Balancing a transportation problem: Is it really that simple?

Transcription

1 Original Article Balancing a transportation problem: Is it really that simple? Francis J. Vasko a, * and Nelya Storozhyshina b a Mathematics Department, Kutztown University, Kutztown, Pennsylvania 19530, USA. vasko@kutztown.edu b Computer Science Department, Kutztown University, Kutztown, Pennsylvania 19530, USA. *Corresponding author. Abstract The transportation problem (TP) is discussed in all operational research textbooks. Although the TP can be formulated as a linear programme, owing to its special structure, it can be solved more efficiently than just using the standard simplex algorithm. Typically, the first step in solving a TP is to balance the problem. If total supply is not equal to total demand, then a dummy column (supply greater than demand) or dummy row (demand greater than supply) is created. Although a few papers have discussed the importance of how the dummy column (row) is handled, this has been done only with reference to using one particular heuristic (Vogel s approximation method or VAM). Furthermore, the impact on solution quality based on how a heuristic processes the dummy column or row has not been empirically quantified. The closer an initial heuristically determined basic feasible solution is to the optimal solution, the fewer the required iterations of the modified transportation simplex to determine the optimal solution. The purpose of this article is to empirically quantify the importance of how five TP heuristics (northwest corner method, the greedy heuristic, VAM, Russell s method and the maximum demand method) process the dummy column (row) in a balanced TP. OR Insight (2011) 24, doi: /ori ; published online 18 May 2011 Keywords: balanced transportation problem; greedy heuristic; Vogel s approximation method; Russell s approximation method; maximum demand heuristic Received 2 January 2011; accepted 5 April 2011 & 2011 Operational Research Society Ltd OR Insight Vol. 24, 3,

2 Vasko and Storozhyshina Introduction and Background To begin, we will state the classical transportation problem (TP) mathematically as follows: Minimize z ¼ X i subject to X j X c ij x ij j x ij ps i for i ¼ 1; 2;...; m ðoriginsþ X x ij Xd j i x ij X0 for j ¼ 1; 2;...; n ðdestinationsþ where c ij is the cost of shipping one unit from origin i to destination j, s i is the supply available for shipment from origin i, d j is the amount required to be shipped to destination j and x ij is the amount shipped from origin i to destination j. Small TPs are typically represented by a table (or tableau) with one row of the table for each origin and one column for each destination. The far right column contains the supply amounts (s i ) for each origin and the bottom row contains the destination requirements (d j ) for each column. The intersection of each supply row and destination column is a cell that contains the cost of shipping one unit from that supply row to that destination column. Operational research textbooks typically assume that the problem is balanced, that is, S i a i ¼ S j b j. If the problem is not balanced, a dummy supply origin (if demand exceeds supply) or demand destination (if supply exceeds demand) can always be added to balance the problem. If a dummy column is added (far right in the table), then the costs for this column are typically set to zero. The reasoning is that items shipped to the dummy demand point are not actually shipped. Less popular is to assign inventory holding costs to the dummy column. If a dummy row is added (bottom row in the table), ideally, the dummy row costs should reflect the penalty incurred for unmet demand. In real-world applications, TPs are seldom if ever balanced, and quantifying the cost of unmet demand is usually difficult to do. Once the TP is balanced, textbooks typically discuss several heuristic methods; for example, the northwest corner method, the greedy heuristic, Vogel s approximation method (VAM), and, sometimes Russell s method that are used to generate basic feasible solutions (BFSs) without having to introduce artificial variables. Typically, the best heuristically determined BFS generated is then used as a starting point for the modified transportation simplex algorithm (see Winston, 2004). 206 & 2011 Operational Research Society Ltd OR Insight Vol. 24, 3,

