZABILE PENTRU O PROBLEMĂ DE TRANSPORT (Rezumat) Articolul prezintă o problemă de transport şi rezolvarea acesteia printr-o nouă metodă euristică. S-a definit modelul matematic, algoritmul ce implementează metoda a fost exemplificat pas cu pas pe o instanŃă a problemei şi performanŃele acestuia au fost evaluate comparativ cu cele furnizate de alŃi algoritmi citaŃi în literatura de specialitate, pe seturi de date cu dimensiuni diferite.