Dhaka Univ. J. Sci. 63(1): 1-7, 015 (January) A Computer Oriented Method for Solving Transportation Problem Sharmin Afroz and M. Babul Hasan* Department of Mathematics, Dhaka University, Dhaka-1000, Bangladesh (Received : 10 April 013; Accepted: 9 June 014) Abstract In this paper, an algorithm and its computer oriented program have been developed for solving transportation programming (TP) reducing it into a linear program (LP). After formulating it into linear programming problems the number of variables becomes large. It then, becomes more difficult and time-consuming if it is done manually with simplex method. By using the computer program the solution can be found in a shorter time. It will be shown that a TP with a large number of variables can be solved in few seconds by using this method. A number of numerical examples are presented to demonstrate the method developed in this research. Key Words: LP, TP, Computer program I. Introduction Linear programming (LP) problems can generally be stated as follows: Optimize: zz = cc 1 xx 1 + cc xx + + cc nn xx nn Subject to: aa 11 xx 1 + aa 1 xx + + aa 1nn xx nn = bb 1 aa 1 xx 1 + aa xx + + aa nn xx nn = bb : : aa mm1 xx 1 + aa mm xx 1 + + aa mmmm xx nn = bb mm xx ii 0, where i= 1, n bb jj 0, where j= 1, m The characteristics of standard form are given below: (i) The objective function is of the optimization i.e. maximization or minimization type. (ii) All constraints are expressed as equations. (iii) All variables are restricted to be non negative. The right hand side constant of each constraints is nonnegative. II. Transportation In transportation problems (TPs) the objective is to transport various amount of a single homogenous commodity that are initially stored at various origins, to different destinations in such a way that the total transportation cost is minimum. The distinct feature of TPs is that origins and destinations must be expressed in terms of only one kind of unit. These will be shown in this Section. Transportation Table A specimen of the TP table of m-sources, n-destinations transportation table is given below- Table 1. Transportation Table Destination vv jj 1. n supply method during all iterations leading to the current simplex table. Here i= 1,... m vv jj = multiple of original row m+j that has been subtracted (directly or indirectly) from original row 0 by the simplex method during all iterations leading to the current simplex table. Here j= 1,... n. Examples of TPs In this Section, it will be shown how the TPs minimize the cost using methods of solving TP with the help of numerical examples given as follows. Numerical Example 1 Find the feasible solution of the following transportation problem. Warehouse W 1 W W 3 W 4 Supply Factories F 1 F F 3 Solution: By using North-West corner rule the problem can be solved as follows: Iteration 1 V 1 =14 V =5 V 3 =35 V 4 = -15 Supply 14-10 45-0 5 6//0 u 1 =0 4 5-51 65 5 35-70 55 8/3/0 u =0 u 3 =30 Iteration 14 5 45 5 6 65 5 35 55 8 35 3 65 15 16 4 7 6 13 5 9 35 5 3 * V 1 =14 V =5 V 3 =35 V 4 =37 Supply 3 3 65 13 4/0 7/5/0 6/3/0 13/0 15 16/1 3/0 uu ii 1 Source m where, c 11 c 1 c 1n s 1 c 1 c. c n s....... c m1 c m.. cc mmmm ss mm d 1 d.. d n uu ii = multiple of original row i that has been subtracted (directly or indirectly) from original row 0 by the simplex u 1 =0 u =0 u 3 = - 4 14 5-51 65 5-43 35 3 3-10 45 3 5 * 6 35-5 65 15 13 4/0 7/5/3/0 6/0 13/0 6//0-18 55 8/6/0 16/13/0 * Author for correspondence. e-mail: mbabulhasan@yahoo.com
Sharmin Afroz and Babul Hasan Iteration 3 u 1 =0 u =3 u 3 =10 V 1 =14 V =-7 V 3 =3 V 4 =5 Supply The basic feasible solution is given below: xx 11 = 4, xx 14 =, xx =, xx 3 = 6, xx 3 = 5, xx 34 = 11 Thus the minimum value is 4 14+ 5+ 5+6 35+5 3+11 15 = 506 Numerical Example Find the feasible solution of the following transportation problem. Cost per truck load Plant V A V B V C Supply UU ww UU xx UU yy Solution: Similarly, by using North-West corner rule we can solve the problem and find the solution as follows: The basic feasible solution is given below: xx 1 =76, xx 1 =1, xx 3 =41, xx 31 =51, xx 3 =6; Thus the minimum cost is 8 76+1 16+41 16+51 8+6 16 = 44 III. Formulation of LP Models In this Section, the formulation of TP into LP will be discussed. By considering Z the total distribution cost and xx iiii (i=1,,..., m ; j=1,,..., n) the number of units to be distributed from source i to destination j, the LP formulation of TP becomes: mm nn Minimize, Z = cc iiii xx iiii Subject to nn jj =1 mm ii=1 14 4-3 5-4 45 5-19 65 5 35 6-11 35 3 5-5 65 15 11 4/0 7/5/0 6/0 13/11/0 ii=1 jj =1 xx iiii ss jj, (i = 1,..., m) (Supply constraints) xx iiii dd ii, (j = 1,... m) ( constraints) xx iiii 0, (i= 1,,..., m ; j = 1,,..., n). 6//0-18 55 8/6/0 16/11/0 4 8 8 76 16 4 16 6 8 16 4 77 7 10 41 Applying the process we can formulate the given TP to the LP problems as follows. Formulation of Numerical Example 1 Minimize, ZZ = 14xx 11 + 5xx 1 + 45xx 13 + 5xx 14 + 65xx 1 + 5xx + 35xx 3 + 55xx 4 +35xx 31 + 3xx 3 + 6xx 33 + 15xx 34 subject to, xx 11 + xx 1 + xx 13 + xx 14 6 xx 1 + xx + xx 3 + xx 4 8 xx 31 + xx 3 + xx 33 + xx 34 16 xx 11 + xx 1 + xx 31 4 xx 1 + xx + xx 3 7 xx 13 + xx 3 + xx 33 6 xx 14 + xx 4 + xx 34 13 and xx iiii 0, (i= 1,,3 ; j = 1,, 3, 4). IV. Algorithm for the Program In this Section, the combined simplex algorithm for solving LPs is presented. Step 1: Define the types of constraints. If all are of types go to step. Otherwise go to step 3. Step : Substep 1: Express in standard from. Substep : State with an initial basic feasible solution in canonical form and set up the initial table. Substep 3: Use the inner product rule to find the relative profit factors cc jj as follows cc jj = cc jj zz