3 Balancing a transportation problem Two papers that discuss balancing the TP in the context of using VAM are Goyal (1984) and Ramakrishnan (1988). Both Goyal and Ramakrishnan advocate using the maximum shipping cost as the shipping costs used in the dummy column. Neither of these papers provides any empirical evidence supporting their methodologies. In addition, both of these papers only discuss their approaches as modifications to VAM. However, the goal of both of these methodologies is to discourage making shipping assignments to the dummy column (row) until last. In the TP literature, it is more common to assume that total supply exceeds total demand. Therefore, we will limit our discussion to the case when total supply exceeds total demand and we will set all dummy column costs equal to zero. In other words, we will be studying how processing the dummy column by five heuristics (northwest corner method, the greedy heuristic, VAM, Russell s method and the maximum demand method) impacts the quality of the solutions obtained by the different heuristics. The quality of the heuristic solutions is relevant, because the closer an initial heuristic solution is to the optimal solution, the fewer the required iterations of the modified transportation simplex to determine the optimal TP solution. However, if a heuristic for generating an initial BFS requires excessive computational effort compared with the effort required by the modified transportation simplex to perform a simplex iteration, then the benefit of fewer simplex iterations may be nullified. This will be discussed later when we discuss the implications of this work for solving large TPs efficiently (Section Implications for Practitioners ). Specifically, we will consider two ways (cases) of processing the dummy column by each of the five heuristics. In Case 1, the dummy column gets no special treatment it is viewed the same as the other columns by each of the five heuristics. In Case 2, no assignments are made to the dummy column until last. Case 2 has the same goal as the methodologies of Goyal (1984) and Ramakrishnan (1988). In the next section, we will briefly review the five heuristics that will be empirically tested. Next, we will discuss the experimental design used to generate 4320 test problems that were used to test the two cases on each of the five heuristics. This will be followed by the empirical results, implications for solving large TPs efficiently, and finally some conclusions and recommendations will be offered. Heuristics Tested The three heuristics that are universally discussed in operational research books are: the northwest corner method, the greedy heuristic and VAM & 2011 Operational Research Society Ltd OR Insight Vol. 24, 3,

4 Vasko and Storozhyshina (for example, see Winston, 2004 or Hillier and Lieberman, 2010). In addition, a few operational research books include Russell s approximation method (for example, see Hillier and Lieberman, 2010). VAM first appeared in a paper by Reinfeld and Vogel (1958). Russell s approximation method first appeared in a paper by Russell (1969). Northwest corner method (Winston, 2004) To find a BFS by the northwest corner method, we begin in the upper left (or northwest) corner of the transportation table and assign x 11 its largest possible value, minimum {s 1, d 1 }. If x 11 ¼ s 1, cross out the first row of the transportation table; this indicates that no more basic variables will come from row 1. Also change d 1 to d 1 s 1.Ifx 11 ¼ d 1, cross out the first column of the transportation table; this indicates that no more basic variables will come from column 1. Also change s 1 to s 1 d 1.Ifx 11 ¼ s 1 ¼ d 1, cross out either row 1 or column 1 (but not both). If you cross out row 1, change d 1 to 0; if you cross out column 1, change s 1 to 0. Continue applying this procedure to the most northwest cell in the table that does not lie in a crossed-out row or column. Eventually, you will come to a point where there is only one cell that can be assigned a value. Assign this cell a value equal to its row or column demand, and cross out both the cell s row and column. Greedy heuristic (Winston, 2004) To begin the greedy heuristic, find the variable with the smallest shipping cost (call it x ij ). Then assign x ij its largest possible value, minimum {s i, d j }. If x ij ¼ s i, cross out row i of the transportation table. Also set d j ¼ d j s i.ifx ij ¼ d j, cross out column j of the transportation table. Also set s i ¼ s i d j.ifx ij ¼ s i ¼ d j, cross out either row i or column j (but not both). If you cross out row i, set d j ¼ 0; if you cross out column j, set s i ¼ 0. Next, choose from the cells that do not lie in a crossed-out row or column the cell with the minimum shipping cost and repeat the procedure. Continue until there is only one cell that can be chosen. In this case, cross out both the cell s row and column. Vogel s approximation method (Winston, 2004) Begin by computing for each row (and column) a penalty equal to the difference between the two smallest costs in the row (column). Next, find the row or column with the largest penalty. Choose the variable in this row or column that has the smallest shipping cost. As described in the greedy heuristic, make this variable as large as possible, cross out a row or column, and change the supply or demand associated with this variable. Now recompute new penalties (using only the cells that do not lie in a crossed-out row or column), and repeat the procedure until only one uncrossed cell remains. 208 & 2011 Operational Research Society Ltd OR Insight Vol. 24, 3,

