Dual Based Procedures for Un-Capacitated Minimum Cost Flow Problem

In this article, we devise two dual based methods for obtaining very good solution to a single stage un-capacitated minimum cost flow problem. These methods are an improvement to the methods already developed by Sharma and Saxena [1]. We further develop a method to extract a very good primal solution from a given dual solution. We later demonstrate the efficacies and the significance of these methods on 150 random problems.


Introduction
Un-capacitated min cost flow problem is a special case of min cost flow problem in which arc capacities are assumed to be infinite.Weintraub [2] developed a variant of negative cycle algorithm which searched for the most negative cycle and subsequently introduced it into the feasible flow at each iteration.Later a strongly polynomial time algorithm for min cost flow was developed by Tardos [3] with a computational complexity of O(m 4 ).Enhanced capacity scaling algorithm can be used to solve Transshipment problem with computational complexity of O(n log (n) S(n,m)) (Ahuja et al. [4]).
Tardos [3] developed cost scaling algorithm with the computational complexity of O(n 3 log n).In this algorithm, dual optimality conditions are relaxed to form e-optimality conditions.Thus the best primal based methods solve un-capacitated min cost flow problem in O(n 3 log (n)).Recently Juman [5] has presented a heuristic with O(n 3 ) running time to solve un-capacitated transportation problem, and is shown to perform better than VAM.
Successive shortest path algorithm was developed by Busakar and Gowan [6].This algorithm maintains dual feasibility at each step and iteratively achieves primal feasibil-ity.Edmonds and Karp [7] proposed the first polynomial time algorithm by modifying the method to calculate shortest paths, to solve min cost flow problem with computational complexity of O((n + m) log U).Dual simplex for network flow was first analyzed by Hegason and Kennington [8].Plotkin and Tardos [3] improved the pivoting strategy with (m 2 log n) bound over the pivoting strategy proposed by Orlin [9].This improves the number of pivot steps required in dual simplex algorithm.This algorithm runs in O(m 3 log(n)) time.Ali et al. [10] have demonstrated that an efficient execution of each pivot in dual based algorithm requires less iterations as compared to primal based algorithms.This holds true even for the re-optimization process.However, computational effort required per pivot may be higher.Sharma and Sharma [11] have given a new dual based procedure that has obtained solutions within 85% of the optimal.
Sharma and Saxena [1] have posed the transshipment problem differently.We use the formulation proposed by Sharma and Saxena [1].We then modify the dual based methods developed by them to obtain better solutions with the same complexity of O(n 2 ) and O(n 3 ) respectively.We further devise a method to obtain a good primal solution from the dual solutions already obtained.Empirical results on the random 150 problems are given in Appendix 1.

Problem Formulation
We next present the mathematical formulation of the primal problem and dual problem respectively.

Constants of Problem
k D refers to the demand at the k th demand node, while k d is the demand at market k as a fraction of total market demand.Hence we have where K is the total number of demand nodes.Similarly i S refers to units available for transportation at the source node i and If the problem is balanced, then we have ∑ ∑ , I is the total number of supply nodes.J is total number of transshipment nodes.
node to j and j to k respectively.

Decision Variables
x is the number of units transported from node i to node j and j to k respectively.We also have

Primal (P)
In this formulation we assume flows only in the forward direction.Equation (1) ensures that entire supply is transported to meet the demand, which is valid for the balanced problem.Equation (2) ensures that the total demand is met by the supply.Equations (3) and ( 4) ensure that individual supply and demand constraints are satisfied, while Equation (4) ensures that no inventory is built at any transshipment node.

Dual of the Problem (DP)
In this section we present the dual of the problem P. We associate 1, 2, , , the dual variables corresponding to (1), ( 2), ( 3), ( 4), ( 5) respectively.We first state the dual of the problem as DP and then divide it into two parts as DP-source and DP-sink for computational simplicity. DP Maximize : 1 2 , 0 V and j W unrestricted in sign.

Few Theoretical Results
We start with development of the heuristic for the dual solution, and then move on to develop the heuristic for the primal.Computational attractiveness of these results will be demonstrated in the later sections through empirical testing.Well known dual based approaches (Orlin [9], Plotkin and Tardos [3] and Ali et al. [10]) can be used for our solution to get an advanced start while solving the transshipment problem.We begin by defining the set SPS which is as under-SPS = {SP ik :SP ik is the shortest path between i and k}.Problem (TP) Minimize : and Theorem 1: Optimal solution of problem TP is equal to optimal solution to problem P.
Proof: Since upper value of the flow is unbounded, hence optimal flow for a pair of source node and sink node will be on SP ik .This ensures that any further reduction in the objective value is not possible.Therefore problem TP gives the optimal solution to problem P. Hence proved.