jj = cc jj (inner product of cc BB and the column corresponding to xx jj in the canonical system). Substep 4: If all cc jj o, the current basic feasible solution is optimal and stop calculation. Otherwise select the non-basic variable with most positive cc jj to enter the basis. Substep 5: Choose the outgoing variable form the basis by minimum ration test. Substep-6: Perform the pivot operation to get the table and basic feasible solution. Substep 7: Go to substep 3. Step 3: At first express the problem in standard form by introducing slack and surplus variables. Then express the problem in canonical form by introducing artificial variables if necessary and form the initial basic feasible solution. Go to substep 3. Step 4: If any cc jj corresponding to on basic variable is zero, the problem has alternative solution take this column as pivot column and go to substep 5. Solving the above LP, we see that the optimal basis contains an artificial variable in second iteration. The LP has no feasible solution which is treated by our program efficiency; our program treats this type of LP effectively and shows the basis with artificial variable. Computer Program
A Computer Oriented Method for Solving Transportation Problem 3
4 Sharmin Afroz and Babul Hasan ----------------------------------- ];
A Computer Oriented Method for Solving Transportation Problem 5 Numerical Examples In this Section, the TPs presented in Section II will be solved using combined simplex method after converting these problems to LP discussed in Section III. Numerical Example 1 After putting corresponding values for numerical example presented in section III we obtain the results. The final table is shown
6 Sharmin Afroz and Babul Hasan Table. Final solution for Numeric Example 1 Table 3. Final solution for Numeric Example Since the program is for maximization problem, the cost vectors are given using negative sign. So the optimal result is negative but it is same with the result which we find using simplex method & methods for solving TP. Numerical Example Similarly, after running the program again we find the results. The final table is shown:
A Computer Oriented Method for Solving Transportation Problem 7 Table 4. Another final solution for Numeric Example Since, the program is for maximization problem, the cost vectors are given using negative sign. So the optimal result is negative but it is same with the result which we find using other methods. V. Discussion and Comparison TP are one of the most important and successful applications of quantitative analysis to solve business problems. Generally, the purpose of this problem is to minimize the cost of transporting goods from one location to another which can meet the needs of each arrival. But the methods of solving TP sometimes create difficulties. Lot of calculations were performed to find out the optimum solution. Mistakes may be committed in manual calculations. Most of the methods are time consuming also. After converting these problems into LP and using computer based program which is discussed in this paper, the difficulties have been removed. Using the computer program the desired solution can be found out easily. But the execution time was short. In a short time the solution was found with the help of computer program. It, therefore, be conclude that the computer program is the best process for finding the solution. VI. Conclusion In this paper, we present an algorithm and its computer oriented program written in the programming language Mathematica for solving TP after converting the TP into LP. After formulating the TP into LP the number of variables increases. Then it becomes more difficult and time-consuming if we use manual calculation with simplex method. But using the computer program we can find our solution in a short time. The number of variables is not matter at all in this program. Even for a large number of variables a few seconds is needed for finding solution. In one word, we can say that the solution of TP by converting to LPs and applying the computer technique is one of the best ways. References 1. Asaduzzaman, S.M. and M. Babul Hasan, 009. A Computer Oriented Combined simplex method for solving linear programming problems, Dhaka University, Journal, 57(), 199-03.. Dentzig, G. B., 196. Linear Programming and Extension, Princeton University Press, Princeton, N. J, 99-314. 3. Eugere, D., 001. Schaum s Outline- Mathematica, McGraw-Hill, New york, 64-90. 4. Frederich S. Hillier, Gerald J. Lieberman, 1995. Introduction of Operations Research, Sixth edition, McGraw-Hill, New York, 350-404. 5. Ganesh Chandra Ray, D., Md. Elias Hossain, 008. Linear Programming, 3 rd edition, Titas Publications, Dhaka-1100, 119-9. 6. Gass, S. L., 004. Linear Programming Method and Application, fifth edition, McGraw-Hill Book Company, New York, 319-350. 7. Gupta, P.K., D.S. Hira, 005. Problems in Operations Research Principles and Solution, S.Chand& Company LTD., New Delhi-110055, 406-484. 8. Gupta, P.K., Man Mohan, 1997. Linear Programming and Theory of Games, Eighth edition, Sultan Chand & Sons. New Delhi-11000, 343-46. 9. Ravindran, A., Don T. Phillips, James J. Solberg, 1987. Operations Research Principles and Practice, second edition, John Wiley & Sons, New York, 01-50. 10. Reeb, J. and S. Leavengood, 00. Operations Research (Transportation Problem: A Special case for Linear Programming Problems), Oregon State University, EM8779, $ 3.50, 610-695. 11. Wolfram, S., 000. Mathematica, Addition Wesley Publishing company, Menlo Park, California, New York, 70-80. 1. Winston, W.L., 1993. Operation Research application and Algorithms, third edition, International Thomson Publishing, California, USA, 361-414.