5 Balancing a transportation problem Set this variable equal to the supply or demand associated with this variable, and cross out the variable s row and column. Russell s approximation method (Hillier and Lieberman, 2010) For each source row i remaining under consideration, determine its R i, which is the largest unit cost c ij still remaining in that row. For each destination column j remaining under consideration, determine its K j, which is the largest unit cost c ij still remaining in that column. For each variable x ij not previously selected in these rows and columns, calculate D ij ¼ R i þ K j c ij. Select the variable having the largest value of D ij (ties may be broken arbitrarily). Maximum demand heuristic (Pargar et al, 2009 without using VAM to break column ties) This heuristic allocates to the column with the maximum demand, the maximum possible units to the lowest-cost cell in the column. If the demand for the column is not met, then units are allocated to the next lowest-cost cell in that column until the column demand is met. In the case of ties, that is, more than one column with maximum demand, then the column with the lowest unit cost (c ij ) is selected. The original maximum demand heuristic used VAM to break column ties, but Storozhyshina et al (2011) showed that breaking ties using the lowest unit cost gave just as good results and requires less computational effort. Experimental Design We randomly generated 4320 problems using the experimental design framework used by Storozhyshina et al (2011). This design is an extension of the framework used by both Kirca and Satir (1990), as well as Mathirajan and Meenakshi (2004). Following Storozhyshina et al (2011) and Mathirajan and Meenakshi (2004), we restrict our problems to full density TPs, that is, it is feasible to ship from any source to any destination. The details are as follows: Problem size (m supply points n demand points): Nine problem sizes are tested. The problem sizes are (4 5), (4 10), (8 10), (10 10), (10 20), (10 40), (10 60), (10 80) and (10 100). Cost structure (C ij : i ¼ supply point 1, 2, y, m and j ¼ demand point 1, 2, y, n): Problems with 12 cost ranges (R) are tested. The mean cost is taken to be equal to 500. The ranges used are R ¼ (20, 50, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000). For each range, the costs are randomly & 2011 Operational Research Society Ltd OR Insight Vol. 24, 3,

6 Vasko and Storozhyshina generated from the following uniform distribution: U(C ij : [mean cost R/2, mean cost þ R/2]). Supply and demand structure (S i and D j ): The mean demand is equal to 100. Given the mean demand, the mean supply is expressed as mean supply ¼ K[(n mean demand)/m], where K is the degree of imbalance between total supply and total demand. The mean supply values are generated for four imbalance coefficients, K ¼ (1, 2, 5, 10). The D j and S i are then generated from the uniform distributions of U(D j : [75, 125]) and U(S i : [0.75 mean supply, 1.25 mean supply]). For each combination of (m n), R and K, 10 problem instances were randomly generated, yielding a total of 4320 [ ] problem instances. All heuristics were coded in Visual Basic and solved on a Hewlett- Packard notebook PC with Intel Core 2.10 GHz CPU and 4.00 GB of RAM. For each problem instance, the best heuristic solution was used by the transportation simplex (also coded in Visual Basic) to obtain the optimal solution to that problem. Empirical Results The performance of the five heuristics was measured using average relative percentage deviation that was used in Mathirajan and Meenakshi (2004), as well as in Storozhyshina et al (2011). Specifically, for given parameter values, that is, cost range, problem size and supply imbalance, the percentage that a heuristic value deviated from the optimal value averaged over 10 problem instances is calculated. The computational results are summarized in Table 1. Table 1 clearly demonstrates that making dummy assignments last (Case 2) is a major benefit to both the maximum demand and the greedy heuristics. In fact, the higher the imbalance between supply and demand (K41), the more beneficial it is for these heuristics to make assignments to the dummy column last. This benefit is not surprising as, for both the maximum demand and the greedy heuristics, the higher the supply available the more the dummy column will indiscriminately consume the supplies if the dummy allocations are not left until last. In fact, except for the northwest corner method that gives the worst (and same) overall results in both cases, in Case 1 the maximum demand heuristic gives the next worst overall results; but in Case 2, the maximum demand heuristic gives the best results. For the VAM and Russell heuristics, the benefit of making assignments to the dummy column last is small. This is regardless of the value of K, that is, the imbalance between supply and demand. In fact, 210 & 2011 Operational Research Society Ltd OR Insight Vol. 24, 3,