Heuristic to Solve Dual of the Problem (H1)
DP-source and DP-sink are equivalent in structure to DRP1 in Sharma and Murlidhar [12].Sharma and Murlidhar [12] have given an efficient algorithm to solve DRP1 which can be modified to solve DP-source and DP-sink.
Step 1. DP-source and DP-sink can be rewritten as under DP-source 1 Maximize : 1 V and j W unrestricted in sign.
Step 2. Find ( ) ∀ all i, j, k and 0 j W = and remove all the redundant constraints in DP-source and DP-sink (Equa- tions ( 8) and ( 9)).In case of tie, only one equation is retained while others are eliminated.This reduces the DP-source and DP-sink to the following form: DP-source Maximize: d represent the least cost transportation route between source and transshipment node and transshipment node and sink respectively.
Step 3. We sort the values of * k d in an increasing order and re-index such that ( ) 1, , max , .
Step 4. Since . Solution to the problem is given by (

∑∑
We repeat the whole procedure for different increases in values of W j ∀ all j and retain the best solution.
It may be noted that when we increase/decrease the value of W j ∀ j, DP-source increases while DP-sink decreases as per the structure of DP-source and DP-sink.Actually all four possibilities are there for a general case.Our algorithm here intends to balance value of W j for the best trade-off possible.
Proof: Complexity of algorithm is dominated by step 2 which can be solved in O(n2) time.

Heuristic to Solve Dual of the Problem (H2)
In the previous algorithm, we tinkered with value of j W along ik SP .There is no rea- son as to why we should not tinker with the values of , , ( ) Step 0: Set W j = 0 ∀ j = 1, …, J.
Step 2: j = j + 1; if j > J then stop or else go to step 3.
Step 3: Increase value of W j in steps and compute for each value of W j : max_value of DP-source and DP-sink.
Proof: Complexity of the step is heuristic is dominated by step 3 which can be completed in O(n 3 ) steps.

Development of the Heuristic to Obtain a Good Primal Solution (H3)
In this section we will develop a primal heuristic by utilizing the complimentary slackness condition.This heuristic extracts a good primal solution from a good dual solution by utilizing complimentary slackness condition.Let us denote the solution of DP by . Slack S so and S si is defined as following: . If S so = 0 and S si = 0, then 0 ij X ≥ and 0 jk X ≥ .X ij and X jk can assume a positive value if for the corresponding i and k, we have then flow along this has to be zero if complimentary slackness property is not to be violated.As we are working with good dual solution (and not optimal dual solution), we may have to send positive flow along a path (i,k) even if SP ik > 0. But the heuristic so described tries to minimize DN ik and hence keep complementary slackness violations as low as possible to get good primal solution.If at the end of execution of algorithm DN = 0, then we have the optimal primal solution.In this way DN ik is similar to Kilter number (ref OUT-OF-KILTER algorithm (a primal dual approach) [13] for solving general min-cost-flow problem).We find shortest path from every source node 'i' to every sink node 'k' using these slacks as weights, and then make the allocations according to shortest path available.Detailed heuristic is described as under.
Step 1: Compute S ik = S so + S si ∀all j and particular i and k.
And S ik' = S so' + S si' ∀j' ≠ j for the same i, k.
If S ik' < S ik then S i1k1 = S ik' , Repeat the step ∀i, j and k.
Step 2: Find i and k: d k > 0 and b i > 0.
Proof: Complexity is dominated by the step 1 which is sorting and can be solved in O(n 2 ).

Results and Discussion
We have solved 150 random problems of varying sizes using methods proposed in this article and the ones developed by Sharma and Saxena [1].We performed one tail paired-test and F-test on the results.Results of paired t-test are as follows.In terms of duality gap, Subroutine S3(O(n 3 )) performs better than subroutine S2(O(n2)) with the statistical significance of 0.00722 (p-value) in Sharma and saxena [1].Similarly in terms of duality gap, H2(O(n 3 )) in this paper performs better than H1(O(n 2 )) with a statistical significance of 0.000419 (p-value).H2(O(n 3 )) in this article performs better than S3(O(n 2 )) form Sharma and saxena [1] with a statistical significance of 2.94E−15.For F-test, F-statistic was calculated to be 58.31 as against the f-critical value of 2.62.P-value was calculated to be 4.05E−32.In terms of computational time, no significant difference is registered between these methods, however methods in this paper perform slightly better than those proposed in Sharma and saxena [1].This is largely due to the fact that we calculate shortest path between the source nodes and sink nodes in contrast to shortest path individually between source and transshipment nodes and transshipment nodes and sink nodes respectively.This method is better computationally.

Conclusion
In Submit or recommend next manuscript to SCIRP and we will provide best service for you: Accepting pre-submission inquiries through Email, Facebook, LinkedIn, Twitter, etc.A wide selection of journals (inclusive of 9 subjects, more than 200 journals) Providing 24-hour high-quality service User-friendly online submission system Fair and swift peer-review system Efficient typesetting and proofreading procedure Display of the result of downloads and visits, as well as the number of cited articles Maximum dissemination of your research work Submit your manuscript at: http://papersubmission.scirp.org/Or contact ajor@scirp.org fined as source slack and sink slack respectively ( ) , so si S S in the later sections.Next we describe this heuristic in detail.
S so and S si be the source and sink slacks respectively, DN ik = SP ik × X ik .DN ik is then referred to as deviation number.If SP ik = 0, then we can send a positive flow along this arc without violating the Complimentary slackness property.However if SP ik > 0,