7 Balancing a transportation problem Table 1: Average relative per cent deviation Heuristics Overall K=1 K=2 K=5 K=10 Case 1: Dummy gets no special treatment Northwest corner Greedy VAM Russell MD Case 2: Dummy assignments are made last Northwest corner Greedy VAM Russell MD Note that some of the results in Table 1 first appeared in Storozhyshina et al (2011). for K ¼ 1, the VAM results are better for Case 1 versus Case 2. This is easy to understand for VAM, as the dummy column has a zero penalty and, for the rows, the largest cost will give the largest penalty. The dummy column impacts Russell s method in an analogous manner. Finally, for the northwest corner method, as the dummy column is located farthest east in the transportation table, the dummy column assignments are made last in both Cases 1 and 2. Implications for Practitioners Historically, there has been a real disconnect between heuristics discussed in operational research textbooks for obtaining initial BFSs to the TP and actual, highly efficient computer codes for solving large TPs. This disconnect was pointed out very clearly in both Glover et al (1974) and Gass (1990). Essentially, the opinion expressed in these papers is that the inordinate amount of time required by VAM to find an initial feasible solution does not warrant its use. Furthermore, it is their belief that VAM should be relegated to hand calculations, if that. To generate an initial BFS, Glover et al (1974) use a modified row-minimum rule, in which the rows are cycled through in order, each time selecting a single cell with the minimum cost to enter the basis. The cycling continues until all row supplies are exhausted. As recommended in Storozhyshina et al (2011), using highly efficient network codes such as the ones by Bertsekas (1991) designed specifically to solve TPs or even his more general minimum cost network flow problem (MCNFP) codes require very little time to solve large TPs. Specifically, & 2011 Operational Research Society Ltd OR Insight Vol. 24, 3,

8 Vasko and Storozhyshina Vasko et al (1994) solved slab (semifinished pieces of steel) assignment to customer order problems using one of Bertsekas s MCNFP codes to solve TP subproblems. They were able to solve 300 origin by 3000 destination TPs in under a minute on a 386 PC (25 MHz). Conclusions and Recommendations The first step in solving the TP, which is discussed in all operational research textbooks, is to balance the problem. Once the TP is balanced, the next step is to heuristically determine a BFS to the TP. This solution is then used as a starting point for the modified transportation simplex algorithm. How to heuristically process the dummy column or row created by the balancing process is typically not discussed in operational research books and has had very limited discussion in the operational research literature. In this article, we test two cases for processing the dummy column. Case 1 is that the dummy column gets no special treatment and Case 2 is that dummy assignments are made last. On a large set of test problems (4320 problems), we have shown that, for the maximum demand and the greedy heuristics, there is a major difference in the solution quality based on how the dummy column is processed. For VAM and Russell s heuristic, the difference in solution quality is small, and for the northwest corner method the difference is nonexistent. On the basis of these results, for K ¼ 1, for hand calculations only, we recommend using VAM to generate an initial BFS and the processing of the dummy column, if it exists, is not important because the mean demand equals the mean supply (although, on the basis of random sampling, we do not expect total supply to exactly equal total demand). For K41, and for large problems for which K ¼ 1, it is very important to process the dummy column last and the best heuristic to use is clearly (because of both computational efficiency and solution quality) the maximum demand heuristic with the greedy heuristic coming in second. About the Authors Francis J. Vasko is professor of Mathematics at Kutztown University of Pennsylvania. Before coming to Kutztown University in September 1986, he worked for more than 8 years as an employee in the Research Department at Bethlehem Steel, solving a variety of real-world applications in operations research. He then served as a consultant to Bethlehem Steel Corporation from 212 & 2011 Operational Research Society Ltd OR Insight Vol. 24, 3,

9 Balancing a transportation problem September 1986 to March Since 2003, he has served as a consultant to several corporations focusing on the solution of production planning, resource allocation and cutting stock problems. His current research focuses on using a mixture of combinatorial optimization techniques in order to more accurately model and solve important real-world applications in production planning, strategic planning and resource allocation. He earned a BS in mathematics from Kutztown University in 1974, an MS in mathematics in 1976, an MSIE (Operations Research) in 1978 and a PhD in Operations Research in 1983 all graduate degrees were from Lehigh University. Nelya Storozhyshina is currently a graduate student at Kutztown University of Pennsylvania. She earned her BS and MS in Acoustical Engineering from Kiev Polythechnic Institute in Ukraine in 2001 and 2003, respectively; and BS in Computer Science from Kutztown University in She is a full-time student at the university and is working as a graduate assistant. As a GA, Storozhyshina is carrying out research in using surrogate constraint normalization for set covering problems. After graduation, she hopes to pursue a career in the fields of computer science and operational research. References Bertsekas, D.P. (1991) Linear Network Optimization Algorithms and Codes. Cambridge, MA: MIT Press. Gass, S.I. (1990) On solving the transportation problem. Journal of the Operational Research Society 41: Glover, F., Karney, D., Klingman, D. and Napier, A. (1974) A computation study on start procedures, basis change criteria, and solution algorithms for transportation problems. Management Science 20: Goyal, S.K. (1984) Improving VAM for unbalanced transportation problems. Journal of the Operational Research Society 35: Hillier, F.S. and Lieberman, G.J. (2010) Introduction to Operations Research, 9th edn. New York: McGraw-Hill. Kirca, O. and Satir, A. (1990) A heuristic for obtaining an initial solution for the transportation problem. Journal of the Operational Research Society 41: Mathirajan, M. and Meenakshi, B. (2004) Experimental analysis of some variants of Vogel s approximation method. Asia-Pacific Journal of Operational Research 21: Pargar, F., Javadian, N. and Ganji, A.P. (2009) A heuristic for obtaining an initial solution for the transportation problem with experimental analysis. The 6th International Industrial Engineering Conference, Sharif University of Technology, Tehran, Iran. Ramakrishnan, C.S. (1988) An improvement to Goyal s modified VAM for the unbalanced transportation problem. Journal of the Operational Research Society 39: & 2011 Operational Research Society Ltd OR Insight Vol. 24, 3,

10 Vasko and Storozhyshina Reinfeld, N.V. and Vogel, W.R. (1958) Mathematical Programming. Englewood Cliffs, NJ: Prentice-Hall. Russell, E.J. (1969) Extension of Dantzig s algorithm to finding an initial near-optimal basis for the transportation problem. Operations Research 17: Storozhyshina, N., Pargar, F. and Vasko, F.J. (2011) A comprehensive empirical analysis of 16 heuristics for the transportation problem. OR Insight 24: Vasko, F.J., Creggar, M.L., Stott, K.L. and Woodyatt, L.R. (1994) Optimal assignments of slabs to orders: An example of appropriate model formulation. Computers & Industrial Engineering 26: Winston, W.L. (2004) Operations Research: Applications and Algorithms. Belmont, CA: Thomson. 214 & 2011 Operational Research Society Ltd OR Insight Vol. 24, 3,